ANALYSIS AND DESIGN OF SLOW WAVE STRUCTURE FOR BACKWARD WAVE OSCILLATORS A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF MIDDLE EAST TECHNICAL UNIVERSITY BY DOĞANCAN ESER IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN ELECTRICAL AND ELECTRONICS ENGINEERING JANUARY 2019 Approval of the thesis: ANALYSIS AND DESIGN OF SLOW WAVE STRUCTURE FOR BACKWARD WAVE OSCILLATORS submitted by DOĞANCAN ESER in partial fulfillment of the requirements for the degree of Master of Science in Electrical and Electronics Engineering Department, Middle East Technical University by, Prof. Dr. Halil Kalıpçılar Dean, Graduate School of Natural and Applied Sciences Prof. Dr. Tolga Çiloğlu Head of Department, Electrical and Electronics Engineering Prof. Dr. Şimşek Demir Supervisor, Electrical and Electronics Eng. Dep. METU Examining Committee Members: Prof. Dr. Sencer Koç Electrical and Electronics Engineering, METU Prof. Dr. Şimşek Demir Electrical and Electronics Engineering, METU Prof. Dr. Asım Egemen Yılmaz Electrical and Electronics Engineering, Ankara University Assoc. Prof. Dr. Özgür Ergül Electrical and Electronics Engineering, METU Assoc. Prof. Dr. Avni Aksoy Accelerator Technologies Institute, Ankara University Date : 31.01.2019 I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work. Name, Surname: Doğancan Eser Signature : ABSTRACT ANALYSIS AND DESIGN OF SLOW WAVE STRUCTURE FOR BACKWARD WAVE OSCILLATORS Eser, Doğancan M.S., Department of Electrical and Electronics Engineering Supervisor: Prof. Dr. Şimşek Demir January 2019, 80 pages High power microwave is an emerging area that finds different applications in radar, directed energy weapons, plasma science, satellite communications, particle physics and medicine fields. Some examples of high power microwave devices are klystron, vircator, cyclotron, magnetron, travelling-wave tube amplifier (TWTA) and backward-wave oscillators (BWO). In this work, we focus on backward-wave oscillator that is one of the high power microwave sources. Backward-wave oscillator is a real high power microwave sources that convert electron beam energy to waves. It is also called as Cherenkov devices because emission process is analogous to Cherenkov radiation which occurs when the electron velocity exceeds the light velocity in that medium. Beam-wave interaction is achieved in the part called as slow-wave structure. Wave velocity is reduced below the electron beam velocity in this part. This part can be designed with periodic obstacles, dielectric or metamaterial. However, using dielectric is not preferred due to dielectric breakdown for high power microwave devices. Instead of that metallic periodic obstacles or metamaterials are used. This part affects the operation frequency, conversion efficiency, output power and compactness. As a result of these, slow wave structure is the crucial part of the BWO. In this work, we analyzed and designed the slow-wave structure for high power backward-wave oscillators. First, the unit cell of the slow wave structure simulation is performed in terms of dispersion diagram using commercial 3D electromagnetic solver. Modes of the SWS are also investigated for both empty circular waveguide and SWS. Interaction impedance which is the parameter for conversion efficiency is extracted for the unit cell. Phase velocity and group velocity of the wave in this medium is also simulated. Eight unit cells are combined to measure the propagation of the wave in that structure. Combined 8 unit cells are fabricated and measured in terms of S parameters. This measurement validates the dispersion diagram of the unit cell since it resonates for the desired mode and frequency. Hot test simulation of the slow wave structure is also investigated using commercial 3D particle solver. In this simulation, annular electron beam is used and validated SWS is used for hot test simulation. In order to confine the annular electron beam analytical magnetic field is applied. At the end, we observed the operation frequency as the 2.49 GHz which is in the passband region of the validated SWS. Keywords: Slow wave structure (SWS), Backward-wave oscillator (BWO), Metamaterial, Dispersion