SLAC-R-95-455 UC-414

RADIO-FREQUENCY PULSE COMPRESSION FOR LINEAR ACCELERATORS*

Christopher Dennis Nantista

Stanford Linear Accelerator Center

Stanford University, Stanford, CA 94309

January 1995

Prepared for the Department of Energy

under contract number DE-AC03-76SF00515

Printed in the United States of America. Available from the National Technical

Inf&mation Service, U.S. Department of Commerce, 5285 Port Royal Road, Springfield, Virginia 22161.

*Ph.D. thesis, Uniyersity of C.alifornia, Los Angeles.

DEDICATION

To my father, Vincent, for prodding me to finish,

to my mother, Maureen, for not,

and to my wife, Mikayo, for giving me a reason.

. . . 111

TABLE OF CONTENTS

. . . Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..I~I

1 Table of Contents .......................................................... iv

Acknowledgements ...................................................... ..vi i

Vita ....................................................................... ..i x

Publications and Presentations ........................................... .x

Abstract of the Dissertation ............................................. .xii

1. Introduction ............................................................ ..l

2. SLED ...................................................................... .

SLED Theory ............................................................ .7

SLED in Use ............................................................ 12

. -

3. Binary Pulse Compression ............................................ .17

BPC Theory ........................................................... ..18

Single-Source Operation ................................................. 23

System Development ................................................... ..2 5

The SLAC 3-Stage BPC ................................................. 29

4. SLED-II ................................................................ .41

SLED-II Theory ........................................................ .43

GainandEfficiency .................................................... ..4 8

Multiple Staging .......................................... ..~............4 9

Comparison With SLED ................................................ .52 s -

Tuning .................................................................. 56

iv

.

5. The 3-dB Directional Coupler ........................................ .63

Function ............................................................... ..6 3

Design Problem ........................................................ ..6 6

Semi-Analytic Approach ............................................... ..7 0

Development ........................................................... ...7 6

,Testing and SUPERFISH Correction .................................... .81

6. Circular Waveguide Offsets ............................................ .88

TEsi-TMr-i Bend Mixing ............................................... .89

Generalized Telegraphist’s Equations .................................... .94

Application of the Theory .............................................. 101

7. Other Components .................................................... 110

The “Flower-Petal” Mode Converter .................................... 110

Delay and Transfer Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Circular 90” Bends ................................................... ..12 0

Vacuum Pumpouts/Mode Filters ....................................... .122

Irises, Tapers, and Shorts ............................................... 126

8. SLED-II Experimental Results ...................................... .131

Low-Power Experiment ................................................. 131

High-Power Experiment ................................................ .140

Accelerator StructureTest .............................................. 146

9. Variations On a Theme ................. : ............................. .155

Amplitude-Modulated SLED ........................................... .155

Phase-Modulated SLED ................................................ 158 s -

SLED-II With Disc-Loaded Delay Lines ................................ .162

V

Ramped SLED-II for Beamloading Compensation ....................... 167

Others ................................................................ ..17 1

10. Conclusion .......................................................... ..17 4

References ............................................................... ..17 8

Appendix: An Equivalent Circuit Model .......................................

for Traveling-Wave Structures .................................. 183

vi

ACKNOWLEDGEMENTS

The physical limitations of time and space prevent me from recounting in detail

how each of the following people or groups have been of help to me in the prepa-

ration of this dissertation, the performance of the research which went into it, and

the general pursuit of a doctoral degree. I hope all will nevertheless understand my

sincere gratitude. I am indebted to my advisor, Professor David B. Cline, and to

Professor Ronald D. Ruth of the Stanford Linear Accelerator Center for the oppor-

tunity they’ve provided me. I’ve greatly benefited from the guidance of Professor

Perry B. Wilson and Professor Norman M. Kroll. The advice and assistance of my

SLAC supervisor, Dr. Theodore L. Lavine, are much appreciated. I would also like

to acknowledge and thank:

Z. David Farkas, Albert Menegat, Prof. Roger H. Miller, Dr. Arnold E. Vlieks,

Dr. Sami G. Tantawi, Dr. Juwen Wang, Dr. Thomas Knight, Harold A. Hoag,

Henry Deruyter, Dr. Kwok Ko, Dr. Ron Koontz, John Eichner, Chris Pearson

and the SLAC Machine Shop, Chuck Yoneda and the SLAC Vacuum Group, Rich

Callin, Karen Fant, Dr. Roger Jones, Terry Lee, Craig Galloway, Rod Loewen,

Dr. George Spalek, the SLAC Accelerator Theory & Special Projects Dept., the

SLAC Klystron & Microwave Dept., the SLAC Publications Dept., Dr. Eric Nel-

son, Xin-Tian (Eddie) Lin, Joseph Scott Berg, S. Rajagopalan, Penny Lucky and

_the UCLA Physics Dept., Jim Kolonko and the UCLA Center for Advanced Ac-

celerators, Prof. James B. Rosenzweig, Prof. Chan Joshi, Prof. William Slater,

vii .-

I ‘T. -. .

Prof. Harold Fetterman, Dr. Melvin Month and the US Particle Accelerator School

staff, the Advanced Accelerator Group at Argonne National Laboratory, the Accel-

erator Physics community, and Dr. David Sutter and the Department of Energy,

who made this research possible.

. . . VI11

February 21, 1964

June 1986

September 1986-December 1987

March 1988

January 1988-October 1994

June-September 1988

September 1989-October 1994

January 1992

March-September 1994

VITA

Born

New York, New York

B.A., Physics

Cornell University Ithaca, New York

Teaching Assistant

University of California, Los Angeles Los Angeles, California

M.S., Physics University of California, Los Angeles

Los Angeles, California

Graduate Student Researcher

University of California, Los Angeles

Los Angeles, California

Graduate Student Researcher Argonne National Laboratory

Argonne, Illinois

Graduate Student Researcher

Stanford Linear Accelerator Center

Stanford, California

Teaching Assistant for USPAS

RF Systems for Electron Linacs

and Storage Rings

University of Texas, Austin Austin, Texas

Engineering Physicist

Stanford Linear Accelerator Center

St anford, California

ix

. ^

PUBLICATIONS AND PRESENTATIONS

Z.D. Farkas et al., “Radio Frequency Pulse Compression Experiments at SLAC,”

presented at the SPIE Symposium on High Power Lasers, Los Angeles, CA, January

20-25, 1991; SLAC-PUB-5409.

N.M. Kroll et al., “A High-Power SLED II Pulse Compression System,” presented at

the 3rd European Particle Accelerator Conference, Berlin, Germany, March 24-28,

1992; SLAC-PUB-5782.

T.L. Lavine et al., “Binary RF Pulse Compression Experiment at SLAC,” presented

at the European Particle Accelerator Conference, Nice, France, June 12-15, 1990; SLAC-PUB-5277.

T.L. Lavine et cd., “High-Power Radio-Fequency Binary Pulse-Compression Ex-

periment at SLAC,” presented at the IEEE Particle Accelerator Conference, San Francisco, CA, May 6-9, 1991; SLAC-PUB-5451.

T.L. Lavine, C.D. Nantista, and Z.D. Farkas, “SLED-II Delay-Line Length Stabi-

lization,” Next Linear Collider Test Accelerator (NLCTA) Note #16, February 3,

1994.

C. Nantista, “An Equivalent Circuit Model of the 30-Cavity Structure,” SLAC

Advanced Accelerator Studies (AAS) Note 75, September 1992.

C. Nantista et al., “High-Power RF Pulse Compression With SLED-II at SLAC,”

presented at the IEEE Particle Accelerator Conference, Washington, D.C., May

17-20, 1993; SLAC-PUB-6145.

C. Nantista, N.M. Kroll, and E.M. Nelson, “Design of a 90” Overmoded Waveguide Bend,” presented at the IEEE Particle Accelerator Conference, Washington, D.C.,

May 17-20, 1993; SLAC-PUB-6141.

C.D. Nantista and T.L. Lavine, “Mechanical Tolerances on Circular Waveguide

and Flanges for the SLED-II Delay Lines,” Next Linear Collider Test Accelerator (NLCTA) Note #17, February 3, 1994. e -

C.D. Nantista, and T.L. Lavine, “Analysis of Mechanical Tolerances on Circular

X

Waveguide and Flanges for the Low-Loss High-Power Transmission Lines,” Next

Linear Collider Test Accelerator (NLCTA) Note #21, April 1, 1994.

C.D. Nantista, “SLED-II Adjustable Short Power Dissipation,” Next Linear Collider

Test Accelerator (NLCTA) Note #22, April 6, 1994.

C.D. Nantista, “Ramping SLED-II for Beam-Loading Compensation,” Next Linear

Collider Test Accelerator (NLCTA) Note #27, August 19, 1994.

. J. Norem et al., “The Development of Plasma Lenses for Linear Colliders,” presented

at the IEEE Particle Accelerator Conference, Chicago, Illinois, March 20-23, 1989.

J.M. Paterson et al., “The Next Linear Collider Test Accelerator,” presented at the

15th International Conference on High Energy Accelerators, Hamburg, Germany, July 20-24, 1992; SLAC-PUB-5928.

R.D. Ruth et al., “A Test Accelerator for the Next Linear Collider,” presented at

the ECFA Workshop on e + - Linear Colliders, Garmisch-Partinkirchen, Germany, e

July 25-August 2, 1992; SLAC-PUB-6293.

R.D. Ruth et al., “The Next Linear Collider Test Accelerator,” presented at IEEE Particle Accelerator Conference, Washington, D.C., May 17-20, 1993; SLAC-PUB-

6252.

- A.E. Vlieks et al., “Accelerator and RF System Development for NLC,” presented . _ at the IEEE Particle Accelerator Conference, Washington, D.C., May 17-20, 1993;

SLAC-PUB-6148.

J.W. Wang et al., “High Gradient Tests of SLAC Linear Collider Accelerator Struc-

tures,” presented at the LINAC 94 Conference, Tsukuba, Japan, August 21-26,

1994; SLAC-PUB-6617.

P.B. Wilson et al., “Progress at SLAC on High-Power RF Pulse Compression,” pre- sented at the 15th International Conference on High Energy Accelerators, Hamburg,

Germany, July 20-24, 1992; SLAC-PUB-5866.

_

xi

ABSTRACT OF THE DISSERTATION

Radio-Frequency Pulse Compression

for Linear Accelerators

bY

Christopher Dennis Nantista

Doctor of Philosophy in Physics .

University of California, Los Angeles, 1994

Professor David B. Cline, Chair

Recent efforts to develop plans for an electron-positron linear collider with

center-of-mass energy approaching a TeV have highlighted the need for sources

capable of delivering hundreds of megawatts of peak rf drive power at X-band

frequencies. This need has driven work in the area of rf pulse compression, which

enhances the peak power available from pulsed rf tubes by compressing their output

pulses in time, accumulating the available energy into shorter pulses.

The classic means of rf pulse compression for linear accelerators is SLED. This

technique is described, and the problem it presents for multibunch acceleration

explained. Other pulse compression schemes, capable of producing suitable output e -

pulses are explored, both theoretically and experimentally, in particular Binary

xii

Pulse Compression and SLED-II. Th e merits of each are~considered with regard to

gain, efficiency, complexity, size and cost.

The development of some novel system components, along with the theory

behind their design, is also discussed. The need to minimize copper losses in long

waveguide runs led to the use of the circular T&r propagation mode in over-moded

guide, requiring much attention to mechanisms of coupling power between modes.

The construction and commissioning of complete, high-power pulse compression

systems is reported on, as well as their use in the testing of X-band accelerating

structures, which, along with the X-band klystrons used, were developed at SLAC

in parallel with the pulse compression work.

The focus of the dissertation is on SLED-II, the favored scheme in some cur-

rent linear accelerator designs. In addition to our experimental results, practical

implementation considerations and design improvements are presented. The work

to date has led to detailed plans for SLED-II systems to be used in the Next Linear

Collider Test Accelerator, now under construction at SLAC. The prototype of the

upgraded system is near completion. .

Descriptions of various rf pulse-compression techniques besides the aforemen-

tioned three, including those pursued at institutions other than SLAC, are included

to give a broad taste for the field and a sense of future possibilities.

. . . x111

1. INTRODUCTION

This dissertation treats the subject of the temporal compression of guided

pulses of radio-frequency power, which, by conservation of energy, makes it possible

to obtain peak power levels higher than are available directly from microwave tubes

such as klystrons. It does not deal directly with the production of rf power by

klystron tubes or by other means. Details of and recent advances in that science

are well documented elsewhere. Nor does it deal substantially with the conversion

of rf power to particle beam energy in accelerator structures, although familiarity

with accelerator concepts is assumed. The focus is on an area of accelerator physics

. _ inbetween the klystron and the structure, the manipulation of rf energy. While this

subject has not traditionally commanded the greatest attention, its importance for

the future of the accelerator field, particularly on the high-energy frontier, is gaining

wide recognition. Before launching into the particulars of rf pulse compression

techniques, I present the following motivation.

Colliding beam machines have led to great advances in elementary particle

physics in the past quarter century. In order to complete the confirmation of the

Standard Model, the nature of exact SU(2) s y mmetry breaking must be determined.

It is believed the source will be found at center-of-mass energies well above the rest

energies of the W and 2 bosons. Lepton colliders offer “cleaner” opportunities than

hadron machines for probing this domain. As synchrotron radiation loss limits the

practical energy of a circular electron-positron collider to about that of LEP II

(200 GeV), th ere has been great interest in recent years in building a linear collider,

similar to the SLC but with separate electron and positron linacs, with a center-

of-mass energy of 500GeV to 1TeV (5 to 10 times that of the SLC). At SLAC,

a design has been developed for such a machine, referred to as the Next Linear

Collider (NLC). Th e experimental program of the NLC is to include [l]:

- studies of top quark and its interactions

- detailed examinations of the interactions of gauge bosons

- searches for neutral scalars (Higgs) or other new particle states

- search for new phenomena

To keep the linacs of such a collider to a reasonable length, much of the in-

ternational accelerator physics community has decided to aim for an accelerating _

. - gradient of 50-100 MeV/ m, a factor of 3-6 above that of the SLC. The amount of

rf power required to reach these gradients in accelerator structures is reduced by

the choice of an X-band operating frequency of 11.424GHz, four times the SLC

rf frequency. The design calls for peak power in the range 60-240 MW per meter

of accelerator. The lack of established high-power X-band technology has resulted

in much R&D toward a suitable power source. The following list is a sampling of

concepts pursued.

some contenders: e -

- relativistic klystron [2]

2

- gyroklystron [3]

- choppertron [4]

- cluster klystron [5]

- two-beam accelerator [6], [ 71

Some of these sources have achieved modest success and/or are still being

developed. However, foreseeing that the more exotic concepts were not likely to

produce practical, reliable devices on a reasonable timescale, SLAC has taken a

conservative approach, which is to:

1. Develop conventional klystron tubes at this X-band frequency capable of de-

livering 50-100 MW pulses of several times the structure filling time.

2. Use an rf pulse-compression scheme to exchange pulse length for higher peak

power.

An intensive program of X-band klystron development is ongoing at SLAC.

Although klystrons have been in use for decades, the extension of the technology

* _ into the frequency and power ranges desired has presented several problems to be

addressed. These include gun arcing, output cavity breakdown, and rf window

failure. An initial series of experimental klystrons of 100MW design have been

produced [8]. V arious output circuit designs were tried and, although the goal

power and pulse width were not reached, some tubes achieved moderate success and

much was learned. Current efforts are towards producing reliable 50 MW klystrons

for the NLC Test Accelerator under construction. A prototype tube has met the

specifications of 50 MW at a 1.5 ns pulse width. The 100 MW program will later be

resumed. X-band klystron development has also been pursued by other laboratories e -

and companies, such as KEK and Toshiba in Japan, The Institute for Nuclear

I : .

Physics in Russia, and Haimson Research Corporation in Palo Alto, California.

The difficulties encountered by the klystron developers underscores the need

to incorporate a pulse-compression scheme if the required peak power per feed is

to be realized in an accelerator. Pulse compression is not new to linear accelerator

operation, as will be seen in the next chapter. However, as will also be seen, the

standard technique, SLED, is not useful for current purposes. The next generation

linear collider will have toQroduce a luminosity of 3 x 1O33 cm-2s-1 to achieve its

physics goals. To relieve space charge and background problems, a long train of

bunches will be accelerated on each pulse. Final focus tolerances require that these

bunches be very uniform in energy. The exponential power spike produced by SLED

pulse compression is unsuitable. Uniform acceleration of long bunch trains requires

a pulse of constant amplitude.

.

In what follows, I examine the theory and implementation of rf pulse com-

pression: Alternatives to SLED are presented, the most attention being given to

SLED-II, the scheme included in SLAC’s current high-gradient linac design. A good

part of the dissertation also deals with the design of rf components necessary for

the development of a practical pulse-compression system. Results from experiments

with prototype systems and their use in accelerator tests are presented. Essential

issues are explored and future plans given.

One method of pulse compression not considered in the coming chapters should

be mentioned here, due to its familiarity. Chirping is a technique widely used in the

fields of radar and lasers. For example, it allows a long pulse to be amplified before

being compressed to the desired width when the energy density at that shorter pulse e -

width is too great for the amplifier to handle. With chirping, a continuous variation

in frequency (or equivalently phase) is given to the pulse at generation. When

this pulse is passed through a dispersive medium, the resulting variation in group

velocity causes the tail of the pulse to gain on the head, resulting in a shortening

of the pulse length. While this technique could be applied to rf pulses, it was

ruled out for accelerator use for a couple of reasons. First, there was the difficulty

and expense foreseen in developing and producing highly dispersive transmission

lines for multimegawatt pulses which would not dissipate too much of the power.

Second, there was the problem of achieving amplitude and phase stability across

the compressed pulse as well as a precise design frequency.

The focus has instead been on non-dispersive methods of compressing constant

frequency, tailored rf pulses with passive microwave circuits. Unlike the exotic

power generation devices mentioned earlier, this conservative focus avoids the com-

plications of beam-rf interaction.

5

2. SLED

The conceptual predecessor of the pulse compression method currently favored

for next generation linear collider designs at SLAC and in Japan is known as SLED

[9]. Invented in 1973 at the Stanford Linear Accelerator Center and developed in the

years following, SLED is the brainchild of Perry Wilson and David Farkas. They

received a 1991 IEEE Particle Accelerator Technology prize for their invention.

Originally an acronym for SLAC Energy Doubler, it later came to stand for SLAC

Energy Development. Although the author was not involved in work on SLED, its

bearing on this thesis warrants the inclusion of a brief exposition of its theory and

application.

SLED is a pulse compression scheme in which rf energy builds up in high Q

resonators during most of a klystron pulse’s duration and is then largely extracted

during the last fraction of the pulse. The idea emerged from the observation in

testing super-conducting cavities that immediately after the input power is cut, the

power emitted from a heavily over-coupled cavity approaches four times that input

power. Normally, this power would travel back toward the source. In the SLED

network, a pair of resonant cavities are excited symmetrically through a four-port

3-dj3 directional coupler. This causes the power reflected from the resonators to

be directed away from the source, through the fourth port, so that it can feed an

-

6

accelerator. This use of a 3-dB coupler is explained in Chapter 5. Furthermore, it

was realized that reversing the phase of the input, rather than switching it off, gave

an even greater power multiplication, with a theoretical limit of nine. It should be

noted, however, that the increased peak power afForded by SLED is accompanied

by a sharp exponential decay, so that the average power within a compressed pulse

is generally much lower than the maximum.

SLED THEORY

Figure 2.1 shows a diagram of a SLED circuit. It works as follows: An inci-

dent rf wave from the klystron, of amplitude Ei,, reflects off the waveguide-cavity

interface, giving an initial output wave of amplitude* -E;,. As the cavity fills

exponentially, it emits through the coupling iris a wave E, opposite in phase to the

interface reflection. The output wave is the superposition of these two backward

waves.

. _ E out = Ee - Eirz-

Since the cavity is strongly overcoupled, the emitted wave amplitude will surpass

the incident amplitude, causing the total output wave to pass through zero. After a

positive amplitude has built up, the phase of the incident wave is suddenly shifted

by 7r radians. This immediately changes the sign of the interface reflection, so

that it adds to, rather than subtracting from, the emitted wave. As the emitted

wave cannot change instantaneously, due to the finite filling time of the cavity, the

* With their common sinusoidal time dependance suppressed, it is convenient to

_speak of waves 180” out of phase with each other in terms of positive and “negative”

amplitudes.

7

-, -. .

To Accelerator

TE 015

In

From Klystron

Figure 2.1 SLED pulse compression system as implemented on the

Stanford Linear Accelerator.

8

- _. -. .

output wave experiences a sudden amplitude increase of 2 Ei,. The emitted wave

then drops steeply as the cavity attempts to charge up to the opposite voltage,

yielding the characteristic spiked output. The incident wave is turned off after

the compressed pulse has reached its desired duration, usually the filling time of

a traveling-wave accelerator section. By this time, the energy stored in the cavity

should be largely depleted for good efficiency. The output amplitude then drops by

E;, and decays toward zero. These waveforms are illustrated in Figure 2.2(a).

A more analytic description of the SLED mechanism is obtained by appealing

to conservation of power. We begin with

. .

Pin = due

pout + PC + -p (2.1)

where P;, and Pout are self-explanatory, PC is the power dissipated in the cavity

walls, and UC is the electromagnetic energy stored in the cavity. Let k be the pro-

portionality constant relating the square of the field to power flow in the waveguide.

Then

. _ Pi, = kEfn7

and

P out = k(Ee + rEin)2,

where I? is the reflection coefficient of the waveguide-cavity interface. Now, from the

definition of the cavity coupling coefficient, p,

power (with no incident wave), we can write

T1 kE,2

as the ratio of emitted to dissipated

J-C

“,a’

Finally, from the definition of unloaded Q, we have

u Qo, &ox c=- c=- w w P’

9

-

.

I I I I I I I I I I I I I I I I I I-

2- ( > a E

E* 2 1 - r Y- - - _y.L-. ‘\,

‘.ie \

E I ./’ \ /

0 / I I I

I I

-l- L - J rl 1 I I I I I I I I I I I I I I Ill

0 1 2 t1 3 t2 4

I I I I I I I I I I I I I I I I l-l

5

(b) P out

- I I I ! ’

0 1 2 t, 3 t, 4

t b-4

Figure 2.2 SLED field and power waveforms. A sign change in the field s - plots is indicative of a phase reversal. Eout is the difference of E, and Ei,.

G, is the effective gain in the compressed pulse.

10

:. -. .. - ;

with w the resonant frequency, assumed equal to the drive frequency, so that

due 2kQo E dEe -= -- dt w@ e dt *

At time t = 0, which we’ll define as the instant the input pulse reaches the

cavity, UC is zero, as the cavity cannot fill instantaneously. Therefore E,, is zero.

Substituting the above expressions into equation (2.1), then, tells us that /I’[ = 1.

To account for the 7r phase shift associated with a reflection, we set I’ = -l.*

Equation (2.1) can now be recast as

J% = (E, - Ei,)2 + ;Ef + $Ee$f$,

From this we have the following first order, non-homogeneous differential equation

for the emitted wave.

dEe - LE, = -E;,.

dt + 2QL

UP

Qo

QL here is the loaded Q, given by QL = Qs/( 1 + ,B). Defining T, = 2&L/w and

o = 2p/(I + p), it is useful to rewrite this as

T dEe cx + Ee = aEin- (2.2)

It is seen that T, is the loaded cavity time constant and that a! gives the steady-state

emitted field.

As indicated before, the input to SLED is a constant amplitude pulse with the

phase reversed towards the end. For simplicity let’s use unit field and say

1, o<t<t1

Ei, = -1, t1 <t<t2

0, t2 < t

* This is true to the extent that the cavity can be treated as a lumped resonant

qirc-uit. In reality, it’s a very good approximation for small coupling. I’ll return to

this point in the section on SLED-II.

11

I

‘T. -

With this piecewise-constant driving term and the condition E, = 0 at t = 0, the

solution to equation (2.2) is

E,(O < t < tl) = a(1 - CtlTe),

E,(tl 5 t < t2) = a [ (2 - e-t’+-(t-tl)lT, - l] ,

E,(t > t2)= a (24tl/~~)~-(t2-t,)/T, _ 1

[ 1 e-(t-t2)/Tc.

We may now express the normalized output field, E, - Ein, as

Eout(O 5 t < tl) = cy(1 - CtiT,) - 1,

Eout(tl 5 t < t2) = a [ (2 - e-tl/T”)e-(f-tl)/T, - l] + 1,

EOut(t > t2) = ol[(2-e-'l/~=)~-(tZ-'l)/Tc _ +-('-'2)/T'.

The output field initially builds up toward a - 1. After the phase reversal and

accompanying amplitude jump, it begins falling exponentially toward -(o - 1)

with time constant T,. These equations for the fields describe the time plots of

Figure 2.2, which also shows the input and output power waveforms.

From equation (2.4), we see that the maximum field intensity with a SLED

. _ system is magnified by a factor of

E mad = E,&) = a (1 - .-“lir,) + 1. (2.5) -

SLED IN USE

The energy gained by a particle beam in traversing an accelerator structure -

powered with a SLED pulse is not boosted by as much as the peak field. This is

due to the pulse’s exponential decrease in amplitude and the finite filling time of

such structures. The pulse profile is reflected in the accelerating gradient along the

structure. We must take this into account in evaluating the benefit of SLED.

12

One can define an efficiency of energy compression for SLED as the ratio of the

energy in the compressed pulse to the total input energy. That is

1 t2

rl ec = - J t2 t1

Eout(f)2 dt.

For accelerator rf, however, we are more interested in how SLED affects the energy

gain of a charged particle beam. For a constant-impedance accelerator structure,

this is measured by the average across the compressed pulse of the field magnification

weighted by the attenuation up to the appropriate position in the structure. The

voltage across the structure filled with the compressed pulse is

Vci = J

L

E(z, t) dz = ~~e-=+~

0 J

t2

Eout(t)PQt dt,

11

where u9 is the group velocity and cy the attenuation per length in the structure.

