visible-light tomography of tokamak plasmas
TRANSCRIPT
Visible-light tomography of tokamak plasmas
Citation for published version (APA):Ingesson, L. C. (1995). Visible-light tomography of tokamak plasmas. Eindhoven: Technische UniversiteitEindhoven. https://doi.org/10.6100/IR450408
DOI:10.6100/IR450408
Document status and date:Published: 01/01/1995
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VISIBLE-LIGHT TOMOGRAPHY OF TOKAMAK PLASMAS
PROEFSCHRIFT
ter verkrijging van de graad doctor aan de Tech
nische Universiteit Eindhoven, op gezag van de
Rector Magnificus, prof.dr. J.H. van Lint, voor
een commissie aangewezen door het College van
Dekanen in het openbaar te verdedigen op
maandag 18 december 1995 om 16.00 uur
door
Lars Christian lngesson
geboren te Ljungby (Zweden)
Dit proefschrift is goedgekeurd door de promotoren:
prof.dr.ir. D.C. Schram
en
prof.dr. F.C. Schüller,
en de copromotor: dr. A.J.H. Donné.
CJP-GEGEVENS KONINKLIJKE BIBLIOTHEEK, DEN HAAG
Ingesson, Lars Christian
Visible-light tomography of tokamak plasmas I Lars Christian Ingesson. - (S.I. : s.o.] Proefschrift Technische Universiteit Eindhoven. - Met Jit. opg.- Met een samenvatting in het Nederlands. ISBN 90-386-0117-4 Trefw.: tomografie I tokamak I plasma's
The work described in this thesis was carried out as part of a research programme of the "Stichting voor Fundamenteel Onderzoek der Materie" (FOM) with financial support from the "Nederlandse Organisatie voor Wetenschappelijk Onderzoek" (NWO) and EURATOM. It was
carried out at the FOM-Instituut voor Plasmafysica in Nieuwegein, The Netherlands. This thesis was partly funded by FOM and by "Stichting het Burgerweeshuis Meppel."
In tfte worfá accoráing to tfte positivist, tlie inspiring tliing a6out scram6Ceá eggs is tliat any way you turn tliem tliey're sunny siáe up.
In tlie worlif accoráing to tfte q_istentia[ist, tlie liopefess tliing a6out scram6Ceá eggs is tliat any way you turn tliem tliey're scram6ff.:f.
Tom Robbins [Robb80]
Abstract
One of the most proruising ways to genera te electrical power in the next century is by nuclear fusion. This could be a safe and clean souree of electricity for which the fuels are abundant.
Most current research into nuclear fusion is directed towards magnetic confinement of ex
tremely hot plasmas in so-called tokamak devices. Befare fusion reactors will become operational, still many probierus need to be solved, both in fundamental and technological areas.
Among the farmer ones are the reasans for enhanced transport (which degrades the confine
ment) and processes of plasma-wal! interaction.
To address these questions, the RTP tokamak in the FOM-Instituut voor Plasmafysica in
Nieuwegein, The Netherlands, a limiter tokamak with major radius 0.72 mand minor radius 0.164 m, is equipped with several high-resalution diagnostics. One of them, the 80-channel
visible-light tomography diagnostic, is described in this thesis. The design of the system, its
characterization, and measurements are addressed. The system is suited for studying fluctuations and to deterrnine the local density of species that give rise to radiation, which is important
both for transport studies and plasma-wal! interaction.
The aim of this tomographic diagnostic is to resolve the local ernission of visible light in the
plasma from line-integrated measurements in one poloidal cross-section. The system views the plasma from five directions with 16 channels each. The detectors are sensitive in the wave
length range 300-1100 nm and optica! filters can be used to select a narrow range. In this
thesis ernission in the hydragen Ha line and continuurn radialion are studied. Corrections for
angle-of-incidence effects on interference filters have tobetaken into account. Viewing dumps are used to prevent reflections on the vessel walls, which would complicate the interpretation.
The bandwidth of the electranies is 200kHz, which enables the resolution of fluctuations. To
achieve this high temporal resolution, in combination with a high spatial resolution, optica! imaging systems close to the plasma are used. To correctly interpret the measurements taken by
the system, the i rnaging properties have been studied extensively, for exarnple in the framework of the so-called weighting matrix. The system has been absolutely calibrated for the wavelength ranges studied.
The most important analysis tooi of the measurements has been the tomographic inversion to obtain the local emissivity in the plasma. Two inversion techniques have been employed: a
constrained optirnization method, and a newly developed iterative projection-space reconstruction technique. Both methods have been tested by phantom calculations. In the case of relatively smooth and symmetrie phantoms good reconstructions are obtained.
Abstract
Nearly all emission profiles in the visible range in RTP are found to exhibit asymmetries. In
particular the asymmetrie profiles in Ho: light have been studied. Variations of at least a factor
of four in emissivity at the edge of plasma over varying poloidal angles are observed. The
asymmetrie peaks usually occur in the same places, but when the plasma is moved or the toroidal magnetic field is reversed the positions may change drastically. No convincing agree
ment has been found between the measurements and several possible causes for the asymme
tries that are suggested in the literature, such as local recycling from the limiter or wal! and
drifts. The findings of asymmetrie peaks by this high-resolution system might have implications for the understanding of plasma-wal! interaction, transport in the edge, and the interpreta
tion of emission measurements with less spatially-resolving systems on other tokamaks.
A quantitative analysis has been carried out of absolutely calibrated Ho: measurements. The
thickness of the radiating layer has been determined, as wel! as the neutral hydrogen density
inside and outside the asymmetrie peaks, and the partiele confinement time. Although an atomie
collisional-radiative model is used in the calculations, the influence of molecular processes is also considered. Quantitatively their influence is not profound, but the H! ions, appearing as
intermediale particles in the molecular processes, rnight have important consequences for the
localization of the emission.
From continuurn measurements the effective ion charge in the centre has been derived. In high
density plasmas the value found is in reasonable agreement with the value derived by other methods. In the wavelength range used, a significant amount of non-continuurn radialion influ
ences the determination at both the edge and in the centre, in particular in low-density plasmas.
Measurements during MHD activity have revealed that the Ho: emission is significantly influenced by magnetic island structures at the edge of the plasma, and that its relationship with the
local electron density is complex. Although the island structures rotate, the Ho: ernission seems to be related to the fluctuating electron density in the asymmetrical neutral hydrogen density peaks, but with varying dependences in different Iocations in the edge. lt is also found that the
neutral hydrogen density changes on the time scale of 0.1 ms. This indicates that neutral hydrogen density fluctuations rnight also occur in other fluctuating phenomena.
Incoherent fluctuations in ernission have also been studied. Evidence has been found of cen
timetre-sized structures at frequencies between I and 100 kHz. The system enables the study of corre lations between channels viewing from different directions, unlike systems on most other
tokamaks. The first results with this approach are prornising: significant correlations are found.
In actdition to the description of the essential aspectsof the visible-light tomography diagnostic and the physical results obtained, brief overviews are given of: radiative processes contributing
in the visible speetral range, the mathematica! background of tomography and various tomographic reconstruction methods.
vi
Samenvatting
Eén van de meest veelbelovende technieken om in de volgende eeuw elektrische energie te pro
duceren is kernfusie. Kernfusie kan een veilige en schone energiebron worden, waarvoor de
brandstoffen in overvloed voorradig zijn . Het meeste hedendaagse kernfusie-onderzoek is gericht op magnetische opsluiting van zeer hete plasma's in zogenaamde tokamaks. Voordat
fusiereactors operationeel kunnen worden, is nog veel onderzoek vereist, zowel op fundamenteel als op technologisch gebied. Voorbeelden van gebieden van fundamenteel onderzoek zijn: plasma-wand wisselwerkingsprocessen en de oorzaken van verhoogd transport (hetgeen
de opsluiting verslechtert).
De RTP tokamak in het FOM-Instituut voor Plasmafysica in Nieuwegein, een limiter tokamak
met grote straal 0.72 men kleine straal 0.16 m, is voor dit onderzoek uitgerust met een uitgebreide verzameling diagnostieken. Eén van deze hoge-resolutie diagnostieken, het 80-kanaals zichtbaar Jicht tomografie-systeem, wordt in dit proefschrift beschreven. Aandacht wordt
besteed aan het ontwerp van de diagnostiek, de karakterisatie en metingen. De diagnostiek
maakt het mogelijk om fluctuaties te bestuderen en om de lokale dichtheid van de atomen en
ionen die straling uitzenden te bepalen, hetgeen van belang is voor de studie van transport en plasma-wand processen.
Het doel van deze tomografische diagnostiek is het reconstrueren van de lokale emissie van
zichtbaar licht binnen in het plasma uit verscheidene lijn-geïntegreerde metingen in een poloïdale
doorsnede. Daartoe wordt het plasma uit vijf richtingen met16kanalen elk geobserveerd. De detectoren zijn gevoelig in het golflengtegebied 300-1100 nm; met optische filters kan een kleiner gebied gekozen worden. In dit proefschrift wordt de emissie in de Ha-waterstoflijn en
continuümstraling bestudeerd. In het geval van interferentiefilters zijn correcties voor de hoekafhankelijke transmissie noodzakelijk, hetgeen is bestudeerd. Het systeem is uitgerust met viewing dumps die reflecties aan de wanden voorkomen. De bandbreedte van de elektronica is
200kHz, hetgeen het oplossen van snelle fluctuaties mogelijk maakt. Voor een correcte inter
pretatie van de metingen zijn de afbeeldingseigenschappen van het systeem uitgebreid onderzocht. Het systeem is absoluut gekalibreerd in de onderzochte golflengtegebieden.
Tomografische reconstructie van de metingen is de meest gebruikte analyse-techniek. Twee ver
schillende reconstructie-methoden zijn gebruikt: een optimalisatie-techniek met randvoorwaarden en een nieuw ontwikkelde methode voor iteratieve reconstructie van de projectie-ruimte.
Beide methoden zijn in simulaties getest: wanneer de emissieprofielen relatief glad en symmetrisch zün, worden goede reconstructies verkregen.
Vrijwel alle emissieprofielen van zichtbaar licht in RTP vertonen asymmetrieën. Vooral de asymmetrische profielen van Ha-emissie zijn bestudeerd. Voor verschillende poloïdale hoeken
Samenvatting
worden variaties in emissie aan de rand van het plasma van tenminste een factor vier
waargenomen. De asymmetrische pieken verschijnen meestal op dezelfde plaatsen, maar kunnen drastisch veranderen wanneer het plasma wordt verplaatst of wanneer het toroïdal.e magneetveld wordt omgekeerd. Tot nog toe is geen overtuigende overeenstemming gevonden met in de literatuur gesuggereerde mogelijke oorzaken, zoals lokale recycling aan de wand en drift.
Het ontdekken van de asymmetrische pieken met dit hoge-resolutie-systeem kan gevolgen
hebben voor het begrip van plasma-wand wisselwerking en voor de interpretatie van emissie
metingen met lagere-resolutie-systemen op andere tokamaks
Absoluut gekalibreerde Hcx-metingen zijn quantitatief onderzocht. Zowel de dikte van de stralende Jaag, de neutrale waterstofdichteid binnen en buiten de asymmetrische pieken, als de deeltjesopsluitingstijd zijn onderzocht. Hoewel een atomair botsings-stralingsmodel is gebruikt
voor de berekeningen zijn ook de consequenties van moleculaire processen onderzocht. De
invloed van moleculaire processen is niet zo groot in quantitatieve zin, maar de H!-ionen, die als tussenstap in de moleculaire processen geproduceerd worden, zouden consequenties kunnen hebben voor de plaats waar de straling wordt uitgezonden.
Uit continuüm-metingen is de effectieve ionenlading in het centrum van het plasma afgeleid. In
hoge-dichtheicts plasma's wordt een waarde gevonden die in redelijke overeenstemming is met
waarden die op andere wijzen worden afgeleid. In het gebruikte golflengtegebied is er een sig
nificante hoeveelheid niet-contiuümstraling die de bepaling van de continuümstraling in zowel de rand als in het centrum bemoeilijkt, met name in lage-dichtheids plasma's.
Metingen tijdens MBD-activiteit laten zien dat de Hcx-emissie grote invloed ondervindt van magnetische-eiland structuren aan de rand van het plasma, en dat er een complexe relatie met de
lokale elektronendichtheid bestaat. Hoewel de eilandstructuren roteren, lijkt de Ha-emissie
vooral gerelateerd aan de fluctuerende elektronendichtheid op de plaats van de asymmetrische pieken in neutrale-waterstofdichtheid, met verschillende afhankelijkheden in verschillende posities. Verder is gevonden dat de neutrale-waterstofdichtheid varieert op een tijdschaal van 0.1 ms, hetgeen betekent dat fluctuaties in neutrale-waterstofdichtheid ook in andere fluctu
erende fenomenen kunnen optreden.
Ook incoherente fluctuaties zijn bestudeerd. Er zijn aanwijzingen gevonden voor structuren in
de grootte-orde I cm bij frequenties 1-100kHz. Met het huidige systeem is het ook mogelijk, in tegenstelling tot systemen op de meeste andere tokamaks, om correlaties tussen signalen van verschillende kijkrichtingen te bestuderen. De eerste resultaten zijn bemoedigend: significante
correlaties zijn aangetoond.
Naast de beschrijving van de essentiële onderdelen van de zichtbaarlicht tomografie-diagnostiek
en de verkregen resultaten worden ook korte overzichten gegeven van stralingsprocessen die bijdragen in het zichtbare gebied, de wiskundige achtergrond van tomografie en verscheidene
tomografische reconstructie technieken.
V lil
Contents
1 Introduetion ................................................................... . 1 1. 1 Nuclear fusion research and plasma physics . .... . ...... . . . .............. . . ... . .. . . . . ......... 1
1 . 1. 1 Thermonuclear fusion ..... . . . .................... . .. .. ......... . .. . .. . .... .. ....... ..... 2
1 . 1 . 2 The tokamak ............... . .. . ...... .... . . ................ . ..... . ... . ... . .... .. . ......... 3
1.1.3 Plasmas in tokamaks .. .......... .... .......... .. .... ........ .............. ...... ... .. .... 5 1.1.4 The Rijnhuizen Tokamak Project .... ........ .......................... .... .. ........... 8
1.2 Tomography .............. .. ....................................................... ......... .. ... 10
1. 2. I A short history of tomography .................. . . . . . . ... . .................... . ...... 10 1.2.2 Tomography in plasma physics research ... ... .. . .... ................... . ....... ... . 12
1. 2 . 2. 1 Specific probieros and opportunities of tomography in plasma
physics ...................... . ....................... .... . . .. ... .. . . .. .... .. . .. 12 1.2.2.2 X-ray tomography ....................... .............. .. .. ... .... ........ .. . 13
1.2.2.3 Visible-light tomography ................. . ..................... ...... ...... 13 1.2.2.4 Other types of tomography ............... ... ............. ........ .......... 14
1.3 Visible-light tomography on RTP ..... .. . ........ . . . .. . . ... . ... . . .. ...................... . . . . 15
1.3.1 Motivation to study visib1e light. ... ......... .. .. ... . ....... ... ...... . ...... .. .... .... 15
1.3.2 Choices for the diagnostic .. .......... ...... .. .. .... .. .. .. .. .. .... .. ..... ............ .. 16 I. 3. 3 Description of the main features of the diagnostic ........ .. .... .. ........... .... ... 17 I . 3.4 Consequences of the choices and new aspects of this diagnostic ... . . . . . ...... . . 18
1.4 This thesis .................... .... .... . ..... . ................... . .. . .. . .................. . ..... . . 19 1.4.1 Outline ................ ....... ......................... .. ................................. 19
1. 4. 2 Publications related to this thesis . . .... ..... . ..... .. ... . .. ............. . ......... ..... 20 1.4.2.1 1 ournals . ............... ... . ... . ..... .. .... . .. ... . . .. . .......... .. .. . .. . .. . .. .. 20
1.4.2 .2 Conference proceedings ............ . ...... . .......... . .. . . . . . ...... . ....... 20
2 Radiation processes in tokamaks ........................................ 23 2. 1 Line radialion ................. . ... . ... . ..................... . .......... .... ........ . . . . . . . .. ... . 23
2.1. 1 Saba and corona! equilibrium, and line emission .............. .... .... .. ..... .. .. .. 23 2. 1. 2 Emission from hydragen . . .. .... . . . . . . . . . .. .. . . ................. . .. . . .. . . . .. .......... 26
2.1.3 Emission from impurities ....... .. .. ............... .. ... .... ................ .......... 28
Contents
2.1.4 Transport, opacity and other complicating effects .. ... ... .......... .... ...... ... ... 30 2. 2 Charge-exchange recombination radlation .... . ... . . ... .. . ... . . .. . . . .. .. . .. . . . . ..... . .. . . .. .. 31 2.3 Bremsstrahlung and recombination radiation .. .... .... ... .... .. .. .. .. ......... .... .. ........ 33 2.4 Sumrnary ...... ... ....... . .... . . . ................ . .... . ... ................ . . . . . . ..... . ........... 35
3 Tomography and other analysis methods ............................. 3 7 3. 1 Basic mathematica! aspects of tomography ..... . . ........................... . . .............. 37
3. I . I The Radon transform .. . . ..... .. ...... .. ... ... . ....... ... . ... ............ . ............. 37 3.1 .2 The inverse Radon transform ..... . ........ . . ......... . .. .. .. . . .. .. . .... . . . .. . . . .. . .. . 39 3.1.3 Discrete descriptions ofthe Radon transform and its inverse ... .. ........ .. ...... 39 3 .1.4 IH-posed problerns . ........................ .... .................. ...... ................ 40 3.1 . 5 Properties of the Radon transform ....... .. ............ .. ...... .. .................... 41
3.1.5 .1 Properties of projection space .. ....... .... .. .... ......... .. ................ 41 3.1.5.2 Projection theorem ........ .. .... . ........... . .. . .. .... .. . .... ......... . .... . 43 3.1.5.3 Filtered backprojection .. ................ ... ........... ........ .. ... .. ... .... 44
3.1 .6 Terminology ...... ... ....... . .... . ...... . . . ............ . . .. ... . . . . ...................... 44 3.1 .7 Noise .......... ..... ................ .... ......................... .. . .. .......... .. ....... 46
3. 2 Some tomographic inversion methods ....... . . .............. . . . . . . .. . . . .. .. ... ... .. . . .... ... 48
3.2.1 Filtered back projection ........ .......... .. .. ........ .. .. .. .. .. .. ...... .. .... .......... 49 3.2.2 Fourier techniques . .. .. .... ....... ....... .. .. .. ...... ..... . ... . ...... .. .. .. ....... .. ... 50 3.2.3 Series expansion methods ........ .. .... ..... .... .. .......... .... .. .... .... .. ...... .. .. 50
3.2.3.1 Description of the Cormack method .............. ........ ................ . 51 3.2.3.2 Considerations when using the Cormack method and extensions ..... 52
3.2.4 Algebraic reconstruction techniques (ART) ................ .. ...... .. ............... 53 3.2.5 Optimization methods other than ART ......... ..... ...... .. .... .... .... .. ........... 55
3.2.5.1 Smoothness . . ... .. ... .. .......... . . . ........ .. .............. ........... . . . ... 56 3.2.5 .2 lterative versus non-iterative methods .... ...... ... .... .. .......... . .. .. .. 57 3.2.5 .3 Reconstruction method by Fehmers .. .. .. .... ......... .................... 57 3. 2. 5 .4 Maximum entropy .......................................... ................. 5 8
3.2.6 Application oftomography methods in plasma physics . . . . .... . ... . .............. 58 3.2.6.1 Other methods used in plasma physics ...... .... .... .. .. .... ............. 58 3.2.6.2 General considerations on tomography methods for the visible
light tomography system on RTP ....... .. . .. . . . . ..... . .......... . . . ...... 59 3.2.7 General remarks on implementing and testing algorithms .. .... .... .. .. .. .... .. ... 60 3.2.8 Some variations to straightforward tomography .. .. .. ...... .... ...... ....... .. .... 61 3.2.9 Some properties of kemels, apparatus functions and weight matrices in
conneetion with tomography ...... .. .......... .. .. ........... .. .... .. .. ............... 62 3. 3 An iterative projection-space reconstruction algorithm .... . .... . .... .. . . . .. . . .. . . ... . ... .. . 63
x
Contents
3. 3. I Introduetion and points of attention for the visible light tomography system
on RTP ........... .. .. ..... ... ...................... ... ............ .......... ........... . 63 3. 3. 2 Description of the reconstruction algorithm in projection space ..... . . . . .. . .... .. 65
3. 3. 3 Results for reconstructions in projection space ..................................... 68 3. 3. 3 .I Simulations on a fan-beam system ...................... . . ...... ... .. ..... 69
3.3 .3.2 Simulations on the visible-light tomography system ... ... ... ... . . ...... 71 3.3.4 Conclusions .. .. . ..................... . ......... .. . . . . . . ......... . . ....... .. . ... ... ... ... 72
3.4 Analysis methods other than tomography ... . ......... . ..... .. .......... . ........ . ... . ...... 73
3.4.1 Parametrization methods .. .... ........... . ........ . . . .......... . ........... . .... . . .... 73
3.4.2 Singular value decomposition .............................................. .. ........ 74 3.4. 2. 1 Principles of singul ar value decomposition .......... ...... ... . .......... 7 4
3.4.2.2 Biorthogonal decomposition ............. .. .................. ... .. ... ...... 75
3.4.3 Correlation analysis ............... ...... ...................................... ... ...... 77
4 The system for visible-light tomography on RTP .................. 7 9 4.1 Design criteria ... . ....... . ..................... . ... . . ..................... .. . . . ........... . . . .... 79
4. 1 . l Overview of the design .... .. ........ ... ......... ... ... .... .. ... . .. ....... ............. 79
4.1.2 Design tools: ray tracing ... ... .. ...... ... .... .... ..... . ........... .. .... ............... 82 4.2 Hardware .... ............................ ... .. ............... ..... ......... .. ....... . . . .......... 82
4.2.1 Windows ...... ..... ..... .. ................ .. . .. ... .. . .. . ..................... . . ... ...... 83
4.2.2 Optica! imaging systems ....... ... ..... ... .. .. .. .. .. ...... .............. .. .. .. .. .. .. ... 83 4 .2.3 Design ofviewing dumps and shields .. ............ .. ............... ........ .. .. .... 85 4.2.4 Detectors and electronics ... . ..... ................. . . . ...... . .. ... ..... . . .. ............ 87
4.2.4.1 Detector. .... .. ......... .... ................. . ................... .... . .. ... .... 88
4 .2.4.2 Transimpedance amplifier .. ... .. . .. .. ... ..... .. .. ... . . ... .... .. ... ......... 88 4 .2.4.3 Magnetic and electronic shielding .. .. .. ... .. .. .... . .. ............ .. ... .... 90
4.2.5 Data acquisition and processing ... . . . . .. . . .... . . . . . ... .. ... . .. ...... . . .. . . . . .. . .. . . . . 91 4.2.6 Positioning .... .. . . ... . ....... .. .. . .. ....... .. . .. ... . . .. . .... . . . . ...... ... ..... .... . ..... 92
4.3 The application of optica) filters ..................................... ... .......... .. ........... 92 4.3.1 Interterenee filters ........ . . . . ... ... .. . ............. . . . .. .. . ................... .. . ...... 93
4. 3. 2 Calculation of transmission of interterenee filters for arbitrary solid angles .... 93 4.3.3 Calculations on interterenee filters with a model transmission curve ...... . .. .. . 94 4. 3.4 Calculations of effective transmission curves for actual interterenee filters ..... 96 4.3.5 Considerations for other types of filters .. . . .. ... . . . . ...... ....... . ... .. . . ........... 97
4. 3. 6 Implementation of filter effects in ray-tracing calculations .. . ... .. ............ . ... 98 4.3.7 Conclusions ............ ... . .. . . ................... . ........ .. . ........ . ... . ..... . ....... 99
4.4 Sununary ... ..................... .. ... . ............................ . .. . .. . ............... ........ 99 Appendix 4.A Technica! details of the design .. .. ............ . ................................ . 101
4.A.1 Details of the windows ... . . .. . .. .... . . . . ..... . ...... .. .. . . .. . . .. .. . .. .. . . . . . ... .. . . . . 101
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Contents
4 .A.2 The blackening of viewing dumps and shields .............. ... . .. . . .. . . .......... 101 4.A.3 Details ofthe imaging systems .. ... . ... . . . .... ............ .. .... .... ... ... .. . . . .. .. . 101
4 .A.4 Details of the shielding of the cameras ...... ......... ... ........ ... .. ..... ......... . 102 4.A.5 Details ofthe positioning .. ....... .. . . ....... . .. .......... . . . . .. . . .. . . . . . . . . . . . .. . . . .. 103
5 Characterization of the RTP visible-light tomography system. 105 5. 1 The coverage of projection space .............. .... .. . . ............... . .. .. . ... . . ... ......... 105
5 .1 .1 The coverage of projection space for the visible-light tomography system . . . . 106
5 .I . 2 The relationship between the weight matrix and projection space .. ..... . . . ..... 107 5. 2 Description of the system by the weight matrix and by an approximation by line
integrals ........... ... . ....... . .. . ........ ... . . . . .... .. .. ......... . . . . . . . ... . . .. . .. . ............. 109
5. 3 Calculation of the weight matrix ..... . .......... . ... . .. ...... . ... . ... . ... . . . .......... .. .... . 110
5.4 Measurement of the weight matrix ... .. .... ....... .. ..... .. ... ..... .. .......... ......... ... .. 112
5.4.1 Description of the ex perimental set-up and the measurements . . . . ... . .. . . .. . . . . . 11 3 5.4. 2 V alidation of the measured weight matrix . . . ........... . ... . . .. . . .. . .. . . . . . . . ...... 11 3 5.4. 3 Calculation of the coverage of projection space from the measured weight
matrix ...... . . . ........ . . . .. . .. ...... . ... . ............. . .... .. . . . ...... . ...... . . . ..... .. . 118 5.5 Sealing factor. .. .. ... . ... . ... ........ . ..... .. .. .. . . ........ . . ...... ... .. ... . . .. .... ..... ........ 119
5.6 Calibration ...... . . .... ..... . . . . . .......... . . .... ........ . ............ . . . .... . .... ... . ......... . . 120
5.6.1 Geometrie part of weight matrix ... ........ ..... ................ .... ...... ....... .... 121 5. 6. 2 A spects of calibration ......... . . . . . . . . . . . . .. . . . . . . ....... . .. . . . .. . . .. . .. . . . .... . . . .... 121 5.6.3 Absolute calibration for Ha filters . .. . . . . . .... .. ... ..... . . . .... . . . . . .. . . .. . ... . ..... 122
5.6.3.1 Corrections for Ha. filters . ... . .................. . .. .. . . . ... .. . ..... . ...... 123 5. 6. 3. 2 Absolute calibration .... ... ....... . . . .............. .. . ..... . . ........ . ..... . 124
5.6.4 Absolute calibration for continuurn filters .................. . .... . . . .. . . ..... . ..... . 125 5. 7 Sununary . . . . ....... .. ..... . ... . . ... .. . . . .. .... . . . . .. . . . . ... .. ... .... . . . . ... . ......... . . ... . . . . . 126
6 Simulations and measurements of simple emission profiles . ... 127 6.1 Phantom calculations for tomography ........ .. ..... ........... ........... ..... ........ .... . 127
6. 1. 1 Noise estimate .... . . .. . . . . ..... . ..... ... ......... ........... . . .. . . .. . . . . ............... 127 6.1.2 Some aspects of phantom calculations ............. .. ... .. ... ... .... .............. .. 129
6.1.3 Comparison of the IPR and CO tomographic inversion methods .... . . . . . . . ... . 131 6 .1.4 The influence of phantom position on the quality of reconstructions .... . . . .... 135 6. 1.5 Conclus io ns .. .. ... . .. . .. .. ... . . . ... .. . . .. . .. . .. .......... . . . . .. ... ... . . .......... . ..... 138
6.2 Reconstructions of measurements ofsimple plasmas .. . ..... . . . .. . .. ... . .......... .. ... . . 138 6.2.1 Glow-discharge cleaning plasma ... . .. .... .. ........ . .. . .. . . .. .... . . . . . . . . .......... 138
6 .2 .2 Continuurn radialion . . ........... . ...... . . . .. .. . .............. .. ... . ..... . ....... . .... 141 6 .2 .3 ECRH-startup plasma . . . . . .. .. . . . .. . .. .. .. .... . .. . . . . . . . .... . . .. . ....... . . ... .. . ..... 142
6.3 Sununary .. . . . . . ... . ............... . ... . . .. . . .. . .. . .. . . .... ... . .. . . .. . . . . . .. . . . .. . .. . . .. . . . . . . . . 145
xii
Contents
7 Measurements of stationary asymmetrie emission profiles .... 14 7 7.1 Ha emission ................................................................... . ..... . ..... .. . .. 147
7.1.1 Emission profiles for different plasma conditions ................ .......... ....... l47 7. I. 2 Absolute emissivity and neutral hydragen density .. . ............. . ............... 151
7. I . 2. I Es ti mate of local emissivities from tomographic reconstruction .. .. .. 151
7.1 .2.2 Thickness of the radiating layer .......... ............. .. .. .............. .. 152
7. 1. 2 . 3 Asyrnrnetries . . . .. .. ....... . ...... . .... . . . . . ................ . . . . . . ........... 153 7.1.2.4 Neutral hydrogen density ........ .. ... .. ................................... 155 7 .1. 2. 5 Partiele confinement time ....... .. . . .......... . ... . ........ ... ........ . .... 156
7 .1.2.6 Processes contributing to Ha emissivity ................... .. .......... . 157
7.1.3 Start-up of discharge .................................................................. 160
7 .1.4 Summary of Ho: measurements .. . ...... . . .. .. . .. . ..... .... ......... . .... . . ........ . 164
7.2 Continuurn emission . .. .. .... . . ... ... . ... .... .. . . .. ........ . ... ......... . . . . ....... . ... .. . .. .. 164
7. 2. 1 Measurements of continuurn radiation ........... . . .... .. ...... . ..... ... . . .... . ... .. 165 7 .2.2 Tomographic reconstructions and determination of Zeff ............ . .... .. ... .... 166 7. 2 . 3 Summary of continuurn measurements .... .... . . . . . ...... . .. ......... .. . . .......... 170
7. 3 Total visible emission .. .. ... .. .... . . ...... .. ............ . ... . ......... . ... . . ....... .. . . . ...... 170
7 .3.1 Plasma position dependenee oftetal radialion .. .................... ... . .... . .... .. 171
7. 3. 2 Camparisen of the contributions from different wavelength ranges .... . . .. . .. . 171
7.4 Discussion on asymmetries .... ...... ....... .. ... .. .... .... ......... ......... ... .. .... ..... ... 173 7.4.1 Asymmetries in the literature ............. .... .................... ..... ... ... .. .... .. . 173
7.4 .2 Causes for toroidally asymmetrie partiele distributions ........................... 174 7. 4. 3 Causes for toroidal1y symmetrie, poloidally asymmetrie partiele
distri bution s ... . ............ ... .......... .... ... .. ................ .. ..... . ...... .. ...... 176 7 .4.4 Concl usions ... .. ... .. . . .. . . . . .. ..... ... . .. .. . .... . . . ....... .. .. . .. . .. . . ... .. . ... ..... . . 178
8 Measurements of MHD activity ........................................ 179 8.1 Introduetion MHD island structures .................... ... ................... . .... ......... 180
8.1.1 Theory .. . ............ . ...... . ........ . .............. . ... . ..... . . . ........ . .. . . .......... 180
8. 1. 2 Moti vation to study MHD island structures .... ..... ..... ...... ... ... .. . ........... 182
8. 1.3 Diagnostics observing MHD activity .. . ................ ....... .. .. ... .. .... .. ..... . 182 8. 1.4 Rotation .... ...... ..... .... .. . .. .............. .. .. .... .................... .... ....... .. . 184
8.2 A first look at the measurements . . . . ......... . . . .. . . . . . . . . ... ..... . ..... . . . ........ .. .... .. . . 187 8.3 Phantoms for simulations ................................................. .. .. ........ ........ 189
8.4 Analysis by tomographic reconstructions ........ ... .. ......... ................ .. ... ........ 193
8. 5 Analysis by correlation techniques .. ..................... .. ................................. 197 8. 6 Analysis by SVD .............. .. . .. ......... .. ........ .. .. . .. .... ........... .. ..... .... .. .. . .. 199 8. 7 Analysis of edge channels ....... . . . . . .... . . .. . . ... .. . . ... . . . ... . .. ..... . .. . . . . . . . .. . .. . . . . ... 202
xiü
Contents
8.8 Summary and conclusions ... ........ .. ... .......... ................. .................. .. .. . .. 203 Appendix 8.A Chopper spike remaval by SVD ..... ... ..... ... .. .... .. ....... .... .... ...... ... 204
8.A.1 Characterization of chopper spikes .............. .... .......... ............ ...... .. .. 204 8.A.2 SVD filtering of chopper spikes ..... ........ .... ..... .... .. .... ...... ......... .. .... 206 8.A.3 Conclusions ... ... .. . ............ ... .. . ........ .. ........ .. . .. . .. ....... .. ........... .. . 207
9 Measurements of fluctuations .......................................... 209 9. 1 Auctuation measurernents . . ........... ... . .. . .. ..... . . . . .. ...... . ... . . . .. . .. ........ ..... . ... 209 9. 2 Chord-averaged fluctuations of visible ernissivity ... .. ............. . .... . . . .. . . ... . ....... 210 9. 3 Analysis of visible-light measurements .............. . .... ... .... . ..... . ...... . . . ........... 212
9. 3. 1 Spatia-temporal structures ... . ..... .. ....... .. . . .... . ... . .. . ....... . ... .. . ........... 213 9.3.2 Temporal Fourier analysis ............ .. ... . ... . .... .. ....... ... ..... ... ............ . 216
9 .3.3 Correlation analysis . . ............. . .... ....... . . . ....................... . ............ . 217
9 .3.4 Spatial Fourier analysis .. .............. .... ......... ...... ...... . ...... ...... ....... .. 220 9.4 Conclusions ...... . .... . ..... ... . . ... .... . .. ........... .. ........ .. . . ............... . ..... . . ... . 222
10 Conclusions and recommendations ................................... 223 10.1 Conclusions . . . .... . . . ... . . . . ............ . ... . .. . . .. . . .. ....... . . . .. . ... . . .. . . ... . ..... . ..... 223 10.2 Recomrnentations .... . .. . ... ................ . ..... . . .. .. .. . .... . . ......... . .. .. ...... . .... . . 225
1 0.2.1 Improvements of the system .. .... ....... .. .... ............. . .. ...... .. . .. .. . .. .. 226 1 0.2.2 Improvements of the diagnostic methad .. .. ... . ............ .. ..... . .. . ......... 226 10.2.3 Suggestions for future measurements and analysis .. . . . .. . . .. . ...... . ........ 227
References ...................................... . .................................. 2 2 9
Acknowled gements .... ......... . .................................. . ... ... ......... 241
Curriculum vitae ....... . .. . ............................. . . . .... . ..... . . . .......... 2 4 2
xiv
Introduetion
1 The subject of this thesis is the visible-light tomography diagnostic which has been constructed for the Rijnhuizen Tokamak: Project (RTP). lts design and tools for obtaining and interprering
measurements are discussed, as wel! as various results. The purpose of the diagnostic is to
determine the spatial distribution of the emission in various parts of the visible spectrum: reconstructions from line-integrated measurements by a number of detectors viewing in one
plane are obtained by means of tomographic techniques. The determination of spatial character
istics with high temporal resolution can contribute to the understanding of the eh araeter of the dynamics of fusion plasmas. Three aspects that can be studied by means of this diagnostic are:
(1) the spatial distribution of line radiation and continuurn radiation, which gives information about the spatial distri bution of neutral particles (such as atomie hydrogen) and charged parti
cles (such as impurities); (2) the spatial-temporal behaviour of macroscopie instahilities in the
plasma; and (3) smali-scale fluctuations in the plasma. The methodology described in this thesis is partly applicable to fields other than nuclear fusion research, such as low temperature plasma physics.
In section 1.1 a brief overview is given of the field of nuclear fusion and tokamak physics,
emphasizing the importance of spectroscopie diagnostics fora greater understanding of physical
processes in tokamaks. Subsequently, insection 1.2, background is given on the application of
tomography inside and outside plasma physics, focussing on tomography systems for measurements in the visible range. The aim and main features of the visible-light tomography diag
nostic on RTP are discussed insection 1.3. Finally, section 1.4 summarizes the structure and objectives of this thesis.
1.1 Nuclear fusion research and plasma physics
Nuclear fusion is an area of physics currently receiving widespread attention. The ultimate aim
of the research is to establish a reliable, safe and inexhaustible energy souree that might contribute to the salution of the world energy problem. Research into nuclear fusion also provides interesting fundamental plasma physics. In this section a brief introduetion is given on ther
monuclear fusion, on the tokamak, i.e. the most important type of device to create the conditions for fusion , and on background of plasma physics. Furthermore, the objectives and char-
Chapter 1 Introduetion
acreristics of the RTP tokamak are presented. This is the device in the FOM-Instituut voor
Plasmafysica on which the research described in this thesis was carried out.
1.1.1 Thermonuclear fusion
When light nuclei are brought close together, they fuse, yielding new nuclei lighter than the
total mass of the initia! nuclei and, according to Einsrein's equation E = mc2, a surplus of
energy. Here, Eis the energy released, m the mass difference between the original nuclei and
the fusion products, and c the speed of light. The nuclear fusion process is the opposite of nuclear fission, in which a heavy nucleus falls apart into lighter nuclei, also producing energy.
Nuclear fusion is the energy souree of the sun and the stars. The fusion reaction that has a high
cross-sectionat the lowest temperature, i.e. the easiest one to achieve on earth, is the reaction between the hydragen isotapes deuterium (D) and tritium (T). The DT fusion reaction produces
a 4He nucleus and a neutron with a combined energy of 17.6 Me V. Deuterium is present in nature, whereas tritium has to be bred from lithium by nuclear reactions. Large reserves of both deuterium and lithium exist on earth. The DT mixture is often called "fuel" and the fusion reaction "burning," although no chemica! processes are involved.
Because the positively charged nuclei repel each other, the Coulomb force between them has to
be overcome befare the probability of fusion is sufficiently high. This requires the nuclei to be
heated to temperatures T higher than 108 K, equivalent to an energy kT> 10 keV, k being the Boltzmann constant. In plasma physics it is customary to omit the factor k in writing and to
express the temperature directly in electron volts (eV), i.e. in units of energy. At these temperatures the fuel is fully ionized and, therefore, in the plasma state. To produce enough energy in a reactor it is necessary to confine a dense enough plasma for a long enough time so that the
energy produced by fusion balances the energy losses. The required reactor product nTE of the ion density n and energy confinement time TE is 4 x 1020 m-3 s. Combined with the required value for the ion temperature T, this is often expressed as the value of the triple product nTET> 5 x 1021 m - 3 s keV.
Different methods to achieve thermonuclear fusion are being pursued in various experiments all
over the world. U neontrolled fusion on earth has been achieved in hydragen bomb explosions, where the necessary conditions to trigger the reaction are reached by compression, achieved by exploding an atomie fission bomb. To enable the useful extraction of energy from nuclear fusion, it is necessary to control the process. One way is to implade a smal! sphere filled with DT fuel by uniformly illurninating it with extremely intense laser or ion beams; this is called inertial confinement fusion. Another approach is muon-catalyzed fusion, where the electrans of
the fuel atoms are replaced by muons, causing the atoms to be closer to each other and requiring only little energy to overcome the fusion harrier. However, the most successful method, so far, is the magnetic confinement of a fuel mixture in the plasma state. Although different magnetic
2
Nuclear fusion 1.1
configurations are being considered, most work is going into the tokamak configuration,
described in more detail in the next subsection.
Over the past decades much progress has been made to achieve fusion conditions: roughly one
order of magnitude impravement in nrET every five years. Recently, breakeven has been
reached in principle, i.e . heating and energy production would have balanced if a DT mixture
had been used. Less than one order of magnitude increase from the present record value is
needed to reach ignition, i.e. a self-sustained burning plasma. Realistic values of the main
parameters for an ignited plasma are: n around 1020 m-3, TE of the order of a second, and T more than 10 keV. Unfortunately, achieving this goal requires a largerand more expensive
machine than the present ones. The International Thermonuclear Experimental Reactor (ITER),
a joint effort of the European Union, the United States, the Russian Federation and Japan, is
designed at present to reach the conditions for net energy production. ITER will be built mainly
on the basis of extrapolations by empirica! sealing laws obtained from large tokamaks around
the world. Greater understanding of the fundamental physics of plasmas in fusion devices
could imply that ignition could also be achieved in devices smaller than ITER by improved
control of operational parameters. This is a field where smal! tokamaks can contribute signifi
cantly. Larger tokamaks focus more on addressing teehoical challenges, such as how to extract
the energy from the plasma, the study of the fusion reaelions and the transport of the fuel and
"ash," and the reliable operation of reactor-type devices. At present most experiments do not
address fusion itself, but the conditions to make it possible. Hence only a few major tokamaks
use, or are scheduled to use, DT mixtures; tokamaks generally use deuterium or hydragen as
filling gas, to prevent radioactive contamination of the equipment.
1.1.2 The tokamak
The plasma in a tokamak is contained in a torus-shaped vacuum vessel. Magnetic fields are
used to confine the plasma and to achieve conditions for fusion, i.e. to separate it from the
material wal! and to compress it. The particular magnetic configuration of a tokamak was
invented in the late 1950s in Russia; its name is an acronym of TOpOIUallbHaR KAMepa 11
MArHMTHaR KaTywKa: toroidal chamber and magnetic coils. Since its invention, tokamaks
have been built all over the world in many different sizes and configurations. The world's
largest tokamak is JET in England, a joint European project. ITER, too, will be a tokamak.
The operation principle of a tokamak and the definition of coordinates is depicted schematically
in Fig. l . I . The principal magnetic field in a tokamak is the toroidal fie ld B~, which is pro
duced by a number of coils around the vessel, its direction being denoted by the toroidal angle
Ij>. Due to the toroidal shape, the coils are closer together on the inside and so produce a
stronger field there than on the outside: roughly B~"" !IR, where Ris the major radius of the tokamak. The inside, small R, is therefore sametimes referred to as high-field side (HFS), and
the outside, large R, as !ow-field side (LFS). The centre of the plasma is at R = Ro.
3
Chapter 1 Introduetion
Because of its non-uniform strength, the toroidal field is oot sufficient to confine the plasma, i.e. to balance the outward plasma pressure by an inward magnetic pressure. The non-uniformity and toroidal curvature cause a drift of the charged particles in the plasma, which is in opposite directions for ions and electrons. To compensate forthese effects, a second magnetic field is produced by a transfarmer inducing a toroidal current in the plasma, the plasma acting as the secondary winding of the transforrner. The transfarmer eperation requires a time-varying
current in the primary winding, which means that for a constant unidirectional plasma current and without additional measures the tokamak can only operate in pulses. The current can also be driven by non-inductive mechanisms, such as the injection of neutral beams or radio-frequency waves. The plasma current induces a poloidal magnetic field Be. In each poleidal crosssectien of the plasma, a system of coordinates can be defined [Fig. l.l(b)] : either polar coordinates (r,8), r =a being the minor radius, or Cartesian coordinates (R,Z), R being horizontal and Z vertic al. Together, B41 and Be give rise to a helical field-line structure as indicated in Fig. 1.1 (a), which in the ideal case form nested toroidal magnetic flux surfaces. Further coils are
( a )
r-----R
4
colls tor plasma posltion control and colls of prlmary winding transtormar
-, l [ TBV ' plasma
column
poleidal coils helical field lines
Figure 1.1 (a) Schematic representation of the eperation principle of a
tokamak device and (b) definitions of the coordinates in a poloidal plane. The
plasma current /p and toroidal, poloidal and venical magnetic fields, BtJl. Be and Bv. respectively, are shown in the directions that are usuaily used in
RTP. The coils for the horizontal and vertical plasma position control and the
primary winding of the transformer in reality consist of several coils.
Nuclear fusion 1.1
needed to position the plasma. A vertical field maintains the horizontal position by counteracting
the outward expansion of the plasma column due to the toroidal configuration. The vertical
position is maintained by a horizontal field . The current and the vertical and horizontal positions are controlled by feedback systems.
The toroidal plasma current Ohmically heats the plasma because the plasma is resistive. The
resistivity is proportion alto Te - 312, the electron temperature; therefore, the efficiency of ohmic
heating decreases with temperature. To reach ignition, additional healing is needed by injecting
beams of energetic neutral atoms into the plasma, or by Iaunching radio-frequency waves that are resonantly absorbed by the plasma.
The plasma is enclosed by a vacuum vessel. The purity of the plasma has to be strictly main
tained because impurities in the plasma would cause an increase of radiation and hence Ioss of
power. Preferably, contact between the plasmaand the material wall should be minimized to
avoid both the cooling of the plasmaand the contamination of the plasma byerosion from the wal I. In one approach to minimize the contact, the radius of the plasma is Iimited by a material contact at some point, defining the Jast closed flux surface. The material contact, called limiter, is either at one toroidal position or along the toroidal circumference of the tokamak at one
poloidal position. In an alternative approach, called divertor, the magnetic contiguration pro
duces an X point defining a separatrix, inside which the last closed flux surface is Iocated.
Plasma particles are diverted to target plates designed to absorb their energy. The divertor approach is used to test reactor conditions and the required extraction of fusion power and impurities. Outside the last closed flux surface the density is not necessarily zero, even though there is no toroidal confinement This part of the plasma is called scrape-off-layer (SOL). The vessel is usual!y made of stainless steel which is coated by a film of, for example, baron,
which has the property that it effectively retains impurities in the wall. Very cleanplasmascan
be obtained in this way. The vessel is filled with the operation gas before a plasma discharge is made by inducing a toroidal voltage by the transformer. During the discharge the density is
reguiared by puffing gas into the vessel. The gas diffuses into the plasma and ionizes. More directly, pellets of, for example, hydragen ice cao be shot into the plasma, depositing particles by evaporation closer to the core of the plasma. loos lost by the plasma are neutralized when
hitting the wall , and can enter the plasma again. This process is called recycling. Plasma-wall
interaction is an important research topic in the fusion community.
1.1.3 Plasmas in tokamaks
This subsection addresses some fundamental properties of tokamak plasmas related to studies by the visible-light tomography diagnostic. A more thorough treatrnent of these matters is given
in chapter 2 as well as in the chapters about ex perimental results.
5
Chapter 1 Introduetion
A plasma is a col! eetion of ionized atoms and free electrons. The low-density plasmas that occur
in tokamaks are quasi-neutra!, which means that the electron density ne is related to the ion
density n; by nee= L;Z;n;e, where Zie is the charge of the ions, e the electron charge, and the
summation goes over the ion species i. An important parameter of a plasma that describes its purity is the effective ion charge Zeff, defined as
( 1. 1)
where quasi-neutrality has been invoked in the last step. Zeff is an important quantity because it strongly influences the resistivity of and the radiation emitted by the plasma.
At temperatures occurring in the centre of tokamaks all but the heaviest impurity i ons are fully ionized. At the edge of the plasma the i ons can be in all possible ionization states. A variety of
radiation processes occur in the plasma. The collisions of the electrans with ions cause the
electrans to emit bremsstrahlung, which is continuurn radiation if the electron remains free after the collision. This radiation extends from the microwave to the x-ray speetral region. If the electron is bound after the collision, there is also a continuurn contribution called recombination radiation. The amount of bremsstrahlung and recombination radialion depends on the density,
temperature and purity of the plasma. Ions that are not fully ionized emit line radiation when
excited electrans decay to lower energy states. The line raillation produces lines in the spectrum
from the infrared to the x-ray speetral region. Fully or partially ionized ions can capture an electron from neutral atoms and emit line-radiation when the electron decays. This process is
called charge-exchange recombination. The power loss in tokamaks due to radialion can be significant and is therefore detrimental to the performance. At the same time the radiation is a valuable diagnostic tooi to probe plasma parameters in the interior of the plasma. The radiation
processes that are important for this thesis are discussed in more detail in chapter 2. Here it is sufficient to say that: (1) recombination radialion is usually negligible in the visible spectrum, and therefore Zeff can be derived from the bremsstrahlung measured in a line-free part of the spectrum; and (2) that line-radiation is roughly proportional to nenz, where nz is the density of the ion species with charge Z.
Another important type of raillation is electron-cyclotron radiation. The electrans and ions in the
plasma gyrate around the magnetic field lines. The cyclotron frequency ~ is
(L)c JqJB' (1.2) m
where q is the charge, B the magnetic field and m the mass of the particle. Radialion resonant with this frequency is in the radio-frequency range. Due to the curved path the particles experi
ence a constant acceleration, which causes the electrans to emit electron-cyclotron radiation (ECE) at this frequency or a higher harmonie. For an optically thick plasma, i.e. a plasma that is locally in thermal equilibrium at these frequencies, the plasma is a black body emitter. The
6
Nuclear fusion 1.1
intensity of ECE is proportional to the temperature, and therefore at the cyclotron frequency a local temperature can be measured, because at different major radii the magnetic field and hence
the frequency varies. This is the principle of the ECE diagnostic. The plasma can also be heated
by launching RF waves of this frequency or a higher harmonie into the plasma. The energy is transferred to the electrans by the resonance with the cyclotron motion, giving rise to electron
cyclotron resonant heating (ECRH).
Apart from the gyro-motion, the main motion of electrans and ions is bound to the magnetic
field lines, and, because the high temperature plasmas are largely collisionless, the transport is
mainly parallel to the field lines. Therefore, the temperature of both electrans and ions can be different in parallel and perpendicular directions, and also, the temperatures of electrens and
ions are not necessarily the same. Because of this fast transport along the field lines, most plasma parameters such as density, temperature and pressure are, to a very good approxima
tion, independent of the position on a toroidal flux surface. However, due to collisions and the
gyro-motion there is some perpendicular, i.e. radial, transport of particles. There is also a heat
flux connected to the partiele collisions. For electroos the thennallosses turn out to be several orders of magnitude greater than can be expected from collisions. Other mechanisms must be
responsible for this so-called anomalous transport, which at present is an important research topic. Electrostatic and magnetic fluctuations are possible candidates for causing enhanced
transport.
As noted in the previous subsection, the magnetic configuration of a tokamak ideally consists of nested toroidal flux surfaces (see Fig. 1.2). A useful quantity descrihing the magnetic field-line
structure is the normaJized toroidal pitch q of the field !i nes:
rBif! q =RoBe· (1.3)
Here, use has been made of the large aspect ratio approximation, i.e. Ro >>a. Perturbations are likely to be amplified on flux surfaces with rationat q, i.e. where field lines close onto themselves. Such instahilities at the edge of the plasma can
cause the plasma to disrupt, i.e. to end abruptly, and hence q is also referred to as safety factor. Perturbations in a
resistive plasma can cause the break-up of the magnetic field line structure, which, due to symmetry conditions in the tokamak, reconnect to form island structures between
the nested tori (see Fig. 1.2). These island structures are
often referred to as MHD islands or MHD activity, since
Figure 1.2 Schematic representation of the flux surfaces in a circu
lar tokamak. Some field I i nes on the surfaces are drawn, of which the
pitch has been exaggerated. A lso two island structures are shown.
7
Chapter 1 introduetion
they are described by the theory of resistive magnetohydrodynamics (MHD), which treats the
plasma as a single fluid. MHD islands and other MHD phenomena are observed in all tokamaks. Between the islands stochastic regions might appear which break up the flux
surf aces. Evidence has recently been found for this break-up of flux sulfaces into filamentary structures. In the centre of the plasma in the Rijnhuizen Tokamak Project large local variations
are observed in electron temperature and pressure mainly during additional heating, but also
during ohmic plasmas [LopS94]. Fluctuations at the edge are routinely measured in many
tokamaks. Clear evidence for filamentary structures at the edge of the plasma has been found by
looking at the visible radiation [ZweM89].
The current in the plasma causes motion of electrans and ions, which can be considered as two
fluids. The ion fluid carries most momentum. Because both the electrans and the ions carry the current, the fluid motion is nat uniquely determined by the current. Processes such as a rotatien, caused by a radial electric field which may exist due to different radial transport properties
of electrans and ions, can give rise toa toroidal rotatien of the fluid, whereas the poloidal rotation is strongly damped. Because of low resistivity in the centre of the plasma, due to the high
temperature, the magnetic structure is likely to move with the fluid. Hence, magnetic structures
may rotate toroidally.
1.1.4 The Rijnhuizen Tokamak Project
The aim of the ex perimental programme of the Rijnhuizen Tokamak Project (RTP) is the study of transport mechanisms in tokamak plasmas. Th is is possible even in a small tokamak because
the physical processes in the plasma are similar to those in large tokamaks.
Table I .I gives the key-parameters of RTP. To study the transport processes, RTP has equip
rnent to perturb the plasma: electron-cyclotron resonance heating and a pellet injector for hydragen pellets. RTP is a limiter tokamak, having a top-down limiter and a circular limiter in one poloidal plane. The stainless steel vessel is regularly boronized.
Overviews ofthe diagnostics programme are given in Refs. [Donn91, Donn94]. The following diagnostics are available on RTP:
• Various magnetic piek-up coils to measure the magnetic field outside the plasma. From these measurements the plasma position and current can be derived, which are used for feedback
control of these quantities. A lso, information about fluctuations and structures in the mag
netic field is gathered this way.
• A 19-channel far-infrared interremmeter [LamK90]. The phase shift due to the electron density of the plasma is measured along 19 parallel beams, yielding the line-integrated electron density. The diagnostic has been extendedtoa polarimeter, where the line-integrated Faraday
8
Nuclear fusion 1.1
rotation of the polarization gives information about the poloidal magnetic field component
parallel to the beam.
• A 20-channel heterodyne electron-cyclotron ernission (ECE) radiometer to measure the local radiation temperature of the plasma [Ge!H95]. The radiation temperature has to be corrected for optica! thickness effects, and elaborate calibrations are needed before measurements can
be interpreted. Recently the diagnostic has been extended to include electron-cyclotron
absorption measurements.
• Thomson scattering. A laser pulse scatters on the electrons in the plasma and the Doppier
broadening of the scattered light gives in formation on the Jocal electron temperature, and the intensity of the scattered light on the electron density. On RTP one laser pulseperplasma discharge is available. There is a single point system and a multiposition system [ChuB94],
the latter measuring the local quantities at 180 positions along a verticalline.
• An 80-channel x-ray tomography diagnostic. Line-integrated bremsstrahlung and recombina-
Table 1.1 Main parameters of RTP.
Quantity
Major radius
Minor radius: top-down limiter circular poloidallimiter vessel wa11
Plasma current
Loop voltage
Toroidal field at Ro
Number of coils
Pulse duration
safety factor at edge
Electron density
Ohmic input power
ECR input power
ECR frequency in centre of plasma for usual field
Electron temperature: Ohmic healing ECRH
Ion temperature (typically during Ohmic heating)
Effective ion charge
Energy confinement time: Ohmic healing ECRH (low power)
Symbol
Ro
a
B!p
fee
Te
T;
Zeff
Value
0.72 m
0 .164 m 0.180 m 0.235 m
40-145 kA
1.5- 3 V
1.9- 2.4 T
24
< 500 ms
2.2-7
0.3MW
0 .86 MW
60GHz
< 1 keV < 4 keV
= 0.7 Te
> 1.5
< 6 ms < 7 ms [KonH94]
9
Chapter 1 Introduetion
tion radiation in the soft x-ray range (1-lOkeV) is measured along 80 chords distributed
over five directionsin one poloidal plane [CruD94].
• Recently, some spectroscopie diagnostics have been installed: a visible spectrometer, vacuum and extreme ultraviolet (vuv and xuv) spectrometers, and a five-channel multi-layer mirror
spectrometerforsome impurity lines in the soft x-ray range.
• Other diagnostics include: a bolometer, a soft x-ray putse height analyzer, a Fourier-trans
form Michelson interferometer for ECE, a four-channel pulsed-radar microwave system and
a nine-channel ECRH transmitted power measurement.
1.2 Tomography
Tomography is a metbod to study the internal structure of objectsin a non-destructive way. The
word originates from the Greek word "LÓI!OÇ which means "cut," indicating that the internal
structure is obtained in cross-sections of the object. In this section a short history of the application of tomographic techniques is given, together with some basic background. The
application of tomography to plasma physics is discussed, with an emphasis on the application
of visible-light tomography on tokamaks. An introduetion to visible-light tomography on RTP is given in section 1.3. The mathematica! background relevant for this thesis is given in
chapter 3.
1.2.1 A short history of tomography
X rays have long been used as a diagnostic tooi in, for example, medicine. Common x-ray
photographs show only a projection of the absorption coefficient of the object under observa
tion. In the l920s a metbod of taking x-ray photographs was developed where the photographic
film and the x-ray souree are displaced in opposite directionsin parallel planes, with the patient in between. This produces a sharp image of only one parallel plane in the object, while details of all other planes are smeared out. This technique, that made it possible to study details of
cross-sec ti ons of bodies, was called, among other things, tomography.
When many projections (i.e. line integrations) are made of an object from many directions, it is possible to reconstruct mathematically the internal structure of the object. Application of such reconstructions from projections became feasible only after the advent of powerful computers. Image reconstructions have been made in astronomy fora long time using similar mathematica!
techniques. The first application of such techniques to resolve internal structures was by x-ray imaging in medicine in the 1960s. This technique is called computerized tomography (CT),
indicating that the result is obtained by numerical techniques. Another name is CAT, which stands for computer assisted tomography, computer aided tomography or computerized axial tomography. CT quickly became an important and widely-used diagnostic. In 1979 the Nobel
10
Tomography 1.2
prize in medicine was awarded to G.N. Hounsfield and A.M. Cormack fortheir contributions
to the development of x-ray CT.
After the first application of x-ray CT, the technique was adopted for many different purposes:
for other types of radiation in medicine, and in science and industry. The comrnon feature is
that a quantity of the object to be studied is measured along many viewing lines, or approximate
viewing lines. Hence, the measurements are line integrated and the mathematica} techniques of
CT are used to approximately determine the local quantity. The quantity to be determined is
generally the local absorption coefficient of the object for a certain type of radiation, or the
power emitted locally by the object. The former is called active CT: a souree outside the object generates radiation and the transmitted power is measured. The latter is called passive or emis
sion CT because the object itself generates the quantity to be studied and no external souree is
required. Examples of active CT are medica! x-ray CT and interferometric prohing of the elec
tron density in a plasma, whereas the measurement of decay products of radio-isotapes injected
into a parient and the measurement of the radialion emitted by a plasma are examples of passive
CT. In genera!, CT is applied to cross-secrions of the object: the arrays of detectors and, in the
case of active CT, the souree are in one plane. This plane can be scanned through the object to
obtain a three-dimensional understanding of the internal structure, but truly three-dimensional
CT is also possible with by viewing the object from many directions.
In the simplest case of CT, the quantity is integrated along straight lines. Even in this case,
complications such as scatter, diffraction and re-absorption of emitted radiation can occur. Furthermore, the finite widths of the inlegration paths sametimes cannot be ignored. Also
refraction can severely complicate the problem: it causes the inlegration to be along curved
lines, the pathof which depends on the internal structure to be resolved.
For different applications many of these complications have been solved so that approximate
internal structures of objectscan be obtained. Examples of the application of CT in medicine are the transmission of x-rays, positron emission tomography (positrons resulting from radioactive
decay), and certain applications of the transmission of ultrasound and nuclear magnetic reso
nance. Si mi lar techniques can be applied for nondestructive testing of the internal structure of
materials. Examples are the study of internal structures in rocks, and the checking of fuel rods
for nuclear power plants. Th ere are also many applications of CT -techniques in geophysics (seismology), atmospheric or ionospheric studies and astronomy. The applied mathematica!
techniques have much in comrnon with those used in image processing. More detailed descrip
tions and references to the many fields where CT is applied can be found in, for example, Refs.
[Herm80, Dean83, PikP83].
Although the expression "computerized tomography" is more precise, in this thesis the word "tomography" will be used, which is guite common practice in plasma physics. The word
"tomographic reconstruction" is used as synonymous with "approximation of the internal
11
Chapter 1 Introduetion
structure by CT from many line-integrated measurements, possibly combined with additional information." All tomography described in this thesis is computerized, but in some other fields it is possible to obtain tomographic reconstructions by analogue means (for instanee by opties)
[BarS77; BarS81, Ch. 8].
1.2.2 Tomography in plasma physics research
Tomography is applied to measurements in both low and high-temperature plasma physics. In this subsection attention is given only to high-temperature plasma physics, as this is the field of
interest for this thesis. Much of what is said in this sec ti on, however, is aiso applicable to lowtemperature plasmas. First, the application of tomography to tokamaks is descri bed, foliowed
by a short overview of soft x-ray emission tomography, tomography of theemission in the visible region, and other applications.
1.2.2.1 Specific problems and opportunities oftomography in plasma physics
A plasma is in general a turbulent medium with many structures. Tomography is a way to
resolve the in tema! structure of plasmas from line-integrated measurements. Many of the interesting plasma phenomena have a high frequency, requiring time-resolved measurements to be properly diagnosed. This is a major difference with mostother applications of tomography,
where the object under study does not change in time and large amounts of data can be collected
with, for example, only one moving array of detectors. Even though in medica! tomography time-resolved measurements are made (for example of the heart), a temporal resolution of the
order of tentbs of seconds is usually sufficient, whereas in plasma physics typical time scales are of the order of milliseconds and microseconds. Furthermore, access to plasmas is, in gen
era!, restricted: coils and other structures around a tokamak, as wellas the limited number and size of ports, prevent the viewing of the plasma from all directions. Forthese reasoos the detectors need to be fixed. The number of detectors is mainly determined by their cost. In plasma physics there is, in genera!, several orders of magnitude less data available for a tomographic
reconstruction at one point in time than, for example, in medica! tomography, which severely
limits the spatial resoiution that can be achieved. One advantage is that in plasma tomography temporal information can be taken into account; methods to take into account temporal informa
tion are, however, not abundant in the literature. Some of these methods are mentioned in chapter 3 (mainly subsection 3.2.2). In tokamak plasmas, refraction and re-absorption are negligible in the ranges of visible radialion and x rays.
Due to limited access and expensive detectors and electronics, in many applications of multichannelline-integrated measurements on fusion devices all channels view the plasma from only
one side. To reconstruct the local emissivity of the plasma, assumptions about symmetry are needed: a so-called Abel inversion is done, which is a tomographic inversion under the assumption of circular symmetry. This has been widely applied to all wavelength regions where
12
Tomography 1.2
radiation is emitted by the plasma. More views are needed to obtain more detailed information
by tomographic reconstructions withno or fewer assumptions about symmetry. Some of these
applications of tomography are briefly reviewed.
1.2.2.2 X-ray tomography
X-ray tomography has become a standard diagnostic on many tokamaks. It is commonly used
to study different types of instahilities in the plasma, for example the sawtooth instability (an
instability which causes a recurrent, fast loss of confinement in the care of the plasma, limiting
the obtainable central pressure). The x-ray emission is a function of plasma density, temperature and impurity concentrations [Hutc87]. These plasma parameters are, at most conditions,
roughly constant on magnetic flux surfaces, and therefore the surfaces of constant emissivity are closely related to the magnetic surfaces. Until some years ago x-ray tomography systems on
tokamaks had up to three viewing directions with a limited number of detectors, e.g. Refs.
[CamG86, GraS88]. This limited number of viewing directions allowed to resolve only the main features of the plasma [GraS88], and made it difficult to study for example the spatial structure of MHD phenomena. Attempts have been made to increase the effective number of
viewing directions by assuming rigid rotation and consictering the measurements at several
points in time; references are given in subsection 3.2.2.2 where this approach is discussed.
Recently, on several tokamaks systems have become available that have five or more indepen
dent views, for example on: TdeV [JanD92], RTP [CruD94] , Alcator C-MOD [GraW91], JET [AlpB94], ASDEX Upgrade [BesM94] and TCV [AntD95]. In theory such systems, withof the order of 100 measurements, can achieve a higher resolution. However, when compared to
medica! tomography with of the order of los measurements, it is clear that specific tomographic
reconstruction methods are needed in plasma physics that take into account the limitations
resulting from the small number of detectors and viewing directions. This can be done by using
a priori knowledge and by making more stringent assumptions than in medica! tomography, for example a bout the smoothness of the emission pro files. Therefore, even with the modern multi
camera systems on tokamaks, the amount of detail that can be extracted from the measurements is in general limited. Methods are being developed that can extract quantitative information about certain features from tomographic reconstructions, for instanee [TanB95]. X-ray radia
lion is mainly emitted from the centre of the plasma, where the temperature is high. To study structures in the edge of the plasma other techniques than the measurement of x rays have to be utilized.
1.2.2.3 Visible-light tomography
Emission in the ultraviolet (uv), visible or infrared (ir) part of the spectrum takes place in par
ticular at the edge of the plasma, and can therefore give inforrnation about structures at the edge. While x-ray tomography has been widely applied as a plasma diagnostic, emission tomography in the near-ir, visible and near-uv is not very common. Usually multichannel measurements
13
Chapter I Introduetion
from only one view combined with an Abel inversion, assuming cylindrical symmetry, are employed to obtain an impression of the radial profiles. Examples of such determinations of line radiation profiles are described in Ref. [SucH78] and for the determination of Zeff in Refs. [FooM82, Kad080, GuiM94, ParA93]. In the past, visible-light tomography has been used to study spatial profiles of line-emission [MyeL78, KutL88, Kut091] and bremsstrahlung [SugG91], as wellas the total speetral emission in a broad wavelength range [Ho!N86]. For these studies on the emissivity in a poloidal plane either a very limited set of detectors or a moving detection system was used, resulting in a low spatial or temporal resolution. More recently, systems with more viewing directions and detectors have become available. Such systems have different objectives and performances, for example to measure the emission profiles of impurity lines [TerS95, Kurz95] or of Ha radiation [KurS95, Kurz95, IwaY93] (Ha radiation is the name for the Balmer speetral line of hydrogen at a wavelength of 656.3 nm, which results from the transition of an electron from the n = 3 to the n = 2 level, n being the principal quanturn number; it is in general the strongest contribution to visible radiation). In some systems the spatial resolution is very high, while the temporal resolution is low [Tak092]. On many tokamaks multichannel systems are installed for spectroscopie measurements, for line-radiation, determination of Zeff, measurements of di vertor plasmas, etc. Tomography is usually not possible with such systems unless detailed assumptions about the expected emission profiles are made. Therefore, there is no extensive literature about detailed tomo
graphic studies of emitting structures in the visible region.
The systems mentioned above view the plasma in a poloidal plane. It is also possible to reconstruct the local emissivity from a tangential view, for example a measurement with a twodimensional camera. To invert these line-integrated measurements to local emissivities certain assumptions are needed about the symmetries in toroidal direction, e.g. constant emissivity along field lines. Techniques similar to tomography are used for the inversion. This has been applied for Zeff measurements in, for example, Ref. [SchS88], while for the x-ray region applications of a tangential view is described in Refs. [TakT86, lwaT87, DurC90, GoeA92].
1.2.2.4 Other types of tomography
Besides tomography in the x-ray and visible speetral ranges on fusion devices, tomography is also done with bolometer arrays, see e.g. Refs. [Hart90, FucM94, Nav095], neutron and y-ray cameras [Ada193], and interferometers [BroJ92, HowW93]. The algorithms that are used in tomography can also be applied to certain other inverse problems (for example some types of deconvolution), in which case the spatial variabie in tomography might be replaced by some other quantity.
14
Visible-light tomography 1.3
1.3 Visible-Iight tornography on RTP
1.3.1 Motivation to study visible light
Various phenomena in tokamaks can be studied by visible-light tomography. The knowledge of
the local emissivities in, for example, a poleidal plane can be very useful to understand the
physical background of various plasma phenomena. Generally, only line-integrated measure
ments can be taken of theemission of an object, and therefore tomographic inversion of these
measurements is needed to probe the internal structure of an object.
As described in subsection 1.1.3 there are several contributions to the emissivity in the visible
wavelength range: continuurn radiation (bremsstrahlung) and line radiation (of hydragen and
impurities). Structures in the plasma appear differently in these different components since they
are caused by different physical processes. Possible fields of application of visible-light tomog
raphy are:
( 1) It can complement x-ray tomography systems in the study of MHD phenomena, because in
smal! tokamaks such as RTP many of the interesting structures are in regions where the x
ray emissivity is low. On RTP the x-ray and visible-light tomography systems are located
in approximately the same poloidal cross-section. Therefore results from bath diagnostics
can in principle be interpreted without the necessity to take into account time delays or curvature of field lines. Unfortunately, at this time part of the data acquisition has to be shared
between the two diagnostics, so that bath full systems cannot be used at the same time.
(2) The turbulent plasma is thought to contain many small self-organized structures, such as
filaments and vortices. These structures are expected to have si zes of the order one centi
metee in RTP. Tomography is a way to spatially resolve structures. Structures might, however, be too small to be resolved by tomography with a limited number of detectors. In
such cases multi-channel measurements along narrow chords from many different direc
tions can give useful information.
When selective optica! filtering is used, further possibilities are:
(3) The emission profile of Ha radialion can give information on the neutral hydragen density and on recycling mechanisms of particles from the wal!.
(4) Line radiation from impurity atoms gives information on the densities ofthose species in a
eertaio ionization stage.
(5) From the bremsstrahlung the local value of Zeff, the effective ion charge of the plasma, can be derived, if also the local electron density and temperature are known. Usually on toka
maks only an average Zen is known (from line-integrated measurements for example),
15
Chapter 1 Introduetion
while knowledge of the Zeff profile would be very useful to understand the current distribution, the impurity distribution, et cetera.
In this thesis measurements for all applications, except (4), are discussed.
1.3.2 Choices for the diagnostic
The visible-light tomography system on RTP was designed to be used for all applications men
tioned in the previous subsection. Different wavelength ranges can be selected by optica) filters.
The requirements for the diagnostic determine the position of the diagnostic, and the spatial and temporal resolution. These are described in the following.