Diagram, Interaction Impedance ÖZ GERİYE GİDEN DALGA OSİLATÖRLERİ İÇİN DALGA YAVAŞLATICISI TASARIMI VE ANALİZİ Eser, Doğancan Yüksek Lisans, Elektrik ve Elektronik Mühendisliği Bölümü Tez Yöneticisi: Prof. Dr. Şimşek Demir Ocak 2019 , 80 sayfa Yüksek güçlü mikrodalga uygulamaları son yıllarda radar, yönlendirilmiş enerji silahları, plazma bilimi, uydu haberleşmesi, parçacık fiziği ve tıp alanlarında bir çok uygulama alanı bulmuştur. Klistron, virkatör, kiklotron, magnetron, yürüyen dalga amplifikatörü (TWTA) ve geriye giden dalga osilatörü (BWO) yüksek güçlü mikrodalga kaynaklarının bazı örnekleridir. Bu tezde yüksek güçlü mikrodalga kaynaklarından biri olan geriye giden dalga osilatörü üzerinde durulacaktır. Elektron enerjisini dalga enerjisine çeviren geriye giden dalga osilatörü bir mikrodalga kaynağıdır. Bu cihazlar aynı zamanda Cherenkov cihazları olarak da bilinmektedir. Bunun nedeni Cherenkov ışıması ile olan benzerliğinden gelmektedir. Elektron hızı aynı ortamda bulunduğu ışığın hızını geçerse Cherenkov ışıması meydana gelmektedir. Elektron ve dalganın etkileşim gösterdiği yerler dalga yavaşlatıcı yapısı olarak bilinmektedir. Çünkü bu yapılarda dalganın faz hızı elektron demetinin hızının altına düşer. Bu işlevi gören yapıları tasarlamak için periyodik olarak konan engeller, dielektrik veya metamalzemeler kullanılabilir. Dielektrik kullanımı dalga yavaşlatıcı yapısı için uygun olsa da dielektrik kırılımından dolayı yüksek güçlü mikrodalga kaynaklarında tercih edilmez. Dielektrik malzeme yerine periyodik olarak eklenen metal engeller veya metamalzemeler olarak da bilinen yapılar kullanılabilir. Dalga yavaşlatıcı yapısı osilasyon frekansı, dönüşüm verimliliği, çıkış gücü ve kompaktlık gibi birçok parametreyi etkilemektedir. Bu tezde geriye giden dalga osilatörleri için dalga yavaşlatıcısı tasarımı yapılmıştır ve bu yapı analiz edilmiştir. İlk olarak bir dalga yavaşlatıcı yapısının birim hücresinin dağılım diyagramının öngörümlemesi 3B elektromanyetik çözümü yapan bir yazılımda yapılmıştır. Boş bir dalga kılavuzundaki ve yavaşlatıcı dalga yapısındaki modlar incelenmiştir. Dönüşüm verimliliğini doğrudan etkileyen bir parametre olan etkileşim empedansı da yine aynı programda incelenmiştir. Tasarımın bu yapısında dalga negatif bir grup hızına ve elektron demetine yakın bir hıza sahip olması gerekir. Bu nedenle dalganın faz ve grup hızlarının öngörümlemesi 3B elektromanyetik tasarım programında yapılmıştır. Bu tasarımda dalgaların yayılımını ölçmek için 8 birim hücreden oluşan bir yapı birleştirilmiş ve üretilmiştir. Bu üretim dağılım diyagramının doğrulanmasında kullanılmıştır. Ölçümlerde rezonans frekansları ve modu öngörümlemelerle yakın çıkmaktadır. Dalga yavaşlatıcı yapısının elektron demeti içeren öngörümlemesi de ayrıca yapılmıştır. Halka biçiminde bir elektron demeti yavaşlatıcı dalga yapısının içerisinde gözlenmiştir. Halka biçimindeki elektron demetini bir arada tutmak için analitik manyetik alan uygulanmıştır. Son olarak 2.49 GHz frekansında osilasyon yapan bir geriye giden dalga osilatörü gözlenmiştir. Anahtar Kelimeler: Dalga yavaşlatıcı yapılar, Geriye giden dalga osilatoru, Metamalzeme, Dağılım diyagramı, Etkileşim empedansı to the memory of Turkish soldiers gone but never forgotten ACKNOWLEDGMENTS First of all, I would like to express my deepest gratitude to my family Mihriban, Abutter and Müge Eser and special thanks to person who will share my life Esin Gizem Uslu. They are irreplaceable people of my life. Words are not sufficient to express their value for me. I appreciate your patience and support throughout the thesis. I want to thank God for them. I cannot express enough thanks to my advisor, Prof. Dr. Şimşek Demir for his patience, inspiration, encouragement, excellent guiding and caring throughout the thesis. Besides being an excellent instructor, he is a perfect engineer for me. I learned a lot from him. The one I need to thank most for giving me such an opportunity. I would like to thank my first technical leader Mehmet Altuntaş and my colleagues Alper Ünal, Alp Manyas, Umut Aytaç Mutlu, Muhammed Acar and Ozan Koca. I would like to thank Kazım Sömek and Anıl