The relevant efficiency is the ratio of the compressed to input pulse lengths times

the square of the ratio of this gain to the unSLEDded gain.

t2 -t1 qci = -

Jl12 Eout(t)eQVgt dt 2

t2 St:’ * eavg t dt 1

This efficiency is lower than qec. It is also lower than what one would get with a

flat pulse of the average field amplitude, since the weighting factor favors the low

end of the pulse.

Many accelerators, including the SLAC linac for which SLED was developed,

use constant-gradient structures. Therefore, the following analysis and the efficiency

to be defined are generally referred to rather than those given above. In a constant

gradient structure, the group velocity is linearly increasing from the input end to the

output end to maintain field strength (f or a flat input pulse) as power is dissipated

in the walls. Let us define g such that e -

13

where L is the length of the accelerator section. Letting the waveform described

above enter the section at t = 0, we get the accelerating voltage for a particle

entering at time ti (neglecting beam loading and group velocity during transit) by

integrating the field.

L L

v= J E(z, t;) dz = J

E (ti - At(z)) dz,

0 0

where

&(z) = J

= dr’ = 0 dz’)

*ln(l - :z). PJgo

When any of the three field discontinuities is in the structure, the voltage integral

is discontinuous at

Zn = !j [l - (1 - g)(t-wy

where t, is 0, tl, or t2.

The above integration yields an accelerating voltage like that shown in Figure

2.3. For a particle entering between tl and t2, the voltage is given by

V(f, < ti < t2) I ; { (a - 1) [g - 2 + 2(1 - g)(t’-tl)‘Tf] - ae-ti /Tc

C [I - (1 - s)c]

+ 2ae-(ti-tl)lTc

C

[l _ (1 _ g)C(ti-tl)/5

I>

.

Here Tf is the structure filling time, assumed equal to t2 - tr,and

L C=l-----= Tf

svgo Tc ’ ’ T, ln(1 - g)

Setting L = 1 normalizes the voltage to that of a structure filled with an unSLEDded

unit-amplitude pulse. The maximum, at ti = t2 when the compressed pulse fills the

structure, is then

V Q evTflTC(2 - estllTc) [l - (1 - g)c] - (y + 1. max = -

SC P-6)

14

~II~‘I”“l”“I”“I’~~ 0 1 2 t1 3 t2 4

ti (PSI

Figure 2.3 Acceleration versus injection time, normalized to the accel-

eration obtained without SLED.

The theoretical limit of V,,, is seen to be three. In reality, however, this limit

cannot be approached with finite-Q SLED cavities. This is because of the competing

requirements of large ,B and large Z!‘,. There are also practical limitations on tr.

The effect of SLED is equivalent to boosting the klystron power by a factor of

(accelerating voltage with SLED)2

Gp = (accelerating voltage without SLED)2 = ‘,“M’ (2.7)

as indicated in Figure 2.2(b). If we define a compression ratio, C,, as the ratio of

the klystron pulse duration to the filling time, the compression efficiency is given

bY

GP qc=c.

l- (2.8)

As with the constant-impedance structure, this efficiency is somewhat degraded e -

from the efficiency of energy compression by the pulse shape.

15

I :

A SLED microwave network, attached to each klystron, has been in use on the

SLAC linac since 1979, boosting the linac energy by about 60%. It utilizes drum-

shaped resonant cavities oscillating at 2.856 GHz in the T&is mode, with a Qs of

105. Each is coupled to a 3-dB hybrid in rectangular S-band waveguide through a

circular aperature such that /3 = 5. This gives a QL of 16,667, and a time constant

. Z’, of 1.86 ps. A klystron pulse of duration tz = 3.5 ps is used to power a structure

of filling time 0.82~s so that t r = 2.68~s and C, = 4.27. Finally, the slope of the

structure group velocity is described by g = 0.681. These are the values used in

obtaining the plots of Figures 2.2 and 2.3. Plugging them into equation (2.6) yields

an energy gain factor of 1.613. This is equivalent to increasing the klystron power

by a factor of G, = 2.6 with an efficiency of vc = 0.61 (qec N 0.72). Implementation

of SLED was a major step in achieving the goal of 50 GeV per beam in the Stanford

Linear Collider.

A similar system known as LIPS (LEP Injector Power Saver) serves the injector

linac of the Large Electron-Positron collider at CERN in Geneva, Switzerland. The

SLED concept has also found application at other laboratories around the world,

including DESY in Hamburg, Germany, IHEP in Beijing, China, INP in Novosi-

birsk, Russia, and KEK, in Tsukuba, Japan, where a recent model uses double,

side-wall irises, rather than a single, end-wall iris, to allow greater peak power flow

.

(380 MW) without rf breakdown [lo].

16

3. BINARY PULSE COMPRESSION

In operation of the SLC, three bunches (two of electrons and one of positrons)

are accelerated along the linac per rf fill. The slope just before the peak of Figure 2.3

is used to counteract the effect of beam loading, and sufficiently uniform acceleration

is achieved. A next-generation linear collider will require long trains of bunches per

rf fill to achieve sufficient luminosity while keeping space charge effects down. This

fact, along with the very tight energy stability requirements of the final focus system,

make standard SLED unusable. A flat rf pulse is needed.

A second method of rf pulse compression, referred to as Binary Pulse Com-

. _ pression, was proposed and developed at SLAC to meet this need for a flat or

constant-amplitude pulse. The name derives from the fact that the peak power

is, for a perfect system, doubled in successive stages, while the pulse width (dura-

tion) is halved. Invented by Z.D. Farkas [ll] in 1986, the Binary Pulse Compressor

(BPC)* offers a flat output pulse with theoretical ideal gain of 2N, with N the

number of stages employed, and an intrinsic efficiency of 100%. Achievable gains

and efficiencies are somewhat lower due to component losses. Recall, however, that

with SLED even a perfect system has its efficiency degraded by power being emit-

e *- Also referred to in the literature as Binary Power Multiplier (BPM) and Binary

Energy Compressor (BEC).

17

1. -. .

ted before and after the compressed pulse, dropping off away from its maximum

at a compression ratio of three. When used to power an accelerator section, the

effective SLED efficiency is further reduced by the shape of the pulse. Binary Pulse

Compression does not have these short-comings. The flat output pulse means the

compression efficiency relevant for acceleration is equal to the energy compression

efficiency. It is, however, limited to power-of-two compression ratios. Like SLED,

BPC utilizes passive microwave circuit elements and phase coding of the low-power

input rf. Unlike other methods, it (generally) requires two sources, and produces

two outputs, which may be combined if desired.

BPC THEORY

BPC operation is straightforward. Phase patterns are encoded in low-power

drive signals. Klystrons, or other pulse amplifiers, then convert these signals to

high power rf. The switching times must remain short compared to the compressed

pulse width. Each BPC stage successively combines and splits the phase-coded rf

input pulses into two output pulses of half the duration and approximately twice

the power (minus component losses). This is achieved using 3-dB couplers and low-

loss delay lines as follows: Rf power is fed simultaneously into two isolated ports

of a 3-dB coupler. The reference planes (phases) are chosen such that same-phase

inputs are combined in one of the two remaining ports and opposite-phase inputs

are combined in the other. In either case, the signals cancel at the remaining output

port. Halfway along the input pulses, a 180” phase shift in one input relative to the

other switches the power from one combiner port to the other. The combined leading

half travels along a folded delay line equal to its length, allowing the compressed

18

pulses to arrive simultaneously at the next stage or in the loads (accelerator feeds).

Each delay line is therefore half as long as that of the previous stage. Two stages

of this process are illustrated in Figure 3.1. Pluses and minuses indicate phases of

0 and x, respectively, relative to a common reference.

- Hl - ’ H2 t

Figure 3.1 2-stage illustration of the Binary Pulse Compression process.

. _

Proper BPC phase coding for the two input pulses is determined by the number

of pulse compression stages to be applied. The pulses are divided into time bins

whose duration is determined by the delay line length of the final stage, which in

turn is determined by the structure fill time or desired compressed pulse width. Each

time bin is then assigned a phase of 0 or 7r, according to the following algorithm:

Let the leading edges of the input pulses be to the right (i.e. let time increase to the

left), and let the code for input two be written below the code for input one using

the plus/minus notation. Also assume that the output, whichever port it’s directed

to, has the phase of the top input. Start by writing one plus bin for each input.

Bring the lower code up behind the upper code, thereby doubling the number of

bins in the latter. Below this, write the new lower input code as identical to the

_ -

19

:. -. . - ;

upper code in its leading half and opposite to the upper code in its trailing half.

Repeat the above two steps, carrying them out N times to obtain the coding for N

stages of BPC. For example, below are the resulting 2N-bin codes for one, two, and

three stages. The output of the N ” stage of a system is the input for N - 1 stages.

1 stage

-1

2 stages

11

3 stages

input one: + - + + - + + +

input two: - + - - - + + +

Between couplers, one pulse travels a much longer distance through a delay

line than does the other pulse. While the delay lines should be designed to give as

little attenuation as is feasible, they will no doubt result in some power difference

between the pulses as they enter the next stage. Let us consider how this affects e -

the power-directing function of the directional coupler. If the two input powers are

20 .-

PA and PB and the two output powers are PC and PO, a 3-dB coupler should give

(neglecting insertion loss)

Pc=[~~+~~]2=pA~PB+~~,

PD= [&&~~]2=pA;pB-&&

(3-l)

or vice versa, depending on the phase coding. For equal input powers, this results

in all the power being combined in one or the other output port. For unequal input

powers, the power distribution given by the above equations is plotted versus PB/PA

in Figure 3.2. It is evident that the power-directing function of the 3-dB coupler is

not very sensitive to input level mismatches. We can have fifty percent more power

into one port than the other and still get 99% of the total power properly directed.

Of course, if one of the’ two sources dies, then the output power will be cut in half.

In equation (3.1), it is assumed the two paths between couplers differ by exactly

an integral number of guide wavelengths, so that the relative phase is preserved. If

they do not, the second terms on the right must be multiplied by the cosine of the

additional phase difference, 64. For PA = PB = P, we get * _

PC = P(l + cos@) = 2Pcos 2 0

-,

4 P-2)

PD = P(1 - COs64) = 2Psin2 2.

Phase error is therefore a greater concern than amplitude imbalance in BPC oper-

ation. The delay lines must give not only the proper time delay (pulse width), but

also a proper phase delay, and a means of tuning them over a range of a wavelength

must be included in any BPC system.

With power direction thus controlled, the chief causes of inefficiency are ohmic

losses in individual components, reflections, and mode conversion, where possible.

Denote the average power combining efficiency of a coupler, or hybrid, and short

21

.

WI i3 1.0

5

2 0.8 3 B El p: 0.6

8

s 04 k * 2

E z 0.2

4 E

E 0.0 0 0.2 0.4 0.6 0.6 1

RATIO OF INPUT POWERS

Figure 3.2 Sensitivity of the power-directing function of 3-dB couplers

in BPC operation to input power mismatch. Pout is the properly directed

power, and P,,, is the residual, or misdirected, power.

connecting line as qh. Let one minus the additional loss to a pulse travelling through -

. _ delay line m be called q&,,. We can then write the following expression for the

expected power gain of an N-stage BPC.

G, = 77hN fi (I+ %d - m=l

(3.3)

For all 7’s equal to one, this gives 2N, as mentioned earlier. The compression ratio,

or ratio of input to output pulse length, is

c T = 2y (3.4)

and the compression efficiency is given again by

. -

rlC GP =-

cr *

22

(3.5)

I :. -. .. - ;

Note that this gain and efficiency represent an average for two output pulses, made

coincident. Also, since losses in long delay lines should be predominantly due to

wall loss, we can approximate qd,,, by

?j’d, N (f?-2ao’T)

2N-m

, (3.6)

where cu and vg are respectively the attenuation constant and group velocity in the

delay lines, and T is the time bin duration. This reflects the fact that the delay of

the mth line is 2N-mT.

Binary Pulse Compression is best suited to high frequencies, where accelerator

section fill times are short. The physical lengths and the losses of delay lines become

reasonable in such cases. It would not be appropriate at the SLC frequency, but

merits consideration at the higher frequencies of some future linear collider designs.

SINGLE-SOURCE OPERATION

. _

_ -

As high-power rf sources at X-band were (and still are) in the process of being

developed, P. E. Latham of the University of Maryland foresaw in 1988 the desir-

ability of being able to test a high-power BPC with only one available source. He

presented the insight [12] that, with an additional delay line and a modest sacrifice

of efficiency, a single source could be used.

With power levels in the multi-megawatt range, it is not possible to introduce

the coding after splitting the source output. The most obvious approach would be

to produce a pulse twice as long as desired, encoded with the first phase pattern

followed by the second. This single pulse could then be split with a 3-dB coupler

into two identical pulses. A delay of one pulse by half its length would then render

23 .-

the appropriate phase patterns coincident and ready for BPC compression. This

approach, however, throws away half of the energy, the non-overlapping halves of

the two pulses.

Latham shows that with a little elegance, one can create the proper overlap

pattern for a given compression ratio with significantly less waste. For example, the

following pattern is created by splitting a single eleven-bin pulse and delaying the

bottom pulse by three time bins.

+-++-+++---

+-++-+++---

Tuning the delay line to give the whole bottom pulse an effective 7r phase shift,

we get the following pulses, whose overlap is precisely the three stage input shown

before.

+-++-+++--- _ -

- +--+---+++

Here 3/11, or 27%, of the energy is wasted. There is also a slight loss of power

due to attenuation in the extra delay line. Again, single-source operation is use-

ful-for system development, particularly at high power. Its inefficiency does not

recommend it for long term or standardized use.

24

SYSTEM DEVELOPMENT

In the late 1980’s, development began at SLAC of a high-power binary pulse

compressor, to be driven by klystrons of the XC series [S], being designed to deliver

a hundred megawatts at 11.424 GHz. A compressed pulse duration of about seventy

nanoseconds was desired.

. _ Figure 3.3 Electric field patterns for (a) the T&s mode in rectangular

waveguide and (b) the T&l mode in circular waveguide.

For a system to be practical, losses must be small enough to allow the achieve-

ment of reasonable efficiency. Standard X-band power transmission is done in rect-

angular WR90 waveguide, with dimensions 0.9” x 0.4”. It is single-moded, operating

in the dominant T&o mode (Figure 3.3(a)), with a loss at 11.424 GHz of 0.1 dB/m.

With a group velocity of 0.82c, this translates to 24.6dB/ps, far too much loss . -

for the required delays on the order of 0.1 ps. It was therefore deemed necessary

25

(avoiding superconductivity) to use overmoded waveguide. The best mode for such

waveguide transmission is the TE 01 mode in circular guide (Figure 3.3(b)). This is

the same mode used as a standing wave in the SLED storage cavities. While not

the dominant mode in circular waveguide, it quickly becomes the least lossy mode

as the radius is increased. Because its magnetic field has no azimuthal component

at the wall, its attenuation constant shares with those of the higher TEo, modes a

unique form,

R, Wf I2

a=arloJiqqpy (3.7)

where R, is the surface resistance, 70 the impedance of free space, and a the guide

radius. The fields are attenuated due to wall loss as e--a’. Since the cutoff frequency

is inversely proportional to the radius, (Y is seen to drop off approximately as u-’

well above cutoff. For delay lines, one is more interested in the attenuation per unit

delay time, given by

cvt = o!vg = - (3.8)

. _ which is exactly proportional to a -3 above cutoff. The attenuations per unit length

and per unit time of a TE 01 wave at 11.424 GHz are plotted as a function of waveg-

uide diameter in Figure 3.4. Some other circular modes are shown for comparison.

Another benefit of the T&l mode is its azimuthal symmetry. There is no polar-

ization which needs to be maintained between components. Also, the absence of

longitudinal currents makes the mode insensitive to short gaps such as occur at

flange joints. Finally, the fact that the electric field lines don’t terminate on the

walls helps avoid arcing at high power levels.

Circular waveguide of inner diameter 2.81” (WC281) was chosen for the delay s -

lines. This has a theoretical loss of 4.06 x 10s3 dB/m and 1.08 dB/ps. Another

26

z 0.05

2 z 0.04

3 p 0.03

s

g 0.01

E PI 0.00

P= 2

2 E 0

- ;. .

’ I ’ ’ ’ ’ I . I I I I I I I I I

I I I I I I

2 3 4 5

DIAMETER (inches)

II I I I I

2 DIAMETER (inches)

Figure 3.4 Attenuation per meter of waveguide and per microsecond of

delay for some of the lower propagation modes of circular copper waveg-

e - uide as a function of waveguide radius. The frequency is assumed to be

11.424GHz. Note the T& behavior.

27

advantage of large diameter for high power is better molecular conductance, which

eases demands on the pumping. For good conductivity and good vacuum properties,

the guide was drawn from oxygen-free copper. Other components, where mode

conversion is more of a danger, were made with a less over-moded 1.84” diameter.

These include 180” bends and 3-dB hybrid couplers.

To allow delay lines to fold back and meet the shorter transmission lines at

the next coupler, horseshoe-shaped 180” bends were designed and fabricated from

corrugated circular guide by Charles Moeller and John Doane at General Atomics

Corporation, San Diego, California. The corrugations split the degeneracy of the

T&r mode with the Ti’Urr mode by strongly affecting the latter’s wave impedance.

This permits a bend design with minimal mode-conversion loss. The corrugations

are rounded, presenting no sharp edges, and the curvature of the bend is varied in a

sine-like fashion, being maximum at the apex. The bends are coated with fiberglass

for rigidity to maintain their design shape.

. _ The 3-dB couplers were also designed and manufactured by Charles Moeller

at GA. They are unique devices with two circular ports and two rectangular ports.

With the couplers placed side by side, this permits the short connections to be made

in WR90, while the delay line connections are made in circular guide. They were

machined in halves from copper blocks about a meter in length. A WC184 circular

guide is connected by an adiabatically tapered longitudinal slot to the side wall of

a WR90 guide. H-bends in the WR90 guide near the ends place the rectangular

ports on the top of the coupler. The halves are joined with closely spaced bolts.

Ten-inch-long tapers connect the 1.84” components to the 2.81” delay line e -

waveguide. These were bored from steel. Their nonlinear profile was designed by

28

:. -. .

John Deane to suppress conversion to the TE 02 mode, whose cutoff diameter is

passed through in the taper.

At the circular input and output ports of a such a binary pulse compressor,

transitions are needed between the circular mode and the rectangular mode used

in sources and loads. We used adapted Marie-type mode transducers [13]. The

rectangular end of each is an adapted magic-T, with the two cross-bar arms folded

up in the same direction. The common wall of these arms is discontinued in the

main body of the device. This converts between the T&s mode at the out-of-plane

port and a TE 2s mode in wide guide. This new rectangular cross-section is then

adiabatically transformed into an “X” shape and finally into a 1.84” circle, leading

the electric fields into the T&r configuration.

The results of a low-power, two-stage experiment performed with these com-

ponents were reported in 1989 [14]. A 312ns pulse was compressed by a factor of

four to 78ns with a peak power gain, averaged between the two outputs, of ap-

proximately 3.2. This corresponds to an efficiency of N 81%. Tuning was done via

. a phase shifter in the WR90 connection between hybrids. The gain was within a

couple of percent of what was expected from calculations based on the measured

insertion losses of the components and theoretical waveguide attenuation.

THE SLAC 3-STAGE BPC

In 1990, a three-stage, high-power binary pulse compressor [15] was constructed

and commissioned at SLAC with components described above. A schematic layout

of the system is shown in Figure 3.5. An initial delay line permitted operation with

‘a single klystron, and a fourth 3-dB coupler enabled us to combine the outputs into

29

: .

a single pulse to achieve maximum power for accelerator structure tests.

Good vacuum is necessary to prevent rf breakdown in systems carrying multi-

megawatt pulses. Residual gas is easily ionized by the electric field levels attained.

The pulse compressor was thus designed and prepared for operation under high

vacuum (10e7 Torr) as well as high power.

The delay lines were fitted with foot-long slotted waveguide sections inside

vacuum manifolds to facilitate evacuation. The vacuum manifolds (not shown in

Figure 3.5) are large steel cylinders with pairs of diametrically opposed flanged

ports on their sides. They are placed perpendicular to the delay lines to accomodate

multiple waveguides sections. One manifold with four slotted feed-throughs pumped

lines Dl and D3 simultaneously. Another pumped lines D2 and D4, close to the

hybrids, and a third smaller one pumped the end of line D2, just before its turn-

around.

The- 180” turn-arounds were braced and mounted on aluminum translation

tables. Sliding waveguide sleeves in vacuum bellows were attached to their ports.

Powerful motors, also mounted on the tables, then allowed them to be slid back and

forth like a trombone with about an inch of leeway. This meant about two inches

of possible path length adjustment, varying the electrical length over more than a

guide wavelength. Remotely controlled, these motors provided the mechanism for

tuning each delay line’s electrical length in steps of three degrees of phase to achieve

the proper relative phase at the next hybrid. -

The bulk of the delay lines was formed from sections of 2.81”-diameter waveg-

yide ranging in length between ten and twenty feet, to the ends of which stainless

steel flanges were welded. The four hybrids, of the type described in the last section

30

WI390 D4 70 ns

2 Output Klystron

Figure 3.5 Layout of the 3-stage BPC in single-source configuration.

The Hl-H4 are 3-dB hybrids, the T’s are tapers, the B’s are corrugated,

translatable bends, and the M’s are Mar% mode transducers.

31

were encased in vacuum-tight steel jackets. The short connections between them

were made in rectangular guide, with directional couplers incorporated for diag-

nostics. Copper gaskets, crushed between knife edges on the conflat-type flanges,

provided a vacuum seal at all joints between components.

In general, components were vacuum baked or chemically cleaned to remove

contaminants from their walls before installation in the BPC. Exceptions were

the delay line waveguide sections. The inner walls of these were scrubbed with

Scotchbrite and lint-free tissue soaked in acetone, using a makeshift, drill-driven

rotary tool. Recall that in the TE 01 mode electric fields do not terminate on the

walls. Furthermore the power density, and consequently the field strength, is lower

in the large-diameter w.aveguide than in other components. These facts reduce the

danger of rf breakdown and ease the vacuum requirements in the WC281 runs. Sol-

vent cleaning was thus deemed sufficient vacuum preparation them. It would have

been inconvenient and expensive, in any case, to have divided the delay lines into

sections short enough to accomodate other cleaning techniques.

Before construction, the insertion losses of the various components were mea-

sured to the best of our ability using an HP 851OC Network Analyzer. These are

given in Table 3-A in both decibels and percent. The hybrid losses (Hl-H4) were

found to be about 5% greater for power combined in the rectangular output port

than for power combined in the circular port. The delay line losses (Dl-D4) are cal-

culated, not measured, using the theoretical WC281 loss for ideal copper plus 20%

for surface roughness. They do not include the bend loss, which is listed separately.

The WR90 loss is also theoretical.

The entire BPC system extended two hundred feet from the bend of delay line

32

Table 3-A

component insertion loss

Mode transducer 0.07 dB

Taper (1.84”-2.81”) 0.01 dB

Bend,bellows,& tapers 0.1 dB

Hl to circular port 0.12 dB

Hl to rectangular port 0.35 dB

H2 to circular port 0.11 dB

H2 to rectangular port 0.35 dB

H3 to circular port 0.10 dB

H3 to rectangular port 0.32 dB

H4 to circular port 0.08 dB

H4 to rectangular port 0.33 dB

Dl (WC281) 0.27 dB

D2 0.37 dB

D3 0.18 dB

D4 0.09 dB

WR90 connection 0.06 dB

percent

2%

0.2%

2%

2.8%

7.7%

2.6%

7.8%

2.2%

7.1%

1.9%

7.4%

6%

8%

4%

2%

1.4%

Measured losses of individual components.

. _ Dl to the bend of delay line D2 and was assembled above the klystron stands in

the SLAC Test Laboratory. The hybrids and one manifold sat atop a specially

built platform in test bay 8, and the delay lines were supported on a framework of

uni-strut running the length of the building.

Two 150 liter/s ion pumps and one 150liter/s turbo pump attached to the

pump-out manifoIds and several 8 liter/s ion pumps on rectangular waveguide sec-

tions were used to evacuate the system to the 10V7 Torr range. The total volume

and surface area of the vacuum system were approximately 50 cubic feet and 600

square feet, respectively. Pump down and subsequent powering up while maintain-

ing the vacuum took several weeks of single-shift operation at 6Opps. The power

33

.

level had to be raised gradually, because the rf created such effects as heating and

multipactoring that enhanced the outgassing rate of the system walls. Had the

R&D mission of the BPC allowed for more rigorous component cleaning and baking

and stricter vacuum technique, the time required for rf processing of the surfaces

may have been reduced considerably.

.

0 a m Modulation ?,

Generator

Detector

Figure 3.6 Electrical schematics of BPC (a) input circuit and (b) power

measurement circuits.

The input to the system was prepared as indicated in Figure 3.6(a). A signal

generator provided an rf signal at the design frequency of 11.424 GHz. This was

passed through a biphase TTL-controlled phase shift keyer (PSI<), which impressed

the proper bin coding. As only one X-band klystron was available at SLAC at

the time, the single-source phase-modulation pattern shown earlier was used. The

pattern length of 770ns consisted of eleven time bins of 70 ns each, eight of which

entered into the compression. A TWT (T raveling Wave Tube) then amplified the

signal to the kilowatt level necessary for driving the klystron, and the klystron,

powered by a synchronized modulator, raised it to the multi-megawatt level. A

splitter was not necessary, since the klystron (XCl) had two symmetric output

34 .-

ports. WR90 runs, designed to be equal in total length, connected one output to

the first hybrid and the other, through a mode transducer, to the three bin initial

delay line.

Power measurements were made using 55-dB directional couplers and tunnel-

diode detectors connected to a fast oscilloscope. Six such couplers were built into

the system, at the two inputs, at the two outputs (after converting at the circular

one to WR90), and at the short rectangular connections between stages. These

are shown in the electrical schematic of Figure 3.6(b). To eliminate error due to

calibration of the nonlinear crystal detectors, careful relative power measurements

were made by switching the same crystal diode between diagnostic ports and using

an accurate variable attenuator to obtain on it the same voltage readout. Accuracy

is within about five percent.