The plasma in a tokamak rotates in the toroidal direction. This means that any feature in the
plasma that is not localized toroidally will pass every poloidal plane. Features in the plasma that
do not change shape during one rotation and are symmetrical over the flux surfaces wiJl not
give rise to any changes when observed in one poloidal cross-section. Some perturbing features, however, follow the field Jin es and therefore have helical symmetry, resulting in an apparent rotation of the structure in the poloidal plane due to the toroidal rotation. For many phenomena it is therefore sufficient to study the local emission in one poloidal cross-section to obtain information about the entire plasma. The diagnostic was designed to measure in one
poloidal cross-section, enabling tomographic reconstructions of the emissivity in this crosssection. For a full determination of the toroidal structure of the features observed, however, additional simultaneous measurements at other poloidal cross-seelions might be needed. There
are also phenomena in tokamak plasmas that depend on the toroidal position, for example the ablation of a pellet injected at one toroidal position.
For tomographic inversion a large number of measurements from many viewing directions is needed. The spatial resolution necessary to visualize small structures such as filaments by standard tomography would require of the order of 1000 detectors distributed overtensof viewing directions. With many measurements in time it might be possible to obtain information about structures smaller than can be resolved by standard tomography if assumptions about the tem
poral evaJution (for example rigid rotation) can be made. To resolve structures in the plasma or to perfarm correlation analysis between channels, viewing chords with a width at most compa
rable with the dimensions of these structures are needed so that the structures are not smoothed out in the integration over the width of the chord.
Many phenomena such as turbulences take place on time scales slow er than I o-s s, and therefore this was taken as a requirement for the temporal resolution of the diagnostic. The required large bandwidth of the deleetion electranies makes it necessary to measure all detectors in paral
lel (i.e. no multiplexing for example). Therefore the number of detectors is mainly determined by the cost of the detectors, parallel electtonics and data-acquisition system. Because of the required bandwidth, the amplification factors in the detection system cannot be very high (high
16
Visible-light tomography 1.3
amplification makes the electranies slower). Tagether with the integration over only narrow
chords, this might result in a too low sensitivity if a conventional pin-hole system is used,
especially in combination with narrow optica! filters. Therefore imaging systems have been
designed, resulting in more signa! and a better signal-to-noise ratio. The imaging systems have
the further advantage that they can give the required narrow viewing chords. With the current
number of detectors, 80, the required coverage with sufficient spatial resolution cannot be
achieved if the viewing chords are distributed evenly over the plasma. Because most visible
light is ernitted from the plasma edge, smal! structures can be resolved most successfully if the
viewing chords are more concentraled at the edge.
In order to obtain a good signal-to-noise ratio the amount of light that i:; observed by a detector
for a given ernissivity can be further increased by viewing a larger ernitting volume. The size of
the volume in the poloidal plane is already deterrnined by the required resolution. The thickness
of the cross-sec ti on, i.e. the tomidal ex tent of the measuring volume, can be optirnized to yield
as much signa! as possible. For this optimization an estimate is needed of the maximum allow
abie thickness in toroidal direction in relation to the temporal resolution and toroidal velocity of
structures in the plasma. Typical toroidal veloeities of the fastest struc:tures to be measured in
RTP are of the order of 104-1 os mis, corresponding to a frequency of the order of 10 kHz. At
a sampling frequency of 500 kHz or 1 MHz an average measurement is obtained over 5-10 cm
in the toroidal direction, giving the maximum extent in toroidal direction corresponding to the
sampling frequency.
1.3.3 Description of the main features of the diagnostic
The diagnostic was designed to be implemented on diagnostic ports of the vacuum vessel of
RTP which have to be shared with the soft x-ray tomography diagnostic. The resulting lirnited
space, tagether with the requirement of good resolution and mainly measuring the edge, deter
mined the distri bution of the views of the plasma. The plasma is viewed from five directions
(five cameras) with 16 channels each. The cameras view only part~: of the plasma from as
diverse angles as possible. For tomography, however, a good coverag<! of the plasma is essen
tial, and as a campromise a two-fold overlap was chosen (i.e. every region in the plasma is
seen by at least two detectors, and there are as few non- or singly-covered regionsas possible).
Two-fold coverage is the minimum to do tomography without assumptions about symmetry; it
enables an approximate reconstruction of the emission profile. For tomographic inversions
knowledge about theemission from the plasma centre is also needed, and therefore one camera
views through the centre. The system is sensitive to light in the range 300--1100 nm. For rea
sans of simplicity and lack of space, it is preferabie to have the detectors outside the vacuum
vessel. This, however, requires vacuum windows, but also makes it possible to put changeable
optica! filters in front of the detectors to select the wavelength range that is observed.
17
Chapter I introduetion
Figure 1.3 The centrallines-of-sight of all de
tectors of the visible-light tomography system in
a poleidal cross-section of the tokamak. The op
tica) components are indicated. Close to the mir
rors, detectors and lens these central chords lose
their meaning and it is more appropriate to con
sider all possible rays as is done for Fig. 4.1.
The bandwidth for the electronics was chosen to be 200 kHz. Four of the cameras have imaging systems consisting of two spherical mirrors, while the fifth
camera has a lens system instead. The
optica! imaging systems with mirrors are
positioned close to the plasma, inside the vacuum vessel. These imaging systems also allow the plasma to be viewed from
more extreme angles than otherwise would have been possible. The coverage
of the system is depicted in Fig. 1.3.
In Table 1.11 the main features of the system are summarized. In chapter 4 a detailed description of the diagnostic is
given.
detector
Table 1.11 Main features of the visible-light tomography system.
Total number of detectors
Number of viewing directions
Bandwidth
Wavelength range of sensitivity
80
5 with 16 detectors each
200kHz
300-1100 nm
1.3.4 Consequences of the choices and new aspects of this diagnostic
The visible-Iight tomography system on RTP is a versatile diagnostic for different wavelength ranges, unlike most other systems discussed in subsection 1.2.2. Furthermore, the system
consists of a relatively large number of views and detectors compared to most other systems operational at the time of its design. The light is colleeled by optica! imaging systems close to the plasma, which is a new concept for tomography systems on tokamaks. In other systems
somelimes Ienses are used, but usually relat.ively far from the plasma so that their viewscan be
18
Visible-light tomography 1.3
considered as parallel beams. Imaging systems such as the present one collect more light, but
make the interpretation of the measured signals more difficult. A proper cal i bration of the sys
tem is complicated, and, because of incomplete views of the plasma by each camera and differ
ences between the imaging systems, the signals cannot be processed directly by tomographic inversion algorithms. This means that the characteristics of the imaging systems require much
attention.
It should be stressed that cameras with narrow viewing chords observing only part of the
plasma are not the best choice for tomographic inversion. A clear distinction should be made
between "optical resolution:" the effective width of the viewing chords; and "tomographic resolution:" the smallest structure that can be reliably inverted. Better tomographic inversions are
possible if all cameras view the entire plasma, whereas in such a case with wider chords many
fluctuating phenomena that could be studied by for example correlation analysis are smoothed
out. The current system is a campromise between the requirements for tomographic inversions
and for other analysis methods of various phenomena.
The advantages, disadvantages and characterization of the system are described in chapters 4
and 5, while a new tomographic inversion methad to handle the probieros caused by the chosen
campromise fortheset-up is introduced insection 3.3.
A way to make optima] u se of the system is to not only consicter the measurements at one point
in time as in conventional tomography, but to consicter measurements at many times simultaneously. This can be done by a step-by-step refinement investigating several phenomena with dif
ferent space and time scales, for example first investigating a time averaged signal and subse
quently studying smaller rnaving structures that cause variation of signals in time. The emissiv
ity of the plasma has, however, been found to show more variation than expected and to give
rise to asymmetries in the profiles that complicate tomographic reconstructions due to the rela
tively smal! number of channels and the limited coverage. Therefore it tums out that it is already difficult to obtain information for just one point in time from a specialized tomographic inver
sion method, making it hard to extract smal! structures even if time is taken into account. More
advanced methods, for example combining correlations and tomography in a more direct way,
remaio to be developed.
1.4 This thesis
1.4.1 Outline
The objective of this thesis is to describe the visible-light tomography diagnostic on RTP and
the results achieved with it. For this, the aim of the diagnostic is described and some background of plasma physics that is used to interpret measurements is given. Much attention is
given to tomographic inversion methods, partly because many aspects from different methods
19
Chapter 1 Introduetion
are used in the method developed for the diagnostic, but also because it is hard to find brief reviews of different methods in the literature. The main part of the thesis is devoted to the description of the diagnostic, its application, and the interpretation of measurements.
The aspects which were introduced in this chapter are discussed in a more detailed way in the next chapters. Chapter 2 focuses on the ernission of visible radiation in plasmas. In chapter 3 an overview of tomography and tomographic inversion methods is given, a new tomography method is introduced, and other analysis methods are discussed. Chapter 4 describes the design of the visible-light tomography diagnostic for RTP. In chapter 5 the system is characterized so that it can be used for tomographic inversions, of which examples are given in chapter 6 for simulated and actual ernission profiles with a simple shape as a validation of the tomography methods . In the subsequent chapters attention focuses on measurements on RTP: stationary and slowly varying phenomena in chapter 7, MHD phenomena in chapter 8, and fluctuations in chapter 9. Finally, a summary and proposals for the future are given in chapter 10.
1.4.2 Publications related to this thesis
1.4.2.1 Joumals
L.C. Ingesson, J.J. Koning, A.J.H. Donné and D.C. Schram, "Visible light tomography using an optica) imag
ing system", Rev. Sci. Instrum. 63, 5185 (1992). (Related to chapters 4 and 5.)
L.C. Ingesson, V.V. Pickalov, A.J.H. Donné and RTP-Team and D.C. Schram, "First tomographic reconstruc
tions and a study of interference filters for visible light tomography on RTP," Rev. Sci. Instrum. 66, 622
(1995). (Related to chapters 4, 5 and 6.)
V. V. Pickalov, L.C. Ingesson, "An iterative projection-space reconstruction algorithm for visible light tomog
raphy on the RTP tokamak," to be published. (Related to chapter 3.)
L.C. Ingesson, A.J.H. Donné, D.C. Schram, "Visible-light tomography on the RTP tokamak," to be published.
(Related to chapters 4, 5, 6 and 7.)
L.C. Ingesson, "Approximations in the description of the physical properties of imaging systems for tomogra
phy algorithms," to be published. (Related to chapter 5.).
1.4.2.2 Conference proceedings
L.C. Ingesson, J.J. Koning, A.J.H. Donné, D.C. Schram, "Visible light tomography on RTP," in Diagnostics
for contemporary fusion experiments (Proceedings of the Workshop, Varenna, 1991), Ed. P.E. Stolt et al.
(Socièta Italiana di Fisica, Bologna, 1991), ISSP-9, pp. 901. (Related to chapters I, 2, 3, 4 and 5.)
V.V. Pickalov, L.C. Ingesson, A.J.H. Donné and D.C. Schram, "Tomography algorithms for visible light
tomography on RTP," Proceedings ofthe 1992Jnternational Conference on Plasma Physics, Innsbruck, 29 June
-3 July 1992, Ed. W. Freysinger et al., Europhysics Conference Abstracts Vol. 16C (EPS, 1992), Part IJ, pp.
1143. (Related to chapters 3 and 5.)
20
This thesis 1.4
L.C. Ingesson, V. V. Pickalov, A.J.H. Donné, D.C. Schram, "An iterative projection-space reconstruction algo
rithm for visible light tomography on the RTP tokamak," Oral presentation at the 1993 International Sympo
sium on Computerized Tomography, Novosibirsk, 10- 14 August 1993 (unpublished). (Related to chapter 3.)
L.C. Ingesson, V. V. Pickalov, A.J.H. Donné, D.C. Schram and RTP-Team, "First results with the visible light
tomography system on RTP," Proceedings of the 20th EPS Conference on Controlled Fusion and Plasma
Physics, Lisboa, 26-30 July 1993, Ed. J.A. Costa Cabral et al. Europhysics Conference Abstracts Vol. 17C
(EPS, 1993), Part III, pp. 1147. (Related to chapters 3 and 5.)
L.C. Ingesson, V.V. Pickalov, A.J.H. Donné, D.C. Schram and RTP-Team, "Observation of MHD phenomena
with the visible light tomography system on RTP," Proceedings of the 21st EPS Conference on Controlled
Fusion and Plasma Physics, Montpellier, 27 June-I July 1994, Ed. E. Joffrin et al., Europhysics Conference
Abstracts Vol. 18B (EPS, 1994), Part III, pp.1316. (Related to chapters 3 and 8.)
L.C. Ingesson, A.J.H. Donné, D.C. Schram and RTP-team, "Poloidally asymmetrie emission of visible light in
RTP," Proceedings ofthe 22nd EPS Conference on Controlled Fusion and Plasma Physics, Boumemouth, 3-7
July 1995, Ed. B.E. Keen et al., Europhysics Conference Abstracts Vol. !9C (EPS, 1995), Part VI, pp.337.
(Related to chapter 7.)
21
Radiation processes tokamaks
• ID 2
In this chapter a brief overview is given of the different racliative processes that occur in tokamak plasmas. Attention is given to those processes that contribute to theemission in the visible
part of the spectrum and to the relationships between these processes: line radiation, charge
exchange recombination radialion and bremsstrahlung. More detailed descriptions of these processes and their influence on tokamak operation can be found in numerous review articles and
textbooks, for instanee in Refs. [DeMM81, DeMM84, Hutc87, Moni91] . Allemission processes described hereare assumed to radiale isotropically.
2.1 Line radiation
Line radialion is emitted when an excited electron, bound to an atom or ion, decays to a lower
energy level. Due to the high temperatures atoms and ions are easily excited or ionized, mainly
by collisions with electrons. Line radialion is emitted from the infrared to the x-ray speetral ranges. From the measured spectrallines one can derive the neutral hydrogen density, the type
of the impurities which are present in the plasma and the density of their ionization states. This
section discusses some mode Is for the description of population densities of energy levels and charge states, and how the population mechanisms of excited states are related to the emission
in spectrallines.
2.1.1 Saha and coronal equilibrium, and line emission
To predict the emission by line radialion of hydrogen and impurity species, a model is needed to describe the population of the ground state and excited energy levels. For the impurities also a description is required for the population of their charge states, which are reached by successive ionizations of the neutral atom. This charge state distri bution and the dis tribution over the energy levels depend on the electron temperature and density through the electron collisionality.
In thermal equilibrium the population density nzj of i ons with chargeZin the energy level j is
related to nz,i of level i by the Boltzmann factor :
nz; 8z; -E JT --' =--' e IJ e , (2 . 1) nz.J 8Z,J
Chapter 2 Radialion processes
Figure 2.1 Sketch of the atomie processes included in the corona model,
indicated by the symbols used fortheir rates: A ij is the spontaneons decay
rate, Cu the collisional excitation rate, a2 the (radialive) recomt-ination S rate, and Sz the collisional ionization rate. Wiggly lines indicate radiative z transitions.
where Te is the electron temperature, Eu= Ei-EJ the energy difference between the levels, and the statistica! weights g of
the levels arise due to the degeneracy of quanturn states in each
energy level. The ratio of ground state densities of different ionization states is given by the Saba equation [Hutc87]
aZ+,
~A ij
c,i Ai,
nz,1
nz+l,l ne 2gz+l.l (2n:m; Te )3/2 e -xz,j!Te, (2.2)
nz,J 8z,J h
where ne is the electron density and XZJ the i anization potential of level j of charge state Z.
These distributions over energy levels do only apply in local thermal equilibrium. This is usually not applicable to tokamak plasmas, except for very high energy levels. Therefore, other
models have to be used. However, aften it is convenient to express these rnadeis as deviations
from the equilibrium densities.
In the corona! model, which applies for low-density plasmas, a balance is assumed between
collisional ionization and radiative recombination, and between collisional excitation and spon
taneous decay. The basic assumptions are that all upward transitions, i.e . a transition to a
higher energy level, are collisional, and that all downward transitions are radiative. Here the
plasma is assumed to be optically thin, so that there is no upward transition by absorption, and
it is assumed that at the prevailing low electron densities collisional de-excitation is negligible.
Figure 2.1 schematically shows the transitionstaken into account in the corona! model. Tostart
with, only the processes inside a particular ionization state wil! be considered; in subsection 2.1.3 the distribution over ionization states of impurities will be discussed.
As a consequence of the corona! model, only smal! numbers of atoms and ions are in their
excited levels compared to those in the ground states. The steady-state population density of the
excited level i in corona! equilibrium is related to the ground state density by
(2.3)
where Cli is the collisional excitation rate between the ground leveland level i, AiJ is the spontaneous radiative decay rate from statest in level i to statesin level} of the species under consid-
The lolal emission by lhe transition from levelito level} is obtained by summing A;'j·n; · over all g; stales
i' and &j stales j'. Here, nr is the population density of the statesof level i, that are assumed to be equally
populated, n;· = n;lg;. which is valid if the states are degenerate and the system is isotropic. The resulting total
24
Line radiation 2.1
eration. The ernissivity ifJ (power emitted per unit time per unit volume per unitsolid angle) of
a spectralline ernitted in an optically thin plasma by a transition between level i to j is given by [Cowa81]
(2.4)
which is valid, irrespective of the model used to obtain nz,i· For corona! equilibrium, using Eq.
(2.3), this gives
z _ hv Aii t:ij - 47t Ii-1 c,i ne nz.,.
A;p p;l
(2.5)
It follows that the ernissivity is proportional to the electron density and the ground-state density
of the ion in charge state Z, the latter depending strongly on temperature. For light elements the
values for Ai) are well-known and are tabulated, for example in Ref. [WieS66]. The Cli has to
be modelled to properly take into account the temperature dependence. Different expressions are
used in the literature [DeMM81, Isle84]. A convenient expression for the rate of excitation by
collisions between electrans and (charged) ionsis
C··= 158 10-11 fji g -Eij!Te ( 3-1) ,1 . x !Te m s ,
E;j"IJ Te (2.6)
where Eij is the energy difference between the levels i and j, and both Eij and Te are given in eV. The absorption oscillator strength/ii is related to Ai) and is usually given in the sametables
as Ai). The meaning of the Gaunt factor g is discussed in section 2.3; it is a function of EijfTe.
The value of the Gaunt factor in Eq. (2.6) for relevant values of Eij and Te is discussed, for
instance, in Ref. [DeMM81]. Fora rough estimate g == 0.2 might be taken. In the derivation of
Eq. (2.6) a Maxwellian velocity distribution of the electrans is assumed.
The corona! model is often applicable to tokamak plasmas, although only marginally for hydragen [DeMM84] . Especially at high densities and low temperatures it can be necessary to take
radiative, dieleetronie and charge-exchange recombination into account in the model because
excited levelscan be significantly populated. Such models are called collisional-radiative mod
els, which have been reviewed in the literature, for example in Ref. [DeMM81]. Models for the
ra te coefficients can be found in Ref. [Hutc87, pp. 203] . The extensions to the corona! model for hydragen are discussed in the next subsection. Gradients in plasma parameters and the
resulting transport, and time-dependent phenomena, affect the equilibrium. These effects are
discussed in subsection 2.1.4.
decay rate can be interpreled as an average over the initia! stales and a sumover the final ones [ShoM68].
Hence, the total number of transitions between levels i andj is equal to A;jn;, i.e. without a factor g;, where
A;j is the average decay rate per state found in tables.
25
Chapter 2 Radialion processes
2.1.2 Emission from hydrogen
For hydragen plasmas extensive calculations on collisional-radiative rnadeis have been per
formed by Johnson and Hinnov [JohH73] and Drawin and Emard [DraE77]. One important extension to the corona! model that is taken into account by these rnadeis is the loss of population density in a given level by collisiona1 excitation to higher levels. The extension to Eq. (2.3)
reads
Cline nz,l = ( L~~\ Aij )nz,; + ( L j tc.i Cij ne )nz,;. (2.7)
The second term on the right-hand side can become important at high electron densities.
Regimes where the second term dominates are sametimes referred to as the excitation saturation
phase (ESP). A consequence of Eq. (2.7) is that in ESP theemission is not proportional to ne.
as it was in Eq. (2.5). A further extension to Eq. (2.7) that is relevant when high levels are
significantly populated is the inclusion of the population of a given level by the radiative cascade from the higher levels. In tokamaks this extension is usua1ly not required.
In collisional-radiative rnadeis it is customary to express the population densities nz,i in terms of the ground state value nz, 1 and their Saba equilibrium values. From Eq. (2.2), the Saba
equilibrium population density np for hydragen in state p is given by
[ 2 ]3/2 * _ 2 2 h Xp / Te nH P- nep e ,
' 2rrmeTe (2.8)
where Xp the i anization poten ti al of state p. The factor p2 originates from the degeneracy of the
energy levels, i.e. the statistica! weights gp in Eq. (2.2).t Quasi-neutrality ni "'ne has been assumed, where ni is the density of ionized hydrogen. • The subscript H will be dropped as
long as we consider atomie hydrogen. In steady state the equation for the population density of
hydragen atoms in energy state p is given by
* • nl nP = ro,p nP + r1,p nP ';;*'
I (2.9)
where ro and r1 are the excitation coefficients which have been tabulated for a wide range of densities and temperatures by Johnson and Hinnov [JonH73] and Drawin and Emard
[DraE77]. The values of the coefficients in these two references differ slightly, butnotmore than a factor of two [DraE77]. The ro coefficient is related to the population of the levelp by,
for example, recombination. Recombination is a small effect and therefore ro "' I . As wiJl be shown in subsection 7.1.2.4 where experimental H<X results are discussed, the system is far
For hydrogen 8 H,p = 2p2, and 8Htp = I (no electron, so only one possible state), and hence in Eq. (2.2)
8H,p/(2gHt l) = P 2
* Quasi-neutrality implies ne ~ 'I;n;, i.e . a sum over all species. For small Zeff the sum 'I;n;"' nH+.
26
Line radialion 2.1
trom Saha equilibrium. Therefore, it is found that the second term in Eq. (2.9) dominates. The
r1 coefficient is related to all processes contributing to the population of level p. The depen
denee of r1,p on electron density, as calculated by Johnson and Hinnov, is displayed forsome
temperatures in Fig. 2.2. In this example p = 3 was chosen because it is related to the Ha emission discussed in chapter 7. For the electron densities occurring at the edge in RTP, ne of
the order 1019 m-3, the plasma is found to be in a transition between the coronal modeland
ESP. AJthough the deviations from the coronal model seem smal!, these coefficients include all
relevant processes and not only a model for the collisional excitation as in the combination of
Eqs. (2.3) and (2.6). Therefore using Eq. (2.9) is preferable. The excitation coefficients are
only weakly dependent on the temperature for Te> x1, as is clear from Fig. 2.2.
Opacity of the plasma can have a significant influence on population densities. Even if the
speetral transition under consideration is optically thin, optically thick lower transitions have a
significant influence on the population densities in all levels. This has the consequence for the
ground state density determined from the measured nH,p: in the optically thick case one order of magnitude lower a ground state density may be obtained. The value of r1 (and r0), therefore, depends on opacity. As will be discussed in sec ti on 2.1.4, opacity is, in genera!, negligible in tokamak plasmas. This is verified in subsection 7.1.2.4 where Eq. (2.9) is used to calculate the
ground state density from Ha measurements.
Collisional-radiative calculations not only give the r 1 (and ro) coefficient of the excited states, but also the ionization rate coefficient S and recombination rate coefficient a (see Fig. 2.1). On the assumption that the r1 coefficient and S dominate the equations for excited levels and i0n-
1 o-3
ESP 10-4
"'- 1 0--5 ..._~
---11 eV 10-6 --- - ---44 eV
-- --- 176 eV
10-7 '
1016
Figure 2.2 The coefficient r 1,3 for hydragen from a collisional-radiative model as a function of electron
density ne for lhree electron temperatures in the optically thin case. (Values taken from Ref. [JohH73].)
27
Chapter 2 Radialion processes
c 0 ö ..c 0..
ö I ..... 10 Q) 0.. (fJ
c 0
~ N c 0
10 T (eV)
e
n = 1018 m-3 e
n =1019 m-3 e
n = 1020 m-3 e
100 woo
Figure 2.3 The number of ionizations per emitted Ha photon from an collisional-radiative model as a
function of electron temperature for three electron densities. (Adapted from Ref. [JohH73].)
ization, the ionization rate is proportional to the population density of a given energy level, and by Eq. (2.4) proportional to the emitted light in a given speetral line. Figure 2.3 shows the
number of ionizations per emitted Ha pboton obtained from such a calculation. In the case of
hydrogen this ratio is fairly independent of electron temperature for Te> 10 e V, which implies that the Ha emission is a useful quantity to determine the number of ionizations in the plasma. This will be used in subsections 7.1.2.5 and 7.1.3 to estimate quantities such as the rate at
which the plasma is replenisbed by ionizations of neutrals entering the plasma (the so-called recycling) and the partiele confinement time. For electron densities ne < I Ql8 m-3 the ratio is
nearly independent of ne. whereas for ne > 1020 m-3 it is approximately proportional to ne-
So far, only atomie processes have been considered. lt is however not entirely correct to
assume that all hydrogen ions originate from atoms entering the plasma. The major part of hydrogen enters the plasma as molecules and various processes of ionization and dissociation
produce atomie hydrogen, but also hydrogen ions directly. This influences the amount of
emission. Furthermore, because excited hydrogen is produced by some molecular processes that are nottaken into account by the collisional-radiative models discussed, the r coefficients might be influenced if these processes are taken into account. The influence of molecular processes is discussed in more detail in subsection 7.1.2.6. Furthermore, the presence of excited hydrogen molecules gives rise to radialion in molecular bands.
2.1.3 Emission from impurities
Although emission from impurity species are not studied in detail in this thesis, in the discussion a bout the total emission in the entire visible range impurities play a role, which justifies a
28
Line radiation 2.1
1.0 4+ 6+
(!) (.) c <tl "0 c :::J .0 <tl (!) > :;::: <tl
Q) .... 0.1
10 1 o2 103
T (eV) e
Figure 2.4 The carbon equilibrium ionization state distribution as calculated by a collisional-radiative model
neglecting transport. The curves represent the relative abundances and are designated by the charge of the ion i za
ti on state to which they correspond. (Adapted from Ref. [CarP83].)
discussion of their main properties. Impurity radialion can also, in principle, be studied by the
visible-light tomography system by employing optica! filters selecting suitable spectrallines.
This operation is foreseen for the future.
So far, only the population densities of hydrogen atoms have been considered. In the case of
multi-electron atoms, not only the population densities of the energy levels in a given ionization
state play a role, but also the distribution over the various ionization states. Because the relax
ation times for excitation and radiative decay (of the order of nanoseconds) are much shorter
than the relaxation times for ionization (of the order of microseconds) and recombination (of the order of seconds), the excited levels are, at each instant, in equilibrium with their ground level.
Therefore, the ra te equations for the popuiatien of the ground levels (by i anization and recom
bination) and of the excited levels may be solved separately.
As a consequence of the model, only negligible numbers of i ons are in their excited levels com
pared to in the ground states. Inside an ionization state the corona! model usually is appropriate, and Eqs. (2.5) and (2.6) can be used to relate the spatially resolved emission of a given speetral
line, obtained by tomography, to the ground state density of the corresponding ionization state.
If measurements are available for each ionization state, summation of all the obtained nz,l over Z gives the total profile of the impurity under consideration. The tata! impurity content is an
important quantity determining plasma parameters such as Zeff and the total power ernitted as
line radialion and other radialion processes [JenP77]. Often measurements are only available of one or a few spectral lines, in particular in spatially-resolved measurements. Therefore calcula
tions are needed to yield the other ionization states Z.
29
Chapter 2 Radialion processes
The ground state densities nz, 1 of the ionization states of an impurity species can be solved
from a system of coupled equations descrihing the time dependenee or the steady state of nz, 1
on ne. nz+l,l and nz-1,1; the coefficients of these equations are the ionization rate S and
recombination rate a. Electron-temperature dependent models for the ionization and recombination rates are used in such calculations. Hence the fractional abundances of the ionization states
are functions of Te and ne. As an example the Te dependenee of the popuiatien of the ionization
stales of carbon are shown in Fig. 2.4. The dependenee on ne is weak. Charge state distribu
tions for other elementscan be found in, for example, Refs. [Summ79, CarP83]. According to
the conventional notation, the spectrum of an ionization charge state Z of an atom is indicated
by a Roman number for Z+ I on the right of the symbol of the species. He nee, as an example, a designates the spectrum of neutral carbon, CII of singly ionized carbon, et cetera.
Because the radial temperature profile in a tokamak peaks in the centre of the plasma, the tem
perature axis in Fig. 2.4 can be replaced by radius. This implies that the various ionization
states of impurities wil! be found in shells around the centre of the plasma, and that therefore a
given speetral line of an ionization state will be emitted mainly from within the corresponding shell, giving rise to hollow emission profiles. Complicating effects such as transport and
charge-exchange processes are discussed in the next subsectien and section 2.2, respectively.
2.1.4 Transport, opacity and other complicating effects
Several complicating effects might have to be taken into account when the models described in subsections 2.1.2 and 2.1.3 are applied to tokamak plasmas. In this subsectien the effects of
transport, time dependence, opacity and line broadening are briefly considered.
In the models discussed in subsections 2.1.1 to 2.1.2 the plasma is assumed to be homoge
neous. In genera!, for the ground state it is necessary to include transport in the rate equations
because the plasma bas steep gradients in ne and Te. and hence in nz. Transport causes the ions to move to regions with significantly different plasma parameters within the transition times
between ionization states. Transport not only bas an influence on the spatial dependenee of the
popuiatien densities, but also on the magnitude of population densities: the transport equations
and population density equations should be coup led. Modelling of transport is, however, diffi
cult because transport in tokamaks is not yet fully understood [Moni91]. In the core it is usually
sufficient to take into account only the radial flux of impurities for the higher ionization states because the impurities are spread evenly over each flux surface due to the fast transport parallel
to the magnetic field lines. But at the edge it might be necessary to include diffusion parallel to
the magnetic field lines because the smearing out over a flux surface is not fast enough [Moni91]. In the Jiterature often asymmetriesin theemission in the edge plasma are reported. In
chapter 7 some asymmetries observed on RTP wil! be discussed. In this thesis the effects of transport are only considered in a qualitative way. The main effect of transport is an inward
broadening of the shells of ionization states. In particular, for hydrogen this means that there is
30
Line radiation 2.1
an appreciable neutral density in regions with temperatures where it would not be expected from
the model described in subsection 2.1.2.
The situations discussed are steady state. In some cases the lifetimes of the charge states are
insufficient to guarantee that an equilibrium population is reached. In such cases, and when the
plasma parameters change in time, time-dependent calculations are necessary, see for instanee Ref. [CarP83].
Usually, the emitted spectrallines are broadened, for example by the Doppier shift due to the
high temperature. Moreover, the relatively strong magnetic fields present in tokamaks can be
expected to influence the spectrum through its effect on the details of the atomie processes (for
example through Zeeman splitting). In genera!, the influence of magnetic fields on the popula
tion characteristics of the excited levels of ions is masked by the relatively high frequency of
electron collisions [McWS84]. For the temperatures at theledge of the plasma where most lines
in the visible range are emitted and for the fields in RTP the linewidths are considerably smaller
than I nm. This is neglig i bie compared to the width of the optica! filters used in the visible light
tomography system (several nanometres).
On fusion devices it is commonly assumed that the plasma is optically thin in the visible range,
i.e. re-absorption of radiation is negligible. For most speetral !i nes this is a very good approxi
mation [DeMM81, Hey94]. Only for the Lyman aline ofH (wavelength 121.6 nm) the plasma
can be opaque for high neutral densities. This is the reason why, instead, the Balmer lines Hc:x
(wavelength 656.3 nm) or Hp (486.1 nm) are used for diagnostic purposes on tokamaks. A
further advantage of the these lines is that they are in the visible range where measurements are
less complicated than in the ultraviolet.
2.2 Charge-exchange recombination radiation
As indicated in the previous subsection, various processes exist for the recombination of elec
trons and ions, which have to be taken into account in the rate equations. The main types of
recombination are: radiative, charge-exchange and dielectronic. The radiative recombination, in
which a free electron recombines to a bound state, is important to understand the radiative prop
erties; it gives rise toa continuurn radiation because the possible energy levels of the free electron are continuous. The description of radiative recombination can conveniently be combined
with that of bremsstrahlung, which is done in the next subsection. Charge-exchange recombi
nation gives rise to line radiation and the process should be included in the collisional-radiative
models described insection 2.1. It, however, has a different dependenee on plasma parameters
and gives rise to radiation in different locations than the line radiation discussed insection 2.1,
which justifies the brief discussion of charge-exchange recombination radialion in this subsec
tion. Other recombination processes are not discussed here because they are less relevant in the visible range; descriptions can be found in Refs. [Hutc87, DeMM81, Isle84].
31
Chapter 2 Rad.iation processes
The charge-exchange pracess with neutral hydrogen, giving rise to charge-exchange recombination radiation, is described as follows:
(2.10)
where A indicates an impurity species, Z its charge and the asterisk that the ion is in an excited
state. Any neutral atom wiJl give rise to charge exchange recombination with other ions, but
because hydrogen is by far the most abundant in plasmas the pracess with hydrogen wiJl dominate. After the charge-transfer pracess of Eq. (2.10) the excited ion will decay and emit line
radiation. Roughly speaking, the excited state of the recombining ion, i.e. the one capturing the
electron from hydrogen, wiJl have a binding energy corresponding to the binding energy of the electron captured from hydrogen. The cross-section of the process with excited hydrogen is
considerably larger than with hydragen in the graund state. Due to the relatively low binding energy of hydrogen, mainly higher levels of the impurity ion are populated by charge exchange.
Furthermore, recombination tends to occur to states with appreciable orbital momenturn (i.e.
high levels), which due to transition selection rules has the effect that, instead of decaying directly to the ground state, a cascade of transitions to intermediale levels is necessary. This
gives rise to emission of many spectrallines in the visible and near ultraviolet. In high-resolution spectroscopy the contributions to a line by normal line radiation and charge-exchange
recombination can be distinguished from their different linewidths, the charge-exchange contribution being braader because it is emitted from a hotter region in the plasma.
Charge exchange recombination spectrascopy is usually used in conneetion with neutral beam
injection (heating beams or diagnostic beams), see for example Ref. [BoiH89]. With the neutral
beam there is a large concentration of energetic neutral hydragen along the beam path, giving
rise to charge-exchange radialion fram both the core and the edge plasma regions crassed by the
beam. Several plasma parameters can be determined from the measurement of the chargeexchange lines, the most important ones being the impurity ion concentration and temperature. Also without neutral beams charge-exchange recombination can contribute significantly to speetral lines, mainly from the edge where the neutral hydrogen density is high. Even in the
centre smal! concentrations of neutral hydrogen can exist, giving rise to non-negligible chargeexchange recombination radialion due to the relatively large crass-sections [Isle84]. The neutral
hydrogen density in the central part of the plasma can be temporarily increased by injecting pellets of hydrogen ice into the plasma, which can be expected to give rise to impurity chargeexchange emission. Charge-exchange recombination spectroscopy is called active if neutral beams are employed and passive if it results from the neutral hydragen density of the plasma. In active charge-exchange recombination spectroscopy often the contri bution of the passive component has to be taken into account as wel!.
The total number of charge-exchange recombination events, and hence the emanating line radiati on, is proportion al to nH nz,p. where nH is the neutral hydrogen density. The charge-exchange recombination rate coefficient is proportional to the velocity-averaged charge-exchange cross-
32
Charge-exchange recombination 2.2
section, which depends on the beam energy or local temperature. The charge-exchange process is relatively more important at low densities than the other recombination processes [lsle84]. Since the emission of a particular spectralline depends on the cascade process giving rise to it, detailed calculations are needed that take into account all processes involved. The experience on tokamaks shows that the contribution of charge exchange is significant compared to that of electron excited impurity line-radiation [Isle87, MagH94], complicating the interpretation of low-resolution speetral measurements in which different line-widths of the various contributions cannot be distinguished.
2.3 Bremsstrahlung and recombination radialion
Processes contributing to radiation can be classified by the state of the electron before and after the radiating event. The line radiation described in section 2.1 is caused by a bound-bound transition. Radiation can also occur when a free electron is accelerated in the electric field of a charged particle. This process is a free-free transition if the electron is free after the encounter and the resulting radiation is called bremsstrahlung. The free electron can also be captured, which is the case of free-bound recombination radiation. Bath the free-free and free-bound pro
cesses can bedescribed consistently, and hence the term bremsstrahlung is sametimes used to include the Jatter. Because the free electron has a continuurn of possible energy states, free-free
and free-bound transitions give rise to a continuurn emission. The continuurn extends from the microwave region (the plasma frequency) up to the x-ray region (where the pboton energy is of the order of the electron temperature or more).
The bremsstrahlung is usually expressed by a classica! (non-relativistic) formula with quanturn mechanica! corrections. The quantum-mechanical effects and some classica! effects are described by the Gaunt factor G. Fora Maxwellian velocity distribution the expression for the emission per unit volume per unitsolid angle per unit frequency ist [Hutc87]
( ) = _z2[LJ3 87t (2me ]
112 - hv/Te E v v nen, r;; 2 3 e
47tt:0 3-v3mec 7tTe
x [g- + G ..t X i eXJTe + ~ G Z2XH ~eZ2XHII2Te] (2.11) ff 11) ~12 l •
n Te l=n+1 l Te
the first term inside the square brackets being the free-free contribution, the second recombination to the lowest unfilled shell (n), and the third to all other shells. The subscript v indicates
In the literature, see for example Ref. [Kad080], this expression is somelimes given in cgs units, whereas the
equivalent formula with the physical constaniS eva)uated [cf. Eq. (2.12)] is given in SI or derived units, which
can be very confusing. Care should be taken in evaluating the formula that is given in cgs units, since e2 in
cgs units is equivalent to e2/(41t Eo) in SI.
33
Chapter 2 Radiation processes
that the emissivity t:v is per unit frequency to distinguish it from t: in Eq. (2.4), which is
implicitly integrated over the spectralline. The meaning of the symbols is as follows: ni is the
density of the ion in charge state Z, f{) the permittivity of vacuum, me the electron mass, lfff the Maxwell-averaged free-free Gaunt factor, Ç the number of unfilled positions in the lowest shell, Xi the ionization potential of the ion, and XH the Rydberg energy (13.6 eV). The term between square brackets multiplied by Z2 is sametimes referred to as the x-ray enhancement factor. The
gaunt factor Gn = 0 for hv< z2XHfn2 because in that case no recombination is possible.
A sufficient condition for the recombination radialion to be negligible is h v << z2XH because only high states n contribute. The contribution of high n states, however, rapidly becomes negligible because of the n-3 dependence. For singly ionized species this requirement is
À.>> 87 nm, which is the case in the visible range. For Te>> z2XH recombination radialion is negligible for all frequencies. Therefore, it is a good approximation in the visible range to
neglect the recombination contribution [Kad080, FooM82] . lf Eq. (2.11) is summed over all
ion species and charge states, recognizing the term Li ne ni z7 = n ~ Zeff [from Eq. ( 1.1)] and evaluating the fundamental constants, the following expression is obtained for the emissivity per unit volume per unitsolid angle per unit frequency:
2z -_ 5 0 10-54 ne effgff -hv/Te (W -3 -1 H -1) t:v - . x Ir e m sr z , -y Te
(2.12)
where ne is in m-3, and Te and hv in eV. Except at the extreme edge of the plasma, hv<< Te is valid in the visible range. Therefore, the exponential is approximately I, and t:v has only a weak dependenee on Te. In this range the bremsstrahlung is also virtually independent of
frequency. If t:v(v) is expressed in wavelength À., the right-hand side in Eqs. (2.11-2.12) should be multiplied by d v/dl\.= (cl)}). The Gaunt factor lfff is approximately constant in the
visible range. Gaunt factors for various ranges of parameters have been calculated by Karzas
and Latter [KarL6l]. In the literature usually lfff"" 3 or 4 is taken in the visible range [Kad080, FooM82, SchS88].
If the absolute value of the local emissivity can be determined in the plasma, and if local values
for ne and Te are known, in principle Zeff can be determined from Eq. (2.12). The bremsstrahlung must be measured in a line-free part of the spectrum, which turns out to be hard
to find on a tokamak. Besides emission lines from ions in the plasma, also continuurn radiation from molecules at the edge and in the scrape-off layer of the plasma mask the bremsstrahlung. Tomography might be a salution to approximate the absolute values of emission in the interior
of the plasma, so to speak looking through the edge.
34
Summary 2.4
2.4 Summary
An overview has been given of the radiation processes in tokamak plasmas that contribute in the
visible part of the spectrum. Si nee the plasma is optically thin in this wavelength range the
emissivity of line radialion is proportional to nenz; however, for electron densities ne > 1020 m-3 the dependenee on ne becomes less strong and for high electron densities the depen
denee on ne disappears. The emissivity by charge-exchange processes is proportional to nH nz. These radiative contributions might be difficult to distinguish, although for passive charge
exchange the emissivity will give a smaller contribution than strong speetral lines in the line
radiation that is caused by electron collisional excitation. The continuurn emissivity of
bremsstrahlung is a function of electron density, temperature and Zeff. These processes can be
studied with the visible light tomography system by selecting parts of the spectrum. The magni
tudes of various contributions for plasma conditions measured in RTP are discussed in chapter
7.
35
Tomography and other analysis methods 3 This chapter introduces mathematica! aspects of tomographic inversion methods and describes the mathematica! analysis techniques used in this thesis. The depth of discussion depends on
how much is needed in this thesis. In the first section an overview is given of different basic
mathematica! aspects of tomography, definitions of tomography terminology are given and methods to take into account the influence of noise are discussed. The basic aspects of several tomographic inversion methods are introduced in section 3.2. Although the most important
methods are described, it is nota complete review. The new tomographic inversion technique developed for the visible light tomography project is discussed extensively and examples of its
application are shown insection 3.3. Finally, insection 3.4 some other techniques than tomog
raphy to analyse multichannel measurements are described: parametrization techniques, singular value decomposition and correlation techniques.
3.1 Basic mathematica) aspects of tomography
In this sec ti on the Radon transform, its inverse and several of its properties are introduced. The
Radon transfarm describes line integrals of functions. Tomographic inversion of idealline-inte
grated measurements is in principle the calculation of their inverse Radon transform. Rigoureus
mathematica! derivations can be found in the books by Natterer [Natt86]. Deans [Dean83] and Helgason [Helg80]. A more popular treatment can be found in, for example, Herman [Herm80]. Some properties of ill-posed problems are discussed, terrninology in tomography is
introduced, and aspects of noise that are relevant for tomographic inversions are mentioned.
3.1.1 The Radon transform
Let (x,y) be the coordinates of a plane, and ga function defined on this plane. The Radon trans
farm! of g in two dimensions is defined as:
f = IJ(g = Lg(x,y) ds, (3 .1)
for alllines L, where <l(_designates the Radon transferm operator and s denotes the path along L. This type of mapping and its inverse was first studied by Radon [Rado 17]. One can identify g with, for example, the local emissivity in a cross-sectional plane through a plasma, La view-
Chapter 3 Tomography and other analysis methods
y ing line through the plasma, andfthe measurement.
The Radon transform can be extended to higher
dimensions: then the integration takes place over
hyperplanes [Natt86, chapter 2]. The extension of
the line-integration to higher d.imensions is a differ
ent generalisation and is known under the name x
"\--'----+---x ray transform, and has different properties [Natt86,
Figure 3.1 Definition of the coordinates.
chapter 2]. lt is clear that the two-dimensional x-ray
transform and two-dimensional Radon transform
are identical. Only the two-dimensional Radon
transform wil! be considered in this thesis. Apart
from the obvious application to tomographic prob-
lems, the Radon transform has certain advanta
geous properties that make it useful in, for example, solving certain types of partial differential
equations [Dean83].
Equation (3.1) can be written more explicitly after defining of a new system of coordinates (see
Fig. 3.1 ). Here inlegration along straight lines is assumed, whereas in practice the !i nes could
be curved, for example in the case of refraction (then the ray path could even depend on g,
making the problem very difficult to solve). For the present application the lines can be consid
ered to be straight since refraction is negligible. Fora physical function g, one can always find
a circle with radius a outside which g(x,y) = 0. A straight line can be described by two
parameters; in this thesis p and Ç are chosen with the convention that p lies in the interval [-a,
a] and Ç in [0, 1t]. An alternative often used in the literature is the use of coordinates p and 1/J, with pin [0, a] and 1/J in [0, 27t]. The functionsfand L in Eq. (3.1) depend on the parameters p
and Ç. The parameters (p,Ç) of all possible lines form the coordinates of a space, conventionally
called Radonspace or projection space. In this thesis the latter name wil! be used. The equation
for the line is p +x sin Ç- y cos Ç = 0, and the spatial coordinates can be written as
x= -psinÇ + scosÇ,
y = p cos Ç + s sin Ç,
where sis a parameter denoting the length along the line. One can write Eq.(3.1 ) as
f(p,Ç) = J!~-psin Ç + scosÇ,pcos Ç + ssin Ç) ds, (3 .2)
or
f(p,Ç) =IJ g(x,y) 8(p +x sin Ç- ycos Ç) dxdy, (3.3)
where 8 is the Dirac delta function . The edge of the area where g(x,y) -:t. 0 can betaken as the
limitsof integration in Eq. (3.3).
38
Mathematica[ aspects 3.1
3.1.2 The inverse Radon transform
The calculation of the inverse of the Radon transform, i.e. the salution of the unknown function
g from the knownf, belongs toa class of problems known as Fredholm inlegral equations of the ftrst kind
f(w) = J K(w,z) g(z) dz, (3.4)
where w and z are variables that in the case of the Radon transfarm denote (p,Ç) and (x,y),
respectively. The kemel K describes the properties of the measuring system: for example the
fact that the viewing chords rnight not strictly be lines, but have a eertaio width, or that attenuation occurs by re-absorption of emitted radiation. The kemel is closely related to the point spread function or instrument function; which will be discussed further in subsection 3.2.9.
The Radon transfarm of Eqs. (3.1 )-(3.3) can be considered to have a kemel equal I on the
viewing line, and 0 elsewhere.
An inversion formula for the Radon transfarm was derived by Radon in 1917. In most textbooks on computerized tomography derivations are given on different levels of abstraction [Herm80, Natt86, Dean83, Helg80]. The inverse Radon transfarm in Cartesian coordinates
(x,y) is:
1 in I~ I a g(x,y)=[1(-1f](x ,y)=--2 2 dÇ . Ç Ç :J f(p,Ç)dp. 1t o -~ p+xsm - y cos up
(3.5)
Sufficient conditions for the existence of such an inverse are that g is continuous and bounded, and that there is a region outside which g is zero. Furthermore ()jldp should be continuous
[Herm80]. There are many equivalent formulae that can be obtained by partial inlegration and
coordinate transforms.
lf the function g(x,y) has cylindrical symmetry and is expressed in polar coordinates (r,e) as g(r,e) = g(r) , the problem of Eq. (3.1) is known as an Abel integral equation, and its inverse
can be written as [Hutc87, p. 124]:
I Ja 1 d g(r) = -- ~ f(p) dp. (3.6) 1t r "V P -r dp
3.1.3 Discrete descriptions of the Radon transform and its inverse
lf the function g is deftned on a grid, one can obtain a discrete version of the Radon transform:
!; = L Kijgj, (3.7) 1
where the summation is over all cells of the grid and Kis the kemel of Eq. (3.4), here in matrix form . As indicated in Eq. (3.7) the functionfhas also been made discrete, and therefore Eq.
39
Chopter 3 Tomography and other analysis methods
(3.7) is a system of linear algebraic equations, or equivalently a matrix equation. Inversion of this system of equations gives an approximation of the inverse Radon transfonn.
In the case of line-integrals, Kij is the lengthof the chord i through cellj. In more complicated measurement systerns integration over the width of the chord and other properties can be taken into account in K.
3.1.4 111-posed problems
Although an explicit formula fortheinverse Radon transfarm exists [Eq. (3.5)], it is ill-conditioned because of the bad behaviour of the integrand. The same is true for Eq. (3.6), where integrand has a singularity at p = r. The inlegral overpin Eq. (3.5) is a Cauchy principal value
integral. Furthermore, the complete Jj!Jp [or f if Eq. (3.5) is integrated by parts] should be
known in analytic form to use the inversion formulae directly, which is not realistic in practice
where fis only measured at discrete points.
The inverse problems described by Eqs. (3.1), (3.4) and (3.7) belang to a class of problems that are known as ill-posed problems. Same authors call these problems improperly posed or incorrectly posed. The main property of these problems is that their solutions are not stabie to
small changes in the initial data. This means that two functionsfthat are very close can result in
solutions g arbitrarily far apart, where the distance is understood as the norm in the function
space used. Because noise is always present in experimental data and in numerical simulations noise is caused by round-off errors, the inversion of ill-posed problems is badly behaved.
Properties of ill-posed problems are discussed by, for example, Tikhonov [Tikh77] and Turchin [TurK71].
Because in practice the function to be inverted is always sampled at a discrete set of measuring
points, the ill-posedness is even worse. This can be seen as an incomplete data problem, although in the literature this term is usually reserved to problems where measurements from eertaio directions are missing, and not to problems where the missing data are the gaps caused by the discretization.
Methods exist to imprave the stability of ill-posed problems, which are commonly called regu
larization. It is very important to properly take into account an estimate of errors in the measurement, such as noise. Many such methods are intuitively meaningful, and were in use already long befare the theory of ill-posed problems and their regularization was developed. Regularization methods are for example the usage of a priori information such as an initia! estimate of the solution, boundary conditions, imposed smoothness, or other lirnits to the class of allowed solutions (such as parametrizations). Also filtering can be used, for instanee by lirniting
the number of terms in an expansion, or in iterative salution methods using relaxation parameters and stopping the iterative process after a eertaio number of iterations. The regularizations can be described by a regularization parameter. This parameter depends on the problem, the
40
Mathematica! aspects 3.1
method of solution, and the noise involved. In the literature ways to find a proper regularization
parameter are descri bed, see for example the overview and references in Refs. [Tikh77,
Natt86]. The most common way to find a proper value for the regularization parameter, however, is by trial and error.
Examples of regularization are given when different algorithms for tomographic inversion are discussed in section 3.2.
3.1.5 Properties of the Radon transform
Some properties of the Radon transfarm are discussed in this subsection. In the literature many more properties are discussed, see for example Natterer [Natt86] and Deans [Dean83]. Here the
emphasis is put on basic properties and relations, and other properties that are needed later.
In Fig. 3.2 the processof the Radon transfarm is shown in a descriptive way. All possible line
integrals of g(x,y) in real space result in all points off(p,{> in projection space. Fortheinverse operation,fwill in general only be known in a limited number of points in projection space.
For the Radon transform, transferm pairs of functions exist in the same way as for the Fourier
transform. Several examples of Radon transferm pairs are given in Ref. [Dean83]. The Radon
transferm is a linear operator.
3.1.5.1 Properties of projection space
If one takes a point souree at the point (x0 ,y0), i.e. g(x,y) = 8(x-x0) 8(y-yo), and considers
this in po lar coordinates r = -../(x2+y2) and e = arctan(y/x), then with the help of Eq. (3.3) one
y
i
.aL-----------~------------~
·a 0
-x a
a .------------.------------, (b)
·a c__ __________ __,_ ____________ ...J
0 112 1t 1t
- Ç (rad)
Figure 3. 2 Contour plots of (a) a function g(x,y) and (b) its Radon transformflp,Ç). One viewing chord with
parameterspand Ç (line) in real space (a) and its representation (point) in the projection space (b) are shown.
41
Chapter 3 Tomography and other analysis methods
can determine that.fl:p,Ç) is zero everywhere, except on the Jine
p-r0 sin(80 - Ç) = 0.
For the function.fl:p,Ç) the following properties hold:
f(p,Ç + 1t) = f( -p,Ç),
f(p,Ç +27t) = f(p,Ç).
(3 .8)
(3.9a)
(3.9b)
Equation (3.9a) describes a periodicity, which we cal! Möbius periodicity or Möbius property
because of the resemblance of projection space on Ç in [0,7t] with a Möbius band (i.e. a twisted
band of which the ends are connected).
For parallel beam systems, which are imaging systems where the line-integrals of g are taken
along parallel Jines (i.e. lines with equal Ç) for pin [-a, a], several useful properties that relate different views (i.e. different values of Ç) can be derived. For Ç = 0:
a a {t;Q J~(p,O) dp = f_j_~-~g(s,p) dsdp = fJ g(x,y) dxdy = E , (3 .10)
where Eq. (3.2) has been used. The last integral is the two-dimensional integral over g, being,
for example, the total emissivity E from a cross-section, which is independent of the orienta
tion; thus Eq. (3.1 0) is valid for any Ç. This property will be referred to as conservat ion of total
emissivity for projections.
Equation (3.1 0) is the zeroth moment of both f(p ,Ç) and g(x,y). Si mil ar expressions can be
x= JJ xg(x,y)dxdy I E
p = f p f(p, f;)dp I E
Figure 3. 3 Schematic representation of the relation between the two-dimensional centre-of-mass of g, and the
centre-of-mass of the projections f
42
Mathematica! aspects 3.1
derived for the first and highermomentsas wel!. This is known as the two-dimensional consis
tency theorem [PriW90]. The interpretation of the first moment is that the projection of the two
dimensional centre-of-mass of g(x,y) in direction Ç corresponds to the one-dimensional centre
of-mass in pof the projectionfip,Ç). This property is exemplified in Fig. 3.3. Because the twodimensional centre-of-mass is a point in real space, it follows from Eq. (3.8) that the one-dimensional centres-of-mass foltow a sine in Ç in the projection space.
3.1.5.2 Projection theorem
Close relationships exist between the Radon and Fourier transforms. Fourier transfarms play a central role in many tomographic inversion methods, and many properties, such as the inverse
Radon transform, can more easily or more generally be derived with Fourier transfarms than by other methods. Basic properties of the relation between the Radon and Fourier transfarms have
been discussed by Deans [Dean83, section 4.2]. One important property is the projection theo
rem, equivalently called F ourier slice theorem, projection slice theorem or central slice theorem
[Herm79, p. 149; KakS88; section 3.2; Dean83, section 6.2]. If the two-dimensional Fourier transform is denoted by l"f"2, and the one-dimensional Fourier transform with respect to the p
variabie by P1 (with corresponding frequency coordinate p), the projection theorem in operator
form reads:
(3.11)
No te that, like the left-hand si de, the right-hand si de of Eq. (3.11 ) depends on two variables: p and Ç. The interpretation of this theorem is depicted in Fig. 3.4: the one-dimensional Fourier transfarm of a projection f at angle Ç is equal to the two-dimensional Fourier transfarm of g
V
(b)
[P2g](u, v) = [P2g](psinÇ,pcosS} =
[l"f"dJ(p,Ç)
Figure 3. 4 Schematic representation of the Fourier slice theorem Eq. (3 .11 ). (a) Real space [coordinates (x,y))
with a projection at angle Ç. (b) In the Fourier space [frequency coordinates (u,v)] the two-dimensional Fourier
transform of g along the indicated line is equalto the one-dimensional Fourier transfarm of fwilh respect to the p
variabie (corresponding frequency coordinate p).
43
Chapter 3 Tomography and other analysis methods
along the line through the origin perpendicular to the viewing direction denoted by Ç. For
inversion this means that if 'Ftf is known for a number of angles, 'FUJ is known along equally
many lines, and its discrete inverse two-dimensional Fourier transfarm (possibly after interpo
lation) gives an approximation of g.
3.1.5.3 Filtered backprojection
An operator related to the inverse Radon transfarm is the backprojection operator CB, which is
defined as:
[CBJ](x,y)::; J/c-xsinÇ + ycosÇ,Ç) dÇ. (3 .12)
Backprojection thus is the inlegration of a functionf in projection space along the sine repre
senting the point (x,y) in real space, i.e. it adds a contribution of alllines going through the
point for all projections. This is equivalent to (unweighted) smearing out (i.e. "backprojecting")
the val u es of f equally along the !i nes over which they have been measured, and adding the contributions from the different projections in each point. Backprojection is not sufficient to
invert the Radon transform, but it produces blurred images of the proper solution with some
limes recognizable features.
Backprojection in combination with other operators can produce the inverse Radon transform.
This is calledfiltered backprojection. In many books about tomography, e.g . Refs. [Herm80,
p. 97; Dean83, p. 134], it is shown that:
Cf( -Ij::;- 2~ CJ3J{(J)j, (3.13)
where ID is {)f()p; and Jfthe Hilbert transfarm with respect to the first variable, fora function
q(p,Ç):
[Jfq](p' ,Ç) = _..!_f~ q(;,Ç) dp. 1t -~ p -p
The operators J{ID constitute a convolution, and therefore filtering operation.
3.1.6 Terminology
(3 .14)
The terminology used in the tomography community can be confusing. Various terms for the
same concepts are used by different authors. Here I wish to clarify the terminology used in this
thesis.
The problem of tomography is often described as the reconstruction of images from projections
(see for example Ref. [Herm80]). It can also bedescribed as the inversion of the Radon trans
farm (which is only valid for line integrals). Hence the equivalent terms used are: tomographic
reconstruction, tomographic inversion and Radon inversion, although the last term is only cor-
44
Mathematica! aspects 3.1
(a) (b)
Figure 3. 5 Schematic representation of (a) a parallel beam system and (b) a fan beam system.
reet for pure line integrals. To avoid confusion with the term reconstruction of projection space introduced in section 3.3, the term tomographic inversion will be used here to indicate the
process of determining an approximate emission profile from the measurements. Here profile is
used to indicate that the emissivity is considered to be a function of two variables in a poloidal
cross-section of the tokamak. Profile can also be used to indicate a cross-section of this func
tion of two variables, or a radial profile (when there is only a dependenee on radius).
Tornogram is often used to indicate the solution (or its image) of the tomographic inversion
process. In this thesis it is equivalent to the reconstruction of the emissivity profile.
Parallel beam system is used to designale a deleetion system where detectors are positioned
(often equidistantly) on straight lines, each detector measuring the line-integrated quantity along
straight lines perpendicular to the line on which the detectors are situated [Fig. 3.5(a)]. The deleetion therefore takes place along sets of parallellines. In this thesis parallel beam system is
used to designale a system with perfect line integrals. In practice however, it is often easier
either to use an approximate point souree (in active tomography) with the detectors arranged on
a straight line or circle on the opposite side of the object, or (in passive tomography) to measure
the object with an array of detectors that are much smaller than the object itself. The dis tribution
of I i nes in such a case is fan shaped, and such systems are therefore called fan beam systems [Fig. 3.5(b)].
Projection space is the space of the coordinates of the line-integral functionfi:p,Ç) (see subsec
ti on 3.1 .1 ). The image of this function is often called sinogram. The explanation of this term is
that the Radon transform of a 8-function results in a sine-shape in projection space [see Eq.
(3.8)] . Any function in projection space can therefore be thought of as built up from sine functions. It follows that the si nogram is the Radon transform of the tomogram. A projection is the
function f(p,Ç) for fixed Ç. It corresponds to the measurement with parallel beams from one
45
Chopter 3 Tomography and other analysis methods
direction (angle Ç), which sametimes is called camera from the practical point of view. In con
figurations different from parallel beams, for example fan beams, camera is aften used as a synonym of viewing direction, and even projection is used for the measurements of each cam
era.
Usually the entire object is covered byeach camera or projection. In the system under consideration in this thesis this is not the case, however. Here, viewing direction will be used to des
ignale the incomplete projections from approximately one direction. This corresponds to the
physical concept of a camera: a box with several detectors and their electronics.
A viewing direction consists of several detectors. The regions viewed by each detector will be
designated by viewing chord, which indicates that it has a certain width. Often the chord has a kind of fan-shape, and therefore can be called viewing fan as well ( or viewing cone in the three
dimensional case). If an approximation is madetoa Jine, which is an average over the width of
the chord, this line is called viewing line.
The usual way to test the performance of algorithms for tomographic inversion is to first
assume a function g, and to calculate f from the known properties of the system under consid
eration. Then the inversion ofjis calculated by the algorithm and it is compared with the original g. As discussed in subsection 3.1.4 the performance is very dependent on the properties of
noise. Therefore, in a simulation a realistic level of random noise should be added to the vector
f The test function gis conunonly called phantom (for example phantom em.ission profile), and the calculated vector f pseudo-measurements. The phantom can also beseen as the exact solu
tion of the inverse problem, and could even be called exact tomogram.
The performance of an algorithm depends on the properties of the system and on the phantom.
Systems and algorithm can be optim.ized for the type of expected functions g by computer simulations. Apart from theoretica! and mathematica! considerations, simulations are usually the only way to really test the performance, and to check the reliability of tomograms. This performance is further discussed in subsection 3.2.7. Usually structures appear in the tornogram that were not present in the phantom. Such structures that are caused by the system or the algorithm are
called artefacts.
3.1.7 Noise
As indicated before, noise has a large influence on the salution of ill-posed problems. It is
important to classify different types of noise. Noise can be defined as the value that should be added to the exact value J to match the measured value:
j(t) = ](t) + Eo + E(t), (3.15)
where t:(t) and Eo designale errors dependentand independent of time, respectively. Because usually in tomography only time slices (i.e. measurements at one point in time) are considered,
46
Mathematica[ aspects 3.1
the two types of errors cannot be distinguished, and both contribute to noise in tomographic
inversions, albeit in different ways. Roughly, E(t) can be considered as random noise, and t:o as the systematic errors, which will be called noise from systematic errors.
The random noise spreads around the noiseless values J with a certain distri bution function. Usually a Gaussian distribution is assumed in simulations (Gaussian noise) because most
sourees of noise can be considered nearly Gaussian (quantization effects, for instanee due to
photons, are not noticeable in the present system), and because it is easy to handle and to im
plement. The random noise can be divided into noise with an absolute level and noise with a level proportional to the signal. The former arises, for example, from amplifier noise from the electranies and electronic piek-up, and it is herecalled absolute noise. The latter type of noise
originates from the light souree itself and is called relative noise.
Noise from systematic errors is caused by, for example, uncertainties in the positions of the
cameras. Usually this type of errors does not have a random distribution among the detectors, but has comparable values of t:o for the various detectors. Such a deviation from the assumed values should be taken into account in simulations and has the effect of noise on the tomographic inversion.
It is sometimes said that random noise is high-frequency noise (i.e. measurements by adjacent
detectors deviate much) and noise from systematic errors is low-frequency noise (i.e. not much
variation between adjacent detectors). The identification of random noise with E(t) and systematic errors with E{) is not strict. Drift in time of amplifier offsets, for example, can have the char
acter of noise from systematic errors if the drift is similar for various detectors. Calibration
errors can have both the character of systematic errors, and of random noise for different detec
tors, caused for example by the measuring accuracy during calibration. Furthermore, relative noise can be caused by fluctuations in the signal (when the detection system is signallirnited for
example, or from fluctuations from the light souree itself), but also by spatial aliasing errors from structures that cannot be resolved by tomographic inversion.
When noise sourees are considered for tomography systems on tokamaks often only calibration errors, electronics noise and positioning errors are considered. These usually are quite smal!,
and therefore the amount of noise is often underestimated. Usually signals show "plasma
noise", apparently due to plasma fluctuations of which the precise nature may not be clear. lt might be caused by fluctuations in time or by small structures. Small structures can have a noisy effect if they cannot be resolved by the tomographic inversion method, for instanee
aliasing effects if considered in the spatial frequency space. Furthermore, geometrical effects, due to, for example, overlapping or crossing viewing chords measuring the same part of the
object with different sensitivities, can cause deviations that have the effect of noise for the
inversion algorithm.
47
Chapter 3 Tomography and other analysis methods
The influence of noise on reconstruction errors in tomography is discussed in subsection 3.2.7. In some tomographic inversion methods the time-depen .<'··1Ce of signals is taken into account, in which case the description given above becomes more ccmplicated. A methad that can dirninish the influence of the random noise is indicated in subsectt_'.,. 3.2.3.2.
3.2 Some tomographic inversion methods
A large variety of numerical schemes exists for tomographic inversions. Some methods, for example, discretize the evaluation of Eq. (3.5), and are therefore called direct Radon inversion methods. Many different possibilities exist already for these direct Radon inversion methods, for example by writing the inversion formula in a different way by partial integration, or by the way the singularities are handled. Different methods and implementations have different prop
erties that rnight be advantageous for a given implementation. Simulations can help to find the best implementation.
In this section only the methods relevant for this thesis are discussed: methods that are used in this thesis, and for comparison methods that are used in sirnilar probierus in plasma physics. Des pi te this restriction, most important classes of tomographic inversion algorithms are covered in this sec ti on. Only the main properties of the methods are discussed because detailed informa
tion about different implementations can be found in textbooks and a large number of publications. General overviews of methods can be found in most textbooks referenced here, and in review articles by for example Rangayyan [RanD85], Pickalov [PikP83], Lewitt [Lewi83] and
Censor [Cens83].
The methods covered here are: (1) Filtered backprojection (FBP), the most popular methad in medica! tomography which is based on Eq. (3.13) (subsection 3.2.1 ). (2) Methods such as Fourier techniques and the Gerchberg-Papoulis method, which are discussed because some ideas from these methods are used in section 3.3 (subsection 3.2.2). (3) Series expansion methods, where both si des of Eq. (3.1) are expanded in orthogonal sets of functions, of which the components are Radon inverses. The most well-known such method is the Cormack methad which is commonly applied in x-ray tomography in plasma physics (subsection 3.2.3). (4) Pixel methods that arebasedon Eq. (3.7). The well-known iterative algebraic reconstruction technique (ART) and all its variations belang to this class (subsection 3.2.4). Also certain (non-iterative) implementations of maximum entropy methods are pixel techniques, as well as other iterative and non-iterative optirnization methods (subsection 3.2.5). Some other methods used in plasma physics are mentioned and a justification for the choice of reconstruction methocts for visible-light tomography system is given (subsection 3.2.6).
In the literature various classifications of tomography methods are made. A distinction that is often made is between transfarm methods and series expansion methods [Lewi83]. In transfarm methods the inversion step is done analytically before the numerical discretization, such as
48
Tomographic inversion methods 3.2
in FBP and Fourier methods, whereas series expansion methods discretize the problem befare
inversion, such as in ART and most optimization techniques (pixel methods). The Cormack
methad could be classified in either of the two. Here a more functional distinction is made to describe the aspects of different methods that are important for this thesis.
All methods, except the pixel methods, make use of the approximation that the measurements
are integrations along exact ti nes. Modifications to take into account the width of the viewing
chord, as wel! as modifications to take into account re-absorption and refraction, are possible in
some methods. Some remarks about these approximations are made in subsection 3.2.7, along with remarks about methods totest the implementations.
In subsection 3.2.8 some variations to straightforward tomography are introduced that make use of the temporal dependenee of signals. Finally, in subsection 3.2.9, some properties of the
weight matrix [the kemel of Eq. (3.7)] are discussed and its relation to the point spread func
tion.
3.2.1 Filtered back projection
Filtered backprojection (FBP) methods arebasedon Eqs. (3.13) and (3 .14). Because of the convolution involved, these methods are somelimes called convolution methods. If the inverse
Radon transfarm is rewritten in the form of Eq. (3.13), it is relatively easy to discretize and the
singularities are not difficult to handle. This methad is popu1ar because the quality of reconstructions is in general very good compared to other methods, because it is relatively simple to
implement, and because it only needs little memory and computer time. FBP methods are usually used for probieros with many detectors and viewing directions.
Equation (3.13) can be derived in the Fourier domaio by making use of the Fourier slice theorem Eq. (3.11), in which case Eq. (3.13) cao be written as [KakS88, p. 64]:
g(x,y) = l1t dÇJ~ F(p, Ç)lple21tip(- xsin Ç+ycosÇ) dp, (3.16) 0 -~
where F(p,Ç) is the one-dimensional Fourier transfarm of the projectionf(p,Ç) at angle Ç. Note that the outer inlegral is the backprojection of Eq. (3.12). The operators J{Cf) have been replaced
by the filtering function lpl in the frequency domain. This function, which can be seen as a weighting of the projection per radius, originates from the Jacobian of the transfarm from rectangular frequency coordinates to polar coordinates for the two-dimensional Fourier transform.
FBP methods differ in the ways in which Eq. (3.13) or Eq. (3.16) are implemenled (i.e. with
or without Fourier transforms), the ways in which singularities are treated, which techniques are used to reduce artefacts, and how the problem is regularized [KakS88, Herm80]. Additional
filtering by a window function on lpl is usually used to limit the frequency to within the Nyquist limit, which is determined by the number of detectors per viewing direction. To circumvent the
problem of singularities in the Hilbert transform, some methods use the inverse Fourier trans-
49
Chapter 3 Tomography and other analysis methods
form of a filter in frequency space, which therefore appears as a convolution in real space. A
filter that approximates lpl in the frequency domaio is the widely applied Shepp-Logan filter
[PikP87, p.ll9]:
H(p) = \Pisinc(p I 2p0 )rect(p I 2p0 ), (3.17)
where po is the Nyquist limit and the functions sine and reet are defined as
. _ sin 1tX _ {I, lxl < 112 smc(x)-~· reet( x)- O, lxl > 1 12 .
Note that rect(x) is the window function. The inverse Fourier transfarm T)(p) of H(p) can now
be convolved with the projection and then backprojected to give the approximate inversion for
mula:
g(x,y) =I ct.; J2(p,Ç)T](-xsinÇ + ycos.;- p) dp, (3.18)
where the inner inlegral is a convolution.
Apart from regularization by additional filtering, further regularization can be done by spline
smoothing of the raw data before applying FBP. The FBP method used in this thesis uses the
Shepp-Logan filter and regularization by spline smoothing of the raw data [PicM84].
3.2.2 Foorier techniques
Fourier techniques for tomography arebasedon the Fourier slice theorem Eq. (3.11 ). The one
dimensional Fourier transfarm of the projections gives the two-dimensional transfarm G of g
along lines. The Fourier transfarm of g can be found in the entire plane (u,v) of Fourier space
by interpolation, after which g is obtained by the inverse Fourier transform. The mathematica!
implemen talion of this method is notstraightforward [Herm80, section 9.2].
Especially if information about partsin Fourier space or real space is missing (e.g. incomplete
data or the spatial frequency being bounded in the projections), it can be useful to iterate be
tween Fourier space and real space [KakS88, p. 311] to take advantage of all information in
both spaces. In real space, for example, the boundary of the reconstruction and positiveness of
the salution can be imposed easily. Iteration between the spaces ensures that this information
from real space also is taken into account in Fourier space, while in Fourier space other infor
mation is present and, for example, spatial filtering can be applied. This iteration is called the
Gerchberg-Papoulis method [DefM83 and references therein].
