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Perturbed-wall microwave mode converters and bends in overmoded circular waveguide

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P e r t u r b e d - w a l l m i c r o w a v e m o d e converters a n d b e n d s in
o v e r m o d e d circular w a v e g u i d e
Luo, Gwo-Huei, Ph.D.
The University of Wisconsin - Madison, 1992
UMI
300 N. Zeeb Rd.
Ann Arbor, MI 48106
A dissertation entitled
PERTURBED-WALL MICROWAVE MODE CONVERTERS AND
BENDS IN OVERMODED CIRCULAR WAVEGUIDE
submitted to the Graduate School of the
University of Wisconsin-Madison
in partial fulfillment of the requirements for the
degree of Doctor of Philosophy
by
Gwo-Huei Luo
Degree to be awarded- December 19
May 19 9 2
August 19_
Approved by Dissertation Readers
<2ks
January 15, 1992
^^L^C^-t
Majoy Professor
/
1
Date of Examination
Ciu^
Dean, Graduate School
PERTURBED-WALL MICROWAVE MODE CONVERTERS AND
BENDS IN OVERMODED CIRCULAR WAVEGUIDE
by
GWO-HUEI LUO
A thesis submitted i n partial fulfillinent of t h e
r e q u i r e m e n t s for the degree of
DOCTOR OF PHILOSOPHY
(ELECTRICAL ENGINEERING)
at the
UNIVERSITY OF WISCONSIN-MADISON
1992
ii
ABSTRACT
PERTURBED-WALL MICROWAVE MODE CONVERTERS AND
BENDS IN OVERMODED CIRCULAR WAVEGUIDE
Gwo-Huei Luo
Under the supervision of Professor Ronald J. Vernon
At the University of Wisconsin-Madison
High-power microwave tubes producing from several kilowatts up to
many megawatts often generate modes which are not suitable for plasma
heating, radar applications, or long distance transmission. To obtain a more
desirable mode, a sequence of mode converters is often used with various
high-power microwave sources such as gyrotrons and backward-wave
oscillators. This thesis considers the theory, design, fabrication, and testing
of smooth deformed-wall mode converters for this purpose.
Applying perturbation theory and Maxwell's equations, we can
determine the coupling coefficients for various geometric deformations. A set
of coupled-mode equations is employed to describe mode coupling due to the
waveguide deformation.
The design, fabrication, and testing of a single-period 8.6 GHz
TM 0 1 -TE 1 1 serpentine mode converter in 4.76 cm diameter waveguide is
discussed. The measured conversion efficiency is better than 99%, which is
iii
determined from the far-field open-end waveguide radiation pattern. A
design technique for 90° waveguide bends for TM 01 and T E n modes is
developed. Analytical solutions for a set of three coupled-mode equations is
developed for constant curvature and sinusoidal profile waveguide bends.
A design for a 110 GHz TE 0 7 -TE 1 5 2 helical-multifoil mode converter
is discussed. It has a computed conversion efficiency of 99.7% in a 1.5 cm
radius waveguide. Far-field radiation patterns from the TE 0 7 -TE 1 5 2 mode
converter are presented and discussed.
TE01-TE03-TEQ5-TE07
sequential varying-radius mode converter has
been developed to obtain the TE 07 mode for use with the TE 0 7 -TE 1 5 2
helical-multifoil mode converter. A tapered transducer is also considered for
the TE 15 2 -TE 1 5 i whispering gallery mode conversion. A very interesting
property related to step tuning has been found for the TE 1 5 2 -TE 1 5 1 mode
converter. A mode converter to produce a 60% TE 22 2 /(£ and 40% TE 22 4
7150° mode combination also has been discussed and designed for use with
the radial extraction of the electron beam in new generation gyrotrons.
Approved;
Date
Ronald J. Vernon, Professor
Electrical and Computer Engineering.
iv
ACKNOWLEDGMENTS
The author is thankful to Professor Ronald J. Vernon for his guidance
and supervision during this research program. The author is particularly
grateful for Professor Vernon's effort to read, confirm, correct and give
support in preparation of this dissertation. The author is also grateful to
Professor J.E. Scharer, J.B. Beyer, W.N.G. Hitchon and all other members
of the graduate faculty who contributed encouragement and guidance. The
author wishes to thank Professors J.C. Sprott, L. McCaughan and J. Booske
for reviewing the dissertation. The author is indebted to Dr. J. Mclean for
many useful suggestions in writing and carefully examining the first draft
of this dissertation.
This research work was supported by the U.S. Department of Energy
through contract DE-FG02-85ER52122. The appreciation also extended to
Dr. T.V. George for his encouragement, and support during the entire project.
The computing work was supported by the National Energy Research
Supercomputing Center (previous known as National Magnetic Fusion
Energy Computer Center).
Thanks are extended for the helpful discussions and the assistance of
his colleagues Dr. M. Buckley, J. Lorbeck, P. Sealy and J. Shaffii.
The
author is also grateful to D. Casper and M. Blankstain in helping him carry
v
out experiments and to F. Schelriff for the use of his data acquisition
program. The assistance of Mr. W. Meier of the ECE machine shop made it
possible for this work to be completed.
Finally, the author's parents should be given the most credit for this
work because of their belief and trust. Education has opened his eyes and
mind.
It was the motivating force in completing his degree.
Great
appreciation is also extended to his wife, Chiou-Yueh, for her encouragement,
patience and support and also to his two sons, Howard and Wylie, who bring
a delightful life to him.
vi
List of Contents
Page #
Abstract
ii
Acknowledgments
iv
List of Contents
vi
Chapter 1. INTRODUCTION
1. High Power Microwave Tubes and Output Modes
1
2
2. Optical Ray Picture for Mode Conversion
10
3. Mode Converters and Coupled Mode Equations
13
4. Outline of This Disseration
15
Chapter 2. SERPENTINE MODE CONVERTER
19
1. Derivation of the Coupling Coefficients for Serpentine Mode
Converters
20
2. Single-Period Mode Converters
32
3. Design of the 8.6 GHz Serpentine Mode Converter
34
4. Test Results for the 8.6 GHz Serpentine Mode Converter
42
5. Design of a 3.0 GHz Serpentine Mode Converters
53
Chapter 3. AZIMUTHAL DEFORMATION OF CYLINDRICAL
WAVEGUIDE
59
1. Derivation of Coupling Coefficients
60
2. Azimuthal Deformations and the Selection Rule
68
3. An Elliptically Shaped Polarization Converter
76
4. Helical-Multifoil Type Mode Converter
81
vii
Chapter 4. APPROXIMATE SOLUTION OF COUPLED MODE
EQUATIONS
85
1. A Set of Two Coupled-Mode Equations
87
2. The Scaling Law and Bandwidth of Mode Converters
93
Chapter 5. WAVEGUIDE BENDS AND THREE-MODE COUPLED
EQUATIONS
99
1. Waveguide Bends for Degenerate Modes
101
2. Waveguide Bends for Non-degenerate Modes
106
3. Waveguide Bends with Constant Curvature
110
4. Waveguide Bends Using a Sinusoidal Profile
114
5. The Design of TM 01 and T E n Waveguide Bends at 8.6 GHz . . 118
Chapter 6. A MODE CONVERSION SYSTEM TO OBTAIN A ROTATING
TE 1 5 2 MODE
125
1. Rotating Modes in a Linear Structure
127
2. Rotating Modes in a Helical Structure
133
3. A TE 01 -TE 07 Varying-Radius Mode Converter
136
4. The Design of the Helical Multifoil Mode Converter
140
5. Fabrication and Test Results for the TE 01 -TE 07 and
TE 0 7 -TE 1 5 2 Mode Converters
150
Chapter 7. WHISPERING GALLERY MODE CONVERTER DESIGN . 164
1. The TE 1 5 2 to TE 1 5 x Mode Converter
164
2. Mode Converter Performance for Different Modes with the
Same Gyrotron Cavity
174
3. An Uptapered Mode Converter from the TE 15 2 to TE 1 5 x
Mode
175
viii
4. An Uptapered Mode Converter to Produce Specific Output
Mode Combinations
180
Chapter 8. CONCLUSIONS
193
APPENDDC A. Coupling Coefficients and Attenuation Constants
198
APPENDDC B. Integration Formulas for Bessel Functions
203
APPENDDX C. Perturbed Field Components in a Deformed Waveguide 205
APPENDIX D. The Selection Rule
208
APPENDDC E. Fabrication of the Helical-Multifoil Mode Converter . . . 213
APPENDDI F. Fabrication Errors and Measurement Errors for the HelicalMultifoiled Mode Converter
REFERENCES
219
223
1
Chapter 1
Introduction
The exploration of new concepts for high-power microwave, millimeterwave and sub-millimeter wave sources has progressed very rapidly in recent
years.
The search for new high-power sources, and transmission and
launching systems has been driven by the introduction of new technologies and
the requirements of present and future applications. The urgent requirement
of high-power, millimeter-wave tubes for Electron Cyclotron Resonance
Heating (ECRH) of plasmas has driven tube research to produce nontraditional output modes. For example, some gyrotrons produce an output in
a high-order rotating T E m n mode where m » n, for example the T E 1 5 2 mode.
Hence, to use the output from a high-power microwave source for a specific
application, the output mode and transmission system become very important
issues in conjunction with the microwave tube research and the high-power
application, especially in the plasma fusion area.
Some high-power microwave tubes have an output in a mode that may
not be suitable for the desired application or for transmission. In such cases,
it may be necessary to convert these output modes to another mode which is
more appropriate for direct application or long distance transmission. The
mode conversion process for high-power tubes in both the millimeter-wave
2
frequency range (60 to 300 GHz), and microwave frequency ranged to 60 GHz),
is a challenging research area.
The work presented here is part of a research program at the University
of Wisconsin for the development of mode conversion and transmission systems
for high-power microwave tubes and especially the particular components to
be used in such systems. This research is supported by the U.S. Department
of Energy for the work relating to gyrotrons and by Harry Diamond Laboratory
for the work relating to high-power Backward Wave Oscillator (BWO). In our
work on components relating to gyrotrons, we interface with a group at MIT
responsible for basic gyrotron research and another at Varian Associates which
is responsible for the development of commercial high-power gyrotrons.
1. High-Power Microwave Tubes and Output Modes
Our
main concern will be with high-power BWO's with operating
frequencies of 3.0 and 8.6 GHz, and gyrotrons with operating frequencies above
100 GHz. Although the BWO is one of the older members of the microwave
tube family, it is still a widely used RF power source, and its applications are
continually growing. Gyrotrons are a relatively new efficient source of highpower millimeter waves and have been proven valuable for Electron Cyclotron
Resonance Heating of fusion plasmas [1].
This dissertation will first discuss mode converters for the high-power
BWO's, and then gyrotrons. The high-power BWO's with which we will be
3
concerned generate the TM 01 mode. The radiation pattern for the TM 01 mode
has a central null which is surrounded by an azimuthally symmetric lobe with
an electric field polarization in the direction of the polar angle unit vector aG.
This mode pattern is not useful for radiating into space or feeding a
conventional reflector. However, we can use a serpentine mode transducer to
convert these modes into a linearly-polarized T E n mode, which will generate
a more useful radiation pattern. Moreover, we can then use a corrugatedwaveguide mode converter to produce an HE 1 1 mode which has a field pattern
very close to that of a Gaussian distribution.
Figure 1.1 shows the field
patterns of many of the modes that are discussed in this report.
The first generation of gyrotrons typically had output power capabilities
in the range of 100-200 kilowatts. The operating frequencies of various models
ranged from approximately 8 GHz to 150 GHz.
The output from these
gyrotrons was commonly in a TE 0 n mode. The TE 0 n modes are not normally
suitable for plasma heating or radiating into space with a useful pattern. They
all have a null in their radiation patterns on the waveguide axis, are
azimuthally polarized, and have relatively low maximum gain. Therefore,
these modes were also commonly converted to the TE1]L mode in a smooth-wall
circular waveguide and then sometimes to an H E n mode in a corrugated
waveguide.
A TE 0 n mode output is commonly converted into a linearly-polarized
mode through a sequence of one or more mode converters. One typical mode
TE 0 2
TE 0 1
TEU
TE
TE15il
TMQ!
15,2
'
E
Fig. 1.1. Waveguide mode patterns of the important modes that
will be discussed in the dissertation.
HEn
TE 1 2
5
converting system uses the following sequence: TE 0n -TE 01 , TE 01 -TE 11( and
then T E n - H E n . This type of mode conversion system, shown in Fig. 1.2, has
received considerable attention in the last several years [2] - [4]. Since the
TE 0 1 mode in a highly overmoded waveguide has the lowest ohmic loss of any
mode in a smooth wall waveguide, it is well suited for high-power long-distance
transmission.
Even the first generation gyrotrons with TE 0 n mode outputs have
reached output power levels of 200 kilowatts and above from a single tube.
However, the controlled fusion research program is in need of further advances
in microwave average-power capabilities. It has been estimated [1] that 20
MW of ECRH power in the frequency range between 100 GHz and 300 GHz
will be required for a duration of several seconds.
This power could be
supplied by multiple sources, but the power level for each tube, for economic
reasons, should be in the megawatt range.
The development of a high-power, high-frequency gyrotron is therefore
of great importance. The new generation gyrotrons are intended to generate
output power in the megawatt range and to operate at frequencies above 100
GHz. A major problem associated with the high output power gyrotron,
however, is mode competition. Higher output power requires larger cavities
and higher order operating modes. Because the density of modes increases
with the radius, the separation between the intended operating mode and
spurious modes decreases and mode competition becomes a very serious
Gyro Iron
Lossy W a l l
Toper
Converter
Section
2.5"dio.
2.5-l.094"dia.
B (from A)
Mode
Converter
A (to B)
Mode
TEQn -
TEQ|
To R e f l e c t o r
System
Mode
Polar iz otion
Converter
Conver ter
Horn
TE
0.
~TEU
TE
««
"LP
-TE««
o
T E
EP
U
-
H E
H
_ITTnTTrrrTn
Fig. 1.2. Conventional mode conversion system for a first
generation TE 0n mode gyrotron.
Oi
7
problem. Fortunately, a class of high-azimuthal-index rotating TE m n modes
with m » n, can mitigate this problem and are commonly used in the new
generation of gyrotrons. These modes have their power concentrated near the
waveguide wall and hence are frequently called whispering gallery modes (WG
modes). Two of the WG modes being used in current gyrotron designs are the
TE 15>2 and the TE 22>2 .
The high-loss properties of whispering gallery modes makes a
conventional microwave transmission system impractical for long-distance
power transmission. Hence, it would be of value to develop new types of mode
transducers to convert the WG mode into a low-loss H E n mode in a corrugated
waveguide, a TE 0 n mode in a circular waveguide, or a polarized collimated free
space beam. One type of mode converter which appears to be appropriate to
consider for this task is the helical-multifoil mode converter.
A helical-
multifoil mode converter can convert a rotating TE m n mode into a TE 0 n mode
with large radial index. A design of a helical-multifoil mode converter has
been examined using computer simulation. The simulation indicated that the
design exhibited a very good conversion efficiency
within a single
perturbation period.
We define the conversion efficiency for a mode converter as the ratio of
the output power in the desired mode(s) to the total input power to the
converter. The mode purity for a mode converter is defined as the ratio of the
total output power in the desired mode(s) to the total output power.
8
Possible designs for a TE 0n -TE 01 varying radius mode converter have
been discussed by Thumm and Kumric [5]. Here three kinds of designs have
been considered. One, directly converts the TE 0 n to the TE 0 1 mode. This
method requires a large number of perturbation periods which seriously
restricts the conversion efficiency bandwidth. Another, converts the TE 0 n to
TE 0 n - 1 to TE 0 1 mode step by step. This method requires a large perturbation
wavelength which may not be practical to integrate into a mode conversion and
transmission system. A reduced radius method also has been tried with
shorter and fewer perturbation periods, but the power handling capacity is
drastically reduced.
The high power handling requirement is the main
difficulty of converting a TE m n whispering gallery mode to the TE 0 1 mode for
long distance transmission.
An alternative to waveguide mode conversion is using a quasi-optical
antenna which can directly convert a whispering gallery mode into a polarized
collimated free-space beam. This was originally discussed by Vlasov et al. [6,
7] who presented a design with a conversion efficiency of about 80%.
Therefore, this type of quasi-optical antenna is referred to as a Vlasov antenna.
The polarized collimated free-space beam emerging from the Vlasov antenna
is then transmitted over long distances using a series of optical mirrors, or is
coupled directly into a corrugated waveguide. One compelling reason to use
the quasi-optical antenna is that it is possible to integrate the quasi-optical
antenna into the gyrotron where it also serves to separate the spent electron
9
beam and the generated electromagnetic wave. Hence, the importance of the
conversion efficiency of the quasi-optical mode transducing antenna is of
particular importance for the future prospects of ECRH. The design of new
generation quasi-optical antennas is currently being studied by other
colleagues in our group [8, 9].
The use of this quasi-optical antenna allows the transformation of a
whispering gallery mode into a mode which exhibits a radiation pattern that
is well suited for ECRH in plasmas. The first generation Vlasov launchers for
TE m 2 modes often have a double humped main lobe, whereas Vlasov launchers
for T E m l modes appear to produce a more nearly Gaussian beam. Thus, it
may be desirable to have available designs for TE m 2 -TE m l type mode
converters to transform the TE m 2 mode to the more desirable T E m l mode for
the first generation Vlasov launcher designs. Even though the new generation
two-reflector Vlasov launcher has alleviated the double humped main lobe
problem [8], the design of TE m 2 -TE m l varying-radius WG mode converters are
still worth discussing in this program.
A low-power whispering-gallery mode generator is not available as
commercial equipment. Therefore, initially only those laboratories having a
high-power gyrotron could generate appropriate TE m n modes. This severely
restricts research and experiment on quasi-optical launchers. For low-power
testing of improved high-power quasi-optical antenna systems, a pure rotating
whispering-gallery mode is required. Two methods have been proposed in our
10
group [10] to generate whispering gallery modes. One of these uses a helicalmultifoil mode converter to convert a commercially available TE 0 1 mode to the
desired TE m n mode.
Because of the strict tolerance requirements, it is
expensive to manufacture even a single helical-multifoil mode converter. The
other proposed method, using a rectangular waveguide wrapped around a
circular waveguide with a series of azimuthal holes on the surface of the
circular waveguide, is first being designed to generate a TE 6 2 mode in order
to test the concepts involved. This device is beyond the scope of this work.
Using a varying-radius type mode converter, we can convert the TE 0 1
mode, which is commercially available, to the TE 0 7 mode. Then a helical
multifoil mode converter will be used to convert the TE07 mode into TE 15 2
mode. The reason the TE 0 7 mode has been chosen as an intermediate mode,
is the TE07 and TE 1 5 2 mode have the closest roots of the first derivative of the
Bessel function. Therefore, the coupling between these two modes is much
larger than to other spurious modes and the beat wavelength between TE 07
and TE 1 5 2 is of a reasonable length that can be manufactured, although
perturbation tolerances require very precise machining.
2. Optical-Ray P i c t u r e for Mode Conversion
Many high-power microwave and millimeter-wave tubes generate a mode
that cannot be utilized directly. Hence, it is often desirable to use the proper
perturbed structure to design a mode converter to meet the requirement of the
11
application. Actual waveguide mode converters are required to convert a
specified fraction or all of the power incident in a particular mode of the
waveguide to some desired mode without excessive power loss to other modes.
A very basic picture of the mode conversion process can be explained
from a ray-optical picture. In Fig. 1.3 ray theory illustrations show two of the
possible perturbations which give mode coupling; a) radius irregularity b)
curvature of the waveguide axis. Fig. 1.3 provides both an initial physical
understanding as well as an introduction to the concepts and the terminology
of mode converters in general. It may be observed that, in both cases, the
incident optical-ray no longer maintains the same angle with the axis with
respect to the perturbation inside the circular waveguide. In electromagnetic
wave theory, this corresponds to a change in the propagating mode for the
electromagnetic wave due to the wall or axis perturbation.
The ray-optical picture is a very simple picture with intuitive appeal, but
it is not so complete a description as that provided by electromagnetic theory.
Especially when the waveguide is not highly overmoded, the asymptotic
approximation to the electromagnetic wave of the optical-ray theory will not
be very accurate. In two of our designs, one for an 8.6 GHz and another for a
3.0 GHz system, there are respectively 6 and 3 propagating modes. An opticalray theory won't be able to give us an accurate design picture for such a case.
A set of coupled mode equations can be setup for these perturbed cylindrical
12
(a)
Fig. 1.3. The illustration of optical-ray theory to explain the
physical meaning of mode conversion a) wave propagates in a
straight waveguide, b) wave propagates through a radius
irregularity waveguide, c) wave propagating in a curved axis
waveguide.
13
waveguides and the power flow between those propagating modes can be found
by solving the coupled mode equations.
3. Mode Converters a n d Coupled Mode E q u a t i o n s
Many phenomena occurring in physics and engineering can be viewed
as coupled-mode processes. This includes the kinetic energy exchange between
two pendulums connected by an elastic string, the directional couplers of
microwave technology, and the energy exchange between an electron beam and
slow-wave electromagnetic wave in a traveling wave tube.
The coupled-mode formalism also is a very powerful tool in the design
of mode converters, where it is helpful in understanding and analyzing a
variety of important devices.
waveguide
bends,
These range from the waveguide taper,
corrugated-waveguide
converters,
and
smooth-wall
converters, up to recently developed multifoil mode converters.
With a series expansion of the electromagnetic fields in terms of
waveguide modes under the boundary conditions of the particular structure,
Maxwell's equations may be converted into coupled-mode equations or
generalized transmission line equations. This technique was pioneered by
Schelkunoff [11]. The coupled-mode equations, in general, consist of infinitely
many first order-ordinary coupled differential equations, two for each mode,
one for the forward- and one for the backward-travelling components. From
the discussion of [3], we can find that the backward coupling is so small in the
14
problems which will be considered here that we can omit it in our design
procedure for small waveguide perturbations. Only coupling to propagating
modes will be considered.
As we solve a typical set of two coupled-mode equations, we find that if
the wave number of a purely periodically-perturbed structure is equal to the
beat wave number between the input and the desired output mode, we will
have strongest coupling between these two modes. In a numerical simulation
procedure, we can use this fact as a starting point in the design of a purely
periodically-perturbed mode converter.
As a more compact design is
attempted, we can reduce the number of perturbation periods and the total
length of the mode converter by increasing the perturbation amplitude to
retain the total coupling effect. For a larger perturbation amplitude, stronger
power coupling per unit length into the desired mode occurs but the spurious
modes are also more strongly excited.
A perturbed waveguide structure a with constant beat wave number
cannot usually meet both the requirements of compact design and high
conversion efficiency. New techniques to improve mode converter design have
been developed in our group. We have modified the constant beat wave
number structure to a beat wave number changing as a the function of position
to keep synchronization of conversion between the input mode and desired
output mode and reject the unwanted spurious modes.
Conventional gyrotron designs use an uptaper from the small radius
15
cavity to the collector whose output radius is determined by a collector
diameter large enough to dissipate the energy generated by the spent electron
beam.
From a simple calculation, we can find that the beat wavelength
between two non-degenerate modes which are well above cutoff is proportional
to the square of waveguide radius. Because of the long beat wavelength and
additional modes entering the coupled-mode equations, it becomes very difficult
to get good conversion efficiency with mode converters of reasonable length for
large waveguide diameters. Thus we proposed to combine the uptaper with the
mode converter. With this combination, we predict that the new design will
shorten the overall length of the mode converter significantly.
4. Outline of Dissertation
Chapter 2 presents mode converter designs that have been developed by
the author for high-power BWO's with operating frequencies of 8.6 and 3.0
GHz. The designs for these mode converters have only a single perturbation
period. No other single perturbation period serpentine type devices were
known to the author at the time of their design, but a single perturbation
period varying-radius type mode converter, TE 02 -TE 01 , has been designed and
tested by M. Buckley [3]. For serpentine type mode converters [2] - [4], the
number of perturbation periods has always been three or larger in the past.
In Chapter 2, we will present single-period designs for TM 0 1 -TE n serpentine
mode converters, for 8.6 and 3.0 GHz, and test results for the 8.6 GHz mode
16
converter will be discussed. The coupling coefficients for the serpentine mode
converter will also be derived.
In Chapter 3, we derive the general form for coupling coefficients for
varying-radius mode transducers which leave the azimuthal index unchanged
(i.e. TE mn -TE mq ). Using these coupling coefficients, we can design an uptaper,
downtaper, or varying-radius type mode converter with the proper perturbation
structure. A similar derivation can be used to obtain the coupling coefficients
for different perturbation structures (for example a screw-type rotating-mode
mode converter or a polarization mode converter).
After these coupling
coefficients are calculated, they can be used to set up the coupled-mode
equations and employed in numerical methods to optimize conversion
efficiency.
In Chapter 4, we will use a perturbation method to solve the coupled
mode equations.
A special profile, sinusoidal coupling function, will be
discussed for two coupled-mode equations. The scaling law and the bandwidth
for the purely periodic coupled function will be presented.
Chapter 5 will discuss waveguide bends. In a microwave transmission
system, we will need to change the transmission direction.
