tel aviv universityprimage.tau.ac.il/libraries/theses/exeng/free/1509386.pdftel aviv university the...
TRANSCRIPT
TEL AVIV UNIVERSITYTHE IBY AND ALDAR FLEISCHMAN
FACULTY OF ENGINEERING
LOW-VOLTAGE FREE-ELECTRON LASERS
AND RELATED DEVICES
Thesis submitted for the degree of
“Doctor of Philosophy”
By
Rami Drori
Submitted to the Senate of Tel Aviv University
June 2001
TEL AVIV UNIVERSITYTHE IBY AND ALDAR FLEISCHMAN
FACULTY OF ENGINEERING
LOW-VOLTAGE FREE-ELECTRON LASERS
AND RELATED DEVICES
Thesis submitted for the degree of
“Doctor of Philosophy”
By
Rami Drori
This research work was carried out under the supervision of
Prof. Eli Jerby
Submitted to the Senate of Tel Aviv University
June 2001
I would like to thank my wife Liora for her support and encouragement
I dedicate this thesis to her and to my beloved children
Shaked and Aviv
for all the time that should have been theirs
and to my parents
Gal’ya and Nathan
Acknowledgements
I thank Prof. Eli Jerby for his guidance and support.
I thank graduate students Avi Shahadi and Michael Korol for many fruitful
discussions.
I am indebted to Prof. Gil Rosenman and Dr. Dima Shur for their cooperation in
utilizing part of this thesis.
I thank Mr. Alon Aharony who, as part of his M.Sc. thesis, obtained some of the
presented results.
א
Abstract
This thesis presents a novel operating regime of free-electron lasers (FELs) at radio
frequencies (RF), and studies of related technologies and schemes. The latter have been first
employed in cyclotron-resonance masers (CRMs). These studies include development of a
ferroelectric-cathode electron-gun and the incorporation of helix structures. Both may
consequently be used in FEL devices. The experimental and theoretical studies were carried
out at our laboratory as part of the activity of developing low-voltage compact free-electron
based electromagnetic (EM) generators.
The operating range of the FEL is extended in this thesis toward its lowest frequency and
accelerating voltage (< 1 GHz, < 6 kV). Due to this new and unusual operating range we
propose the acronym FER where “R” stands for Radio. The FER employs a non-dispersive
transmission line cavity that supports TEM waves having extremely long wavelengths. The
FER devices presented here obey the basic physical rules of the mature FEL devices but
operate in a new and unusual regime in which the radiation wavelength λ is much longer than
the wiggler period wλ , i.e. wλ>>λ .
An experimental study of the FEL tunability by a variable dielectric loading has been carried
out using the FER device. The dielectric load was implemented by a variable amount of
distilled water poured into glass pipes inside the interaction region. The fluid-loaded
microwave generator in general is demonstrated here for the first time.
The fluid-loaded FER has demonstrated a record of an FEL-type device, operating in the
lowest voltage of 420 V and radiating the lowest frequency of 266 MHz. The low frequency
ב
of the FER signal enables its monitoring without detection using a fast digital-oscilloscope.
This ability provides a clear insight into FEL time-domain phenomena.
A one-dimensional steady-state electron dynamics and an amplification model of the FER in
the linear regime were derived. Analysis of the electron trajectories shows that employing an
axial magnetic field is necessary for guiding the electrons through the interaction structure.
Due to the axial magnetic field, a cyclotron interaction is excited in certain conditions, in
addition to the FER interaction. For the CRM interaction, the wiggler acts as a distributed
kicker that imparts to the electrons the required transverse velocity components. Results of
parametric analysis (parameters such as accelerating voltages and magnetic field amplitudes)
agree with the experimental observations.
In a complementary research, an electron gun that employs a high-dielectric ceramic
(ferroelectric) cathode has been developed and investigated. In this cathode the electrons are
emitted from plasma excited on the ceramic surface by a voltage-pulse of the order of 1 kV
applied on the ceramic in a nanosecond time scale.
In order to study the ferroelectric-cathode electron-gun in a microwave generator, it was
applied first in a device without a wiggler. The operating parameters were adjusted to obtain
CRM interactions near the cutoff of a hollow cylinder cavity. Interactions in this operation
regime, observed in the experiment around 7 GHz, tolerate energy-spread of the electrons.
Improvement of its performance may lead to the employment of the ferroelectric-cathode
electron-gun in both CRM and FEL devices.
Practical cold cathodes may have a major contribution to the compactness and cost reduction
of free-electron EM sources. Ferroelectric cathodes are cheaper than the thermionic cathodes,
and they do not require heating or pre-activation processes and their vacuum requirements are
ג
less strict. According to the best of our knowledge, high-dielectric-ceramic cathodes had not
been utilized before in any EM radiation generator.
A broad and tunable bandwidth CRM was studied also in the framework of this thesis.
Traveling-wave-type oscillations independent of the axial magnetic-field strengths have been
observed in a single-helix device. One of the interactions was observed between the electrons
and both forward and backward spatial modes simultaneously. This interaction, applicable in
oscillator schemes, may be more efficient in comparison to interaction with only one spatial
mode.
In order to get cyclotron interaction, we proposed to employ a bifilar–helix structure, in which
the transverse component of the electric field on the axis is much stronger than in the
single-helix structure. In the bifilar-helix based CRM experiment, a stand-alone electron beam
assembly was used. This electron-beam assembly enables the conducting of experiments with
various interaction structures without breaking the vacuum. Preliminary results of the
bifilar-helix experiment exhibit both CRM oscillations and amplification. The bifilar helix can
be used in future FER schemes as a combined wiggler and slow-wave structure.
Typical FELs and CRMs operate at high accelerating voltages which dictate a certain scale
and cost of these devices, which might be impractical (i.e. large and expensive) for many
applications. This thesis presents studies that may lead to the development of new low-cost
and compact free-electron devices for medium power applications in the RF and in the
microwave ranges.
ד
List of publications and presentations carried out in relation to this thesis:
Papers
1. A. Shahadi, E. Jerby, M. Korol, R. Drori, M. Sheinin, V. Dikhtiar, V. Grinberg, I.
Ruvinsky, M. Bensal, T. Har’el, Y. Baron, A. Fruchtman, V. L. Granatstein, and G.
Bekefi, “Cyclotron resonance maser experiment in a non dispersive waveguide,” Nucl.
Instrum. and Methods in Phys. Res., Vol. A358, pp. 143-146, 1995.
2. R. Drori, E. Jerby and A. Shahadi, "Free-electron maser oscillator experiment in the UHF
regime," Nucl. Instrum. and Methods in Phys. Res., Vol. A358, pp. 151 - 154, 1995.
3. A. Shahadi, E. Jerby, Li Lei, and R. Drori, “Carbon-fiber emitter in a cyclotron-resonance
maser experiment,” Nucl. Instrum. and Methods in Phys. Res., Vol. A375, pp. 140-142,
1996.
4. R. Drori, E. Jerby, A. Shahadi, M. Einat, M. Sheinin, "Free-electron maser operation at 1
GHz/1keV regime," Nucl. Instrum. and Methods in Phys. Res., Vol. A375, pp. 186-189,
1996.
5. E. Jerby, A. Shahadi, R. Drori, M. Korol, M. Einat, I. Ruvinsky, M. Sheinin, V. Dikhtiar,
V. Grinberg, M. Bensal, T. Har’el, Y. Baron, A. Fruchtman, V. L. Granatstein, and G.
Bekefi, "Cyclotron resonance maser experiment in a non-dispersive waveguide," IEEE
Trans. Plasma Science, Vol. 24, pp. 816-824, 1996.
6. R. Drori and E. Jerby, "Free-electron-laser type interaction at 1 m wavelength range,"
Nucl. Instrum. and Methods in Phys. Res., Vol. A393, pp. 284 - 288, 1997.
7. R. Drori, M. Einat, D. Shur, E. Jerby, G. Rosenman, R. Advani, R. J. Temkin, and C.
Pralong, “Demonstration of microwave generation by a ferroelectric-cathode tube,” App.
Phys. Letters, Vol. 74, pp. 335-337, 1999.
8. R. Drori and E. Jerby, “Tunable fluid-loaded free-electron laser in the low electron-energy
and long-wavelength extreme,” Phys. Rev. E., Vol. 59, pp. 3588 - 3593,1999.
9. A. Aharony, R. Drori and E. Jerby, “Cyclotron resonance maser experiments in a bifilar
helical waveguide,” Phys. Rev. E., Vol. 62, pp. 7282-7286, 2000.
ה
Conferences and workshops presentations (the name of the presenter is underlined)
1. A. Shahadi, R. Drori, and E. Jerby, "Cyclotron-resonance maser experiments in first and
second harmonics,” The 18th IEEE Convention in Israel, Tel Aviv, March 7-8, 1995 (oral
presentation).
2. R. Drori, A. Shahadi, and E. Jerby, "Observation of free-electron laser at the UHF
regime," The 18th IEEE Convention in Israel, Tel Aviv, March 7-8, 1995 (oral
presentation).
3. R. Drori, E. Jerby, and A. Shahadi, “Extremely long wavelength free-electron laser
experiment,” SPIE, Vol. 2557, pp. 270-281, 1995 (poster presentation).
4. A. Shahadi, R. Drori, and E. Jerby, “Cyclotron-resonance maser experiments in first and
second harmonics,” SPIE, Vol. 2557, pp. 339-346, 1995 (poster presentation).
5. R. Drori, E. Jerby, and A. Shahadi, "Free-electron maser operation at 1 GHz / 1 keV
regime," The 17th FEL Int'l Conf., New York, USA, Aug. 1995 (poster presentation).
6. A. Shahadi, E. Jerby, Li Lei, R. Drori, "Carbon-fiber emitter in a cyclotron-resonance
maser experiment," The 17th FEL Int'l Conf., New York, USA, Aug. 1995 (poster
presentation).
7. R. Drori and E. Jerby, "First operation of a free-electron maser above 1 Meter
wavelength," The 18th FEL Int'l Conf., Rome, Italy, Aug. 26 -30,1996 (oral presentation).
8. R. Drori and E. Jerby, "Low-voltage (750 V) free-electron laser operation at VHF (280
MHz)," Proc. IEEE, pp. 52-54, Jerusalem, Israel, 1996 (oral presentation).
9. R. Drori, D. Shur, E. Jerby, G. Rosenman, R. Advani, and R. Temkin, “Radiation bursts
from a ferroelectric-cathode based tube,” IR and MM Waves Conf. Digest, Virginia, USA,
pp. 67-68, 1997 (oral presentation).
10. R. Drori, “Fluid-loaded FER and CRM experiments,” CRM and gyrotrons research
workshop, Kibutz Ma’ale Hachamisha, Israel, May 18-21, 1998 (oral presentation).
11. R. Drori and E. Jerby, "Fluid-loaded free-electron laser in the long-wavelength
low-voltage extreme," The 20th FEL Int'l Conf., Williamsburg, USA, Aug. 16-21, 1998
(poster presentation).
12. R. Drori and E. Jerby, Low-voltage free-electron lasers,” The 45th annual meeting of the
Israel Physical Society, Tel Aviv University, Israel, March 18, 1999 (oral presentation).
ו
ContentsChapter 1. Introduction…………………………….…………………………..………….. 1
PART I
FREE-ELECTRON LASER AT RADIO FREQUENCIES (FER).…….. 6
Chapter 2. Extremely Long-Wavelength Free-Electron Laser (FER)..……….….…… 8
2.1. FER principle scheme………………....…………………………………………….... 8
2.2. Experimental setup..………………………………………………………………….. 12
2.3. Experimental observations.…………………………………………………………… 15
2.4. Analysis of the FER operation………………………………………………………... 20
2.5. The rising of cyclotron oscillations…………………………………………………… 23
Chapter 3. Tunable FER.……………………………..………………………………….. 28
3.1. Fluid-loaded FER scheme……….……………………………………………………. 28
3.2. Experimental observations..………………..…………………………………………. 31
PART II
RELATED STUDIES.……………………....………………………………. 38
Chapter 4. Ferroelectric-Cathode Free-Electron Electromagnetic Source….………... 40
4.1. Ferroelectric-cathode e-gun.…….…………………………………………………….. 40
4.2. Ferroelectric-cathode Cyclotron-Resonance Maser (CRM)…………………………... 42
Chapter 5. Helix Devices..…….………..………………………………………..………… 47
5.1. Theoretical and technological background.………………………..…………………... 48
ז
5.2. Single-helix experiment...…………………..….………………….…….……………. 50
5.3. Bifilar-helix CRM experiments………………………………………………………. 60
Chapter 6. Summary……………………………………………………………………... 66
Appendix A. Electron Dynamics in FER………………………………………………... 71
Appendix B. Linear Model of FER Amplification……………………………………… 78
References…………………………………………………………………………………. 87
1
Chapter 1
Introduction
Free-electron devices produce electromagnetic (EM) radiation across almost the entire spectrum.
Microwave tubes, such as magnetrons and klystrons, operate efficiently at wavelengths down in
the cm range. Slow-wave devices, as traveling-wave tubes (TWT)-type devices, are used for
various applications at wavelengths ranging down through the mm range. Cyclotron resonance
masers (CRMs) that are known for their high-power capability are used in the mm range.
Free-electron lasers (FELs) operate in the mm range and in shorter wavelengths [1-4].
Both FEL-type and CRM mechanisms are based on resonant interactions between the
amplified EM wave and an electron beam (e-beam). The interaction occurs near the
synchronism condition,
phez
evv1
nµ
ω≅ω (1.1)
where eω is the electron-motion frequency and n its harmonic number, ezv and phv are the
electron axial velocity and the wave phase velocity, respectively. The electron-motion
frequency eω is defined below as the wiggling frequency wω for FELs, and as the cyclotron
frequency cΩ for CRMs. The minus and plus signs of the Doppler-shift term, phez vv , in
the tuning relation (1.1), denote interactions with forward and backward waves, respectively.
In FEL-type devices, the electron trajectory is induced by a wiggler, a magneto-static field of
alternating polarity with a periodicity
2
ww
2kλ
π=, (1.2)
where wλ is the wiggler period. The electron-motion frequency is given then by
wezw kv=ω . (1.3)
In CRMs, the electron trajectory is induced by an axial static magnetic field, B||. The
relativistic electron-cyclotron frequency is defined as
e
||c m
eBγ
=Ω, (1.4)
where e and emγ are the charge and the relativistic mass of the electron, respectively.
FEL-type and CRM devices operate typically with electrons accelerated to energies above
tens of keV. This thesis is devoted to development of FEL-type and CRM devices operating
with low-energy (<10 keV) electrons. This range of accelerating voltages enables the
reduction of device overhead and cost, but limits the available output power and consequently
dictates a relatively low operation frequency range.
Low-voltage FEL-type and CRM schemes can be employed in multi-beam devices [5], a
concept proposed by Jerby [6] for producing high-power EM radiation. The practical
advantages of this concept, studied first as a CRM array [7], are the alleviation of
space-charge effects by utilizing low-current e-beams and the feasibility of high-power
microwave generation by a compact device.
3
PART I of this thesis presents experimental and theoretical studies of FEL-type devices
operating in the radio frequencies (RF) range [8-11]. We proposed the acronym FER
(Free-Electron RF source) for this FEL-type device.
The operation of the FER is presented in Chapter 2. Electron energies below 10 keV dictate
FEL-type interactions with fast waves ( cvph ≈ ) at frequencies lower than 2 GHz (this is
easily derived from Eq. (1.1) for cm 4 w =λ ). This range of frequencies is realized in the FER
by employing a non-dispersive transmission-line cavity having zero cut-off frequency [8-10].
The interaction region of the FER consists of two co-planar metallic strips shielded by a
metallic hollow-tube. The magneto-static field is produced in this device by a combination of a
solenoid and a planar wiggler [12]. Consequently, the electron trajectory is a superposition of a
wiggling motion and a cyclotron motion.
In certain accelerating voltages, a parasitic cyclotron interaction has been excited in addition to
the FER interaction. This interaction is due to the cyclotron motion of the electrons exhibited in
the FER. For the cyclotron interaction the wiggler serves as a distributed kicker.
