SLAC-R-95-455. UC-414. RADIO-FREQUENCY PULSE COMPRESSION. FOR
LINEAR ACCELERATORS*. Christopher Dennis Nantista. Stanford Linear ...
SLAC-R-95-455 UC-414
RADIO-FREQUENCY PULSE COMPRESSION FOR LINEAR ACCELERATORS*
Christopher
Dennis Nantista
Stanford Linear Accelerator Stanford University,
Center
Stanford, CA 94309
January 1995
Prepared for the Department
of Energy
under contract number DE-AC03-76SF00515
Available from the National Technical Printed in the United States of America. 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, to my mother,
Maureen,
for prodding
me to finish,
for not,
and to my wife, Mikayo, for giving me a reason.
...
111
TABLE
... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..I~I
Dedication 1
OF CONTENTS
of Contents
..........................................................
iv
Acknowledgements
......................................................
..vi i
Table
Vita
Publications Abstract
and Presentations
2. SLED
.x .xii ..l
............................................................
.
......................................................................
SLED Theory
............................................................
.7
SLED in Use
............................................................
12
3. Binary
Pulse
Compression
............................................
BPC Theory
...........................................................
Single-Source
Operation
System Development
4. SLED-II SLED-II
Theory
Staging
- Comparison Tuning
..18
.................................................
23 ..2 5
.................................................
29
................................................................
GainandEfficiency Multiple
.17
...................................................
The SLAC 3-Stage BPC
s
........................................... .............................................
of the Dissertation
1. Introduction
. -
..i x
.......................................................................
With
.41
........................................................
.43 ..4 8
.................................................... .......................................... SLED
................................................
..................................................................
iv
..~............4
9 .52 56
5. The 3-dB
Directional
Function
Semi-Analytic
...............................................
Waveguide
Offsets
............................................
TEsi-TMr-i
Bend Mixing
...............................................
Generalized
Telegraphist’s
Equations
Application
of the Theory
7. Other
Components
..............................................
Mode
Circular
90” Bends
Vacuum
Pumpouts/Mode
8. SLED-II
Filters
Results
High-Power
Experiment
Accelerator
StructureTest
9. Variations
- Phase-Modulated SLED-II
With
.81 .88 .89 .94 101 110
110
......................................
..12 0 .122 126 .131
.................................................
131
................................................
.140
On a Theme
Amplitude-Modulated
.......................................
...............................................
and Shorts
Experiment
..7 0
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
...................................................
Experimental
Low-Power
....................................
Converter
Delay and Transfer Lines
Irises, Tapers,
....................................
....................................................
The “Flower-Petal”
s
....................................
Correction
..6 6
...7 6
...........................................................
,Testing and SUPERFISH
.63 ..6 3
........................................................ Approach
Development
6. Circular
........................................
...............................................................
Design Problem
.
Coupler
..............................................
SLED
SLED Disc-Loaded
: .............................
.................
...........................................
................................................ Delay Lines ................................
V
146 .155 .155 158 .162
Ramped Others 10.
SLED-II
for Beamloading
Compensation
.......................
..17 1
................................................................
Conclusion
..........................................................
References
...............................................................
Appendix:
An Equivalent
Circuit Model
for Traveling-Wave
Structures
vi
167
..17 4 ..17 8
....................................... ..................................
183
ACKNOWLEDGEMENTS
The physical limitations
of time and space prevent me from recounting
how each of the following
people
ration of this dissertation,
the performance
the general pursuit of a doctoral sincere gratitude. Professor
tunity they’ve Perry B. Wilson
Dr.
me.
Deruyter,
Dr.
Callin,
Fant, Dr.
SLAC
Spalek,
Klystron
son, Xin-Tian
Physics
celerators,
Prof.
Roger Dr.
Center for the oppor-
from the guidance
of Professor
The advice and assistance
Roger
Jones,
of my
I would also like
Lin, Joseph
Dept.,
Thomas
Dr.
Knight,
Theory
Publications
Scott Berg, S. Rajagopalan,
James B. Rosenzweig,
and the UCLA Prof.
Group, Rod
Projects Dept.,
A. Hoag,
Chris Pearson
Vacuum
& Special
E. Vlieks,
Harold
John Eichner,
and the SLAC
the SLAC
Jim Kolonko
Arnold
Terry Lee, Craig Galloway,
Accelerator Dept.,
H. Miller,
Ron Koontz,
Shop, Chuck Yoneda
& Microwave
_the UCLA
Ko, Dr.
the SLAC
(Eddie)
Prof.
Juwen Wang,
Kwok
Machine
George
Linear Accelerator
M. Kroll.
my
David B. Cline, and to
L. Lavine, are much appreciated.
Menegat, Dr.
and the SLAC
Dr.
Professor
understand
and thank:
Farkas, Albert
Karen
to my advisor,
Norman
Dr. Theodore
Sami G. Tantawi,
Henry
I hope all will nevertheless
I’ve greatly benefited
and Professor
SLAC supervisor,
Z. David
degree.
of the research which went into it, and
D. Ruth of the Stanford
provided
to acknowledge
or groups have been of help to me in the prepa-
I am indebted
Ronald
in detail
Dr.
Loewen,
Dept.,
the
Eric Nel-
Penny Lucky and
Center for Advanced
Chan Joshi,
Rich
Prof.
William
Ac-
Slater,
vii .-
I
‘T.
-.
.
Prof. Harold Fetterman, staff, the Advanced erator Physics
Dr. Melvin Month and the US Particle Accelerator
Accelerator
community,
Group at Argonne
and Dr.
David
who made this research possible.
...
VI11
National
School
Laboratory,
the Accel-
Sutter and the Department
of Energy,
VITA
February
Born New York, New York
21, 1964
B.A., Physics Cornell University Ithaca, New York
June 1986
September
March
January
1986-December
1988
1988-October
June-September
September
January
1987
M.S., Physics University of California, Los Angeles, California
Los Angeles
Graduate Student Researcher Argonne National Laboratory Argonne, Illinois Graduate Student Researcher Stanford Linear Accelerator Center Stanford, California
1994
Teaching Assistant for USPAS RF Systems for Electron Linacs and Storage Rings University of Texas, Austin Austin, Texas
1992
March-September
Los Angeles
Graduate Student Researcher University of California, Los Angeles Los Angeles, California
1994
1988
1989-October
Teaching Assistant University of California, Los Angeles, California
Engineering Physicist Stanford Linear Accelerator St anford, California
1994
ix
Center
.
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. “High-Power Radio-Fequency Binary Pulse-Compression T.L. Lavine et cd., periment at SLAC,” presented at the IEEE Particle Accelerator Conference, Francisco, CA, May 6-9, 1991; SLAC-PUB-5451.
ExSan
T.L. Lavine, C.D. Nantista, and Z.D. Farkas, “SLED-II Delay-Line Length Stabilization,” Next Linear Collider Test Accelerator (NLCTA) Note #16, February 3, 1994. C. Nantista, “An Equivalent Circuit Model of the 30-Cavity Structure,” Advanced Accelerator Studies (AAS) Note 75, September 1992. “High-Power RF Pulse Compression With SLED-II C. Nantista et al., presented at the IEEE Particle Accelerator Conference, Washington, 17-20, 1993; SLAC-PUB-6145.
SLAC
at SLAC,” D.C., May
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, C.D.
1993; SLAC-PUB-6141.
Nantista
and T.L.
Lavine,
“Mechanical
Tolerances
on Circular
and Flanges for the SLED-II Delay Lines,” (NLCTA) Note #17, February 3, 1994. e -
Next Linear Collider
C.D.
of Mechanical
Nantista,
and T.L.
Lavine,
“Analysis
X
Waveguide
Test Accelerator
Tolerances
on Circular
Waveguide and Flanges for the Low-Loss Linear Collider Test Accelerator (NLCTA)
High-Power Transmission Note #21, April 1, 1994.
C.D. Nantista, “SLED-II Adjustable Short Power Dissipation,” Test Accelerator (NLCTA) Note #22, April 6, 1994. C.D. Nantista, “Ramping Collider Test Accelerator
Lines,”
Next
Next Linear Collider
SLED-II for Beam-Loading Compensation,” (NLCTA) Note #27, August 19, 1994.
Next Linear
.
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. “A Test Accelerator for the Next Linear Collider,” presented at R.D. Ruth et al., Germany, the ECFA Workshop on e + e - Linear Colliders, Garmisch-Partinkirchen, July 25-August 2, 1992; SLAC-PUB-6293. presented at IEEE R.D. Ruth et al., “The Next Linear Collider Test Accelerator,” Particle Accelerator Conference, Washington, D.C., May 17-20, 1993; SLAC-PUB6252.
. _
and RF System Development for NLC,” presented A.E. Vlieks et al., “Accelerator 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 tures,” presented at the LINAC 94 Conference, Tsukuba, Japan, August 1994; SLAC-PUB-6617.
Struc21-26,
P.B. Wilson et al., “Progress at SLAC on High-Power RF Pulse Compression,” presented at the 15th International Conference on High Energy Accelerators, Hamburg, Germany, July 20-24, 1992; SLAC-PUB-5866.
xi
-
_
ABSTRACT
OF THE
Radio-Frequency
DISSERTATION
Pulse Compression
for Linear Accelerators
bY
Christopher Doctor University
of Philosophy of California,
Professor
Recent
efforts
center-of-mass capable
to develop
energy
frequencies. enhances
hundreds
Los Angeles,
. 1994
David B. Cline, Chair
a TeV
have highlighted
of megawatts
linear
collider
of peak rf drive power
the peak power available from pulsed rf tubes by compressing
at X-band
technique
is described,
and the problem
explained. e -
Other pulse compression both
which
their output
the available energy into shorter pulses.
The classic means of rf pulse compression
are explored,
with
the need for sources
This need has driven work in the area of rf pulse compression,
pulses in time, accumulating
pulses
in Physics
plans for an electron-positron
approaching
of delivering
Dennis Nantista
for linear accelerators
it presents
for multibunch
schemes, capable of producing
theoretically
and experimentally,
xii
is SLED. This acceleration
suitable output
in particular
Binary
Pulse Compression gain, efficiency,
and SLED-II.
complexity,
The development behind
Th e merits of each are~considered
size and cost.
of some novel system
their design, is also discussed.
waveguide
much attention
The construction systems
of coupling
The focus of the dissertation rent linear accelerator
mode in over-moded power between
high-power
designs.
considerations
klystrons
Test Accelerator,
pulse compression accelerating
used, were developed
is on SLED-II, In addition
the favored
at SLAC
scheme in some cur-
to our experimental
and design improvements
results, practical
are presented.
The work
systems to be used in the Next Linear
now under construction
at SLAC.