From the
discussion of Chapter 2, we find that a bend in the waveguide axis will cause
mode coupling. Hence a careful examination of waveguide bends is necessary
for a complete transmission system. Two coupled-mode equations for the
degenerate, TE 0 n and TM l n , modes will be discussed as an introductory part
17
of this chapter. Three coupled-mode equations have been solved for a constant
curvature bend and a sinusoidal profile. 90° waveguide bends for 8.6 GHz
TM 01 and T E n modes have been designed with very good calculated
transmission efficiency.
A design chart for constant curvature waveguide
bends has been developed for the degenerate modes (TE 01 , TM U ) and
(TE 02 , TM 12 ).
Chapter 6 will focus on how to generate a whispering gallery mode from
a TE 0 n mode. The new generation gyrotron has a rotating TE m n mode output,
and there is no commercial device to generate such a rotating mode. A helicalmultifoil mode converter has been designed and constructed for this purpose.
First, we convert a commercially available TE 0 1 circular waveguide mode,
operating at 110 GHz with .199" diameter, to TE 07 mode, with 1.1811"
diameter. Then the helical-multifoil mode converter is used to convert the
TE 0 7 mode to the TE 1 5 2 mode. A non-rotating structure for rotating modes
also has been discussed in this chapter.
Chapter 7 describes varying-radius mode converters for a new gyrotron
being developed with an operating frequency of 140 GHz. In this chapter, we
discuss a conventional varying-radius mode converter and an uptapered TE 15 2T E 1 5 1 mode converter at the 140 GHz operating frequency. The step tuning
properties of the varying-radius mode converter will also be discussed. Use of
a combined uptaper and mode converter will also be discussed for the TE 22 2
mode at 110 GHz. Finally, a mode converter design producing the proper
18
amplitude and relative phase for a combination of two or three modes will also
be addressed for use with the radial extraction method, where the electron
beam is diverted through an azimuthal gap in a gyrotron with minimum affect
on the mixed modes, at 110 GHz.
Chapter 8 is a summary of work on theory, design, and measurement for
the research that has been done for past several years and possible extension
of the research for future work is suggested.
19
Chapter 2
Serpentine Mode Converter
Certain types of high-power backward-wave oscillators (BWO's)
currently under development generate their output in the TM 01 transverse
magnetic mode in a circular waveguide. This mode has a transverse electric
field in the waveguide which is completely radial and produces an open-end
radiation pattern entirely in the 0 (the polar angle) direction with a null on the
waveguide axis. This radiation pattern is not optimum for most applications
and it is often desirable to convert the TM 01 mode into the T E n transverse
electric mode which has its maximum on the waveguide axis, is approximately
linearly polarized, and has a higher maximum directivity than the TM 01 mode.
Our main interest here is in certain BWO's operating at 3 and 8.6 GHz
with megawatts of output power and the output being in the TM 01 mode in a
circular waveguide. In this chapter, we will discuss the design of a serpentine
mode transducer to convert the TMQ1 mode to the T E n mode, which is much
more suitable for communication, radiating into space, or other applications.
First, we will derive the coupling coefficients for serpentine type mode
converters following Morgan's method [11]. Then, we will use numerical
techniques to obtain an axial profile which optimizes the conversion efficiency
for the 8.6 and 3.0 GHz TM ni -TE 1n converters. The theoretical conversion
20
efficiencies for the 8.6 and 3.0 GHz serpentine mode converters which were
designed are 99.4% and 99.5%, respectively. Test results for the 8.6 GHz
serpentine mode converter will be presented.
1. Derivation of t h e Coupling Coefficients for Serpentine Mode
Converters
In the general case, an initial or input mode will be designated by modal
indices mn, i.e. TE m n and TM mn , and a mode that is being coupled to will be
designated by modal indices pq, i.e. TE pq and TML-. To simplify the notation
in the equations which follow, we will arbitrarily order the initial modes to be
considered and label them with a single index i which represents both the
modal indices mn as well as the TE and TM designation. Similarly, we will
represent the modes being coupled to by a second single index j which will
represent both the pq and the TE and TM designation for these modes. The
same mode ordering scheme will be used for both the i and the j indices.
The coupled mode equations for a serpentine-type mode converter were
derived by Morgan [11] using techniques pioneered by Schelkuneff [12]. We
will closely follow the method used by Morgan. Here we will take (u,v,w) to
be the coordinate variables and (epe2,e3) the metric coefficients of the
orthogonal curvilinear coordinate system. In a general orthogonal curvilinear
coordinate system (u,v,w), the element of length is
21
dl2 = e\du2
+ e\dv2
+ e2dw2
.
The curved waveguide can be described in these coordinates where, according
to Fig. 2.1,
u = r, i> = <}), w = £
and
Cj = 1, e 2 = r, e 3 = 1 + —cosc{) .
In straight circular-cylindrical waveguide, the waveguide radius, a, is
constant along the longitudinal axis. The serpentine mode converter also has
a constant radius but along a deformed longitudinal axis. In a cylindrical
waveguide, the electromagnetic field in the waveguide can be derived from the
scalar functions \|/x- and \|/t' given by:
\|/j = NiJmikirisinimfy)
(2.1)
Vj- = iV"jJm(^jr)cos(m({))
(2-2)
and
where Nt is the normalization constant, kp is the root of the corresponding
Bessel function for TM m n modes or the root of the first derivative of the
corresponding Bessel function for TE m n modes respectively. Also we will use
p0 as the free space wavenumber. The propagation wavenumber for mode i
can be written as P{- = ((302 * k2)^.
We assume the waveguide field varies with
COScp
Fig. 2.1* The curvilinear coordinate system we have chosen for
the serpentine waveguide.
=1
23
distance z along the waveguide axis only through the phase factor, exp(±j$jZ).
A time dependence of the form e ^ is assumed, where j = (-1)^.
For cylindrical waveguide, The \j/ function satisfies the transverse wave
equation:
*y = J_[JL(fiiV)
+
e 1 e 2 du ei du
=
±(fiiv>]
(2.3)
dv e 2 dv
-k2y
where the separation constant k is also known as the cutoff wavenumber.
The transverse field components in the curved waveguide can be written
in terms of the straight cylindrical waveguide field functions ^ and \yt':
EU
= E v* t-^r
l
exdu
K'Hv^^.i
+d
^
= E Vi Eiu
e2ov
i
dJ^L] = E Vi Eiv
e%dv
etdu
2
exdu
p2e2dv
(2.5)
t
/
Hu - -E h [ i ? - - dt fi-^L] = - £ ! , Hiu
i
(2.4)
(2.6)
i
where d t is called the TE to TM amplitude ratio and Vi and It are referred to
24
as the modal voltage and current parameters. Vi and /; will be determined
later by solving the coupled mode equations. This will become clear at the end
of this section.
Maxwell's equations in the curvilinear coordinate system (u, v, w), as
shown in Fig. 2.1, can be expressed as:
1
[J-fegB^) - JLfejjEy)] = -jcoutfu
e<£% dv
dw
1
eg i
1
exe2
1
e^
1
eg i
1
eje 2
[J-ieiEu)
dw
(2.8)
- JLfe,^,)] = -j(aiiH0
du
(2.9)
[JL(e2Ev) - -l(«i^u)] = -JWHW
du
dv
( 2 -!0)
[^-fea^w) ~ 4 - ^ 2 ^ ) ] = J<B*EU
3u
3u;
(2-1D
[JL( e i tf u ) - JLfegff,,,)] =7CoeEu
dw
du
(2.12)
[JLiefl,,)
3w
- JLiexHu)] =jmEw
dv
.
(2.13)
Substituting Eqs. (2.4) through (2.7) into Eqs. (2.10) and (2.13) and using the
properties of Eq. (2.3), we obtain the longitudinal field components in the
curved waveguide:
25
h
Hw = jae E Vidi-L
#u> = . W E
'
J
i ~
(2.14)
y'.
(2.15)
Vi
Po
The quantities dt and kt must be chosen so that the boundary conditions are
satisfied. The boundary conditions at the wall of the waveguide are taken to
be
at waveguide wall.
Ew ~ &v ~ °
The condition Ew - 0 requires that Jm(kp) = 0. Since the mode components
are linearly independent, the only solution which satisfies E \
u
= 0 has
wall
every individual term inside the summation is set equal to zero at the wall.
Hence we have,
di*
e2dv
exdu
r-a
Since the each individual term of the field inside the curved waveguide is set
to satisfy the boundary condition, the individual field term is called the local
normal mode of the curved waveguide. Since any two local normal modes
26
satisfy the orthogonality condition over the transverse corss section of the
curved waveguide
*
J (Eit x Hjt) • ds
v
t-j
•> e 2 3u
e29u
e2dv
ei3u
« 2 e29i>
exdu
e29i>
Q2 exdu
= / [ l ? ^ + J0 iw ff,J ds = 5y .
(2.16)
The quantity o^.- is the Kronnecker delta. For i = j , iVj is determined so that
the above equation is satisfied.
All the quantities in Eqs. (2.4) - (2.7) and Eqs. (2.14) - (2.15) have now
been determined except ^ and V,-, the modal current and voltage parameters.
To find relations for them, Eqs. (2.4) - (2.7) and Eqs. (2.14) - (2.15) are
substituted for the field components in Maxwell's equations, Eqs. (2.8) - (2.13).
To derive the telegraphist's equations [11], we multiply Eq (2.8) by,
»2
-v /
-e 3 ( — L . - djJ JL _—L.) ,
e2dv
and multiply Eq. (2.9) by
Q2 exdu
27
+
e 3 (-—- + dj - i _^L) ,
exdu
Q2 e2dv
add the resulting equations and then integrate this sum over the cross section
of the waveguide. Then we have
Jr-e 3 (i^L
e2dv
- dj%^.){J-JL^-(e3Ew)
«2 exdu eg^ dv
exdu
- ^ te2E„)] •jtopffj
dw
Q2 e2du egi dw
du
=J f J ^ - ^ ( e 2 E y ) + J L j ^
e2
dw
ei dw
c2
du
+ —Hjv\_-—-(ezEw) + jeieacou/f„ ] ds .
Using the orthogonality condition, Eq. (2.16), the wave equation, Eq. (2.3), and
the curvilinear coordinate system used by Morgan [12], shown in Fig. 2.1, we
can obtain the first telegraphist's equation
i +jlUj
(jLUJ
+ MiE/i
Clio
=
+
HjvHiv)ds
|*
(HJUJL(Z,EW)
J
[WjJIiu
e2dv
+
HjvJU&Jds
exdu
.
(2.17)
We have used the specific metric coefficients for i.epe2,e3) in the derivation of
28
the first generalized telegraphist's equation, Eq. (2.17), and the second
generalized telegraphist's equation, Eq. (2.18) below.
To derive the second telegraphist's equation, we multiply Eq. (2.11) by
- e3e(if
* d*h
,
du
e dv
x
2
and multiply Eq. (2.12) by
- e 3 (—J— - dj—J—)
e2dv
exdu
,
add the resulting equations and then integrate the sum over the waveguide
cross section. Thus, we have
Jf - e 3 ( ^ . +
exdu
dJ^yUuL^eaHv,)
e2dv e2e3 dv
-e3(^- - dj^iJ-iMe^)
e2dv
exou egx dw
= J(±EjJu^.(e2Hv)
e2 dw
- ±Ej*(
eiHu)
ex J dw
1
- JLfeaff „)] - jmEu)
dw
- -jUegPJ] - jaeH0 ) ds
du
- ±JEju[*(e3Hw)
e2 dv
- je#<M>Eu}
d
+ —Ejv[—.(e3Hw) + jetfaiopHJds .
ex J du
Now, using the orthogonality condition, Eq. (2.16), the wave equation,
29
Eq. (2.3), and the curvilinear coordinate system shown in Fig. 2.1, we obtain
the second generalized telegraphist's equation
dl;
£. +
dw
jmVj * jm E Vt JZ, (EjuEiu + EjvEiv)ds
i
We may recast the telegraphist's equations in a more convenient form in terms
of amplitudes of forward (a.) and backward (&.•) traveling waves is obtained by
using the following substitution in Eq. (2.17) and (2.18)
Vj - fz~(aj
+
bj) ,
_ (fij - bj)
where Z, = i!^H. for TE modes and Z, =
J
J
RP/
P/ for TM modes. Hence, we can
cop
rewrite Eq. (2.17) as
dw
dw
COE\
k2
+
= >uE Mat - &i)_L feu-4^ "e2- 4 r - ^ <fc
ei
*• Po
v^r
30
" JW fe ~~:^ai
~ bi^HjuHiv
+
HjvHiv)ds
.
(2.19)
and Eq. (2.18) as
i
da: db,-
k2
= J^T,di%{ai2
i
a
i
* bih/zi'((JEj
- Eju-JL-Kv/ds
ur±J
e2dy
exdu
- >e^E V^a*+ Wj/iu + V « i ^
•
(2.20)
J
We can set
2
2
IA) = f[^L Jf;B 4 - ^ V i ) + ~ Hjv - 4 - ( ^ ) J ds
Po
^
Po e i
- J§( # , „ Hiu
2
+
Jfy, H t y ) ds ,
(2.21)
2
{si = Jft^i. E,.U d< -e 4 - ( ^ ) - ^i- ^ df -4-(^)] ds
Po
2 ^
p2
eiaw
- J $ ( £,„ £ J U + EJo Eiv ) ds .
Eq. (2.19) and Eq. (2.20) may be simplified to
(2-22)
31
r—(^-+^)+_^p/_
dw dw
_
=
j2Jm_L_(ai-bi)
(A) , (2.23)
coet
and
- J _ ( ^ - _ i ) + y c o e / z 7 ( a , + 6.) = £jcoe/z~(a l -+&;) \B) .
r^~ dw dw
j
(2.24)
If we multiply Eq. (2.23) by 1/(Z/*, and Eq. (2.24) by (Zjj*, add the resulting
equations, and then the sum divide by 2, we have
^L+JVjaj'j-EiK^+Kjfit)
.
(2.25)
If we multiply Eq. (2.23) by 1AZ/*, and Eq. (2.24) by (Zjf*, subtract the
resulting equations, and then divide the difference by 2, we have
5 - M - -J E «#>.- -fyd•
< 2 - 26 >
The coupling coefficients in Eqs. (2.25) and (2.26) are
K$ = ±
P
Q (A) + ^ L ( B }
2
.
(2.27)
s/iyj"
The coupling coefficients for certain modes are listed in Appendix A. In
addition, the normalization constants of the \\f functions are derived in
Appendix A.
For a small perturbation, the backward coupling can be
neglected, i.e. we can neglect Eq. (2.26) and bt in Eq. (2.25). In numerical
32
design procedure, we will include at least the first and second order coupled
modes in our calculation. The third order or higher order coupled modes are
small and may not be included in the numerical simulation.
2. Single Period Mode Converters
The traditional philosophy of designing a mode converter [2,4,13] is to
use a large number of perturbation periods to suppress the spurious modes.
A mode converter with a small perturbation amphtude and a large number of
perturbation periods can suppress the excitation of spurious modes. However,
the large number of perturbation periods gives a narrower bandwidth. Also,
in some cases, it is impractical to design a mode converter with a large number
of periods when the input mode and desired output mode have a long beat
wavelength. For example, the beat wavelength between the TM 01 and T E n
modes for the 8.6 GHz BWO mode converter with 4.76 cm diameter waveguide
is 46.36 cm.
From the discussion of reference [14], we know that a continuous
change of perturbation wavelength, in addition to the change of perturbation
amplitude, can improve the suppression spurious modes.
Hence we have used a structure that gives us this kind of advantage.
In the purely periodic perturbation structure, a profile described by
f{z) = a 0 + e 0 [1 - cos(p\jZ)]
33
is used.
We have replaced the constant perturbation or structure wave
number (3S in this equation by the expression
H(z) = PgZ + frisintPsZ) + 62sin(2ps2)
where Ps, bv and b2 were varied from period to period. To obtain the best
results, in some cases the axial variation was further modified by replacing
e0{l - cos[#(z)]}
with
e 0 [l + c x cos(paz)](l - cos[#(z)]} .
(2.28)
The small perturbation of the harmonic term, H(z), accounts for the
difference in phase constant between the input and desired output modes.
This kind of profile has proved valuable in the design of varying-radius type
mode converters [3]. The theoretical calculation in the varying-radius type
mode converter has been intensively investigated [3]. The author has used the
profile described in Eq. (2.28) to design a mode converter of the serpentine
type.
Our
TMQJ-TEJ^
serpentine mode converter design used only a single
perturbation period. At the time this work was first reported in conference
proceedings [15,16], to the author's knowledge, the only single-period mode
converter design in the open literature was for a varying-radius TE 02 -TE 01
mode converter [17] for which the conversion efficiency was about 97%. Since
34
that time another compact design has been reported for a serpentine-type
TMQJ-TE-Q
mode converter [18], in which a changing phase of coupling
coefficients technique has been used to accomplish the complete power
transformation. Our focus will be on an 8.6 GHz design for 4.76 cm output
diameter.
3. Design of the 8.6 GHz Serpentine Mode Converter
The optimized perturbation amplitude and wavelength for a purely
periodic 8.6 GHz
TMQJ-TE-Q
serpentine mode converter is shown in Fig. 2.2.
Table 2.1 lists the conversion efficiency, output mode purity, and the loss of the
optimized amplitude and wavelength for the pure cosine profile.
The
perturbation wavelength of the pure cosine profile approaches the beat
wavelength as the number of perturbation periods increases. This will become
clear in Chapter 4 as we solve two coupled-mode equations for two modes for
the case of a sinusoidal perturbation. From Table 2.1 we also find the mode
purity and the ohmic loss increase as the number of perturbation periods
increases. At the same time the conversion efficiency remains at about the
same value.
There are only six modes that can propagate in the 8.6 GHz serpentine
mode converter. We achieve a good optimized conversion efficiency, 98.3%,
with a 2-period purely periodic mode converter. This gives us the hint that a
single-period serpentine mode converter for the 8.6 GHz BWO is an achievable
Fig. 2.2. Graph of the perturbation wavelength and amphtude v.s.
number of periods for the optimized 8.6 GHz purely periodic mode
converter.
Table 2.1
Conversion Efficiency, Mode Purity, and Ohmic Loss in an Optimized Pure
Periodic Perturbation Profile for 8.6 GHz TM 0 1 -TE n Mode Converter
number of periods 1
2
3
4
5
10
efficiency
98.1%
98.3%
98.1%
98.4%
98.7%
97.5%
mode purity
98.4%
98.7%
98.8%
99.3%
99.8%
99.8%
ohmic loss
0.26%
0.48%
0.73%
0.93%
1.17%
2.23%
37
goal. In a 60 GHz TE 01 -TE 11 serpentine mode converter previously designed
[14], there are 72 modes which can propagate. Here the TE 01 -TE 12 and T E n TE 2 1 mode coupling is so strong that we need more perturbation periods to
suppress the spurious modes.
A serpentine-type
TM^-TE-Q
mode converter has been designed for a
frequency of 8.6 GHz in a 4.76 cm diameter waveguide. Figure 2.3 shows the
mode coupling diagram. In the numerical design procedure, we only included
the first and second order coupling in our calculation.
The third order
coupling, which is between TE 2 1 and TE 31 , has been omitted because of the
small amplitude of the excited TE 2 1 mode. It is possible to obtain a mode
conversion efficiency above 99% with a single "adjusted" period of about 50 cm.
The mode content along the converter is shown in Fig. 2.4. Here the modal
amplitudes are normalized so that the fraction of power in each mode is given
by the square of the amplitude. The perturbation profile we used can be
expressed as in Eq. (2.28) and the details of the structure are
tabulated in
Table 2.2. The computed conversion efficiency of this design is 99.5% with
0.27% ohmic loss, which is mainly due to the TM 01 mode. The conversion
efficiency versus frequency response diagram for the single-period TM 0 1 -TE n
serpentine mode converter is shown in Fig. 2.5. In comparison to the 2-period
and 3-period pure periodic structures, the optimized single adjusted period
design has much better frequency response.
^
L
TE
31
TM 01
Fig. 2.3. Couphng diagram of a TMQJ-TE-Q single-plane serpentine
mode converter at 8.6 GHz with a diameter of 4.76 cm.
co
00
00
O'
CO
<Dd'
B
a.
6
d
d
o
d
0.0
0.2
0.3
0.4
0.5
z(m)
Fig. 2.4. Mode content along an optimized 8.6 GHz TM 0 1 -TE n
serpentine waveguide with a 99.5% conversion efficiency.
0.9
c
£
0.8
w
c
o
c/3
>
0.7
/
4 — single
period
— 3-period
_ _ _ / — i . — 5-period
/
/
c
o
U
/
0.6
/
/
/
1
0.5
J
7.6
7.8
• i.
8
•/•
\
\
-i
8.2
i
i
i
I
i
8.4
i
i
,
I
i
8.6
i
i
L
i :
8.8
i
i
L_!
9
i
i L \i
I i i i i I J_I
9.2
9.4
,\
L-lJ
L-J
I
I
9.6
L
9.8
Frequency (GHz)
Fig. 2.5. Comparison the conversion efficiency of single-period 3period, and 5-period 8.6 GHz serpentine mode converters.
o
Table 2.2
The Optimized Structure for the 8.6 GHz TM 0 1 -TE n Mode Converter
The detail structure of the serpentine mode converter is
fCz)=a0+e0[l+e1cos(fJ1z)]{l-cos[P2Z+e2sin(p2z)+E3sin(2P2z)]}.
a0
value
EQ
ex
.0238 .0237 -.02
&2
e
3
-.14 -.02
The output result
total length
52.5 (cm)
total loss
0.27%
conversion efficiency
99.4%
Pi
4K
P2
2TC/.525
42
4. Test Results for t h e 8.6 GHz Serpentine Mode Converter
Two versions of the above design for the 8.6 GHz
TM^-TEJJ^
mode
converter were fabricated via the following procedure: 1) A numerical
controlled milling machine was used to fabricate a split-block version of the
serpentine mode converter, shown in Fig. 2.6. It is required that the mode
converter operate under vacuum and that would be difficult with the split
block converter because of its form. The following steps were used to produce
an exact inner profile of the mode converter machined in step 1) such that the
second version did not have the longitudinal slit and can be evacuated. 2) The
assembled split-block converter was filled with liquid low melting-point metal
to make a mandrel for electroforming. 3) The liquid mandrel was allowed to
cool and the split block separated.
4) The hardened mandrel was
electroformed with copper and the low melting point metal was melted out.
5) Flanges were attached to each end of the electroformed mode converter.
Both the split block and electroformed mode converter were tested at
low power (200 mW). A schematic drawing of the measurement system is
shown in Fig. 2.7. Figure 2.8 shows the coordinate system used for the
measurement system. This measurement system was set up with the help of
M. Blankstein and D. Casper. In order to excite the TM 01 mode, we used a
straight-wire probe at the center of a circular waveguide as a mode generator
as shown in Fig. 2.9. The probe was fed by a coaxial line. The theoretical and
TM01-TE11
Mode Converter
•2.1250
Fig. 2.6. Drawing of split-block 8.6 GHz T M o r T E n serpentine
mode converter.
Attenuator
Isolator
Modulator
TMQJ
Receving
Horn
Klystron
Generator
Mode converter
\
Rotating Arm
Rotator
Position
Plotter
Computer
Voltage
Detector
<e
Fig. 2.7. Far-field radiation pattern measurement system for the
8.6 GHz serpentine mode converter.
E Plane
Electric field
polarizatio
at center
waveg
H Plane
Receiving horn
H Plane arc
E Plane arc
Fig. 2.8, Coordinate system used in the measurement of the
fields radiating from the waveguide aperture.
4^
en
Modified Type N (F) Panel Mount
Cylindrical Waveguide
J,WGUSCBCOOeO»MwJ
MMAAMWV*
aaaaaaaaaofMw
Fig. 2.9. The 8.6 GHz TM01 mode generator.
4^
CD
47
measured T M ^ E 0 radiation patterns from the T M ^ generator are shown in
Fig. 2.10. Theoretically, there is no E^ associated with the TM 01 mode.
We used the mode generator to obtain the T M ^ mode for the input into
the
TMQJ-TE-Q
mode converter. The relation between the input TM 01 mode,
the linearly polarized T E n output mode, and the plane of the axial waveguide
perturbation is shown in Fig. 2.11. The computed amplitude of each mode
which can propagate in the waveguide is listed in Table 2.3. We used the
method given by Silver [19] to obtain the radiation pattern for the mode
combination predicted to be present. In this method, the aperture field is
assumed to be the same as it would be if the waveguide were of infinite length
and reflection from the aperture is neglected.
Table 2.3
The Phase and Amphtude of the Mode Predicted to be
Present in the 8.6 GHz Mode Converter
™oi
TE
Amplitude
.0062
.9952
Phase
-.340
-1.504
n
TE
2i
™n
T%
.0173
.0431
.017
2.636
-1.24
1.67
Figure 2.12 shows the measured and theoretical E-plane far-field
radiation pattern from the electroformed serpentine mode converter. There is
very good agreement between the measured and theoretical results for the E-
o
o~
o
" O-
T
CQ X
|
-30 0
Intensity
£3 ° -
f ^
:
measured
o
o_
T
1
1
o
o
m
I
-90.0
1
:
-60.0
-30.0
0.0
theoretical
!
1
30.0
60.0
Theta (degrees)
Fig. 2.10. Radiation pattern from the TM 01 mode generator.
Using Silver's method [11], we can find that there is no E^
component for the TM family.