If we assume insertion losses combine linearly, Table 3-A leads us to expect

a gain of about 5.7 for the system, down from the ideal of eight. Measurements

were made at different input power levels. The system had to be tuned carefully,

one stage at a time, to obtain proper phasing. Table 3-B compares measured gains

and efficiencies for different parts of the system to those calculated from Table 3-A

and to ideal gains. These first measurements were done at low power (- 1 kW)

using just the TWT. Gains are defined as the sum of the output powers divided

by the sum of the input powers. Since power measurements could only be directly

made at the 55-dB couplers, power levels at other positions were inferred from

these measurements and the measured or calculated transmission properties of the

components. The last two rows include the final hybrid to combine the power into

one output. The initial delay line is not considered part of the system for gain

35

Table 3-B

Section

Stage 1

Stage 2

Stage 3

Stages 1-3

Combined circular

Combined rectangular

Ideal Expected

2 1.76 (88%)

2 1.82 (91%)

2 1.85 (93%)

8 5.93 (74%)

8 5.82 (73%)

8 5.50 (69%)

Measured

1.75 f 0.1 (88%)

1.79 f 0.1 (90%)

1.75 f 0.12 (88%)

5.47 f 0.27 (68%)

4.76 f 0.19 (60%)

4.92 f 0.36 (62%)

Gains and efficiencies of the BPC.

calculations. Its loss is corrected for.

Agreement between expected and measured gains is seen to be generally quite

good. The combiner gives more loss than expected, particularly through the circular

port. A mechanical obstruction was suspected, but it was never disassembled. At

higher powers, efficiency seemed to decrease slightly. At 20 MW input, the 3-stage

gain was 5.2. Reflected power at various points in the system were found to be less

than four percent of the forward power at one kilowatt and less than one percent

during the high. power measurements. Figure 3.7 shows the signals seen at each

stage at one kilowatt input, clearly demonstrating pulse compression. The power

outside the main pulse comes from the three throw-away bins. Glitches indicate

a phase reversal. Figure 3.8 shows the input and output signals for 15MW input

power with a close-up of the flat compressed pulse.

-

In this configuration, the Binary Pulse Compressor was operated with a maxi-

mum input power of 25 MW, and a maximum output power of 120 MW was achieved

[16]. No evidence of electrical breakdown was observed. The width of the com-

36

I -.. -. . - ;. ^

. L

output

Klystron output

200ns/div

. - Figure 3.7 RF power waveforms measured at each stage of pulse com-

pression for single-source BPC.

37

Figure 3.8 Input and output pulses for BPC and enlarged compressed

pulse at 15 MW input.

38

pressed pulse was about 60ns, rather than the ideal of 70ns. This shortening is

attributable to a rise time determined by the switching speed of the electronics and

to inaccuracies in the delay line lengths, resulting in imperfect overlaps.

In 1992, another X-band klystron became available, and the BPC was recon-

figured to operate with two sources, as intended [17]. This involved the elimination

of delay line Dl from the system and the synchronization of the two klystrons.

Each klystron drive was given its proper three-stage phase pattern and was now

only eight bins or 560ns long. Figure 3.9 shows measured waveforms at different

stages for this configuration. The amplitude and phase traces of the compressed

pulse are also shown. The slight phase variation may be due to the klystron output.

The input power levels were quite unbalanced, 10 MW for one and 34 MW for the

other. From equation (3.1), we therefore expect to lose an extra 8% to misdirection.

The maximum output measured was 175 MW, yielding a gain of four, somewhat

below expectations. Power was most likely lost to mode conversion at joints and

less accurate tuning.

. _

The SLAC S-stage BPC has been put to use for high-power tests of an X-band

accelerator structure [18]. Th ese were performed in a concrete bunker built behind

the klystron test stand and dubbed the Accelerator Structure Test Area (ASTA).

The output of the combiner was fed via a WR90 run into a 30-cell travelling wave

structure, in which an accelerating gradient of lOOMV/m was reached with no rf

breakdown. Studies of the dark current captured in this section at different gradients

and for different pulse rise times were also performed.

39

- _ .,

.

-I+4a c -

-167.2 - -298 100

ns/div 702

25 ns/div

Figure 3.9 (a) Detected power at each stage and (b) amplitude and

phase of the combined compressed pulse for the BPC operating with two

sources.

40

4. SLED-II

We’ve seen that the SLED method of pulse compression can present problems

associated with the exponential nature of its waveform. BPC overcomes these prob-

lems by providing a flat compressed pulse, but has its own drawbacks. It should

be obvious from the description and diagrams of the last chapter that Binary Pulse

Compression can be a very space-consuming operation. The SLAC BPC is quite

bulky due to the lengths of delay line necessary even for its relatively short 70ns

pulse. Folding the lines up with more 180” bends might help with this problem,

but would increase the power loss, as would using disk-loaded delay lines to reduce

the group velocity. Super-conductivity may provide a solution, albeit an expensive

one. Fortunately, there is another pulse compression technique which avoids the . _

short-falls of those hitherto described. SLED-II* offers a flat pulse while occupying,

for higher compression ratios, considerably less space than a BPC (though still far

more than SLED).

The concept of SLED-II was originally presented by A. Fiebig and C. Schieblich

[19] in 1988. It was later developed independently by P. B. Wilson, Z. D. Farkas,

and R. D. Ruth [20] at SLAC, as an extension of the SLED idea. The revolutionary

improvement is the replacement of the pair of resonant cavities by a pair of long

resonant delay lines. These can be low-loss circular waveguides, such as those used

* Also referred to in the literature as Resonant Line SLED (RELS).

41 .-

in the BPC, which are shorted at one end and iris-coupled to isolated ports of

a 3-dB coupler at the other. As the emitted field changes only at discrete time

intervals given by the round-trip delay time, a flat output pulse of -that duration is

obtainable. This makes SLED-II useful for the uniform acceleration of long bunch

trains. It also opens the possibility of multiple stage compression.

Incident = 1

( w I iris ,

Figure 4.1 (a) Phasor triangle and (b) reference planes for partial re-

flection at a waveguide iris.

s -

42

SLED-II THEORY

Consider a waveguide interrupted by an iris with a reflection coefficient of

magnitude s. Continuity of the fields at the iris hole requires that the phasors

representing the incident, reflected, and transmitted waves form a triangle [21] as

in Figure 4.1(a). The transmitted field immediately to the right of the iris must

equal the vector sum of the incident and reflected fields immediately to the left.

Since by conservation of energy the magnitude of the transmission coefficient is

dm, the triangle is a right triangle. Call the phase angle of the reflection 8.

We can write R = se

ie

T = &-&(B--‘r/2),

with

8= % + sin-l s.

Now, defining the plane of the iris as z = 0, take reference planes on either side of

the iris at

zp = *v+4

2P ’ (4.1)

as shown in Figure 4.1(b) w h ere ,B is the propagation constant of the wave (e-‘p’

dependence). With respect to these planes, the reflection and transmission coeffi-

cients, symmetric with respect to direction, become

T’ = iJ1-sz,

(4.2)

If a short is placed an integer number of half-wavelengths beyond the right reference

plane, it will return the transmitted wave with an overall phase shift of 7r radians.

Transmission back to the left plane adds another 7r/2, for a total of 27r, putting

43

it completely out of phase with the reflected wave. On the other hand, the wave

now reflected to the right will gain another r phase shift from R’, ending up in

phase with the initial right-travelling wave between the iris and short. The field

emitted from an iris-coupled delay line at resonance is thus out of phase with the

iris reflection, as indicated in Figure 4.2(b). I n what follows, reference to the above

_ l defined planes will be understood.

Two such delay lines are incorporated in a SLED-II system, as shown in Figure

4.2(a). This pairing allows one to use a 3-dB hybrid to direct the power away from

the source, just as in SLED. A three-port circulator cannot be used because the

ferrite’s anisotropy could not be maintained at the high field levels anticipated. The

pulse compression, however, can be understood in terms of a single resonant delay

line.

Let -s be the iris reflection coefficient. Let td be the back-and-forth delay

time, given by

tp2L, Vi2 (4.3)

where L is the delay line length and vg the group velocity in the line for the chosen

mode and operating frequency. L is chosen to give the desired td, presumably

determined by the fill-time of an accelerator structure to be powered. Finally, let

2r represent the round-trip field attenuation parameter of the lines. (In general,

27. !x 2aL, where o is the attenuation constant of the waveguide.) The compression

ratio, C,, is constrained to be an integer for SLED-II. An input pulse of duration

C, x td is thus fed into the system. During the nth time interval of duration td the

field emitted from the delay line, through the iris, for constant input amplitude E;,

-

44

out

1 t 1 3-dB

Ag td -

Coupler L=NT=VgT

In

Irises Resonant Delay Lines

(a) SLED-II Diagram

II

- U

i + e

(b) Field Illustration

Figure 4.2 (a) Schematic of a SLED-II pulse compression system and

(b) illustration of waves just outside the delay lines.

45

is

J%(n) = 0, n = 1,

E,(n) = (1 - s2)e-2r [l + ses2’ + s2e-4r + -- - (se-2r)n-2] Ei, n > 1. (4.4)

After each delay time, a new term is added to the geometric progression in brackets,

as the bouncing pulse front impinges once more on the iris. Since se-2r < 1, these

equations can be combined as

E,(n) = (1 - s”) e-2r

1 - se-2r [I - (se-2r)“-1] Ei, n = 1,2 ,... Cr. (4.5)

Superposition with the outer iris reflection yields the output wave

E,,t(n) = E,(n) - s&, n=1,2 C,-1, , **- (4.6)

until the final time bin. At time t = (C, - l)td, the phase of the input pulse is

shifted by 7rIT, so that the waves add constructively and we get a compressed output

pulse of duration td with amplitude

* - Ep = Eout(G> = E&‘r) + 4

(1 - s”) e-2r =

1 - se-2r [ ( 1 - se-2r)Cr-1] + s (4.7) -

For a given C, and r, s can be chosen to maximize Ep. For the ideal case of a

lossless system,

%=1+2s-s Ei

CT-1 _ p, (lossless).

The derivative with respect to s yields as an approximate analytic solution for the

optimal reflection coefficient [22]

1 e -

=

[ 1 Fpzip so

&-l/2

1 7 C, 2 3 (lossless).

46

.-

- .

cl .d

Y 2 0 w"

-1

L -_---___-- -I L I I

1 I I

I I I I

I ’ I I

I I I I I 1-i (

-1 0 1 2 3 4 5 6 7 8

r------------

I I I

-1 0 1 2 3 4 5 6 7 8

t/t,

Figure 4.3 SLED-II output amplitude and power waveforms. Dashed

s - lines indicate the input, and a sign change in the field plots represents a

phase reversal. Here C, = 6, s = 0.684, and e-2r = 0.98995.

47

In the general case, one just optimizes the exact expression for Ep numerically to

find the best s value.

Again we define a power gain and compression efficiency

and G

qc = 2. Cl-

Typical SLED-II waveforms are shown in Figure 4.3. The output field ampli-

tude begins at -sEi. Successive steps bring it up past zero as the emitted compo-

nent builds. Finally the phase flip yields a jump in amplitude, extracting much of

the stored energy. One delay time after the flip, the power is cut, since the flattop

can be maintained no longer. The output drops, and the remaining energy trickles

out of the delay lines.

GAIN AND EFFICIENCY

The theoretical limit on G, for SLED-II is nine, since Ep of equation (4.7) _

. approaches 3Ei with r = 0 as C,. + 00 and s + 1. Realizable systems with losses

and finite compression ratios will, of course have somewhat lower capabilities. As

with SLED, but not BPC, there is an inherent inefficiency of this method due

to power emerging before and after the compressed pulse. Unlike SLED, there is

no additional inefficiency due to pulse shape. Table 4-A lists the optimized iris

reflection coefficient for a lossless system, the compressed pulse field enhancement,

the ideal gain, and the intrinsic efficiency Q for several compression ratios.

-

The effect of component losses may be considered as the efficiency qh of round-

t_rip-reflected transmission through the 3-dB hybrid circuit (not included in the last

section’s efficiency definition) and an efficiency 7~ due to the delay line loss. The

48

2

3

4

5

6

7

8

9

10

11

12

15

18

Sopt E,IEi

0.500 1.25

0.549 1.63

0.607 1.85

0.651 2.01

0.685 2.12

0.711 2.20

0.733 2.27

0.752 2.32

0.767 2.37

0.781 2.41

0.792 2.45

0.820 2.53

0.841 2.59

Table 4-A

Gideal rli

1.56 0.781

2.66 0.887

3.44 0.860

4.02 0.804

4.48 0.746

4.84 0.692

5.15 0.644

5.40 0.601

5.62 0.562

5.82 0.529

5.98 0.499

6.39 0.426

6.68 0.371

Ideal SLED-II parameters.

latter, as an effect on the overall gain, depends on the compression ratio as well

as the attenuation parameter. In Figure 4.4, 71 is plotted versus round trip power

transmission in the delay lines, e -4r, for various compression ratios.

The overall efficiency of pulse compression can then be thought of as

This formulation is useful for predicting the performance of actual SLED-II systems

with different components and compression ratios.

MULTIPLE STAGING

s - The fact that SLED-II produces a flat, or constant amplitude, compressed

pulse makes it possible to consider using that pulse as the input for another pulse

49

. rjL

0.85

0.80

0.75 0 .Q 0.92 0.94 0.96 0.98 1

Round Trip Power Transmission (e-4T)

Figure 4.4 Efficiency factor ql, due to delay line loss, as a function

of round-trip power transmission (field attenuation squared) for various

compression ratios including the limit curve.

. -

compressor. Two or more SLED-II stages can be cascaded as shown in Figure 4.5.

The length of the final stage’s delay lines determines the output pulse length. Since

each preceding stage requires a delay time equal to the desired input of the following

one, more than two stages are probably not desirable. The delay times and input

pulse length in terms of the compression ratios and output length tf are:

td t2) = tf ,

$1) = ptj12) = ptf, d (4.9)

t in e -

= C(‘)ty) = Cw@)tf. f r

Ct2) should be the smaller of the two compression ratios to minimize the length of f

50

:. -. .

the first stage delay lines. The phase coding of the input is more complicated than

for a single stage. The coding of the second stage must be superimposed on each

ty’ interval of the first stage coding, requiring a total of 2(Ci1’ T 1) phase flips.

The waveform patterns for a (6 x 3) two-stage system are shown in Figure 4.6.

v&p)

2

Stage 1

Figure 4.5 A two-stage SLED-II pulse compressor.

. _ As we’ve seen, a good range of gains are available with a single SLED-II system,

depending on the available input pulse length. h1ultiple staging, however, does offer

a way to overcome the gain limit and/or improve efficiency. For example, consider

a Cr = 6 system followed by a C, = 3 system. The ideal (neglecting component

losses) gain would be, from Table 4-A, 4.48 x 2.66 II 11.92. The intrinsic efficiency

would be 0.662. Compare these numbers to the lossless gain and efficiency for an

equal length input pulse in a single C, = 18 system, 6.68 and 0.371 respectively. Of

course the losses in the delay lines and the added hybrid must be taken into account

-and the benefits weighed against the added cost and bulkiness of the system in any

real consideration of multiple-sta.ge SLED-II pulse compression.

51

I - ,. -. .

IF

0 _ _.,,._.. ._.,, .._......,...... . . . . . . . .

-1 r -

.

0 5 t/p lo l5 2o

Figure 4.6 2-Stage SLED-II waveforms: the phase-coded (sign indicates

phase) input for a 6 x 3 system and the resulting output at each stage.

Two time scales are indicated.

COMPARISON WITH SLED

It is interesting to understand how the SLED-II theory relates to the SLED _

theory developed a couple of chapters ago. They are really the same phenomenon

described in different regimes of resonator dimensions and coupling. Note that both

equations (4.5) and (4.7) yield a theoretical upper limit on the peak output field

of three times the input field. If we simultaneously decrease the delay time and

52 .-

.

increase the reflection coefficient of a typical SLED-II system, the steps begin to

approach a smooth line. Leaving the input power on for a fixed time after the phase

flip while the delay time shrinks, we get many steps in the compressed pulse itself,

which begins to resemble the smoothly decaying SLED pulse. This is demonstrated

in Figure 4.7.

I I I I I I I I I I I I I I I I I I I I I

I I I I I I I I I I I I I I I I-

O 10

3t,

30 40’ 50

Figure 4.7 Output waveform of a SLED-II system with small coupling

driven by a pulse many delay times long and continuing for several delay

times after phase reversal. Note resemblance to SLED waveform.

To tie the two theories together, let’s consider how the emitted field approaches

steady state in SLED and SLED-II as their respective resonators charge up. Com-

paring the ratios of emitted fields at time t = ?Ztd (before the phase flip) to their

53

asymptotic limits, we get from equations (2.3) and (4.5)

SLED SLED-II

1 _ e-ntd/Tc - 1 _ (Se-27)n = 1 _ e--n(2r--ln ~1,

or

Tc, - td

27-ins’ (4.10)

This gives us the equivalent time constant of the delay line when viewed as a reso-

nant cavity. Now consider a short, low-loss delay line discharging. The energy lost

to the walls in one round trip (not emitted) is

AU = (e-4r -l)&

21 (e -4s - 1) U, lAU/ << Uo

N -4ru.

Now, with

dU AU -4ru --

dt = td N td ’

we can write

Q. = w” Wtd

-dU/dt - 47’

where w is the rf angular frequency. Finally, using

~QL 2Qo T,=-=

W 4 + P) ’

we can derive the correspondence

p - +,

e _

(4.11)

(4.12)

where p is the cavity coupling coefficient.

54

These relationships show how the SLED-II parameters can be recast into SLED

parameters as we move from the regime where one description is appropriate to

that where the other is; that is, as our resonant elements become appropriately

considered lumped circuits rather than transmission lines. The two regimes can be

distinguished as

SLED : g <cl, c (4.13)

SLED-II: $ - 1. C

The SLED-II parameters 7, s, and td give rise to the SLED parameters Qs and /?.

The number of parameters is one less in the SLED picture because we eliminate the

discretization of time.

In the frequency domain, the distinction between SLED and SLED-II can be

cast in terms of the mode density of the resonator spectrum. The longer the res-

onator, or the higher the longitudinal index of the operating mode, the denser the

spectrum is around that mode’s frequency. This allows the resonator to respond

more faithfully to the Fourier spectrum of a flat pulse.

. _ One more point should be made here. Recall that in the section on SLED

theory, the reflection coefficient amplitude of the cavity coupling iris was treated

as unity. Though a justified approximation for the SLED regime, this is not the

exact physical value. Intuitively, it cannot be, as it allows for no power to enter

and fill the cavity. The above comparison lets us resolve this dilemma. Any real

cavity, having finite dimensions, can be considered to have a non-zero delay time for -

a wave bouncing between the coupled face and the opposite face. From the above

equivalent expressions for Qe and /3, we can write

s -

Ins = 4 4 L --td = ---,

2&o Qo vg

55

where L is the cavity length and vg the group velocity of the traveling-wave mode

corresponding to the standing-wave, which depends on the transverse cavity dimen-

sions. For the cavity described in reference [9], we have Qo = 1.08 x 105, p = 5,

and w = 27r x 2.856 x 10gsV1, and can calculate td = 2.87ns. These values give

s 2! 0.9988, very close to one, as claimed.

.

TUNING

For a SLED-II pulse compression system to work properly, the delay lines must

indeed be resonant. As with cavities, this means that the spacing between the short

at one end of each line and the iris reference plane (see theory section) at the other

must be an integer number of half guide wavelengths. Successive reflections within

the line will then be in phase. Any SLED-II system, just like any BPC, should

therefore include a means of tuning the lines. This is most conveniently done by

using movable plungers as shorts.

. _ Let’s consider the effect of detuning on the SLED-II operation. Let the shorts

be displaced from the nearest resonant positions by distances AZi, where i (= 1 or

2) indicates the particular delay line. Each successive round trip of the rf pulse then

has its field contribution phase shifted by a detuning angle

9i = -2p Ali, (4.14)

Take first the case where both lines are detuned by the same amount. Let

41 = 42 = 4. As long as symmetry is maintained, the 3-dB hybrid will direct

all the power away from the source. Since detuning effectively adds an imaginary

counterpart to the attenuation parameter, it is easy to see that equation (4.7)

56

becomes

G Ei=

C1 - ‘“1 e-2r+i’ [I _ (se-2r+iqh)C,-l] + s 1 - Se-2r+i4 9 (41 = $42 = 0). (4.15)

The amplitude and phase of the compressed pulse for C, = 6,12, and 00 and given

values of T and s are plotted versus 4 in Figure 4.8. The effect of detuning can

be illustrated graphically with a phasor diagram. Figure 4.9 demonstrates how

successive reflections in the detuned line contribute to the emitted field phasor.

Also shown is the resultant output when the iris reflection is added just before and

just after the phase flip. In a tuned line, all the fields would lie along the real axis.

I- I, .\ -I 5 1

(Ep/Ei)2 ’

I

-1.5 -1 -0.5 0 0.5 1 1.5

Detuning Angle $I (radians) s=O.80, e-2T=0.98

1.0

0.5

0.0

-0.5

-1.0

Figure 4.8 Compressed pulse relative power and phase as a function of

common delay-line detuning angle for compression ratios of 6 (dot-dashed),

12 (dashed), and 00 (solid).

57

.-

1.0

0.8

0.6

0.4

0.2

0.0

Real Axis C,=9, s=O.80, e-27=0.98, $I= 15”

Figure 4.9 Phasor representation of detuned SLED-II.

Figure 4.10 shows the locus of steady state emitted field values traced out as 4 is

varied from 0 to 7r and illustrates how it is approached for various angles. This is just

. _ the coefficient before the brackets in equation (4.15). It describes a circle of radius

(1-s2)e-2T/(1-s2e-4’ ) with its center on the real axis at (1-s2)se-4r/(1-s2e-4r).

This is similar to the steady state response of a cavity driven off resonance. The

latter is given by E,/Ei = ocos$ei$, where T/J = tan-l [(w,Z - w2)Q~/wow]. It

passes through the origin, while Figure 4.10 does not. Using equivalent SLED-

II parameters as derived in the last section, we get for the SLAC SLED system

se-2r = se-wtd/=i?o N 0.9986. The ratio of the center coordinate to the radius

in the expressions above approaches unity, justifying the treatment of SLED as a

cavity, or lumped element.

Now consider the case where the two shorts are displaced equal distances in

58

0.6

0.4

0.2

0.0 0 0.5 1 1.5

Real Axis C r+m, s=.O80, e-2T=0.9&3

Figure 4.10 Phasor illustration of the asymptotic emitted field for a

SLED-II detuned in the common mode at detuning angles of 5”, 15”, 30”,

45”, 90”, and 150”.

opposite directions. 41 = -42 = 4, so that the two emitted fields are complex

. _ conjugates of each other. The forward pulse is the sum of the fields from the two

lines. A reflected wave, equal to the difference of the fields from the two lines, is

_ - sent back toward the source with maximum amplitude E,. Thus we find

(41 = -42 = $),

(4.16)

(41 = -42 = 4).

The phase of either pulse is independent of 4, as the phase effects of the oppositely

detuned lines cancel each other. The amplitudes are plotted versus 4 in Figure 4.11

for the mentioned compression ratios.

In general, there will be a combination of the above two effects. Define a

59

\ \

\ (E&J/E \

\ \

\

-.. \ ,\

v

“0 0.2 0.4 0.6 0.8

Detuning Angle # (radians) s=O.BO, c?=0.98

Figure 4.11 Compressed pulse relative power and reflected (misdirected)

pulse relative power as a function of differential delay-line detuning angle

for compression ratios of 6 (dot-dashed), 12 (dashed), and 00 (solid).

common detuning angle 6 and a differential detuning angle 6 as

*= d1+42

& sA2'

2 -

(4.17)

60

The compressed and maximum reflected amplitudes are then given by

EP l( z=s 1 _ s2) e-2r+iB

[ e i6 1 - (se -2r+qe+q c,-1

>

1_ se-2T+i(B+6)

El- -= Ei fc 1 - S”> e-2r+ie

-i6 1 - (se -2r+i(e-6) q-1

+e >

1 - Se-2r+i(B-6) 1 +s, [ i6 1 - (se -2r+i(e+6) c,-i (4.18)

e > 1 - Se-2r+i(O+6)

4 1 - (se -2r+i(e-6) G-1

-e >

1 _ Se-2r+i(B-6) 1 , where the two terms in the brackets represent the contributions due to each delay

line individually. Implicit in all this is the assumption that the emitted field steps do

not become significantly out of synch between the two lines, i.e. that 2C,\Sl/(pus) <<

td or equivalently IAll - Al21 << L/C,..

The above analysis suggests.a method of tuning a SLED-II system. First the

relative positions of the shorts should be adjusted by moving them in opposite

directions to eliminate the reflected field (or minimize it in an imperfect system).

When this is achieved, 6 will have been set to zero (modulo 27r), according to

equation (4.18). Th e output is then given by equation (4.15) with 4 = 8.

Next the shorts should be moved together to set 6 to zero. This can be done by

visually maximizing Ep on a scope. There is, however, a more sensitive way. Notice

in Figure 4.8 that the amplitude of E,/Ei forms a broad peak, while its phase passes

sharply through zero. Monitoring this phase is the best way to tune out 8. This

requires a reference. As equations (4.6) and (4.4) indicate, the output during the

initial time interval (n = 1) depends only on s, and is thus independent of the short

gas-itions. Its phase relative to the compressed pulse should be X. Phase-comparing

both the first and final time intervals of the output SLED-II waveform to a reference

. _

61

: . -

signal from the source provides a good means of zeroing the phase and maximizing

the amplitude of the compressed pulse [23].

.

. _

62

5. THE 3-dB DIRECTIONAL COUPLER

In the operation of all three methods of rf pulse compression developed in the

preceding chapters, an essential element was a 3-dB directional coupler, also called

a, hybrid junction. Its function here is to direct the flow of electro-magnetic power.