The prototype
of the
.
system is near completion. of various rf pulse-compression
tioned three, including
modes.
work.
to date has led to detailed plans for SLED-II
Descriptions
losses in long
on, as well as their use in the testing of X-band
in parallel with the pulse compression
upgraded
copper
propagation
of complete,
which, along with the X-band
implementation
along with the theory
The need to minimize
to mechanisms
and commissioning
is reported
structures,
Collider
components,
runs led to the use of the circular T&r
guide, requiring
with regard to
techniques
those pursued at institutions
besides the aforemen-
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
pulses of radio-frequency
the subject
of the temporal
power, which, by conservation
compression
of guided
of energy, makes it possible
to obtain peak power levels higher than are available directly from microwave It does not deal directly
such as klystrons. klystron
elsewhere.
of rf power to particle with accelerator . _
inbetween subject
concepts
the klystron
is assumed.
techniques,
Standard
commanded
beam
launching
machines
It is believed
on the high-energy
into the particulars
have led to great advances In order to complete
the nature of exact SU(2)
with the conversion although
familiarity physics
While this
its importance
for
frontier, is gaining
of rf pulse compression
in elementary
particle
the confirmation
of the
sy mmetry breaking must be determined.
the source will be found at center-of-mass
energies of the W and 2 bosons.
in that science
of rf energy.
the greatest attention,
by
motivation.
in the past quarter century. Model,
the manipulation
field, particularly
I present the following
Colliding physics
Before
structures,
of rf power
The focus is on an area of accelerator
and the structure,
has not traditionally
recognition.
Nor does it deal substantially
beam energy in accelerator
the future of the accelerator wide
Details of and recent advances
tubes or by other means.
are well documented
with the production
tubes
energies well above the rest
Lepton colliders offer “cleaner”
opportunities
than
hadron
machines
practical
for probing
energy
this domain.
of a circular
electron-positron
similar to the SLC but with separate energy
of 500GeV
to 1TeV
a design has been developed Collider
to about
program
-
detailed examinations
-
searches for neutral scalars (Higgs)
-
search for new phenomena
of the interactions
physics
that of LEP II a linear collider,
linacs, with a center-
referred
At SLAC,
to as the Next Linear
of the NLC is to include
studies of top quark and its interactions
accelerator
and positron
loss limits the
(5 to 10 times that of the SLC).
-
ternational -
collider
for such a machine,
Th e experimental
(NLC).
electron
[l]:
of gauge bosons
or other new particle states
To keep the linacs of such a collider
.
radiation
th ere has been great interest in recent years in building
(200 GeV),
of-mass
As synchrotron
to a reasonable
community
has decided
length,
much of the in-
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
the choice
to reach these gradients
of an X-band
rf frequency.
operating
frequency
The lack of established
in much R&D
toward
some e -
of 11.424GHz,
a suitable
high-power
power source.
pursued.
contenders:
relativistic
structures
is reduced
by
four times the SLC
The design calls for peak power in the range 60-240 MW per meter
of accelerator.
concepts
in accelerator
klystron
[2]
2
X-band
technology
The following
has resulted
list is a sampling
of
_
-
gyroklystron
[3]
-
choppertron
[4]
-
cluster klystron
-
two-beam
Some
accelerator
[6], [71
of these sources
developed. produce
[5]
However, practical,
conservative
foreseeing
reliable
approach,
1. Develop
have achieved
modest
success
that the more exotic
devices
on a reasonable
and/or
concepts
timescale,
are still being
were not likely to SLAC
has taken a
which is to:
conventional
klystron
tubes at this X-band
frequency
livering 50-100 MW pulses of several times the structure 2. Use an rf pulse-compression
scheme to exchange
capable
of de-
filling time.
pulse length for higher peak
power. An intensive Although * _
program
klystrons
of X-band
klystron
development
have been in use for decades,
the extension
into the frequency
and power ranges desired has presented
addressed.
include
failure. produced
These
gun arcing,
output
An initial series of experimental V arious
[8].
output
circuit
cavity
klystrons
designs
is ongoing
of the technology
several problems
breakdown,
of 100MW
were tried and,
Current efforts are towards producing
for the NLC Test Accelerator specifications resumed. e -
under construction.
of 50 MW at a 1.5 ns pulse width.
X-band
and companies,
klystron development such as KEK
to be
and rf window design have been
although
power and pulse width were not reached, some tubes achieved moderate much was learned.
at SLAC.
the goal
success and
reliable 50 MW klystrons
A prototype
tube has met the
The 100 MW program
will later be
has also been pursued by other laboratories
and Toshiba
in Japan,
The
Institute
for Nuclear
.
:
I
Physics
in Russia, and Haimson
The difficulties to incorporate
encountered
.
standard
bunches
will be accelerated
is unsuitable.
a pulse of constant
However,
I examine
the scheme included in SLAC’s
part of the dissertation
to SLED
the theory
SLED-II,
are presented,
by SLED
of long bunch trains requires
of rf pulse com-
the most attention
current high-gradient
of a practical
pulse-compression
system.
being given to
linac design.
also deals with the design of rf components
A good
necessary
for
Results from experiments
tests are presented.
Essential
and future plans given. of pulse compression
not considered
here, due to its familiarity.
fields of radar and lasers. For example, being compressed e -
a long train of
require that these
and implementation
systems and their use in accelerator
One method
problems,
to achieve its
power spike produced
Uniform acceleration
Alternatives
issues are explored
The next generation
of 3 x 1O33 cm-2s-1
The exponential
pression:
with prototype
as will also be seen, the
amplitude.
In what follows,
the development
the need
peak power per feed is
on each pulse. Final focus tolerances
bunches be very uniform in energy. pulse compression
underscores
SLED, is not useful for current purposes. a luminosity
California.
is not new to linear accelerator
To relieve space charge and background
goals.
be mentioned
developers
Pulse compression
as will be seen in the next chapter. technique,
in Palo Alto,
scheme if the required
linear collider will have toQroduce physics
Corporation
by the klystron
a pulse-compression
to be realized in an accelerator. operation,
Research
in the coming chapters should
Chirping is a technique
widely used in the
it allows a long pulse to be amplified before
to the desired width when the energy density at that shorter pulse
width is too great for the amplifier to handle.
With chirping,
a continuous
variation
in frequency
(or equivalently
this pulse is passed through velocity
is given to the pulse at generation.
a dispersive
medium,
While
ruled out for accelerator and expense
foreseen
this technique
and producing
of achieving
variation
in group
in a shortening
to rf pulses,
it was
highly
dispersive
transmission
too much of the power.
and phase stability
across
pulse as well as a precise design frequency.
tailored
power generation
When
First, there was the difficulty
amplitude
The focus has instead been on non-dispersive frequency,
be applied
pulses which would not dissipate
there was the problem
the compressed
could
use for a couple of reasons.
in developing
lines for multimegawatt
plications
the resulting
causes the tail of the pulse to gain on the head, resulting
of the pulse length.
Second,
phase)
rf pulses with passive devices mentioned
of beam-rf
methods
microwave
of compressing
circuits.
earlier, this conservative
interaction.
5
constant
Unlike the exotic
focus avoids the com-
2. SLED
The conceptual
predecessor
for next generation
of the pulse compression
received
SLED is the brainchild
a 1991 IEEE
Originally
currently favored
linear collider designs at SLAC and in Japan is known as SLED
[9]. Invented in 1973 at the Stanford Linear Accelerator years following,
method
Particle
an acronym
of Perry Wilson
Accelerator
Technology
for SLAC Energy Doubler, Although
Center and developed
Energy
Development.
bearing
on this thesis warrants the inclusion
and David Farkas.
in the They
prize for their invention.
it later came to stand for SLAC
the author was not involved
in work on SLED, its
of a brief exposition
of its theory and
application. SLED is a pulse compression resonators during
scheme in which rf energy builds up in high Q
during most of a klystron
the last fraction
testing super-conducting
of the pulse.
pulse’s duration
The idea emerged
cavities that immediately
power emitted from a heavily over-coupled power. network,
Normally,
be directed
coupler.
from the observation
cavity approaches
cavities are excited
through
6
four times that input
the source.
symmetrically
This causes the power reflected
away from the source,
in
after the input power is cut, the
this power would travel back toward
a pair of resonant
3-dj3 directional
and is then largely extracted
the fourth
In the SLED
through
a four-port
from the resonators
to
port, so that it can feed an
-
accelerator.
This use of a 3-dB coupler is explained
in Chapter
5. Furthermore,
was realized that reversing the phase of the input, rather than switching an even greater power multiplication, noted,
however,
that the increased
by a sharp exponential is generally
with a theoretical
it
it off, gave
limit of nine. It should be
peak power afForded by SLED is accompanied
decay, so that the average power within a compressed
pulse
much lower than the maximum.
SLED THEORY Figure 2.1 shows a diagram dent rf wave from the klystron, interface,
giving
exponentially, interface
of a SLED circuit. of amplitude
an initial output
it emits through
reflection.
Ei,,
An inci-
reflects off the waveguide-cavity
wave of amplitude*
the coupling
The output
It works as follows:
-E;,.
iris a wave E,
wave is the superposition
As the cavity
opposite
fills
in phase to the
of these two backward
waves. .
E out
_
Since the cavity the incident positive
is strongly
amplitude,
amplitude
by 7r radians.
=
Ee
overcoupled,
Eirz-
the emitted
causing the total output
wave amplitude
wave to pass through
has built up, the phase of the incident
This immediately
changes
that it adds to, rather than subtracting wave cannot
-
change instantaneously,
* With their common
the emitted
zero. After a
wave is suddenly
the sign of the interface
from,
will surpass
wave.
shifted
reflection,
so
As the emitted
due to the finite filling time of the cavity, the
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
then drops yielding
steeply
a sudden amplitude
as the cavity
the characteristic
the compressed a traveling-wave
E;,
attempts
spiked output.
pulse has reached accelerator
should be largely depleted
to conservation
of power.
The incident
off after
duration,
and Pout
constant
usually
The output
the filling time of
amplitude
are illustrated
of the SLED mechanism
then drops by
in Figure 2.2(a).
is obtained
by appealing
We begin with =
pout
+
are self-explanatory,
walls, and UC is the electromagnetic portionality
wave is turned
By this time, the energy stored in the cavity
for good efficiency.
Pin
where P;,
wave
voltage,
its desired
section.
description
The emitted
to charge up to the opposite
and decays toward zero. These waveforms A more analytic
..
increase of 2 Ei,.
PC
+
due -p
is the power
PC
(2.1)
dissipated
energy stored in the cavity.
in the cavity
Let k be the pro-
relating the square of the field to power flow in the waveguide.
Then .
Pi,
_
=
kEfn7
and P out
where I? is the reflection definition
coefficient
of the cavity coupling
power (with no incident
=
k(Ee
+
rEin)2,
of the waveguide-cavity
coefficient,
p, as the ratio of emitted
wave), we can write T1
kE,2
J-C “,a’
Finally, from the definition
of unloaded u
c=-
Qo, w
Q, we have c=-
&ox w
9
interface.
P’
Now, from the to dissipated
-
I
I
I
I
I
2-
2
I
1 -
E* r Y-
0
I / ./’‘.ie /
-l-
rl
I
I
I
I
(a >
E .
I
I
I
I
I
1
I
- - _y.L-.
I
‘\, \ I I I
I
I
I
1
0 I
I
I
I
I
2 I
I
I-
\
I
I
I
E
I I -
L
I
I
I
I
I
I
I
I
I
J I
3
t1
I
I
I
I
Ill
4
t2
I
I
I
l-l
I
I
!