90.0
input
TMQJ
mode
output TE ^ mode
Fig. 2.11-Relation between the plane containing the curved axis
of a TMQJ-TEJJ mode converter and the direction of polarization
of the linearly polarized T E n mode.
measured
. theoretical
-90.0
-60.0
-30.0
0.0
30.0
60.0
90.0
Theta (degrees)
Fig. 2.12. Comparison of the measured radiation pattern and the
theoretical data in the E-plane for the 8.6 GHz serpentine mode
converter. The theoretical mode pattern is computed for the mode
combination predicted to be present.
51
plane radiation pattern.
The measured and theoretical H-plane far-field
radiation patterns are shown in Fig. 2.13. As can be seen in the Fig 2.13, the
agreement between the theoretical and measured results is not as close as we
would expect.
Both the split block and electroformed mode converters
produced essentially identical results.
Since there is discrepancy between theoretical and experimental results
in the H-plane, a completely different method of exciting the T E n mode in a
2.38 cm radius circular waveguide was also used. A rectangular waveguide to
circular waveguide transducer was used to convert a TE 10 mode in a
rectangular to a T E n mode in a 1.46 cm radius circular waveguide. Then an
uptaper was designed to increase this 1.46 cm radius to 2.38 cm radius
waveguide.
The measurement results for the H-plane far-field radiation
pattern are shown as a dotted line in Fig. 2.13 where we can find a very good
agreement with the T E n mode generated from the serpentine mode converter.
Hence, from Figs. 2.12 and 2.13 we can predict that the output mode purity is
better than 99%. The discrepancies between the theoretical and experimental
H-plane results arise from an overly simplified model used in the calculation
of radiation pattern. Sliver's method assumes the aperture field is the same
as the field in the interior of the waveguide and that reflection from the
aperture is negligible.
A comparison of measured results and Silver's
approximation is given in reference [19] for the case where the radiated mode
not is far above cutoff. There is better agreement between the measured and
o
o
theoretical
measured (converter)
measured (TE U taper)
o
o
-90.0
-60 0
-30 0
0.0
30.0
60.0
90.0
Degree
Fig. 2.13. Comparison of the measured radiation patterns and the
theoretical data in the H-plane for the 8.6 GHz serpentine mode
converter. The theoretical mode pattern is computed for the mode
combination predicated to be present.
53
calculated radiation patterns for the E-plane than H-plane in these
measurements just as in the measured results for our case.
In our
experimental results, we further confirmed the H-plane discrepancy exists for
modes, which are not far above cutoff, radiated from an open end waveguide.
The comparison
of the
exact theoretical
result
and
Silver's
approximation has been shown in reference [19] when the radiated mode is not
far above cutoff. The E-plane radiation pattern has better agreement between
the exact and approximate solution than the H-plane radiation pattern does.
That is just what we have in the measured results for our case.
5. Design of a 3.0 GHz Serpentine Mode Converter
A second TM01-TE1]L serpentine mode converter has been designed for
a backward wave oscillator with an operating frequency of 3.0 GHz with a
TMQJ
output mode in a 10.85 cm diameter waveguide. It was a requirement
for this mode converter that the perturbation amplitude exceed the waveguide
radius to avoid x-ray transmission from the BWO through the waveguide.
Again it is desired to convert this TM 01 mode to a T E n mode.
There are only three modes, TM 01 , T E n and TE 21 , that can propagate
in this size guide at 3.0 GHz. The beat wavelength between TM 01 and T E n
mode is 75.18 cm. Because the beat wavelength between the input mode and
the desired output mode is so long, the best choice for a TM 0 1 -TE n mode
converter for this BWO would again be a single perturbation-period mode
54
converter even if this were not required to obtain the large perturbation
amplitude. The profile described in Eq. (2.28) still works well in this frequency
and diameter. We include all of these three modes in the calculation. In the
optimization procedure, we find that Eq. (2.28) can be simplified to:
f{z) = aQ + e 0 (l - cos[p> + 6X sin(p>) + b2 sin(2p\;Z)]} .
(2.29)
We obtain a 99.8% computed efficiency and 0.13% ohmic loss. The wall profile
and the parameters of Eq. (2.29) for the 3.0 GHz serpentine mode converter
was shown in Fig. 2.14. The mode content along the converter is shown in Fig.
2.15.
In this design, the perturbation amplitude was only 91.1% of the
waveguide radius. Therefore, the design requirement has not been met. A
new profile was next considered:
fU) = a0 + ek[l - cos(pV*)] •
( 2 - 3 °)
The extra constraint, that the perturbation amplitude be larger than the
waveguide radius, requires us to trade the input and output port alignment for
the additional freedom in adjusting the perturbation amplitude of each half
section independently.
In this case, we relax the constraint of aligning the input port with the
output port to achieve the more important design factor that the perturbation
amplitude exceed the waveguide radius. The perturbation amplitude can then
be chosen to be different for each half period. Table 2.4 gives the detailed
structure for the optimized 3.0 GHz TM 01 -TE 11 mode converter. The computed
Fig. 2.14. The waveguide profile of an optimized 3 GHz TM^T EE,n, sernentine
converter
serpentine mode converter.
en
56
Fig. 2.15. Mode content of an optimized 3 GHz TM 0 1 -TE n serpentine
waveguide with a 99.8% conversion efficiency.
57
conversion efficiency is 99.57% and the ohmic loss is 0.127%. The computed
mode content along the converter is shown in Fig. 2.16.
Table 2.4
Optimized Structure for the 3.0 GHz TM 01 -TE 11 Mode Converter
Section 1: 0 < z < Xx/2
fl[z) = a 0 + ex [1 - cosCP]^ z)]
where a 0 =.05425 (m), e t =.0545 (m), ^=0.805 (m), p x = 2it/Xv
Section 2: 0 < z < XJ2
f(z) = a 0 + £3 [1 - cos(p2 z + TC)] + 0.019
where a 0 =.05425 (m), e 2 =.045 (m), ^=0.765 (m), p 2 =
Overall result
total length:
0.785 (m)
total loss:
0.127%
conversion efficiency:
99.57%
2TCA,2.
Fig. 2.16. Computed mode content along a 3.0GHz T M 0 1 - T E n
serpentine mode converter withoffset input and output ports and a
perturbation amplitude larger than the waveguide radius. Its
computed conversion efficiency is 99.5%.
59
Chapter 3
Azimuthal Deformation of Cylindrical Waveguide
In Chapter 2, we discussed a method to find the coupling coefficients for
a cyhndrical waveguide with a curved waveguide axis. From the derived
coupling coefficients, we can design a waveguide bend or mode converter which
will couple power into those modes with azimuthal index which differ by one.
The couphng coefficients are proportional to the local curvature of waveguide,
or approximately proportional to the second derivative of the serpentine
structure if the perturbation is small.
In this chapter, we will use a simple geometric approach [16] to derive
the boundary conditions for a waveguide with a straight z-axis but having a
radial or azimuthal deformation. Using these boundary conditions, we can
derive the coupling coefficients for azimuthally deformed structures in a
straight-axis waveguide. The simplest case for a straight-axis converter is
varying-radius type mode converter, which is the zero-order deformation. For
this kind of device, only modes with the same azimuthal index as the input
mode will be encountered in the coupled mode equations. This is because the
varying-radius type mode converter is an azimuthally symmetric structure.
An elliptical-waveguide polarization converter is another example of a
straight-axis mode converter. It can be used to convert a linearly polarized
60
mode, for example from the T E l n or TM l n family, to a circularly or elliptically
polarized mode. The polarization converter is an important component in
microwave transmission for use with plasma heating or deep-space satellite
communication.
For the new generation gyrotrons, the output modes are rotating TE m n
modes, with m » n. There are no commercial devices to generate these kind
of "whispering gallery" modes at low power for research purposes. However,
such modes are needed to test at low-power mode converters and transmission
systems to be used with new generation gyrotrons. One of the possible ways
to generate a whispering gallery mode from a TE 0 n mode for low-power testing
is using a helical multifoil mode converter which is one type of azimuthally
deformed cylindrical waveguide. From the reciprocity principle, it can be
shown that it is possible to use the same perturbation structure to convert the
whispering gallery mode into a TE 0 n mode for high-power long-distance
microwave transmission.
1. Derivation of Coupling Coefficients
For a slightly deformed circular waveguide, we can derive the coupling
coefficients by dividing the deformation geometry into transverse and
longitudinal planes [20], To introduce the effects of the deformation of the
waveguide cross-section, we write the total fields as the sum of the fields in
61
the undeformed waveguide (with subscript u) and the change arising from the
deformation (with subscript d):
E = Eu + Erf , H - Hu + H(£ .
(3.1)
For small perturbations of the waveguide wall, the total waveguide field
can also be expressed as a sum over the normal modes (Eit H{) that can
propagate in the undeformed waveguide. As in the discussion of Chapter 2,
we use the forward and backward traveling waves into our calculation. It will
be convenient for us to include them at very beginning of the derivation.
Assuming the fields vary as exp(±jPj z) in the axial direction, we obtain
E = J^AfryEyJteg
+ A;(z)E]e®iZ
(3.2)
and
H = J X t e W ' e * + A:(ZW~#P*
•
(3.3)
i
Now we define a new vector:
V = E x (H*)* + (JE*)* x H .
(3-4>
From the two-dimensional divergence theorem, we have
J v, • V ds = j V • dr dl .
(3.5)
62
The divergence of Eq. (3.5) can be written as the sum of the transverse and
longitudinal components:
v • V = v, • V + a2 • ^ 1 .
t
z
dz
(3.6)
Using the orthogonality properties of the modes in the waveguide given
by Eq. (2.16) and Maxwell's equations for a source free region, we find that the
integration of Eq. (3.6) over the cross section is
f v -Vds
= fv • [E x (Hp* + CE*)* x H] ds = 0
(3.7)
Hence,
fv, - V d s = - \ dz'?Y_ds
(3.8)
= - f JL[E x cirjr + (E*r x m-ds
J
*dz
l
- -S£(EAJBfe-&
+ £ x
f
(£AjH/e~J*/l
l
+
+
A;£,V^) x (H7)*
AjHj-^Hs .
Combining the orthogonality property and the divergence theorem of Eq. (3.5),
we simplify Eq. (3.8) as follows:
63
9A ±
f V - ar dl = * 2. -'c
"'
(3.9)
dz '
So
L = *lf[Ex
(H*)* + (IS*)* x fl] • dr dl
.
(3 10)
'
From Eq. (3.10), we need to know the components of Ed and Hd to solve the
problem. In the transverse (r, <j>) plane (Fig. 3.1) the tangential field on the
deformed surface at r = a + 5 (§, z) may be written in the form
BE,
E^ = EU) - Eur siny + Ed<? cosy + 8 -J±
dr
cos\|/
(3.11)
where cos \\r » 1, sin \)/ = tan \j/, and tan y = -1/a Q5/9<)>), for a small
perturbation. Thus we can simplify Eq. (3.11) as:
Et = Eu, • i i „
• B+ • 8 ^
.
(3.12)
In the longitudinal (r, z) plane, Fig. 3.2, the first order approximation for the
perturbed electric field gives us
dE
Ez = Euz cosy + Eur siny + E^ cosy + 8 — ~ cosy .
(3.13)
dr
Again, cos y ~ 1, sin y = tan y and tan y ~ do79z, for a small perturbation.
Thus,
(r, (p) plane
deformed
undeformed
surface
surface
Fig. 3.1.The waveguide deformation in the (r, §) plane.
(r, z) plane
deformed
•*•—
undeformed
surface
surface
Fig. 3.2.The waveguide deformation in the (r, z) plane.
65
Ez - Euz • EU*L
+Edz + 5^L
.
(3.14)
An expression of the same form can be derived for the H field as follows:
H
* • H»*+ £§*»•*ff^* ^ •
and
The more rigorous mathematical derivation of Eqs. (3.12) and (3.14) are given
in Appendix C.
Now if the waveguide has perfectly conducting walls, the zero-order
term vanishes automatically because it only expresses the boundary condition
on the undistorted guide. Hence we have:
[jMu
x H f + Ef
x Hu) • dr dl = 0
(3.15)
and Eq. (3.10) reduces to
oA'i
L =*if
dz
=
* \ L-a [Ed*H*
[Ed x (H*T + (E*)* x Hd] • dr dl
- EdzH^
+E H
* *
~ E*H*$
dl
•
(3 16)
"
From references [12,21], a circular waveguide with a complex wall
structure may be replaced by a simpler "wall impedance" model resulting in
66
a simplified analysis. Thus, we have the following set of equivalent boundary
conditions at the inner surface of the waveguide wall structure r - a:
Z 0 = i ^ | r = a = 0 and Zz = - E±.\rma
IT
" "
.
IT
*
£1-,
(3.17)
tl*.
From the above boundary conditions, we obtain
( B * • ±™Eur
Y
a d<j>
• S^±)\s
dr
=0
(3.18)
and
( E ^ • *LEur • 5 ^ - ) I.
dz
dr
= - Z z (ff^ + ±™Hur
+
5^)|,
.
(3.19)
From Eqs. (3.18) and (3.19), we find Ed(? and (E & +Z2 #d<!)) in terms of 8, JSU,
H u , and the derivation of En and Hn as
a d<j)
dr
dz
9r
Then, we can substitute Ed. and (JS^ + Zz HdJ into Eq. (3.16):
dAf
~dT - *jfrJ*«PZ'
~ H*(Ed*
+Z
^^)]
dl
67
dAi
dz
ln fa
^ .d , ^__+*_
H .„dE,
^ - ^ . l ^
2-"
"
• «£)*
[ 8 ^ • **„
v
dr
dz
+
Br
Z
2
a 3$
( S ^ •i^H^B
dr
a d(|>
<3-20>
•
From Eqs. (3.2) and (3.3), we write Eq. (3.20) in the form
±Af(z)
= TK^A+^-M-W
dz l
j
* K+rAjiz^W
where K* + and K*' are the couphng coefficients for waves propagating in the
same and opposite directions respectively. They can be expressed as
KI* = £ p« d* (Hii-®lk
-(HfJ[(±8^
*
dr
+
i^p
dz jr
+z ( ±
z
5
+
liisp]
^ ± 1^;)]}
dr
a d* •>'
•+-
.
(3-21)
+
From Eqs. (2.7) and (2.14), we know HZ is real and H~ is imaginary.
These properties have been used in Eq. (3.21). The coupling coefficients can
be found from Eq. (3.21) if we know the deformed structure of the mode
converter. Some of the coupling coefficients have been calculated for the
varying-radius type mode converter and are listed in Appendix A.
68
2. Azimuthal Deformations a n d t h e Selection Rule
It is interesting to find the relation between the azimuthal index change
and the deformation of the structure. To separate the perturbation structure
factor, 8, from the coupling coefficients, we will take out 8 and 3S/3({). Only the
forward coupling will be considered. Therefore,
K*+ =
v
2f
rc
(A 5 + A
•*>
*
38) <fy
2
(3.22)
3d)
where
A, - -%WJ^
•B
A2 = - " [ f f ^
^
- MyEjr • Z ^ t ) ] U
+ H^ZfljJl^,
.
<3 23)
-
(3.24)
Combining the scalar function in Eqs. (2.1) - (2.2) and the field components in
Eqs. (2.4) - (2.7) and Eqs. (2.14) - (2.15), we can extract the azimuthal variable
from the H{ and Ei fields.
We rewrite Eq. (3.23) and Eq. (3.24) as follows:
Ai = S sin(m<)>) sin(p<j>) + C cos(m<j>) cos(p<J>)
(3.25)
A2 = M x sin(m<j>) cos(p<}>) + M 2 cos(m<))) sin(p<|))
(3.26)
where S, C, M lf and M 2 are functions of r which are evaluated at r = a.
Suppose the deformation, a non-rotating structure, can be expressed as:
69
S(z,<]>) = a/(z)cos /({>
(3.27)
For the simplest case, we assume / = 0, giving 8 as a function of z only. As a
result, the deformation of the waveguide is a mode transducer of the varyingradius type. We find from Eq. (3.22) that only those modes with the same
azimuthal index (i.e. m =p) will be coupled. Hence, we can design the T E 1 5 2 TE 1 5 x mode transducer discussed in Chapter 1 using a varying-radius
deformation. The calculated couphng coefficients between the TE m n - TE m q ,
TE m n - TMjn , and TM m n - TM,^ mode pairs are listed in Appendix A.
If / > 2, the cross section of the waveguide in the (r,<})) plane takes the
shape of a multifoil as shown in Fig. 3.3 for the case of / = 5. Also, from Eqs.
(3.22), (3.25), and (3.16), we find that a nonzero integration will result when
I = | m ±p |
(3.28)
where m and p are the azimuthal indices for the mode we are considering. We
will call Eq. (3.28) as the selection rule. Only for those modes satisfying this
selection rule, is there a non-zero coupling coefficient. Figure 3.4 is a threedimensional view of a varying-radius type perturbation, i.e. 1 = 0. Figure 3.5
is a three-dimensional view of a serpentine type perturbation, i.e. 1 = 1.
It is worthwhile to compare the couphng coefficients for the tilted axis
serpentine structure discussed in Chapter 2 and the coupling coefficients for
the offset radius serpentine structure discussed in this chapter. The coupling
coefficient is proportional to the curvature of a serpentine waveguide as
Fig. 3.3. Waveguide cross-section in the (r, <b) plane for S((b) = a
(1 - e cos 5(b).
71
Fig. 3.4. A 2-period varying-radius type mode converter with
s = Ho 4- 6 cos(Apz)j Am ~ U«
p *1
II qq
»
o
CO
1-1 •
+ bx
CO
rs
o
>
>
to
•
"CD
N
O
T3
CO
3.
o
CO o
O>
w
CD
B%
II
CO
3
c+
J-k > - • •
S
3
CD
o
o
pCD
o
o
<
CO
a(0
•-1
cf
73
discussed in Chapter 2 and the coupling coefficient for a fixed axis serpentine
structure is promotional to the first derivative of the profile.
The coupling coefficients we use to setup the coupled mode equations
should be consistent with the manufacturing method. From the discussion of
discrete waveguide transitions [22], there are three basic discrete deformations
as shown in Fig. 3.6. For a varying-radius mode converter, we can use a
discrete diameter change to manufacture the mode converter. For a serpentine
type mode converter, there are two possible approaches for manufacturing the
mode converter: (1) the tilt method, and (2) the offset method. The first
method, the tilt method, is appropriate for manufacturing serpentine mode
converters when the design is based on the couphng coefficients derived using
the waveguide axis-bend method which was presented in Chapter 2.
The second method, the offset method, is appropriate for manufacturing
serpentine mode converters when the design is based on the azimuthally
deformed waveguide method which was discussed in this chapter. The relation
of coupling coefficients between the offset and tilt method for a serpentine
mode converter can be expressed as:
where K^Q is the coupling coefficient for the continuous offset and K^ is the
couphng coefficient for the continuous tilt case. In both cases we have
r
aO
7
a0
(a)
aO
a^~
(b)
3
aO
al
_L_
T
(c)
Fig. 3.6.Three basic discrete deformation a) tilt deformation b)
offset deformation c) diameter changed deformation.
^3
75
assumed that the ends of the waveguides are parallel and the waveguide
perturbation is small.
The different descriptions of coupling coefficients correspond to using
different sets of normal modes to describe the field within the deformed
waveguide. For purpose of illustration, consider the case of deviation from
straightness of the guide axis. The coupling coefficient K^ corresponds to the
normal modes of the deformed guide with cross section perpendicular to the
axis of the undeformed guide. The coupling coefficient K^ corresponds to the
normal modes of the deformed guide with cross section perpendicular to the
axis of the deformed guide.
When a varying-radius type mode convertor is designed using the
technique presented in this chapter, the appropriate manufacturing method is
the step diameter change method.
For a helical structure, we can write the deformation as
8(z,<j)) = afe) cos(Ap\2 ± /(b) .
This can be decomposed into sine and cosine perturbations as:
5Xz,<b) = a/(z)[cos(A(5z)cos(Z(b)Tsin(A(iz)sin(Z(b)]
= a/ccos(Z<b) + a/ssin(/(b) .
Appendix D discusses the coupling effects due to the sine and cosine
perturbations.
76
3. An Elliptically Shaped Polarization Converter
For the ECRH current drive experiment, it is desired to launch the EM
wave with the wave vector at oblique angle relative to the confining magnetic
field. Therefore an elhptically polarized incident wave at the plasma edge is
optimum for coupling to the plasma. Thus the transformation of a linearly
polarized wave to an elliptically, or circularly polarized wave is important.
To simplify the problem and consider the basic concept of polarization
conversion, initially only the co-polarized and cross-polarized T E n modes will
be treated in a uniform elhptical waveguide and it will be assumed that all of
spurious modes can be neglected.
Consider 2£0 as the amplitude of a linear polarized electric field incident
on the waveguide as shown in Fig. 3.7. E 0 can be decomposed into two
components
£(z=0) = dx E0cos(b + dy EQsmty = dx Ex(0) + dy Ey(0)
where Ex(0) is parallel to the major axis dx of the guide cross section. E (0) is
parallel to the minor axis dv, and <b is the angle between EQ and the major
axis. We will call E (0) the co-polarized mode and Ex(0) the cross-polarized
mode. Since the co-polarized and cross-polarized modes travel with different
phase velocities in the elliptical waveguide, we can write
= a 0 (l-f- e cos2<p)
Fig. 3.7. An elliptical deformation (solid) from a circular
waveguide (dotted) for a TE l m or TM l m polarization converter.
78
E(z) = dx Ex(0)e'J^
+ dy
EyiO^'^
= [ dx Ex(0) + dy Ey(Q)e-JAfc ] e~J^
where Ap = B - $x, and f5x and B are the phase constants of the cross-polarized
and co-polarized modes. If the incident linear polarized wave is aligned at 45°
from the major axis (<b = ±45°), we have Ex(0) = Ey(Q). The wave will become
progressively more elhptical until, at ABz =rc/2,it is circularly polarized. The
wave will return to linear polarization, but rotated by 90° at ABz = n. The
polarization continues to alternate between linear and circular polarization as
shown in Fig. 3.8. This is simplified situation, where we assume that only one
mode, the T E n , can propagate in the waveguide.
Since we operate the T E n mode in an overmoded waveguide, many
spurious modes will be excited. Furthermore, the abrupt change between a
circular and elliptical waveguide will cause unexpected spurious modes
excitation. Hence a gradual transition from circular waveguide into elliptical
waveguide will be necessary. We can use coupled mode equations to design a
polarization converter with the coupling coefficients between the ordinary and
cross polarization of TE m n and T M ^ waveguide modes. The setup of the
coupled mode equation will be further discussed in Chapter 6 for the elliptical
polarization and helical multifoil mode converters.
From Appendix D, we can find that, for TE m n or TM m n polarization
79
00.
<
CV2
u
cd
cu
p}
• d
fl
CD
CD
&
0)
T3
r—t
CO
00.
<
o
•co
1—H
«s
>
03
-2
ri
o
a •43
o
•43
r—i
QQ.
<
C\2
crj
N
o
&H
CO•
CO•
hb
Pn
•ccrj
r—H
o
a,
u
a
3u
Vi
•|H
O
T3
a
a$
80
converters, the only multifiol wall perturbation which can excite the cross
polarized mode is of the form
a/((b^;) = afe) sin(2m<b) .
(3.29)
The simplest deformation for which Eq. (3.29) applies is for I = 2 or m = 1, i.e.
an elliptical deformation. From Appendix D, we can find that for a multifoil
mode converter with an even number of flutes, i.e. I is an even number, a nonzero coupling coefficient can be found between the co-polarized and crosspolarized modes with the same azimuthal index, where m = 1/2. For example,
the elliptically deformed circular waveguide can be closely described by the
1 = 2 azimuthally deformed waveguide. Hence, we can use the elliptically
deformed waveguide to design a T E l n polarization converter.
In a practical case study, we will focus on a 60 GHz T E 0 1 - H E n mode
conversion sequence as discussed in Chapter 1.
In the multistep mode
conversion process shown in Fig. 1.2, the TE 0 l mode is azimuthally polarized
and the hybrid H E n mode in the corrugated waveguide is only slightly
birefringent under elliptical deformation [23]. The polarization conversion is
best achieved in the T E n mode. In the alternative multi-step mode conversion
process [4]: TE 02 -TE 01 -TM 11 -HE 11 the polarization conversion can be obtained
in the T M n section.
The slightly elliptically deformed circular waveguide will excite the
cross-polarized mode from a linearly polarized input mode with its polarization
81
not oriented along the major or minor axis of the elhptically deformed
waveguide. For example, an input co-polarized T E n mode, denoted by a
superscript o, will couple to the cross-polarized T E n mode, denoted by
superscript x. Certainly, the first order couphng should include the TE 3 n and
TM 3n modes into the calculation if the input mode is T E n . Also there is
second order mode coupling between the TE 3 n and TM 3n mode families and the
TE 5 n and TM 5n mode families. The first and second order coupling diagram
for a 60 GHz with 1.098" diameter T E n polarization converter was shown in
Fig. 3.9.
There are eighty modes that can propagate in this diameter
waveguide.
Since the current gyrotron program goals are to have operating
frequencies above 100 GHz, gyrotrons are no longer producing a TE 0 n mode
output so the conversion sequence: T E ^ - T E ^ H E ^ or TEgn-TMj^-HEj^ is not
a critical issue for this program any more. Hence, we here only discussed the
polarization converter concept above. No practical design for the polarization
converter will be presented here.
4. Helical-Multifoil Type Mode Converter
Those modes which satisfy the requirement of Eq. (3.28) with i £ 3 will
couple with each other through a high order azimuthally deformed structure.