Korkmaz for their support in mechanical design. I would also thank to Semih Küçük and Büşra Timur for their support. I want to thank to Neslihan Yasav for her support. I would like to thank my friends I learned something Akif Yılmaz for his patience and support, Aysel Akgemci, Can Köksal, my oldest friend Cemil Kocaman, Çağrı Güven, Gökay Aytekin, Gökay Yurdakul for king game, Hande Özcan, İbrahim Akalın, Harun Avcı, Kadir Bacanak, Mahmut Altınay the best man, Mevlüt Kara, Orcun Bayram, Suay Aytekin, Şükrü Kaşıkcı, Tunahan Erkoyuncu, special thanks to Turan İlhan for enjoyable time and Zeki Yılmaz. I would also thank to my first teacher Saadet Pınarbaşı for her effort on teaching. I would also special thank to my grandfather Mustafa Doğan Eser who has great place in my life. I will live your name. I would also thank to my grandmothers Fatma Eser, Gülsüm Tokuş. Also, I would thank to my grandfather Kazım Tokuş I never saw. I would also thank to rest of my family. This work is founded by PRF Arge A.Ş. I am grateful to PRF Arge for their supports. TABLE OF CONTENTS ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v ÖZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv LIST OF ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . xix CHAPTERS 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 High Power Microwave Devices . . . . . . . . . . . . . . . . . . . . 5 2.2 Backward-Wave Oscillators . . . . . . . . . . . . . . . . . . . . . . 12 2.2.1 Conventional Slow-Wave Structure BWO . . . . . . . . . . . 15 2.2.2 Metamaterial Slow-Wave Structure BWO . . . . . . . . . . . 17 3 THEORY OF OPERATION . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.1 Modes on Circular Waveguide . . . . . . . . . . . . . . . . . . . . . 21 3.2 Group Velocity and Phase Velocity . . . . . . . . . . . . . . . . . . . 23 3.3 Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.4 Interaction Impedance . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.5 Electron Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.6 Backward Cherenkov Radiation . . . . . . . . . . . . . . . . . . . . 26 3.7 Cherenkov-Cyclotron Instabilities . . . . . . . . . . . . . . . . . . . 27 3.8 Introduction to Metamaterial . . . . . . . . . . . . . . . . . . . . . . 28 4 DESIGN AND SIMULATION OF SLOW WAVE STRUCTURE . . . . . . 33 4.1 Simulation Environment : CST Studio Suite . . . . . . . . . . . . . . 33 4.1.1 Time Domain Solver . . . . . . . . . . . . . . . . . . . . . . 33 4.1.2 Frequency Domain Solver . . . . . . . . . . . . . . . . . . . . 35 4.1.3 Eigen Mode Solver . . . . . . . . . . . . . . . . . . . . . . . 35 4.1.4 Particle-in-cell Solver . . . . . . . . . . . . . . . . . . . . . . 36 4.2 Cold Test Simulation of SWS . . . . . . . . . . . . . . . . . . . . . 36 4.2.1 Mode Analysis of Circular Waveguide . . . . . . . . . . . . . 40 4.2.2 Analysis of Rectangular SWS Loaded Circular Waveguide . . 41 4.3 Validation of SWS . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.4 Hot Test (PIC Solver) Simulation of SWS . . . . . . . . . . . . . . . 57 5 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 LIST OF ABBREVIATIONS ABK Annular Beam Klystron BWO Backward Wave Oscillator CAE China Academy of Engineering CHAMP Counter-electronics High Powered Microwave Advanced Missile Project CRM Cyclotron Resonance Maser FEL Free Electron Laser HPM High Power Microwave LHM Left Handed Media MILO Magnetically Insulated Transmission Line Oscillator NINT Northwest Institute of Nuclear Technology NUDT National University of Defense Technology PIC Particle in Cell PPML Plasma Pulsed-Power and Microwave Laboratory RBWO Relativistic Backward Wave Oscillator RPM Recirculated Planar Magnetron SWS Slow Wave Structure TWT Traveling Wave Tube CHAPTER 1 INTRODUCTION High power microwave (HPM) devices are devices that can provide more than 100 MW power in the centimeter and millimeter wave range of frequencies between 1-300 GHz. At high power levels, vacuum tube devices are the only candidate since the solid state transistor can not provide sufficient power. It is an emerging area as a new technology. HPM devices have application areas in radar systems, directed