’
5
(b)
P out
-
0
1
2
I
t, 3
t,
4
t b-4
Figure s
2.2
SLED field and power waveforms.
- plots is indicative
of a phase reversal.
Eout
G, is the effective gain in the compressed
10
A sign change in the field
is the difference of E, pulse.
and Ei,.
:.
-.
-
..
with w the resonant frequency,
;
assumed equal to the drive frequency,
so that
2kQoE dEe --
due -= dt
e
w@
dt
*
At time t = 0, which we’ll define as the instant the input pulse reaches cavity,
UC is zero, as the cavity cannot
Substituting To account Equation
the above expressions
into equation
for the 7r phase shift associated
(E,
-
From this we have the following for the emitted
Ei,)2
is zero.
E,,
then, tells us that /I’[ = 1.
with a reflection,
+
;Ef
+
we set I’ =
-l.*
$Ee$f$,
first order, non-homogeneous
differential
equation
wave. dEe
-
dt
QL here is the loaded
+
LE, 2QL
=
UP -E;,. Qo
Q, given by QL = Qs/( 1 + ,B). Defining
T,
=
2&L/w
and
+ p), it is useful to rewrite this as T dEe cx +
It is seen that T, emitted
(2.1),
Therefore
(2.1) can now be recast as
J%=
o = 2p/(I
fill instantaneously.
the
Ee
= aEin-
is the loaded cavity time constant
(2.2)
and that a! gives the steady-state
field.
As indicated
before, the input to SLED is a constant
phase reversed towards the end. For simplicity Ei,
=
1, -1, 0,
amplitude
pulse with the
let’s use unit field and say
o
se-2T+i(B+6)
1 -
(se
+e
1 -
-2r+i(e-6)
c,-1
+s, 1 c,-i ,1 >
q-1
Se-2r+i(B-6)
El-= e-2r+ie Eifc1-S”> [
i6 1 - (se -2r+i(e+6) > e 1 - Se-2r+i(O+6) 4
-e
1 - (se 1 _
-2r+i(e-6)
>
G-1
Se-2r+i(B-6)
where the two terms in the brackets represent the contributions line individually. not become
Implicit in all this is the assumption
significantly
td or equivalently
.
_
When
of the shorts
to eliminate
this is achieved,
equation
(4.18).
that the emitted field steps do
method
should
the reflected
of tuning a SLED-II
be adjusted
is then given by equation
Ep
on a scope.
in Figure 4.8 that the amplitude sharply
through
zero.
requires
a reference.
Monitoring As equations
First the
them in opposite
(4.15)
system).
27r), according
to
with 4 = 8.
to set 6 to zero. This can be done by
There is, however,
of E,/Ei
system.
it in an imperfect
6 will have been set to zero (modulo
Th e output
visually maximizing
by moving
field (or minimize
Next the shorts should be moved together
a more sensitive way. Notice
forms a broad peak, while its phase passes
this phase is the best way to tune out 8. This (4.6) and (4.4) indicate,
the output
initial time interval (n = 1) depends only on s, and is thus independent gas-itions.
r/d:
E(O)*
=
0,
X
=
H(O)*
We also obtain,
= 0 at y = &d/2,
E,
Ep)f
’
of course, the
must each obey the wave equation
V2f
With
E
E =iPoaHz
and
whereqo
invariant
+k,fikx,
_
ik efikx,
z
70
72
(5.10)
we obtain
the following
-. .
-.
.
.
Ef)*
=
-
Fk,
cos
and HP)*
=
“c
cos
70
where p stands for any positive
integer and k,
boundary
= E
condition
to combinations metallic
at AB
is E,
=constant.
that vanish at x = 0, yielding
wall boundary
condition
=
J(2442
by thetwo
cash k,x
at the symmetry
boundary
E,
k2.
Our assumed
This limits the higher-order
conditions,
variation.
fields.
fields
If we apply a
plane of the coupler,
J% = 0 at x = 1. We are thus limited to the zero-order are determined
-
we need
The two coefficients
so that ,ikl
Ey
=
$
JEp)+
(eik~~l~~ikl
+
g
ceik[
_
,-ikl
jEr)-
= E sink(Z-x) sink1 It follows that E,
’
= 0 and H
=
-iE
cosk(Z-
f
sinkl
TO
__
We can now obtain
an expression
x)
for the admittance
’ per unit length at AB
from
yr= SA”E;Hzdy l&WY12’
evaluated
at x = 0. This yields
Y,” = --2
cot kl,
(5.11)
770d
where
the superscript
condition
reminds
us that this is for the symmetric
at x = 1. To find the admittance
mode
boundary
for the slot fields that correspond
to
73 .-
antisymmetric dE,/dx
=
0
modes,
we set up a magnetic
wall at the symmetry
at x = 1. We then obtain in a similar fashion
y; = - ’
rlod
tankl.
(5.12)
Now we consider the circular section, shown in Figure 5.3(c). -
above Equation
(5.10) become
. rlo
8% a$
-“iiy-
E+ = iyf$.
and Of the cylindrical kp)eiP+,
wave equation
since the Neumann
such fields yields for E4
The relationships
here E,=
Jp(
plane, requiring
functions
an expansion
E4
=
we are limited
solutions,
iv0
to those of the form
blow up at the origin.
Expanding
H,
in
of the form
2
A,
Ji(
kp)eiP$.
p=-00
We determine
the constants
-.
E~(u,
A,
#) =
from the boundary
iv0
2
condition
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.
iv0
Ji(ku)Ap
=
=-
& E 27r
The solutions
are
J J ,-id@j 02T
f(d)e-‘P+dd
+
-$
p = =
0;
P#O*
74 .-
.-,
-.
.
The field components
are thus
where we’ve used J-,
= (-1)”
Jp to change the summation.
for H, and the fact that E4 = E along the boundary
Now, using this solution
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 modes
of our coupler
mittance
looking
circular section.
wave numbers
cross section
of the symmetric
by setting
-Y,”
and antisymmetric
or -Y[“,
respectively,
out from the slot, equal to Y,“, the admittance
looking
TE
the adinto the
That is, we solve
(5.14)
for the symmetric
modes and
-tankZ=
sin 1c, lr
(5.15)
75
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) -N-
Jp4
ku +
-
(ku)3 2P2(P
P
+
ku
J2( k,r) cos 24 sin 68 - ANi~,Ji(k,r)sinq5sinM.
-BN2$ C
C
where I’ve used the Bessel fuction identities J:(s)
90
= Jo(z) - $Ji(z)
and JnS1(x) -
‘-.
-.
Since longitudinal longitudinal
E fields completely
determine the power in TM modes and
H fields determine that in TE modes, we recognize in the first term of
each of the above equations the perturbed amplitude of the T&f11 and TEol waves. The second term in the E,t equation shows there is some coupling of TMlI to TMzn modes. (Note that since the k, here is not that appropriate
for Titcfzl, other modes
must come in to match the boundary conditions, although they may be evanescent). The second and third terms in the expression for H,I show coupling between TIMII and TEzn modes and between TEol and TEI,
modes respectively.
For reasons of
phase slippage, mentioned earlier, coupling to these other modes will, in general, be small. Hence we shall ignore these extra terms and concentrate on the TEo~-TM~, coupling. The i’s in the above equations tell us that the waves couple 90” out of phase. Let us, then, remove the phases from the amplitudes and set
* _
A = I4
A=d
B = PI,
B = 23.
Let’s also use the fact that 2
and
d- F
N1
=
N2
= -?-
k,2 jh2 Jo(jh2)d@i$
fi
to replace Nz/Nl With
with dw
(6.3)
k,2 jb2 Jo(jh2)6i$
= firlo.
these changes, we have:
23’= BcosSe+dkc
sinM=B+d----
d’=dcosS8-a-
sin68 21 A - 23ik,
dk
91
&kc
he 68,
(64
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
dd=JZlc,
ade
.
From
(6.5)
P = A2 + lS2 = 1,
we get, by conservation of power,
dP-
2/p
de
Substituting
--j-g + 28; dD= 0.
the above expression for dZ?yields
dd= --JY ik,
1-d
or
Integrating
gives
s_im~larly, we get B = sin
92
de,
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
k e) cos(fikc 4 As the TEol
&
(6.6)
9
and TM~I waves travel around a gentle curve, the power exchange
between them can be seen as a simple rotation vector through an angle proportional
in mode space of the above power
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 transporting the low-loss TE o1 mode around bends, has been recognized since the 1940’s [27]. (Parenthetically, Several strategies
the same relationship exists between all TEo,-T$~I,
for overcoming *it have been devised.
pairs.)
We shall return to this
subject in a later chapter. Our present concern is with the design of an offset, which consists of two equal bends in opposite
directions.
Fortuitously,
coupling will not be so troublesome
the above analysis suggests that mode
in such a device.
We can simply allow some
power to be transfered into the TM11 mode in the first bend and count on it returning to TEol in the second bend. The product of the two opposite rotations return the power vector to its incident orientation. e -
However, we have ignored effects
such as coupling to other modes, and the difference in the attenuation
93
should
constants of
is convenient to normalize the field patterns in each mode to unit power flow. We can then represent the amplitudes of the forward and backward travelling as A$ and A,,
n
waves
so that
and PA4
=
c
l4m”
-
c
(6.8)
IA,(412~ n
n
Each mode is described by a transverse scalar function Tn(p, 4) satisfying
1 d2Tn P2 -I---=
and the boundary
w-9
-kciTn, a2
condition
Tn(p
0
, for TM modes
= %d> = 0
,for TE modes,
=
$Tn(p
a, 4)
=
(6.10)
where a is the waveguide radius. Replacing the subscript m with the usual pair of azimuthal and radial subscripts and using (nm)
to signify TM modes and [nm] to signify TE modes, the solutions
are: *
T(nm)
T[nml
* For cross-polarized
=
N(nm)
=
N[nml
Jn(X(nm)
Jn(X[nm]
E)
f)
sin
~0s
n4
nd,
modes, make the substitutions:
95
sin +
cos, cos +
- sin.
-
If the expressions for the transverse fields in terms of V’s and I’s in equations (6.13) are expanded longitudinal equations,
in components;
they can be substituted,
field expressions, into equations (6.14). multiplied
by appropriate
factors,
Various pairs of the resulting
are then subtracted
over the guide cross-section with the help of the T orthogonality tedious mathematical here.
Eliminating
approximations
manipulations
the V,(,)‘s
and 1,1,1’s involves some matrix
c
dz
dIna -=dz
where the double-indexed
language
Af’s
the appropriate
c
manipulation
and
equations of the form
Kn,nKa7
and admittance
pairs of T functions. mode amplitudes,
and, using equations
and A-‘s.
The
z7n,nIn
n
n
impedence
work in terms of microwave . -
relations.
that assume
dKn -=-
volving
and integrated
at this point are not worth tracing in detail
One arrives finally at generalized telegraphist’s
* _
along with the
coefficients
are integrals in-
As it is more convenient
for us to
we will shed the transmission
(6.12), express the results directly
line
in terms of the
We have:
dA+
.
m Cn [Ck,nAz +C,,,A,] dz =--2 dAmdr e
-
= +‘C
[CtE,nAX + C$,nAi] n
where the coupling coefficients are as follows:
99
,
(6.15)
For identical indices, (6.16) c,,,
= 0.
