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J.Nicolas MEDSI’16
Optics and mechanics of mirror benders
Josep NicolasClaude RugetCarles Colldelram
J.Nicolas MEDSI’16
1. Basics of x-ray mirror optics2. Characteristics of elliptic mirrors3. Mirror bender calculations and metrology4. Mechanical design of mirror benders
J.Nicolas MEDSI’16
1. Basics of x-ray mirror optics§ Focusingwithreflectiveoptics§ Surfaceerror,aberration
J.Nicolas MEDSI’16
0 5 10 15 20 250
0.2
0.4
0.6
0.8
1
Photon Energy (keV)
Tran
smis
sion
0 5 10 15 20 250.999
0.9995
1
1.0005
1.001
Photon Energy (keV)
Refra
ctio
n in
dex
re(n
)
Transmission of1mmofSilicon
Refractive indexfor Silicon
Refractive index of x-rays
J.Nicolas MEDSI’16
Total reflection
102 103 10410-8
10-6
10-4
10-2
100Au
Photon Energy (keV)
Refra
ctio
n in
dex
re(n
)
db
n=1-d-ib𝑛" sin 𝛼" = 𝑛( sin 𝛼(
𝜃* = 2𝛿�
J.Nicolas MEDSI’16
Mirror 400mm(toroidal) Mirror 1200mm(plane elliptic)
Amirror inits vessel
J.Nicolas MEDSI’16
Anopticalsystemimages(stigmatically)an objectpointA,ontoanimagepointA’ ifalltheraysthatdepartfromA,andreachtheopticalsystem,gatheratA’,independently oftheirdirection.
objectpointImagepoint
Opticalsystem
Examplesoffocusingsystemsare:Apinhole,lenses(refraction),mirrors(reflection),Fresnellenses(diffraction).In(almost)allthecasesthefocusingelementdeviatesthedirectionoftheincomingrays.
To focus (v.)
J.Nicolas MEDSI’16
Rays vs Waves
Rays arethelinesthatindicatethedirectionofpropagationofenergy,directionofvariationofphase
X-Raysarewaves(andphotonsaretheirquanta),andtobepreciseoneshoulddowavefront propagation.
Geometricaloptics,isanapproximation,basedonpropagationofrays.
AWavefront isthesurfaceofpointsatthesameopticalpathdistancefromtheemissionpoint.(forus,atthesamedistance),i.e.phaseisconstantacrossasurface
“Raysareperpendiculartowavefronts”
Rays
Wavefront
J.Nicolas MEDSI’16
hν=24eV– λ=50nm Raytracing
6m1m
Pointsource Toroidal mirror100x10mm2
100μm
Althoughraytracingcannotaccountfordiffractionorinterferenceeffects,itreproducesaccuratelytheimagesifenergyishighenough(shortwavelength)
hν=123eV– λ=10nm
Ray tracing vs wavefront propagation
J.Nicolas MEDSI’16
6m1m
130x6μm2
gaussian sourceToroidal mirror100x10mm2
100μm
Thesourcesizeoftheelectronbeamreducesthecoherenceofthebeam,andreducesthecontrastofwaveeffects.
hν=24eV– λ=50nm Raytracinghν=123eV– λ=10nm
Effect of the source size
J.Nicolas MEDSI’16
object
mirror
image
Ellipsoidal mirror
Focusing with mirrors
“Thesumofthedistancestothetwofociisconstantforallthepointsofanellipse”
J.Nicolas MEDSI’16
Stigmatic imagingAnellipsoidalmirrorfocusespoint-to-pointstigmatically(oranastigmatically)
Ellipsoidal mirror
Localcurvatureisnotconstant
Sagittalcurvaturevariesfromupstreamtodownstream
J.Nicolas MEDSI’16
Stigmatic imagingAnellipsoidalmirrorfocusespoint-to-pointstigmatically(oranastigmatically)
Ellipsoidal mirror
Localcurvatureisnotconstant
Sagittalcurvaturevariesfromupstreamtodownstream
J.Nicolas MEDSI’16
The approximation ofthe toroidal mirror tothe ellipsoid isaccurate only inthe centerofthe mirror.
Rays far from it arenot properly focused.
Toroidal aberration
Aberrations
Toroidal
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u"
v"
h"
p"
q"
α"
β"
M"
A"
B"
Aberration computing
ℎ 𝑢, 𝑣 =𝑢(
2𝑅 +𝑣(
2𝑟
𝑂𝑃𝐷 𝑢, 𝑣 = 𝐌 − 𝐀 + 𝐁 −𝐌
Approximatetorusequation
Opticalpathdifference
J.Nicolas MEDSI’16
𝐀 −𝐌 = 𝐴= −𝑀=( + 𝐴? − 𝑀?
( + 𝐴? − 𝑀?(�
𝐹AB 𝑝, 𝑞, 𝛼, 𝛽, 𝑅F =1
𝑛!𝑚!𝜕AKB𝐹𝜕𝑢A𝜕𝑣BL M,M|O,P,Q,R,ST
𝐹UV 𝑢, 𝑣|𝑝, 𝑞, 𝛼, 𝛽, 𝑟F = W W 𝐹AB 𝑝, 𝑞, 𝛼, 𝛽, 𝑟F 𝑢A𝑣BX
BYM
X
AYM
Image conditionWe need to compute the optical distance between object and mirror, and between mirrorand image.
It can be expressed as a Taylor series
J.Nicolas MEDSI’16
Main aberrations• F20 meridional defocusing*
• F02 sagittal defocusing
• F30 primary coma*
• F12 astigmatic coma
𝐹(M =cos( 𝛼2
1𝑝 +
1𝑞 −
2𝑅 cos 𝛼
𝐹M( =121𝑝 +
1𝑞 −
2 cos 𝛼𝑟
𝐹"( =sin𝛼2
1𝑝 −
1𝑞
1𝑝 +
1𝑞 −
cos 𝛼𝑟
𝐹\M =sin𝛼 cos( 𝛼
21𝑝 −
1𝑞
1𝑝 +
1𝑞 −
1𝑅 cos 𝛼
F20 F30 Will appear again on the tutorial
J.Nicolas MEDSI’16
acos211
Rqp=+
Meridional focusing
a
Sagittal focusing
p
q
R
a
p
q
r
-1 -0.5 0 0.5 1-200
-100
0
100
200
300
incidence angle variation (mrad)
foca
l dis
tanc
e va
riatio
n (m
m)
SagittalMeridional
The basic equation for paraxial optics is the focus condition
rqpacos211
=+
Varying the incidence angle
J.Nicolas MEDSI’16
dZ
dP
dzI
When one changes the pitch of a focusing mirror, one steers the beam, but also changesthe incidence angle (therefore the focusing).