3.2.3 Series expansion methods
In series expansion methods both sides of Eq. (3.1) are expanded in a series of orthogonal
functions. Here the Cormack method, which uses a particular set of orthogonal functions, is
discussed, tagether with some extensions. The explicit formulae of the expansion are given to
clearly show the relationships between them, although these expressions are not used in this
50
Tomographic inversion methods 3.2
thesis. These expressions of the Cormack metbod show in a convenient way the order of the
expansion that can be used for tomographic reconstructions, which is important inforrnation for
all tomographic inversion methods.
3.2.3.1 Description ofthe Cormack methad
The expansion of g into orthogonal functions is usually done in polar coordinates (r,e). In
projection space it is now more convenient to use the range of the angle 1/J as defined in subsec
ti on 3.1.1, i.e. [0, 21t], insteadof angle Ç, and to choose p always positive. The radial coordi
nates rand p are normalized to the radius of the reconstruction area. Circular harmonies keep
their form under Radon transform,t and are therefore a suitable choice for expansion in the e and 1/J. In the two-dimensional case the expansions of g(r,e) andj{p,I/J) into circular harmonies
are:
and
g(r,e)= Lgm(r)eim8
m=O
f(p,I/J)= Lfm(p)eim!Jl .
m=O
(3 .19)
(3.20)
Note that in this definition the functions gm(r) andfm(p) are complex and thus give two parts,
one for cos mx and one for sin mx (x is e or 1/J). It can be shown that the "radial" parts gm(r)
andfm(p) farm a Gegenbauer transfarm pair [Natt86, Dean83]. lffm(p) is known, its Radon
inverse can be written as [Corm63, Dean83]:
(r) = _.!.~J' f m(P) T m(P Ir) r dp gm d ( 2 2 )1 / 2 • 1trr pp - r
(3 .21)
where T m(x ) = cos(m arccos x) are the Chebyshev polynornials of the first kind. Many proper
ties can be derived for different expansions of the radial components into series of orthogonal
functions [Cha V81], but one is of particular interest: the gm(r) is expanded in Zernike polyno
rnials Rmt(r) as:
where Rmt(r) is defined as:
gm(r) = Lam/Rmt(r),
1= 0
(3.22)
t The angular harmonies are special cases in two-dimensional space of spherical harmonies that have useful
properties for the Radon transform, see Ref. [Dean83, pp. 151].
51
Chapter 3 Tomography and other analysis methods
Ll (-1/(m+2l-s)!rm+21-2s Rmf(r) =
s!(m + l-s)!(l- s)! s=O
(3.23)
The Radon transfarm of the Zernike polynomials are Chebyshev functions of the second kind
Un(x) = sin(narccosx) for n = 1, 2, ... and Uo(x) = arcsin x . Hencefm(p) is expressed as: 00
fm(p)= Lam! m+~l+ 1 Um+2l+l(p). (3.24)
1=0
Equations (3.19), (3.20), (3.22) and (3.24) define the tomographic inversion method known as the Cormack method [Corm64]. Because there is only a finite amount of data available, the
problem needs to be regularized. This is done by limiting the number of functions in the
expansions. The maximum number of m for which there is information in the data is deter
mined by the number of projections, and the maximum number of I by the number of detectors
in each projection [GraS88, Howa88, KakS88, chapter 5 therein], which is equivalent to the Nyquist limit. The way of the implementation by Granetz [GraS88] is to make a least squares
fit to the measured data in projection space to Um+2l+J(p) and the harmonie functions cosmif>
and sin mif> todetermine the coefficients aml· Because of the discretization of fm(p) along p,
Eqs. (3.20) and (3.24) can be combined in a matrix equation from which the coefficients aml
can be solved. This can be done, for example, by determining a regularized inverse of the
matrix equation by truncated singular value decomposition (see subsectien 3.4.2). The fit in projection space can also be made by Fourier expansion of j(p,1p) in the 1/> direction and then
calculating the amt from the projections of the f m(p) on the U m+21+ 1 (p ), i.e. inversion of Eq. (3.24) [DneL90]. Once the coefficients have been determined, the inversion to obtain g(r,8) is
obtained by substitution of the coefficients aml into Eqs. (3.22) and (3.19).
3.2.3.2 Considerations when using the CornUI.ck methad and extensions
Although the Cormack method was developed for medica! tomography, the properties are not
advantageous to resolve all small structures because of the choice of orthogonal functions. In
plasma physics, and mainly in soft x-ray tomography, the choice of circular harmonies is how
ever well suited for application to MHD-phenomena. Especially with a few number of projee
tions still reasanabie results can be obtained from the limited amount of information. With more projections the resolution can however be disappointing. Very asymmetrie and localized emis
sion profiles cannot be reconstructed well with only few components. Furthermore a low level
of emissivity in the centre of the plasma (as is the case with visible light) cannot be modelled
wel! by a fini te series of such radial functions.
Usually the number of harmonies that can be taken into account has to be limited further than the Nyquist limit to imprave the reconstructions. Aliasing from structures in the plasma with
higher components than the maximummand I cause distortions in the reconstructions. A way
52
Tomographic inversion methods 3.2
to delermine the number of components that should be taken into account, and even be
weighted by their reliability, has been developed and applied [DneL90, LyaT93]. This method
delermines the reliability of the various coefficients by a statistica! analysis in time, after which
a weighted smoothing is applied to the coefficients. This reduces the influence of noise on the
signals and noise caused by the tomographic inversion method.
Because Zernike polynomials are not well behaved at the edge of the reconstruction area (where
they have a value unity, whereas the emissivity is expected to go to zero), some authors have
tried other expansions inslead of an expansion into circular functions and Zernike polynomials.
In general an expansion of g(r,e) is made into circular functions and Bessel functions (called
Fourier-Bessel expansion), e.g. Refs. [Naga87, NagB90, WanG91]: Bessel functions give a
good distribution in radius where the different components are maximum and minimum, and
are zero at r = I. In this case no analytica! expression for fm(p) is obtained, but the fit has to be
done numerically.
To imprave the capabilities of the Cormack methad for application on non-eireular plasmas,
methods have been developed to transfarm flux coordinates to circular ones [NagE92]. or to
describe the expansions into functions given in flux coordinates [Fuch94]. Also, the number of
viewing directions has been artificially increased by assuming rigid rotation of the plasma
structures: the measured profiles at different instances in time are rotated to represent one point
in time, e.g. Refs. [NagE92, NagT81, Büch91].
3.2.4 Algebraic reconstruction techniques (ART)
The methods discussed in this subsection arebasedon the discrete Radon transfarm Eq. (3.7)
and the salution of this system of equations in an iterative way. In this subsection iterative
methods of the type algebraic reconstruction techniques (ART) are described. Non-iterative
methods and other iterative optimization techniques to solve such a matrix equation are presenled in subsectien 3.2.5.
Strictly speaking, methods based on Eq. (3.7) are also series expansion methods where the
function K describes the discretization into cells. Usually square cells are taken. The function
K, or matrix Ku, is also called weight matrix. It contains all information about the contribution
of the emissivity to a certain detector. As said in subsection 3.1 .3, for perfect line-integrals the
weight matrix elements contain the length of viewing chord through each cel!. For more com
plex systems the weight matrix elements can accurately describe the imaging system.
The basic iterative methad to solve Eq. (3.7) for tomographic probierus is basedon Kaczmarz's
methad in rnathematics and is called the algebraic reconstruction technique (ART), more
specifically called additive ART. Its basic formula is:
53
Chapter 3 Tomography and other analysis methods
(0,0) l; 0
Figure 3. 6 Graphical representation of the convergence of ART, applied to three equations (solid I i nes) with
two unknowns 8l and 8l· The origin has been taken asthestarting point and the iterative projections to the lines
described by Eq. (3.25) are indicated by dashed lines. In this example À.== I. Note that the system of equations is
inconsistent and a single solution does not exist.
(3.25)
where j and k indicate one of the N pixels, i the detector, l the iteration, À! is a relaxation
parameter (which may depend on the iteration) and IIKIIi = (L,;K2.) 1,'2 the Euclidian norm. In o !)
Eq. (3 .25) the iterations are only indicated symbolically. The iterations l are several times over
all detectors D, and i= lmodD. Allpixelsjare updated in each iteration. Such a methad is called a row-action methad because in each iterative step only one row of the matrix equation is
considered, and the matrixKis not changed during the iterations [Cens81].
The interpretation of Eq. (3.25) is indicated in Fig. 3.6 fora case of three detectors and two
pixels. In a consistent underdetermined system ART converges to the minimum norm salution
(i.e. llgll minimum), if the initia! salution gO is properly chosen [Herm80, p. 188; Natt86, p. 1361138/170], for example gO= (0, ... ,0). In the case of noise (when the !i nes in Fig. 3.6 have
essentially a non-zero width corresponding to the noise level) such a unique salution is not
defined: the system is inconsistent. A measure of convergence, in the case that there is no solu
tion, is the residual norm RI:
(3.26)
54
Tomographic inversion methods 3.2
where 11·11 is again the Euclidean norm. If this norm is minimum, the least-squares salution has
been obtained. It is common that after converging, the residual norm at a certain iteration starts
to increase. The basic regularization of the ART methad is to break the iterations at this point
(this is justified in [Natt86, p.89]). Another regularisation is the choice of the relaxation parameter À in Eq. (3.25). It can be shown that, for Àt ~ 0 in successive iterations, ART converges [Natt86, p. 1361138; CenE83].
Many variations exist on Eq. (3.25). The order of the equations can for example be sorted in
such a way that successive equations are "far apart" as to speed up the convergence. In ART the
values of the pixels are updated each iteration l (this is economical in the usage of memory space), but it can also be done every D iterations (an average update of all changes made during
the iterations over all detectors); this is called SIRT (Simultaneous lterative Reconstruction
Technique) [KakS88, HerL76]. In a technique called SART (Simultaneous ART) the basis
functions are chosen differently from the pixels to approximate the line integrals to minimize
digitization effects, and some other optimizations are done (also simultaneous updates of all pixels) [KakS88]. Many different ways have been found to take into account noise in the measurement: for example in additive ART [Herm80, section 11.3], ART2 [Gord74] and ART3 [Gord74, Cens81]. Multiplicative ART (MART) [Cens81] uses a multiplicative formula instead of the additive one of Eq. (3 .25):
l+l = À(_L_JjJ.l(ij l. ; (Kgl); ;
(3.27)
where À and J.l are relaxation parameters. Sametimes the exponent is omitted. Overviews of
ART-methods and the original references can be found in Refs. [Herm80, Gord74, KakS88,
HerL76, Cens81].
Improved reconstructions can somelimes be obtained by applying certain processes between the iterations l. Such processes can be: smoothing and the setting to zero of negative pixel values
(which would correspond to negative emissivities and hence are unphysical).
3.2.5 Optimization methods other than ART
Under the proper conditions ART can find the least squares salution (also called minimum norm solution) of Eq. (3.7), but it doesnottake into account noise. Other ART-Iike methods try to solve it for given noise. In this subsection other more general ways to optimize the solu
tion g of
Kg =f + t: (3.28)
are discussed, where t: denotes the noise. One can express the optimum by the inequality
(3 .29)
55
Chopter 3 Torrwgraphy and other analysis methods
where again 11·11 denotes the residual norm as defined in Eq. (3.26). An introduetion on this
problem can be found in Ref. [Herm80, section 6.4)] . Equation (3.29) deLermines a set of
solutions. A criterion is needed to obtain the most probable salution of these.
3.2.5.1 Smoothness
Such a criterion is, for example, requiring smoothness on the solution, if information about the
smoothness is available a priori. With a smoothness operator Q one can write the minimization problem as:
min{ll!- Kgll2 + a(g,Qg)}, (3 .30)
where (-,.) is the se al ar product and a the regularization parameter which deLermines how important the smoothing is with respect to the norm. Equivalently, by tak:ing the derivative
d/dg, Eq. (3 .30) can be written as:
Kt(Kg-f)+aQg =O, (3 .3 1)
where t denotes the adjoint operator. Introducing weighting by the measurement error inf, thus
multiplying Kg andfby a matrix W which has the inverse of the standard deviations squared on the diagonal, Eq. (3.31) can be written as:
Ha f ,--A--------.. ,.......-'-.,
(Kt WK+ aQ )g = KtWJ , (3.32)
where a new operator Ha and transformed measurements J have been introduced. Thus a new
equation of the type of Eq. (3.7) is obtained, with K replaced by Ha, andfby }, which might
be better behaved than Eq. (3.7), and which can be optimized fora [Sa!G84; PreP82, p. 128].
The smoothness operator is related to the derivatives of the function it acts on, thus linking the
values of the tunetion in neighbouring pixels. A rigourous definition of a smoothness operator Q in a plane with Cartesian coordinates (x,y) is [SaiG84]:
(g,Dg) = Jf[ c,g' + c,(~!)' +c3( ~ )']dxdy, (3 .33)
where the coefficients c determine the weights of the different components. Higher order derivatives can also be included. Different smoothness propertiescan be imposed in the x and y
direction (which might be a physical property of the problem being descri bed) by taking c 1 and c 2 unequal. The derivatives in matrix form are given by, for example, (Jg!Jx ); = (gi+l- 8i-1)!2h, where h is the pixel size. The matrix Q is then found by inspeetion from Eq. (3.33). Also other definitions of the smoothness matrix exist. Smoothness operators are
discussed in e.g. Refs. [Herm80, section 12.3, Wi!E82].
56
Tomographic inversion methods 3.2
3.2.5.2 Itemlive versus non-ilerative methods
Eguation (3.32) can be solved in a variety of ways, both by iterative methods and non-iterative
ones. Because Ha is better behaved than K, it might be possible to solve it by matrix inversion, which is non-iterative, for example by truncated singular value decomposition or a method that
will bedescribed in subsectien 3.2.5.3. Other non-iterative optirnization methods are described
in, for example, Refs. [Herm80, chapter 13; WiiE82]. Evidently, ART-Iike schemes can be
applied to Eg. (3.32). There arealso ether iterative ways to solve matrix equations. One such
method, called Richardson's method, is described by Herman [Herm80, chapter 12]. SIRT
also, strictly speaking, belongs to this class. Another way to construct an iterative solution method is to make a non-stationary differential equation from Eq. (3.31) by replacing the right
hand side by dg/dt [VemF74; PreP82, p. 91]. The stationary solution (i.e. dgldt = 0) of Eg.
(3.32) is found by iterating the digitized equation:
g1+1 = g1 - rKtW(Kg1 - j)- raf2g1, (3.34)
where ris the relaxation time appearing from the digitization of dg/dt.
If the dimeosion of Kis very large, iterative methods are usually preferable, since they use a
minimum amount of memory space and are suitable for sparse and unstructured K. In the case
of the visible-light tomography diagnostic the di mension of Kis relatively smal! and both itera
tive and non-iterative methods can be considered. An advantage of non-iterative methods is that
fora fixed weight matrix and constraints the inverse needs to be found only once and repetitive tomographic inversions are obtained very quickly by matrix multiplication.
3.2.5.3 Reeonstruc/ion methad by Fehmers
The non-iterative constrained optimization algorithm by Fehmers [Fehm95] is one ofthe meth
octs used for tomographic reconstructions in this thesis. In this method the smoothness is imposed by thesecondorder derivative matrix f2. The object function o(g) and eenstraint func
tion c(g) are defined by
o(g) = llf2gf • c(g) = IIKg- !112 -llt:ll2 ·
(3.35)
In the function o(g) also low emissivity outside the plasma can be imposed. The object function o(g) is minimized under the eenstraint c(g) = 0, yielding the solution g. This means that the
smoothest function is found for which the misfit equals the noise level. The optimization is
done by a Lagrange multiplier method, which results in the salution of a generalized eigenvalue
problem. The most calculation intensive part of this eigenvalue problem depends only on K and
Q . The Lagrange multiplier, which is related to the regularization parameter in Eq. (3.31), is thereafter quickly solvable for any f and E, from which the solution g is easily calculated. In this
thesis this reconstruction algorithm is referred to as the Constrained Optimization (CO) method.
57
Chapter 3 Tonwgraphy arul other analysis methods
3.2.5.4 Maximum entropy
If smoothness is not known a priori, it can be implemented by the maximum entropy approach.
The entropy of g is defined as:
(3.36)
where the sum goesover the pixelsj. In a method developed by Frieden [Frie72] and applied to
tomography in plasma physics by Holland and Navratil [HolN86, HolF88, HolP90] Eq. (3.36)
is maximized under the constraints of the measurements in the form of Eq. (3.7) and noise
properties. The problem is then solved by the method of Lagrange multipliers. Maximum
entropy can also be applied in iterative methods. In fact it has been proved that MART gives the
solution with maximum entropy [Lent76]. Another iterative method by Minerbo (MENT)
[Mine79, MinS80] is derived in a different way from MART, but is sirnilar [without the expo
nent in Eq. (3.27)]. Other applications of maximum entropy in tomography can be found in
Refs. [GulN86, Nila82, Kemp80, Cott90] and more background information in Ref. [ShoJ80] (here the minimum cross-entropy is derived in the case a priori information exists in the form of
an approximate solution).
3.2.6 Application of tomography methods in plasma physics
3.2.6.1 Other methods used in plasma physics
The tomographic inversion methods discussed above form the main classes. For specific prob
lems (e.g. little or incomplete data) variations have been made, or new methods have been
invented. Without the prelension of being complete, a short overview is given of some interest
ing methods that have been applied to diagnostic systems on tokamaks.
As mentioned before, the Cormack method (subsection 3.2.3) has been widely applied in plasma physics, as have maximum entropy methods (subsection 3.2.5.4). Pixel methods such
as ART generally do not give satisfactory results [DécN86, IwaT87], unless many measure
ments are available [TakT86] . Other pixel methods which are based on minimization have,
however, proved successful [Wi!E82, FucM94]. Expansion methods other than in pixels or the
functions of the Cormack method have been developed, which often use minimization [ZolK92] or matrix inversion to find the coefficients (in Ref. [IwaT87] splines are used). Some
applications assume circular or elliptic equi-ernissivity contours and are therefore very related to
Abel-inversion methods, or an expansion on such contours [Smeu83]. Other methods not
directly related to the ones discussed here are described in, for example, Refs. [I wa Y89,
IwaY93].
Considerations about the reliability and the obtainable resolution are given in Refs. [Nila82,
Howa88].
58
Tomographic inversion methods 32
3.2.6.2 General considerations on tomography methadsjor the visible light tomography sys
tem on RTP
To choose a proper method for tomography, the properties and lay-out of the system should be taken into account. In this subsection several methods are considered.
The Cormack methad is the one most widely applied in tomography on tokamaks, at least on
measurements in the soft x-ray region. In the visible range theemission can, however, be so
inhomogeneous that it cannot properly be resolved by the basis-functions of the Cormack
method. As stated in subsectien 3.2.3.2, the number of angular and radial harmonies that can be used depends on the number of views of the plasmaand the number of detectors per view. For the present system, where between 1.5 and 2 full views are possible, this means that at
most only four angular harmonies (l, cos(}, sin(} and cos 2(}) can be resolved. For the Cor
mack methad this is a strict limitation to the angular resolution, otherwise it would become
unstable. Furthermore the methad is not very suitable for the hollow profiles [LouN83] which often occur in the visible wavelength range.
Many algorithms for tomographic inversion require a regular grid in projection space. These
methods can be used after interpolation from measurements on an irregular grid to a regular
grid, as wiJl bedescribed in subsectien 3.3.2. By interpolation the number of interpolated mea
surements can be increased to a number that can give good results with tomographic inversion
methods that are not specifically made for sparse data, such as FBP and ART. Simulations with ART for an 80-detector system have given reconstructions that were not satisfactory, whereas after the interpolation in projection space (i.e. more virtual measurements) the results improve.
In the interpolation in projection space to many virtual detectors a priori information about
smoothness and other properties can be taken into account, thus giving a smooth function that
can be tomographically inverted. Insection 3.3 the justification for using such a methad for the visible light tomography system is given, logether with a description and examples.
For sparse-data systems minimization methods are also good candidates, and because of the small number of detectors they do not need to be iterative. A minimization methad is success
fulty applied in this thesis. Also maximum entropy methods can be considered. To delermine which method gives the best result fora given system fora certain kind of emission profiles (e.g. peaked, hollow, large structures, small structures) numerical simulations are required taking into account a realistic level of noise.
Different tomography algorithms need different numbers of computations to receive a result.
For the investigations presented in this thesis the time-constraints are not very strict: usually
only a small number of time slices is considered. The algorithm described in section 3.3 requires a processing time of the order of a minute on a workstation.
59
Chapter 3 Tomography and other analysis methods
3.2.7 General remarks on imptementing and testing algorithms
The description of the numerical implementation of the tomographic inversion methods de
scribed in sections 3.2.1-3.2.5 is beyond the scope of this thesis. It is important to mention, however, that usually algorithms are most suited for parallel beam systems because the sets of
parallel beams from the different viewing directions form a square grid in projection space, which facilitates numerical integration over p and Ç. If the measurements are taken in a different
configuration, for example fan beams, there are several possibilities to cope with the situation. (1) The algorithm can be madefora non-rectangular grid. (2) The method can be rewritten fora
different system in analytica! form [KakS88, section 3.4; Herm80, section 10.1]. (3) The roeasured grid is transformed to a rectangular grid, for example, by interpolation [Herm80, section
10.5].
The residual norm as defined in Eq. (3.26) in the case of pixel methods, or as l!f- ~11 in the
general case, is an indicator of the quality of the tomographic inversion. Usually the Euclidean norm is used [Eq. (3.26)]. To make the reconstruction error estimate independent of the magnitude of the values, the relative residu al norm af,
(j - liJ -1{gll f- 11111 '
(3.37)
is considered. This is a quality indicator that can be calculated for any problem, since it only
requires the measurements and the result of inversion. As stated in subsection 3.1.6, usually the only way to test the performance of the combination of a certain system, algorithm and object is by simulations. With simulations a more objective estimate of the quality can be ob
tained by camparing the inversion of the pseudo-measurements with the phantom. A relative "reconstruction error" a8 in this case is:
a Jlg- goll g llgoll ' (3 .38)
where g0 designales the phantom and g the result after tomographic inversion. This quantity gives insight into artefacts and into areas and structures that are smoothed away. Finally, the
relative norm of the measurement error 8 is defined as 8 = l!f-Joll/l!foll, where Jo is the noiseless (pseudo-) measurement (from the exact solution go). Note that this is a norm over all detectors, which is different from the measurement error of one detector. In the case of measurements, only an estimate of 8 can be given.
To obtain a proper impression of the performance of an algorithm the simulations should include noise: different algorithms behave very differently in the presence of various levels of
noise, and the regularization should be optimized to a certain noise level. Therefore, estimates
are needed of the noise, or possible unknown systematic errors, as described in subsection 3.1.7 .
60
Tomographic inversion methods 3.2
The optimization of parameters in the algorithm can be done by minimizing the reconstruction
error (either a1or Cig). In such a way, for example, the optimum value fora parameter such as
the number of iterations in ART -like methods can be obtained, and the optimum regularization parameter can be found. It should be noted, however, that in methods such as in Eq. (3.31) cr1 is nat minimized itself [as in Eq. (3.29)], but that an additional property (i.e. smoothness) has
been added. Minimizing CIJ could result in a= 0, which is undesirable. In such cases the
optimum a is reached when dj"' 5J.. In general it is proper to optimize a problem for a certain
class of g by minimizing eig. The optima! Cig and CIJshouid be close in simulations, because in subsequent inversionsof measurements CIJis the only quality indicator available. In subsectien 3.3.3 this is investigated for the methad developed insection 3.3.
The reconstruction error depends on the initia! noise in the data, the measuring system and the
inversion algorithm. A noise amplification factor can be defined as ag/8. As a rule of thumb it
can be said that well-posed problems have a noise amplification factor of approximately 1,
whereas for well-regularized ill-posed problems noise amplification factors of 1.5-3 can be
achieved. The latter number is on average, for re1atively smal! signals the reconstruction error can be much larger. This is the noise amplification factor for random noise, for systematic errors it can still be around 1. Little can be said in general about the different behaviour of the noise amplification factor for relative and absolute noise: it depends on the specific situation and
requires simulations to frnd out.
From simulations it can be determined whether line integrals are sufficiently good approxirnations for the system used. If deviations are found possible corrections can be tried in the simu
lations.
As stated before, reconstructions usually give artefacts: structures that have been introduced by
the tomographic inversion methad in combination with a given system. The sensitivity to cer
tain types of artefacts can be checked by doing simulations with the expected type of tomograms. A special type of artefact, called the Gibbs phenomenon, aften occurs at sharp gradi
ents: the salution shows oscillations at the point of the sharp gradient, see for example Ref.
[Chap87, section 10.2]. It is also common that the reconstruction shows an increase along the viewing directions (especially with few cameras), and that there is a spreading out into regions
not sampled wel! and far from the boundaries.
3.2.8 Some variations to straightforward tomography
Usually tomographic inversions are made of the raw signals (possibly multiplied by calibration and sealing factors). If signalsf can be separated in a time-independent (de) part J and a time
dependent (ac) part J ,j = J + j, it can be interesting to invert J and j separately. This separation can give more insight in the fluctuation structures and changes ing. In principle the Radon transfarm is linear, but the inversion methad need nat be completely linear, for example
61
Chapter 3 Tomography and other analysis methods
by the inclusion of smoothing and the preventing of negative values in the solution. Therefore,
inversion of J and J might not give exactly the g and g that result from the separation of g.
One can also compare thefi:t) at different times t. The values rneasured at consecutive times can, for example, approximate the derivative df!dt because
f(t +LH)- f(t) dj ==-
l':.t dt (3.39)
Because of linearity the inversion of dfldt could give information about dgldt .
In the previous examples some processing tak.ing into account the temporal dependenee of the measured quantities is done before tomography. A different approach that uses singular value
decomposition is discussed in subsectien 3.4.2.2, and some methods were indicated in subsectien 3.2.3. In again a different approach, the time can be considered as a third coordinate in
actdition to the two spatial ones, and three-dimensional tomography can be done. Because the
measurements are line integrals in a three-dimensional space, the problem is to find the inverse x-ray transfarm from lines in purely parallel planes (a plane corresponding toa point in time). Nevertheless this can be a useful procedure because usually a strong correlation can be assumed
between structures in successive time slices, and this correlation would partly be taken into account by the tomography method, that is: a smooth transition between successive time slices
is achieved. Such a method has not yet been applied. A Cormack method where also a Fourier
expansion is made in time has been described in Ref. [KraK88].
3. 2. 9 Some properties of kernels, apparatus functions and weight matrices in conneetion with tomography
The point spread function is defined as the function resulting from an operatien on a point
souree in two-dimensional space [i.e. on a delta function O(x) O{y)]. For linear operators each function can be seen as a summatien of many point sourees and hence the collection of point spread functions for all points (x,y) describes the operator completely. For the Radon transferm
it was shown in subsectien 3.1.5 that the point spread function has a sine-shape in projection space. In the case of the Radon transfarm with a kemel [Eq. (3.4)], it is clear that the point spread function equals f(p,Ç) = K(p,Ç,x,y), and K(p,Ç,x,y) can be considered to be the point
spread function for all points. Usually the non-ideal K will tend to widen the response function f(p,Ç), such that a point souree results in a sine-shaped band in projection space instead of a sine-shaped line. Because the function K describes the measuring system completely, it is also
referred to as apparatusfunction (or instrumentfunction). Ifthe souree is nota point souree but has a spatial extent, a similar widening occurs. For a square souree this is derived in Ref. [Dean83, pp. 62].
The weight matrix is the discrete counterpart of the point spread function, where the contribution is averaged over pixels. The weight matrix elements contain the relative contribution of
62
Tomographic inversion methods 3.2
each pixel to a eertaio detector with respect to the other pixels. A minor difference in usage
between weight matrix and point spread function is that usually the point spread function is
considered from the pixel point of view (i.e. to which detectors does a pixel contribute),
whereas the weight matrix can be considered as descrihing which pixels contribute to which detectors. The latter gives more insight; it can be depicted in a plot (see section 5.3, Fig. 5.2).
A measurement of a system with a eertaio point spread function can be considered as a mea
surement with an ideal system convolved with the point spread function. The influence of the
point spread function (or kemel) on the Radon transfarm and its inverse is discussed in Ref.
[SmiK85]. Insteadof pixels, alsoother basis functions can be used to result in the expansion Eq. (3.7). For example, it has been proposed to choose the set of point spread functions as
basis functions [BuoB81].
3.3 An iterative projection-space reconstruction algorithm
An Iterative Projection-space Reconstruction (JPR) algorithm that takes into account the features of the visible light tomography system on the RTP tokamak has been developed by V.V. Pick
alov and the author. The reconstruction method and results are described in this section. The
tomographic inversions are done by a regularized scheme of the Fittered Back Projection
method. A scheme in which iterations are done between the projection-space reconstruction and
tomographic inversions to imprave the results is demonstrated. The algorithm and simulations arealso described in Ref. [MelP95] .
3. 3.1 Introduetion and points of attention for the visible light tomography system on RTP
The visible light tomography system on the RTP tokamak has some properties that make it difficult to apply standard tomographic reconstruction techniques directly to its measurements.
These features include the non-parallelism and the distri bution of the relatively few viewing
chords, and the width and other properties of the viewing chords. These properties of the sys
tem are discussed in detail in chapter 5.
To solve the problem associated with the distribution of viewing chords, an iterative scheme for reconstruction of the projection space has been developed. The purpose of this scheme is the
interpolation and smoothing of the signals from the irregular grid in projection space, given by
the viewing directions of the detectors, to a regular one. In Fig. 3.7 the positions of the detec
tors of the system in projection space are shown, forming an irregular grid. A regular grid, also
indicated in Fig. 3.7, corresponds to a parallel beam system. A methad to do a similar recon
struction in projection space has been described by Prince and Willsky [PriW90]. Other techniques (for example Akima [Akim78]) to interpolale from an irregular to a regular grid have
been considered as wel!, but were not used here because information known for the system and
63
Chapter 3 Tomography and other analysis methods
~ x• x 0 ,g 0 .x >I' x 0 0 x •x 'l<
0 "\ 0 0 0
Xo 0 0 oX ~ oX
x 0 ~ x 0 :l< X' x 0 0 y• x 0 ox )Q x 0 0
0 0
1 0
x ((. 0 0 x 0 0 .-x X' 0 x ~ x 0 : x
~0 0 0 0 ol 0 0 Xo
o XX
x
oX o•
0.. 0 x 0 0 0 x 0 0 0 0 ct< 0
x 0 0 x 0 •x 0 'I! 0 x 0 0 0 0 0 0
x 0 0 x 0
// Xo 0 x
x 0 ::x 0 0 ~ x . 0 0
~ xo 0 . x• ~ 0 •x
0 0 0 0 0 0
"" Xo 0 0 x 0 d<
~ x 0 x 0 ~ -1
0 0 0
0 rt/2 1t
s (rad)
Figure 3. 7 The points of measurement in projection space of the actual visible light tomography system on
RTP (more details in chapters 4 and 5) (solid circles) and the regular grid corresponding to measurements by a
parallel beam system (open circles) (here a typical regular grid of 11 projections with 27 detectors each is de
picted). For comparison also the coverage of the projection space of a fan beam system with 5 cameras with each
15 detectors is depicted (crosses). The origin of each fan is located at three times the reconstruction radius. The p
axis has been scaled to the radius a of the reconstruction area (a= 0.19 m for the actual system).
smoothing cannot be implemenled as easily as in the metbod presented here. The interpolated
signals on the regular grid can be tomographically inverted by standard techniques for parallel
beams. Here the FBP technique as described in subsection 3.2.1 is used.
The pseudo-measurements on the regular grid correspond to line-integrals, whereas the actual
system does not measure exact line integrals because of effects such as different sensitivities, calibration factors, width of viewing chords and other properties. Therefore, to be comparable
to the pseudo-measurements, the measurements f should be scaled to values that would be roea
sured by line integrals, taking into account these effects. Since these effects are contained in the
weight matrix, the power measured by each detector can be expressed as:
f; =IJ W;(x,y) g(x,y)dxdy"' Lj W;jKj• (3.40)
where Wi is the weight function for detector i, and Wij the weight matrix of Eq. (3.7) (the
weight matrix is more thoroughly discussed in sections 3.2.9 and chapter 5). The power /i measured along a pure line-integral can be expressed, according to Eq. (3.3), as:
]; =IJ g(x,y) 8(p; +x sin Ç;- ycosÇ;)dxdy. (3.41)
64
Projection-space reconstruction algorithm 3.3
The sealing factors; from measurements of the actual system to ones that would be measured
by pure line-integration is:
]. S; = _1_.
!; (3.42)
With a known sealing factor s; the scaled measurement /; can be obtained from the measure
mentJi. Note that the sealing factor depends on theemission profile g, i.e. g acts as a weight
function for W;(x,y). For a constant emission profile g(x,y) = 1 inside the plasma and 0 out-
side, one obtains the approximate relation:
J L S · = _1_"' --'-
' !; L;Wij' (3.43)
where L; is the chord length through the plasma. The consequences of the sealing are discussed
in detailinsection 5.5.
3.3.2 Description of the reconstruction algoritbm in projection space
The main function of this reconstruction method in projec
tion space is the interpolation of the scaled signals from the
irregular grid to a regular one. The sealing is done by mul
tiplying the signa! with an appropriate sealing factor, e.g.
calculated from an a priori assumed emission profile.
The relation between two-dimensional functions determined
on the irregular and regular grid is contained in a linear sys
tem of equations which describes a bi-linear interpolation to
the four regular grid points surrounding each irregular grid
point. Following the definition of quantities in Fig. 3.8, Op and oç denoting distances in projection space, one obtains for the weights of interpolation:
yielding the relation
a1 =(1-oç)(l-op),
a2 = (1- oç)op,
a3 = oç oP'
a4=oç(l-op);
4
]; = ~>mÎm m=l
p
t
Figure 3. 8 Definition of the sym
bols for the bi-linear interpolation be
tween an irregular and a regular grid.
(3.44)
(3.45)
65
Chapter 3 Tomography and other analysis methods
between the scaled measured signa! by detector i, /i, and the interpolated val u es J m in the sur
rounding regular grid points m. For all regular grid points, this can be extended to the system of coupled equations
]=A], (3.46)
where A includes all Gim for all detectors i with measurement/i. and where mis renumbered to all regular grid points, with interpolated measurements J m· The bi-linear interpolation described in Eqs. (3.44) and (3.46) is the simplest interpolation possible, i.e. only between the ciosest
neighbours. This could be extended in the same scheme to more neighbours or to, for example, bi-cubic splines, which would make the matrix A more complicated. Simp Ie linear interpolation together with the steps discussed in the next paragraphs proved adequate.
The system of equations (3.46) is solved iteratively in a way very similar to the Algebraic
Reconstruction Technique (ART) in tomographic inversions. The difference with straightfor
ward ART is the application of extra steps inside the iterations. In each iteration the following steps can be taken: (1) interpolation, (2) smoothing, (3) application of boundary conditions, ( 4) application of a priori information and constraints and (5) tomographic inversion and backcalculation. Some of these steps can be switched off or are not done in all iterations. Each step is discussed below. The main (outer) iteration is designated by iteration number l. All iterative
steps described are summarized symbolically in Fig. 3.9.
Step 1. For iteration l + I the interpolation is done for the regular grid points m which surround the irregular grid points by one iteration of the ART method [Eq. (3 .25)] over all D
0 g
f
Sealing
I g~d, I l; S; =I I Wg· j l) )
-1 I /; =s;/;
t (J
f
J
'I f Gerchberg- f--Papoulis-like step (Sb)
j~ = fldl
I g ~------ - --------------
lnterpolation by one iteration of ART (I) smoothing (2) extra conditions (3,4)
- 1 ' I ' l +l f,f -7 f
l+l-7l ~ 'I f
(J · f
Tomography FBP (Sa)
'I I f -7 g
• • Figure 3. 9 Schematic representation of the iterative procedure. The symbols and numbering defined in the text
are used. The dashed arrow has nol been implemenled yet.
66
Projection-space reconstruction algorithm 3.3
detectors. The iterative ART-formula in this case is: ~ A//'-1
All' AII'-1 fr-(Aj· )r J,;, = f,;, +Àl IIAII7' al'm·
(3.47)
The iteration is over l' = I .. . D, which corresponds to i and l in Eq. (3.25). Before the iterations, JL·~=O is set to J~, and afterwards Jlin1 to JU;;D. The residual norm according to Eq. (3.26) now is R 1 = { Li U i- (A}l)i] 2 } 112.
Step 2. Subsequently a smoothing is done. The tunetion of the smoothing, apart from smoothing itself, is to spread out the interpolation outside the regular grid points that surround measuring points. In principle the smoothing can be included into the matrix A as was described in subsectien 3.2.5, but here a simp Ier way was chosen. The smoothing on J is dorre by window smoothing, usuallyin a moving 3 x 3 window, i.e. the average over the points in this window is taken as the new value for the central point. Alsoother types of smoothing have been tried in simulations, such as independent two-dimensional spline smoothing; these gave similar results.
Step 3. Boundary conditions applied are:
f(p,!~) = 0 for l%121, (3.48)
i.e. zero emission outside the reconstruction area (radius a), and the Möbius periodicity of Eq. (3.9). The first boundary condition is reached by putting the edge values to zero on every iteration. The Möbius periodicity is in principle already implemented in steps (1) and (2), but could also be reached by taking the average of the val u es of J (p, Ç) at Ç = 0 and Ç = 1t.
Step 4. An initia! solution Jo from a priori information about the expected solution or from previously made calculations can betaken into account at the beginning of the iteration. The Jt can also be modified by forcing them to comply with constraints such as the conservation of emissivity [Eq. (3.10)]:
(3.49)
where Eis the total emitted power. Also highermomentscan be treated in this way. Equation (3.49) could be applied by sealing the integral over each projection to the average value E of all integrals. For the first moment it would be possible to shift the centre-of-mass of each projectiontoa sine shape as described in subsection 3.1.5. From simulations with phantoms it tumed out that application of the conservation or emissivity eenstraint worked wel!. However, in the case of ex perimental data, in which case more inconsistencies are present, it was however not useful to improve the result of reconstructions.
Step 5. Steps 1--4 do the required interpolating and smoothing processin projection space, after which any tomographic inversion metbod for parallel beams can be used to obtain the eerresponding tornogram (step 5a in Fig. 3.9). The number of pseudo-detectors after interpolation
67
Chopter 3 Tomography and other analysis methods
can be sufficiently large for such methods so that no further assumptions are needed in the tomographic inversion. Simulations have however shown that if only steps 1-4 are applied, the
resulting sinogram does not yet contain completely consistent data for tomographic inversion. This can be improved by a scheme similar to the Gerchberg-Papoulis iterative scheme between actual space and Fourier space as described in subsection 3.2.2. This Gerchberg-Papoulis-Iike scheme iterates between projection space and real space. At this stage of the iteration the jl is tomographically inverted to result in gl. From this gl the pseudo-measurements along the lines
corresponding to the regular grid points in projections space can be backcalculated [with Eq.
(3.41)] which replace the previous values Jl (step Sb in Fig. 3.9). These new values are consis
tent with all properties of projection space, and further iterations can make these backcalculated values converge to the measured values. For the tomographic inversion the regularized FBP
method described in subsection 3.2.1 is used.
Because all properties of projection space are contained in this method, this step 5 automatically
implements steps 3 and 4, and also does some smoothing. Because step 5 does not need to be applied on every iteration, implementing steps 3 and 4 can still be useful.
If the sealing factor of Eq. (3.42) depends on g, also a new sealing factor can be calculated from gl with Eqs. (3.40)-(3.41) and be applied to the measurements f to obtain new scaled
measurements fl. This step has not been implemented, because the dependenee of the sealing
factor on g is not very large and only affects the edge channels, as is demonstraled in section
5.5
The iterations are stopped when the residual norm does not decrease any more: i.e. when RI+! ~RI.
3.3.3 Results for reconstructions in projection space
Simulations with the algorithm described in the previous subsection have been carried out on various assumed emission profiles. In this subsection one phantom has been chosen to illustrate the importance of several parameters for different systems. Illustrations are given of both the reconstruction in projection space and subsequent tomographic inversion. In the simulations all dimensions are normalized to the radius of the reconstruction region. The phantom is bounded
by two circles, one with radius 0.9 and the other with 0.5. The larger circle is centred at the centre of the reconstruction, whereas the smaller one is shifted by 0.2 in the x direction. The emission between the circles is taken as unity and elsewhere zero. Figure 3.10 gives the phan
tom and its sinogram. The assumed Gaussian noise has 1% variance, relative to the pseudo
measurement. In step 2 of the algorithm regularized cubic spline smoothing was used in both p
and Ç direction. Border values of the sinogram are taken as zero and the conservalion of emis
sivity [Eq. (3.49)] is applied. Simulations with the phantom described are discussed in this
68
Projection-space reeonstruc/ion algorithm 3.3
(a) g(x,y) 1
0.5 0
-1
x/a
f(p,ksi)
-1 y/a pi a
Figure 3.10 (a) Tornogram and (b) si nogram of the phantom used for the simulations.
section, whereas simulations with various other phantom profiles to test the applicability of
tomography on the visible light tomography system are described in chapter 6.
3.3.3.1 Simulations on afan-beam system
The algorithm was first tested on a fan-beam system with five evenly distributed cameras hav
ing 17 detectors each. Such a system has a reasonably uniform coverage of projection space
(crosses in Fig. 3.7). The relaxation parameter Al has been varied; it was taken constant, i.e.
independent of the iteration l. In Fig. 3.11 the tornogram reconstruction error (Jg is given as a
function of iteration l for various X Figure 3.12 shows the tornogram reconstruction error (J8 ,
the si nogram reconstruction error (i.e. the difference in each grid point between the sinogram
of the phantom and the reconstructed sinogram) and residual norm (Jf (i.e. the difference be
t ween the measured point and the reconstructed sinogram) for A=2.0. It is clear that the con
vergence of all three quantities is camparabie and that the non-decreasing residual is a good cri
terion for breaking the iteration procedure. Theoretically, ART as in Eq. (3.47) only converges
if A= 0 ... 2 [Herm80, pp. 184; Natt86, section V.3]. Here, convergence is found also for
A> 2; the resulting reconstruction error quantities at the point of breaking of the iterations are
given in Table 3.1. The optimum varies for different phantoms and for the present phantom it is
Table 3.1 Reconstruction error quantities for simulations for 2::; A::; 3.
A. tornogram error si nogram error residual norm
2 .0 36.1 11 4.7
2.1 36.0 11 4.6
2.2 35.9 11.0 4.5
2.4 35.8 11.0 4.3
2.5 35.7 11.0 4.8
2.6 48.3 23 .5 23.9
3.0 77.1 59.5 47.6
69
Chapter 3 Tomography and other analysis methods
140
120
1 DO
~ 0 80 '00>
60
40
20 0 20 40
iteration
-------- 0 .5
60 80
Figure 3.11 Tornogram reconstruction error erg as function of iteration I for various relaxation parameters .:1.
for the fan-beam system. The iterations were broken at the point where the residual norm did not decrease any
more.
around A= 2.4. For largervalues the quality of the reconstruction quickly deteriorates and the
reconstructions errors oscillate between iterations (see Fig. 3.11 ). Table 3.1 shows that the improverneut of the optimum A value over A= 2 is smal!; hence A= 2 or slightly largeris a good choice.
Figure 3.13 shows the reconstructed sinograrn and tornogram forA= 2.0. These results have
been compared with a different non-iterative method: a modified version of FBP with SheppLogan filter [BroV88]. The modification consistsof bi-linear interpolation of the fan-beam data
into parallel beams and a regularized spline smoothing after it. By the interpolation the desired number of projections and viewing chords can be obtained. It has been found that this method gives reconstructions with the same accuracy as implementations of the exact formula of FBP for fan beams, but that it is faster [BroV88]. For comparison the same number of projections and viewing chords as for the interpolation in projection space was used. The result is very
similar to Fig. 3.13; the reconstruction errors for the tornogram and sinogram (calculated from the reconstructed tomogram) were 37.8% and 11.8%, respectively. This shows that the reconstructions of the new projection-space reconstruction method are of comparable quality as the ones of the modified FBP method, even slightly better. The main advantage of the new methad
is that it is applicable to irregularly distributed detection points in projection space, whereas the other methad is only suitable for fan-bearn systems.
70
Projection-space reconstruction algorithm 3.3
100rr-,--~.-.-..--~-.-• .--~l-.-.l--~-.-l,--,
-----+--- tornogram error 80 -
----- sinogram error
~ ~
60 --e-- residual norm -
"--0
40 11 ,__ .._ (J) -
20 ~ ~
-
" 0 0 10 20 30 40 50 60 70
iteration Figure 3.12 Tornogram reconstruction error, si nogram reconstruction error and residual norm as a function of
iteration for a simulation with À= 2.0 for the fan-beam system.
3.3.3.2 Simulations on the visible-light tomography system
Figure 3.14 shows the reconstruction of the same phantom for the coverage of projection space
of the visible-light tomography system (see Fig. 3.7) for À= 2.0. The reconstruction is reasonably good: the shape of the tomogram, e.g. the position and the shape of the hole, corresponds wel! to the phantom. Due to the less uniform coverage and 1ack of measurements in
some parts the quality of the reconstructions is less than for the fan-beam system: tornogram
error 43%, sinogram error 17% and residu al norm 14%. The convergence of the reconstruction
errors for this system is slower than the convergence shown in Fig. 3.12 for the fan-beam system: it takes approximately 20 iterations to approximate the asymptotic value and the iteration is braken off after 97 iterations.
The application of the Gerchberg-Papoulis-like iterations have an important improving effect on
the reconstructions for the visible-light tomography system, whereas the effect for the fan-beam system is marginal. For the same parameters as before the omission of the Gerchberg-Papoulis
Iike scheme results in a better residu al norm ( 10% ), but a much worse tornogram reconstruction error (53%) and a less symmetrie tomogram. The reason for this is that without the GerchbergPapoulis-like scheme a better fit to the measurements is obtained in reconstruction space, but
that the resulting sinogram is not self-consistent. With the Gerchberg-Papoulis-like scheme a consistent si nogram is obtained which yields a better tomogram, whi1e the fit of the si nogram to
the measured pointscan be worse. The marginal effect of the Gerchberg-Papoulis-like scheme for fan-beam reconstructions in contrast to the visible-light tomography system can be ex-
71
Chapter 3 Tomography and other analysis methods
( a ) g(x,y) 1
0.5 0
-1
x/a
-1 y/a p/a
Figure 3.13 (a) Tornogram and (b) si nogram of the reconstruction of the phantom with A.= 2.0 for the fan
beam system.
( a ) g(x,y) 1
0.5 0
-1
x/a
( b ) f(p,ksi)
-1 y/a -1 p/a
Figure 3. 14 (a) Tornogram and (b) si nogram of the reconstruction of the phantom with A. = 2.0 for the visi
bie-light tomography system.
plained by the more symmetrie coverage of projection space that does not allow inconsistencies by interpolations to appear.
For the application of the algorithm to actual measurements the sealing described in subsection 3.3.1 has tobetaken into account. This is done in the simulations described in chapter 6.
3.3.4 Conclusions
A new method has been developed to reconstruct projection space, from which tomographic inversions can be made. For fan-beam systems the method works very well, i.e. comparable to other methods. For the visible light tomography system the results of reconstructions are also good, but due to the less uniform coverage of projections space the reconstructions show more smoothing and more artefacts. For the latter system the Gerchberg-Papoulis-like scheme yields a significant impravement of the results. The relaxation parameter for the ART-like iterations for the reconstruction of projection space has been optimized, its optima! value being À= 2 or slightly larger.
72
Other analysis methods 3.4
3.4 Analysis methods other than tomography
In this sec ti on some methods related to tomography are mentioned, and some aspects of analysis methods such as singular value decomposition and cross-correlation are discussed. The last
two methods are used in chapters 8 and 9.
3.4.1 Parametrization methods
Inslead of inverting an inverse problem, one can also try to obtain the salution by applying the forward calculation of Eqs. (3.1 ), (3.4) or (3.7) on trial solutions and minimizing the discrepancy between the calculated pseudo-measurement and the measurement. By standard minimiza
tion techniques a least-squares salution of the problem can be obtained in a class of functions
that is described by parameters. A fast method to obtain the parameters is function parametriza
tion [MilL91], in which, from a class of functions and parameters, a database of solutions is
made from which the set of parameters giving the best match with the measurements is determined. A similar approach is the training of a neural network with such a database. It is impor
tant that the forward calculation describes the measurement process wel!, that the parametrization is adequate to describe the physical quantity measured, and that the method properly takes
into account noise. Such methods are generally only applied with relatively few parameters, less than one usually would like to reconstruct in tomography. lt is important for parametriza
tion methods that the parameters are approximately independent, i.e. the discrepancy tunetion that is to be minimized is not nearly independent of certain combinations of parameters. This is
difficult to satisfy for parametrizations of physical functions and line integrals thereof. For example: the measurements and the line integrals of the parametrized functions have to he scaled
by a linear parameter, whereas most integrals also depend approximately linearly on the
parameter descrihing the width of the function; the result is that two parameters that describe clearly distinct features of the function do hardly have distinct influences on the line integrals. This feature is related to the ill-posedness of the inversion problem. Because most visible
emission profiles in RTP have very complicated shapes that follow from tomographic inversions and other analysis techniques (see chapters 7 and 8), which would require a large
number of parameters to describe properly, parametrizations havenotbeen attempted for the measurements described in this thesis.
Instead of the described "forward" parametrization method with functions that have a clear physical meaning other more well-behaved and regularized methods related to parametrizations are used in tomography. In the strict sense every pixel method basedon Eq. (3.7), which is based on a grid, can be regarded as a parametrization method where the values in all pixels form
the parameters. A similar equation is obtained when the problem is expanded in orthogonal functions and the expansion coefficients are to be solved from the system of equations, as is the
case for the Cormack method in subsection 3.2.3 or other basis functions replacing the grid
73
Chapter 3 Tomography and other analysis methods
[HanW85]. For such systems of equations occurring from parametrizations the inverse can be
obtained by methods similar to the ones described in subsections 3.2.4 and 3.2.5.
3.4.2 Singular value decomposition
Singular value decomposition (SVD) is a powerful method with several applications. Two
applications are described here: it can be used to calculate a regularized inverse operator and
characterize the ill-posedness, and it can decompose experimental data into components with
different spatial and temporal behaviour, thus making it possible to investigate separate compo
nents and to filter out, for example, noise.
3.4.2.1 Principlesof singular value decomposition
SVD can be done on any m x n matrix A, and it results in a decomposition in the matrices U,
S, and V:
(3.50)
where VT is the transpose matrix of V. In various textbooks where SVD is described, different
definitions are used for the matrices. With the definitions according to the NAGt routine
F02WEF the matrix multiplication can bedescribed in the following way:
(3.51)
where the numbers m and n on the left and above each matrix indicate its dimensions. The
matrix U is row orthonormal, V is orthonormal and S is diagonal with non-negative values. The nonzero elements Sk of S are called the singular values, which are by convention sorted in ascending order. SVD is unique, except when some of the singular values are equal. The SVD
is the analogue of the similarity transformation which diagonalizes a square matrix, and is related to the eigensystem of the matrix AA T The largest singul ar value divided by the smallest
gives the condition number of the matrix AAT. If the condition number is largeit means the
matrix, if applied in a matrix equation, is ill-conditioned and might result in a bad solution. A method todetermine the ill-posedness of a problem by singular value decomposition of operators is described by Natterer [Natt86, p.91] .
Because the matrixSis diagonal, the backcalculation toA is simply
t NAG-Library Mark 15 (Numerical Algorithms Group Ltd., Oxford, UK).
74
Other analysis methods 3.4
(3.52)
Equation (3.52) is useful because if there are many smal! values Sk, an approximation of A can
be obtained by truncating the summation, neglecting the small values. For a matrix equation LpijXj =bi it can be shown that the x can be solved in a least squares sense by calculating
xj = Ikvjk ..lurbi. sk
(3.53)
where for all zero Sk the llsk has been set to zero. For a square non-singularA the x is the solution; for a singularA the resulting x is the salution with smallest Jength llx!!2; whereas if there is no salution the x with smallest residuali!Ax- bil is found [PreF89]. Due to these properties the matrix vs-1 uT is a kind of inverse of A, and is sametimes called the pseudo-inverse
of A. The result of solving ill-conditioned matrix equations might be improved by setting large
values llsk to zero. Truncating SVD applied to discrete ill-posed problems [such as Eq. (3.7)]
is equivalent to Tikhonov regularization [HanS92]. As has been said, SVD implicitly seeks to
minimize the norm of the solution; furthermore it can be shown that the truncated sum is the best approximation in the least squares sense obtainable with that number of components. The
norm of the rest termafter truncation is smaller than the smallest singular value that is not truncated [AubG91, theorem 1.12].
3.4.2.2 Biorthogonal decomposition
An interesting application of singular value decomposition to temporally resolved multichannel
measurements is the decomposition into, for example, spatial and temporal components. Recently it has been applied totokamaks [Nard92, Fuch94, DudP94, Dudo95, BesM94]. It is
called biorthogonal decomposition by some authors. The framework is outlined in Refs.
[AubG91, Nard92, DudP94, Dudo95] . Here only the application to the multichannel visible light tomography diagnostic is considered, i.e. only analysis of spatia-temporal behaviour.
For this purpose the time traces of a multichannel diagnostic are combined into a matrix A, here it is assumed that each time trace forms a column of the matrix. Singular value decomposition
decomposes the matrix into components. The columns of V (i.e. rows of VT) are related to the
spatial behaviour of the measurements, they form an orthonormal basis, and are called principal
axes, temporal eigenveetors or topos [notation: vk(Xj) = VJk· where Xj is the detector number] . The projections of A along V, i.e. the product US, give the time evolution of the signa! along the corresponding principal axis, and are called principal components, spatial eigenveetors or
chronos [notation: Sk uk(ti) = Sk U ik· where ti is discrete time; usually "chrono" indicates only uk(ti)]. The orthogonal basis can be expected to describe better the features of the signa! compared to other possible bases, such as the Fourier basis [Nard92]. Furthermore, the basis is linked to both the spatial and temporal behaviour, decomposing the measurements into the most
significant features that appear in both space and time. Unlike Fourier analysis, SVD rnight piek
75
Chapter 3 Tomography and other analysis methods
out modes with varying frequency in time. The singular values give the weight, or importance,
of the various components. Principal component analysis is related to SVD, but it only decom
poses the measurements into one of the matrices U and V, and the singular values, and he nee is
not sufficient to fully reconstruct the signal from the components. It should be noted that SVD is a deterministic tool, rather than statistica!.
Usually, there are only a few large singular values for measurements, and many smal! ones.
This means that there is much redundant information in the multichannel signals. The compo
nents corresponding to large singular values describe significant features in the object measured
(i.e. features observed simultaneously on several locations or for a sufficiently long time),
whereas the ones corresponding to smal! values are related to uncorrelated random and noisy
processes. If it is known which components correspond to noise, these can be fittered out by
truncating the summatien ofEq. (3.52) [DudP94, Dudo95]. Usually, it is clear from the values
of the singular values which of them correspond to the main components. In the Jiterature crite
ria are described to determine which singular values correspond to stochastic processes and which to deterministic processes. One such criterion is the Akaike information criterion (AIC)
[Akai74, WaxK85, WonZ90] which is used both in SVD [Dudo95] and in tomography to
delermine the order of expansion in the Corrnack method [Naga87, IwaT87]. This criterion was
not useful for the analysis in this thesis. An important advantage of SVD over other analysis
methods is the linkage between spatial and temporal behaviour: separate components can be
analyzed in their behaviour in both space and time. For example, tomographic reconstructions
can be made of one topo [Nard92, DudP94]. Furtherrnore, it uses a spatia-temporal criterion,
i.e. correlations in bath space and time, to distinguish random processes, unlike, for example,
Fourier analysis, which only finds correlations in time or space (i.e. Fourier analysis is a
monovariate method, handling only one variabie at the time while freezing the other ones). SVD
can distinguish modes that have a distinct spatial and temporal behaviour, but is unable to distinguish modes which have different properties in time but the same properties in space, and vice versa, resulting in components describing mixed modes [Nard92]. A disadvantage com
pared to Fourier analysis is that the resulting orthogonal basis in SVD does not necessarily cor
respond to physical properties, si nee, for example, modescan be mixed. If there are only a few
significant components, the signa! can almost be fully reconstructed from these. Storing only
the main components might be an efficient data reduction method, albeit nota reversible one.
SVD is not scale invariant, i.e. it is sensitive to the sealing of the data and the size of the matrix
A. Therefore it is appropriate to scale the data befare processing in order for the singul ar values
to have a smaller spread [DudP94, AubG91], i.e. the matrix A to be better conditioned. In the
literature this sealing is usually achieved by subtracting the temporal average and normalizing
the signals to the standard deviation of each time trace or the noise level. If this is not done the first two topos would be related to the average values and the first order deviations from the
average, respectively. For SVD such a process should be applied with care, because it is not a
76
Other analysis methods 3.4
natura! thing to do since averaging overspace or time may not yield identical results [AubG91
p. 696].
A complication in interpreting the topos and chronos resulting from SVD is the line-integrated nature of visible-light emission measurements. In the line-integrals an averaging is done, and
behaviour from different positions is contained in the same signa!. Therefore the topos and chronos might be difficult to interpret at certain circumstances, and might even be misleading in, for example, the case of mixed modes. SVD always has to be applied with care.
3.4.3 Correlation analysis
Correlation techniques are widely applied in physics. Therefore only a description is given of
the formulae that are used in chapters 8 and 9.
Spatia-temporal measurements s(x,t), where x denotes position and t the time, can be analysed
by a variety of techniques. The spatial, temporal and spatia-temporal Fomier transfarm S are
given by
and
S(k,t) = f_7~x,t)e·ik·xdx,
S(x,w) = f_7~x,t)e-imtdt,
S(k,w) = f_~~J!~x,t)e·i(kx+mt)dxdt,
(3.54)
(3.55)
(3.56)
respectively, where k is the wave vector and w the angular frequency. The wave vector and
angular frequency are related to the wavelength A and frequency fby:
lkl = 2A.n, w = 2nf. (3.57)
Because usually f ins te ad of wis used when descrihing the temporal behaviour, S and related
quantities will be expressed inf. The (auto) power spectrum P;;(j) of the signal s;(t) is given by
P;;(f) = S;'(f)S;(f), (3.58)
where the asterisk denotes the complex conjugate. In a similar way the cross-power spectrum
Pij(j) between two signals can be defined.
The temporal cross-correlation function pij(t) between two signals s;(t) and sj(t) is defined as
J~~ s;(t)sj(t + r)dt p (r) = , (3.59)
IJ (j. (j .
' J
77
Chapter 3 Tomography and other analysis methods
where ris the time-lag and <Jij denotes the standard deviation of the signal i or j. The correlation Pii of the signal with itself is called the auto-correlation. Due to the convolution theorem for Fourier transfarms Eq. (3.59) can be expressed in termsof the cross-power spectrum. Therefore, instead of evaluating the inlegral of Eq. (3.59), in many applications the signals are Fourier transformed and multiplied: the inverse Fourier transfarm then giving the cross or autocorrelation. A sirnilar definition as in Eq. (3.59) can be given for the spatial correlation.
The cross-correlation gives information on the time scale over which the fluctuations in two signals correlate, but it does not give information at which frequencies the correlated fluctuations occur. For this purpose the coherency ICij(j) is a useful quantity:
P;/f) IC;j(f) = ~P;;(f)Pjj(f) (3.60)
All quantities in Eq. (3.60) should be smoothed to give the coherency a meaning, for otherwise
by definition Eq. (3 .58) it would be I for allf
The Fourier transfarm and the correlation functions, and the discretization thereof, have to be applied with care. Examples of complications are: the lirnited time-windows of measurements, and averaging processes that can be implied to imprave the accuracy of the spectrum by reducing the speetral resolution. For the application of these analysis techniques on fluctuation measurements on tokamaks and a description if the complications, see for example Ref. [Thei90].
The measurements of the visible-light tomography system being line integrals rather that local measurements poses some problems for the interpretation of the results of correlation analysis. Not only is it incorrect to express the measurements as functions of position x, but also the cross-correlations can be expected to be reduced due to the averaging of fluctuations over the chord. Furthermore, because the viewing chords are not parallel, the interpretation is complicated further. These problems are dealt with in detail in chapter 9 and to a lesser extent in chapter 8.
78
The system visible-ligh t on RTP
for tomography 4
The construction of the system for visible-light tomography on RTP is described extensively in this chapter. The main parts of the system are discussed with an emphasis on parts of the
design that required calculations: the general lay-out, the positions of shields and viewing
dumps, and the application of optica! filters. Some measurements related to the effectiveness of
viewing dumps and electro-magnetic shield.ing are d.iscussed. The design criteria are recalled in
section 4.1 and the main hardware features are described in section 4.2. Special attention is
given to the optica! filters that are used in the system in section 4.3. A summary of the main characteristics of the system is given in section 4.4. In the main text only properties of the sys
tem that are relevant for this thesis are presented. Some technica] details are presenled in an
appendix.
4.1 Design criteria
The main design criteria and their justification stemming from the requirements for the diagnos
tic have already been discussed in section 1.3. In subsection 4.1 .1 a brief overview is given of the technica! aspects of the design. In section 4.2 they are discussed in more detail. Furthermore, in subsection 4.1.2 design tools, such as ray-tracing codes that have been developed, are
presented.
4.1.1 Overview of the design
The design criteria discussed in section 1.3 give the following requirements. In total 80 chan
nels are taken with a bandwidth of the electronics of 200 kHz and a photon yield as high as possible. Severe geometrical constraints are imposed by port size and the presence of the x-ray tomography system, which limit the possible viewing directions to five. The coverage of the plasma is mainly two-fold with emphasis on the edge. Because of the constraints, the coverage
of the plasma is not symmetrie. To enable the resolution of smal! structures (for example by
correlation analysis), narrow viewing chords are used.
The criteria are fulfilled by optica! imaging systems. Imaging systems can collect more light than is possible with pinhole systems with a camparabie resolution (see section 5.3). Four of the five imaging systems consist each of two spherical mirrors close to the plasma, the fifth
CIUlpter 4 The systemfor visible-light tomography
Table 4.1 Dimensions of the RTP vessel, tomography portand x-ray tomography system. All
radii are given with respect to the centre of the vessel.
Distance from flanges to vessel centre
Port Dimensions (the same on top and bottorn of the vessel)
Vacuum vessel
Plasma
X-ray tomography system
460 mm
220 mm high, 50 mm in toroidal d.irection and 340 mm in poloidal direction.
Wall at a radius of 240 mm
Top-down limiter at a radius of 164 mm
2 tubes on each !lange, 3 tubes 60 mm wide and 30 mm deep, one tube slightly smaller, all reaching through the port to 200 mm from the vessel centre
viewing direction has a lens system instead. Mirrors have the advantage of no chromatic aber
rations when a large wavelength range is measured.
The four optica! imaging systems with mirrors and corresponding cameras are, logether with
the soft x-ray tomography system, attached to two flanges that are mounted on two ports of
RTP (top and bottom). A schematic of the poloidal cross-section of the vessel with the compo
80
80
nents and the views of the plasma is
shown in Fig. 4.1. The soft x-ray
tomography system inside the vacuum
vessel consists of tubes for each camera
as well as structures outside the vessel,
which limit the space available for the
visible light tomography system both
inside and outside the vessel. Some key
dimensions of the port, the vessel and the x-ray tomography system are given in
Table 4.1. One of the flanges with some
of the structures attached to it is depicted
in Fig. 4.2. The visible light tomography
!!c:"!!!l."--7""1tir" DE Figu re 4.1 A polo i dal cross-section of the
tokamak vessel and the visible light tomography
system. Rays traeed from each third detector ele
ment are drawn to indicate the coverage of the
plasma region (shaded area). The detectors are
indicated by DA to DE. The numbers correspond
to the edge detectors of the cameras; these num
bers and the ones in between will be used to des
ignale particular detectors. The various viewing
dumps, shields and filters are indicated.
Design criteria 4.1
camera that observes the plasma from the LFS has the imaging system of lenses outside the vacuum window.
The rnirrors are not masked by apertmes or walls. Therefore, measures have to be taken to prevent light from reaching the detector directly or via only one of the rnirrors: fora proper imag
ing only light reflected by both rnirrors should reach the detectors. To block the direct light, several shields have been installed, the design of which is discussed in subsection 4.2.3. Some of the shields also block light that could otherwise reach the detectors via reflections on the many reflective surfaces of the port and flange. Furthermore, viewing dumps have been designed to prevent reflections of the vessel wall, which would give a diffuse contri bution from the entire plasma cross-section and would complicate the interpretation of the measurements.
Optica! filters can be inserted between the imaging systems and the detectors. The angular distribution of the rays incident on the filters can cause deviations in the transmission properties of the filters for different detectors. In section 4.3 these deviations are investigated for different
types of filters: interference filters, coloured glass and grey filters.
Figure 4.2 Photograph of top flange of the visible-light tomography system. The mirrors and filter holders
are visible. The tubes are part of the soft x-ray tomography system. The shields and camera box es have notbeen
mounted yet.
81
Chapter 4 The systemfor visible-light tomography
4.1.2 Design tools: ray tracing
For the design of the optica! imaging system, viewing dumps and shields, ray-tracing programs
have been developed. The optica! imaging systems, consisting of spherical mirrors in a Z-configuration, cannot be described by analytica! formulae in a simple way, but can be simulated
adequately by ray-tracing calculations. The only components that need to be considered in ray
tracing calculations of this system are curved reflecting surfaces, apertures and obstructions.
The ray tracing can be done with straight lines between all objects, also in the plasma region
because the plasma is optically thin in this wavelength range.
The design of the mirror positions and the study of properties of the imaging can be done by
two-dimensional ray tracing, i.e. rays only in a poloidal cross-section. This is because the
essential imaging properties of the system are in the poloidal plane. However, the imaging sys
tems are in fact three-dimensional. For accurate calculations, a three-dimensional ray-tracing
code has been developed. It has been used to design viewing dumps and shields. In the two
dimensional ray-tracing code the rays are traeed between components in a pre-described order, whereas in the three-dimensional case the order in which different components are reached is
determined during the tracing itself.
The ray tracing is used in section 4.2 to determine from which part of the plasma light reaches
the detectors. This is most easily done by reversing the rays and tracing rays that start on the
detector in all possible directions. The ray tracing is used in chapter 5 to delermine the amount of light that reaches each detector from every plasma position, and is instead done starting in the
point in the plasma that is under consideration to determine which of the detectors is reached by
each ray.
In the ray-tracing calculations the effects of slabs of optica! material (such as the vacuum win
dows and optica! filters) have been neglected. The effects of the thickness of the slabs, a translation of the exiting ray with respect to the incident ray, can be implemenled in the present ray
tracing codes, both the two-dimensional and three-dimensional one. In subsection 4.3.6 it is
discussed how the effects of optica! filterscan be implemenled fully in the three-dimensional
ray-tracing code.
4.2 Hardware
In this sec ti on the important features of the hardware of the visible light tomography system are
discussed. The following components are described: vacuum windows, the optica! imaging
systems with mirrors and lenses, the detector, amplifiers and shielding against interference, the
data acquisition and the starage of data. Furthermore, some attention is given to the accurate determination of the positions of all components. In the appendix to this chapter technica!
information is given on the components presenled in this section.
82
Hardware 4.2
4.2.1 Windows
Vacuum windowsare needed totransmil the light from the vacuum vessel to the detectors out
side the vessel. To match the speetral range of the detectors, sapphire windows have been chosen. The sapphire windows are attached to tubes, the inner walls of which are covered by foils
of chemically blackened stainless steel to prevent unwanted reflections on the inside of the
tubes. The construction of the vacuum windows is discussed in more detail in Appendix 4.A.l;
the blackening process in Appendix 4.A.2.
4.2.2 Optica] imaging systems
The optica! imaging systems for the visible light tomography system have been designed in
such a way that within the limited space available, they collect as much light as possible from
narrow viewing chords from five viewing directions distributed around the plasma.
For four of the cameras, mirrors are used inside the vacuum vessel. The mirrors have a reflection coefficient of approximately 90% in the entire wavelength range covered by the detectors
(300-1100 nm). The quartz mirrors have a reflective coating of silver and a proteelive coating
to withstand the conditions inside the vessel and to facilitate cleaning. During operation, how
ever, a dark layer is deposited onto the mirrors which is impossible to remove completely with
various cleaning agents. Also the harsh conditions inside the vessel, such as possibly the high
microwave ECRH power launched into RTP from a neighbouring port, cause damage to the mirrors, such as cracks and tiny craters. This means that some parts of the mirrors have a lower
reflection coefficient. At present no estimation is available and the deterioration of the reflection
of the mirrors is ignored. It is expected that further damages are less likely to occur after the
instaBation of shields (see subsection 4.2.3 fora description of the shields), which proteet the
en trances of the ports.
One camera is equipped with a lens system insteadof mirrors: a camera objective is positioned
in front of the detector array, looking through a vacuum window. The window is slightly too
small with respect to the imaging system, so that it has the effect of an aperture (the window
size is limited by the flange size). This lens system is of preliminary nature and might bere
placed by a mirror system in the future. The advantage of the lens system is that it collects more
light than the mirror system because it views a larger volume in toroidal direction . However, it has less spatial resolution in the poloidal plane.
The considerations mentioned in sectien 4.1, i.e. emphasis on coverage of the edge, narrow
viewing chords and at least two-fold coverage of allpartsof the plasma, resulted in the design
depicted in Fig. 4.1, where the components of the system and rays from every third detector are
shown. The mirrors are spherical and the focus of the imaging has beenchosenon the opposite side of the plasma. Not much variation in light yield is expected between configurations with
focus on different positions because the mirrors are close to the plasma. Each detector is 2 mm
83
Chapter 4 The system for visible-light tomography
wide. For the focus on the opposite side of the plasmaand a required effective viewing chord
width of about 10 mm, the sphericaJ mirrors were chosen with "focaJ lengths" of 562 mm (MI)
and 155 mm (M2). Focal length is here understood as half the radius of curvature, which is
only valid for light rays close to the optica! axis relative to the radius of curvature. The mirrors have been cut to reetangles (60 mm in poloidal direction and 16 mm in toroidaJ direction) to fit
into the vacuum vessel port. The result is that in the toroidal direction a slab of 16-30 mm is
imaged onto the detectors, and in poloidal direction effectively a strip of 10 mm width.