,O being the wave number and (Y the attenuation
constant of the mode.
For coupling between different modes,
C&),(,)
=
;
JaZ(m),(n)
f
k2,,,)~(n)~~~,+(m)~(n)]
,
1
c&),[n]
=
c$],(m)
=
(6.17)
~kz(m,,I.l
k2Z[ml,[nl
-
[J&E]
7
kc~rn~kc[n~~~rnl,~n~
lPiJ%i
f
E[m],[n]
P[m]P[n]
7
\i--1
in which,
T(m),(n) _ -
and
y[m],[n]
For ease of expression subscripts. letter
=
kc(m)kc(n)
=
Icc[m] IcC[n]
J J
s tT(rn)T(n)dSy
s tT[m]T[nldS*
in the above, I’ve freely varied the specificity
A single, undelimitered
letter stands for a general mode; a delimitered
stands for a mode of the appropriate
delimitered
of the
knd
(“(
)“=TM,“[
pair distinguishes the two indeces of the mode.
]“=TE);
and a
It is also important
to
note that all expressions are for and all sums to be taken over only the modes that e can propagate
at the given frequency in a guide of radius a.
100
APPLICATION
OF THE
We can now apply these generalized designing and modeling inner diameter
telegraphist’s
the performance
of the waveguide
modes of power propagation.
THEORY equations to the problem of
of a circular waveguide offset.
The 1.75”
emerging from the 3-dB coupler allows fourteen
Six of these, however, are cross-polarizations.
Since
the symmetry of the planar offset implies coupling to no more than one polarization of any field pattern,
we can eliminate
these, and are left with eight modes.
Of
these, one is the TEol input mode, and another is the Tkf11 mode, whose coupled amplitude
we were led a couple of sections back to expect to vanish at the end of
an offset. That leaves six potential
sources of trouble.
Now, a useful thing to notice about the results of the last section is that the coupling coefficients between modes all involve &integrals
of the form (Remember
c 0; cos 4.):
J
2m
SOSC$cos rnqicos nq5dqi = i
0
J
02* [cos(m + l)$ + cos(m - I)$] cos nq5d4
or
J
27r
cos q5sin mq5sin n4 dq5= i
0
J
02r [sin(m + l)$ + 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. s
-
That means considering
TE21, TM21, and Tit&.
Assuming
the
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 diameter
waveguide
in 1.75”
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 value
coefficient
0.982/b
%~ll)
0.819/b
C~~lslll
-0.864/b
%s211
0.967/b
%w)
C[~l],(ll)
O 0.0187/b
%,[,,)
0.0718/b
%M211
-0.286/b
C-(111.(211
Curvature coupling.
The propagation curvature. parameters e -
constants act as “self-coupling”
They are complex in this treatment.
independent
of the
Their values for the aforementioned
are listed in Table 6-B. The imaginary
are easily calculated with the following formulae.
102
coefficients,
parts, or attenuation
constants
Table 6-B coefficient
%I
SOlI
Pm
real part
imag. part
=
= Crm(m-l)
(m-l>
166.1
0.0050
%>01)
166.1
0.0096
Cowl
224.6
0.0038
%1,P11
196.1
0.0088
%m
62.61
0.0256
“Self coupling”
aY[n,4
(propagation
constants).
Xfn m] PI.“, (ka)z + ( R, k
= 2
--a(n3m) - qOa
Here, R, is the surface resistivity
Xfn,ml - n2>
of the guide walls, given, for ideal copper, by
= 2.61 x lo-‘a
and f is given in Hz.
twenty percent to the calculated
(6.18)
P(n,m)
R, =
where 0 is the conductivity
n2
fi,
In the table above, I’ve added
o’s to allow for an increase in resistivity
surface roughness at centimeter wavelengths
due to
[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,
directions,
they move in and out of phase more rapidly
for less interaction. e -
since the phase velocities
for such waves are in opposite along the guide, allowing
If a bend is gentle enough, one should therefore
neglect the backward waves without
much loss of accuracy.
103
be able to
This will be verified
-
;.
(6.15) fr om the last section then reduce to
later. Equations
-dAm = -2 ’
dz
c
C+
m,n
A
(6.19)
n
n
for forward waves, where the mode superscripts have been dropped. specify initial conditions .
We can now
and integrate the above set of coupled, first-order
ential equations numerically
differ-
to evaluate different offset designs and examine what
occurs along them.
/
\
Ib I I
I
I
/
/
0
/
/ , /
/ /
0 /
//
I
offset
/
\0
\‘-:_1
\
a-
--
.----
/
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: the- offset, some percentage
Apl] = 1; all other A’s = 0.) In the middle of
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
=Pm -PnAPm,n
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 2lr lb = lAP[ll],[Ol]l This
cyclic phenomenon
- = 10.74cm = 4.229”.
(6.20)
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
curvature,
and hence the maximum
the pipe.
angle, was determined
The radius of
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 highpower 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 machining.
slot profile and better
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
thereby be more closely met, and the deformation *
that the specifications
could
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 s -
in the TElol] mode is given in the last column of the
table. The power distribution
along each offset is shown in Figures 6.2-6.4.
106
0.0030 1.0 0.0025 0.8 0.0020 0.6 0.0015
0.4
0.0010
0.2
0.0
0.0005
0.0000 0
10
5
15
15
10
z (inches)
z (inches)
Figure 6.2
5
0
Power distribution
in coupled modes along the axis of offset
OSl.
0.0030
0.0025 0.8 0.0020 0.0
0.0015 0.4
0.0010
0.0005
0.0000 0
5
10
15
20
25
Power distribution
5
10
15
20
25
z (inches)
z (inches)
Figure 6.3
0
in coupled modes along the axis of offset
107
0.0030
0.0025
0.0020
0.0015
0.0010
liLb&i 0.0005
T&l
0
5
15
10
0.0000
5
10
15
z (inches)
z (inches)
Figure 6.4
0
Power distribution
in coupled modes along the axis of offset
os3.
Table 6-C design
of&et
arc length
radius of curvature
OS1 OS2
2.125”
4.229”
0.9971
7.53”
8.457”
0.9951
OS3
2.537”
8.457”
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. e “shooting”
As a check, I used the numerical method called
[31] to solve the two point boundary value problem of equations (6.15)
108
.
: I
with the coefficients boundary
in Tables 6-A and 6-B, the geometry
of the offsets, and the
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 to those gotten by direct integration T&r,
.
TI&r,
powers converge very accurately
of equations (6.19).
The reflected powers in
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 heart of the device.
Combined,
components.
The focus on them is further justified
are the parts for which I am most responsible.
they represent the
by the fact that they
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
necessary to transfer . -
the power from the fundamental
circular T&l
The reverse procedure
mode.
110
rectangular
system, it is mode to the
is necessary to feed the compressed
pulse into the accelerator
input coupler or rectangular
waveguide
Binary Pulse Compressor,
these tasks were accomplished
load.
For our
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
insertion loss is largely due to wall currents. An idea for developing
two percent
a more compact
and less lossy mode converter was thus greeted with enthusiasm. In 1991, a small KU ciates, Inc. Through
band mode converter
[32] came to our attention.
developed
Its operating
a program of theory, experimentation,
by Microwave
frequency
is N 35 GHz.
and numerical modeling,
adapted the unique design of this so-called “flower-petal”
Asso-
SLAC has
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 1.6” within the device, chosen to render the T&i
port.
The diameter of
mode safely cutoff, is expanded
in a nonlinear taper to the desired 1.75”. In the rectangular portion, septum,
perpendicular
to the electric field, bifurcates
tapered up in steps, to form two parallel waveguides.
a knife-edged
the guide as its height is 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, 1.6” circular guide at our frequency. at the irises, which is longitudinal e geometry
can propagate
in the
Power is coupled through the magnetic field
in the rectangular
guides. The symmetry
of the
thus limits the coupling to modes in the circular guide for which Hz is
111
t
I I Zt 1 I
II I
I I 1
!I.
- ErrgEE 4
Figure
7.1
l-w
TE ,0-T& flower-petal mode converter.
symmetric with respect to the x-z plane. This precludes the Th.&,lmode 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 to TEll
may also be expected.
mode purity was achieved. out reflections.
However, with the dimensions optimized
A post was placed near the rectangular
very good
port to match
An insertion loss of about 0.7% was deduced from measurements.
A pair of flower-petal
transducers, when tested for power handling capability
resonant ring, withstood
150MW
without sign of breakdown. (-
across the y-z plane, some coupling
5%) compared
at a pulse width of severalhundred
in a
nanoseconds
These mode converters are rather narrow in bandwidth
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 explained delivered * _
in the chapter on Binary Pulse Compression.
mode have been
To maximize
the power
in our compressed pulse, we decided to use circular waveguide
for the SLED-II
~delay lines, but also for transporting
not only
power between the klystron
and the compression
system, whose proximity
klystron
We further planned to plumb with circular guide from the
test gallery.
was limited
by the layout of 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 transfer lines mentioned ,above, WC175
in our experiment.
(1.75” i.d.) was the choice.
For the
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. is,down
a
As seen in Table 7-A, the theoretical
factor of five from that of the standard rectangular
For the delay lines, low loss is more of a priority.
wall loss for WC175 guide.
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 wc175,
of SLED-II.)
Its theoretical loss is down a factor of five from that of
and its suitability
had already been demonstrated
upgrade, we would replace the delay line waveguide the wall loss by, coincidentally, per meter is accompanied Table 7-A. Appoximately . _
in the BPC
In a future
with larger WC475, reducing
one final factor of five. The reduction in attenuation
by an small increase in group velocity,
as indicated
in
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
also be considered.
must
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 written
joints are likely to be the chief source of mode coupling.
a paper dealing with continuous distortions,
waveguide
cross-section
Morgan has
or gradual deformations
of
[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. discontinuities
Coupling
Let us consider briefly the effect of discrete
at joints, over which we have more control.
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 discontinuity
ple power only between propagating
of a given azimuthal
symmetry
m will cou-
modes whose azimuthal indeces differ by that
number.
Let us consider four types of small discontinuities in 4.75” circular waveguidediameter changes, axis offsets, tilts, and slight ellipticity backwards wave coupling, small compared to forward
mismatches.
We’ll include
coupling, only for diameter
changes. Formulae for the relative amplitudes of parasitic modes excited from TEol by each of these discontinuities,
appropriately
parametrized,
along with the relevant modes for WC475 at 11.424GHz. the same as that used in the last chapter.
e -
and the coupled mode respectively.
115
are presented below,
The general notation
is
The subscripts 0 and n indicate TEol
diameter change: The propagating
modes coupled by diameter changes are TEG,
TE,f,, TE&,
and TE$4.
XnXO
rn = (-1)” (A .
- Po>JmFl
P-1)
&a0 -p
where 6~0 is the change in radius.
offset. L The propagating
modes coupled by offsets are TE11, TE12, TE13, and TE14.
2 rn
=
5
V-2)
&&:~po)&!Tl)
where 6ar is the offset distance.