Special cases. 1:1 magnifiction. Flat mirrors
Varying the incidence angle
For a fixed figure Two motors (pitch, z)control four optical parameters (focus,Beam angle, spot position, footprintcenter / coma)
J.Nicolas MEDSI’16
Figure error: slope
J.Nicolas MEDSI’16
Figure error for coherent sources (a case)
Position [ m]-5 0 5
Inte
nsity
[a.u
.]
0
0.2
0.4
0.6
0.8
1
1.2
Just consider the caseinwhich the mirror is perfectalmost everywhere except for astep inthe center.
-slope erroris very small,
- There is nolighton axis due todestructiveinterference between the two sides ofthewavefront
For diffraction limited imaging (FELs),oneneeds tocontrolthe wavefront distortion,which is given by the residualheight error.
J.Nicolas MEDSI’16
Figure error: height error
Δ𝑤 = 2Δℎ𝑐𝑜𝑠𝛼
Interference effects inimage formation aregiven by the Strehl ratio(actualpeakintensity tothe errorfreepeak intensity)
Δ𝑤
Δℎ
𝑆 ≅ 1 −2𝜋𝜆 Δ𝑤 f
ghi
(
Δℎ fghi
≤ 𝜆1 − 𝑆�
4𝜋𝑐𝑜𝑠𝛼ExampleLCLSII – S=0.97- 2.1mrad -13keVè 0.4nmrms
J.Nicolas MEDSI’16
designed for the resonant inelastic X-ray scattering (RIXS)experiments at MAX IV require an aperture length of 280 mmwith 100 nrad r.m.s. slope error. In addition, elliptical-cylinder-shaped focusing optics of 20 nrad r.m.s. residual slope devia-tion up to a length of 1000 mm are proposed to focus photonsat the Single Particles, Clusters and Biomolecules (SPB)experimental station at the European XFEL (Mancuso et al.,2013). In contrast to such long focusing mirrors, the length ofKirkpatrick–Baez (KB) mirror substrates at sources likePETRA III or MAX IV will be in the range 100–200 mm(Kalbfleisch et al., 2010; Johansson, 2014). Such opticalcomponents of elliptical and hyperbolic shape are alreadyavailable today (Matsuyama et al., 2012; Siewert et al., 2012)and allow diffraction-limited focusing of hard X-ray photonswithin nanometre focus size (Mimura et al., 2010). All theseoptical components used in tangential focusing have fairlylarge radii of curvature. On the other hand, sagittal focusing,which was proven to preserve brilliance very well, especially inthe soft X-ray regime, results in highly curved surfaces ofcylindrical, toroidal or ellipsoidal form setting new challengesfor manufacturing.
Dedicated metrology instrumentation of comparable accu-racy has been developed to characterize such optical elements.Second-generation slope-measuring profilers like the Nano-meter Optical component measuring Machine (NOM)(Siewert et al., 2004; Yashchuk et al., 2010; Nicolas & Martinez,2013; Assoufid et al., 2013) allow the inspection of reflectiveoptics up to a length of 1.5 m (Alcock et al., 2010) with anaccuracy better than 50 nrad r.m.s. It has taken the place of thewell known Long Trace Profiler-II (LTP) (Takacs et al., 1987)as a fundamental tool for the inspection of optics. It should benoted that metrology does not only provide characterizationof optical components, but such data can be directly used inrealistic beamline modelling (Samoylova et al., 2009).
2. On the precision of optical elements to guide andfocus X-rays
Synchrotron optical components are long shaped (up to 1.3 m)and used under the grazing-incidence condition (Wolter, 1952)
which makes them difficult to measure and thus to manu-facture. The impact of shape imperfections at optical compo-nents in the long spatial frequency error regime (from about1 mm to aperture length) on the imaging performance of anX-ray focusing system is related to the induced local phaseshifts in the reflected beam. The latter distort the wavefrontand cause the converging beam to have phase errors.Assuming a small source and a large distance between sourceand mirror, the acceptable p.v. mirror height variations !h arelimited to a few single nanometres only,
ð2!="Þ !hðxÞ!! !! sin # # 1; ð1Þ
with # being the grazing angle and " the wavelength of theX-ray beam (Samoylova et al., 2009). A further criterion for abeamline performance is the r.m.s. wavefront distortion whichshould be $r.m.s. < "/14 or better. This leads to a conditionknown as the Marechal criterion (Marechal, 1947), where theacceptable root-mean-square height error for a number ofoptical components over all spatial frequencies present withinthe residual surface errors is given by
!hrms $"
14ffiffiffiffiNp
2#; ð2Þ
where N is the number of reflecting surfaces in the system(Siewert et al., 2012). Clearly, the requirements on surfacequality become linearly more difficult to achieve withdecreasing X-ray wavelength, hence the difficult challenge ofmaking hard X-ray reflective optics of sufficient quality.Practically, mirrors of sub-nm r.m.s. figure quality for thelong-, mid- and high-spatial figure error need to be manu-factured and hence measured. These conditions have been themotivation to improve deterministic finishing technology aswell as metrology capabilities. To demonstrate the currentstate of metrology, we will discuss the inspection of a super-polished focusing mirror pair for beamline P06 at PETRA III.Finishing technology like ion beam figuring (IBF) (Schindleret al., 2003; Thiess et al., 2010) and elastic emission machining(EEM) (Yamauchi et al., 2002) allow the substrate topographyto be controlled on an atomic scale. Of course, this is onlyrealistic if precise topography data are available. Mirrorsfinished by use of EEM have recently shown <1 nm r.m.s.figure accuracy on a length of up to 350 mm (corresponding to50 nrad r.m.s. slope deviation) (Siewert et al., 2012) and haveallowed diffraction-limited focusing at the CXI experiment atLCLS (Boutet & Williams, 2010). However, upcoming opticslike the 1 m-long KB-focusing mirrors (Kirkpatrick & Baez,1948) at the European XFEL with a required residual slopeerror of 20 nrad r.m.s. (this corresponds to about 1 nm p.v.figure error!) will represent a challenge for both metrologyand finishing technology. To reach such a quality, the grav-itational sag and clamping forces need to be accounted for inthe mirror substrate so as to provide the required figure shapewhen mounted at the beamline. In addition, diffraction-limited sources translate into very high power densities onoptical components and additional care is required to preservethe mirror shape as well as possible, even by means of activeoptics. Whereas heat-load-induced deformations are very
new science opportunities
J. Synchrotron Rad. (2014). 21, 968–975 F. Siewert et al. % Characterization of ultra-precise X-ray optical components 969
Figure 1Improvement of mirror quality in terms of slope error during the last twodecades (based on data and measurements performed at the BESSY-IIOptical Metrology Laboratory).
