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

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89

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90

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91

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maser experiments,” Nucl. Instrum. and Methods in Phys. Res., Vol. A358, pp. 327 -

330, 1995.

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patent # 5,998,773.

[33] M. Einat and E. Jerby, “Anomalous and normal Doppler effect in a stripline-loaded

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92

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93

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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.

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Boston, 1992; and references therein.

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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 ..………………………………………………………………………………