The Neolithic of Europe: papers in honour of Alasdair Whittle

c 2017 Marcus Pearlman ALL RIGHTS RESERVED BOISE STATE UNIVERSITY GRADUATE COLLEGE DEFENSE COMMITTEE AND FINAL READING APPROVALS of the dissertation submitted by Marcus Pearlman Dissertation Title: Simulation of a Crossed-Field Amplifier Using a Modulated Distributed Cathode Date of Final Oral Examination: 15 February 2017 The following individuals read and discussed the dissertation submitted by student Marcus Pearlman, and they evaluated his presentation and response to questions during the final oral examination. They found that the student passed the final oral examination. Jim Browning, Ph.D. Chair, Supervisory Committee Wan Kuang, Ph.D. Member, Supervisory Committee Kris A. Campbell, Ph.D. Member, Supervisory Committee Yue Yin Lau, Ph.D. External Examiner The final reading approval of the dissertation was granted by Jim Browning, Ph.D., Chair of the Supervisory Committee. The dissertation was approved by the Graduate College. ACKNOWLEDGMENTS I would like to thank the Boise State University department of Electrical and Computer Engineering for financial support of this research. I thank the U.S. Air Force Office of Scientific Research for their financial support under Grant FA955012-C-0066. I also express my gratitude to TechX Corporation for their support of our work with Vsim and in particular David Smithe for his help. And, of course, I am immensely grateful to the review committee, Wan Kuang, Kris Campbell, Jim Browning, and Yue Ying Lau for their time and effort in validating this dissertation. I also want to thank Janos Cserna for helping develop much of the experiment, specifically the X-Y stage used to measure the dispersion. I also wish to thank Kyle Straub and Tyler Rowe for their part in developing the experimental setup, and for their collaboration. v ABSTRACT Current crossed-field amplifiers (CFAs) use a uniformly distributed electron beam, and in this work, the effects of using a spatially and temporally controlled electron source are simulated and studied. Spatial and temporal modulation of the electron source in other microwave vacuum electron devices have shown an increase in gain and efficiency over a continuous current source, and it is expected that similar progress will be made with CFAs. Experimentally, for accurate control over the electron emission profile, integration of gated field emitter arrays (GFEAs) as the distributed electron source in a crossed-field amplifier (CFA) is proposed. Two linear format, 600 and 900 MHz CFAs, which use GFEAs in conjunction with hop funnels as an electron source, were designed, modeled in VSim, and built at BSU. The hop funnels provide a way to control the energy of the electron beam separately from the sole potential and to protect the GFEA cathode. The dispersion of the meandering microstrip line slow wave circuit used in the device and the electron beam characteristics were measured and validated the simulation model, but experiments failed to show electron beam interaction with the electromagnetic wave due to insufficient current from the available cathode. To complete the research, a working CFA built at Northeastern University (NU) was modeled. The NU CFA was a linear format, device operating at 150 MHz, with 10 W of RF input power, and typically 150 mA of injected beam current. The electrically short device (6 slow wave wavelengths long) achieved 7 dB of gain. After validating the Vsim model against the experimental results, an electrically longer version (9 wavelengths) vi was simulated with both an injected beam and distributed cathode. To model the distributed cathode computationally efficiently, where the emitted electron energy can be controlled separately from the sole potential, a new electron injection method was developed, using a divergence-free region. Static electron emission profiles showed no improvement over the injected beam model but the temporally modulated cathode was found to significantly improve the performance. It was found that the temporal modulation could improve the small-signal-gain from 13 dB for an unmodulated source to 25 dB with an injected current of 150 mA and 0.1 W of RF drive power. This improvement is only likely to be observed for higher power devices (>10 kW) because of the additional RF drive power required by the GFEA, however. For larger RF drive powers, the improvements to gain become much smaller. With an RF drive power of 10 W, the modulated cathode showed 9 dB of gain, and the injected beam variant showed 8 dB. The signal-to-noise ratio (SNR) using the modulated cathode was consistently at least 15 dB higher than the SNR of the unmodulated cathode. This reduces the likelihood of excitation of unwanted modes. Even though this device showed small improvements to gain at large RF drive powers, it is proposed here that improvements to maximum power in higher power devices are likely, due to the inherent mode-locking mechanism of the modulated cathode, but this still needs to be confirmed. Previous research studying the effects of a modulated cathode in a magnetron and the improvements to the SNR shown here, show promise in this regard. vii TABLE OF CONTENTS ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv LIST OF ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxii 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Device Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Research Objectives and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Overview of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1 Crossed-Field Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Wave Velocities and Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Slow Wave Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.1 Helix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.2 Meander Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 Electron Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.5 RF-Beam Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 viii 2.6 2.7 2.8 2.5.1 Interaction Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.5.2 Beam Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.5.3 Theoretical Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Electron Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.6.1 Thermionic Cathodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.6.2 Emitting Sole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.6.3 Field Emitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.6.4 Hop Funnels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.7.1 COMSOL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.7.2 SIMION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.7.3 Vsim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 State of the Art in MVEDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.8.1 Cylindrical Emitting Sole CFAs . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.8.2 Linear Format Injected Beam CFAs . . . . . . . . . . . . . . . . . . . . . . 63 2.8.3 Linear Format CFA at Northeastern University . . . . . . . . . . . . . 66 2.8.4 Simulation of a Distributed Cathode in a Rising Sun Magnetron 69 2.8.5 Field Emitter Use in Microwave Vacuum Electron Devices . . . . . 70 3 Research Overview 3.1 ...................................... 75 Proposed Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.1.1 Injected Beam Configuration Experiment . . . . . . . . . . . . . . . . . . 76 3.1.2 Meander Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.1.3 Cathode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.1.4 Distributed Cathode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 ix 3.1.5 3.2 Sole/Hop Funnels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Research Chronology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4 CFA Experiments and Measurements . . . . . . . . . . . . . . . . . . . . . . . . 4.1 4.2 85 Full CFA Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.1.1 Vacuum Chamber and Electromagnets . . . . . . . . . . . . . . . . . . . . 85 4.1.2 CFA Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.1.3 Slow Wave Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.1.4 GFEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Meander Line Dispersion Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.2.1 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.1 COMSOL Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.2 SIMION Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.3 Vsim Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.3.1 The VSim Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.3.2 Create the geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.3.3 Non-Uniform Grid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.3.4 Uniform Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.3.5 Injected Beam Cathode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.3.6 Distributed Cathode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.3.7 Distributed Cathode With Spatial and Time Varying Current . 138 6 Experimental and Simulated Results and Discussion of BSU CFA 146 6.1 CFA Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 x 6.2 6.3 Meander Line Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6.2.1 S-Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6.2.2 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 6.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Beam Optics: SIMION Comparison With Experiment . . . . . . . . . . . . . . 158 6.3.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.3.2 SIMION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.4 Vsim Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 7 Experimental and Simulated Results and Discussion of Northeastern University CFA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 7.1 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 7.2 Electron Optics in Vsim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 7.3 7.4 7.2.1 Cathode Placement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 7.2.2 Beam Electrode Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 7.2.3 Cathode to Sole and Anode to Sole Voltage Study . . . . . . . . . . . 176 Injected Beam Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 7.3.1 Optimum Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 7.3.2 Beam Current Sweep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 7.3.3 Bandwidth Sweep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Injected Beam Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 7.4.1 Resolution Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 xi 7.5 7.4.2 RF Power and Beam Power Gain Study . . . . . . . . . . . . . . . . . . . 188 7.4.3 Gain vs. Circuit Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 Distributed Cathode Approximation Studies . . . . . . . . . . . . . . . . . . . . . 204 7.5.1 Cathode Approximation 1: Raised Cathode . . . . . . . . . . . . . . . . 205 7.5.2 Cathode Approximation Two: Segmented Cathode . . . . . . . . . . 207 7.5.3 Cathode Approximation Three: Raised Cathode With Approximated Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 7.6 7.7 7.8 Distributed Cathode Studies: Static . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 7.6.1 Uniform Emission Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 7.6.2 Linear Emission Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 7.6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Distributed Cathode Studies: Time Varying . . . . . . . . . . . . . . . . . . . . . 222 7.7.1 Sine Wave Emission Profile Results . . . . . . . . . . . . . . . . . . . . . . 223 7.7.2 Injected Beam Using Sine Wave Profile . . . . . . . . . . . . . . . . . . . 225 7.7.3 Square Pulse Emission Profile Results . . . . . . . . . . . . . . . . . . . . 226 7.7.4 General Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 7.7.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 7.7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 Implementation on Higher Power CFA Designs . . . . . . . . . . . . . . . . . . . 244 7.8.1 Direct Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 7.8.2 Alternate Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 8.1 Model Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 8.2 General Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 xii 8.3 Distributed Cathode Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 8.3.1 Static Current Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 8.3.2 Time-Varying Current Distributions . . . . . . . . . . . . . . . . . . . . . . 256 8.4 Method Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 8.5 Modulated Distributed Cathode Importance . . . . . . . . . . . . . . . . . . . . . 259 8.6 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 A CFA Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 A.1 Measurement and Control Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 A.1.1 CFA Ground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 A.1.2 Earth Ground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 A.1.3 Opto-Isolators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 A.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 xiii LIST OF TABLES 1.1 Summary of Microwave Vacuum Electron Devices . . . . . . . . . . . . . . . . 3 3.1 Slow wave specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.1 Summary of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.2 Slow wave circuit and CFA dimensions of the NU experiment and the VSim adjustments. Bold listed elements are parameters which are altered in the VSim simulation to align well with the coarse grid. . . . . 107 xiv LIST OF FIGURES 1.1 Range of applications of MVEDs in comparison with solid state devices. Reproduced with permission from [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 2 Cylindrical, injected beam, non-reentrant, backward wave crossed field amplifier. [17] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Cylindrical, emitting sole, reentrant, forward wave crossed field amplifier. [17] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Linear format injected beam crossed field amplifier [2]. . . . . . . . . . . . . 13 2.4 Dispersion diagram for waves traveling in either direction in a rectangular waveguide [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.5 Dispersion diagram for waves traveling in either direction in a periodically loaded rectangular waveguide [2]. 2.6 . . . . . . . . . . . . . . . . . . . . . . . . 16 Dispersion diagram for periodically loaded rectangular waveguide showing multiple harmonics [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.7 Helix (a) and view of helix cut at each x and unrolled (b).[7] . . . . . . . . 19 2.8 Backward wave interaction at two turns per wavelength. [7] . . . . . . . . . 20 2.9 A microstrip type meander line, showing a conducting meander circuit over a dielectric material and a ground plane. xv . . . . . . . . . . . . . . . . . . . 21 2.10 Electron forces and trajectories for (a) a constant electric field and no magnetic field, (b) a constant magnetic field into the page and no electric field, and (c) a constant electric field perpendicular to a constant magnetic field into the page. . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.11 Electron trajectories for crossed electric and magnetic fields for various initial velocities (u0 ) [2]. ωc is the cyclotron frequency defined by ωc = qB/m, where q is the particle charge, B is the magnitude of the magnetic field, and m is the particle mass. . . . . . . . . . . . . . . . . . . . 25 2.12 Voltage vs. magnetic field showing the operation region of magnetron [2], which is the same for a CFA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.13 Motions of electrons due to the RF field in a rotating coordinate system of a CFA. Electrons in positive RF potentials move towards the sole as they give up energy, and move clockwise into the decelerating region (the region between positive and negative RF potentials where the electric field points clockwise). Electrons in negative RF potentials gain energy, and cycloid right back into the sole. Electrons in the decelerating regions remain in the region but move towards the sole as they give up energy. [17] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.14 Plot of Pierce theory efficiency as a function of beam current for the NU CFA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.15 Energy Level Diagram near the surface of a metal [2] . . . . . . . . . . . . . . 35 2.16 Fermi-Dirac distribution for various temperatures. . . . . . . . . . . . . . . . . 36 2.17 A typical secondary electron yield curve for an arbitrary material [2]. . 38 2.18 Diagram showing the ’multiplication’ of electrons on the surface of an emitting sole cathode[2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 xvi 2.19 Gated field emitter diagram showing the field enhancement near the needle tip [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.20 Energy level diagram at the surface of a material with and without an applied electric field [24]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.21 The current density vs. the gate emitter voltage of the gated field emitters fabricated by Guerra et. al. [25] . . . . . . . . . . . . . . . . . . . . . . . 46 2.22 Hop funnels used in [37], showing the operation during (a) full electron transmission and (b) no transmission using the Lorentz 2E [60] simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.23 Hop funnel structure with sole electrode using Lorentz 2E [60] . . . . . . . 48 2.24 Yee grid showing the position of the various field components. Electric field components are on the middle of the edges and magnetic field components are on the center of the faces. . . . . . . . . . . . . . . . . . . . . . . 56 2.25 FDTD simulation flow [71]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.26 Double helix coupled vane slow wave structure commonly used in CFAs. [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.27 The current power capabilities of published CFA data. [2]. . . . . . . . . . 61 2.28 (a) conventional CFA comparison with a (b) cathode-driven and a (c) hybrid variant. [85]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.29 (a) Short and (b) long Kino electron gun schematics [87]. . . . . . . . . . . 64 2.30 Northeastern CFA schematic in Browning et. al. [14, 15] . . . . . . . . . . . 67 2.31 Northeastern CFA Gain vs. frequency plot in Browning et. al.Vas = 1250 V, B = 5.2 mT [14] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.32 Northeastern CFA Gain vs. Beam current plot in Browning et. al. VAS = 1200 V, B = 5.5 mT [15] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 xvii 2.33 Northeastern CFA Gain vs. circuit with and without an electron beam in Browning et. al. Prf = 10 W, VAS = 1200 V, B = 5.5 mT[15] . . . . . . . 69 2.34 GFEA matching circuit used in the TWT work [8], proposed by Calame [97] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.35 (top) The transparent cathode configuration with 6 cathode strips and (bottom) a solid cathode [98]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.1 Schematic representation of the injected beam CFA design with dimensions, not to scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.2 The diagram showing the meander line microstrip. A metal line meanders over a dielectric with thickness Hd over a ground plane. . . . . . . 78 3.3 Schematic representation of the distributed cathode CFA design, not to scale. Electrons injected into the hop funnels are extracted though slits in the sole electrode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.4 Diagram outlining the research flow of the three CFA designs. The BSU experimental work was used to validate the simulation model, but all work on the BSU CFAs were terminated after determining the design was unfit. Results from the Northeastern CFA experimental work were also used to validate the simulation model, and the design was used for the distributed cathode studies. . . . . . . . . . . . . . . . . . . . . 82 4.1 Photograph of the electromagnets and the chamber system where the CFA experiments are run. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.2 Schematic representation of the CFA design, not drawn to proportion. 4.3 Top down view of the CFA structure without the slow wave circuit. . . 88 xviii 87 4.4 Photograph of slow wave circuit SW2. A rectangular copper wire meanders on top of a Teflon dielectric which is on top of an aluminum ground plane. The copper wire is fixed to the ground plane by polypropylene screws. The input an output ports are SMA connectors which are connected to the copper wire by silver paste. . . . . . . . . . . . . 89 4.5 Top down view of the CFA structure without the slow wave circuit and the end hats to show the PixTech cathode and the gate and emitter connections. 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Photograph of the standing wave measurement setup. The slow wave circuit sits on top of an x-y stage, and a coaxial cable connected to a spectrum analyzer on one end and the other end is placed right over the slow wave circuit with the center conductor exposed. . . . . . . . . . . . 