In SLED and SLED-II, it is used to direct power reflected from the resonant storage

elements away from the source and towards a load. In Binary Pulse Compression, it

is used at each stage to direct the combined power of two phase-coded input signals

into the output port appropriate for given time bins. In this chapter, I briefly review

the characteristics of this important microwave component. I then detail the design,

. _ fabrication and t-esting of a new 3-dB coupler which meets the special requirements

of our SLED-II system.

FUNCTION

A directional coupler is a four-port microwave junction with the following char-

acteristics: If power is incident on one port, that power is coupled out through two

of the other ports, but not through the remaining one. .Furthermore, there is no

reflection at any input port when the other ports are all terminated by their char-

acteristic impedances. Power incident in port 1 or port 3 is distributed between

63

ports 2 and 4. Likewise, power incident in port 2 or port 4 is distributed between

ports 1 and 3. Ports 1 and 3 are uncoupled, as are ports 2 and 4.

There exist a number of different types of directional coupler. They are gener-

ally comprised of two waveguide sections sharing a common wall in which there is

one or more aperatures or slots. If power Pi is incident on one port and we denote

the power emerging from the far end of the adjacent guide as PC, the coupling is

defined in decibels as

c = lOlog$ C

Directional couplers with a high value of C, or very little power transfer, are used to

measure power flow in a waveguide without much perturbing that power flow. They

can discriminate between forward and backward waves in the main guide. A crystal

detector is placed on the appropriate port of the coupled guide, and the other port

is terminated in a matched load. For diagnostics, we employ 20-dB couplers in

our low power tests and combinations giving on the order of a hundred dB signal

attenuation in our high power experiments.

From the given properties of a directional coupler and the requirement of sym-

metry, we can say several things about its scattering matrix, which, acting on

a four-vector of’complex input amplitudes, gives the amplitudes of the outgoing

waves at the four ports. Assuming no losses in the junction adds the condition of

unitarity. It can then be shown that the general form of the scattering matrix for an

ideal directional coupler, with appropriately chosen reference planes, is as follows

0 0 Cl Cl 0 0 ic, ic,

& 0

& "d 0 i i "d

ic2 0

ic2 0 I ic2 0 Cl ' ic2 0 Cl '

\ic, 0 Cl 0 / ic2 0 Cl 0 ) e -

where Cl and C2 are real numbers such that C2 = dm. The coupling can be

64

written in terms of these parameters as C = -20 log C2.

For our pulse compression applications, we require directional couplers that

split power equally between the two output ports. That is, we require Cl = C2 =

l/4, or 3-dB coupling. The scattering matrix for a 3-dB coupler is

010 i

s =$ ( 1 0 i 0 ) 0 iO1' i 0 ‘1 0

(5.2)

Let us see how this device operates in our systems.

In SLED and SLED-II, power is fed into one port, say port 1. For unit power,

the output is given by

Now the load on each output port reflects the wave with a factor which we previously

designated as Eout. These reflected waves become our new inputs for ports 2 and 4.

Assuming mechanical symmetry, so that they pick up no additional phase difference

in this reflection, we finally get as ouput

Thus the reflected waves cancel at port 1 and combine at port 3, assumed matched

to a transmission network, and the power directing function of the coupler is ac-

complished.

e - For Binary Pulse Compression to work with the phase coding described in

Chapter 3, a slight modification is necessary. Let us move the reference planes out

65

I ‘:. -. -‘.

by three quarters of a guide wavelength at ports 3 and 4, keeping the others fixed.

Calling the port displacements Zi, we define the matrix

The scattering matrix of the 3-dB coupler with respect to the new reference planes

is then given by 010 1

w

For in-phase and out-of-phase unit inputs at ports 1 and 3, the outputs are seen to

be, respectively, 1

nI 0 h s;== o

0 ( ) 0

and 1 0

Al s “l = ; . 00 0 Jz

The power is combined and directed through port 2 or port 4 depending on the

_ - phase relation of the inputs, as desired.

DESIGN PROBLEM

The SLED coupler is constructed from single-moded, rectangular waveguide.

The hybrids used in the Binary Pulse Compressor are more complicated, coupling

,a rectangular guide to an over-moded circular waveguide. They were designed and

fabricated by General Atomics. For SLED-II, it was decided that the resonant delay

66 .-

lines as well as the power transport lines from klystron to hybrid and hybrid to load

(structure) would operate in the TEol circular waveguide mode in order to minimize

losses. We thus desired a hybrid with four circular ports. Knowing of no existing

coupler of this type, we undertook to design one to suit our needs.

We chose to design the component in 1.75” diameter guide, which has several

propagating modes at 11.424 GHz, but far fewer than the larger guide to be used in

the delay lines. While we could not avoid being overmoded, since TEol is not the

fundamental mode, we thought it prudent to work less overmoded when manipulat-

ing the rf, trading some ohmic efficiency for greater safety from mode conversion.

Our coupler is essentially two parallel lengths of such waveguide whose interiors are

joined by a long, shallow, longitudinal slot of appropriate length. The geometry

is similar to that of the General Atomics hybrids used in the Binary Pulse Com-

pressor, except that one of the guides in the latter is rectangular. The coupling of

power through the slot can be viewed in the following way, analogous to the coupled

oscillator problem. . _

If we consider the cross section as a single waveguide, there are two normal

modes which in the limit of zero slot width are combinations of a TEol wave in each

circular guide. The mode in which the electric field lines in each half are oriented

in the same sense, clockwise or counter-clockwise, we refer to as the symmetric

mode; the one in which they are oppositely oriented, we call the antisymmetric

mode (See Figure 5.1). The slot perturbs these two modes differently, splitting

their degeneracy and resulting in a difference in their propagation constants, ‘ci

and kg”. e - Now, power entering one port, in the TEol mode, can be considered a

linear combination of equal amplitudes of the above-mentioned modes such that

67

.

Symmetric mode

Antisymmetric mode

Figure 5.1 The two orthogonal TEsr-like coupler modes.

the fields add in the input guide and cancel in the adjacent guide. As the waves . _

propagate along-.the coupler, the symmetric and antisymmetric components will

shift in phase with respect to each other. This phase shift at a distance z from the

_ - onset of coupling is given by

z

Ak, dz’ =

/

,‘(k;(l’) - k;(z’)) dz’. (5.4)

Writing the propagation constants as functions of position reflects our freedom to

adiabatically vary the slot width. If the slot is kept narrow enough that the cutoff

perturbations remain small, a good approximation of $ can be obtained from [24]

(5.5)

68

where k,” and k,” are the cutoff wavenumbers as functions of position, k is the free-

space wave number o/c, and k, is the T&l cutoff wave number in the unperturbed

circular guide.

Input fields Output fields

-- -b

-m--w -+)

3 -dB Coupler

Figure 5.2 Phtior illustration of 3-dB coupler operation. The dashed

phasors at each port represent the two coupler modes, and the solid phasors

are their vector sum. The lower mode on the left is shifted by 7r/2 on the

right.

Assuming no reflection or losses, we have at the end of the slot the same two

equal-amplitude 2% 01 combinations, but with a phase shift &,t. Indicating the

waveguide of the input port by the subscript A and the other by the subscript B,

we get, from unit input, the following output powers at the forward end of the

coupler 1

PA = - 4 11 + ce’” I’ 4 tot

= cos2 - 2

1 PB = - qll _ @tot I2 = sin2 ~tot -

2 -

w-9

Our aim of equal power division means we want $tot = 7r/2. This is clearly seen e -

from the phasor representation in Figure 5.2.

69

SEMI-ANALYTIC APPROACH

The problem of designing a coupler that will give us this desired phase shift re-

quires our knowing the mode splitting caused by a slot of given dimensions. We can

obtain good results with a calculation method used by Norman Kroll in dealing with

magnetron side resonators [25]. Th e method involves combining electro-magnetic

field theory with a circuit analogy and matching admittances at a boundary.

We first take advantage of the symmetry of the problem by dividing the cross

section halfway through the connecting slot and considering only one half. Different

boundary conditions will be applied at this dividing line to obtain symmetric and

antisymmetric modes. Our goal is to find the cutoff spectrum of the guide. This is

equivalent to finding the longitudinally invariant modes of a resonator of the same

cross section and arbitrary length. Since we’re interested in the modes which arise

from T&l (with adiabatic introduction of the slot), we’ll consider only modes with

transverse electric fields.

Consider an arbitrary resonator cross section with an arbitrary boundary AB

across a single opening (See Figure 5.3(a)). W e can define the voltage of an equiv-

alent lumped circuit with terminals at points A and B as Jf 3 . x. Rather than

defining a current, we can write the admittance as

Iv* 2P*

(5.7)

P being the power, 5 ‘VI*. In terms of the fields across the opening, we write

p=$?x7i’*.+ida. 8

The admittance per unit length can then be expressed as

(5.8)

70

- ;. .

with the integrals along the same boundary. If we assume a distribution for E along

AB, we complete the boundary conditions of the metallic walls. We can then solve

for the fields of modes which approximate that distribution.

Figure 5.3 Geometries of the theoretical development.

Let us apply this technique separately to the two parts of our cross section, the

rectangular slot and the circular guide. For each, we’ll assume a constant tangential

electric field E along the boundary Al?. In one case this boundary will be a straight

l;ne and in the other an arc, but for a narrow slot, they will coincide closely enough

for our method to be applicable. Using the above expression for admittance makes

-

71

us less sensitive to such field distribution assumptions than we would be using I/V

We take the simpler rectangular section first, shown in Figure 5.3(b). With a

time dependence eiwt understood, Maxwell’s equations for harmonic fields in free

space are as follows:

Combining these with our assumption of a transverse, longitudinally invariant E

field tells us that %! is purely longitudinal and that

and E Y

=iPoaHz k ax ’

where qo = dc ~0 is the impedance of free space. We also obtain, of course, the

condition that these field components must each obey the wave equation

V2f + k2f = 0. (5.10)

With the boundary conditions E, = 0 at y = &d/2, we obtain the following

parallel-plate waveguide solutions for k > r/d:

E(O)* = 0, X

Ep)f = +k,fikx,

H(O)* ik efikx, _ z

70

72

- . . -. . . -

Ef)* = Fk, cos

and

HP)* = “c cos

70

where p stands for any positive integer and k, = J(2442 - k2. Our assumed

boundary condition at AB is E, = E =constant. This limits the higher-order fields

to combinations that vanish at x = 0, yielding cash k,x variation. If we apply a

metallic wall boundary condition at the symmetry plane of the coupler, we need

J% = 0 at x = 1. We are thus limited to the zero-order fields. The two coefficients

are determined by thetwo E, boundary conditions, so that

Ey = $ (eik~~l~~ikl

,ikl

JEp)+ + g ceik[ _ ,-ikl jEr)-

= E sink(Z-x)

sink1 ’

It follows that E, = 0 and

H f

= -iE cosk(Z- x)

TO sinkl ’

_ _ We can now obtain an expression for the admittance per unit length at AB from

yr= SA” E;Hz dy l&WY12’

evaluated at x = 0. This yields

Y,” = --2 cot kl,

770d (5.11)

where the superscript reminds us that this is for the symmetric mode boundary

condition at x = 1. To find the admittance for the slot fields that correspond to

73 .-

antisymmetric modes, we set up a magnetic wall at the symmetry plane, requiring

dE,/dx = 0 at x = 1. We then obtain in a similar fashion

y; = - ’ tankl. rlod

(5.12)

Now we consider the circular section, shown in Figure 5.3(c). The relationships

- above Equation (5.10) become here

. rlo 8% E,= -

-“iiy- a$

and E+ = iyf$.

Of the cylindrical wave equation solutions, we are limited to those of the form

Jp( kp)eiP+, since the Neumann functions blow up at the origin. Expanding H, in

such fields yields for E4 an expansion of the form

E4 = iv0 2 A, Ji( kp)eiP$.

p=-00

We determine the constants A, from the boundary condition

-. E~(u, #) = iv0 2 ApJi(ka)eiPd = f(4),

p=-co

where a is the guide radius and

From Figure 5.3(c), we see that II, = arcsind/2a. The solutions are

iv0 Ji(ku)Ap = & J 02T f(d)e-‘P+dd

E + =- J 27r -$

,-id @j p = 0;

= P#O*

74 .-

.-, -. .

The field components are thus

where we’ve used J-, = (-1)” Jp to change the summation. Now, using this solution

for H, and the fact that E4 = E along the boundary AB, we obtain the admittance

per unit length from

Using a circle superscript to denote this admittance, we conclude

(5.13)

We find the cutoff wave numbers of the symmetric and antisymmetric TE

modes of our coupler cross section by setting -Y,” or -Y[“, respectively, the ad-

mittance looking out from the slot, equal to Y,“, the admittance looking into the

circular section. That is, we solve

(5.14)

for the symmetric modes and

sin 1c, -tankZ= -

lr (5.15)

75

_ . -tan kl closest to the TEol root are shown, indicating respectively the solutions

for the antisymmetric modes, where I’ve used d/2u = sin+. With a fixed at 0.875”,

solutions were found numerically for various values of 1 and d. It was necessary to

truncate the sum on the right at a value n >> ku and rewrite the rest of the sum

using

Jp( ku) ku + (ku)3 -N- Jp4 - P 2P2(P + 1)’

ku << p,

to obtain

pgl (y)2 $$$ =kupzl (y)2 5

+ $pzl (T)2 p2[pl+ q.

The first sum on the right can be replaced for small II, using the approximation

The second sum converges rapidly and can thus be appropriately truncated.

In Figure 5.4, the right side of equations (5.14) and (5.15), labeled F($, ku), is

plotted as a function of ku for given cross section dimensions. A vertical asymptote

exists at the cutoff of each circular TE mode. The intersections with cot kl and w

for k,9 and kz. One is seen to be shifted down and the other shifted up from the

circular guide cutoff.

DEVELOPMENT

e - To provide an initial test of our design theory, we built a resonator with a cross

section typical of our planned coupler and a depth of one half guide wavelength

76

I . - ;

-tan kl

-4 I I I I I I I I I I I

3.0 3.8 4 4.2

ka

Figure 5.4 The.function F($, k ) a versus La. The asymptote at 3.8317

corresponds to the TEol cutoff in circular guide. The abscissas of the indi-

cated intersections with cot kl and - tan kl give the cutoffs of the symmet-

ric and antisymmetric modes in the coupler, respectively. (Here 1c, = 3.87”

and-Z/u = 0.10056.)

for the uncoupled circular guide at 11.424 GHz. This was machined from a copper

block and a covering plate was prepared with small holes for coupling probes. Using

a Hewlett Packard HP 8510C Network Analyzer with electric and magnetic field

probes, we measured a spectrum of resonant frequencies. We were able to identify

all the modes by comparison to calculations for a cylindrical resonator with no slot.

Figure 5.5 is a plot the network analyzer response showing the peaks identified

as the two TEoll-like modes. These are shifted to either side of 11.424GHz by

the-degeneracy-splitting effect of the slot. The bump in between is the TMlIl-like

mode. Amplitudes are irrelevant, being functions of our probe coupling efficiency.

77 .-

START ll.e897weee m-lx - 11.456966666 mx

Figure 5.5 Resonance plot for a &/Zdeep resonator of the coupler cross

section. The two spikes to the right are the split T&1-like modes. (The

large resonance near the center is T&,20.)

The TEoll mode splitting is measured to be Ajr N 45MHz. Using the depth of

the resonator and cutoff frequencies computed by our semi-analytic technique, I

predicted Afr II 48MHz. Theory and experiment agreed within a few percent.

This was satisfactory considering the possible effects of machining errors and probe

perturbation. A computer program based on the theory of the last section was

considered accurate enough to proceed with a first design and set the dimensions

of a prototype coupler. The indication was that if the technique was inaccurate it

would lead to under-coupling, from which we could recover.

From the computed cutoff wavenumbers and our design frequency we obtain e -

the guide wavenumbers of the two modes for given slot dimensions. Their difference

78

is the rate of phase slip, d#/dz. F or a slot width d = 3mm, assumed average, and

our chosen radius, the rate of phase slip was plotted as a function of the slot half-

depth 1. This revealed a series of alternating upright and inverted “U’s, above and

below zero, respectively, with a periodicity of a couple of centimeters. Negative

values indicated that the symmetric mode had the faster phase velocity. Switching

between branches occurred when an asymptote of either cot kl or tan kl crossed the

T&r asymptote of F($, ka). As th ese “U”s approached vertical asymptotes, we

could theoretically make d$/d z as high as desired and thereby keep the coupler

length short. However, the slope with respect to I would also become quite high,

leaving us very vulnerable to machining errors. A slight deviation from the design

value of I could lead to. a large error in the total phase slip. Aiming for accuracy, I

therefore decided to work at a point where

From consideration of wall loss in the slot, material cost, and adiabaticity, the

shortest such slot depth was chosen.

To avoid reflections and power transfer to other normal modes, the cross-section

must be varied slowly on the scale of the guide-wavelength (X, 1: 3.78 cm). The slot

would be adiabatically introduced and terminated by linearly tapering the width d

from zero at the ends to a maximum at the center. A conservative tapering rate

of one millimeter per decimeter of length was chosen. My program and the goal of

4 = 7r/2 th en d t e ermined the coupler length and maximum slot width, d,,,. Table

5-A lists the design dimensions for the first prototype 3-dB coupler.

79

I .

Table 5-A

a 1 2.2225cm 1

1 1 1 0.2235cm 1 .

Hybrid dimensions.

s is the slot length, and the other parameters were previously defined. (Note: The

separation of the circular guide axes is 2a + 21.)

Drawings were made and submitted to the SLAC machine shop, and the cou-

pler was fabricated as shown in Figure 5.6 (except with a diamond-shaped slot).

_ -

It was machined-from a long copper block, which was first bisected. The cross-

section, divided along the plane of symmetry connecting the circle centers, was

then machined out of the two halves. The level of the ridge between the semi-

circular troughs in each half was given a longitudinal slope to provide the varying

slot width. A row of bolt holes at one inch intervals was provided along either side

of the sections, and they were brought together with an alignment accuracy of a

few mils. Special rectangular flanges were machined for the ends to receive closely

spaced circular waveguides. The close spacing of the ports made it necessary to

use-circular waveguide offsets to render them easily accessible. The design of these

accessories is described separately in the next chapter.

80

Dmax =‘5.29 mm f I I

II II

It n

WC-I 75 Off sets

2L = 0.44 cm

Cross section

-a = 2.223 cm Figure 5.6 Diagram of the circular-waveguide 3-dB coupler with modi-

fied slot. The offsets are described in the next chapter.

TESTING AND SUPERFISH CORRECTION

Marie mode converters and unbrazed, inexpensively bent offsets allowed us to

conduct several cold-test experiments on the prototype hybrid, utilizing both the

Network Analyzer and a pulse generator. Slight tapers, conical absorbing loads,

and rectangular waveguide extentions were among the other neccessary equipment.

Despite various problems associated with testing an over-moded, circular waveguide

component with ports several feet apart, we were able to demonstrate that its

-

81

.

behavior was fairly successful.

EF -3.0 dE3

-3.223 dl3

STN?T 11 STOP

.000000000 a-k 12.B GHZ

Figure 5.7 Original 3-dB coupler transmission plot as a function of

frequency for (a) the opposite port of the same guide and (b) the opposite

port of the adjacent guide. The intersection indicates equal power division

at 1-1.04 GHz.

Frequency sweeps, however, showed that it was not tuned as accurately as

we’d hoped. Equal power division at the output ports occurred at a frequency of

11.04 GHZ, rather than at 11.424 GHz. This is evidenced by the Network Analyzer

plots of Figure 5.7. The rising plot is the straight through transmission. The falling

plot is the adjacent arm coupling. This behavior is predicted by the approximate

expression of Equation (5.5), which indicates that the phase slip, and thus the

coupling, decreases with increasing frequency, At 11.424 GHz, the ratio of output

powers is about 0.78. This corresponds to an eight percent undershoot in 4. As the

82

I : .

I

L I I I I I I I I I

-100 -50 0 50 100

Time (ns) Figure 5.8 Pulsed power transmission measurements of original coupler

at 11.424GHz. In descending order, there are the input pulse, the direct

transmission, and the adjacent-line transmission.

. same expression has 4 proportional to the mode splitting, this is in rough agreement

with the seven percent discrepancy in our resonator experiment. Results of pulsed

power a transmission experiment, shown in Figure 5.8 revealed the same unequal _ -

power division at 11.424 GHz and equal division at 11.04 GHz. They also show

faithful transmission of accidental sharp features of the pulse, indicating that the

coupler is broad band enough not to distort fast-risetime pulses.

Fortunately, our conservative design left us in the correctible position of having

a slot that was too narrow. It needed to be opened wider, redesigned and rema-

chined to give more coupling. To first order, we could simply assume our program

overpredicts by eight percent, no doubt due to our small rC, approximation, and aim

83

for a phase slip eight percent high. Alternatively, I decided to use the field solver

SUPERFISH in arriving at a new slot profile.

Figure 5.9 SUPERFISH plots of the electric field patterns for the (a)

symmetric and (b) antisymmetric modes of our T&r coupler.

. .

Symmetry allowed me to model only one quarter of our coupler cross section,

permitting a finer mesh. I ran the code for both the electric wall and magnetic

wall boundary conditions on the edge representing the plane inbetween the circular

guides, thus finding the symmetric and antisymmetric solutions. I used eight differ-

ent values of the slot width d, ranging from zero to eight millimeters. SUPERFISH

solves for both the electric field pattern, important here for mode verification, and

the cutoff frequency of a mode. A full field plot of the modes in question for a

particular slot width is presented in Figure 5.9. From the cutoff data accumulated,

I calculated a set of values for Ak, at 11.424 GHz. I then used these with a cubic

84

157

156

155

154

153

152

151

0.6

0.5

-c “0.4

. “,.,

E 20.2 a _ -

0.1

0.0

I I I I I I I I I I I I I l llll

0 0.2 0.4 0.6 0.8

d (cm)

I I I I I I I I I I I I I I I I I I I I I

--

I / \ l- , \ 1

/

d / /

/

/ /

/ /

\ \

\ \

\ \

I I I I I I I I I I I I I I I I

0.2 0.4 0.6 0.8 1

Distance Along Slot (m)

Figure 5.10 (a) Th e wavenumbers of the TEol-like modes in the coupler e

- as functions of slot width and (b) the design slot width and resulting

integrated phase slip as functions of distance.

85

I :

spline interpolation routine in a new slot design program. We decided not to in-

crease the maximum slot width, but rather to increase the linear rate at which the

slot opened from either end until it reached that width and then level it off. This

resulted in a new design with a 10.04cm middle section of constant slot width, as

shown in Figure 5.6. The dimensions listed in Table 5-A remain unchanged. Fig-

ure 5.10(a) shows the guide wavenumbers as functions of slot width d, and Figure

5.10(b) shows the design slot width and predicted phase slip as functions of distance

.

z along the slot.

-3.0

$z - -3.6

11.0 _ -

11.5

Frequency (GHz)

12.0

Figure 5.11 Corrected 3-dB coupler transmission plot as a function of

frequency for (a) the opposite port of the same guide and (b) the opposite

port of the adjacent guide. Much of the excess loss is due to additional

components of the experimental setup.

After the slot modification was completed, a repeat of our Network Analyzer

86

tests showed the equal division point to be at about 11.5GHz, much closer to our

design frequency. The transmission plots are shown in Figure 5.11. The multiplic-

ity of required components, whose losses had to be accounted for, and the evident

oscillatory dependence on frequency (and presumably on mode transducer spacing),

indicating parasitic modes, made accurate transmission measurements virtually im-

possible. The output power ratio at 11.424 GHz, however, seemed to measure fairly

consistently at 0.985 f 0.005. Th e average one way transmission loss in the cou-

pler, not counting the offsets, was measured to be approximately 0.11 f 0.04dB

(2.5 f 0.9%). Th e round-trip transmission loss, measured at adjacent ports with a

shorting plate covering the far ports, was determined to be about five percent. The

isolation of adjacent ports was seen, with conical loads placed at the far ports, to

be better than -4OdB:

This design has also been used for the CERN Linear Collider (CLIC) program

in a version scaled down by Lars Thorndahl to function at 30 GHz.

87

6. CIRCULAR WAVEGUIDE OFFSETS

As mentioned earlier, the design of our circular guide 3-dB coupler left us the

problem of devising a way to separate the waveguides emerging from adjacent ports.

This is necessary to allow flanging and tapering to the larger diameter delay lines.

While bends and offsets are relatively straight forward design problems in single-

moded, rectangular waveguide, the situation is not as simple in overmoded circular

guide. Our offsets must not only transmit power, but must also preserve the T&r

mode purity. Power transfered into other propagating modes is of no use in SLED

II and, if not reflected in the final mode converter, would only interfere with the

. operating mode signal.

The T.&r and other standard circular modes are normal modes only of straight,

smooth-walled circular waveguide. In curved guide with circular cross-section, the

curved geometry or altered boundary leads to a different set of solutions to Maxwell’s

equations which look like hybrid combinations of the familiar modes. They can only

be approximated in terms of familiar mathematical functions using perturbation

* theory. Rather than attempting to solve for these modes, we can, for modest cur-

vature, work in the basis of the straight guide modes provided we account for their

no longer being orthogonal by developing a theoretical model for power coupling

between modes.

88

TEol--TMl1 BEND MIXING

By far the most troublesome mode in navigating bends is the TMii mode.

The problem arises because the TE 01 and TMii modes are degenerate. That is,

they have the same cutoff frequency, and thus the same guide wave number. Con-

sequently, if power is coupled from one mode into the other, however gradually,

the effect is cumulative, since the relative phase of the waves is constant along the

waveguide. Other modes continuously slip (or gain) in phase relative to the TEol

wave, and this provides a limiting mechanism for how much power can transfer

into them. Essentially all of the power, however, can transfer into a TMii wave at

proper intervals along a curved guide, as we shall see.