Two-dimensionaJ ray tracing was used to delermine and optimize the imaging properties of the
Z-configuration of the mirrors. This configuration yields narrow chords inside the plasma region viewed. Detrimental effects on the imaging by this configuration, for instanee no sharp
focus, are nat very important since light is collected from all positions along the chord. The ray
tracing also reveals the effect of vignetting, i.e. the detectors at the edges of the array receive
less light than the central ones because for the edge detectors only parts of the finite-sized mir
rors contribute to the imaging. The properties of the imaging system are studied in more detail in chapter 5.
Because of the narrow ports and the presence of the x-ray tomography system, the viewing
planes of all cameras do not coincide exactly: the planes of cameras B and C are slightly tilted
and the one of E is shifted in toroidal direction with respect to the other planes. The largest
deviations from the common plane are less than I o/o of the tokamak circumference. Th is deviation is negligible when it is compared with the requirements described in subsectien 1.3.2,
where the averaging effect of the maximum sampling frequency was discussed in conneetion
with the properties of tokamak plasmas. Therefore, all cameras are considered to be in the same
tomidal position. Details about the orientation of the viewing planes are given in Appendix
4.A.3. In the three-dimensional ray-tracing calculations these positions are taken into account
properly, whereas in the two-dimensional ray tracing the systems have to be projeeled onto the same poloidal plane, ensuring correct positions of the rays with respect to the vessel and main
taining the three-dimensional distances of the mirrors. Therefore, the positions of the mirrors
and detectors in Fig. 4.1 are projections onto the poloidal plane and the location of the rays is
an approximation.
The alignment and determination of the imaging properties are important to interpret the measurements correctly. The alignment of the mirror systems is described in Appendix 4.A.3. A
laser beam and a small light souree can be used to delermine the orientation of the viewing
chords. It is, however, difficult to accurately delermine the imaging properties in this way. Two
approaches to delermine the imaging properties of the system are discussed in this thesis. The
first approach uses ray-tracing calculations. For such calculations it is important to know the
positions and orientations of the mirrors very accurately. These positions can be determined on a computerized measurement bench, measuring reference points that are attached to the mirrors.
The alternative approach to characterize the system is by measuring the weight matrix.. Both
84
Hardware 4.2
methods are described in more detail in sections 5.3 and 5.4, respectively. The latter approach is the one actually used.
4.2.3 Design of viewing dumps and shields
As indicated in subsectien 4.1.1, viewing dumps and shields have been installed to diminish reflections on walls and other structures, and to prevent light from reaching the detectors di
rectly without imaging by the mirrors. The shields and view dumps have been blackened, which is described in Appendix 4.A.2. Because the system is essentially three-dimensional,
three-dimensional ray tracing has been used to determine the required sizes and positions of viewing dumps and shields.
In Fig. 4.3 rays traeed from the detectors of one camera into the plasma are shown: rays re-
vessel
' plasma radlus
direct rays imaged rays
Figure 4.3 Rays from all detectors of camera B that reach into the vessel. In the figure all rays are projeeled
in perspeelive onto the poloidal plane. Besides the rays, the flange, openings for the vacuum windows, mirrors,
detectors, port walls, vessel wall (partly shaded), one of the two tubes of the x-ray tomography system on this
!lange (partly shaded), and the plasma extent (dashed line) are drawn. In this ray-tracing calculation shields and
apertures were taken into account. The different types of rays are indicated.
85
Chapter 4 The system for visible-light tomography
t
-R
Figure 4.4 Rays from the upper cameras
intersecting a horizontal plane at the opening
of the top port (where the port is attached to
the main vessel). The plane is viewed from
the centre of the plasma, looking up. The
opening for the rays that are imaged by the
mirrors is drawn. Direct rays outside this hole
are blocked by the shield.
flected by the mirrors as wel! as rays reaching the detectors directly. From ray-tracing calculations it could be determined where shields and apertures should be put to block direct light. A
more accurate way todetermine the positions of the shields is by calculating the points of inter
section of the rays with a given plane. In Fig. 4.4 such a collection of points is shown on the
position of one of the shields that is mounted on the tubes of the x-ray tomography system on
the si de of the port dosest to the plasma. The contour of the intersection points of rays imaged by the mirrors give the required shape of the aperture of the shield, such that most direct rays are blocked by the shield and properly imaged rays are let through. Two of such shields have
been mounted on both flanges (see Fig. 4.1), with the additional function to prevent light from reaching the port under different angles, which could reflect against the port walls and reach the
detectors. Smaller shields have been mounted on the holders of the minors, because not all
direct light could be blocked by the shields in the port. For all cameras these shields have been
successful to block direct light, except for camera B where half of the detectors still see a smal! amount of direct light (see Fig. 4.4). The effects of this direct light are discussed in subsection
5.4.2.
Measurements with the system installed on a dummy tokamak section have been used to evalu
ate the effect of reflections on the walls. The set-up forthese measurements are discussed in subsection 5.4.1. To study the contribution of reflections on the walls to the signals, separate
measurements of the background of retlected light and nonreflected light are needed. This can be done by measuring the light from a smal! source: some detectors will see the souree directly,
while other ones wil! only measure the retlected light. An estimate of the total background light
produced by a constantemission profile can be obtained by multiplying the measured reflected light averaged over all detectors by the number of times that the light souree would fit into the plasma [Fig. 4.5(a)]. An estimate of the nonreflected light can be obtained by multiplying the
signa! of the detectors seeing the souree by the number of times the souree fits into the viewing region [Fig. 4.5(b)]. Measurements taken with the souree in different positions yield comparable results. With viewing dumps the reflected light is just above the minimum detectable level (the intensity of the souree is limited by the requirement that the detectors seeing nonreflected light should not saturate so that a comparison between nonreflected and background light is possible). An upper limit for the contribution of reflected light to the total signa! of 10% is found (partly containing a contribution from noise). Without viewing dumps (the tokamak
86
Figure 4.5 Schematic depietion of
the plasma, the sou ree, and the regions
contributing to the reflected light and
nonreflected light.
walls were simulated by sheets
of aluminium) the observed
background level was about two
to three times as high for the
viewing directions with mirrors,
while for direction E (with the lens system) it was even more.
The absence of the viewing
rafleeled
Hardware 4.2
nonraflactad
V• •;,! ragion contributing light
dumps would therefore have large effects on the interpretation of measurements.
The positions of viewing dumps have been determined in a similar way as the shields. The
places where the rays interseet the vacuum vessel delermine where viewing dumps should be installed. Viewing dumps for all cameras have been designed, and form three black shields that
are attached to the vessel wall (see Fig. 4.1). The tubes of the x-ray tomography system are
closer to the plasma than the viewing dumps, and could not be blackened because the blacken
ing could contaminate the plasma. Some detectors of the cameras A and C see parts of these
tubes, and could therefore suffer from reflections (more is said about this in subsectien 5.4.2).
4.2.4 Detectors and electronics
A schematic of the electronic system of the visible light tomography system is given in Fig.
4.6. Because of the required bandwidth, all electranies is parallel, meaning that each detector
has a system as in Fig. 4.6. In this subsectien a description is given of the detectors, the ampli
fiers, and the shielding of the electronic systems.
The detectors and electranies are positioned just outside the port of the tokamak, where there is
little space and where electrical interterenee levels are high. To avoid this complex solution, the
light could have been transferred by an optica! fibre bundie and have been detected in a well
shielded surrounding far from the tokamak. This would have had the additional advantage that
separate photodiodes could have been used which have a Jower junction capacitance. Fibres, however, were not chosen because of their increased absorption in the short wavelength range
and because of the coupling losses. With hindsight however, the disadvantages of fibres prob
ably would have been of little importance when compared with the efforts that were needed for
the chosen configuration.
87
Chapter 4 The systemfor visible-light tomography
camera box with copper and tin-piale shielding data acquisition room
+15V
diodearray element
r------------------------------ · I I
R= 750k0
500
I I I
I I I I I I
I ~3Qm I L--------j
main amplifier
Line-driver 1-----,--j
transimpedance amplifier 1
I I I
ADC computer
'-----' I I
------------------------------~ l--------~-------J
Figure 4.6 Schematic of the electtonic system of the visible light tomography system.
4.2.4.1 Detector
As detectors an array 1 with 35 silicon pin photodiodes is used in each camera; 32 of the diodes
are connected in pairs to form 16 detector elements, each 2 mrn wide (total area 2 x 4.4 mm2).
The remaining three detectors are grounded to avoid cross-talk on the neighbouring detectors. The current i produced in a photodiode by incident light power P is
i= 1)e P (4.1) hv '
where 1) is the quanturn efficiency, e the electronic charge, and h v the photon energy. The
detectors are sensitive in the wavelength range 200--1100 nm with a maximum quanturn efficiency of 88% at 720 nm, corresponding toa sensitivity (i.e. 1)elhv) of 510 mAIW. Figure 4.7
shows the speetral dependenee of the quanturn efficiency and sensitivity of the detector. The wavelength range of the system is limited to 300 nm on the short-wavelength side due to the imaging opties. The detector arrays are contained in 40 pin DIP type ceramic cases and are proteered from the surroundings by quartz glass windows. The detectors are used in reverse bias,
at which the junction capacitance is relatively low for this type of detectors: for the two detector elements combinedit is 120 pF.
4.2.4.2 Transimpedance amplifier
The current produced by the photodetector is to be measured, and therefore to be transformed
into a voltage. The bandwidth of the detection system is determined by the RC time-constant of the first stage of the measuring system, which in its turn is determined by the capacitance of the
detector and the input-impedance of the electronics. A transimpedance amplifier is used instead
I Hamamatsu S2313-Q35
88
Hardware 4.2
100 0.6
;R ~ 80 0.5
>. (/) (I) () 0.4 :::1 c (/)
Q) 60 ;::+ 'ü ;::· :f; 0.3 ~ Q)
E 40
~ :::1 0.2 c ct! :::1 20 rr 0.1
0 0 200 400 600 800 1000 1200
wavelength (nm)
Figure 4.7 Speetral dependenee of the quanturn efficiency and sensitivity of the detectors. These quantities
have been derived from a curve on the data sheet of the supplier (see Footnote 1).
of a resistor (the amplifier schematically depicted in Fig. 4.6), because the transimpedance
amplifier has an input-impedance that is approximately the value of the resistor divided by the
amplification value. The maximum possible current-to-voltage amplifïcation, which is deter
rnined by the resistor, is a trade-off between signal-to-noise ratio and the required bandwidth.
The main characteristics of the transimpedance amplifier are listed in Table 4.11. The input of
the amplifier is a field-effect transistor resulting in a high input-impedance. The bandwidths of
Table 4.11 Specifications of the transimpedance amplifiers
Quantity
Transimpedance resistor (R in Fig. 4.6)
Effective voltage amplification factor (A in Fig. 4.6)
Bandwidth
Response
Typical signa I when ent i re visible range observed
Maximum signa! (saturation level)
Noise level
Cross-talk for large input signals at frequencies lower than bandwidth
Typical number of photons
AC-coupling time constant
Time constant of !ow-pass filter of power supply
Value
750 kû
= 100
200kHz
0.4 V/',J.W for red light
IV
2V
<I mV
< I mV
IQil/s
10 s
10 s
89
Chapter4 Ihe systemfor visible-light tomography
the 80 amplifiers are slightly different from each other because electronic components have not
been matched and because the length of connecting cables varies. Because of the varying
bandwidths, differences in phase shifts may occur that should betaken into account when high
frequency (> 100kHz) phenomena are studied. A !i ne-driver with 50 Q output-impedance is applied to generale a current for the output voltage that can be transporled over a 30 m long co
axial cable to the shielded data-acquisition room. The transimpedance amplifiers and line
drivers of four detector elements are assembied on one single electronic print. Cross-talk
between channels has been minimized by extra shielding. The line-drivers are ac-coupled with a
time constant of 10 s to avoid the influence of drift. A plasma discharge in RTP lasts for at most 0.5 s and therefore the ac-coupling has a negligible influence on the measurements.
All detector elements of one array have a common cathode, to which a positive voltage is
applied. This voltage from a power-supply is filtered by a !ow-pass filter to avoid piek-up ef
fects. The 16 detector elements are each connected by twisted wires with the corresponding
transimpedance amplifiers. In this scheme closed circuits in which voltages might be induced by magnetic fields cannot be avoided. Putting the cables too closetoeach other to reduce piek
up would cause increased cross-talk between channels.
4.2.4.3 Magnetic and electronic shielding
The electranies of each camera is supported and electrically shielded by a capper box. Each
camera box had to be specifically designed to fit into the limited space available. The boxes are grounded properly as is described in Appendix 4.A.4. The noise level that is measured depends
strongly onshielding and grounding. At RTP noise levels of less than 1 mV peak-to-peak are
obtained when RTP is not in operation. This level is one to two orders of magnitude higher
than the signal-to-noise ratio that can be calculated if it were signallimited, dark-eurrent limited
or caused by resistor noise. Therefore it must be caused by the amplifier or by interference, or by a combination of the two.
Because of the strong and fluctuating magnetic fields near RTP, magnetic shielding is also re
quired. The main cause of interfere nee from the rnagnetic fields are the chopper controlled feed
back systems on RTP: the power needed to generale the fields that control the horizontal and
vertical position of the plasma is pulsed by switching on and off thyristor circuits (with switching frequency around I kHz). The switching causes time-dependent magnetic fields. The
feedback system for the plasma current is also chopper controlled, but has been modified
recently to smooth the effect of the chopper spikes. The varying magnetic fields during the
chopper spikes result in piek-up problems in many diagnostic systems. In the visible light
tomography system the most likely place where piek-up can occur is in the wires from the
detector array to the pre-amplifiers: the fluctuating fields can induce a voltage in the loop made up by the power supply wire and the signa] wires of all the detector elements. This induced
voltage is amplified by the amplifier and gives a large disturbance on the measurement. Experi-
90
Hardware 4.2
ments have been carried out, changing the position and orientation of one camera, resulting in a
rough estimate of the voltage induced in a loop at the position of the camera. The condusion is
that a voltage induced in the detector wires can fully explain the piek-up observed during measurements with the tomography system.
Magnetic shielding has been applied to reduce the piek -up. Because of the aperture in the box,
essential to let light through for the detectors, the shielding cannot be perfect. The very limited space around the cameras prevents the usage of thick layers of shielding. Possible shielding
methods and materials are discussed in Appendix 4.A.4. The solution chosen is to make a tin
piale box inside the copper box, which reduces piek-up. However, the shielding problem bas notbeen solved satisfactorily, and when interpreting measurements the chopper spikes have to
betaken into account. The amplitudes of the piek-up vary greatly between detectors and differ for different plasma conditions. In Appendix 8.A the characteristics of the the spikes are given
and a metbod is described to remove the chopper spikes by software from the signals. It would
be better, however, to avoid the chopper spikes of the position control by a smoothing similar to the one that is done for the plasma current controL
4.2.5 Data acquisition and processing
For each deleetion channel an amplifier is needed that can amplify or attenuate the signals to the proper level to make u se of the maximum range of the analogue-to-digital convertor (ADC), see
Fig. 4.6. The amplification or attenuation factors (x 118 ... x 16 with steps of x 2) can be set by computer and are automatically taken into account in the RTP data-acquisition programs.
Each channel bas its own ADC, which is VME based [HarW93]. These ADCs have a resolu
tion of 12 bits and a maximum sampling frequency of 1 MHz. However, usually 500kHz is used to measure during the entire pulse length of RTP discharges (0.5 s), since the available
memory for 8 channels is 4 Mbyte. Each module of 8 channels bas a local processor, which can carry out calculations on the signals. Because the amount of data per discharge is too large
to be stored on disk, only part of the discharge can be stored with fast sampling (a time-window). The decision on the time-window mustbetaken before the discharge. Data for the entire
discharge is stored at a slower sampling. The signals are stored, after appropriate digital filtering by the local processor, in the RTP DOM4 database [Hare93]. If large high-frequency fluctuations are present in the signals, they should be passed through analogue filters (in the rnain
amplifier module) before the ADC to avoid aliasing.
For further processing by tomography and other analysis programs, a program bas been written
to read the data in a selected time range from the DOM4 format and to automatically subtract offsets (introduced by the main amplifiers and ADCs), tobselect channels that are broken or saturated, and to do processing such as averaging and ac/de separation of the signals. Since
91
Chapter 4 The system for visible-light tomography
usually some detectors are deselected, the analysis programs need to keep track of actual detec
tor number for each signal, which complicates the data-handling in the programs.
4.2.6 Positioning
For tomographic processing a proper alignrnent and knowledge about the positions of the detectors is needed, because any uncertainty or wrong assumption introduces errors in the recon
struction. In the visible light tomography system the situation is complicated by the mirror sys
tem of which the properties have to be known accurately. Reference points have been attached
to all components of the system, i.e. mirrors, cameras, camera holders and flanges, so that they can be mounted in exactly the same position every time.
The way the positions of the mirrors can be measured was described in subsection 4.2.2. How the positions of the detectors can be measured, and how the relative positions of the flanges
have been determined, is described in Appendix 4.A.5. As was indicated in subsection 4.2.2.
the measurements of the mirror positions and detectors have not been used, but instead the imaging properties have been determined by measuring the weight matrix. The deviations in the
position in the ports from the design values are smaller than could be taken into account in the
set-up for the measurement of the weight matrix, and hence are nottaken into account in the tomographic inversions in this thesis. For more accurate tomographic studies in the future it
might, however, be necessary to more properly take into account the positions of all compo
nents.
4.3 The application of optical filters
To select eertaio wavelength ranges, for instanee one speetral line, optica! filters can be installed. This is done outside the vacuum vessel, between the vacuum window and the detector array, to avoid complications caused by using and changing filters in vacuum. Different kinds of filters are used. Interference filters can select narrow wavelength regions around speetral lines, whereas coloured glass filters can block large parts of the spectrum. Furthermore, grey
filters can reduce the light level independent of the wavelength.
Light from the plasma is incident onto the filters in a large angular range up to 16°. For narrowbandpass filters such as interference filters the transmission characteristics are angle-dependent. Also for coloured-glass and grey filters the transmission characteristic with respect to normal incidence is changed slightly by the increased path-length (thickness) travelled by a ray at oblique incidence. These effects in interference filters are studied in detail to delermine how nar
row the transmission bandwidth can be for this appl!cation. The effect of coloured-glass and
grey filters on the transmission in the present system is also discussed.
92
Optiealfliters 4.3
4.3.1 Interference filters
Effects that occur in interterenee fLiters for oblique incidence of a parallel bearn of light are: the
shifting of the peak of transmission toshorter wavelengtbs [DriV78, Liss68, LisW59], the widening of the transmission band [DriV78], and the decrease of the peak of transmission.2
The shifting of the peak is the most important effect, and is given by the approximate formula:
192 L1À.(19)=--2 À.0 ; (4.2)
2nerr
where 19 is the angle of incidence with respect to the normal, llÀ.( 19) the angle dependent wavelength shift, À.o the central wavelength of the transmission curve at normal incidence and nerf the effective refractive index of the dielectric material in the filter. In the literature different, but for
the quoted range of validity equivalent, formulae can be encountered. The quoted range of
validity varies, however, and bere Eq. (4.2) is assumed tobevalid for 19 < 20° [Liss68]. The
width of the transmission curve is determined at the half value and is called full width half maximum (FWHM). For the calculations presenled in this section nerf= 2.1 bas been taken, a value close to the ones of commercially available filters for the wavelength under consideration: À.= 656.3 nm (the Ha.-line). For an angle of incidence of 16° this gives a LlÀ. of approxi
mately 6 nm. The filter to be used should transmit the required spectrallines for all angles of
incidence between 0° and 16°, and corresponding shifts of the peak transmission. This suggests that a filter is needed that is shifted 6 nm to a Jonger wavelength with respect to the spectralline, and is at least 6 nm wide. This consideration does, however, nottake into account the effect of
inlegration over the solid angle of the viewing cone, which is discussed below.
4. 3. 2 Calculation of transmission of interf erenee filters for arbitrary solid angles
The inlegration over a solid angle changes the characteristics of filters compared to parallel bearns incident under an angle. From the literature approximations of transmission curves of
interference filters are known for circular viewing cones [DriV78, Liss68, LisW59, Lind67]. Often these approximations are for smoothly shaped (for instanee Gaussian [Lind67]) transmission curves. A rule of thumb is that the central wavelength for a transmission curve inte
grated fora cone has a shift that is approximately half the shift calculated from Eq. (4.2) for the maximum semi-angle in the cone. In the present system the solid angle that is viewed by a
detector has a rectangular shape (because the mirrors are rectangular), and for most detectors it is asymmetrie. In the poloidal direction, detectors in the middle of the array view between ±11 o ,
while detectors at both edges of the array view between --6° and + 16°, and +6° and -16° respectively. In the toroidal direction all detectors view between approximately ±3°.
2 No suitable description has been found. In Refs. [LisW57] and [Lind67] neglecting this effect is justified.
93
Chapter 4 The systemfor visible-Light tomography
The effective transmission curve T(À) of the filter for light incident from a given sa lid angle can be calculated from [Lind67, LisW59]
r21t r19< QJ) J, dq> J, diJ r(À, iJ) sin iJ
T(À) = o 121t o ld(QJ) , dq> diJ sin iJ
0 0
(4.3)
where iJ again is the angle of incidence with respect to the normal, q> the rota ti on angle around
the nonna!, r(À, iJ) the transmission curve for parallel beams incident at angle iJ, and iJ( q>)
defines the boundary of the solid angle. The denominator normalizes the transmitted light to the amount of light flux incident from the entire solid angle. In the cylindrical-symmetric case of a
viewing cone, the integral over q> gives 2n:, and only a one-dimensional integral is to be calculated for iJ( q>) = iJmax. the maximum semi-angle of the cone. In the rectangular-cone case the
two-dimensional integral of Eq. (4.3) has to be calculated with the boundary iJ( q>) depending on the shape of the mirror.
In the next subsection results are presented on basis of a model transmission curve r(À, iJ) to determine which FWHM and central wavelengthof an interference filter gives smal! deviations
in transmission between central and edge channels. Similar calculations for the measured curve
of an actual filter used in the system are presented in subsection 4.3.4 todetermine the perfor
mance. In subsection 4.3.5 the same calculation methad is used for other types of filters.
4. 3. 3 Calculations on interference filters with a model transmission curve
In the calculations the model transmission curve depicted in Fig. 4.8(a) was assumed, which is
an idealized shape of normal commercially available filters (for which the top is quite flat; here it
was taken slightly peaked to be able to delermine the wavelength of peak-transmission by numerical means). In the literature aften a Gaussian shaped filter is used for this type of calculations, but this is nota realistic representation of modem interference filters . The shape of the
curve was assumed to remaio equal for different angles of incidence, while the characteristic quantities, i.e. the shift of the transmission curve and the FWHM, are taken to depend on the angle [DriV78]. The latter has only a minor effect. The decrease of peak-transmission is nat
taken into account.2 The peak for normal incidence is taken as 100%, whereas in real filters a characteristic value is of the order of 70%. This is, however, not important since only relative shapes of the transmission functions are of interest. The calculations are for unpolarized light and polarization effects were nat taken into account. In Fig. 4.8 the numerically calculated3 effective transmission curves are shown for a filter with a FWHM of 5 nm: for a central
3 For the inlegration of the model transmission curve the routine DOIDAF from the NAG-Library Mark 15 is
used (Numerical Algorithms Group Ltd., Oxford, UK). Forsome calculations this routine is unstable outside
the wavelength region of interest for the conclusions in this section.
94
Optiealfliters 4.3
100
--(a) . . \ ',
80 (b) . ' --- . '' ~
. . \ ·. ------· (c) .. . ' e..... '• . ,, ------- (d) :I .
·.\'·, c 60 .. . Q } .. en '\ en .. . Ë
~ • en 40 ·~ c CU /; .. ~ ·~
20 / .. '\
I' • 0 650 652 654 656 658 660
wavelength (nm)
Figure 4.8 Transmission curves for a model interference filter with a FWHM of 5 nm: (a) the model curve for
normal incidence, the effective curves for (b) centraland (c) edge detector elements, and (d) the effective curve for
a detector with a circular viewing cone with a maximum semi-angle of 8°.
detector (b), an edge detector (c), and for an imaginary detector viewing with circular solid
angle (a cone with a semi-angle of 8°) (d). This circular cone results in a widening of the curve
comparable to that for the actual detectors. Compared to the case with circular viewing cone, the curves for both detectors have a decreased transmission for long wavelengths, and are wider at
short wavelengths. Full transmission of the same speetral line is possible for both detectors
(and all detectors in between), if the central wavelength Ào of the filter is chosen properly. For filters with a FWHM larger than 5 nm, the relative effects become smaller, and therefore
selection criteria for the filters are less stringent. This, of course, has to be weighted against the disadvantage of a larger contribution by background continuurn radiation and possibly by spectrallines in this range. Filters more narrow than 5 nm, however, seem impractical because the widening amounts to several nanometres on the short-wavelength side and tberefore no
impravement in wavelength selection is made. A filter with a FWHM of 5-10 nm seems to be an acceptable choice for the present optica! system. For comparison, the typical linewidth of the
Doppler-broadened Hcx-line emitted at the edge of the plasma is 0.1 nm, while lines emitted from the centre will be several times wider.
A high accuracy in the specifications of the filters is very important for the present system.
Unlike in most applications, a tilt of the filter to fine-tune the peak-transmission wavelength is
not possible because this wou1d always have an actverse effect for some of the detector elements. The Ha-filters with a FWHM of 10 nm that were purchased on basis of the above cal-
95
Chopter 4 The systemfor visible-light tomography
Ól (IJ
~
15
<D 10 0 c (IJ -o () .!;;
ö ~ 5 c
"'
635 640 645 650 655 660 665
wavelength (nm)
Figure 4.9 Contour plot of the transmission function -r(Ä, lJ) of an actual Ha interference filter for parallel
beams incident at angle lJ, obtained from linear interpolation from experimental transmission curves for four
angles 1J (0°, 5°, 10° and 15°). The labels of the contours give percentages of transmission.
culations4 unfortunately did not all meet these specifications, with the result that corrections
must be made in the calibration factors of each detector. The main probierus with some of the supplied filters is that Ào is not shifted to a Jonger wavelength than that of the Ha line and that the peak transmission decreases considerably for non-normal incidence.
4. 3. 4 Calculations of effective transmission curves for actual interf erenee filters
From the transmission curves supplied by the rnanufactueer for parallel beams incident at several angles, the transmission function 'l(.lt., 19) can be derived by interpolation (Fig. 4.9). The angle-dependent wavelength shift is evident from the figure, and also a decrease in peak
transmission values can be observed. The linear interpolation in the 19 direction has been done along parabolic paths determined by the wavelength shift from Eq. (4.1 ), where the neff was determined from a curve fit to the determined central wavelengths, as a function of the angle of
incidence. For the filter discussed here neff"' 1.8 ± 0.1 was found. In Fig. 4.10 the same type of curves as in Fig. 4.8 have been calculated5 for the actual filter. Similar effects as before
4 Corion Corp. , SI0-656-S
96
The NAG inlegration routines described in footnote 3 did not converge properly for the new linearly
interpolated transmission function, possibly because of the non-continuous derivatives in the function around
the measured points which are caused by the linear interpolation. Therefore, slower inlegration was done by
summation over intervals chosen adequately small for the required accuracy.
Optiealfliters 4.3
70
60 --(a)
~ ~
50 --- (b)
------- (c) c 40 .Q ------- (d) (/)
.!Q 30 E
(/) c co 20 .:=
10
0 630 635 640 645 650 655 660 665 670
wavelength (nm)
Figure 4.10 Transmission curves for an actual Ha interference filter with a FWHM of 10 nm: (a) the mea
sured curve for normal incidence, the effective curves for (b) centraland (c) edge detector elements, and (d) the
effective curve fora detector with a circular viewing cone with a maximum semi-angle of 8°. The vertical line
indicates the position of the Ha line.
can be observed. The filter would have been reasonable if Ào had been large enough and the
tolerances had been smal!, however, contrary to the model filter even in that case there is a significant difference in transmission between central and edge channels. A further effect not present in Fig. 4.8 is the decrease of the peak transmission factor for the edge channels, probably
caused by the decrease in peak value of r(À, 6) for large t?. This is the main cause for the deviations from the model calculations. The calculations for the actual filters, showing a maxi
mum difference in effective transmission values of 5% at the Ha wavelength between central
and edge detectors for the filter deviating most from the specifications, ind.icate that for correct interpretation of measurements these differences should be taken into account.
4.3.5 Considerations for other types of filters
In filters such as coloured glass filters and grey filters, the filtering properties are distributed homogeneously over the glass volume. In such a case two angle-of-incidence effects play a role: increased distance travelled through the filtering medium and different reflection coefficient
at the surface for oblique angles of incidence. The latter effect is negligible for the angles of incidence under consideration. The thickness effect, however, has an influence and is studied
here. The increased thickness because of an angle of incidence is diminished by the fact that refraction reduces the angle inside the medium. Therefore, inslead of the external angles given
97
Chapter 4 The system for visible-light tornog raphy
in subsectien 4.3.2, the intemal angles calculated from SneJ's6 Iaw are used: the approximate
poloidal angles for central detectors are ±7°, for the edge detectors+ 10.5° and -4° (and +4° and
-10.5° for the other edge), and ±2° in toroidal direction for all detectors.
In a grey-filter the relation of the transmission coefficient for thicknesses d1 and d2 is: dz
r(d2)=[r(d1)pï. (4.4)
The thickness of the filtering medium d2 for an internal angle of incidence t} is related to the
thickness of the filter (at normal incidence) d 1 by d2/d1 = 1/cost}. The effect of the integration
over the solid angle viewed by the detectors has been studied for grey-filters [with Eg. (4.3)].
In Fig. 4.11 the relative change of the effective transmission coefficient with respect to that at
normal incidence as a function of the transmission coefficient is depicted for central detectors,
edge detectors and, for comparison, a detector with a circular viewing cone. For grey-filters
applicable in the visible light tomography system ( r> I 0%) the difference between central and
edge detectors is less than 0.5%, which is negligible with respecttoother effects in the system.
Coloured-glass filters are filters with a wavelength-dependent Iight-absorbing medium. For a fixed wavelength the same effects as described for grey-filters occur. For the coloured-glass
filters used forsome measurements in this thesis the angle-of-incidence effects arealso negli
gible (see discussion in subsectien 5.6.4).
4.3.6 Implementation of filter effects in ray-tracing calculations
The above calculations give estimates for the variation in effective transmission coefficients for
the different detectors of an detector array. The assumed angles of incidence are, however, only
estimates, and therefore the exact angles of incidence of all detector elements should be known
befare Eg. (4.3) is used. Actually, also an inlegration over the detector surface is needed. These
exact angles of incidence can be found by ray tracing.
Insection 5.2 the way to characterize the imaging system by ray tracing is described. The range
of angles of incidence, and hence the effective transmission by the filter, depends on the posi
tien where the radiation is emitted. It is possible to take into account all effects of interference
filters, coloured glass filters and grey filters for each ray, for instanee by consictering the frac
tion that is transmitted of the light flux represented by the ray. At present this is not done, and
the effective transmission coefficient as given by Eq. (4.3) is used fora given wavelength.
6 In English literature the name of the Dutch physicist Willebrord Snel van Royen (Snellius in Latin) is often
misspelt with double I.
98
~ 0 c~
.Q Q)
~ g ·- Q) E-c (f) ·c (.) ~.f: ~"(ij ·- E <ll ~ e>o c c (1l 0 .c~
Ut) <ll Q) > a. :;::;cn ~~ ~.c
~
0
-1 1-
..... ..... !-'""
-2 I-
. . -3 .
0 .1
Optica[ filters 4.3
I I
- .· ..... .... ..... .... . .
-------(a)
-.- (b)
------- (c)
I I
10 100
transmission at normal incidence (%)
Figure 4.11 Relative change of the effective transmission coefficient with respect to that at normal incidence
as a function of the transmission coefficient for: (a) a circular viewing cone with maximum int~rnal semi-angle
of 5°, (b) centraland (c) edge detector elements.
4.3. 7 Conclusions
In this section the application of different types of filters in the visible light tomography system
was investigated. Emphasis was put on the difference in transmission between various detector
elements resulting from the effects of various angles of incidence on the transmission curve.
For properly chosen interference filters (i.e. not too narrow) no corrections are needed. Even
for small deviations from the specifications (such as in the Ha filters currently in use) a 5% dif
ference in transmission level for different detector elements occurs. In coloured-glass and grey
filters no significant effects occur.
The calculations show that it is useless to use interference filters with a FWHM smaller than
5 nm. Therefore the application of this diagnostic is limited to intense lines emitted by the
plasma, that have no or insignificant neighbouring lines within a few nm from their wave
length.
4.4 Summary
The main components of the visible light tomography system on RTP have been discussed.
Important aspects of the system are views from different directions around the plasma, imaging
by mirrors, the presence of viewing dumps and shields, fast electranies and the possibility to use optica! filters. The system operates satisfactorily, although electro-magnetic piek-up per-
99
Chapter 4 The systemfor visible-light tomography
turbs some !ow-signa! measurements. If optica! filters are used, various effects should be taken into account.
The main advantages and disadvantages of this system are summarized here. The imaging systems delermine in narrow viewing chords and collect more light than a pin-hole system with similar properties. This amount of light can be used to obtain a good signal-to-noise ratio at a large bandwidth. Some parts of the system, e.g. the windows and mirrors, are fragile and might need replacement in the future. The electronics and data-acquisition of the detector chan
nels are in parallel because of the required bandwidth and sampling frequency. Because of high costs of parallel processing and limited access to the tokamak the number of views of the plasma is five, with 16 detectors each. The coverage is such that in principle tomography is possible and that fluctuations at the edge of the plasma can be measured. Viewing dumps and shields have been installed to greatly reduce problems that would arise in the interpretation of measurements. Electronic piek-up by the electtonics is a problem that has been reduced by
shielding, but has notbeen eliminaled completely. The usage of optica! filters in the system has been thoroughly studied and criteria for filters have been set.
100
Technica[ details of the design Appendix 4.A
Appendix 4.A Technical details of the design
Details of the visible light tomography system that are not directly relevant for the results in this thesis are given in this appendix. Aspects are discussed of the windows, the blackening of the
viewing dumps and shields, the lay-out of the imaging system, the electro-magnetic shielding
of the cameras, and the positions of the cameras and ports.
4.A.l Details of the windows
Because there is very little space on the flanges, the windows are brazed onto tubes that are
welded to the flanges. The tubes consist of a part of titanium to which the sapphire is brazed,
and a part of stainless steel to which the titanium is brazed. The high-temperature brazing was
done in a vacuum furnace. Titanium is used because sapphire can be brazed to it, and to com
pensate for the different thermal expansion coefficients between sapphire and stainless steel.
The tubes and windows are circular, except at the flange where the tube is reetangolar to save space. Their size is such that most light imaged by the mirrors will reach the detectors.
The window construction is very fragile, and the windows cannot be repaired if a severe leak
appears because all structures that are welded to the flange (such as the tubes for the x-ray
tomography diagnostic) make it inaccessible for tools. Aftera leak occurred that could be fixed
by vacuum glue, a new design has been made for the windows, where the windows are
clamped between a gold and a he!icoflex ring. Because of the very limited space, only three
bolts are available for the clamping, which complicates the application of a uniform force. The
new windows work satisfactorily in tests and are now manufactured for back-up flanges in case
the present ones break. In the new design the optica] filters are mounted onto the damping
mechanism, whereas in the present design they are inserted into holders mounted between the
window and the camera.
4.A.2 The blackening of viewing dumps and shields
All of the shields and viewing dumps have been made black. The shields are made of anodized
aluminium, whereas the viewing dumps are stainless steel plate that have been chemically
blackened [HooK86]. These blackened metal plates have been extensively tested on vacuum compatibility at elevated temperatures that can be expected in RTP (200°C). The foils inside the
window tubes have been treated in the same way as the viewing dumps.
4.A.3 Details of the imaging systems
The optica! systems of cameras A and D !ie in the same vertical plane, while camera E is shifted 15 mm toroidally. The other viewing planes are slightly tilted. For camera B the planeis tilted go around the vertical axis, giving a maximum displacement of 35 mm with respect to the plane
101
Chapter 4 The system for visible-light tomography
of camera A and D. For camera C the mirrors are in a Z-configuration in the toraidal direction,
resulting in less narrow chords in the poloidal plane. The plane of camera C is tilted I o with
respect to the long ax.is of the mirrors resulting in virtually the same plane as cameras A and D.
Small deviations in imaging occur in camera B because with the current positioning mecha
nisms the mirrors cannot be aligned correctly, giving a smallloss of light.
An impravement to the system could be made by replacing the damaged quartz mirrors by alu
minium mirrors, which are less fragile. The quality of aluminium mirrors is sufficient for the
imaging requirements of this system. Aluminium mirrors also could be mounted more robustly
onto the holders.
The mirrors are mounted onto positioning mechanisms by clamps. The positioning mechanisms
are attached to holders that are mounted onto the flanges. The positioning mechanisms are
adjustable by screws araund two ax.es. Alignment is done by imaging an long light source,
which is parallel to the the poloidal plane and perpendicular to the viewing direction, onto all
detector elements of one camera. Proper alignment is found by iteratively putting the souree far
from and close to the mirrors, which in each step are adjusted to properly image the light souree
onto the detectors.
4.A.4 Details of the shielding of the cameras
A copper box surraunds and supports the electronics of a camera. It is electrically insulated
from the supporting holders that are attached to the flanges that are mounted on ports of RTP.
The box is attached to a translation stage so that it can be moved to change the focus slightly.
The material of the bolders and translation stages is non-magnetic, in order not to be affected by
the magnetic fields around the tokamak. The copper box is conneeled to the shielding of the
data-acquisition room via copper bellows and tubes. The grounding of the electranies and the
coaxial cables that transport the signals are also connected to the shielding of the data-acquisi
tion room, independently. This ensures proper graunding and electrical shielding.
The box es are made of 1 mm thick copper plate. The conducting Cu acts both as electronic and
magnetic shielding. With further layers of material, of which the thickness and permeability has
been optimized with respect to the Cu, the remaining electramagnetic interference can be
reduced. The shielding effectiveness for fluctuating magnetic fields depends on the product 6J.lf and the thickness [Whit80, Riki87], where <J and J1 are the conductivity and permeability of the
metal, respectively, andfthe frequency of the fluctuating magnetic field. Shielding is required
against fluctuations near 10kHz, which is an oscillation occurring after switching on and off
the chopper of the feedback systems described in subsection 4.2.4.3. Cu has a high <J, but a
relatively low J.l, and therefore has a shielding efficiency that is several orders of magnitude
smaller than that of iron, which has a lower <J but a much higher J1 [Whit80]. j.l-Metal, which
has a very high permeability, loses the high permeability at high magnetic fields (>0.5 T
102
Technica! details of the design Appendix 4.A
[Whit80]). Ordinary tin-plate has proved as effective as ,u-metal for the camera boxes, which
are positioned close to the toroidal magnetic coils of RTP that produce a magnetic field of the
order of 2 T inside the vacuum vessel. Experiments have been done with different layers of
shielding. Thicknesses of 1-2 mm yielded significant reductions of piek-up, to approximately
50% of the original value. Because of Jack of space in most places of the cameras only 1 mm
thick metal could be applied inside the copper box: in the cameras of the top flan ge ,u-metal is
used, in the camerasof the bottorn flan ge tin-plate.
4.A.S Details of the positioning
The positions of the detector arrays can be determined in a similar way as the mirrors, while the
position of the detector elements with respect to the ceramic cases on which they are mounted
can be measured under a microscope without having to remove the proteelive quartz glass.
The determination of the positions of the mirrors and detectors is only relative to reference
points on each flange. To compare measurements from both flanges and the side camera, their
relative position and orientation should be determined as well. For this, the position and orien
tation of the counter flanges of the tokamak vessel have been measured. The measurement,
which is complicated by the fact that the counter flanges are not completely flat, indicates that
the counter flanges are reasonably parallel (largest rotation 0.2 degrees). The upper and lower
counter flanges are shifted approximately 1 mm in radial direction, which is approximately
compensated for by the mounting of the flange, and 5 mm in toroidal direction (which is not
important for the interpretation).
When the flange is mounted on the counter flange, a gold wire (ring) is used as a vacuum seal.
The surface of both the flanges and counter flanges has been polisbed to achieve a good seal.
The bottorn flange has a shallow groove for the wire to facilitate the mounting. The position of
the flange with respect to the counter flange is ensured by two reference plates on each flange.
When mounted, the flanges are rotated approximately 2 degrees with respect to each other,
which might give a deviation of the l mm inthelines of sight if this rotation is nottaken into
account. The thickness of the wire after mounting has been measured by inserting thin metal
foils between the flanges until the gap was completely filled. The thickness of the gold wire,
which might vary because of a non-uniform application of force by the bolts, is constant within
0.2 mm, which is negligible. The thickness of the wires is approximately 0.85 mm for the top
flange and 0.70 mm for the bottorn flange, which increases the distance between the flanges,
when taking into account the distance between the counter flanges, 1.3 mm with respect to the
design di stance of 920 mm.
The position of the side port is not well aligned perpendicularly to the other flanges. This is not
important because only the vacuum window is mounted on it. Because the imaging properties
of the lens system of the side camera are not well known and it has a lower spatial resolution
103
Chapter 4 The system for visible-light tomography
than the other cameras, the orientation of the side camera has notbeen determined accurately . Besides, the side camera is not mounted directly onto the vacuum vessel but to the support structure, which means that if the vessel moves during a plasma discharge the measurements by the side camera are not well correlated to the other carneras, which are directly mounted on the vessel. These uncertainties about the si de camera could be a souree of errors.
Movements of the vacuum vessel when pumped vacuum and during operation have not been studied (all measurements on the positions were carried out without vacuum). Also the camera bolders could move slightly during a discharge, because there is not enough space to make them very stiff.
104
Characterization of the RTP visible-Iight ton1ography system 5 The properties of a system need to be well understood before it can be used for tomography. The characterization of some important features of the visible-light tomography system introduced in chapter 4 is described in this chapter. The detectors and imaging systems have to be calibrated, the relative orientation of the viewing chords has to be determined, and the effects
caused by the finite width of the detectors have to be studiêd. Same properties have been calculated by ray tracing, while other properties have been measured. In th.is chapter these results are
d.iscussed and compared.
More specificalJy, the coverage of projection space of the system is presented and its relation to the weight matrix is derived insection 5.1. Various aspectsof the weight matrix are discussed in section 5.2, results of a calculated weight matrix are presenled in section 5.3, and various
aspects of a measured weight matrix for the system are described in section 5.4. From the measured weight matrix, a sealing factor is determined insection 5.5. The calibration of different parts of the system, including an absolute calibration for Ha and continuurn filters, is discussed insection 5.6. In conclusion, section 5.7 summarizes the main results described in this chapter.
5.1 The coverage of projection space
An important characteristic of a tomography system is its coverage of projection space. This section discusses this on the basis of calculations by two-dimensional ray tracing and the considerations of subsectien 4.1.2. Three-dimensiona] ray tracing does not yield extra information
in this case, because projections of the rays on the poloidal plane have to be made anyway to give their (p,Ç) coordinates. Furthermore, a relationship between the weight matrix and the coverage in projection space is derived. The determination of the average viewing chord of a detector can better be done by a different method, which is discussed in subsection 5.4.3. However, the latter metbod to calculate the averages provides less insight than the calculated complete coverage.
Chapter 5 Characterization ofthe system
5 .1.1 The coverage of projection space for the visible-light tomography system
The calculation of the coverage of projection space is straightforward. From each detector element, a large number of rays is traced. The calculated path through the plasma after reflections
on the mirrors gives the (p,Ç) coordinates. For the cameras with mirror imaging systems (A
D), this results in the coverage of projection space that is depicted in Fig. 5.1. Each dot repre
sents one ray, each cl oud of points represents a detector element. The shapes of the clouds of points are due to the properties of the optica! imaging systems. For a pin-hole system, the clouds have the shape of a parallelogram, more or less like the shapes for camera C. Sufficient inforrnation about the lens system was not available to do a similar calculation for the side camera (E). For completeness, the averages of the viewing chords for the side camera are given in
Fig. 5.1.
Figure 5.1 shows that the projection space is nonuniformly sampled by the system. Further-
120- ~\ -J?
A ./.10 71I 60- .J13 ,.P64 74I f-'IÎ16 ~61
Ê o A5a ??I .s 0- A55 sof E f-
Q_ ~52
• ~49 -60-
,,48 ,.,.17
68 E , ,.20 c \\' 45
-120- I65 \ ' 42 ,.23 f-
\\'39 'f.26 '36 B f29
33 f32 -180 I I I I I
0 30 60 90 120 150 180 !; (0)
Figure 5.1 The coverage of projection space of the cameras with mirror systems (A-D) as determined by two
dimensional ray tracing. Twenty rays were traeed from five positions on each detector element considered. The
dots represent the (p,Ç) coordinates of each ray drawn in Fig. 4.1, i.e. for each third detector to clearly show
which region is covered by the detectors; the intermediale detectors would only fill the gaps between the clouds of
points belonging to the detectors considered. The points representing the average viewing chord of every third de
tector of the side camera E are given as well, with error bars indicating their approximate ex tent in pand Ç direc
tions. The numbers for each cl oud of points refer to the detector numbers introduced in Fig. 4.1.
106
The coverage of projection space 5.1
more, the two-fold coverage of the plasma results in measurements for lpl values at only three
(lp I< 40 mm) or four ( 40 mm < lp I< 140 mm) angles. As was pointed out in subsection
3.2.3 about the Cormack method, this means that only three or four angular harmonies can be
resolved due to the Nyquist limit, i.e. only the angular functions 1, cos Ç, sin Ç, and possibly either cos 2Ç or sin 2Ç. In subsection 3.2.6.2, it was indicated that other tomographic inversion
rnethods might resol ve structures of higher angular harmonies if the orientation of the structures
is favourable, i.e. the significant features of these structures appear at points in projection space
covered by the system.
5.1.2 The relationship between the weight matrix and projection space
The shape of the cl oud of points belonging to a detector is related to the weight matrix in a way
that can be derived as follows. The relationship between theemission function g(x,y) and the
line-integrated value f(p,Ç), i.e. a continuous form of Eq. (3.40), is
f(p,Ç)= IJ W(p,Ç,x,y)g(x,y)dxdy, (5.1)
where the weight function W(p,Ç,x,y) can be written as
W(p,Ç,x,y) = w(p,Ç) b(p+xsinÇ- ycosÇ). (5.2)
In Eq. (5.2) the factor w describes the attenuation in the imaging system by the (p,Ç) depen
dence, e.g. an angle dependent filter or non-uniform reflection on the mirrors (caused by darn
ages and contamination on the mirror surface). Withoutlossof much generality, these compli
cating effects can be neglected, assuming w(p,Ç) = I. The discrete weight function W;(x,y)
belonging to a detector i is determined by integrating over all rays (p,Ç) that reach that detector,
yielding the measured power/;
/; = IJ f(p,Ç)dpdÇ = IJ IJ g(x,y) b(p+xsinÇ- ycosÇ) dxdydpdÇ =
{p.Ç }; {p.Ç};
IJ g(x,y) IJ 8(p+xsinÇ- ycosÇ) dpdÇdxdy, (5.3)
{p.Ç};
where {p,Ç} i denotes the (p,Ç) values belonging to detector i. The change of integration order
in Eq. (5 .3) is possible because {p,Ç} i only depends on the detector, but not on x and y. Com
parison with Eq. (3.40), that is
/; =IJ W;(x,y) g(x,y) dxdy, (5.4)
gives as a definition for the weight function Wi(x,y) for detector i
107
Chopter 5 Choracterization ofthe system
W;(x,y)= JJ 8(p+xsinÇ-ycosÇ)dpdÇ=
{p.~};
I: dÇ JJ!;(p,Ç) 8(p+xsinÇ- ycosÇ) dp,
where Il;(p,Ç) denotes the window function
Il;(p,Ç) = {0, (p,Ç)"' {p,Ç};. I, (p,Ç) E {p,Ç};
Evaluation of Eq. (5.5) yields
W;(x,y) = D1;(-xsinÇ + ycosÇ,Ç) dÇ = dÇ,
(5 .5)
(5.6)
(5.7)
where dÇ is the angle between the points where the curve p =-x sin Ç + y cos Ç crosses the
boundary where Il;(p,Ç) = 1. It is interesting to nnte that the two-dimensional weight function W;(x,y) is fully determined by Il;(p,Ç), whereas the opposite is not the case. This is because
W;(x,y) is integrated over the detector; hence it only contains information to deterrnine the aver
age (p,Ç) coordinates for a detector.
Equation (5.7) indicates a way to calculate the weight matrix in the two-dimensional case. The
weight matrix element of the point (x,y) in the two-dimensional case corresponds to the angle
spanned by the detector seen from that point, which is equivalent to dÇ. This corresponds to
what was described in subsection 3.2.9 as the fraction of the locally emilted power that reaches
the detector. However, for a more exact calculation of the weight matrix, the three-dimensional
properties of the system should be taken into account. This can be done by determining the
solid angle that the detector element i spans seen from a point (x,y) through the imaging sys
tem, which means that Eq. (5.7) is not sufficient to determine the weight function W;(x,y). In practice it is simpler in the two-dimensional case and necessary in the three-dimensional case to
delermine dÇ directly in (x,y) space, insteadof in projection space, by calculating the angle or
solid angle spanned by the detector. This is described for the two-dimensional case in section
5.3.
From Eq. (5.7) and the statements in the previous paragraphit follows that the di mension of the two-dimensional weight function W; is that of angle, whereas in the three-dimensional case the
di mension is that of solid angle. The di mension of emissivity g is W m- 3 sr-1, while that of f is W. If the weight matrix is written as in Eq. (3.40), j; = LjWij g1, it contains the volume of
the cell over which W;(x,y) has been integrated for the discretization. Hence, the dimension
used is sr m3. Because the function g is taken only in the plane (x,y), all quantities are consid
ered after projection onto this plane: consequently, two-dimensional tomography can be used
and most formulae remain applicable.
108
The coverage of projection space 5.1
The average (p,Ç) coordinates for each detector can be defined as the centre-of-mass, calculated
from the first moments, of the function f(p,/;) llt(p,/;) . Because usually f(p,/;) is a smooth function and the area in projection space covered by one detector is relatively smal!, this average
will be approximately equal to the centre-of-mass of the function IT(p,Ç). However, it is not straightforward to calculate this centre-of-mass from the cloud of points for an imaging system
because in general the calculated points will not be distributed uniformly over the cloud. There
fore a different approach has been used where the (p,Ç) coordinates of the main axis of the
weight matrix elements belonging to a detector are calculated (see subsection 5.4.3).
5.2 Description of the system by the weight matrix and by an approximation by line integrals
The description of the measuring system by the weight matrix is a powerful method. The
meaning of the weight matrix has been discussed in the previous section and in subsection
3.2.9. In this section some aspects are considered about the advantages and the limitations of this description. The approximation of the measuring system by line integrals is also discussed.
There are several reasens to de termine the weight matrix. (1) The weight matrix contains all
information about the system. With the imaging system with mirrors in Z-configuration, com
plex imaging properties are expected, which can be studied by consictering the weight matrix.
(2) Accurate phantom simulations can be done with knowledge of the weight matrix because such a simulation takes into account chord-width effects. (3) Many tomographic inversion methods require the weight matrix, and it is better to use the actual one instead of the weight
matrix of line integrals, i.e. the lengthof the line through each cel!. (4) It is complicated to accurately measure the actual (p,Ç) coordinates of each detector, whereas they can easily be derived
from the weight matrix.
The weight matrix contains by definition all geometrie information about the measuring system,
and can be extended to contain calibration factors as well (see sectien 5.6). Insome cases not
all that information is necessary. In optical systems, in general, the étendue is conserved. The étendue of an emitting surface is defined as the solid angle that is imaged, integrated over the
area that is emitting [Long73]. If theemission extends to outside the viewing cone of the imag
ing system, the viewing cone delermines the area. This means that, in an ideal optica! system for a uniform spatial emission with dimensions larger than those of the viewing cone of the system, the same amount of light is received from all parallel planes inside the emitting region.
lt is convenient to consicter planes of uniform emission perpendicular to the central viewing line, where the central viewing line corresponds to the one with the average (p,Ç) values in the
previous section. Because the étendue is constant for all planes, only the line-integral of the spatial variation of emissivity along the central viewing line needs to be considered. In tomog
raphy systems it is often assumed that theemission is uniform in planes over the area bounded
109
Chapter 5 Characterization ofthe system
by the viewing cone, so that it is allowed to use line-integrals in the interpretation and tomographic imaging. The line-integral assumption is forther discussed insection 5.5 in connee
tion with the sealing factor between the actual measurements and line-integrals, where it is
shown that especially at the edge the assumption of line integrals is nat permitted and that the range of validity depends on the assumed emission profile. The étendue has the property of the inverse of the sealing factor [Eq. (3.42)].
The Jine-integral assumption also makes it possible to do a relative cabbration between detectors
by putting an extended uniformly emitting light source, i.e. larger than the viewing cone, perpendicularly to the central viewing line of each detector and campare the measured values. The distance to the light souree is unimportant. This approach is nat used in this thesis, because
sufficient in formation was obtained from the measurement of the weight matrix.
5.3 Calculation of the weight matrix
The weight matrix elements cao be calculated by ray tracing. From points in the plasma uniformly distributed rays are traeed in either two or three dimensions todetermine the angle or solid angle, respectively, spanned by the detectorseen through the imaging system. In the three
dimensional case, a scheme has been developed to trace only the rays on the contour of the
detector, from which the solid angle can be determined. For the calculations presented in this
sec ti on the pixel si ze is chosen such that the important features are visible. If the weight matrix is to be calculated for tomography applications, the pixel size is determined by the number of
pixels that is reasanabie in tomography. In genera!, the latter pixel size is large compared with
the imaging properties of the system: the chord-width only fits into two to four pixels. lt was found that more accurate results are obtained if the angle or solid angle is calculated for several
points in each pixel and then averaged. The three-dimensional calculations are rather time-consuming and have nat been used for the full determination of the weight matrix. The calculations might be speeded up by a Mante-Cario method, in which the integral over the three dimensions
110
of the cell and two angular dimensions is calculated.
In Fig. 5.2 a typical example is shown of
the weight-matrix elements descrihing one detector. Figure 5.3(a) shows the same for a larger region. The shape is deter-
Figure 5.2 Three dimensional graph of the
elements of the weight matrix describing the
properties of one detector (A8). The cells corre
spond to a grid in the poloidal plane. The height
gives the value of the weight-matrix element.
Calculations ofthe weight matrix 5.3
Figure 5.3 Contour plots of weight-matrix elements of a detec
tor. (a) Of the same detector as in Fig. 5.2, but fora larger part of
the plane. The part shown in Fig. 5.2 is indicated by the square. (b)
Of a detector in a pin-hole system. The contour levels in (a) and (b)
are not the same.
mined by the properties of the imaging system, i.e.
mirrors in Z-configuration. For this detector the main
part of radialion colleeled comes from the 10 mm wide
peaked region, whereas the radiation emitted in the shoulder contributes less. The peaked part betonging to
an adjacent beam wiJl overlap with the low region, so
that the peaked parts of the beams under consideration
are close, but separated. The width of the peaked part is
g N
somelimes referred to as re salution of the system in this thesis, or as optica! resolution to dis
tinguish it from the resolution obtainable by tomography. The height increases while the width decreases with disrance from the detector because the focus is on the far side of the plasma. The
étendue in this case is the value of the weight matrix integrated along lines perpendicular to the
main viewing chord, because the weight matrix represents the angle spanned by a detector. The
étendue is therefore independent of the di stance, as it should.
The weight matrix has also been calculated for a pinhole system with similar properties as the optica( imaging system and meeting the same constraints: detectors at the same disrance from
the plasmaand similar resolution. The result is shown in Fig. 5.3(b). The beam of the pinhole
system is more symmetrie and the height of the beam decreases with the distance by the same
amount as the width increases. A quantitative comparison shows that for similar resolution the
optical imaging system collects more than one order of magnitude more light than the pinhole
system.
An example of the solid angle spanned by detector elements found by three-dimensional ray
.. mirror
Figure 5.4 Typical example of the solid angle spanned by two detector elements (shaded areas) with respectto
the first mirror seen from the plasma (outer contour). The coordinates of this plot are: horizontally the angle in
the poloidal plane and vertically cos IJ, where IJ is the angle with the toroidal axis.
111
Chapter 5 Characterization ofthe system
E E
20
c 0 .... ()
Cll L.
'0 jij "0
10 0 L. 0 .....
Figure 5.5 Contour plot of the weight
function values of one detector (AS) as a
function of the tomidal coordinate and the
coordinate along the major axis of the
tokamak, obtained from three-dimensional
ray tracing. It can be considered as a cross
section through the middle of the square in
Fig. 5.3 along the horizontal axis, with
the addition of the toroidal coordinate.
tracing is shown in Fig. 5.4. Two
regions repcesenting the solid angles of two detector elements, with a
narrow band in between, are de
picted with respect to the first mirror seen from the plasma. For the central detectors the entire mirror sur
face is used for imaging. A typical 0'-----'--------'---_J_----'-----'
-12o -115 - 11 o -105 -1 oo -95 solid angle of a central detector is
R (mm) 2 x 1 Q-3 sr. The edge detectors
only use part of the mirror (because
of vignetting in the poloidal direction) and hence have smaller weight matrix values. This is considered in the following two sections. The weight function has been studied by three-di
mensional ray tracing. In Fig. 5.5 the weight function of one detector is calculated as a function of two coordinates (of which one is in the toroidal direction). This is an extension to Fig. 5.3 in
the toroidal direction. From the depicted weight function values, the étendue can be calculated in two dimensions. The étendue is typically 3 x I o-7 sr m2 for a central detector, while the value for a typical peak weight matrix element is 4 x 1 o-s sr m x M m 2, where M is the
area in the poloidal plane of a sufficiently smal! pixel.
5.4 Measurement of the weight matrix
Because of the restricted accessibility of the tokamak it is difficult to position an appropriate light souree inside the vacuum vessel in accurately known positions to measure the weight matrix. Therefore, the complete set-up of visible-light tomography was mounted on a table with
the poloidal plane parallel to the table. This was done after the alignment of the mirrors, befare the diagnostic was mounted onto the tokamak, so that the measurements reflected the actual situation. The measured weight matrix is compared with the calculated one, and the (p,Ç) coordi
nates of the detectors are calculated from it.
112
Measurements of the weight matrix 5.4
5.4.1 Description of the experimental set-up and the measurements
For the measurements the flanges of the system were mounted on a table in the same relative
positions as on the tokamak. The exact orientation and position of the port flanges, and the thickness of the vacuum seal (a gold wire, see Appendix 4.A.5), were nottaken into account
because the accuracy of the set-up was of the same order of magnitude as the corrections that
would have been needed (a millimetre). The position of the side camera is not known very
accurately. Furthermore, the vacuum window acts as an aperture for that camera, which was
taken into account by a reflecting me tal tube of the same size, but without the window. There
fore, the weight matrix elements of the side camera have a larger uncertainty than the ones of
the other cameras. Black cardboard was used as viewing dumps. This set-up was also used to
determine the influence of the reflections on the walls with and without viewing dumps, as was
described in subsection 4.2.3.
The cameras were connected to the main amplifiers like in the measuring system at RTP. The
same ADCs were not available, so that a different system of ADCs was used . This system transferred the data to a personal computer after which the data was stored on the central com
puter in the usual DOM4 format. Unfortunately the ADCs did notmeet up to the specifications:
relative calibration factors had a larger spread than specified. Due to later modifications the
original calibration has been lost, resulting in uncertainties in the measured values.
To measure the weight matrix on a grid in the poloirlal plane, a light souree is required of the size of the grid, that is long in the tomidal direction (i.e. Jonger than the toroirlal extent of the
viewing cones). A constantly emitting souree could not be used for this system because the
electronics and ADCs are ac-coupled. Tests have been made with a thin long glass beaker, filled
with milk diluted in water and above which a stroboscape flashes, as has been used fora simi
lar purpose in Ref. [Ho!N86]. It proved difficult to obtain sufficient scattered light from the beaker and to suppress reflected light from other surfaces. Therefore, a different approach was
chosen with a fluorescent light tube. The tube had a diameter of 18 mm, and theemission
oscillated at 100Hz. The souree was placed successively in all grid points inside the plasma
region. The size of the souree determined the grid, with pixel size 14 x 14 mm2 and 26 x 26
pixels for the total plasma region. The oscillation of the light intensity was measured, after
which the amplitude was measured by making a fit (the signals of all detectors had the same phase and shape). These amplitudes for all detectors measured at all grid points constitute a
relative weight matrix between the detectors. No absolute calibration was done in this set-up.
5.4.2 Validation of the measured weight matrix
In this subsection the relative values of the measured and calculated weight matrices are compared. Linking the values by absolute calibration is discussed in section 5.6. In Fig. 5.6 meas
ured weight-matrix elements of two detectors are compared with the ones calculated by two-
113
Chopter 5 Characterization ofthe system
(a} (b}
Figure 5.6 Contour plots of (a) the measured and (b) calculated weight-matrix elements for two detectors (D52
left and C40 right). The circle represents the plasma boundary (radius 170 mm). The '"wobbly'" shape and the
appearance of local maximaand minima in both (a) and (b) are digitizing effects of the grid [square cells of 14 x
14 mm2 for (a), and 7 x 7 mm2 for (b)] and have no physical significance. Noise effects of background light were
suppressed by filtering away all elements smaller than 2% of the maximum values.
dimensional ray tracing. The similarity is good. Because of the size of the test source, in the measured case a slightly smoothed matrix is obtained with wider viewing beams than in the cal
culated case. The grid size in the calculated case was chosen such as to clearly display the important features.
For some detectors the measured weight-matrix elements deviate from the ones expected, showing additional structures. Only one main viewing chord is expected (cf. Fig. 5.6, where the elements of two detectors are shown), whereas in some cases an additional structure, a Jess pronounced second viewing chord, is observed. This structure is attributed to two effects; for botheffects an example is given in Fig. 5.7. In both cases light reaches the detector also from a different direction than the main viewing direction. The detectors A14, AIS and A16 [Fig.
5.7(a)] see light that is reflected on one of the x-ray tomography cameras. The detectors B 17,
B18, B19 [Fig. 5.7(b)], on the other hand, collect light that reaches the detectors without imaging by the mirrors. The latter effect can occur because, as described in subsection 4 .2.3, the shields for this camera could not be made such as to shield all direct light. The contributions of botheffects are smaJJ, but nol negligible: the weight-matrix elements are of the order of 10% of the maximum weight-matrix elementsof the main viewing chord. In tomography methods
that make use of this measured weight matrix these effects of reflections and direct light are automatically taken into account.
114
Measurements of the weight matrix 5.4
( b)
Figure 5.7 Contour plots of the measured weight matrix for two detector elements: (a) Al6 and (b) 818. The same remarksas for Fig. 5.6(a) apply here.
The calibration factors of the ADCs used for the measurement of the weight matrix are nol
available, as was mentioned in the previous seclion. To obtain a reasanabie result when the
weight-matrix elements of the various detectors are compared, the deviations caused by the
ADCs have to be known. It is expected lhat for a given smooth ernission profile the measure
ments of adjacent detectors !ie on a smooth curve, and that for each detector the calibration fac
tor has a random spread around an average value. Therefore, the pseudo-measurements, calcu
lated for a phantom ernission profile and the measured weight matrix, should !ie on a smooth
curve. The deviations can be corrected for by deterrnining a corrective multiplication factor. The
smoolhness has been imposed by requiring thatthe pseudo-measurements !ie on a polynornial, which is fitled through lhe pseudo-measurements per camera. The correction factor found (up
to 7%) produced smooth pseudo-measurements for a variety of phantoms, and are used in all
further applications of the measured weight matrix.
Other uncertainties in the weight matrix measurements are caused by noise and systematic er
rors, such as errors in the positioning of the light souree and fluctuations in the intensity of the light souree for measurements at different positions. These errors cause random fluctuations on
the measured weight-matrix elements. Because the weight matrix is used either to calculate
pseudo-measurements, an inlegrating step that averages out random fluctuations, or for tomog
raphy, where regularisation is done by smoolhing, these errors wil! nol have a large influence.
However, noise in the measurement of the weight-matrix elements that should be zero caused
noisy deviations in calculated pseudo-measurements. Relatively many such elements had negalive values, probably indicating an error in lhe fitting procedure to the measured oscillation of
the light source. Therefore, all values smaller than 2% of a typical maximum value fora central
115
Chopter 5 Characterization ofthe system
--(a)
------- (b)
-; ----- (c)
~ (ij c
.2> Cf)
A B D
detectors of viewing direction E
Figure 5.8 (a) Measurements of all channels (indicated per camera) of a physical phantom: a light souree with
diameter 55 mm at the centre of the plasma region. As a numerical phantom a uniformly emitting disk with the
samediameter at the same position as the phantom was taken, from which the pseudo-measurement could be cal
culated with (b) the measured weight matrix and (c) line integrals. The height of the numerical phantom was
taken such as to result in pseudo-measurements of camparabie signa] level as the actual measurements. For the
comparison with line integrals to be possible the signals of measurements and pseudo-measurements for the
measured weight matrix were scaled as described insection 5.6.
detector have been set to zero, resulting in better calculated pseudo-measurements. The grid,
which is relatively large compared with the width of the viewing chord, also has an influence
on the quality of pseudo-measurements. It is difficult to estimate the influence of such effects,
but the two checks discussed below show that the weight matrix is reliable. The total error in
the pseudo-measurements calculated from the measured weight matrix are estimated to be approximately 3% (it is forsure that they are smaller than the correction factor):_ Th is es ti mate
does not include uncertainties due to differences of positions on RTP compared to on the
dummy set-up where the matrix was measured.
In the sameset-up as for measuring the weight matrix, measurements have been done with a
larger light source, a physical phantom. The light souree was the same tube light with a circular diffusively transmitting paper around it. This light souree is mainly a surface emitter, and nota
volume emitter as is required for a proper simulation. lt was expected that for the relatively
smal! size of the souree (diameter 55 mm) tbis distinction would not be important. A measure
ment for the physical phantom is shown in Fig. 5.8 (a). A simulation has been done with a numerical phantom with the same properties (but being a volume emiuer), giving very similar
results [Fig. 5.8 (b)-(c)]. For channels with high numbers in camera A and low numbers in camera B there is a clear distinction between the pseudo-measurements calculated from the
weight matrix and from line integration. These channels were not expected to measure anything
116
0
Q) 1.5r-l'\ .... 00 0~ § 00 ..... ooo .... o 00 .~ al 1.o.o ••• ·~. ·., oo ••• ····~ .~ ..... : •.. ·o ?".. .r .,. '\, m 0.5 :.._ •• • •• ~. 0 •• •
- •• • •• 0. ••
g I ··.,l<t'• 0.0 I I
A 8 c D detectors of viewing direction
-
-
Measurements ofthe weight matrix 5.4
<9" 0 .... ~ o,,•' ~oo ,~o o.. 0 +
V •"' .P./ .• 0 • .. • • • + • ~ .. . . . .., . .
<> ~ .
detectors of viewing direction E
Figure 5.9 Pseudo-measurements calculated with (a) the two-dimensionally calculated weight matrix and (b)
the measured weight matrix, for three different cylindrically symmetrie phantom profiles: +=constant, • =
parabol ie and o = hollow. The relative values of the phantoms are such that the line inlegral for p = 0 gives the
same va lues. The va lues of both graphs have been normalized per camera to the average signa! values for the con
stant profile. Viewing direction E is missing in (a) because the current ray-tracing code can only calculate the
weight matrix for the mirror systems, not for the lens system.
for the phantom in this position. This effect can be attributed to the reflections and direct radia
lion that occur for these detectors, as described earlier in this section (recall Fig. 5.7). The
effect is not negligible and is contained properly in the measured weight matrix. Figure 5.8 also
indicates that the physical phantom simuiales a volume emitter well, that the measured weight
matrix gives good results calculating pseudo-measurements, and that the sealing factors which are determined insection 5.5 work well to obtain measurements similar to line-integrals.
The measured and two-dimensionally calculated weight matrices have been used to calculate
pseudo-measurements for three ctifferent emission profiles (Fig. 5.9). The overall shapes of the
curves are similar and show the same behaviour for both cases. However, some effects have to
be explained. Most of the unsmoothness in the case of the measured weight matrix has been removed by applying the correction factor, butsome small deviations remain. For the measured
weight matrix, averaging is done over larger pixels, so that slightly wider and more flat projeetions might result. Three other effects might play a role. (1) For the calculated case the design
positions of the mirrors were taken, which differ from the actual positions. (2) The ray tracing
was done for two dimensions whereas, especially for the viewing directions that are tilted out
of the poloidal plane (B and C), a three-dimensional calculation is necessary. (3) Reflections
from pi aces where no viewing dump was possible (the channels on the right si de of the part of
the detector axis labelled by "A"), and light not imaged by the mirrors reaching the detectors
(the channels on the left side of "B"), arenottaken into account in the calculation of the weight
matrix and give a significant increase in the measured case. lt can be concluded that there is
good agreement between the calculated and measured weight matrix. Because the measured weight matrix contains all information for the actual system, its u se is preferabie over that of the
two-dimensionally calculated one for the design positions.
117
Chapter 5 Characterization ofthe system
A further remark that is needed with respect to the measured weight matrix is that only the ele
ments inside a radius of 170 mm were measured, which is only slightly larger than the plasma region (a= 164 mm). The edge channels of camera B, however, look outside this region and their weight matrix was therefore not determined properly. In future measurements of the weight matrix a larger area should be covered.
5. 4. 3 Calculation of the coverage of projection space from the measured weight matrix
As was indicated in subsectien 5.1.2, it is nat straightforward to calculate the coverage of projection space from ray tracing. lt is also difficult to determine it accurately by a direct measurement, for example by accurately determining the path of a laser beam which is shone in from
the detector position. Fortunately, it is relatively easy to derive the coverage of projection space
from the weight matrix.