93 5.1 COMSOL model for SW2 showing the generated mesh . . . . . . . . . . . . . 95 5.2 SIMION CFA configuration from the side (normal to the x − y plane). Electrons cycloid from right to left in this model. . . . . . . . . . . . . . . . . . 96 5.3 Vsim Geometry with electrons. The RF wave is input on the edge of the domain, within the coaxial port. The RF wave travels within the dielectric region between the ground plane and the green meander line. Electrons are emitted from the cathode region, and cycloid right due to the crossed electric and magnetic fields. The electrons interact with the RF wave and give up their energy to amplify the RF wave. . . . . . . . 97 xix 5.4 View of the dimensions of the slow wave circuit and ports from the top view, normal to the y-axis. The green meander line comes down (in the y-direction) through the outer conductor of the coaxial cable, and then meanders above the dielectric (not shown) and ground plane shown in red on the x − z plane. 5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 View of the VSim model, showing the dimensions of the slow wave circuit and ports from the side view, normal to the Z-axis for the NU CFA study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.6 Top down view (normal to y-axis) of (a) the meander line geometry with a good alignment with the grid and (b) the corresponding Ey field of a generic run . The green section denotes locations where Ey = 0 which corresponds to conductor, and the blue part is vacuum. . . . . . . . 104 5.7 Top down view (normal to y-axis) of the meander line geometry with a poor alignment with the grid (a) and the corresponding Ey field of a generic run (b). The green section denotes locations where Ey = 0 which corresponds to conductor, and the blue part is vacuum. . . . . . . . 105 5.8 Top down view (normal to y-axis) of the input coaxial cable geometry (a) and the corresponding Ey field of a generic run (b). The green section denotes locations where Ey = 0 which corresponds to conductor, and the blue part is vacuum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 xx 5.9 The boundary conditions used to control the electron beam injection and cycloid trajectory. The electrons are emitted at a potential 200 V more positive than the sole so that the cycloiding electrons do not easily collect on the sole. The beam electrode is placed there to control the beam injection into the region between the anode and sole. The end hats are outlined with a dotted line and are at z = 0 and the upper edge of the z domain. Periodic boundaries are at the edges of the x and z domain. The periodic BCs allow for smaller model by keeping smooth electric fields at the edges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.10 Corresponding potentials of the beam optics in the VSim model. . . . . . 109 5.11 View of the dimensions model of the NU CFA with particles from the top view, normal to the y-axis. Electrons are shown in blue dots to demonstrate space charge spreading the beam towards the z edges and to show the end hats reflecting the beam back towards the center. . . . 110 5.12 Vsim particle boundary conditions showing the the dielectric and beam electrode, which are not particle sinks, and boundary absorber, cathode, sole, and end collector, which are particle sinks. . . . . . . . . . . . . . . 113 5.13 Vsim non-uniform mesh on the X-Z plane. The green section is the meander line, red is the ground plane, The white circles are the space between the inner and outer conductor of the coaxial cable, and the black lines are the mesh. In X, regions which coincide with the circuit is 2 cells wide and regions between the circuit is 2 cells wide. In Z, the circuit region is 2 cells wide, but in the center, the length of the cells is increased. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 xxi 5.14 Vsim non-uniform mesh on the Y-X plane. The green section is the meander line, red is the ground plane, The white circles are the space between the inner and outer conductor of the coaxial cable, and the black lines are the mesh. In X, regions which coincide with the circuit is 2 cells wide and regions between the circuit is 2 cells wide. In Z, the circuit region is 2 cells wide, but in the center, the length of the cells is increased. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.15 E-field diagnostic showing the charge accumulation in the non-uniform grid model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.16 Cathode/Sole approximation one. The ERF = 0 region is placed right above the sole electrode so that electrons can be emitted into the interaction region from a potential less negative than the sole. The ERF = 0 sets the electric fields equal to zero to prevent accumulation of charge at the electron emission location. . . . . . . . . . . . . . . . . . . . . . . 125 5.17 The first configuration of the cathode/sole approximation 2. Cathode potentials are pink, and sole potentials are green. Only the first three cathode potentials emit electrons. Electrons in this case have an ’erratic’ trajectory as they leave the cathode. Also, the cycloid radius is a multiple of the cathode separation length, and many electrons just squeeze right back through the cathode potential at the right two cathode potential locations and are collected on the cathode/sole region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.18 A close up on the cathode/region of cathode approximation two, showing the dimensions. This view also shows the ’erratic’ electron trajectories as they leave the cathode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 xxii 5.19 The second configuration of the Cathode/Sole approximation 2. Electron emission points are offset to the right of the cathode potentials (pink) in between the cathode and the sole. The cathode separation is optimized so that cycloid radius is offset from cathode segments down the tube and electrons can be repelled back into the interaction region. 128 5.20 A simple example of the divergence free region current propagation from the injection point to the domain edge at (a) t = t0 , (b) t = t1 , (c) t = t2 . Red dots indicate cells where the divergence is not equal to zero, and green dots are divergence free points. Index notation is used where Jij indicates the current density at cell number i in the x-direction. The divergence free region in this example is 3 cells high, and takes 3 timesteps for the current originating from the third row of cells to propagate to the y = 0 edge. . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.21 The y-component of the electric field overlaid with cycloiding electrons in a (a) standard vacuum region and in a (b) divergence free region. The electron beam spread at x = 7.0 cm is shown in the figures to emphasize the difference in electron beam trajectories. . . . . . . . . . . . . . 134 5.22 The y-component of the electric field overlaid with electrons electrons in (a) a standard vacuum region and (b) in a divergence free region . . 136 5.23 The x-component of the electric field overlaid with electrons (a) in a standard vacuum region and (b) in a divergence free region . . . . . . . . . 137 5.24 Three spatial profiles: linear profile with positive slope and no DC current in blue, linear profiles with negative slope and 50% DC current in magenta, and a uniform profile in green. xxiii . . . . . . . . . . . . . . . . . . . . . 140 5.25 The sine wave electron emission profile compared to the ERF x field with φof f set = 0 rad at ωt = φt . In this case the profile peaks are in the accelerating regions of the RF wave (out of phase). . . . . . . . . . . . . 141 5.26 The square pulse electron emission profile compared to the RF electric field in the x-direction (ERF x ) at ωt = φt . In this case the pulse is “out of phase” with the RF field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.27 Proposed resonant circuit to minimize consumed power by the GFEA. 144 5.28 Calculated electron current density pulses from the MIT GFEAs [25] for a sinusoidal input from (a) Vg = 20 − 50 V and (b) Vg = 35 − 50 V. 145 6.1 S-parameters of SW2 from both simulation (COMSOL) and measured (network analyzer) showing the cutoff frequency. . . . . . . . . . . . . . . . . . 148 6.2 Measured electric field intensity for SW2 for frequencies (a) 500 MHz and (b) 1.15 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 6.3 COMSOL simulated electric field intensity for SW2 for frequencies (a) 500 MHz and (b) 1.15 GHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 6.4 2D spatial FFT of the measured and simulated electric field intensity for SW2 for frequencies (a) 500 MHz and (b) 1.15 GHz. . . . . . . . . . . . . . 151 6.5 (a) Voltage through time of the VSim dispersion model using an impulse signal and periodic boundaries in x, and (b) the FFT of that signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.6 Measured and Simulated dispersion diagram for SW2 using (a) xcomponent of the spatial FFT , and using (b) the FFT along the center in the x-direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 xxiv 6.7 Measured and simulated retardations of SW2 using the FFT along the center in the x-direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.8 Experimental I-B Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 6.9 SIMION I-B Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 6.10 Sample Trajectories for three different magnetic fields. With lower magnetic fields, more electron trajectories collect on the slow wave circuit, at high magnetic fields, most of the current travels down the tube to the end collector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.11 The SW2 model used in VSim with electrons shown in blue and the slow wave circuit in red . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 6.12 Gain along the length of the SW2 circuit compared to the predicted gain of Pierce theory for Prf = 1 W, Ibeam = 20 mA and 150 mA, and a circuit whose length is 2.5 times as long as the experimental circuit. 165 7.1 (a)Dispersion and (b) the retardation vs. frequency of SW3 as calculated from the VSim model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 7.2 Vsim model showing the dummy electrode used to control the beam injection in the NU CFA and the cathode placement dimensions. 7.3 . . . . 171 (a) Gain and (b) the measured currents on various electrodes vs. the position of the cathode for Ibeam = 150 mA, B = 5.2 mT, and Vas = 1250 V. The Position at 0 m corresponds to directly underneath the first period of the slow wave circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 7.4 Electron trajectories corresponding to (a) 1.3 cm cathode offset placement and (b) 4.0 cm cathode offset placement. Electrons are shown in blue and the slow wave circuit is red. . . . . . . . . . . . . . . . . . . . . . . . . . . 173 xxv 7.5 (a) Gain and (b) the measured currents on various electrodes vs. the potential of the beam electrode for Ibeam = 150 mA, B = 5.2 mT, and Vas = 1250 V. The Position at 0 m corresponds to directly underneath the first period of the slow wave circuit. . . . . . . . . . . . . . . . . . . . . . . . . 175 7.6 Electron trajectories for VAS = 1550 V, and different values of VCS , (a) 200 V, (b) 300 V, (c) 400 V, and (d) 500 V. . . . . . . . . . . . . . . . . . . . . . . 177 7.7 (a) Gain vs. Vcs for various Vas and (b) Gain vs. Vas for various Vcs . The retardation is held constant by adjusting the magnetic field to keep the V /B ratio constant. 7.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 (a) Gain vs. Magnetic Field for Ibeam = 150 mA, Vas = 1250 V, Vbo = −200 V and (b) the corresponding currents from VSim. . . . . . . . . . . . . 180 7.9 (a) Gain vs. injected beam current and comparison to experimental data found in Browning et al. [14, 15] and (b) the corresponding currents from VSim. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 7.10 Efficiency vs. injected beam current and comparison to experimental data found in Browning et al. [14, 15] . . . . . . . . . . . . . . . . . . . . . . . . . . 184 7.11 (a) Gain vs. Frequency and comparison to experimental data found in Browning et al. [14] and (b) the corresponding currents . . . . . . . . . . . . . 185 7.12 (a) Gain vs. RF power for Ibeam = 150 mA and (b) the corresponding efficiency vs. RF input power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 7.13 Signal-to-Noise ratio for different powers . . . . . . . . . . . . . . . . . . . . . . . . 192 7.14 The input and output voltage with Ibeam = 150 mA for (a) Prf = 10 mW and (b) Prf = 0.1 mW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 7.15 Gain vs. RF power for Ibeam = 100 mA . . . . . . . . . . . . . . . . . . . . . . . . . 195 xxvi 7.16 (a) The Gain vs. the SW circuit length with Prf = 1 W and Ibeam = 150 mA and (b) the corresponding simulated efficiency . . . . . . . . . . . . . 197 7.17 (a) The Gain vs. the SW circuit length with Prf = 10 W and Ibeam = 150 mA and (b) the corresponding simulated efficiency . . . . . . . . . . . . . 198 7.18 Gain along the length of the circuit for both VSim simulation (red), and Pierce theory (blue). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 7.19 The AC coupled RMS y -component of the electric field, in red, and the moving average, in blue, along the length of the circuit for (a)0.625 cm, and (b) 1.25 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 7.20 Two electron trajectories with different cycloid radii. The blue trajectory has half the cycloid radius than the red trajectory. Er1 and Er2 are the estimated energy extracted from the smaller and larger radii, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 7.21 A variation of the cathode approximation method one to test the ERF = 0 region effects on gain. This variation emits electrons similarly to the injected beam configuration but there is the ERF = 0 region. Electrons are allowed to enter the region but no RF fields are calculated there. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 7.22 (a) The gain and the corresponding (b) currents vs. the conducting boundary region thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 7.23 Electron trajectories for 7 emitters, VAS = 1650 V, B = 6.8 mT, and different values of Vcs , (a) 200 V, (b) 300 V, and (c) 400 V. 7.24 (a) The gain and the corresponding (b) currents vs. . . . . . . . . . 209 the cathode separation for Vas = 1650 V and B = 6.7 mT. . . . . . . . . . . . . . . . . . . . . 211 xxvii 7.25 (a) The gain and the corresponding (b) currents vs. the magnetic field with Nc2c = 11 cells for the cathode approximation 2. . . . . . . . . . . . . . . 212 7.26 The two setups to determine the effect of the divergence free region. Emission from a (a) conducting region and from (b) the divergence-free region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 7.27 Gain vs. emission length, Le , for the both divergence-free (DF) cathode approximation and the raised emitting conductor. . . . . . . . . . . . . . . . . 214 7.28 Optimization of the CFA parameters using the Divergence-free region showing the (a) gain and (b) currents vs. Vas . Vcs = 200 V and the magnetic field is optimized for each voltage point to maximize gain. . . . 216 7.29 (a) The gain and the corresponding (b) SNR vs. the RF input power for the injected beam configuration wit a 1.5 cm emitter length and the distributed cathode configuration with 10 cm and 20 cm emitter lengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 7.30 (a) The gain and the corresponding (b) SNR vs. the cathode length for the distributed cathode approximation with Prf = 0.1 W . . . . . . . . 219 7.31 (a) The gain and the corresponding (b) currents vs. the beam current for the uniform emission profile with Le = 20 cm and Prf = 0.1 W. . . . 220 7.32 (a) The gain and the corresponding (b) SNR vs. the phase difference between the beam profile and the RF wave for Le = 30 cm, Prf = 1 W, and Ibeam = 150 mA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 7.33 (a) The gain and the corresponding (b) SNR vs. the phase difference between the beam profile and the RF wave for Le = 30 cm, Prf = 1 W, and Ibeam = 150 mA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 xxviii 7.34 (a) The gain and the corresponding (b) SNR vs. the phase difference between the beam profile and the RF wave for Le = 30 cm, Prf = 1 W, and Ibeam = 150 mA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 7.35 The (a) phase and the (b) gain along the circuit for the square pulse profile with φof f set = 0 rad, Le = 30 cm, Lp = 1 cm, Prf = 1 W, and Ibeam = 150 mA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 7.36 The (a) phase and the (b) gain along the circuit for the square pulse profile with φof f set = π rad,Le = 30 cm, Lp = 1 cm, Prf = 1 W, and Ibeam = 150 mA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 7.37 (a) The gain and the corresponding (b) SNR vs. the phase difference between the beam profile and the RF wave for Le = 30 cm, Prf = 1 W, and Ibeam = 150 mA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 7.38 (a) The gain and the corresponding (b) output power vs. the RF input power on the circuit for both the modulated current and the injected beam with Ibeam = 150 mA. The modulated cathode uses Le = 30 cm. The RF input power on the x-axis does not include the modulated cathode power. In (a), the red line includes a 1 W modulated cathode power and the blue line includes a 0.1 W modulated cathode power. . . 230 7.39 (a) The efficiency and the (b) SNR vs. the RF input power on the circuit for both the modulated cathode in red and the injected beam in green with Ibeam = 150 mA. The modulated cathode uses Le = 30 cm. The RF input power on the x-axis does not include the modulated cathode power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 xxix 7.40 The RF input signal contribution to (a) gain and the corresponding (b) output power vs. the RF input power on the circuit for both the modulated and injected beam cathode. The RF input power on the x-axis does not include the modulated cathode power. . . . . . . . . . . . . . 232 7.41 (a) The gain and the corresponding (b) output power vs. the beam current for both the modulated and uniform current distributions with Prf = 0.1 W. In (a), the red line includes a 1 W modulated cathode power, the blue line includes a 0.1 W modulated cathode power, and the green line shows the uniform current case. The modulated cathode uses Le = 30 cm and the uniform current uses Le = 20 cm. . . . . . . . . . . 233 7.42 (a) The efficiency and the (b) SNR vs. beam current on the circuit for the modulated distribution in red and the uniform distribution in green. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 7.43 (a) The currents for the modulated cathode compared to (b) the currents of the uniform cathode for the beam current sweep. . . . . . . . . . . 234 7.44 Electron trajectories for the square pulse emission profile for Vas = 1550 V and Le = 1 cm as time progresses. . . . . . . . . . . . . . . . . . . . . . . . 235 7.45 Electron trajectories for various beam injection types and RF input powers with Vas = 1550 V and Ibeam = 150 mA. 7.46 Diagram of the shielded cathode slit concept. . . . . . . . . . . . . . . . . . . 236 Lateral gated field emitters on each side of the slit emit electrons and are pushed out through the slit in the sole electrode by the pusher electrode [112]. . . . 247 xxx 7.47 Diagram of a CFA using end hat assisted injection. The majority of the electron current is supplied by a traditional thermionic or secondary emitting cathode (violet arrows), and modulated electrons are injected in at the end hats (blue arrows). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 7.48 Diagram of the GFEA assisted secondary emitting cathode. GFEA locations (these could be hop funnel or the shielded cathode slits) emit electrons in phase with the RF wave with energies so that they collide with the secondary emitting sole at the optimum energy (Emax ) to emit the maximum amount of secondary electrons (δmax ). . . . . . . . . . . . . . . . 