Theoretical studies of FELs with a planar wiggler and an axial guide field have been carried out for
operation regimes that differ from that of the FER [13,14]. The steady-state electron dynamics and
the gain-dispersion relation of the FER in the linear regime have been derived and presented in
Appendices A and B. The experimental observations agree very well with the results of the
theoretical analyses. Phenomena that arise in these analyses, such as electron drift and the
dependence of the gain on the guide field for TEM waves, are predicted also by Refs. [13] and
[14].
A new concept of frequency tunability by a variable dielectric-loading is the scope of Chapter 3.
This new concept is demonstrated using the FER device [11]. The dielectric load is implemented
4
in this experiment by distilled water. By varying the amount of the distilled water along the tube,
the FER operating frequency is tuned in a range of ~4%. To the best of our knowledge, the
fluid-loaded microwave device is demonstrated for the first time here.
Studies of related technologies and schemes that may be employed in compact low-voltage
free-electron devices is the scope of PART II. This part presents the utilization of a
ferroelectric cathode in a CRM and the operation of low-voltage helix based devices.
Cold cathodes have major contributions to the compactness and the cost reduction of
free-electron EM sources. They are less costly than the thermionic cathodes, heating and
activation processes are not needed, and the vacuum requirements are less strict in the devices
in which they are employed. Chapter 4 presents a study of an electron gun (e-gun) that
employs a high-dielectric ceramic (ferroelectric) cathode [15]. In the ferroelectric cathode the
electrons are emitted from the plasma of a surface flashover, which is generated by high
voltage stress applied to the ceramic in a nanosecond time scale [16].
Cold cathode electrons might have energy spread that is too high for obtaining efficient
FEL-type interaction [2]. However, CRM operating near its cut-off as in the gyrotron device,
tolerates energy-spread of the electrons [17]. Hence we study first the employment of an
e-gun with ferroelectric cathode in a CRM, operating in this regime [18]. According to the
best of our knowledge, high-dielectric-ceramic cathodes had not been utilized before in any
EM radiation generator.
The incentive of the experiments presented in Chapter 5 was to develop a low-voltage
free-electron device that combines fast-wave interactions (FEL, CRM) with helix structures.
The helix structure supports slow-wave modes and in particular it can be matched in an
5
amplifier scheme in a wide frequency range [19]. Interaction with slow-wave enables the
reduction of the accelerating voltage for a certain interaction frequency.
A single helix scheme immersed in a solenoid magnetic field was employed first. In this
experiment only TWT-type interactions have been observed. In a unique interaction, the
electrons interact simultaneously with both forward and backward waves. This kind of
interaction has the potential for an efficient oscillator operation. In order to get cyclotron
interaction, a bifilar-helix structure was proposed. The transverse electric field supported by
this structure presents better conditions for the excitation of cyclotron interaction. Indeed,
both oscillator and amplifier CRM operations have been observed in bifilar-helix CRM
experiments [20].
The thesis is concluded in Chapter 6 with a brief summary on the main results and proposed
directions for further research work. The main achievements of this work are:
(i) Operating of FEL-type devices in extremely long-wavelengths,
(ii) Proving the feasibility to tune free-electron devices by using a variable dielectric
medium,
(iii) First employment of a ferroelectric cathode in a free-electron EM source,
(iv) A wide-tunability band low-voltage CRM in a bifilar helix.
Low-voltage CRMs that employ ferroelectric-based e-guns are under further studies [21,22].
This new technology together with the schemes that are presented here first have the potential
to be implemented in future devices.
6
PART I
FREE-ELECTRON LASER AT RADIO
FREQUENCIES (FER)
FELs have been studied in the last decades in a wide spectrum of wavelengths. The
worldwide activity in the field of the FEL experiments is shown in Fig. I.1, which summarizes
the data presented in Refs. [23] and [24]. Most of the FEL research activity is devoted to the
generation of EM radiation in the millimeter, infrared and shorter wavelength ranges.
Electron Energy
Rad
iatio
n W
avel
engt
h
FERActive FELProposed FEL
1 keV 1 MeV 1 GeV
1 nm
1 µm
1 mm
1 m
Fig. I.1. Worldwide FEL-type experiments. Our contributions are marked by squares in the
left extreme. The curved line is calculated by Eq. (1.1) for cm 4 w =λ .
7
Short-wavelength FELs [25-27] operate with relativistic e-beams in free-space optical modes
( cvez → cvph = ). The EM radiation wavelength in these devices derived from relation
(1.1) is given by
2w
2γ
λ=λ , (I.1)
where it is much shorter than the wiggler period ( wλ<<λ ). FELs operating in the microwave
and in the millimeter wave ranges are known as FEMs or ubitrons [28-30]. They employ
mildly relativistic e-beams (tens of keV up through MeV) in hollow metallic waveguides
)cv( ph > and radiate in wavelengths that are of the order of the wiggler period ( wλ≈λ ).
The development of FEL-type devices operating with extremely low-energy electrons
(< 6 keV) [8-11] is described in Chapter 2 and Chapter 3. In these experiments the EM
wavelength is much longer than the wiggler period ( wλ>>λ ), and it reaches the UHF
band. Therefore, we proposed the acronym FER (free-electron RF source) for this FEL-type
device operating in a new regime. The main characteristics of the FEL, FEM, and the
proposed FER, are summarized in Table I.1.
Table I.1. Operating regimes of FEL, FEM and the proposed FER.
Device e-beam energy wavelength EM guide λλλλ vs. λλλλw
FEL relativistic microns free space(non-dispersive) wλ<<λ
FEM mildlyrelativistic millimeters wave-guide
(dispersive) wλ≈λ
FER non relativistic meters transmission line(non-dispersive) wλ>>λ
8
Chapter 2
Extremely Long-Wavelength Free-Electron Laser (FER)
FEL-type operation at frequencies lower than 1 GHz is realized in this work in a waveguide
supporting TEM waves. The low frequency operation is enabled by the use of a
non-dispersive TEM-mode transmission line (note that typical FEMs use hollow waveguides
which introduce cutoff frequencies). The principle scheme of the FER is described in Section
2.1. A non-relativistic e-beam (< 6 keV) travels in this experiment in a non-dispersive
transmission-line cavity along a planar wiggler and an axial magnetic field. An experimental
arrangement and observations are presented in Sections 2.2 and 2.3. Clear and reproducible
FER oscillations are observed in the first three longitudinal modes of the cavity (0.28 GHz at
1 kV, 0.56 GHz at 2 kV, 0.83 GHz at 6 kV). Due to its low frequency, the FER signal is
monitored directly without detection using a fast digital-oscilloscope. This ability provides a
clear insight into FEL time-domain phenomena. Analysis of the FER operation is described in
Section 2.4. The experimental observations agree very well with the theoretical results using
the derivations that are presented in the Appendices. Rising of cyclotron oscillations, in
addition to the FER oscillations during the same e-beam pulse, is described in Section 2.5.
The CRM interaction was excited due to the presence of the axial magnetic field, whereas the
wiggler acted as a distributed kicker.
2.1. FER principle scheme
A principle scheme of the FER is illustrated in Fig. 2.1. Two parallel striplines stretched along
a rectangular waveguide (WR187) provide the non-dispersive transmission line that supports
9
quasi-TEM modes. Two mirrors at both ends of the 53 cm long transmission-line form a
cavity. A five-layer, coaxially fed, folded-foil copper wire forms a planar wiggler
cm) 4( w =λ [12]. The wiggler, shown in Photograph 2.1, is tapered at both ends by changing
gradually the number of layers for adiabatic entrance and exit of the e-beam. Although large
currents are required in the wiggler winding, this type of wiggler has the advantages of easy
manufacturing and modifications. A low-energy e-beam (<6 keV) is confined by an axial
magnetic field down the cavity through the apertures embedded in the mirrors. The
cross-section of the FER interaction region is depicted schematically in Fig. 2.2.
z
Metal stripsElectronbeam
B||
BW
xy
DipoleProbe
IW
IW
Folded-foil wiggler
Fig. 2.1. Schematic of the FER device.
Two dipole-probes (shown in Fig. 2.1) are inserted into the narrow waveguide wall in order to
sample the EM signal evolved in the cavity. The different locations of the probes enable a
distinction between the longitudinal cavity modes, as shown in Fig. 2.3. Probe A is located 3
cm from the cavity mirror. Probe B is installed in the middle of the cavity's length, thus it
10
samples only the odd longitudinal modes. Photograph 2.2 shows the folded-foil wiggler
constructed on the FER cavity and the two connectors to the RF probes embedded in the
narrow cavity wall.
e-beam
Wires
Rectangular tube
Wiggler
Solenoid
Fig. 2.2. Cross-section scheme of the FER interaction region.
Probe A Probe B
Wigglingelectrons
Secondmode
Thirdmode
Fundamentalmode
Fig. 2.3. The FER scaling; the first three axial modes of the cavity with respect to the electron
wiggling motion, and the positions of the dipole probes.
11
Photograph 2.1. The tapered end of the folded-foil wiggler.
Photograph 2.2. The folded-foil wiggler constructed on the FER cavity and the
two RF connectors.
12
2.2. Experimental setup
The experimental setup is illustrated in Fig. 2.4. The e-beam is generated by a thermionic
cathode (Spectra-Mat, STD200) heated through an isolating transformer. The electrons are
dumped at the exit of the interaction region onto a collector, which is also used to measure the
e-beam current. The cathode holder is shown in Photograph 2.3, and the anode used also as a
cavity mirror is shown in Photograph 2.4. A home-made triple-pulser, its principle described
in Ref. [31], generates the solenoid, the e-gun, and the wiggler pulses. The e-gun pulser and
the high-current wiggler pulser are triggered at the peak of the ~20 ms solenoid pulse, as is
illustrated in Fig. 2.5.
Fig. 2.4. FER experimental set-up.
This experimental setup operates in a single-pulse mode. The wiggler and the solenoid
currents are measured in the pulser on a 50 mΩ resistor and the e-gun voltage is measured by
a high voltage probe (Textronix P6015A). An optional zener-diode is connected to the e-gun
in order to get flat-top voltage pulse. Fig. 2.6 shows a block diagram of the RF diagnostic
circuitry. The sampled signal is attenuated and split for power and spectral measurements.
Collector
Triplepulser
Iw
-H.V.
K~
Isol.
Zener diode(optional)
Isolating o-ring
To vacuumpump
-e50 ΩΩΩΩ
To scope
13
The signal power is detected by a calibrated crystal detector and the frequency variation along
the pulse is measured by a Frequency-Time Interval Analyzer (HP5372A). The FER output
signal is observed also directly (without any detection) by a fast digital oscilloscope at a
sampling rate of 1G sample/s.
0
2
4
6
8
10
0 5 10 15 20 25Time [ms]
Mag
nitu
de a
rb u
nits
e-gun
wiggler
solenoid
Fig. 2.5. A timing diagram of the solenoid, the wiggler, and the e-gun pulses.
14
F E R
Fast Digital Oscilloscopes
e current
Frequency-Time Analyzer
RF Attenuator
Detectore-gun voltage
DC Block
Fig. 2.6. Block diagram of the FER diagnostic circuitry.
15
Photograph 2.3. The cathode holder.
Photograph 2.4. The anode at the entrance to the FER cavity.
16
The operating parameters of the FER experiments [9,10] are listed in Table 2.1.
Table 2.1. FER - experimental parameters
Energy 0.4 – 1 [keV]
Current 0.1 –0.2 [A]
Electron beam:
Pulse width 1-2 [ms]
Wiggler period 4 [cm]
Number of layers 5
Wiggler strength ~0.4 [kG]
Magnetic field:
Uniform solenoid 1-2 [kG]
Rectangular tube 1.87 x 0.87 [inch2]
Strip-line cut 3 x 1.9 [mm]
Distance betweenstrip-lines
10.2 [mm]
Waveguide:
Cavity length 53 [cm]
2.3. Experimental observations
The interaction with the fundamental longitudinal mode of the cavity is demonstrated in Fig.
2.7. The detector output of the corresponding RF signal is shown in Fig. 2.7a. The e-gun
voltage pulse shown in Fig. 2.7b, produced by using the Zener diode, has a maximum voltage
of 760 V and a pulse width of ms .51 . The interaction’s spectral evolution, measured by
the frequency and time-interval analyzer, is shown also in Fig. 2.7b, and a coherent
oscillation at 275 MHz ( cm 109 =λ ) is clearly observed. This frequency corresponds to the
fundamental longitudinal mode of the cavity.
17
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Det
ecto
r Out
put a
rb u
nits
(a)
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.5 1.0 1.5 2.0Time [ms]
EG
un V
olta
ge k
V
274.2
274.7
275.2
275.7
276.2
Freq
uenc
y M
Hz
(b)
Fig. 2.7. The FER operation at the fundamental cavity mode; (a) The RF detector output.
(b) The e-gun voltage variation and the measured frequency.
A sweep in the e-gun voltage as shown in Fig. 2.8a yields a sequence of oscillations in the
first three longitudinal modes. The simultaneous detector outputs of Probe A and Probe B are
shown in Fig. 2.8b. The measured frequencies of the three pulses shown in Fig. 2.8a, are 0.85
GHz, 0.55 GHz, and 0.28 GHz, in agreement with the third, second, and first longitudinal
cavity modes, respectively. The mode identification is confirmed by the coupling of the
second mode to Probe A and not to Probe B (located in a null of the even modes). In addition,
18
the fundamental mode coupling to Probe B is larger than to probe A, in accordance with their
different locations (see Fig. 2.3).
0
1
2
3
4
5
6E
Gun
Vol
tage
kV
0.2
0.4
0.6
0.8
1.0
Freq
uenc
y G
Hz
(a)
0
1
2
3
4
5
0.5 1.0 1.5Time [ms]
Det
ecto
r Out
put
arb
uni
ts Probe B
Probe A
(b)
Fig. 2.8. The FER operation at the first three cavity modes; (a) The e-gun voltage sweep and
the measured frequencies of the first three cavity modes. (b) The RF detector outputs
sampled by probe A and Probe B. (See probes’ locations in Fig. 2.3).
An accumulation of over one hundred experimental shots is presented in the form of
a frequency-voltage map in Fig. 2.9. Each circle represents measured frequency and e-gun
voltage at the peak of each RF pulse. Three groups of circles are clearly observed in the
frequencies of the first three longitudinal modes. The curved line shows the theoretical FEL
19
tuning relation (1.1) for a backward wave. The slightly higher e-gun voltages in the
experimental results may be attributed to energy and angular spreads of the e-beam.
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7E-Gun Voltage [kV]
Freq
uenc
y G
Hz
Fig. 2.9 A frequency-voltage diagram of the FER device; experimental results (circles) and
the theoretical FEL tuning curve (1.1).
The ability to monitor the signal directly without any detection is demonstrated in Figs. 2.10
and 2.11. Fig. 2.10 shows a direct oscilloscope measurement, on a nanosecond time scale,
of the FER oscillation at the fundamental mode. A signal built-up is shown in Fig. 2.11 on a
microsecond time scale. A narrower time-window, focusing on the excitation phase only
would enable more detailed information in addition to the signal envelope depicted in Fig.
2.11.
20
-2
-1
0
1
2
0 5 10 15 20Time [ns]
Am
plitu
de a
rb u
nits
Fig. 2.10. An experimental observation of the fundamental FER mode oscillation by a digital
oscilloscope.
-8
-4
0
4
8
0 2 4 6 8 10Time [µs]
Am
plitu
de a
rb u
nits
Fig. 2.11. The FER signal built-up as observed by a digital oscilloscope; A stable frequency
measurement (see Fig. 2.7b), using a Time-Interval Analyzer, begins where it is
marked by the dotted line. The irregular curve stems from the insufficient sampling
rate.
21
2.4. Analysis of the FER operation
Applying an axial magnetic field was essential for guiding the electrons through the
interaction region. This experimental evidence is proved by solving the electron equations of
motion, Eqs. (A.3) in Appendix A, as a function of the accelerating voltage and axial
magnetic field. For each accelerating voltage, the electron trajectory has been computed for
axial magnetic fields ranging from zero up to the value for which the electron does not
acquire transverse displacement that exceeds the transmission-line width (10.2 mm) along the
cavity (53 cm). Fig. 2.12 shows the computed values of the minimum guiding magnetic fields,
which enable the transportation of the electrons through the FER cavity. These computed
values agree well with the axial magnetic field that has been applied in the experiments (1-2
kG).