Figure 3.10 shows a 3-dimensional picture of a rotating fifteen-foil azimuthally
deformed waveguide.
n=1..5
n=1..4
n=1..3
TMin TM?n
TM§n TM3n
TM§ n
TE?n TEi n
TE§ n TE3Xn
n=1..4
TE§ n TEsn
n=l..5
TM5 n
n=1..4
Fig. 3.9.Coupling diagram of a 60 GHz T E n input mode at a =
1.098" for elliptical deformed waveguide.
GO
83
Fig. 3.10. A lM>-period helical multifoil mode converter with
a = a 0 + e cos(Am(b + A0z) with Am = 15.
84
High-azimuthal-index mode generators are not commercially available.
However commercial devices for converting the TE 10 mode in a rectangular
waveguide to the T E n or TE 0 1 mode in a circular waveguide are available.
Hence we can use a helical multifoil mode converter to generate the desired
high-azimuthal-index mode from a T E n or TE 0 1 mode for use in "whispering
gallery" type Vlasov launcher experiments. From the reciprocity theorem, we
see that the same structure can also be used to convert the "whispering
gallery" mode into a low-azimuthal-index mode.
For high-power long-distance transmission, it is best to choose a loworder TE 0 n mode, due to the inherent low loss properties of these modes, as
the output modes from the helical multifoil type mode converter. There will
be more discussion in Chapter 5 about the difficulty of transmitting whispering
gallery modes through a waveguide bend. These will increase the possibility
of integrating the helical multifoil mode converter directly into the gyrotron to
produce a TE 0 n mode from the output window of gyrotron. Then, either a
conventional method of transmitting the TE 0 n or a Vlasov launcher to convert
the TE 0 n mode into a Gaussion like mode can be used. A detailed discussions
of the relation between coupling coefficients and the perturbation structure,
and the design of a helical multifoil mode converter will be presented in
Appendix D and Chapter 6 respectively.
85
Chapter 4
Approximate Solution of Coupled Mode Equations
Many phenomena occurring in physics and engineering, for example the
kinetic energy exchange between two pendulums connected by an elastic string
and the energy exchange between an electron beam and the slow-wave
electromagnetic wave in a traveling tube, can be viewed as coupled-mode
processes. The coupled-mode formalism is also a very powerful tool in the
design of mode converters, where it is helpful in the understanding and
analyzing of a variety of important devices. These range from waveguide
tapers, waveguide bends, corrugated waveguide converters and smooth wall
converters, up to the recently developed multifoil mode converters.
In this chapter, a set of two coupled mode equations, representing an
input mode and a single desired output mode in circular waveguide, are solved
in a series form by the method of successive approximation. We will assume
that the circular waveguide is lossless, the backward couphng is small enough
to be neglected, and the couphng coefficients can be derived for different wall
deformations as discussed in previous chapters.
In general, the size of the waveguide we have used to design mode
converters is highly or moderately overmoded so that the waveguide is able to
handle large amounts of power. Conventional microwave methods used in
86
single-mode waveguide components such as bends and twists cannot be used
in overmoded waveguides because it is not possible to prevent the transfer of
power into the spurious modes. The actual number of modes involved in the
numerical calculation is determined by the operating frequency, waveguide
size, and the selection rule derived in Chapter 3. A large number of modes can
be included in the numerical simulation without great difficulty. However,
because of the limited access to computation facilities, in most cases only the
first and second order coupling are considered without losing too much
information in the mode converter design.
It seems unlikely that the general case involving a large number of
modes will be understood before the two-mode case is fully understood. For
the general case where the coupling coefficient is function of position, the twomode coupled equations can be reduced to a single second-order nonlinear
equation.
Only under some specific circumstances can we find an exact
solution to the second-order nonlinear equation. Hence in this chapter, an
approximate method for solving a set of two coupled mode equations will be
derived for a general couphng function. Examples of sinusoidal perturbation
structures will be solved for two-mode coupled equations. Furthermore, we
will discuss the scahng law and frequency response, or bandwidth, for a set of
two coupled mode equations.
87
1. A Set of Two Coupled Mode Equations
Complete characterization of a mode converter requires numerical
integration of the coupled mode equations for all interacting modes. However,
some insight and a basic scaling law can be obtained by considering only the
two modes of major interest, but allowing a mismatch between the beat
wavenumber of the involved modes and the structure
perturbation
wavenumber of the mode converter.
For a lossless waveguide, a geometric imperfection (e.g. straightness
deformation and radius imperfection) will induce mode coupling. We can use
the coupled mode equations to describe the mode converter system. The two
coupled mode equations can be written as follows:
f l l = -JhAiiz)
dz
^ 1 = K^z^iz)
dz
+ K12(z)A2(z)
,
(4.1)
- JPW*)
.
W.2)
We will consider here only those cases for which p^ and P2 a r e constant or
nearly constant in z. $x and p 2 a r e completely independent of z for serpentine
mode converters and bends and are only weak functions of 2 for varying-radius
mode converters with small radius perturbations in highly overmoded
waveguides. For a lossless waveguide, the total power is conserved
88
*JAib)A'to)
+ A2(z)A*(z)] = 0 .
From this restriction it can be shown then
K12 = -K21
.
It is convenient to introduce the following change of variable:
Axfe) * axfejT-*1* ,
A2(z) = S 2 ( z ) e *
(4 3)
-
.
^
By combining Eqs. (4.3) and (4.4) with Eqs. (4.1) and (4.2), we can further
simplify Eqs. (4.1) and (4.2) into the following form:
dK
I = Kiz^izW^*
,
(4-5)
dz
dK>
I
dz
= -mz)a: 1 (z)e"- /APl2Z ,
where K(z) is the coupling function and A$12
=
<4-6)
Pi " $2-
Assuming the input signal at z - 0 has unit magnitude and zero phase,
and the desired output mode has zero magnitude at z = 0:
3^(2 = 0) = 1 , X2(z = 0) = 0 .
(4.7)
We will use A1(n^(z) and A2(n^(z) to be the nth approximation to the solution of
Eqs. (4.5) and (4.6). Let the initial approximation be given simply by the
initial conditions of Eq. (4.7)
89
A~m(z) = 1 , A~2(0)(z) = 0 .
The above equation represents zero order coupling. This means the coupling
is so small that there is no energy transformation between these two modes.
Now, following Picard's method [24] we can obtain the successive
approximation as follows:
X1(0)(z) = 1 , A~2$p) = °
J1(1)(z)= 1 + ^zK(s)ejA^^m(s)ds
=1
(4 8)
^2(l)(2) - "jfX*(s)e ~Ml^l(0is^s
= -^K'is^'^ds
(4.9)
^l(2)(2) = 1 + j[^(s)^'Apl2S7r2(1)(s)rfs
m
= 1- £ia9)e *d8£K*(t)e
^2(2)^) = -^*(s)e"- / ' APl2S ^ 1(1) (s)ds
~Ml^dt
(4.10)
90
= -fclC'is^^ds
X1(3)(2) = 1 + j^Kis^1^
=1 -
(4.1D
X 2(2) (s)ds
fcKis^^ds^K^t^^dt
(4.12)
^2(3)(z) = -j[*ins)e ~ JApl ^Z 1(2) (s)rf S
^K^^dt^K^uK
X 1(n) (z) = 1 + ^^^Kn-lpWs
~M^du\
(4.13)
(4.14)
91
Sawfe) = -fcK^-M^^pVs
(4.15)
Suppose we have weak coupling between mode At and mode A2 through
a sinusoidal coupling function with wavenumber Ps. Then we can write the
couphng function K(z) as
K(z) = K sin(ps z)
where K includes the perturbation amphtude of the perturbed structure and
the coupling coefficient for the proper modes, which has been derived for
various deformation structures in previous chapters.
From the second order approximation, Eqs. (4.10) and (4.11), we can find
the amplitude of mode A2 can be calculated as:
A2fe) = ~^K\s)e
-J*h*ds = - K ' J ^ s i n t p ^ ~Ml2Sds
K • e--jApi;*
.[/ApxasinCpaZ) + pscos(psz)]
-K*ps
and the power in mode A2 can be expressed as
2
|A2|-
|K|2(Ap12sinp82 sinAp122; + pscosps2 cosAp122 - $s)'
(Ps " AP? 2 ) 2
92
iK^CpgCOSpgZ sinAp 12 z - A p ^ s i n p ^ cosAp 12 z) 2
+
;
.
2
(4.1b)
(Ps - AP^ 2 )2
We can see from Eq. (4.16) that when Ps = A$12, maximum power transfer
between modes At and A2 occurs. This can be taken as a rule of thumb and
is known as the coherence condition. We can take the Ps -> Ap /2 limit of Eq.
(4.16) with the LTIospitaTs rule to find out the maximum power in mode A2.
Here, we choose the way in solving Eq. (4.11) again with the constraint of the
coherence condition, we can express the power in mode A2 as
|A 2 | 2 = |K*j[*sinAP12s e~Mi*
ds\2
|K|2[(2Ap12z)2 + 2 - 2cos(2AP12z) - 4Ap12zsin(2Ap12z)]
16AP?2
An extreme value of IA 2 1 2 can be found at A$12z = nn.
For a mode converter design, we need to maximize the power of mode
A2 at the output port.
From Eqs. (4.16) and (4.17), we have found the
expression in terms of variables pa, the beat wavenumber of the waveguide,
andL, the total length of the converter, which will give the maximum of power
transfer from mode Aj to mode A2.
Purely periodic coupling functions have been used to optimize an 8.6
GHz
TMQ-^TEJJ
serpentine mode converter in Chapter 2.
It has been
demonstrated there that as the number of perturbation periods increase, the
93
optimum value of p s approaches AP between the TM 01 and T E U modes. This
has been clearly illustrated at Fig. 2.3 and Table 2.1.
For a mode converter, the design objective is a complete power transfer
from the input mode to the desired output mode. If we assume total power
transfer to the desired output mode, reducing the number of perturbation
periods will increase the required amplitude of the perturbations. This will
become more clear as we proceed into next section.
2. The Scaling Law a n d Bandwidth of Mode Converters
From the solution of Eq. (4.17) we can define the actual conversion
efficiency, q, as A€)A2 . Suppose there is complete power transfer from mode
A1 to mode A2. A relation between the total length, L, of the mode converter
and the coupling coefficients can be found as
Ax(0)
where all the power is assumed to be in mode A1aiz
4
= 0. When p s = AP22, all
of the energy is transferred to mode A2 at zt = 211 K I, the length of the ideal
two-mode converter. If we consider a varying-radius mode converter between
the TE m n and TE mQ modes, zt can be expressed as
94
-1
zt =
2a 0 X 2
2
(x? -- m2)Px
2
%2
%1
<*i Pi - P2
(x 2 --
m 2 )p 2
(%2--
"12)P2
- m 2 )Pi
where a^ is the perturbation amplitude and aQ is the average radius for the
mode converter. For m = 0, the converter length is
_ 2a03/p^2"
z
t =
a
l
X1X2
which we call the scaling law for the azimuthally symmetric TE 0 n mode and
all TE m n modes with x n 2 »
m2.
The step tuning property of the varying-radius mode converter,
discussed in Chapter 7, can be explained qualitatively in terms of the complete
power transfer phenomenon scaling law for two-modes theory.
Next, we will determine the bandwidth of a mode converter. First, we
assume that the beat wavenumber between the two coupled modes equals the
perturbation wavenumber Ps of the mode converter
Pio(coo) " P2o(coo) = Ps at
co = co0
(4.18)
where
P/iO^o)
co^ue -
(h.)2
a0
(4.19)
P10(co0) and P20(co0) are the wavenumbers for mode A1 and A2 respectively at
95
operating frequency co0. This Pn0(a>) can be expanded in a Taylor series as a
function of co around co0 as:
fW©> • P^^lcoo
+
^
k
^ " W 0) + | - ^ ^ 0
( C 0
" W 0) 2 *.-
where co0 is the center operating frequency. Using this expansion, we can
rewrite Eq. (4.19) as:
Pn0(fi>> = P n f l W
+
a
,
"
+
- •
Pnotoo)
Now Eq. (4.18) can be expanded around co
co0ue
Pio(co) - p20(co) = p10(co0) - P2o(co0) + (co - co0)Pio^o)
- (co - c o o ) - ^ ! - = Ps + 5
P20(w0>
(4.20)
where
s - Ar, „ «o ^ 2 0 ^ 0 ) " Pio(« 0 )]
o = Aco COgUe
Pio(©o)P2o(®o)
with Aco = co - co0.
When the frequency changes, the beat wavenumber changes according
to Eq. (4.20). If we substitute Eq. (4.20) into Eq. (4.16) and assume the total
length of the mode converter remains constant as we change the operating
frequency, Eq. (4.16) can be rewritten as follows
96
|A2(z=L)|2
q =
*
—
lA^O)!2
2|K|2P2
-—il - cosoL) .
(5 2 + 28ps)2
After some algebraic manipulation, we can find the relation between the
conversion efficiency and the frequency difference from co0 by the amount of Aco
q =
|A2(z=L)|2
lAife-O)!
2
=
4|K|2ps2
2
(8 + 28ps)
.
2
sin
26L
—
2
4|K| 2 L 2 sin 2 ^Acoa
1
'
2
2 2
iVVAco a (Acon - 2)2
, _
(4.21)
where
Q -
"W"
Pl0(°)0)p20(a)0)
From Eq. (4.21) we find; as the number of perturbation periods, N,
increases; the bandwidth of the mode converter will become narrower, as
shown in Fig. 4.1. Also, as the frequency approaches the center frequency,
Aco —» 0, the conversion efficiency approached (KL) /4, regardless the number
of perturbation periods. From Eq. (4.21), it can be shown that
41 KL |2[:^L(2AcoQ - Aco2Q2)sinaV7iAcoQ) - 4(Q - Acoa 2 )sin 2 (^Acoft)]
dr\ _
2
2
2 2 3
dA(0
N V(2Acoa - Aco Q )
For a large number of perturbation periods, N, the maximum conversion
efficiency will occur at Aco = 0.
97
o
a
Qi
•r-l
o
Ifi
<3i
frequency
Fig. 4.1. Theoretical frequency response diagram for a pure
sinusoidal coupling function.
98
A smaller number of perturbation periods will give us a better frequency
response. However, a larger perturbation amplitude is required to compensate
for the shorter converter length. This will potentially increase the coupling to
spurious mode and the backward coupling. Two moderately overmoded mode
converters have been designed and are discussed in Chapter 2. The theoretical
simulation , assuming the backward coupling can be neglected, gives a very
good conversion efficiency for both single-period mode converters. One of the
single-period devices has been built and tested. The reflected power can not
be detected in low-power measurements.
With the limitation of the space in plasma heating devices, we need to
design mode converters to be as compact as possible. Some researchers [5,13]
use the method of reducing the waveguide radius to enhance the coupling and
reduce the overall design length. The smaller of the waveguide radius, the
shorter the beat wavelength between the modes being considered, allowing a
shorter converter to be designed. However, the power handling capability will
be reduced. To optimize the performance of a mode converter it is often
necessary to offset one variable by trading it off against another, in order to
meet the overall system requirement and performance.
Mode converter
performance may frequently be optimum when the number of perturbation
periods is minimized subject to the constraint of some minimum conversion
efficiency.
99
Chapter 5
Waveguide Bends and Three-Mode Coupled Equations
In the discussion presented in Chapter 1, we used a simple optical-ray
picture to explain that any bend in a circular waveguide will result in the
bounce angle changing and coupling to other modes. In Chapter 2, we adopted
the method of Schelkunoff and Morgan to derive the couphng coefficients, from
which it can be seen that modes with azimuthal index which differ by one will
couple with each other through a small curvature of the waveguide axis.
In most microwave transmission systems, a change in the direction of
energy transmission is necessary.
For a conventional
single-mode
transmission system, any excited spurious modes are evanescent and attenuate
very quickly after the waveguide bend. The only thing we need to avoid is a
sharp bend, which will increase the overall reflection coefficient in the
transmission system. Hence, a gentle waveguide bend can be used in a singlemode transmission system without conversion loss or reflection.
In microwave and particularly millimeter wave transmission systems
designed to carry hundreds of kilowatts or perhaps megawatts of power,
overmoded waveguides are often used to handle the large amounts of power.
For example, the TE 0 1 mode has been used in high-power microwave
transmission systems, because it has low loss in an overmoded waveguide.
100
However without careful design, an overmoded waveguide bend will couple
significant power from the TE 0 1 into spurious modes. The most strongly
coupled spurious modes for a TE 0 n mode in a waveguide bend are the TM l n
modes.
Because the TM l n mode and the TE 0 n mode have the same eigenvalue,
they are referred to degenerate modes. Hence, they propagate with the same
phase velocity in a circular waveguide and we expect that there will be strong
mode coupling to the TM l n mode as a TE 0 n mode propagates through a
waveguide bend. A careful analysis needs to be performed before we design
a waveguide bend for these modes. Otherwise, methods need to be developed
to destroy the degeneracy between TE 0 n and TM l n modes. For example, a
slightly elhptically deformed waveguide can destroy the degeneracy between
TE 0 n and TM l n modes. Or a dielectric lining close to the waveguide wall
changes the TM l n mode phase constant appreciably with almost no change to
TE 0 n mode [25]. The phase velocities are now different and, despite curvature
couphng, mode conversion stays small. Generally speaking, any method which
affects the TM l n and TE 0 n mode differently will remove the degeneracy and
become a potential solution for degenerate mode waveguide bends.
For the first generation gyrotron, the output mode was usually a TE 0 n
mode. Therefore a design method for waveguide bends for degenerate modes
is valuable for the first generation gyrotron. In new generation gyrotrons, with
an operating frequency at 110 GHz or above, the output mode is usually a
101
whispering gallery mode. In case of the high-power backward wave oscillators
with operating frequencies of 8.6 GHz and 3 GHz, we consider the T E n and
TMQJ modes. In both cases, none of those modes are degenerate. Hence our
main interest here will focus on waveguide bends for the nondegenerate
modes.
A brief derivation for degenerate modes will be treated as an
introductory part in this chapter.
1. Waveguide Bends for Degenerate Modes
Two-mode coupled equations can be expressed as:
f^i
dz
= -jpjA! + K12A2
^ 1 = K21A, - JPaA, ,
dz
(5.1)
(5.2)
where Px and P2 are the propagation constants for mode Ax and A2
respectively, and K12 and if21 are the coupling coefficients between mode Ax
and mode A 2 . Now we introduce new variables A x and A2 defined by
-./Pi*
A x = ~A~ie
and
A2--Kx®*
.
For degenerate modes, such as the TM l n and TE 0 n modes, we have Px = P2.
In this case, we can further simplify Eqs. (5.1) and (5.2) as:
102
^ 1 = K12A2
dz
(5.3)
and
dAo2
= *
jr^TT.
=
12X
For a lossless mode converter K12 = -K21
(5-4)
and for a serpentine type
deformation, K12 is imaginary, i.e. K12 =jK. Hence, the solution for Eqs. (5.3)
and (5.4) are:
Xj. = cos (Kz)
and
~A~2 = sin (Kz) .
When Kz = n(2N+l)/2 there is complete power transfer from A x to A 2 . That
is there will be total power transfer for a uniform waveguide bend with radius
of curvature R and bend angle 9 t when
R @1 t - — (22V+1) .
2K
Writing K = Kr/R, the bend angle can be expressed as:
8, - JL(22V + 1) ;
i.e., the power transfer is independent of the curvature of the waveguide bend
103
for the case where only two modes are considered. Only the bend angle will
affect the power distribution.
We have assumed that the backward coupling can be neglected in the
coupled-mode equations.
Under this assumption the curvature of the
waveguide bends will be small such that the backward coupling can be
omitted. Even though the curvature of the waveguide bend does not affect the
power distribution, the reflection coefficient is determined by the curvature of
the waveguide bends in the transmission system. Therefore, in the case of
degenerate modes, we can only have discrete angles to have highest
transmission efficiency for a particular waveguide diameter and fixed
operating frequency. The coupling coefficient between the degenerate TE 0 n
and TM l n modes can be expressed as follows:
\/2x„
R
Hence, the maximum transmission efficiency will occur at an angle 6 t where
Qt = lllHL
7dV .
Po«
where N is an integer. Figures 5.1 and 5.2 show P0a vs. 0 t for (TE 01 , T M n )
and (TE 02 , TM12) degenerate mode pairs respectively. We can choose the
waveguide diameter properly to get the maximum efficiency for the required
bend angle. However, in general, we won't have the freedom to choose the
104
COL
et
Fig. 5.1. The design diagram for a waveguide bend for a TE
or T M n degenerate mode.
105
9.
Fig. 5.2. The design diagram for a waveguide bend for a TE 0 2
or TM 1 2 degenerate mode.
106
diameter we want for waveguide bends. Therefore, a lined waveguide or
slightly elliptically deformed waveguide can be a practical solution for the
design of waveguide bends for degenerate modes.
2. Waveguide Bends for Non-degenerate Modes
In this section we make some assumptions to solve a set of three-coupled
mode equations for waveguide bends. For waveguide bends, the wave numbers
for propagating modes are constant over the entire bend length. It will be
assumed that the ellipticity waveguide cross section of the bend can be
neglected. This will also be assumed for a serpentine type mode converter. A
second assumption for a waveguide bend is that there is only weak coupling
between the input mode and spurious modes. A third assumption we make is
that the coupling function is constant or a known function.
We have designed waveguide bends for T E n and TM 01 modes at 8.6
GHz with 2.38 cm radius. From the coupling diagram, Fig. 2.3, we find that
the T E n and TM 01 modes are strongly coupled to each other. If we had only
the two coupled modes, the waveguide bend for the T E n mode would serve as
well for the T M ^ mode bend also. Actually, for a short bend design in which
additional modes propagate, the optimum waveguide curvature is not the same
for T E n and TM 01 waveguide bends. For a T E n waveguide bend the TE 2 1
mode is the next most important mode to take into consideration (after the
107
T M ^ mode). For a TM 01 waveguide bend the TM 11 is the next most important
mode to take into consideration (after the T E n mode).
The third mode plays an important role in the waveguide bend. Suppose
there is a weakly-coupled third mode entering into our calculation. We can
write the three-mode coupled equations as follows:
fli = - M
dz
dA2
dz
• Kl2A2
,
K21A1 - jfaA2 + K23A3
(5.5)
,
(5.6)
and
^1
dz
= K32A2
- jfoA*
>
(5 7)
-
where we assume A 2 is the input mode and Ax and A 3 are decoupled spurious
modes. From conservation of energy, we can find the coupling coefficient must
satisfy Kmn = -Knm .
For a waveguide bend, Kmn
is imaginary, and
K
— K
^mn
nm'
If the radius of curvature is sufficiently large most power will remain
in the input mode A 2 . Thus, we can introduce new variables, A1/A2 and A3/A2,
which we will call the spurious-mode amplitude ratios. From the chain rule,
we have:
108
d_r
dz
A2
\
(,
^
d_
dz
1 dAx
Ax d
A2 dz
A2 dz
\ dA3
A3 d
A2 dz
^2
A2 dz
(5.8)
(lnA2)
(lnA2)
and
cflnA2
i dA2
= -7
A1
~ K12-r-
-r-
~ JP2
+ A
A3
(5.10)
23-r~
dz
A2 dz
A2
A2
Substituting Eqs. (5.5) - (5.7) into Eqs. (5.8) - (5.10), we can find:
f
\
Ai
dz , 2
A
( . >B
= -jftii
A2
+
K12
K 12
A2
•o Ai
" JP2-7-
AXA2
v
+ A
23
=-
(5.11)
and
r
_d_
dz
\
~ ^23 " 7p3-r~ ~ ^12
7T- + ^P2-r- " ^23
\
(5.12)
Ao z
)
Suppose there is only weak coupling between A1 and A 2 , and between
A2 and A 3 , such that
(
A
-0 ,
A2
^
^3
A
v 2,
- 0 and f ^ . - 0
A
A
2
2
In this case, we can further simphfy Eqs. (5.11) and (5.12) as follows
(5.13)
109
d_
dz
VA2,
^
dz Ao
+
= -J(Pl - P2)-r±
A
2
£
= -JCPs " P2)-r
A
+
2
*12
(5.14)
(5.15)
#23
Solving Eqs. (5.14) and (5.15), we find
A
r-Jf**'
- ^ P ^ ' '
A2
(5.16)
dz'
and
±1 - X ^ * ' ^ * '
(5.17)
where Ap12 = p x - P2 and Ap32 = p 3 - p 2 .
From Eqs. (5.10), (5.16) and (5.17), we have
J* lnA2 = 4 ^ 1 2 " JP2
dz
A2
- j f c ^ " .
- K12^K12e~J>*m*~
dz' -32
•
+
#23—
A2
K23^K2#
-jfofa/1*"^,
^ (5.I8)
Equation (5.18) is the general expression for the input mode A 2 regardless of
the form of the coupling function and the propagation wave number provided
A j and A2 remain small. Equation (5.18) can also be used in varying-radius
uptaper design, in which the beat wavenumber is function of z.
110
3. Waveguide Bends with Constant Curvature
For a serpentine type mode converter or a waveguide bends, the beat
wave numbers Ap12 and AP23 are not functions of position. If the coupling
function is constant, that is, a constant-curvature bend has been chosen, then
we can directly integrate the right hand side of Eq. (5.18)
*JnA2 = i q 2 2 j y y A p l 2 f e - z V - jpa
2
Kl3fr~JAMz~zl)dz'
+
2
= _EE_fe-->Ap^ _ D _ fa + ^nJe-M& _ 1} .