Some reference values of figure error
Siewert etal.JSR21,968-975(2014)
Quickly evolved inthe recent past:
• Development ofmirror metrology
• Deterministic polishingtechniques:
Most (usual)companies candelivermirrors better than 0.5µradrms,onwhatever figureandlength.
For flats erroris sub-nanometer,andslope errorbelow 100nrad,cost anddelivery time.
Sagittal curvature always difficult
J.Nicolas MEDSI’16
Causes of figure error• Fabrication error(figuring,
polishing metrology):wellbelow 0.5µradrms
• Gravity sag.
• Clamping mechanics.
• Stressinduced by coolingscheme.
• Thermal deformations.
• Bendererrors,ellipseapproximatino errors
• Coating stress!Bakeoutdeveloped stresses (?)
mirror
Cooling pads
Cooling tube supports
mirror
Cooling pads
Cooling tube supports
Mirroralone:R=331.039m /0.163μrad RMSInholder:R=338.303m /0.533μrad RMS(Firstiteration)
J.Nicolas MEDSI’16
MisalignmentsNoerror
Error∆q=0.5% Pitcherror50µrad Yaw error50µrad
6m1m
PointsourceEllipsoidat3deg100x10mm2
FOVis 100µm
J.Nicolas MEDSI’16
By decoupling verticalandhorizontalfocusing,one canobtain aberration freefocusing byusing high quality mirrors polished flat,with low slope error,andmechanically bent ontoplano-elliptic figure.
Focused beam
50μm
10μm
Kirkpatrick-Baez configuration
J.Nicolas MEDSI’16
• Decouple horizontalandverticalplanes
• Astigmatic sources or images
• Nosagittal curvature
• Allow different stripes
• Moreaccurate metrology
• Relaxes alignment requirements (nosagittal curv.)
• If bender
• Start from flator spherical mirrors
• Adaptive focus andprimary comma
• Several foci
• Focus independent ofalignment
Some KB advantages
J.Nicolas MEDSI’16
J.Nicolas MEDSI’16
2. Characteristics of elliptic mirrors§ Geometryoftheellipse.§ Height,slope,curvature.§ Polynomialapproximation.
J.Nicolas MEDSI’16
𝑎 =𝑝 + 𝑞2
Description of an elliptic mirror
Majorsemi-axis
𝑐 =12 𝑝( + 𝑞( − 2𝑝𝑞 cos2𝛼�
Halfdistancebetweenfoci(lineareccentricity)
𝑒 = 1 −2
𝑝 + 𝑞 𝑝𝑞 cos( 𝛼�
eccentricityMinorsemi-axis
𝑏 = 𝑝𝑞� cos𝛼
pq
A
A’
x
y
α
c
a
b ψ
tan𝜓 =𝑞 − 𝑝𝑞 + 𝑝 cot𝛼
Orientation
J.Nicolas MEDSI’16
Equations of the ellipse
αp
q
A(-psinα,pcosα)
m
A’(qsinα,qcosα)
x
y
𝐫t − 𝐫B + 𝐫tu − 𝐫B = 𝑝 + 𝑞
𝑥 + 𝑝sinα ( + 𝑦 − 𝑝cosα (� + 𝑥 − 𝑞sinα ( + 𝑦 − 𝑞cosα (� = 𝑝 + 𝑞
Vectorequationoftheellipse
Implicitequationoftheellipse(asafunctionofcoordinates)
J.Nicolas MEDSI’16
Explicit equations of the ellipse
𝑦 𝑥 = −𝑥 𝑝( − 𝑞( sin 2𝛼 − 4 𝑝 + 𝑞 cos𝛼 𝑝𝑞 − 𝑝𝑞� 𝑝𝑞 − 𝑥 𝑝 − 𝑞 sin 𝛼 − 𝑥(�
𝑝( + 6𝑝𝑞 + 𝑞( − 𝑝 − 𝑞 ( cos2𝛼
Concavebranchoftheexplicitsolutionoftheellipse
• Notethaty(0)=0• Thisisanexact solution!
𝑑𝑦𝑑𝑥 𝑥 = −
2 𝑝 + 𝑞 cos𝛼 𝑝 − 𝑞 sin 𝛼 −𝑝𝑞� 𝑝 − 𝑞 sin 𝛼 + 2𝑥𝑝𝑞 − 𝑥 𝑝 − 𝑞 sin 𝛼 − 𝑥(�
𝑝( + 6𝑝𝑞 + 𝑞( − 𝑝 − 𝑞 ( cos2𝛼
Slopefunctionoftheellipse
• Notethaty’(0)=0• Thisisanexact solution!