The average main direction of the viewing beam of a detector can be determined from the first and secend moments of its weight-matrix elements, which determine an equivalent ellipse [Teag80]. The main axis of the equivalent ellipse can be considered to be the main viewing
chord, which determines the coordinates of the detector in projection space. In principle it
should be weighted by the emission profile, but for the same reasans as mentioned at the end of subsectien 5.1.2 this effect is not important. For those detectors that see reflections or direct
light {p,Ç} i consists of two, or more, disconnected parts. The calculation of the equivalent ellipse in this case would result in a weighted average between the {p,Ç} i regions, which has
no meaning. Therefore, themaindirection of the weight-matrix elements has been selected by
0.20 I I I
0.10 - ~\, of" E 0.00-
a. 49
-. ~/
80•" -
-0.10 ~E cj" 17
~\, • 65
33 -0.20 I I
-
0 45 90 135 180
s (0)
Figure 5.10 The coverage of projection space for the visible-light tomography system calculated from the
measured weight matrix.
118
Measurements of the weight matrix 5.4
taking into account only values larger than a threshold of 11% of the maximum value of the elements.
This way of determining the coverage of projection space was tested with the calculated weight matrix, which resulted in points lying in the clouds in Fig. 5.1. For detectors viewing the edge
of the plasma the metbod did not work properly, because only a few matrix elements that are
inside the plasma region are not enough to determine accurately the main direction of the
equivalent ellipse. The coordinates in projection space of these detectors were found by
extrapolation, which seems reasonable because all other points of a camera lie approximately equidistantly on a straight line. Figure 5.10 shows the coverage of projection space that was determined in this way. These are the coordinates used for the tomographic inversions of measurements on RTP.
5.5 Sealing factor
The sealing factor Si for detector i was derived in subsectien 3.3.1, and it is given by Eq. (3.42):
S · = Ji I !i • (5.8)
lts purpose is to scale the measurement.fi toa Jine inlegral measurement/i. so that measurements of different detectors can be compared. In general it depends on the emission profile. If
the étendue is constant along the viewing chord and if theemission profile is approximately flat in the perpendicular direction over the width of the non-zero weight-matrix elements, the seal
ing factor is independent of the emission profile. Then it only corrects for effects such as
vignetting that occurs for the edge detectors, and is given by Eq. (3.43), i.e. fora flat profile
with unit emission.
The sealing factor has been determined for several emission profiles, by calculating.fi from the measured weight matrix and /i from the line integrals. This is shown in Fig. 5 .11. The values
for the differentemission profiles correspond well, except for channels close to the edge (low channels of camera A, high channels of B, and low channels of C), justifying the assumption
that the central detectors can be approximated by line integrals with a fixed sealing factor independent of the emission profile. An example of the sealing process with this assumption is given in Fig. 5.12. Figure 5.12(a) shows unscaled pseudo-measurements for three different cylindrically symmetrie profiles. The points of different cameras are scattered. The high values correspond to detectors of the si de camera (low 5caling factors in Fig. 5.11 ), which receive
more light than detectors of the other cameras. This is because the side camera views the plasma through a circular lens system while the other cameras use rectangular mirrors that limit the aperture in the toroidal direction . In Fig. 5.12(b) the values have been scaled by the sealing
119
Chapter 5 Characterization ofthe system
..... 0 ü $ 0> c ca (.) Cl)
x
A
x
0
x • +
x
detectors tor viewing direction E
Figure 5.11 The sealing factor calculated for all detectors for various emission profiles: +=constant, • =
parabolic, o = hollow, and x= reconstructed glow discharge profile discussed in subsection 6.2. I. The units of
the sealing factor are arbitrary because they depend on the measured weight matrix.
factor detennined for a flat profile, and they are camparabie to the line-integrated values shown
in the same figure. The points of different cameras now !ie on curves, as they should for cylin
drically symmetrie profiles. There are however deviations from the line-integrated values, which show that the applicability of the measured weight matrix and sealing factor is limited.
Several reasans for the deviations of the sealing factor at the edge can be given. The fini te width
of the viewing chord probably is the most important one, causing the measurements to be incompatible with line integrals. However, assuming strip integrals instead of line integrals did not result in improved sealing factors. The effect cannot be ascribed to the poor detennination of the measured weight matrix for edge channels, because simulations with the calculated weight matrix show a similar behaviour. To compare signals from different detectors without
using the weight matrix directly, the sealing factor that has been obtained is the most reliable. Channels for which the sealing factor cannot be determined properly, unfortunately cannot be
interpreted, and cannot be taken into account in tomographic inversion methods that require the measurements to be scaled.
5.6 Calibration
The weight matrix contains by definition [Eq. (5.7)] the geometrie properties of the imaging
system. Calibration factors are needed to describe the response of the detectors. The definition of the weight matrix can be extended to include the cal i bration factors of the detectors and electronics. The measured weight matrix has not been measured absolutely to contain the geometrie .
120
::lrT-r,-ro-ro-TörT-r~rT-r,-ro-r,
!i ..
50 100 p(mm)
150
(a)
50 100 p(mm)
Calibration 5.6
( b)
150 200
Figure 5.12 Pseudo-measurement calculation of various cylindrical symmetrie phantom emission profiles
with the detectors ordered aecording totheir p value (a) befare and (b) after sealing. The profiles, which are nor
malized such that the total emissivity is the same for all, are: += constant, • = parabolic, and o = hollow. In
(b) also the line-integrated pseudo-measurements for the sameprofiles are plotled as lines. The sealing for all
profiles is by the sealing factorfora flat profile.
information, it only contains the relative values between detectors. In this section the absolute
values of the measured weight matrix and the calibration of the detectors are discussed. Fur
thermore, the system is absolutely calibrated for Ha filters and continuurn filters.
5.6.1 Geometrie part of weight matrix
The measured weight matrix can be scaled to the weight matrix which contains the absolute
geometrie information about the system by comparing its values to the ones obtained by three
dimensional ray tracing. The sealing should be the same for all detectors. In the ray tracing the
actual positions of the minors and detectors should be known. The comparison was made for a
central detector which has approximately the same viewing chord as the chord resulting from the calculations. Because of noise in the measurement of weight matrix elements, it is advanta
geous to compare integrated quantities such as the étendue. For this purpose the expected signal
in one detector for a unity emission in the entire emission region calculated from the measured
weight matrix is scaled to the calculated étendue (section 5. 3) multiplied by the chord length.
Due to the effects mentioned, the uncertainty in this calibration procedure is estimated to be
lû% in the absolute value; the relative uncertainty between channels in smaller (see subsection 5 .4.2).
5.6.2 Aspects of calibration
The output of the measuring system is a voltage (i.e. dimension offi is V). Hence the dimen
sion of the calibration factors is V /W, and the di mension of the weight matrix that contains both
the geometrie in formation and the cal i bration factors is V sr m3 W-1. The measured weight matrix contains a relative calibration between channels, and thus also the relative sensitivity of
the detectors and electranies (for the wavelengtbs of the light source), and the differences in
121
Chapter 5 Characterization of the system
reflection coefficient of the mirrors for different detectors. To properly compare the measured
weight matrix with the calculated one, the calibration factors are needed. These effects also
explain smal! differences in the sealing from the measured weight matrix to the absolute geo
metrie weight matrix and contribute to the uncertainty given above (this was taken into account
in the estimation of the uncertainty). For an absolute calibration the absolute sensitivity of the
detectors has to be taken into account. The speetral sensitivity of all detectors is assumed to be
equal to the curve supplied by the rnanufactueer (Fig. 4.7) within the uncertainty of the geomet
rie weight matrix. The calibration factor depends on the wavelength region studied and the fil
ters that are used. This is discussed for Ha filters and continuurn filters in the following two
subsections. lt is also necessary to take into account the transmission values for the optica! fil
ters and the average reflection coefficient of the mirrors.
In cases where the measured weight matrix is not used (nor sealing factors that are derived from
it), a relative calibration between channels might be needed. In principle the étendue of each
channel can be measured with a uniform light source, and also the relative calibration of detectors and electronics can be done by putting them ju st in front of a uniform light source. Such a
light souree is, for example, an inlegrating sphere, a sphere with a diffusely reflecting inside
wal! into which light is shone. The light exits the sphere, after multiple reflections, through an
opening as an approximately uniform emitter. Attempts have been made to do such measure
ments, but they were notaccurate enough. A light emitting diode (LED) has been used instead
to verify that the relative sensitivity of neighbouring detectors in combination with their ampli
fiers varies less than 1%. Because the di stance between the LED and detector array could not be
kept absolutely constant, these measurements were inadequate to delermine whether there is a
gradual change in sensitivity over the lengtb of the array. A more accurate calibration is needed
if the measured weigbt matrix is not used.
The frequency dependent transfer function of the electronics bas been determined individually for several detectors. The bandwidth (i.e. -3 dB point) was found to vary between 150kHz
and 200kHz. Because the bandwidth is not the same for all detectors, the exact transfer func
tion should be known if signals with frequencies higher that 100kHz are to be compared, both
in amplitude and phase.
5.6.3 Absolute calibration for Ha filters
In the case of Ha filters, and other interference filters, the transmission coefficient is an impor
tant factor that bas to be taken into account because the maximum value is usually less than 70%. The transmission coefficient for each detector is considered first; thereafter the sensitivity
of the detector at the Ha wavelength is discussed.
122
Calibration 5.6
5.6.3.1 Correctionsjor Hafilters
In subsection 4.3.4 it was shown that the angle-of-incidence effects in the visible-light tomog
raphy system are nat negligible for the Ha filters used. The significanee of angle-of-incidence effects varies for different detectors because of their different angles of incidence. The fact that
some of the filters do nat have the required shift of central wavelength (to a Jonger wavelength)
and that the maximum transmission for parallel beams under non-normal incidence decreases
significantly make corrections necessary. Furthermore, corrections are needed because the
transmission properties of the filters used for the different cameras vary between filters, resulting in different corrections for each of the cameras, bath in absolute value for the entire camera and in relative value between the detectors of the camera.
The corrections are based on calculations similar to the ones in subsection 4.3.4. The average
transmission factor for the angles of incidence on a number of detector elements of each camera
are calculated by means of the transmission curves for the actual filters (such as Fig. 4.9) . Interpolating between the channels of a camera gives an estimate of the proper transmission values (Fig. 5.13). For each camera for which ray-tracing calculations are available, i.e. cam
eras A-D, the error in relative transmission values for the detectors of one camera are estimated to be 1% to 2%. Due to uncertainties in the absolute values of the transmission curves supplied
by the manufacturer of the filters, the relative error between cameras A-D is estimated to be 2%
to 3%. For camera E the range of angles of incidence is not known as accurately as for the other cameras, but it is considerably larger due to the relatively large exit pupil of the camera objective. Tomographic reconstructions that take into account the average transmission values at the
70 I I I I
~ 60 I- ~H:nniHinl!!!fulfiiHlJI.tin -~
c
!flll~lllj 0 Cl)
50 I-(/) .E (/)
c <13 ~ 40-
30 _l_ I I
A B c 0 E detectors of viewing direction
Figure 5.13 Estimated detector averaged transmission values of the Ha filters for all channels of the visible
light tomography system. Error bars give the relative errors between channels of one camera.
123
Chnpter 5 Chnracterization ofthe system
Ha wavelength for the angles of incidence found for the detectors of camera E have been checked for consistency. This was done for several differentemission profiles and the reconstructions were compared with reconstructions made without tak.ing into account the measure
ments of camera E. However, due to the uncertainties, the error bars on the estimated transmission values of camera E are large.
5.6.3.2 Absolute eaUbration
Absolute calibration factors are needed todetermine the absolute emission profile. The sensitivity of the detectors and amplification factors of the electronics have been discussed in subsections 4.2.4.1 and 4.2.4.2. The output signa! is the voltage over the transimpedance, which is
produced by the photodiode current. Tak.ing Eq. (4.1) and the quanturn efficiency for the Ha line from Fig. 4.7, the output voltage U (in volt) for an incident light power P (in watt) is given
by
U= (3.4 ±0.1) x 105 P. (5.9)
The transmission factors of the interterenee filters and reflection coefficients of the mirrors are taken into account in the tomographic reconstructions, as well as the es ti mate of the absolute value of the geometrical part of the measured weight matrix as described in subsection 5.6.1.
3
~ > 2
li)
0
~ ·:; +=' ëi5 c a> (/)
0 2.5 3 3.5 4 4.5 5
frequency (1014 Hz)
1200 1000 900 800 700 600 wavelength (nm)
Figure 5.14 Speetral sensitivity of the detectors of the visible-light tomography system corrected for the spec
tral transmission coefficient of the continuurn filters used and the reflection coeffic ient of the mirrors.
124
Calibration 5.6
5.6.4 Absolute calibration for continuurn filters
For the continuurn measurements, coloured-glass filters for a wide speetral range are used.
Because the measurement is over a wide speetral range, the speetral dependenee of the transmission of the filters and the detector sensitivity have to be taken into account.
Table 5.1 Transrnittance of the coloured glass filters used. The coloured glass filters usedt
Wavelength (run)
695
> 740
656 (Ha)
Transmittance (%)
46
91
< w-2
transruit light which has a wave
length À> 695 nm (see table 5.1 for the characteristics). The an
gle-of-incidence effects discussed
in subsection 4.3.5 can be neglected for these filters. This is
possible because the internal transrnittance, i.e. the transrnittance without surface effects such as reflection, is larger than
99.9% for long wavelengths, for which the effects according to Fig. 4.11 are very smal!. The angle-of-incidence effects are large at the wavelengtbs where transmission goes from a very low to a high value, but for practical purposes these effects will only change the shape of the
transmission curve in a negligible way.
The filter defines the lower wavelength of the speetral range used, while the upper wavelength
is determined by the sensitivity of the detector, which was described in subsection 4.2.4.1
(Fig. 4.7). Figure 5.14 shows the speetral sensitivity of the detectors, which has been corrected for the transmission factor of the filters and the reflection coefficient of the rnirrors. Note that
these corrections were not included in Eq. (5.9) (because the correction for the filter m the Ho: case is different for each detector). For electron temperatures Te>> h v, which is the case in a
large part of the plasma for frequencies in the visible range, the bremsstrahlung emissivity is
approximately independent of frequency [see Eq. (2.12)]. Therefore, the integral over frequency range, which is measured by the detector, gives for the sensitivity
..!!..._ = (2.8 ± 0.3) x 1019 V Hz, Pv W
(5.10)
where Pv is the power received by the detector per unit frequency. If g is the tomographically
inverted local value for the Iine-integrated measurements U, the ratio g!Ev. where Ev is the frequency dependent local ernissivity, is equal to U!Pv because the order of integration over frequency and along the chord can be changed. The consequence of this observation is that Eq. (5.1 0) can be used to de termine the absolute emissivity of any part of the plasma where the
emissivity is independent of the frequency, and that this determination is not perturbed by the
t RG695 filter, rnanufactured by: Schot! Glaswerke, Mainz, Gerrnany
125
Chapter 5 Characterization of the system
line integral through a part of the plasma where this assumption is not valid. Because in the
edge the assumption of independenee from the frequency is not valid, the absolute emissivity in the edge cannot be determined other than by a deconvolution over the frequency. Such a decon
volution is only possible if both the speetral sensitivity (Fig. 5.14) and the frequency dependenee of the emissivity are known. If the radialion is known to be purely bremsstrahlung, Eq.
(2.8) and knowledge of the electron temperature have to be used for the Jatter. Unfortunately,
however, additional radialion in the selected speetral range from line-radiation of impurities and
molecular radialion can be expected at the edge, complicating the absolute determination of the
emissivity.
The contri bution of frequency independent radialion to the total range to which the detector is
sensitive, i.e. without the continuurn filter, can be ca1cu1ated in the same way. This calculation
yields
(5 .11)
5.7 Summary
Ray-tracing calculations have given a good understanding of the imaging properties of the sys
tem. The system has been characterized satisfactorily by determining the coverage of projection
space and by measuring the weight matrix. Si x of the 80 channels suffer from either direct light or reflections. These effects are properly taken into account in the measured weight matrix. The
measured weight matrix has not been determined properly for some channels looking at the
edge of the plasma, limiting the number of channels that can be used in tomographic inversion
methods that make use of the weight matrix. The sealing factor, which is needed to campare the
signals of the various channels , has been determined adequately for channels with
p < 140 mm. For channels with p < 120 mm it is virtually independent of the shape of the emission profile, as long as it is smooth. For the channels look.ing at the edge of the plasma no
satisfactory sealing factor could be found, meaning that these measurements cannot be consid
ered in analysis methods which make use of the sealing factor. Concluding, some edge channels can neither be used in analysis methods employing the weight matrix (two channels, and
three additional channels that were not functioning at the time of the measurements of the weight matrix), nor in methods that require the measurements to be scaled (four channels).
Absolute cal i bration factors have been determined for the usage of Ha and continuurn filters.
126
Simulations and nteasurements of simpte emission profiles 6 The reliability of tomographic inversion methods needs to be studied befare they are used for the interpretation of measurements. In subsection 3.3.3 phantom calculations were presented for the inversion methad developed by Pickalov and the author. In this chapter more phantom calculations that are relevant for plasmas are presented and compared with similar calculations
for the inversion methad by Fehmers (see subsection 3.2.5.3). This is done in section 6.1. After the phantom calculations, in section 6.2 attention is directed towards emission profiles
actually measured in RTP. The reliability of reconstructions of symmetrie and simple emission profiles is studied. "Simple" refers to the shape and symmetry of the plasma emission.
6.1 Phantom calculations for tomography
Jn this section tomographic invers i ons are made of phantoms to de termine the reliability of the the inversion methods when applied to the visible-light tomography system. Firstly, an estimate is made of a realistic level of noise that should be taken into account in the phantom calcula
tions. Subsequently, practical aspects of the phantom calculations are discussed. Then, two
methods for tomographic inversion are compared by means of phantom calculations. The methods that are compared are: the reconstruction in projection space developed by Pickalov and the
author (section 3.3), which is referred to as the Iterative Projection-space Reconstruction (IPR) method; and the constrained optimization metbod developed by Fehmers (subsection 3.2.5.3,
[Fehm95]), which is referred to as the Constrained Optimization (CO) method. After the comparison, effects caused by the non-uniform coverage of the system are discussed on basis of
phantom calculations with localized phantoms at several positions in the reconstruction region. Finally, a summary is given of the importantaspectsof the tomographic inversion methods.
6.1.1 Noise estimate
As discussed befare (subsection 3.2.7), a realistic level of noise should betaken into account in
phantom calculations. In the CO methad an estimate of the noise level should be given to the algorithm tagether with the (pseudo-)measurements.
6 Simple emission profiles
> 2.0 (ij c 0> ïii :E g 1.0 Q)
15 ïii ">
0.0 0 50 100 150 200 250
time (ms) 1.25
~ 1.20
(ij c 1.15 0> ëii :E 0> 1.10 Q)
:0 ïii 1.05 ·:;
1.00 150.5 150.7 150.9 151.1 151.3 151.5
time (ms)
Figure 6.1 (a) Typical timetrace of a signa! measured by one detector of the visible light tomography system.
(b) Blow-up of the signals of three adjacent detectors.
Typical time traces of measurements by the visible light tomography system are shown in Fig. 6.1. The temporat fluctuations in the signals have, on average, a relative standard deviation of typicatty 4%. These fluctuations are mainly proportional to the signa! level, and are much larger
than the noise of the electronics, which in its turn is much larger than the photon noise. The electronic noise is usuatty smaller than the bit level of the ADCs, which is apparent in the sig
nals before the plasma discharge. Therefore, only relative noise is considered in the phantom
calculations of this chapter. However, if the fluctuation level is estimated from projections, i.e. camparing the signa! levels of detectors at one point in time, a smalter value is obtained because the projections are smoother. These observations indicate that the fluctuations in time and space
128
Phantom calculations 6.1
are caused by plasma behaviour on a scale of several centimetres, such that adjacent detectors see the same fluctuating structure. Statistica] analysis has shown that the measured projections
are smoother than projections obtained from phantoms with 3% relative noise. Because fluctua
tions between adjacent channels are a souree of noise in tomographic inversions, 3% relative noise is assumed in the phantom calculations. The differences between temporal fluctuations
and spatial fluctuations between detectors under various plasma conditions are discussed more
thoroughly in chapter 9. Uncertainties in the measured weight matrix arenottaken into account explicitly because comparison of the projee ti ons obtained from phantoms are smoother than the 3% relative noise.
The result of the tomographic inversions seems not to be very dependent on the amount of
noise that is used in the calculation of the pseudo-measurements, if it is below 5%. In the case
of the IPR method differences in reconstruction errors of the order of 1% (absolute value) are observed for different levels of noise on the pseudo-measurements. A possible explanation for
this insensitivity to noise is the large amount of smoothing that is applied to interpolale in projection space. Furthermore, the reconstruction errors for many phantoms are so large that this difference due to noise is not significant.
6.1.2 Some aspects of phantom calculations
In the phantom calculations the values of an analytica] phantom emission profile are multiplied
by the weight matrix to obtain the pseudo-measurements for all detectors [Eq. (3.7)]. Noise is added to these pseudo-measurements. The pseudo-measurements are then tomographically
inverted, yielding a tornogram and a sinogram. The tornogram is compared with the phantom, giving the tornogram error ag according to Eq. 3.38, whereas the sinogram is compared with the pseudo-measurements, giving the residual norm at according to Eq. 3.37. The tornogram
can be used as a phantom to obtain the backcalculated measurements, i.e. the values that would
be obtained if the reconstructed tornogram were the true emission profile. These backcalculated values should in principle be equal to the values of the reconstructed sinogram at the coordi
nates corresponding to the detectors, with the difference that the sinogram represents measurements for line-integrals, whereas the backcalculation takes into account the weight matrix and sealing. The backcalculated measurements are helpful in determining the quality of the tomog
raphic inversion and can give information on which parts of the tornogram are reliable and which parts are nol. In the CO method the sinogram does not play a role in the inversion process, and it is therefore calculated from the tomogram. In this case the misfit between the meas
urements and unscaled pseudo-measurements is relevant; the misfit can be different from af because of discretization errors and errors induced by the sealing. In the CO method the misfit is an input-parameter of the inversion, and hence not a good qualifier for the quality of the
reconstruction.
129
6 Simple emission profiles
All phantom calculations in this chapter have been carried out with the measured weight matrix.
Calculations with line integrals and various grid sizes to study grid effects gave essentially the
same results with deviations in the reconstruction errors smaller than 1%. However, if the
measured weight matrix is used, the sealing process of the pseudo-measurements introduces an error afbetween the pseudo-measurements and the exact sinogram of the phantom of 3% to 5%
(which is also the difference between the misfit and O"J). This can be regarded as a souree of
errors additional to the added noise, which, as stated at the end of the previous subsection, has
on1y a minor influence on the ag and O"J of the tomographic inversion. In the calculation of ag and O"J minor deviations (smaller than 1 %) could resu1t from grid effects and interpolations
between different grids. To conclude, ag and O"Jof the tomographic inversions are estimated to have an absolute accuracy of 2 to 3%.
It should be noted that phantom calculations that make use of the weight matrix do not give any
in formation on the influence of systematic deviations of the measured weight matrix from the
actual system. On1y variations between the channels are known to besmaller than 3%. There
fore phantom calculations (this section) are not sufficient and validatien by means of actual measurements is neededas well (section 6.2).
All phantoms represent the emissivity of light in one poloidal cross-sec ti on of the tokamak. The
orientation of phantoms and tomograms is taken according to the usu al depietion of the system
as in Fig. 4.1: the axis of major radius R pointing to the right, and the axis of the vertica1 coor
dinate Z pointing upwards. For the sinograms the (p,Ç) coordinates as defined in subsectien 3.1.1 are used. For all phantoms the plasma (minor) radius, i.e. the radius outside which no
radiation is emitted, was taken as 0.17 m. The tomographic inversion by the IPR method is
done in a reconstruction area with radius 0.19 m. This radius was taken because the tornogram
showed many artefacts for phantoms with a steep gradient in emission at the edge if 0.17 m
was taken as the radius of the reconstruction area. These artefacts are caused by the steep gradients or, in the case of measurements, a non-zero emission just outs ide the plasma. In the CO
method the tomographic inversion is done with the weight-matrix elements that are known; the
corresponding grid is extended by one cel! in all directions, yielding a region of camparabie size
to the reconstruction area mentioned above. For comparison with the IPR method, the same
area with radius 0.19 m is shown in all figures of phantoms and tomograms, assuming zero
emission outside the area where the weight matrix is known. Usually a contour plot is sufficient
to understand the geometry of the phantom or tomogram, but in cases where Iocal minima and
maxima exist it is preferabie to also show a three-dimensional graph. In the contour plots used,
dashed contours indicate negative values. Furthermore, in each example only the relevant or
significant reconstruction information is given, i.e. a selection is made from the phantom, the
exact sinogram, the reconstructed tomogram, the reconstructed sinogram, and the pseudo and
backcalculated measurements.
130
Phantom calculations 6.1
( d) -:::1 ·--s---IPR ~ ---&---co «i c Ol ëii '0 (!)
(ij :::1 E (ii
A E detectors of viewing direction
Figure 6.2 Tomographic inversion of a parabolic phantom. (a) Phantom, (b) tornogram of inversion by the
IPR method, (c) tornogram of inversion by the CO method, (d) pseudo-measurements (the I cr-noise level is indi
cated by error bars) and backcalculated measurements for both tomograms.
6 .1. 3 Comparison of the IPR and CO tomographic inversion methods
In order to compare the performance of the tomographic inversion methods of IPR and CO, several phantoms were reconstructed by both methods. The phantoms were idealized shapes of
emission profiles that are expected in a tokamak experiment. In this section the measured weight matrix is used to calculate the pseudo-measurements (with 3% relative noise) and back
calculated measurements. The IPR method was used with the Gerchberg-Papoulis-like step, the
optimum relaxation parameter found in subsection 3.3.3 (A= 2.25) and window smoothing. Somelimes non-negativeness is imposed, i.e. negative emissivity values are set to zero in each
iteration.
131
6 Simple emission profiles
( a ) g(x,y)
20 10 0
-1
xla
( d ) g(x,y)
20 10 0
-1
xJa
-1
-1
-:::::s
~ ~ c Cl ëii "0 Q)
§ :::::s
E ëii
y/a
y/a
' '. '. ' . ' . .
A
-1
-1
-1
-1
B c
( C ) g(x,y)
20 10 0
-1
xJa
y/a
y/a
' •••• __ B
--- phantom --- phantom (noisy) ·· -s- · · IPR .. ·&· .. co (3%)
-- -- -- -co (6%) C0(1%)
D E
detectors of viewing direction
-1 y/a
Figure 6.3 Tomographic inversion of a hollow phantom. Contour plots and three-dimensional plots are shown
for: (a) phantom and (b) tornogram of inversion by the IPR method. Tornogram of inversion by the CO methad
with (c) 3% assumed noise, (d) 6% and (e) I%. (f) Pseudo-measurements (noise level indicated by error bars) and
backcalculated measurements for all tomograms.
132
Phantom calculations 6.1
( d)
--- phantom
--- El - --IPR
---~---co
A E
detectors of viewing direction
Figure 6.4 Tomographic inversion of a localized phantom with steep edges. (a) Phantom (top) and exact sina
gram (bottom), (b) tornogram and si nogram of inversion by the IPR method, (c) tornogram and sinogram of
inversion by the CO method, (d) pseudo-measurements (noise level indicated by error bars) and backcalculated
measurements for both tomograms. In the CO method 2% of noise was assumed because this gave a better result
than 3%.
!33
6 Simp ie emission profiles
The reconstruction errors <rg and <rtfor several phantoms are given in Table 6.I. The phantoms
are: (1) a flat emission profile, which is a test case not expected in reality; (2) a parabol ie profile, chosen to represent the peaked ernission profiles expected in the case of continuurn
radiation; (3) a hollow profilechosen to represent the profiles expected in the case of line-radiation; (4) a localized emission profile; and (5) a profile simulating MHD activity. For the three
most interesting profiles, tomograms, sinograms and backcalculations are shown in Figs. 6.2-
6.4.
The reconstruction errors are relatively large when compared with reconstructions of
measurements with a more uniform coverage of the object. In all cases studied the CO methad gives as good as orbetter results than the IPR method: the <rg is considerably smaller for most
phantoms. The <rJ for the CO methad is 2%-4% larger than the misfit, which is an input parameter of the methad and usually chosen as 3%, but considerably smaller than af for the
IPR method. The CO method gives good results, for example a good hollowness of the hollow
phantom (Fig. 6.3), whereas the IPR methad applies much smoothing to achieve a reconstruc
tion. If the noise level is overestimated in the CO method, the result is more smoothed [Fig. 6.3(d)], whereas with too low an estimated level of noise artefacts appear [Fig. 6.3(e)]. The
reason that the reconstruction is not so good for too low an estimated noise value is that the reconstruction tries to make a fit to the noisy data [see Fig. 6.3(f)]. With the CO methad some
limes a smaJier reconstruction error can be obtained by slightly underestimating the amount of
noise. Especially when the object to be reconstructed is smalt compared with the reconstruction radius, and is therefore only seen by a small number of cameras, oversmoothing becomes
apparent in the IPR method: the non-zero values are spread over a large area and the peak value
is reduced. Although the IPR methad oversmooths the results, the shapes and positions of the
tomograms are reasonable. Both the IPR and the CO methad have difficulties to reconstruct
Table 6.1 Reconstruction error for several phantoms reconstructed by the IPR and CO tomographic methods. In the phantoms 3% relative noise was assumed.
Tornogram error ag (%)
Phantom type Fig. IPR co flat 29 29
parabalie 6.2 32 10
hollow 6.3 36 23 35 38
localized 6.4 63 45
MHD 40 27
* In the CO methad the misfit is an input parameter of the inversion.
134
Residual norrn cr1 and misfit(%)
IPR (af) CO (misfit)*
12
12
14
40
12
3
3
3 6
2
3
Phantom calculations 6.1
steep gradients, and have a tendency to give non-zero emission outside the plasma region. The
MHD phantom is discussed in detail in chapter 8.
A lirnitation of the IPR method is that it needs the signals to be scaled, which introduces errors.
For example, the sealing factor obtained from a flat profile is in principle not applicable in the
case of localized ernission, resulting in larger errors when it is applied. At present, the sealing
factor is not modified iteratively in the method. Furthermore, the effects of reflections and direct
light, which are properly taken into account in the weight matrix and hence in the CO method,
rnight give rise to artefactsin the IPR method. This asks fora more detailed discussion in sec
tion 6.2.
6. 1. 4 The inOuence of phantom position on the quality of reconstructions
The position of a phantom can have an influence on the quality of the reconstruction. In general
this is caused by the discreteness of the viewing directions, resulting in non-uniform sampling
of the object. Such effects can be expected to be particularly important for the visible light
tomography system with its non-uniform coverage of the plasma. These effects are studied in
this subsection, by means of reconstructions by the IPR method. The phantom used is an off
axis Gaussian emission profile rotated around the centre of reconstruction. A Gaussian is a
smooth profile, which results in better reconstructions than discontinuous profiles such as a
step function. To obtain a clear distinction between the Gaussian phantoms at different posi-
30o/o~~--~--~--L-~--~--~--L-~--~--~~
0 60 120 180 phantom angle
240 (deg.)
300 360
Figure 6.5 (a) Tornogram reconstruction error a8 and (b) si nogram reconstruction error CYJ for reconstructions
by the IPR method of Gaussian phantoms rotating at a radius of 0.05 m (solid line) and 0.15 m (dashed line)
from the centre of reconstruction for several discrete angles.
135
6 Simp ie emission profiles
(a)
0
( c)
0 45 90 135
~ (deg.)
.. ••
/
180
tions; the width of the Gaussian was taken smalt compared with the radius of reconstruction, but large enough to enable reasonable reconstructions under advantageous conditions: the 1/e width of the Gaussian was TJ/e = 0.04 m while the radius of reconstruction was 0.19 m.
The reconstruction errors a8 and aJfor Gaussian phantoms at several polar angles and two radii from the centre of reconstruction are depicted in Fig. 6.5. The tornogram error a8 and residual norm af show the same behaviour. The errors for larger radii are Jarger than forsmaller radii. For a radius of rotation 0.05 m there is a large variation in the quality of reconstruction. This can be explained by considering the coverage of the plasma. In Fig. 6.6 the phantoms, reconstructions and the sinogram of the phantom are shown for two phantoms at radius 0.05 m, that in Fig. 6.5 gave the smallestand Jargest error, respectively. From the sinograms it can beseen that forthebest case [Fig. 6.6(f)] the phantom is measured by many detectors of three cameras, whereas in the worst case [Fig. 6.6(c)] the phantom is only measured by some edge detectors
of the cameras.
136
PhLlntom calculations 6.1
( d )
( f )
-a. 0
0 45 90 135 180
~ (deg.) Figure 6.6 (a) Gaussian phantom at radius 0.05 m at the angle with the largest reconstruction errors (60°), (b)
corresponding reconstruction by the IPR method and (c) the si nogram of the phantom with the positions of the
detectors indicated by dots. (d-f) are as (a-c), respectively, but for a phantom at the sameradius at the angle with
the smallest reconstruction error ( 180°).
For some positions reflections and direct light affect significantly the pseudo-measurements if the object is small; if the object is larger the influence becomes relatively less important. The present IPR methad does nottake into account the weight matrix (i.e. not iteratively modifying
the sealing factor); therefore, artefacts appear in the tomograms. Improved results for localized emission profiles are expected if the weight matrix is properly taken into account. However, calculations with line integrals instead of the measured weight matrix, i.e. circumventing prob
lems concerning the sealing factorand reflections and direct light, show that only Ojis reduced significantly and e5g remains roughly the same. Besides, for emission profiles extending over a
large part of the plasma volume these distortions are of less importance, as can be inferred from
the reconstructions discussed in the previous subsection.
137
6 Simple emission profiles
6.1.5 Conclusions
For phantoms the CO methad seems preferabie compared to the IPR methad because it yields
more accurate results. The IPR methad gives results of which the position and shape is in reasonable agreement with the phantom. The CO method, however, gives less smoothed recon
structions with reliable magnitude. The CO method requires an accurate estimate of the noise
level because too Jow an assumed misfit leads to artefacts. If a larger misfit is assumed,
smoother profiles are obtained. The IPR methad is hampered by the necessity of sealing and by
the fact that for reconstructions of localized profiles reflections and direct light can have a darn
aging effect on inversion. These effects might be taken into account in the iterative scheme as described in subsectien 3.3.2, but this has notbeen implemenled yet. The reliability of both the
IPR an CO methods is tested on actual plasmas in the following section.
6.2 Reconstructions of measurements of simple plasmas
Phantom calculations are important to delermine the potential of the measuring system and
tomographic inversion methods. In the simulations, aspects of the system are taken into account
which might differ from the actual conditions when the emission of plasmas is measured.
Therefore, it is important to validate the tomographic inversions for phantoms as well as for
measurements on plasmas with approximately known emission profiles. Because the usual
emission profiles of tokamak discharges show asymmetries, three types of sim pier plasmas are studied in this section: glow-discharge cleaning plasmas, the continuurn emission in the begin
ning of a discharge, and ECRH-startup plasmas.
6.2.1 Glow-discharge cleaning plasma
A glow-discharge cleaning (GDC) plasma is a low-temperature plasma (Te "" lOeV, ne "" JOI 4
to JQIS m-3 [Wint88]) that is used to clean the innerwallof the tokamak vessel. The plasma is
made by applying a de voltage of several hundreds of volts between an electrode inserted into
the vessel and the vessel wal!. Under these conditions the electrans travel through the entire
138
Reconstructions ofmeasurements 6.2
vessel before reaching the wal!. The experiments described in this subsection were carried out
with He as filling gas. Theemission profile ot the GDC plasma is expected to be approximately
cylindrically symmetrie. The measurements were taken in the entire wavelength range to which
the detectors of the visible light tomography system are sensilive.
Because the electranies of the visible light tomography system is ac-coupled, it is not suited to
study steady-state phenomena, such as the GDC plasma. The simplest way to obtain a time-
::l
~ (ij c Cl ëii
" Q)
(ij (.) en
-::l ~ (ij c Cl ëii
" Q)
(ij (.) en
0
( c) .0·
--- Signal /.· ;': ,,
,}
· · -e· · · Polynomial fit in p
···El··· Backcalculated IPR
· · · "'· - - Backcalculated CO
A B c D E
detectors for viewing direction
• )f( liC )0( ~- X X >o<X>O<:!!i: ><!( ( d) )0()0()0( ~~
~
~I:!JJ! ..___ x* ~
0 o of!<P~io••t~ * 0coco o oCbf~(~i 1~'iP~~ l(
i~ EIJ~ ~· x Signal !::!ft!~ •
0 Polynomial fit 0 ' • 0 Backcalculated IPR 1@1!11 ~= x
llllt§8! Backcalculated CO 0
88~ x Abel inversion
50 100 150 lpl (mm) and radius (mm)
r 0 ()
~ CD 3 Ciï en :c::· ~
?i' c ~
200
Figure 6.7 Reconstructions of a GDC plasma. (a) Tomographic inversion by the IPR method. (b) Tomog
raphic inversion by the CO methad (2% misfit assumed). (c) For each detector the scaled signals, polynomial fit,
and backcalculated measurements for the tomographic inversions with the IPR an CO methods. (d) The same as
in (c), but with absolute p as variable. Here also the Abel inverse of the polynomial fit is given with the radius
as variable. The error bars in (c) and (d) give the estimated error in the measurement.
139
6 Simple emission profiles
dependent phenomenon without the need of special triggering equipment was to measure the
switch-off of the GDC plasma; the switch-off being triggered by a switch-on of one of the
magnetic fields of the tokamak. The signa! level was of the order of I m V ("' 2 x 10 I 0 pho
tons/s). A 50Hz perturbation with an amplitude of 0.5 mV, of which the origin is not known, could be filtered out. Because, apart from this perturbation, the surroundings were relatively
noise-free compared to normal tokamak operation, the noise was in the order of the bit-level of
the sampling (15!!V). Therefore, a reasonably accurate measurement was possible: an average
relative accuracy of 4% was achieved, although for most channels it was of the order of 2%.
Various GDC discharges gave similar results.
The measurements have been studied by three different inversion methods: (1) cylindrical
symmetry is assumed and a radial polynornial fit to the scaled signals is analytically Abel in
verted [Eq. (3.6)], (2) tomographic inversion by the IPR method, and (3) tomographic inver
sion by the CO method. The results are shown in Fig. 6.7. Both the tornogram by the IPR
method [Fig. 6.7(a)] and the tornogram by the CO method [Fig. 6.7(b)] show asymmetries; the ones in Fig. 6.7(b) being more pronounced. From the backcalculated measurements [Fig. 6 .7(c)] it is clear that the CO reconstruction is good, whereas the IPR reconstruction
( C5J= 16%) is oversmoothed. The overall structures are sirnilar, and the reconstruction errors
are camparabie to the ones in the symmetrie cases in subsection 6.1.3. For the CO reconstruc
tion a misfit of 2% was assumed, whereas with 3% an almast symmetrical profile could be
obtained. The deviations larger than the error bars [see Fig. 6.7(c)] are thought to be real: with a misfit of 2% these deviations are foliowed better than with 3%, and therefore 2% was chosen
for Fig. 6.7(b). Uncertainties in the weight matrix can be interpreled as uncertainties in the
measurements. Therefore, it rnight be required to assume the overall error in the measurements
to be larger than the indicated error bars, but these errors will be smoothed out in the recon
struction because they do not represent significant deviations by several adjacent channels. The
shape of the right-upper corner of the profile in Fig. 6.7(b) is caused by the edge of the reconstruction area. In Fig. 6.7(c) also the polynornial fit in p to the scaled measurements is drawn,
showing in which cameras the asymmetries become apparent. In Fig. 6.7(d) the same informa
tion as in Fig. 6.7(c) is plotted, but as a function of p (thus assuming cylindrical symmetry),
giving insight into the radius at which the deviations from the cylindrically symmetrical profile
assumed for the fit occur. The variation between the measured points and the fit is much larger than the estimated errors (see error bars). The sealing factor for p values for which the largest
deviations occur is expected to be accurate (see Fig. 5.11 ). Therefore, it is eertaio that the emission profile is asymmetrical. The Abel-inverse of the fit is also shown, indicating that the
average ernission profile is slightly hollow. The Abel-inverled profile was used to calculate one
of the curves of the sealing factor in Fig. 5.11.
The question that remains to be answered is whether the structures observed in the CO methad are reliable, or whether they are artefacts caused by uncertainties in the weight matrix and scal-
140
Reconstructions ofmeasurements 6.2
ing factors. The result of the CO method depends strongly on the assumed misfit. Therefore, a number of simple plasmas have been studied. Furthermore, artefacts are usually structures that
appear along the viewing directions, which is not the case in Fig. 6.7(b).
6.2.2 Continuurn radiation
As described in chapter 2, the continuurn emission profile is expected to be approximately
cylindrically symmetrical and peaked, provided the plasma parameters are symmetrical and the
electron density is peaked. The profiles measured on RTP in a wavelength range that is expected mainly to contain continuurn radiation usually show asymmetries, except at the beginning of a discharge. The physical aspects of the emission measured when continuurn filters are
applied are considered insection 7.2. In Fig. 6.8 reconstructions of a reasonably symmetrical
profile are shown. The reconstruction error with the IPR metbod [Fig. 6.8(a)] is f3J= 14%,
ëii c Cl ëii
---Signa!
·· · B · ·· IPR
· - -~---co
detectors of viewing direction
E
Figure 6.8 Tomographic reconstructions of continuurn visible light emission at the start of a discharge
for (a) the IPR method, and (b) the CO method. (c) The backcalculated measurements for both methods.
141
6 Simpte emission profiles
whereas for the CO methad [Fig. 6.8(b)] a misfit of 2% was assumed. The reconstructions are
very similar in shape and also the backcalculations [Fig. 6.8(c)] are good. If a small misfit is
assumed (misfit< l %) in the CO method, artefacts appear mainly at the right-upper side, similar to these for the GDC discharge reconstructions above.
6.2.3 ECRH-startup plasma
Electron-cyclotron resonance heating (ECRH) is sametimes used to induce breakdown at the start of a discharge in RTP. Usually a sharp peak is observed by most visible light channels at the start of the discharge; aften some channels saturate for a short time. The area from which light is emitted quickly changes shape and eventually fills the entire plasma region. The inten
sity of the light varies strongly between channels and shots. For the present system purpose it is more suitable to study a discharge with a plasma created by ECRH without a plasma current
because these plasmas do not fill the entire reconstruction region and are not necessarily hol
lew. Several such occasions have been measured where the emissivity changes in a smooth way. In the case under consideration the ECRH radiation is injected in 0 -mode from the !owfield side and in X-mode from the high-field side, both at the fundamental frequency. The reso
nance plane is at R = 0.74 m (the vessel centre is at R = 0.72 m). The emissivity was measured in the total wavelength range to which the detectors are sensitive.
The increase in electron density and emission is seen 0.3 ms after the switch-on of ECRH; the emission reaches a steady-state in 0.5 ms, whereas the density only increases appreciably somewhat later and reaches the steady state in I ms. Measurements on similar discharges by
Thomson scattering indicate a local electron density and temperature in the order of
1 x 1019 m-3 and 200 eV, respectively. The shape of the line-integrated density profile does
not change much during the density increase. The emissivity seems to undergo two phases: at first the shape of the line-integrated emission profile does notchange much, but when the density increases appreciably an upward and outward movement of the main emission is observed. The emission starts at very low electron densities. It is not known which processes contribute
to this emission since the measurements were not speetrally resolved, but it is expected to be mainly Ha radiatio11 (cf. subsectien 7.1.3 about Ha emission inthestart-up of a discharge).
Only one example of theemission profile in the first phase is given because in the secend phase it becomes quite asymmetrical, with possibly emission from outside the reconstruction area. In Fig. 6.9 the line-integrated density is given fora time where the visible emission has reached a steady state; the density during the first phase of increase in emission is too low to give a reli
abie measurement. From these line-integrated measurements the centre-of-mass can be calculated, which according to the consistency theerem ( subsectien 3.1.5.1) corresponds to the cen
tre-of-mass of the (local) density profile. The centre-of-mass is located at R = 0.78 m, i.e . shifted to the !ow-field side with respect to the resonance layer.
142
0.55 0.6 0.65 0.7 0.75
R (m)
Reconstructions ofmeasurements 6.2
0.8 0.85 0.9
Figure 6.9 Measured line-density of the electrans during ECRH without plasma current for various major
radii. The curve is a spline fit between 13 measured points. The major radii at which the centTe-of-mass of the
curve lies and where the resonance layer is are indicated.
In Fig. 6. 10 the tomographic inversions of the visible light emission by the IPR and CO meth
ods are shown, having ar= 30% and an assumed misfit of 4%, respectively . The IPR recon
struction is oversmoothed, as apparent from the backcalculated signa! [Fig. 6.10(c)]. From the CO reconstruction it is clear that the additional emission on top of the smooth profile comes
from the right-upper side. With a larger assumed misfit this asymmetry does not disappear. It is
not clear whether the "pushing outward" of emissivity is an artefact, which also appeared
slightly in the GDC and continuurn case, due to Jack of information for the reconstruction. If the CO tornogram is used to backcalculate pseudo-signals by line integrals instead of the weight
matrix, too high pseudo-measurements are obtained at the edge. Furthermore, the outermost channels of viewing direction B which are nol used in the reconstruction (the weight matrix and
sealing factors are not known forthese channels) show a signa! level that is lower, although not
zero, than the value that would be expected from such an increased emissivity at the edge of the
plasma. This contradiction is unresolved, and hence the exact nature of the asymmetry cannot
be found by the current system and analysis methods. A possible explanation for the appearance of the increased emission in the right-upper side is tl;lat camera C receives more light in
reality than is described by the measured weight matrix. Tijis might, for example, be caused by
a reflecting surface that was not present in the set-up in which the weight matrix was measured.
Too low weight-matrix elements for camera C would cause a inconsistency in the totalemission
in the reconstruction, which might be resolved by the algorithm by putting the surplus emission
in a place where the other cameras do not see it. Simulations have shown that for emission profiles with much emission at the edge, for example a flat profile, similar structures as in Fig.
6.10(b) are obtained if it is assumed that camera C measures sometensof percents higher val-
143
6 Simpte emission profiles
ëä c Cl ëiï
--- Signal
---a--- IPR
---~---co
_, ' I
I I I \
\
' I \
detectors of viewing direction
~
( c ) co co ~ ~ -.J 0 w OJ
'ii 1\) 01 èn 3 C/)
E
Figure 6.10 Tomographic reconstruclions of the visible light emission during ECRH without plasma current
for (a) the IPR method, and (b) the CO metbod (4% misfit assumed). (c) The backcalculated measurements for
both methods.
ues than calculated from the weight matrix. Fora peaked profile the effect is less pronounced: a
signa! increased by 20% on camera C does hardly have any effect of the reconstruction,
whereas with an increase of 50% large artefacts occur in the centre of the reconstruction. Be
cause the effect is different for different phantoms, it is difficult to estimate the increase in sig
na! on camera C that might explain all observed artefacts, and therefore it is not possible to correct for it. Because the weigh matrix elements of camera E are known less accurately than for
the other cameras (see subsectien 5.4.1) it could be expected that this is the souree of the large
tornogram values in the upper-outer side, for example if the weight matrix elements have been
overestimated. Th is, however, is not the case, because in reconstructions similarly large values
are obtained when the detectors of camera E are nottaken into account in the reconstructions.
The visible light measurements of the ECRH-startup plasma and the corresponding tomog
raphic inversions cannot be fully explained by the spatial density distribution. Figure 6.9 indi-
144
Reconstructions of measurements 6.2
cates that there is no measurable density for major radius R < 0.66 m at which some emission
is found. Furthermore, the density at large R cannot explain the asymmetry in emissivity (the
centre-of-mass of the CO reconstruction lies at 0.80 m). The asymmetries found in various
types of plasmas and physical explanations based on measurements in different wavelength ranges are discussed in chapter 7.
6.3 Summary
The tomographic inversions of phantoms and measurements described in this chapter show that tomography with few channels viewing only parts of the plasma from a smal! number of direc
tions is difficult. The viewing configuration of the visible light tomography system is too
complicated to interpret the measured signals directly in terms of the line-integrated emission
profile, and tomography is needed to give the overall structure. For simple profiles, both
phantoms and measurements, satisfactory results are obtained by tomographic inversion, which
cao be used in the interpretation of measurements. Conclusions on asymmetries can be drawn from the difference between the tomographic inversion and the measured signals. Two different
tomographic inversion methods have been considered in this chapter. When possible both the
IPR and CO methods are used in the analysis of data presenled in this thesis, and their differ
ences are considered. The CO methad has become available only recently. Therefore, forsome
studies only the IPR metbod has been used.
For the interpretation of data the following considerations have to be taken into account. The
IPR metbod gives oversmootbed reconstructions of which the overall shape reflects the main
structure of the profile where smaller structures may have been smoothed away. Because of
this, details in these reconstructions cannot be trusted. The CO methad gives more detailed
results. The CO reconstructions show that minor changes in the measurements or the assumed
misfit can have profound implications on the reconstructed profile. This means that even though
structures are apparent in the reconstruction, these might be artefacts caused by underestimating
the noise, by uncertainties in the weight matrix, or by insufficient information due to the non
homogeneaus sampling in projection space. The estimated misfit is an important parameter to
control the amount of smoothing; the :nisfit should carefully be determined from comparison
between the backcalculated signa! and the measured signa!. When a very smal! misfit is assumed ( < 1%) for measurements, the metbod has the tendency to give emission outside the
reconstruction area, probably because the measured weight matrix does not accurately enough
describe the system. This reveals that the application of the CO methad to the visible light
tomography system and eertaio types of measurements might give unreliable reconstructions.
Therefore, the results that show a large asymmetry can only be trusted in so far as that they
indicate that there is an asymmetry, but cannot reveal the 11recise structure of this asymmetry. Because the two methods have different characteristics, it is advantageous to use both methods
145
6 Simple emission profiles
and campare the results, which can give an indication about the reliability of features appearing
in the reconstructions.
A non-uniform and non-complete coverage of the plasma can in principle be improved in tomographic inversion algorithms by assumptions about symmetry and time evatution of the
plasma. However, in chapter 7 it wil! be shown that the plasma emission in most cases is
asymmetrie, and taking into account the time evolution is not simple (see subsection 3.2.7). Therefore, such attempts would be complicated and require many assumptions about the be
haviour of the emission, which has notbeen attempted in this thesis.
146
Measurements of stationary asymmetrie
profiles • • entiSSIOD 7 In this chapter stationary and slowly varying phenomena in the visible light emission in RTP are studied. Emission profiles have been measured with Ha and continuurn filters, and without optica! filters. Measurements with these two filters and without filters are discussed in sections
7 .1, 7.2 and 7 .3, respectively. Most emission profiles exhibit asymmetries. These asymmetries
are quantified from tomographic inversions. Possible explanations are discussed in sec ti on 7 .4.
7.1 Ha emission
Hydrogen is the main constituent of plasmas in RTP and, therefore, measurement of the emis
sion from excited levels of hydrogen atoms is an important diagnostic tooi. As described in
section 2.1 , the Ha line is often used for diagnostic purposes in tokamaks. From the emissivity
of this line one can derive quantities such as the neutral hydrogen density and the souree rate, i.e. the rate at which the plasma protons are replenisbed by ionization of neutral atoms diffusing
into the plasma.
On RTP the Ha emission has been studied under different conditions. Theemission profiles are
asymmetrie and can be divided into a number of classes. These profiles, and the evolution of
the emissivity during the start-up of a discharge are the topic of this section. To derive the locaJ . emissivities from the raw signals, the properties of the interference filters and the calibration
factors have tobetaken into account in the way as described in subsection 5.6.3.
7.1.1 Emission profiles for different plasma conditions
A number of different shapes of emission profiles of Ha light have been observed under various conditions. Here, Ohmic discharges are analysed. Figure 7 . I shows the scaled visible light tomography measurements forsome typical discharges. For the line-integrated measurements
the scaled values are given, which take into account the calibration factors and represent the quantity that would be measured along ideallines-of-sight. The scaled vaJues are equivalent to the measured power divided by the étendue of each detector. The signa! levels scale roughly
linearly with the electron density, as can be expected from Eq. (2.9) and Fig. 2.2. The average scatter of measured points around a linear fit with density is 30% for each camera. To investi
gate differences in the emission profiles, the ratios between the maximum signa! of the various
Chapter 7 Measurements of stationary asymmetrie emission profiles
cameras have been studied. Especially the ratios between cameras A and B (displaying roughly in/out asymmetry), E and B (up/down asymmetry), and Band C showed variation for different plasma conditions.
Significantly different values of these ratios were obtained for the different operational condi
tions: standard conditions, reversed tomidal magnetic field B~. and an inward shift of the
plasma of élR = -30 mm. Table 7.1 gives the discharge parameters of typical examples of
~E
1,_ (/)
~
6
5
4
I
'
""
I I I
.... -·~ •o ~ -0
~
«i 3 - .~ • -~ 0 ~ 0~ ~ 2~ 0 ~~ -
~1~ -~ Q I~ --.~ 7
A 8 C D E
detector of viewing direction
Figure 7.1 Scaled measurements of Ha light for different discharges, of which the main parameters are sum
marized in Table 7.1. The scaled signals are the ones that would be measured by pure line integrals, taking into
account the transmission factor of the filters.
I N
0.0
-0.19+---'---....---_j__----1
-0.19 0.0
R(m)
0.19 -0.19 0.0 0.19
R(m)
Figure 7.2 Tomographic reconstructions by the (a) IPR and (b) CO methods of the Ha emissivity profile as it
is usually observed (standard conditions, open circle in Fig. 7. 1 and Table 7.1). The emissivity has been abso
lutely calibrated; each contour corresponds to 2 W m-3 sr- 1•
148
Ha emission 7.1
these cases. Standard conditions here refer to standard plasma position and current and field direction; the plasma current and densities were varied, however. In standard condüions a set of
discharges with a low current (/p"' 65 kA) at intermediale densities (6 x 1018 m-2 < Jne dl < 11 x 1Ql8 m-2) exhibited ratios deviating from the ones of the other discharges, for which
the ratios were relatively independent of plasma parameters. However, for all discharges for
standard conditions the shapes of the tomographic reconstructions did notshow significant dif
ferences. As indicated, the magnitude showed deviations of 30% between discharges. No clear
dependenee of the deviation on the electron density and temperature has been found. Therefore,
most probably the neutral hydragen density nH varied between discharges. However, wall
depletion of hydrogen by He glow-discharge cleaning did not have a clear effect on the general emissivity.
In Figs. 7.2-4 typical examples of tomographic reconstructions are shown for the various cases. The reconstructions by the IPR and CO methods correspond reasonably wel! and point
out the main features of the emission profiles. The profile obtained for standard conditions (Fig. 7.2) has most emission at the edge of the plasma, as is expected for speetral lines of
Figure 7.3 Tomographic reconstructions by the (a) IPR and (b) CO methods of the Ha emissivity profile
measured during reversed tomidal magnetic field ( diamond in Fig. 7.1 and Table 7.1).
Figure 7.4 Tomographic reconstructions by the (a) IPR and (b) CO methods of the Ha emissivity profile
measured when the plasma was displaced inwards (cross in Fig. 7.1 and Table 7.1).
149
Chapter 7 Measurements of stationary asymmetrie emission profiles
Table 7.1 Main parameters of the characteristic discharges shown in Fig. 7.1. The symbols
refer to Fig. 7.1. The temperature is the central temperature determined from Thornson scattering and ECE, and q0 is the safety factor at the edge of the plasma.
Symbol Discharge number time(ms) /p(kA) qa Jnedl (xJQ18 m-2) Te (keV) Special features
• r 19940620.009 ISO 66 6.2 6.5 0.6
0 rl9941005.051 220 87 4.6 9.7 1.0
• r19940708.017 200 79 5.2 2 .7 reversed BtJ>
x r19941005.041 100 56 7.1 2.7 0 .9 óR=-30mm
hydrogen. lt has significant asymmetrie pea.ks to the upper and lower outer side. The IPR re
construction is smoothed: in the actual emission profile there is more emissivity in the positions indicated by the CO method.
When the tomidal magnetic field is reversed (Fig. 7.3) there are changes in the emissivity profile: more emissivity to the !ow-field side (LFS), and furthermore, a downward shift of emis
sivity on the high-field side (HFS). The shift of the plasma centre to the HFS by 0.18 a (Fig. 7.4), a being the minor radius, also results in a change of the emissivity profile. As can be
expected there is a large increase on the HFS. It is striking that the emissivity is mainly from the upper half of the plasma. These observations are discussed in more detail in section 7 .4.
lt is reasanabie to assume that the level of non-Ho. radiation that is transmitted by the filters is negiigibie. The filter transmits iess than 1% of the total continuurn radiation in the range of the
detector, which is notsignificant (see subsection 7.3.2). Spectroscopie measurements on RTP
have shown two CU iines close to the Ho. line (at wavelengths of 657.8 nm and 658.3 nm), but also that these are significant only during disruptions. These lines are negleeled because they are severely attenuated by the filter characteristics and angle-of-incidence effects (see subsection 5.6.3 .1).
In most reconstructions a significant emissivity is found in the central parts, where, because of a low neutral density, a very iow visible emissivtty is expected. The contribution by
bremsstrahlung is too smal i to be significant. Furthermore, because the temperature in the centreis of the order of I keV most impurities wil! be fuily ionized and notproduce visible iine radiation. Because no spectroscopie measurements are avaiiable forthese discharges, a contri
bution from iine radiation cannot be ruied out completely. Because the significant emissivity in
the centre found by the tomographic reconstructions does notshow up equaliy in both reconstruction methods, it is iikeiy to be an artefact. lt is a known probiemof tomographic inversions
of very holiow profiles that too high leveis in the centre are obtained. Another common feature is that most reconstructed steady state Ho. emissivity profiles are holiow, and sametimes even have an unphysical "negative" emissivity in the CO reconstruction at R = a/2 in the midplane.
150
Ha emission 7.1
The fact that it is smal! and sametimes negative suggests that it is an artefact, but it is remark
able that it results from both reconstruction methods. lt is very unlikely that the non-zero re
constructed emission in the centre results from an inaccurate calibration of the Ha filters be
cause the effect of the Ha filters is not large enough to cause it, which has been verified by simulations. A possible explanation is that due to the sharp gradient in emissivity at the upper outer side the central emissivity undershoots: the correct emissivity from the centre of the
plasma might be an average between the hollowness and the slight peak in the centre. The
emissivity in the centre obtained from :hese reconstructions can therefore only be considered to be an estimate. At other positions a better es ti mate is possible.
In subsectien 7.1.2 a quantitative description is given of the profile that is measured during
standard conditions. The es ti mate of the local emissivity from the reconstructions is discussed
in subsection 7 .1.2.1. Possible reasans for the asymmetries are discussed in subsection 7.1.2.3 and section 7.4.
7.1.2 Absolute emissivity and neutral hydrogen density
In this subsection the absolute Ha emissivity and quantities derived from it, such as the neutral
hydragen density and the partiele confinement time, are discussed for one particular tomographic reconstruction. The asymmetries observed are discussed and different processes
contributing to the radiation are considered.
7.1 .2.1 Estimate oflocal emissivitiesfrom tomographic reconstruction
The absolute Ho: emissivity profile that was chosen for detailed study is during a typical dis
charge for standard conditions (the second in Table 7.1). The tomograms by two reconstruction
methods are given in Fig. 7 .2. An estimate of the local emissivities is found by taking into account the differences between the two reconstruction methods and the problems with artefacts discussed in subsection 7 .l.I.
The differences in emissivity levels from the IPR and CO reconstructions are considerable,
which is caused by the different effects that the asymmetrie profile has on the methods. First of all, the IPR methad oversmooths the reconstruction of asymmetrie profiles, as was shown in
Chapter 6 and is evident from the backcalculated signals. Therefore, the peak in the upper outside should be at least a factor of two higher. On the other hand, the peaks obtained by the CO methad are too high. This beoomes clear when the backcalculation is done by line integrals,
which results in too high values, in a similar way as was described in subsection 6.2.3. Furthermore, the negative values in the reconstruction have a strong effect on the backcalculated
values forsome cameras, which indicates that the peak at the lower outside is too high. Balanc
ing these effects, i.e. too Iow values by the IPR methad and too high values in the main asymmetries by the CO method, and camparing the found values by the backcalculated line integrals
and measurements, reasanabie absolute emissivity values can be estimated.
151
Chapter 7 Measurements of stationary asymmetrie emission profiles
7.1.2.2 Thickness of the radiating layer
From the tomographic reconstruction in Fig. 7.2 it is clear that the H~ emission not only comes
from the SOL, where the temperature is low enough for H not to be ionized, but also from an edge layer of several centimetres inside the limiter radius. Tomographic reconstructions are
known to, in genera!, smooth hollow profiles. In particular for the asymmetrie emissivity
measured, it is therefore justified to have a closer look at the thickness of the radiating layer.
Camera B is the onJy viewing direction observing the far edge of the plasma in detail; it is also
the camera that mainly measures the largest asyrrunetric peak. Assuming the plasma to be lo
cally cylindrically symmetrical, the line-integrated signals of this viewing direction can be Abel inverted. Because the Abel inverse inlegral [Eq. (3 .6)] is over p values outside the radius of
which the local emissivity is required, it is possible to do the inversion of the limited view of
camera B to obtain the approximate local emissivity of the region covered. This is not possible
for the other cameras because these do not view the far edge. The Abel inverse was obtained by
fitting a polynomial to the points and analytically inverting the polynomial . For the Abel inversion the unscaled measurements had to be used because the sealing factor for the edge channels,
which play a major role in the inversion, are unknown (see sectien 5.5). The measurements and
their Abel inverse are shown in Fig. 7.5. The figure also shows the measured points after seal
ing to illustrate the significanee of effects contained in the weight matrix, e.g. chord width and
- 0 ~--'Q'·-~· ••
:J
~ 0 ••• '
,.__ ~--·-·· .. c··
(ij c Ol (i)
"0 ~ :J (/) <U <D E
~ ,. - 0
0
0 100 120 140 160
p, r (mm)
180
c
--- 0 200
Figure 7.5 The line-integrated signals of camera B and their Abel inverse. The measurements (solid circles)
have been corrected for the angle dependenee of the Ha filter and are given as a function of impact parameter p.
The measured points have been fitted by a polynomial curve (dashed line). The open circles represent the meas
urements after sealing by the sealing factorfora flat profile. The solid line is the analytica! Abel inverse of the
fitted curve, i.e. an approximation of the local emissivity as a function of minor radius r .
152
Ha emission 7.1
vignetting of the imaging system. For p < 160 mm the effect is small, whereas for larger p the
width of the chord plays a significant role. Figure 7.5 shows that the thickness of the radiating
layer is a bout 4 cm, and if the chord-width effects could be taken into account by sealing it
would probably be approximately 1 cm thinner. This is smaller than the width in the tornograpbic reconstruction ( == 7 cm), which is probably due to more smoothing in the tornograp bic
reconstructions. The determined width is approximately the same as the 5 cm found in TFR
(a= 20 cm) [TFR75]. A thickness of the neutral density layer inside the minor radius of 4 cm
has been determined in RTP by an independent metbod [Heij95]: measurements of edge elec
tron density with the pulse radar reflectometer on RTP show that in the edge the density drasti
cally deviates from the profile expected from profile consistency [Kado87]. which is ascribed to
the neutral density layer. The thickness of the neutral density layer inside the minor radius (a:::
0.164 m) found by the reflectometer measurements is larger than the one found from the Abel
inverse.
The width of the radiation layer inside the plasma where Te> 50 e V can be explained by the fini te ionization and charge exchange rates and the energies with which atoms and molecules
enter the plasma: the atoms can travel into the plasma edge befare they dissociate and are ion
ized. The ionization rate is higher than the charge-exchange rate, and therefore ionization domi
nates. The ionization rate <GV>ion. where a is the i anization cross section, of neutral hydragen at electron temperatures 50eV <Te< lOOeV is of the order of I0-15 to I0-14 m3 s-1 .
Assuming a torus entering the plasma with veloeities VH of the order of 104 mis, the mean free
path for ionizations of atoms,
(7. 1)
can be calculated to be of the order of several cm, which is in agreement with the thickness of
the radiating shell in the plasma. In principle most hydragen enters the plasma as molecules, and therefore the rates of all molecular ionization and dissociation processes, which will be dis
cussed in subsection 7 .1.2.6, should be taken into account. These processes, ho wever, do not
lead to significant changes in the estimate. Because Eq. (7 .1) is dependent on ne and only little
knowledge is available about VH in RTP, only this rough comparison is possible.
Furthermore, it should be noted from Fig. 7.5 that the Abel inverse of the peak shows that there is little emission outsider= 180 mm, indicating that the large emission in that position result
ing from the CO methad must be an artefact.
7.1 .2.3 Asymmetries
The tomographic reconstructions revea1 that the Ha emission is asymmetrie in the poloidal plane where it is measured. Only the presence of neutral hydragen can give rise to Ha emission. There are no indications that the other parameters on which the Ha emissivity depends, ne
and Te. are asymmetrie inside the plasma. Therefore, the only reasanabie explanation for the
153
Chapter 7 Measurements of stationary asymmetrie emission profiles
asymmetry in the emissivity is an asymmetrie neutral hydrogen density distribution in both the
SOL and plasma edge. This must be caused by an asymmetrie intlux of atomie hydrogen and
sourees thereof. The main influences of the asymmetries on the calculation of the total Ha
emission are discussed in this subsection. Insection 7.4 the asymmetries are studied more in
detail.
It is known from the literature [Isle84] that significantasymmetriesexist both toroidally and
poloidally in tokamaks, mainly for hydrogen and the lower ionization stages of impurities. The
main souree of asymmetries is the presence of a limiter, where due to heating by the contact to
the plasma a continuous flow of impurities is created and hydrogen ionscan be recycled (i.e. neutralized on the limiter and retuming into the plasma as a hydrogen atom or molecule). The
toroidal asymmetries as a function of limiter position have been studied in Refs. [VinR74,
Razu84] (see Fig. 7.6), where a large dependenee of emissivity on the distance from the posi
tion of measurement to the limiter was observed (up toa factor of 10 difference in emissivity).
The toroidal ex tent of the increased emissivity is typically about 10 to 20 cm, which cocresponcts roughly to Àion· However, the decay in emission as a function of toroidal distance to
the limiter is not exponential (see Fig. 7.6), indicating that in toroidal positions far from the
limiter other sourees for recycling exist or that other processes than atomie ionization occur near
the limiter. The forrner explanation is usually given in the literature. An alternative explanation
is given in subsection 7.1.2.6.
ç 0.80 ëii c (!)
"E 0.60 '0 (!)
.!::! (ij 0.40 E 0 c
0 . 00 L._c__L__L__L__L_L.._~L__L__L.._L__L.._.l-.1.-.l-J.__J.__J.__.L.......J
0.00 0.05 0.10 0.15 0.20 toroidal distance to limiter (m)
Figure 7.6 The toroidal dependenee of line-integrated Hp emission on the distance toa poloidallimiter as pre
senled in Refs. [VinR74] (TM-3 tokamak; solid circles) and [Razu84] (T-1 0 tokamak; open circles). The dashed
line shows the exponential decay rate expected for Àion = 6 cm. The regions bounded by horizontal and vertical
!i nes and designated by A and B show the simplification of the toroidal dependenee of emission used to estimate
the influence of theemission close to the limiter ( region A) to the total emission ( region B).
154
Ha emission 7.1
The poloidal cross-section of the visible-Iight tomography diagnostic is located 120° toroidally
from the limiter, which consists of a circular limiter and two movable locallimiters on the top
and bottorn (poloidal extent 10 cm) that are at a smaller minor radius than the circular limiter.
The main concern for the calculation of the partiele confinement time is the question whether the
observed asymmetry in Ha light can be expected to appear in every poloidal plane. Unfortunately noother spatially resolved measurement of Ha radialion is available on RTP and the
souree of the asymmetrie in flux is not known. Because the main toroidal asymmetry is expected
to be caused by the limiter, it is assumed that the power emitted in each poloidal cross-section away from the limiter is the same as in the cross-section of the visible-light tomography diag
nostic, which is relatively far from the limiter. Assuming the typical values of a 5 to 10-fold
increase in a layer 20 cm thick in the toroidal direction around the limiter [Razu84] (area A in
Fig. 7.6), which is 5% of the circumference of RTP, a 20% to 50% increase of totalemission
could be caused by the limiter, compared to the value of the total emission based on the emissivity measured at a position far from the limiter (area B in Fig. 7.6).
7.1 .2.4 Neutral hydrogen density
From the tomographic reconstruction in the poloidal cross-section viewed by the system (Fig. 7.2) the emissivity in the asymmetrie peak is estimated to be 20 W sr1 m-3. Emissivities out
side the main peak range from 5 to 10 W srl m-3. The main peak radiates about 40% to 60% of the total power in the poloidal cross-section. The absolute emissivity can be related to the
ground-state density of neutral hydragen by the model discussed in subsectien 2.1.2. Equation
(2.4) relates the emissivity to the density of the excited leveland the spontaneous emission rate
coefficient Ai)- For the Ha line, which results from the transition between the levels 3 to 2, A 32 = 4.41 x 107 s-1 [WieS66]. This results in nH,3 "" 2 x 1013 m-3 for the peak. To use
the model the electron temperature and density are needed. Unfortunately no accurate temperature measurements in the edge of the plasma and the scrape-off layer (SOL) are
available on RTP. Extrapolation from reliable temperature measurements in the central part
(ria< 0.7), gives close to the edge, where the Ha emissivity peaks, 30eV <Te< lOOeV, which is in the range of typical values found as edge temperature on other tokamaks [StaM90].
Using Xp = XHip 2 for hydrogen, XH = 13.6 eV, and ne"" (1.0 ± 0.5) x 10 19 m-3, Eq. (2.8) gives the Saha equilibrium density 108m-3< nj < 109m-3, which is four orders of magni
tude smaller than the measured n3.