energy weapons, satellite communications and medicine. There are different types of high power microwave devices such as magnetron, travelling wave tube amplifier, klystron and backward wave oscillator. They are different from solid state transistor and vacuum tube transistor since operation principle is different and microwave circuit is the crucial part of the system. The basic operation of all high power microwave devices resembles each other. An electron gun is used to produce electron beam. This electron beam is guided by external magnetic field. Otherwise, electrons hit the wall of the microwave circuit repelling each other which result in lower efficiency for devices. Moving electron interacts with the electromagnetic waves and transfers their energy to electromagnetic waves. This interaction is called as beam-wave interaction and occurs in microwave parts of the high power microwave devices. One of the high power microwave devices is the backward wave oscillator (BWO). It comprises of electron gun, slow wave structure (SWS), output antenna and applied magnetic field. Electron gun is the part of the microwave devices that provides electron beam in vacuum tubes. Emitted electron mechanism of the electron gun can be the thermionic emission or field emission. Emitted electrons guided by the magnetic field since the free electrons repel each other. BWO is also called as the O-type devices since electrons move parallel to magnetic field. Moving electrons parallel to magnetic field transfers their energy to electromagnetic waves. The condition that electrons transfer their energy to electromagnetic wave is the synchronism condition. Phase velocity of the electromagnetic wave should be close the axial velocity of the electron to provide synchronism condition. This process resembles the Cherenkov radiation which occurs when the electron velocity exceeds the light velocity in that medium. Beam wave interaction occurs in the slow wave structure part of the BWO. It is the crucial part of the BWO since it affects the parameters like conversion efficiency, operation frequency, compactness of the device. Phase velocity of the electromagnetic wave is reduced to the axial electron velocity in this part. Otherwise, synchronism condition can not be provided since the phase velocity of electromagnetic wave is greater than the speed of light in the empty waveguide. It makes the interaction impossible. Loading dielectric can be method to slow wave structure however it is not appropriate for high power microwave devices because of the dielectric breakdown. So, full metal structures are needed as slow wave structure. Metallic slow wave structures are the periodic obstacles that slow the phase velocity of the electromagnetic wave. There are different types of SWS. SWSs are classified as conventional and metamaterial slow wave structure in this thesis. Works before the metamaterial invention is considered as conventional slow wave structures and design includes the metamaterial approach is considered as the metamaterial slow wave structures. In this thesis, operation principle of the BWO is investigated. First of all, unit cell of the SWS is designed in order to fit the TM mode dispersion diagram in between 2.4-2.5 GHz because lower S band can be accepted as sweet spot considering beam power transfer. In addition, electronic components may have slot which is suitable for wavelength in these frequency to penetrate into the electronic components. Dispersion diagram of the unit cell is observed for lowest order modes. Assuming the beam wave interaction occurs with the TM mode under the strong axial magnetic field, SWS is designed to provide TM mode propagation. Combined 8 unit cells are simulated. TM mode propagation is observed for the complete SWS excited by the axial pin. In order to validate the TM mode propagation, combined SWS is fabricated and excited with the axial pin as in simulation. Fabricated SWS is measured in terms of the S parameters. Passband is observed in between 2.43-2.72 GHz with the correlation of dispersion diagram. Validation of the SWS which is suitable for TM mode propagation is achieved