J.Nicolas MEDSI’16
Curvature and imaging conditionCurvatureoftheellipse
• Notethatitisnotconstant
1𝑅 𝑥 =
𝑑(𝑦𝑑𝑥( 𝑥 =
𝑝𝑞� (𝑝 + 𝑞) cos𝛼2 𝑝𝑞 − 𝑥 𝑝 − 𝑞 sin 𝛼 − 𝑥( \ (⁄
1𝑅M
=~𝑝 + 𝑞) cos𝛼
2𝑝𝑞
Curvatureatthecenteroftheellipse
2𝑅M cos𝛼
=1𝑝 +
1𝑞
• Itisthefocusingcondition(mer)
J.Nicolas MEDSI’16
Taylor expansion of the ellipseDefinition
with
Inthecaseofanellipse,thecoefficientsarefunctionsofp,q andα
𝑦 𝑥 = W𝐸A𝑥AX
�YM
𝐸A = 1𝑛!𝑑A𝑦𝑑𝑥A�=YM
𝐸( =cos𝛼4
1𝑝 +
1𝑞 =
12𝑅M
𝐸\ =sin 2𝛼16
1𝑞( −
1𝑝(
𝐸� =E\(
4E(5 − csc( 𝛼 + E(\ sec( 𝛼 𝐸� =
E\\(7 − 3 csc( 𝛼)4E((
+ 3E((E\ sec( 𝛼
𝐸� =E\�(21 − 14csc( 𝛼 + csc� 𝛼)
8E(\+ E(E\((7 − csc( 𝛼) sec( 𝛼 + 2E(� sec� 𝛼
J.Nicolas MEDSI’16
Aberrations, revisited
𝐸\ =sin 2𝛼16
1𝑞( −
1𝑝(
𝐹(M =cos( 𝛼2
1𝑝 +
1𝑞 −
2𝑅 cos 𝛼
𝐹\M =sin 𝛼 cos( 𝛼
21𝑝 −
1𝑞
1𝑝 +
1𝑞 −
1𝑅 cos𝛼
𝐸( =cos𝛼4
1𝑝 +
1𝑞
𝐹(M = 2cos𝛼cos𝛼4
1𝑝 +
1𝑞 −
12𝑅
𝐹\M = −2cos𝛼sin 2𝛼16
1𝑞( −
1𝑝(
Remember:
Defocusisgivenbythequadratictermoftheellipsepolynomial
IfF20=0Primarycomaisgivenbythecubictermoftheellipsepolynomial
𝐹(M = 2cos𝛼 Δ𝐸(
Remember: 𝐹\M = −2cos𝛼 Δ𝐸\
J.Nicolas MEDSI’16
3. Mirror bender calculations and metrology
§ Euler-Bernoulliequation§ Rectangularfootprintcase§ Customfootprintcase
J.Nicolas MEDSI’16
Elastic beam theoryEuler-Bernoulliequation
Secondmomentofinertia
• z(x)isthedeformationinducedbytheload.[m]
• M(x)isthedistributionofbendingmomentsalongthemirror
• EistheYoung’smodulusofthebody.[N/m2]
• I(x)isthesecondmomentofinertiaofthemirrorsubstratesection.[m4]
𝐼 𝑥 = � 𝑧(𝑑𝑥𝑑𝑧����F�A
𝐼?? =𝑏ℎ\
12
Forarectangularsection
𝐸 · 𝐼 𝑥𝑑(
𝑑𝑥( 𝑧 𝑥 = 𝑀 𝑥 𝑀 𝑥 = W 𝑥 − 𝑥F 𝐹F
�
F|=T�=
Bendingmoments
𝑏ℎ
J.Nicolas MEDSI’16
Integration (rectangular mirror footprint)
𝑧 𝑥 = 𝑐� + 𝑐"𝑥 + � �𝑀 𝑥′𝐸𝐼 𝑥′ 𝑑𝑥′
="
�� (⁄𝑑𝑥"
=
�� (⁄ 𝑧 𝑥 = 𝑐� + 𝑐"𝑥 + 𝐸(𝑥( + 𝐸\𝑥\
GeneralSolution toEuler-Bernoulli’equation For constant section mirror the solutionis athird degree polynomial
𝑀 𝑥𝑥
𝐸( =3𝑎𝐸𝑏ℎ\ 𝐹� + 𝐹�
𝐸\ =2𝑎
𝐸𝑏ℎ\𝐿 𝐹� − 𝐹�
𝐹�𝐹�
𝑎 𝐿 − 𝑎 𝐿
J.Nicolas MEDSI’16
𝐸( =3𝑎𝐸𝑏ℎ\ 𝐹� + 𝐹�
𝐸\ =2𝑎
𝐸𝑏ℎ\𝐿 𝐹� − 𝐹�
Rectangular mirror footprint• The sumofforces controls the main curvature (andthe
defocusing)
• The difference between the forces controls theeccentricity (andprimary coma)
120100
Eccentricity (E3) [m-2]
806050
10-6
-4-3
3210
-1-2
100
Force Downstream [N]
120
Force Upstream [N]
100
Curvature (E2) [km-1]
806050
0.0260.0240.022
0.020.0180.0160.014
100
Force Downstream [N] Force
Upstream [N]
𝐸(𝐸\
=𝐸(,���𝐸\,� �
+ 𝐸(� 𝐸(�𝐸\� 𝐸\�
𝐹�𝐹�
Measurement No1 2 3 4 5 6
Fit E
rror [
rad
rms]
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
FU=50.1 NFD=50.0 N FU=100.0 N
FD=49.9 N FU=99.9 NFD=100.0 N
FU=50.1 NFD=99.8 N
FU=120.3 NFD=99.7 N
FU=60.3 NFD=80.3 N
Accuracy of the linear fitE2E3
fitbetterthanabout50nrad rms(metrologylimited– tooquick)
J.Nicolas MEDSI’16
High demagnification ellipsesTheerrorofthecubicpolynomialapproximationisgivenby:
𝑦¡¢¢£¢ 𝑥 =
−𝑥 𝑝( − 𝑞( sin 2𝛼 − 4 𝑝 + 𝑞 cos𝛼 𝑝𝑞 − 𝑝𝑞� 𝑝𝑞 − 𝑥 𝑝 − 𝑞 sin 𝛼 − 𝑥(�
𝑝( + 6𝑝𝑞 + 𝑞( − 𝑝 − 𝑞 ( cos2𝛼 −cos𝛼4
1𝑝 +
1𝑞 𝑥( −
sin 2𝛼16
1𝑞( −
1𝑝( 𝑥\
Position [mm]-200 -150 -100 -50 0 50 100 150
Hei
ght E
rror [
nm]
-100
0
100
200
300
400FU=7 N, FD=269 N,Error:6.267 μ rad
For ALBA’s LOREAbeamline HFM10:1,400mmlong,this erroris inthe order of6.3urad rms
𝑦¡¢¢£¢ 𝑥 = 𝑦¡¤U*¥ 𝑥 − 𝐸(𝑥( − 𝐸\𝑥\
Forhighdemagnificationellipses,higherorderaberrationsaresignificant
J.Nicolas MEDSI’16
Varied width mirrors
Byinsertingtheexactcurvatureprofile,wehavethedefinitionoftherequiredmirrorwidth.