1.0
1.2
1.4
1.6
1.8
2.0
0 1 2 3 4 5 6 7
E-beam Voltage [kV]
Min
inum
Gui
ding
Fie
ld
kG
Fig. 2.12. Computed values of the minimum axial magnetic fields needed for electron guiding
along the FER cavity.
The electron equations of motion, Eqs. (A.3) in Appendix A, are solved numerically using a
MatLab solver for the FER parameters ( kG 2 B kG, 0.4 B cm, 4 ||ww ===λ ) and for an
22
electron injected on axis x0= y0= z0=0 and accelerated by 0.75 kV. Projections of the orbit onto
the x-z and y-z planes are shown in Figs. 2.13a and 2.13b, respectively. The later is broader
because of the focusing effect of the wiggler in the x-z plane. The transverse wiggling
accompanies a transverse drift trajectory as it is shown in App. A.
-0.4
-0.2
0.0
0.2
x m
m
(a)
-0.1
0.0
0.1
0.2
0.3
0 10 20 30 40 50z [cm]
y cm
(b)
Fig. 2.13. Projection of the simulated electron orbit onto (a) x-z plane and (b) y-z plane.
The ratios between the wiggling and the cyclotron frequencies are shown in Fig. 2.14 for the
first three FER longitudinal modes and the actual applied magnetic fields in the experiment.
The corresponding axial velocities are computed by Eq. (A.12) in Appendix A. In these cases
wezc kv>Ω , hence the electron orbits are classified here as group II orbits [3].
23
0.0
0.1
0.2
0.3
0.4
1.0 1.2 1.4 1.6 1.8 2.0
B || [KG]
w /
c
0.45 kV
1.9 kV
5 kV
Fig. 2.14. The ratio between the wiggling and the cyclotron frequencies for wiggler field of
0.4 kG and accelerating voltages: 0.45 kV, 1.9 kV and 5 kV; The axial velocities
are the solutions of Eq. (A.12) in Appendix A.
Numerical solutions of the Pierce-type equation, Eq. (B.12) in Appendix B, are shown in
Figs. 2.15a and 2.15b. In order to obtain oscillations, the gain per round should satisfy the
condition, 1Q
QG−
> . The Q-factor of the FER cavity was measured to be Q~200. For this
value the gain should be G > 1.005. The computed gains, as shown in Figs. 2.15, are
sufficient to obtain the observed FER oscillations. As mentioned previously, the interactions
are observed in energies higher than the predicted values. This is due to energy and angular
spreads of the e-beam where the theoretical model assumes cold e-beam.
24
0.98
0.99
1.00
1.01
1.02
0.2 0.3 0.4 0.5 0.6
Gai
n
(a)
0.98
0.99
1.00
1.01
1.02
3.0 3.5 4.0 4.5 5.0 5.5 6.0
E-beam Voltage [kV]
Gai
n
(b)
Fig. 2.15. Gain vs. e-beam voltage for FER radiation at (a) 280 MHz and (b) 850 MHz;
The e-beam currents are 100 mA and 200 mA, respectively. The axial magnetic
field is 1.5 kG and the wiggler field is 0.4 kG.
2.5. The rising of cyclotron oscillations
Cyclotron oscillations may rise as a parasitic effect in this FER scheme. The magnetostatic
field in the FER experiments is produced by a solenoid and a planar wiggler. Consequently,
the electron trajectory is a superposition of a wiggling motion and a cyclotron motion that
25
may result in two corresponding synchronism conditions, (Eq. (1.1)). The FER resonance
condition is,
cv1kv
ez
wezFER µ
=ω (2.1)
and the cyclotron resonance condition is
cv1
ez
cCRM µ
Ω=ω . (2.2)
These interactions are expected in different electron energies and longitudinal cavity modes.
The RF coupling and the diagnostic circuitry of the FER experimental device, in which CRM
interactions were observed [8], are illustrated in Fig. 2.16. Parameters of the FER
experimental device are listed in Table 2.2.
LO 1 LO 2 IF 2 IF 1
Splitters
Collector
Oscilloscopes
Fig. 2.16. The RF diagnostic circuitry of the dual FER-CRM experiment
26
The signal is coupled out through two coaxial connectors (SMA, 50 Ω) embedded in the
collector as shown in Fig. 2.16. This wire forms a loop antenna that couples the signal to
another small antenna. The sampled signal is split into two arms for power and frequency
measurements. In the frequency measurement arm, the signal is split again and mixed with
two local oscillator (LO) signals, generated by two external RF oscillators. This arrangement
enables the simultaneous heterodyne measurement of the FER and the CRM spectral contents.
Table 2.2. Dual FER and CRM experimental parameters
Energy < 1 [keV]
Current ~0.2 [A]
Electron beam:
Pulse width ~1 [ms]
Wiggler period 2 [cm]
Number of layers 3
Wiggler strength 0.2-0.3 [kG]
Magnetic field:
Uniform solenoid 2 [kG]
Rectangular tube 0.9 x 0.4 [inch2]
Wires diameter (circular) 1.9 [mm]
Distance between strip-lines 11 [mm]
Waveguide:
Cavity length 75 [cm]
Two distinct radiation bursts are observed during the electron energy sweep. The e-gun
voltage variation vs. time is shown in Fig. 2.17a. Two microwave pulses, denoted by A and B,
are observed in the detector output trace shown in Fig.2.17b in two different e-gun voltage
levels, 2.7 kV and 0.9 kV, respectively. The double heterodyne detection provides a
simultaneous measurement of the center frequencies of the two microwave pulses. Each mixer
output correlates clearly with each RF pulse. The results show that a 4.93 GHz mixer output
27
coincides with Pulse A, and a 0.80 GHz mixer output coincides with Pulse B. A parametric
analysis shows an agreement between the tuning relations in Eqs. (2.1) and (2.2) and the
operating conditions of Pulses A and B, respectively. Hence, Pulse A corresponds to the
cyclotron resonance interaction, and Pulse B corresponds to the FER interaction. Similar
signals have been obtained in over one hundred shots in this experiment.
0
1
2
3
4
EG
un V
olta
ge k
V
(a)
0
5
10
15
20
25
0 0.5 1 1.5 2
Time [ms]
Det
ecto
r Out
put
mV
(b)Pulse A(CRM) Pulse B
(FER)
Fig. 2.17. Typical experimental measurements of two radiation bursts in a single shot; (a) The
e-gun voltage. (b) The RF detected power.
A summary of runs performed in this experiment with various LO frequencies in the range of
0.4 GHz to 6 GHz is presented in Fig. 2.18 in a form of a frequency-energy map. The circles
28
and squares denote observations of mixer outputs in Pulse A and Pulse B, respectively, with
various LO frequencies. Two distinct groups are clearly observed in the map. The spectral
content of Pulse A is centered around 5 GHz (±1 GHz) with an electron energy centered
around 3 keV. The other group in the map, related to Pulse B, is centered around 1 GHz and 1
keV. The average values of the two groups (5 GHz at 3 keV for Pulse A, and 1 GHz at 1 keV
for Pulse B) agree well with the corresponding tuning conditions of Eqs. (2.1) and (2.2).
Hence, Pulses A and B obtained in many shots are related to the CRM and FER interactions,
respectively. The spectral line widening of the radiation observed, might be related to various
causes including the axial velocity spread of the electrons, the solenoid field non-uniformity
at its ends, and the narrow spikes observed in the detected power. The different electron
energies in which the FER and the CRM interactions occur may enable the operation of each
mode separately by a proper voltage tuning.
0
1
2
3
4
5
6
7
0 1 2 3 4 5Electron Energy [keV]
Freq
uenc
y G
Hz
FER
CRM
Fig. 2.18. A frequency-voltage map of the FER and CRM bursts in a 0.4 – 6.0 GHz LO scan.
29
Chapter 3
Tunable FER
The fluid loading of microwave tubes was proposed first in [32] as an inherent component of
the electron-wave interaction. The possibility of implementing this concept for frequency
tuning has been demonstrated in the FER device and is presented in this chapter. The variable
dielectric loading is implemented by distilled water in glass pipes situated on both sides of the
strip-line structure of the FER. Section 3.1 describes the fluid-loaded cavity and the effect of a
variable dielectric loading on it. The experimental arrangement and observations are
presented in Section 3.2. Clear oscillations are observed at the fundamental cavity mode at
0.27 GHz for electrons energy down to 0.4 keV. By varying the fluid dielectric loading, the
FER operating frequency is tuned in a range of 10 MHz. In this experiment, power levels up
to 3 Watts with electronic efficiencies of ~ 3 % were detected (electronic efficiency stands
here for the ratio between the signal output and the e-beam power levels). To the best of our
knowledge, a fluid-loaded microwave source (of any kind) is demonstrated for the first time
in the framework of this thesis.
3.1. Fluid-loaded FER scheme
A principle scheme of the fluid-loaded FER is shown in Fig. 3.1. This device is implemented
by placing glass pipes filled with variable amounts of distilled water on both sides of the FER
strip-lines structure [11]. The terminals of the two U-shaped glass pipes are shown in
Photograph 3.1. This strip-lines structure forms a non-dispersive waveguide and also provides
a metallic protection to the glass pipes from electrons bombardment, hence preventing their
30
electrical charging. A similar structure has been used for the first time in the dielectric-loaded
CRM conducted in our laboratory [33].
z
Metal stripsElectron beam
B||
BW
xy
Dipole Probe
IW
IW
Water pipes
Folded-foil wiggler
Fig. 3.1. A principle scheme of the fluid-loaded FER.
Photograph 3.1. The terminals of the two U-shaped glass pipes.
Two mirrors with holes at both ends of the transmission-line form a 53 cm long cavity. A
low-energy e-beam is injected into the cavity and interacts with the quasi-TEM wave in the
31
presence of the fluid-load. An RF probe is located in the middle of the wide-wall axis in order
to sample the first cavity mode (the probe connected to one of the strip-lines can not be
embedded in the narrow wall, due to the glass-pipes placed between the narrow wall and the
strip-line).
The effective dielectric-coefficient of the transmission-line cavity in its quasi-TEM
fundamental mode (i.e., εeff c vph= ( / )2 ) is determined by the amount of the distilled water
in the pipes. As a result, the effective length of the cavity and hence the axial wave-number of
the EM wave can be varied. The FEL tuning relation, Eq. (2.1), is given for this scheme by
c/ezveff1w
ε
ω≅ω
µ, (3.1)
where effε and consequently ω are controlled externally by the amount of the water in the
tube. In view of Eq. (1.1), the phase velocity phv is varied by an external means in this
device.
The resonance frequency of the cold cavity (i.e., without an e-beam) is measured by a Vector
Network Analyzer (HP8714B) in a one-port scattering analysis mode. A probing
frequency-swept signal is injected into the cavity, while its reflection coefficient is measured.
A minimum in the reflection trace indicates a resonance frequency. Fig. 3.2 shows power
reflection measurements at the cold cavity resonance frequencies in two extreme levels of
distilled water in the glass-pipes. The minima in the two reflection traces show the
corresponding resonance frequencies, 266 MHz and 276 MHz, for full and empty glass-pipes,
respectively. The fundamental resonance frequency of the cavity without the glass-pipes is
285 MHz.
32
The effective dielectric coefficient εeff of the transmission-line, operating in the fundamental
quasi-TEM mode, is deduced from the variation in the resonance frequency caused by the
water loading, i.e.
2
)d(Rf0Rf
)d(eff
=ε (3.2)
where d is the quantity of water in the pipes, and fR0 and )d(Rf are the cavity resonance
frequencies without the glass-pipes, and with pipes filled with the actual amount of water,
respectively (i.e., fR0 =285 MHz).
0.0
0.2
0.4
0.6
0.8
1.0
260 265 270 275 280
Frequency [MHz]
Cav
ity R
efle
ctio
n
Full withdistilled water
Emptyglass-pipes
Fig. 3.2. The (cold) FER cavity resonance-frequency measurements; power reflection traces
for full and empty glass-pipes.
3.2. Experimental observations
The experimental setup and the diagnostic circuitry are similar to those used in the FER
experiment described in Chapter 2. The operating parameters of the fluid-loaded FER
experiment are listed in Table 3.1.
33
Table 3.1: Fluid-loaded FER - experimental parameters
Energy 0.4 – 1 [keV]
Current ~0.2 [A]
Electron beam:
Pulse width ~2 [ms]
Wiggler period 4 [cm]
Wiggler strength 0.2-0.4 [kG]
Magnetic field:
Uniform solenoid 1-2 [kG]
Rectangular tube 1.87 x 0.87 [inch2]
Strip-line cut 3 x 1.9 [mm]
Distance between strip-lines 10.2 [mm]
Cavity length 52.6 [cm]
Waveguide:
Effective dielectric range 1.06-1.15
Results of resonance-frequency measurements in a range of water loads and the
corresponding dielectric coefficient εeff , Eq. (3.2), are presented in Fig. 3.3. A range of
εeff = 1.06 to 1.15 is obtained by varying the water loading between empty and full pipes,
respectively.
A typical FER pulse is shown in Fig. 3.4. The effective dielectric coefficient in this example
is εeff = 111. . The detected output signal and the corresponding e-gun voltage pulse are
shown in Figs. 3.4a and 3.4b, respectively. The actual frequency during the pulse, measured
by the Frequency-Time Interval Analyzer (HP5372A), is also presented in Fig. 3.4b.
34
265
267
269
271
273
275
277
279
0 20 40 60 80 100
Water Volume [cm ]
Res
onan
ce F
requ
ency
MH
z
1.06
1.08
1.10
1.12
1.14
1.16
Effe
ctiv
e D
iele
ctric
Coe
ffic
ient
εeff
3
Fig. 3.3. Resonance-frequency tunability in a range of water loads, and the corresponding
effective dielectric coefficient εeff (from Eq. (3.2)).
It is noted that radiation frequencies for increasing voltages are slightly higher than those
measured for the same decreasing voltages. In addition, amplitude fluctuations are clearly
seen in the trace of the detected output power in Fig. 3.4a. The frequency of this amplitude
modulation (AM), counted with respect to the instantaneous voltage gradient, is presented in
Fig. 3.5. For each amplitude fluctuation period (∆T), the instantaneous frequency (1/∆T) and
the corresponding temporal voltage gradient (∆V/∆T) are computed. Both FM and AM
frequency measurements (Figs. 3.4b and 3.5, respectively) show similar ranges of FER
frequency variations (~ 0.3 MHz) during the voltage pulse. This may hint that both effects are
associated, possibly through the interrelations between the FER phase-shift and the varying
electron-energy, and the cavity resonance. These effects require further theoretical and
experimental studies.
35
0
5
10
Det
ecto
r Out
put
arb
uni
ts (a)
0.0
0.2
0.4
0.6
0.8
1.0
0 1 2
Time [ms]
EG
un V
olta
ge
kV
269.8
270.3
270.8
271.3
271.8
Rad
iatio
n Fr
eque
ncy
MH
z(b)
Fig. 3.4. A typical FER output signal with a medium dielectric loading, εeff = 1.11;
(a) The detected FER output signal; (b) The e-gun voltage pulse and measured
frequency during the pulse (HP5372A measurements).
Increasing the fluid loading reduces both the FER oscillation-frequency and the operating
voltage. Fig. 3.6 shows the RF detector output, the e-gun voltage, and the RF frequency, for
εeff = 115. . The oscillations are excited in the leading and trailing edges of the voltage pulse,
from 420 V to 640 V. The RF frequency shift during the pulse (measured by the HP5372A)
follows the voltage sweep.
36
050
100150200250300350
-800 -600 -400 -200 0 200 400 600
Temporal Voltage Gradient [V/ms]
AM
Fre
quen
cy K
Hz
Fig. 3.5. The amplitude modulation (AM) frequency with respect to the voltage gradient.
0
200
400
600
800
1000
0 0.05 0.1 0.15Time [ms]
EG
un V
olta
ge
Vol
t
264
265
266
267
268
Rad
iatio
n Fr
eque
ncy
MH
z
Detector Output [arb. units ]
Fig. 3.6. A typical FER output with a full dielectric loading, εeff = 115. , at low voltage
(≥ 420 V). The FER output-signal frequency ( ) sweeps with the e-gun voltage
increase.
37
The tunability of the fluid-loaded FER is shown in Fig. 3.7, as observed in many runs with
different dielectric loads. The inverse dependence of the FER oscillation frequency on the
dielectric loading is almost linear in this range.