250 A.1 Measurement setup schematic. There are two potentials at which signals are measured, earth ground and CFA ground. Recording earth ground measurements is easily done by LabVIEW data acquisition (DAQ) Crate. CFA ground based measurements transmit the measurement signal through analog opto-isolaters. Control of CFA ground based currents and voltages is done by a CFA ground based microcontroller which communicates to the earth ground computer through digital opto-isolators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 A.2 RF system flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 A.3 RF system flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 A.4 Opto-isolator schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 xxxi LIST OF ABBREVIATIONS MVED – Microwave Vacuum Electron Device FEA – Field Emitter Array GFEA – Gated Field Emitter Array pFED – Printable Field Emission Display SEM – Scanning Electron Microscopy SEY – Secondary Electron Yield UHV – Ultra High Vacuum I-V curve – Transmitted Current Vs. Hop Voltage CFA – Crossed-Field Amplifier NU CFA – Northeastern University Crossed-Field Amplifier BSU CFA – Boise State University Crossed-Field Amplifier RF – Radio Frequency (commonly used to describe electromagnetic waves in general) CFA1, CFA2 – Refers to the two Crossed-Field Amplifiers created at Boise State University which use the slow wave circuits SW1 and SW2 respectively. SW1, SW2 – The first and second slow wave circuits designed at BSU TWT – Traveling Wave Tube MIT – Massachusetts Institute of Technology xxxii PIC – Particle In Cell MUMPS – Multifrontal Massively Parallel Sparse direct solver 3D – Three Dimensional FDTD – Finite Difference Time Domain PEC – Perfect Electric Conductor CFL – Courant-Friedrichs-Lewy NU – Northeastern University Sim. – Simulation Exp. – Experiment SNR – Signal-To-Noise Ratio LTCC – Low Temperature Co-Fired Ceramic BSU – Boise State University FFT – Fast Fourier Transform PML – Perfectly Matched Layer ES – electrostatic BC – Boundary Condition EM – Electromagnetic DC – Direct Current (commonly used to specify a constant voltage, ie. DC bias voltage) SW – Slow Wave HV – High Voltage xxxiii RMS – Root Mean Squared PSD – Power Spectral Density xxxiv 1 CHAPTER 1 INTRODUCTION Microwave vacuum electron devices (MVEDs) are devices that utilize free electrons in vacuum to interact with the geometry, electric and magnetic fields, and radio frequency (RF) waves to create an amplifier or an oscillator. There are many different MVED device configurations, but these generally fall under three categories: O-Type (linear), M-Type (crossed-field), and fast-wave devices . Linear devices are configurations where the electron is focused by a magnetic field aligned with the beam. These type of devices extract kinetic energy from the beam to produce gain or oscillations. Examples of these devices are gridded tubes [1–5], klystrons [1, 2, 4, 6] and traveling wave tubes (TWTs) [1–4, 7–10]. Crossed-field devices are configurations with a static electric field perpendicular to a static magnetic field, and the electrons follow a cycloidal trajectory with the average drift velocity in the direction perpendicular to both fields. These devices extract potential energy from the beam to get gain. Examples of these are Magnetrons [1–4, 11–13] and Crossed-Field amplifiers (CFAs) [1–4, 13–19]. Fast-wave devices use the cyclotron frequency for operation; a gyrotron [1–4, 20] is one example of many gyro devices. MVEDs fall into a niche of high power and/or high frequency devices. Solid-state devices have not yet achieved the power density capabilities of MVEDs. Figure 1.1 shows a chart comparing MVEDs with solid state devices [2]. Each MVED 2 configuration has its own advantages and disadvantages with regards to several figures of merit: bandwidth, gain, maximum power, efficiency, size, and signal to noise ratio. A summary of the advantages and disadvantages of each device is difficult to compile because of the diversity within each group and the application. An attempt is made here, and Table 1.1 summarizes the advantages and the applications of common MVEDs. Generally, all the figures of merit listed for the device type cannot all be maximized simultaneously for one particular device because interconnectivity of the figures of merit. For example, generally gain is inversely proportional to the bandwidth, so the maximum listed gain listed for that device cannot exist simultaneously with the maximum bandwidth. Also, power at higher frequencies is much lower than power at lower frequencies, so the maximum listed power is generally not associated with the maximum listed frequency. Therefore, the specifications listed in the chart Average Power (W) are rather subjective. 10 6 10 4 Vacuum Devices Dominate Reg i on 10 1 10 of C om p etit 2 ion Solid State Dominates –2 0.1 1 10 Frequency (GHz) 100 1000 Figure 1.1: Range of applications of MVEDs in comparison with solid state devices. Reproduced with permission from [2]. 3 Table 1.1: Summary of Microwave Vacuum Electron Devices Device Gridded Tubes Figures of Merit • Low frequencies • Low power • Low cost • Compact size Applications Klystrons • High Gain (> 60 dB), • Low Bandwidth at low powers (< 5 %) • Moderate bandwidth at high powers (10 − 15 %) • High Power (> 100 MW pulsed, > 1 MW CW) , • Frequency Range (≈ 1 − 100 GHz) • Large size • Particle accelerators • Deep space communications • TV broadcasting • Medical imaging [21] TWTs • Moderate to High Gain (30 − 50 dB), • High Bandwidth (2 octaves) at low powers (< 200 kW) • Moderate bandwidth (< 15 %) at high powers • Low to moderate power (1 MW CW), • Frequency (< 100 GHz) • Radar • Satellite communications CFAs • Low Gain (< 20 dB), • High power (> 10 MW pulsed, > 400 kW CW), • High efficiency (. 70 %), • Frequency Range (< 20 GHz) • Moderate bandwidth (< 15 %) • Compact size • Radar • Electronic Counter Measures • Noise Generators • Particle Accelerators 4 Device Figures of Merit Magnetrons • Low bandwidth, • High Power (< 1 GW pulsed, < 100 kW CW), • High Efficiency (up to 90%) Gyrotrons • High power (< 2 MW CW pulsed, > 400 kW CW), • Moderate efficiency (< 50%), • High frequency (10 GHz − 1 THz) • Compact size Applications • Microwave Ovens • Radar • Medical imaging [21] • Particle accelerators [22] • Nuclear Fusion • Radar • Medical imaging No MVED meets the needs of all applications, and each choice has compromises. The focus of the dissertation is on crossed-field amplifiers, and the most notable features of these devices are the high power capability with decent bandwidth in a compact size. The klystron outperforms the CFA in terms of gain and maximum power, but the device is very large at lower frequencies. So for more portable applications, the CFA is more practical. The disadvantages of CFAs are the low gain and relatively high noise, which limits the CFA in several applications. Improving the gain and noise characteristics would make the CFA much more appealing for a variety of applications, and this is the ultimate goal of this dissertation. 1.1 Device Concept The goal of this research was to demonstrate a linear format CFA which uses gated field emitter arrays (GFEAs) [23–27] as the electron source to spatially and temporally vary the injected electron current density in order to maximize efficiency, gain, and 5 bandwidth and to minimize noise. Linear format in this case is not to be confused with O-type devices where the beam and the magnetic field are in the same direction, but refers to the linear geometry. The original goal of this research included an experimental component, but the CFA experiments showed no gain. Some experimental components are still presented here, but the focus is primarily on simulation. Current crossed-field devices use thermionic or secondary electron emitting cathodes, with the exception of some magnetrons [28, 29] and the A6 magnetron which uses a transparent cathode [11, 12] where explosive emitters [30, 31] are used. In the transparent cathode work, current is emitted from explosive emitters at discrete locations in the geometry, but each source has the same approximate current. That work showed an increase in magnetron oscillator performance as discussed later. In this dissertation, the cathode will emit from discrete GFEA sources but with varying currents, in an amplifying configuration. By spatially varying the currents in the interaction region, better performance with respect to efficiency and gain is expected. Gated field emitters are much more efficient electron current sources than typical thermionic cathodes [2, 24, 32]; they have higher modulation frequency capability, and they have the advantage of easy spatial control. The use of GFEAs in MVEDs is not a new concept [8, 9, 11, 12, 33], but in general, it has not been implemented due to emission current and reliability constraints. In this work, a technique of using hop funnels [34–38] to integrate GFEAs into a MVED device will be discussed [39], but this approach was not implemented due to experimental problems as explained later. Three different simulation softwares were used in this work: SIMION [40], COMSOL [41] , and VSim [42]. The simulations modeled three different CFAs in total: two variations were designed and built at Boise State University (BSU) and one was built and studied at Northeastern University (NU) [14, 15]. The electron trajectories 6 were modeled in SIMION and VSim and were compared with experimental results of the BSU CFAs. The slow wave circuit dispersion characteristics were modeled in COMSOL and VSim and were compared with the experimental BSU CFAs. The full CFA simulation was modeled in VSim, but because the BSU CFAs produced no gain, the simulation results were compared with experimental measurements of a very similar CFA used at NU in order to validate the simulation model. Each of these CFAs use the same basic design, but with different dimensions. The same basic VSim model can be used to simulate each design. The simulation results confirmed the poor performance of the BSU CFAs, and the focus of the work was shifted to the NU CFA design. An in depth study of the general CFA operation and physics was performed and was confirmed with theory and the results of the NU work. A variation of the NU model, the spatial modulation is performed using a distributed cathode with spatial- and time-varying electron current distributions, was developed to study the effects on gain, signal to noise ratio (SNR), and efficiency. 1.2 Research Objectives and Contributions Current CFAs demonstrate high power capability with good bandwidth in a compact size, but they have low gain and relatively high noise. To extend their usefulness, a considerable amount of effort has been put forth to improve gain and the noise characteristics of CFAs. The objective of this research is to see if it is possible to use a spatially and temporally modulatable cathode in place of traditional secondary emitting and thermionic cathodes to improve gain and lower noise in CFAs. The focus of this dissertation is mainly on simulation work. One of the original 7 goals was to perform original GFEA experimental work, but because of experimental problems, the experimental work was only used to validate the simulation model. The primary contributions of this research are: 1. Development of a linear format CFA simulation model in VSim [42] which implements a distributed cathode. 2. Comparison of different static distributed electron emission profiles. 3. Demonstration via simulation of improved performance using a temporally modulated electron emission profile. This is the main contribution to CFA research. The small-signal-gain and the signal-to-noise ratio of the device dramatically improves using this modulated cathode implementation and has the potential to increase the maximum output power of the device by improving mode-locking. The first phase of the research validates the VSim simulation model against theory, experiments, and simulation results from COMSOL and SIMION. Experiments determining the slow wave circuit behavior and electron beam trajectory of the BSU CFA were performed at BSU and compared against the simulations. The full CFA operation simulation using VSim was compared against experimental results performed at NU on the NU CFA. These results are also compared with and confirm the current observations in CFA literature. The next phase of the research studied the effects of different static electron emission profiles from a distributed cathode. These electron current distributions were studied with respect to gain, SNR, and efficiency. There are many published research efforts which use many different cathode types and implementation, but currently there is no published study comparing different electron emission profiles in the same device. 8 The third phase of the research studied the effects of different modulated electron emission profiles. Both an injected and distributed beam were studied with respect to gain, SNR, bandwidth, and efficiency. Currently there are no publications which utilize a modulated electron beam in a CFA. The last phase was to investigate the performance and the plausibility of this distributed cathode approach for real CFA applications. The model studied in this research is a low power device, and in order to achieve the goal of implementing a high power and high gain CFA, the modulated cathode must be achievable in a higher power device. 1.3 Overview of the Dissertation Three different CFA designs are described here. All are of the same type but with different dimensions and operating frequencies. The first CFA designs were simulated and were tested via experiment. These designs showed no electron-RF wave interaction experimentally, and the results were corroborated via simulation. The research effort shifted to a design used in previous work at Northeastern University [14], which showed moderate gain (7 dB) in that work. The bulk of the contributions of this work use the NU CFA design and are performed via simulation using Vsim. A general background of CFAs is presented in Chapter 2. There are two important components of the CFA: the slow wave structure and the electron beam. A general background on these two concepts and their function is given. A general overview of the numerical methods used in simulation software used in this research is also presented. Also, a detailed overview of the NU CFA design studied in this work is given. 9 In Chapter 3, the proposed CFA and research objectives are explained in detail. The general research chronology is described, starting from the BSU CFA experimental and simulation work, to the NU CFA simulation work, and to the distributed cathode simulation work. Chapter 4 outlines the experimental design for the BSU CFA. Three experiments were performed: one experiment to measure the slow wave circuit dispersion characteristics, another experiment to determine the electron beam trajectory characteristics, and one to test the full electron beam interaction with the RF wave. Chapter 5 describes the SIMION, COMSOL, and VSim simulation models in detail. A small portion of the research was performed using SIMION and COMSOL, so only a brief overview of those simulations are given. The bulk of the research was performed with VSim, and a very thorough overview of that software and the models are provided. The injected beam VSim model variation, which is used to validate the VSim model against experiments, is discussed in great detail. The distributed beam VSim simulation models, which is the focus of this research, are also discussed in great detail. The spatially- and time-varying distributed cathode methods are presented here, and the approximation to implement the distributed cathode is explained. Chapter 6 presents the results of the experiments (using the experimental setup described in chapter 4) and simulations (using the simulation models described in chapter 5) performed on the BSU CFA. The experimental results matched up with simulation rather well, and the simulation results confirm and determine the reason for the poor performance of the BSU CFA designs. This result justifies the switch of the CFA design to the one used at NU. Chapter 7 presents the simulation results of the NU CFA design. Detailed studies of the device physics and of different methods to optimize gain are presented and 10 compared with theory. These results are compared and validated against the NU experimental results. Finally, the contributing factor in this research, using a variation of the NU design, the different spatially- and time-varying distributed cathode results are finally presented and analyzed. The benefits of the distributed cathodes are clearly outlined, and a thorough discussion of the impact of these results is given. Chapter 8 summarizes the notable results in these studies and draws conclusions. The benefits and costs of the distributed cathode studies are all outlined, and the possible applications are given. Details on the next step of this research are also provided. 11 CHAPTER 2 BACKGROUND 2.1 Crossed-Field Amplifiers There are several aspects in which CFAs can be grouped: mode of operation (forward or backward wave), electron beam source (injected beam or emitting sole), geometry (cylindrical or linear), and beam collection (reentrant beam or end collector). Figure 2.1 shows an example of a cylindrical, injected beam, backward wave, non-reentrant CFA [17]. The geometry is obviously cylindrical where the sole electrode is surrounded by the RF circuit which also acts as the anode. A static electric field is created by the potential difference between the sole and RF circuit. The electron beam is injected at one point, labeled cathode, thus classifying this as an injected beam. The grid and accelerating anode control the electron trajectories upon entering the interaction space. Electrons cycloid clockwise around the sole, interact with the RF wave which is traveling counter clockwise along the RF circuit, and are collected on the collector. The collector at the end of the interaction space prevents electrons from ’reentering’ at the beginning of the interaction space; thus this is a non-reentrant device. The electron beam travels in the opposite direction of the group velocity of the RF wave which classifies this a backward wave device. The attenuator on the delay line is used to prevent amplification of other modes in the circuit. 12 RF In Sole Collector RF Out Accelerating Anode Grid Cathode B RF Circuit and Anode RF Attenuator Figure 2.1: Cylindrical, injected beam, non-reentrant, backward wave crossed field amplifier. [17] Figure 2.2 shows an example of a cylindrical, emitting sole, forward wave, reentrant CFA. In the emitting sole design, electrons are emitted from the entire center electrode, similar to the magnetron. The center electrode can be a thermionic cathode, a secondary emitting cathode (this is sometimes called a cold cathode), or a combination of both. Electrons travel clockwise and interact with the RF wave which is also traveling clockwise. Electrons that do no give up all their potential energy and collect on the RF circuit reenter the interaction region. These electrons are essentially ’recycled’ and can give up any leftover energy from the previous rotation. This improves efficiency of the device over the end collector technique, but these electrons can cause feedback from output to the input. To minimize this feedback, a drift space is included to remove the modulation of the electron spokes. 13 Figure 2.2: Cylindrical, emitting sole, reentrant, forward wave crossed field amplifier. [17] Figure 2.3 shows a linear format, injected beam, forward wave CFA. Electrons are emitted from the cathode into the interaction region. The electrons travel down the tube and interact with the RF wave on the slow wave circuit, give up their potential energy, and collect on the delay structure. Electrons that do not collect on the delay line are collected by the end collector. Obviously, linear format CFAs are non-reentrant. This linear format design is very similar to the design used in this research. The first iteration in the research is very close to this injected format design. The desired final design will emit electrons from the sole and is explained in more detail later. Figure 2.3: Linear format injected beam crossed field amplifier [2]. The linear format, non-reentrant CFAs are very similar to the cylindrical format non-reentrant designs. The mechanisms are the same for both geometries, but the 14 main difference between the theory of these designs are that the governing equations for electron motion are either in cylindrical or Cartesian coordinates. Because the design in this research is of a linear format, the equations presented in this work are for the linear format. All the CFA variations consist of two main parts: the slow wave circuit, also called delay line, and the electron beam. The slow wave circuit retards the phase velocity of the RF wave in order to easily interact with the electron beam. The electron beam travels close to the speed of the RF wave and forms “bunches” or “spokes”, as they are called in cylindrical geometries, due to focusing from the RF wave. If the electron beam is slightly faster than the RF phase velocity, potential energy is extracted from the beam and provides amplification of the RF wave. The slow wave structures are very dispersive, and an extensive overview on the subject is provided. First a discussion on wave velocities and dispersion and then a general overview of different slow wave circuits are presented. The motion of the electron beam is also very important. The cycloidal trajectory of the electron is discussed. Because field emitters and eventually hop funnels are proposed for our CFA, a brief overview of GFEAs and hop funnels is provided. 