Integrating again from z = 0 to z = L, the length of the converter, we obtain
lnA2 = - _^L{K 1 2 L + J ^ - ^ P i ^ - I?L{K23L + ZH^-^P^}
jAp 12
jAp 12
jAp 23
jAp 23
- JWL - _ | _ - __|_ .
Ap12
Ap23
(5.19)
We can separate the real and imaginary parts of Eq. (5.19).
V
V
V2
M
lnA2=-M2-±2L-^3L)LH-^e ~ ^
AP12 AP23
^
f2
+^2Le
Ap23
jA
V2
V2
~ ^) - f}L - _f?±
Ap22
Ap23
Ill
TC2
lnA2 = ~M2L-
12
A
Pi2
TC
L-
32
TC
TC
L+^lsin(Ap 1 2 L)4-_ilsin(AP 3 2 L)]
AP32 A p 2 2
Ap2^
[IE. COS(AP12L) + •£** cos(Ap32L) - -EE. - J ^ l ]
Ap 2 2
Ap 2 2
Ap 2 2
Ap 2 2
= j I(L) + K(L) .
I(L) is the imaginary part of lnA2 and %(L) is the real part of the lnA2. Hence
the magnitude of the A 2 can be expressed as
TC
L)
\A2\ = e^
TC
TC
TC
= exp[_^.cos(Ap 12 L) + —|_cos(AP32L) - _ J L - _ | - ]
A
A
A
A
Pl 2
P32
Pl2
P32
.
The imaginary part of the A 2 only contributes to the phase variation of the
mode A 2 . If the maximum amplitude of A 2 is desired at the end of a
waveguide bends, of length L, the extremum can be found from
JL|A 2 | = 0 => JL**M = e*MjL%(L) = 0 => AjR(L) = 0 .
dL
dL
dL
dL
Since the couphng coefficient is proportional to the waveguide bend curvature,
K12 = K12IR
4.X(D
dL
= K12$/L,
and we have
- - 4-WX - COS(AP 1 2 L)] (Kl2<1,)2
dL
(Ap12L)2
+
[1 - cos(Ap 32 L)] ^ f
)
(Ap32L)2
112
JLSRCL). dL
(<|)Kl2)2
2
sin(Ap 12 L)
2(<|)Kl2)
+
2
L Ap 12
-
Ap ^
3
[1 - cos(Ap12L)]
(<|)K32)
sin(AP32L) + 2((})K32) [1 - cos(AP32L)] = 0 .
L 2 Ap 32
Ap2^3
(5.20)
Multiple local, minimum or maximum, solutions for IA21 can be found from
the above equation corresponding to different lengths, L.
The second
derivative of IA2I can be expressed as:
-*L|A2| = ^(L\d^(Df
dL2
dL
+
eimjP&VL) .
dL2
(5.21)
A solution to Eq. (5.20) should be chosen such that the second derivative of
\A21 with respect to L is less than zero, corresponding to a maximum of IA 2 1.
The second derivative of WL) can be expressed as:
rf
^TO)=-i^cos(AP12L)+4(^Kl2)^in(Ap12L)+6(0Kl2)^cos(AP12L)-l]^
dL2
L2
L 3 AP 12
AP2^4
((l)Kq9)2
Y 32
2
L
4 ( 0 Kq 9 ) 2
cos(Ap32L)+
K
Z2
6(<t)Kq9)2
l ' sin(Ap32L)+ w 3 2 ' [cos(AP32L)-lj X5.22)
L Ap 32
Ap2^4
3
We assume K32 « K12 and AP12 < Ap32, which is valid for the 8.6 GHz
T M ^ or T E n waveguide bend. Hence, Eq. (5.20) reduces to
113
o
sin(Ap12L) =
-(1 - cosAP12L) .
Ap12L
The solution for L for a maximum IA 2 1 is then given by
L =
2nn
where n = 1, 2, 3... .
APi2
This is the same conclusion we will discuss in Sec 5.
Since the relative spurious amplitude ratio is affected by the curvature
of the waveguide bend, we need to estimate the amplitude of spurious mode
to confirm that the approximation in Eq. (5.13) is valid.
The spurious
amplitude ratio for the waveguide bend is:
.* - £ * * e ±
dz'
= -*Kl2 (1 - e ^ 1 2 z)
APi2
Hence
Alt
_,*12
I max
A2 • —
AP 12
(5.23)
i ^3 i
(5.24)
Similarly
^2
l max
i ^32
AP32
From Eqs. (5.23) and (5.24), we can choose the curvature of the waveguide
114
bend properly such that the spurious mode amphtude ratio will not exceed 0.2.
Then, the approximation we made in Eq. (5.13) will be valid. Using Eq. (5.20)
combined with Eqs. (5.23) and (5.24), we can find the solution for the
waveguide bend with specific bend angle 9 for which IA21 is a maximum and
the spurious mode amplitude ration does not exceed 0.2. Generally, as L
increases; (giving a smaller spurious mode amplitude ratio), there will be more
half periods in the mode content diagram along the waveguide bend. At the
same time, we will experience a narrower bandwidth and larger ohmic loss for
the waveguide bend. The optimized length depends on the design requirement
and which parameters can be traded for the most important parameters.
4. Waveguide Bend Using a Sinusoidal Profile
In the previous section we considered waveguide bends of constant
curvature for a nondegenerate major mode. Many other bend profile curvature
functions are, of course possible. We will consider a second bend profile in this
section, the sinusoidal curvature distribution.
fXz) = ilaniJE.)
2L
L
The same techniques can be applied to other curvature functions. The integral
of the coupling function along the waveguide bend is equal to the total bend
angle <j). That is, trial functions have been normalized to the bend angle <j).
115
* = j f A*') dz'
where L is the arc length parameter.
For the sinusoidal curvature distribution, we can solve the set of three
coupled-mode equations as
JnA2 = K^K^^-^dz'
dz
-fa
•
K32jjK32e-J^-z,>dz(5.25)
and we can separate the structure factor and the coupling coefficient as
K12(z) = K 12 /(z) = K 1 2 111 sin(J^)
£Li
(5.26)
Li
and
K32(z) = K 32 flz) = K 3 2 | * sin(2^) .
2J-i
(5.27)
Li
The angle needed for the bend is frequently predetermined. Hence, the
angle $ is a predetermined value. If we substitute Eqs. (5.26) and (5.27) into
Eq. (5.25) we have:
az
2L
L
* K322(?±)28m(™)
Solve Eq. (5.29) we have
M
L
(z s i n ( J E ) e " - f t * ^ dz'
(5.29)
116
lnA(L) = %(L) + j I(L)
where
so.) - v&tn**1
2
L
* " t o * * • K322d^£^ltC0S(fe2L)
(& - (£)2)2
2
L
(P 2 2 -(i) 2 ) 2
Li
JL/
and
/a) = K l 2 2 ( ± ) 2 ( « ) 2 — E J * — - a?
2
2
*
L
2[(i) -p 2 ]
L
P32^
_ ^ TC ^2
- p2
(^-("JV
L
+ K 2 ( 4> )2( " )2
322
2
sinPi2L
Li
2[(£) 2 - P 2 2 ]
L
sin
P32^
tp 2 2 -(£) 2 ] 2 '
We can find the extreme points from
4-|A
2 (L)| = ° •
dL
Following a similar procedure to that discussed in the previous section, the
first derivative of WL) is
d
2 ,4>tf -H 4 Pi 2 {(Pi2L 2 - irSfeinpujL + 4p 1 2 L(l
m n
1Z
dL
+
2 A 2
K 3 2 (_)
2
+
cosp12L)}
2
m
( P ^ T2 2 _- JL
K2)3
2
> 2 T 22
-7i4p32{(P^2L
- ^„)2s i n p 3 2 L + 4p 3 2 L(l + cosp 32 L)}
—
__
m2
(F
32
r2
L*
-
, to. 30)
„2N3
n^
which we set equal to zero. Solving above equation, we can obtain multiple
118
,Ai,
A2
Kl2
,
(
£)2
2L
^ 2 M
„ „ D T C „ N 2 ^ rv2 • 2 ^
(J±.m - cos-^)
+ pfsin^
_ p2 >|
(5.31)
and similarly,
A
i 3,
A
2
K !H
, 2 3 2L
(iL) 2 (l - cosJL*)2
L
( «.)2 _ p2 >| L
L
p2sin2iL3
+ Wl
L
.
(5.32)
Eqs. (5.31) and (5.32) give us other constraints in the design of the
waveguide bends which limit the spurious mode ratio.
5. The Design of T M ^ a n d T E n Waveguide Bends at 8.6 GHz
In the 8.6 GHz transmission system, our main interest is in the
waveguide bends for the T M ^ and T E n modes which are non-degenerate. The
waveguide bends for TM 01 and T E n modes have been studied in a constant
curvature and sinusoidal profile for a 4.76 cm diameter waveguide at 8.6 GHz.
For the T M ^ waveguide bend, the strongest coupled spurious mode is the
T E n mode. For the T E n waveguide bend, the most serious spurious mode is
the TM 01 mode. As previously discussed the bend curvature will be the same
for these two major modes, if we consider only two coupled modes.
The coupling characteristics of these constant curvature bends for TM 01
and T E n modes was investigated numerically using all six propagating modes.
The coupling diagram is the same as shown in Fig. 2.3. In the numerical
119
simulation, the radius of curvature was chosen so that the output mode purity
was highest for the bend angle we desired. In this case our main concern is
a 90° bend with a curved length of approximately two of the beat wavelengths
between the T E n and TM 01 modes. Figure 5.3 shows a three-dimensional
picture of a 90° waveguide bend. Figure 5.4 shows the mode content along a
constant-curvature waveguide bend for a TM 01 input mode. The transmission
efficiency for the TM 01 mode is 99.43% with .57% ohmic loss for a 90° bend
and 99.68% for a 45° bend. Figure 5.5 shows the mode content along a
constant-curvature waveguide bend for a T E n input mode. The transmission
efficiency is 99.56% with .33% ohmic loss for a 90° bend and 99.43% for a 45°
bend.
A numerical simulation for a whispering gallery mode has also been
tried. A TE 1 5 2 mode was generated at 110 GHz in a circular waveguide of
3 cm diameter. The coupling diagram is shown in Fig. 5.6. Both first and
second order coupling have been included in the numerical integration. Figure
5.7 shows the mode content along a constant-curvature 90° waveguide bend for
the TE 1 5 2 mode. The overall length is 1.256 m with 99.4% of output mode
purity and 6.23% of ohmic loss. We do not claim that this is an excellent
design, but this exploration indicates a potential difficulty in using the
whispering gallery mode as a major propagating mode.
120
Fig. 5.3. Three-dimensional picture of a 90° waveguide bend.
Amplitude
a" H **l
**•
P ©
et- i-»
O
e
e-»-
t3
ct-
en
ifc.
s
o
B
o n
3
o rt-
•a
P
2.
$0
S
(ft
P
e-t-
CD
O
3
(S3
Oi
00
o
•-<.
3
00
05
§
N
o CO
o o
ca o
co C
t?5 CO
o
3
n$
B
CO
CO
o
CO
to
TE 11
co
o-
0)
B
•r-i
'3,
<3 •*
TM01
d
Fig. 5.5. Mode content along an 8.6 GHz 90° bend with 99.8%
T E n output mode purity and .33% ohmic loss for a T E n mode
input.
n
n
TE
14,n™14,n
TE
13,n™13,n
TE
16,n™16,n
TE
17,n™17,n
Fig. 5.6. The coupling diagram of a TE 15 2 waveguide bend at
110 GHz with a diameter of 3 cm.
p
z (m)
Fig. 5.7. Mode content along a 110 GHz 90° bend with 99.4%
TE 1 5 2 output mode purity and 6.2% ohmic loss for a TE 1 5 2
mode input.
125
Chapter 6
A Mode Conversion System to Obtain
A Rotating TE i g > 2 Mode
The output modes from newer gyrotrons currently under development
are rotating whispering gallery modes, TE m n modes with m » n. Whispering
gallery modes are not good for long distance transmission, as indicated in
previous chapters, or for directly launching into a plasma. Several types of
mode conversion and transmission systems have been proposed. One of the
more promising is a Vlasov antenna type system. This antenna system uses
a special termination of the circular waveguide to feed a series of reflectors
which produce a linearly-polarized Gaussian-like beam. This beam can be
focussed into a corrugated circular waveguide for HE X1 mode transmission or
can feed a quasi-optical transmission system of reflectors. A second promising
transmission system is the slotted coaxial waveguide converter proposed by
Moeller [27]. This conversion system has the advantage that it is entirely
within a waveguide and again feeds an H E n mode in a corrugated waveguide.
However, neither of these approaches has been entirely successful at this time.
The low-power testing of these and other proposed mode conversion and
transmission systems requires the appropriate whispering-gallery mode input,
commonly a TE 15 2 or TE 22 2 mode at this time. There are no commercial
126
mode converters which can produce such modes from the dominant TE 10 mode
in a rectangular waveguide. It is the purpose of this chapter to discuss the
design of such a low-power mode conversion system. Two such systems will
be discussed here; the first produces a TE 15 2 mode and the second a TE 2 2 2
mode. Our major concern will be with the TE 1 5 2 system.
Both of the low-power whispering-gallery mode generating systems
described above begin with a commercially available rectangular TE10-mode-tocircular-TE01-mode converter. In the TE 1 5 2 system the circular waveguide
TE 0 1 mode is converted to a TE 07 mode with a varying radius converter. The
TE 07 mode is nearly degenerate with the TE 15 2 mode and a helical multifoil
converter can be used to convert the TE 07 mode to this desired output. A
similar sequence, TE 0 1 - TE 0 9 - TE 2 2 2 , is used in the second case. Mode
converters of this type had been proposed and designed by Thumm and Jacobs
[281 previously. However, their designs were in comparatively small diameter
waveguide and used extremely small perturbations, the maximum radius
difference between the peak and the valley of the multifoil being only .003".
Such small perturbations require extremely close tolerances in order to obtain
good mode purity at the output.
Our design uses a much larger radius
waveguide and larger perturbations, a peak to valley radius difference of about
.021." This requires less demanding tolerances. The larger diameter also
raises the possibility that such a helical-multifoil mode converter could also be
used at high power to convert to a TE 0 n mode for low-loss transmission.
127
In Chapter 3, we presented a brief discussion of multifoil mode
converters and polarization converter.
A selection rule and coupling
coefficients have also been presented in general form in Chapter 3. Coupledmode equations will be used to describe the mode coupling in a multifoil mode
converter in linear and helical structures. Design and experimental results
will be discussed for TE 07 -TE 15 2 hehcal multifoil mode converter at 110 GHz.
1. Rotating Modes in a Linear Multifoil Structure
The serpentine-type mode converter discussed in Chapter 2 and the
waveguide bends discussed in Chapter 5 are basically linear structures, where
the waveguide axis perturbation is located in a single plane. Varying-radius
type mode converters are azimuthally symmetric structures which keep the
azimuthal mode index and the polarization properties between the input mode
and desired output mode unchanged.
The new generation high-power gyrotrons generate rotating TE m n
modes. Hence, a rotating mode in a linear and a helical structure needs to be
carefully examined. In this section, a rotating input mode will be considered
in a linear structure.
We assume a linear multifoil waveguide wall
perturbation can be written as:
a(z,§) = QQ + 8(z,<j>) = a 0 + 8/c(z) cos(Z<{>)
(6.1)
128
where we restrict ourselves to a fixed-axis multifoil mode converter, that is
I > 2. From the derivations of Chapter 3 and Appendix D, we can find that
there is no coupling between different polarizations of the same mode in a
linear multifoil mode converter. Hence, we have
K
x
=0 =K
x
(6-2)
where the first index of Kllc refers to the input mode 1, the second index with
x above the index is represents coupling to the mode which is cross-polarized
to mode 1, and third index c indicates that the deformation is a cosinusoidal
perturbation. A similar representation is used for mode 2.
Assuming the linear perturbation can be expressed by Eq. (6.1), we can
rewrite the coupled mode equations for rotating modes in a linear multifoil
structure as follows:
^ 2 . = -JPjAio
dA\x
dz
dA2o
dz
+
KvuAto
_ __.aA A
+ KvX
A
_
+. IT
A
~ -Jh lx
A
M
= -jftjAto
i2iA2o
+
,
K&A*
xx
+ Kv
.
(6-3)
liA2x
(QA^
TSX
(6.5)
XilcAio * K£,AU
™*L - -jfeA* • 4 l c A l o • J £ » A t o .
(6.6)
where the subscript o represents one possible polarization of mode 1 (or 2) and
129
the subscript x represents the cross (orthogonal) polarization of mode 1 (or 2).
From the discussion of Appendix D, we have
K
12c ~ K12c ~ °
an(
K
^
l2e ~
xx
l2c
K
Hence, Eqs. (6.3)-(6.6) can be further simplified as
£ £ • - -JPxAto • K12cA^
dz
dA
H = -JM^
dz
dA
dz
= -./Ma,
+
+
,
(6.7)
K^A^
,
(6.8)
K21cAl0
,
(6.9)
J ^ j ,
.
(6.10)
and
dA^
I = -jpaAjfc
az
+
If we multiply Eq. (6.8) by ±j and add the resulting equation to Eq. (6.7), we
have
-f-(A lo ±jAlx)
az
= -JPxCA^
±./A 1 X )
+ K12c(A2o ±jA2x)
.
(6.11)
If we multiply Eq. (6.10) by ±j and add the resulting equation to Eq. (6.9), we
have
130
4-(A2o ±JA2x) = -.TfcCAao ±JA2x) + i^ 21c (A l0 ±.jAJje) .
dz
(6.12)
From Eqs. (6.11) and (6.12), we can find two important conclusions for
a hnear multifoil structure. First, it is obvious that if the input mode is a
rotating mode we will get a coupled mode which is also rotating and with the
same rotation.
Second, it is important to note that we won't obtain a
stationary mode from a rotating mode with a linear cosinusoidal deformation.
With the substitution of variables
Ax = (Ay, ± JA^1*
,
and
Aa - CA* ± j A ^ y * * ,
Eqs. (6.11)-(6.12) can be simplified as follows
JLA~1 = KucA'p'W*
dz
,
(6.13)
4*2~K2u£i*~m
dz
•
(6.14)
and
Suppose the coupling coefficients are real, K12c = ~K21c, and there is a uniform
deformation throughout the mode converter. We find the solution of Eqs.
(6.13M6.14) is
131
A 1 (z)=c2 ,/
A2(z)=e ^
{A1(0)[coshz-^jsm-Lsz]-2jA2(0)-J2.sin±z}
2
s
2
s
(6 15)
'
2
{-2A1(0)^£sin^+A2(0)[cosi^v^sinlz]}
s
2
2
s
2
(6 16)
-
where
= ^ApJa + 4K22
Often, we have A±(0) = 1 and A 2 (0) = 0. Equations (6.15) and (6.16) can be
further simplified as
Ax(z) = e 2
[cos(lz) - j^Lsm(±z)]
2
s
2
,
and
A22(z) = -2c
2
_Hsin(£^) .
s
2
Maximum power transfer from A-± to A 2 occurs at sL = f2n + IJ re. Complete
power transformation occurs only i£4K12 » A$i2 and sL = (2n + 1) n. This is
one of the special cases for which we can find an exact solution for the coupled
mode equations.
For a sinusoidally perturbed linear structure, if the number of the flutes
of the mode converter is equal to twice the azimuthal index of the input mode,
132
we can use this kind of sinusoidal perturbed waveguide as a polarization
converter to convert a stationary mode into a rotating or partially rotating or
elliptically polarized mode, or vice-versa. For example, a 2n-foil waveguide can
be used as a polarization converter for TE n m mode. More specifically, we will
use an elhptical polarization mode converter as an design example.
The
deformation is
8s(z,(j>) = 52/s(z)sin(2Z()))
where 1 = 1.
Consider a T E n input mode which is operated at 60 GHz in waveguide
with a radius of 13.9 m. We wish to convert a stationary T E n mode into a
rotating T E n mode. Suppose the deformation is so small that they have the
same propagation constant. The coupled-mode equations for the polarization
mode converter can be written as
iJ-Aiofe) = ffijJfettLcfe) ,
dz
(6.17)
and.
dAu(z)dz
= Knx3(z)Al0(z)
.
A solution for the elliptical polarization converter can be expressed as
Al0(z) = cos[jQzK11xs(z')dz']
and
,
(6.18)
133
Au(z)
= sintj^jJzVz7]
where we have assumed A lo (0) = 1 and A lx (0) = 0. We obtain a rotating, T E n
mode, if
{j^K^z'Wz'l
is satisfied.
= 1 + 2nn where n = 0, 1, 2, ...
The couphng mechanism for the 60 GHz T E n polarization
converter is shown in Fig. 3.9. All of the modes presented in Fig. 3.9 should
be considered in the numerical simulation. A transition in which tapers a
circular waveguide tapers into an elhptical waveguide is an essential element
of the polarization converter. Taper sections should be included at the input
and output port of the polarization converter to avoid an abrupt shape change
which would excite unwanted spurious modes.
2. Rotating Modes in a Helical-Multifoil Structure
For a low-power testing, we need to generate a rotating TE m n mode from
TE 0n mode to test Vlasov launchers or other WGM conversion systems. If a
linear multifoil structure is employed, the TE 0 n mode can only be converted
into a stationary TE m n mode. A helical-multifoil structure is necessary to
convert a TE 0 n mode to a rotating TE m n mode.
For long distance transmission, we can also convert a rotating
whispering-gallery mode into a TE 0 n mode which have low loss when they are
134
well above cutoff. From the reciprocity principle, the same helical multifoil
mode converter, which is designed to generate a rotating TE m n mode, can also
be used as TE m n -to-TE 0 n mode converter.
Hence, our focus will be on a helically perturbed structure. A helical
multifoil structure can be written as following
a(z,<|>) = a,Q + &Xz,<]>) = a 0 + 8/(z) cosdfy ± A(3z)
(6.19)
= a 0 8/c(z) cos(Z<t>) * 5/s(z) sin(Z<J))
where
8/cCz) = 8/(z)cos(Apz) and 8/s(z) = 8/(z)sin(A(3z) .
The plus and minus signs indicate different sense of rotation.
Careful
selection of the plus or minus sign is necessary to get the appropriate sense of
rotation.
If we consider only two coupled modes, the coupled-mode equations for
a helical structure can be written in the form
dAl0
dz
= -JPi-Aio
+ K
uAix
* K ^
+ K
ilAix
* K^A^
+
# 1 2 ^ + K12cA
,
2o
(6.20)
135
dA-te
dz
= -jfoAix
+ K&^TX
+
dA^
dz
= ~3^2o
+
^ r**4„
12^2*
•K^22sA2x
+
+ JSTx^Ai, + i^i2sA2o
+
K2xlsAu
+
+
^12^20
^K?L
1 2 ^A„
2* .
-^22<rA2x
+
+
+ K
2lAlo
K ^ ^
(6.21)
+
K
2lAlo
,
(6-22)
and
dAfr
dz
= -JfeAte
+
^22sA2x
+
* K^A^
-K22(?A2x
+
**«Ata
^21^10
+ K
21<Alo
,
(6.23)
where we use A l 0 as the A x mode with ordinary polarization and A l x as the
cross-polarized AJL mode and a similar representation for the A 2 mode. The
first subscripts on the coupling coefficients correspond to the related modes,
the third subscript indicates either a sin or cos wall perturbation, and the
superscript indicates the polarization.
From Appendix D, we find that some of coupling coefficients in
Eqs. (6.20M6.23) will vanish. We then have
dA-,
*
dz
= - J P , A I O + tfufcAa,
+
K^A^
,
(6.24)
136
+
***
=
-JkAu
*
*S*A*
*5A* •
dz
dA
22. = -jPjjAa,
dz
+
X 21c A l0 + J C ^
dA *L = - J M ^
dz
+
K j ^
+
K£AU
, and
.
(6 25)
-
(6.26)
(6.27)
3. TE 0 1 -TE 0 7 Varying-Radius Mode Converter
The strongest coupled TE 0 n mode with TE 15 2 mode can be found
from the table of roots of Bessel function. From Table 6.1, we can find that the
strongest coupling is between TE 07 and TE 15 2 mode. TE 07 has been chosen
as the intermedins mode to excite TE 15 2 mode. A 110 GHz rectangular-TE 10 circular-TE 01 mode converter is commercially available however a TE 01 -TE 07
varying radius mode converter had to be designed in order to supply the TE 07
mode for the TE 0 7 -TE 1 5 2 hehcal multifoil mode converter. Several possible
approaches could be used to design the TE 01 -TE 07 mode converter: (1) direct
conversion from the TE 0 1 to the TE 07 mode; (2) sequential conversion from the
TE 01 through each adjacent mode, TE 01 -TE 02 -TE 03 -...TE 07 ; and (3) sequential
conversion through alternate modes, TE 0 1 -TE 0 3 -TE 0 5 -TE 0 7 . The first of these
methods requires that the TE 07 mode be propagating and thus a large uptaper
is required before the direct conversion can take place. Then the coupling is
very weak and many perturbation periods will be necessary in a guide where
137
Table 6.1 Coupling Coefficients and Beat Wavelength
Between the TE 0 n and TE 15 2 Modes
The coupling coefficients between TE 0 n and TE 1 5 2 modes
n
2
3
4
5
6
7
8
9
10
C ls -45.48 -75.09 -121.41 -213.93 -528.02 2328.4 407.9 236.77 179.02
The beat wavelength between TE 0 n and TE 1 5 2 modes
n
2
3
4
5
6
7
8
9
10
AX .0129 .0145 .0176 .0245 .0492 .1787 .0258 .0121 .0068
138
seven or more modes are coupled. In method 2 it is unnecessarily complicated
to optimize six sections to obtain a high mode purity at the output of each.