265
267
269
271
273
275
277
1.06 1.08 1.1 1.12 1.14 1.16
Effective Dielectric Coefficient , εeff
Out
put R
F Fr
eque
ncy
M
Hz
Fig. 3.7. The fluid-loaded FER tunability; the dots indicate center-frequency
measurements in many pulses for different dielectric loading conditions.
The electron axial-velocity vez computed by Eq. (A12) of Appendix A for data measured in
different experimental runs, is shown in Fig. 3.8 (i.e., the actual wiggler and solenoid fields
and the electron energy in each run are substituted to Eq. (A.12) to find vez ). Each run is
represented by the minimal and maximal electron velocities in which radiation is observed.
The experimental results for vez are larger than those predicted analytically by Eq. (2.1) for a
backward-wave interaction. However, the tendency of the axial velocity to decrease as the
dielectric loading increases is similar in both experimental and analytical lines in Fig. 3.8.
The FER electronic efficiency (i.e., the ratio between the coupled output signal and the
e-beam power) is shown in Fig. 3.9. The efficiency tends to increase with the dielectric
38
loading, and it exceeds an average of ~2%. It should be noted, however, that the RF probe in
this experiment was used for frequency measurements and relative signal detection. The
efficiency can be improved by optimizing the Q-factor.
0.03
0.04
0.05
0.06
0.07
1.06 1.08 1.10 1.12 1.14 1.16
Effective Dielectric Coefficient
Nor
mal
ized
Axi
al V
eloc
ity
Fig. 3.8. Maximal ( ) and minimal ( ) normalized axial velocities ( c/ezv ) in many FER
pulses. The solid line shows the result of the analytic tuning relation (2.1) for a
backward wave (the forward wave interaction results in a smaller axial velocity).
0.0
1.0
2.0
3.0
1.06 1.08 1.1 1.12 1.14 1.16Effective Dielectric Coefficient , εeff
Effic
ienc
y
Fig. 3.9. The FER output-coupling efficiency with respect to εeff .
39
PART II
RELATED STUDIES
Our studies on low-voltage FERs presented in the first part of this thesis have been
accompanied by studies on various low-voltage components and devices. Since they are of
interest for free-electron sources they are presented here as related devices. One study is on
e- gun that employs a cold cathode made of a ferroelectric material and the other is on a broad
tunability bandwidth CRM.
Considering the FEL or CRM devices overhead, one should take into account the electrons
generation technology. Thermionic emission is based on heating of the emitting surface in
order to provide the electrons energy to overcome the potential barrier (work function) and to
leave the surface. Current densities up to the order of 100 A/cm2 for long pulses or continuous
operation can be produced [2]. However, high temperature (above 1000oC) is required, and if
the emitter is exposed to air (“poisoned”) a drastic reduction in its emissivity occurs.
Reduction of the device overhead can be achieved by using cold cathodes. One type of such
cathode is the high-dielectric (ferroelectric) ceramic cathode [15,16]. In this cathode the
electrons are emitted from the plasma excited by a voltage pulse of the order of 1 kV applied
to the ceramic in a nanosecond time scale. The lifetime of the plasma is of the order of few
microseconds. The current density provided by these cathodes may exceed 100 A/cm2.
Ferroelectric cathodes can operate in poor vacuum conditions, at room temperature, and with
low voltages. Heating and pre-activation are not needed and they are easy to fabricate and
handle, as compared to thermionic or field-emission cathodes.
40
Cold cathode electrons might have energy spread that is too high to obtain efficient FEL-type
interaction [2]. However, CRM interaction near cut-off, as in the gyrotron device [17],
tolerates energy spread of the electrons. The first implementation of a ferroelectric cathode in
a CRM device [18] is the subject of Chapter 4.
The development of a broad tunability bandwidth, low-voltage (< 10 kV) CRM is presented in
Chapter 5. The CRM device employs a helix structure. In a single-helix scheme, only
TWT-type interactions have been observed. A bifilar-helix has been proposed in order to take
advantage of both its transverse electric field components on axis and broad bandwidth
characteristics. This proposal initiated a bifilar–CRM experiment [20]. Preliminary results of
the bifilar-CRM exhibit both CRM oscillations and amplification. The bifilar helix can be
used in future FER schemes as a combined wiggler and slow-wave structure.
41
Chapter 4
Ferroelectric-Cathode Free-Electron EM Source
High-dielectric (ferroelectric) ceramic cathode is utilized in this study for the first time in
microwave generator. The cathode, developed by Rosenman et. al [16] is employed in a
low-voltage (<10 kV) CRM device. In this cathode, electrons are extracted from plasma
excited on the ceramic surface by applying very short (nanoseconds) rise-time pulse of the
order of 1 kV. The e-gun scheme and operation are described in Section 4.1. The
experimental arrangement and observations are presented in Section 4.2. The
ferroelectric-cathode CRM is demonstrated at ~7 GHz, near the cut-off frequency of a hollow
cylindrical cavity. According to the best of our knowledge, high-dielectric-ceramic cathodes
had not been utilized before in any EM radiation generator.
4.1. Ferroelectric-cathode e-gun
The ferroelectric-cathode e-gun is illustrated in Fig. 4.1. The cathode is made of
high-dielectric (εr ~ 4,000) PLZT 12/65/35 ceramic plate (~ 1 cm2 area and 1 mm thickness)
[16]. A conductive silver paint contact (6 mm diameter) is deposited on the rear surface of the
ceramic plate. A brass washer is glued to the emitting surface as a ring electrode and its
external and internal diameters and thickness are 6.0, 3.4, and 0.2 mm, respectively. A
stainless steel grid (52 µm wire diameter, 460 µm period) is mounted directly onto the brass
washer front providing a volume for the free plasma expansion.
42
A positive trigger voltage pulse of (≤ 1 kV), with a rise time of tens of nanoseconds, is
applied between the rear contact and the grounded washer. As a result, plasma is excited on
the ceramic surface having a lifetime of the order of few micro-seconds and expanding in a
velocity of ~1 cm/µs. Typical current densities provided by these cathodes are up to 100 A/cm2
[15]. A constant accelerating voltage that does not depend on the trigger pulse level is applied
between the grounded cathode and the anode.
Fig. 4.1. A scheme of the ferroelectric-cathode e-gun
voltage is applied between the grounded catho
In the scheme illustrated in Fig. 4.2, the accelerating vo
pulse only. The latter configuration has been employed in
(~10 cm long) with several wave-guides (hollow cylinder
helix-loaded cylinder (~50 cm long) [18]. Due to the way
applied to the e-gun in these experiments, the width of the
unstable profile were determined by the trigger voltage
quality of these current pulses was not sufficient to
interactions. Both negative and positive pulses trigger the p
Cavity
+-
Cathode grid
K A
High-dielectricceramics
Fastswitch
+
H.V.where a constant acceleration
de and the anode.
ltage is applied during the trigger
our laboratory in a compact device
, helix, double strip-lines) and in a
in which voltages (1-2 kV) were
current pulses (<500 ns) and their
pulses. It is possible, that the low
obtain stable and reproducible
lasma generation. However, it was
43
found that when a positive trigger voltage is applied to the rear surface of the cathode the
electron energy spread is smaller [34].
Fig. 4.2. A scheme of the ferroelectric-cathode e-gun where the accelerating voltage
is applied during the short trigger pulse only.
4.2. Ferroelectric-cathode CRM
In order to study the ferroelectric-cathode e-gun in a microwave generator, it was applied first
in a CRM. The reason of this choice is the wide energy acceptance of the CRM in the
gyrotron mode. A scheme of the CRM experimental device is shown in Fig. 4.3, the
microwave diagnostic setup is depicted in Fig. 4.4 and the device parameters are listed in
Table 4.1. The waveguide consists of a stainless steel cylinder biased by an accelerating DC
voltage up to 9 kV. The spacing of the accelerating gap is 2.5 cm in order to avoid a voltage
breakdown during the current pulse. A disc with a hole in its center enables the passage of the
e-beam and forms a microwave partial mirror. The cyclotron orbits of the electrons along the
cavity are induced and confined by the solenoid magnetic field. The end of the drift-tube is
free of magnetic field hence the e-beam is dumped onto the inner wall of the cylinder and its
current is measured by a Rogovsky coil. A Teflon vacuum window transmits the microwave
Cavity
+-
K A
Fastswitch
High-dielectricceramics
Cathode grid
44
signal into a WR90 adapter where it is coupled by a diagnostic system, which consists of a
band-pass filter and a calibrated crystal detector.
Table 4.1 Ferroelectric-cathode CRM – experimental parameters
Energy <10 [keV]
Current ~0.5 [A]
Electron beam:
Pulse width ~1 [µs]
Uniform solenoid 2.4-2.6 [kG]Magnetic field:
Kicker ~10 [kA turns]
Stainless steel cylinder diameter (inner) 26 [mm]Waveguide:
Length 60 [cm]
Fig. 4.3. A scheme of the ferroelctric-cathode CRM.
Fig. 4.4. The microwave dia
Solenoid
Microwave
+ H.V.Current
Coupler
E-gun
Transparent isolationand vacuum seal
Isolationand vacuum seal
Kicker
Partial mirror
To vacuum pump
Evacuated drift tube
Isolation
Matched
Microwave
B(6.1- )Attenuator Detector
Ferroelectric-cathodeCRM
.P.F.9.4 GHz
gnostic setup.
To scope≈≈≈≈
45
The feasibility of a CRM operation with a ferroelectric cathode is demonstrated clearly near
the waveguide cutoff frequency. The TE11 cut-off frequency, fco = 6 9. GHz, is found by
transmission measurements shown in Fig. 4.5.
0.0
0.2
0.4
0.6
0.8
1.0
6.5 6.75 7 7.25 7.5Frequency [GHz]
Tran
smis
sion
Mea
sure
men
ts
Fig. 4.5. The waveguide (cold) transmission measurements.
The CRM interaction mechanism is verified by measuring the microwave output power as a
function of the solenoid magnetic field, B0. The latter determines the electron cyclotron
angular-frequency meB||
c ≅ω . The CRM operating condition near cutoff is ω ω≈ c where
ω is the EM radiation angular-frequency. Consequently, the CRM operating condition should
be coc f2π≥ω≈ω , or ( ) co|| f em 2B π≥ . This operating condition is verified experimentally
as evidence of the CRM type of interaction. As is seen clearly in Fig. 4.6, the microwave
output is obtained only when the cyclotron frequency is larger than the waveguide cut-off
frequency. Cyclotron interactions near cut-off tolerate e-beam energy spread [17].
46
0
10
20
30
40
50
6.5 6.75 7 7.25 7.5
Cyclotron Frequency [GHz]
Mic
row
ave
Out
put P
ower
dB
m
Fig. 4.6. CRM operation near cutoff; The CRM output power vs. the cyclotron frequency
determined by B||.
Typical current and microwave output detector traces are shown in Figs. 4.7a and 4.7b,
respectively. Currents of 0.4 A and ~1 µs pulse-width are measured. The microwave
output-power exceeds 25 W in many shots when the e-beam is accelerated to 9 kV. The
electronic coupling efficiency (i.e. the ratio between the microwave-coupled power and the
electron-beam power) exceeds 1 %. These preliminary results provide a basis for future
studies [21,22].
47
Fig. 4.7. A typical CRM output; (a) Electron current and (b) microwave detector output.
0
0.1
0.2
0.3
0.4
Col
lect
ed C
urre
nt A
(a)
0
50
100
150
200
250
0 0.4 0.8 1.2Time [ms]
Det
ecto
r Out
put
mV
(b)
48
Chapter 5
Helix Devices
This chapter presents preliminary development stages of broad-tunability bandwidth
low-voltage CRM (< 10 kV). The helix has broad bandwidth characteristics, hence it has been
chosen as the waveguide for this device. However, in helix-based amplifier schemes some
effort has to be made to suppress oscillations resulting from backward-wave interaction. CRM
oscillations are expected in a single-helix based device due to the presence of the axial
magnetic field and the transverse components of the electric field adjacent to the helix wire.
The characteristics of the single-helix structure are described in Section 5.1. The single-helix
experiment is presented in Section 5.2. TWT-type oscillations are observed in two frequency
ranges (around 2 GHz and at 4.11 GHz).
In order to obtain CRM interactions, a bifilar-helix scheme was employed. In this structure
the transverse electric field components on the axis are larger than in the single-helix scheme.
Preliminary results of the bifilar-CRM exhibit CRM oscillations tunable in the frequency
range of 2.4 - 8.4 GHz and an amplification at 5 GHz. In the CRM oscillator experiment, the
frequency is controlled mainly by the axial magnetic field. The bifilar-CRM experiment,
described in Section 5.3, utilizes a stand-alone e-beam assembly developed in our laboratory.
The stand-alone e-beam assembly enables the changing of interaction structures without
breaking the vacuum.
49
5.1. Theoretical and technological background
Several slow-wave CRM schemes have been proposed and developed in order to increase the
interaction bandwidth and to reduce the e-beam energy [33,35-40]. The slowing down of the
EM-wave phase-velocity is obtained by loading the wave either by dielectric material or by a
periodic structure [41-45]. However, these schemes are typically narrow bandwidth ones. A
common type of a slow-wave circuit having extremely broadband characteristics is the helix
[46] that can be made with a bandwidth of over two octaves. Helix-loaded gyrotron device
employing relativistic e-beam has been studied before in Ref. [47].
Consider a helix wound of a perfect conducting wire extended infinitely in free-space. The
parameters describing the helix, shown in Fig. 5.1a, are the pitch p, the diameter 2a and the
pitch angle ψ [ )a2/p(tg 1 π=ψ − ], the angle formed between the windings and a plane normal
to the axis of the system. Each mode in the helix contains the entire set of spatial harmonics
wave components [19]. The axial wave-number βm of each spatial harmonic is given by
p2m0mπ+β=β , (5.1)
where m is the harmonic order. The fundamental zone of a typical k-β (Brillouin) diagram for
a helix extending infinitely in free space is shown in Fig. 5.1b. The phase velocity of all the
spatial harmonic components is less than the velocity of light. This leads to the existence of
forbidden regions in which mode solutions are not allowed. The EM phase-velocity of the
fundamental mode is given by, )sin( cvph ψ= , and the boundaries of the propagating region
are determined for the basic zone by β±=k 0.
50
a
p
ψ
(a)
k
π/pβ
(b)
k=βπ/p
ωc/ve
-π/p
m=0m=0
m=-1 m=+1e-beam line
AB
Fig. 5.1. (a) Helix; (b) k-β diagram for a helix.
The k-β diagram is modified slightly (not shown in the diagram) near the forbidden regions
due to interactions between the free-space helix modes and the modes that exist in the
combined helix-cylinder structure [48,49]. The m=1, 2, … etc. are spatial orders with positive
phase velocities, whereas the m=-1, -2, … etc. are spatial orders with negative phase
velocities. The group velocities of the spatial harmonics of a given mode are identical, since
they are all associated with the same wave. Since there are spatial harmonics having phase
and group velocities in opposite directions, the helix is a useful structure in backward-wave
oscillators [50]. However, this characteristic can also be a disadvantage in helix-type
traveling-wave amplifiers, unless some effort is made to suppress oscillations resulting from
51
backward-wave interaction. The electron velocity is tuned near synchronism with one of the
spatial harmonic components of each mode. The intersections of the e-beam line in the k-β
diagram (points A and B in Fig. 5.1b) represent possible CRM interactions with spatial
harmonics of a backward wave.
5.2. Single-helix experiment
The single-helix experimental device is shown in Fig. 5.2, and its parameters are listed in
Table 5.1. The helix is formed by a copper wire, which is inserted into a glass tube in order to
keep its shape (see Photograph 5.1). A stainless steel cylinder forms the EM shielding and
vacuum chamber. The two ends of the helix are connected to coaxial (50 Ω) semi-rigid cables,
inserted and sealed into two copper gaskets, through which the signal is coupled in and out.
Table 5.1 Single helix CRM - experimental parameters
Energy < 10 [keV]
Current ~0.2 [A]
Electron beam
Pulse width ~2 [ms]
Uniform solenoid 1-2 [kA turns]Magnetic field
Kicker ~10 [kG]
Diameter 9.6 [mm]
Pitch 15 [mm]
Wire diameter 1.5 [mm]
Helix
Length 50 [cm]
Glass tube diameter (inner/outer) 12.2 / 17.2 [mm]Shielding
Stainless steel tube inner diameter 21.3 [mm]
52
Triple pulser
Ikicker-H.V. Isol.