2.2 Wave Velocities and Dispersion There are many ways to represent dispersion. A plot of phase velocity vs. frequency is the easiest to visualize and understand, but it shows limited information. Usually dispersion is represented as frequency vs wave number. This later view, although less intuitive, is much more informative and contains all the information about the slow wave circuit. This approach is discussed here. The phase velocity is represented by 15 the ratio ω/β, and the group velocity is the slope of the curve ∆ω/∆β, where ω is the angular frequency and β is the wave number. An example of the dispersion in simple rectangular conductive wave guide [2] is given in Fig. 2.4. Note that as the frequency approaches the cutoff frequency, ωc , the wave number becomes infinitely small. The wavelength becomes very large as the group velocity slows down, and propagation is cutoff. Below this frequency, there is no propagation. As the frequency increases, the slope of the line approaches the speed of light. It is also important to note that waves can travel in both directions with identical characteristics, so the diagram is symmetric about the ω axis. ω Slope = c β Figure 2.4: Dispersion diagram for waves traveling in either direction in a rectangular waveguide [2]. For periodic structures, there are frequency harmonics and spatial harmonics called Hartree harmonics [43]. Many different modes can exist on the circuit, and the modes are periodic with respect to the period of the device. A detailed description of this can be found in Gilmour [7], but a summation is given here. Because the dispersion is directly related to the periodicity of the device, the wave number is now represented with βL, where L is the period of the structure. The dispersion diagram for a periodically metallic loaded waveguide is shown in Fig. 2.5. The periodic nature of the loaded waveguides causes periodic reflections. Sometimes these reflections add 16 in phase, resulting in propagation, and sometimes they are out of phase, resulting in cutoff. In this cutoff case, when the half-wavelength becomes equal to the period of the structure, βL = π, the reflected waves add out of phase with the forward waves, and propagation stops. If the wave number is allowed to continue to increase, a backward wave mode is excited. The space in the diagram where β > π are called the Hartree harmonics. ω Propagation Stops Backward wave -2π -π 0 βL π 2π Figure 2.5: Dispersion diagram for waves traveling in either direction in a periodically loaded rectangular waveguide [2]. There are also harmonics in frequency as well. Fig 2.6 shows the dispersion diagram for the periodically loaded waveguide including the frequency harmonics. There are three regions highlighted in the figure to explain frequency effects. When applying a waveform from the source to the load in the positive direction in the periodically loaded waveguide, at a frequency below ωc1 , the group and phase velocity are both positive. This corresponds to region (a) in Fig. 2.6. As the frequency increases above ωc1 , propagation stops, which corresponds to the cutoff region. Increasing the frequency above ω2 , the group velocity is still positive, but the phase velocity becomes negative. This corresponds to region (b) in Fig. 2.6. This backward mode exists until the second cutoff region occurs. 17 ω –2π π 0 β π 2π Figure 2.6: Dispersion diagram for periodically loaded rectangular waveguide showing multiple harmonics [2]. It is an important concept to note that power is still transmitted through the device during a backward wave. Power is transmitted in the same direction as the group velocity. The only time where power transmission does not occur is in the cutoff regions. When applying a signal from a source towards a load, the group velocity will always be positive, except during cutoff, and it is the phase velocity which changes direction. This concept will be reiterated during the discussion of the slow wave structure used in this work. 2.3 Slow Wave Circuits Many different slow wave circuits have been developed and studied over the years [7, 44–47]. A very common one used in TWTs is the helical structure. Crossed-Field devices rarely use helical slow wave circuits; however since this type of circuit has 18 similar characteristics to the meander line used in this research, a summary is given here. 2.3.1 Helix The helical structure is a conductive coil that resembles a spring. The phase velocity can be estimated by simple geometry. The phase velocity is proportional to the distance traveled in the direction of interest over the total distance the wave traveled. Fig. 2.7(a) shows the helix, and Fig. 2.7(b) shows the helix cut at the points marked as x and unrolled. Using the view of the helix in Fig. 2.7(b), it is easy to see the phase velocity along the pitch from a wave propagating along the wire at the speed of light, c, is given by p vp = c sin ψ = c q p2 + (2πa)2 (2.1) where p is the pitch, a is the helix radius, c is the speed of light, and ψ is the angle of the helix. The approximation of the phase velocity is rather constant, or non-dispersive, across a wide range of frequencies; hence TWTs which use this circuit have high bandwidth. Deviations from this approximation are due to effective changes in inductance and capacitance when changing the frequency. There are also deviations due to the periodicity of the device when the slow wave wavelength λsw = p, 2p. When λsw = 2p, there are two simultaneous modes, a backward wave on the edge of the structure and a forward wave down the center of the helix [2]. This phenomenon is also observed in the meander line circuit studied in this research. 19 x (a) x x x 2a p ψ c (b) c sin ψ 2πa ψ Figure 2.7: Helix (a) and view of helix cut at each x and unrolled (b).[7] To visualize this backward wave mode, consider Fig. 2.8. Consider an electron traveling left to right and a backward wave traveling right to left along the helical line. In Fig. 2.8(a) the electron feels a force. In Fig. 2.8(b) the electron has traveled to the right 180◦ , and the backward wave has traveled to the left 180◦ , and the electron feels no force. In Fig. 2.8(c), the electron has traveled to the right another 180◦ , and the backward wave has traveled to the left 180◦ , and the electron feels another force. This mechanism causes causes backward wave oscillations (BWOs) in TWTs. The frequency of backward wave oscillations depend on the beam transit angle from turn to turn, and thus the BWO frequency can be tuned by the beam voltage. These BWOs are caused by electron beams, but the backward wave mode exists when driving high RF frequencies on the helix circuit where λsw < 2p. 20 F + + e (a) 0 0 0 0 0 0 F=0 0 e (b) + + F + + e (c) 0 0 0 Figure 2.8: Backward wave interaction at two turns per wavelength. [7] 2.3.2 Meander Line The meander line slow wave circuit is used in this research. Figure 2.9 shows an example of a microstrip meander line circuit. The circuit ’zig zags’ back and forth in order to retard the phase velocity. Any type of zig zag pattern can be used, but the one shown in Fig. 2.9 is the same as is used in this research. The conductor forms a transmission line over a dielectric and a ground plane. This type of circuit has not been very common for use in crossed field devices, but some interest has increased as of late [48–50]. Because of the flat geometry, one can use well established semiconductor fabrication techniques to manufacture meander lines, which would allow for the creation of tiny slow wave structures (<1mm) to allow for higher frequency applications. 21 Figure 2.9: A microstrip type meander line, showing a conducting meander circuit over a dielectric material and a ground plane. Two qualities of the circuit determine the predicted phase velocity: effective dielectric and the geometry. The microstrip has dielectric and a ground plane on one side of the meander line and air on the other. This results in an effective dielectric constant. The equation for the effective dielectric of a microstrip is given in most electromagnetic textbooks and is shown in Eq. (2.2). Hd is the dielectric height, WL is the circuit line width, Lp is the pitch length, Wsw is the width of the slow wave circuit, HL is the meander line height, and εr is the permitivity constant for the dielectric. This equation assumes air or vacuum on one side. This effective dielectric determines the velocity retardation, Rv , of the wave traveling on the microstrip via Eq (2.3). The phase velocity retardation due to the geometry, Rgeom , is shown in Eq. (2.4). Combining Eq. (2.3) and (2.4) gives the total retardation of the circuit in Eq. (2.5). 22 εef f = εr − 1 εr + 1 + p 2 2 1 + 12Hd /WL Rv = (2.2) c 1/ µ0 ε0 εef f (2.3) 2L + Lp Lp (2.4) √ Rgeom = Rtot = Rv Rgeom (2.5) There are a few design considerations in the choice of width and length. In order to get good interaction with the electron beam, the length of the circuit must be long; at least six slow wave wavelengths is the arbitrary threshold used here based on prior work [14]. In fact, the longer the circuit, the greater the gain of the device [7], up to a limit. The second design consideration is operating below cutoff where no backward wave modes exist. This is to prevent mode competition within the circuit. Similar to the helix circuit, a backward wave mode begins at βLp = π, and cutoff is at βLp = 2π. To minimize the complexity of the design, the operation of the meander line will be where λsw < 2Lp such that βLp < π. Operation is possible at Lp < λsw < 2Lp ; however complicated precautions are needed to prevent amplification of this backward mode, similar to the precautions used in helix TWTs [2]. 23 2.4 Electron Motion A CFA contains both static magnetic and electric fields. The force electrons feel from both the magnetic and electric field is described by the Lorentz equation F = q (E + v × B) where q is the electric charge of the particle, E is the electric field, v is the particle velocity, and B is the magnetic field. To visualize this force, Fig. 2.10 shows an electron in a constant E-field, constant B field, and a combined E and B field which are perpendicular to each other. With a constant E field, the electron feels a force in the opposite direction and travels parallel to the field lines, shown in Fig. 2.10(a). With a constant B field, the electron feels a force perpendicular to both the magnetic field and the velocity (U0 ) of the electron, and travels in a circle on the plane perpendicular to the B field, shown in Fig. 2.10(b). In a crossed electric and magnetic field, the electron feels forces that are more complex and change throughout the trajectory. The electron travels in a cycloidal trajectory with the average velocity (Uavg ) perpendicular to both fields, shown in Fig. 2.10(c). 24 E F F B (a) y or ject U0 Tra (b) B Uavg E (c) Figure 2.10: Electron forces and trajectories for (a) a constant electric field and no magnetic field, (b) a constant magnetic field into the page and no electric field, and (c) a constant electric field perpendicular to a constant magnetic field into the page. The shape of the cycloidal trajectory is altered by the initial kinetic energy of the particle. Fig. 2.11 shows the trajectories of electrons with various initial velocities[2]. 25 yo + uo c Ey y 3 uo 2 Bz Ey yo uo r = Bz uo 2 Ey Ey = 0 2 uo 3 uo c = uo Bz = uo 3 x 0 0 uo c 2 uo c 3 uo c 4 uo c Figure 2.11: Electron trajectories for crossed electric and magnetic fields for various initial velocities (u0 ) [2]. ωc is the cyclotron frequency defined by ωc = qB/m, where q is the particle charge, B is the magnitude of the magnetic field, and m is the particle mass. A CFA has a static electric field and a perpendicular magnetic field. A cycloidal electron trajectory is observed in CFAs as shown in Fig. 2.10(c). There are 3 parameters of this trajectory that are important: average velocity, the cycloid height, and peak kinetic energy. The average velocity, which is given in Eq. (2.6), or guiding center motion is important because the beam must be able to travel close to the same velocity as the RF wave on the circuit for proper interaction. The height of the trajectory is important because the trajectory must not hit the anode so the electrons can travel down the tube (Hull cutoff condition) [51]. The kinetic energy is important because the velocity of the particle in the direction of wave propagation must be equal to the velocity of the RF wave at at least one point in the trajectory (Hartree condition) [1]. 26 v= E×B |B|2 (2.6) The Hull cutoff condition describes the threshold voltage or magnetic field for which all the injected current goes to the anode in a crossed field device. For an electron born at the cathode with zero velocity in a linear format, the Hull cutoff voltage (Vhc ) and magnetic field (Bhc )are given by Eqs. (2.7) and (2.8) respectively. d is the anode-cathode distance, B0 is the magnetic field, e is the electron charge, m is the electron mass, and V0 is the voltage between the anode and cathode. Vhc = Bhc 1 e 2 2 B d 2m 0 (2.7) r m 2 V0 e (2.8) 1 = d The Hartree condition describes the voltage and magnetic threshold at which interaction with the RF wave can occur. For interaction to occur, the velocity on the electron hub surface must be greater than the phase velocity of the wave. If the velocity of the electrons at the hub surface is less than the phase velocity of the RF wave, energy will be transfered to the electron beam and no amplification can occur. The Hartree condition for a linear format device is given in Eq. (2.9), where β is the wave number of the RF wave and ω is the angular frequency. Vh = ωB0 d m ω 2 − β 2e β (2.9) The operation of crossed-field devices is between the Hull cutoff and the Hartree line. Figure 2.12 graphically shows the operation region of a magnetron in terms of voltage and magnetic field; this is the same operation region for CFAs. 27 Figure 2.12: Voltage vs. magnetic field showing the operation region of magnetron [2], which is the same for a CFA. 2.5 RF-Beam Interaction Figure 2.13 shows the electron beam interaction with an RF wave for a CFA in a moving frame of reference with the electrons [17]. Electrons in the positive regions above the Hull cutoff voltage, drift towards the anode and give up potential energy to the wave. Electrons in the negative regions below the Hartree voltage gain potential energy and drift towards the sole. These electrons remove energy from the RF wave. If more electrons give up energy rather than remove energy, amplification will occur. Also, these electrons form spokes during this process as can be seen in Fig. 2.13 where the two trajectories converge. 28 Figure 2.13: Motions of electrons due to the RF field in a rotating coordinate system of a CFA. Electrons in positive RF potentials move towards the sole as they give up energy, and move clockwise into the decelerating region (the region between positive and negative RF potentials where the electric field points clockwise). Electrons in negative RF potentials gain energy, and cycloid right back into the sole. Electrons in the decelerating regions remain in the region but move towards the sole as they give up energy. [17] The beam and the RF field interaction on the circuit is described by Pierce theory [10]. The reference discusses in detail the interaction in a TWT but also applies similar principles to crossed-field devices. The reference is quite thorough, but for additional insights see [7, 43, 52]. In summary, theoretical gain is predicted by relating two impedances: the interaction impedance, Z, and the beam impedance, ZB . These impedances relate power to a voltage or electric field and are explained in the following sections. 2.5.1 Interaction Impedance The interaction impedance relates the input power on the circuit to the x-component of the electric field created at the beam location in the interaction region. Eq. 2.10 is the formula for calculating the Pierce interaction impedance, Zp , where Ex is the peak value x-component of the electric field at the beam location, λsw is the slow 29 wave wavelength, and Pin is the RF input power on the circuit. With a higher Ex at the beam location, the higher the impedance, and the greater the interaction. A higher Pierce interaction impedance gives a higher device gain. Zp = Ex2 λsw 8π 2 Pin (2.10) The Pierce interaction impedance shown here is the interaction impedance for a TWT, and a different variant is used for crossed-field devices. Eq. 2.11 shows the equation for the interaction impedance for a crossed-field device [52], where Ex and Ey are the x and y components of the electric field at the beam location, y0 is the distance away from the slow wave circuit, β is the wave number,and the ∗ denotes the complex conjugate. Far away from the circuit, Z = Zp . Note that closer to the circuit, the interaction impedance for crossed-field devices is smaller than the Pierce interaction impedance. The interaction impedances are on the order of tens of Ω.
j Ex Ey∗ − Ex∗ Ey λ2sw Z= = Zp coth βy0 16π 2 Pin (2.11) It can be difficult to analytically determine the interaction impedance of a slow wave circuit, but measuring it is relatively simple. The interaction impedance is found by applying an RF signal to the slow wave circuit terminated into a match load and measuring the x− and y-components of the electric field amplitude along the interaction region at the predicted beam location. These measurements can be used in Eq. 2.11 to determine the interaction impedance. With a highly cycloidal beam, the beam location varies along the trajectory. In the highly cycloidal case, the beam location is approximated somewhere within the trajectory. This method to 30 measure the interaction impedance can also be used in simulation. 2.5.2 Beam Impedance The beam impedance, ZB , is simply the voltage loss of the beam over the beam current. The voltage drop of the beam, often called the beam voltage, is the cathode to anode voltage, Vca . Eq. (2.12) gives the beam impedance. Note that as Vca increases, the beam impedance increases, and as Ibeam increases, the beam impedance decreases. It is found that a lower impedance leads to higher gain and efficiency and to a more compact device, while high impedance leads to greater stability [53]. Section 2.5.3 describes the effect of beam impedance on the gain and efficiency of the device. ZB = 2.5.3 Vca Ibeam (2.12) Theoretical Gain Both the theoretical gains for the TWT and the CFA are presented here. The gain for a TWT is show in Eq. (2.13) where gain parameter, C, is shown in (2.14). The gain for the CFA presented here is from [52] and is shown in Eq. (2.15) where N is the length along the interaction space in slow wave wavelengths and D, also called the gain parameter, is shown in eq. (2.16), β is the wave number, and d is the anode to sole distance. Both of these equations for gain do not include space charge effects. These equations are only valid in the small-signal-regime of the circuit, before saturation occurs. The gain for the CFA also assumes the beam is non-cycloidal which means the beam is injected at the exact velocity where it moves with a straight trajectory and a constant velocity. 