The propagating
modes coupled by tilts are TMII,
TE11, TEN, TEn,
and
T&4. I? - ilca&), n- JZXO
TMn,
(7.3) others,
where 68 is the small tilt angle.
ellipticitv:
e -
The propagating
modes coupled by ellipticity
.TE23, and TE24.
116
mismatches
are TE21, TE22,
- d,i,)/4
where 6a2 = (d,,,
for one side of the joint, the other being considered
circular.
Each of these types of discontinuity
extracts
a pure TEol wave given by the sum C II’,l” calculation
a small fraction
of power from
over the coupled modes.
A bit of
gives the combined fractional power loss at a WC475 joint as
6P
-
P
=. 0.676 ba; + 1.18 6,; + 3.36 6~; + 32.3 6e2,
(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
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
are
. 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 larity specification
+ (0.010”)2 = 0.0106”. The perpendicu-
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 =
(Sa2)rms
(sqrm8 Using these in equation 10m4 at each joint. equivalent
=
J /-
;Ax2
+ Ay2 N 0.012” t 7.6)
iA?.
N 0.0087”
= A0 N 0.001 rad
(7.5), we get an rms fractional
power loss of 5.08 x
This attenuates the TE 01 wave by about 0.00221 dB, which is
to the resistive wall loss in approximately
we can meet the above specifications keep joint mode-conversion
2.8m of 4.75” waveguide.
If
and avoid resonances, we should be able to
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;
Despite fewer propagating in equation nuity.
(7.5),
Notice,
+ 9.78Se2.
modes, the first three coefficients
are larger here than
b ecause a given error represents a greater fractional
however,
that the last coefficient
is considerably
disconti-
smaller here.
In-
creased guide diameter leaves one more vulnerable to tilts. This is clear from the adependence of the TEol/TMl1 in the denominator
coupling and arises from the extra factor of (pn -PO)
for the other tilt-coupled
reduced by increasing a, the propagation
waue. Their characteristic
modes. As the cutoff frequencies are
of the modes approaches that of a plane
angles of reflection off the waveguide
thus draw closer together.
118
wall diminish and
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. operation,
When we began
we were puzzled to find our gain down from the cold tests by nearly a
factor of two.
We’d expected
nothing this drastic.
some additional
A diagnostic
loss due to the transport
line, but
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
As k approaches
formulae share a factor p,1’2.
approaches
zero.
coupled amplitude mode conversion
Although
the equations
*
frequencies are very close to 11.424 GHz.
* Coupling
WC175
a severe sensitivity
to
has two modes whose cutoff
They are TE41 (11.416 GHz),
and TEl2
coefficients which remain finite in passing through cutoff can be ob-
tained by considering e wavenumber,
cannot be applied in this regime (The
cannot excede unity.), they do indicate to modes near cutoff.
ken, or vice versa, P,.,
the finite conductivity
of the walls and using the adjusted
given by /3;c = p2 + 2(i + l)@a
[37], as pointed out by Lawson [38].
119
The latter is considered the main culprit, as it is generated by m = 1
(11.445 GHz). distortions,
which should greatly dominate m = 4 distortions.
It thus turns out that our diameter choice of 1.75” was unfortunate. catastrophe
in the smaller components,
but in the long transport
We avoided
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. sandwiched
between the indistinguishable
2.81” seems dangerously mental in our experience. WC293
cutoffs of the modes mentioned
above.
close to the TE 13 cutoff, but this hasn’t been too detriNevertheless,
for power transport,
our plans for the future involve going to
well away from any cutoffs.
clear, the nearest mode being the non-threatening
CIRCULAR In transporting
1.75” is the worst spot, being
WC475
is safely in the
TM43.
90” BENDS
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
T&l
mode.
procedure
for over-moded
circular waveguide,
particularly
One could adjust the diameter to give one complete
in the
back-and-forth
transfer of power between TEol and the degenerate TM11 in the bend. According to equation
(6.6),
heavily over-moded,
h owever, the required diameter
is 3.56”.
The bend would be
and it would be impossible to maintain a small guide radius
120
1
-I-
L
0
1
0.5
1 1.5
2
2.5
WAVEGUIDE DIAMETER (inches)
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.
formed from approximately
four feet of 1.75”-diameter
longitudinal
corrugations
(azimuthal
grooves)
degeneracy.
In producing
such corrugated
The bends they supplied are aluminum tube, where again
are used to split the TEol-TMl1
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.
121
Of the four
bends in hand, the average measured insertion loss is about expensive due to the elaborate
machining process.
A different design for a TEol 90” bend was developed than corrugations,
we use a pair of partial
longitudinal
the plane of the bend, to remove the problematic .
adiabatically bend.
2%. They are rather
at SLAC
[39]. Rather
septa, perpendicular
degeneracy.
to
These are to be
introduced in straight sections before and after the circular arc of the
The guide radius, radius of curvature,
so that the propagating
and septum dimensions are chosen
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 tool in fixing our parameters.
It is a finite-element
[40] was an essential
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. review. .
_
It is currently under patent
It should be less lossy and may be cheaper to manufacture than the corru-
gated bend.
Another
is being pursued. two flower-petals.
idea, inspired by the success of the flower-petal
It is simply to use a rectangular-guide
mitred
transducers
bend between
This design is very compact and will most likely be used in the
NLCTA.
VACUUM Like the BPC,
PUMPOUTS,‘MODE
our SLED-II
which e - require evacuation.
FILTERS
system involves long runs of circular waveguide
Short pumpout sections were designed for 1.75” waveg-
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 conversion. Even if good azimuthal symmetry
were maintained
through tight machining
and assembly tolerances, power could be lost to higher-order which propagate loss unaccounted
TEon modes, none of
in the smaller guide. Such coupling likely contributed to the extra 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.
lines would be relatively
For our first SLED-II
prototype,
the delay
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
of the electric
due to the self-closing configuration
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
magnetic fields that are excited by the TEol mode in the waveguide.
longitudinal
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 operating mode.
Our pumpouts have sixteen l/8”
gaps, 3/4” in depth and separated
by disks l/4 ” thick.
While low leakage currents).
the pumpouts
are designed to preserve the TEol
of power from modes with azimuthal
magnetic
mode,
they do al-
fields (or longitudinal
This is a useful feature, as it helps to remove parasitic modes excited
123
-
by imperfections misalignments. excitations.
in the transmission
system such as waveguide
dents and flange
This reduces the danger of resonant power loss due to successive
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
Each pumpout
mode converters.
was enclosed in its own coaxial vacuum manifold,
4.5” in di-
ameter, with conflat flanges and a pumping port on the side.. Unfortunately, was found to pose a danger to their rf behavior.
this
Two of the four manifold encased
pumpouts exhibited much higher insertion losses (a few percent) than had been previously measured. This is no doubt due to excitation
of one or more disks could couple power from TEol,
space. A slight misalignment through
a parasitic
of a resonance in the annular
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 was partially
compensated
the two on the delay line side. This loss of pumping by an accidental gap between the offsets and the hybrid
body, inside the hybrid manifold, other pumpouts’was
on that side. Evidence
of negligible
loss in the
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.
sOur- present plans are to separate the functions into different components.
Pumping
of pumping
and mode filtering
will be done through sections perforated
124
with
Figure 7.3
Demonstration
in circular waveguide e
- waveguide
of mode filter removal of resonance spikes
between mode transducers.
The short section of
used for the upper plot was replaced by an equal-length
filter for the lower plot.
125
mode
-
many small round holes distributed perturbation
around the waveguide
so as not to present
asymmetries on the azimuthal order of any propagating
filters with one or four gaps or grooves are being designed. 2.93” waveguide,
our new choice for power transport.
modes. Mode
They will operate in
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, The partially
reflecting
signed for waveguide
TAPERS,
SHORTS
irises necessary for SLED-II
were also de-
to avoid transfering
The iris coupling is done at the smaller
absence of longitudinal
TEon modes, which are all
power into higher-order
cutoff at this size but not in the delay line guide. *
operation
of 1.75” diameter, to be placed at the ends of the offsets on
the delay line side of the 3-dB coupler. diameter
AND
Azimuthal
symmetry
and the
electric fields prevent conversion to other TE modes or to
TM modes, respectively. We planned to operate our first SLED-II twelve.
prototype
at a compression ratio of
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. at the time the iris drawings were submitted eassembled optimum
reflection
value is, however,
quite broad.
126
Our delay lines were not yet to the machine shop. The
Had we waited and measured
.
:
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 two-port
microwave
junctions
and an S-matrix technique for symmetric,
[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
the backwards signal.
Although.the
measurements
picked out
agreed, the latter method was
slightly more accurate due to the uncertainty caused by the non-unity VSWR former setup without
an iris. The reflection
of the
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”, e -
tapers
nonlinear,
General Atomics
from the Binary Pulse Compressor to complete the transition.
127
They also provided
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
with the delay lines being assembled from multiple segments. erator application
tolerance, especially Furthermore,
accel-
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 temperature
and contraction.
The
of the lines will be affected both by internal rf heating and by fluctua-
tions of the ambient temperature. to temperature,
A good approximation
for the phase sensitivity
taking into account expansion of both length L and radius a, is
given by
(7.7)
A4 = 2Pol L( k/P)2 KAT, 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 controlled
shorting
plungers attached
vacuum feed-t hroughs . These consist of 3/4”
to motor-
thick, aluminum disks,
copper plated and supported on the ends of steel rods. Aluminum was used to minimize torque on the rods, because perpendicularity . guide axis is essential for mode preservation.
128
of the shorting surface to the
A tilt in a reflecting short is equivalent
'
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]
H,=-
where A+ and A-
Jl(X[Ol]Pl4
xv0 k
(A+ aJo (xpq
-A-),
>
are the normalized forward and backward wave amplitudes.
similar expression for E+ contains a factor (A+ + A-), short A-
= -A+.
Thus the tangential
requiring that at a perfect
magnetic field on the shorting face is
J1(X[01]P/4
7f
.
The
Pa
aJoX[ol]
The relative ohmic loss, normalized AP -=
to the propagating
17ftan12
P
Rs
=
8
ds
P[Ol]
a J,2(X[Ol]Pl4P
2
{
N
Inml
=
fi$&+
‘En
Jn(xInml>
1
,n#O ,n=O
=
The cutoff wave numbers in equation .(6.9) are given by
k,, = 2. a The 2’ functions are normalized
J
s
such that
qTn . qT,dS
= kc;
where S is the cross-section of the waveguide. .
_
satisfy orthogonality
Js
T;dS = 1,
The T’s and their derivatives
(6.11)
also
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 yields the reversed Poynting
of In reflects the reversed magnetic
field, which
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. of orthogonal p,4, and z.
curvilinear z therefore
The natural set
coordinates to use are the “bent cylindrical becomes the longitudinal
coordinate
coordinates”,
along a curved axis,
whose radius of curvature we’ll call b. q5is 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,
and
e4 = f-3
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)
where Vz(n)(z) and Iz~nl(z) are to be determined. Maxwell’s
4)
n
equations, in this coordinate
*
system, take the following
form:
(6.14)
* In straight guide, we would have simply
98
:. I
Mwable shafts
III signal IN ‘10
kolator IsolatorAttenuator
Amplifier
Si. Generator
PWLSER
I
SCOPE
Figure 8.1
reproduce . _
Schematic of 3-dB coupler network transmission test.
the same voltage.