Method 3 was the approach chosen and the mode conversion was combined
with an uptaper.
The output diameter of the 110 GHz commercial TE 0 1 mode generator
is 0.199". In the design procedure used, first the waveguide was uptapered to
the size in which the TE 0 3 mode can propagate. Then TE 0 1 -TE 0 3 mode
conversion was accomplished while the waveguide diameter was further
increased. TE 03 -TE 0 5 and then TE 0 5 -TE 0 7 mode conversion was accomplished
while gradually increasing the diameter. The final section was an uptaper to
the required output diameter for the helical multifoil. In the design of the
TE 0 5 -TE 0 7 mode converter, we assumed that there was no coupling to the
TE0a mode which was below cutoff.
Figure 6.1 shows the mode content along the TE 01 -TE 03 mode-converter
and uptaper from an input diameter of .199" to output diameter 0.676" for this
section. The computed output mode purity of the TE o 3 mode for this converter
and uptaper is 99.5%.
Using a similar technique, we have designed a
TE 0 3 -TE 0 5 mode converter with the input from TE 0 1 -TE 0 3 mode converter
with an overall TE 0 5 mode purity of 99.4%. Figure 6.2 shows the computed
mode content along this mode converter. Finally, a TE 0 5 -TE 0 7 mode converter
has been designed with input diameter of 0.773" and output diameter of
1.1811". Figure 6.3 shows the computed mode content along the TE 0 5 -TE 0 7
\
TE 01
\
Oi
d-
TE 02
'•
00
d
;
*
.
\
:
\
d-
d
•.
.—._
*.
i.»**
*•
s
k
*•
•
-
/
d-
:
'''••
/
••••..
/
-
/
;
:
i
:
/
o
d-
! \r
0.00
0.02
-
\
\
TEo, « \
• > «\
« •• \
I\.J\:
\
•
•>
*
\
\
......
/ ; * . / .•«"
.* •
0.06
\
0.00
\
>
."- .
..
* "•- «
• .
*
*
*
'
. « • «
\
•
'
.
i
. . .
*.
J.
\."./\'.
.-.1
-
/•..•»'--.,A
.*••• 'J •••:.•*• .«•'
•
•
f
\
\
i
-,•
»
J ;\....r0.04
/
''•
•»
1W
: \;
\
:
:
/*'•
•
d-
'
:
I
:
<
M
© -
\
\
J
:
,•"'.,
/
/
\{
/
:
/
<n
d-
' • , , , ,
\
:
.•
irt
d
in***
/
•••....
rf
°3
-
t-
co
'•
TE
.,-,/
'•
i
*?
,_,
o.io
0.12
0.14
0.16
:(m)
Fig. 6.1-Graph of mode amplitude versus longitudinal distance
for a 110 GHz TE 0 1 -TE 0 3 varying-radius mode converter. The
output mode purity is 99.5% with .32% ohmic loss
o
z(m)
Fig. 6.2. Graph of mode amplitude versus longitudinal distance
for a 110 GHz TE 0 3 -TE 0 5 varying-radius mode converter section.
The overall output mode purity from a TE 0 1 input to section 1
is 99.06%.
o
z(m)
Fig. 6.3. Graph of mode amplitude versus longitudinal distance
for a 110 GHz TE 0 5 -TE 0 7 varying-radius mode converter section.
The overall output mode purity from a T E 0 1 input to section 1
is 99.2%.
142
mode converter. The overall computed output mode purity for TE 0 1 -TE 0 7
uptaper converter is 99.2% with 1.5% of ohmic loss. The total length of the
TE 0 1 -TE 0 7 converter is 30.41 cm.
This TE 0 1 -TE 0 7 varying-radius mode
converter is shorter than Kumric's [5] design and yet has fewer perturbation
periods with excellent mode purity.
Comparing the overall conversion
efficiency and the overall length with Kumric's [5] design and Levin's[29]
design, we have a much better result. The details of the structure for our
optimized TE 0 1 -TE 0 7 mode converter have been listed in Table 6.2. The wall
profile for the uptaper converter is shown in Fig. 6.4. Figure 6.5 shows a
mandrel for the TE 0 1 -TE 0 7 varying-radius mode converter.
4. The Design of the Helical Multifoil Mode Converter
From the selection rule, we find that a 15 period helical-multifoil mode
converter is needed to accomplish the T E 0 7 - T E 1 5 2 mode conversion. The
fabrication method for helical-multifoil mode converter is discussed in
Appendix C. Figure 6.6 shows the couphng diagram for the TE 0 7 -TE 1 5 2 mode
converter. The coupled-mode equations for the rotating TE 1 5 2 mode generator
in a helical structure have been discussed in a previous section of this chapter.
Appendix D discusses which coupling coefficients vanish and the sign of the
couphng coefficients due to the sin and cos components of the perturbation
structure.
In the numerical simulation, all of those propagating modes satisfying
143
Table 6.2 Details of the Structure of the TE 0 1 - TE 07 Mode Converter
f|(z) = a ^ {1 + (-l) psi EQJ [1 - c o s d ^ + e^ sin kj Zj +6^ sin 2kj zi)]}
where kj = 27c/^ and 0 <, kfc < n.
Section 1: TE 0 1 - TE 0 3 mode converter: input radius a 0 = 0.002563 (m).
i
1
2
3
4
5
6
7
8
9
.63
.11
.09
.11
.09
.12
.091
.078
.098
Eli
0.0
0.4
0.3
.18
.25
.47
.15
.38
.08
e
0.0
-.35
-.35
-.35
-.35
0.5
-.04
0.1
.08
.05
.032
.03
.031
.031
.0278 .028
0
0
1
0
1
0
0
1
0
.0104
0.
0.
0.
0.
0.
0.
0.
0.
E
0i
2i
\
PSj
Ai
.0282 .03
where flj represents the length of a flat section immediately after the section.
Section 2. TE 0 3 - TE 05 mode converter: input radius a 0 = 0.008585 (m).
fj(z) = aj.j {1 + (-l) psi EQ! [1 - c o s d ^ + exi sin kj Zj +6^ sin 2kj Zj)]}
where kj = 270^ and 0 <, kjZj < K.
i
1
2
3
4
5
6
7
8
9
10
11
e oi
.061 .053 .061 .054 .062
.053 .061 .053
.061 .053
.063
e^
-.52
-.08
-.14
-.10
.46
-.5
-.12
.42
.25
-.32
-.04
e 2i
-.22
.32
-.12
-.04
.38
.33
-.43
-.42
.08
.06
.25
\
.0122 .013 .0122 .0122 .0122 .0123 .0121 .0123 .0122 .0124 .0119
PSj
0
1
0
1
0
1
0
1
0
1
0
144
Table 6.2 (continued)
Section 3. TE05 - TE 0 7 mode converter: input radius a 0 = 0.009811 (m).
f^(z) = a ^ [1 + (-l) psi e oi [1 - costft^ + exi sin kj Zj +6^ sin 2\ Zj)]}
where \ = 2id\ and 0 < kjZj < K.
i
1
2
3
4
5
6
7
8
9
10
11
e oi
.055
.046
.051 .042
.051 .042
.051 .045
.051 .045
.051
eu
0.
.40
.30
-.30
-.30
.40
.10
0.0
.20
-.20
.25
e 2i
0.0
.40
-.30
.00
-.40
.25
-.25
-.30
.30
.25
-.05
\
.077
.0079 .0079 .0079 .0079 .0082 .0082 .0082 .0082 .0082 .0082
psj
fli
0.0
0
1
0
0.0
0.0
0.0
1
0
0.0
1
.0037 0.0
0
1
0.0
0
0.0
1
0
0.0
0.0
6
9
12
15
zaxis (inches)
(a)
5A3--
2-
i
I
V
1
I
1
3
1
i
1
1
1
I
I
1
I
)
1
6
>
9
)
I
1
I
I
I
I
12
zaxis (inches)
(b)
Fig. 6.4. a) Scaled wall profile b) Exaggerated wall profile
diagram of a varying-radius TE 01 -TE 07 mode converter for use
at 110 GHz.
I
t
i
(
»
15
146
#
o
u
CD
O
5 . ' "_ ,*- » . V < . » " ^
a
o
S
H
!
O
V
O
0)
-tJ
=#<M
W
CO
tub
•r-t
^TE30>1 TE30,l
TEi5,n
TEi5)n
n=1..5
n=1..10
n=l..ll
n=1..4
Fig. 6.6. Coupling diagram of a TE 0 7 -TE 1 5 2 helical-multifoil mode converter at 110
GHz with a 3 cm diameter.
148
the selection rule were included in the calculation except the TE 30 i modes.
A circular waveguide, with a 3 cm diameter, was gradually perturbed to a 15
period helical multifoil waveguide then tapered back down to a smooth circular
waveguide. This avoids the sudden change of the waveguide shape which
would excite unwanted spurious modes. Figure 6.7 shows the mode content
along the TE 0 7 -TE 1 5 2 helical-multifoil mode converter. Table 6.3 shows the
details of the structure of the helical-multifoil converter. Figure 6.8 shows a
picture of the mandrel of the TE 0 7 -TE 1 5 2 helical-multifoil mode converter.
The peak-to-valley perturbation amplitude of the helical-multifoil converter
is .0214". The local machine shop can only achieve a ±.0005" tolerance. A
discussion of how the peak-to-valley amplitude of the helical multifoil
converter was measured is described in Appendix F.
The 110 GHz T E 0 7 - T E 1 5 2 helical-multifoil mode converter with
diameter of 3 cm has been designed and fabricated.
In theory the mode
conversion can be achieved in a relatively short length, 17.7 cm, and with very
high conversion efficiency, 99.5%. For a purely rotating mode, the power
rotation ratio is zero.
The computed power rotation ratio of the output
rotating TE 1 5 2 mode is .0024.
A 110 GHz TE 0 9-TE 2 2 2 helical-multifoil mode converter with 3 cm
diameter has also been designed. The computed mode content along the axis
is shown in Fig. 6.9. The parameters of this single beat- wavelength helicalmultifoil mode converter are shown in Table 6.4.
Z(m)
Fig. 6.7.Graph of mode amplitude versus longitudinal distance
for a 110 GHz TE 07 -TE 15 2 helical multifoil mode converter. The
output mode purity is 99.5% with 0.614% ohmic loss.
Table 6.3 Optimized Waveguide Profile for T E 0 7 - T E 1 5
2
Helical Multifoil Mode Converter
Section 1. Increasing perturbation amplitude (0 <z <L1):
a(z,§) = a 0 + eiZCOs(Am§ - Apxz)
a 0 = 0.015, Lx = 0.059, e x = 0.0046, Apx = -^L,
Am = 15
Section 2. Center section constant perturbation amplitude ( 0 < z < L2 ):
a(z,(J)) = a 0 + e1L1cos(Am<{) - Ap^z-APj!/-^
L 2 = 0.059, Ap2 =
2K
.315
Section 3. Decreasing perturbation amplitude ( 0 < z < L3 ):
a(z,§) = a0+(e1L1-e2z)cos(Am(j)-A(33z-AP2L2-AP1L1)
L3 = 0.059, AP3 = _ | ! L , e 2 = e j f i
.olO
Total length
: 17.7 cm
Computed mode purity
: 99.5%
Computed ohmic loss
: .614%
L<2
151
OJ
-M
U
ir
>
o
0)
o
a
U3
O
as*
O
a
a
B
• I-t
\3\
vi*
K
CO
CO
TE 09
CD
O '
CD
CD-
o-
CO
oCM
o-
o
o.
EsSffifflfeea.
0 00
O.Oi
0 02
0.03
0 04
0 05
0 08
0 07
0 08
Z(m)
Fig. 6.9. Graph of mode amplitude versus longitudinal distance
for a 110 GHz TE 09 -TE 22 2 helical multifoil mode converter. The
output mode purity is 99.58% with 0.595% ohmic loss.
Table 6.4 Optimized Waveguide Profile for TE 0 9 -TE 2 2 > 2
Helical Multifoil Mode Converter
Section 1. Increasing perturbation amplitude (0 <z <L1):
a(z,§) = CIQ + ejZCOs(Am<t> - A(3jz)
a0 = 0.015, Lx = 0.025, e1 = 0.0092, Apx =
2n
, Am = 22
.1065
Section 2. Center section constant perturbation amplitude ( 0 < z < L2 ):
a(z,(j>) = a 0 + e1L1cos(Am§ - A P ^ - A P ^ )
L2 = 0.025, Ap2
27t
.1065
Section 3. Decreasing perturbation amplitude ( 0 < z < L3 ):
a(z,<])) = a 0 +(e 1 L 1 -e 2 z)cos(Am(t)-AP 3 z-AP 2 L 2 -Ap 1 L 1 )
L3 = 0.025, AP3 = - J *
.1065
Total length
: 7.5 cm
Computed mode purity
: 99.58%
Computed ohmic loss
: .595%
e2 =
zx-1
L»2
154
Since these hehcal-multifoil mode converters are compact and have very
high computed conversion efficiency, it may be possible to integrate such a
device directly into a gyrotron. A combination of TE 0 n modes may have better
transmission efficiency and lower mode conversion across a gap structure than
whispering gallery modes do. (See Chapter 7.) This might improve the
efficiency of the radial extraction method. Also a low loss TE 0 n mode might
be a desired output from the gyrotron for the purpose of long distance
transmission.
5. Fabrication and Test Results for the TE 0 1 - TE 0 7 and TE 0 7 - TE 1 5 ) 2
Mode Converters
The simulation results for the TE 0 1 - TE 07 varying-radius mode
converter have a 99.2% TE 0 7 output mode purity. More than 3,000 points have
been used in the simulation procedure. The numerically controlled lathe used
for machining the mandrel can only accept about 1,200 points. Thus the
smaller set of data points as used by the lathe was also used to resimulate the
mode converter. Due to the choice of an inappropriate step size for the lathe
data set, the mode converter can only generate 97% TE 07 mode purity of the
and 1.04% of power will remain in TE 03 mode. With a better choice of the
fabrication step size, the output mode purity of the TEQ7 mode can reach
98.5%.
The input diameter of the TE 0 1 - TE 07 mode converter is only .199" and
155
the output radius of the converter is 1.181". Due to the large difference
between the input and output diameter, there is difficulty in holding the mode
converter rigid during the machining process. This may also have contributed
to some mechanical error.
An open-end waveguide radiation pattern method has been used to
examine the mode purity of TE 0 7 and TE 1 5 2 modes. The far-field radiation
region begins at r = 60 cm from the mode converter aperture. A receiving horn
was placed 100 cm away from the mode converter. Figure 6.10 shows the
measured results and the theoretical far-field pattern of the TE 0 7 mode. In
the measurement procedure we identified that there are T E l n modes, which
should be filtered out by the mode filter, at the input of the TE 0 1 -TE 0 7 mode
converter. Hence, we rotated the mode filter such that the T E l n modes were
minimized, which is not zero, in the plane we took the measurement. From
Fig. 6.10 we can estimate the power distribution in the spurious modes. The
largest spurious mode is TE03 with about 2% of the output power.
A
recalculated theoretical far-field radiation pattern which includes the most
important spurious modes TE 01 , TE 02 , TE 03 , and TE 04 modes, and measured
results is shown in Fig. 6.11. The estimated measured mode purity of the
TE 07 mode is 96%. The detail mode purity of those spurious modes were list
in Table 6.5.
The generated TE 0 7 mode was used to test the TE 07 - TE 1 5 2 helical
multifoil mode converter. Since the output of TE 0 1 - TE 0 7 mode converter is
Position
(degrees)
Fig. 6.10. Comparison of the measured E^ radiation pattern and the theoretical data
ik mode converter. The theoretical mode
for the 110 GHz TE 01 -TE 07 varying-radius
patterns are computed for the pure TE 07 mode.
Position (degrees)
Fig. 6.11. Comparison of measured and theoretical E . component of the radiation
pattern from 110 GHz TE 0 1 -TE 0 7 varying-radius mode converter. The theoretical
mode pattern is computed for the mode combination predicted to be present.
158
Table 6.5 Power distribution at TE 07 and spurious modes
TE 0 1 TE 02 TE 03 TE 04 TE 07 T E n
.01
.003 .022
.004
.96
.0002
not a pure TE 07 mode, we used the estimated power distribution in all
propagating modes as an input to resimulate the output from TE u 7 - TE 1 5 2
mode converter. The spurious modes excited by the varying radius converter
simply pass through the helical multifoil mode converter without significant
mode conversion. Figure 6.12 shows the radiation pattern from the helicalmultifoil mode converter. From Fig. 6.12 the measured E. radiation pattern
of TE 0 7 -TE 1 5 2 helical multifoil converter shows an asymmetric pattern. If we
rotate the helical multifoil mode converter, the main peak changes from -40°
to 40° or vice versa. We also notice that within range -20° < 0 < 20°, the
pattern remains the nearly same no matter how we rotate the waveguide
converter. This implies that the low order TE 0 n modes were not significantly
affected by the helical multifoil mode converter.
From the asymmetric radiation pattern of Fig. 6.12, we conclude that a
small amount of TE 07 mode remained at the output of the helical multifoil
mode converter. Using the theoretical far-field radiation pattern we have
estimated that there is 6% ~ 7% of the TE 07 mode at the output of the helical
multifoil mode converter. The waveguide field components, inside the circular
x>
O
T3
C
Position
(degrees)
Fig. 6.12. Comparison of E^ components of the measured radiation pattern at two
different orientation angles for the 110 GHz T E ^ - T E '15,2
j c o helical-multifoil mode
converter,
160
waveguide for the TE 03 , TE 07 , and TE 1 5 2 modes are shown in Fig. 6.13. Using
this plot, we can find that the fields of TE 1 5 2 are negligible inside a .4 cm of
radius. However, E^ for the TE 07 mode is maximum near the center of the
waveguide and Ex for the TE 0 3 mode is also large fairly close to the center.
Hence, an 8 inch long tapered wood cylinder with a diameter of 0.35" has been
inserted into a straight section of waveguide to absorb the TEQ7 mode following
the helical multifoil mode converter.
A significant improvement in the
symmetry of the TE 15 2 radiation pattern can be seen in Fig. 6.14.
The azimuthal radiation pattern, Fig. 6.15, also has been taken to
estimate the rotation ratio. From Fig. 6.15, we can estimate that about 85%
of the power is in the right-hand rotating mode and 15% of power in the lefthand rotating mode.
0.005
0.004
"
I
'
0.003
0.002
0.001
/
*
i
: :
»
..•••>
i
\ • •-. /
v
'
/
/
\
"- »
-0.004
\
\
/
/
1
0
- TE03
—
--TE07
- E # of TE15,2
—Ep of TE15,2
1
•
:
\ /
-0.005
!
1
(
1
I
1
.-••'
\
\
i
'
- \
\
\
\ — '
. '
..•••"
\
\
px
-0.002
r-
\
/ '•-. \
\ \ .-•' /
/ 'v. . \ ... \ X J
1
I i
^
/ \
' / "VI
-i
\
i
I
-0.001
\
/ i
t—H-
i
0
-0.003
C\..
i
v
\
ks
1
0.005
0.01
0.015
radius (m)
Fig. 6.13. The waveguide fields components for the TE03, TE07, and TE15 2
modes at 110 GHz inside a circular waveguide with 3 cm diameter. All
mode are normallized to carry the same power.
Position (degrees)
Fig. 6.14. Comparison of Ex components of theoretical pure T E 1 5 2 mode and the
measured radiation pattern for the 110 GHz TE 0 7 -TE 1 5 2 helical-multifoO mode
converter with an 22 cm wood inserted along the circular waveguide axis. The
waveguide diameter is 3 cm and the wood cylinder diameter is .8 cm.
0
-2
-4
-6
-8
10
12
14
16
18
20
22
24
26
28
30
flU
ft A
A1 A A1 A 'A M
/ ] U l\ rfv l l \ '/l 1/ \ / i : f \ f\
i"t J J"( 1 "'1T i 11 J"\
liiilfj'liM'fl
1
1 l 1 1:1 \ 1 1
A
A
in
A
l\A\d\.l\. i.l.i.UivlL
.J.-i-J-4-4-4-4—Vi"i-J--j
111 31 i'i I f l J i ,4
j 1 ...I ...ll... J....I....' 1
"• h
|••)••
111 LI
A
y\ J\ 'A i
1
...i.
\ ij 1 A j i
ii
nfirf l
Id 1
\ 'A 'l\
if
l\.2.\.J..\..i\
i 13 : i! I1
•••VII
"Vl—iJ"-""17 — T "'
if V ! S 1
theoretical
: — ——
i
150
i
1
-100
1
1
i
-50
0
50
Position (degrees)
1
measured
i
100
i
i
150
Fig. 6.15. Comparison of the measured azimuthal radiation pattern and the
theoretical data for the 110 GHz TE 0 7 -TE 1 5 2 helical-multifoil mode converter with
a 22 cm wood cylinder inserted along the circular waveguide axis.
co
164
Chapter 7
Whispering Gallery Mode Converter Design
The first generation Vlasov launchers for rotating TE m 2 modes often
give a double humped main lobe. Thus it may be desirable to have available
designs for TE m 2 -TE m j waveguide mode converters to transform the TE m 2
mode to a TE m 1 mode which is better suited for current Vlasov Launcher
designs.
A new generation of internal reflector Vlasov Launchers may overcome
this problem of double humps in main lobe for TE m 2 mode. However, these
varying-radius type mode converters for whispering gallery mode are still
worthwhile to mention as they are part of designs that have been performed
within the author's Ph.D. program.
Furthermore, there are some very
interesting characteristics that have been found in the step tuning and the
mode combination of the varying-radius type mode converter. The following
sections will discuss several types of varying-radius mode converters which are
designed for whispering gallery modes.
1. The TE 1 5 4 J to TE 1 5 f l Mode Converter
According to the selection rule developed in Chapter 3, a TE m 2 -TE m ±
mode converter can be designed by using a variation of the waveguide radius,
165
A/ = 0. The coupling coefficients were calculated in Chapter 3 and are listed
in Appendix A for reference. First generation gyrotrons typically have an
output mode with an azimuthal index equal to zero (i.e. TE 0n ). In the design
of varying-radius TE 0 n type mode converters, we do not need to consider the
couphng between the TE 0 n and TM 0n modes because these two mode families
are not coupled by purely varying-radius wall perturbations [20]. The major
spurious mode for a TE 0n -TE 0 nml varying-radius type mode converter is the
TEQ n + 1 mode. This is due to the very small difference of beat wavelength
between T E ^ - T E g ^ and TE 0 n -TE 0 n + 1 .
Hence, for any profile that can
successfully suppress the TE 0 n + 1 mode in the design procedure, the conversion
efficiency can be expected to be good. This is the reason that a two or even
single perturbation period device can be built for the TE 0n -TE 0 n . x type mode
converter.
The major output modes of the new generation gyrotrons are
TE m n modes where m * 0. In this case, the TE m n and TM
modes will be
coupled together in varying-radius type mode converters. Hence, we divide the
couphng into three categories: (a) TE mn -TE mq , (b) TE m n - T M ^ , and (c)
TM^-TMjnq, where m = 15 for the design case to be discussed. Our design is
for an operating frequency of 140 GHz and an output port diameter of 1" as for
the MIT 140 GHz gyrotron. The calculated coupling coefficients are listed in
Table 7.1. The coupling coefficient between the TE 15 2 and TE 1 5 3 modes is
larger than that between the TE 15 2 and TE 15 x modes, and the beat
wavelengths for these two mode pairs are nearly the same. Therefore, the
Table 7.1
Couphng Coefficient and Beat Wavelength of the 140 GHz
Varying-radius Mode Converter
(TE 1 5 2 -TE 1 5 1 )
(TE15>2-TE15>3)
(TE 15)1 -TE 153 )
Ky
4.05023
5.95163
2.47325
AA.
.02511
.02312
.01203
Ky and Ak are the coupling coefficient and beat wavelength,
respectively.
167
mode competition between the TE 1 5 3 and TE 15 1 mode is expected to be quite
strong.
The mode content along the axial direction of a purely sinusoidal
periodic-perturbation mode converter is shown in Fig. 7.1. Here we can find,
as expected, that the TE 1 5 3 mode is strongly coupled to the TE 15 2 . Figures
7.2 and 7.3 show that as the number of perturbation periods increases, the
mode purity for the TE 1 5 j output increases. However, we lose energy to the
waveguide wall because of the high losses associated with whispering gallery
modes. Fabrication also becomes more difficult and expensive for transducers
with many periods.
Next we will consider a perturbation structure of the same form as that
discussed in Section 2.2, a profile with a harmonic term inside the cosine
perturbation, to account for the variation of the beat wave number along the
converter. A highly efficient perturbation structure of this type has been
developed for the TE 0 n - TE 0 n_1 mode converter by Buckley [3]. We use a
similar perturbation structure in the 140 GHz gyrotron mode converter design,
but we need to consider not only the coupling between modes in the TE 1 5 n
family but also need to include the TM 15 n mode family. The coupled-mode
equations are much more complicated here than for the case that has been
discussed by Buckley. Surprisingly, we found that a similar perturbation
structure is also valid in this high-azimuthal-index mode converter. However,
168
J
16,2
*
*
TE 15,3
."t...,
...