The value of n3 as determined from the measurements can be related to the ground state density
nl> which will approximately correspond to the total neutral density nH. This has been done by solving Eq . (2.9), i.e. the collisional-radiative model. Assumptions in the model of Eq. (2.9) are that the plasma is quasi-stationary and homogeneous. Because of gradients in temperature and density, and the neglect of transport, results from the model have to be considered with care. Because nj << n3 only the r 1 term in Eq. (2.9) is significant, as indicated in subsection 2.1.2; i.e. atomie recombination is negligible. Taking the r1 for an optically thin plasma
155
Chapter 7 Measurements of stationary asymmetrie emission profiles
[JohH73], Eq. (2.9) yields n1 "'4x IQ16m-3. It should be checked whether opacity plays a role, as the values of r0 and r1 depend on opacity. As indicated insection 2.1, particularly the Lyman a line, related to the transition n = 2 ~ n = 1 where n is the principal quanturn number, might exhibit some opacity in tokamak conditions. Opacity becomes appreciable at ground state densities of the order of 6 x 1017 m-3 [Hey94] and therefore the RTP plasma is clearly optically thin. If Te= IOeV is assumed insteadof lOOeV, tbe resulting n 1 is about a factor of 2 higher, showing that the calculation is not very sensitive to the temperature.
The found ground state density is the peak value of the emissivity. Over the same minor radius it varies by a factor of 4. The collisional radiative model can also be used to calculate the number of ionizations in the plasma, which is applied in the next subsection to calculate the partiele confinement time. In the model used to give the coefficients for Eq. (2.9) and in the assumptions used to link the hydrogen density to the emissivity, atomie hydrogen is considered to be the only contribution to the Ha radiation. In subsection 7.1.2.6 the possible influence of processes of molecular hydrogen is considered.
7.1.2.5 Partiele confinement time
The partiele confinement time rp can be derived from
N rP = dN'
r--dt
(7.2)
where Nis the total number of H+ i ons in the plasma and ris the total influx of H+ i ons. For hydrogen the number of ionizations due to electron collisions occurring per emitted Ha pboton is approximately a constant close to 10 forTe> 10 e V at the densities in the region where most Ha light is emitted [JohH73] (see Fig. 2.3 and the discussion in subsection 2.1.2). Therefore, the total flux of neutral hydrogen entering the plasma is proportional to the total number of Ha photons emitted by the plasma, and the partiele confinement time can be deduced from the total emitted Ha light [StaM90, Cohe86]. For small Zeff the number of H+ ions is approximately equal to the total number of electrons.t In the case studied, the rate of change in the number of i ons or electrons, dN/dt, was very small and could be neglected.
An assumption in the calculation of rp from the Ha emissivity is that all atoms emitting Ha light actually enter the plasma, whereas theemission from the SOL comes from atoms that might not enter the plasma. Therefore, only the emission from inside the limiter radius should be taken into account. Tomography resolves the Iocation of tbe emission and therefore yields sufficient information to determine theemission from inside the limiter radius, which is not possible if
t For example, if Zerr= 2 and c6+ is the only impurity, then ne = 1.04 nH+. For heavier impurities the
influence is smaller, so that the 4% error made by assuming ne = nH+ is a reasonable upper limit.
156
Ha emission 7.1
only a few chordal measurements are available as is the case on many tokamaks. The emission fram ins i de the limiter radius is 60% to 70% of the total.
Assuming tora i dal symmetry, the total Ha power radiated fram the plasma inside the limiter
radius in the discharge under consideration is 30 W, which with 10 ionizations per photon cor
responds to an influx of araund 1021 atoms/s. With a tata! number of 9 x 1018 electrans, this
yields Tp = 10 ms, which is of the same order of magnitude as the energy confinement time
usually found in RTP (TE< 6 ms [KonH94]). The prabable increased emitted power at the
limiter (see Fig. 7 .6) will decrease Tp. Estimates of this effect are difficult to make because no
information is available on the magnitude and ex tent of the enhanced radiation close to the lim
iter, but with the estimated values in subsectien 7 .1.2.3 Tp is likely to be between 20% to 50%
times shorter. Furthermore, other processes such as dissociative ionization of H2 molecules
should be taken into accountfora proper deterrnination of Tp [StanM90], which is discussed in
the next subsection.
The found value of Tp in RTP is in reasanabie agreement with the values found in other toka
maks [TFR75, Kurz95]. It can further be validated by camparing the influx with the neutral
hydragen density. The tata! influx of neutral hydragen is equal to
Tin= nHvHS, (7.3)
where S is the surface through which the neutrals enter, i.e. the surface of the plasma column
(S ""4n:2R0a). Assuming nH"" n1, and using the !in from the Ha measurements and an aver
age n 1 (subsection 7.1.2.4), VH can be calculated to be around 104 rn/s, which is a reasanabie
value.
7.1 .2.6 Processes contributing to Ha emissivity
So far, it has been assumed that hydragen atoms are only excited by electron collisions with
atomie hydrogen. Most of the particles entering the plasma are, however, hydragen molecules.
The H2 molecule is lost by the following processes [McNB84, McNe89]
H2+e ~H*+H+e (dissociative excitation), (7.4a)
H2 +e ~ H* +H++ 2e (dissociative ionization), (7 .4b)
H2 +e ~Hi+2e (ionization), (7.4c)
and
H2+ W ~ Hi+H* (charge exchange), (7 .4d)
where H* denotes a product atom that may be excited and therefore may emit Ha radiation.
Electron impact on H and Hi, and charge exchange of H give the other channels to excited
hydragen atoms:
H+e ~ H*+e
H!+e ~ H*+H
(atomie excitation), (7 .4e)
(dissociative recombination), (7 .4f)
157
Chapter 7 Measurements of stationary asymmetrie emission profiles
H! + e -t H* + H+ + e (dissociative excitation), (7 .4g)
and
(charge exchange). (7 .4h)
Atomie excitation [Eq. (7.4e)] is the process that has been considered so far. Note that because
of dissociative ionization of either H2 [Eq. (7.4b)] or H! [Eqs. (7.4c) or (7.4d) followed by
Eq. (7.4g)] a part of the intlux never has the opportunity to give rise to Ha emission, contrary
to what is assumed if only atomie excitation and ionization are taken into account. McNeill et al.
[McNB84] claim that other processes such as radiative recombination, dissociation of Hj and charge exchange are unimportant as sourees of H*. Although the molecular charge-exchange reaction of Eq. (7 .4d) has a small cross-section according to the older literature [FreJ7 4 ], recent experiments show that the cross-section of charge exchange with vibrationally excited H2 might be significantly larger and could give a contribution. However, this finding only affects the
branching ratio of the production of H!, but not significantly the number of emitted Ha photons per entering molecule. Molecular hydragen recycling from the wall might not be vibrationally excited befare it undergoes one of the reactions of Eqs. (7.4a-d) [McNe89] , unlike
hydragen that is puffed into the vessel.
The relative importance of the reactions Eqs. (7 .4a-h) is determined from camparing their
cross-sections. While most cross-seelions have been tabulated [JanL87, FreJ74, Jone77], some cross-seelions for the reaction that yields H* in the n = 3 state are not known, and therefore have to be scaled from states for which the cross-seclion is known. Recent experiments, for example in starage rings, have given more accurate cross-seelions for collisional processes of
vibrationally excited H! [Mitc95] and branching ratios of the final hydragen states after dissa
cialive recombination of H! [ZajA95]. McNeill et al. [McNB84, McNe89] have estimated the
relative contributions of the various processes to Ha radiation. It is found that about 2/3 of all H2 becomes H!, but excitation of product H atoms remains the main souree of Ha: only about 20% of the Ha photons are produced by electron impact on molecules (H2 and H!). This seeming discrepancy can be explained as fellows. Apart from dissociative recombination and excitation of H!, also dissocialive ionization (H! +e-t H+ + H+ + 2e ) neects to be taken
into account for the total Ha production, because it reduces the number of excited hydragen aloms. As indicated in subsection 2.1.2, the molecular processes influence the r coefficients in Eq. (2.9) because they give rise toother channels to excited atomie hydragen that arenottaken into account in the Johnson and Hinnov calculations.
While the molecular processes have no dramatic consequences for the total Ha production because most reaelions lead to H*, they can have significant consequences for the location of Ha emission. The Iifetime of H! derived from the cross-seelions of the dissociative processes is of
the order of several microseconds. Therefore, the charged H! has sufficient time lo be transporled along the field lines in the SOL or plasma and give the Ha radialion in a completely different location than where the H2 molecules enter the SOL. This influences the understanding
158
Ha emission 7.1
of the asymmetries in the Ho: emission. Furthermore, it could explain the non-exponential
decay in Fig. 7.6: insteadof being caused by recycling on other surfaces than the limiter, it
might be related to the transport of Hi i ons from the limiter.
De pending on the amount of recycling, which differs between tokamaks, the production of Ho:
photons, taking into account the molecular processes, is 0.04 to 0.08 photons per entering
molecule [McNB84, McNe89]. In these calculations charge exchange [Eq. (7.4h)] is taken into
account, which contributes about 30% of the radiation. Whereas most non-charge-exchange
radialion is emitted outside the last-closed flux surface, which is in agreement with the findings
in subsection 7 .1 .2.2, the charge-exchange radiation is emitted al most entirely some centimetres
inside. The values found are not highly sensitive to the ne and Te profiles. The dependenee on
recycling, in particular on the sticking (recombination) probability of atoms on the wal!, results
from the inclusion in the model of the fact that a part of products from H2 do notmove toward
the plasma interior, but towards the wal! where they might either stick or recycle.
The results from the inclusion of the reactions Eqs. (7.4a-c) and (7.4e-h) can be compared with
the previous calculations which used the Johnson and Hinnov model. The number of Ho: pho
tons produced per entering molecule derived from the molecular model (0.04 to 0.08), corre
sponds to 0.02 to 0.04 photons per produced proton (each molecule is the souree of two pro
tons), i.e. 25 to 50 atomie ionizations per emitted Ho: photon. This is higher than the 10 ioniza
tions per emitted pboton predicted by the Johnson and Hinnov model, which means that the rp found in subsection 7 .1.2.5 is somewhat too large. Given the accuracy of the roodels and the
measurements, these deviations are of limited influence on the interpretation of the measure
ments. Nevertheless, they indicate that it is likely that rp < re, which is only possible if the
partiele fluxes are large only at the edge, while the central plasma transport is dominaled by heat
conduction. The ground state density of atomie hydrogen calculated in subsection 7 .1.2.4
assumes hydrogen to enter the SOL and plasma as atoms. However, in reality molecular hy
drogen is the dominating species flowing into the SOL and plasma and it is the souree of atomie
hydrogen. Therefore, it is not very useful to compare the atomie hydrogen density calculated by
the molecular model with the one of the atomie model. From the model that takes into account
molecular processes and the density of excited state n = 3, which is directly derived from the
measurement, the H2 and H densities can be determined if transport processes are taken into
account. This has not been attempted.
Further processes that might require consideration are gas puffed into the vessel and charge
exchange of atomie hydrogen with impurities. It bas been suggested by McNeill [McNe89] that
molecules recycled from the walt are in their ground vibrational state, whereas during gaspuff
ing (a gas jet) H2 is vibrationally excited. This increases the rate of electron-impact dissociation
into ground state hydrogen atoms, whereas the rate of dissociation into excited states changes
only little [McNe89]. Thus, dissociative excitation into excited states and molecular ionization
both become less probable compared to dissociation into ground state atoms, resulting in less
159
Chapter 7 Measurements of stationary asymmetrie emission profiles
Ha emission than in the case of no jet. The Ha emission from pellets of hydrogen ice injected into the plasma originates from a plasma cloud surrounding the pellet which is in local thermal
equilibrium, for which very different models apply [McNe89]. This topic, however, is beyond the scope of this thesis. The amount of emission can also be influenced if charge exchange [Eq. (2.10)] of atomie hydrogen with impurities, for example C3; is significant. Such a process
reduces the number of hydrogen atoms that are available for excitation, consequently reducing
the emission. lt increases the ratio of ionizations to the number of emitted Ha photons, and
therefore reduces 'l"p further. The cross-sections for such reactions are of the same order as
electron collisional excitation and effects of several percent can be expected for impure plasmas.
The details of such reactions depend on the amount of impurities and the distri bution of the various ionization stages. Due to lack of knowledge on these details and the limited importance
in relatively clean plasmas, charge-exchange with impurities was not taken into account in the present study.
7.1.3 Start-up of discharge
The start-up phase of a discharge is the period in which the plasma is formed and the plasma
current is ramped up. During the formation of the plasma the neutral hydrogen gas is heated and radiates much Ha light: usually a sharp peak is observed during the first milliseconds of the
discharge. Because the neutral hydragen density is uniform at the beginning of the start-up
phase, a reasonably uniform emission profile can be expected. Measurements of this process have been described, for example, by the TFR group [TFR75]. In some cases in RTP an
asymmetrie distribution of emissivity is observed during the start-up phase.
As in subsection 7 .1.1, a wide variety in magnitude of emissivity occurs, but the shapes of the
emission profiles seem to be very distinct for the two current ramp-up rates of current that have
been studied: 3.9 MNs and 2.1 MA/s. Figure 7.7 shows time traces of representative channels for both classes of start-up behaviour. The fast ramp-up [Fig. 7.7(a)] gives a much faster increase and decay of the Ha emission than the slow one [Fig. 7.7(b)], and different channels
exhibit different temporal behaviour, whereas for the slow ramp-up all channels increase at approximately the same rate. After the Jatter observation it is not surprising that the emission profile for the fast ramp-up is very asymmetrie at the beginning and changes shape, whereas for
the slow ramp-up it grows symmetrically. Figure 7.8 shows the tomographic reconstructions at several points in time forthefast ramp-up, and Fig. 7.9 shows the reconstruction at one time for comparison (at other times it is roughly the same) for the slow ramp-up.
160
Ha emission 7.1
5
4 30 -~
E ...... 3 20 Cll "0 -~ 2 ~ (ij c 10 Cl ëii
ö I 0 0
-1 0 2 4 6 8
5
4 30 ~-E
...... 3 20 Cll "0 -~ 2 ~ (ij 10 c Cl ëii
ö I 0 0
-1 6 8 10 12 14 16
time (ms)
Figure 7.7 Time traces of two representative Ha signals of the visible--light tomography diagnostic (solid and
open circles correspond to channels AIO and 824, respectively) for (a) a fast and (b) a slow current ramp-up. The
vertical lines indicate the points in time for which tomographic reconstructions are given in Figs. 7.8 and 7.9.
The plasma current is also given ( dasbed line).
The tomographic reconstructions of the fast ramp-up by both the IPR and CO methods show
that the Ha emission starts at the upper inside of the plasma region and spreads out to fill the
entire plasma roughly uniformly when the emission peaks. For the slow ramp-up approxi
mately the same emission profile is obtained as during the peak of the fast ramp-up, but the shape remains the same during a few milliseconds. The electron density shows similar
asymmetries as the Ha emission, although the interferometer diagnostic can only distinguish a higher density on the HFS in the fast ramp-up case and reasonably symmetrie profiles in the slow ramp-up case. In the fast ramp-up the density increases faster than in the slow ramp-up
161
Chapter 7 Measurements of stationary asymmetrie emission profiles
Figure 7.8 Tomographic reconstructions of the Ha emissivity by the IPR (a---<l) and CO (e-h) methods of the
fast current ramp start-up phase of a discharge. The contour levels do nol represent the sameabsolute levels in
the different reconstructions due to the large variation in absolute emissivity during the start-up (see Fig. 7.7).
case, but at a slower rate than the Ha emission. In the fast ramp-up case the electron density
profile is flat from the LFS edge of the plasma to R"" Ro + a/2 and falls off steeply for larger R. After the peak, the Ha emissivity shows a transition to the asymmetrie steady-state emissivity profiles described in subsections 7 .1.1 and 7 .1.2.
For the fast ramp-up a second increase in emissivity is observed [around t = 8 ms in Fig. 7 .7(a)]. Th is increase does not appear in discharges with a slightly slower ramp-up. The effect
of this second peak on the shape of theemission profile is smal!. During the second increase in the Ha emission the electron density profile changes shape drastically and is peaked around R"" Ro- a/2. After this time both the density profile and the Ha emissivity reach their steady state values. Apparently the way the plasma is formed has little influence on the emission pro
file during the current plateau phase of the discharge.
The total number of ionizations can be calculated from the total Ha emission in the same way as in subsection 7 .1.2.5. Because the temperature in the starting plasma can be assumed to be higher than lOeV [TFR75] the number of ionizations per emitted Ha photon is close to 10 according to the Johnson and Hinnov model. Equal total Ha emissivity in each poloidal crosssection is assumed for the calculation of the total number of Haphotons emitted, which seems reasonable in the start-up phase. Figure 7.10 shows the total number of electrous Ne in the
torus, as determined with the interferometer, the rate of change in the number of electrous dNJdt and the total ion production rate. The number of ionizations and dNeldt are equal within a factor of 2, which indicates that the accuracy of the determination of number of ionizations is
162
Ha emission 7.1
Figure 7.9 Tomographic reconstruction of the Ha emissivity by (a) the IPR and (b) CO methods during the
slow ramp-up start-up phase of a discharge.
of this order since dNJdt cannot be larger than the number of ionizations. The discrepancy dis
appears if the molecular processes discussed in section 7 .1. 2. 6 are taken into account, because
more ionizations are predicted. After t = 3.5 ms the partiele confinement time calculated from Eq. (7 .2) has a reasonable value of 3 ms and rises to the value calculated in subsection 7.1.2.5.
These findings are in agreement with Ref. [TFR75].
During the start-up phase of the discharge the plasma is not wel! confined because the magnetic
flux surfaces have notbeen fully created yet. Therefore, most electroos are lost, mak.ing a high
creation rate necessary (yielding the peak in Ha ernission). The asymmetry can be explained by
00- a. z
0 2 2 lb-~ dN /dt 6 e""- 6 ' en :J c i'j' 0
~ ...... tî ionizations 6 " Q) - :J (ii
~-/ CJl -0 x ......
:';, ' Q) ' .0 ...... 0 E
. ·- .. ' . .. .... 0 ~ 0 ::::l CJl c .!... -0 2 4 6 8 10
time (ms)
Figure 7.10 Total number of electrans (solid line), rate of change of the number of electrons dNe/dt (dotted
line) and the number of ionizations derived from the total Ha emission (dashed line) for the discharge in Fig.
7.7(a) .
163
Chapter 7 Measurements of stationary asymmetrie emission profiles
this bad confinement the stray fields have a large influence on the plasma distribution at the start of the discharge. The slow ramp-up seems to give a better start-up of the plasma, but it is
not: the losses are much higher and the breakdown (i.e. when the current starts flowing) takes place much later. Visible-Iight tomography might help to optimize the stray fields to obtain a more symmetrie and more efficient start-up.
7.1.4 Summary of Ha measurements
The spatial distribution of Ha emission has been derived from absolutely calibrated measurements. A good agreement is found with measurements on other tokamaks and model calculations in the literature. The results of the calculations with the Johnson and Hinnov model are not altered drastically if all significant processes giving rise to Ha radiation are taken into ac
count. The measurements during the start-up phase of a discharge indicate that those corrections are necessary to obtain consistency. The channels of Ha production, from H2, Hor H!, do not
differ much in their yield, but the portion that goes through the H!, which is estimated to be
20%, can have an influence on the location of Ha emission quite different from the other channels. The high spatial resolution of the visible light tomography system is unique and therefore
provides a valuable extension to results in the literature. In particular the detailed determination of asymmetries could have important consequences for the interpretation of measurements of
Ha emission in the literature, where usually symmetry is assumed. Due to the presence of the
dis ti net asymmetries, even the coverage of the visible-light tomography system on RTP is only
sufficient to resolve the main structures. Fora better quantitative understanding of the asymmetries, a higher resolution, more edge diagnostics and more spectroscopie information would be
advantageous. Measurements of other hydrogen lines, for example, would give in formation on
the population of the energy levels, which could give more insight in the popuiatien meehanisrus and verify the model used to calculate the ground state density of neutral hydrogen. It
could also be useful to study the asymmetries in other working gases, such as He. Possible causes for asymmetries are discussed in more detailinsection 7.4.
7.2 Continuurn emission
Bremsstrahlung is the second contribution to the plasma emission in the visible range of the
spectrum. It has a dependenee on plasma parameters different from that of line radialion of hydrogen and impurities, and can therefore be used to determine other quantities. It is continuurn radiation, hence any suitable part of the speetral range to which the detector is sensitive can be used to measure it.
In this section the determination of Zeff from continuurn measurements is discussed. Again
significant asymmetries are observed in most cases, whereas a peaked smooth profile was
164
Continuurn ernission 7.2
expected from the dependenee on the plasma parameters. Possible explanations for these
asymmetries are given.
7.2.1 Measurements of continuurn radiation
For the measurement of continuurn radiation preferably a part of the spectrum should be chosen
in which contributions from line radiation are negligible. In the literature a narrow window (0.5
to 3 nm) around, for instance, the wavelength À= 523 nm is used [Kado80, SchS88,
GuiM94]. There are indications that even in such narrow wavelength rangesthereis a signifi
cant contri bution from molecular quasi-continuurn radiation and that therefore the detennination
of Zeff from an absolutely calibrated line-integrated measurement is doubtful. The advantage of
tomography is that it can determine the emission in the centre without being completely dis
torted by contributions in the edge. With a narrow filter as is usually used for visible
bremsstrahlung measurements the visible light tomography system on RTP would not receive a
sufficient amount of light. Furthermore, a speetral overview of the light emitted in RTP was not
available, so that such a narrow wavelength range could not be chosen properly. Forthese rea
sons a large wavelength interval was chosen: 695 nm < À< 1200 nm. This interval does not
contain the most important hydragen Balmer lines which account for the main emission, and
most impurity lines are expected to have shorter wavelengths. The maximum wavelength was
limited by the sensitivity of the detector, while the minimum wavelength was limited by a
coloured-glass filter. This filter is referred to as continuurn filter. The main advantage of the
filter is that a sufficient amount of light is received and that the importance of individual speetral
lines to the total signa! is reduced. The calibration of the system witb this filter was discussed in
subsection 5.6.4.
A number of discharges has been diagnosed with the continuurn filter. Most radiation is ex
pected to originate from Bremsstrahlung [Eq. (2.12)], which for high temperatures gives an
emissivity proportional to nlt-ffe. If the density and temperature profile shapes are the same
for all discharges that are compared, it is possible to replace the emissivity in this proportional
ity by the line-integrated emissivity, the density by the line-integrated density, and the tempera
ture by the maximum temperature in the plasma. Figure 7.11 shows this dependence. The
channel displayed in Fig. 7.11 is chosen such as to circumvent the peaks at the edge found in
the tomographic reconstruction. All channels exhibit roughly the same features, but because
many look through a highly radiating region at the edge the approximate abso1utely measured
power cannot be calculated by Eq. (2.12). Figure 7.11 shows that above a threshold value the
line-integrated signa! indeed is roughly proportional to [fnedZYNTe,max. which is indicated by
a line through the origin fitted to the points. The puffing of Ne during the discharge seems to
influence neither the magnitude nor the shape.
165
Chapter 7 Measurements of stationary asymmetrie emission profiles
";"-10 N
I C)J E 8
";"
Ui 5: 6 •
IJ)
";" 0
4 • (ij •• c .21 2 • Cl)
• • without Ne
"0 o with Ne Q)
(ij 0 ()
Cl) 0 1 2 3
[f ne dl ]2 j ~Te,max (1 037 m -5 e y-1/2)
2 Figure 7.11 Line-integrated signals of a typical detector (channel 052) as a function of [fne dl] ;,) Te,max for
several discharges, at one time in each discharge. The measured power has been calculated from Eq. (5.10), as
suming the emissivity to be independent of the frequency, which wil! give correct values after tomographic in
version for the centre of the plasma. The line is a fit through the origin and the points above 0.5 x J037 m·Sey·I/2.
An interesting observation can be made if profile consistency between the plasma parameters
[Kado87] is assumed to be valid. The bremsstrahlung emission is proportional to Zeffn~!T12 which can also be written as Zeff ne (ne Te)!f3/.2 The plasma pressure Pe is given by ne Te, and the current density by j = E/1] = VL f321Zeff, where Eis the electric field, VL the loop voltage
and 1J the plasma resistivity (a weak nonlinear dependenee of 1J on Zeff has been neglected).
Therefore, the bremsstrahlung emission profile is approximately proportional to ne(r) Pe(r)/j(r).
Due to profile consistency the ratio Pe(r)lj(r) is constant for given plasma parameters. There
fore, the bremsstrahlung emission profile is proportional to the electron density. Because the
absolute value of the ratio Pe(r)/j(r) depends on the plasma parameters, for example on the con
finement time and on Zeff, rewriting the bremsstrahlung formula to a proportionality to
ne(r) Pe(r)!j(r) is mainly helpful to derive the density distribution from the shape of theemission
profile, but not the absolute value of the density.
7. 2. 2 Tomographic reconstructions and determination of Zerf
The tomographic reconstructions of a time slice of a medium-density discharge are shown in
Fig. 7 .12. For most discharges reeonstruc ti ons with si mi lar shapes are obtained. Both recon
struction methods show approximately the same asymmetries, outer-upper side, inner-upper
side and outer-lower side, which are oversmootbed by the IPR metbod whereas the CO metbod
gives too high values at the edge. In the centre both methods give ( 1.8 ± 0.4) x I o- 14
166
Continuurn emission 7.2
( b ) 0.19
I 0.0 I N N
-0.19+---__l __ ,--_J...._ __ --l -0.19+----L--'--,.--------l
-0.19 0.0
R(m)
0.19 -0.19 0.0
R(m)
0.19
Figure 7.12 Tomographic reconstruction of the conlinuum emissivity by the (a) TPR and (b) CO methods in a
medium density discharge. The emissivity has been absolutely calibrated on the assumption of constant contin
uurn radialion in the wavelength region viewed; each contour corresponds to 2 x JO-Is W m-3 sr-I Hz-I .
W sr- 1 m-3 Hz-I for this specific example. The Zeff calculated from Eq. (2.12) using the
local values of the electron density and temperature ne = 8 x 1019 m-3 and Te= 0.6 keV as determined by the interfere meter and Thomson-scattering diagnostics, respectively, is
Zeff = 4 ± l. The gaunt factor g ff= 3.2 ± 0.2 [KarL61] has been used in this calculation (see
section 2.3). RTP does nothave a routine measurement of Zeff; for camparisen it therefore has to be derived from other measured quantities.
A methad to calculate the central Zeff for comparison is from the resistivity. Using the Spitzer resistivity in the centre of the plasma [Wess87] (no neoclassical correction is needed in the cen
tre because there are no trapped particles) and deriving the current density from the safety factor
qo in the centre, Zeff is 312
Z = 60 6 VL qo Te(O) eff · '
B~ N(Zerr )In A (7 .5)
where Te is in keV and the other quantities in SI units. In Eq. (7.5) VL is the loop voltage in the centre (the average value of the loop voltage measured on the HFS and LFS is taken), Bq> the
tomidal magnetic field, In A is the Coulomb logarithm and N(Zeff) the Spitzer-Härm correction for non-hydragenie plasmas. Equation (7.5) has been validaled fora wide range of Zeff in RTP
by Konings [Koni95]. The value for q0 is found from a current density profile calculation based on the Te profile measured by Thomson scattering.
The value of Zeff found from the continuurn radialion is a factor of 2 to 4 higher than the one found by other methods. Similar results for Zeff, i.e. a factor of 2 to 4 too high, in the centre are obtained for points close to the line drawn in Fig. 7 .11. This clearly indicates that the emissivity in the centre determined with the visible light tomography diagnostic for a large part is bremsstrahlung and that the diagnostic gives reasanabie results. Other discharges that do not lie
167
Chapter 7 Measurements of stationary asymmetrie emission profiles
close to the line indicated in Fig. 7.11, which is mainly at low densities, yield Zeff"" 10 from the tomographic reconstruction, whereas the Zeff deterrnined by other methods remains low. This is remarkable in the discharges where Ne was puffed into the plasma. For some dis
charges the tomographic reconstruction has different features than the ones for points close to the line in Fig. 7.11, but there is no indication that artefacts different from those in Fig. 7.12
are present that could explain a too high value.
Assuming that mainly bremsstrahlung is ernitted from the central part, which is required to
explain the measured ernissivity, a reasonably smooth emission profile is expected because of
the smooth electron density and temperature profiles (Fig. 7.13). It is very unlikely that the Zeff
profile is strongly asymmetrie, since consistent results between density and soft x-ray meas
urements have been obtained fora symmetrie Zeff profile [Cruz93]. Therefore, the asymmetries
in the central parts of tomographic reconstructions must be due to artefacts: the low regions close to the centre have ernissivities lower than the bremsstrahlung ernission that corresponds
with the local density and temperature for Zeff= 1. The artefacts are most likely caused by the strong ernission at the edge that cannot be reconstructed properly and therefore gives rise to artefacts. This rnight also explain the too large value found in the centre. However, it is remark
able that both reconstruction methods yield sirnilar artefacts. When the line-integrated measurements are compared with line-integrals of the calculated bremsstrahlung profile for the meas
ured ne and Te profiles and several values of Zeff, it is found that most channels see more non-
8
6 '?-E
a> ~0 4
Q)
c 2
600
400 (1) -i
200
limiter radius
L-~~---L--~--~~--~--~--~~~o 0 -200 -100 0 100 200
minor radius (mm)
Figure 7.13 Electron density and temperature measured by the interferometer and Thomson scattering, respec
tively. The temperature is measured along a verticalline, whereas the density is obtained from Abel inversion of measurements along parallel verticallines. The profiles show no indication of asymmetries that would invalidate
the application of Abel inversion and the plotting of both signa Is along the same axis (minor radius).
168
Continuurn emission 7.2
bremsstrahlung than bremsstrahlung. If Zerr is assumed to be constant in the plasma, it is found
that the maximum Zeff consistentwithall measurements is Zeff"' 3, for which value the meas
ured signals for some channels would be caused entirely by bremsstrahlung. Therefore, it
seems plausible that indeed the artefacts of the reconstruction slightly raise the reconstructed values in the centre. Due to the occurrence of significant artefacts in the reconstructions it is
meaningless to attempt to calculate a Zeff profile, despite the fact that the assumption about
symmetrie ne and Te profiles is justified (Fig. 7.13).
It is relevant to not only campare the ernissivity in the centre, which is a factor of 2 to 4 higher in the measurement than can be explained from bremsstrahlung, but also the total ernission. The total radiation is a factor of 3 to 5 higher than what would be expected from the measured den
sity and temperature profiles if it were pure bremsstrahlung. This is a larger deviation than in the centre, and can therefore be ascribed to the radiation at the edge. For the discharges with
lower densities the total radiation is increasingly higher than the calculated bremsstrahlung. This
is in agreement with the deviation in Fig. 7 .11. In the low-density discharges there is a signifi
cant number of suprathermal e1ectrons. Equation (2.12) for the bremsstrahlung ernission is only valid for Maxwellian velocity distributions. An accelerated electron can radiate photons with
energies up to its own energy, hence suprathermal electroos also contribute to the radiation in
the visible range. However, the suprathermal electroos have a density that is several orders of magnitudes smaller than the density of thermal electrons. Furthermore, the (non-Maxwell-aver
aged) gaunt factor is only a weak function of energy and frequency. Therefore, the suprathermal electroos are only likely to contribute a significant portion of the bremsstrahlung at frequencies corresponding to the energies higher than the thermal energy. The curved pathof the suprathermal electroos in the magnetic fields causes them to ernit synchrotron radialion as well,
which is mainly in the microwave and infrared speetral regions. Only for very energetic electroos visible radialion is emitted, which, however, for the relativistic veloeities is ernitted in a
narrow cone in the direction of motion (i.e. tangentially). Because the visible light tomography system views the plasma poloidally and the imaging system in positioned in the narrow ports, it
is unlikely that the synchrotron radialion is measured. It is more probable that the high Zeff val
ues found at low densit1es are caused by artefacts due to the high edge emission than by effects caused of the suprathermal electrons. Because the bremsstrahlung scales with n; whereas line
radiation scales with ne. at Jow densilies the radialion from the edge is relatively larger than the
bremsstrahlung.
The large amount of radiation at the edge cannot be bremsstrahlung. The posilion, and the simi
larity with the asymmetries in the Ha emission profiles, suggest the radialion to be line radiation. No speetral overview is available in the speetral range observed with the continuurn filter.
It is likely that the radialion is of molecular origin, i.e. from emission of excited H2. because
the shapes of the asymmetries are roughly similar to those of the Ha emission. For molecular radiation the radialing edge layer can be expected to be thinner than in the case of Ha. With the Abel inversion metbod used for Fig. 7.5 the thickness of the layer has been calculated, and
169
Chapter 7 Measurements of stationary asymmetrie emission profiles
indeed it is more narrow than 3 cm: for r < 0.15 m there is hardly any contribution at all. Other
possible sourees of radialion are speetral !i nes of atomie hydragen and impurities. In the wavelength range of the continuurn filters no intense speetral I i nes of impurities are expected because they must be in very high excitation states to radiate such long wavelengths. Hydragen only emits a few Paschen lines at wavelengtbs in this range. Using the collisional-radiative model
Eq. (2.9) and the ground state density found in subsection 7.1.2.4 it can be calculated that these
hydragen lines Logether emit less than 1% of the power of the Ha.line, which is insufficient to
explain the radiation at the edge. No such estimate is available for impurities, and therefore
these cannot be entirely discounted. Theemission at the edge might also be caused by insufficient rejection of the Ha. light by the filter ( transmission < lQ-2% ), which for low densities becomes relatively more important compared to bremsstrahlung. It would be valuable if the
measurements were repeated in He plasmas, with appropriate filters not viewing He lines, because the edge radialion would not be present if it is caused by hydrogen. Furthermore,
because Zeff is higher, more emission would be obtained at the sameelectron density.
7.2.3 Summary of continuurn measurements
The measurements of bremsstrahlung are complicated by a significant amount of asymmetrie non-bremsstrahlung at the edge of the plasma. Given the artefacts due to radlation at the edge
and the relatively large error bars, the reconstructed values of the bremsstrahlung emission in
the centre are reasonable, except for plasmas with a low density. The findings are in agreement with studies of Zeff in the visible range in the literature (see subsection 1.2.2.3 for references),
which usually find an increase of emission at the edge, even when very narrow filters are used.
The large asymrnetries and artefactsin the reconstructions of the present study, however, might
partly be due to the large bandwidth of the filter, and due to the smal! si ze of the tokamak with
a, therefore, relatively more influential edge region.
Reasonably symmetrie emission profiles have been observed during the first millisecond of discharges (see subsection 6.2.2). However, the emissivities in these instances seem too large to be accounted for by solely bremsstrahlung. At these conditions the temperature and density
measurements are unreliable and noisy, and therefore it is difficult to make definite statements.
7.3 Total visible emission
Many measurements taken on RTP have been without optica! filters: the total radialion in the
entire speetral range to which the detectors are sensitive is measured. It is not possible to do an absolute calibration in this case, unless the relative importance of the different contributions is known: in the inlegration over the speetral range of the detectors the speetral sensitivity of the detector has to be taken into account. The measurement of the total visible emission is useful to visualize emitting structures in the plasma and to obtain a sufficiently high signal-to-noise ratio
170
Total visible radiation 7.3
to properly measure fluctuations, although for a good physical understanding speetrally resolved measurements are needed. In this section measurements of the total emission are discussed and a comparison is made of the various contributions.
7.3.1 Plasma position dependenee of total radiation
The shapes of emission profiles measured in the entire visible range under normal circum
stances are very similar to the ones of Ha, only the magnitude is larger and the central value is raised due to the bremsstrahlung and the width of the radiating layer at the edge is thicker. Figure 7.14 shows reconstructions by the IPR method for different vertical shifts during the same discharge. An upward displacement over a distance allO [Fig. 7.14(a)] hardly influences the emission profile [cf. Fig. 7.2(a)], whereas a similar downward displacement [Fig. 7.14(b)] alters theemission profile significantly. The results for more extreme displacements (up to a/5)
are similar. The significanee of these findingsis discussed insection 7.4.
7.3.2 Comparison of the contributions from different wavelength ranges
To interpret the total emission it is important to know the relative importance of the different contributions. The measurements with the Ha and continuurn filters can be used to determine the contribution of those speetral ranges to the total. The comparison has been made for some typical channels (looking through the central part of the plasma) and for the total emission as determined from the tomograms. The absolute values of the Ha measurements in Figs. 7.1 and 7.2 and the continuurn measurements in Figs. 7.11 and 7.12 can be compared ifthe latter are integrated over the bandwidth of the continuurn filter: 2.5 x 10'4 Hz to 4.5 x IOl4 Hz (see
Fig. 5.14).
To estimate the contribution of the total continuurn radiation from an extrapolation of the measurements with the continuurn filter some assumptions are needed because little other speetral
Figure 7.14 Tomographic reconstruction by the JPR method of the .emissivity in the entire speetral range to
which the detectors are sensitive for two different plasma positions: (a) !!.Z = + 15 mm and (b) !!.Z = -18 mm
(both corresponding to approximately 1110 or the minor radius).
171
Chapter 7 Measurements of stationary asymmetrie emission profiles
information is available. Assuming the emissivity to be independent of frequency, as is approx
imately the case for bremsstrahlung, the sensitivities obtained in subsection 5.6.4 can be used
for the range of the continuurn filter and the complete spectrum. The former yields the sensitiv
ity (U/Pv)cont.filter [Eq. (5.10)], the latter (U/Pv)total [Eq. (5.11)).t Here, U is the output sig
na) of the detection system, an Pv the incident light power per unit frequency . Consequently,
the contribution of continuurn radialion in the entire speetral range is expected to be
(U!Pv)totatl( U/Pv)cont.filter"' 4.6 times the one measured with the continuurn filter. In subsec
tion 7.2.2 it was shown that this assumption cannot be justified because only 20% to 40% of
the total radialion measured with the continuurn filters can be accounted for by bremsstrahlung,
but the estimate can at least be used to gain insight in the relative importance of different contri
butions to the radiation. For the Ha contri bution such an estimate is not needed, provided the
Ha line is the only line contributing in the bandwidth of the filter, which was justified in sub
section 7. l.I.
To campare the measurements with Ha. continuurn and no filters, discharges with similar properties are needed. Because the filters cannot be changed on a shot-to-shot basis the dis
charges considered are not entirely reproducible. A statistica! study of the visible light tomogra
phy measurements on many similar discharges has shown that even for similar discharges, i.e.
similar plasma parameters, significant differences in visible emission can occur. Therefore, the
comparison of the different filters is necessarily on single discharges and not on an average
over many discharges, and the comparison can only be rough. Although the measurements vary
considerably, the major shapes of the tomographic reconstructions are usually similar for the
widely different plasma conditions, as was discussed for Ha measurements in subsection
7 .1.1. Some discharges with si mil ar den si ties and temperatures were chosen for the compari
son. The comparison of the total emissivities calculated from the tomograms showed that the
con tribution of Ha is roughly two times that of the continuurn radiation. It also appears that the
measured Ha radiation and estimated total continuurn radialion tagether approximately account
for the total measured radiation. The comparison of individual representative channels yielded
variations on this conclusion, for example Ha and continuurn tagether resulting in more than
the measured total radiation, probably due to the dependenee on the position of asymmetries
and the larger sensitivity to small changes in plasma conditions of individual channels than the
total emissivity.
As follows from the absolute emissivity measurements with continuurn filters described in sub
section 7 .2.2, a significant part of the radiated power observed is emitted at the edge and cannot
t In Eqs. (5.1 0) and (5.11) the reflectivity of the mirrors and other Jasses are taken into account. In the case of
Ha radiation, the calibration factor [Eq. (5.9)) and corrections for the filter transmission and other losses have
been separated because the corrections vary between detectors. Proper care of these differences has been taken
when camparing the quantities of Ha and continuurn measurements.
172
Total visible radiation 7.3
be bremsstrahlung. Therefore, the approximations used to estimate the total continuurn emission are rather rough. The agreement in the comparison with the total radiation shows that the
contributions of radiation other than bremsstrahlung, i.e. line radiation and molecular radiation,
must give the same contribution in the window of the continuurn filter and the window for
shorter wavelengths. In the latter window significant contributions from impurity lines can be
expected. The collisional-radiative model of Eq. (2.9) and the found ground state density of
hydragen makes it possible to estimate the power emitted in the hydragen Balrner lines H~, Hy.
etc., that !ie in the visible wavelength range with wavelengtbs shorter than Ha (the Paschen
lines are negligible). The combined contributions of these lines can be expected to be about
10% of the power emitted in the Ha line.
7.4 Discussion on asymmetries
The asymmetries in the emission profiles are linked to asymmetrie distributions of neutral hy
drogen atoms and hydragen molecules, and, possibly, impurity ions. Because the souree of the radialion in the measurements discussed in this chapter is only certain in the case of Ha emis
sion, the discussion of asymmetries wil! focus on hydrogen. It is of great importance to understand the causes of the asymmetries in the emission profiles because they are linked to a nonuniform creation of ions, which will have a large influence on transport processes in the
edge. Asymmetrie emission profiles for line radialion from hydragen and light impurities in the
visible range, similar to the profiles described in this chapter, have aften been reported in the literature. Results from the literature are briefly reviewed in this section. Possible explanations for the asymmetrie emission profiles can be divided into two groups: explanations that require
the poloidal asymmetries to be toroidally symmetrie, and explanations that assume the cause of the asymmetry to be toroidally localized. Because no information on the Ha emissivity at other
toroidal positions is available in RTP, approximations such as in subsectien 7.1.2.3 are required. This, however, does not help to distinguish between the two groups. In this section possible explanations for the toroidally asymmetrie and toroidally symmetrie distributions are
discussed.
7 .4.1 Asyrnrnetries in the literature
Asymmetries in the edge of plasmas have been discussed in several review articles, see for instance Refs. [StaM90, Isle84]. Often, the asymmetries are ascribed toa localized source, for example the recycling of hydragen and impurities at the limiter. But also other localized sourees can exist outside the limiter radius. As pointed out in sec ti on 2.1, transport of i ons along the
field lines is not sufficiently fast to spread out the localized influx over the entire flux surface,
which is often given as the explanation for the asymmetrie emission profiles of hydragen and lower ionization stages impurities. Other effects, such as drift, are also given as explanations of
173
Chapter 7 Measurements of stationary asymmetrie emission profiles
the asymmetries. The causes for the asynunetries are studied in detail in the following subsec
tions; the emphasis of this subsection is on observations on other tokamaks.
On most tokamaks the edge diagnostics do not have a sufficiently high spatial resolution to study the asymmetries in detail. Therefore, the information available is usually only sufficient to indicate that there is a top-down asymmetry, e.g. in Refs. [BrauS83, TerM77]. More detailed spatial studies of the emission of speetral lines have been pursued by a number of authors, for
instanee Suckewer et al. [SucH78]. In general it is found that the distri bution of emission from
higher ionization stages are more symmetrie than that of lower ionization stages, which is in agreement with the faster parallel transport in the centre. Suckewer et al. [SucH78] found that in the PL T tokamak, apart from the dependenee on i anization stage, the emission for discharges
with low edge temperatures are more symmetrie than for those with a high edge temperature.
The positions of the asymmetries varied for different discharges and no clear single explanation was found. It has been found that the asymmetries of the impurities were opposite to those of
the working gas [SucH78], in the samedirection [Razu84] or varying [BraS83]. In many in
stances the direction of the asymmetries has been found to depend on the direction of the magnetic field. The direction of the asymmetries has been found to be in the direction of the ion V B
drift [Razu84] or in the opposite direction [BraS83]. Kuteev et al. have reported on impurity measurements with a two-view tomography system on the FT-2 tokamak [Kut091, KutL88],
with similar results: profiles for Ov and Cv symmetrie, and for Cm, On and Om asymmetrie. Profiles with a uniform narrow ring with one sharp peak on the inner side was found for Ovr and H~. The apparent localized souree is ascribed to the plasma column touching the wal!.
The toroidal distri bution of theemission from light impurities and the dependenee on the limiter has been studied in Ref. [AIIM8l] (on the Alcator A tokamak). The difference of the temporal
evatution of the emissivity between different locations could be explained by a model based on the relative magnitudes of the transport rates along and across the field ti nes.
In divertor machines it is natura! that the emissivity peaks near the divertor. Hence, asymmetries reported on di vertor machines seem not to be relevant to the understanding of asymmetries
on limiter machines. However, poloidal asymmetries on other locations than the divertor [Kurz95] might have causes related to those of asymmetries in limiter machines.
7.4.2 Causes for toroidally asymmetrie partiele distributions
The most obvious causes for an poloidally asymmetrie Ha emission profile are localized in
fluxes of H2 and H. Therefore, the recycling of hydragen from the limiter, wall and possible other objects has to be considered, which, in general,would result in toroidally asymmetrie distributions.
Most recycling occurs at the limiter, which is the only material contact with the plasma. When no gas is puffed into the tokamak, the limiter is a souree of an asymmetrie influx of both hy-
174
Discussion on asymmetries 7.4
drogen and impurities. This asymmetrie influx, however, should mainly affect the emission
close to the limiter. The visible-light tomography diagnostic is located 120° away from the top
down limiter in RTP. Neutral hydragen cannot be expected to remain localized for such a dis
tanee ( 1.5 m), but the portion of Ha radialion originating from H! i ons can. It is uncertain, however, whether this portion can completely explain the totalemission in the asymmetries.
RTP has top-down limiters which extend 10 cm in the poloidal direction, which is 10% of the
poloidal circumference of the plasma, thus giving two localized sourees of recycling. However, the main asymmetry is in the wrong direction fol!owing the helical field lines from the limiters
(cf. the field lines in Fig. 1.1). In RTP the range of possible plasma currents can vary qa be
tween 2.2 and 7. At 120° toroidal angle the poloidal angle of rotation of the helix is 20° to 50° counter clockwise in the usual frame of reference, which is not a significant change: no clear
change of the position of the asymmetries has been observed when the plasma current is varied.
The smaller asymmetrie peaks in the outside-lower and inside-upper side could be associated
with the transport along the field lines of H! ions produced at the limiter. This explanation is made less convincing after the observation that a shift of the plasma column changes the emission profile in a way different from what would be expected. An upward shift [Fig. 7 .14(a)] hardly changes the profile, whereas a downward shift [Fig. 7.14(b)] produces a higher peak in
the inside-upper corner, rather than in the outside-lower corner where it would be expected. An
inward shift (Fig. 7.4) gives an emission profile similar to the one of the downward shift, which is rather surprising. No definite conclusions can be drawn from this complicated be
haviour, which is also the finding on some other tokamaks, for example Ref. [SucH78] and
unpublished studies.
Whereas the minor asymmetrie peaks could in principle be explained by recycling from the lim
iter, the main asymmetry at the outer-upper side has no such obvious source. Apart from the limiter, the plasma is not expected to be in contact with other materials. The nearest structures relatively close to the plasma are the soft x-ray tomography cameras at a minor radius of 0.2 m,
i.e. 3 cm from the plasma, in the same poloidal cross-section. Another structure protruding into
the vessel is the ECRH launcher, which is located in the midplane, 15° toroidally from the
tomography system. Because the peaks appear at different positions notclose to one of the soft x-ray cameras, nor the midplane, when the plasma position is changed, an increased recycling
at these structures does not seem to be a likely explanation. Therefore, a recycling souree for the main asymmetry is unlikely to be the full explanation for the asymmetries.
Another possible souree of nonlocal recycling on the vessel wal! is an poloidally asymmetrie outflow of ions from the plasma. The poloidally asymmetrie outflow could be caused by the drift mechanisms that are discussed in the next subsection. When the ions collide with the wal!,
they lead to a local degassing and sputtering [Razu84], and hence recycling. The recycling, and therefore influx, could be toroidally asymmetrie in this case because the distance to the vessel wal! varies, because of the ports, the welding joint between vessel sections, and equipment that
175
Chapter 7 Measurements of stationary asymmetrie emission profiles
is present in the vessel. However, it is unlikely that even a localized recycling on the wall could
give rise to the localized influx observed. The neutral hydrogen molecules and atoms that leave the wal! in all directions would spread out while traversing the 7 cm to the plasma edge.
A different phenomenon that could be relevant for toroidally asymmetrie partiele distributions is
the occurrence of halo currents. Halo currents flow in the SOL between contact points on the limiter when there is a difference in potential between the points on the limiters (caused, for
example, by drift). Due to I x B forces, which because of the strong toroirlal magnetic field
are large for even small halo currents, the halo current can influence the plasma flow. This in
fluence is asymmetrie in the toroidal direction because the limiters are localized.
7. 4. 3 Causes for toroidally symmetrie, poloidally asymmetrie partiele distributions
Toroidally symmetrie distributions that are poloidally asymmetrie require the cause to be present in every poloidal cross-section and not to follow the helical shape of the magnetic field Jin es. In this sec ti on the effects of drift, marfes and charge-exchange processes are discussed.
The B x V B drift of impurity i ons has been found to be a qualitative and quantitative accept
able explanation of the asymmetries in emission from impurities by a number of authors. Terry
et al. [TerM77] proposed a formula for the vertical VB drift distance of an impurity ion diffus
ing along a field line. Numerical simulations basedon neoclassical transport theory for ions in the Pfirsch-Schlüter regime were done by Burrel and Wong [BurW79]. Brau et al. [BraS83] described calculations where anomalous radial transport and atomie physical processes were
included, and the previous model was extended to the banana regime (in the plateau regime the
agreement wîth experiments was poor), and gave a simplîfîed kinetic formulation that gives
approximate distances. There are also findings that arenotconsistent with these models on drift [SucH78]. The models only explain the asymmetriesof the impurity content as natura! consequence of neoclassical transport. The relationship with asymmetries of neutral hydrogen is
unclear. The agreement of the models with impurities in the far edge is poor and is ascribed to asymmetriesin the boundary conditions [BurW79, BraS83]. Therefore, it is difficult to investi
gate the consequences of these models quantitatively for the H<X measurements on RTP.
Qualitatively the effects of drift can be understood as follows. V B drift gives rise to a separation of charges. As discussed in subsection 1.1.2 the helical magnetic field is necessary to limit this effect. However, the separation of charges gives rise to an electric field , which logether
with the toroirlal magnetic field causes an E x B drift in the horizontal equatorial plane. This results in a double vortex structure on both sides of the equatorial plane. Because the plasma
rotates poloidally, the vortex structure can deform, giving rise toa poloidally asymmetrie flow structure, both top/down and in/out asymmetrie. The shape of theemission profile [for example Fig. 7.2(a)] seems consistent with such a deformed double-vortex flow process. However, it is
176
Discussion on asymmetries 7.4
not clear which processes between the outflowing ions and the neutral atoms could be respon
sible for the neutral density to be asymmetrie. As indicated in the previous subsection, the
asymmetrically outflowing ions could cause an asymmetrie recycling from the wal!.
The main test to check whether drift is a possible explanation for the measured asymmetries, is
to re verse the tomidal magnetic field : the asymrnetry should change from top to bottorn or vice
versa because the drift is reversed. The positive results in the literature were summarized in
subsectien 7 .4.1 . The main findings are that the drift can explain the asymmetries in emission
of impurities, and that the Ho: emission follows this pattem or opposes it. However, neutral
hydragen does oot suffer from drift, and therefore only Hi ions could be involved to influence
the Ha emission, or charge exchange as is discussed below.
Reversing the field in RTP [Fig. 7.3] has a significant effect on theemission profile, butnota
complete shift of the asymmetry to the bottorn si de. To fully distinguish between the effects of
local recycling at the limiter and the effect of drift, the current should be reversed as well in
order not to alter the field lines. Although the current was not reversed during the experiments
in RTP, the measurements show that drift cao at best only be a partial explanation for the
asymmetriesin RTP.t
Insufficient parallel heat conduction in combination with the strong temperature dependenee of
impurity radialion is responsible for the marfe phenomenon in tokamaks [NeuS86]. A marfe is
a quasi-steady-state instability (it is stabie for several energy confinement times) which is
poloidally localized with a strongly enhanced density, reduced temperature and high radiative
losses. Marfes are observed at high densities as precursors of radiative disruptions in most
major tokamaks . If the radial heatlossin a tokamak were poloidally symmetrie, there would be
no preferential poloidal angle for this instability. However, experimentally they are only ob
served at the HFS, which suggests a peaking of the radial heat loss on the LFS. The presence
of the asymmetries in RTP, however, seems virtually unrelated to the plasma density, and,
moreover, the position is at the LFS. Although classica! marfes do not seem to be present in
RTP, marfe-like processes cannot be excluded.
A possibly drift-related phenomenon observed in RTP that drastically changes position when the field is
reversed, occurs in low-density plasmas with a relatively high Zeff, in which suprathermal electrans are
present, with intense electron-cyclotron resonance heating (ECRH) (power> 200 kW). Extremely localized
peaks in theemission are observed, seen by only one or two channels per camera. The observations from
different directions are not compatible with emission inside the plasma, but are insuffïcient to accurately
point out where the radiation is coming from. The positions of the localized emission change drastically
when the toroidal magnetic field is reversed. This indicates that the effect might be related to the toss of
toroidally trapped i ons or electrans in the field ripple, which is caused by the discretenessof the toroidal-fïeld
coils, due to drift [BasR92, CarB94]. This means that the phenomenon is toroidally asymmetrie. In similar
discharges the soft x-ray tomography system observes localized emission as wel!. However, these are bursts
of emission mainly after ECRH is switched off.
177
Chapter 7 Measurements of stationary asymmetrie emission profiles
It has been suggested that charge exchange recombination of impurity i ons with a poloidally
asymmetrie neutral hydrogen profile could explain asymmetries in the impurity emission
[Hoga82]. Simulations have shown that in that case the asymmetriesin impurity and hydragen
emission profiles should be opposite [BraS83]. However, this does not explain the reason for
the asymmetry in the neutral hydragen profile. As was said in subsection 7.1.2.6, charge ex
change between impurity ions and neutral hydrogen is not expected to be a dominant process
since the impurity density is too low, but in principle it could reduce the number of neutral
hydragen atoms. A poloidally asymmetrie distri bution of impurity ions or drifting impurities
could therefore give rise to an asymmetrie Ha emission, but it is unlikely to be a significant
effect. No noticeable effect of varying Zeff has been observed on RTP.
7 .4.4 Conclusions
A local influx of H2 and H seems to be the reason for the asymmetrie emission of Ha radiation.
The eau se of the asymmetrie influx is not fully understood, but the explanation for the asymme
tries in the emission profiles measured on RTP can partly be sought in ionic drift and the
possibility that the transport of Hi ions along the magnetic field lines can have an influence.
More investigations are needed for a greater understanding of the measurements and the con
flicting reports in the literature. In particular observations of Ha radialion in other toroidal loca
tions would be useful to determine whether the asymmetries are toroidally symmetrie. The
measurements on RTP clearly show that the asymmetries can be very profound and not only
top-down, as is inferred on most tokamaks from measurements by spectroscopie diagnostics
with less spatial coverage. The presence of asymmetries, albeit found in a small tokamak on
which the asymmetriescan be expected to be relatively large due to the relatively thick edge and
SOL, suggests the need for a more careful evaluation of speetral measurements and of the ap
plicability of Abel inversion on multi-chord data on other tokamaks. The importance of the
found asymmetries is that the densities of the ionization states of impurities are influenced by charge-exchange processes with the asymmetrie neutral hydragen profile, which can have a
strong effect on the transport properties of hydragen and impurities in the edge and SOL.
Numerical models should explicitly take these asymmetries into account. The asymmetries also
complicate the interpretation of other phenomena measured by the visible-light tomography
system on RTP, which are discussed in chapters 8 and 9.
178
Measurements of MUD activity 8 Magnetohydrodynamic (MHD) activity in RTP has been studied with the visible-light tomography system. Due to perturbations, the magnetic flux surfaces are broken up into smaller struc
tures that show up as island structures in a poloidal cross-section. The magnetic island struc
tures influence the plasma parameters such as density and temperature. Due to toroidal rotation
of the island structures and the helical shape of the field lines, diagnostics, which generally
view in a poloidal cross-section, abserve oscillations in the measured quantities. This is re
ferred to as MHD activity. The motivation to study the MHD activity is that many types of instabilities in tokamaks are related to these phenomena. Modifications of the magnetic field line
structure also have an influence on transport, which is an important issue in present tokamak
research.
In this chapter measurements of visible light emission during the occurrence of so-called
m = 2, n = l island structures (i.e. two islands in the poloidal plane) are discussed. Mainly measurements of Ha light are described because of its direct physical meaning. lt proved to be
difficult to obtain a good understanding of the processes giving rise to the oscillations in ernis
sion: tomographic reconstructions do not give detailed inforrnation about the emitting struc
tures, and comparison of the phases of a large number of channels shows a very complex
behaviour. Despite these complications, possible explanations for the observed phenomena are found by various analysis methods.
In section 8.1 an introduetion is given on MHD islands and the study thereof. Section 8.2 dis
cusses and compares the measurements of several diagnostics. The phases of the visible-light
signals exhibit a complicated behaviour and are studied in a variety of ways in the following
sections. The analysis of Ha emission forms the main part of this chapter. In section 8.3 numerical phantoms to simulate the expected emissivity are described and the magnitude of the
oscillations is discussed. Two different types of phantoms are discussed. In section 8.4 the
phantoms are used to do simulations totest the applicability of tomography, and tomographic
reconstructions of the Ha emission during MHD oscillations are presented. The phase relations
between the various signals are studied by correlation techniques in section 8.5, and singular
value decomposition (SVD) is applied with the samepurpose in section 8.6. These methods all contribute to the understanding of the visible-light ernission. However, due to the asymmetrie
emission profile and the lay-out of the system these analyses give insufficient information about
Chapter 8 Measurements of MHD activity
the exact topology of the oscillations. Hence, as the final method, some raw signals are corre
lated with the other diagnostics insection 8.7. The results are summarized insection 8.8. The appendix discusses the application of SVD to remove the chopper spikes from the visible-light tomography signals.
8.1 Introduetion MHD island structures
The main properties of MHD is land structures are introduced in this section. The various diagnostics used are discussed and rotation of the magnetic structures in the toroidal direction is studied. The toroidal rotation is required in this chapter to relate measurements by different diagnostics to each other.
8.1.1 Theory
The plasma consists of electrons and ions; it can therefore be described by fluid equations for electrons and ions. The electron and ion fluid veloeities can also be combined to give a single fluid description, which is done in MHD theory. The fluid equations of MHD contain the electric and magnetic fields that are present in a tokamak. This theory can describe certain types of instahilities which usually bring about a change of the magnetic topology.
The safety factor q, which was introduced in subsection 1.1.3, plays an important role because the magnetic topology is particularly susceptible to perturbations at radii where q has rational values. Recalling Eq. (1.3), q can be expressedas the ratios
rBtP m q = RoBe = -;;· (8.1)
where, as before, ris the minor radius, Bip the toroidal magnetic field, Ro the major radius and Be the poloidal magnetic field. If q is rational, m andnare integers: mis the number of toroidal turns and n the number of poloidal turns a field line makes before connecting to itself. The safety factor is a function of r and usually has a roughly parabolic shape that starts slightly below 1 on the axis and in RTP is 2.5 to 8 at the edge. The safety factor at the edge of the plasma r =a, qa, can be evaluated:
(8.2)
Because the plasma has a finite resistivity, the plasmafluid and the magnetic field structure are not necessarily connected. Island structures, i.e. nested flux surfaces, can form when perturbations, for example smal! displacements of field lines, are amplified on field !in es that conneet to themselves, i.e. that !ie on a rational q surface. Such perturbations that have the form exp(imx), where x = e- !!... 1/J is the coordinate perpendicular to the field line (i.e. x has a constant value
m
180
Introduetion MHD islands 8.1
Figure 8.1 (a) The flux surfaces at q = 2 in a poloidal cross-section with the arrows indicating the perturbed
part of the poloidal magnetic field (i .e. the field at the rational surface has been subtracted) after reconnection.
The dot-dashed line indicates the rational surface, and the dashed line the separatrices. The usual direction of lp
and B~ is indicated as well, as it was given in Fig. l.I. (b) The toroidal topology of the flux tube of the m= 2,
n = I is land structure.
on the field line), are resonant on the surface q =mln. For the island topology breaking and reconnection offield lines is necessary, which is thought to be possible due to the finite resis
tivity, i.e. the so-called tearing modes. The topology of the m = 2, n = 1 magnetic island
structure is shown in Fig. 8.1. The separatrices indicate the 0-points (the islands) and the X
points. In the case of the m = 2, n = I structure one is land makes one poloidal turn during two toroidal turns . Note that after one toroidal turn the islands swap position, which means that
both islands in a poloidal plane constitute one flux tube.
The island width wis given by [Wess87, WhiM77]
w=4 rq,B, , (
A Jl/2 mq B8
(8.3)
where the prime designales the derivative with respect tor, and iJ, is the amplitude of the radial
magnetic field perturbation that causes the island. The island size in RTP for the type of discharges studiedis typically of the order of 5 cm [Mi!L93]. This is larger than the resolution of
the tomography system and significantly larger than the ion Larmor radius, i.e. the radius of the gyration of ions around the field Jines, comparable to the gyration of electrous discussed in
subsection 1.1.3.
The process of reconnection can repeat itself and create nested flux surfaces inside the islands. When islands on separate rational surfaces grow and start to overlap, this creates stochastic
areas between the closed flux surfaces. The existence of a stochastic layer around the separatrix is expected to strongly enhance radial transport because plasma regions that were separated
before the reconnection become conneeled via the X-points
181
Chapter 8 Measurements of MHD activity
8.1.2 Motivation to study MHD island structures
The study of MHD activity is motivated by the importance that it has fortransport and for instabilities in the plasma. Examples of such instahilities are the m = I, n = I instability that is related to the post and precursors of the sawtooth crash [Kado92]. and the formation of m = 2,
n = 1 islands that can cause disruptions, i.e. unwanted violent endings of discharges that can
cause damage to large tokamaks, if the q = 2 surface is close to the edge of the plasma. A
greater understanding of these phenomena is important to improve the operation of tokamaks.
Because the main plasma parameters are approximately constant on flux surfaces, the magnetic islands will have an influence on the temperature and density of the plasma, and through these also on the emissivity of the plasma. Because visible-Iight tomography is particularly sensitive
in the edge of the plasma, it is a good candidate to study MHD structures there. Furthermore,
because the tomography system on RTP has a better coverage than other diagnostics and narrow viewing chords, it might determine the spatial structure better and possibly characterize the stochastic regions. Measurements on RTP have shown that the MHD activity is very prominent
in the measurements taken by the visible-light tomography system. It is interesting to study the influence of the island structures on the neutral hydragen and impurity densities (impurities
could, for example, accuruulate in the islands), which is in principle possible by visible light
tomography.
8.1.3 Diagnostics observing MHD activity
MHD activity is observed by the various diagnostics that are discussed in this subsection. The
magnetic structure rotales toroidally with a frequency of the order of 10kHz, which wil! be
discussed in detail in subsection 8.1.4. Due to the helical structure the MHD activity manifests
itself as temporal oscillations of the quantities measured. Diagnostics that also have a spatial resolution can resolve more information about the spatial structure. Measurements from different diagnost.ics can be compared because of the toroidal rotation.
The most important diagnostics in RTP that abserve the temporal behaviour of island structures at the edge of the plasma are: interferometry (electron density), ECE radioroetry (electron temperature), magnetics (poloidal and radial magnetic field), visible-light tomography (visible-light emission) and recently also reflectometry (electron density). Soft x-ray tomography is very use
ful to study the internal structure of the plasma. However, this diagnostic does not yield in for
mation about phenomena that occur near the edge of the RTP plasma, because x-ray emission is weak there. The meaning of measurements of MHD activity by the various diagnostics is dis
cussed below.
The interferometer measures the line-integrated electron density along parallel verticallines in one poloidal plane. For cylindrically symmetrie density profiles these measurements can be
Abel inverted to give the local electron density. The Abel inversion can be corrected fora hori-
182
Introduetion MHD is lands 8. I
zontal shift. The island structures in the density profile, however, cannot be resolved by Abel
inversion. Due to the synunetric set-up of the diagnostic the direction of the rotation cannot be
determined. When taking into account the measurements by other diagnostics and the raw data
of the interferometer, simulations can be used to fit parameters to the raw measurements, thus quantifying the density inside the islands as wellas the island size [Larnm91, MilL93].
ECE radioroetry is used for local determination of the radlation temperature. The channels that
measure the ECE from radii where the island structure perturbs the electron temperature see the
oscillation. Because this diagnostic also measures in a symmetrical way, with the measuring points located on a horizontal line in the midplane, the direction of rotation cannot be determined from this diagnostic either. Furthermore, positions of the island structures will be close
to the edge of the plasma where the electron density is low. Therefore, the plasma is not opti
cally thick and the determination of the temperature is more complicated.
Twelve Mirnov pickup coils are located in one poloidal cross-section, measuring dBefdt. The
signals are integrated electronically to yield Be. The coils are mounted inside a stainless-steel ring at radius r = 0.213 m and have a -3 dB bandwidth of approximately 10kHz. Th is bandwidth is based on calculations of the shielding by the ring and by comparison of the temporal
response to changes in the field with a faster coil. Due to the limited, not accurately known,
bandwidth, the measurements of high frequency fluctuations by the coils can only be compared
with other diagnostics after taking the phase shift into account. However, the relative phases of the coils can be compared because they are all enclosed in the same ring which should give rise to approximately equal phase-delays for all coils. The coils measure the magnetic field pertur
bation caused by the islands at the coil radius, which is attenuated with respect to the magnetic
field perturbation in the island itself. An incremental current is associated with the magnetic
island (i.e. flowing through the 0-point, forming the flux tube of the island). The direction of the incremental current depends on the sign of the magnetic shear s = rq'lq. A positive shear results in an incremental current in the opposite direction of the main plasma current [Wess87].
The role of the shear for the incremental current becomes clear when consictering the direction
of the equilibrium magnetic field gradient in a plane perpendicular to the magnetic field lines, i.e. it cannot be derived from Fig. 8.1 (a). The direction of the incremental current de termines
whether an increase or decrease with respect to the equilibrium field is measured outside the 0-
point of the island: a positive shear leads to a reduced magnetic field. Comparing the measurements of the various coils in one poloidal plane, the m number of the mode can be derived,
provided a sufficient number of coils is available. If coils are positioned at a sufficient number of toroidal locations, also the n number can be found. Unfortunately, in RTP coils are only
available in one poloidal plane, and hence the n has to be guessed and verified with the meas
urements of other diagnostics. In RTP there are also 12 saddle coils that measure dB rldt at approximately the same toroidal position.
183
Chapter 8 Measurements of MHD activity
The ernissivity measured by the visible-light tomography diagnostic results from various pro
cesses that depend on the electron temperature and density, the neutral hydragen concentration,
impurities and the influence of the magnetic islands on these quantities. Due to the five viewing
directions a higher spatial resolution is possible than with the other diagnostics.
The measurements in different poloidal cross-sections can be compared by taking into account
the helical structure of the magnetic field, provided the m and n of the mode are known. Due to the toroidal rotation of the magnetic structure, the various diagnostîcs abserve fluctuations that
are related to the islands passing by. Therefore, the analysis of the different measurements of
most diagnostics is most easily done in the time domaio by calculation of the phase difference between signals. Due to the spatial resolution of the magnetic coils a spatial Fourier analysis is possible on the coil signals. Local measurements in one poloidal plane cannot distinguish the n
number: all m modes with the samen, and the same toroidal frequency , wiJl show the same
frequency in the signals. For line-integrated measurements the situation can be more compli
cated: chords viewing inside the rotation radius of the island can see a double frequency due to each island passing the chord twice each rotation. The chord width and finite island size smooth out this effect and usually only a slight modulation of the signa! is seen on top of the main
oscillation.
8.1.4 Rotation
As has been pointed out previously, the magnetic structure rotales toroidally. To campare signals of different diagnostics the rotation direction must be deterrnined and taken into account. Because most momenturn is carried by the ion fluid, only ions wiJl be considered to derive the
rotation. The fluid equation for ions is:
m; n{ ~+(u· V)u] = qn;(E+u x B)- Vp; -R, (8.4)
where m; is the mass of the ion, n; the density, u the fluid velocity, q the charge, E the electric field, B the magnetic field, Pi the pressure, and R represents the momenturn loss due to friction with electrons. For a steady state, neglecting the radial contributions to (u · V)u and the fric
tion, the radial force balance in a toroidal geometry gives approximately
I Jp u8 Bq, -uq,B8 = E, ---'.
qn; Jr (8.5)
The first term on the right-hand side corresponds toE x B drift while the second term corresponds to ion diamagnetic drift (also called gradient-driven drift). A radial electric field comes
about due to ambipolar diffusion: ions diffuse faster than electrons, resulting in a negative potential in the centre of the plasma [SigC74] . In a toroidal geometry the fluid cannot flow
freely in the poloidal direction: it is damped due to the strong poloidal viscosity [HirS81]. Therefore, the E x B and diamagnetic drifts wil! result in mainly a toroidal rotation of the
plasma. Because both E, and Jp!Jr are negative, they workin opposite directions in Eq. (8.5).
184
Introduetion MHD is lands 8.1
To the zerothorder the two drifts are of the samemagnitude [SigC74]. Usually it is found that
the Er contribution dominates (first order effect), which results in a counter-current rotation in Ohmic plasmas. The tomidal rotation velocity caused by the diamagnetic drift can be estimated
to be of the order 2 x 104 m/s in RTP. This corresponds to a rotation frequency of a bout 10kHz, which is what is actually found in RTP. The Ex B drift is expected to dominate, indicating that Er> 104 V/m, which is a value that is also found in other tokamaks.
The magnetic field structure moves with the plasma fluid. In partienlar in the centre, where the
resistivity is low, the magnetic structure tends to rotate with the fluid velocity. More towards the edge, the magnetic flux surfaces and perturbations thereof might move related to the local fluid velocity, or they might loek to the modes on central flux surfaces and be dragged along at
the same velocity as the central rota ti on. Both phenomena have been observed and are described
in the literature, for example in Ref. [VriW95] where coupled m = 2 and m = I modes are
found in Ohmic plasmas, but uncoupled modes in plasmas with neutral beam injection. Note
that the two phenomena result in either different or the same frequency of modes with different m numbers, and that the locking occurs in the toroidal geometry: a diagnostic in one poloidal
plane will abserve many different orientations of the different modes with respect to each other
when it rotates toroidally. A further argument against poloidal rotation of the magnetic structure, apart from the negligible poloidal fluid rotation due to damping, is the observation that
energy is needed to compress and expand the islands during the poloidal rotation.