31 G = −9.54 + 47.3CN C3 = Zp 4ZB G = −6 + 54.6DN D2 = [dB] |Z| (βd) ZB (2.13) (2.14) [dB] (2.15) (2.16) There are a few aspects to note on the gains of the device. The initial gain for the TWT and CFA is −9.54 dB and −6 dB respectively. Also, the gain increases with the length of the device. The slope of the gain is determined by gain parameters C and D for the TWT and CFA, respectively. Note that the ratio between pierce interaction impedance and beam impedance (Zp /ZB and |Z|/ZB ) are small quantities, less than 1. The gain parameter for the TWT, C, is related to this ratio by the 1/3 power, and the gain parameter for the CFA, D, is related to this ratio by the 1/2 power. The 1/3 power of a small quantity is larger than the 1/2 power; therefore the gain of the CFA will be less than that of a TWT given similar circuit impedance, current, and voltage. The gain of the device only depends on the interaction and beam impedances. The gain does not, however, depend on the RF input power of the device. The RF input power will determine where on the circuit the gain saturates. With higher RF input powers, the gain saturates in a shorter length as the energy of the beam depletes. A greater interaction impedance produces a steeper slope. This is because a high 32 interaction impedance means there are higher electric fields at the beam location for a specific power. Also, a larger beam current results in a greater gain because more current provides more energy to extract. These two observations are intuitive, but an unexpected observation is that with higher anode to sole voltages (Vca ), the gain decreases. This is unexpected because with higher Vca , there is higher beam power and thus more available energy to extract. No physical explanation is given in the references why increasing Vca decreases the gain. A conceptual attempt at describing the relationship between gain and the beam voltage is provided here. As the beam voltage is increased while keeping the RF wave voltage constant, the relative modulation of the beam caused by the RF wave is diminished. Conceptually, it may be easier to use a TWT example. The beam velocity in a TWT is determined by the beam voltage. With increasing beam voltages, the modulation of the kinetic energy of the electrons relative to the overall beam energy is diminished. With lower relative modulation, the gain decreases. In a CFA, the explanation becomes obscured by the fact that the average kinetic energy of the beam remains unchanged as potential energy is given up to the RF wave and by the fact that the cycloid trajectory constantly shifts between potential and kinetic energy; but the kinetic view of the electron in the cycloid trajectory offers insight into the relative modulation of the electrons during the cycloid trajectory. There are two interesting observations about the efficiency predicted by Pierce theory. Increasing the beam current not only increases gain but also increases the efficiency of the device. Also, increasing the RF input power increases efficiency. Because the gain is constant for different RF powers, by increasing the RF power, more power can be extracted from the beam for the same length, thus increasing efficiency. The relationship between beam current and efficiency is not immediately 33 apparent from the gain equation. Rearranging the terms, Eq. (2.17) shows efficiency. This equation is also difficult to visualize, so Fig. 2.14 shows a plot of the efficiency for the parameters similar to the ones used in the NU CFA for this research where the anode to cathode voltage Vca = 1250 V, the input RF power Pin = 1 W, the interaction impedance Z = 7.4 Ω, the anode to cathode distance d = 0.416λ, and the CFA length in slow wave wavelengths N = 9. Note that the method to determine the interaction impedance is explained in Sec. 7.4.3. The efficiency increases as the beam current increases. Of course, the limit of this efficiency occurs when the power extracted from the beam gets close to the total power of the beam. Note that the efficiency is very low as is the case for this device from experiment. ef f = Pin  100.6 10K √ Ibeam Vca Ibeam −1  where K = 54.6N r Zβd 2Vca × 100 [%] (2.17) 34 8 Efficiency [%] 6 4 2 0 0 100 200 Beam Current [mA] 300 Figure 2.14: Plot of Pierce theory efficiency as a function of beam current for the NU CFA. 2.6 Electron Sources There are two common electron sources used in MVEDs: thermionic and emitting sole cathodes. A less common cathode is a field emitter which is proposed here for reasons described later. A general introduction to each of these sources and a comparison is given here. In order to integrate the GFEAs into the CFA, the use of hop funnels is also proposed, so a brief introduction of hop funnels is given. 2.6.1 Thermionic Cathodes Thermionic cathodes utilize heat in order to emit electrons. Typical cathode temperatures are & 1000◦ C[1, 3], and current densities are as high as ≈ 100 A/cm2 [1, 2]. By increasing the temperature of the cathode, the number of electrons with sufficient energy to escape increases. The phenomenon is quantum mechanical and requires the use of the energy level diagram for explanation. Fig. 2.15 shows the energy level 35 diagram near the surface of a metal. The parabolas represent the energy levels of adjacent atoms. The Fermi level defines the top of the conduction band. The work function is defined as the difference between the energy level of the vacuum and the Fermi level of the metal. In order for electrons to escape the metal, the electron must have an energy greater than the work function of the material. Electron Energy eϕ E = Eo Conduction Band Cathode Vacuum x=0 x Figure 2.15: Energy Level Diagram near the surface of a metal [2] The energy of the electrons is defined by the Fermi-Dirac distribution, which is a function of temperature. The average number of fermions in a single-particle state i is given by the Fermi-Dirac distribution, shown in Eq. (2.18) where ǫi is the energy of the particle, k is the Boltzmann constant, T is the absolute temperature, and µ is the total chemical potential. Fig 2.16 shows the Fermi-Dirac distribution for various temperatures. At temperatures of 0◦ K, the energy distribution describes that all electrons occupy an energy state below the Fermi level which is where ǫ/µ = 1. At temperatures greater than 0◦ K, 50% of electrons occupy a state below the Fermi level and 50% occupy a state above. Only high energy electrons which are greater than the work function, for example ǫ/µ > 3, of the material, which occupy the tail, are emitted into vacuum. A higher temperature results in a the ’thicker’ tail, and it becomes more probable that electrons occupy this very high energy state. 36 n̄i = 1 e(ǫi −µ)/kT + 1 (2.18) Figure 2.16: Fermi-Dirac distribution for various temperatures. Using the work function and the Fermi-Dirac distribution function, the current density, J, of a thermionic cathode is given in Eq. 2.19. A0 is the thermionic emission constant, T is the temperature, e is the electron charge, φ is the work function of the material, and k is the Boltzmann constant. This equation is known as the Richardson-Dushman equation. As the temperature of the cathode increases, so does the current until the space charge limit is reached. The voltage at which space charge limits the current from the cathode is known as the Child-Langmuir Law and is shown in Eq. (2.20). P is the perveance and defined in Eq. (2.21), where A is the emission area, d is the anode-cathode distance, e is the electron charge, m is 37 the electron mass, and ε0 is the permitivity. The space charge limited regime is caused by the negative charge from the emitted electrons depressing the electric field near the cathode, where at a certain charge density the electric field suppresses any additional electron emission. The space charge limited regime the electron emission density is limited by the electric field near the cathode. The Richardson-Dushman equation describes electron emission in the temperature limited regime, the regime where electron emission is limited by the temperature of the cathode. Note that the space charge limit is not only a limit on thermionic cathodes but on all cathodes and is very important in the CFA operation in this dissertation. J = A0 T 2 e−eφ/kT (2.19) I = P V 3/2 (2.20) √ r 4 2 A e A P = ε0 = 2.33 × 10−6 2 9 md d (2.21) There are two ways to increase thermionic electron emission from a material: increase the temperature of the device or lower the work function of the material. The maximum temperature and the work function are determined by the material, so the choice in material for thermionic cathodes is important. The choice boils down to a balance between the work function and the melting point of materials. Unfortunately, many low work function materials also have low melting points [3]. 38 2.6.2 Emitting Sole Emitting sole cathodes utilize secondary electron emission to create the electron beam. This requires a discussion about secondary emission yield (SEY) first and then of the emitting sole cathode. 2.6.2.1 Secondary Electron Emission Secondary electron emission is a phenomenon in which an electron collides with the surface of a material and one or more ’new’ electrons are emitted. There are two primary variables that determine the number of electrons emitted per electron collision: incident energy and angle of incidence of the primary electron. A typical plot of the secondary emission yield (SEY), δ, versus the primary electron energy is shown in Fig. 2.17. Secondary Emission Coefficient, 2.0 max = 2.0 1.5 1.0 0.5 E I = 53 eV E max = 350 eV E II = 1,710 eV 0 0 400 800 1200 1600 Primary Electron Energy (eV) Figure 2.17: A typical secondary electron yield curve for an arbitrary material [2]. There are four primary points of interest on the curve: 1. The lowest primary energy at which the SEY is unity called the first crossover energy, EI 39 2. Maximum secondary electron yield, δmax 3. The primary energy, Emax , where the maximum secondary electron yield occurs 4. The highest primary energy at which the SEY is unity called the second crossover energy, EII . The shape of the curve is universal for all materials, metal or insulator; however the four primary qualities do change from material to material. The two important qualities that are of importance to emitting sole cathodes are δmax and Emax . Emitting sole cathodes desire a very high SEY in order to get the most current. Emax is important because many emitting sole designs do not use a thermionic cathode and require the RF field to initiate the current. A low Emax requires a lower RF power to initiate the current. The first crossover energy is important to the operation of electron hop funnels as discussed later. The angle of incidence of the primary electron is also important to the SEY. Shallower impact angles yield more secondary electrons. At shallower angles, the primary electron stays closer to the surface and gives up its energy to electrons closer to the surface, and then these electrons have a higher chance to escape the material due to the smaller distance of the escape path. The energy of the emitted secondary have relatively low energies (. 10 eV) [2]. Any electrons that are emitted near the energy of the primary electron are due to electron backscatter, which is a highly elastic collision. The low energy electrons are referred to here as secondary electrons, and electrons with energies near the primary electron energy are referred to as backscattered electrons. 40 2.6.2.2 Emitting Sole Operation The cold cathode emitting sole operation is represented in Fig. 2.18. By applying energy to an emitting sole device, electrons are slammed into the cathode. From that one electron collision, multiple secondary electrons can be formed, which also can strike the cathode and produce their own secondary electrons. In this way, starting with one electron and some input energy, many more electrons can be generated over a short distance. < 1ns Figure 2.18: Diagram showing the ’multiplication’ of electrons on the surface of an emitting sole cathode[2]. Emitting sole cathodes can be used in conjunction with an alternate cathode source or by itself. A thermionic cathode can be used to initiate the beam, and any electrons which gain energy from the RF wave, collide with the sole and produce new electrons which will increase the current density and improve efficiency. Note that electrons which collide with the cathode are in an unfavorable location to give up energy to the RF wave. A few mechanisms exist in which these out of phase electrons shift their phase to a favorable location to give up energy to the RF wave, which is explained in [54]. Emitting sole designs, which do not use an alternate source, initiate the current using the input RF signal to accelerate ’stray’ electrons into the cathode. 41 The choice in cathode material is important. A material with high δmax is desired, and it must be able to withstand the heating due to the electron back bombardment. Also, the surface should be inert in order to maintain consistent operation for the lifetime of the CFA. Slight changes in surface can cause significant changes in the SEY properties of the material, and special precautions must be made in order to maintain the desired SEY. Diamond film on a molybdenum substrate has extremely high δmax (δmax > 20) but degrades rapidly to a yield of unity due to ion and electron bombardment, which makes their use questionable [2]. More common values of δmax of materials used in CFAs are around 2, such as Beryllium oxide (δmax = 2 − 2.8 depending on the surface quality) [2, 55], platinum (δmax = 2.2) [56, 57], and Tungsten Oxide (δmax = 2.3) [2, 58]. 2.6.3 Field Emitters Field emitters utilize strong electric fields along with sharp geometric contours to emit electrons from a conducting material into vacuum via the tunneling effect [7]. There are two general types of field emitters: gated field emitters and field emitters with no gate. Fig. 2.19 shows the configuration of a gated field emitter. A non-gated field emitter is one without the control electrode in the figure. Ungated emitters operate in a diode configuration in which the anode to tip voltage causes field emission. In gated field emitters, the control electrode is used to control the emission current, which makes them more flexible for use as a cathode. At a certain threshold, when the electric field at the emitter tip is on the order of 109 − 1010 V/m, the emission current increases rapidly due to the tunneling effect. At very small geometries, only a modest voltage (≈ 100 V) is needed to create the electric fields necessary for field emission. 42 Anode Electron Flow Control Electrode Emitter Equipotentials Figure 2.19: Gated field emitter diagram showing the field enhancement near the needle tip [2]. The current density near the emitter tip is in the 106 to 1012 A/cm2 range [33]. By having an array of 107 tips/cm2 theoretical current densities of 1000 A/cm2 are possible. Maintaining this high current density over a broad area, however, is difficult to achieve, and current densities of 20 A/cm2 at 120mA [9] are currently achievable. Recent results have demonstrated current densities of 100 A/cm2 using silicon tips by Guerrera et. al. [25–27]. These new GFEAs from Guerrera are used as a model of the GFEAs used in this dissertation, so a summary of that work and a discussion on relevant characteristics will be discussed in Sec. (2.6.3.2). But first, a discussion on the physics and the issues of FEAs is presented. 2.6.3.1 FEA Physics and Practical Issues To understand the relationship between the electric field and electron emission from the surface of the material, the energy level diagram of the material at the surface is used. Figure 2.20 shows the energy level diagram at the surface of a material with and without an applied electric field. 43 Figure 2.20: Energy level diagram at the surface of a material with and without an applied electric field [24]. The dashed line is the energy level diagram for the material with no electric field. Without the electric field, an electron must overcome the potential barrier given by the work function of the material, φ. By applying an electric field, a lowering of the potential barrier is observed, ∆φ. With high electric fields, the energy barrier becomes very narrow. Because of the wave like nature of the electron, there is a probability that the electron exists in the vacuum side of the barrier even though it does not have sufficient kinetic energy to escape, which is called the tunneling effect. At a certain threshold, when the electric field is on the order of 109 − 1010 V/m, the current increases rapidly due to this effect. The current density (J) is found by using the shape of the energy barrier, integrating the probability function of an electron tunneling through the barrier, and multiplying by the electron supply function. The current density due to an applied electric field is shown in Eq. (2.22). This equation is known as the Fowler-Nordhiem equation. 44 J = 1.42 × 10 −6 E 2 φ exp  10.4 φ1/2  exp  −6.44 × 107 φ3/2 E  (2.22) There are many benefits of the gated field emitter over thermionic and emitting sole cathodes. The low transconductance, low capacitance, and small gate-emitter gap allow for a fast modulation of the electron beam. Modulation frequencies of 10 GHz are currently achievable [32]. Also, because there is no need for a heating assembly or very high voltage supplies, the efficiency of GFEAs are higher, and smaller assemblies are possible. Another advantage is that the construction of a cathode with highly resolved spatial control and integration in an MVED is much easier than using a thermionic cathode. For these reasons, GFEAs are proposed in this work. The main issue with GFEAs are their susceptibility to ion back bombardment and oxidation. Even at very low pressures for conventional tubes (10−7 − 10−9 Torr), desorbed gas neutrals from electron bombardment/heating are ionized by the emitted electrons which then accelerate back towards the cathode and can damage the emitter tip. This lowers the field enhancement factor which degrades the performance. Oxidation also can occur on the emitter tips. This phenomenon increases the work function of the emitter which also degrades the performance. Oxidation, however, is reversible to a certain extent [24]. Beam uniformity and beam convergence is another concern with FEAs. Creating an array of consistent emitter tips is difficult. Because of this, the current densities vary from tip to tip and cause a non-uniform current density. Also, due to the emission characteristics and the lack of focusing at the emitter tips, a high transverse velocity component in the electron beam is expected. High current densities at the emitter tip can cause some space charge defocussing as well. With the current GFEAs, integration with MVEDs is difficult, mainly due to the 45 low lifetime of the emitters at high current densities. Even so, lower power devices have been described [59]. Also, considerable improvements have been made in GFEA research in the past 20 years, and GFEA improvements are expected. 2.6.3.2 Discussion of Silicon GFEAs Fabricated at the Massachusetts Institute of Technology Field emitters developed at MIT [25–27] have shown many desirable characteristics for use in MVEDs and are a promising candidate for current modulation. Current densities > 100 A/cm2 with gate-to-emitter voltages < 75 V have been demonstrated with lifetimes > 100 h. The low gate-to-emitter voltage is of special note because it provides an easy way to modulate the current with relatively low power. The I-V curves for various array sizes are shown in Fig. 2.21. The current varies exponentially with voltage. With a change of roughly 15 V, the current varies by about 2 orders of magnitude. 46 Figure 2.21: The current density vs. the gate emitter voltage of the gated field emitters fabricated by Guerra et. al. [25] 2.6.4 Hop Funnels Hop funnels are structures made from insulating material which use secondary electron emission to transmit current[36–38]. Fig. 2.22 shows a diagram of an electron hop funnel and the simulated electron trajectories for two different hop electrode voltages. Fig. 2.22a(a) shows a case where the hop funnel transmits current with a hop electrode voltage of 750 V. Electrons emitted from an electron source move up due to the electric field created by the hop electrode and collide with the funnel wall. 47 Secondary electrons created from this collision travel up the wall due to the electric field, collide with the wall, and create secondary electrons of their own. This cycle repeats until no secondary electron is created or until the electron leaves the funnel exit. Fig. 2.22b(b) shows an unfavorable condition where transmission does not occur where the hop electrode is 0 V. The electric field created by the hop electrode does not accelerate the emitted electrons above the energy at which the first crossover occurs on the Secondary Electron Yield (SEY) curve. These electrons produce less than 1 secondary electron per collision and eventually charge the funnel wall negatively. This negative charge accumulates and counteracts the electric field of the hop funnel and repels the electrons back towards the cathode. (a) (b) Figure 2.22: Hop funnels used in [37], showing the operation during (a) full electron transmission and (b) no transmission using the Lorentz 2E [60] simulation. The majority of the electrons leaving the funnel are born at the potential of the hop funnel wall [37]. The average kinetic energy of the electrons when they are born is approximately 5 eV, determined by secondary electron emission. Using this property of hop funnels, one can control the energy of the electrons leaving the source. Also, by adding another electrode, one can control the potential outside the funnel, while 48 having separate control of the energy at which the electrons are born. Figure 2.23 shows this design. The hop electrode controls the energy at which the electrons are born, and the sole electrode controls the potential which is seen from anywhere above the funnel structure. This is the design proposed for the distributed cathode in the CFA. By biasing the hop electrode less negative than the sole, electrons cycloidal in the interaction region are born at a potential less negative than the sole. Without the RF wave, this prevents the cycloidal electrons from being collected on the sole. Figure 2.23: Hop funnel structure with sole electrode using Lorentz 2E [60] 2.7 Simulation Three simulations are used in this work: COMSOL [41], SIMION [40] and Vsim [42]. COMSOL is a finite element solver; SIMION is an electrostatic particle trajectory code; and Vsim is a finite difference particle in cell (PIC) code. This software will be discussed in more detail in this section 2.7.1 COMSOL COMSOL is a very user friendly multi-physics solver which uses the finite element method. One can build complex geometries, create complex mesh, and solve many 49 different problems all in one program. To build a model, one must create the geometry, mesh the geometry, and then choose the appropriate solvers. COMSOL has many different features to create the geometry. These are very similar to many cad type programs, and the features are not discussed in detail here. General features include: workplanes, rectangles, spheres, etc. The meshing algorithm allows individual control of regions or boundaries and of the mesh technique. There are two types of mesh: structured and unstructured. The unstructured mesh used here consists of tetrahedrons in the region with triangles on the boundaries. The mesh can also be non-uniform, meaning that the mesh can change size in order to resolve very small features without the cost of resolving regions which need fewer elements. Three important meshing parameters are the minimum mesh size, the maximum mesh size, and the growth rate. The minimum mesh size prevents the generation of too many elements which limits the memory usage. The maximum mesh size limits the error of the model and ensures that the features of the geometry of the RF wave can be resolved. The growth rate parameter limits the change in size of the adjacent elements. Mesh quality can be controlled from these mesh parameters. Having too large of a mesh and having large growth rates introduces error. The mesh should be the smallest size and the smallest growth rate within computational memory constraints. COMSOL has many different solvers that can be used to study a problem: timedependent, stationary, or eigenfrequency solver. The time dependent solver is used to see phenomena development in time. The stationary solver determines the steady state of the phenomena. The eigenvalue solver determines the natural harmonic oscillations of a time dependent problem. COMSOL was used to determine the 50 dispersion and the standing wave pattern of the meander line slow wave circuits used in the CFA designs. Because only the steady state solution is needed to determine the standing wave pattern, the stationary solver is used. There are many different algorithms that can be used. The algorithms are broken up into two categories: direct solvers and iterative solvers. Direct solvers directly solve the system of linear equations using a LU factorization (lower upper factorization) method. These solvers generally use a lot of memory and can be slow. Iterative solvers start with an initial ’guess’ of the solution and iteratively make new ’smart guesses’ that are hopefully closer to the actual solution. When the error between the iterations becomes smaller than a convergence criterion, the solution is considered to be found. Iterative methods use less memory and can be faster. There are many different iterative solvers, and the difference among them is in the method they use to ’guess’ the next iteration. The study performed in this work is a frequency sweep which uses a stationary iterative solver. The algorithm used in this work is the default, which is a biconjugate gradient stabilized iterative (BiCGStab) method [61]. This method uses the multifrontal massively parallel sparse (MUMPS) direct solver [62] to create the initial guess. 