Zener diode (optional)
RF connector
To helix
Coaxial semi-rigid cable
To vacuum pump
Thermionic Pierce Gun
Collector
Kicker Solenoid
K
Glass tube Support
Isolating o-ring
Copper gasket
Vacuum seal
Heat p.s.
~ Stainless steel cylinder
Fig. 5.2. A scheme of the single helix-based experimental device.
Photograph 5.1. The helix structure.
53
The e-beam is generated by a thermionically emitting Pierce-type e-gun constructed in our
laboratory [51]. The e-beam is injected into the interaction region undergoing a cyclotron
motion due to the axial magnetic field. A magnetic kicker obtains the initial electron rotation,
which is a necessary condition for amplification in a normal-Doppler cyclotron interaction.
The e-beam is dumped onto a collector where it is measured using a 50 Ω resistor.
An impedance mismatch exists in each transition between the helix ends and the coaxial lines.
This mismatch feature is the source of the oscillating behavior of transmission-loss
measurement trace of the helix, shown in Fig. 5.3. The maximal points in the trace coincide
with the resonance frequencies of the cavity. The transmission loss measurement accuracy,
determined by the frequency step used in the Network Analyzer, is ±3 MHz.
-40
-30
-20
-10
0
0 1 2 3 4 5Frequency [GHz]
Tran
smis
sion
Los
s dB
Fig. 5.3. Transmission loss measurement of the helix structure.
Fig. 5.4 illustrates schematically the microwave circuitry and its diagnostic. The signal is
coupled out at each end of the waveguide and travels along a coaxial line in an outer path to
the opposite end. This microwave circuit enables a feedback mechanism in addition to the
54
feedback caused due to the transition mismatch in the waveguide ends. A low-pass filter
separates the signal occurred by the electrons that hit the helix from the EM signal. The
waves that travel to both directions in the outer path are coupled through 20 dB directional
couplers for power and frequency measurements.
Fig. 5.4. Diagnostic circuitry of the device with the outer microwave path.
Typical results obtained in many shots using the scheme presented in Fig. 5.4 are shown in
Figs. 5.5 – 5.8. The detector outputs shown in Fig. 5.5a, are sampled by the two directional
couplers and they are on the same amplitude scale (both signals attenuated by the same
50 Ω
Fast Digital OscilloscopesFrequency-TimeAnalyzer (HP5372A)
Attenuators
DetectorsE-gunvoltage
-e
Low-pass /
high-pass filter
50 Ω
Helixcurrent
Collectorcurrent
LO
-20 dB-20 dB
Splitter
DC block
≈≈≈≈
DC block
Microwave portMicrowave port
Collector
55
value). The e-gun voltage variation and the measured IF frequencies of the signals are shown
in Fig. 5.5b. The values denoted in parentheses are the actual frequencies of the signals
determined in measurements with various LO frequencies. The IF frequencies shown in the
figure are for LO frequency of 2.2 GHz.
0
5
10
15
Det
ecto
r Out
put
arb
units
backward
forward
(a)
0
1
2
3
4
5
6
7
0.0 0.2 0.4 0.6 0.8 1.0
Time [ms]
EG
un V
olta
ge
kV
150
250
350
450
550
650
750
IF F
requ
ency
MH
z
(b) (1.586 GHz)
(1.693 GHz)
(1.797 GHz)
(1.913 GHz)(1.948 GHz )
Fig. 5.5. Detected oscillations obtained in the experimental scheme shown in Fig. 5.4;
(a) Detector output traces; (b) E-gun variation and frequency measurements for
fLO=2.2 GHz.
The spectral contents of the microwave pulses detected in both the leading and the trailing
edges of the e-gun voltage pulse are the same. In the leading edge of the e-gun pulse, the
56
measurement of the signal at 1,797 MHz is missing, possibly because of its low-level
magnitude. One can observe that as the e-gun voltage increases the interaction occurs with
waves having higher frequencies.
The IF frequency of the two microwave signals excited in the top of the e-gun voltage pulse
is not presented in Fig. 5.5b because their spectral content was out of the low pass-band filter
(< 2 GHz) of the Frequency-Time Interval Analyzer. These signals, evolved in opposite
microwave paths, are exceptional also because their detected power levels are not the same as
is the case for all the other observed signals. Their spectral content is determined by using LO
frequencies around 4 GHz as is shown in Figs. 5.6a and 5.6b. In both of these figures the
measured e-gun variation and the IF frequency are shown during the microwave pulse. When
the LO frequency is 4.0 GHz, the IF frequency follows the e-gun voltage sweep, as is shown
in Fig. 5.6a. On the other hand, the IF frequency sweeps in the opposite direction of the e-gun
sweep when the LO frequency is 4.5 GHz, as is demonstrated in Fig. 5.6b. Hence, the
frequency of these signals is in the range of 4,114-4,118 MHz (for each e-gun voltage value
the IF frequencies for the two LO frequencies adds up to the difference between them - 500
MHz in the presented example).
The frequencies of the microwave pulses are supposed to coincide with the resonance
frequencies of the waveguide. This is indeed the case for all of the detected microwave
signals except for the one having the frequency of 1,948 MHz (denoted in Fig. 5.5b). This
phenomenon is demonstrated with the help of Figs. 5.7a and 5.7b representing zoomed
windows of the transmission loss measurement of the waveguide, as presented in Fig. 5.3.
57
3
4
5
6
7
EG
un V
olta
ge k
V
110
112
114
116
118
120
122
124
IF F
requ
ency
MH
z
(a)
3
4
5
6
7
0.2 0.3 0.4 0.5 0.6Time [ms]
EG
un V
olta
ge k
eV
378380382384
386388390392
IF F
requ
ency
MH
z
(b)
Fig. 5.6. E-gun voltage variation and frequency measurements for (a) fLO=4.0 GHz and
(b) fLO=4.5 GHz.
One possible explanation for the frequency mismatch of the signal at 1.948 GHz, is that the
outer microwave path is responsible for the oscillation mechanism of this signal. This
feedback mechanism is probably stronger than the inner microwave path, since the output
power of the microwave signals associate with it are higher than the power levels of the other
microwave signals (see Fig. 5.5b).
58
-8
-6
-4
-2
0
1.5 1.7 1.9 2.1
Tran
smis
sion
Los
s dB 1.587 1.705 1.816 1.934
(1.586) (1.693) (1.797) (1.913)2.039
(a)
-25
-20
-15
-10
-5
4 4.1 4.2 4.3 4.4 4.5Frequency [GHz]
Tran
smis
sion
Los
sdB 4.114
(4.117)
(b)
Fig. 5.7. Transmission loss measurements in the frequency ranges of (a) 1.5-2.1 GHz and (b)
4-4.5 GHz; radiation frequencies (in parentheses) and the corresponding resonance
frequencies are denoted.
The results presented above are explained by the dispersion diagram, Fig. 5.8. Three spatial
harmonics are presented in the diagram: the first and the second spatial harmonics of the
backward mode (delivers microwave energy from the collector end to the e-gun end of the
waveguide) and the first spatial harmonic of the forward mode (delivers microwave energy in
the opposite direction). The width of the fundamental cell in the diagram is 2π/p≅ 419 m-1. The
59
light lines determine the forbidden regions where the phase velocity of the fundamental
spatial harmonic is c 0.4 sin(26.44) c vph ≅= .
0
1
2
3
4
5
0 200 400 600 800Axial Wavenumber [1/m]
Freq
uenc
y G
Hz
2.2-9.2 kV
6.0-6.4 kV
2π/p
Light lines
m=+1
m=+2
m=+1
Fig. 5.8. Dispersion diagram of the single-helix device.
The e-beam lines in Fig. 5.8 determine the range of energies, 2.2 - 9.2 keV, in which
interactions have been observed. Interactions in the frequency range around 2 GHz occur
with the first spatial harmonic of the backward mode. Interactions at 4.11 GHz occur between
electrons having energies of 6.0-6.4 keV and with two spatial harmonics carrying EM energy
in opposite directions simultaneously. The two spatial harmonics are the first spatial
harmonic of the forward mode and the second spatial harmonic of the backward mode. The
fact that the signal is amplified on both directions may explain the domination of this
interaction over that with the single spatial harmonic.
60
A different microwave scheme used is shown in Fig. 5.9. In this scheme an inner feedback is
produced by shorting the e-gun end of the waveguide. The measurement of the current
(collected at the helix and the collector) and the microwave diagnostic is similar to the
scheme shown in Fig. 5.4.
Fig. 5.9. Diagnostic circuitry of the device shorted at the e-gun end.
Figs. 5.10a and 5.10b present results obtained in the experimental scheme shown in Fig. 5.9
using a zener diode in order to obtain flattop e-gun voltage. For e-gun voltage level of 6.4 kV,
the spectral content is in the range 4,118 – 4,124 MHz and for e-gun voltage level of 9.4 kV,
50 Ω
Fast Digital OscilloscopesFrequency-TimeAnalyzer (HP5372A)
Attenuator
DetectorE-gunvoltage
-e
Low-pass /
high-pass filter
Microwaveport
50 Ω
Helixcurrent
Collectorcurrent
LO
Short
DC block
Splitter
≈≈≈≈
Collector
61
the spectral content is in the range 2,372 MHz, which is in agreement with the mechanisms
explained above.
2
3
4
5
6
7E
Gun
Vol
tage
kV
0
5
10
15
20
Det
ecto
r Out
put
arb
uni
ts
4.12 GHz @ 6.4 kV
(a)
6
7
8
9
10
0.0 0.2 0.4 0.6 0.8
Time [ms]
EG
un V
olta
ge
kV
0
1
2
3
Det
ecto
r Out
put
[arb
. uni
ts ]
2.37 GHz @ 9.2 kV
(b)
Fig. 5.10. Detector output and e-gun voltage variation for the flat-top voltages (a) 6.4 kV and
(b) 9.2 kV.
5.3. Bifilar-helix CRM experiments
In order to obtain CRM interaction we proposed to introduce a bifilar-helix structure as a
waveguide [20]. This structure enhances the transverse EM wave components as compared to
62
the single helix scheme. These transverse components, shown in Fig. 5.11, are essential for
the CRM operation.
+
-
E E-e
Fig. 5.11. The bifilar helix; an odd-mode is depicted.
The experimental device is based on a stand-alone e-gun assembly developed in our
laboratory. The concept behind this experimental bench, illustrated in Fig. 5.12, is to use a
stand-alone e-beam assembly as an insert in various interaction structures. This concept
enables modifications of the interaction region without breaking the vacuum. The e-beam is
generated in a thermionic cathode that is placed inside a vacuumed glass-tube. The cathode is
heated through an isolating transformer. Negative high-voltage is applied to the cathode and a
grounded ring can be placed outside the glass-tube in front of the cathode plane to serve as an
anode. The e-beam is dumped on a collector and its current is measured using a 50 Ω resistor.
-H.V.
Cathode AnodeGlass tube
Collector
Tovacuum pump
-e
~
(8-12 V)
50 W
Fig. 5.12. Scheme of the stand-alone e-beam experimental bench.
63
Both CRM oscillations and amplifications have been observed. The oscillator device is shown
schematically in Fig. 5.13. The helices are formed by winding a copper tape (0.5” wide)
around the glass tube of the stand-alone e-beam assembly. The two tapes are shortened at the
cathode end. Photographs 5.2 and 5.3 show the collector and a segment of the glass tube with
a bifilar winding of copper tapes, respectively.
KickerSolenoid Cylindrical cavity
!H .V .
Cathode Anode
Collector-e
50 Ω
Cut-off section
MW output
Fig. 5.13. A scheme of the bifilar CRM oscillator.
Photograph 5.2. The collector at the end of
the glass tube.
Photograph 5.3. The bifilar copper tapes
wound on the glass tube.
64
A metallic cylinder with cut-off sections in its both ends forms a shielding cavity. The
aperture of the grounded shielding is used as an anode. Three synchronized power supplies
(the apparatus used in all the previous experiments reported here) generate the solenoid, the
e-gun, and the kicker pulses. The axial magnetic field confines the e-beam down to the
collector and induces the cyclotron motion. The kicker imparts transverse velocity to the
electrons. The e-beam interacts with the EM wave and is dumped onto a collector where its
amplitude is measured using a 50 Ω resistor. The microwave signal evolved in the cyclotron
tube is detected by a coaxial probe at the collector end.
In the amplifier scheme, shown in Fig. 5.14, the solenoid serves as the EM shielding having a
large inner diameter in order to decrease the fringe electric field. This may enhance the filling
factor of the interaction. Impedance match sections are formed on both ends of the bifilar
structure. EM signal in the range 3.0 - 7.5 GHz, is injected at the e-gun end and is extracted
from the collector end.
50 Ω
Solenoid
Collector
-eMicrowave
Fig. 5.14. The bifilar-CRM amplifier matching scheme.
The CRM oscillator frequency is tuned by varying the solenoid field. The CRM tunability,
shown in Fig. 5.15, is demonstrated over three octaves in this experiment, at the frequency
65
range of 2.4-8.4 GHz in agreement with the CRM tuning relation (1.1). The oscillations
frequency is higher than the cyclotron frequency due to the Doppler shift.
2
3
4
56
78
9
2 3 4 5 6 7 8 9Cyclotron Frequency [GHz]
Freq
uenc
y G
Hz
Cyclotron frequency
Radiation frequency
Fig. 5.15. The bifilar-helix CRM tenability range; the circles indicate frequency
measurements in different axial magnetic strengths. The solid line indicates the
cyclotron frequencies.
A typical result of the bifilar-helix CRM amplifier is shown in Fig. 5.16. A signal at a
frequency of 4.93 GHz is injected into the waveguide while applying axial magnetic field of
1.65 kG. This magnetic field value corresponds to cyclotron frequency of 4.62 GHz. The
difference between the frequency of the amplified signal and the cyclotron frequency is
attributed mainly to the Doppler-shift. The “rabbit ears” picture is the outcome of
synchronism conditions that are met twice, in the leading and in the trailing edges of the
e-gun voltage pulse. As is expected, signal absorption is observed near the two-synchronism
conditions.
66
0
1
2
3
0.0 0.5 1.0 1.5 2.0Time [ms]
Gun
Vol
tage
kV
0
1
2
3
4
Det
ecto
r Out
put
arb
uni
ts
Fig. 5.16. Bifilar-helix CRM amplification measurement.
67
Chapter 6
Summary
This thesis presents an experimental study of novel schemes of low-voltage FEL and CRM
devices. The FEL-type mechanism is demonstrated here in a new regime, wλ>>λ . It
radiates at UHF with electrons accelerated by less than 6 kV. To the best of our knowledge,
this is a first demonstration of an FEL-type operation in the UHF range. The FER (R for radio
frequencies) interactions occur with quasi-TEM modes in a non-dispersive transmission-line
cavity. Clear and reproducible oscillations are observed in the first three longitudinal modes
of the cavity (0.28 GHz at 1 kV, 0.56 GHz at 2 kV, 0.83 GHz at 6 kV). In high
voltages ) kV 3 ( > , due to the presence of the axial magnetic field, a cyclotron interaction is
observed, in addition to the FER interaction along the same e-beam pulse. For the CRM
interaction, the wiggler acts as a distributed kicker that rotates the electron beam.
A one-dimensional steady-state electron dynamics of the FER and its gain-dispersion relation
in the linear regime were derived. Analysis of the electron trajectories verifies the need of an
axial magnetic field for transportation of the e-beam through the interaction region. Taking
into account the low Q factor of the cavity, the solutions of the gain-dispersion equation show
that the gain per round obtained in the FER is sufficient to obtain oscillations.
The demonstration of FEL tunability by a variable dielectric loading has been carried out
using the FER device. The variable dielectric load was implemented by distilled water. To the
best of our knowledge, fluid-loaded microwave device is demonstrated here for the first time.
By varying the amount of the distilled water the FER operating frequency is tuned in a range
of ~4%. This concept could be relevant, in general, as a method to tune FELs in both
68
oscillator and amplifier schemes. The variable dielectric, demonstrated here by the fluid
loading, can also be implemented by solid materials also, either by a variable geometry or by
a ferroelectric material. Compared to the known method of FEL tuning by electron-energy
variation, the controlled dielectric loading alleviates the need to change the accelerator and
electron-optics settings.