by the measurements. In addition, SWS under the electron beam is simulated using the CST Particle Studio. Annular electron beam is used with DC electron emission model. Inner and outer radius of the beam are 5 and 2.2 mm. 2 T analytical magnetic field is applied to guide the electron beam. Phase space plot of the electrons is observed to investigate the velocity and energy of electrons. Finally, operation frequency of the oscillation is observed as 2.49 GHz. 10.7 MW peak power is achieved at the output of the BWO for 450 kV and 100 A electron beam with %23.8 peak efficiency. Based on the above mentioned content, this thesis includes 5 chapters. The first part is the introduction part which defines the thesis content briefly. Second part is devoted to explain the previous work in literature. In addition to the work on BWO including both metamaterial based and conventional SWS, second chapter also includes works on other high power microwave sources such as TWT, klystron, magnetron. Third chapter gives theoretical background to design SWS. In the chapter four, CST solvers used in this thesis mentioned briefly. Design details and simulation results are also given in the forth chapter. Final chapter is devoted to give conclusion, future work. In conclusion, backward wave oscillator simulation is achieved for the 450 kV, 100 A electron beam under the 2 T magnetic field with %23.8 peak efficiency. Peak output power is observed as 10.7 MW and oscillation frequency is 2.49 GHz. CHAPTER 2 LITERATURE REVIEW 2.1 High Power Microwave Devices High power microwave systems are an emerging area as a new technology. It is defined as systems that exceed the 100 MW peak power at the frequency range 1 GHz to 300 GHz [1]. It has many applications in radar systems, directed energy weapons, satellite communications, medicine and laboratory sources for susceptibility and vulnerability testing of electronic systems. This subject is one of the hot research topic studied in United States of America, China, Russia, Western Europe, Japan, Taiwan, India, South Korea, and Singapore. In these countries, most of the research is devoted for high power microwave weapons. The Counter-electronics High-Powered Microwave Advanced Missile Project (CHAMP) is a successful and contemporary example of HPM weapons. It is a joint concept technology developed by Boeing and US Air Force, Directed Energy Directorate at Kirtland Air Force Base. The aim of this project is to develop air-launched directed energy weapon to damaging electronic systems with electromagnetic pulse. This system is tested on October 2012 by Boeing [9]. Besides that, studies [12]-[15] are done in Plasma, Pulsed Power, and Microwave Laboratory (PPML) in University of Michigan for new device which is termed as Recirculated Planar Magnetron (RPM) [15]. Possible advantages of RPM for airborne applications are discussed in these studies. 3D printed plastic anodes electroplated and thermal sprayed with copper is discussed in [17] for RPM. Northwest Institute of Nuclear Technology (NINT), National University of Defense Technology (NUDT), and China Academy of Engineering (CAE) are places where working on HPM devices. Main research topics are on to increase the power efficiency, overcome to pulse shortening phenomenon, increase the pulse duration and operation in low magnetic field. Magnetically insulated transmission line oscillator (MILO) and Triaxial Klystron Amplifier studied in [18] and [19]. Relativistic backward-wave oscillator (RBWO) is studied in NINT [20]. Advantages of RBWO are high power efficiency, high output power and high repetition rate. Russia also revealed its microwave cannon [10] in Jun 2015, which is supposed to disable drones and warheads at a distance of up to six miles. In Japan, researches are continuing in the fields that are plasma heating by Gyrotrons, accelerators by Klystrons, and Free Electron Laser (FEL) [1]. The history of microwave vacuum tubes starts from the first microwave generation by German physicist Heinrich Hertz in 1888. Actually, Hertz helped to general acceptance of James Clerk Maxwell’s theoretical observation that claims electromagnetic