𝐸𝑏 𝑥 ℎ\
12𝑑(𝑦𝑑𝑥( 𝑥 = 𝑀M + Δ𝑀
2𝑥𝐿
𝐸𝑏 𝑥 ℎ\
12𝑝𝑞� (𝑝 + 𝑞) cos𝛼
2 𝑝𝑞 − 𝑥 𝑝 − 𝑞 sin 𝛼 − 𝑥( \ (⁄ = 𝑀M + Δ𝑀2𝑥𝐿
𝑏 𝑥 = 𝑏M1𝑏M𝑀M +
1𝑏MΔ𝑀
2𝑥𝐿
24 𝑝𝑞 − 𝑥 𝑝 − 𝑞 sin 𝛼 − 𝑥( \ (⁄
𝐸ℎ\ 𝑝𝑞� (𝑝 + 𝑞) cos𝛼
Deformationequationincludescurvature:
Thesolutiondependsontheforcesonecanapply.
Oftenb(x)isjustapproximatedbyaloworderpolynomialvariation
𝑀M =12 𝑎"𝐹" + 𝑎(𝐹(
Δ𝑀 =12 𝑎"𝐹" − 𝑎(𝐹(
J.Nicolas MEDSI’16
Examples of varied profile mirrors
-200 0 200
-20
0
20
-200 0 200
-20
0
20
-200 0 200
-20
0
20
-200 0 200
-20
0
20
Mirrorwidthprofilefora10:1demagnificationmirror,fordifferentbendingmomentcombinations.
J.Nicolas MEDSI’16
Model of deformation
𝐸𝑏 𝑥 ℎ\
12𝑑(𝑦𝑑𝑥( 𝑥 = 𝑀M + Δ𝑀
2𝑥𝐿
Deformationofthemirror,inthiscase,isNOTapolynomial.
𝐸𝑏 𝑥 ℎ\
12𝑑(𝑦𝑑𝑥( 𝑥 = 𝑎
𝑥𝐿 −
12 𝐹� − 𝑎
𝑥𝐿 +
12 𝐹�
𝐸𝑏 𝑥 ℎ\
12𝑎𝑑(𝑦�𝑑𝑥( 𝑥 =
𝑥𝐿 −
12
𝐸𝑏 𝑥 ℎ\
12𝑎𝑑(𝑦�𝑑𝑥( 𝑥 = −
𝑎𝑥𝐿 −
12
𝑦 𝑥 = 𝐹�𝑦� 𝑥 + 𝐹�𝑦� 𝑥
Position [mm]-150 -100 -50 0 50 100 150
Hei
ght E
rror [
nm]
-40
-30
-20
-10
0
10
20Base Functions
hU(x)hD(x)
…butitisstillalinearcombinationoftwofunctions.
J.Nicolas MEDSI’16
Position [mm]-150 -100 -50 0 50 100 150
Res
idua
l Hei
ght [
nm]
-5
0
5
10
15
20
Residual Error#1: 0.330 urad#2: 0.385 urad#3: 0.401 urad#4: 0.378 urad#5: 0.377 urad#6: 0.341 urad
Metrology of varied width mirror benders
Measurement No1 2 3 4 5
Fit E
rror [
N rm
s]
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
FU=50.1 NFD=50.1 N
FU=50.4 NFD=249.6 N
FU=50.4 NFD=399.3 N
FU=199.8 NFD=400.7 N
FU=201.0 NFD=250.0 NAccuracy of the linear fit
FUFD
Afterfittingdatatotheproposedbaseoffunctions,residualfigure(polishingerror)isequalforallbendingpositions
J.Nicolas MEDSI’16
4. Mechanical design of mirror benders§ Functionalitiesofbender§ Parasiticforces§ Cyl benders,U-type,4-point,scissor,
bimorph,…
J.Nicolas MEDSI’16
A zoo of mechanical solutions
(b) cantilever spring bender
(a) “s” spring bender
Mechanical design approaches
M. R. Howells, et al., Theory and practice of elliptically bent x-ray mirrors, Opt. Eng.39(10), 2748-61 (2000)
Beam theory applied to a bent structure:
xL
CCCC
x
yxEI 2121
2
2
2)(
!!
+=
"
"
E is Young’s modulus of the mirror material
I(x) is the moment of inertia as a function of
position along the beam, or mirror
C1
and C2 are the end couples producing
bending moments
12)(
3
0
hbIxI ==
bird-like shapeanti-clastic bending
Slide: 9ACTOP08, Trieste 10 October 2008A. Rommeveaux
KB development at ESRF
0
2
4
68
10
12
14
2001 2002 2003 2004 2005 2006 2007 2008
Nu
mb
er o
f b
end
ers
Different configurations: ¾ Single bender (13)¾ KB 170-170 (8)¾ KB 170-96 (8)¾ KB 170-300 (3)¾ KB 300-300 (5)
Since 2001: 61 benders tested and pre-shaped at the metrology lab22 with multilayer coatings
ID23 5 x 7 µmID27 2 x 4 µmID13 0.3 x 0.4 µmID22 76 x 84 nmID19 45 nm in Vertical
Graded multilayer *96 mm 170 mm
300 mm
Spot size
* O.Hignette
et al., AIP Conf Proc.-
January 19, 2007 -
Volume 879, pp. 792-795
Spot size achieved is generally close to geometrical limit (i.e.
not limited by optics aberrations)
Slide: 5ACTOP08, Trieste 10 October 2008A. Rommeveaux
LTP characterization of Long mirror benders
1.