The FER devices presented here obey the basic physical rules of the mature FEL types,
though they operate at extremely low voltages and long wavelengths, at the lowest end of the
FEL operating spectrum (266 MHz at 420 V). Interactions with quasi-TEM waves occur also in
short wavelength FELs that are very large-scale and expensive experiments. Hence, for scientific
and educational purposes, low-cost FER can be used as a tabletop platform to study and
demonstrate fundamental FEL physics (even in small laboratories). The low frequency of the
FER signal enables its monitoring without detection, using a fast digital-oscilloscope. This
ability provides a clear insight into FEL time-domain phenomena. The FER concept is
believed, therefore, to be a useful extension of the FEL family in both scientific and practical
terms.
For practical purposes, an FER version with enhanced efficiency and power may add another
range of applications to the diverse FEL family. The FER tunabilty could be a useful feature
for these applications. This tuning method is applicable by other means also to shorter
wavelength FELs. Practical applications in the UHF include a wide range of industrial
processes, radio communication, radar, RF accelerators, and plasma processing [53].
Operation regimes and concepts demonstrated in this thesis may lead to the development of
new devices. The dual FER and CRM operation may be exploited to develop an FER-CRM
device operating in two frequencies. The fluid-loading concept may be used not only for
69
frequency tuning but also for the fluid heating [32]. For this application the dielectric loss of
the fluid is exploited where as an example for water at 1 GHz its level is around 15 dB/m
[53]. The design of such a device should be optimized in order to get the maximum heating
(dielectric loss) while maintaining the oscillation conditions.
Reduction of free-electron sources overhead can be achieved by using cold cathodes in
appropriate applications. They are cheaper than the thermionic cathodes, heating and
pre-activation processes are not needed, and the vacuum requirements are less strict in the
devices in which they are employed. One type of such a cathode is the high-dielectric
(ferroelectric) ceramic cathode. In this cathode the electrons are emitted from a plasma that is
excited on the ceramic surface by a voltage pulse of the order of ~1 kV, applied on the
ceramic in a nanosecond time scale. The lifetime of the plasma is of the order of few
micro-seconds and typical current densities provided by these cathodes are up to 100 A/cm2.
Ferroelectric cathodes can operate in poor vacuum conditions at room temperature, and with
low voltages. Ferroelectric cathodes do not need heating and pre-activation and they are easy
to fabricate and to handle, as compared to thermionic or field-emission cathodes. A
ferroelectric-cathode has been employed in the CRM device operating at ~7 GHz, near the
cut-off frequency of a hollow cylindrical cavity. The use of ferroelectric cathodes may
advance the microwave tube technology for various applications. According to the best of our
knowledge, high-dielectric ceramic-cathodes have not been utilized so far in any EM radiation
generator. This experiment is followed in our laboratory by studies of new CRM schemes, in
which unique features of ferroelectric cathode can be utilized exclusively. Ferroelectric
cathodes can be used in a low repetition-rate or single-shot compact CRMs. They can be
easily fabricated in various shapes for producing specified cross-sectional profiles of the
electron beams.
70
Development of broad tunability bandwidth, low-voltage (< 10 kV) CRM employing helix
structures has been carried out. In a single-helix device TWT-type oscillations have been
observed in two frequency ranges (around 2 GHz and at 4.11 GHz) that do not depend on the
axial magnetic-field strengths. Oscillations in the frequency range of 2 GHz are attributed to
interactions with a first spatial harmonic of the backward mode. Oscillations at 4.11 GHz are
attributed to simultaneous interaction with both a first spatial harmonic of the forward mode
and a second spatial harmonic of the backward mode. The latter interaction dominates in the
competition with the former one possibly due to the amplification of the EM wave to both
directions in the oscillator. In order to get cyclotron interaction, we proposed to conduct an
experiment based on a bifilar-helix. In addition to its broad bandwidth characteristics this
structure supports transverse electric field components on axis, a necessary feature for
cyclotron interaction. The bifilar–CRM experiment has been conducted using a stand-alone
e-beam assembly developed in our laboratory. The stand-alone e-beam assembly enables the
changing of interaction structures without breaking the vacuum. The bifiar-CRM exhibits
both oscillator and amplifier operations. CRM oscillations have been observed over two
octaves, from 2.5 GHz up through 8.4 GHz where the frequency is controlled by the axial
magnetic field. A net CRM amplification of 16 dB (peak) was achieved around 5 GHz.
Typical FELs and CRMs operate at high accelerating voltages which dictate a certain scale
and cost of these devices which might be impractical (i.e. large and expensive) for many
applications. This thesis presents studies that may lead to the development of new low-cost
and compact free-electron devices for medium power applications in the RF and in the
microwave ranges. A Related activity is carried out by a European consortium intending to
develop low-cost FEL for industrial applications [54].
71
Low operating voltages limit the available output power and dictates a certain operation
frequency range. However, high-power EM radiation can be obtained by utilizing low-voltage
schemes in an array like the CRM-array proposed by Jerby et al. [5]. The practical advantages
of this concept are the alleviation of space-charge effects by utilizing low-voltage, low-current
e-beams and the feasibility of high-power microwave generation by a compact device. The
implementation of these compact high-power devices needs further studies.
72
Appendix A: Electron Dynamics in FER
Steady state electron trajectories and velocities in the combined wiggler and axial magnetic
field assuming the operation parameters of the FER are derived. The particular configuration
of interest is the propagation of an e-beam through an external magnetic field as is illustrated
in Fig. A.1.
-e B||
Bw
y
z
x
E
Fig. A.1. Configuration under study.
The external magnetic field consists of a wiggler field and a solenoid guide field,
[ ]||wwwwww0 B)zkcos()yksinh(Bz)zksin()ykcosh(ByB ++= (A.1)
which satisfies 0BB 00 =⋅∇=×∇ . Here, Bw and B|| denotes the wiggler and the axial
magnetic field amplitudes, respectively. We assume here that the magnetic field exhibited by
the e-beam is negligible in comparison to the external magnetic field.
The Lorentz force-equation of an electron having a charge –e and a rest mass em is
)BP(dtPd
F 0000
0 ×η−== (A.2)
73
where 0P is the electron momentum, e00 me γ≡η , and 0γ is the relativistic factor. As a
result, the motion equations of an electron are
)zksin()ykcosh(vBdt
dv
Bvdt
dv
)zksin()ykcosh(vBBvdt
dv
wwx0w0z0
z0x00y0
wwz0w0z0y00x0
η−=
η=
η+η−=
(A.3)
In steady-state const0 =γ , hence dzdv
dtdz
dzd
dtd
z0== where z0v is the z component of the
steady-state electron velocity. As a result, the motion equations of an electron are,
)zksin()ykcosh(PBdz
dPv
BPdz
dPv
)zksin()ykcosh(PBBPdz
dPv
wwx0w0z0
ez
z0x00y0
ez
wwezw0z0y00x0
ez
η−=
η=
η+η−=
(A.4)
where z0B is the z component of the external magnetic-field, and ezv ( ezP ) denotes the
average value of the axial velocity (momentum). The average value can be represented by the
root-mean-square over one wiggler period given by ∫λ
λ≡
w
0
2)z(z0
w
2ez dzv1 v . Differentiating
the x component of Eq. (A.4) as a function of z results in,
)zkcos()ykcosh(kPB
dzdP
B)zksin(kP)zkcos(dz
dP)yksinh(B
dz
Pdv
wwwezw0
y0||0wwy0w
y0ww02
x02
ez
η+
η−
−η−=
(A.5)
74
For transverse components of the electron velocity, which are very small in comparison to the
axial velocity, the transverse electron displacements from cavity axis of symmetry are much
less than the wiggler period and ezwy0w P)ykcosh( P)yksinh( << . Substituting dz
dP y0 from
Eq. (A.4) at Eq. (A.5) results in,
)zkcos()ykcosh(kvkdz
vdwwwwx0
2z02
x02
Ω=+ (A.6)
where ez
z0z0 v
kΩ
≡ , z00z0 Bη≡Ω , and w0w Bη≡Ω . The solution of Eq. (A.6) is given by,
)zkcos(v)zksin(B)zkcos(Av wwxz0z0x0 ++= (A.7)
where )ykcosh(kv
vk)ykcosh(
kk
kv w2
w2ez
2z0
2ezww
w2w
2z0
wwwx
−Ω
Ω=
−
Ω≡ . It is evident, therefore, that
the resonance wezz0 kv≈Ω should not be approached too closely.
Differentiating Eq. (A.7) as a function of z and substituting it into the x component of Eq.
(A.4) results in,
( ) ( )
)zksin()ykcosh(k
)zksin(vkk
)zkcos(dz
zkdk
1B)zksin(dz
zkdk
1Av
wwz0
w
wwxz0
wz0
z0
z0z0
z0
z0y0
Ω+
+−=(A.8)
where ( )
)zksin()yksinh(zk1dz
zkdk
1www
z0
wz0
z0 ΩΩ
−= . After imposing initial conditions (at
z=0) on equations (A.7) and (A.8), we arrive at the following expressions for the electron
velocity,
75
[ ])zkcos()zkcos(v)zksin(v)zkcos(vv cwwxc)0(y0c)0(x0x0 −+−= (A.9a)
ΩΩ
+−
+
ΩΩ
−
+
ΩΩ
−=
)zcksin()zksin()ywksinh(zwkcw)zcksin()zwksin(
wkck
wxv
)zckcos()zksin()ywksinh(zwkcw1)0(y0v
)zcksin()zksin()ywksinh(zwkcw10(x0vv
w
w
w)y0
(A.9b)
where we are using the approximations ezvc
ckz0kΩ
≡≅ and ||0cz0 Bη≡Ω≅Ω . These
approximations are valid for the FER since ||www B)zkcos()yksinh(B << ( 2.0BB
||
w < and
4.0yk w < ).
Now, dz
rdvdtrdv ez== or
ez
00vv
dzrd
= . Applying this operator to Eqs. (A.9) results in,
[ ])zkcos()zkcos(vv
)zksin(v
v)zkcos(
vv
dzdx
cwez
wxc
ez
)0(y0c
ez
)0(x00 −+−= (A.10a)
( )
( )
ΩΩ
+
+
ΩΩ
−
+
ΩΩ
−=
)zksin()zksin()yksinh(zk)zksin(- )zksin(kk
vv
)zkcos()zksin(yksinhzk1v
v
)zksin()zksin(yksinhzk1v
vdz
dy
cwwwc
wcw
w
c
ez
wx
cwwwc
w
ez
)0(y0
cwwwc
w
ez
)0(x00
(A.10b)
1vv
dzdz
ez
ez0 =≅ (A.10c)
76
After integrating Eqs. (A.10) over z we arrive at
[ ] )0(0c
c
w
wwc
c
yc
c
x0 x
k)zksin(
k)zksin(
1)zkcos(k
)zksin(k
x +
−α+−
α+
α= (A.11a)
( )
( ) +
+−−−−
ΩΩ
+
−α=
Σ
Σ
∆
∆
Σ
Σ
∆
∆k
)zksin(zk
)zksin(z
k
)zkcos(1
k
)zkcos(1yksinhk2
kzkcos1
y
22wwc
w
c
cx0
( )
( ) +
−+−
ΩΩ
−
α
Σ
Σ
Σ
Σ
∆
∆
∆
∆k
)zkcos(zk
)zksin(k
)zkcos(zk
)zksin(yksinhk2
kzksin
22wwc
w
c
cy
( ) ( )
( ) )0(022wwc
w
2w
wc
c
cw
yk
)zksin(zk
)zksin(z
k
1)zkcos(
k
1)zkcos(yksinhk
2
k
)zkcos1(kk
1zkcos
+
−+
−−
−ΩΩ
+
−+
−α
Σ
Σ
∆
∆
Σ
Σ
∆
∆
(A.11b)
tvz ez0 = (A.11c)
where ez
)0(x0x v
v≡α ,
ez
)0(y0y v
v≡α ,
ez
wxw v
v≡α , cw kkk −≡∆ and cw kkk +≡Σ .
In order to examine the validation of the above derivation, the electron trajectories (Eqs.
(A.11)) are computed and compared with the solution of the motion equations (Eqs. (A.3)).
Electron orbit projections on the x-z and y-z planes for electron energy of 5.5 keV are
presented for comparison in Figs. A.2a and A.2b. The trajectories computed by using the
77
results of the above derivation show good resemblance to the solutions of the equations of
motion apart from the electron drift along the interaction region.
-0.10
-0.05
0.00
0.05x
[cm
]
(a)
-0.1
0.0
0.1
0.2
0.3
0.4
0 10 20 30 40 50
z [cm]
y [c
m]
(b)
Fig. A.2. Projections of electron orbit onto (a) x-z plane and (b) y-z plane for accelerating
voltage of 5.5 kV; comparison between solutions of Eqs. (A.3) (green) and (A.11)
(blue).
As the accelerating voltage decreases, the drift decreases as well. This drift results from the
combined presence of the uniform axial magnetic field and the wiggler field [13]. The
transverse drift can cause the beam to move into a region of a weaker EM field and even
strike the waveguide structure. This is not the case for the FER since much more drift is
allowed before the electrons could strike the strip-lines and the electric field that exists
78
between the two strip-lines is assumed to be constant. The ripple exhibited by the electron in
the x-z plane is due to the presence of the axial magnetic field. The periods of this ripple fit
the values of the corresponding cyclotron wavelengths mm 7.7k2 cc =π=λ .
The average value of the axial velocity is derived in Ref. [14] and is given by,
20
22w
2c
2w
2c
2w
2
2ez 1-1
)(2
)(1
c
v
γ=
ω−Ω
ω+ΩΩ+ . (A.12)
This result is based on the one-dimensional representation of the external field given by
||ww0 Bz)zksin(ByB += (A.13)
which is a valid expression for the FER. This is justified by imposing the relevant FER
operation parameters ( 2.0BB
||
w < and 4.0yk w < ), on Eq. A.1. The solution of the electron
equation of motion results then in the following expressions for the transverse components of
the electron velocity [14],
)zkcos(vv wwxx0 = , (A.14a)
)zksin(kk
vv ww
cwxy0 = (A.14b)
where 2w
2ez
2c
2ezww
wxkv
vkv
−Ω
Ω≅ , cezc k v≡Ω and wezw k v≡ω .
Finally, expression (A.13) is obtained by averaging the transverse velocity over one wiggler
period , ∫λ
⊥⊥ λ≡
w
0
2)z(
w
2 dzv1 v using expressions (A.14) and utilizing energy conservation.
79
Appendix B: Linear Model of FER Amplification
Derivation of a one-dimensional linear model of the FER gain mechanism is presented here.
Theoretical studies of FELs with a planar wiggler and an axial guide field have been carried out
for operation regimes that differ from that of the FER [13,14]. In the particular configuration of
interest, depicted in Fig. B.1, the e-beam is affected by wiggler and axial magnetic field and
interacts with an electromagnetic plane wave defined by xem ExE = and yHyH = . An
electrostatic field, ebE , is associated with the space charge of the e-beam.
Fig. B.1. The configuration under study.
The electron equation of motion is
( )
⋅−×+η−= Ev
cvBvE
dtvd
2 , (B.1a)
the equation of continuity is
0t
J =∂ρ∂+⋅∇ , (B.1b)
and the Maxwell’s equations are
Electronbeam
B||
BW
zx
y
Eem
Eeb
80
tHE 0
em∂∂µ−=×∇ (B.1c)
1JtEH +∂∂ε=×∇ (B.1d)
0
1ebEερ
=⋅∇ (B.1e)
where the first order components, represented by the subscript 1, are induced by the
interaction between the e-beam and the EM radiation. The current density is vJ ρ= where the
electron velocity, 10 vvv += , and the electrons density, 10 ρ+ρ−=ρ , are the sum of their
steady state and first order components. Hence, the first order of the current density is given
by 01101 vvJ ρ+ρ−= .
As shown in App. A, the derived steady-state trajectories do not exhibit the transverse drift
which is computed to be less than 1 mm over the entire operation range of the FER. However,
since the transverse electric field is assumed to be transversely uniform, the small drift along
the FER cavity can be neglected and the derived steady state trajectories and velocities are
used here.