Subtracting
the measured loss of the WR90
nents and mode launchers and the difference in coupler calibrations the insertion loss of the shorted hybrid-offset
assembly.
compo-
then yielded
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
Figure 8.2
I
I
I
I
I
I
I
1
I
I
I
I
1
I
I
of round-trip
transmission,
to that used for the hybrid. As the directivity to be poor, we used a three-port
I
I
I
I
r]h, of shorted
assembly in the vicinity of the operating
The round-trip loss of each line was individually
I
11.428
11.426 11.424 11.422 FREQUENCY (GHz)
Measurements
3-dB coupler/offsets
.
I
11.42
frequency.
measured with a set-up similar
of our diagnostic couplers was found
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.
small losses, we used the following
Rather than rely on direct measurements of 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 s -
operation,
but smaller, is seen. Taking the limit of equation (4.5) for large n and solving for
134
-
the round-trip
field attenuation, e-2r
we find =
Ee(
4Ei
(84
1 - s2 + sEe(m)/Ei’
The reflection coefficient had been previously determined, the square root of our relative power measurement.
and the field ratio is just
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
the shorting piston motors remotely.
so that we could drive
The motor controllers were hooked up to a
PC, and a program written to move the shorts simultaneously, or opposite
directions,
As mentioned
with a chosen step size.
in either the same 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 to the input and- output converters
offsets.
and the remaining
was added a PSK
To these were attached
instrumentation
were connected
the flower-petal
mode
of Figure 8.1. To our drive circuit
(ph ase shift keyer) triggered
to reverse the input phase 75ns
before the end of a C,-bin pulse. Figure 8.3( a ) sh 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. output pulse amplitude.
It exhibits the expected
For comparison,
step behavior
Figure 8.3(b) illustrates
and flat
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
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I
I
I
I
I c)
I I
a
0
-m i -- y-y
,-y$y=y-y --
-468
0 I,,,
II,,
,
-------
III,
-,
532
100 ns/div
III,
1
II,,
5-L
c,=12 s=O.805
4-
w qh=O.22
e-2T=0.976 3-
2-
1:
0
200
400 t
Figure 8.3 e
.600
600
1000
bs>
(a) Lo w- p ower measurement and (b) prediction of the output
- power waveform
for our SLED-II
prototype
twelve.
136
with a compression ratio of
.
: I
the non-linearity
of the crystal detector.
The waveforms in Figure 8.4 are to scale, normalized They
were obtained
by multiplying
function of the crystal detector. in the thirteenth
the recorded scope traces by the calibration
In 8.4(a), there is no phase modulation.
bin corresponds to E,(13)2.
8.4(b) shows full SLED-II
The finite risetime of the crystal used, evident shift, and sensitivity
of its calibration
the apparent degradation
to the input power level.
The peak operation.
in the input glitch at the phase
at higher voltage
levels both contribute
of the compressed pulse flatness.
The amplitude
to
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.
_-
-
ratios (-
Forseen SLED-II
5) with significantly
applications
will employ
much lower compression
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
pression ratios.
our system, we measured the gain at different
com-
Our data is presented in Figure 8.5 for C, = 2-16. The solid line
137
-
-
I 1
2.5 .
. .
I I
. ’
I !
. .
I I
*
-Input
--Put
2
1.5
-.. .” ~
1
0.5
F --A-
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
0.7
0.8
0.9
1
1.1
Time (ps)
0 .
-
Figure 8.4 (a) without
0.1
0.2
0.3
0.4
Input and normalized
0.5 0.6 mt? w
output pulses of the SLED-II
and (b) with phase switching.
138
system
:.
.
I
Gain, s=O.805 -2
_ FIT O2
RND.
Gain With Theor.
“’
Hybrid
s --
Measured
TRIP
4
LOSSES:
6
Compression
Figure 8.5
SLED-II
prototype
ohmic losses considered,
Wall Losses Gain
Hybrid-8%,
0
Delay Line &
Delay
10
Lines-4.8%
12
14
Ratio
ideal gain, predicted gain with theoretical
and measured gain as functions of compression
ratio. The line through the measured data is a theoretical fit based on the s
-
given total round-trip losses in the coupler and offsets and the delay lines. These parameters
are consistent with our experimental
determinations.
The reflection coefficient is taken to be 0.805 in all three curves.
139
_
16
Rectangular Wave!guide\
LOad
Circular
90” Bend
Mode Transducer ~u’lZIZl Circular
Figure 8.6
’ Drive/Phase Shifter
Schematic of prototype
high-power
141
SLED-II
system.
: I
As mentioned
in the preceding chapter, the circular waveguide
nally made in WC175.
Sensitivity
run was origi-
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 system.
The klystron voltage
range, power was fed into the
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
molecules from the copper.
outgassing,
As the power was gradually
gassing had to be accomodated
or desorption
of gas
raised, the increased out-
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.
s
-
When we’d reached a few megawatts
out of the klystron and close to 20MW
peak compressed power the data in Figure 8.7 was taken. The diamond at C,. = 12
142
5
0
4 d ‘l-l
cl I E
3
$ 2 -i
1 ot 1
1
B 2
3
4
0
--
Ideal (lossless)
#
--
Measured
0
--
Typical Measurement
I
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I
5
6
7
8
9
Compression Figure 8.7
Megawatt-level
Gain, s=O.805
Gain
1 10
11
12
Ratio
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. modulator _-
Irregularities
in the shape of the klystron pulse, due to the
pulse shape and to small reflections,
pulse shapes made gain measurement
as well as imperfect
a rather imprecise procedure.
compressed 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
than a couple of tenths of a dB. The overall indication our power was lost in the waveguide
connection
to be accurate to better was that a few percent of
between the klystron
and pulse
compressor, as expected. s
- While
XC2 was being used elsewhere in an experiment
development,
an experimental
X-band
accelerator
143
related
to klystron
structure was installed
in the
.
.
:.
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 compressed 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 vacuums of different parts of our experiment. the structure while maintaining
SLED-II
The former had allowed us to connect to under vacuum. In each case, the window
was removed and its holder replaced with a WR90 single extended
spool piece. This resulted in a
vacuum envelope, except for the klystron.
of the klystron was split by a magic-T
The output waveguide
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 .
_
The splitting halved the amount of
power each window had to handle. The precise mechanism of window failure is not well understood,
_-
of the tube.
but progress is being made in their development
The maximum output power we reached with this SLED-II High-power
waveforms
[44].
system was 154 MW.,
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.
signal reflected toward the klystron, pulse, and the SLED-II
horn top to bottom are: the
the klystron output,
the modulator
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 modulator
The broad inverted curve is
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.
ture, along with the amplitude variation, development
of broader-bandwidth
in the future by the
klystrons.
ACCELERATOR The compressed pulse produced high-power
will be ameliorated
This fea-
STRUCTURE
TEST
by our prototype
rf tests on a 75cm, constant-impedance
SLED-II
was used to run
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)
The ASTA
structure.
b-earn line [46] is shown in Figure 8.9. As the electron gun had not
been commissioned
at the time of this experiment,
structure were disconnected for this experiment.
and the pair of quadrupole
The full instrumentation
field-emitted
magnets were turned off
will be operated
longer, modified structure and the upgraded SLED-II An in-line Faraday
all elements upstream of the
in late 1994 with a
system.
cup gave a measure of the accelerated
dark current, or
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 copper. _-
The
radiation
from field-emitted
second and third waveforms
are the compressed
Finally,
and leaving the structure, respectively.
electrons
the bottom
slamming
into
rf pulse entering
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
s
* The example parameters used in the Appendix -
X-band structure containing thirty cells.
147
are for a different, but similar
Electron 01
Gate Velve
/
W!r
Scintilletor
/
AaaleratorL \\\stNoture
Figure 8.9
.
Schematic of the ASTA
input pulse and the corresponding
waveforms),
Lens
beamline.
output pulse computed for the 75cm structure.
In addition to the expected 52ns delay and attenuation experimental
Prdhmchsr
(not evident in uncalibrated
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 plained.
To be effectively
of the traveling
accelerated, an electron must be captured in an rf bucket
wave. That is, it must approach the phase velocity
mental space harmonic) the wave.
of the -dark current pulse can also be ex-
F’rom equation
(of the funda-
quickly enough to keep up with the accelerating
phase of
(10) of reference [47], one can see that the minimum rf
148
:
.
Figure 8.10
Waveforms
top are: detected
radiation,
from high-gradient
structure test.
From the
the structure input rf, the structure output
rf, and the dark current (Faraday cup signal).
149
field amplitude for capture, normalized
to mc’/e,
&thresh.=$--(d%T-
where k is the free-space wavenumber,
is
JGj-PPP),
& = 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 longitudinal
momentum,
At our frequency, 61.2 MV/m.
zero
so, for dark current, the above expression reduces to
the threshold
gradient
for capture from rest (at ,f3p =
1) is
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 thresh.-g1-a). -
(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 program
of the phase of voltage oscillations in adjacent cells, the ECM
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 s its - variation would appear to enhance this effect. The phase velocity dip is due to the higher-frequency
content of the pulse, and, as one would expect,
151
the dark
1.0 0.8 !-
I
I
-I
0.6 0.4 0.2 0.0
I
I
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I
I
0
I
I
I
I
50
25
I
I
I
I
I
I
I
I
I
I
I
I
100
75
I
I
I
I
I
I
I
125
I
I
I
I
I-
1.04 1.02
\ .... . . ./!..\..... \I%
P r
0.98
_-
I
; I
0.96 I
I
I
;
I
10
.
Equivalent
tion in the structure.
/
\
/‘,A
\/
I ’ /I I
I
I
-
J I
I
30
20 t
Figure 8.11
’
’
III
I
O
I
I
I
I
I
I
40
(4
circuit model simulation of x-f pulse propaga-
(a) the power (field squared)
s cells, with dispersion evident. velocity
. ...
I’.
. El
.
04
(b) the field amplitude
in the tenth cell.
150
in the first and last and the local phase
I50
I
:
5.00
z a
.
E
1.00
E 5
0.50
z 2
-
0.10
5 Figure 8.12
Energy
20
15 10 ELECTRONENERGY (MeV) distribution
25
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 spectrometer tatively
8.12 shows the energy distribution at two different
of dark current measured by the
average accelerating
with simulations performed
gradients.
by Seiya Yamaguchi
These agree quali-
[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
uniform capture from along the length of the structure. is greater towards the input end, so is the probability
152
indicate fairly
While the local gradient
of electrons originating
there
.
.-
:
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.
with the dates they were taken.
Evident
Two sets of data are presented, along 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
our data, we expect a dark current at lOOMV/ m of about 0.5mA. current of more than an amp in the NLC, background and beam loading.
With
this is considered tolerable
from
a design
in terms of
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,
events in the structure. . _
-
a
hundred,
we were able to observe occasional
The dark current would jump breifly by a factor of about
and the relative
signal amplitudes
indicate where the breakdown occurred. klystron, 154MW
breakdown
out of SLED-II
(G,
from our scintillator
We eventually
reached 34MW
array would out of the
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. demonstrate work-into
an application
These results have been included here to
of the pulse compression development
the overall linear collider program.
tor), SLED-II
system, and accelerator
The X-band klystron (with modula-
structure together
153
and to tie that
represent a prototype
of
.,*.r.,.*.