disturbance travels in the free space with velocity of light. Researches were expanded to make Maxwell’s equations understandable by "The Maxwellians" who are George Francis FitzGerald (1851-1901), Oliver Lodge (1851-1940) and Oliver Heaviside (1850-1925) [11]. The interest in radio science in the 20th century developed rapidly. In 1930s, higher frequency operation is observed connecting the cavities to electrical circuits. The first Klystron is produced in 1937 by Russell and Sigurd Varian [21] and microwave tube was born. "Klystron" is an ancient Greek verb referring to the action of waves breaking against a shore. After that, activity in microwave tube area followed by the invention of Magnetron, Backward-Wave Oscillator (BWO) and Traveling Wave Tube (TWT) during World War II. In the 1950s, gyrotrons came into the literature as a new device as a result of the efforts to control thermonuclear fusion for energy production which requires a detailed understanding of the interaction between particles and waves. In the 1970s, solid state devices dominate research areas and researches on microwave tubes are curtailed. From 1990s to now, that has changed and high power microwave tubes appeared in the research area. General components of HPM devices are prime power system, pulsed power system, microwave source and antenna. HPM devices need intense electrical pulses in the order of MV. This can be achieved by using capacitor banks that can store low voltage and slowly rising signal and provide high voltage, fast rising signal. Drivers part composed of prime and pulsed power system should be well-matched to microwave sources for efficient energy transfer. Driver parts are used to generate electron beam to provide electromagnetic radiation in microwave source part. There are three basic kinds of electromagnetic radiation by charged particles that are Cherenkov or Smith-Purcell radiation, transition radiation and bremsstrahlung [22]-[23]. Cherenkov radiation occurs when electron velocity is greater than phase velocity of electromagnetic waves. Transition radiation occurs when a charged particle passes through inhomogeneous media, such as a boundary between two different media. Klystrons are the examples for the HPM devices which transition radiation occurs. Bremsstrahlung radiation occurs when electrons oscillate in the external magnetic field or electric field. Cyclotron resonance masers (CRMs), free-electron lasers and vircators are the examples for bremsstrahlung devices. Alternatively, HPM sources can be classified into the 3 categories that are O-type, M-type and space charge type. For O-type devices, electrons move parallel to strong magnetic field. BWO and TWT are the examples for this type of device. For M-type devices, electrons drift perpendicularly to crossed electric and magnetic fields. Magnetron, MILO are the examples of the M type devices. In the last case, current exceeds the space charge limit and formation of virtual cathode is observed. Vircator is a good example to understand the radiation principle of this type of devices. Klystron is an O-type devices which composed of electron gun, drift tube and collector. RF input signal is coupled to cavity with the same resonant frequency of the cavity fundamental mode TM010. Depending of the phase of the fields excited by input RF signal, electrons passing through cavity are accelerated or decelerated by electric fields in the cavity. This results in bunches formation in the beam and it travels along drift tube. Magnetron is a M-type Cherenkov device utilizing crossed electric and magnetic fields to generate microwave signal. Traveling wave tube amplifiers (TWTs) utilize axial magnetic field to focus the electron beam and use a slow wave structure such as a helix or coupled cavities. 