Assuming non-relativistic electrons as is applicable for the FER, 2.0cv< , Eq. (B.1a) reduces
to
[ ]BvEdtvd 1 ×+η−= (B.2)
Here the static magnetic field is approximated by, ||www0 Bz)zksin()ykcosh(ByB += .
81
Linearizing Eqs. (B.1) and (B.2) and assuming that all transverse oscillations are negligible,
i.e. z
y
,x ∂
∂<<∂∂
∂∂ , we obtain
tH
zE y
0x
∂
∂µ−=
∂∂
(B.3a)
x1xy Jt
Ez
H+
∂∂
ε=∂
∂− (B.3b)
tzJ 1z1
∂ρ∂
−=∂∂
(B.3c)
[ ])zkcos()zkcos(vvJ cwwx1x10x1 −ρ+ρ−= (B.3d)
ez1z10z1 vvJ ρ+ρ−= (B.3e)
[ ])zksin()ykcosh(BvBvHvEvz
vt wwwz1||y1yez0xx1ez −+µ−η−=
∂∂+
∂∂ (B.3f)
||x1y1ez Bvvz
vt
η=
∂∂+
∂∂ (B.3g)
[ ][ ])zksin()ykcosh(BvH)zkcos()zkcos(vEvz
vt wwwx1ycwwx0zz1ez +−µ+η−=
∂∂+
∂∂
(B.3h)
0
1z qz
Eερ
=∂∂
(B.3i)
where it is assumed that the e-beam enters the interaction region with no transverse velocity
components, i.e. 0vv )0(y0)0(x0 == and the x component of the steady state velocity
induced by the wiggler is given by [ ])zkcos()zkcos(vv cwwxx0 −= . The reduction factor q
82
of zE in Eq. (B.3i) is due to the existence of transverse component of the e-beam field, ebE ⊥ ,
and is given by [19],
0
1zz
ebz
q1
zE
zE
E1
zE
ερ
=
∂∂
=
∂∂⋅∇
+∂∂ ⊥⊥ . (B.4)
The first-order differential equations (B.3) are solved by applying the Fourier transform in the
time domain, and the Laplace transform in the space domain [52]. After imposing the
following initial conditions: 0EJv )0(z)0(z1)0(1 === we obtain,
y0)0(xx H~jE~E~s ωµ−=− (B.5a)
x1x)0(yy J~E~jH~H~s −ωε−=− (B.5b)
1z1~jJ~s ρω−= (B.5c)
( ) ( )( ) ( ) ( )( )I1
I1
wx11
11
wxx10x1
~~2
v~~2
vv~J~ +−+− ρ+ρ−ρ+ρ+ρ−= (B.5d)
ez1z10z1 v~v~J~ ρ+ρ−= (B.5e)
( ) ( ) ( )( )1z1
1z1w
wy1cyez0xx1ez v~v~)ykcosh(
j2v~H~vE~v~svj +− −
Ω+Ω−ηµ+η−=+ω (B.5f)
( ) x1cy1ez v~v~svj Ω=+ω (B.5g)
( ) ( ) ( )( ) ( ) ( )( )( ) ( )( )I
yI
ywx
0
1x1
1x1w
w1y
1y
wx0zz1ez
H~H~2
v
v~v~)ykcosh(j2
H~H~2
vE~v~svj
+−
+−+−
+ηµ
+−Ω−+ηµ−η−=+ω
(B.5h)
83
10
z~qE~s ρ
ε= (B.5i)
Here s is the variable of the Laplace transform. The superscript indices of the transformed
functions denote shifts in the Laplace space such as ( ) ( )wjks1 f~f~ µ≡± and ( ) ( )cjks
I f~f~ µ≡± .
Combining Eqs. (B.5a) and (B.5b) results in the transformed wave equation,
( )( )( ) x10)0(xxzz J~E~E~jksjksj ωµ−=−+− (B.6)
where εµω= 022
zk . Solving Eqs. (B.5c)-( B.5i) leads to the following expression for x1J~ ,
( )
( ) )0(x12c
2ezww
1)y,x(0
x12c
2
2ww
wx1)y,x(0
x1
E~Qv)ykcosh(
4j
E~Q)ykcosh(
jsv4
J~
Ω+Ω
ΩΩ
Θω
ηρ
−
Ω+Ω
ΩΩ+
Θω
ηρ=
−
−(B.7)
where, ( )
( )( )1
wx2c
2
1ww
1 sjv )ykcosh(
Q −−
−Ω+Ω
ΩΩΩ≡ .
Here ( ) ( ) 2q
211 ω+Ω≡Θ −− is the resonance parameter, ezsvj +ω≡Ω is the tuning parameter
and the plasma frequency is defined by )y,x(2p
0
)y,x(02q gq
gqω=
ε
ηρ≡ω .
The transverse profile of the electron current distribution is defined by y)(x,0)y,x(0 g ρ=ρ
where gsection cross beam theinside 1
section cross beam theoutside 0)y,x(
=
and =ρ0 constant.
84
In obtaining expression (B.7), we assumed normal-Doppler resonance, ( ) 01 ≈Θ − , hence
small quantities have been neglected. Inserting expression (B.7) in the transformed wave
equation (B.6) yields the following relation in the Laplace space,
( )( )( ) ( ) ( ))0(x3x2112)y,x(
2p
)0(xxzz E~Q E~QQc4
g E~E~jksjks +
Θ
ω−=−+− − (B.8)
where
( ) wx2c
2
2ww
2 jsv )ykcosh(
Q −Ω+Ω
ΩΩ≡ and
2c
2ezww
3v )ykcosh(
QΩ+Ω
ΩΩ≡ .
Assuming the following electric field and its transforms,
zjk)y,x()z(x zeAE −Φ= , )y,x()0()0(x AE~ Φ= , ( )
)y,x()y,x()jks(x AAE~z
Φ≡Φ= ++ leads to
the following dispersion relation
( ) ( ) ( )( )
( )( )
( ) )y,x()0(3112)y,x(
2p
)y,x(2112)y,x(
2p
)y,x()0()y,x(zz
AQQc4
g AQQ
c4
g
AAjksjks
ΦΘ
ω+Φ
Θ
ω−
=Φ−Φ+−
−+
−
+
(B.9)
Applying the integral operator ∫Φy,x
)y,x( dxdy to Eq. (B.19) results in
( ) ( ) ( )( )
( )( )
( ) f)0(3112
2p
f2112
2p
)0(zz
FAQQc4
FAQQc4
AAjksjks
−+
−
+
Θ
ω+
Θ
ω−
=−+−
(B.10)
85
where Ff is the filling factor defined by ∫
∫
Φ
Φ
≡
y,x
2)y,x(
y,x
2)y,x()y,x(
fdxdy
dxdyg
F .
In the linear regime the wave-number kz undergoes a small perturbation, hence we define in
the Laplace space a corresponding variable sjks z ′+−= where zks <<′ is real. The
dispersion relation as a function of s′ is given by
( )[ ]
( )[ ]
ω′′+ω−′−θ′
=
ω′′+ω−′−θ′′
f2
2p312
q2
ezz)0(
f2
2p212
q2
ezz)s(
Fc4
QQvsjsjk2A
Fc4
QQvsjsjk2A
(B.11)
where ( ) ezwz vkk +−ω≡θ ,
( )( )[ ] ( )( ) ( )( )( )wzwx
wwezwzezz12
c2
ezz1jkjksjv
)ykcosh(vjkjksjvjksjvjksjQ−−′−
Ω−−′+ω−′+ωΩ+−′+ω≡′−
( )( ) ( )( )[ ] ( )zwx12
c2
ezz2
ezzww2 jksjvvjksjvjksj)ykcosh(Q −′−Ω+−′+ω−′+ωΩ≡′−
and
( )( ) ( )( )[ ] ez12
c2
ezz2
ezzww3 vvjksjvjksj)ykcosh(Q−
Ω+−′+ω−′+ωΩ≡′ .
Finally the Pierce type gain-dispersion equation for the amplitude growth rate is given in the
Laplace s space by,
86
( ) ( )
( ) ( ))0(
f2p
z
212ez2
c2
z2q
2
f2p
z
312ez2
c2
z2q
2
)s( A
FˆkQQ
8ˆksˆˆsˆs
FˆkQQ
8ˆksˆˆsˆ
A
θβ
+
Ω−−−ω
θ−−θ
θβ
−
Ω−−−ω
θ−−θ
= . (B.12)
Here, assuming 1y)cosh(kw ≈ ,
( ) ( ) ( )wzez
wxwwz
12c
2z1 kks ˆkksˆˆksˆQ ++
ββ
−Ω−−−ω
Ω−−−ω≡
−,
( ) ( ) ( )
Ω−−−ω+
ββ
−Ω−−ω≡ 2c
2zz
ez
wxwz2 ˆksˆks ˆksˆQ ,
( ) wz3 ˆksˆQ Ω−−ω≡ ,
cvwx
wx ≡β and c
vezez ≡β . All the operating parameters in Eq. (B.12) are dimensionless
and normalized as follows:
θτ=θ is the resonance parameter,
τΩ=Ω is the tuning parameter,
τΩ=Ω c,wc,wˆ is the wiggling and the cyclotron frequencies,
τω=θ ppˆ is the space-charge parameter,
Lsjs ′= and kLk = are the wave-numbers, where the electron time of flight along the
interaction length L is ezvL=τ . Near resonance 0kksˆ wz ≈−−−ω , 1Q simplifies to
( )wzez
wx1 kksQ ++
ββ
−≡ and the Pierce type equation, Eq. (B.12), simplifies to the 5th order
equation in s .
87
Gain curves have been obtained for a number of axial magnetic fields, ||B , with all other
parameters held constant. The curves for FER radiation frequency of 280 MHz are shown in
Fig. B2. We note that as the axial magnetic field amplitude decreases the gain increases. This
behavior is reported in Ref. [13] also for TEM waves with axial magnetic field amplitudes
above the resonance, wezc kv=Ω . The explanation can be attributed to the coupling between
the x-directed electric field and the x-directed wiggling motion. The x-directed wiggling
motion has large amplitudes near resonance and decreasing amplitudes as the axial magnetic
field increases.
0.90
0.95
1.00
1.05
1.10
0.2 0.3 0.4 0.5 0.6
E-beam Voltage [kV]
Gai
n
B|| = 1.0 kGB|| = 1.25 kGB|| = 1.5 kG
Fig. B2. Gain vs. e-beam voltage for FER radiation at 280 MHz as a function of axial
magnetic field strength; The e-beam current is 100 mA, and the wiggler field is 0.4
kG.
88
References
[1] V.L. Granatstein and I. Alexeff, Ed., High-Power Microwave Sources, Artech House,
Boston, 1987; and references therein.
[2] J. Benford and J. Swegle, High-Power Microwaves, Artech House, Boston, 1992; and
references therein.
[3] H. P. Freund and T. M. Antonsen Jr., Principles of Free- electron Lasers, Chapman and
Hall, London, 1992; and references therein.
[4] H. P. Freund and G. R. Neil, "Free-electron lasers: Vacuum electronic generators for
coherent radiation," Proc. IEEE, Vol. 87, pp. 782-803, 1999; and references therein.
[5] E. Jerby, A. Kesar, M. Korol, Li Lei and V. Dikhtyar, “Cyclotron-resonance-maser
arrays,” IEEE Trans. Plasma Sci., Vol. 27, pp. 445-455, 1999.
[6] E. Jerby (unpublished).
[7] M. Korol and E. Jerby, “Linear analysis of a multibeam cyclotron-resonance maser
array,” Phys. Rev. E., Vol. 55, pp. 5934 - 5947,1997.
[8] R. Drori, E. Jerby and A. Shahadi, "Free-electron maser oscillator experiment in the
UHF regime," Nucl. Instrum. and Methods in Phys. Res., Vol. A358, pp. 151-154, 1995.
[9] R. Drori, E. Jerby, A. Shahadi, M. Einat and M. Sheinin, "Free-electron maser
operation at 1 GHz/1keV regime," Nucl. Instrum. and Methods in Phys. Res., Vol.
A375, pp. 186-189, 1996.
[10] R. Drori and E. Jerby, "Free-electron-laser type interaction at 1 m wavelength
range," Nucl. Instrum. and Methods in Phys. Res., Vol. A393, pp. 284 - 288, 1997.
[11] R. Drori and E. Jerby, “Tunable fluid-loaded free-electron laser in the low
electron-energy and long-wavelength extreme,” Phys. Rev. E., Vol. 59, pp. 3588 -
3593,1999.
89
[12] A. Sneh and E. Jerby, “Coaxially fed folded foil electromagnet wiggler,” Nucl. Instrum.
Methods Phys. Res., Vol. A285, pp. 294-298, 1989.
[13] Y. Z. Yin and G. Bekefi, “Dispersion characteristics of a free electron laser with a
linearly polarized wiggler and axial guide field,” J. Appl. Phys., Vol. 55, pp. 33-42,
1984.
[14] H. P. Freund, R. C. Davison, and G. L. Johnston, “Linear theory of the collective
Raman interaction in a free-electron laser with a planar wiggler and an axial guide
field,” Phys. Fluids B2, pp. 427-435, 1990.
[15] G. Rosenman, D. Shur, Ya. E. Krasik and A. Dunaevsky, “Electron emission from
ferroelectrics,” J. Appl. Phys., Vol. 88, pp. 6109-6161, 2000; and references therein.
[16] D. Shur, G. Rosenman, Ya. E. Krasik and V. D. Kugel, “Plasma-assisted electron
emission from (Pb,La)(Zr,Ti)O3 ceramic cathodes,” J. Appl. Phys., Vol. 79, p. 3669,
1996.
[17] V. A. Flyagin, A. V. Gaponov, M. I. Petelin and V. K. Yulpatov, “The gyrotron,” IEEE
Trans. Microwave Theory and Tech., Vol. MTT 25, pp. 514-521, 1977.
[18] R. Drori, M. Einat, D. Shur, E. Jerby, G. Rosenman, R. Advani, R. J. Temkin, and C.
Pralog “Demonstration of microwave generation by a ferroelectric-cathode tube,” App.
Phys. Letters, Vol. 74, pp. 335-337, 1999.
[19] R. E. Collin, Foundation for microwave engineering (McGraw-Hill Book Company,
NY, 1966).
[20] A. Aharony, R. Drori and E. Jerby, “Cyclotron resonance maser experiments in a bifilar
helical waveguide,” Phys. Rev. E., Vol. 62, pp. 7282-7286, 2000.
[21] M. Einat, E. Jerby and G. Rosenman, “A microwave gyro amplifier with a ferroelectric
cathode,” Accepted for publication in IEEE-MTT.
90
[22] M. Einat, E. Jerby and G. Rosenman, “High repetition-rate ferroelectric-cathode based
gyrotron,” Submitted for publication in App. Phys. Letters.
[23] W. B. Colson, “Short wavelength free electron lasers in 1996,” Nucl. Instrum. Methods
Phys. Res., Vol. A393, pp. 6-8, 1997.
[24] H. P. Freund and V. L. Granatstein, “Long wavelength free-electron lasers in 1996,”
Nucl. Instrum. Methods Phys. Res., Vol. A393, pp. 9-12, 1997.
[25] L. R. Elias, W. M. Fairbank, J.M. J. Madey, H. A. Schwettman and T. I.
Smith, “Observation of stimulated emission of radiation by relativistic electrons in a
spatially periodic transverse magnetic field,” Phys. Rev. Lett., Vol. 36, pp. 717-720,
1976.
[26] M. Billardon, P. Elleaume, J. M. Ortega, C. Bazin, M. Bergher, M. Velghe, Y. Petroff,
D. A. G. Deacon, K. E. Robinson, and J. M. J. Madey, “First operation of a storage-ring
free-electron laser,” Phys. Rev. Lett., Vol. 51, pp. 1652-1655, 1983.
[27] G. N. Kulipanov, V. N. Litvinenko, I. V. Pinaev, V. M. Popik, A. N. Skrinsky, A. S.
Sokolov, and A. N. Vinokurov, “The VEPP-3 storage-ring optical klystron: lasing in the
visible and ultra violet regions,” Nucl. Instrum. Methods Phys. Res., Vol. A296, pp. 1-3,
1990.
[28] R. M. Phillips, "The ubitron, a high-power traveling-wave tube based on a periodic
beam interaction in unloaded waveguide," IRE Trans. on Electron Devices ED-7, pp.