.I...
.,.*
7Scm Section
October 9,1993
Pulse Length: 75ns lo3
L
’ ’ a * * ’ ’ ’ Ba ’ ’ ’ L ’ ’ ’ 4 50 80 70 80
Average Accelerating Gradient (MV/m) Figure 8.13
Dark current amplitude
dient in the 75cm accelerating conditioning.
as a funcion of accelerating
structure before .
154
gra-
and after prolonged
rf
9. VARIATIONS
ON A THEME
In this chapter, I wil1 explore some modifications
to the SLED/SLED-II
While they may not all be practical, they should at least be of theoretical Moreover,
they point to possibilities
compression.
for further development
The section on Ramped-SLED-II
actually implementing
in the NLCTA.
idea.
interest.
in the area of pulse
describes a technique which we are
I will include, for completeness, highlights of
foreign pulse compression work.
AMPLITUDE-MODULATED Through
modulation
SLED
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
of the emitted
field is canceled by.a rise in the direct, or iris-reflected,
field.
dive 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
where Z’, = 2Q~/w and a = 2@/(1+ e -
+
Ee
=
OEin,
,B). The Laplace Transform of this
155
gives
or (1 + sTc)Fe(s) where Ee(tl)
is the amplitude
the time of phase reversal).
= afin
(9.2)
+ TcEe(tl),
at the beginning
of the input modulation
(i.e.
at
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)
=
t1 < t < t2.
C,
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 l-a+sT,
rn
=
C C/Z s - (a - l)/Tc - s[s - (a - 1)/T,]’
Taking
the inverse Laplace
E,(tl),
we find the required input amplitude modulation
Ei,(tl
< t < t2) = 5
(9.4)
Ee(tl)-
Transform
+
and substituting
1 _ e-tl/Tc _ - C o-1
the above expression for to be
,,(--1)(+-WTc >
.
(9.5)
In equation (9.5), C is an arbitrary constant representing the desired amplitude magnification. an effective
The effective gain of the SLC SLED system is 2.6, corresponding
to
C of 1.612. If we desire the same gain with a flat output pulse, the
‘156
I
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I
I
I
I
. . . * . -. . *
2
r
I
.
-
-, . . -*-
-
-. I
E-in
1
I_’
1
-.
1
-
*.I i! I -
0
-1
Figure 9.1 modulated
1
3
1
0
Input (solid) and output (dashed) waveforms for amplitudeSLED.
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
input field
as
eiwt factor, consider the real and imaginary
driving independent
parts of the
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
(9.6)
Eout&) dout(t) = tan-l Eoutl(t). 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 -=
6 (LYEini- ECi) - %,
dt
For an input of normalized
Einl
. With
i = 1,2.
c
constant amplitude varying in phase as 4(t),
= cos
4(t),
and
Einz = sin 4(t).
these equations, our condition for flatness becomes
(Eel - ~0s4) $( c
crcosd-E,,)+sin~$-
+(Eez -sin 4)’
1
d$ dt
olsin4--I&,)-cos$z
P-7)
1
=O.
The emitted fields in this equation are themselves functions of time and of the phase variation.
Their solutions are
[J
t
E,.(t)
= aemtlTc
$
C
E,,(t)
et’lT= cos qS(t’)dt’ + (1 - e-tl’Tc
t1
J’ et’lTc sin $(t’) dt’.
= ae-tlT~ f C
51
159
)I (9.8) ,
I
:
After the real cavity field is driven steadily for time tl,
let the input phase be
shifted by -7r/2, rather than the standard X. Inserting this along with the emitted field values Eel(tl)
= cy (1 - eVtliTc)
solve for the time derivative completing
and Ee2(tl)
= 0 into equation (9.7), we can
4 (defined as the right-hand limit at the discontinuity),
our initial conditions on as 4(h)
=
2,
((tl)=-$[l-e;tl,Tc
+a(l-e-'l'Tc)]'
If (9.7) has been met at all points between relationship
and t, we have also the desired
tl
IEout(t>12 =IEout(b)12,
which we can write as
(Eel - cos 4)” + (I& - sin 4)” = o2 Combining and E,,.
t [Jt +J
1 - eet-liTc
>
+ 1.
equations (9.7) and (9.10), we can eliminate the terms quadratic in E,, The resulting equation, written explicitly
et’lTc cos $(t’) dt’ + T, (1 - emtllTc
in terms of r$(t), is
)I[$(a
- l)cosd+
$sin+
C
t1
et’jTc sin qS(t’)dt’
$(o
- l)sin4
1
- Jcos 41 = [o (1 - e-‘1/TC)2 + l] et.
t1
Although
‘(9.11) 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 some value t > tl, s
as a polynomial
to find an approximate
solution, out
expansion in (t - tl)/T,.
- Figure 9.2 shows the results of such a search. The coupling parameter
SLAC
linac SLED
of the
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 and for successive higher order corrections up to fourth
constant variation of &tl),
order. Reasonable flatness, with less than one percent amplitude droop, is achieved = 0.35T,
for about t2 - tl
at a gain of G = 2.68. If we end the pulse there, the
efficiency is nc = 0.51. IIll
1.5 -
1.0
2
IlIt
.
.
.
.
IIII
.
.
.
IllI
.
-
1111
. . . \ . . .
--
-
. .
.-
.-
-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
ations of successively higher order polynomials solid curve is for $in = -r/2
This could be improved
for input phase vari-
in X = (t - tl)/T,.
The
- 2.582X - 1.6X2 - 2.5X3 -5X4.
by using higher Q (e.g.
superconducting)
cavities.
SinceT, = ~&L/W = (2/w)Qo/(l + P), raising Qs allows us to raise p without changing the time constant. e -
As p increases, o approaches
2, and the gain and
efficiency in the above example approach 3.41 and 0.65, respectively.
161
Figure 9.3 demonstrates
.
another problem with this approach.
the output pulse also varies, whereas accelerator
applications
The phase of
generally
constant phase. Since a constant rate of phase slippage is equivalent frequency, the linear part of this variation may be compensated
call for a
to a lowered
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 that the efficiency
might improve the above picture slightly, but it is clear
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
An undesirable feature of SLED-II
DELAY
LINES
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 multi-level -
-
will need an rf station about every lo-20
scheme for stacking overlapping
One method
delay lines.
we’ve considered for shortening
loaded delay lines to reduce the group velocity.
162
feet, this necessitates a
the system involves using discOne could use the same circular
-
. :
-1
Figure 9.3 amplitude
Phase-modulated
(negative
SLED
output.
The solid curve is the
sign substituted for 7rphase), the dashed curve is the
phase, and the dot-dashed adjusted by detuning.
curve is the phase in the compressed region
The pulse could be terminated
at t/Tc = 1.85 for
reasonable flatness.
waveguide mode, but introduce periodically This structure is equivalent
spaced discs with central coupling irises.
to a series of cylindrical
resonant cavities inductively
coupled through their end plates. The flow of electromagnetic by this modification,
and we can achieve the same delay in less physical length.
The behavior of such a delay line has been theoretically the equivalent-circuit s cells appropriately
energy is slowed down
investigated
model applied to structures in the Appendix,
modified to present a complete or partial reflection.
163
by use of
with the end The chain of
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 waveguide delay lines, despite the reduced length, due to the loss on the discs. ignore the presence of the coupling holes (H, l
(6.18) and (7.8) to derive the following
If we
= 0 on axis), we can use equations
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 .
_
1,386,800 Qo
=
P-7)
1 + 117.1/n ’
To make this worthwhile, it is desirable to decrease the group velocity by about an order of magnitude. us/c = 0.1035.
The delay lines can be shortened
Since X,/2 =
to 12’9”
by making
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 with these parameters. switching
system using loaded delay lines
A compression ratio of five and a ten nanosecond rise and
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
750
500 t
Figure 9.4
Equivalent
form from a SLED-II
e
-
1000
1250
(r-4
circuit model simulation of output power wave-
system employing fifteen-cell disc-loaded delay lines.
Firstly, the loss introduced by the discs lowers the gain by about ten percent.
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
(9.8)
2’=2Qo.
Degradation
of the gain is inevitable
Secondly,
dispersion
significant power variation
distorts
when delay lines are loaded.
the waveform,
giving
rise to an overshoot
and
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 low loss is highly overmoded.
structure used for
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 absorberloaded circumferential e -
gap next to each disk to damp modes with longitudinal
rents. (This idea recalls some of our mode filter designs for waveguide.)
166
cur-
Although
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 campactness, 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.
over-moded,
version, however,
super-conducting
delay lines may yet find application
RAMPED
SLED-II
disc-loaded
in BPC or SLED-II
FOR BEAM
LOADING
In a less
or periodic-structure
powering of accelerators.
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
from the accumulated NLCTA
longitudinal
structure, or equivalently
wakefields of the preceding
bunches.
For the
design parameters, this would amount to a 25% droop in energy across the
beam pulse. One means of achieving this compensation the structure is completely
is to inject the bunch train before
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. s -
As the bunch train enters, beam loading compensation
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 modulating
wave. wave.
system by properly
the phase of the input pulse. The compressed pulse, as we have seen,
can be represented reflection
ramped rf pulse with a SLED-II
as a combination
of several phasors, one from the direct iris
and one for each of the delay-line reflections that make up the emitted
Each of these phasors has its origin in a particular
pulse. By manipulating
time bin of the input
the phase of the time bins, we can control the phasors. I
I
1
I
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I
1
I
I
I
I
I
fl
I
I
I
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I
I
I’
I 0
I
I
I 0.5
1.5
1
Real Figure 9.5
Phasor diagram of output field contributions
time bins and their sum during a ramped SLED-II
from five input
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.
168
Let’s
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
E *=lmFzl 4
El
(9.9)
’
+E2
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.
E,in/Ep
These differ by only six percent and allow us to go as low as
= 0.172.
To determine the desired functions &(t)
and 42(t),
we have the following
quirements:
El cos q&(t) + E2 cos 42(t) = EC, + AE;,
s El sin&(t)
+ E2 sin&(t)
169
= 0.
re-
.
:
I
3.0
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
2.5 / I I
2.0
z 1.5 \ 1
--Ii ‘xl .
\
I
I
1.0
I‘\ ’ \
i‘, ’ \
2.5
3
w”
I
I
I
I
I
I
I
_
200
y
150
T
100
0.0
50
I
0
I
I
I
I
I
I
I
I
750
I
I
I
I
1000
I --100 1250
bs>
1’ I ” ’ I I ’ lb’ I I I ’ I I I ” ’ I I’4 \
1
-50
I \ I I
1 L-
100 50 0
----
$. s
1
\ :\ i
0
h \ I---
I--, I
I
I
250
I
I
I
I
I
I
I
500
I
750
I
I
I
I
I
1000
I
I
7
-100
y
-150
-200 1250 I
t.. (ns)
Figure 9.6 s
,” P
1.5
0.5 _-
I
--
7
500
250
0
1.0
. _
/
’ / L/
2.0
..