Let’s examine what happens to the fields of the two modes in question when the

waveguide curves. We’ll let our z-axis rotate by a slight amount, 68 ,in the positive

y-direction and project the incoming fields onto this new z’ axis. Let A and B

represent the (complex) amplitudes of a TE 01 and a TM11 wave, respectively, in

the guide, whose combined power is unity.

P = IAl2 + IBI” = 1

The field components are given by:

89

TEol

H, = ANJo(k,r)

(6-l)

H4 = BN2F [Jo(k,r) - C

H, = 0,

where Ni and N2 are normalization factors with appropriate dimensions and k, and

,f? are, respectively, the common cutoff and guide wave numbers for the two modes.

Now, along the slightly rotated axis we get:

E,! = E, cos Se + Ed cos #J sin 68 + E, sin q5 sin 68

* _

= BN2 J1 (k,r) cos q+ cos se + ANY ~Jl(k,r)cos q5sinSfJ

- BN2&Ji(kCr)sin2~sin6B+ BkzgJo(kCr)sin2qSsin60 c = Bcos68+A$~sin68) N2JI(k,+os~

2 c

- BN2 gJz(kCr)sin2q5sin68 C

and, similarly,

H,I = Acos68 + BNw N2 iweo sin 68 Ni Jo( kcr) 1 c >

-BN2$ J2( k,r) cos 24 sin 68 - C

ANi~,Ji(k,r)sinq5sinM. C

where I’ve used the Bessel fuction identities J:(s) = Jo(z) - $Ji(z) and JnS1(x) -

90

‘-. -.

Since longitudinal E fields completely determine the power in TM modes and

longitudinal H fields determine that in TE modes, we recognize in the first term of

each of the above equations the perturbed amplitude of the T&f11 and TEol waves.

The second term in the E,t equation shows there is some coupling of TMlI to TMzn

modes. (Note that since the k, here is not that appropriate for Titcfzl, other modes

must come in to match the boundary conditions, although they may be evanescent).

The second and third terms in the expression for H,I show coupling between TIMII

and TEzn modes and between TEol and TEI, modes respectively. For reasons of

phase slippage, mentioned earlier, coupling to these other modes will, in general, be

small. Hence we shall ignore these extra terms and concentrate on the TEo~-TM~,

coupling.

The i’s in the above equations tell us that the waves couple 90” out of phase.

Let us, then, remove the phases from the amplitudes and set

A = I4 A=d

B = PI, B = 23. * _

Let’s also use the fact that

and

N1 =

2 k,2

d- F jh2 Jo(jh2)d@i$

N2 = -?- k,2 fi jb2 Jo(jh2)6i$

to replace Nz/Nl with dw = firlo.

With these changes, we have:

(6.3)

23’ = BcosSe+d- kc

sinM=B+d---- &kc he

d’=dcosS8-a- dk

sin68 21 A - 23 ik, 68, (64

91

where k = WC = we is the free-space wave number and primes indicate ampli-

tudes in the slightly rotated frame. Going now to differentials, we can write:

dD= &Ad@

-Ic ade dd=JZlc,

.

From

P = A2 + lS2 = 1, (6.5)

we get, by conservation of power,

dD dP - 2/p de

--j-g + 28; = 0.

Substituting the above expression for dZ? yields

dd= ik, 1-d de, --JY

or

Integrating gives

s_im~larly, we get

B = sin

92

Finally, solving for the integration constants in terms of the initial conditions,do

and Z?o, and using (6.5), we find:

d=docos(&8) -Bosin(&8)

B = &cos(&8) +dosin(&e)

We can write these results in matrix form as

cos(fikc k e) 4 & 9

(6.6)

As the TEol and TM~I waves travel around a gentle curve, the power exchange

between them can be seen as a simple rotation in mode space of the above power

vector through an angle proportional to the angle of the bend. The proportionality

constant is determined by the ratio of the guide radius to the free-space wavelength.

This relationship between TE 01 and Tikfll, and the resultant problem in trans-

porting the low-loss TE o1 mode around bends, has been recognized since the 1940’s

[27]. (Parenthetically, the same relationship exists between all TEo,-T$~I, pairs.)

Several strategies for overcoming *it have been devised. We shall return to this

subject in a later chapter.

Our present concern is with the design of an offset, which consists of two equal

bends in opposite directions. Fortuitously, the above analysis suggests that mode

coupling will not be so troublesome in such a device. We can simply allow some

power to be transfered into the TM11 mode in the first bend and count on it return-

ing to TEol in the second bend. The product of the two opposite rotations should

return the power vector to its incident orientation. However, we have ignored effects e -

such as coupling to other modes, and the difference in the attenuation constants of

93

is convenient to normalize the field patterns in each mode to unit power flow. We

can then represent the amplitudes of the forward and backward travelling n waves

as A$ and A,, so that

and

PA4 = c l4m” - c IA,(412~ (6.8)

n n

Each mode is described by a transverse scalar function Tn(p, 4) satisfying

1 d2Tn

-I---= P2 a2 -kciTn, w-9

and the boundary condition

Tn(p = a, 4) = 0 , for TM modes

$Tn(p = %d> = 0 ,for TE modes, (6.10) -

where a is the waveguide radius.

Replacing the subscript m with the usual pair of azimuthal and radial subscripts

and using (nm) to signify TM modes and [nm] to signify TE modes, the solutions

are: *

T(nm) = N(nm) Jn(X(nm) E) sin n4

T[nml = N[nml Jn(X[nm] f) ~0s nd,

* For cross-polarized modes, make the substitutions: sin + cos, cos + - sin.

95

If the expressions for the transverse fields in terms of V’s and I’s in equa-

tions (6.13) are expanded in components; they can be substituted, along with the

longitudinal field expressions, into equations (6.14). Various pairs of the resulting

equations, multiplied by appropriate factors, are then subtracted and integrated

over the guide cross-section with the help of the T orthogonality relations. The

tedious mathematical manipulations at this point are not worth tracing in detail

here. Eliminating the V,(,)‘s and 1,1,1’s involves some matrix manipulation and

approximations that assume

One arrives finally at generalized telegraphist’s equations of the form

dKn -=-

dz c z7n,nIn n

dIna -=- dz c Kn,nKa7

n

* _ where the double-indexed impedence and admittance coefficients are integrals in-

volving the appropriate pairs of T functions. As it is more convenient for us to

work in terms of microwave mode amplitudes, we will shed the transmission line

language and, using equations (6.12), express the results directly in terms of the

Af’s and A-‘s. We have:

. -

dA+ . m = --2 C [Ck,nAz + C,,,A,] dz n

dA- m = +‘C [CtE,nAX + C$,nAi] , dr

(6.15) n e -

where the coupling coefficients are as follows:

99

For identical indices,

c,,, = 0.

,O being the wave number and (Y the attenuation constant of the mode.

(6.16)

For coupling between different modes,

C&),(,) = ; JaZ(m),(n) f k2,,,)~(n)~~~,+(m)~(n)] ,

1

c&),[n] = c$],(m) = ~kz(m,,I.l [J&E] 7

(6.17)

k2Z[ml,[nl - kc~rn~kc[n~~~rnl,~n~

lPiJ%i f E[m],[n] P[m]P[n] 7

\i--1

in which,

T(m),(n) = kc(m)kc(n) s tT(rn)T(n)dSy J _ -

and y[m],[n] = Icc[m] IcC[n] J s tT[m]T[nldS*

For ease of expression in the above, I’ve freely varied the specificity of the

subscripts. A single, undelimitered letter stands for a general mode; a delimitered

letter stands for a mode of the appropriate knd (“( )“=TM,“[ ]“=TE); and a

delimitered pair distinguishes the two indeces of the mode. It is also important to

note that all expressions are for and all sums to be taken over only the modes that e -

can propagate at the given frequency in a guide of radius a.

100

APPLICATION OF THE THEORY

We can now apply these generalized telegraphist’s equations to the problem of

designing and modeling the performance of a circular waveguide offset. The 1.75”

inner diameter of the waveguide emerging from the 3-dB coupler allows fourteen

modes of power propagation. Six of these, however, are cross-polarizations. Since

the symmetry of the planar offset implies coupling to no more than one polarization

of any field pattern, we can eliminate these, and are left with eight modes. Of

these, one is the TEol input mode, and another is the Tkf11 mode, whose coupled

amplitude we were led a couple of sections back to expect to vanish at the end of

an offset. That leaves six potential sources of trouble.

Now, a useful thing to notice about the results of the last section is that the

coupling coefficients between modes all involve &integrals of the form (Remember

c 0; cos 4.):

J 2m

0

SOS C$ cos rnqi cos nq5 dqi = i J 02* [cos(m + l)$ + cos(m - I)$] cos nq5 d4

or

J 27r

0

cos q5 sin mq5 sin n4 dq5 = i J 02r [sin(m + l)$ + sin(m - l)$] sinn$ d#.

It follows that, in a circular waveguide bend, coupling occurs only between modes

with azimuthal indeces that differ by exactly one (i.e. for m = n f 1).

Thus, of the propagating modes, TEol couples directly only to TM11 and TEll.

Since the TMlI mode can achieve considerable amplitude, depending on the bend

angle, we should consider coupling through it to be of the same order as direct

TEol coupling. That means considering TE21, TM21, and Tit&. Assuming the s -

amplitudes of these modes remain small (as we’ll see they do), we needn’t concern

101

I

: - ;

ourselves with the higher-n modes, TE31 and TE41. Finally, it can be shown that

the TMl1 mode that TMol couples to is of different polarization than that which is

coupled in from TEol, so TMol won’t be brought into the picture. We are left with a

total of five modes interacting in the bends of our offset: TEol, TMII, TE11, TE21,

and TM21.

Using the formulas given in the last section, one finds, at 11.424GHz in 1.75”

diameter waveguide with radius of curvature b, the values given in Table 6-A for

the relevant coupling coefficients. Note from their definitions that Cm,n = Cn,m.

Table 6-A

coefficient value

%~ll) 0.982/b

C~~lslll 0.819/b

%s211 -0.864/b

%w) 0.967/b

C[~l],(ll) O

%,[,,) 0.0187/b

0.0718/b

-0.286/b %M211 C- (111.(211

Curvature coupling.

The propagation constants act as “self-coupling” coefficients, independent of the

curvature. They are complex in this treatment. Their values for the aforementioned

parameters are listed in Table 6-B. The imaginary parts, or attenuation constants e -

are easily calculated with the following formulae.

102

Table 6-B

coefficient

%I SOlI

%>01)

Cowl

%1,P11

%m

real part imag. part

= Pm (m-l> = Crm (m-l)

166.1 0.0050

166.1 0.0096

224.6 0.0038

196.1 0.0088

62.61 0.0256

“Self coupling” (propagation constants).

( Xfn m] n2 aY[n,4

= 2 PI.“, (ka)z + Xfn,ml - n2 > R, k ---

a(n3m) - qOa P(n,m)

(6.18)

Here, R, is the surface resistivity of the guide walls, given, for ideal copper, by

R, = = 2.61 x lo-‘a fi,

where 0 is the conductivity and f is given in Hz. In the table above, I’ve added

twenty percent to the calculated o’s to allow for an increase in resistivity due to

surface roughness at centimeter wavelengths [30].

As in Table 6-A, the coupling coefficients between forward and backward waves

(C-‘s) are generally small compared to those between modes travelling in the same

direction. Furthermore, since the phase velocities for such waves are in opposite

directions, they move in and out of phase more rapidly along the guide, allowing

for less interaction. If a bend is gentle enough, one should therefore be able to e -

neglect the backward waves without much loss of accuracy. This will be verified

103

- ;.

later. Equations (6.15) fr om the last section then reduce to

dAm ’ - = -2 dz c

C+ A m,n n

n

(6.19)

for forward waves, where the mode superscripts have been dropped. We can now

specify initial conditions and integrate the above set of coupled, first-order differ-

. ential equations numerically to evaluate different offset designs and examine what

occurs along them.

I b / ,

/ / 0

\ /

/ 0

I I / /

\

/

/ offset / / 0 I

\‘-:_1 \ a- -- .---- I

/

/ I /

I /’

Figure 6.1 Geometry of the circular waveguide offsets.

Our offset should consist of two opposite curves with a possible straight section

between them as in Figure 6.1. We feed power into one end in the TEol mode. (That

is, we start with initial conditions: Apl] = 1; all other A’s = 0.) In the middle of

the- offset, some percentage of the power will be in the TM11 mode, depending on

the maximum angle of the axis. The power will also interact with the TEl1 mode,

104

but the phase of the interaction will shift by 27r in a distance 27r/ApIll],Io1], where

APm,n = Pm -Pn- If the change along z of the TE 01 amplitude is not too rapid on this length scale

and the curvature is constant, each differential TEI1 wave excited will be canceled

by one of opposite phase by the end of this distance. A small amount of power

will beat in and out of the TEpl] mode. The maximum, normalized to unit TEpl]

power, is easily seen to be

~lllnlaX = lAp],,,12 = (~C~O~~,[~~~IAP~~~I,~O~~)~ = (7.84 x 10-4m2)b-2,

and the beat length, to first order, is

lb =

2lr -

lAP[ll],[Ol]l = 10.74cm = 4.229”. (6.20)

This cyclic phenomenon suggests that the the arc length of each curve be

adjusted to equal an integral number of beat lengths. This should make the TEL~~I

amplitude tend to vanish at the middle and at the end of the offset, minimizing

* _ power loss to this mode.

The first offsets we had made, OSl, were designed to have one beat length in

each arc. They were made by bending WC-175 waveguide, and the bender required

a straight section of 8.5” between the arcs for gripping the pipe. The radius of

curvature, and hence the maximum angle, was determined by a desired offset of

2.125”. This is enough to permit connection to components joined by 6”-diameter

flanges. These offsets have been used in our initial tests of the 3-dB coupler.

A second set of offsets,OS2,was designed and fabricated when it was envisioned

_that we would use one of the BPC vacuum manifolds to pump out the first high-

power SLED-II prototype. The ports of this manifold are seventeen inches apart, so

105

- .

this second design was significantly larger and less conservative than the first. Each e

arc is two beat lengths long. It was realized in this design that the length of the

straight section could be adjusted so as to bring the TE[21] amplitude generated in

the first arc back to zero in the second arc. The length chosen is 3~/A&~],(ii). The

first arc brings TEi21~ to about the summit of a beat. For it to turn back down in

the second arc, the change in the sign of the curvature must canceled by an overall

7r phase shift between it and its generating mode.

A third offset design, OS3, was arrived at for use with a second 3-dB coupler

model, which was to be slightly shorter, with a smoother slot profile and better

machining. This offset is close in size to OSl, giving slightly more separation. Like

the coupler, offsets of this design were to be machined out of copper blocks, rather

than bent like the previous ones. It was expected that the specifications could

thereby be more closely met, and the deformation of cross-section and scratching

better avoided. This allowed us to do away with the straight section altogether.

Compactness was a greater consideration for this design, and the benefit of a straight

section seen in the last design was shown to be cancelled by the added wall losses.

* _

The features of the three offset designs are given in Table 6-C. The formalism

developed above was used to model their expected performance, and the theoretical

fraction of power transmitted in the TElol] mode is given in the last column of the s -

table. The power distribution along each offset is shown in Figures 6.2-6.4.

106

1.0

0.8

0.6

0.4

0.2

0.0 0 5 10 15

z (inches)

0.0030

0.0025

0.0020

0.0015

0.0010

0.0005

0.0000 0 5 10 15

z (inches)

Figure 6.2 Power distribution in coupled modes along the axis of offset

OSl.

0.8

0.0

0.4

0 5 10 15 20 25

0.0030

0.0025

0.0020

0.0015

0.0010

0.0005

0.0000 0 5 10 15 20 25

z (inches) z (inches)

Figure 6.3 Power distribution in coupled modes along the axis of offset

107

liLb&i T&l

0 5 10 15

z (inches)

0.0030

0.0025

0.0020

0.0015

0.0010

0.0005

0.0000 0 5 10 15

z (inches)

Figure 6.4 Power distribution in coupled modes along the axis of offset

os3.

design of&et

OS1

OS2

OS3

arc length

2.125” 4.229”

7.53” 8.457”

2.537” 8.457”

Table 6-C

radius of

curvature

0.9971

0.9951

0.9968

offset parameters

One can see from the mode power plots that very little of the power ever leaks

out of the degenerate modes. It is easy to believe that the power coupled to the

backward modes is truly negligible. As a check, I used the numerical method called e -

“shooting” [31] to solve the two point boundary value problem of equations (6.15)

108

I : .

with the coefficients in Tables 6-A and 6-B, the geometry of the offsets, and the

boundary conditions:

A+, = 1, m = [Ol]

0, 7-n # WI atz=O

and

Ai = 0, all m at end of offset.

After- a few cycles, the values of the transmitted powers converge very accurately

to those gotten by direct integration of equations (6.19). The reflected powers in

T&r, TI&r, and TM21 are found to be on the order of 10S4’.

. _

109

7. OTHER COMPONENTS

.

In the preceding two chapters, I described the design of two important elements

of our SLED-II system, the 3-dB coupler and the offsets (s-bends) that allow us to

attach its ports to larger diameter components. Combined, they represent the

heart of the device. The focus on them is further justified by the fact that they

are the parts for which I am most responsible. There are, however, several other

components which we found it necessary to design or obtain in order to implement

the SLED-II concept in a high-power system. These can be seen by looking ahead

to Figure 8.6. They include mode converters, 90” bends, vacuum pumpout/mode

filters, irises, tape-rs, delay lines, and adjustable shorting plungers. I will touch on

each of these in this chapter in order to convey a complete picture of our microwave

circuit.

THE “FLOWER-PETAL” MODE CONVERTER

Power is extracted from the output cavity Lf X-band klystrons into rectangular

WR90 waveguide. To feed this into our low-loss circular waveguide system, it is

necessary to transfer the power from the fundamental rectangular mode to the . -

circular T&l mode. The reverse procedure is necessary to feed the compressed

110

pulse into the accelerator input coupler or rectangular waveguide load. For our

Binary Pulse Compressor, these tasks were accomplished by means of Marie-type

mode converters, as described in Chapter 3. At 27”, these devices, while’comparable

in size to other components, are considered somewhat long. Their two percent

insertion loss is largely due to wall currents. An idea for developing a more compact

and less lossy mode converter was thus greeted with enthusiasm.

In 1991, a small KU band mode converter developed by Microwave Asso-

ciates, Inc. [32] came to our attention. Its operating frequency is N 35 GHz.

Through a program of theory, experimentation, and numerical modeling, SLAC has

adapted the unique design of this so-called “flower-petal” transducer to a scaled-up

11.424 GHz version [33].. Th e g eometry and dimensions of our compact device are

shown in Figure 7.1.

The circular port is at a right angle to the rectangular port. The diameter of

1.6” within the device, chosen to render the T&i mode safely cutoff, is expanded

in a nonlinear taper to the desired 1.75”. In the rectangular portion, a knife-edged

septum, perpendicular to the electric field, bifurcates the guide as its height is

tapered up in steps, to form two parallel waveguides. Each of these is coupled to

the circular guide through a pair of oval shaped irises in its side wall at alternate

45” angles. Seen from the circular port, these suggest the mode converter’s name.

Beyond the irises, the rectangular guides are terminated with carefully placed shorts.

A total of ten modes, counting different polarizations, can propagate in the

1.6” circular guide at our frequency. Power is coupled through the magnetic field

at the irises, which is longitudinal in the rectangular guides. The symmetry of the e -

geometry thus limits the coupling to modes in the circular guide for which Hz is

111

t

I Zt

I I I 1 I 1 II I !I.

- ErrgEE 4 l-w

Figure 7.1 TE ,0-T& flower-petal mode converter.

symmetric with respect to the x-z plane. This precludes the Th.&,l mode as well

i .as one-of each of the polarization doublets. .The rectangular shorts are half a guide

wavelength from the midplane between the irises (i.e. at z = A,/2). If we assume

a standing wave null at the latter position (x = 0), we have the condition that H,

must be anti-symmetric with respect to the y-z plane. This eliminates three of the

remaining five circular modes, leaving only T&l and T&,1. At the 45” planes,

the transverse H fields of the T&l mode are azimuthal, while those of the T&I

mode are radial The radial orientation of the iris holes thus discriminates strongly

against>oupling to T&l, particularly if the iris is thick. -. ,

112

As the power-flow is not truly symmetrical across the y-z plane, some coupling

to TEll may also be expected. However, with the dimensions optimized very good

mode purity was achieved. A post was placed near the rectangular port to match

out reflections. An insertion loss of about 0.7% was deduced from measurements.

A pair of flower-petal transducers, when tested for power handling capability in a

resonant ring, withstood 150MW at a pulse width of severalhundred nanoseconds

without sign of breakdown. These mode converters are rather narrow in bandwidth

(- 5%) compared to the Marie variety, but they suffice for accelerator use, where

they can be designed for a fixed operating frequency.

DELAY AND TRANSFER LINES

The benefits of using the circular TEol power transmission mode have been

explained in the chapter on Binary Pulse Compression. To maximize the power

delivered in our compressed pulse, we decided to use circular waveguide not only

* _ for the SLED-II ~delay lines, but also for transporting power between the klystron

and the compression system, whose proximity was limited by the layout of the

klystron test gallery. We further planned to plumb with circular guide from the

output port of the hybrid down through the roof of the concrete bunker on which it

sat for our accelerator structure tests. The more WR90 we could avoid using, the

less ohmic loss we would have in our system. For the BPC, we had used rectangular

guide for power transfer.

Two sizes of circular waveguide were incorporated in our experiment. For the

transfer lines mentioned ,above, WC175 (1.75” i.d.) was the choice. This has the

diameter used as a standard in our component designs. We could therefore connect

113

it directly to the components without using tapers. The runs would no more than

a few meters long, so the savings to be gained by going to larger guide were not

considered worthwhile. As seen in Table 7-A, the theoretical wall loss for WC175

is,down a factor of five from that of the standard rectangular guide.

For the delay lines, low loss is more of a priority. Energy is effectively stored

in them for several bounces, and, as we saw in Chapter 4, their efficiency strongly

affects the efficiency of the SLED-II process. Having it on hand, we decided to use

WC281 from the dismantled initial delay line of our Binary Pulse Compressor. (The

rest of the BPC was left intact as a possible backup until the successful high-power

demonstration of SLED-II.) Its theoretical loss is down a factor of five from that of

wc175, and its suitability had already been demonstrated in the BPC In a future

upgrade, we would replace the delay line waveguide with larger WC475, reducing

the wall loss by, coincidentally, one final factor of five. The reduction in attenuation

per meter is accompanied by an small increase in group velocity, as indicated in

. _

Table 7-A. Appoximately 8% more waveguide is thus needed for a given delay. The

more relevant attenuation per microsecond is given in the last column of the table.

_ - Table 7-A

Guide dB/m %7/c dB/PS

WR90 0.100 0.819 24.6

WC175 0.0216 0.693 4.50

WC281 0.00406 0.894 1.08

WC475 0.00078 0.964 .225

Waveguide characteristics at 11.424 GHz.

In reality, one cannot expect delay line losses to fall as rapidly and simply as

114

: .

the theoretical wall loss. The effect of mode coupling caused by imperfections must

also be considered. Long sections of waveguide are bound to have curvature, dents,

and some distribution of cross-sectional distortions. Furthermore, the length of our

delay lines requires that they be constructed out of multiple sections. Discontinuities

at imperfect joints are likely to be the chief source of mode coupling. Morgan has

written a paper dealing with continuous distortions, or gradual deformations of

waveguide cross-section [34]. Certain aspects of his method, however, such as the

arbitrary assumptions he must make about the statistical distribution of distortions,

render his results of limited applicability. Let us consider briefly the effect of discrete

discontinuities at joints, over which we have more control.

Coupling coefficients for discrete coupling mechanisms can be derived, analo-

gous to those for the continuous mechanism of curvature presented in Chapter 6.

Doane [35] p resents a general recipe for calculating the coupling caused by various

discrete distortions. A

ple power only between . _

discontinuity of a given azimuthal symmetry m will cou-

propagating modes whose azimuthal indeces differ by that

number.

Let us consider four types of small discontinuities in 4.75” circular waveguide-

diameter changes, axis offsets, tilts, and slight ellipticity mismatches. We’ll include

backwards wave coupling, small compared to forward coupling, only for diameter

changes. Formulae for the relative amplitudes of parasitic modes excited from TEol

by each of these discontinuities, appropriately parametrized, are presented below,

along with the relevant modes for WC475 at 11.424GHz. The general notation is

the same as that used in the last chapter. The subscripts 0 and n indicate TEol e -

and the coupled mode respectively.

115

diameter change:

The propagating modes coupled by diameter changes are TEG, TE,f,, TE&,

and TE$4.

rn = (-1)” XnXO

(A - Po>JmFl .

where 6~0 is the change in radius.

&a0

-p P-1)

offset. L

The propagating modes coupled by offsets are TE11, TE12, TE13, and TE14.

2

rn = 5 &&:~po)&!Tl)

where 6ar is the offset distance.

V-2)

The propagating modes coupled by tilts are TMII, TE11, TEN, TEn, and

T&4.

I? - ilca&), n- JZXO

TMn,

others,

where 68 is the small tilt angle.

(7.3)

ellipticitv:

The propagating modes coupled by ellipticity mismatches are TE21, TE22, e -

.TE23, and TE24.

116

where 6a2 = (d,,, - d,i,)/4 for one side of the joint, the other being considered

circular.

Each of these types of discontinuity extracts a small fraction of power from

a pure TEol wave given by the sum C II’,l” over the coupled modes. A bit of

calculation gives the combined fractional power loss at a WC475 joint as

6P - =. 0.676 ba; + 1.18 6,; + 3.36 6~; + 32.3 6e2, P (7.5)

where the ban’s are in inches and Se is in radians. Waves excited at one joint will

interfere with those excited at others. It is wise to vary joint spacings to decrease

the likelihood of resonant build-up of parasitic modes. This can occur if regular

* _ spacing happens to be 1 = 2rm/(/l, - ,&I) for some coupled mode and integer m.

Mode filtering can also be beneficial. Assuming the coupled mode excitations are

suppressed or, on average, add in quadrature, we can calculate a line joint loss as

the sum of the individual joint losses. .