231-241, 1960; see also in Nucl. Instrum. Methods Phys. Res., Vol. A272, pp. 1-9, 1988.
[29] T. J. Orzechowski, B. R. Anderson, J. C. Clark, W. M. Fawley, A. C. Paul, D.
Prosnitz, E. T. Scharlemann, S. M. Yarema, D. B. Hopkins, A. M. Sessler and J. S.
Wurtele, “High-efficiency extraction of microwave radiation from a tapered-wiggler
free-electron laser,” Phys. Rev. Lett., Vol. 57, pp. 2172-2175, 1986.
91
[30] M. Cohen, A. Eichenbaum, M. Arbel, D. Ben-Haim, H. Kleinman, M. Draznin, A.
Kugel, I. M. Yakover, and A. Gover, “Masing and single-mode locking in a
free-electron maser employing prebunched electron beam,” Phys. Rev. Lett., Vol. 74,
pp. 3812-3815, 1995.
[31] V. Grinberg, E. Jerby and A. Shahadi, “Low-cost electron-gun pulser for table-top
maser experiments,” Nucl. Instrum. and Methods in Phys. Res., Vol. A358, pp. 327 -
330, 1995.
[32] E. Jerby, "Liquid heating in interaction region of microwave generator", US
patent # 5,998,773.
[33] M. Einat and E. Jerby, “Anomalous and normal Doppler effect in a stripline-loaded
cyclotron resonance maser oscillator,” Phys. Rev E, Vol. 56, pp. 5996-6001, 1997.
[34] D. Shur, G. Rosenman and Ya. E. Krasik, “High perveance ferroelectric cathode with
narrowed electron energy spread,” to be published.
[35] K. R. Chu, A. K. Ganguly, V. L. Granatstein, J. L. Hirshfield, S. Y. Park and J. M.
Baired, “Theory of slow-wave cyclotron amplifier,” Int. J. Electron. vol. 51, pp.
493-502, 1981; and references therein.
[36] H. Guo, L. Chen, H. Keren and J. L. Hirshfeld, “Measurements of gain for slow
cyclotron waves on an annular electron beam,” Phys. Rev. Lett., Vol. 49, pp. 730-733,
1982.
[37] V. K. Jain, V. K. Tripathi and R. K. Patnaik, "Excitation of electromagnetic modes in a
dielectric-loaded waveguide via cyclotron resonance interaction," IEEE Trans. Plasma
Sci., Vol. PS-14, pp. 31-34, 1986.
[38] T. N. Kho and A. T. Lin, “Slow-wave electron cyclotron maser,” Phys. Rev. A, Vol. 38,
pp. 2883-2888, 1988; and references therein.
92
[39] K. Ganguly and S. Ahn, “Nonlinear theory for the slow wave cyclotron interaction,“
Phys. Rev. A., Vol. 42, pp. 3544-3554, 1990.
[40] K. C. Leou, D. B. McDermott and N. C. Luhmann, Jr., “Dielectric-loaded wideband
gyro-TWT,” IEEE Trans. Plasma Sci., Vol. 20, pp. 188-196, 1992.
[41] E. Jerby and G. Bekefi, “Cyclotron maser experiments in a periodic-waveguide,” Phys.
Rev E, Vol. 48, pp. 4637-4641, 1993.
[42] E. Jerby, G. Bekefi and A. Shahadi, "Observation of frequency modulation in a
traveling-wave cyclotron free electron maser," Nucl. Instrum. and Methods in
Phys. Res., Vol. A341, pp. 115-118, 1994.
[43] E. Jerby, “Linear analysis of periodic-waveguide cyclotron maser interaction,” Phys.
Rev. E, Vol. 49, pp. 4487-4496, 1994.
[44] E. Jerby, A. Shahadi, V. Grinberg, V. Dikhtiar, M. Sheinin, E. Agmon, H. Golombek, V.
Trebich, M. Bensal and G. Bekefi, “Cyclotron maser oscillator experiment in a
periodically loaded waveguide,” IEEE J. Quantum Elec., Vol. 31, pp. 970-979, 1995.
[45] Y. Leibovich and E. Jerby, “Cyclotron-resonance maser in a periodically-loaded
quadropole transmission-line,” Phys. Rev. E, Vol. 60, pp. 2290-2296, 1999.
[46] J. R. Pierce, Traveling-Wave Tubes (Van Nostrand, NY, 1950).
[47] H. S. Uhm and J. Y. Choe, “Gyrotron amplifier in a helix loaded waveguide,” Phys.
Fluids, Vol. 26, No. 11, pp. 3418-3425, 1983.
[48] J. R. Pierce and P. K. Tien, “Coupling of modes in helixes,” Proc. I.R.E., vol. 42, pp.
1389-1396, 1954.
[49] S. Sensiper, “Electromagnetic wave propagation on helical structured (a review and
survey of recent progress),” Proc. I.R.E., vol. 43, pp. 149-161, 1955.
93
[50] R. Kompfner and N. T. Williams, “Backward-wave tubes,“ Proc. I.R.E., vol. 41, pp.
1603-1611, 1953.
[51] E. Jerby, E. Agmon, H. Golombek and A. Shahadi, High-Power Microwave Laboratory
at Tel-Aviv University, Internal Report.
[52] E. Jerby, Ph.D. Thesis.
[53] J. Thuery, Microwaves: Industrial, Scientific, and Medical Applications, Artech House,
Boston, 1992; and references therein.
[54] A. I. Al-Shamma’a, J. Lucas, R. A. Stuart and P. J. M. Van Der Slot, “European
thematic network project for an industrial free electron laser at 10-100 GHz,” Proc. 22nd
International Free Electron Laser Conference, Durham, North Carolina, USA, pp.
16-21, 2000.
אוניברסיטת תל אביב
הפקולטה להנדסה על ש+
אבי ואלדר פליישמ2
התקני לייזר אלקטרוני) חופשיי) במתח נמו!
חיבור לש+ קבלת התואר "דוקטור לפילוסופיה"
מאת
רמי דרורי
הוגש לסנט של אוניברסיטת תל אביב
סיו2 התשס"א (יוני 2001)
אוניברסיטת תל אביב
הפקולטה להנדסה על ש+
אבי ואלדר פליישמ2
התקני לייזר אלקטרוני) חופשיי) במתח נמו!
חיבור לש+ קבלת התואר "דוקטור לפילוסופיה"
מאת
רמי דרורי
עבודה זו נעשתה בהדרכת
פרופ' אליהו ג'רבי
הוגש לסנט של אוניברסיטת תל אביב
סיו2 התשס"א (יוני 2001)
dxe`il מוקדש באהבה ובהוקרת תודה על תמיכה ועידוד לרעייתי
aia`e cwy +מוקדש לילדיי האהובי
ozpe dilb ולהוריי
תקציר
תזה זו מציגה תחו+ פעולה חדש של לייזר אלקטרוני+ חופשיי+ (FEL) בתדר גלי הרדיו וטכנולוגיות
וסכמות משלימות שמומשו תחילה במייזר תהודת הציקלוטרו2 (CRM). פיתוחי+ משלימי+ אלו שעשויי+
להיות משולבי+ בעתיד בהתקני FEL כוללי+ תותח אלקטרוני+ המבוסס על קתודה קרמית פרואלקטרית
ושילוב של מבני גל לולייני (helix). המחקר בוצע באוניברסיטת תל אביב כחלק מפעילות המעבדה לפיתוח
מקורות שולחניי+ של קרינה אלקטרומגנטית שמקורה באלקטרוני+ חופשיי+ המואצי+ על ידי מתחי+
נמוכי+.
תחו+ הפעולה של ה – FEL הורחב בתזה זו לכיוו2 תדרי קרינה ומתחי הפעלה נמוכי+ ביותר
(1GHz, < 6 kV >). תחו+ פעולה חדש ולא רגיל זה זיכה את ההתק2 בכינוי FER כאשר ה – R מייצגת
את תחו+ גלי הרדיו בו הוא פועל. ה ; FER עושה שימוש במהוד של מולי: גל לא;דיספרסיבי התומ: בגלי+
מישוריי+ (TEM waves) בעלי אור: גל גדול במיוחד. התקני ה – FER המוצגי+ כא2 פועלי+ על פי אות+
מנגנוני+ פיסקליי+ על פיה+ פועלי+ כל התקני ה – FEL אול+ בתחו+ פעולה חדש בו אור: הגל של
. wλ>>λ , wλ wiggler ; גדול בהרבה ממחזור ה λ הקרינה
כיוו2 תדר של FEL באמצעות העמסה דיאלקטרית משתנה הודגמה בעזרת ה – FER. ההעמסה
הדיאלקטרית מומשה על ידי מי+ מזוקקי+ שהוכנסו בכמויות משתנות בתו: צינורות זכוכית לאזור
האינטרקציה. מקור קרינה של גלי+ אלקטרומגנטי+ המועמס על ידי מי+ מודג+ במסגרת מחקר זה
לראשונה.
ה – FER מועמס המי+ פעל בתדרי+ ובמתחי ההאצה הנמוכי+ ביותר (MHz at 420 V 266). תדר
הקרינה הנמו: של ה – FER מאפשר דגימה ישירה שלו בסקופי+ דיגיטלי+ מהירי+ ולכ2 חקירה של
.FEL – תופעות זמניות של קרינת ה
FER – במסגרת המחקר פותחו משוואות תנועה של האלקטרוני+ במצב יציב ומודל חד;ממדי של פעולת ה
בתחו+ ההגבר הליניארי. ניתוח של מסלולי האלקטרוני+ מלמד שהפעלת שדה מגנטי צירי הוא תנאי
הכרחי להעברת האלקטרוני+ דר: אזור האינטרקציה ב ; FER. אודות לנוכחות השדה המגנטי הצירי,
נצפית בתנאי+ מסוימי+ ג+ קרינת ציקלוטרו2 בנוס> לקרינת FER. ה – wiggler משמש אז להענקת
רכיבי המהירות הרוחביי+ לאלקטרוני+ לאור: ההתק2, רכיבי+ החיוניי+ לקרינת ציקלוטרו2. תוצאות
ניתוח פרמטרי (תלות במתחי האצה ועוצמות שדה מגנטי) של המודל התאורטי תואמות לפרמטרי ההפעלה
של הניסוי.
במסגרת מחקר משלי+ של טכנולוגיות פותח תותח אלקטרוני+ המבוסס על קתודה קרמית בעלת מקד+
דיאלקטרי גבוה, הקתודה הפרואלקטרית. בקתודה זו האלקטרוני+ מואצי+ מתו: פלסמה הנוצרת על
המשטח הקרמי כתוצאה מהפעלת פולס מתח של כ ; kV 1 למש: מספר ננו;שניות.
תותח האלקטרוני+ ע+ הקתודה הפרואלקטרית מומש תחילה בהתק2 ללא wiggler. נתוני ההפעלה של
ההתק2 כוונו לקבלה של קרינת CRM קרוב לתדר הקיטעו2 של מהוד גלילי. אינטרקציות בתחו+ פעולה
זה, שנצפו בניסוי סביב תדר של GHz 7, רגישות פחות לפיזור האנרגיה של האלקטרוני+. המש: הפיתוח
.CRM – ו FEL של תותח אלקטרוני+ זה עשוי להוביל לשילובו בעתיד בהתקני
לקתודות קרות כמו זו תרומה רבה להקטנת ממדי+ והפחתת עלות של מקורות קרינה אלקטרומגנטית
המבוססי+ על אלקטרוני+ חופשיי+. ה2 לא דורשות חימו+ או הפעלה מוקדמת, דרישות הואקו+ שלה+
פחותות מאלו של הקתודות התרמיות וה2 זולות יותר מה2. למיטב ידיעתנו, קתודות פרואלקטריות לא
מומשו בהתקני קרינה אלקטרומגנטי+ לפני מימוש+ במסגרת תזה זו.
פיתוח של CRM רחב סרט המבוסס על helix והמופעל במתחי האצה נמוכי+ (kV 10>) בוצע ג+ כ2.
תנודות של גלי+ נעי+ (TWT) שאינ+ תלויי+ בשדה המגנטי הצירי נצפו בהתק2 בעל מבנה helix יחיד.
אחת האינטרקציות שנצפו הינה בי2 האלקטרוני+ ובי2 שני אופני+ מרחביי+ בו;זמנית הנושאי+ אנרגיה
אלקטרומגנטית בשני הכיווני+. לסוג כזה של אינטרקציה, המתאי+ לפעולה כמתנד, יש פוטנציאל להיות
יעיל יותר מאשר אינטרקציה המתרחשת ע+ אופ2 אחד בלבד.
אינטרקצית ציקלוטרו2 מתרחשת בי2 רכיבי המהירות הרוחביי+ של האלקטרוני+ ובי2 הרכיבי+ הרוחביי+
bifilar-helix של השדה החשמלי. על מנת לקבל אינטרקצית ציקלוטרו2, המלצנו לכ2 על שימוש במבנה של
בו השדה החשמלי הרוחבי על הציר חזק יותר מאשר במבנה helix יחיד. בניסוי זה, נעשה שימוש בהתק2
שבו אלומת אלקטרוני+ מיוצרת ונעה בתו: צינור זכוכית שאוב. התק2 זה מאפשר ביצוע ניסויי+ ע+ מבני
גל שוני+ ללא שבירת ואקו+. תוצאות ראשוניות של ניסוי זה הדגימו אכ2 פעולת CRM ג+ כמתנד וג+
כמגבר. מבנה של bifilar-helix יכול לשמש בסכמות עתידיות של FER בו;זמנית כמבנה גל איטי וכ –
.wiggler
FELs ו – CRMs פועלי+ בתחו+ הגלי+ המילימטרי+ ובאורכי גל קצרי+ יותר ע+ אלקטרוני+ המואצי+
על ידי מתחי+ גבוהי+ המאלצי+ סדר גודל ועלות להתקני+ אלו שעלול להיות לא ישי+ (גדול ויקר) להרבה
שימושי+. התזה הזו מציגה עבודת מחקר העשויה להוביל לפיתוח התקני FEL ו – CRM שולחניי+ זולי+
הפועלי+ ע+ מתחי האצה נמוכי+ ולפיתוח שימושי+ תואמי+ בתחו+ גלי הרדיו והמיקרוגל.
תוכ2 הענייני+
1 פרק 1. מבוא…………….…………….....…………..………………………….……………
חלק א
6 …………………………...……(FER) לייזר אלקטרוני) חופשיי) בתדר גלי רדיו
8 …………………………………....(FER) (פרק 2. לייזר אלקטרוני) חופשיי) באורכי גל גדולי
8 ....………………………………………………………………...… FER – 2.1. מבנה מתנד ה
12 2.2. מער: הניסוי.............……………………………………………………………………...
15 2.3. הצגת תוצאות הניסוי.……………………………………………………………………...
20 ………………………………………………………………… FER – 2.4. ניתוח של פעולת ה
23 2.5. התעוררות תנודות של קרינת ציקלוטרו2 . ...…………………………………………………
28 פרק FER .3 בעל תדר מתכוונ7.…………………………………………………………………
28 FER .3.1 מועמס מי+.........................………………………………………………….………
31 3.2. הצגת תוצאות הניסוי..…………………………...…………………………………………
חלק ב
38 מחקר משלי)....................................…………....……………………………
40 פרק 4. מקור קרינה אלקטרומגנטית של אלקטרוני) חופשיי) ע) קתודה פרואלקטרית...……...….
40 4.1. תותח אלקטרוני+ המבוסס על קתודה פרואלקטרית………………………………………….
42 CRM .4.2 ע+ קתודה פרואלקטרית…………..…………………………………………………
47 ………………………………………..…………..... helix פרק 5. התקני) המבוססי) על מבני
48 5.1. רקע תיאורטי וטכנולוגי..….…………………..……………………………………………
50 5.2. ניסוי ע+ מבנה helix יחיד ………………...…………………….…….………………….…
60 ...……………………………………………………...bifilar-helix ע+ מבנה CRM 5.3. ניסויי
66 פרק 6. סכו) .…………………………………………………………………………………
71 ..…………………………………………… FER – נספח א. משוואות התנועה של האלקטרו7 ב
78 …………………………………………....………… FER – נספח ב. מודל ליניארי של הגבר ב
87 רשימת מקורות ..………………………………………………………………………………