Mirror intrinsic slope error characterization (out of bender)2.
Mirror clamped on mechanics: •
Evaluation of deformation induced by mechanics and correction if
possible•
Adjustment, optimization of gravity compensators•
Verifying safety switches, range of curvature 3.
Hysteresis cycles4.
Long term stability
Anti-twist Correction
Twist Simulation
before twist correction
after twist correction
Anti-twist mechanisms
Anti-twist correction
should not stress the
mirror substrate
in other directions
Visualization and correction of
surface twist with ZYGO GPITM
with virtual pivot
A – stabilizing bracket
Cylindrical bender ESRF-Scissor bender
S-spring bender
Canti-lever bender
Four-point bender
SLAC HiRes
J.Nicolas MEDSI’16
Functionalities of a bender
1. To introduce the forces that deform themirror to the required ellipse withintolerances.
2. To minimize parasitic forces thatintroduce undesired deformations
3. To introduce forces that deform themirror to compensate gravity sag orother errors
Mechanical design of benders
J.Nicolas MEDSI’16
Functionalities of a bender
1. To introduce the forces that deform themirror to the required ellipse withintolerances.
2. To minimize parasitic forces thatintroduce undesired deformations
3. To introduce forces that deform themirror to compensate gravity sag orother errors
Mechanical design of benders
• Fixtheapplicationpointanddirection
• Demagnify motordisplacementstonanometerresolutiondeformation
J.Nicolas MEDSI’16
Apply controlled forces
• Fixtheapplicationpointanddirection
• Demagnify motordisplacementstonanometerresolutiondeformation
Flexurehingespointtothe“rotation”point.Mirrorisglued(monolithic)
Lever(+bending?)demagnifiesmotorlinearmotiontoanangleconstraintonthemirrorends.
J.Nicolas MEDSI’16
• Fixtheapplicationpointanddirection
• Demagnify motordisplacementstonanometerresolutiondeformation
Pivotincenterofcontactsurfaceensuresforceisnormaltothesurface,andfixesapplicationpoint
(Notvisible)Forceisexertedbyahellicoidal springincompression
Apply controlled forces
J.Nicolas MEDSI’16
• Fixtheapplicationpointanddirection
• Demagnify motordisplacementstonanometerresolutiondeformation
Demagnificationofmotordisplacementbyelasticbendingofthecantilever
Anti-twist Correction
Twist Simulation
before twist correction
after twist correction
Anti-twist mechanisms
Anti-twist correction
should not stress the
mirror substrate
in other directions
Visualization and correction of
surface twist with ZYGO GPITM
with virtual pivot
A – stabilizing bracket
Apply controlled forces
J.Nicolas MEDSI’16
• Fixtheapplicationpointanddirection
• Demagnify motordisplacementstonanometerresolutiondeformation
Demagnificationofmotordisplacementbyarigidlongleverarm
Slide: 9ACTOP08, Trieste 10 October 2008A. Rommeveaux
KB development at ESRF
0
2
4
68
10
12
14
2001 2002 2003 2004 2005 2006 2007 2008
Nu
mb
er o
f b
end
ers
Different configurations: ¾ Single bender (13)¾ KB 170-170 (8)¾ KB 170-96 (8)¾ KB 170-300 (3)¾ KB 300-300 (5)
Since 2001: 61 benders tested and pre-shaped at the metrology lab22 with multilayer coatings
ID23 5 x 7 µmID27 2 x 4 µmID13 0.3 x 0.4 µmID22 76 x 84 nmID19 45 nm in Vertical
Graded multilayer *96 mm 170 mm
300 mm
Spot size
* O.Hignette
et al., AIP Conf Proc.-
January 19, 2007 -
Volume 879, pp. 792-795
Spot size achieved is generally close to geometrical limit (i.e.
not limited by optics aberrations)
Apply controlled forces
J.Nicolas MEDSI’16
Functionalities of a bender
1. To introduce the forces that deform themirror to the required ellipse withintolerances.
2. To minimize parasitic forces thatintroduce undesired deformations
3. To introduce forces that deform themirror to compensate gravity sag orother errors
Mechanical design of benders
• Avoidtwist• Avoidforcesalonglongitudinal
axis
J.Nicolas MEDSI’16
Control parasitic forces
• Avoidtwist• Avoidforcesalonglongitudinal
axis
Anti-twist Correction
Twist Simulation
before twist correction
after twist correction
Anti-twist mechanisms
Anti-twist correction
should not stress the
mirror substrate
in other directions
Visualization and correction of
surface twist with ZYGO GPITM
with virtual pivot
A – stabilizing bracket
Virtualpivottomanuallyadjustthetwistofoneendofthemirrorduringmetrologytests
J.Nicolas MEDSI’16
Torsion (twist) adjustment
Anti-twist Correction
Twist Simulation
before twist correction
after twist correction
Anti-twist mechanisms
Anti-twist correction
should not stress the
mirror substrate
in other directions
Visualization and correction of
surface twist with ZYGO GPITM
with virtual pivot
A – stabilizing bracket
Therearebenderswhicharetorsionfree,andotherswhichallowcorrectingit
J.Nicolas MEDSI’16
• Avoidtwist• Avoidforcesalonglongitudinal
axis
Rolledarticulation(ononesideonly)toallowthemirrorsettleitstwistfreely.
Bendingforceactuatorsallowpitchandrolltoavoidintroducingundesireddeformations
Control parasitic forces
J.Nicolas MEDSI’16
Functionalities of a bender
1. To introduce the forces that deform themirror to the required ellipse withintolerances.
2. To minimize parasitic forces thatintroduce undesired deformations
3. To introduce forces that deform themirror to compensate gravity sag orother errors
Mechanical design of benders
• Gravitysagcompensation
J.Nicolas MEDSI’16
Gravity sag compensation
• Gravitysagcompensation
Manuallyadjustablesprings,forgravitysagcompensation
J.Nicolas MEDSI’16
Functionalities of a bender
1. To introduce the forces that deform themirror to the required ellipse withintolerances.
2. To minimize parasitic forces thatintroduce undesired deformations
3. To introduce forces that deform themirror to compensate gravity sag orother errors
Adaptive optics
• Gravitysagcompensation• Adaptiveoptics
Piezo– electriccorrctorsembeddedinthemirrorbulk
PiezoCeramics
Silicon
Silicon
HVelectrodes
J.Nicolas MEDSI’16
• Gravitysagcompensation• Adaptiveoptics
Nanometerresolution–stiff-pushing/pullingmotors
Adaptive optics
J.Nicolas MEDSI’16
• Gravitysagcompensation• Adaptiveoptics
Force-stabilized– motorizedcorrectors.