2.2 Backward-Wave Oscillators Backward wave oscillator is one of the microwave vacuum electronic devices (MVED). It comprises of mainly 3 parts which are electron gun, slow-wave structure and output antenna. In 2.2.1 (Conventional SWS), sinusoidally corrugated, trapezoidal, and solid rectangular corrugations are analyzed. In 2.2.2 (Metamaterial SWS), Left Handed Media (LHM) with negative permittivity and permeability are explored, enabling reversed Cherenkov radiation, subwavelength operation, and enhanced coupling impedance. CHAPTER 3 THEORY OF OPERATION 3.1 Modes on Circular Waveguide A hollow circular waveguide supports TE and TM waveguide modes. For TM modes, Hz = 0, and Ez is the solution of the wave equation: ∇²Ez + k²Ez = 0 (3-1) In cylindrical coordinates: (∂²/∂r² + (1/r)(∂/∂r) + (1/r²)(∂²/∂φ²) + kc²)ez = 0 (3-2) ez(r, φ) = (A sin(nφ) + B cos(nφ)) Jn(kcr) (3-3) With PEC boundary at r = a, Ez(a, φ) = 0, giving: kc = pnm / a (3-4) βmn = √(k² - (pnm/a)²) (3-5) fcnm = kc / (2π√(με)) (3-6) 3.2 Group Velocity and Phase Velocity Phase velocity is expressed as: Vp = ω / k (3-7) Group velocity is expressed as: Vg = ∂ω / ∂k (3-8) 3.3 Dispersion Relation Brillouin diagram (ω-β diagram) represents the relationship between angular frequency and propagation constant. f = ∑_{m=0}^∞ am cos(mβp) (3-9) 3.4 Interaction Impedance To characterize beam-wave interaction: K0 = |Ezm|² / (2 βzm² W vg) (3-10) where Ezm is the axial electric field, βzm is the propagation constant of the m-th spatial harmonic, W is energy stored per waveguide period, and vg is group velocity. 3.5 Electron Beam & 3.6 Backward Cherenkov Radiation Cherenkov synchronism condition: ω - kz vz = 0 (3-11) 3.7 Cherenkov-Cyclotron Instabilities Condition: ω - kz vz = n Ωc / γ (3-12) where mv²/r = qBv (3-13), ω = v/r = qB/m (3-14), and γ = 1 / √(1 - vz²/c²) (3-15). 3.8 Introduction to Metamaterial Drude-Lorentz constitutive relations: ε(ω) = 1 - (ωp² - ω0e²) / (ω(ω + jΓe) - ω0e²) (3-18) μ(ω) = 1 - Fω² / (ω(ω + jΓm) - ω0m²) (3-19) n(ω) = √(ε(ω)μ(ω)) (3-22) W = (∂(ωε)/∂ω) E² + (∂(ωμ)/∂ω) H² (3-23) CHAPTER 4 DESIGN AND SIMULATION OF SLOW WAVE STRUCTURE 4.1 Simulation Environment : CST Studio Suite CST Studio Suite modules utilized: Time Domain Solver (FIT), Frequency Domain Solver, Eigen Mode Solver (AKS and JDM), and Particle-in-cell (PIC) Solver. 4.2 Cold Test Simulation of SWS Unit cell dimensions (Table 4.1): x_mtm = 19.25 mm, y_mtm = 20.00 mm, gap_mtm = 13.00 mm, t_x_mtm = 2.00 mm, R_trap = 6.00 mm, p = 25.00 mm, d = 7.00 mm, t_z_mtm = 1.00 mm. 4.2.1 & 4.2.2 Mode Analysis & Rectangular SWS Cut-off frequencies for 40 mm diameter hollow waveguide: TE11: 4.40 GHz, TM01: 5.75 GHz, TE01: 7.29 GHz. Loading with SWS shifts the TM-like operating passband to 2.43 - 2.72 GHz (Mode 3). Group velocity is negative (-0.02c to -0.075c), proving backward wave propagation. Normalized phase velocity ranges from 0.4c to 0.9c. Interaction impedance ranges between 10 Ω and 160 Ω across the operating band. 4.3 Validation of SWS An 8-unit cell structure was fabricated (CNC machining and laser cutting) and excited by an axial pin connector. Measured S11 was below -10 dB in the 2.43-2.72 GHz range, with clear resonance peaks correlating directly with the dispersion diagram and cold-test simulation. 4.4 Hot Test (PIC Solver) Simulation of SWS Hot test parameters: 450 kV, 100 A DC annular electron beam (outer radius 5 mm, inner radius 2.2 mm), guided by a 2 T axial magnetic field. The output waveguide has 110 mm diameter (TM01 cutoff = 2.10 GHz). The system produced stable oscillations at 2.49 GHz, yielding 8.3 MW in Mode 3 (10.7 MW total peak power including Mode 1) with 23.8% peak electronic efficiency. CHAPTER 5 CONCLUSION Vulnerability of electronic systems has increased because of miniaturization and non-metal packaging, heightening interest in High Power Microwave (HPM) systems. In this thesis, a novel metamaterial-based slow wave structure for backward wave oscillators was designed, simulated, fabricated, and validated. Key Achievements: 1. Designed a subwavelength metamaterial SWS operating in the 2.43-2.72 GHz frequency band for TM-like mode. 2. Reduced transverse waveguide radius to 20 mm (which is evanescent for empty waveguide at this band). 3. Verified negative group velocity and slow phase velocity confirming backward Cherenkov interaction conditions. 4. Validated cold-test performance via experimental S-parameter measurements on an 8-cell fabricated prototype. 5. 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