The manufacturer’s specifications on our WC475 waveguide indicate an outer

diameter accuracy of 0.007” and a wall thickness accuracy of 0.010”. The accuracy _

in inner radius is then Ar N d(0.0035”)2 + (0.010”)2 = 0.0106”. The perpendicu-

larity specification for flange brazing gives A@ N 1 mrad. The guide axis should be

qentered on the flange o.d. to within Ax N 0.007”. The joining of flanges introduces

an additional alignment error Ay N 0.010”. Assuming a uniform distribution within

117

these error bars, we get the following rms values for many joint discontinuities:

(~~O)rnas = J ; Ar z 0.0087”

(Sal)rma = J

;Ax2 + Ay2 N 0.012” t 7.6)

(Sa2)rms = /-

iA?. N 0.0087”

(sqrm8 = A0 N 0.001 rad

Using these in equation (7.5), we get an rms fractional power loss of 5.08 x

10m4 at each joint. This attenuates the TE 01 wave by about 0.00221 dB, which is

equivalent to the resistive wall loss in approximately 2.8m of 4.75” waveguide. If

we can meet the above specifications and avoid resonances, we should be able to

keep joint mode-conversion losses below wall losses with average pipe lengths of 9.3’

or more.

For WC281, the fractional power loss at a joint is

6P/P = 1.366ai + 4.696a: +9.30&a; + 9.78Se2.

Despite fewer propagating modes, the first three coefficients are larger here than

in equation (7.5), b ecause a given error represents a greater fractional disconti-

nuity. Notice, however, that the last coefficient is considerably smaller here. In-

creased guide diameter leaves one more vulnerable to tilts. This is clear from the a-

dependence of the TEol/TMl1 coupling and arises from the extra factor of (pn -PO)

in the denominator for the other tilt-coupled modes. As the cutoff frequencies are

reduced by increasing a, the propagation of the modes approaches that of a plane

waue. Their characteristic angles of reflection off the waveguide wall diminish and

thus draw closer together.

118

The preceding analysis was done to get a feeling for this process of power

loss and to provide a rationale for calling for fabrication tolerances [36]. Ohmic

losses, mode conversion due to continuous distortions, and the limited sensitivity of

our measuring ability preclude any attempt to verify mode conversion predictions.

Besides, the amount of mode conversion can vary by orders of magnitude depending

on the exact transverse and longitudinal profile of the waveguide.

The equations above also shed light on a problem we ran into when we first

attempted a high-power test of our SLED-II system. Having successfully cold tested

it, we connected it to the klystron via a twelve foot run of WC175. When we began

operation, we were puzzled to find our gain down from the cold tests by nearly a

factor of two. We’d expected some additional loss due to the transport line, but

nothing this drastic. A diagnostic autopsy revealed that the loss was due to this

addition to the system. The problem was solved by replacing the run with a section

of WC281 with appropriate tapers brazed to its ends.

To understand this phenomenon, one must consider the mode spectrum of the

guide in light of the above mode conversion analysis. Notice that all of the coupled

amplitude formulae share a factor p,1’2. As k approaches ken, or vice versa, P,.,

approaches zero. Although the equations cannot be applied in this regime (The

coupled amplitude cannot excede unity.), they do indicate a severe sensitivity to

mode conversion to modes near cutoff. * WC175 has two modes whose cutoff

frequencies are very close to 11.424 GHz. They are TE41 (11.416 GHz), and TEl2

. _

* Coupling coefficients which remain finite in passing through cutoff can be ob-

tained by considering the finite conductivity of the walls and using the adjusted e -

wavenumber, given by /3;c = p2 + 2(i + l)@a [37], as pointed out by Lawson [38].

119

(11.445 GHz). The latter is considered the main culprit, as it is generated by m = 1

distortions, which should greatly dominate m = 4 distortions.

It thus turns out that our diameter choice of 1.75” was unfortunate. We avoided

catastrophe in the smaller components, but in the long transport line mode con-

version got us. One can imagine power coupled to TEl2 forming a standing wave

between the flower petal and the 90” bend. The separation of such elements pro-

vided by the line creates a greater density of possible resonances than is present

within the shorter components. Figure 7.2 shows a spectrum of cutoff diameters

for circular waveguide modes at our fixed frequency. 1.75” is the worst spot, being

sandwiched between the indistinguishable cutoffs of the modes mentioned above.

2.81” seems dangerously close to the TE 13 cutoff, but this hasn’t been too detri-

mental in our experience. Nevertheless, our plans for the future involve going to

WC293 for power transport, well away from any cutoffs. WC475 is safely in the

clear, the nearest mode being the non-threatening TM43.

CIRCULAR 90” BENDS

In transporting power from the klystron to the pulse compressor and from the

pulse compressor to the accelerator structure, it is necessary to negotiate some 90”

bends. Two are indicated in Figure 8.6. As explained in the preceding chapter, this

is a non-trivial procedure for over-moded circular waveguide, particularly in the

T&l mode. One could adjust the diameter to give one complete back-and-forth

transfer of power between TEol and the degenerate TM11 in the bend. According

to equation (6.6), h owever, the required diameter is 3.56”. The bend would be

heavily over-moded, and it would be impossible to maintain a small guide radius

120

0 0.5 1 1.5 2 2.5 WAVEGUIDE DIAMETER (inches)

-I-

L

1

1 Figure 7.2 Spectrum of circular waveguide propagation mode cutoff

diameters at 11.424 GHz. The TEol cutoff is marked.

to bending radius ratio while keeping it compact. It is not likely that one could

sufficiently suppress conversion loss to other propagating modes.

As we had done for the 180” bends of the BPC, we called on General Atomics

Corporation to provide us with these components. The bends they supplied are

formed from approximately four feet of 1.75”-diameter aluminum tube, where again

longitudinal corrugations (azimuthal grooves) are used to split the TEol-TMl1

degeneracy. In producing such corrugated tubing, General Atomics uses a special

machining rig to cut the grooves in the inner surface. The dimensions are designed

to minimize conversion loss. Grooves in the outer surface of the thick wall, offset

from the inner ones, allow flexibility without distortion of the circular cross-section.

The curvature of the bends is that of a half sine wave. Support braces connecting

collars on the ends, like a cord of an arc, maintain the design shape. Of the four

121

bends in hand, the average measured insertion loss is about 2%. They are rather

expensive due to the elaborate machining process.

A different design for a TEol 90” bend was developed at SLAC [39]. Rather

than corrugations, we use a pair of partial longitudinal septa, perpendicular to

the plane of the bend, to remove the problematic degeneracy. These are to be

. adiabatically introduced in straight sections before and after the circular arc of the

bend. The guide radius, radius of curvature, and septum dimensions are chosen

so that the propagating modes excited by incoming TEol at the beginning of the

bend recombine with the same relative phase at the end of the bend. Mode purity

is expected thereby to be preserved. The computer code YAP [40] was an essential

tool in fixing our parameters. It is a finite-element field solver capable of solving

for modes with non-integer azimuthal index (azimuthal being around the arc of the

bend).

We have not yet built a test model of our bend. It is currently under patent

review. It should be less lossy and may be cheaper to manufacture than the corru-

gated bend. Another idea, inspired by the success of the flower-petal transducers

is being pursued. It is simply to use a rectangular-guide mitred bend between

. _

two flower-petals. This design is very compact and will most likely be used in the

NLCTA.

VACUUM PUMPOUTS,‘MODE FILTERS

Like the BPC, our SLED-II system involves long runs of circular waveguide

which require evacuation. Short pumpout sections were designed for 1.75” waveg- e -

uide similar to those used in the BPC. It was decided that pumping would be done,

122

at least initially, only at this smaller radius. In the larger, more over-moded guide

of the delay lines, pumping slots would present a greater danger of mode conver-

sion. Even if good azimuthal symmetry were maintained through tight machining

and assembly tolerances, power could be lost to higher-order TEon modes, none of

which propagate in the smaller guide. Such coupling likely contributed to the extra

loss unaccounted for in the BPC. An additional consideration was the fact that

multiple bounces of the wavefront occur in the delay lines, which would compound

any detrimental effects of pumpouts. For our first SLED-II prototype, the delay

lines would be relatively short. The conductance from their far end to the other side

of the irises was calculated and thought to allow sufficient pumping. The delay line

vacuum is somewhat forgiving due to the self-closing configuration of the electric

fields.

Each pumpout section consists of a set of copper disks supported, with spacers,

on three rods. These support rods intersect the disks near their outer perimeter so as

not to perturb the interior fields. The gaps between disks must be short compared to * _

the free-space wavelength (1.033”) to cut off gap modes with azimuthally symmetric

longitudinal magnetic fields that are excited by the TEol mode in the waveguide.

The depth of the gaps must provide sufficient attenuation to such evanescent modes

between the inner and outer radii. We don’t wish to extract power from the oper-

ating mode. Our pumpouts have sixteen l/8” gaps, 3/4” in depth and separated

by disks l/4 ” thick.

While the pumpouts are designed to preserve the TEol mode, they do al-

low leakage of power from modes with azimuthal magnetic fields (or longitudinal

currents). This is a useful feature, as it helps to remove parasitic modes excited

123

-

by imperfections in the transmission system such as waveguide dents and flange

misalignments. This reduces the danger of resonant power loss due to successive

excitations. For this reason, we have often referred to the pumpouts as mode filters.

Figure 7.3 demonstrates the effect of a mode filter in removing sharp resonances in

a section of circular waveguide between two flower-petal mode converters.

Each pumpout was enclosed in its own coaxial vacuum manifold, 4.5” in di-

ameter, with conflat flanges and a pumping port on the side.. Unfortunately, this

was found to pose a danger to their rf behavior. Two of the four manifold encased

pumpouts exhibited much higher insertion losses (a few percent) than had been pre-

viously measured. This is no doubt due to excitation of a resonance in the annular

space. A slight misalignment of one or more disks could couple power from TEol,

through a parasitic mode, into the annular cavity mode. Perhaps the carefully

aligned disks had become cocked when the pumpouts were baked to prepare them

for use under vacuum. We had planned to install one on the end of each hybrid

offset, but had to eliminate the two on the delay line side. This loss of pumping * _

was partially compensated by an accidental gap between the offsets and the hybrid

body, inside the hybrid manifold, on that side. Evidence of negligible loss in the

other pumpouts’was observed.

It is clear that our pumpout design would be improved by machining in a single

piece, to assure better alignment of the disks, and by the inclusion in the coaxial

manifold of some lossy lining, compatible with high vacuum, to absorb the power

extracted from parasitic modes and lower the quality factors of harmful resonances.

Our present plans are to separate the functions of pumping and mode filtering s -

into different components. Pumping will be done through sections perforated with

124

Figure 7.3 Demonstration of mode filter removal of resonance spikes

in circular waveguide between mode transducers. The short section of

e - waveguide used for the upper plot was replaced by an equal-length mode

filter for the lower plot.

125

-

many small round holes distributed around the waveguide so as not to present

perturbation asymmetries on the azimuthal order of any propagating modes. Mode

filters with one or four gaps or grooves are being designed. They will operate in

2.93” waveguide, our new choice for power transport. In the four-gap scheme, the

spacings are chosen to suppress reflection and conversion to the TEo2 mode, which

is not cutoff at this diameter. Lossy material will be included in one gap, recessed

from the waveguide volume.

IRISES, TAPERS, AND SHORTS

The partially reflecting irises necessary for SLED-II operation were also de-

signed for waveguide of 1.75” diameter, to be placed at the ends of the offsets on

the delay line side of the 3-dB coupler. The iris coupling is done at the smaller

diameter to avoid transfering power into higher-order TEon modes, which are all

cutoff at this size but not in the delay line guide. Azimuthal symmetry and the

* absence of longitudinal electric fields prevent conversion to other TE modes or to

TM modes, respectively.

We planned to operate our first SLED-II prototype at a compression ratio of

twelve. This high and inefficient compression ratio was motivated by the desire to

get as much peak power as possible with an available X-band klystron for testing

short experimental structures and was limited by the available pulse length. The

irises were consequently designed to have a reflection coefficient of 0.79, optimal for

a compression ratio of 12 if delay line loss is neglected. Our delay lines were not yet

assembled at the time the iris drawings were submitted to the machine shop. The e -

optimum reflection value is, however, quite broad. Had we waited and measured

126

: .

the delay line loss before optimizing the reflection, we might have gained no more

than half a percent increase in peak power.

The design was done using MAFIA and an S-matrix technique for symmetric,

two-port microwave junctions [41] and was later checked with a mode-matching

code. It entails an azimuthal ridge formed by a step in diameter from 1.75” down

to 1.55”, followed by a step back up 0.080” beyond. Two such irises were machined

out of steel blank-off flanges and plated with copper.

The actual reflection coefficients of the irises were determined in two ways. One

was to insert them between a flower-petal mode launcher and an absorbing load and

to measure Sir with the Network Analyzer. The other was to calculate them from

the initial reflected power when a delay line was excited through them with a signal

generator. A single line and flower petal were used and a circulator picked out

the backwards signal. Although.the measurements agreed, the latter method was

slightly more accurate due to the uncertainty caused by the non-unity VSWR of the

former setup without an iris. The reflection coefficients, s, were both found to be

about 0.805 f 0.005. When the measured machining errors were taken into account

in the S-matrix calculational technique, the predictions ageed well with this slightly

higher value.

With the irises in WC175 and the bulk of the delay lines in WC281, a transition

between these two diameters was required. Short, linear tapers (Z 8”) between 1.75”

and 1.84” were machined out of pieces of WC175 and fitted with flanges. These

allowed us to then employ the 1.84”-to-2.81”, nonlinear, General Atomics tapers e -

from the Binary Pulse Compressor to complete the transition. They also provided

127

.

sufficient distance for any evanescent TEo2 amplitude generated by the irises to

decay before passing through cutoff.

At the other end of the delay lines electrical shorts were required to close the

extended cavity. For reasons made clear in the Chapter 4, mere blank-off flanges as

end-caps would not be suitable. First of all, the delay line lengths must be equal

to within a few thousandths of an inch in order to present equivalent loads to the

hybrid ports. This would be a prohibitively tight construction tolerance, especially

with the delay lines being assembled from multiple segments. Furthermore, accel-

erator application requires that we work at a precise design frequency at which the

delay lines must be resonant. Finally, one must be able to compensate for changes

in the phase delays of the lines due to thermal expansion and contraction. The

temperature of the lines will be affected both by internal rf heating and by fluctua-

tions of the ambient temperature. A good approximation for the phase sensitivity

to temperature, taking into account expansion of both length L and radius a, is

given by

A4 = 2Pol L( k/P)2 KAT, (7.7) '

where K. is the coefficient of thermal expansion, N 1.6 x lo-5/“C for copper. It is

therefore necessary to be able to agilely tune each delay line length.

To this end, we used non-contacting shorting plungers attached to motor-

controlled vacuum feed-t hroughs . These consist of 3/4” thick, aluminum disks,

copper plated and supported on the ends of steel rods. Aluminum was used to min-

imize torque on the rods, because perpendicularity of the shorting surface to the . -

guide axis is essential for mode preservation. A tilt in a reflecting short is equivalent

128

to a tilt of twice the amplitude at a joint. The thickness was necessary to prevent

power from seeping through the 0.050” gap between the disk and the waveguide wall

and exciting the cavity formed behind the disk. Gap modes, being cutoff, would

decay along the edge of the disk.

The face of each shorting plunger was copper plated to minimize ohmic power

loss in the reflection, as the surface resistance of aluminum is about 28% higher

than that of copper. We can calculate the loss in a reflection as follows.

The transverse magnetic field in the TE 01 mode is strictly radial and is given,

from Chapter 6, by

J W[Ol] Jl(X[Ol]Pl4 H,=- -

xv0 k aJo (xpq > (A+ -A-),

where A+ and A- are the normalized forward and backward wave amplitudes. The

similar expression for E+ contains a factor (A+ + A-), requiring that at a perfect

short A- = -A+. Thus the tangential magnetic field on the shorting face is

7f J1(X[01]P/4 Pa

aJoX[ol] .

The relative ohmic loss, normalized to the propagating power, is

AP -= P

17ftan12 ds

= Rs 8 P[Ol] a

< a2J$(xp11) k J J,2(X[Ol] Pl4P dP.

o

Here R, is the surface resistance, approximately 0.0279 s1 for copper at 11.424 GHz.

70 is the impedance of free space, and the integral is over the disk face. Since x[ol~

is a root of Jr, we can use the normalization integral,

-

. -

J

a a2

129

to get

AP 4% P[OII J;(x[oq) -E-P P 70 k Ji(x[oq)’

Finally, from the recursion formula

Jv-l(X) + Jv+1(x) = $J”(X)

we see that

E = 4& P[OII --.

P 70 k (7.8)

In the limit of large radius a, @[or1 + k = w/c, and this reduces to the loss for plane

wave reflection.

. .

For the 4.75” waveguide used in later models, equation (7.8) amounts to ap-

proximately 0.0285% absorption (- 0.00124 dB), q e uivalent to 1.67m of wall loss.

For the present diameter of 2.81”, we get 0.0265% loss (O.O0115dB), equivalent to

0.283 m of additional guide.

When these shorts were inserted into the ends of the delay lines, flanges at their

bases made a vacuum seals with the waveguide flanges. Vacuum bellows allowed

the rods to be moved in and out by stepping motors over a range of one inch. This

is more than half the guide wavelength of l.l”, so we were guaranteed to be able

to tune to a resonance. The stepping motors are capable of moving the plungers in

steps as small half a mil.

. -

130

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8. SLED-II EXPERIMENTAL RESULTS

I’ve described various elements that comprise our first experimental SLED-II

system and their functions. In varying degrees of detail, I’ve presented the theory

behind their design and measurements or predictions of their individual perfor-

mantes . In this chapter I will cover the integration of these components into a

working rf pulse compressor. I will present experimental data from both low-power

(cold test) and high-p ower, high-vacuum measurements. I will then describe tests

of an X-band accelerating structure, including dark current measurements, powered

through SLED-II.

LOW-POWER EXPERIMENT

The prototype SLED-II pulse compression system was assembled on top of the

concrete bunker referred to as ASTA (A ccc erator 1 Structure Test Area), the delay

lines extending above the ceilings of adjacent rooms. Initial tests of the high-power

system were done at atmospheric pressure and power levels of a couple of hundred

milliwatts [42].

The 3-dB coupler, because it was merely bolted together as a cold-test device, s -

had to be placed inside a large cylindrical vacuum manifold, custom made with

131

: .

four pump-out ports, to prepare it for high-power use. The offsets were inserted

through the manifold endcaps and brazed in place so that they abutted the hybrid

ports when the endcaps were bolted on. This whole assembly was mounted on an

aluminum table with braces to prevent the copper offsets from being bent. I!

The first test done on this coupler/offsets combination was a measurement

to determine the difference in electrical lengths between the two offsets on the

delay-line side. If the irises to be placed at their ends were not equidistant from

the coupler slot, the power-directing function of the hybrid would be adversely

affected. The movable shorts were attached to the delay-line offsets and adjusted to

achieve maximum transmission between the remaining offsets, as measured with the

HP Network Analyzer and two mode transducers. The shorts were then carefully

removed, and their extensions measured to differ by 0.060”. To compensate for this

length difference, the irises were machined with their ridges displaced 0.030” from

the flange midplane, to be installed with opposite orientations.

. _ The experimental setup shown in Figure 8.1 was used to measure the round-

trip loss in the coupler/offset assembly. Data was taken over a frequency range

of 100 MHz centered on the design frequency. The relative position of the shorts

was adjusted for maximum transmission. It was also necessary to move the shorts

in unison slightly to reestablish the maximum as the frequency was changed. We

attribute this to a resonance trapped between the shorts and the flower-petal mode

launchers. In the actual system, these components would not be in such close prox-

imity, and the Q of any such resonance would be greatly reduced. To avoid relying

on nonlinear detector crystal calibrations, we moved the same crystal between the s -

input and output directional couplers and used an accurate variable attenuator to

132

X(nm) = jnm = the rnth root of J,

‘I jd”“,,

= the mth root of J’ n X[nml =

m

and **

N en

(nm) = $.

1

JF X(nm) Jn-1 (x(nm)>

N Inml = fi$&+ Jn(xInml> ‘En =

,n#O ,n=o

{ 2 ,n#O 1 ,n=O

The cutoff wave numbers in equation .(6.9) are given by

k,, = 2. a

The 2’ functions are normalized such that

J qTn . qT,dS = kc; J

T;dS = 1, (6.11) s s

where S is the cross-section of the waveguide. The T’s and their derivatives also

. _ satisfy orthogonality relationships.

We can define voltages and currents as follows:

_ - Vn = KAj2&(Az + A,) (6.12)

I, = K,1/2&(Az - A,),

where

K *PO k [nl = iZj$ = &-/lo

** Note: These N’s are different from Nr and N2 of the last section, which carried s -

dimensionality.

96

are the wave impedances for TM and TE modes, respectively. Here the ,8’s are the

guide wave numbers, given by

Pn = dk2 - kc:.

The minus sign in the definition of In reflects the reversed magnetic field, which

yields the reversed Poynting vector, of the backward wave.

The transverse fields for an arbitrary superposition of modes can now be written

in terms of the T functions as follows:

n n ,

n n

=-

_ - We must now consider the effects of our curved geometry. The natural set

of orthogonal curvilinear coordinates to use are the “bent cylindrical coordinates”,

p,4, and z. z therefore becomes the longitudinal coordinate along a curved axis,

whose radius of curvature we’ll call b. q5 is taken to be zero in the plane of the bend,

on the side farthest from the center of curvature. The metric of this system gives a

length element dl such that

d12 = e;dp2 + e$dd2 + etdz2

97

: . .

with

ep = 1, e4 = f-3 and e z=1+5,

where

B = f cos fp

. To complete our set of fields, we can now give the longitudinal fields in this curved

system by

erEr = c kc(n)Vz(n)(z)T(n)(P) 4) n

where Vz(n)(z) and Iz~nl(z) are to be determined. *

Maxwell’s equations, in this coordinate system, take the following form:

* In straight guide, we would have simply

(6.14)

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Mwable shafts

III kolator signal IN

‘10 Isolator Attenuator Amplifier Si. Generator PWLSER

I

SCOPE

Figure 8.1 Schematic of 3-dB coupler network transmission test.

reproduce the same voltage. Subtracting the measured loss of the WR90 compo-

. _ nents and mode launchers and the difference in coupler calibrations then yielded

the insertion loss of the shorted hybrid-offset assembly. The results are shown in

Figure 8.2. The slight variation with frequency is not surprising for an over-moded

component of this size. The average loss is seen to be 7.3 f 2.0%) or N 0.33 dB.

Delay lines, 32’ 7” in length, were assembled, each from three unequal sections

of WC281. The quality of these old waveguide pieces, taken from the BPC, was not

as good as desired. The delay lines were bowed and had discernable dents and bad

welds. In the interest of timely progress, however, they were judged acceptable. An

upgrade was already in our plans. The diameter tapers were attached to complete s -

the delay lines.

133

I :

0.975

0.950

0.925

0.900

0.675

0.850 I I I I I I I I I 1 I I I I 1 I I I I I I I

11.42 11.422 11.424 11.426 11.428

FREQUENCY (GHz)

Figure 8.2 Measurements of round-trip transmission, r]h, of shorted

3-dB coupler/offsets assembly in the vicinity of the operating frequency.

. The round-trip loss of each line was individually measured with a set-up similar

to that used for the hybrid. As the directivity of our diagnostic couplers was found

to be poor, we used a three-port circulator between the input coupler and the mode

launcher. The backward signal was measured by a coupler on the third port, which

in turn was terminated with a load. Rather than rely on direct measurements of

small losses, we used the following procedure. With the an iris inserted and a -

tunable short on the other end, we drove the line on resonance with a pulse long

enough to approach steady-state. We then measured the backwards power during

the time bin just after the input pulse ended. A jump similar to SLED-II operation, s -

but smaller, is seen. Taking the limit of equation (4.5) for large n and solving for

134

the round-trip field attenuation, we find

e-2r = Ee( 4Ei

1 - s2 + sEe(m)/Ei’ (84

The reflection coefficient had been previously determined, and the field ratio is just

the square root of our relative power measurement. We thus calculated the round-

trip power loss in each line both to be, 4.6 f 0.5%) or about 0.20 dB, corresponding

to e-2r N 0.977. This is double the calculated ohmic loss, indicating non-negligible

mode conversion.

The delay lines, complete with irises, tapers, and shorts, were then attached

to the hybrid offsets. Hundred-foot cables .were prepared so that we could drive

the shorting piston motors remotely. The motor controllers were hooked up to a

PC, and a program written to move the shorts simultaneously, in either the same

or opposite directions, with a chosen step size. As mentioned earlier, pump-outs

which we had intended to include either just before or just after the irises had to be

removed from our design due to excessive insertion loss. Pumpouts were connected

to the input and- output offsets. To these were attached the flower-petal mode

converters and the remaining instrumentation of Figure 8.1. To our drive circuit

was added a PSK (ph ase shift keyer) triggered to reverse the input phase 75ns

before the end of a C,-bin pulse.

Figure 8.3( ) h a s ows a digitizing signal analyzer trace of the output of our system

when tuned and driven for a compression ratio of twelve. The input pulse was 900 ns

long, with a phase flip at 825ns. It exhibits the expected step behavior and flat

output pulse amplitude. For comparison, Figure 8.3(b) illustrates the theoretical

SLED-11 output power for parameters in agreement with our measurements which

give the same gain. The relative heights of different time bins differ slightly due to

135

I I I I I I I I I

c) I

0 a

-m

i y-y ,-y$y=y-y ,

1 -- -, -- -------

-468 0 100 ns/div 532

5-L 5-L I,,, I,,, II,, III, III, II,, II,, III, III, II,,

c,=12 c,=12

4- 4- s=O.805 s=O.805

qh=O.22 qh=O.22 w w

e-2T=0.976 e-2T=0.976

3- 3-

2- 2-

1: 1: 0 0 200 200 400 400 .600 .600 600 600 1000 1000

t bs> t bs>

Figure 8.3 (a) Lo w- p ower measurement and (b) prediction of the output e - power waveform for our SLED-II prototype with a compression ratio of

twelve.