I
/
t 3.0
I
1
-0.5 0.0
I
Input and output waveforms for a partially ramped SLED-II
- with C, = 5. The solid lines indicate field amplitude and the dashed lines / phase.
170
z CD 2%
.
: 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 q&t(t)
Es+AEt-
E; - E,2 E. +AEtlAt
At
(9.10)
= sin-’
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 mentioned. .
One is the VPM
in Protvino,
(VLEPP
Russia, by V.E.
Power Multiplier)
[52], developed at Branch INP
Balakin and I.V. Syrachev.
It works on the same
principle as SLED, but requires only one storage cavity and no 3-dB hybrid. VPM
uses an “open”
operated ations.
cavity, shaped like the wall of a squat barrel.
in a traveling-wave
“whispering
gallery”
The fields of this mode are concentrated
The
This cavity is
mode with many azimuthal variclose 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. s
- In an attempt to improve the SLED-like shape of the output pulse, a two-cavity
VPM
was built [53], in which the stacked barrel cavities were coupled through their
171 .-
: I
common open face. The same cavity-doubling theoretically
idea was tried at CERN
[54], both
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 S.Y. Kazakov
means of acheiving
a flat compressed pulse has been proposed
by
[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
initial
spikes and varying
degees of ripple, depending
used.
The flatness achieved suggests that the performance
outputs with small
on the number of cavities 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
teresing linac powering scheme based on Binary Pulse Compression
an in-
[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
172
such systems, all the accelerator
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
low-loss fast rf switch to circumvent the theoretical limitation
on SLED-II
a
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 a next-generation
linear collider,
have seen much progress towards the design of 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, as Binary flat-topped Standard
and their derivatives, and transmissive pulse compression, such
Pulse Compression
(and chirping).
Our goal has been to produce
pulse suitable for the uniform acceleration SLED,
the archetypal
a
of a long train of bunches.
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 a shorter pulse.
for all the energy in a pulse to be compressed into
As the name suggests, it is limited
are s - powers of two. We have successfully constructed
to compression and operated
system capable of working with one or two sources. Experimental
174
ratios which
a 3-stage BPC
results have been
.
:
I
reported showing good agreement with expectations losses due to imperfections experimental
klystrons
based on measured component
and waveguide attenuation.
to power high-gradient
This system was used with
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 system was perceived
as being too massive and bulky to be incorporated
other rf station of a linac. accelerators,
However,
as efficiency becomes more crucial in future
attention may return to this method of pulse compression.
Our recent focus has been on SLED-II, pressed pulse. increasing SLED-II
It shares with SLED
the extension of SLED with a flat com-
an intrinsic efficiency
compression. ratio and, like BPC,
which decreases with
utilizes long waveguide
is, however, less bulky and less complicated
signed and built a high-power SLED-II
than the fatter.
delay lines. We have de-
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. prototype,
at every
our SLED-II
system was fairly successful.
As a first
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 lines with 4.75”-diameter with rectangular s A partially
waveguide of the delay
waveguide and replacing the 90” bends and 3-dB coupler
waveguide components fitted with flower-petal
mode transducers.
upgraded system has already yielded a gain/efficiency
175
improvement
of
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 A short, experimental, Test Accelerator)
hybrid.
X-band linac called the NLCTA
is under construction
at SLAC,
tain an injector, a chicane for bunch manipulation, .
sections.
A 250ns pulse of 200 MW
including the injector, plan to accomplish
in End Station
B. It will con-
rf power is required for each pair of sections,
this by compressing
gradient of 50MV/m.
1.5~s pulses from 50MW
compression ratio of six, for a gain of 4. Three such SLED-II
in Figure
Linear Collider
and four 1.8 m-long accelerator
to achieve the goal accelerating
An isometric drawing of the NLCTA
(Next
rf distribution
10.1. We are currently developing
klystrons by a
systems are required.
system, not to scale, is shown
the electronics
needed to implement
the ramping described in the last chapter for beam loading compensation.
176
We
-
- Figure 10.1
NLCTA
rf station with SLED-II
TO SCALE).
177
pulse compression (NOT
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:. I
APPENDIX: An Equivalent Circuit Model for Traveling-Wave
In this note, I develop an equivalent structure
operating
30-cell X-band demonstrate
Structures*
circuit model for a constant-impedance
in ‘the 27r/3 accelerating
section developed
mode.
I use the parameters
as part of the NLCTA
its usefulness for examining,
R&D
at SLAC.
in the time domain,
of the I then
pulse distortion,
energy gain, and field profile in the structure for various input pulses. Each cavity is represented by .a parallel circuit inductively actual structure ” this discrepancy.
couples capacitively,
coupled to its nearest neighbors.
but the behavior
The
should not be altered by
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 September
distributed
as SLAC Advanced
1992.
183
Accelerator
Studies (AAS)
Note 75,
:.
.
I
The basic equations of the above equivalent circuit model are
I; + I; + I,” + If = 0 IC n dI; Vn=Lz+kLT =
dI:-,
LdI; dt+kLx-
dI;+,
’
from which one easily derives, defining In E I: + Ii,the .
_
following
first-order differ-
ential equations:
din -=
dt The quantities
(A4
L-1 y p) 12’n - k(K-1
R, L, and C are related
to the characteristics
structure by the relationships
L=!&c=
wo& -=-2Q uo&
184
+ vn+l)]
2R
wo& Q w,,R’
of the actual
I
: -.
However, we don’t need to know RSh if we make the following
change of vari-
ables: Tz = The time derivatives
In.
(R/Q)
then become
dVn -=-wo(L+$) dt
dz, -= dt
2(l:
k2) [2vn
- k(V,-1
(A4
+ %+I>],
where we’ve used
-- 2 w” - LC 2
wou wo$v2 +V2/R &=T-l= Equations
(A.2)
= w. RC.
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 impedance
traveling-wave
structure, we need to add the proper transfer
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 -
We define a complex propogation
k(Vn-1
i- Vn+l)] - ZjwV,.
constant 7 = ,O- jo,
so that the relative steady-
state voltages of the cells vary as e--jrpn, where p is the structure period. out Vn, we now have
185
Dividing
.
: I
w2 = (1 :k2)2
[l-kcosTp]+JT.
or
Expanding
cos yp = cos(@p - j op) = cos /3pcash cup+ j sin ,Opsinh cup, we get from the real and imaginary parts of the above equation, respectively,
and W
-= wo . _
kQ sin ,Opsinh cup 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,
Y = coshap, we get
Y(X)
=
&
+4k2Q4
-(=&r~+&-3~~~+4Q~(3
z&q--J
186
1
(A.4)
(A-5)
The group velocity
is given by
‘g=dp = du
w kp sin ,Bp 2(3”(1 _ k2)
& [ 1 ’ + xz
= iwkpsinpp
l-kxy’
w, in our case, is 27r x 11.424 GHz, and, from a calculation SUPERFISH,
Q = 6,960.
follows from the condition
(A-6)
Y+“2
made with the code
The length of each cell, p, is given as 0.00875m. that the phase velocity
This
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 equivalently x = -l/2) velocity
and requiring the above expression for vg to equal the given group
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 attenuation fesult.
by the above equations.
As a check, the
constant at 27r/3 was found to be in agreement with the SUPERFISH
This value, cy = 0.00452, leads to a field (voltage)
by a factor of e-‘Oa = 0.873.
187
at the output end down
Vl = .
Next
we calculate the impedance
of an infinite chain of inductively
resonant circuits acting as a 27r/3-mode traveling-wave
structure.
Referring
coupled to the
above diagram, we can write
Vi = jwLI1 + jwkLI2 v2
= jwLI2 + jwkLI1
-jwkLIl
+ I2 =
jwL + ZI
Z eff =jwL+
1 -=jwC+A++ZI Eliminating
Q=
= -I25
(wkL)2 jwL
+
ZI
eff
Zl between the above two equations and substituting
woRC = 2R/woL,
we get the following equation for Zeff:
188
wi = 2/LC
and
I
.
:
[l-2(-3’+
+2(1-
j$(E)]Z$f
k2)[j(z)2
+ i(z)]WLZeff +(1-
Now, defining Z = R/Q = woL/2, Zeff
li~)wZLZ = 0
we can rewrite this as an equation for g,ff
=
/Z,
with the complex solution
Zeff
whereA=
Z = *
(l-
=
-A[$
+ ifi] - &P[$
+ 61’ - 2A[F
+ $f - l] ,
+5+$!-1
(A.7)
k2), D = (wo/w), and the minus sign is chosen to make the resistive
part positive.
Equating
Z,ff
transfer impedance,
with the impedance of the right inductor, L, in parallel with a Zt, we get
1 -=--Zt
1
1
Zeff
jwL
189
+zt=-=
zt
zeff 1+ jQZeff
Z
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. cells).
with the desired phase shift between
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 resistance and a reactance.
.
of a
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’ s
=
- One detail remains.
w;
C
z
0.00904 Q
N
62.9.
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-
-+I()=-
2 Rt J
t
Rt R
1+--.
The final equivalent circuit model can be represented as follows:
30
2
k
191
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. consecutive
In plot (a), we see the voltages of two
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) sh ows the response of the first cell when the drive current is turned on instantaneously
Here and in Figure
and left on.
envelope, or voltage amplitude.
A.3, we are looking
at the
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, cell. For too sharp a rise, we see wild fluctuations, particularly
or last
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 structure. s is injected.
t,
is again the total,
N is the cell number.
particle traversing the
linear rise time, and ti is the time the particle Again,
192
the behavior
is much smoother for the
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
could be with a modest modification
losses were not included, although they
of my program.
Not only is the energy gain
rendered more uniform by a ramped pulse, but the lost efficiency somewhat compensated
appears to be
by a broadening of the injection window.
This note is presented as an example of the usefulness of the concept of equivalent circuits, as an educational results or future applications
exercise for myself, and in the hope that these
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 velocity
Dispersion
diagram, normalized
phase velocity
as functions of Bp/n, and real and imaginary
194
and group
parts of &(w/wo).
-1.0 tI,,"I""I""l""r""ll 37.9 37.925
37.95
37.975
36
38.025
t b) 1.2
C”“l”“l”“l”“I
““I
““4
1.0
.
_
0.6
0.4
0.2
0.0
0
25
50
75
100
125
150
t (4
-
-Figure
A.2
Voltage in adjacent cells oscillating
voltage envelope in the first cell.
195
in the 2~/3 mode and
.
-
i 1 , :!
‘&
1.
0 = ‘3 s
- Figure A.3
I. .,I ...l....I....'. 1, e x x x 2 2
Propagation
SU~=“~
‘&
SU()I
='$
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
and after structure filling.
N is the cell number.
197
t,
at times
ti,
during
l
i
+. .** +\ J +** o,,,.‘....l....‘....‘.t.. l .
:
0
20
40
60
60
100
lNJEC3lON TlNE (as)
fI .* .
20
(““““““b,
..
: +* : .+
10
:
t
2 +. l
f
l f
0
0
l....l....l....l...F, 20
60 M
e
-
Figure A.5
60
. .
166
(111)
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