For the diagnostics looking in one poloidal plane the toroidal rotation is transformed into a
poloidal rotation due to the helical shape of the flux tubes associated with the islands. In Fig.
1.1 the directions of lp and Brp for normal conditions in RTP are illustrated, resulting in the helical shape sketched in Fig. 8.1(b). As indicated in subsection 8.1.3, the poloidal rotation
direction cannot be determined by diagnostics that have one symmetrie view. The magnetic coils and the tomography systems yield, in principle, sufficient information for this determina
tion. Figure 8.2 shows the phases between all coils and a reference coil, which corresponds to a clockwise poloidal rotation of an m = 2 mode. The m = 2 nature of the oscillation is appar
ent from the 4n: phase transition over 360° of coils in Fig. 8.2. The spatial mode behaviour is displayed in Fig. 8.3. The spatial structure suggests that the island width is a functions of an
gle, for example wider at the LFS. This observation is only relevant for quantitative studies of
the MHD islands. The clockwise poloidal rotation in this discharge results from a co-current toroidal rotation (i .e. clockwise when seen from the top) , which becomes apparent from Fig. 8.1 (b) consictering the helical curvature of the field !i nes.
The curve in Fig. 8.2 shows an unsmooth behaviour at the coil at 180° (i.e. at the high-field side). Also the spatial structure, such as in Fig. 8.3, shows irregularities at increasing frequencies. Usually, this co i! has a significantly smaller oscillation amplitude than the other coils, whereas the average value is approximately of the same level. Because the irregularities increase
in magnitude for increasing frequencies, a reduced bandwidth of the coil in question compared
185
Chapter 8 Measurements of MHD activity
2n.---.--.---r---r--.---r--,.--.---r---r--.--.~
a> n (/)
ro .s:::. a_
60 120 180 240
angle of coil (deg.)
300
!8
~ ~ 11
360
Figure 8.2 The phase of the signals of all magnetic Mirnov coils with respect to the coil at the !ow-field side,
i.e. 0°, calculated by cross-correlation analysis of the time·traces. The phase is calculated from the time-lag
between the signals and is normalized to 27t for one period (which corresponds to one toroidal rotation). The rota
tion frequency for this discharge was 6.3 kHz. The angles of the coils indicate the poloidal angle of the coils with
respectto the !ow-field side one in the counter-clockwise direction.
to the other coils seems to be a plausible cause, giving rise to a larger phase shift than the other
coils at higher frequencies. Another irregularity is that the slope of the curves in Fig. 8.2 is not
equal to 2 (when both phase and angle are expressed in the same dimension), but larger. Also
this effect becomes larger for higher frequency. The apparent increase is probably also caused
by the bandwidth: the bandwidth of the coils at the HFS being larger than the other on es. It has
notbeen possible to verify the bandwidths of the coils. However, it seems plausible that, because at low frequencies only minor irregularities occur and at higher frequencies the other
diagnostics do notabserve significant changes, the irregularities are not genuine.
Sametimes the mode content of the oscillations is analysed by spatial Fourier analysis of the
signals of the various coils [MiiL93]. It is assumed that the mth harmonie corresponds to MHD
mode m. Figure 8.2 shows that the apparent rotatien is not perfectly harmonie. Furthermore,
po lar representations of the measurements show a more complicated behaviour than is expected
for simpte modes. Therefore, higher Fourier modes will contain part of the lower asymmetrie
MHD modes, especially for higher frequencies. This methad has not been applied because it
does not give any additional information to Figs. 8.2 and 8.3. However, in combination with
magnetic measurements at various toroirlal positions the methad can be very useful to determine
the m and n numbers of the main mode or modes present in the plasma.
186
I I I
•
I
.",..-/ fÎ
I ,é
•
Afirst look at the measurements 8.2
.-----·-----
• :...---
' _._- J
' \ .. I
Figure 8.3 Representation in a polar plot of
the poloidal magnetic mode structure for two
time slices that are separated halfan asciila
tion period (dashed lines). The orientation is
the same as for tomograms. The circle repre
senis the radius at which the coils are located
and the radial distance of the curves from the
circle is proportional to the asciilation part of
the measured magnetic field, i.e. the circle is
the zero level. The amplitude is in arbitrary
units. The points represent the angles at
which the coils measure. These measurements
correspond to the curve in Fig. 8.2. - - -· .....
_____ .J
.._
8.2 A first look at the measurements
The MHD activity occurs at high plasma currents, i.e. low qa, after a steep ramp-up of the current. This sec ti on presents the measurements of the various diagnostics during MHD activity in
a typical discharge. The Ohmic discharge was diagnosed by the visible-light tomography sys
tem using Ha filters . Table 8.1 lists the parameters of this discharge. Measurements with the continuurn filters and without filters have also been taken, but these are more difficult to interpret. In particular in the case of the continuurn filter the processes causing the emission from the
edge are poorly understood (see subsectien 7 .2.2).
Figures 8.4(a-c) showsome characteristic time traces of channels of the visible-light tomogra
phy system. Virtually all channels see the oscillation, but the amplitudes vary greatly, absolutely as weJI as relatively. Forsome channels the crests or troughs are split into two. This also
happens in simulations and is explained by the fact that chords viewing inside the radius of
rotatien see each island passing twice each rotation. If emission from the central part of the plasma is significant, also m = 1 islands could have a perturbing effect. The oscillation ampli
tudes and effects such as the peak doubling are greatly influenced by chord-width effects, which vary between detectors. The tomographic reconstructions of the time-averaged signals are similar to the reconstructions of Ha emission in Fig. 7.2. The magnitude of the fluctuations and the relation to the fluctuations in electron density is discussed in sectien 8.3.
187
Chapter 8 Measurements of MHD activity
:·:f~ !.::f~
2.6~--_L----------~------------~--------~
(\J I E
ï
j
ë .~
111 .. J: :t~ }::::f~
2.5L---~----------~----------_L--------~
:I rel
." .. L
Ql
1- :~ 1-
0.122V\JVWVJ(f)
0.121 ~
aJ 0.120
142.50 142.75 143.00
time (ms)
Figure 8.4 (a-c) Three typical time traces of scaled Ha signals (channels E76, 823 and 829, respectively; 829
is oot scaled because of lack of sealing factor) . (d-f) Time traces of representative measurements of fne dl (chord through island at HFS R = -0. 12 m), the radialion temperature in the island (LFS, R = 0.127 m) and B () by
Mirnov coils (in horizontal planeon LFS). (Discharge rl9941004.021.)
188
Afirst look at the measurements 8.2
Table 8.1 Plasma parameters of the main discharge under consideration m this chapter (rl9941004.021).
Quantity Symbol
Plasma current I P
Toroidal magnetic field Bq,
Safety factor at edge qa
Electron density in centte ne,max
Maximum relative electron density fluctuation in island* iiJïï.e
Electron temperature in centte Te,max
Maximum relative temperature fluctuation in island Tefie
Toroidal rotation frequency f
• Derived from line-integrated measurements, see section 8.3.
Value
133 kA
1.92 T
2.7
2 x 1019 m-3
40%
0.8 keV
30%
7.8 kHz
In Fig. 8.4(d-f) typical signals of the interferometer (Jne dl), the ECE radiometer (Te) and one magnetic pickup coil (at 0°) are shown, respectively. For this chapter mainly the relationship between the electron density and the visible-Iight measurements is relevant. The spatial structure of the line-integrated density is displayed in Fig. 8.5. To campare the phases of the signals the
toroidal distance between the diagnostics has to be taken into account. In subsection 8.1.4 the toroidal rotation was found to be in the negative Ij> direction [see Fig. l.l(a)]. The toroidal distanee from the visible-Iight tomography diagnostic downstteam to the interferometer is 45°. (The Mirnov coils are 112.5° upstream from the interferometer, and the ECE diagnostic 285°.)
8.3 Phantoms for simulations
Numerical phantoms for the emission in a poloidal cross-section have been constructed for simulations to test the analysis methods used in this chapter and to campare the results with those of measurements. The only emission in the visible range in RTP that is sufficiently understood to enable rnadelling is Ha radiation. Assuming the classica! image of purely atomie processes, according to Eqs. (2.4) and (2.9) for Iow densities the local emissivity is proportional to the electron density ne and the neutral hydragen density nH. Totest the analysis methocts used in the following sections a phantom was constructed assuming a cylindrically symmetrical nH distribution, because a simpte phantom is most instruclive for testing purposes. Another, more realistic, phantom was constructed on basis of the asymmetrie tomograms of time-averaged measured Ha emission. A possible fluctuating neutral hydragen density should also be included in a realistic phantom, but it is unknown how the island structures influence the neutral hydragen density. To investigate whether a fluctuating nH is required to explain the
189
Chapter 8 Measurements of MHD activity
~ 0.6 <)I E
Ol
0 .... 6 0.4 .?:-ëii c: Q) "0
"0 Q) 0.2 (ij ..... Cl Q) -c ·a; c 0.0
-0.2 -0.1 0
R-RO (m)
0.1 0
0.2
(/)
~ï c: a (0 a.
ro :::J
I» :::J a. :::J
:c
Figure 8.5 Envelopes of the measured line-integrated electron density profile (solid lines), envelopes of the
simulated local densities (dashed line) and the simuialed static neutral hydrogen profile (dash-dotted line).
measurements, first a static neutral hydrogen density is assumed so that all fluctuations in
phantom emission are caused by fluctuations of the electron density.
The electron density fluctuations are determined from the line-integrated electron density in a
poloidal cross-section measured by the multi-channel interferometer (see Fig. 8.5). The time
averaged signals can be Abel inverted to give the time-averaged density profile. The time-aver
aged density profilene was obtained by fitting line-integrals of the parabolic function
(8.6)
to the interferometer measurements. Here n2 is the peak density. The fluctuating part of the
density 'iie was modelled by two rotating positive or negative Gaussian distortions in radial and
angular direction, with 1/e widths Tt fe and elle [Lamm91]. The amplitude n2 of the density
di stortion is found by adapting it until the fluctuation amplitude of the model corresponds to that
of the line-integrated density measurements. Typically, 'ii2me = 0.4 is found to account for the
8% modulation found in the line-integrated signals. The radius of the eentres of the m = 2 is
lands, which corresponds to the radius at which q(r) = 2, is found from the chord that shows
the maximum fluctuation. The poloidal rotation frequency is taken as half the frequency of the
measurements, because in the case of the m = 2 mode there is only half a poloidal rotation
during one toroidal rotation. Typical values for the parameters are lisled in Table 8.II.
Also islands with an m = 3, n = 1 structure have been modelled to investigate the effects of
distortions of more complex behaviour than m = 2 islands on the analysis methods and to find
190
Phantoms 8.3
Table 8.11 Typical values of the parameters of the phantom of the Ha ernission during MHD activity (n = 1, m = 2 islands).
Quantity Symbol Value
Minor radius (outside which no emission) a 0.17 m
Power in time-averaged electron density a 0.6
Relative amplitude of electron density perturbation ~ 40%- 100%*
Radius of centre of is land rq=2 0.125 m
Radial 1/e width of island r11e 0.02 m
Angular 1/e width of is land el/e 52°
Power in nH density ç 10
Characteristic width of the nH layer À. 0.05 m
Relative nH in centre of plasma to that at the edge n~ln~ 0.3
Toroidal rotation frequency I 8 kHz
The lower limit is the value that agrees with the electron density measurements . However, to properly simulate the measured Ha signals larger va lues had to be used, as is described in the text.
out in how far such complex structures can be resolved by, for example, tomography. When
island structures with different mode numbers are simulated simultaneously, care should be taken of the rotation. As discussed in subsection 8.1 .3 the toroidal rotation frequency is the same for all modes if the modes are coupled, and modes with equal n result in equal frequency
of the measurements. Because in one toroidal rotation the islands of the m = 2 mode rotate
over 180°, and the islands of the m = 3 over 120°, it is clear that the modes have different poloidal rota ti on frequencies, which is the frequency of interest in the phantom.
For the first phantom the neutral hydrogen density nH is taken to be static and cylindrically symmetrie. In a second phantom a more realistic asymmetrie neutral hydrogen density wiJl be
used. The model used for the radial dependenee of the neutral hydrogen density in the first
phantom is
(8 .7)
where n~ is the density that is assumed in the centre to account for the light that seems to be coming in the reconstructions of sec ti on 7.1 and to imprave the correspondence of the simula
tions to the measurements, and n~ is the density at the edge. Typical values for lhe parameters are given in Table 8.11. Figure 8.5 depiets the ne and nH profiles used for the typical phantom
ernission profile shown in Fig. 8.6. In all simulations presented in this chapter 3% relative
noise was added to the pseudo-signals.
191
Chapter 8 Measurements of MHD activity
g(x,y) Figure 8.6 Contour and three-dimensional
plot of a typical phantom emission profile.
The MHD activity gives rise to signifi
cant fluctuations between the X and 0-points of the islands for both the elec
tron density and the temperature (see
Table 8.1). The fluctuations in the line
integrated visible-light signals are, by
comparison, even larger: some channels see as much as 40%, and 60% has
been observed occasionally. It is hard
to explain the fluctuation of the line
integrated visible-light signa! by the Jo-
cal variations in ne and Te. As wil! be
discussed in chapter 9 [Eq. (9.2)], assuming that the deviations from the corona! model are not too severe, the fluctuation in emission is mainly due to the electron and, possibly, neutral hy
drogen density fluctuations. Only for low temperatures ( < 10 eV) the dependenee on tempera
ture fluctuation is dominant, but such low temperatures are not reached in the plasma region
where the temperature fluctuation is large. Simulations with the phantom described above could
only give the measured fluctuation levels of Ha radiation by assuming far larger electron density fluctuations than determined from the interferometer measurements. Possible explanations
are that (I) nH is fluctuating as well, enhancing the effect of the electron density fluctuation; and
(2) that the asymmetrie peaks in emission cause the effect of the averaging process of the line
integrals to be smal!, resulting in almost as large fluctuations in the line-inetegrated measure
ments as in the local fluctuations.
The latter explanation has been verified by constructing a second phantom on basis of the
tomographic reconstruction of the time-averaged signals. The assumption is that the tornogram
divided by the local time-averaged densities (Abel inverted time-averaged interferometer meas
urements) gives local values that are proportional to nH. The time-dependent pseudo-signals are
then obtained by multiplying the static values by the time-dependent ne values obtained in the
way described before. The first, symmetrie phantom gives pseudo-sigoals which spatially behave quite different from the measurements. The parameters were fitted by doing several
simulations, but no good correspondence was achieved. Optimizing the parameters by
parametrization methods, as described in subsection 3.4.1, has not been possible due to the
Jack of correspondence between the phantom and the measurements, and because most parame
ters are not independent. The second phantom gives pseudo-signals that are more sirnilar to the
measurements. It is remarkable that, although neutral hydragen density nH only extends some
192
Phantoms 8.3
centimetres into the plasma and nH is smalt at r q = 2. quite large fluctuations are obtained, both in the measurements and in the phantoms based on tomograms.
Although reasanabie agreement was found by simulations, the second explanation cannot fully
account for the measurements. First of all, although no measurements of the electron density near the edge are available, the density fluctuation is expected to be smal! some centimetres
outs i de r q = 2. The fluctuation level obtained from the phantoms based on the tomograms is still Iower than in the measurements and this phantom does not give fluctuations close to the edge of
the plasma. Some channels viewing outside the last closed flux surface see an appreciable fluctuation. Furthermore, the phase of the channels viewing outside r::: 0.15 m [Fig. 8.4(c)] is inverted with respect to the channels viewing inside this radius [Fig. 8.4(b)]. Temperature
fluctuations are not Iikely to play a role in this region. Therefore, a contribution from fluctua
tions in nH caused by the rotating island structures seems to he the only possible explanation. These observations are further discussed insection 8.7.
8.4 Analysis by tomographic reconstructions
The coverage of the tomography system is, in principle, marginally sufficient to resolve two island structures in the poloidal cross-section. This is because the system on average only has a
two-fold coverage of the plasma, which as discussed in subsections 3.2.3.1 and 3.2.6.2 is
only sufficient to resolve a smal! number of angular harmonies. These Iimitations are studied in this section on basis of phantom reconstructions. Tomographic reconstructions of measure
ments are also studied to extract as much reliable information as possible.
The time-averaged measured emission profiles are similar to the ones described in chapter 7. As
discussed in subsection 3.2.8, it is possible to do the tomographic reconstructions for the time
averaged and fluctuating parts separately, provided the reconstruction metbod is Iinear. The time-averaged part wiJl be referred to as de, the fluctuating part as ac. Comparisons of phantom calculations where the ac and de components have been separated before and after the tornog
rapbic reconstructions have shown that nearly identical results are obtained, which means that nonlinearities in the Iterative Projection-space Reconstruction (IPR) metbod have little influ
ence.
Figure 8.7 shows reconstructions by the Constrained Optimization (CO) method of the ac part of the signals for two orientations of the is lands in the phantom. The IPR metbod gives similar
results, but with different artefacts. The two examples shown represent the best [Fig. 8.7(a)] and worst [Fig. 8.7(b)] cases. Similar to what was found in subsection 6.1.4 about the influence of the phantom position on the result, for eertaio angles reconstructions are obtained with
the structures in the right places, whereas for some ether angles the coverage of the system is insufficient. The main condusion from these phantom calculations is that for smooth well-determjned island structures distorted reconstructions are obtained where in some cases the is-
193
Chapter 8 Measurements of MHD activity
Figure 8.7 (a-b) Tomographic reconstructions by the CO methad of the ac part of a phantom (c-d) of two
islands. Two orientations are shown; (a) corresponds to (c), and (b) to (d). The dashed lines indicate contours of
negative values.
lands even appear in wrong positions. This means that a very careful analysis is required for the
interpretation of reeonstruc ti ons of actual measurements, where more complicated structures are
expected due to, for example, the asymmetries in neutral hydragen density : details in the reconstructions cannot be trusted and structures might even be displaced or not present at all.
Reconstructions of phantoms with combined m = 2 and m = 3 structures show even more
artefacts, indicating that these reconstructions are more unreliable.
Despite the probierus in interpretation found in the phantom calculations, tomographic recon
structions of the ac component of measurements have been made (Fig. 8.8). Some interesting observations can be made, which are the same for the phantom simulation. The reconstructions
of successive time slices show a smooth transition, and after one period approximately the same
reconstruction reappears, giving some credibility to the reconstructions. The reconstructions are
reproducible, i.e. not sensitive to small variations. Due to the asymmetrie distribution more
complex structures can be expected than in the phantom calculation with m = 2 islands, which
already had significant artefacts. Therefore, the details in the reconstructions are not reliable. However, trends, for example a rotation, should be visible if successive time slices are
examined. No clear rotation of the emission profile is apparent from the tomographic recon-
194
Tomographic reconstructions 8.4
( e )
ao oa
Figure 8.8 Tomographic reconstructions by the CO method of the ac part of the Ha signals. Four time slices
are shown to illustrate !he main observations that can be made: (a) starting time slice (I= 0 T, T being !he pe
riod), (b) the next time-slice (I= 0.05 T). (c) time slice half a period (t = 0.47 T). and (d) after one period (I=
0.98 T). The dashed lines indicate contours of negative values. (e) Schematic plots of the orientation of !he posi
live electron density perturbations (indicated by ellipses) in the samepoleidal cross-section, as determined from
!he interferometer. The order of the plots in (e) corresponds to that of (a-d).
structions, except for some pairs of time slices. This has also been examined more objectively
by analysing the difference between successive time slices, which is related to the time deriva
tive. The reconstruction of time slices separated by half a period, however, show clear signs of
inversion of the acemission [see for example Fig. 8.8 (a) and (c)]. Therefore, the tomographic
reconstructions seem to indicate an asciilating emissivity rather than rotating structures.
Reconstructions by the IPR and CO methods, which differ significantly, both lead to the same conclusion, as do reconstructions for different plasma conditions (for example qa > 3) and dif
ferent optica! filters. Without filters the tomographic reconstructions are different, but they show the same type of oscillations. The relation of the peaks and valleys in the ac emission
profile with the positive and negative electron density perturbations is not simple, as can be
inferred by camparing Figs. 8.8(a-b) with Fig. 8.8(d). Sametimes a peak in emissivity corresponds to a peak in ne. but usually not for both islands at the same time. For time slices be
tween Figs. 8.8(a) and (c), and between (c) and (d), the reconstructions become relatively flat,
195
Chapter 8 Measurements of MHD activity
which is not expected from the rotating islands with an ne perturbation. There seems not to be a
clear correspondence between a positive or negative fluctuation in ernission and the island, but
there is astrong dependenee on the position of the islands. No clear onderstanding of the pro
cesses involved in the visible light fluctuations during MHD activity emerges from the tornograpbic reconstructions alone. Therefore, other analysis methods have to be employed in the
following sections. lt can be expected that the asymmetrie nH profile plays a major role.
Furthermore, a possible time dependenee of nH has to be investigated.
The sparse coverage of the plasma by the system could in principle be extended in tomographic
inversion algorithms by assumptions about symmetry and time evolution of the plasma, for example rigid rotation, as indicated in subsection 3.2.3.2. Harmonie poloidal rotation of the
magnetic structure is a good candidate to be studied by this "rotational tomography" method.
Due to magnetic shear, the apparent poloidal rotation in a poloidal cross-section is not likely to
be rigid. Therefore, the assumption of rigid poloidal rotation can only be applied if the fluctua
tions are present in a narrow shell, as is the case for Ha light, or if only one m mode is present.
Even though the island sizes can be expected to be different on the HFS and LFS due to the difference in magnetic field, with the relatively large number of views only a rotation over a smal!
angle needs tobetaken (roughly 360° divided by the number of cameras). Despite the seem
ingly beneficia! conditions, unfortunately it is impossible to use the rotational tomography
technique for the visible-light tomography diagnostic due to the large asymmetries that are
always observed on RTP in the visible light.
1.0
0.5 r:::: 0
~ 0.0 Qj .... .... 0 ()
-0.5
-1.0 0 0.1 0.2 0.3 0.4 0.5
timelag (ms)
Figure 8.9 Auto-correlation of channel E76 (solid line) and cross-correlation between channels E76 and 823
(dashed line).
196
Correlation analysis 8.5
8.5 Analysis by correlation techniques
The phases between the various signals can be studied in a variety of ways. In this section a method based on the cross-correlation is used. Analysis of cross-correlations is a convenient
method because large correlations occur between channels (see Fig. 8.9). The formulae for the
auto and cross-correlation have been discussed in subsection 3.4.3, Eq. (3.59). The cross-cor
relation function PiJ( r) is a smooth oscillating function which has the same period T as the original signals, and the time-lag rm of the first maximum gives the time-lag between the two
signals, 2rr.rm/T giving the phase difference. Accurate deterrnination of rm and T could be obtained by averaging over several periods, which compensates for the finite sampling frequency.
Frequency doubling effects in the signals are, in genera!, smoothed out in PiJ· and are therefore nottaken into account. The amplitudes of the signals and the cross-correlations differ between
channels, which is partly due to instromental effect such as the averaging over different chcrd
widths and different lengths of chord through islands, and partly due to the geometry of the islands themselves, for instanee their different sizes on the LFS and HFS.
In appendix 8.A it is shown that it has notbeen possible to filter out the chopper spikes in the
Q) (/) ro
..r::. a.
0 :l 0
~ 8 • 0 § 0
0
~ s 0 0 0 s
11 0
1[
0 :;: . •a t + ~
+ 0 ~ ~ ~ ëi
~~ i ~ .::::1
' ~
I \ . . " o~L-~~_L~~~~~~~~~_L~~~
0 90 180 270 360 ll<J> (0)
Figure 8.10 Phase difference as a function of angle difference 1!.1/J between viewing chords as calculated from
their cross-corre lation. The drawn !i nes show the dependenee that is expected for rotating m = 2 island structures,
and + symbols are points obtained for a symmetr ie phantom simulation, which indeed I ie on the I i nes. Solid
circles give the points obtained for the measurement of Ha radiation , and open circles are the result of a
si mulation with the phantom based on the CO reconstruction of the time averaged measurements (see section
8.3). Both the po ints of the measurement and of the second phantom deviate from the lines and show many
common features. Cross-correlations were calculated between all chords with !lp< 1 cm, and points with
Pi/<m) < 0.2 were disregarded.
197
Chapter 8 Measurements of MHD activity
Ha measurements analysed in this chapter. Between chopper spikes an unperturbed time win
dow exists of 0.5 to 1 ms. The oscillation has 2 to 10 periods in this interval. Accurate crosscorrelations could be calculated in the cases with more than 5 periods. The auto and cross
correlations in Fig. 8.9 are typical results. The decay of amplitude in the "oscillations" in Fig. 8.9 is mainly due totheshort time window: in cases where the chopper spikes are relatively small and the correlations can be calculated in a Jonger time window the decay is less steep,
indicating that the correlation of MHD oscillations is strong over many milliseconds. Fourier
analysis, which can also be used to calculate the phase between two signals, did not give such accurate results in these short time windows.
Although clear correlations exist between most of the signals, the interpretation of the time-lags
is not evident because the measurements are line integrals. For instance, the phase difference between a central chord, and one at the edge that looks under a different angle, has no direct
meaning. For a rotating structure at a fixed radius, all chords viewing tangentially to the same
radius are expected to have a clear phase difference corresponding to the difference in angle at
which they view. Therefore, the phases of channels with equal impact parameter p within /),p = 1 cm have been compared. Basically the approach is only correct if p 2': r q = 2, in the case of
m = 2 islands; but simulations have shown that, due to the fini te width of the chords and the averaging process in PiJ· meaningful results are also obtained for smaller p. The phase differences obtained can be plotted against the viewing angle difference !:;.tj> (see Fig. 8.10). Points
for which Pij('<m) < 0.2 are disregarded. For the symmetrie phantom of section 8.3 the points close tothelines drawn in Fig. 8. 10 are obtained, as is expected (the scatter in phase difference
is smaller than 15°). The 47t phase difference over !:;.1/J = 27t, i.e. the drawn line making a jump
at !:;.tj> = 1t and !:;.1/J = 27t, is due to the fact that the m = 2 islands need two periods to make one full poloidal rotation. The method is not sensitive toa shift of the rotation centre of 15 mm, nor
to the size of the chosen t:;.p, as has been verified by simulations. However, if m = 2 and
m = 3 are included, distortions occur and many points lie outs i de the !i nes, the number of these points depending on the relative magnitude of the m = 3 islands.
The phase differences found by the same method for the Ha measurements are also shown in Fig. 8.10. The result is very different from what is expected for simple symmetrical rotating
structures. Only m = 2 islands are expected to be present. In the case of measurement of the
total radiation similar results are obtained with even larger a spread over the plane. No clear dependenee on the p values of the channels included in the analysis could be found . Si mulations show that a si mi lar spread and similar "shapes" in the distri bution of points are obtained
for phantoms with a static non-symmetrie nH distribution, as illustrated in Fig. 8.10 for a simulation of a phantom based on the CO reconstruction of the time-averaged signals during the same discharge. Although the points for the simulation are not in the same positions as the ones for the measurement, there is a clear similarity in spread over the plane. This suggests that the spread of the points corresponding to the measurments is, at least partly, attributable to the
198
Correlation analysis 8.5
asymmetriesin nH. The deviation of the simulation from the measurements could result from:
(I) the assumption of static nH being wrong, i.e. a modulation in nH playing a role, (2) the
reconstruction not being perfect, (3) the rnadeling of ne not being exact, and (4) ne being so
large at the island position that the Ha emission is in the excitation-saturation phase (see Fig. 2.2), i.e. only proportional to nH and not to ne.
From the study of cross-correlations and phase differences between channels it can be concluded that large correlations exist between channels, but that these are not only related to a
rotation of light-emitting structures, but can be explained by oscillations in emission due to
perturbations in electron density rotating through an asymmetrie background of neutral hydrogen gas. This oscillating behaviour is in agreement with the observation that many channels are close in phase (many points are concentrated in the corners of Fig. 8.1 0), and the observation
of oscillating emitting structures from the tomographic reconstruction. The same type of behaviour is obtained by the simulation of rotating ne islands in the asymmetrie nH background
determined from the tomogram, showing that a brightening of Ha emissivity in the concentra
tions of nH when a positive rotating ne perturbation passes by is a reasonable explanation. The fluctuations can be further enhanced by the actual fluctuations in nH that were found in section
8.3. However, it should be remarked that, although a wide distribution of phases between channels is found in Fig. 8.1 0, for some cameras the phases between channels of the same
camera show phase shifts that seem correlated to rotating structures.
8.6 Analysis by SVD
Singul ar value decomposition (SVD), or biorthogonal decomposition, has proven to be a useful tooi in analysing MHD activity measured by piek-up coils and soft x-ray measurements
(Nard92, DudP94, Dudo95, BesM94, Fuch94]. It can separate temporaland spatial orthogonal
components in the data (see subsection 3.4.2) which might, if the conditions are favourable, each have a distinct physical meaning. In appendix 8.A the method is applied in an attempt to
filter out the chopper spikes from the visible-Iight signals. In this section the information contained in the first few components is studied to find out whether it can contribute to the interprelation of the measurements.
MHD activity in the total emitted light has been studied by SVD. It has not been possible to study the MHD activity measured in Ha light by SVD. As is shown in appendix 8.A due to the low signa! levels the amplitudes of the fluctuations are relatively smal! compared to the chopper spikes and therefore the chopper spikes show up already in the second component. In this sec
ti on it is found that at least the two first unperturbed components are necessary for analysis.
Between two chopper spikes there were too few time slices to take advantage of the properties of SVD: that it gives time-averaged main components that are present in the signals. This is one of the drawbacks of the SVD method: the outcome of SVD depends solely on the signals that
199
Chapter 8 Measurements of MHD activity
102
Q) ::J (ij >
101 .._ .!!! ::J Ol c ëii
10°
10-1
0 20 40
order
60
Figure 8.11 The singular values of the measurements during large MHD activity.
80
are input, and cannot be influenced other than by choosing the time interval of the signals prop
erly. Before applying SVD, the time averages are subtracted from the signals and the signals are
scaled to the standard deviations (see subsection 3.4.2.2 for the justification of this treatment).
Figure 8.11 shows the singular values of a time window during lar~e MHD activity of measurements without optica! filters. The first two components have singular values that are of the
same magnitude, which is much larger than the other values. This indicates that their behaviour
might be coupled. The absolute magnitude of the singular values depends on the signals and the number of channels and time slices, and is not of interest. The levelling off of the curve be
tween order 10 and 20 is related to stochastic noise, which could be filtered out by disregarding
all components of high order, as was pointed out in subsection 3.4.2.2. The influence of noise on the singular values has been discussed by Dudok de Wit [Dudo95].
Figure 8.12(a) shows the first two chronos (tempora! components). These are approximately harmonie signals with a phase shift of 90°, which is typical of travelling wave phenomena, sueh
as rotating island structures [Nard92]. However, the fact that the measurements are line-inte
grated signals, instead of loeal ones, eomplicates the interpretation. Simulations show that the ehronos resulting from SVD are similar for both the symmetrie phantom and the phantom based
on the reconstrueted time-averaged asymmetrie emissivity. Therefore, the harmonie signals
found by SVD are not neeessarily due to rotation, but ean very well be explained in the same way as the fluetuating tomographie reeonstructions and the phase analysis: rotating perturba
tions in electron density giving rise to fluctuations in the emitted light due to the asymmetrie nH
distribution, and nH having a temporal dependenee.
200
86.4 86.6 86.8 time (ms)
detector of viewing direction E
87
• •' •' . '
SVD 8.6
Figure 8.12 (a) First two chronosof the SVD of large MHD activity measured without optica( filters . (b) The
corresponding topos. Succeeding components are discussed in Appendix 8.A (Fig. 8.15).
Figure 8.12(b) shows the topos (spatial components) corresponding to the chronos discussed. Because the cameras have partial views inslead of whole views, the topos are difficult to inter
pret (cf. the line-integrated profiles in chapter 6). In principle it is possible to make a tornog
rapbic reconstruction of a topo, of which the sealing to the standard deviation of the signa! has been undone. In the literature tornograp bic reeonstruc ti ons of topos have proven to be useful to
localize MHD phenomena in the x-ray emission [Nard92, DudP94]. The orientation of the
island structures found depends on the starting time of the time window on which SVD is applied. Because of the limited number of cameras in the visible-light tomography system, the
tomographic reconstructions are as difficult to interpret as the ones discussed in sectien 8.4. The results for simulated phantoms show that the shapes of the topos are quite varying for different circumstances. lt is concluded that the structures are too complex to be analysed usefully
by SVD from measurements by the current system.
The higher components show signs of the period doubling (see Fig. 8.15 which is discussed in
the appendix). This observation uncovers a problem of the application of SVD. It is clear that due to the line-integrated character of the measurements the period doubling is an effect that cannot be separated in the physical sense, although SVD separates them as orthogonal
"mathematica!" modes. This indicates that even the largest componentsof SVD might nothave
201
Clulpter 8 Measurements of MHD activity
a distinct physical meaning and that one has to be very careful with application of SVD to line
integrated measurements. SVD applied on the measurements of MHD activity only confirms
what had been found by other methods, but does not contribute considerably to the understanding.
8.7 Analysis of edge channels
The analyses so far have shown that it is plausible that the emitting structures do not rotate, but
rather brighten and dim when a rotating structure passes by. This can be studiedinmore detail by looking at the raw signals of channels that view near the main asymmetrie peaks in the emission.
As is clear from section 8.5, the correlation analysis gives complex results, and limiting oneself to the edge channels cannot resolve the entire problem. Three cameras view the asymmetrie
peaks, which are associated with an increased neutral hydragen density, with approximately tangent views to the q = 2 surface and outside this surface. The location of positive electron
density perturbations, which are related to the location of the island structures, was determined
from the interferometer measurements and the helical field lines. For the peaks on the LFS, up and down, it is found that the emission increases at the location of a positive electron density
perturbation. However, in the peak on the HFS the results point clearly at the opposite: the
emission is maximum when the electron density is minimum. This is a behaviour that one would not expect, and the differences between the peaks on the HFS and LFS are puzzling. lt
should be remarked that the neutral hydragen density at the radius of the island is very smal!
(see subsection 7 .1.2.2), and therefore the emissivity perturbation is mainly induced at the edge of the island. The electron density perturbation is different on the LFS and HFS (see Fig. 8.5):
it is smaller on the LFS. Therefore, the dependenee on electron density would be expected to be largeston the HFS, which clearly is not the case. Possibly the electron density does not play a major role after all, but it is a neutral hydragen influx modulated by the passing magnetic is
lands that is important. In section 8.3 it was found that it is hard to explain the asciilation amplitude of the visible light emission by the electron density perturbations alone. Further indications
that the neutral hydragen density is modulated is given by a couple of channels viewing the far edge (p > 0.15 m) [Fig. 8.4(c)] which are in opposite phase with channels inside this radius. Theemission in the edge is minimum when the positively perturbed electron density passes by. Although both the electron density and temperature have not been determined in the far edge,
they are not expected to asciilate much. The emissivity fluctuations at different radii and positions, therefore, seem to have different causes. The most plausible explanation for the fluctua
tion in the far edge is that the neutral hydragen density is significantly influenced by the magnetic island structures, for example by enhanced partiele transport around the magnetic islands [Mari95]. Another explanation is that the magnetic field fluctuation might influence the amount of recycling from the wall or limiter. For the limiter this could result in asymmetries in the
202
Analysis of edge channels 8. 7
fluctuation of emissivity, for example by H! molecules created near the limiter. In the case that
the light emitted in the entire visible range is measured, fluctuations of camparabie magnitude as
for Ha are seen, but with an even more complicated relative phase behaviour.
8.8 Summary and conclusions
During MHD activity large oscillations in visible light emission are observed in RTP. The
measurements have been analysed by several methods. Most attention has been devoted to the
Ha emission during the presence of m = 2, n = 1 islands. The magnitude of the asciilation amplitude of Ha emission cannot be caused completely by the electron density perturbations.
The reasans for this are that the maximum electron density perturbation is not sufficient to
explain the fluctuating emission, that the maximum density perturbation occurs at a radius
where the neutral hydragen density is expected to be smal!, and that the phase behaviour be
tween electron density and emission is complicated, as is discussed below. Temperature effects
are only expected to play a role at the far edge, where, ho"Vever, the temperature is hardly perturbed. The spatial distri bution of the ac part of theemission has been analysed by a number of
methods. Tomographic reconstructions give a rough indication of the structures and seem to
indicate that the emitting structures do not rotate, but oscillate. Correlation analysis to deterrnine
the relative phases between the channels and SVD are consistent with this observation, as fol
lows from phantom calculations. It is found that the results of correlation analysis and SVD
applied to visible light signals are hard to interpret, partly due to their line-integrated nature, and partly due to the inhomogeneous lay-out of the system. Simulations with phantoms that assume
an asymmetrie static neutral hydragen density based on tomographic reconstructions of the
time-averaged Ha emission give similar results as analysis of the measurements. The results
show that the structures that emit visible light do not correspond spatially with the island struc
tures observed by other diagnostics, whilst there exists a clear correlation in time. It is found,
by camparing the phases of channels viewing the edge of the plasma, that in some parts the
emission increases with increasing electron density, but on one location it has the opposite
behaviour. Furthermore, in the far edge of the plasma, the phase is seen to invert, which indi
cates that the neutral hydrogen density might be significantly affected by the MHD activity.
Analysis of discharges for different conditions, for example different qa, and different optica!
filters have not improved the onderstanding of the phenomena.
The asymmetrie peaks found in the Ha emissivity severely complicate the interpretation of the
measurements by the visible-light tomography system. Furthermore, the oscillating structures
are too complex to be resolved in detail by tomographic reconstructions with this system. The
asymmetries are found to influence the structures of the emission, caused by the rotating island
structures. In the future measurements of impurity !i nes during MHD activity might give in formation about the accumulation of impurities in the islands.
203
Chapter 8 Measurements of MHD activity
An interesting observation is that the ion-neutral friction coefficient Rio is proportional toni nH
and due to quasi-neutrality to nenH. Consequently, Rio is proportional to the Ha emissivity.
Therefore, the Ha emissivity is a measure of the ion-neutral friction, which is part of the
friction term in Eq. (8.4). This observation could berelevant for the understanding of plasma rotation.
The main finding of the influence of MHD activity on the Ha emission is that it cannot be fully
explained by the electron density perturbations, and therefore the neutral hydragen density at
the edge appears to be significantly influenced by the magnetic island structures in the plasma.
Three indications that nH is influenced are: ( 1) the magnitude of the Ha oscillations cannot be fully explained by fluctuations in the electron density, (2) the phantom basedon the time-aver
aged asymmetrie emission profile only gives partial agreement in the phase differences between
the channels, (3) in some locations the asciilation cannot be explained by a fluctuating electron
density, and (4) oscillations in emission occur also outside the last-closed flux surface. Coher
ent fluctuations in visible light measurements such as during MHD activity are largely influenced by the asymmetrie neutral hydragen profile. In the next chapter stochastic fluctuations as measured by the system are analysed.
Appendix S.A Chopper spike removal by SVD
As was indicated in subsection 4.2.4.3, the visible-Iight tomography diagnostic suffers from piek-up from the chopper-controlled position feedback system. Even though a significant
reduction of piek-up was achieved by electromagnetic shielding, the chopper spikes are clearly visible when the signa! level is low, for instanee when optica! filters are used. In this appendix
the chopper spikes are characterized, the influence on data analysis is discussed, and a tech
nique to filter out the chopper spikes from the signals without losing useful information is proposed. Different cases are studied where the remaval is successful, and where it is marginal or
impossible. The condusion of the study is that the SVD filtering only works properly under certain strict conditions and that it is only useful in a limited number of cases. Unfortunately
this was not the case for the Ha measurements. Therefore, no chopper spike filtering could be
applied for these.
8.A.l Characterization of chopper spikes
The chopper spikes are significant forms of interterenee on the signals of the visible-light
tomography system. The chopper spikescan be averaged out for steady-state analysis, and, for temporal analysis in short time windows, the window can be chosen such as to lie between
successive chopper spikes. The chopper spikes severely hamper the analysis of the signals by correlation analysis where Jonger timewindowsare necessary. Often, the chopper spikes are so close tagether (frequency 1 to 2 kHz) that there is hardly any unperturbed part for meaningful
204
Appendix 8.A
analysis (0.1 to 0.7 ms unperturbed). The chopper spikes have amplitudes of I 00 m V in the
most affected channels (for most channels it is less than 20mV), where the signatievel is of the order of 20 m V in the case of optica! filters, and of the order of I V without filters. The
magnitudes and temporal behaviour of the spikes vary between discharges and even in discharges, and large differences in magnitude and character exist between various channels.
Therefore, filter methods based on recognition of non-ebanging characteristics have failed. The
chopper spikes are partly fittered out by the digital filtering for low sampling frequencies, but in
cases where the signals have to be studied at high sampling frequencies the chopper spikes are major obstacles in the analysis.
Here it is described how singular value decomposition (SVD) can remove the effects of the
chopper spikes in the signals. The idea that chopper spike removal could be possible is based on the fact that the chopper spikes have features in the signals very distinct from features
sterruning from the plasma emissivity: they happen at the same time on all channels, they show
a peak and oscillations, and each channel measures roughly the same characteristic relative amplitude for different conditions which facilitates the recognition of their features in the "spatial" structure. In subsection 3.4.2 the methods of singular value decomposition and filter
ing were descri bed. Because of the d.istinct features of the chopper spikes in the signals, they are, under favourable conditions, expected to show up in a smal! number of singular compo
nents that only contain little information about the plasma emissivity. Filtering out these com
ponents would accomplish chopper-spike removal. If the main singular components are unintluenced by the chopper spikes, these components that contain the main information about the signals can be used for the analysis. For the results shown in this appendix SVD was applied
after average subtraction and sealing of the signals to their standard deviation, as discussed in
subsection 3.4.2. As a consequence, the first component shows the most significant tluctua
tions, instead of the average values. Usually the SVD is doneon data consisting of all channels
and 500 to 2000 time slices.
To find the characteristics of the chopper spikes and how these show up in the SVD, a quiescent discharge with exceptionally large chopper spikes was analysed. Figure 8.13(a) shows a
time trace of one channel that measures a small signa!. The wiggle starting ju st before t = 208.5 ms is related to the switch-on of the chopper, the wiggle at t"" 208.65 ms with the switch-off; and similar for the second chopper spike shown. In this discharge the unperturbed time between the chopper spikes in which the signals can be analysed is very short. Figure 8. l3(b) shows the first chrono of the SVD. The chrono, which is an average over all channels contributing to this component, shows the same behaviour as the time trace in Fig. 8.13(a).
Figure 8.14(a) shows the amplitudes of the chopper spikes for most channels, as determined
from the raw signals. The first topo [Fig. 8.14(b)], which gives the "spatial behaviour" of the chopper spike, is clearly related to the amplitudes. In the case of external interference such as chopper spikes, the spatial behaviour does not give relations in the plasma emission as seen by
205
Chapter 8 Measurements of MHD activity
0.1 ~ (ij c: Cl 0.0 (ii Q)
:0 (ii ·:;
-0.1
~ ( b)
:i ~ 0 0.0 c: e .c: 0
208.5 209.0 time (ms)
Figure 8.13 (a) Timetrace of one channel during the presence of chopper spikes (indicated by black bars). (b)
First chrono obtained by SVD.
the channels, and therefore are not really "spatial". An interesting observation is made. The
topos sametimes have negative values. From the conesponding time traces it was found that
indeed the sign of the spikes was reversed, which is nottaken into account in Fig. 8.14(a). This can be explained as follows. In subsection 4.2.4.3 the origin of the piek-up was dis
cussed: most probably it is the voltage induced in the wires between the pre-amplifier and the
detectors by the fluctuating magnetic field. The varying magnitude in piek-up between channels is probably due to the varying lengths of the wires and the varying area perpendicular to the
field that is spanned by the wires. Differences between camera boxes are due to varying lengths
of cables, differences in shielding and positioning with respect to the field. The reversal of sign is probably due to a twist of the wires.
8.A.2 SVD filtering of chopper spikes
In the example in the previous subsection the chopper spikes showed up in the first components
of the SVD because they formed the main perturbation. However, the effect of the chopper spikes is not confined to one component. Because the chopper spikes are violent processes
containing several different frequencies (the power spectrum contains some wide-band win
dows) which influence different detectors in unequal ways, also all succeeding components contain effects of the chopper spikes. Therefore, the only possibility for useful filtering is when
the first components contain other phenomena than the chopper spikes and are not perturbed by
it. This is only the case if there are large fluctuations in the plasma, as is the case with MHD activity, as is illustrated in Figs. 8.12 and 8.15. The third and fourth chronos and topos (Fig. 8.15) do notshow as distinct features as in the previous subsection, probably due to mixing of
206
Appendix 8.A
~ 0.3
(a) Q) 0.2 'C :::J
·"" a. E <ll 0.1 Q)
..>< 'ä.
0.0 Cf)
:i ~ 0 0 a. .9
A B c detector tor viewing direction
Figure 8.14 (a) Peak-to-peak magnitude of chopper spikes as determined from the raw measurements for the
detectors (some channels are missing). (b) First topo obtained by SVD.
the chopper-related phenomena with plasma phenomena (such a frequency doubling). How
ever, the third and fourth chronos show peaks that are correlated with the choppers.
When the fluctuations are not large enough, as for the measurements with optica! filters, SVD
does not properly distinguish between the plasma phenomena: the chopper spikes show up already in the second component, often mixed with features of the plasma oscillation. If the
chopper spikes are large, components with a behaviour similar to that in the previous subsec
tionis observed. lt has also been found that in the case of pellet injection during MHD activity,
the sudden increase of signals on all channels, which is similar to the chopper spikes, mixed
these phenomena and made the chopper spikes show up in the first component. The ability to
separate modes is not significantly influenced by the sealing of the signals to their standard
deviation, nor to the chosen starting time (the phase of the chronos is determined by starting
time of the selected time window).
8.A.3 Conclusions
It has been shown that the chopper spikes usually show up in a number of SVD components of
the data and that different phenomena in the plasma mix into those components. This means that
filtering out only the chopper spikes without losing other information is virtually impossible.
Depending on the specific signals, the chopper spikes may notshow up in the main compo
nents. In such a case the first components contain the main information about the signals and
can be used for the analysis; filtering out the components containing the chopper spikes and all
components with smaller singul ar values can be useful to obtain smoother signals with the most significant information. However, if the signallevel is low, the chopper spikes may appear in a
207
Chapter 8 Measurements of MHD activity
~
:::J
~ 0 0 c 0 ..... .c: 0
86 86.5 87 87.5 88 time (ms)
( b)
::J
~ 0 0 0.. 0 -
A E
detector of viewing direction
Figure 8.15 (a) Third and fourth chronosof the SVD of large MHD activity measured without optica! filters.
The duration of the choppers being switched on is indicated by the black bars. (b) The corresponding topos. The
first two components are shown in Fig. 8.12; note that the time base is different.
dominant component. This is also the case when sudden perturbations, such as an injected pel
let, occur in the plasma. Unfortunately the signa! level was low in the case of the Ha measure
ments, and therefore no chopper spike filtering could be applied for these measurements. This
study shows that, although the application of SVD is simple, great care should be taken and
both spatial and temporal components should be studied. Furthermore distinct physical phe
nomena could scramble into the same components. The usefulness of SVD depends greatly on
the specific properties of the signals.
208
Measurements of fluctuations 9 Due to its high spatial and temporal resolution, the visible-light tomography system on RTP is able to visualize small structures that are related to fluctuations in ernissivity. The purpose of
this chapter is to give a brief overview of the literature of optica! measurements of fluctuations
(section 9.1), to describe how fluctuations in electron density and temperature affect the ernis
sion (section 9.2), and to demonstrate the existence of fluctuations of the order of 20% in the
visible light emissivity and to describe their main properties by a number of analysis methods
(section 9.3).
9.1 Fluctuation measurements
Fluctuations in various plasma parameters have been measured in tokamak plasmas. The fluc
tuations are assumed to be related to the turbulence that is thought to play an important role in transport in plasmas. Therefore, the study of fluctuations could lead to a greater understanding
of the transport [Liew85]. The fluctuations of electron density and temperature in the edge can
be locally measured by Langmuir probes, which are applied on many tokamaks. Other measurement techniques from which information about fluctuating plasma parameters can be derived
are microwave scattering, heavy ion beam probes, reflectorneters and radiation detectors.
Radialion detectors, for instanee photodiodes, have been used to measure fluctuations in Ha light [ZweM83, ZweM89, Thei90, HurF92, HurR95] and ultra-violet radiation [WenB95,
BraT95]. Typical results from radiation measurements show that the fluctuations are braadband
with short auto-correlation times, as is typical of turbulence. Typical correlation lengths are several centimetres. Spatially resolved measurements have shown that the fluctuations in time
are related to spatial filamentary structures in the plasma [ZweM89].
Measurements of emitted radialion are necessarily integrated along the viewing chord, whereas
the local fluctuation is the desired quantity. Because spectralline emission is from thin shells at
the edge of the plasma (see subsection 2.1.3), a spatiallocalization of the measurements can be obtained by focussing on the shell. Such an approximation is not valid for the RTP visible-light
tomography system, which has narrow chords through the entire plasma. Recently, correlation analysis has also been applied to line-integrated soft x-ray signals [PázG95]. In these cases the
correlations between two signals are related to a local fluctuation in a complex way because
fluctuating emission from all positions along the paths contributes to the signals [Pázs94,
Chapter 9 Measurements offluctuations
BraT95]. Therefore, the local fluctuations are partly averaged out in the signals, and the cocre
lation in the crossing point is contaminated by correlations between fluctuations along both
chords. For the heavy ion beam probe, where losses occur along the beam path (i.e. a line inte
gral), these effects have been studied by Ross et al. [RosS92]. These effects are also consid
ered in the next section.
9.2 Chord-averaged fluctuations of visible ernissivity
In this section the dependenee of fluctuations of the visible light emissivity on plasma parame
ters is studied and the effect of the chord-averaging process of measurements is considered. As
has been discussed in chapter 2, there are three contributions to visible light: line-radiation,
charge-exchange and bremsstrahlung. Assuming that fluctuations are solely caused by varia
tions in electron density and temperature, ne and Te. respectively, i.e. that the densities of atoms
and impurity ions are roughly uniform over the length and time scales of the fluctuations, one
can derive the first order effects on the local emissivities. In chapter 8 it was found that the
neutral hydrogen density fluctuates on a time scale of 0.1 ms, and therefore possibly needs to
be taken into account as wel!.
Ha radiation is the most significant contribution to theemission in the visible range. For hy
drogen a collisional radiative model is required to describe the popuiatien densities. It is found
from Eq. (2.5) and (2.9) that the local emissivity is proportional to r1 nH, where the coefficient
r 1 depends on ne (nearly linearly for low densities, see Fig. 2.2) and Te. and where nH is the
neutral hydrogen density. For the estimate of the influence of fluctuations of ne and Te on the
emiss"ivity, the functional form of the dependenee is important, but not the act u al values.
Assuming that the dependenee on Te is of approximately the same forrn as in the corona! model,
an estimate can be obtained by a description that also includes impurity ions.
The local emissivity g of line radialion from the transition from energy levelp to level q related
to the ground state density nz is approximated by Eqs. (2.5) and (2.6) on the assumption of
corona! equilibrium:
( T ) nenz -Elp/Te g ne, e oe Ir e
Elp "V Te (9.1)
where Elp is the energy difference between the levelp and the ground level. The constant of
proportionality depends on the levelspand q. The electron density and temperature can be
written as ne= ne+ fie and Te= f e + fe. respectively, where i'ie and fe are the fluctuating
(ac) parts and and ne and fe the steady-state (de) values. The first order effect of these fluctua-
tions on the relative fluctuation of the local emissivity gig is given by
I= iie + !e (~P _ _!_)· g ne Te Te 2
(9.2)
210
Chord-averaged fluctuations of visible emissivity 9.2
Jf tluctuations of nz are included, these are found to have the same dependenee as ne. Equation
(9.2) shows that the f e dependency of gig becomes very large for Te< E lp• i.e. it is only
large in the scrape-off Jayer (SOL) where Te< 10 eV. Therefore, the Te dependenee of the
fluctuations in the emissivity is Jess important than the ne dependence, both for hydragen and
impurity i ons. For charge-exchange recombination the dependenee on Te is more complicated to
model; however, it is expected to be a weak dependence. The ïîe dependenee of gig for change
exchange recombination radiation is the same as for Iine radiation.
Because the excitation and radiative decay processes take place on a nanosecond time scale, the
fluctuations in ernission can follow the fluctuations in density and temperature, which are on the
rnicrosecond time scale. The ionization processes that give rise to a change in ion densities take
place on a rnicrosecond time scale (see subsection 2.1.3), i.e. the sametime scale as the tluc
tuations that are studied, which makes the dependenee on ne and Te more complicated and re
quires transport to be taken into account. From the results of chapter 8 it can be expected that
structures rnaving in time scales of 50 I!S, and possibly much shorter, can give rise to fluctua
tions in the neutral hydragen density nH.
For bremsstrahlung a sirnilar dependenee can be derived from Eq. (2.12):
i= 2 iïe + !e [~V _ _!_J. g ÏÎe Te Te 2
(9.3)
which is valid for g in a frequency interval near v. Equation (9.3) is sirnilar to Eq. (9.2), but it
has different consequences. Firstly, the local emissivity is proportional ton~ and is therefore
peaked in the centre of the plasma where h v/Te is negligible. In the visible range h v""' 1 e V,
which means that an enhancement of Te fluctuations can only be expected in the SOL for even
lower temperatures than for Iine radiation. This means that the Te dependenee can be expected
not to be significant. Because of the ernission from the entire plasma and the strong dependenee
on ne. continuurn radiation is a good candidate for fluctuations studies. The low intensity level
and perturbations by other radialion in the edge (see section 7.2), however, have to be coped
with.
Equations (9.2) and (9.3) describe the relative fluctuations of the local ernissivity. There is no
simple relationship between the relative fluctuations JIJ of the measured power with gig be
cause the fis obtained from a line integral. The line integral also averages over the positive and
negative contributions of g in different positions. According to Eqs. (9.2) and (9.3) gig= F(ne.Te) where F designales the functions given in these equations (the following remarks
remain valid if F also depends on nH or nz). Therefore, JIJ can be written as
J _I F(ne ,Te)g(ïïe,fe)ds
f- I g(ne,fe)ds (9.4)
211
Chapter 9 Measurements offluctuations
where the Iine inlegral is taken as in Eq. (3.1) and ne and Te are functions of the position over
which is integrated. It is clear that the contri bution of the relative local fluctuations in ernissivity are weighted against the time-averaged ernissivity in the same point. Therefore, in general,
]IJ-:t: J F ds. However, there are situations in which Eq. (9.4) can be simplified to ]IJ "' J F ds. The ernission of spectrallines is confined to cylindrical shells in the plasma. Fora chord viewing the shell perpendicularly the line-integral will only be over the thickness of the shell,
making the measurement almost local. This is particularly true if the focus of the imaging is in
the shell on one side of the plasma, whereas on the other si de the fluctuations are averaged over
a large volume. Therefore, correlations in the signals are related to structures in the smal! volume near the focal point. For the RTP visible-light tomography system the chords are narrow over the entire cross-sectien and most chords do notview the radiating shells perpendicularly.
Furthermore, in subsectien 7 .1.2.2 it was shown that the thickness of the radiating layer of Ha
light is a bout 4 cm, which is 24% of the minor radius of RTP. Therefore, the simplification of
Eq. (9 .4) is not justified for most channels of the visible-light tomography system.
9.3 Analysis of visible-light measurements
In this sectien the fluctuations in the visible-light measurements are studied by a number of analysis methods: analysis of the spatie-temporal structures in the signals, temporal Fourier
analysis, correlation analysis and spatial Fourier analysis. In subsectien 6.1. 1 it was indicated that the temporal fluctuations that are measured by the system are related lo spatial structures
(Fig. 6.1 ). The structures that are found to give rise to the fluctuations are too small to be resolved by tomography: the structures are smoothed away. Furthermore, although tomographic
reconstructions of the fluctuating part of the signals J is possible in principle, the averaging of
the fluctuations over the chord width is different for different detectors and cannot be taken into account properly in tomographic reconstructions (the inverse eperation of the averaging is not possible). As was discussed in the previous section, it is not useful to tomographically invert
]IJ.
The chopper spikes that were characterized in Appendix 8.A form a major problem in the present analysis. For statistica! analysis a large number of time slices is needed, but time windows
of only some tenths of rnilliseconds exist in which the measurements are unperturbed by chopper spikes. Therefore, the temporal analysis has been confined to the channels with high signa!
levels relatively unperturbed by chopper spikes, or to time windows carefully selected between chopper spikes. To obtain a good signal-to-noise ratio, the measurements selec ted for analysis are in the full speetral range to which the detectors are sensitive. Furthermore, the analysis has been restricted to an Ohmic discharge during the flat top, sampled at the highest sampling frequency possible: 0.5 MHz. Characteristics of the discharge are given in Table 9.1.
212
A na lysis of visible-Light measurements 9.3
9.3.1 Spatio-temporal structures
Before turning to a quantitative analysis of the fluctuations in the visible-Iight signals in the
following subsections, some observations can be made about the "spatio-temporal" behaviour of the signals. Figure 9.1 (a) shows a contour plot of the relative standard deviation CTreJ , i.e . the
standard deviation of each signa! divided by the time-averaged signals, as a function of time and
channel number. The average c:Trel for many plasma conditions for a sampling frequency of
( a ) 153.65
153.60
en 153.55 s <D E 153.50
:;:::
153.45
153.40
153.35
detectors of viewing direction
( b) 30 ( c) 30
l1 ·§ 20 ..9:
.!!! c: 'ö 20 ..9:
Q)
E 10 :;:::
Q)
E 10 :;:::
0 0 detectors detectors
Figure 9.1 Relative standard deviation of the visible-light signals as a function of time and channel number.
(a) A typical measurement. (b) A si mulation with structures of sizes that just can be resolved by the system. (c)
A simulation with struc tures significantly smaller than the resolution. In (a) the channels are numbered consecu
tively; the separalions between cameras show up as discontinuities along four vertical lines. Some channels of
cameras Band C were nottaken into consideration, hence the zero band. In (b) and (c) only the values for one
viewing direction are shown. Note that to enhance the visibility of the structures the grey scale is nonlinear.
2 13
Chapter 9 Measurements ofjluctuations
Table 9.1 Plasma parameters of the main discharge under consideration in this chapter (rl9941208.0 17).
Quantity Symbol Value
Plasma current lp 55 kA
Safety factor at edge qa 7.7
Electron density in centre ne,max 3.2 x 1019 m-3
Electron temperature in centre Te,max 0.7 keV
0.5 MHz is about 4%, which is significant (the system noise is less than 1 %). For varying
plasma conditions no major differences in behaviour have been observed, but this has notbeen
studied systematically. A large number of structures is apparent from Fig. 9.1 (a). Horizontal features are related to fluctuations seen on several channels at the sarne time, as was also appar
ent from Fig. 6.1. Features that are at angles in the time-channel space can be related to moving structures in the plasma. The angle is related to the velocity perpendicular to the viewing direction. Because the cameras view the plasma at different angles and because the viewing chords
of one camera are not perpendicular, the interpretation of the features is complicated. To in vestigale the significanee of the features simulations have been done.
To simulate fluctuating structures in the local emissivity, random Gaussian noise was added to the phantom. Note that this is different from the noise added in the simulations in chapter 6,
where the noise was added to the pseudo-measurement. In the present simulations the latter
noise would result in uncorrelated pseudo-signals between channels. The fluctuations added to the phantom represent structures present in a given time slice. The fluctuations, and therefore
structures in the emission, are uncorrelated in time, which will cause narrow horizontal bands
in time-channel plots. The size of the fluctuating structures added in the simulations is determined by the grid size. For the measured weight matrix the grid size is fixed. Therefore, simulations have also been done with line integrals, i.e. zero chord width, on an adjustable grid. Table 9.Il lists the three grids that have been used. The cylindrically symmetrie hollow phantom of Fig. 6.3(a) was chosen for the simulations because it approximates the shapes of actual
emission profiles.
Figures 9.1 (b) and ( c) show O'rel for two simulations, one with a grid corresponding to the measured weight matrix (which roughly agrees with the optica! resolution of the system), and one with a much finer grid, respectively. Figure 9.l(b) shows that adjacent channels are correlated, whereas the correlation has disappeared in Fig. 9.1(c). Adding 20% Gaussian fluctua
tions to the phantom resulted in chord-averaged fluctuations similar to the ones measured ( O'rel "'4% ) . The simulations [Fig. 9.l(b-c)] demonstra te convincingly that fluctuating emitting structures in the plasma can be responsible for the features observed in Fig. 9. l(a).
214
Analysis of visible-light measurements 9.3
Table 9.11 Wave vector corresponding to the size of structures in simulations.
Simulation type Grid Structure size (m) Wave vector (m-1)
Line integrals 50x50 6.8 x IQ-3 9.2xl02
Measured weight matrix 26x26 1.4 x Io-2 4 .5xl02
Line integrals !Ox!O 3.4 x IQ-2 1.9xl02
Figure 9.2(a) shows the time-averaged signa! and its standard deviation of a typical RTP dis
charge (see Table 9.1). The values of O'rel are displayed in Fig. 9.2(b). The average O'rel is about
4%, but large deviations between channels occur between 2% and 7%. Simulations have been
carried out to check whether these differences between channels can be explained by geometri
cal effects or whether they must result from different fluctuation levels of theemission in differ
ent positions in the plasma. Figure 9.2(b) shows also the results from the simulation of the
c 0
:.;:::; ctl
Q)s > Q) ·- "0 0 5 êiî ?F-- "0-~ ~
"0 c ctl Ui
!· \
\. \
~ detectors of viewing direction
0 I ::r ,. 0
/ a. / l)J
(i)
e~
'?> c
E 0
Figure 9.2 (a) Typical time-averaged scaled signa! for all detectors. The error bars indicate the standard devia·
tion. (b) The relative standard deviations for the signa! of (a) (solid line) and fora simulation using the measured
weight matrix and assuming 20% of Gaussian noise in the phantom (dashed line). The dot-dashed line represents
the inverse of the area given by the chord width and the chord length through a phantom.
215
Chapter 9 Measurements ofjluctuations
phantom described above with the measured weight matrix. Some features, such as the be
haviour in cameras A, B and E, are reproduced by this simulation. Although the simulation was
done with a hollow phantom, the edges rnight nothave been steep enough to simulate the actual
emission profile. Differences between channels in the averaging process along the viewing
chord and over the chord width rnight be the cause for the observed variation in <Yrel· The averaging is approximately proportional to the measuring area, i.e. chord length multiplied by chord
width. Therefore, the fluctuation level can be expected to be inversely proportional to the area.
The relative chord widths were determined from the measured weight matrix by the moments
metbod described in subsection 5.4.3. The ernission profile was approximated by a radiating
ring between radius 0.14 < r < 0.18 m (see Fig. 7.5), and the chord length through this ring
was calculated. The resulting inverse chord area is also shown in Fig. 9.2(b). The dependenee
of cameras C and D is approximated reasonably well, but the other cameras show less corre
spondence. Of course, the asymmetries in the ernission profiles found in chapter 7 complicate
the comparison of the simulations with the ex perimental values and is probably a reason for the
discrepancies between simulations and measurements. It appears that most of the differences in <Yrel between the channels can be explained by chord-averaging effects. Simulations with chords
having zero width show less structure between channels, and hence the chord width seems to
be a significant effect. It can be concluded that the fluctuation level seems to be roughly propor
tional to the emission level. Because changes in <Yrel are observed for varying plasma condi
tions, a contribution to the differences between channels by position dependent fluctuation
levels of theemission in the plasma cannot be ruled out completely. No detailed study of the
dependenee on plasma conditions has been undertaken yet.
9.3.2 Temporal Foorier analysis
Figure 9.3 shows the power frequency spectrum [Eq. (3 .58)] of a visible-Iight signa] during
the discharge analysed in this section. A signa! was selected that did not suffer from chopper spikes. The power spectrum [Eq. (3.58)] is broadband: no distinct Jocalized features are ob
served at any frequency. Incoherent noise would result in a 1if slope in the spectrum. In Fig.
9.3 a line with slope 1/f is drawn through the spectrum above 100kHz, which probably corre
sponds to the noise level of the electronics. This indicates that between 1 and I 00 kHz, where
the spectrum is significantly above the 1/fline, braadband turbulent fluctuations give rise to an increase in the power spectrum. This is in agreement with the literature, where it is found that
the fluctuations have mainly frequencies below 100kHz [ZweM83, HurH92]. The power
spectrum of signals with appreciable chopper spikes show braadband structures on top of the
spectrum shown in Fig. 9.3. This means that chopper spikes cannot be fittered out in the fre
quency domain without also losing the effects of plasma turbulence.
216
::J
~ E ::J .... ü a.l 0. (J)
(i; ~ 0 0.
10°
10-1
10-2
1 o-3
10-4 102
Analysis of visible-light measurements 9.3
1 o4·
frequency (Hz)
Figure 9.3 Power spectrum of a visibk-light signa! during an approximateiy constant phase during an Ohmic
discharge. The line indicates the 1/fslope.
9.3.3 Correlation analysis
To characterize the fluctuations that give rise to the braadband power spectrum of Fig. 9.3, temporal correlation analysis has been applied. Again, only channels notsuffering from chop
per spikes could be analysed well, because the chopper spikes give rise to significant correlations between channels that cannot be distinguished from the correlations of unperturbed signals. Equation (3.59) was used to calculate the auto-correlation of signals and the cross-earre
lation between signals. Figure 9.4 shows some typical auto and cross-correlations of channels of one camera. Both the auto and cross-correlation drop to a value smaller than 0.2 within a
time-lag of 40 ~s, which apparently is the correlation time of the fluctuations. Assuming the
correlations to be caused by structures along the field lines with a toroidal rotatien frequency of 5 x I 04 m/s (see subsectien 8.1.4 ), the toroidal correlation length is at maximum 2 m. These values are of the same order as for measurements on other tokamaks [ZweM83, HurH92,
Thei90]. Small oscillations for larger time-lagscan be observed, but it is unclear whether these are related to structures that are coherent on a time scale of 0.5 ms.
Also spatial correlations have been studied by calculating the temporal cross-correlations between channels. The cross-correlations between all channels and one channel not suffering
from chopper spikes were calculated. In this way the chopper spikes should not contaminate the
cross-correlations too much. The cross-correlation Pij(O) at time-lag zero with channel E80 are shown in Fig. 9.5. Channel E80 views approximately through the centre of the plasma and intersects most chords. For nearly all channels the cross-correlation at time-lag zero is the maximum, i.e. Pu(O) == max[Pij(T)]. Only a few channels show a slight increase toa maximum
217
Chapter 9 Measurements offluctuations
1.0
0.8 - - - - - - - cross E76 E65
0.6 ---auto E80 c 0 0.4
'@ ------- cross ESO E76
äi 0.2 ..... ..... 0 ü
0.0
-0.2
-0.4 0.00 0.05 0.10 0.15 0.20 0.25
time lag (ms)
Figure 9.4 Typical auto and cross-correlations for some channels.
at a positive or negative time-lag of less than 10 !-LS (see, for example, the curve for the cross
correlation between channels E80 and E76 in Fig. 9.4), which has been neglected.
To enable general statements, correlation plots like Fig. 9.5 have been calculated fora number
of channels of all cameras. Care had to be taken of chopper spikes: the time between chopper spikes is sufficient to calculate a rough estimate of Pij(O). Significant correlations between most
channels are observed. The cross-correlatioris are particularly large [PiJ(O) > 0.5] for adjacent chords and for chords that cross in the edge of the plasma (for example some channels of
camera D in Fig. 9.5), where most light is emitted. Chords that cross in the centre of the plasma
have PiJ(O)"" 0.4, whereas chords that do not cross inside plasma have lower, but still significant, correlations. The edge channels of camera B that view outside the limiter radius do
not correlate with any channel (see Fig. 9.5), which indicates that, although the relative
fluctuations level of those channels is larger than for the other channels (but not caused by electronic noise), the fluctuations in the SOL are not correlated with those inside the limiter
radius. In Fig. 9.5 a steep drop in correlation between channel E80 and nearby channels is
seen, changing to a different behaviour around channel E72. Because the viewing chords are not parallel, and therefore closer on the front side of the plasma than on the opposite, only an
estimate of the correlation length in radial or poloidal direction can be given: (6 ± 2) cm. In
radial direction lhe determination is lirnited by the width of the neutral hydrogen layer. Si mi lar estimates are obtained from correlations with other channels than E80. Assuming that the
structures are along the field lines and moving with the plasma, their size suggests that the toroidal mode number m [Eq. (8.1)] is between 12 and 25. The safety factor at the edge, qa. for lhis discharge was 7.7, which means that the corresponding poloidal mode number nis around
2 to 3. The found correlation length is not resolved in w, and therefore can vary between
218
Analysis of visible-light measurements 9.3
1.0
0.8 c 0
~ 0.6 ê 0 (.)
0.4 CIJ CIJ e (.)
0.2
0.0 E t channel E80 detectors of viewing direction
Figure 9.5 Cross-correlations of all channels with channel E80 at time-lag zero.
different frequency ranges. The results are in agreement with data from other tokamaks
analysed by similar techniques. From the coherence at different frequencies on the Caltech
tokamak Zweben et al. [ZweM83] found that the poloidal correlation length is only 2 cm at
I 00 kHz, but between 5 and 7 cm for lower frequencies (20 and 50kHz). In ASDEX somewhat shorter correlation lengths have obtained [Thei90].
Correlation analysis can quantify the correlation between different measurements. For line-inte
grated measurements the cross-correlation between two signals can give information about
fluctuations in and near to the overlap region of the two chords. In the overlap region the corre
lations are related to fluctuations of emissivity, whereas near the overlap region the fluctuations are due to small moving coherent structures. For line-integrated measurements the amplitude of
the correlation resulting from a small overlap volume is relatively smal! because it is attenuated
by the averaging process along the viewing chord. An estimate for the coherence between local
and line-integrated measurements is given in Refs. [BraT95, RosS92]; the coherence between
two line-integrated measurements will be even smaller. The significant correlations found in the
measurements therefore indicate that the local fluctuations must be very large. In subsection 9.3.1 fluctuations with a standard deviation of 20% were found to agree with the measure
ments, which means that fluctuation amplitudes of up to 60% are frequent.