136

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the non-linearity of the crystal detector.

The waveforms in Figure 8.4 are to scale, normalized to the input power level.

They were obtained by multiplying the recorded scope traces by the calibration

function of the crystal detector. In 8.4(a), there is no phase modulation. The peak

in the thirteenth bin corresponds to E,(13)2. 8.4(b) shows full SLED-II operation.

The finite risetime of the crystal used, evident in the input glitch at the phase

shift, and sensitivity of its calibration at higher voltage levels both contribute to

the apparent degradation of the compressed pulse flatness. The amplitude of the

first bin in both plots is seen to be about 0.6 N S2qh, as expected.

The maximum gain measured for C, = 12 and the corresponding efficiency

were

G, N 4.85,

qc N 4.85/12 N 0.404.

The ideal efficiency for this compression ratio is only vi = 0.499. As mentioned -

. _

_ -

earlier, this low efficiency reflects the greater priority we gave to high-gain in this

experiment. Forseen SLED-II applications will employ much lower compression

ratios (- 5) with significantly higher efficiencies.

From our previous measurements, we have the hybrid efficiency as r]h P 0.925.

Finally, from Figure 4.4 we find, for C, = 12 and the measured delay line loss of

4.5%) ~1 N 0.887. Equation (4.8) would thus predict qc = ?j)jqhq[ N 0.409. Given the

uncertainty in qh and 72, and even in Q (Recall that s is not exactly optimized.),

our measurement is well within the error bars of this prediction.

-

. - To further characterize our system, we measured the gain at different com-

pression ratios. Our data is presented in Figure 8.5 for C, = 2-16. The solid line

137

-

2.5

2

1.5

1

0.5

0

I . I . I . I 1 . I ’ ! . I *

. --Put -Input

~ -. . .”

F --A- 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

Time (ps)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 mt? w

. - Figure 8.4 Input and normalized output pulses of the SLED-II system

(a) without and (b) with phase switching.

138

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:. .

Gain, s=O.805

2 “’

-- Gain With Theor. Delay Line &

Hybrid Wall Losses

s -- Measured Gain

_ FIT RND. TRIP LOSSES: Hybrid-8%, Delay Lines-4.8% _

O- 2 4 6 0 10 12 14 16

Compression Ratio

Figure 8.5 SLED-II prototype ideal gain, predicted gain with theoretical

ohmic losses considered, and measured gain as functions of compression

ratio. The line through the measured data is a theoretical fit based on the

given total round-trip losses in the coupler and offsets and the delay lines. s -

These parameters are consistent with our experimental determinations.

The reflection coefficient is taken to be 0.805 in all three curves.

139

Rectangular Wave !guide\ LOad

Circular

90” Bend

Mode Transducer ~u’lZIZl Circular

’ Drive/Phase Shifter

Figure 8.6 Schematic of prototype high-power SLED-II system.

141

I :

As mentioned in the preceding chapter, the circular waveguide run was origi-

nally made in WC175. Sensitivity to mode conversion at this diameter was demon-

strated by the excessive power loss ‘we observed - far greater than could be at-

tributed to resistive wall losses in the input line. Consequently, this waveguide was

replaced with a run of WC281 and appropriate tapers.

The system was fitted with vacuum pumps at various locations. The hybrid and

delay lines were wrapped with electrical heating tape and insulated with aluminum

foil. These sections were then heated to around 150” C during initial pump-down.

Such in-place baking accelerates the outgassing of water vapor from walls, allowing

the system to be evacuated more rapidly. The tape and foil were later removed.

. _

When vacuum readings reached the 10 -8-10-7 range, power was fed into the

system. The klystron voltage had to be brought up slowly to its design value of

440 kV. A variable attenuator on our klystron drive signal allowed us to dial up the

power level. The rf has the effect of stimulating outgassing, or desorption of gas

molecules from the copper. As the power was gradually raised, the increased out-

gassing had to be accomodated by the pumps. To protect the system, and especially

the klystron, from being damaged by rf breakdown, trips were set on the vacuum

gauges to shut the klystron off whenever the pressure became too high somewhere.

Although various parts of the system were troublesome in turn during rf processing,

the vacuums tripped most frequently near the rf windows and the mode converters.

This is expected from the greater surface-area:to-volume ratio. Multipactoring in

the hybrid was observed at a certain power range, but was processed past.

When we’d reached a few megawatts out of the klystron s -

peak compressed power the data in Figure 8.7 was taken. The

142

and close to 20MW

diamond at C,. = 12

5

4

d ‘l-l

$

-i

3

2

1 B

1

0 cl E

I

0 -- Ideal (lossless) Gain, s=O.805 # -- Measured Gain 0 -- Typical Measurement

ot I I I I I 1 1 2 3 4 5 6 7 8 9 10 11 12

Compression Ratio

Figure 8.7 Megawatt-level measurements of net gain of SLED-II proto-

type and input transport waveguide as a function of compression ratio.

indicates an average gain of about 4.5 that we saw, the datum from the particular

. set being a bit better. Irregularities in the shape of the klystron pulse, due to the

modulator pulse shape and to small reflections, as well as imperfect compressed

pulse shapes made gain measurement a rather imprecise procedure. The ten per- . _ -

cent error bars in the figure reflect this and the fact that the necessary diagnostic

couplings, on the order of 50dB, could not be expected to be accurate to better

than a couple of tenths of a dB. The overall indication was that a few percent of

our power was lost in the waveguide connection between the klystron and pulse

compressor, as expected.

s - While XC2 was being used elsewhere in an experiment related to klystron

development, an experimental X-band accelerator structure was installed in the

143

:. .

ASTA bunker, and rectangular waveguide was plumbed through the roof to connect

it to SLED-II. This structure, with high-power loads connected to its twin output-

coupler waveguides, then served as the new load for our pulse compressor as we

processed it up to higher power levels.

During this procedure, two ceramic rf windows failed. At around 80 MW com-

pressed power, a window at the output of SLED-II broke, and at around lOOMW,

another near the structure gave out. These had been included to separate the vac-

uums of different parts of our experiment. The former had allowed us to connect to

the structure while maintaining SLED-II under vacuum. In each case, the window

was removed and its holder replaced with a WR90 spool piece. This resulted in a

single extended vacuum envelope, except for the klystron. The output waveguide

of the klystron was split by a magic-T hybrid into two arms and then recombined

with another hybrid. Between the magic T’s, in each arm was a pair of rf windows,

four in all, for double protection of the tube. The splitting halved the amount of

. _ power each window had to handle. The precise mechanism of window failure is not

well understood, but progress is being made in their development [44].

_ - The maximum output power we reached with this SLED-II system was 154 MW.,

High-power waveforms are shown in Figure 8.8. The shape of the klystron output

pulse shows features reflecting the SLED-II pulse shape, particularly in the first and

last time bins. This is indicative of imperfect power direction, which can arise from

unequal division in the 3-dB coupler, unequal iris reflection coefficients, or unequal

rf phase lengths between the irises and the coupler. Though we had adjusted the

latter, a vacuum leak during baking had made it ‘necessary to remove one endcap

of the hybrid manifold and reassemble it, which may have shifted things slightly.

144

.

,

200ns/div

20ns/div

Figure 8.8 High-power SLED-II waveforms. horn top to bottom are: the

signal reflected toward the klystron, the klystron output, the modulator

pulse, and the SLED-II output. The lower plot is an expanded view of the

compressed pulse.

145

I : .

The signal reflected toward the klystron is also shown in the figure. Its magnitude

was on the order of a percent of the klystron output. The broad inverted curve is

the modulator pulse.

A phase bridge was used with a reference signal from the frequency generator

to get a measurement of the phase stability of the compressed pulse. This revealed,

l after an initial overshoot, a f5” ripple followed by a fairly flat region. This fea-

ture, along with the amplitude variation, will be ameliorated in the future by the

development of broader-bandwidth klystrons.

ACCELERATOR STRUCTURE TEST

The compressed pulse produced by our prototype SLED-II was used to run

high-power rf tests on a 75cm, constant-impedance structure in the ASTA bunker

[45]. Table 8-A lists some of the characteristics of this disc-loaded, copper structure,

a prototype for the NLC(TA) structure.

* The ASTA b-earn line [46] is shown in Figure 8.9. As the electron gun had not

been commissioned at the time of this experiment, all elements upstream of the

structure were disconnected and the pair of quadrupole magnets were turned off

for this experiment. The full instrumentation will be operated in late 1994 with a

longer, modified structure and the upgraded SLED-II system.

An in-line Faraday cup gave a measure of the accelerated dark current, or

field-emitted current, emerging from the section, and a spectrometer, consisting of

a 1.6 T variable bending magnet and a collimating slit and Faraday cup in a 45” line,

allowed us to analyze the electron energy spectrum. In addition, directional couplers s -

on the input and output waveguides of the structure were used for rf diagnostics,

146

Table 8-A

Structure Parameters.

and scintillating crystals connected by optic fibers to shielded photomultiplier tubes

were used to measure radiation along the side of the structure.

. _ Some data from this experiment is shown in Figure 8.10. The top waveform

is just the bremsstrahlung radiation from field-emitted electrons slamming into

copper. The second and third waveforms are the compressed rf pulse entering

_ - and leaving the structure, respectively. Finally, the bottom waveform represents

the dark current measured in the Faraday cup.

Computer simulation of the structure can help us to understand the features of

our experimental data. This was done with a program based on an equivalent-circuit

model as described in the Appendix *. Figure 8.11(a) shows the rf envelopes of an

* The example parameters used in the Appendix are for a different, but similar s -

X-band structure containing thirty cells.

147

Gate Velve Electron 01

/ W!r Scintilletor /

Aaalerator L \\\-

Prdhmchsr stNoture Lens

Figure 8.9 Schematic of the ASTA beamline.

. input pulse and the corresponding output pulse computed for the 75cm structure.

In addition to the expected 52ns delay and attenuation (not evident in uncalibrated

experimental waveforms), we see the distinct increase in transient amplitude oscil-

lations observed experimentally. This ripple is due to dispersion in the disc-loaded

structure.

The large spike near the beginning of the -dark current pulse can also be ex-

plained. To be effectively accelerated, an electron must be captured in an rf bucket

of the traveling wave. That is, it must approach the phase velocity (of the funda-

mental space harmonic) quickly enough to keep up with the accelerating phase of

the wave. F’rom equation (10) of reference [47], one can see that the minimum rf

148

:

.

Figure 8.10 Waveforms from high-gradient structure test. From the

top are: detected radiation, the structure input rf, the structure output

rf, and the dark current (Faraday cup signal).

149

field amplitude for capture, normalized to mc’/e, is

& thresh. =$--(d%T- JGj-PPP),

where k is the free-space wavenumber, & = T+/C is the normalized phase velocity,

and p is the initial electron momentum normalized to mc. ,Electrons emitted off the

copper surfaces of the structure may be considered to start with essentially zero

longitudinal momentum, so, for dark current, the above expression reduces to

At our frequency, the threshold gradient for capture from rest (at ,f3p = 1) is

61.2 MV/m. Finally, to see the effect of small variations of the phase velocity from

the speed of light, we let pp = 1 - r and find, to order e1j2,

&O -g1-a). thresh. - (8.2)

The threshold field-for capture from rest drops sharply, and thus the amount of dark

current one can expect to capture near the expected threshold in a short section

increases sharply, as pp drops below its nominal value of one.

By comparison of the phase of voltage oscillations in adjacent cells, the ECM

program can be used to monitor local phase velocity. Figure 8.11(b) shows pp at

the tenth cell as the leading edge of the rf, with an 8ns risetime, passes. Although

in steady state & = 1, dispersion causes a dip as the structure fills, which accounts

for the spike on the dark current pulse. The voltage in the tenth cell is also shown,

and its variation would appear to enhance this effect. The phase velocity dip is due s -

to the higher-frequency content of the pulse, and, as one would expect, the dark

151

. El

1.0

0.8

0.6

0.4

0.2

0.0

1.04

1.02

!- I I -I

I I I I I I I I I I I I I I I I I

0 25 50 75 100 125

I I I I I I I I I I I I I I I I I I I I-

P

I’.

04

r \ . ./! ..\..... . . . . . \I% . . . . .

0.98

0.96 _ -

; I ; ’ / \ /‘,A - ’ I \/ J

I I I I I III I I

’ /I I I I I I I I I I I I-

O 10 . 20 30 40 50

t (4

Figure 8.11 Equivalent circuit model simulation of x-f pulse propaga-

tion in the structure. (a) the power (field squared) in the first and last

s cells, with dispersion evident. (b) the field amplitude and the local phase

velocity in the tenth cell.

150

I :

5.00

z a E 1.00

. E 5 0.50

z - 2

0.10

5 10 15 ELECTRON ENERGY (MeV)

20 25

Figure 8.12 Energy distribution of measured dark current from the

75 cm structure at gradients of 64 MV/m and 69 MV/m.

current spike was seen to decrease with increasing rf risetime and corresponding

narrowing of the Fourier spectrum.

Figure 8.12 shows the energy distribution of dark current measured by the

spectrometer at two different average accelerating gradients. These agree quali-

tatively with simulations performed by Seiya Yamaguchi [48] at Orsay, in France,

except at the very low end, where electrons were not stiff enough to all reach the

Faraday cup. The relatively flat regions in these energy distributions indicate fairly

uniform capture from along the length of the structure. While the local gradient

is greater towards the input end, so is the probability of electrons originating there

152

: . .-

being intercepted.

In Figure 8.13, the total dark current, as measured by the in-line Faraday cup,

is plotted as a function of average gradient. Two sets of data are presented, along

with the dates they were taken. Evident from these is the beneficial effect of rf

processing of the surfaces in reducing field emission over time. The field gradients

we were able to produce in this structure did not extend far enough above the

capture threshold for a Fowler-Nordheim plot to be meaningful. Extrapolating from

our data, we expect a dark current at lOOMV/ m of about 0.5mA. With a design

current of more than an amp in the NLC, this is considered tolerable in terms of

background and beam loading. Quadrupole magnets along the linacs will overfocus

local dark current, whose energy is much lower than the beam’s, preventing it from

accumulating.

During initial rf processing, we were able to observe occasional breakdown

events in the structure. The dark current would jump breifly by a factor of about

- . _ a hundred, and the relative signal amplitudes from our scintillator array would

indicate where the breakdown occurred. We eventually reached 34MW out of the

klystron, 154MW out of SLED-II (G, N 4.5), and 131 MW into the structure. This

corresponds to a maximum accelerating gradient of 90 MV/m at the input end -and

an average gradient of 79 MV/m.

This experiment was part of an ongoing series of high-power X-band structure

tests which utilize rf pulse compression. These results have been included here to

demonstrate an application of the pulse compression development and to tie that

work-into the overall linear collider program. The X-band klystron (with modula-

tor), SLED-II system, and accelerator structure together represent a prototype of

153

.,*.r.,.*. .I... .,.*

7Scm Section

October 9,1993

Pulse Length: 75ns

lo3 L ’ ’ a * * ’ ’ ’ B a ’ ’ ’ L ’ ’ ’ 4 50 80 70 80

Average Accelerating Gradient (MV/m)

Figure 8.13 Dark current amplitude as a funcion of accelerating gra-

dient in the 75cm accelerating structure before and after prolonged rf

conditioning. .

154

9. VARIATIONS ON A THEME

In this chapter, I wil1 explore some modifications to the SLED/SLED-II idea.

While they may not all be practical, they should at least be of theoretical interest.

Moreover, they point to possibilities for further development in the area of pulse

compression. The section on Ramped-SLED-II describes a technique which we are

actually implementing in the NLCTA. I will include, for completeness, highlights of

foreign pulse compression work.

AMPLITUDE-MODULATED SLED

Through modulation of the input amplitude after the phase reversal, it is pos-

sible, in theory, to achieve a constant amplitude output pulse from ordinary SLED

cavities. The shape of the input pulse may be tailored so that the exponential dive

of the emitted field is canceled by.a rise in the direct, or iris-reflected, field. To

derive the required pulse shape, we begin with equation (2.2) for the emitted field

amplitude of a cavity driven on resonance, repeated here for convenience.

T dEe cx + Ee = OEin,

where Z’, = 2Q~/w and a = 2@/(1+ ,B). The Laplace Transform of this gives e -

155

or

(1 + sTc)Fe(s) = afin + TcEe(tl), (9.2)

where Ee(tl) is the amplitude at the beginning of the input modulation (i.e. at

the time of phase reversal). If we normalize fields to the constant amplitude of the

input from t = 0 to ti, we have

Ee(tl) = a (1 - e-tl/Tc) .

We require that the total output field during the compressed pulse be constant.

Eout(t) = Ee(t) - Ein(t) = C, t1 < t < t2.

The Laplace Transform of this condition is

Eliminating Fe(s) between equations (9.2) and (9.3), we get the following expression

for Fin(s).

F, (s) = Tc(Ee(tl) - c> - c/s rn

l-a+sT,

= Ee(tl)- C C/Z (9.4)

s - (a - l)/Tc - s[s - (a - 1)/T,]’

Taking the inverse Laplace Transform and substituting the above expression for

E,(tl), we find the required input amplitude modulation to be

Ei,(tl < t < t2) = 5 C + 1 _ e-tl/Tc _ -

> ,,(--1)(+-WTc .

o-1 (9.5)

In equation (9.5), C is an arbitrary constant representing the desired amplitude

magnification. The effective gain of the SLC SLED system is 2.6, corresponding to

an effective C of 1.612. If we desire the same gain with a flat output pulse, the

‘156

2

1

0

-1

I I I I I I I I I I I I I I I_’

. . . * . -. . . * -,

. . r - -*- -

1 - I

-. -.

E- *. I - in 1 i-

! - I -

0 1 3

1

Figure 9.1 Input (solid) and output (dashed) waveforms for amplitude-

modulated SLED. The dotted curves indicate the waveforms in normal

SLED operation.

. _

required input modulation is as shown in Figure 9.1, where a negative field indicates

a K phase shift with respect to the initial input.

The standard SLED waveforms are also shown for comparison. If we calculate

efficiency as the effective power gain times the compressed pulse width divided by

the integrated input energy, we find that with the new method it goes up from 0.61

to 0.65.

In the implementation of this technique, however, practical considerations are

likely to limit what values C can take on with a real power source. Notice that at

the end of the input waveform in Figure 9.1 the amplitude exceeds the level during

157

I :. . .-

dividing out the implicit eiwt factor, consider the real and imaginary parts of the

input field as driving independent SLEDS, represented by the real and imaginary

parts of the cavity fields. Superposition allows us to do this. We solve the standard

SLED equation for each system to get Eoutl(t) and EO,ll,(t) and then use

Eout&) dout(t) = tan-l Eoutl(t).

(9.6)

We desire a constant output amplitude compressed pulse, so we write

Using EOuti = J?& - Eini, along with equation (9.1), appropriately indexed, we see

that

dEouti -= dt

6 (LYEini - ECi) - %, i = 1,2. c

For an input of normalized constant amplitude varying in phase as 4(t),

. Einl = cos 4(t), and Einz = sin 4(t).

With these equations, our condition for flatness becomes

(Eel - ~0s 4) $( d$ crcosd-E,,)+sin~$- c dt 1 P-7) + (Eez - sin 4)’ olsin4--I&,)-cos$z =O. 1

The emitted fields in this equation are themselves functions of time and of the phase

variation. Their solutions are

E,.(t) = aemtlTc $ [ J t

C t1

et’lT= cos qS(t’) dt’ + (1 - e-tl’Tc )I , (9.8) E,,(t) = ae-tlT~ f J’ et’lTc sin $(t’) dt’.

C 51

159

I :

After the real cavity field is driven steadily for time tl, let the input phase be

shifted by -7r/2, rather than the standard X. Inserting this along with the emitted

field values Eel(tl) = cy (1 - eVtliTc) and Ee2(tl) = 0 into equation (9.7), we can

solve for the time derivative 4 (defined as the right-hand limit at the discontinuity),

completing our initial conditions on as

4(h) = 2,

((tl)=-$[l-e;tl,Tc +a(l-e-'l'Tc)]'

If (9.7) has been met at all points between tl and t, we have also the desired

relationship

IEout(t>12 = IEout(b)12, which we can write as

(Eel - cos 4)” + (I& - sin 4)” = o2 1 - eet-liTc >

+ 1.

Combining equations (9.7) and (9.10), we can eliminate the terms quadratic in E,,

and E,,. The resulting equation, written explicitly in terms of r$(t), is

[J t

t1

et’lTc cos $(t’) dt’ + T, (1 - emtllTc )I [ $(a - l)cosd+ $sin+ C 1

J t + et’jTc sin qS(t’) dt’

t1 $(o - l)sin4 - Jcos 41 = [o (1 - e-‘1/TC)2 + l] et.

‘(9.11)

Although somewhat simplified, this is still an intractable transcendental equation

offering little hope of a direct solution. The fact that we have the initial value and

the slope at tl however, suggests an attempt to find an approximate solution, out

to some value t > tl, as a polynomial expansion in (t - tl)/T,.

s - Figure 9.2 shows the results of such a search. The coupling parameter of the

SLAC linac SLED cavities was used, giving cy = 5/3, and the switching time was

160

I

:.

taken to be tl = 1.5 T,. I attempted to flatten the output amplitude out to t = 2 T,.

The compressed amplitude is plotted in 9.2(a) f or a simple -7r/2 phase shift, for a

constant variation of &tl), and for successive higher order corrections up to fourth

order. Reasonable flatness, with less than one percent amplitude droop, is achieved

for about t2 - tl = 0.35T, at a gain of G = 2.68. If we end the pulse there, the

efficiency is nc = 0.51.

IIll IlIt IIII IllI 1111

1.5 - . .

. -- - . . . . . . . . \ .

1.0 - . . . - . - 2 .-

.

w -

0.5 -

0.0 * 0 0.1 0.2 0.3 0.4 0.5

(t-tl)/Tc Figure 9.2 Compressed pulse output amplitude for input phase vari-

ations of successively higher order polynomials in X = (t - tl)/T,. The

solid curve is for $in = -r/2 - 2.582X - 1.6X2 - 2.5X3 -5X4.

This could be improved by using higher Q (e.g. superconducting) cavities.

Since T, = ~&L/W = (2/w)Qo/(l + P), raising Qs allows us to raise p without

changing the time constant. As p increases, o approaches 2, and the gain and e -

efficiency in the above example approach 3.41 and 0.65, respectively.

161

Figure 9.3 demonstrates another problem with this approach. The phase of .

the output pulse also varies, whereas accelerator applications generally call for a

constant phase. Since a constant rate of phase slippage is equivalent to a lowered

frequency, the linear part of this variation may be compensated by using a SLED

system and input pulse of slightly higher frequency than desired in the output. For

optimal phase stability out to 0.35 Tc in the compressed pulse the desired shift is

Sf = 0.326/Tc, a detuning of only 6.2 x 10 -’ for the SLAC system. This reduces

the total phase variation in this region from 41” to less than 5”, as indicated in the

figure.

Further optimization might improve the above picture slightly, but it is clear

that the efficiency and feasibility of this method depend strongly on how much

amplitude droop and phase variation can be tolerated for a given application. This

approach to SLED pulse flattening has also been explored by V.E. Balakin and I.V.

Syrachev [49] at Branch of the Institute of Nuclear Physics in Protvino, Russia with

similar results.

SLED-II WITH DISC-LOADED DELAY LINES

An undesirable feature of SLED-II is the length of the waveguide delay lines. In

the low-loss 4.75” i.d. guide, the group velocity of the TEol mode is 0.964 c. A delay

line length of about 120’ is thus required for an output pulse of 250ns duration.

Since the NLC will need an rf station about every lo-20 feet, this necessitates a

-

multi-level scheme for stacking overlapping delay lines.

One method we’ve considered for shortening the system involves using disc- - -

loaded delay lines to reduce the group velocity. One could use the same circular

162

: .

-1

Figure 9.3 Phase-modulated SLED output. The solid curve is the

amplitude (negative sign substituted for 7r phase), the dashed curve is the

phase, and the dot-dashed curve is the phase in the compressed region

adjusted by detuning. The pulse could be terminated at t/Tc = 1.85 for

reasonable flatness.

waveguide mode, but introduce periodically spaced discs with central coupling irises.

This structure is equivalent to a series of cylindrical resonant cavities inductively

coupled through their end plates. The flow of electromagnetic energy is slowed down

by this modification, and we can achieve the same delay in less physical length.

The behavior of such a delay line has been theoretically investigated by use of

the equivalent-circuit model applied to structures in the Appendix, with the end s -

cells appropriately modified to present a complete or partial reflection. The chain of

163

resonators was assumed to be operated in the r/2-mode for simplicity and minimal

dispersion.

The loss in loaded delay lines would be greater than the loss of straight waveg-

uide delay lines, despite the reduced length, due to the loss on the discs. If we

ignore the presence of the coupling holes (H, = 0 on axis), we can use equations

l (6.18) and (7.8) to derive the following expression for the unloaded quality factor

of each pillbox cell, and thus of the line:

&CO Q” = 1+ (2/n7r)&z3/$, ’

(9.6)

where

ka

‘O” 7 @%/rlo) ( xol/ka)2

is the quality factor of the waveguide, with no endplate losses, and n is the number

of half-wavelengths contained in the cell @p/z). If we choose an inner diameter

4.75”, to match our current waveguide, we get for a copper structure

. _

Qo = 1,386,800

1 + 117.1/n ’ P-7)

To make this worthwhile, it is desirable to decrease the group velocity by about

an order of magnitude. The delay lines can be shortened to 12’9” by making

us/c = 0.1035. Since X,/2 = 0.5358”, this is 285 guide half-wavelengths. Let

us take the line to consist of N = 15 cavities; then each cavity must be n = 19

half-wavelengths long. That is, the cavities resonate in the TEo,I,19 mode. From

equation (9.7), Q 0 is then found to be 193,600. Th e required coupling between cells

(‘c pc Q/P = 2Nh) is determined from the number of cells and the delay time to

be about k = 0.0036.

