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

код для вставкиСкачатьINFORMATION TO USERS The negative microfilm copy of this dissertation was prepared and inspected by the school granting the degree. We are using this film without further inspection or change. If there are any questions about the content, please write directly to the school. The quality of this reproduction is heavily dependent upon the quality of the original material. The following explanation of techniques is provided to help clarify notations which may appear on this reproduction. 1. Manuscripts may not always be complete. When it is not possible to obtain missing pages, a note appears to indicate this. 2. When copyrighted materials are removed from the manuscript, a note appears to indicate this. 3. Oversize materials (maps, drawings, and charts) are photographed by sectioning the original, beginning at the upper left hand corner and continuing from left to right in equal sections with small overlaps. 4. Most photographs reproduce acceptably on positive microfilm or microfiche but lack clarity on xerographic copies made from the microfilm. For any illustrations that cannot be reproduced satisfactorily by xerography, photographic prints can be purchased at additional cost and tipped into your xerographic copy. Requests can be made to the Dissertations Customer Services Department. UMI Dissertation Information Service University Microfilms International A Bell & Howell Information Company 300 N. Zeeb Road, Ann Arbor, Michigan 48106 Order Number 9218355 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, $(z) = a^l + e j l - cos(Hi(kpz))]} for 0 < ILZ £ ii i=1...5 (7.2) 180 A straight section of waveguide is used to adjust the phase mismatch between the TE 1 5 2 and TE 1 5 x modes at the end of each taper section of Eq. (7.2). We include in the numerical calculation all TE 1 5 n and TM 15 n modes as soon as they become propagating. The input and output ports of the uptapered mode transducer are 1.6 cm and 2.8 cm in diameter, respectively. The number of excited propagating modes on each side of the taper-converter are 3 and 13, but only the first 12 modes are included in the calculation. In a preliminary design, we used a purely cosinusoidal perturbation in each section. Figure 7.7 shows the mode content. The conversion efficiency was approximately 89%. Next we added two harmonic terms into H^z). A preliminary result for the conversion efficiency was 95.8%. Figure 7.8 shows the mode amplitude of several modes along the converter. Table 7.5 lists the perturbation profile for each section of the structure in Eq. (7.2). We believe a conversion efficiency of approximately 98% can be achieved with some small changes in each section. 4. An U p t a p e r e d Mode Converter to P r o d u c e Specific O u t p u t Mode Combinations In the proposed design of a new gyrotron, for which the output power is in the megawatt range, the operation efficiency is about 35%. In addition to handling the generated electromagnetic power, it is also necessary to deal with the residual energy of the spent electron beam. The spent electrons are y~ \ 05 dTE / y i 0 .5 i 0.6 i ""\ •••*% • t • « • • t • i 0.4 AMPL] TUDE 0.7 \ % t « » • * a / ft * » d I - t \ * 1 J 9 TE \ \ a a CO 15,. _J d- • l t 15,2 \ •• • •« \./'"" — N. x\ X t fc CM £ E t F." d~ fc fc" B do d- •• - * r -•* "-• • *. » ; •. • V .V JF\\ *-• : . *; I I - i - . ; 0.00 —' ** • * * • "* * \ " £ ''*(HHItlMIHIIl • .* • ; r-*— f—ii 0.02 ''" "*•' '"""t, "•.* \ \ * ^ - " \ * , 0.03 %$>•.... VO^&ZIIIZA IIMI... *•..••• * 5 0.01 y'""% ./ \ / ,->V^----";':'--"-v*'-^:./\ V" 1»-- P t* c* f * , \\ .t —! 0.04 ; » i '•> • ' 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.

1/--страниц