.•'»» ..."
0.00
..
0.02
,...
• " . . » *
.
> .v;n.
•.
,-. . , . . . ,
" 3
1 '• •• ' I i , . , '
•••
0.04
0.06
0 08
010
0 12
0 14
Z (m)
Fig. 7.1. Mode amplitude of the 8 most strongly coupled modes
along a 5-period optimized purely periodic TE 1 5 2 -TE 15 j mode
converter with 78.8% output mode purity and 1.36% loss.
Z (m)
Fig. 7.2. Mode amplitude of the 8 most strongly coupled modes
along a 10-period optimized purely periodic TE 1 5 2 - TE 1 5 j mode
converter with 91.2% output mode purity and 2.7% loss.
170
Z (m)
Fig. 7.3. Mode amplitude of the 8 most strongly coupled modes
along a 15-period optimized purely periodic TE 1 5 2- TE 1 5 j mode
converter with 92.3% output mode purity and 4.13% loss.
171
we need more than three perturbation periods to obtain good conversion
efficiency.
A three-period design has been tried by the author but produced an
optimized conversion efficiency of only about 93%. However, an optimized
design for a five perturbation-period structure with a 98.6% mode purity has
been achieved. We use the radial variation
f(z)=a0{l + e[l + e ^ o s a i ^ z M l - cos(H(z))]}
(7.1)
where e is the relative perturbation amplitude and e1cos(H1(z)) is a small term
used to suppress coupling to the TE 15 n + 1 mode.
The numerical analysis considers all propagating coupled modes in the
calculation, six TE 1 5 n modes and five TM 15 m modes. Fig. 7.4 shows the mode
content along the mode converter axis, using the perturbation of Eq. (7.1), with
only the first eight modes shown on the figure. The details of the structure
parameters for Eq. (7.1) are listed in Table 7.2. This TE 1 5 2 -TE 1 5 x varying
radius mode converter design has been numerically confirmed by Dr.
M. Thumm at Stuttgart, Germany. The five-period mode converter, which is
5.6" long, results in an output mode purity of 98.6% for the TE 1 5 x mode. This
is greater than the mode purity of 92.3% for a 15 period purely periodic mode
converter.
17
i
0.00
i
1
1
i
1
1
0.02
0.04
0.06
0.08
0.10
0.12
1—
0.14
Z (m)
Fig. 7.4. Mode content of a quasi-periodic T E 1 5 2 - T E 1 5 1 mode
converter with 1.38% loss and 98.6% output mode purity.
173
Table 7.2
Optimized Result for the TE 1 5 2 -TE 15 x Mode Converter
i
*
^=27^
c
ii
c
2i
c
3i
1
.00053
.0282
0.1
0.15
-.24
2
.00040
.0284
0.99
0.03
-.27
3
.00033
.0290
-.04
0.08
-.32
4
.00051
.0286
-.46
0.19
-.29
5
.00040
.0284
-.56
0.17
-.28
The output result
total length
14.26 (cm)
total loss
1.38%
mode purity
98.6%
174
2. Mode Converter Performance for Different Modes w i t h t h e Same
Gyrotron Cavity
It has been demonstrated that it is possible to achieve a stepwise
frequency tunability of a gyrotron by varying its cavity magnetic field and thus
stepping the value of m from one TE m 2 1 mode to the next with the gyrotron
operating at high power [31] (Fig. 7.5). The mode converter design described
in Sec. 1 can be used in the stepwise frequency tunable gyrotron with the
conversion efficiency remaining at a reasonably high level as shown in Fig. 7.6.
This kind of tunability might be very useful in plasma heating applications.
A change in the output modes, however, would affect the Vlasov launcher
output.
The main reason that the mode converter can be used for different
mode-frequency combinations is that the beat wavelength and the coupling
coefficient between the TE m 2 and T E m l modes change only a small amount as
the azimuthal index m and the operating frequency are step changed. This is
shown in Table 7.3. The theoretical resonance frequency can be calculated
from the characteristics of the gyrotron cavity.
We adopt the cavity
parameters of the newly developed 140 GHz MIT gyrotron from Reference [30].
The operating frequency of the gyrotron here is commonly the cutoff frequency
of the cavity waveguide plus about 0.5 GHz. The corresponding conversion
efficiencies for the mode transducer we have designed versus the operating
frequency are tabulated in Table 7.4.
o
6
o
>4
l O ^
modes
o
•
o ID
w
cr q
d
V
/
/
7 //
•
5
Q
w q ^
&-• o
o
m= 4
X
/
<
I—•
.
« «
Q
W
j
02
10.0
15.0
i __ i _
i
20.0
MAGNETIC
•
«
* i
i__i
25.0
FIELD
i_
• i
1
30.0
1
1
I
35.0
(kG)
Fig. 7.5. Predicted frequency versus applied magnetic field for
observed T E ^ modes. ( From S.H. Gold et al [28])
176
130.0
135.0
140.0
145.0
150.0
155.U
Frequency
Fig. 7.6. Step tuning properties of the TE 1 5 2~TE15 t mode
converter.
160 0
no"
177
Table 7.3
CoupUng Coefficient and Beat Wavelength
for the Step Tunning Gyrotron
(TE 1 4 2 -TE 1 4 1 )
operating
freq.
132.2 GHz
(TE 162 -TE 16>1 )
(TE 1 7 2 -TE 1 7 1 )
147.9 GHz
156.2 GHz
Kj!
3.8713
4.2256
4.398
AX
.025499
.02477
.02459
K ^ is the couphng coefficientbetween (TE m 2 -TE m 1), and AX. is the beat
wavelength between (TE m 2 -TE m x).
178
Table 7.4
Resonant Frequency and Mode Purity
mTTi
1
where
fPT?
^13,2,1
1
fTP
L
- ^14,2,1
^15,2,1
f,.
125.3
132.4
139.5
fop
125.8
132.9
q(fop) 79.7%
foptm 1 2 3 ' 9
% P tm93.8%
Tf?
X
^16,2,1
r
PTP
x
^17,2,1
140.0
146.6
147.1
153.6
154.1
94.3%
98.6%
91.6%
80.6%
132.0
140.0
147.9
156.2
97.9%
98.6%
94.0%
89.9%
f^.
is the cavity resonant frequency of the gyrotron
f
is the operating frequency of the gyrotron
q
is the conversion efficiency for the operating frequency
foptm is the frequency corresponding to the maximum
conversion efficiency
Tj optm is the maximum conversion efficiency
179
3. An U p t a p e r e d Mode Converter from t h e TE 15>2 to TE 15>1 Mode
One 140 GHz gyrotron design has a TE 1 5 2 mode output in a diameter
of 3.5" guide. At such a diameter, a TE 1 5 2 -TE 15 1 mode converter is not really
practical because of the long beat wavelength and the many TE 1 5 n and TM 15 n
modes that can propagate.
It would be very difficult to obtain a high
conversion efficiency in any reasonable length. However, the cavity diameters
of such tubes are much smaller, and it may be desirable to use a combination
mode converter-uptaper built directly into these tubes.
The concept of a combination of a downtaper with a varying-radius mode
converter has been shown [32] to be an effective way to shorten the total
length of a TE 0 n mode converter and downtaper combination. The uptaper
design for the TE 1 5 2 mode uses a very similar philosophy to that of the
downtaper design. This type of mode transducer also belongs to the varyingradius type mode converter family, and the coupling coefficients can be
calculated in the same way as we discussed in Chapter 3. We propose a taper
structure that is similar to that of Eq. (2.29). The varying axis profile has
been modified to avoid a slope discontinuity between the uptaper and the
adjacent flat section. The uptaper-mode-converter profile can be described by,
$(z) = a^l + e j l - cos(Hi(kpz))]}
for 0 < ILZ £ ii i=1...5
(7.2)
180
A straight section of waveguide is used to adjust the phase mismatch between
the TE 1 5 2 and TE 1 5 x modes at the end of each taper section of Eq. (7.2).
We include in the numerical calculation all TE 1 5 n and TM 15 n modes as
soon as they become propagating. The input and output ports of the uptapered
mode transducer are 1.6 cm and 2.8 cm in diameter, respectively. The number
of excited propagating modes on each side of the taper-converter are 3 and 13,
but only the first 12 modes are included in the calculation.
In a preliminary design, we used a purely cosinusoidal perturbation in
each section. Figure 7.7 shows the mode content. The conversion efficiency
was approximately 89%. Next we added two harmonic terms into H^z). A
preliminary result for the conversion efficiency was 95.8%. Figure 7.8 shows
the mode amplitude of several modes along the converter. Table 7.5 lists the
perturbation profile for each section of the structure in Eq. (7.2). We believe
a conversion efficiency of approximately 98% can be achieved with some small
changes in each section.
4. An U p t a p e r e d Mode Converter to P r o d u c e Specific O u t p u t Mode
Combinations
In the proposed design of a new gyrotron, for which the output power is
in the megawatt range, the operation efficiency is about 35%. In addition to
handling the generated electromagnetic power, it is also necessary to deal with
the residual energy of the spent electron beam. The spent electrons are
y~
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•
'
1
0.05
'.
.-
'T'-"'—"
0.06
-,.v*r.
..
1
0.07
Z(m)
Fig. 7.7. Mode content of a combined TE 1 5 2 -TE 1 5 x mode
converter and uptaper design with a purely cosinusoidal
perturbation.
0.08
o
—I
0 00
0.01
0 02
I—
0.03
0 04
0 05
0 06
0 07
0 08
Z (m)
Fig. 7.8. Mode contents of a combined TE 15 2 -TE 1 5 1 mode
converter and uptaper design with two harmonic terms added in
Eq. (7.2). The conversion efficiency has been improved up to
95.8%.
Table 7.5
Optimized Numerical Results for the Uptapered Mode Converter
*i
Xi=2ic/ki
c
.05
.0066
.142
-.038 .0045
.05
.0111
-.10
-.07
.0054
.0736
.0174
.00
-.02
.0089
.0678
.0252
-.102
.124
.0095
.0551
.0288
-.117
.178
.00
The output result
total length
: 7.25 (cm)
total loss
: 1.17%
mode purity
: 96.9%
ii
c
2i
flat
184
normally collected in a large-radius section of the gyrotron "waveguide" called
the collector. For a megawatt tube about 2 megawatts of power must be
dissipated in the collector. Thus it is desirable to make the collector radius as
large as possible to decrease the spent electron power density on the collector
waveguide wall. The radius of the collector is limited by the mode conversion
that takes place in the uptaper from the gyrotron cavity to the collector and
the downtaper from the collector to the output window.
One technique that is being considered by Varian Associates to eliminate
this problem is to separate the electron beam from the microwave power by a
method called the radial extraction method.
Here the electron beam is
diverted through an azimuthal gap as shown in Fig. 7.9. In this case it is
important to produce a mode combination at the gap that maximizes the power
transmission across the gap and does not produce serious mode conversion in
this transmission. The optimum mode mixture, which not only has minimum
energy radiated into the gap but also will pass across the gap without too
much mode conversion, has been calculated by J. Nielson [33].
For specific mode combinations, the side lobes of the radiation pattern
from an open-end waveguide can be suppressed to a very low power level [34].
It has been found that there is a loose connection between this phenomenon
and the mode mixtures which gives maximum transmission across an
azimuthal gap, in the waveguide. Thus we can utilize this concept to minimize
the power leakage into a gap in the waveguide. The first order synthesis is to
collector
collector
Fig. 7.9. The proposed gap structure. With proper power
distribution and phase adjustment at the input of the gap, the
power radiated into the gap can be minimized.
186
minimize the surface current with the proper mode amplitude and relative
phase combination. The actual mode matching calculation, which evaluates
the power radiated into the gap and the amplitude and phase of the modes
transported across the discontinuity, Fig. 7.9, has been developed by Varian
Associates [35]. We will use the results computed by Varian Associates to
design an uptapered mode converter which produces the required mode
amplitude and relative phase combination.
There are several choices which can minimize the radiation into a gap
in a circular waveguide. After the gap, we need to design another transducer
to convert the mode mixture to a pure TE 1 5 1 or TE 1 5 2 mode. One TE 1 5 n
mode combination which produces excellent transmission characteristics across
an azimuthal gap uses a 10% T E 1 5 1 7180°, 80% T E 1 5 2 /0°, and 10% T E 1 5 3
/180° combination.
Figures 7.10 and 7.11 show respectively the far field
radiation pattern for the pure TE 1 5 2 mode and for this mode combination with
required phase adjustment.
In the numerical calculation procedure, we could consider all of the
modes above the cutoff frequency, but now we limit ourselves to the first
fourteen modes. Six modes interact at the input side of the transducer, and
more than thirty modes above the cutoff frequency at the output port enter
into the coupled-mode equations. Figure 7.12 represents an uptaper with an
input radius of 1 cm, an output radius of 2.5 cm, and a total length of 7.9
inches. Our preliminary design obtains the mode power combination of 9.2%
187
o
d
o
d
co
_
as
•©•
t
,.v?.'tV/WW»»»ww*.i<w.....«.
'WetntH
**•**»*
o
d
••
CO
q£
O ttf}
Q
o
d
co
—Wll
l
o
d
co
I
*f£{tit{WMi4
*Wrf*««W«wwia»M>3w»4B.i*»«j»o.
o
d
00
001-
,
002-
,
00G-
0 ' 0 i - 0 09-
-90.0
90.0
Fig. 7.11. The far-field radiation pattern for a mixture of 10%
TE15>1> 80% T E 1 5 2 and 10% T E 1 5 3 modes at the output with
proper phase combination.
oo
GO
o-
00
d'
n6,2
o
CD
W d
Q
E-i in
30o
CO
d
©
o
o
0 00
0.02
0 04
0.08
0 08
0 10
012
014
0 18
018
0 20
Z (m)
Fig. 7.12. Mode amplitude of the most strongly coupled modes
along a combination uptaper and mode converter giving a
mixture of TE 1 5 j , TE 1 5 2 and T E 1 5 3 modes at the output from
a TE 1 5 ' 2 mode at the input.
190
TE 1 5 1 , 77.6% TE15>2, 9.2% TE15>3, and 1.5% TM 1 5 2 at an operating frequency
of 140 GHz.
Even though our the prehminary design has achieved
approximately the desired power distribution, the phase of each mode needs
to be adjusted such that the power radiated into the gap is minimized. We
found that a very long straight section waveguide is needed to obtain the
desired phase relation among these three modes, TE 15 x , TE 1 5 2 , and TE 1 5 3 .
Hence it is impractical for us to do further research on the three-modecombination uptaper design using the method we discussed here. The phase
adjustment for a two-mode combination uptaper can be achieved within a
reasonable length, even though the power radiated into the gap is somewhat
larger for the two-mode combination.
Due to programmatic changes, the current proposed MW gyrotron is
being designed for a frequency of 110 GHz and a TE 2 2 2 output mode [35], The
newly proposed uptapered mode converter transfers 40% of the TE 22 2 energy
to the TE 2 2 4 7150° mode, while 60% of the energy still remains in the TE 2 2 2
mode. Figure 7.13 shows the most strongly coupled diodes along such a
combination
of uptaper
and
mode
converter
in
which
we
have
(computationally) converted the input TE 22 2 mode into the TE 2 2 2/TE22 4 m ° d e
mixture. We also have a confirmation of our program from Varian Associates'
mode combination results.
Following the uptapered mode converter is an azimuthal gap structure
which will allow the electron beam to escape from the waveguide without
d'
22,4
0.4
Fig. 7.13. Mode amplitude of the most strongly coupled modes
along a combination uptaper and mode converter giving a
mixture of TE22>2 a n c l TE22,4 m<>des at the output from a TE 22 2
mode at the input.
192
losing too much microwave energy and with small mode conversion across the
gap. In addition, a varying-radius mode converter will be needed in order to
convert the combined modes into a pure TE 22 ! or TE 2 2 2 mode. Then a
Vlasov launcher would be used to transform the TE 22 1 or TE 22 2 mode into a
Gaussian like wave.
193
Chapter 8
Conclusions
An overmoded waveguide transmission system can handle high-power
microwaves, generally above several megawatts, without arcing and at the
same time reduce the ohmic loss drastically below that of a dominant mode
waveguide.
However, the unintentional imperfections in the circular
waveguide will induce mode coupling.
Thus an overmoded waveguide
transmission system should be designed very carefully. Mirrors, Vlasov type
couplers, mitre bends, directional couplers, and other such devices must all
be designed to minimize losses and mode conversion in a high-power
overmoded transmission system. In short, the system between the source
and the plasma or other application is not trivial for the high-power
microwave or particularly millimeter wave transmission hne, and experience
in the millimeter wavelength range is very recent. Techniques here are very
different from single mode transmission systems.
In this dissertation, we have explored several of the key components
for smooth cyhndrical waveguide transmission and mode conversion systems,
for example, tapers, varying-radius and serpentine mode converters,
waveguide bends, and helical multifoil mode converters. The various mode
converters can be used as companion components to high-power microwave
194
generators such that the generator's can be converted to a desired
transmission or radiation mode.
For some high-power BWOs, a TM 01 mode has been generated with
megawatts of output power. It has been demonstrated in Chapter 2 that a
single-period TM^-TE!! serpentine mode converter can achieve very high
conversion efficiency. This departure and improvement over the traditional
philosophy of using large number of periods to suppress the spurious modes.
We have shown that, at least in some cases, it is impractical and also
unnecessary to use a large number of perturbation periods to achieve high
mode purity.
Numerically simulated results for a varying-radius T E n uptaper and
a serpentine-type TM^-TEj^ mode converter have been verified by
experiment at 8.6 GHz in 2.36 cm radius waveguide.
The measured
radiation patterns have very good agreement with the theoretical
calculations, in E-plane.
The discrepancy, in H-plane, between the
theoretical results and the experimental radiation pattern was found to be
due to inaccuracy in the theoretical radiation pattern. This was shown by
also converting from the TE 10 rectangular waveguide mode directly to the
T E n circular waveguide mode. The T E n mode obtained in this independent
manner also has the same measured radiation pattern, in H-plane, as did the
TM 0 1 -TE n mode converter. This confirmed that the aperture field models
195
for the waveguide open-end radiation patterns are over simplified for modes
not far above the cutoff frequency.
The output frequency of gyrotrons can be step tuned by varying the
magnetic field. The TE 1 5 2 -TE 15 x varying-radius mode converter can
accommodate this step tuning to a certain extent. This property has been
investigated in Chapter 7. This could be an important consideration for
varying radius mode converters. The way to improve the bandwidth of a
varying-radius mode converter in response to step tuning is by using fewer
perturbation periods when this is possible.
A serious problem in gyrotron design arises from the limited collector
surface available to dissipate the heat generated by the spent electron beam.
Thus it is an important issue to deal with the spent electron beam in highpower gyrotrons. New collector configurations have been discussed in several
papers [36,37]. Two of the most feasible methods are 1) to integrate a Vlasov
launcher directly into the gyrotron, and 2) to use the radial extraction
method to direct the spent electron beam through an azimuthal gap as shown
in Fig. 7.9.
Chapter 7 discusses a possible mode combination which
minimizes the loss of microwave power through the azimuthal gap of 2)
without significant mode conversion or reflection. This mode combination
can be achieved with a varying-radius mode converter of reasonable length.
196
Some experiments have been carried out on this topic at Varian Associates
with successful results.
We did not fabricate any piece of the varying-radius type mode
converter for whispering gallery modes, but two of the design programs have
been cross checked with Varian Associates and Dr. M. Thumm at Stuttgart,
Germany. We find good agreement with these other two groups. In the
Varian group, varying-radius type whispering-gallery mode converters have
been fabricated and tested and experimental results agree reasonably well
with their numerical simulation.
Our computer simulation has shown that a high efficiency-helical
multifoil mode converter can be achieved with a single-perturbation
wavelength. We have designed such converters for both the rotating TE 15 2
and TE 2 2 2 modes for use in low-power testing of Vlasov launcher systems.
A TE 07 -TE 15 2 helical-multifoil converter has been fabricated and tested.
Open-end waveguide radiation pattern method has been used to examine the
conversion efficiency. From far-field radiation pattern, we estimated that
only 93 ~ 94% power in TE 07 mode has been converted to TE 1 5 2 mode. An
azimuthal radiation pattern also has been taken to examine the rotating
power ratio. We found 85% of power in the right-hand rotating mode and
15% of power in the left-hand rotating mode. The discrepancy between the
computed efficiency and the measured results is probably due to the uneven
197
shape of each period at the cross section of the helical-multifoil mode
converter.
We have explored most of components for a smooth wall cyhndrical
waveguide. Some of those components have been fabricated and tested with
good agreement between theoretical calculation and experimental results.
From these components, we found that if the input mode and desired output
mode are nearly degenerate, we can design a mode converter within a
relative short length and even single-period.
The analytical solutions of a three-mode coupled equations for the
waveguide bend have been solved and the design criteria also has been
specified. For a moderately overmoded waveguide, a compact waveguide
bend is possible but there is some difficulty in designing waveguide bends
with satisfactory transmission efficiency within reasonable length for
whispering-gallery modes.
198
Appendix A
Coupling Coefficients a n d Attenuation Constants
We will use the following notation for both the serpentine and varyingradius mode converters.
P;
: propagation constant for mode i
%i : root of Bessel function or derivative of Bessel function for mode i
a(z) : waveguide radius as a function of position
Ky
a0
: coupling coefficient between mode i and mode j
: average radius of the mode transducer at a fixed z(for example, a0 is
a constant in z for / > 0)
P0
: the free space wave number
R(z) : the local curvature of the axis of the serpentine waveguide
k
i
• *j = Xita0
Dmax(z): The maximum diameter of cross section at position z
Dmin(z)\ The minimum diameter of cross section at position z
1. Coupling Coefficients for Serpentine Mode Converter
Assume the perturbation is small so that the radius curvature of the
waveguide axis is much larger than the radius of the waveguide. The coupling
coefficients between various coupled modes are given below:
199
™0n - TE lq
KU-4U
2
hi
Pi
< *j
< Pi (kf
k?)lx]
- 1]V2
aoR(z)
(A.1)
™ l n " TE2q
Kij = JU
N P;
2
2
2
2
< Pi (k - k )[X - 4f
a z)
(A.2)
^
TMon - TM lq
_ .{fcpjfe? + &*]
+
2
2$$(k
p2[*2 + */] - 2 X l x / i y
-
2 2
(A.3)
k ) aoR(z)
TE 0n - TE lq
_ jKPiPj +
+ kf - k2k2) - kfkjikf
PQX^4
* < / "
2
ki$$(k
+ *?)]
(A.4)
2 2
- k ) sjx] - l fife)
TE l n - TE2q
_ jaoKPiP,- + tf0)(x2x] ~ 2x] ~ 2x?) - ^^yXiX/xf - 4 + $ ]
*0 =
2
2(X, - xfVp^Vx? ~ 1 /x* " 4 R(z)
(A.5)
2. Coupling Coefficients for Varying-Radius Mode Converters
TE
mn ~ TEmq
K _
x? - ™ 2
a'(z)
^
2
2
^
(X,- - X ^ o
X* - m2 <
p,-
\
2
2
m
X,2
2
Xi -m
*J
\ Pi
(A
TEmn
m „ - TM„
mqn
-m$Qa'(z)
(A
*«
V^PiPj yx? " m2
a
o
TM m n - TM mq
K
ij =
a'(z)
5Ci ^
(X- - xf)a0
^Pi
+x2
%J
Pi
(A
N PJFJ
3. Coupling Coefficients for Helical Multifoil Mode Converters
TE™,„
mn — T.E„„
pq
K
u
_ [PiXJXx? - mp) • p yX 2 (x 2 - mp)](D max - D m i n )
4y/2a0^~sjx2
- m2 ^
~ P2 (xf -
2\
x,)
201
TE
- TM
Ku =
4V^a0/pJp~/xf ~ m2
TM™ - TMpq
*0=
(PiX| + WfljwL* - < n >
4v^a0(xf ~ x])Jm
TM™ - TEpq
K
Ay -
PoP^max " DLn>
±fcaj^\x2
- p2
4. Attenuation Constant for the Circular Waveguide
The power flow for a single TE mode in a circular waveguide can be
written in terms of the scalar function of Eqs. (2.1) and (2.2)
P = 1 Z0 p0 P; ^ 2 ffy2da
.
Similarly, the power carried by the TM modes can be expressed as
P ^ Y o P o P ^ - 2 J J ¥ f da .
The surface current density on the wall is equal to the tangential
202
magnetic field. Thus, we obtain the losses for TE modes by integrating the
total tangential magnetic field around the guide boundary:
PL - | Rs j C Pf I* x W l 2 + *,VW* •
Similarly, the losses for TM modes can be written as
pL = jRa
Po y? f i W i 2 ^ •
The attenuation constant for TE modes is
_ Raf [p 2 {ftxwl2 + kf y2 ] dl
a
2Z0%$ik2
Rs Po [( Xi_ +
Z 0 ^ a Poa
m2 j
%2 _ m 2
jfyfda
(for
^^^
modeg)
_
(A ^ g)
The analogous expression for TM modes is
^ fls p2 Y2 j | v, ^ l 2 dZ
2 F 0 p0 pt- *? JJ \j/f da
Rs
Po
Z 0 Pj a
(for TM mn modes) .