Adaptive optics
J.Nicolas MEDSI’16
Position [mm]-200 -100 0 100 200
Heig
ht e
rror [
nm]
-5
-4
-3
-2
-1
0
1
2
3
4
5
AchievedOptimalDifference
Position [mm]-200 -100 0 100 200
Hei
ght e
rror [
nm]
-50
-40
-30
-20
-10
0
10
20
30
40
50
SlopeError: 0.115μradrmsSurfaceError:0.858nmrms
Using 4actuators,500mm
Initialslopeerror 0.87μradrmsInitialsurfaceError:23.2nmrms
Difference tomodel-baseoptimum of0.08nm
Adaptive opticsResults of the ALBA-SENER nanobender
J.Nicolas MEDSI’16
Position [mm]-200 -100 0 100 200
Hei
ght e
rror [
nm]
-10
-8
-6
-4
-2
0
2
4
6
8
10R=2.94kmR=3.56kmR=3.98km
The correction is preserved within the nanometer with variations oftheradius ofcurvature of35%.
Adaptive optics
J.Nicolas MEDSI’16
Acknowledgements
ALBA Claude RugetCarles CollderamIgors SicsPablo Pedreira
SLACDaniele CoccoDaniel MortonCorey HardinMay Ling Ng
Valeriy Yashchuck (ALS)Ray Barrett (ESRF)
SENERIRELEC
J.Nicolas MEDSI’16
Backup slides – gravity sag correction§ Rectangularfootprintcase§ Customfootprintcase
J.Nicolas MEDSI’16
Gravity sag calculation
F1 F2
-ρLbhg
z
ya1
a2
L
1. Equilibriumofforcesandtorques:todeterminealltheforces.2. Functionofmomenta:todeterminetheRHSoftheE-Bequation.3. Integration.4. Boundaryconditions:todeterminetheintegrationconstants.
J.Nicolas MEDSI’16
Equilibrium of forces
¦𝐹" + 𝐹( − 𝑔𝜌𝐿𝑏ℎ = 0
𝑎"𝐹" + 𝑎(𝐹( −𝑔𝜌𝐿(𝑏ℎ
2 = 0
F1 F2
-ρLbhg
z
ya1
a2
L
𝐹" = 𝑔𝑚𝑎( − 𝐿 2⁄𝑎( − 𝑎"
𝐹( = 𝑔𝑚𝐿 2⁄ − 𝑎"𝑎( − 𝑎"
• Allforcesandtorquesmustaddtozero,otherwisethemirrorwouldbeacceleratinglinearlyorangularly.
• Torquesarecalculatedfromtheoriginofcoordinates(notthecenterofthemirror)
J.Nicolas MEDSI’16
Function of moments
RegionI.(onlyweight)
𝐹ª 𝑥 = −𝑔𝜌𝑏ℎ𝑥
𝑎ª 𝑥 =𝑥2
𝑀� 𝑥 = 𝐹ª 𝑥 𝑥 − 𝑎ª 𝑥
= −12𝑔𝜌𝑏ℎ𝑥
(
J.Nicolas MEDSI’16
Function of moments
RegionI.(onlyweight)
𝐹ª 𝑥 = −𝑔𝜌𝑏ℎ𝑥
𝑎ª 𝑥 =𝑥2
J.Nicolas MEDSI’16
𝑀��� 𝑥
= −12𝑔𝜌𝑏ℎ𝑥
(
+ 𝐹" 𝑥 − 𝑎"+ 𝐹( 𝑥 − 𝑎(
Function of moments
RegionII.(weight,F1)
RegionIII.(weight,F1,F2)
𝑀�� 𝑥
= −12𝑔𝜌𝑏ℎ𝑥
(
+ 𝐹" 𝑥 − 𝑎"
J.Nicolas MEDSI’16
Integration (1)• Eachregionhastobeintegratedseparately,becausethefunctionisnot
continuous.• Inallthecasesthefunctionofmomentaisapolynomialoforder2.Whichwhen
integratedtwicebecomesapolynomialoforderfour.
𝑧« 𝑥 = 𝑧M + 𝑧"𝑥 + 𝑧(𝑥( + 𝑧\𝑥\ + 𝑧�𝑥�
𝑑𝑧«𝑑𝑥 𝑥 = 𝑧" + 2𝑧(𝑥 + 3𝑧\𝑥( + 4𝑧�𝑥\
𝑑(𝑧«𝑑𝑥( 𝑥 = 2𝑧( + 6𝑧\𝑥 + 12𝑧�𝑥(
𝑘 =𝐸𝑏ℎ\
12𝐸𝑏ℎ\
12𝑑(𝑧«𝑑𝑥( 𝑥 = 2𝑘𝑧( + 6𝑘𝑧\𝑥 + 12𝑘𝑧�𝑥(
Bydefining: Wehave:
J.Nicolas MEDSI’16
𝐴( = 0𝐴\ = 0
𝐴� = −𝑔𝜌𝑏ℎ24𝑘
𝑀�(𝑥) = −12𝑔𝜌𝑏ℎ𝑥
(
𝐴M, 𝐴"
RegionI.
Implies
2𝑘𝑧( + 6𝑘𝑧\𝑥 + 12𝑘𝑧�𝑥( = 𝑀 𝑥
Integration (2)
Unknown
𝑀��(𝑥)= −𝑎"𝐹" + 𝐹"𝑥
−12𝑏𝑔ℎ𝑥
(𝜌
𝐵( = −𝑎"𝐹"2𝑘
𝐵\ =𝐹"6𝑘
𝐵� = −𝑏𝑔ℎ𝜌24𝑘
RegionII.