2.7.2 SIMION SIMION is a 3D finite difference particle trajectory code for electrostatic fields. The program has a graphical interface in which one can create the geometry and boundary conditions, create particle sources, and post process all in one program. The code also has a very extensive scripting language to build the geometry, inject particles, 51 perform real-time data processing, and perform post processing. This code is used to study the electron optics in the CFA. The simulation only has a static electric field solver, a static magnetic field solver, and a particle trajectory solver. No electromagnetic waves can be modeled. The geometry or the potentials of electrodes can be altered during simulation, but the code still uses the electrostatic solver. Space charge can also be modeled but is not used in this work. 2.7.2.1 Electrostatic Solver To create the static electric fields, SIMION uses potential arrays. The user defines the boundary conditions of each electrode or electrodes and assigns them a number. Each unique number corresponds to its own potential array. Each potential array is solved individually to find the electric field at each point. Because of the additive solution property of the Laplace equation, each separate potential array can be added together to find the electric field in the total geometry. The algorithm to solve for the electric field for each potential array is a dynamically self-adjusting over-relaxation method [63, 64]. This refining algorithm is only called once, unless the user requests otherwise, and the code uses a fast adjust method to alter the electrostatic field upon a change in electrode voltage. After the first solution of a potential array, each point can use a scaling factor to account for a different electrode voltage. After the potential array is updated, it can then be added to the other potential arrays to get the total solution. 52 2.7.2.2 Magnetic Field Solver The code also has potential arrays for static magnetic fields. Magnetic fields are normally represented and measured as gradients, and to utilize a similar solver as the electrostatic, SIMION needs magnetic potentials. The code uses vector magnetic potentials to create magnetic fields. Magnetic ’potentials’ can be defined in the potential array and can solve for the magnetic field in the same way as for the electrostatic solver. This is not the best way to solve for the magnetic field because magnetic poles do not have uniform magnetic potentials. Magnetic fields generated by this solver must be carefully studied for accuracy. An external magnetic field can also be input manually. Since the static magnetic field in the CFA is uniform, one can supply a constant for the magnetic field at all points in the particle trajectory. 2.7.2.3 Particle Push The force on the particles is calculated from the electrostatic field, the magnetic field, and the charge repulsion. Each of the forces are then added together to find the total force. With these forces, the particle trajectories are determined using an adaptive time-step 4th order Runge-Kutta method. It should be noted that the trajectory algorithm is ’blind’ to boundaries and sharp gradient edges. To detect boundaries, on each trajectory calculation, the algorithm detects if an edge is crossed. If the edge is detected, the algorithm tries new time steps until it can approach the wall in an accurate manner. Detecting sharp gradient edges is achieved by testing the coefficient of variation squared for the four Runge-Kutta acceleration terms against an accuracy level [64]. The time step is reduced until all values are less than the upper limit or the minimum time step size has been reached. 53 2.7.3 Vsim The term ’particle-in-cell’ (PIC) refers to a technique of tracking macroparticles in a Lagrangian frame, while the moments of the distribution such as densities and currents are computed simultaneously on stationary Eulerian mesh points. Macroparticles represent the mass and charge of a large number of single particles. This approximation is used to reduce the computational load by simulating less particles. Vsim is a 3D PIC code which uses the finite difference method to determine electric fields and magnetic fields and uses these fields to push the particles. Time-varying electromagnetic fields and electrostatic fields can both be calculated on the finite difference mesh. Any initial space charge in the system due to the charged macroparticles are accounted for in the electrostatic solver. The electric field created by the movements of charged macroparticles are accounted for in the electromagnetic solver. Currents can also be defined in this code, and the electromagnetic fields they create are accounted for in the electromagnetic solver. Charge densities can also be defined and are accounted for in the electrostatic solver. Many different particle dynamics are also included in the code. Particle to particle interactions, secondary electron emission, and photon emission can all be modeled. Monte-Carlo methods are used to model these phenomena [65–67]. These particle interactions are not used in this work. 2.7.3.1 Static electric field solver The static electric field solver uses the finite difference grid. Given all the boundary conditions and charge distribution (ρ) the potential (U ) within the domain can be solved with Poisson’s equation. In vector form, Poisson’s equation is given by Eq. 54 (2.23). The numerical expression is given by Eq. (2.24), where ui,j,k is the potential at a specific grid point location U (i∆x, j∆y, k∆z), and ∆x, ∆y and ∆z are the spacing between the grid locations in the x−, y− and z-directions. Poisson’s equation requires an iterative solver, and there are many algorithms which can be used. The choices are bi-conjugate gradients stabilized (bicgstab) [68], conjugate gradient squared (cgs), generalized minimal residual (gmres) [69], conjugate gradients (cg), or transposefree quasi minimal residual solver (tfqmr). The user has control over many other parameters of these algorithms not shown here. To determine the electric fields, the gradient of U is found. ∇2 U = ρ (2.23) ui−1,j,k − 2ui,j,k + ui+1,j,k ui,j−1,k − 2ui,j,k + ui,j+1,k ui,j,k−1 − 2ui,j,k + ui,j,k+1 + + = ρi,j,k ∆x2 ∆y 2 ∆z 2 (2.24) 2.7.3.2 Electromagnetic Field Solver The electromagnetic solver uses the Yee finite difference time domain (FDTD) scheme[70]. The method solves Faraday’s and Ampere’s laws, shown in Eqs. (2.25) and (2.26), where E and B are the electric and magnetic field vectors and J is the current density. ∂B +∇×E=0 ∂t (2.25) ∂D −∇×H=J ∂t (2.26) 55 In Cartesian coordinates, Eqs. (2.25) and (2.26) are expanded to the following system of scaler equations: − ∂Bx ∂Ez ∂Ey = − ∂t ∂y ∂z − ∂Ex ∂Ez ∂By = − ∂t ∂z ∂y ∂Bz ∂Ex ∂Ey = − ∂t ∂y ∂x ∂Dx ∂Hz ∂Hy = − − Jx ∂t ∂y ∂z ∂Hx ∂Hz ∂Dy = − − Jy ∂t ∂z ∂y ∂Dz ∂Hx ∂Hy = − − Jz ∂t ∂y ∂x The finite difference grid defines electric fields on the middle of the edges, and the magnetic fields in the center of the face. Figure 2.24 shows a grid cell and the various positions of the electric and magnetic field components. These positions are chosen so that the boundary condition for a perfect electric conductor (PEC) on the edge of the cube contains and sets the perpendicular components of the electric field and normal component of the magnetic field to zero. For example, plane surfaces normal to the x-axis contain the points where Ey , Ez , and Hx are defined. 56 Figure 2.24: Yee grid showing the position of the various field components. Electric field components are on the middle of the edges and magnetic field components are on the center of the faces. The numerical calculation flow is shown in Fig. 2.25 [71]. First the electric and magnetic fields are initialized to zero. Using Faraday’s law, the electric fields are updated in the interior of the domain. Then the boundary conditions are updated. These boundary conditions can be static or time-varying depending on the simulation problem. After the electric fields are updated, the magnetic field is updated using Ampere’s Law. This leapfrog approach using the Faraday-Ampere-Faraday-Ampere updating scheme repeats until the maximum timesteps are achieved. 57 Figure 2.25: FDTD simulation flow [71]. The maximum timestep that can be used with the FDTD scheme is limited by the time it takes light to transmit through a cell known as the Courant-Friedrichs-Lewy (CFL) stability criterion [72]. The Courant condition is shown in Eq. (2.27). With timesteps that exceed the Courant condition, the FDTD scheme is unstable. ∆t < q c ∆x1 2 + 1 1 ∆y 2 + 1 ∆z 2 (2.27) The FDTD grid is a Cartesian or cylindrical grid which approximates every boundary with this grid. Curved boundaries, ones which do not align well with the cell edges, are usually approximated by a stair-step method [73]. This method 58 only has a first-order accuracy with grid size. Another method which is available in VSim is the Dey-Mittra [74, 75] cut-cell method. This method has a second order accuracy with grid size. because there are only a few unimportant curved boundaries used in the CFA in this research, the Dey-Mittra method is not used, so a review is not provided. 2.7.3.3 Particle Push Algorithm To model each electron in an beam would be very impractical due to computational and time constraints. Particles in Vsim are modeled as macroparticles, where one macroparticle has the charge and mass of many particles. To move the particles, the code uses the Boris-Push Lorentz force equation [76] ∂γmv = q (E + v × B) ∂t where m, q, v and γ are the mass, charge, velocity of electron and the relativistic factor, respectively. VSim can also model the random interactions between particles or the random production of particles. This uses a statistical approach called the Monte-Carlo collision model [65–67]. No particle to particle collisions are modeled nor are any particles created from collisions in this dissertation, so no discussion is provided on the Monte-Carlo model. The only interaction between particles is through the coulomb force between particles through the electromagnetic solver. 59 2.8 2.8.1 State of the Art in MVEDs Cylindrical Emitting Sole CFAs The first cylindrical CFA was developed by William C Brown with Raytheon [77]. He called the device the Amplitron and it was based on the magnetron. It was basically a magnetron design (cylindrical reentrant design with a center thermionic cathode, and a strapped vane structure slow wave circuit) which was altered to accommodate an input and an output. After this demonstration, many configurations have been studied. A few examples are given in [53, 78, 79]. Cylindrical, reentrant CFAs, which use an emitting sole cathode, are the most common CFAs in use today due to their compact size and the efficient recycling of unspent electrons. These conventional CFAs were explained in the beginning of Chapter 2. There are only a few published modern designs as many of the designs are proprietary and unpublished, but the general design is the cylindrical, reentrant and emitting sole CFA. The emitter materials vary for the application where the desired qualities of the emitting sole are a high secondary electron yield, good heat dissipation, and robust performance [2]. The desired qualities of the thermionic cathode is a low work function, high melting point, and long lifetimes [2]. No one to date, however, has published results showing the incorporation of GFEAs in a CFA. The qualities of the slow wave circuit that are important are bandwidth, coupling impedance, unwanted mode suppression, and power/heat dissipation. The slow wave structures also vary depending on the application. The most common slow wave structure in forward wave cylindrical CFAs is a double helix coupled vane shown in Fig. 2.26 [2]. This structure is made up of two helices and is supported by metal vanes. The metal vanes provide structure for the helices and also help dissipate heat. 60 Helix Helix Vane Figure 2.26: Double helix coupled vane slow wave structure commonly used in CFAs. [2]. 2.8.1.1 Power Capabilities The current power capabilities of CFAs are comparable to that in 1985, and a list of examples is shown in Fig. 2.27. Gain ranges from 7 − 20 dB, efficiencies from 50-75%, and bandwidths up to 10%. Information on the exact design of these high power configurations is difficult to find. Generally, the only information available is the general figures of merit such as gain, output power, efficiency and noise level. A few examples without the design specifications in the lower frequency range (450 MHz − 4 GHz), which is more relevant to the low frequency design proposed here, are found here [80–82]. The only thorough description of the design specifications found was of an X-band (11.424 GHz) crossed-field amplifier with an output of 300 MW[83], which is described in the next section. 61 10 7 2. Pulsed ForwardWave CFAs 10 6 3. Pulsed BackwardWave CFAs Power (W) 1. CW Injected Beam CFAs 10 5 10 4 5. Pulsed SuperPower CFAs 10 3 10 2 0.1 3 4 2 1 e 4. CW Super-Power CFAs 5 b -Tu ave r) row we Mic Po al- rage on nti (Ave nve er Co ronti F 10 8 1.0 10 Frequency (GHz) 100 Figure 2.27: The current power capabilities of published CFA data. [2]. There are three factors that limit the output power of CFAs [2, 17]: 1) a limitation in the available cathode current, 2) the onset of a competing oscillation where the anode-to-sole voltage has reached a region of synchronism, or 3) a limitation in gain of the main amplifying mode when the RF drive power is no longer able to retain lock at the higher output power. The first two limitations can be avoided with appropriate design. Proper cathode design can increase the current to avoid the limitation set by factor 1. To prevent unwanted oscillation, proper junction matching techniques can be used to prevent reflections within the circuit, and selective attenuation techniques for the unwanted frequencies can be used [7, 17, 84]. The third limitation is an intrinsic limit resulting from the basic interaction process and is difficult to avoid. There are a few techniques to minimize the mode interference [7, 17], but mode interference remains the main limiting factor to gain and output power of CFAs. 2.8.1.2 High Power X-Band Crossed Field Amplifier To demonstrate typical electron beam current densities and output powers, the specifications for a 11.424 GHz CFA designed by Eppley et. al. [83] is presented here. The 62 exact dimensions and description of the design is given in that work, but only a few characteristics are described here. This is a cylindrical format, cold cathode emitting sole, backward wave, reentrant CFA, similar to the one shown in Fig. 2.2 except this is a backward wave device. The design in this work was still in development at that time and presents no experimental results, but simulation results showed 300 MW of output power at 65% efficiency with a RF drive power of 6 MW. The cathode current density was 41 A/cm2 , and 2600 A was observed at the anode. This current density is typical in CFAs, and this current density is currently achievable by GFEAs [25–27]. These operating characteristics will be used to explore GFEA use in high power devices in Sec. 7.8. 2.8.1.3 Cathode Driven Crossed Field Amplifiers The cathode driven CFA [85, 86] has the basic format of the conventional cold cathode CFAs, but uses a slow wave circuit on the cathode as well as on the anode. This configuration decouples the output circuit from the input circuit when no electrons are present. The coupling occurs when electrons are present and an RF input signal is applied to the cathode circuit. Figure 2.28 shows 2 different variants of a cathode driven configuration with a comparison to conventional CFAs. Conventional CFAs, shown in Fig. 2.28(a), use a smooth cylinder to emit electrons. The RF electric fields from the anode are weakest at the cathode because it is radially disposed from the anode. The electron cloud at the cathode thus has a weaker frequency-determining component which produces a noise component, typically 50 dB below the output signal. By driving the RF at the cathode, the RF drive signal is highest at the cathode. Figure 2.28(b) shows the input signal on the cathode circuit alone. This showed an improvement to gain, but no improvement to noise. The importance of control over 63 the entire electron trajectory is apparent from the cathode-driven only experiment, so a hybrid approach was developed. Figure 2.28(c) shows a hybrid approach, where the RF drive is applied to both the anode and cathode, providing improved electron trajectory control. The hybrid approach showed the gain improvement observed in the cathode-driven only experiment along with dramatic improvements to the signalto-noise ratio (20 dB/MHz) over conventional CFAs. Figure 2.28: (a) conventional CFA comparison with a (b) cathode-driven and a (c) hybrid variant. [85]. This approach isolates the source power from load reflections. By decoupling the output from the input, the stability of the CFA is improved. This also extends its applications to ones where a variable load is used, such as particle accelerators [86]. 