The visible-light tomography system on RTP is unique in its ability to take fluctuation meas
urements from several directions. However, as the interpretation of Fig. 9.5 and equivalent cal
culations for other channels show, it is quite hard to localize the fluctuations by studying the cross-correlations between channels that are not adjacent. The large cross-correlations between
channels from different viewing directions suggest that a more thorough analysis could be
219
Clwpter 9 Measurements offluctuations
worthwhile. So far, methods to obtain a two-dimensîonal structure by assigning the cross-cor
relation or coherence between the chords to each crossing point (see for example Ref.
[CasC94]) have been unsuccessful to visualize the structures that give rise to the correlations.
9.3.4 Spatial Foorier analysis
In the previous subsection an upper limit of (6 ± 2) cm was found for the size of the fluctuat
ing structures. Fourier analysis in the space domain can give additional information, which will
be called k spectrum, k being the wave number. The number of channels per viewing direction
is 16, which gives only nine k points due to the Nyquist limit. The resolution, therefore, is
rather lirnited, but can still give useful information [Thei90, HurR90].
A complication in the visible-light tomography system is that the chords per viewing direction
are not parallel. The wave number used in the current analysis is deterrnined from Eq. (3.57)
taking into account the difference in impact parameters p of each chord. This results in an aver
age chord separation between the plasma edge close to the camera and the edge far away, and
hence an average k. The k power spectrum is obtained from Eq. (3.54) and Eq. (3.58). It is calculated for many time slices (between chopper spikes) during a non-ebanging phase in the
plasmaand averaged in time. The average gives a better insight into significant features in the
spectra because noise in individual spectra is averaged out. Power spectra have been calculated
for simulations and the measurements (Fig. 9.6).
In the simulations the phantom and noise as described in subsection 9.3.1 was used. The initia!
slopes for k < 100 m-1 behave as can be expected: steeper forsmaller structures (see Table
9.11 for the relation between grid and structm:.: size). The reason that the curve for the finest
grid is less steep than the other ones for k > 100 m- 1 is alîasing of the high spatial frequency
structures to lower frequencîes. Usually, averaging over the chord width acts as spatial filtering
to limit the bandwidth, but this simulation was done with line integrals over chords with zero
width. In the spectrum of the coarsest grid a feature is seen at k"' 200 m-1. This feature is
probably related to the size of the structures in the simulation (see Table 9.11). If no or less
noise is included in the simulations, steeper slopes are obtained for smal! k. However, for large
k, for instanee k > 250 m-1 in the simulation of camera E with the measured weight matrix,
the k spectrum jumps to almast the same value as in the simulation with noise, which indicates
that simulation noise related to the grid size plays a role.
The spectra of the measurements show slopes that are sirnilar for the four cameras. The small
differences in slopes for different cameras are caused by the different p dependenee of chords
per camera, and different angles at which the cameras view the plasma. Similar differences
between camerasin slopes and absolute values (whîch depend on theemission profile) are ob
tained in simulations. The slope of the curve for camera E is sirnilar to the slopes in the simulations, in particular to the coarse grid. The slight bending upward of the curves at high k values
220
:J 1 o-1 ~ E 2 t5 2 QJ 10-0.. Cl)
L.. QJ
~ g_ 1 o-3
0.
measured 6-.
---o---B
·--~--- D
·- -6--- E
-+---- 50x50
------ 26x26 ---10x10
100
Analysis of visible-light measurements 9.3
'o
600
Figure 9.6 Spatial power spectra of simulations (solid lines) and ex perimental data ( dashes lines). The simula
tions show the spectra calculated for camera E for three grid sizes. The ex perimental curves are for cameras A. B,
D and E.
is probably due to aliasing (see for example Ref. [Thei90]), but it is less severe than in the simulation with the fine grid. Cameras A and B show a feature around k = 400 m~1 . Although
some influence from aliasing cannot be totally excluded, it seems likely that this feature corresponds to structures of size 1.6 cm. Because the slopes of experimental data are less steep than simulations without noise and si mi lar to slopes of simulations with 20% relative noise added,
there is evidence that the fluctuations in the measurements are caused by fluctuating structures of the size between 1 and several centimetres. Is it difficult to obtain precise estimates of the sizes, but probably the sizes vary, resulting in a broadband k spectrum.
The spatial spectra studied in this subsection are averages in time, and hence over all frequencies. In the literature where S(k,W) is studied [HurR95, Thei90] a dependenee of the
k spectrum on wis found, which varies between different plasma conditions. The modelling in the simulations in this sectionare only adequate in k, but no correlations exist in time. In the lit
erature modelsof S(k,w) have been descri bed, for example the "random-phase" model by Carl
son [Carl9l].
221
Chapter 9 Measurements offluctuations
9.4 Conclusions
Large fluctuations in the visible light emission have been measured, which are found to be well modelled by Gaussian fluctuations with a standard deviation of 20%. The fluctuations show a braadband temporal power spectrum between 1 and 100kHz. The structures giving rise to the
fluctuations are estimated to have sizes between 1 and 6 cm. Evidence has been found of
structures of 1.6 cm in size. The lower limit result from the observation that there is not much aliasing in the k power spectrum. The opper limit results from the found correlation lengths. The results are in agreement with the literature. It is likely that the large structures, which are relatively large compared to the plasma size (a= 0.164 m), correspond to low temporal
frequencies, and the smal! structures to high frequencies. An S(k,OJ) analysis would be useful to verify this expectation. It could be interesting to correlate the visible-light measurements with
the chord-averaged electron density measurements of the interferometer, which is located toroidally 45° from the tomography system, corresponding to about 0.55 m.
The correlation time for the signa! is about 40 !J.S, which gives a toroidal correlation 1ength of 2 m. The results obtained are similar to what has been found other tokamaks, but the possibil
ity to analyse chords from different viewing directions is unique. Large correlations between
viewing directions occur, especially if the chords cross in the edge region. Fluctuations in the
SOL are found to be uncorrelated with the fluctuations occurring inside the limiter radius. The analysis by cross-correlations between crossing chords is promising, but needs to be worked out further, for example by calculating the spatial coherence.
Usually is is assumed that the fluctuations in emissivity are mainly caused by fluctuations in electron density [ZweM89]. This is plausible because large correlations are found with Lang
muir probe measurements [ZweM83, Thei90]. However, the fluctuations of the neutral hydra
gen density at 10kHz found in chapter 8 justify a more thorough investigation whether fluctuations in the neutral hydragen density are also relevant for the type of fluctuations studied in
this chapter. It would be interesting to delermine whether the fluctuations can be coupled to the recycling.
The visible-light diagnostic is valuable for fluctuation measurements. However, the interpretation is hampered by the chopper spike interference. In the future the analysis could focus on the spatiallocalization of the fluctuations from the measurements from several viewing directions. The current study is not detailed enough to show differences in size of the fluctuating structures between !ow-field and high-field side, which can be expected (for example if the size is related to the ion Larmor radius) . Zweben et al. [ZweM89] found that the poloidal wavelength is related to the safety factor at the edge, i.e. the m number at the edge (see subsection 8.1.1 ). The diagnostic has the potential to study these effects in detail.
222
Summary and recommendations 10 Section 10.1 surnmarizes the features of, and interpretation methods used for the visible-light tomography system on RTP, and the main results of measurements. In section 10.2 a number
of improvements to the diagnostic and analysis methods is described, and experiments that could be interesting are proposed.
10.1 Summary
A diagnostic system for visible-light tomography bas been constructed on the RTP tokamak.
The diagnostic measures the light in the wavelength range 300 to 1100 nm emitted in one
poloidal cross-section. Optica! filters can be used to select a smaller wavelength range. The
system has been absolutely calibrated for two wavelength ranges: to measure the Ha line and
continuurn radiation. The plasma is viewed from five directions with 16 detectors each. The light is collected by imaging systems. The light-gatbering system with optica] imaging systems close to the plasma is efficient compared to pin-hole systems and lens systems far from the
plasma, which are commonly used in similar systems. Due to the imaging a sufficient number of photons is detected to enable the use of fast electranies (bandwidth 200kHz). Conse
quently, the system can be used to study high-frequency fluctuations in emission.
Viewing dumps and shields have been installed to avoid reflections on the tokamak vessel walls
and light reaching the detectors without being imaged by the imaging systems. These preeautions function properly for all but a few channels, which have to be treated with care. The application of interference filters to select speetral lines has been studied. Due to the relatively
large angles of incidence in the imaging systems, the minimum full-width-half-maximum (FWHM) of the interference filter is limited to 5 to 10 nm. For the Ha filters with a FWHM of I 0 nm it was necessary to take into account the angle-of-incidence properties in detail (the
effects are of the order of several percent difference between detectors). The main cause for this is that the maximum transmission in the actual filters decreases significantly for increasing angle of incidence. Many channels of the system suffer from electronic piek-up caused by the toka
mak control systems, which has not been possible to reduce sufficiently by shielding or by filtering techniques. Analysis in this thesis has been restricted to time windows unaffected by the
interference.
Chapter JO Summary and recommendations
The line-integrated measurements by all detectors can be inverted by tomographic techniques to
give the local emissivities. It is necessary to describe the imaging properties properly, because
they have to be taken into account in the tomographic inversion; this description is far more
complicated than for pin-hole or other simple imaging systems. This has successfully been
done by the weight matrix, which has been measured and calculated. An alternative method is
the sealing of measurements to values that would be measured along pure line-integrals. This
works well for most channels, except for channels that suffer from reflections and channels
viewing the edge of the plasma. Relationships between both descriptions have been investigated
in detail, for example the relationship between the weight matrix and the coverage of projection
space. To the knowied ge of the author, it is the fust time that the influence of the finite detector
size on the coverage of projection space has been described in this way.
A new tomographic inversion method that in principle can cope with the complex geometrie lay
out of the system, Iterative Projection-space Reconstruction (IPR), has been developed. The
method depends on a sealing process of the measurements to values expected for line integrals,
which involves taking into account all known properties of the system. An impravement can be obtained by iteratively applying the sealing, since it depends on the reconstruction obtained, but
this does notalter the results dramatically. Another tomography method, a constrained opti
mization algorithm, has been applied successfully as well. The advantage of the latter method is
that the full weight matrix is taken into account.
Both tomographic reconstruction methods work well in conneetion with the system for phan
tom simulations and symmetrie plasmas. The lay-out of the system is not optima! for tomogra
phy: partial views and inhomogeneous coverage due to mechanica! constraints and design crite
ria complicate the application of tomography algorithms. The coverage of the system is suffi
cient to be able to reconstruct the main asymmetrie features that are found in most emission
profiles of plasmasin RTP. Comparison of the results of both tomography methods and comparison of the backcalculated signals with the measurements helps to delermine the reliability of
the features found in the reconstructions. Partial views and inhomogeneous coverage should, if
possible, be avoided in tomography systems. Whole views would increase the usefulness of
the diagnostic since they enables the comparison of raw signals without the need of assump
tions.
The main result obtained from the analysis of measurements taken by the diagnostic is the de
tailed analysis of large asymmetries in the emission profile in various wavelength ranges.
Although asymmetries have been reported on many tokamaks, the characterization from meas
urements from more than one or two directionsis new. For Ha light the thickness of the radlat
ing layer at the edge of the plasma has been found to be 4 cm, the neutral hydrogen density has been determined (4 x I016 m-3), the partiele confinement time was calculated (=- 10 ms), and
the emission during the start-up of a discharge was studied. All results are in agreement with
the literature. For the calculations the Johnson and Hinnov collisional-radiative model was
224
Summary 10.1
used, which is based on purely atomie hydrogen. Because virtually all hydragen enters the
scrape-off layer and plasma as molecular hydrogen, the dissociation and i anization processes of
molecular hydragen should be taken into account. The influence of these on the results, which
is not dramatic, has been discussed . However, because H! ionscan travel appreciable distances
befare yielding Ha radiation, they could play a role in the distri bution of light. The cause of the
asymmetrie emission profiles has not been completely resolved. Apparently the influx of
hydragen is very asymmetrie. It has been found that ionic drift and the transport of H! i ons can
be part of the cause. Nonlocal recycling on other surfaces than tbe limiter doesnotseem likely.
These findings might have implications for the interpretation of speetral measurements on other
tokamaks, which in general are less spatially resolved. Furthermore, if the emission in other
tokamaks is also similarly asymmetrie, this should be taken into account in transport models.
The continuurn light has been used to delermine the effective ionic charge, Zeff, in the centre the
plasma. Values somewhat too high compared toother determinations were found, as is usual
from bremsstrahlung measurements. Also in this wavelength range, which was quite large,
asymmetries in the emission profiles at the edge were found that cannot be due to
bremsstrahlung.
The asymmetries in the emission profiles and the partial views by the system complicate the
quantification and interpretation of measurements. In studies of MHD activity at the edge of the
plasma, which gives rise to large fluctuations in the visible-light emission, it was found that the
rotating island structures of the magnetic field does not cause rotating structures in the emis
sion, but mainly oscillations when the islands pass the asymmetrie peaks in neutral hydragen
density. Most channels are correlated to the electron density fluctuations caused by the rotating
islands, butsome channels have the opposite effect, which means that also the neutral hydragen
density fluctuates significantly. This is particularly the case in the far edge and scrape-off-layer.
The visible-light measurements give evidence for braadband fluctuations in the range 1 to
100kHz, which are related to structures in the plasma with si zes between 1 and 6 cm. In the
literature mainly measurements with channels viewing from only a single direction have been
analysed. The observed correlations between measurements from different directions (cross
correlations larger than 0.6) indicate that it might be promising to extend the study of chapter 9.
10.2 Recommendations
In the process of operating the visible-light tomography diagnostic and the analysis of its meas
urements a number of possible improvements, recommendations for future systems, and sug
gestions for further investigations have come up. The improvements suggested in this section
are divided in upgrades of the system and more long-tenn changes to the diagnostic method.
225
Chapter JO Summary and recommendations
10.2.1 Improvements of the system
Parts of hardware that might be improved or replaced are the windows and mirrors. A new design for remevabie windows has already been made, which was described in Appendix 4.A.l. Most mirrors have been damaged by the harsh conditions inside the vacuum vessel.
Since the shields have been installed the rate of darnaging is expected to be reduced; but in the near future the damage wil! have unknown effects on the measurements. The quartz mirrors
could be replaced by aluminium mirrors, which are less fragile (see Appendix 4.A.3). Further
more, the adjustable mirror holders should be designed more robustly to eosure that the mirrors remain in the same position.
In a new system the positions of the mirrors should be determined more accurately so that
three-dimensional ray tracing can be used to calculate the weight matrix. Alternatively the
weight matrix could be measured again (which is necessary after any change to the system).
The weight matrix should be known in a much larger area than the plasma cross-section to
avoid the probieros encountered with the present measured weight matrix, in which the least accurately known channels view one of the asymmetrie peaks.
The reduction of the electromagnetic interterenee is required for further fluctuation measurements. More shielding does not seem feasible, and therefore the reduction has to come from the
souree of the interference: the plasma position control system. This system is currently being
upgraded. Measurements of fluctuations at frequencies f> 100kHz require a frequency dependen! cabbration of all channels so that the amplitude and phase characteristics of the channels, which differ appreciably, are known.
When interterenee filters are applied, it would be advantageous, i.e. it would reduce uncertain
ties, if filters with a larger FWHM (at the expense of less speetral selection), or filters with a negligible change in peak transmission for non-normal incidence were used.
10.2.2 lmprovements of the diagnostic method
A general statement on "ideal" tomography systems is, additionally toa (nearly) infinite number of viewing directions and chords, that a symmetrie coverage is beneficia!. It is not only benefi
cia! for the quality of tomographic reconstructions, but also facilitates the interpretation of the raw measurements. The interpretation of the raw measurements of the current system was complicated further by the views being only partial. Whole views would enable todetermine the
totalemission from the raw measurements by Eq. (3.1 0), and the centre of mass and orientation of the emitting structure can be derived from the first and higher moments, as follows from the consistency theorem ( subsectien 3.1.5.1 ). This is the case for parallel beams, but also other
symmetrical distributions of viewing chords over whole views would enable approximations. If a symmetrie coverage is not possible, simulations with tomography algorithms should be done to delermine whether the intended configuration can resolve the required structures.
226
Recommendations 10.2
As discussed in subsection 4.2.4 optica! fibres could be advantageous. The loss of light in the
fibre would be compensated by the much lower capacitance of the single photodiodes one could
use because a higher amplification is possible. Because the electronics could be in a well
shielded environment far from the tokamak, a higher amplification would not necessarily lead to
more noise. The use of optica! fibres, however, would somewhat reduce the optica! range that
can be measured. If possible, it would be advantageous to position the fibres, equipped with
smalllens systems, in si de the tokamak vessel, in the shadow of the limiter. This would enable
an uniform coverage of the plasma, and impose virtually no limit on the number of viewing
directions nor on the number of detectors. A smal! feedthrough for all fibres would be the only
required access to the vacuum vessel. Of course, the fibres should be shielded against the hos
ti ie environment.
In the current system the requirements of high spatial and temporal resolution for fluctuation
measurements, and the requirement of the possibility to make tomographic reconstructions of
the plasma emission, led, with technica! constraints and limited access to the tokamak, to a
system that is capable of fulfilling both requirements, but neither of them optimally. The fluc
tuating structures are so small that thousands of fast detectors would be required to resolve
them by tomographic techniques, which is not feasible financially; whereas the emission pro
files that are interesting to study, for example the asymmetries, are nearly stationary. Therefore,
to study the various phenomena more in detail, the separation of the system into two would be a
good solution. The first system would have hundreds of slow detectors, which could be rela
tively cheap because the data-acquisition could be multiplexed, or alternatively CCD camera in
combination with fibres could be used. Fibres seem to be a good choice because they enable
many views without !i mits on access. Due to the relatively large acceptance angle of the fibres
the imaging could at best give chord widths of some centimetres across the plasma, which
could be sufficient. The second system would have some tens of very fast detectors viewing the
plasma with parallel beams from two or three directions. This system would be very suitable
for fluctuation measurements.
10.2.3 Suggestions for future measurements and analysis
Most of the application fieldsof the diagnostic, listed in subsection 1.3.1, have been explored
in this thesis. These application fields were: (I) MHD phenomena, (2) fluctuations, (3) Ha
emission profile, ( 4) impurity profiles, and (5) Zeff profiles. These fields are discussed below.
(1) Although the MHD phenomena show up very distinctly in the signals, the interpretation is
complicated by the asymmetrie distri bution of neutral hydrogen, and is therefore so far of
limited use. It has been found that the neutral hydrogen density is affected by the MHD
activity. It could be interesting to study the effects of MHD activity on impurity radiation.
which could give information on possible impurity accumulation in islands.
227
Chapter JO Summary and recommendations
(2) The diagnostic is capable of resolving fluctuations in the emission, which is related to smalt turbulent structures in the plasma. So far, the basic properties of the fluctuations have been studied. Extensions of the studies to two-dimensional correlations look promising. Furthermore, the visualization of the turbulent structures could clarify the relationship of the structures with the breaking up of the magnetic flux surfaces, with electrostalie fluctuations, et cetera. Measurements of continuurn radiation could give information about fluctuations in the centre of the plasma, if other contributions from the edge can be excluded.
(3) Ha emission profiles have shown large asymmetrie peaks that are not fully understood. Because the interpretation of most other measurements by this diagnostic depend on the asymmetries, and because it is important to understand the transport processes in the edge that can give rise to such asymmetries, a continued study seems advisable. Speetral measurements of the relative emissivities of other hydrogen Iines could help to verify the models used to calculate ground state density of neutral hydrogen. Furthermore, it is important to
determine whether the asymmetries in the emission profile are similar in every poloidal cross-section. Tangential measurements, for example by means of a tv camera, could help to answer that question, especially if such measurement were available in different poloidal cross-sections. Finally, it could be interesting to study the Ha emission during He plasmas to exarnine the recycling in a controlled way.
(4) Line radialion from impurities has notbeen studied yet. The light levels are expected to be low, and therefore would be pertuebed by the electromagnetic interference. Speetral information in the visible range is required to delermine which lines are good candidates to be studied. It would be interesting to verify whether asymmetries also exist in the impurity
emission profiles.
(5) Zeff has been determined with reasonable accuracy in the centre of the plasma, but due to high emission at the edge and artefacts of the reconstructions not much can be said in other locations. A spectroscopie overview of the measured speetral range is required to delermine the origin of the radialion at the edge. Discharges in He plasmas could help to delermine whether the emission is related to hydrogen. If the souree of the radialion is known, perhaps corrections could be possible to obtain Zeff profiles. It does not seem feasible to measure the bremsstrahlung in a much smaller speetral range because the detectors are not sufficiently sensitive.
Other applications of the diagnostic are to studies of th~ ablation of pellets of hydrogen ice injected into the plasma and disruptions in various wavelength ranges.
228
References
[AdaJ93]
[Akai74]
[Akim7S]
[AIIMSI]
[AlpB94]
[AntD95]
[AubG91]
[BarS77]
[BarS SI]
[BasR92]
[Beke66]
[BesM94]
[BoiHS9]
[BraSS3]
J.A. Adamset al., "The JET neutron emission profile monitor," Nucl. Instrum. Methods 329,
277 (1993)
H. Akaike, "A new look at the statistica! model identification," IEEE Trans. Automatic Control
AC-19, 716 (1974)
H. Akima, "A method of bivariate interpolation and smooth surface fitting for irregularly dis
tributed data points," ACM Trans. Math. Software 4, 148 (197S)
S.L. Allen et al., "The intluence of the limiter on euv emissions from light impurities in the Alcator A tokamak," Nucl. Fusion 21, 251 (19S1)
B. Al per et al. , "The JET multi-camera soft x-ray diagnostic," Proceedings of the 21 st EPS Con
ference on Controlled Fusion and Plasma Physics, Montpellier, 27 June-I July 1994, Ed. E. Jof
frin et al., Europhysics Conference Abstracts Vol. !SB (EPS, 1994), Part III, pp. 1304
M. Anion, M.J. Dutch, H. Weisen, "Relative calibration of photodiodes in the soft-x-ray speetral
range," Rev. Sci. Instrum. 66, 3762 (1995)
N. Aubry, R.Guyonnet and R. Lima, "Spatiotemporal analysis of complex signals: theory and
applications," J. Stat. Phys. 64, 6S3 (1991)
H.H. Barrett, W. Swindell, "Analog reconstruction methods for Lransaxial tomography," Proc.
IEEE 65, S9 (1977)
H.H. Barrett, W. Swindell, Radiological imaging, Vol.2 (Academie Press, New York, 19Sl)
V. Basiuk et al., "Profile measurements of localized fast electroos and ions in Tore Supra," Pro
ceedings of the 1992 International Conference on Plasma Physics, Innsbruck, 29 June - 3 July
1992, Ed. W. Freysinger et al., Europhysics Conference Abstracts Vol. 16C (EPS, 1992), Part I,
pp. 175
G. Bekefi, Radiation processes in plasmas (Wiley, New York, 1966)
M. Bessenrodt-Weberpals et al. , "Sawtooth filtering by singular value decomposition," Proceed
ings of the 21st EPS Conference on Controlled Fusion and Plasma Physics, Montpellier, 27
June - I July 1994, Ed. E. Joffrin et al., Europhysics Conference Abstracts Vol. !SB (EPS,
1994 ), Part lil, pp. 1312
A. Boileau et al., "The deduction of low-Z ion temperature and densities in the JET tokamak using
charge exchange recombination spectroscopy," Plasma Phys . Controlled Fusion 31, 779 (19S9)
K. Brau, S. Suckewer, S.K. Wong, "Vertical poloidal asymmetries of low-Z element radialion in
the PDX tokamak," Nucl. Fusion 23, 1657 ( 19S3)
References
[BraT95]
[BroJ92]
[BroV88]
[Büch91]
[BuoB81]
[BurW79]
R.V. Bravenec, H.Y.W. Tsui, O.W. Ross, Y. Wen, "Cross-corre1ation techniques applied to chord
integrated measurements (abstract)," Rev. Sci. Insttum. 66, 460 (1995)
D.L. Brower, Y. Jiang, W.A. Peeb1es, S. Burns, N.C. Luhmann, "App1ication of high-resalution
interferometry to plasma density measurements on TEXT-Upgrade," Rev. Sci. Insttum. 63, 4990 (1992)
A.B. 5pOHH11KOB, IO.B. BocK060HHI1KOB, B.B. nHKaJlOB, H.B. Wapanona, "CpanHeHHe
allfopHTMOn noccraHon!leHHII no neepHLIM npoeKuHnM," 31leKTPOHHoe MoJle11HpOnaHHe 10 (6). 81 (1988) [A.V. Bronnikov, Yu.E. Voskobojnikov, V.V. Pickalov, N.V. Sharapova, "Comparison
of reconstruction algorithms for fan-beam projections," Electronica! Modelling (USSR) 10 (6), 81
(1988) (in Russian)]
R. Büchse, Torrwgraphische Untersuchung interner Disruptionen in dèn Tokamaks ASDEX und
TFTR, IPP IIUI75 (Max-Planck-Institut für Plasmaphysik, Garching, 1991)
M.H. Buonocore, W.R. Brody, A. Macovski, "A Natura! pixel decomposition for two-dimensiona1
image reconstruction," lEEE Trans. Biomedical Engineering BME-28, 69 (1981)
K.H. Burrell, S.K. Wong, "Theoretica] explanation of the poloirlal asymmetry of impurity speetral
1ine emission in collisiona1 tokamak p1asmas," Nucl. Fusion 19, 1571 (1979)
[CamG86] J.F. Camacho, R.S. Granetz, "Soft x-ray tomography diagnostic for the Alcator C tokamak," Rev.
Sci. Instrum. 57, 417 ( 1986)
[CarB94] J. Carrasco et al., "Fast electron ripp1e losses during LHCD on Tore Supra: model and experi
ments," Proceedings of the 2/st EPS Conference on Controlled Fusion and Plasma Physics,
Montpellier, 27 June-I Ju1y 1994, Ed. E. Joffrin et al., Europhysics Conference Abstracts Vol.
ISB (EPS, 1994), Part I, pp. 278
[CarC95] L. Carraro et al., "lmpurity intlux diagnostics in RFX," Rev. Sci. Instrum. 66, 610 (1995)
[Carl91] A.W. Carlson, "The distribution of fluctuation measurements within the randon-phase model," J.
[CarP83]
[CasC94]
[CenE83]
[Cens81]
[Cens83]
[Chap87]
[ChaV81]
230
Appl. Phys. 70, 4028 (1991)
A.W. Carlson, "Two-point estimates of the speetral density function with finite separation and
volume," J. Appl. Phys. 70, 4033 (1991)
P.G. Carolan, V.A. Piotrowicz, "The behaviour of impurities out of corona] equilibrium," Plasma
Phys. 25, 1065 (1983)
T.A. Casper et al., "Fluctuations in high /3p plasmasin DIII-D," Proceedings of the 2/st EPS
Conference on Controlled Fusion and Plasma Physics, Montpellier, 27 June-I July 1994, Ed. E.
Joffrin et al., Europhysics Conference Abstracts Vol. 18B (EPS, 1994), Part I, pp. 190
Y. Censor, P.P.B. Eggermont, D. Gordon, "Strong underrelaxation in Kaczmarz's method for
inconsistent systems," Numer. Math. 41, 83 (1983)
Y. Censor, "Row-action methods for huge and sparse systems and their applications," SIAM
Review 23, 444 (1981)
Y. Censor, "Finite series-expansion reconstruction methods," Proc.IEEE 71, 409 (1983)
D.C. Chapeney, A handbaak of Fourier theorems (Cambridge University Press, 1987)
S. Cha, C.M. Vest, "Tomographic reconstruction of strongly refracting fields and its application
to interferometric measurements of boundary Jayers," Appl. Opt. 20, 2787 ( 1981)
[ChuB94]
[Cohe86]
[Corm63]
[Corm64]
[Cott90]
[Cowa8l]
[CruD94]
[Cruz93]
[Dean83]
[DécN86]
References
C.C. Chu et al., "High spatial resolution multiposition Thomson scattering system on RTP,"
Proceedings of the 21 stEPS Conference on Controlled Fusion and Plasma Physics, Montpellier,
27 June- 1 July 1994, Ed. E. Joffrin et al., Europhysics Conference Abstracts Vol. 18B (EPS, 1994 ), Part III, pp. 1248
S.A. Cohen, "Particle confinement and control in existing tokamaks," in Physics of plasma-wall
interactions in controlled fusion (Proceedings of NATO Advanced Study Institute, 30 July- 10
August 1984, Val-Morin, Quebec), Ed. D.E Postand R. Behrisch (Plenum Press, New York,
1986), NATO ASI Series Vol. 131, pp. 773
A.M. Cormack, "Representation of a function by its line integrals, with some radiological applica
tions," J. Appl. Phys. 34, 2722 (1963)
A.M. Cormack, "Representation of a function by its line integrals, with some radiological applica
tions. II," J. Appl. Phys. 35, 2908 (1964)
G.A. Cottrell, Maximum entropy and plasma physics, JET-P(90)04 (JET Joint Undertaking,
Abingdon, 1990)
R.D. Cowan, The theory of atomie structure and spectra (University of California Press, Berkeley,
1981)
D.F. da Cruz and A.J.H. Donné, "The soft x-ray diagnostic at the RTP tokamak," Rev. Sci.
Instrum. 65, 2295 (1994)
D.F. da Cruz, Soft x-ray emission from tokamak plasmas, Thesis, Rijksuniversiteit Utrecht, 1993
S.R. Deans, The Radon transfarm and some ofits applicalions (Wiley, New York, 1983)
R. Décoste, P. Noël, "Image reconstruction techniques for computed tomography from sparse data:
X-ray imaging on the Varennes tokamak and other applications," SPIE Vol. 661 Optica! Testing
and Metrology (1986), pp. 50
(DefM83] M Defrise, C. de Mol, "A regularized iterative algorithm for limited-angle inverse Radon trans
farm," Opt. Acta 30, 403 (1983)
[DeMM81] C. De Michel is, M. MattioJi, "Soft-x-ray spectroscopie diagnostics of laboratory plasmas," Nucl.
Fusion 21, 677 (1981)
[DeMM84] C. De Michelis and M. Mattioli, "Spectroscopy and impurity behaviour in fusion plasmas," Rep.
Prog. Phys. 47, 1233 (1984)
[DneL90] Yu.N. Dnestrovskij, E.S. Lyadina, P.V. Savrukhin, "Space-time tomography problem for plasma
diagnostic," IAE-5201115 (Kurchatov Institute, Moscow, 1990)
[Donn9l] A.J.H. Donné et al., Plasma diagnostics for the Rijnhuizen Tokamak Project, Rijnhuizen Report
91-207 (FOM Instituut voor Plasmafysica, 1991)
[Donn94] A.J.H. Donné and the RTP-Team, "Diagnostics for the Rijnhuizen Tokamak Project," Plasma
Physics Reports 20, 192 (1994) [From: Fiz. Plazmy 20, 206 (1994)]
[DraE77] H.W. Darwin and F. Emard, "Instantaneous population densities of the excited levels of hydragen
atoms and hydrogen-like i ons in plasmas," Pysica 8SC, 333 ( 1977)
[DriV78] W.G. Driscoll and W. Vaughan, Handbaak of opties (McGraw-Hill, New York, 1978), pp. 8-86
[Dudo95] T. Dudok de Wit, "Enhancement of multichannel data in plasma physics by biorthogonal decom
position," Plasma Phys. Controlled Fusion 37, 117 (1995)
231
References
[DudP94]
[DurC90]
T. Dudok de Wit, A.-L. Pecquet and J.-C. Vallet, "The biorthogonal decomposition as a tooi for
studying fluctuations in plasmas," Phys. Plasmas 1, 3288 (1994)
R.D. Durst, P.G. Carolan, B. Parham, the COMPASS Group, "Tangential soft x-rayNUV
tomography on COMPASS-C," Proceedings of the 17th Conference on Controlled Fusion and Plasi1Ul Hearing, Amsterdam 25-29 June 1990, Ed. G. Brifford et al., Europhysics Conference
Abstracts Vol. 14B (EPS, 1990), Part IV, pp. 1564
[Felun95] G. Fehmers, "A new algoritm for ionospheric tomography," to be published
[FooM82] M.E. Foord, E.S. Marmar, J.L. Terry, "Multichannel light detector system for visible continuurn
measurements on Alcator C," Rev. Sci. Instrum. 53, 1407 (1982)
[FreJ74] R.L. Freeman, E.M. Jones, Atomie col/ision processes in plasi1Ul physics experiments: analytic
expressions for selected cross-secrions and Maxwellian rate coejficients, Report CLM-R 137
(Culham Laboratory, Abingdon, 1974)
[Frie72] B.R. Frieden, "Restoring with maximum likelihood and maximum entropy," J. Opt. Soc. Am.
62, 51 I (1972)
[Fuch94] G. Fuchs, Y. Miura, M. Mori, "Soft x-ray tomography on tokamaks using flux coordinates," Plasma Phys. Controlled Fusion 36, 307 (1994)
[FucM94] J.C. Fuchs, K.F. Mast, A. Hermann, K. Lackner, "Two-dimensional reconstruction of the radia
lion power density in ASDEX Upgrade," Proceedings of the 21 stEPS Conference on Controlled
Fusion and Plasma Physics, Montpellier, 27 June- I July 1994, Ed. E. Joffrin et al., Euro
physics Conference Abstracts Vol. 18B (EPS, 1994), Part UI, pp. 1308
[Ge1H95]
[GoeA92]
[Gord74]
[GraS88]
[GraW91]
[GuiM94]
[GuiN86]
[HanS92]
232
J.F.M. van Gelder, K.C.E. Husmann, H.S. Miedema, A.J.H. Donné, "Heterodyne radiometer at
Rijnhuizen Tokamak Project for electron cyclotron emission and absorption measurements," Rev.
Sci. Instrum. 66,416 (1995)
S. von Goeler et al., "Initia! lower hybrid current drive results and first two-dimensional images of
hard x-ray emission in PBX-M," Proceedings of the 1992 International Conference on Plasi1Ul
Physics, Innsbruck, 29 June- 3 Ju1y 1992, Ed. W. Freysinger et al., Europhysics Conference
Abstracts Vol. 16C (EPS, 1992), Part II, pp. 949
R. Gordon, "A tutoria1 on ART," IEEE Trans. Nucl. Sci. NS-21, 78 (1974)
R.S. Granetz, P. Smeu1ders, "X-Ray tomography on JET," Nucl. Fussion 28, 457 (1988)
R.S. Granetz, L. Wang, "Design of the x-ray tomography system on Alcator C-MOD," in Diag
nostics for contemporary fusion experiments (Proceedings of the Workshop, Varen na, 1991 ), Ed.
P.E. Stolt et al. (Socièta Italiana di Fisica, Bologna, 1991), ISPP-9, pp. 425
R. Guirlet et al., "Routine average effective charge calcu1ation using visible bremsstrahlung emis
sion and comparison with the impurity transport code of Tore-Supra," Proceedings of the 2lst
EPS Conference on Controlled Fusion and Plasma Physics, Montpellier, 27 June - I July 1994,
Ed. E. Joffrin et al., Europhysics Conference Abstracts Vol. l8B (EPS, 1994), Part III, pp. 1280
S.F. Gul!, T.J. Newton, "Maximum entropy tomography," Appl. Opt. 25, 156 (1986)
P.C. Hansen, T. Sekii, H. Shibahashi, "The modified truncated SVD methad for regularization in
general form," SIAM J. Sci. Stat. Comput. 13, 1142 (1992)
[HanW85)
[Hare93)
[Hart90)
[HarW93)
[Heij95)
[Helg80)
[HerL76)
[Herm80)
[Hey94)
[HirS81)
[Hoga82]
[HoiF88]
[HoiN86)
[HoiP90]
References
K.M. Hanson, G.W. Wecksung, "Local basis-function approach to computer tomography," Appl.
Opt. 24, 4028 (1985)
P.C. van Haren, "The DOM4 database system," Comput. Phys. 7, 701 (1993)
K.T. Hartinger, Bestimmung zweidimensionaler Strahlungsverteilungen in Hochtemperaturplas
men des Tolwmaks ASDEX miltels bolometrischer Messungen und Computertomographie, IPP
III/170 (Max-Planck-Institut für Plasmaphysik, Garching, 1990)
P.C. van Haren and F. Wijnoltz, "The TRAMP data acquisition system," Comput. Phys. 7, 638
(1993)
S.H. Heijnen, Putsed-radar rejlectometry, Thesis, Rijksuniversiteit Utrecht ( 1995)
S. Helgason, The Radon transfarm (Birkhäuser, Boston, 1980)
G.T. Herman and A. Lent, "Iterative reconstruction algorithms," Comput. Biol. Med. 6, 273
(1976)
G.T. Herman, Image reconstructionfrom projections (Academie Press, New York, 1980)
J .D. Hey, "Optica! spectroscopy of tokamak plasmas," Trans. Fusion Techno!. 25, 315 ( 1994)
S.P. Hirshman, D.J. Sigmar, "Neoclassical tranport of impurities in tokamak p1asmas," Nucl. Fusion 21, 1079 (1981)
J.T. Hogan, "The effect of low energy electron capture collisions (H0 +C11+) on the partiele and
energy balance of tokamak plasmas," in Physics of electronic and atomie collisions (invited papers
of the XII International Conference in the physics of electron ie and altomie collisions, Gatlingurg,
15-21 July 1981 ), Ed. S. Datz (North-Holland, Amsterdam, 1982), pp. 769
A. Holland, R.J. Fonck, E.T. Powell, S. Sesnic, "Tomographic imaging of MHD activity in
tokamaks by combining diode arrays and a tangentially viewing pinhole camera," Rev. Sci.
Instrum. 59, 1819 (1988)
A. Holland, G.A. Navratil, "Tomographic analysis of the evolution of plasma cross sections,"
Rev. Sci. lnstrum. 57, 1557 (1986)
A. Holland, E.T. Powell, R.J. Fonck, "Image reconstruction methods for the PBX-M pinhole
camera," PPPL-2676 (Princeton Plasma Physics Laboratory, Princeton, 1990)
[HooK86] T. Hooft, T. van der Klis, J.W. Dumortier, Oppervlakte behandelingen van roestvrij staal, 3e uit
gave (Vereniging voor Oppervlaktetechniek van Materialen, Bilthoven, 1986), p. 146
[Howa88] J. Howard, "Tomography and reliable information," J. Opt. Soc. Am. A5, 999 (1988)
[HowW93) J. Howard et al., "The H-1 Multi-View lnterferometer," Proceedings of the 6th International Sym
posium on Laser Aided Plasma Diagnostics, Bar Harbor, 25- 28 October 1993, pp. 154
[HurH92] P.D. Hurwitz, B.F Hall, W.L. Rowan, "Detector array for measurements of high-frequency fluctua
tions in visible and near-UV emission from tokamaks," Rev. Sci . Instrurn. 63, 4614 (1992)
[HurR95]
[Hutc87)
[Isle84]
P.D. Hurwitz, W.L. Rowan, "Application of a three sample volume S(k,m) estimate to optica!
measurements of trubulence on TEXT," Rev. Sci. Instrum. 66, 441 (1995)
I.H. Hutchinson, Principlesof plasma diagnostics ( Cambridge University Press, Cambridge, 1987)
R.C. !sier, "Impurities in tokamaks," Nucl. Fusion 24, 1599 (1984)
233
References
[Isle87]
[IwaT87]
[IwaY89]
[IwaY93]
[JanD92]
[JanL87]
[JenP77]
[JohH73]
[Jone77]
[Kad080]
[Kado87]
[Kado92)
[KakS88]
[KarL61]
[Kemp80]
[Koni95]
[KonH94)
234
R.C. Isler, "A review of charge-exchange spectroscopy and applications to fusion plasmas,"
Physica Scripta 35, 650 (1987)
N. lwama, H. Takami, S. Takamura, T. Tsukishima, "An approach with the Akaike information
criterion to the radialion distri bution reconstruction of toroidal plasma," IEEE Trans. Plasma Sc i.
PS-15, 609 (1987)
N. lwama, et al., "Phillips-Tikhomov regularization of plasma image reconstruction with the gen
eralized cross validation," Appl. Phys. Lett. 54, 502 (1989)
N. lwama, H. Yosida, H. Takimoto, M. Teranishi, "GCV-aided regularizations in sparse-data com
puter tomography with application to plasma imaging," Proceedings ofthe 4th International Sym
posium on Computerized Tomography, August 10-14, 1993, Novosibirsk, Ed. M.M. Lavrent'ev
(VSP, Utrecht, 1995)
C. Janicki, R Décoste, P .Noël, "Soft x-ray imaging diagnostic on the TdeV tokamak," Rev. Sci.
lnstrum. 63, 4410 (1992)
R.K. Janev, W.D. Langer, K. Evans, D.E. Post, Elementary processes in hydragen-helium plas
mas (Springer Verlag, Berlin, 1987)
RV. Jensen et al., "Calculations of impurity radialion and its effects in tokamak experiments,"
Nucl. Fusion 17, 1187 (1977)
L.C. Johnson and E. Hinnov, "Ionization, recombination, and population of excited levels in
hydragen plasmas," J. Quant. Spectrosc. Radial. Transfer 13, 333 (1973)
E.M. Jones, Atomie col/ision processes in plasma physics experiments: analytic expressions for
selected cross-seelions and Maxwellian rate coefficients //,Report CLM-Rl75 (Culham Labora
tory, Abingdon, 1977)
K. Kadota, M. Otsuka, J. Fujita, "Space- and time-resolved study of impurities by visible spec
troscopy in the high-density regime of flPP T-Il tokamak plasma," Nucl. Fusion 20, 209 (1980)
B.B. Kadomtsev, Philos. Trans. R. Soc. London A322, 125 (1987)
B.B. Kadomtsev, Tokamak plasma: a complex physical system (!OP Publishing, Bristol, 1992),
pp. I IO
A.C. Kak and M. Slaney, Principles of computerized tomographic imaging (IEEE Press, New
York, 1988)
W. J. Karzas, R. Latter, "Electron radiative transitions in a Coulomb field ," Astrophys. J. SuppL
55, 167 (1961)
M.C. Kemp, "Maximum entropy reconstructions in emission tomography," Med. Radionuclide
Imaging 1, 3 I 3 (I 980)
J.A. Konings, Electron thermal transport in tokamak plasmas, Thesis, Rijksuniversiteit Utrecht
(1994)
J.A. Konings et al., "Transport analysis in ohmic and ECR heated plasmas in the RTP tokamak,"
Plasma Phys. Controlled Fusion 36, 45 (1994)
[KraK88]
[KurS95]
[Kurz95]
[KutL88]
References
H. Krause, M. Kornherr, ASDEX Team, NI Team, "High resolution sparse channel tomography
for slowly varying rotating SXR profiles," Proceedings of the /Sth EPS Conference on Controlled
Fusion and Plasma Heating, Dubrovnik, 16-20 May 1988, Ed. S. Pe'Sié et al. Europhysics
Conference Abstracts Vol. 12B (EPS, 1988), Part III, pp. 1179
C. Kurz et al., "Deterrnination of Ha emissivities from line-integrated brightness measurements
on Alcator C-Mod," Rev. Sci. Instrum. 66, 619 (1995)
C. Kurz, Tomography of light emission from the plasma edge of Alcator C-Mod, Thesis,
PFC/RR-95-5 (Plasma Fusion Center, Massachusetts Institute of Technology, 1995)
B.V. Kuteev et al., "Two-dimensional optica! tomography of impurities in the FT-2 Tokamak,"
Proceedings of the JSth EPS Conference on Controlled Fusion and Plasma Healing, Dubrovnik,
16-20 May 1988, Ed. S. Pesié et al. Europhysics Conference Abstracts Vol. 12B (EPS, 1988),
Part ill, pp. 1 155
[Kut091] B.V. Kuteev, M.V. Ovsishcher, "Jnvestigations of light impurities transport in tokamak using
smali-view optica! tomography," Proceedings of the 18th EPS onference on Controlled Fusion
and Plasma Physics, Berlin, 3- 7 June 1991, Ed. P. Bachmann et al., Europhysics Conference
Abstracts Vol. !SC (EPS, 1991), Part IV, pp. 241
[LamK90] A.C.A.P. van Lammeren, S.K. Kim, A.J.H. Donné, "Multichannel interferometer/polarimeter
system for the RTP tokamak," Rev. Sci. Instrum. 61, 2882 (1990)
[Lamm91] A. van Lammeren, Electron and current density measurements on tokamak plasmas, Thesis, Rijks
universiteit Utrecht (1991)
[Lent76] A. Lent, "Maximum entropy and multiplicative ART," Proceedings of the SPSE Conference on
Image Analysis and Evaluations, ed. R. Shaw, Toronto 1976, pp. 249
[Lewi83] R.M. Lewitt, "Reconstruction algorithms: transfarm methods," Proc. IEEE 71, 390 (1983)
[Liew85] P.C. Liewer, "Measurements of microturbulence in tokamaks and comparison with theories of tur
bulence and anomalous transpon," Nucl. Fusion 25, 543 (1985)
[Lind67] S.L. Linder, "Optimization of narrow optica! speetral filters for nonparallel monochromatic radia
tion," Appl. Opt. 6, 1201 (1967)
[Liss68] P.H. Lissberger, "Effective refractive index as a criterion of performance of interference filters," J.
Opt. Soc. Am. 58, 1586 (1968)
[LisW59] P.H. Lissberger, W.L. Wilcock, "Propenies of all-dielectric interf erenee filters. II. Filters in paral
lel beams of light incident obliquely and in convergent beams," J. Opt. Soc. Am. 49, 126 ( 1959)
[Long73] R.S. Longhurst, Geometrical and physical opties, Third Edition, (Longman, Harlow, 1973), p. 20,
450, 466
[LopS94] N.J. Lopes Cardozo et al. , "Plasma filamentation in the Rijnhuizen tokamak RTP," Phys. Rev.
Lett. 73, 256 (1994)
[LouN83] A.K. Louis, F. Natterer, "Mathematica! problems of computerized tomography," Proc. IEEE 71,
379 (1983)
235
References
[LyaT93) E.S. Lyadina, C.P. Tanzi, D.F. da Cruz, A.J.H. Donné, "A space-time tomography algorithm for
the five-camera soft x-ray diagnostic at RTP," Proceedings of the 20th EPS Conference on Con
trolled Fusion and Plasma Physics, Lisboa, 26-30 July 1993, Ed. J.A. Costa Cabral et al. Euro
physics Conference Abstracts Vol. 17C (EPS, 1993), Part III, pp. 1151
[MagH94) C.F. Maggi et al., "Impurity 1ine emission due tothermal charge excahnge in JET edge p1asmas,"
Proceedings 2/st EPS Conference on Controlled Fusion and Plasma Physics, Montpellier, 27
June-I July 1994, Ed. E. Joffrin et al., Europhysics Conference Abstracts Vol. 18B (EPS,
1994 ), Part II, pp 826
[Mari95) M.M. Marinak, "Behaviour of the partiele transport coefficients near the density limit in the
microwave tokamak experiment," Nucl. Fusion 35, 399 (1995)
[McNB84] D.H. McNeill et al., "Edge region hydrogen line emission in the PDX tokamak," J. Vac. Sci.
Techno!. A 2, 689 (1984)
[McNe89] D.H. McNeill, "H-alpha photon yield in fuelling of tokamaks," J. Nucl. Mater. 162-164, 476
(1989)
[McWS84) R.W.P. McWhirter and H.P. Summers, "Atomie radialion from 1ow density plasma," in Applied
atomie colfision physics, Ed. H.S.W. Massey et al. (Academie Press, Orlando, 1984), Vol. 2, pp.
51
[MelP95) T.C. MellbHHKOBa, B.B. llHKaJlOO, To.Mo2paljun 1lJ/U3.Mbl (HayKa, HosocH6HpCK, 1995) (in
press) [T.S. Melnikova, V.V. Pickalov, Plasma tomography (Nauka, Novosibirsk, 1995) (in
Russian))
[MilL91)
[MilL93)
[Mine79)
[MinS80]
B.P. van Milligen, N.J . Lopes Cardozo, "Function parametrization: a fast inverse mapping
method," Comput. Phys. Commun. 66, 243 (1991)
B.P. van Milligen et al., "Gradients of electron temperature and density across m = 2 magnet ie
islands in RTP," Nucl. Fusion 33, 1119 (1993)
G . Minerbo, "MENT: a maximum entropy algorithm for reconstructing a souree from projection
data," Computer Graphics and Image Processing 10, 48 ( 1979)
G.N. Minerbo, J.G. Sanderson, D.B. van Hulsteyn, P. Lee, "Three-dimensional reconstruction of
the x-ray emission in laser imploded targets," Appl. Opt. 19, 1723 (1980)
[Mitc95) J.B.A. Mitchell, "Eiectron-molecular ion collisions" in Atomie and molecular prosesses in fusion
edge plasmas, ed. R.K. Janev (Plenum Press, New York, 1995)
[Moni91) P. Monier-Garbet, "Visible and uv emission spectroscopy," in Diagnostics for contemporary
fusion experiments (Proceedings of the Workshop, Varen na, 1991 ), Ed. P.E. Stott et al. (Socièta
Italiana di Fisica, Bologna, 1991), ISPP-9, pp. 331
[MyeL78] B.R. Myers, M.A. Levine, "Two-dimensional speetral line emission reconstruction as a plasma
diagnostic," Rev. Sci. Instrum. 49, 610 (1978)
[Naga87] Y. Nagayama, "Tomography of m = I mode structure in tokamak plasma using least-square-fit
ting method and Fourier-Bessel expansions," J. Appl. Phys. 62, 2702 (1987)
[NagB90] Y. Nagayama et al., "Image reconstructions of ECE and x-ray signals for high beta plasmas on
TFTR," Rev. Sci. Instrum. 61, 3265 (1990)
236
[NagE92]
[NagT81]
[Nard92]
[Natt86]
[Nav095]
[NeuS86]
[Nila82]
[ParA93]
[PázG95)
[Pázs94)
[PicM84)
[PikP83]
[PikP87]
[PreF89]
[PreP82]
[PriW90]
[Radol7]
References
Y. Nagayama, A.W. Edwards, "Rotational soft x-ray tomography of noncircular tokamak plas
mas," Rev. Sci. lnstrum. 63,4757 (1992)
Y. Nagayama et al., "Soft x-ray tomography of the m = 2 magnetic island structure in the JIPP
T-Il tokamak," Jap. 1. Appl. Phys. 20, L779 (1981)
C. Nardone, "Multichannel fluctuation data analysis by the singular value decomposition method.
Application to MHD modes in JET," Plasma Phys. Controlled Fusion 34, 1447 (1992)
F. Natterer, The rnathematics of computerized tomography (Wiley and Teubner, Stuttgart, 1986)
A.P. Navarro, M.A. Ochando, F. Medina, B.P. van Milligen, "Loca1 plasma radialion 1oss rate
determination in tokamaks and stellarators," Rev. Sci. lnstrum. 66, 552 (1995)
J. Neuhauser, W. Schneider, R. Wunderlich, "Thermal instabilities and poloidal asymmetries in
the tokamak edge plasma," Nucl. Fusion 26, 1679 (1986)
R.A. Niland, "Maximum reliable information obtainable by tomography," J. Opt. Soc. Am. 72,
1677 (1982)
H.K. Park, H Adler, "Electron density profile from visible bremsstrahlung array measurements on
TFTR," Proceedings of 6th International Symposium on Laser-Aided Plasma Diagnostics, Bar
Harbor, October 25-28, 1993
I. Pázsit, R.D. GiJl, Diagnostics of tokamak fusion plasma via correlation analysis of soft x-ray
signals, Report JET-P(95)23 (JET Joint Undertaking, Abingdon, 1995)
I. Pázsit, "Density correlations in two-phase flow and fusion plasma transport," 1. Phys. D 27,
2046 (1994)
V.V. Pickalov, T.S. Melnikova, "Tomographic measurements of plasma temperature fields,"
Beitr. Plasmaphys. 24, 417 ( 1984)
V.V. Pickalov, N.G. Preobrazhenskii, "Computer-aided tomography and physical experiment,"
Sov. Phys. Usp. 26, 974 (1983) [Usp. Fiz. Nauk. 141, 469 (1983)]
B.B. llHKanoa. H.r. llpeo6paJKeHCKHH, PeK.OHcmpyK.muaHan mo.Atozpatjiun a
za.wauHa.AruKe u ljiu:JuK.e nAaJ.Atbt (1-layKa, HosocH6HpCK, 1987) [V.V. Pickalov, N.G.
Preobrazhensky, Reconstructive tomography in gas dynamics and plasma physics (Nauka,
Novosibirsk, 1987) (in Russian)]
W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes in Pascal
(Cambridge University Press, Cambridge, 1989), pp. 61
Hr. llpeo6paJKeHCKHH, 8.8. llHKanoa, Heycmoll'luabte Jaaavu auamocmuK.u nAaJ.Mbt
(HayKa, HonoCH6HpcK, 1982) [N.G. Preobrazhensky, V. V. Pickalov, lll-posed problems of plasma
diagnostics (Nauka, Novosibirsk, 1982) (in Russian)]
J .L. Prince, A.S. Willsky, "A geometrie projection-space reconstruction algorithm," Linear
Algebra and lts Applications 134, 151 (1990);
J.L. Prince, A.S. Willsky, "Constrained sinogram restoration for limited-angle tomography," Opt.
Eng. 29, 535 ( 1990)
J. Radon, "Über die Bestimmung von Funktionen durch ihre lntegralwerte längs gewisser Mannig
faltigkeiten," Ber. Verh. Saechs. Akad. Wiss. Leipzig, Math.-Phys. 69,262 (1917)
237
References
[RanD85)
[Razu84]
[Riki87]
[Robb80]
[RosS92]
[Sa1G84]
R. Rangayyan, A.P. Dhawan, R. Gordon, "Algorithms for limited-view computed tomography: an
annoted bibliography and a challenge," Appl. Opt. 24,4000 (1985)
K.A. Razumova, "Results from T-7, T-10, T-Il and TM-4 tokamaks," Plasma Phys. Controlled
Fusion 26, 37 (1984)
T. Rikitake, Magnetic and electromagnetic shielding (Terra Scientific Publishing Company,
Tokyo, 1987)
T. Robbins, Stilllife with woodpecker (Bantam Books, New York, 1980), p. 53
O.W. Ross et al., "Effect of beam-attenuation modolation on fluctuation measurements by heavy
ion beam probe," Rev. Sci. Instrum. 63, 2232 (1992)
M.Kh. Salakhov, I.O. Grachev, R.Z. Latipov, l.S. Fishman, "The regularized algorithm of local
emission reconstruction in the spectroscopie study of an optically thin plasma of arbitrary shape,"
Computer Enhanced Spectroscopy 2, 117 (1984)
[SchS88] D.P. Schissel, R.E. Stockdale, H. St.John, W.M. Tang, "Measurements and implications of Zeff
profiles on the Dlli-D tokamak," Phys. Fluids 31, 3738 (1988)
[ShoM68] B.W. Share, D.H. Menzel, Principlesof atomie spectra (Wiley, New York, 1968), pp. 443 [Note that Eq. (8.12) on p. 444 should read: observed rate = N(al') A(af,af)]
[ShoJ80] J.E. Shore, R.W. Johnson, "Axiomatic derivation of the principle of maximum entropy and the
principle of minimum cross-entropy," IEEE Trans. Inf. Theory IT-26, 26 (1980)
[SigC74] DJ. Sigmar J.F. Clarke, R.V. Neidigh, K.L. Vander Sluis, "Hot-ion distribution function in the
Oak Ridge Tokamak," Phys. Rev. Lett. 33, 1376 (1974)
[Smeu83] P. Smeulders, A Jast plasma tomography routine with second-order accuracy and compei!Sationfor
spatial resolution, IPP 21252 (Max-Planck-Institut für Plasmaphysik, Garching, 1983)
[SmiK85] K.T. Smith, F. Keinert, "Mathematica! foundations of computer tomography," Appl. Opt. 24,
3950 (1985)
[StaM90] P.C. Stangeby, G.M McCracken, "Plasma boundary phenomena in tokamaks," Nucl. Fusion 30,
1225 (1990)
[SucH78] S. Suckewer, E. Hinnov, J . Schivell, Rapid scanning of spatial dis tribution of speetralfine inten
sities in PLT tokamak, PPPL- 1430 (Princeton Plasma Physics Laboratory, Princeton, 1978)
[SugG91] S. Sugimoto, S. Goto, "Spectroscopie plasma tomography with multiple photocollector arrays ,"
Rev. Sci. Instrum. 62, 2138 (1991)
[Summ79] H.P. Summers, Tables and graphs of collisional dieleetronie recombination and ionisation coeffi
cients and ionisation equilibria of H-like to A-like ionsof elements, AL-R-5 (Culham Laboratory, Abingdon, 1979)
[Tak092]
[TakT86]
238
S. Takamura, Y. Oda, S Sakurai, "Plasma edge structure of a tokamak with ergodie magnetic lim
iter," Proceedings ofthe 1992 International Conference on Plasma Physics, Innsbruck, 29 June-
3 July 1992, Ed. W. Freysinger et al., Europhysics Conference Abstracts Vol. 16C (EPS, 1992),
Part 11, pp. 859
S. Takamura, T . Todd, T. Edlington, CLEO Group, "X-ray i rnaging of electron cyclotron heated CLEO plasma by tangen ti al projection," Plasma Phys. Controlled Fusion 28, 1717 ( 1986)
References
[TanB94] C.P. Tanzi, H.J. de Blank, A.J.H. Donné, "A new approach to internal disruption analysis with
the five-camera soft-x-ray diagnostic on RTP," Rev. Sci. Instrum. 66, 537 (1995)
[Teag80] M.R. Teague, "Image analysis via the general theory of moments," J. Opt. Soc. Am. 70, 920
(1980)
[TerM77] J.L. Terry, E.S. Marmar, K.I. Chen, H.W. Moos, "Observation of poloidal asymmetry in irnpu
rity-ion ernission due to VB drifts," Phys. Rev. Lett. 39, 1615 (1977)
[TerS95] J.L. Terry, J.A. Snipes, C. Kurz, "The visible, imaging diode arrays on Alcator C-Mod," Rev.
Sc i. Instrum. 66, 555 ( 1995)
[TFR75] Equipe TFR, "Line radialion in the visible and in the ultraviolet in TFR tokamak plasmas," Nucl.
Fusion 15, 1053 (1975)
[Thei90) G. Theimer, Methoden zur Untersuchung de räumlichen Struktur von Dichtefluktuationen in the
Randschicht von Fusionsexperimenten demonstrien am Divertortokamak ASDEX, Report
IPP TIUI69 (Max-Planck-Institut für Plasmafysik, Garching, 1990)
[Tikh77] A.N. Tikhonov and V.Y. Arsenin, Solutions of ill-posed problems (Winsten, Washington, 1977)
[TurK71] V.F. Turchin, V.P. Kozlov, M.S. Malkevich, "The use of mathematical-statistics method in the
solution of incorrectly posed problems," Sov. Phys. Usp. 13, 681 (1971) [Usp. Fiz. Nauk 102,
345 (1970))
[VernF74] V. Vemuri, F.-P. Chen, "An initia! value method for solving Fredholm inlegral equations of the
first kind," J. FrankJin Inst. 297, 187 (1974)
[VinR74) N.D. Vinogradova, K.A. Razumova, "Role of diaphragm in the partiele balance in a tokamak,"
JETP Lett. 19, 156 (1974) [ZhETF Pis. Red. 19, 261 (1974)]
[VriW95] P.C. de Vries, G. Waidrnann, A.J.H. Donné, F.C. Schüller, "MHD-mode stabilization by plasma
rotatien in TEXTOR," Proceedings of the 22nd EPS Conference on Controlled Fusion and Plasma
Physics, Bournemouth, 3-7 July 1995 (to be published)
[WanG91] L. Wang, R.S. Granetz, "A simplified expression for the Radon transferm of Bessel basis func
tions in tomography," Rev. Sci.Instrum. 62, 842 (1991)
[WaxK85] M. Wax, T. Kailath, "Detection of signals by inforrnation theoretic criteria," IEEE Trans. Acous
tics, Speech and Signa! Proc. ASSP-33, 387 (1985)
[WenB95) Y. Wen, R.V. Bravenec, "High-sensitivity, high-resolution measurements of radiated power on
TEXT-U," Rev. Sci. Instrum. 66, 549 (1995)
[Wess87] J. Wesson, Tokamaks (Clarendon Press, Oxford, 1987)
[WhiM77] R.D. White et al. , "Saturation of the tearing mode," Phys. Fluids 20, 800 (1977)
[Whit80) D.R.J. White, A handhook on electromagnetic shielding materials and petformance, (Don White
Consultants lnc., Grainesville, Yirginia, 1980)
[WieS66]
[WiiE82)
(Wint88)
W.L. Wiese, M.W. Smith, B.M. Glennon, Atomie transition probabilities, Vol. I (Hydrogen
through neon), NSRDS-NBS 4, (National Bureau of Standards, Washington D.C., 1966)
J.H. Williamson, D.E. Evans, "Computerized tomography forsparse-data plasma physics experi
ments," IEEE Trans. Plasma Sci . PS-I 0, 82 ( 1982)
1. Winter, Wandkonditionierung von Fusionsanlagen durch reaktive Plasmen, Jül-2207
(Kernforschungsanlage Jülich, 1988)
239
References
[WonZ90) K.M. Wong, Q.-T. Zhang, J.P. Reilly, P.C. Yip, "On information theoretic criteria for determin
ing the number of signals in high resolution array processing," IEEE Trans. Acoustics, Speech and
Signal Proc. 38, 1959 (1990)
[ZajA95) D. Zajfman et al., "Measurement of branching ratios for the dissociative recombination of cold
HD+ using fragment imaging," Phys. Rev. Lett. 75, 814 (1995)
[ZoiK92] S. Zoletnik, S. Ká1vin, "A method for tomography in arbitrary expansions," Rev. Sci. Instrum.
64, 1208 (1993)
[ZweM83) S.J. Zweben, J. McChesney, R.W. Gould, "Optica! imaging of edge turbulence in the Caltech
tokamak," Nucl. Fusion 23, 825 (1983)
[ZweM89) S.J. Zweben, S.S. Medley, "Visible imaging of edge fluctuations in the TFTR tokamak," Phys.
Fluids B 1, 2058 (1989)
240
Acknowledgentents
First of all I would like to thank Tony Donné for the support to get the visible-light tomography diagnostic operational and this thesis finished, and Daan Schram and Chris Schüller for the
numerous discussions on the work and this thesis. Bearing in mind the quotalion at the begin
ning of this thesis, I am very grateful that both of the promotors are genuine positivists, which
has been a great help when the eggs seemed more scrambled than we had hoped for, and to me
there did not seem to be any way to unscramble them.
Other people directly involved in the visible-light tomography project have been Jan-Jaap Kon
ing, whostarled it, the students Joost de Vries, Gerrit Kolthof and Marc Hoppenbrouwers, and
on the technica! side Wim de Haan and Sirnon Kuyvenhoven. I thank them for all their contri
butions. I am grateful to Valery Pickalov, from whom I have learned a great deal about tomog
raphy, and tagether with whom I managed to get tomographic reconstructions of the measurements. I am also grateful to Gijs Fehmers for making his tomography methad available,
preparing programs for me, and making many reconstructions.
All people of the RTP team deserve thanks for their contributions during the measuring ses
sions for visible-light tomography and in the analysis afterwards. In particular Paul Smeets and
Cor Tito have helped to get the diagnostic mounted on RTP. In the technica! department I thank in particular Peter Wortman, Ad Agterberg, Wil Julsing, Frits Hekkenberg, Wim Kersbergen,
and all other people involved, fortheir contributions to the design and manufacturing of parts of
the diagnostic.
Dirceu Ferreira da Cruz and Cristina Tanzi have surprised me by their ability to stand me when
sharing the office with me. I should give a little bit more attention to Cristina, not only because she suggested to write these acknowledgements herself, but also because of the special relation
ship we have had over the past years and because of all her darnaging comments that made me
see life in a different, but not necessarily correct, perspective. I am also grateful to her for
sharing a Russian and Siberian adventure with me (some weeks including a tomography con
ference). When talking about adventures, I should also thank Rik Tarnmen fora great time in
the enchanting land of New Mexico (during a diagnostic conference).
I also thank Wim Tukker and Piet van Kuyk fortheir help with many tricky figures for this
thesis, Hajnal Vörös for ordering many articles for me, André van Kan and Henk Heslinga for
increasing my disk quota whenever I needed it and their care of the computer systems, Alan
Edwards and Barry Alper fortheir help during my visit to JET, which has inspired the way I
set up the data-handling and analysis programs, Mervi Mantsinen and my parents for just being there, supporting me and standing me during the busy PhD work, all colleagues who make
Rijnhuizen a pleasant place to work and have facilitated my time there, the people who have
painstakingly read my thesis, et cetera, et cetera.
Curriculum vitae
I was born on 30 June, 1967, in Ljungby in Sweden. When I was not yet three years old I
moved, together with my parents, to Meppel in The Netherlands, where I went to primary and secondary school. At a young age I wanted to become an "inventor" and made, as it seemed at that time, many far-reaching inventions that would shake the world. Later I took a more
philosophical approach and started wondering whether humankind was worthy of those great
inventions. Not so much excitement happened in my life until I found a practical application of secondary school chernistry in explosives, bombs, rockets and rnissiles. During several years I
publisbed a popular scientific magazine for my fellow students and teachers.
Despite great applicability, I decided that chernistry was too mundane, and although languages
and philosophy interested me much, physics and rnathematics seemed to be the real thing. So, after my secondary school diploma (VWO) at the Rijksscholengemeenschap Meppel in 1985, I
starled to study Technica! Physics at the Universiteit Twente in Enschede, The Netherlands.
During my study I helped organizing a three-week field trip for physics students to the United States in 1988, visiting many companies, universities and institutes. I also contributed to the
physics department magazine by writing articles about various topics, and was a student mem
her of a comrnittee evaluating the teaching curriculum of the physics department In 1989 I did
my practical training at ABB Corporale Research in Västeräs, Sweden, on simulating optica!
properties of compound semiconductor devices for high-power applications. For my master's thesis in 1990 I developed a Scanning Force Microscope ( also called Atomie Force Microscope) for biologica! applications at the Optoelectronic Engineering group of Prof. B. Bölger and Dr. N. van Hulst at Universiteit Twente.
In 1990 I started working on visible-light tomography, the basis of this thesis, at the FOMInstituut voor Plasmafysica "Rijnhuizen" in Nieuwegein, the Netherlands. My PhD thesis su
pervisors are Prof. D.C. Schram of Technische Universiteit Eindhoven, Prof. F.C. Schüller of Universiteit Utrecht and FOM-Rijnhuizen, and Dr. A.J.H. Donné, who is the leader of diagnostic group of FOM-Rijnhuizen. During my PhD work I visited JET Joint Undertaking for six weeks to workon soft x-ray tomography.
At the time of printing of this thesis I am working on a bolometer tomography diagnostic at JET
Joint Undertaking, Abingdon, United Kingdom.
Stellingen
behorend bij het proefschrift
Visible-light tomography
of
tokamak plasmas
Christian Ingesson
Eindhoven, 18 december 1995
I
De combinatie van tomografie en tijdanalysemethoden voor de bepaling van ruimte
lijke, fluctuerende emissiepatronen staat nog in de kinderschoenen.
11
Om de betrouwbaarheid van tomografische reconstructie-algoritmen te onderzoeken is
het noodzakelijk om simulaties uit te voeren, bijvoorbeeld door verschillende metho
den te vergelijken. Bij voorkeur wordt dit gedaan tijdens het ontwerp van de tomogra
fische diagnostiek opdat het systeem geoptimaliseerd kan worden en eventuele teleur
stellingen voorkomen kunnen worden. Dit proefschrift, hoofdstukken 3 en 6.
m Aibeeldingseffecten in tomografie-systemen, zoals de eindige breedte van de kijkbun
dels, worden vaak ten onrechte verwaarloosd.
IV
Indien de aanwezigheid van de duidelijke asymmetriëen in de in R TP gemeten emissie
profielen algemeen blijkt, dan zijn de in fusie-onderzoek toegepaste transportmodellen
niet adequaat
Dit proefschrift, hoofdstuk 7.
V
De bijdrage van de H; reactiepaden bij de dissociatie van moleculair waterstof zou
van belang kunnen zijn voor de lokalisatie van de Ha. straling in tokamaks aangezien ionen, in tegenstelling tot ongeladen deeltjes, langs de magnetische veldlijnen wegge
voerd worden van de plaats waar ze ontstaan.
Dit proefschrift, hoofdstuk 7.
VI
Mocht een "Theory of everything" ooit gevonden worden, welke mogelijkheid uit filo
sofisch oogpunt omstreden is, dan nog zal deze nauwelijks van belang zijn voor de
beschrijving van de wereld om ons heen.
VII
Het gebruik van Engelstalige leuzen in reclameboodschappen, of het geheel Engelstalig
zijn van die boodschappen, is een belediging voor het Nederlandse volk en dient
afgestraft te worden met een kopersstaking.
VIII
In tegenstelling tot hetgeen velen denken, bevat de Nederlandse taal wel degelijk vele
elementen die het tot een krachtige en mooie taal maken. Helaas raken die elementen
in het hedendaags Nederlands steeds meer in onbruik. Nederlands verdient het om
tegen verwording en verdwijning beschermd te worden.
IX
De stelling "Iemand die één horloge heeft weet hoe laat het is, maar iemand met twee
kan nooit zeker zijn" kan verbijzonderd en uitgebreid worden tot "Iemand die de
beschikking heeft over één zichtbaar licht kanaal kan onder symmetrie-aannamen aller
lei bruikbare fysische grootheden afleiden, maar iemand met 80 kanalen ontdekt dat
deze elkaar tegenspreken en kan daarom slechts minder harde uitspraken doen."
x
Humor zou wel eens een menselijke eigenschap kunnen zijn die meegenomen dient te
worden in de zoektocht naar artificiële intelligentie, aangezien lachen één van de men
selijke reacties is om een logische paradox te doorbreken, waar een algoritme in een
eindeloze lus zou raken.
XI
De tijdplanning van de meeste promoties en van de eerste kernfusiereactor hebben ten
minste één eigenschap gemeen: de datum wordt steeds naar achteren geschoven. De
meeste promoties worden toch tot een goed einde gebracht, dus hetzelfde is niet on
denkbaar voor een kernfusiereactor.