𝐵M, 𝐵"
𝑧® → 𝐴F, 𝐵F, 𝐶F
RegionIII.
𝑀��� 𝑥= −𝑎"𝐹" − 𝑎(𝐹( + (𝐹"+ 𝐹()𝑥 −
12𝑏𝑔ℎ𝑥
(𝜌
𝐶( = −𝑎"𝐹" + 𝑎(𝐹(
2𝑘
𝐶\ =𝐹" + 𝐹(6𝑘
𝐶� = −𝑏𝑔ℎ𝜌24𝑘
𝐶M, 𝐶"
Implies Implies
Unknown Unknown
Notationchange!!!
J.Nicolas MEDSI’16
Boundary conditions1.Continuitybetweenregions+ attbese pointsdeformationiszero
𝑧� 𝑎" = 0𝑧�� 𝑎" = 0𝑧�� 𝑎( = 0𝑧��� 𝑎( = 0
𝑧� 𝑎" = 𝑧��(𝑎")𝑧�� 𝑎( = 𝑧���(𝑎()
2.Derivability(continuityof1stderivative)betweenregions
ThesystemislinearonA0,A1,B0,B1,C0,C1,andcanbesolvedwithuniquesolution(usingMathematica).
è SeeMathematicaNotebook:GravitySagCalculation.nb
J.Nicolas MEDSI’16
−500 0 500−300
−200
−100
0
100
200
300
Position [mm]
Surfa
ce h
eigh
t [nm
]
BesselEnds
Example of gravity sag correctionAmirrormeasuredwithsupportsattheBesselpointsandattheends.Gravitysagisremovedinbothcases.
J.Nicolas MEDSI’16
Gravity sag of arbitrary profile mirror
𝐸𝑏 𝑥 ℎ\
12𝑑(𝑧«𝑑𝑥( 𝑥 = 𝑀« 𝑥 +𝑀" 𝑥 +𝑀( 𝑥
Thewidthofthemirrorisnotconstant
MG:gravity,M1:support1,M2:support2
CalculatingMG
• Massattheleftofx:
• Masscenter:
M1,M2
𝑚� 𝑥 = 𝜌ℎ � 𝑏 𝑥u 𝑑𝑥u=
�±/(
𝑥� 𝑥 = � 𝑥′𝑏 𝑥u 𝑑𝑥u=
�� (⁄� 𝑏 𝑥u 𝑑𝑥u=
�� (⁄³
𝑀F 𝑥 = 𝑥 − 𝑥F 𝐹F𝜃 𝑥 − 𝑥F
𝜃 𝑥 Heavyside Stepfunction
𝐼M 𝑥
𝐼" 𝑥𝐼M 𝑥
J.Nicolas MEDSI’16
Gravity sag of arbitrary profile mirror
𝑑(𝑧«𝑑𝑥( 𝑥 = 12
𝑀« 𝑥 +𝑀" 𝑥 +𝑀( 𝑥 𝐸𝑏 𝑥 ℎ\
• Thewidthofthemirrorisnotconstant.
• Theequationisno-longertheintegralofapolynomial.
• Itcanbeintegratednumerically.
• Allthefunctionsofxbecomevectors(lineararrays)
MG:gravity,M1:support1,M2:support2
𝑓 𝑥 → 𝑓 𝑥A → 𝑓 𝑛Δ𝑥 → 𝑓A
J.Nicolas MEDSI’16
CalculatingMG(x)
• Massattheleftofx:
• Masscenter:
𝑚� 𝑥 = 𝜌ℎ � 𝑏 𝑥u 𝑑𝑥u=
�±/(
𝑥� 𝑥 = � 𝑥′𝑏 𝑥u 𝑑𝑥u=
�� (⁄� 𝑏 𝑥u 𝑑𝑥u=
�� (⁄³
ThesebecomenumericalintegralsIfyoulikeyoucanapplymoresophisticatedintegrationmethods,Simpson’srule,Spline,…
Moment of inertia of the weight
𝑀« 𝑥 = −𝑔𝑚� 𝑥 𝑥 − 𝑥� 𝑥 𝑀« A = −𝑔 𝑚� A 𝑥A − 𝑥� A
𝑚� A = 𝜌ℎΔ𝑥 W 𝑏B
A
BY"𝑥� A = W 𝑥B𝑏B
A
BY"
W 𝑏B
A
BY"
µ
AndthefinalexpressionofMG(x)isalsoavector
J.Nicolas MEDSI’16
Moments of inertia of the supportsCalculatingM1(x),M2(x)
• NeedtoknowtheactualvaluesofF1 andF2 (notjustitsexpressionasafunctionofmirrordimensions).
• TheycanbewrittenincloseexpressionusingtheHeavisidefunctionstepfunction
𝐹" = 𝑔𝑚M𝑥( − 𝑥M𝑥( − 𝑥"
𝐹( = 𝑔𝑚M−𝑥" + 𝑥M𝑥( − 𝑥"
m0:totalmassx0:masscenterofthemirror
𝑀F 𝑥 = 𝑥 − 𝑥F 𝐹F𝜃 𝑥 − 𝑥F
𝑀F A = 𝑥A − 𝑥F 𝐹F𝜃 𝑥A − 𝑥F
J.Nicolas MEDSI’16
Integration• Onceallthefunctionsofxareevaluated,theycanbenumericallyintegrated
• Notethat,sinceweusecloseexpressions,therearenoboundaryconditionsbetweenzones.
• Whichbecomes
• Averageheightandaveragepitchareundetermined.Often,thesearesimplysettoaveragezeroastheydonotcarryinformationaboutthemirrorfigure,butaboutitsposition.
𝑧 𝑥 = 𝑐� + 𝑐"𝑥 + � � 12𝑀« 𝑥′ + 𝑀" 𝑥′ + 𝑀( 𝑥′
𝐸ℎ\𝑏 𝑥′ 𝑑𝑥′="
�� (⁄𝑑𝑥"
=
�� (⁄
𝑧A = 𝑐� + 𝑐"𝑥A +12𝐸ℎ\ W W
𝑀« O + 𝑀" O + 𝑀( O
𝑏O
B
OY"
A
BY"
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