164

Figure 9.4 shows a simulation of a SLED-II system using loaded delay lines

with these parameters. A compression ratio of five and a ten nanosecond rise and

switching time were used. The solid square waveform shows the output power of

a system using unloaded copper waveguides of the same diameter, for comparison.

The dashed waveform is the output of such a non-dispersive system with the same

Qo as the disc-loaded system. A couple of drawbacks of this scheme are immediately

apparent.

,I 0 250 500 750 1000 1250

t (r-4

Figure 9.4 Equivalent circuit model simulation of output power wave-

form from a SLED-II system employing fifteen-cell disc-loaded delay lines.

Firstly, the loss introduced by the discs lowers the gain by about ten percent. e -

Recall from Chapter 4 that the impact of increased lossiness depends on the com-

165

pression ratio. The round-trip delay-line attenuation, e-2r, is related to Qo through

wtd

2’=2Qo. (9.8)

Degradation of the gain is inevitable when delay lines are loaded.

Secondly, dispersion distorts the waveform, giving rise to an overshoot and

significant power variation across the compressed pulse. These ripples are caused

by sidebands which result from the interaction of the Fourier spectrum of the input

pulse with the sharp cutoffs of the structure passband. Their amplitude can be

reduced by increasing the switching time, but at the cost of efficiency. Their period

and persistence can be reduced by increasing the cell-to-cell coupling, and thus the

passband, but this increases the number of cells required to achieve a given delay.

There are conflicting goals in the factoring of the total number of half-wavelengths,

determined by the desired delay line length, between the number within a cell and

the number of cells. On the one hand, one wants a high Qo, and on the other one

wants a -broad passband (6w/w,/2 = k). Th e circuit model and &a formula used

. _ break down in either extreme, both of which approach standard SLED-II.

In addition to these faults, recall that the large, cylindrical structure used for

low loss is highly overmoded. The cells described would have myriad resonances

and, with significant cell coupling, many of their passbands would overlap TEo,l,,

in the dispersion diagram. The density of the spectrum increases with disc spacing,

and the span of each mode increases with coupling, adding to the design conflict.

Power conversion to such modes can seriously affect SLED-II performance.

This problem has been addressed by.T. Shintake [50], who suggests an absorber-

loaded circumferential gap next to each disk to damp modes with longitudinal cur- e -

rents. (This idea recalls some of our mode filter designs for waveguide.) Although

166

this renders the spectrum much sparser, it does not deal with the many TEo,,,~

modes, which would be the predominant parasitic modes in an axi-symmetric line.

From the preceding discussion, it seems clear that, despite the virtue of cam-

pactness, the theoretical limitations, technical complexity, and cost of disc-loaded

delay lines make them generally inferior to simple over-moded waveguide delay lines

for use in rf pulse compression. Space and copper are relatively inexpensive. In a less

over-moded, super-conducting version, however, disc-loaded or periodic-structure

delay lines may yet find application in BPC or SLED-II powering of accelerators.

RAMPED SLED-II FOR BEAM LOADING COMPENSATION

For a long train of bunches to be accelerated uniformly, some means of beam

loading compensation must be incorporated into the operation of an accelerator.

Otherwise the leading bunches will gain more energy than later bunches which suffer

from the depletion of stored energy in the accelerating structure, or equivalently

from the accumulated longitudinal wakefields of the preceding bunches. For the

NLCTA design parameters, this would amount to a 25% droop in energy across the

beam pulse.

One means of achieving this compensation is to inject the bunch train before

the structure is completely filled with rf energy, so that the continued filling cancels

the beam loading effect [51]. If the length of the bunch train is on the order of the

filling time, however, as is expected for the NLC, this is not practical. An alternative

which has been suggested [l] is to prefill the structure with a linearly ramped pulse

which then becomes flat. As the bunch train enters, beam loading compensation s -

is achieved by the ramped portion of the rf pulse leaving the structure and being

167

I :

replaced at the input end by the peak value.

We can create a partially ramped rf pulse with a SLED-II system by properly

modulating the phase of the input pulse. The compressed pulse, as we have seen,

can be represented as a combination of several phasors, one from the direct iris

reflection and one for each of the delay-line reflections that make up the emitted

wave. wave. Each of these phasors has its origin in a particular time bin of the input Each of these phasors has its origin in a particular time bin of the input

pulse. By manipulating the phase of the time bins, we can control the phasors. pulse. By manipulating the phase of the time bins, we can control the phasors.

I’

I I 1 I I I I I I I

I I I I I I I 1 I fl I I I I 0 0 0.5 0.5 1 1 1.5 1.5

Real Real

Figure 9.5 Figure 9.5 Phasor diagram of output field contributions from five input Phasor diagram of output field contributions from five input

time bins and their sum during a ramped SLED-II compressed pulse.

-

It is convenient to combine the individual phasors, by application of the same

phase variation, into two phasors, as shown in Figure 9.5. These two can be rotated

from maximum positive and negative angles at the beginning of the time bin to the e -

real axis at the end of the desired ramp .and kept there for the remainder. Let’s

168

represent the phasors as

The time functions can be chosen so that the imaginary components always cancel

and the real components add to give a linear ramp until the phasors are coincident.

The minimum starting point for the ramp is determined by the relative length of the

phasors. If they are of equal length, there is no lower limit, as they can begin along

the positive and negative imaginary axis. If they are unequal, it is easy to see that

the smallest fraction of the final amplitude we can start with, while maintaining a

sum phase of zero, is

*=lmFzl E

4 El +E2 ’ (9.9)

with the shorter one along the imaginary axis. It may therefore be necessary to aim

for equal magnitude phasors.

For example, let’s consider compression by a ratio of C, = 5 in a system whose

delay lines have e -2Tc = 0.98995 (2% power loss). With an optimized reflection

coefficient of s = 0.651, the contributions to the output field have amplitudes 0.1527,

0.2369, 0.3676, 0.5704, and 0.651. Combining the phasors from the first, second

and fourth bins and those from the third and fifth bins, we can obtain El = 0.960

and E2 = 1.019. These differ by only six percent and allow us to go as low as

E,in/Ep = 0.172.

To determine the desired functions &(t) and 42(t), we have the following re-

quirements:

El cos q&(t) + E2 cos 42(t) = EC, + AE;, s -

El sin&(t) + E2 sin&(t) = 0.

169

I : .

3.0

2.5

2.0

--Ii ‘xl 1.5

. 1.0

-0.5

0.0

3.0

2.5

2.0 . . 3

w” 1.5

1.0 . _

0.5

0.0 _ -

I I I I I I I I I I I I I I I I I I I I _ 200 - --

I /

y 150

/ I T 100

z \ i‘, I‘\ I

I 50

1 \ ’ \ ’ \ I I 1 0

’ /

/

L/ 7 -50

I I I I I I I I I I I I I I I I I I I I --100

0 250 500 750 1000 1250

t bs>

,” P

1’ I ” ’ I I ’ lb’ I I I ’ I I I ” ’ I I’4 100

\

1 I I I

\ 1 L- 1

---- 50 0 $. s

h 7 -100 z \ CD

:\ \ y -150 2% I--, I---

i I I I I I I I I I I I I I I I I I I I -200 0 250 500 750 1000 1250

t.. (ns)

Figure 9.6 Input and output waveforms for a partially ramped SLED-II

s - with C, = 5. The solid lines indicate field amplitude and the dashed lines

phase. /

170

I : .

Here Es 2 Emin is the desired starting amplitude of the ramp, AE = Ep - Eo, and

At 5 td is the desired duration of the ramp. Solving these yields

qqt) = cos-l Es+AEt- E; - E,2

At E. +AEtlAt

q&t(t) = sin-’ (9.10)

A ramp from Eo = 0.28 Ep over 104 ns of a 250 ns rf pulse was shown to compensate

beam loading to the 10m3 level in the NLCTA design. Figure 9.6 shows the solution

for the above system that gives a compressed pulse with these specifications. Note

that the phase variation given to the final input time bin is combined with its normal

180” phase flip. The output is linearly ramped in amplitude but flat in phase.

OTHERS

There are a few other approaches to rf pulse compression that should be men-

.

tioned. One is the VPM (VLEPP Power Multiplier) [52], developed at Branch INP

in Protvino, Russia, by V.E. Balakin and I.V. Syrachev. It works on the same

principle as SLED, but requires only one storage cavity and no 3-dB hybrid. The

VPM uses an “open” cavity, shaped like the wall of a squat barrel. This cavity is

operated in a traveling-wave “whispering gallery” mode with many azimuthal vari-

ations. The fields of this mode are concentrated close to the concave wall, so that

no endplates are needed to confine it. A rectangular waveguide, wrapped around its

equator, is coupled to the cavity by a series of periodic slots. X-band VPM’s, about

one foot in diameter with Qo’s of 2 x 105, have been constructed and operated.

In an attempt to improve the SLED-like shape of the output pulse, a two-cavity s -

VPM was built [53], in which the stacked barrel cavities were coupled through their

171 .-

I :

common open face. The same cavity-doubling idea was tried at CERN [54], both

theoretically and with LIPS cavities. The result was a compressed pulse with a top

like a sideways “S”, a spike followed by a hump. I was able to reproduce the shape

by running my disc-loaded delay line program with only two cells, operating in the

T mode.

Another means of acheiving a flat compressed pulse has been proposed by

S.Y. Kazakov [55]. H is i d ea is to use one main cavity and several, separate correction

cavities coupled consecutively by waveguide, like cascaded SLED’s or VPM’s. The

correction cavities are tuned at different frequencies around the operating frequency

so that, taken together, the resonances simulate a portion of the spectrum of a long

delay line. Simulations of this technique show SLED-II-like outputs with small

initial spikes and varying degees of ripple, depending on the number of cavities

used. The flatness achieved suggests that the performance of disc-loaded delay

lines could also be improved by varying the disc spacing and coupling so that the

N TEoln modes of the combined resonant system also imitate the spectrum of a . _

smooth, shorted waveguide.

H. Mizuno and Y. Otake of KEK, in Tsukuba, Japan, have proposed an in-

teresing linac powering scheme based on Binary Pulse Compression [56]. The idea is

to combine power from two klystrons with a 3-dB coupler and use a phase reversal

in one klystron to direct the leading half of the combined pulse into a delay line and

the trailing half directly to the accelerator. The novelty is that rather than folding

the delay line back on itself, it is used to power a distant upstream section of the

ljnac. The delay time required is less than half the input pulse by the beam travel

time between the fed sections. By interweaving such systems, all the accelerator

172

sections are powered. This is essentially single-stage BPC with a clever distribution

network.

Finally, back at SLAC, there is currently interest in and work on developing a

low-loss fast rf switch to circumvent the theoretical limitation on SLED-II efficiency

[57]. If the reflection coefficient can be changed on a time scale short compared to

the delay time, higher gains become achievable. For C, = 10, for example, changing

s once, before the last bin, can raise the ideal gain from 5.6 to 8.3. Ifs is also changed

after- the first bin, during which it should ideally be zero, this is further raised to

N 9.4.

173

I :

10. CONCLUSION

The past few years at SLAC have seen much progress towards the design of

a next-generation linear collider, including development of rf pulse compression

systems. As the powering of the linacs is likely to be the most expensive aspect of

such an enterprise, the rf system assumes great importance. Because state-of-the-

art X-band klystrons are more limited in peak power than in pulse width, relative

to the desired power source specifications, rf pulse compression may be a necessary

means of achieving sufficient accelerating gradients.

We have examined various schemes for both reflective pulse compression, such

as SLED, SLED-II, and their derivatives, and transmissive pulse compression, such

as Binary Pulse Compression (and chirping). Our goal has been to produce a

flat-topped pulse suitable for the uniform acceleration of a long train of bunches.

Standard SLED, the archetypal rf pulse compressor used on the SLAC linac, was

therefore not an option.

Binary Pulse Compression has the advantage of having no intrinsic inefficiency.

That is, it allows theoretically for all the energy in a pulse to be compressed into

a shorter pulse. As the name suggests, it is limited to compression ratios which

are powers of two. We have successfully constructed and operated a 3-stage BPC s -

system capable of working with one or two sources. Experimental results have been

174

I : .

reported showing good agreement with expectations based on measured component

losses due to imperfections and waveguide attenuation. This system was used with

experimental klystrons to power high-gradient tests of X-band accelerator struc-

tures. An undesirable feature of BPC is the length of low-loss delay lines required.

Each added stage requires twice as much waveguide as the previous one. Our sys-

tem was perceived as being too massive and bulky to be incorporated at every

other rf station of a linac. However, as efficiency becomes more crucial in future

accelerators, attention may return to this method of pulse compression.

Our recent focus has been on SLED-II, the extension of SLED with a flat com-

pressed pulse. It shares with SLED an intrinsic efficiency which decreases with

increasing compression. ratio and, like BPC, utilizes long waveguide delay lines.

SLED-II is, however, less bulky and less complicated than the fatter. We have de-

signed and built a high-power SLED-II system. Use of the TEol circular waveguide

mode for- its low loss required the development of several novel waveguide compo-

nents, which have been described herein in various degrees of detail. As a first . _

prototype, our SLED-II system was fairly successful. It too was used in tests of

accelerator structures, providing peak powers as high as 150 MW with a power gain

_ - approaching five. Along the way pitfalls and areas for improvement were identified

and addressed.

An improved SLED-II system is nearing completion. Many of the modifications

have been mentioned, including replacing the 2.81”-diameter waveguide of the delay

lines with 4.75”-diameter waveguide and replacing the 90” bends and 3-dB coupler

with rectangular waveguide components fitted with flower-petal mode transducers. s -

A partially upgraded system has already yielded a gain/efficiency improvement of

175

N 7% over our previous results, despite doubling of the delay line length for a 150 ns

pulse and accidental damage to the magic-T hybrid.

A short, experimental, X-band linac called the NLCTA (Next Linear Collider

.

Test Accelerator) is under construction at SLAC, in End Station B. It will con-

tain an injector, a chicane for bunch manipulation, and four 1.8 m-long accelerator

sections. A 250ns pulse of 200 MW rf power is required for each pair of sections,

including the injector, to achieve the goal accelerating gradient of 50MV/m. We

plan to accomplish this by compressing 1.5~s pulses from 50MW klystrons by a

compression ratio of six, for a gain of 4. Three such SLED-II systems are required.

An isometric drawing of the NLCTA rf distribution system, not to scale, is shown

in Figure 10.1. We are currently developing the electronics needed to implement

the ramping described in the last chapter for beam loading compensation.

176

- - Figure 10.1 NLCTA rf station with SLED-II pulse compression (NOT

TO SCALE).

177

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180

[40.] Eric M. Nelson, “High Accuracy Electromagnetic Field Solvers for Cylindrical Waveguides and

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181

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182

I

:.

APPENDIX:

An Equivalent Circuit Model

for Traveling-Wave Structures*

In this note, I develop an equivalent circuit model for a constant-impedance

structure operating in ‘the 27r/3 accelerating mode. I use the parameters of the

30-cell X-band section developed as part of the NLCTA R&D at SLAC. I then

demonstrate its usefulness for examining, in the time domain, pulse distortion,

energy gain, and field profile in the structure for various input pulses. Each cavity

is represented by .a parallel circuit inductively coupled to its nearest neighbors. The

actual structure couples capacitively, but the behavior should not be altered by ”

this discrepancy. The ends of the model are impedance-matched at the operating

frequency, a slight transient reflection indicating the mismatched Fourier content of

the pulse. Dispersion in the structure is quite noticeable. The pulse distortion is

greatly reduced as the rise time is increased to above ten nanoseconds.

.

- * Originally distributed as SLAC Advanced Accelerator Studies (AAS) Note 75,

September 1992.

183

I :. .

The basic equations of the above equivalent circuit model are

from which one easily

I; + I; + I,” + If = 0

IC n

dI; dI:-, Vn=Lz+kLT

dI;+, ’ =

LdI;

dt+kLx-

derives, defining In E I: + Ii,the following first-order differ-

. _ ential equations:

din (A4

-= dt L-1 y p) 12’n - k(K-1 + vn+l)]

The quantities R, L, and C are related to the characteristics of the actual

structure by the relationships

L=!&- 2R

wo& wo&

c= 2Q Q -=- uo& w,,R’

184

I : - .

However, we don’t need to know RSh if we make the following change of vari-

ables:

Tz = (R/Q) In.

The time derivatives then become

dVn -=-wo(L+$) dt

dz, -= dt 2(l: k2) [ 2vn

- k(V,-1 + %+I>], (A4

where we’ve used

2 2 -- w” - LC

wou &=T-l=

wo $v2 +V2/R

= w. RC.

Equations (A.2) can be used to numerically model the transient behavior of a

set of coupled resonant cavities. We need to add an appropriate driving current in

the first -cell. We need to know the ratio of the drive frequency to the uncoupled

resonant frequency, ws . We need to know k and Q, and finally, for the problem of

an impedance-matched traveling-wave structure, we need to add the proper transfer

impedance to the first and last cells.

First, we derive a dispersion relation. Combining equations (A.2) and assuming

a time dependence of ejwt, we get

@Xl -= dt2

-w2vn = 2(lw5k2) i2’n - k(Vn-1 i- Vn+l)] - ZjwV,.

We define a complex propogation constant 7 = ,O - jo, so that the relative steady-

state voltages of the cells vary as e--jrpn, where p is the structure period. Dividing

out Vn, we now have

185

I : .

w2 = (1 :k2) 2 [l-kcosTp]+JT.

or

Expanding

cos yp = cos(@p - j op) = cos /3p cash cup + j sin ,Op sinh cup,

we get from the real and imaginary parts of the above equation, respectively,

and

W kQ sin ,Op sinh cup -= wo l-k2 ’

. _ We may consider the first of these equations to be the dispersion relation for

the structure, where (Y is a function of p obtained by combining the two equations.

Defining the variables x = cos pp,

we get

Y = coshap,

Y(X) = & -(=&r~+&-3~~~+4Q~(3 +4k2Q4 (A.4) 1 z&q--J (A-5)

186

The group velocity is given by

du w kp sin ,Bp & ‘g = dp = 2(3”(1 _ k2) [ 1 ’ + xz

Y+“2 (A-6)

= iwkpsinpp l-kxy’

w, in our case, is 27r x 11.424 GHz, and, from a calculation made with the code

SUPERFISH, Q = 6,960. The length of each cell, p, is given as 0.00875m. This

follows from the condition that the phase velocity equal the speed of light at the

given frequency for pp = 2~/3.

We can now determine the proper value for k by setting pp = 27r/3 (or equiva-

lently x = -l/2) and requiring the above expression for vg to equal the given group

velocity of the structure, 0.033~. The result is

k = 0.03705

This completes our dispersion relation, from which we find for the 27r/3 mode

W - = 1.00991 wo

We now have all the parameters needed to describe the behavior of the general

circuit as determined by Equation (A.2). We still have to match the ends of the

chain and drive the structure. The first four plots in Figure A.1 show the dispersion

diagram and the attenuation per cell, phase velocity, and group velocity as functions

of phase advance per cell as determined by the above equations. As a check, the

attenuation constant at 27r/3 was found to be in agreement with the SUPERFISH

fesult. This value, cy = 0.00452, leads to a field (voltage) at the output end down

by a factor of e-‘Oa = 0.873.

187

Vl

=

.

Next we calculate the impedance of an infinite chain of inductively coupled

resonant circuits acting as a 27r/3-mode traveling-wave structure. Referring to the

above diagram, we can write

Vi = jwLI1 + jwkLI2

v2 = jwLI2 + jwkLI1 = -I25

+ I2 = -jwkLIl

jwL + ZI

Z =jwL+ (wkL)2

eff jwL + ZI

1 -=jwC+A++- ZI eff

Eliminating Zl between the above two equations and substituting wi = 2/LC and

Q= woRC = 2R/woL, we get the following equation for Zeff:

188

I : .

[l-2(-3’+ j$(E)]Z$f +2(1- k2)[j(z)2 + i(z)]WLZeff

+(1- li~)wZLZ = 0

Now, defining Z = R/Q = woL/2, we can rewrite this as an equation for g,ff =

Zeff /Z,

with the complex solution

Z Zeff = * =

-A[$ + ifi] - &P[$ + 61’ - 2A[F + $f - l]

+5+$!-1 , (A.7)

whereA= (l- k2), D = (wo/w), and the minus sign is chosen to make the resistive

part positive.

Equating Z,ff with the impedance of the right inductor, L, in parallel with a

transfer impedance, Zt, we get

1 1 1 -=--- Zt Zeff jwL

189

zt +zt=-= zeff Z 1+ jQZeff

All we need then to model an infinite or matched series of traveling-wave struc-

ture cavities is the Q of the cavities, the coupling coefficient, k, and the ratio (w/ws),

where wg is the uncoupled resonant frequency and w is the frequency at which the

structure operates in the desired mode (i.e. with the desired phase shift between

cells). The former number was given, and the latter two were derived from the

group velocity and operating mode by appealing to the circuit model itself. Using

our values in a transfer impedance calculation, we get

Re Zt(2r/3) = 45.601

ImZt(2,p) = 28.287

2, as a function of frequency near ws is shown in the last plot of Figure A.l. Finally,

this corn-plex impedance can be replaced by an equivalent parallel combination of a

. resistance and a reactance. The extra resistor represents power flow, and the extra

inductor represents a slight detuning of the end cells. Their values are found to be

Rt = 0.009073R

_ - L t = 50.40 L,

leading to new end-cell parameters

-1

Q’ = w; C z 0.00904 Q N 62.9.

s - One detail remains. It is useful to adjust the amplitude of the drive current,

IO, so as to normalize the steady-state voltage level in the first cavity. Adding the

190

I :.

rest of the structure as a transfer impedance, the circuit is

Considering the current source and input impedance elements as a generator, we

have the available power

Since we are matched,. this is equal to the power flowing in the load (the chain

represented in one circuit),

1v =--= pf 2RL

Setting V = 1, we then have

;I;&=; ++; v- t

2

J

Rt -+I()=- 1+--.

Rt R

The final equivalent circuit model can be represented as follows:

2 30

191

k k

I : .

This model can obviously be extended to any number of cavities, and the parameters

changed to model different structures and/or modes.

Figure A.2 shows the model in operation. In plot (a), we see the voltages of two

consecutive cells (10 and 11) oscillating with the proper phase relation. Actually,

I measured the phase shift to be 0.6651~ in this plot, but this small error changes

with time and can be attributed to the frequency impurity of the finite driving

pulse. Plot (b) h s ows the response of the first cell when the drive current is turned

on instantaneously and left on. Here and in Figure A.3, we are looking at the

envelope, or voltage amplitude. Transient oscillations as well as a reflection can be

seen. The overall level, however, hovers around one and it can be seen that we are

matched in steady state.

For Figure A.3, a pulse composed of a linear ramp from zero to one followed

by a fifty nanosecond flattop was sent through the structure model. The duration

of the ramp, t,, was varied from zero in steps of five nanoseconds. The fall time

of the input was zero. Each row of Figure A.3 is for a different rise time, and the

columns show the responses of the first cell, the fifteenth, and the thirtieth, or last

cell. For too sharp a rise, we see wild fluctuations, particularly down the line, where

dispersion has had time to distort the pulse. For a more gentle rise of ten or fifteen

nanoseconds, these effects are significantly tamed and the pulse remains fairly flat.

In all of these plots, we can see a small reflection coming in before the end of the

pulse.

Figure A.4 shows field profiles seen by a speed-of-light particle traversing the

structure. t, is again the total, linear rise time, and ti is the time the particle s -

is injected. N is the cell number. Again, the behavior is much smoother for the

192

1

gradual rise.

Finally, Figure A.5 plots the enegy gain of an accelerated particle as a function

of injection time for three ,different input rise times. The particle was inserted on

crest with the velocity of light. Wakefield losses were not included, although they

could be with a modest modification of my program. Not only is the energy gain

rendered more uniform by a ramped pulse, but the lost efficiency appears to be

somewhat compensated by a broadening of the injection window.

This note is presented as an example of the usefulness of the concept of equiv-

alent circuits, as an educational exercise for myself, and in the hope that these

results or future applications of my program, with possible added features, might

prove helpful to the accelerator R&D program at SLAC.

Acknowledgements:

I am indebted to Perry Wilson for his insight and guidance and to Eric Nelson for

* an occasional chat.

193

I

8 d

s - Figure A.1 Dispersion diagram, normalized phase velocity and group

velocity as functions of Bp/n, and real and imaginary parts of &(w/wo).

194

-1.0

1.2

1.0

. _ 0.6

0.4

0.2

0.0

tI,,"I""I""l""r""ll 37.9 37.925 37.95 37.975 36 38.025

t b)

C”“l”“l”“l”“I ““I ““4

0 25 50 75 100 125 150

t (4

- -Figure A.2 Voltage in adjacent cells oscillating in the 2~/3 mode and

voltage envelope in the first cell.

195

- .

i 1 , I. .,I . ..l....I....'. 1, :! e 2 x x 2 x

1. ‘& ‘&

0 = ‘3 SU~=“~ SU()I ='$

s - Figure A.3 Propagation of pulses with different rise times, t,, through

the structure.

196

-. .

l l

+ + +

+ +

+ + +

*

+ + +

* +

+ +

+

* * +

+ +

+

+

l

+ . .

+ +

+

+ 4.

+ +

+ + +

+ l

+ +

+ +

+ + +

+

+

+

s -

Figure A.4 Field profile of pulses with rise times t, at times ti, during

and after structure filling. N is the cell number.

197

l i

: l

.

.** + . +** +\ o,,,.‘....l....‘....‘.t.. J

0 20 40 60 60 100

lNJEC3lON TlNE (as)

(““““““b, I . . 20

10 t

f .* . :

+*

:

.+

:

2 +.

f l l .

f .

0 l....l....l....l...F, 0 20 60 60 166

M (111)

e -

Figure A.5 Energy gain of a particle as a function of injection time for

pulses consisting of linear ramps of duration t, followed by equal flattops.

198