2.8.2 Linear Format Injected Beam CFAs A less commonly used CFA today is the linear format injected beam CFA. These CFAs were explained briefly in the beginning of Chapter 2. These devices use a variety of different slow wave circuits, but the general format is the same as described 64 in the beginning of the chapter. The designs use a Kino type gun [87] to inject the beam current. Figure 2.29 shows both short and long type Kino guns. This gun design allows for dense electron beam in the presence of the magnetic field in CFAs. Another similarity in the linear injected beam CFAs is the use of depressed collectors [7, 88] to collect the electron beam at the end of the tube in order to increase efficiency. Depressed collectors are not discussed in this dissertation, but they increase efficiency of the device by essentially recycling leftover energy in the electron beam collected at the end of the tube and using it for beam emission. (a) (b) Figure 2.29: (a) Short and (b) long Kino electron gun schematics [87]. From the 1950s through the early 1970s much research was performed on these MVEDs because of the high efficiency of the interaction. These devices have since been replaced by TWTs because of their lower cost, higher gain, and greater stability [4]. Even though these MVEDs are not very common today, this is the type used in this dissertation, and a summary of them is given here. Many different Linear format injected beam CFAs have been tested [16, 84, 89–91], and a few important observations are noted here. The exact description of the devices are not given here, but a summary of the 65 general conclusions of those experiments is given. The maximum gain observed among these devices is in the 25 − 30 dB range [84, 90]. Two general limits to getting high gain in these devices are (i) unwanted oscillations outside the matched band and (ii) beam noise. By using selective attenuation on the slow wave circuit and careful circuit termination, the oscillations can be suppressed. Selective attenuation introduces a frequency sensitive loss which attenuates unwanted signals and passes the designed frequency. Gilmour [2] discusses a few attenuation techniques to prevent backward wave oscillations in TWTs. Noise created by the beam is another limit to the gain of the device, and much of the noise is caused by the electron gun design and use of the thermionic cathode. A detailed discussion on noise reduction in CFAs is given by Gilgenbach et. al. [92]. The gain of linear injected beam devices is very sensitive to the beam injection technique [19, 91]. The efficiency of the device depends largely on the cycloiding of the beam in the interaction space, which is highly dependent on beam injection. The experiments by Cooke and Döhler [19, 91] showed that by improving the electron gun optics and making the beam injection “smoother,” significant gains to efficiency are observed. Not only is the beam injection important, but the choice of the beam trajectory itself is important. The beam can be a Laminar flow type or a cycloidal type with varying cycloid radii. An article by Locke [16] developed a theoretical model to model highly cycloidal beams, compared it with experiments, and determined that CFAs which implement a highly cycloidal beam only require 35% of the interaction length of a laminar-beam type for the same output power, gain, and efficiency. The reason given for this improvement is that out-of-phase electrons which extract energy from the RF wave on the circuit are quickly removed from the device by the sole in 66 a highly cycloidal trajectory. These out-of-phase electrons, if they remained in the interaction region, would continue to remove energy from the RF beam, but since they are more easily removed from a highly cycloidal beam, less energy is removed. The research in this dissertation used the highly cycloidal beam. 2.8.3 Linear Format CFA at Northeastern University The CFA described in this section was compared against the simulation results in this dissertation, so a detailed description is given here. In 1991, a group at Northeastern University in Boston, MA developed an injected beam linear format CFA which uses a 150 MHz meandering microstrip line slow wave circuit[14, 15]. This design is used in this dissertation and is shown in Fig. 2.30. The slow wave circuit was comprised of a meandering 1/8 inch diameter copper tube placed on a 1/16 inch thick Teflon dielectric which was placed on a copper ground plane. The circuit was 40 cm long and 25 cm wide with a 1 cm pitch. This slow wave design has a retardation of R = 33. At the operating frequency of 150 MHz, the device length is only 6 slow wave wavelengths long. This length is rather short but is sufficient to see moderate gain. The sole to anode gap was 2.5 cm. Electrons are emitted from a 2% thoriated tungsten filament 10 mil in diameter, and the cathode generates an electron beam about 10 cm wide. A focusing electrode is used to inject electrons into the interaction region as shown in Fig. 2.30. The electron beam is highly cycloidal in order to maximize the interaction over short distances [14–16]. Many of the experiments use an electron beam current of 150 mA. This current is used for most of VSim simulations for ease of comparison with experiment and ease of implementation due to the fact that this current is below the space charge limit of the configuration. 67 Figure 2.30: Northeastern CFA schematic in Browning et. al. [14, 15] The goal of the Northeastern work was to implement in-situ measurements of the electron plasma inside the CFA interaction space during operation. This goal resulted in a low frequency CFA so that the interaction region and RF wavelength were large enough to allow diagnostic probes to be used. Measurements were performed for RF power vs. device length, electron density vs. device length, electron energy distribution of the beam, bandwidth, gain vs. electron beam current, and Langmuir probe current. The measurements of gain vs. frequency and gain vs. beam current are shown in Figs. 2.31 and 2.32, respectively. These two plots will be compared against the VSim results to validate the model. 68 Figure 2.31: Northeastern CFA Gain vs. frequency plot in Browning et. al.Vas = 1250 V, B = 5.2 mT [14] Figure 2.32: Northeastern CFA Gain vs. Beam current plot in Browning et. al. VAS = 1200 V, B = 5.5 mT [15] Another relevant result in this work is the measurements of RF power vs. device length. Fig. 2.33 shows the RF power near the circuit along the length with and without an electron beam. Without the beam, a standing wave pattern emerges whose amplitude fluctuates between 5 and 1 for the entire length. With the beam, a standing wave is also present, but with an increase in amplitude from 0 to 20 cm and then a constant amplitude from then on. From this result, it was concluded that the gain occurs within the 10 − 20 cm length and not the rest of the circuit. This result that the gain occurs only in the first portion of the circuit, if it is true, helps motivate the use of a distributed cathode. Because the gain saturates over relatively short distances, more gain can be achieved by injecting more current after this point. And, in general, the current can be tailored in such a way to help control the gain down the length of the circuit. It should be noted, however, that the RF field is found not to be a good indication of the gain along the length of the circuit based on a simulation study later in this dissertation. 69 Figure 2.33: Northeastern CFA Gain vs. circuit with and without an electron beam in Browning et. al. Prf = 10 W, VAS = 1200 V, B = 5.5 mT[15] 2.8.4 Simulation of a Distributed Cathode in a Rising Sun Magnetron This ongoing research [13, 93–95] focuses on simulation of a rising sun magnetron with a controllable distributed cathode. That work proposed to use gated field emitters as the cathode instead of a thermionic cathode. To test the benefits of using field emitters, simulations were performed using a controllable cathode source. By modulating the injected current to control the spokes in the magnetron device, improved startup times and efficiency were observed and showed reliable dynamic phase control [94]. Startup times were improved from 100 ns for the continuous current case to 40 ns for the modulated cathode case. Efficiencies in the work are not considered the absolute efficiency of the device, but relative comparisons can be made and the efficiency was improved from 80% for continuous current to 95% for modulated current. The work also showed reliable and efficient phase control of the oscillations. And the work showed that the phase can be actively controlled even after the device was oscillating. By shifting the phase of the modulated cathode emission, 70 the spoke locations can be controlled, which controls the phase of the output. The work also determined that only 20% of the total current needs to be modulated in order to get the majority of the benefit to startup time, efficiency, and phase control[95]. Magnetrons are very similar to CFAs, and the promising results in that work indicate that the CFA will benefit from GFEA integration as well. 2.8.5 Field Emitter Use in Microwave Vacuum Electron Devices The use of FEAs in MVEDs has been proposed and implemented for microtriodes [5, 96, 97], klystrodes [6, 32, 96, 97], twystrodes [96], gyrotrons [20], magnetrons [12, 28, 29, 98], and TWTs [8, 9, 99, 100]. The use of GFEAs in gated emission devices such as the microtriode, klystrode, and twystrode is very appealing due to the low transconductance (the ratio of the change in current at the output terminal to the change in the voltage at the input terminal of an active device), short transit times (the time for an electron to travel from the emitter to the gate), and the small package. This approach would allow for high gain, high frequency, small devices. The advantages of GFEAs in TWTs, aside from the improvements to size and efficiency, is the ability to pre-bunch the beam before entering the interaction region. These “emission gated TWTs” can greatly improve the RF performance. 2.8.5.1 FEA Use in CFAs No one to date has published experimental results utilizing FEAs in CFAs, but there has been, however, some theoretical work done by Sokolov et. al. [18, 96] with a distributed FEA cathode in a microelectronic CFA. In that work, the interaction space had to be quite long (tens of wavelengths) in order to accommodate the use of a FEA cathode. This is a disadvantage since the losses in microelectronic lines are 71 much greater. The focus in that work was about the use of two delay line structures to minimize losses in the delay line. The use of two delay lines showed 7 dB improvement over just using one long delay line. 2.8.5.2 FEA Use In TWTs The TWT work by Whaley et. al. [8, 9] is the only published work of a manufactured forward wave device which merits a short discussion. The general design of the TWT remains unchanged from standard TWT designs with only the addition of a GFEA. The main differences are in the beam injection region. Special design considerations were implemented to focus the beam and to protect the GFEA from ion back bombardment. GFEAs have significant beam spread due to the lack of focusing on the emitter tip and space charge defocussing, and the TWT requires good focusing to be optimum. Also, the GFEA has the ability to have independent control over current, decoupled from the accelerating voltage. The focusing technique must be able to work over a wide range of beam currents and acceleration voltages to realize the full potential of GFEAs. Ion back bombardment will degrade the performance of the GFEA and limit the lifetime of the device; therefore protection is necessary. To properly modulate the cathode, a resonant matching circuit, based on a design from Calame et. al. [97], was implemented and is shown in Fig. 2.34. This circuit reduces the necessary RF drive power of the GFEAs, and also allows for a way to maintain the bandwidth of the device. The drive power in this circuit when used as a resonant matching system is given by Eq. (2.28), where Cef f is the GFEA capacitance, ω is the drive frequency in rad/s, V0 is the RF amplitude of the signal, and Qtot is the total quality factor of the system. For higher bandwidth applications, the drive 72 power of the non-resonant design is given in Eq. (2.29), where Ref f is the effective resistance of the GFEA and because Ref f and Cef f are very small, ωRef f Cef f ≪ 1. Prf = ωCef f V02 /Qtot Prf   q 1 2 = ωCef f ωRef f Cef f + 1 + (ωRef f Cef f ) V02 4 (2.28) (2.29) Figure 2.34: GFEA matching circuit used in the TWT work [8], proposed by Calame [97] Whaley et al. [8, 9] have successfully created and operated a 100 W TWT with the use of GFEAs. They developed a new way to focus the beam from GFEAs using multiple lenses. They also implemented a ion shield using a region of positive potential relative to the system between the cathode and the interaction region. They were successful in implementing a 100 W, 5 GHz TWT with a small signal gain of 32.7 dB, a saturated gain of 22.1 dB, and a circuit efficiency of 24%. Life tests of this device were rather short with 150 h of cumulative pulsed operation. 73 2.8.5.3 FEA use in Magnetrons There has been ungated FEA work performed on relativistic magnetrons [29, 101]. Relativistic magnetrons use explosive emission [30, 31] for the electron emission. Explosive emission is a phenomenon observed when a field electron emitter explodes due to very high current density. By using very high voltage pulses (> 500 kV) enormous amounts of currents can be observed (> 5 − 10 kA) from explosive emitters [28]. In relativistic magnetrons, gigawatts of pulsed power can be observed at efficiencies of 20-40%. Many different types of cathodes have been developed and studied [102–106] with a focus on the minimization of plasma formation. The most relevant work to the distributed cathode research in this dissertation is the transparent cathode work [11, 12, 98]. Figure 2.35 shows the transparent cathode configuration in an A6 magnetron [98]. Instead of using a solid cathode as the electron source, a series of six cathode strips are used. Instead of a uniform current originating from a center solid cathode, there are six discrete sources of electrons. The transparent cathode has two main advantages: the azimuthally modulated electron emission (cathode priming) shortens the rise time of power generation and the absence of the solid core allows strong azimuthal electric fields near the cathode, improving the electron beam- RF wave interaction [107]. The number and position of the cathode strips was found to affect the mode and operation of the device. With the number of strips equal to half of the cavities, the π mode grows rapidly; when the number of strips is equal to the number of cavities, the 2π mode is excited. Varying the angle found a difference of 90% of the maximum output power from optimum and unfavorable positions. 74 Cathode Strips Solid Cathode Figure 2.35: (top) The transparent cathode configuration with 6 cathode strips and (bottom) a solid cathode [98]. 75 CHAPTER 3 RESEARCH OVERVIEW The goal of the research is to study new CFA designs which use a controllable, distributed cathode to tailor the electron current injection to improve gain, efficiency, and noise. Originally, experiments and simulations using VSim [42] were proposed to study the effects of the electron current profiles from a distributed cathode, but the experimental design showed no electron beam interaction with the RF wave on the circuit. An extensive investigation of the electron beam trajectories and the dispersion characteristics of the slow wave circuit ruled out these as the source of the problem. It was determined that the maximum current available cathode was less than the minimum current needed for interaction. Because of this, the research focus was shifted to a simulation of a CFA design studied at NU [14, 15]. Two different CFA designs were tested experimentally here at BSU, and three different designs were studied via simulation. This chapter presents the proposed experimental design and outlines the chronology of research. 3.1 Proposed Experimental Design The proposed CFA design is a linear format with a meander microstrip line for the slow wave circuit. GFEAs in conjunction with hop funnels were proposed to implement the controllable distributed cathode. GFEAs provide a simple way to have a controllable 76 distributed cathode. Hop funnels provide the protection for the GFEAs from the high electric fields and current densities of the interaction region and provide a way to control the energies of the injected electrons separately from the sole potential. Two different configurations are proposed in this work: an injected beam and a distributed beam. 3.1.1 Injected Beam Configuration Experiment The injected beam CFA pictorial schematic is shown in Fig. 3.1 along with the dimensions. The electron trajectory is shown as the red cycloidal line. The electrons are emitted from the GFEA and follow the cycloidal trajectory due to the crossed magnetic and electric field. The cycloidal trajectory in the figure is a pictorial representation and not representative of the actual trajectory. The electrons enter the interaction region, interact with the RF wave on the circuit, and collect either on the slow wave circuit or the end collector. The electric field in the interaction region is controlled by the potentials on the sole and slow wave circuit. The magnetic field is controlled by external Helmholtz configuration. An RF wave is input on the slow wave circuit on the left, and if the electron velocity is close to the phase velocity in the interaction region, the RF wave will be amplified at the RF output on the right. Note that the GFEA cathode is below the sole electrode in some parts; this is because the GFEAs available to the group were large (9.5 × 12.5 cm), and this was the best way to fit the cathode in the CFA chamber. The GFEA cathode and slow wave circuit are discussed in the next sections. 77 Figure 3.1: Schematic representation of the injected beam CFA design with dimensions, not to scale. 3.1.2 Meander Line A meander line microstrip circuit is used as the slow wave circuit. The meander line circuit is used because of its ease of manufacture and ease of impedance matching. The practical use of meander lines is limited by the inability of the circuit to dissipate power and by dielectric charging. Because of the lower power operation of this CFA, the meander line is sufficient. Figure 3.2 shows the geometry and dimensions of a generic meandering microstrip circuit. The exact dimensions and parameters of operation of the circuits used in this work are listed in Table 3.1. The circuits were designed to be at least 6 slow wave wavelengths long and to fit in the chamber available to our group. Two circuits were designed and used. The first circuit, SW1, experimentally demonstrated undesirable phase velocities, so a new design, SW2, was developed. Much of the results are redundant between the circuits, so the experimental focus is on the circuit called SW2. 78 Figure 3.2: The diagram showing the meander line microstrip. A metal line meanders over a dielectric with thickness Hd over a ground plane. Table 3.1: Slow wave specifications Name Period Width Line Line Dielectric Effective Estimated Operating (P) (W) Width Height Thickness dielectric Retardation Frequency [mm] [mm] [mm] [mm] [mm] (ǫr ) (R) [MHz] SW1 8 50 1.5 1.8 0.5 1.796 18.09 800-1000 SW2 7 74 1.2 1.8 0.33 1.815 29.83 400-600 3.1.3 Cathode The proposed cathode for this CFA is a GFEA. The GFEA available to the group was a Spindt type gated field emitter array [23] obtained from PixTech Field Emission Displays fabricated in 2001 [108]. The cathode unit was 9.5 × 12.5 cm and the CFA configuration was designed around this constraint. The emission area is about 4 cm2 , and the desired current was on the order of 100 − 200 mA. GFEAs at the time this CFA was designed (2011) had demonstrated current densities of 20 A/cm2 , which would theoretically allow for 80 A of current from the emission area, but space charge 79 limits the current with the electric fields used in the CFA to currents on the order of 100 − 200 mA . At the time of this writing, current densities of 100 A/cm2 have been achieved by GFEAs developed by Guerra et. al. at MIT [25–27]. 3.1.4 Distributed Cathode The distributed cathode configuration includes the same meander line circuit and electron source as the injected beam configuration. The difference is the sole design. The distributed cathode CFA pictorial schematic is shown in Fig. 3.3. In this configuration, electrons are emitted up into the hop funnel, ’hop’ up the dielectric wall, and enter the interaction region. There are multiple injection points in this design, and electron current at each injection point can be controlled by the GFEA. The potentials between the sole electrode and the slow wave circuit control the electric field in the interaction region while the hop electrode controls the electron energy of the electrons. Figure 3.3: Schematic representation of the distributed cathode CFA design, not to scale. Electrons injected into the hop funnels are extracted though slits in the sole electrode. 