(A. 10)
Appendix B
Integration Formulas for Bessel Functions
From mathematical tables [38] we find:
\x Jm(ax)Jm($x)dx =
x[aJm($x)j' (ox) - pVm(ca)J^(px)]
(P2 - a 2 )
and
(xJm(ax)dx = ^ [ J ^ ( c a ) ] 2
+
*l(i -
j£-)J2m(ax)
To calculate the coupling coefficients, we need
Jj J m (coc)j'
(px) x2dx = —-J \Jm(ax)Jm($x) x dx
d$
L^ax22j'm(ov)j'm(§x)
p^ - a
oa
^-TT[(a2
(p^ - err
+
- Px2Jrm(oa)^(px)]
P 2 ^ m ( « ^ < ( ( ^ ) ~ 2apxJm((k)ji((xx)] , (B
and
/^(ox^CPx)*2^ =
^-Jjm(ax)Jm(^x)xdx
1
Pz - a 2
-^
(p z - a 2 ) 2
[ocx2J^(cxx)ji(Px) - px2J^(c<x)J^(px)]
{(a2
WJfixW'jax)
j&
+
P2W^(ccx)J^(px) - ocJ^(Px)Jm(ccx)
+ apx [Jm(ax)J^(px) + J m (pW^(ax)]}
2 ?
[aJm(px)J^(ocx) - pJ m (axy^(Px)] .
(B.2)
(p* - a )
To obtain an expression for the normalization constant, the following integral
must be evaluated:
Tap J^(ca)J^(px) x dx + m2(jm(ax)Jm($x)
J
J
1 dx
x
= 2m |Jm(otx)Jm(Px) _ dx - a mJ \Jm($x)Jm+i(ax) dx
j
x
- pmJJm(ocx)Jm+1(px) dx + apjx Jm+1(ocx)Jm+1(px) dx
- aPjJTO+1(ax)Jm+1(px) x dx +TO\d[Jm(wc)Jm($x)]
= mJm(ca)Jm(px) + ap
a
+
2
-xWmdaxW'm*!®*)
a 2 - P2
-—±
- p2
(B.3)
205
Appendix C
P e r t u r b e d Field Components in a Deformed Waveguide
Consider a perfectly conducting waveguide whose inner surface is
defined by p = a + e 8(<f>,z), where $(§,z) is a continuous function and e is a
small dimensionless parameter, as shown in Fig. C.l. The inward unit vector
normal to the guide surface is given by
n = n0 + £n± = - a p + e t - — - ^ + (_-)a 2 ]
y
ad§ y
dz
up to the first order in e, where &p, &±, and az are unit vectors in the direction
of increasing p, <J>, and z, respectively.
The electric field in the perturbed waveguide may be taken as
E = 220(p,(|>,z) + e l ^ p , ^ ) >
where all fields are assumed to satisfy Maxwell's equations and to vary
harmonically with time. EQ is chosen to satisfy the appropriate boundary
conditions for a conducting circular waveguide of radius a. The perturbation
field E1 is chosen such that in the distorted guide the total field satisfies the
condition,
n x E = 0 at p = a + e6X())^) .
Expand the electric field, in the perturbed waveguide, around the p direction
p=a+£Pj ((p,z)
(a)
2aCl+0
2a<l-£>
7
<b)
Fig. C.l The coordinate system used to derive the boundary
condition for a perturbed cylindrical waveguide.
to
o
207
I?(a+e 8,<j),z) = E(a,$,z)
+ eS(<}>,z).
dE0(p$j)
—={
ip=a •
3p
Up to the first order approximation in e, we have
nxE| p = a + e 5 = (n0 + e»!) x (E0 + eEi)| p = a + E 5
dE
=(n0 + mx) x (E0 + eS—i. + eEi)|Kp = a
dp
= n0 x E0\a^
+ E(»! x
dJE?0
+ 8n 0 x ___ + n0 x E ^ ^
EQ
dp
.
The zero order term vanishes automatically since it merely expresses the
boundary condition on EQ in the undistorted guide.
If we equate the first order term to zero, we can obtain expressions for
the § and z components of E1 at p = a in terms of p, and the components of EQ
and their derivative at p = a. In the <{> direction, we have
« S ^ 2 i • e £ u • E 0yp ^ » - 0 .
dp
dz
In z direction, we have
dp
v
v
ad§
These are the boundary conditions we have derived at Chapter 3.
208
Appendix D.
The Selection Rule
This appendix will provide the basic concepts for designing mode
converters using different azimuthal deformations. The concept may be
considered as being similar to the quantum mechanical formalism for the
electron transition between different energy levels. The eigenmodes of an
electromagnetic wave in a circular waveguide behaves similarly to the atomic
state function.
The transition of electrons inside an atom molecule is
governed by a selection rule. A similar phenomenon can be found in a
circular waveguide with certain wall perturbations. The waveguide mode
selection rule comes from azimuthal perturbations of the perfect circular
waveguide. The details of derivation of the coupling coefficients can be found
from Maxwell's Equations and the proper boundary conditions as discussed
in Chapter 3.
The coupling coefficients between modes of eigenfunctions, tymn and
\|/ , (like an atomic state function) can be expressed as
Kmp = < Vmnl 8 \\fpq > .
Since \|/
varies as
209
and \\f varies as
e-jp*
g-^V
for rotating modes. Optimum conversion between mode m and mode p is
obtained if 8 has component
e-j(m-p)(? e-J(Pm-Pp)*
Since 8 must be a real quantity, we have
8 = 8Cz)cos(Z(}»)cos(AP£) = Sc(z)cos(Z<j>) linear structure
(D.l)
8 = 8Xz)cos(Apz -/<{>) rotating structure
(D.2)
or
=Sc(z)cos(Z(j>) + Ss(;z)sin(Z<{>)
which is the superposition of the cosinusoidal and sinusoidal perturbation.
In Eqs.(D.l) and (D.2), Z = I m - pi and AP = p m - p p . We will name the
following special cases:
Z = 0 -» a varying-radius type mode converter,
Z = 1 -> a serpentine type mode converter,
Z = 2 -> a elliptical polarization converter, and
I > 2 -» a helical-multifoil mode converter.
210
There are two polarizations for each mode we are considering. We will
call the wave function proportional to cos(m({>) the ordinary mode and the
wave function proportional to the sin(m<J>) the cross polarization. In Eq.
(D.l), we have one set of perturbations. Another set of perturbations is
8 = 8s(z) sin(Z<j>).
An index system similar to the discussion in Chapter 6 will be used,
K^
and K^ . The first two indices represent the modes we are considering
and the third index represents the sin perturbation or cosine perturbation.
A superscript x above the m or q represents the cross polarization in mode
m or mode q. 1 ^ is the couphng coefficient for mode m and mode q without
the azimuthal function.
Kmpc = kmp< cos(m<J>) | cos(m-p)<]) | cos (p§)> = —kmp .
(D.3)
K
mpc = kmp< cosMO | cos(m-p)<|) | sin (p<|>)> = 0 .
(D.4)
Kmps - kmp< cos(m<|») | sin(m-p)<)) | cos (p<}>)> = 0 .
(D.5)
K
mps
K
= k
mpc
mp<
= k
c<
mp<
>s(m<|>) | sin(m-p)(}> | sin (p(j))> = --£& m * .
sin m< )
(
t ) I cos(m-p)({) | cos (p<J>)> = 0 .
(D.6)
(D.7)
211
KZpe = KP< sin(m(|») | cos(m-p)^
| sin (p<}>)> = lk^p
.
(D.8)
^mP» = ^^,< sin(m(J)) | sin(m-p)<j> | cos (p<J>)> = JHkxmp .
(D.9)
K
mp* = *C< siiHm<J)) | sin(/n-p)(j) | sin (p§)> = 0 .
(D.10)
From Eqs.(D.3)-(D.10), some of the coupling coefficients vanish due to the
orthogonallity of mode polarization and the waveguide perturbation.
The coupling coefficients between modes with the same azimuthal
index do not vanish for the polarization converter case.
= kxm'm< sin(m<j)) | cos(2m<j>) | sin(m<j>) > = -lkxmxm
KL
K
mmc = kmm< cos(m<|>) | cos(2m(j)) | cos(m(j>) > = lkmm
. (D.ll)
.
(D.12)
C
= kxmm< sin(m(j>) | cos(2m(j)) | cos(m<j)) > = 0 .
(D.13)
KL
= &™< sin(m(})) | sin(2ro<|0 | sin(m<?>) > = 0 .
CD. 14)
mm, = ^mm< cos(m<|>) | sin(2/n(})) | cos(m(j)) > = 0 .
(D.15)
K
Km, - Kxm< sin(m<|>) | sin(2m({)) J cos(m<{)) > = —kxmm .
(D.16)
We can use the above conditions to determine how many modes need
212
to be considered, and how the modes involved in the different deformation
when we design a mode converter.
213
Appendix E.
Fabrication of t h e Helical Multifoil Mode Converter
From Fig. 3.10, the complexity of the helical multifoil structure 15
periods can be seen. The following fabrication procedure has been used: 1)
Mill the helical-multifoil perturbation on the outside of a solid aluminum
mandrel, 2) electroform the mandrel with copper, 3) dissolve the aluminum
mandrel chemically in a solution containing 240g/L of sodium hydroxide, 4) put
flanges on both side of the electroformed copper waveguide.
An improved milling procedure needs to be developed so that we can
produce a more precise replica of our helical multifoil design on the aluminum
mandrel. The cutting tool is assumed to be the shape of hemispherical ball
and the numerically controlled milling machine can exactly control the center
of the milling ball when the ball is cutting on a cylindrical surface. Fig. E.l
shows the relation between the ball and part of the helical surface in the (x,y)
plane.
The surface of the helical structure in some (x,y) plane can be expressed
as:
a(<j>) = a 0 + e cos(Aro 4>) .
Translating the polar coordinate into (x,y) coordinate we obtain
Fig. E.l The coordinate relation between cutting ball and the
cross section of a helical multifoil mode converter.
215
x = a(<j») cos((>
and
y - a(§) sin({) .
The center of the milling ball, in the (x^y^ coordinate, is at the point
xb = r cos<(> + Xj
and
yb = r sincj) + yx .
We want to evaluate (x^y^ in terms of r, <}>, aC^), a'(§). Now suppose the
cutting point at the multifoil surface is (xv yt), then we have
xx = a(ty{) costj^
and
yx = aO^) s i n ^ .
The slope of the line tangent to the surface and the ball that passes through
(xu y7) can be written as
dy i
dx
Ul
_ dy_ d§ •
" cZ<(> d x Ul '
and
- 2 - = a'((j)) sin<|) + a((}>) cos<j)
d<j)
d!x
-— = a '(<))) cos<j> - a(<j>) sin(t> .
cZ0
Hence, we have
dy,
c& '
s
q ^ i ) smfri + a(<l>i) cos(})1
a'^j) cos^j - a^j) sin^j
1)
The line passing through points (x^ yb) and (xu yt) is normal to the tangential
line passing through point (xv y7). Hence we have
216
^ i ~ Vfc dy |
=
_1
(E.2)
xx - xb dx *'
Substituting Eq. (E.l) into Eq. (E.2), we have
. a'(A,) sind), + 0(6,) cosd),
*i - ** - -(yi - yb) - r - r —
,. . . .
aiij)!) cost])! - a ^ j ) sin<])1
The distance from cutting point to the center of the ball must equal the radius
of the cutting ball
(xx - xbf + (yx - yb? = r 2 .
Solving for y ; - yb from above two equations, we have
/
N2
a'ityj sin<j>x + aO^) cost}^
(ya - ybf 1 +
a'(^x) cosc^ - aO^) s i n ^
= r2
j
J
Hence, the center of the milling ball can be found from
1
T
\2
f
xh - a((]))cos(l) + sign(x^> r 1 + a'ityj cos^! - aO^) sin^j
a'(4>!) sin<()1 + aO)^) cos^
(E.3)
1
y 6 = a((}>)sin(}) + sign(yx) r 1 + r
1
a (({>!) sin^j + a(^x) cos^
a ' ^ ) cos^! - aOjjj) s i n ^
y
(E.4)
J
where sign(x;) and sign(y7) are the plus and minus signs which are determined
by the signs of (x^y,). From Eq. (E.3) and (E.4), we can find the center of the
217
milling ball with respect to different <J>X. Fig. E.2 shows the cutting tool
cutting through an extremely exaggerated surface of a multifoil mode converter
mandrel. The radius of the cutting tool is chopen such that the valley of the
x-y cross section will nicely fit the cutting tool. The minimum number of
longitudinal cutting passes for an acceptable cusp height is chosen to be 24
passes per period. The largest cusp has been estimated to be less than .1
thousandth of one inch.
Fig. E.2 a) Diagram showing a cross-section of the helical
multifoil mode converter with "tool cut" circles for the case of 8
cuts per period, b) Expanded view of one period of the top
drawing with the vertical dimension expanded much more than
the horizontal.
219
Appendix F.
Fabrication E r r o r a n d Measurement E r r o r for the
Helical Multifoil Mode Converter
The center position of a milling ball has been calculated in Appendix
E.
The maximum peak-to-valley perturbation amplitude of the helical
multifoil mode converter we have designed in Chapter 6 is only .0214". We
have tried to setup a measurement system for the machined mandrel to
assure the cross section of the helical multifoil mode converter around a
constant-z cross-section satisfied our requirement. The measurement setup
was in the ECE machine shop. A small misalignment of the mode converter
rotating axis on the lathe is expected. A dial indicator has been used to find
out the peak to valley variation for each period.
1. A Misaligned Rotating Center
Assume the rotating center of a cross section of the mode converter has
been misaligned by .002" compared with the original rotating center. We will
call the new rotating center as C and the original center as O. The relative
coordinate system is shown in Fig. F.l. The original cross section perimeter
can be described as
220
r0 = a 0
+ecos{mcp)
2
r=(r^+A -2r0Acos^)
Fig. F.l. The coordinate relation between real rotating center
of a mode converter and a misaligned rotating center.
c
221
r0 = a + ecos(m§) .
Using the new rotation center to describe the profile, we have
r 2 = rl + A2 - 2r,)Acos<t>
= [a0 + ecos(m<}>)]2 + A2 - 2r0Acos<]) .
In the new rotation center, we can find that the angle corresponding to an
extreme value should satisfy
dr2
= 0 = -2raesin(m<{>)[a0 + ecos(m<]))] + 2Asin<|)[a0 + ecos(m<}>)]
3<J)
+2emAsin(m(j>)cos<l> .
The maximum and minimum angles only change a very small amount
compared to the original value. Figure F.2 shows the peak to valley variation
for each flute. If we misaligned the rotating center by .002", an .0008" peak
to valley variation is expected.
0.0219
0.0207 ~
0.0205
120
180
240
360
degrees
Fig. F.2.The peak-valley variation due to the misalignment of
the rotation center.
References
[1]
V.L. Granatstein and I. Alexeff, ed., High-power Microwave
Source, Artech House, p.3-35, (1987).
[2]
U. Rhee, "Design improvements for mode and polarization
converters for 60 GHz gyrotrons," Ph.D. dissertation, U. of
Wisconsin-Madison, (1986)
[3]
M.J. Buckley, "Compact quasi-periodic and aperiodic mode
converters for overmoded circular waveguide," Ph.D. dissertation,
U. of Wisconsin-Madison, (1988).
[4]
M. Thumm, et al, " Very high power mm-wave components in
oversized waveguide," Microwave J., pl03 (1986)
[5]
H. Kumric, et al, "Optimization of mode converters for generating
the fundamental TE 0 1 mode from TE 06 gyrotron output at 140
GHz" Int. J. Electronics, Vol. 64, p.77-94,(1988).
[6]
S.N. Vlasov and I.M. Orlova, "Quasioptical transformer which
transforms the
waves in a waveguide having circular cross
section into a highly directional wave beam," Radiophysics
Quantum Electron., vol. 17, pl48 (1974)
[7]
S.N. Vlasov, et al, "Transformation of whispering gallery mode,
propagating in circular waveguide, into a beam of waves," Radio
Eng. Electron. Phys., vol. 21, no. 10, pl4 (1975).
[8]
J. Lorbeck and R.J. Vernon, "Quasi-optical conversion of the
output from a whispering-gallery mode gyrotron to a free-space
beam with arbitrary power and phase distributions across both
transverse dimensions of the beam," Fifteenth Int. Conf. on
Infrared and Millimeter Waves," Orlando, Florida, p301 (1990).
[9]
P. Sealy and R.J. Vernon, "Equivalence principle and physical
optics modeling of radiation from TE 0 n and TM 0n mode vlasov
launchers," Fifteenth Int. Conf. on Infrared and Millimeter Waves,
Orlando, Florida, p216-218, (1990).
[10] R.J. Vernon Ed. DOE Project Annual Report, 1991.
[11] S.P. Morgan, "Theory of curved circular waveguide containing an
inhomogeneous dielectric," B.S.T.J., vol. 36, pl209-1251 (1957).
[12] S.A. Schellkunoff,
"Conversion of Maxwell's equations into
Telegraphist's equations," B.S.T.J., p955 (1955)
[13] C. Moeller, "Mode converters used in the doublet III ECH
microwave system", Int. J. Electronic, Vol. 53, p.587, (1982)
[14] G.H. Luo, "An investigation of three-period 60 GHz TE 0 1 -TE n
mode converter" M.S. report, U. of Wisconsin-Madison, (1987)
[15] M.J. Buckley, G.H. Luo, and R.J. Vernon, "New compact
broadband high-efficiency mode converters for high power
microwave tubes with TE 0 n or TM 0n mode output," IEEE MTT-S
Int. Microwave Symposium, New York, NY, pp. 797-800 (1988)
[16] G.H. Luo, D.A. Casper, and R.J. Vernon, "Design of a single-period
8.6 GHz TM 0 1 -TE n serpentine mode converter and TM 01 and
T E n bends in moderately overmoded circular waveguide," 15th
Int. Conf. on Infrared and Millimeter Wave, Lake Buena Vista,
FL, pp. 434-436 (1990).
[17] M.J. Buckley and R.J. Vernon," A single period TE 02 -TE 01 mode
converter in highly overmoded circular waveguide," IEEE trans.
MTT, Vol. 39, No.8, pp. 1301-1307, (1991).
[18] D.V. Vinogradov and G.G. Denisov, "Waveguide mode converters
with step type coupling," Int. J. of Infrared and Millimeter Waves.
Vol. 12, No. 2, pp. 131-140, (1991).
[19] S. Silver, ed. , Microwave antenna theory and design, McGrawHill, MIT Radiation Laboratory Series, Vol. 12, p336-338 (9149).
[20] J.L. Doane, "Propagation and mode coupling in corrugated and
smooth wall circular waveguide," Infrared and Millimeter Waves,
vol. 13, pl23 (1985)
[21] J.W. Carlin, S.C. Moorthy, "TE 01 transmission in waveguide with
axial curvature," B.S.T.J., vol 56, pl849-1872 (1977).
[22] H.E. Rowe and W.D. Warters, "Transmission in multimode
waveguide with random imperfections",
B.S.T.J., pl031-1169,
May (1962)
[23] J.L. Doane, "Polarization converters for circular waveguide
modes," Int. J. Electronic, vol. 61, pll09-1133 (1986)
[24] E.L. Ince, Ordinary differential equations. Dover, New York, NY,
1956.
[25] S.E. Miller "Notes on methods of transmitting the circular electric
wave around bends," Proc. of the I.R.E., p.1104-1113, (1952).
[26] K.I. Thomassen "Electron cyclotron wave sources and apphcations
for fusion" J. of Fusion Energy, Vol.9, no.l, p.1-6, 1990
[27] C.P. Moeler and J.L. Doane," Coaxial converter for transforming
a whispering gallery mode to the H E n mode," 15th Int. Conf. on
Infrared and Millimeter Waves, Orlando, FL, pp.213-215, (1990).
[28] M. Thumm, and A. Jacobs, "In waveguide TE 01 -to-whispering
gallery mode conversion using period wall perturbations," 13th
Int. Conf. on Infrared and Millimeter Waves, Honolulu, Hawaii,
pp.465-466, (1988).
[29] J.S. Levine,"Rippled wall mode converters for circular waveguide,"
Int. J. of Infrared and Millimeter Wave, vol. 5, no. 7, p.937-952,
(1984).
[30] K.E. Kreischer, et al, "The design of megawatt gyrotrons for the
227
compact ignition tokamak,"Thirteenth Int. Conf. on Infrared and
Millimeter Waves, Honolulu, Hawaii, pl79 (1988)
[31] S.H. Gold, et al, "High peak power ka-band gyrotron oscillator
experiment," Phys. Fluids, vol 30, pp2226-2238 (1987)
[32] M.J. Buckley, G.H. Luo and R.J. Vernon, "Very short quasiperiodic and
aperiodic mode converters for 60 and 140 GHz
gyrotrons," Thirteenth Int. Conf. on Infrared and Millimeter
Waves, Honolulu, Hawaii, pl21, Dec. 1988.
[33] J. Neilson, private communication.
[34] M. Thumm, et al, "Radiation patterns with suppressed sidelobes
for quasi- optical mode converter," Int. Conf. on Infrared and
Millimeter waves, Honolulu, Hawaii, p463 (1988).
[35] R.J. Vernon, private discussions.
[36] K. Thomassen, "Electron Cyclotron Wave Sources and Applications for
Fusion," Journal of Fusion Energy, pp. 1-6, Vol. 9, No. 1, 1990.
[37] K. Felch, et al, "Gyrotrons for ECH Applications,"Journal of
Fusion Energy, pp. 59-75, Vol. 9, No. 1, 1990.
[38] M.R. Spiegel, ed., Mathematical hand book of formulas and tables,
McGraw-Hill, pl36, 1968.
VITA SHEET
Title of thesis
Perturbed-Wall Microwave Mode Converters and Bends in
Overmoded Circular waveguide
Major Professoi<" Ronald J. Vernon
Major
Electrical Engineering
Minor
Distributed
Name
Gwo-Huei Luo
Place and date
of birth
Taiwan, Republic of China October 28, 1959
Colleges and
Universities:
years attended
and degrees
National Taiwan Normal University 4 yrs. BS in Physics
The University of Wisconsin-Madison 2 yrs. MSEE
The University of Wisconsin-Madison 4 yrs. Ph.D.
Memberships iil
learned or
honorary
societies
IEEE and Sigma Xi
Publications
1. "A Single-Period Serpentine Mode Converter and
Waveguide Bends in a Moderately Overmoded Circular
Waveguide", G.H. Luo, D.A. Casper, R.J. Vernon, to be
published.
(continued on next page)
January 17, 1992
current date
Publications
(continued) 2. "Design of a Single-Period 8.6 GHz TM 01 -TE 11 Serpentine
Mode Converter and TM 01 and TEXx Bends in a Moderately
Overmoded Circular Waveguide", G.H. Luo, D.A. Casper, R.J.
Vernon, 15th Int. Conf. on Infrared and Millimeter Waves,
Lake Buena Vista, FL,Dec. 10-14, 1990, Digest pp. 434-436.
3. "Waveguide Mode Converters for Gyrotrons Producing
High-Azimuthal-Index Rotating Modes", G.H. Luo, R.J.
Vernon, 14th Int. Conf. on Infrared and Millimeter Waves,
German, Oct. 9-13, 1989, Digest pp. 160-161.
4. "Varying-Radius Mode Converter for 140 GHz Gyrotrons",
G.H. Luo, R.J. Vernon, Progress in Electromagnetic Research
Symposium, Boston, Massachusetts, July 25-27, 1989,
Proceedings pp.198-199.
5. "Very Short Quasi-Periodic and Aperiodic Mode Converters
for 60 and 140 GHz Gyrotrons", M.J. Buckley, G.H. Luo, R.J.
Vernon, 13th Int. Conf. on Infrared and Millimeter Waves,
Honolulu, Hawaii, Dec. 5-9, 1988, Digest pp.117-118.
6. "New Compact Broadband TE 0n -TE Q1 . TE 01 -TE 11 and
TE 15 o-TE15 1 Mode Converters for Millimeter Wave
Gyrotron", M.J. Buckley, G.H. Luo, R.J. Vernon, 30th Annual
Meeting of American Physical Society, Division of Plasma
Physics. Hollywood, Florida, Oct. 31 - Nov. 4, 1988.
7. "New Compact Broadband High-Efficiency Mode
Converters for High Power Microwave Tubes with
TE 0 n or TM 0n Mode Outputs", M.J. Buckley, G.H. Luo, R.J.
Vernon, IEEE MTT-S International Microwave Symposium,
New York, NY, May 23-27, 1988, Digest pp.797-800.
8. "New Compact Quasi-Periodic and Aperiodic Mode
Converter for 60 GHz and 140 GHz Gyrotron", M.J. Buckley,
G.H. Luo, R.J. Vernon, 12th Int. Conf. on Infrared and
Millimeter Waves, Lake Buena Vista, Florida, Dec. 14-18,
1987, Digest pp.214-215.
9. "New Compact TE o r TE 0 2 , TE 02 -TE 01 , and TE 0 1 -TE n
Mode Converter for Use with MM-Wave Gyrotrons", M.J.
Buckley, G.H. LuoT R.J. Vernon, 29th Annual Meeting of the
American Physical Society, Division of Plasma Physics, Nov.
2-6, 1987.
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