80 3.1.5 Sole/Hop Funnels The hop funnels were fabricated out of Low Temperature Co-Fired Ceramic (LTCC) [109]. A schematic of the hop funnels/sole is included in Fig. 3.3. The LTCC spans the width and length of the interaction region. Two layers of metal are on the surface of the LTCC structure separated by a dielectric layer. The two metal layers are called the hop and sole electrodes. As explained in Chapter 2, the hop electrode is used to control the energy at which the electrons are born, and the sole electrode is biased more negative than the hop electrode in order to prevent cycloidal electrons from collecting on the sole. 3.2 Research Chronology Because no gain was observed experimentally, the focus of the work shifted to simulation of a design used by a group at Northeastern University [14, 15], which is very similar to the injected beam configuration presented here. The Northeastern design and the simulation model are explained later. Even though the proposed designs showed no gain, valuable data was still obtained for comparison to the simulation for validation. The experimental injected beam design is the simplest control variation to easily test the general function of the CFA. The distributed beam configuration was briefly tested by the group but no results are presented from that experiment. Three different linear format CFA designs are studied in this dissertation. Two were developed at BSU for this dissertation (CFA1 and CFA2 which use slow wave circuits SW1 and SW2, respectively), described above, and one was designed at Northeastern University [14, 15] to perform in situ measurements of the interaction region. Each of the designs contribute to the research but the main contributions 81 come from simulation of a distributed cathode variant of the NU CFA design. This section describes the research flow and major mile markers of the research. Figure 3.4 shows a visual diagram of the research flow. 82 BSU CFA Experimental Work Simulation Work CFA1 CFA2 CFA Model Development Development Development Experiment Experiment FAIL FAIL Dispersion Study NU CFA Experiment Simulation Validation BSU CFA Simulation Investigation Beam Study Northeastern Work Beam Compare Full Run Dispersion Full Run Compare Compare PASS PASS PASS BSU CFA Design Determined Unfit BSU CFA Work Terminated NU CFA Sim Full Run Compare PASS NU CFA OK NU CFA Simulation work Continued NU CFA Simulations Legend NU CFA Characterization Research Flow Injected Beam Distributed Beam General Results Failure Time Varying Static Time Varying Static Research Stages Development Compare And Contrast Analysis Simulation/Experiment Time Varying Improves Design Positive Outcome Negative Outcome Real World Implementation? Figure 3.4: Diagram outlining the research flow of the three CFA designs. The BSU experimental work was used to validate the simulation model, but all work on the BSU CFAs were terminated after determining the design was unfit. Results from the Northeastern CFA experimental work were also used to validate the simulation model, and the design was used for the distributed cathode studies. 83 The original goal of this research was to experimentally and computationally study the effects of different emission profiles using a fully controllable distributed beam in a CFA. The first two designs, CFA1 and CFA2 which use slow wave circuits called SW1 and SW2, respectively, were developed at BSU for this purpose. Injected beam CFA experiments showed no RF interaction with the beam. Three different possibilities were investigated to determine the problem: slow wave circuit dispersion, electron beam trajectory, and insufficient electron beam current. Dispersion measurements were performed and compared with COMSOL and VSim simulations. Dispersion measurements showed a higher phase velocity than predicted from simulation but corroborated the general behavior and trends. The available equipment prevented matching the electron beam E × B velocity to the high phase velocity of SW1, which instigated the development of SW2 used in CFA2. Measurements concluded that SW2 also had a higher phase velocity than predicted from simulation, but allowed for a testable CFA setup with the available equipment. Still no RF wave electron beam interaction was observed. A thorough investigation to determine the reason for the lack of RF wave interaction with the electron beam showed that much more current was required than was capable of the GFEAs used in the experiment. VSim modeling and analytical analysis using Pierce theory both showed a current of 150 mA is needed to observe appreciable gain where only 5 mA of current was available from the PixTech cathodes. The BSU CFAs would never show RF interaction with the electron beam with the available equipment, so the design was shifted to the experimentally verified design from NU. The notable milestones in the BSU CFA experiments and simulations were dispersion experimental validation of the VSim model, simulated confirmation of experimental problem with the design, and the decision to shift to a simulation focus of the NU 84 CFA design. The next phase in the research was the experimental validation of the RF wave interaction with the electron beam in VSim using the NU CFA. VSim simulations were performed to model the exact operating parameters of the NU CFA design. VSim results matched the NU experimental results rather well, and it was determined that VSim model using the NU CFA design was a viable method to test a distributed cathode. After the model was validated against experiments, the focus shifted to characterization of the various cathode implementations. Four different methods were tested: static injected beam, modulated injected beam, static distributed beam, and a modulated distributed beam. Each of these methods was compared, and it was determined that the time varying methods improved the design. This is expected and compelling, but the impact of this result is questionable without a viable method to implement the GFEAs in high power devices. implementation ideas. The discussion then shifts to 85 CHAPTER 4 CFA EXPERIMENTS AND MEASUREMENTS This chapter describes the experiments performed on SW1 and SW2. The full CFA setup with these circuits never showed any gain, but a brief description is given here. Experiments characterizing the electron beam and slow wave circuit dispersion were successful and used to validate the VSim simulation model, so a more detailed description is given for those setups. 4.1 Full CFA Setup The vacuum chamber, electromagnets, the CFA Structure, measurement hardware and LabVIEW software has been built and tested. A summary of the setup is given here. 4.1.1 Vacuum Chamber and Electromagnets The chamber system is shown in Fig. 4.1. The electromagnets surround the chamber and allow for a nine inch sphere of uniform magnetic field inside the chamber. The CFA fits inside this sphere. The pressure of the chamber during operation was in the 10−7 Torr range. 86 Figure 4.1: Photograph of the electromagnets and the chamber system where the CFA experiments are run. 4.1.2 CFA Structure The experimental CFA pictorial schematic is shown in Fig. 4.2. The resistors in the schematic are to measure current to the electrodes and to limit arcs. These resistors are fixed resistors, sized so that the voltage drop is well above the noise level and within the range of the analog to digital converters (∼ 1 V drop at the expected current, 1 kΩ to 1 MΩ at 1 mA to 1 µA currents). The Interaction region is the gray region in between the sole and the slow wave circuit. This region is where the electrons interact with the RF wave. The end hats in the schematic are shown as dashed lines 87 to represent that they are not in the interaction region, but they bound the region on both sides in the z-direction. These electrodes help contain the electron beam within the interaction region. Figure 4.2: Schematic representation of the CFA design, not drawn to proportion. The reason the GFEA extends below the sole electrode is due to the size of the GFEAs available. Section 4.1.4 describes the PixTech cathodes in detail. The CFA design must accommodate this large cathode structure. Fig. 4.3 shows a photograph of the CFA structure without the slow wave circuit. The slow wave circuit in this figure would sit over the sole, overlapping the end hats a little bit. Electrons are emitted from the GFEA and cycloid down, in this view, between the sole and the slow wave circuit. The end hats prevent the electrons from escaping out the sides of the device. To control the emission of the GFEA, Kapton coated wire is fixed to the PixTech cathode by a combination of silver paste and tape and is labeled gate connection in the figure. 88 Figure 4.3: Top down view of the CFA structure without the slow wave circuit. 4.1.3 Slow Wave Circuit Two meander lines were built and studied. The dimensions of each circuit are provided in Chapter 3. Fig. 3 shows one of the two circuits, SW2, built in this work. The circuit is a type of microstrip that meanders at 90◦ angles over a Teflon dielectric which is over a conducting ground plane. Note that the wire height is fairly large for a strip line. This is to minimize the charging of the dielectric during amplification in the CFA configuration. Any electrons that manage to come close to the slow wave structure will be scrapped off by the protruding wire as opposed to striking the Teflon. Of course, with high RF power and amplifications, microstrip slow wave structures encounter high electron bombardment. The meander slow wave structure will most likely not be able to handle high electron bombardment, and this is one limiting factor of the maximum power output. 89 Figure 4.4: Photograph of slow wave circuit SW2. A rectangular copper wire meanders on top of a Teflon dielectric which is on top of an aluminum ground plane. The copper wire is fixed to the ground plane by polypropylene screws. The input an output ports are SMA connectors which are connected to the copper wire by silver paste. 4.1.4 GFEA The main reason the experiment never showed any electron beam interaction with the RF wave was due to the inability of the cathode to supply the necessary current. This section describes the cathode itself, the various problems encountered when using the cathodes, and the reason for the low obtainable current. The electron source used in this research is a Spindt type gated field emitter array [23] obtained from PixTech Field Emission Displays fabricated in 2001 [108]. To remove the cathode from the display assembly, the glass frit seal was broken. Many of the cathodes were damaged during the removal process. Many of the cathodes used in the experiment were cracked from disassembly. The main side effect of a cracked 90 cathode is an increase in electrical short circuits between the gate and the emitter tip which increases leakage current. The increased leakage current limits continuous operation time due to ohmic heating and limits the cathode current control. To limit the leakage current, all the gates and emitters not used were left floating. Ideally these unused sections would be reversed biased to prevent current emission. Because of the complicated network of shorts between gate and emitters a various locations of the cathode, some parts of the cathode would be forward biased due to forward biasing of the active section. From this, unwanted emission sites were active and had to be accounted for in the CFA design. To prevent unwanted current from entering the interaction region or damaging components, usually a piece of metal was placed to intercept the current. Figure 4.5 shows an example of one of the PixTech cathodes laid out on the CFA platform. Notice the crack on the bottom left corner. This is caused by the disassembly process. The lighter color streaked portion on the left side of the cathode is damage caused by arcing when operating the CFA in preliminary work. These preliminary experiments experienced many arcs and damaged many cathodes. The current CFA configuration prevents cathode damage. The gate connections are on the top edge of the cathode, and the emitter connections are along the left edge. the majority of the cathode sits under the sole, the metallic sole platform, and the end hats except for the emitting portion, as shown in Fig. 4.3. 91 Figure 4.5: Top down view of the CFA structure without the slow wave circuit and the end hats to show the PixTech cathode and the gate and emitter connections. 4.2 Meander Line Dispersion Measurements To determine the phase velocity of the circuits, some experiments were performed. These measurements are compared with dispersion measurements simulated in COMSOL and Vsim. The experimental setup is outlined here. 4.2.1 Experimental Two different experiments were performed on the meander lines to determine the dispersion characteristics: (1) measure the standing wave pattern and (2) determine the S-parameters using a Network analyzer. The main goal of these experiments was to find the phase velocity of the slow wave circuit at the operating frequency of the 92 device. A good portion of the dispersion curve is displayed to confirm the experiment and to properly characterize the meander line circuit. 4.2.1.1 Dispersion Characteristics From Standing Wave Measurements An X-Y stage was developed to measure the standing wave above the circuit. Fig. 4.6 shows the experimental setup. The circuit is energized with an RF signal at a particular frequency, and the other end of the circuit is terminated into a short. A shorted load yields a very clean and distinct standing wave pattern. To measure the electric field intensity, a small wire sticking out of a coaxial cable was used as an antenna. This antenna closely resembles a simple monopole, but instead of an infinite ground plane, an outer conductor from the coaxial cable is used. Although not perfectly polarized, the ’monopole’ antenna is more sensitive to electric fields parallel to it, and in this case the y-polarized electric field. A spectrum analyzer is used to measure the field intensity. Stepper motors are used to step in the x and z directions. LabVIEW code was developed to interface with all the equipment, to step the stepper motor, and to take a measurement. An array of electric field intensities can be gathered on the x-z plane right above the circuit to map the standing wave pattern at different frequencies. 93 Figure 4.6: Photograph of the standing wave measurement setup. The slow wave circuit sits on top of an x-y stage, and a coaxial cable connected to a spectrum analyzer on one end and the other end is placed right over the slow wave circuit with the center conductor exposed. To extract the dispersion characteristics from this data, the standing wavelength in the x-direction was found using either a 2D spatial FFT or a 1D spatial FFT along the center of the meander line in the x-direction. From transmission line theory, the traveling wavelength is twice the standing wavelength. In this way the wavelength can be determined for each frequency, and the dispersion diagram can be created. More details on this method are given in Section 6.2.2.1. 4.2.1.2 S-Parameters Using a network analyzer, the S-parameters of the circuit can easily be found. The network analyzer was used to measure the S-parameters up to 3GHz. Because of the periodicity of the circuit, certain cutoff conditions (S21 < −20 dB) should exist as outlined in the Chapter 2. The S-parameters are used to confirm the results of the dispersion diagram created by the standing wave experiments. 94 CHAPTER 5 SIMULATION SETUP Three different simulation tools were used to study different aspects of the CFA design. COMSOL [41] is used to study the dispersion of the meander line slow wave circuit. SIMION [40] is used to study the electron trajectories, and Vsim [42] is used to study the full electron beam and RF wave interaction. The post processing of the results is generally done in MATLAB [110]. This chapter describes the setup of each simulation tool and of the post-processing techniques to analyze the results. 5.1 COMSOL Setup Simulations were performed in COMSOL to determine the dispersion characteristics of the slow wave circuits via the standing wave pattern and the S-parameters. One approximation was used on the port. Instead of a coaxial port, a simple rectangular port at the end of the microstrip was used. This approach was used to minimize the number of elements in the model. Another approximation is that the Teflon screws of the circuit are absent in the model, also to minimize elements. The domain boundary conditions are perfectly matched layers (PMLs). The general slow wave model is shown in Fig. 5.1 along with the mesh. This mesh quality is the best that can be offered due to the memory limitations of the computer, 15 GB. 95 2 Elemets Wide Between Lines 2 Elements Wide On Lines Input Port 1 Element Thick (Te on) Output Port Figure 5.1: COMSOL model for SW2 showing the generated mesh The regions in the mesh that require a small mesh size are in between adjacent wires, the wire itself, and the Teflon. The best mesh achieved within memory constraints is 4 elements between wires, 4 elements on the wires, and 1 element for the Teflon height for SW1. It was found that the minimal requirements are 2 elements between wires, 2 elements on the wires, and 1 element for the Teflon height. The difference in the results from both cases were very minimal. All simulations presented here are with these minimal requirements for mesh size. The ’free space’ meander line simulations were performed. Also, due to the design constraints of the CFA configuration such as the sole and end hats, these were also added into the simulation to see the effect on the dispersive characteristics. The sole and end hats were implemented by using conducting blocks with no actual structural support. The structural support in the real CFA is far removed from the interaction region and is not needed for the simulation. 5.2 SIMION Setup A side view, normal to the x- y-plane, of the 3D CFA setup is shown in Fig. 5.2. The mesh cell size for the electric field calculations is 1 mm. The particle source emits a 96 number a particles from a square patch on the cathode. The energy of the emitted particles is of a Gaussian distribution centered about 60 eV with a standard deviation of 5 eV [108]. The direction of the emitted particles is of a cone distribution with a 60◦ half angle to model the GFEA emission characteristics. Each particle corresponds to a possible trajectory. This trajectory is plotted from cathode to the electrode at which it collects. There are 4 electrodes that are monitored: the sole, anode, end collector, and the emitter/gate. By using many trajectories emitted from the emitter with a distribution of energies characteristic of a FEA, the currents at each electrode can be monitored and compared with experiment. Figure 5.2: SIMION CFA configuration from the side (normal to the x − y plane). Electrons cycloid from right to left in this model. 5.3 Vsim Setup The process of setting up the model went through a few stages and different approaches as a part of this research. Some approaches were abandoned and are irrelevant to the working model; however these approaches are valuable for anyone interested in using Vsim and provide valuable insight; therefore descriptions of these approaches are given here. First, the general model outline is described along with the solvers, geometry, and diagnostics. Then the different grid types are discussed along with their effects on the implementation of the solvers, geometry, and diagnostics. Also, the different types 97 of cathode implementations are discussed: the injected beam configuration and two different types of distributed cathode implementations. 5.3.1 The VSim Model The model requires a creation of the geometry on the grid, an electrostatic field solver, an electromagnetic solver, and a particle push algorithm. Fig. 5.3 shows a 3D view of the injected beam configuration on a uniform grid. The electrostatic (ES) solver is needed to implement the cathode, sole, end hats, and slow wave circuit potential. A separate solver is needed to implement the RF waves on the slow wave circuit. The particles need to interact with both the static and RF electric and magnetic fields (via the particle push algorithm). The electrons also generate electromagnetic fields themselves. Figure 5.3: Vsim Geometry with electrons. The RF wave is input on the edge of the domain, within the coaxial port. The RF wave travels within the dielectric region between the ground plane and the green meander line. Electrons are emitted from the cathode region, and cycloid right due to the crossed electric and magnetic fields. The electrons interact with the RF wave and give up their energy to amplify the RF wave. The following is a summary of the model implementation and is explained in detail in the following sections: 98 • Define the grid and create the geometry • Create electrostatic boundary conditions and solve the electrostatic field • Define RF ports and boundaries and define the electromagnetic solver • Add both the RF and ES fields to create the total electric field • Define the particle parameters, sources, and sinks • Define any diagnostics Table 5.1 shows a summary of all the parameters involved with the implementation of the model. The following sections describe these parameters in detail. These parameters are referred to for the rest of the dissertation, so this table provides a good reference. Table 5.1: Summary of Parameters Category Voltages: Parameter Vas Vcs Vbe Veh Vcathode Vsole Description Voltage from Anode to Sole Voltage from Cathode to Sole Voltage of the Beam Electrode Voltage of the End Hats Potential of the Cathode Potential of the Sole SW Circuit: Lp Wsw Lsw WL HL Hd Lcoax RIcoax ROcoax Length of the Pitch Width of the Slow Wave Circuit Length of the Slow Wave Circuit Line Width Line Height Dielectric Height Coaxial Cable Length Radius of the Coaxial Inner Conductor Radius of the Coaxial Outer Conductor CFA Dimensions: Has Hcbe Height from Anode to the Sole Height from the Cathode to the Beam Electrode Cathode Electrode Length Lc 99 Category Parameter Le We Ls Lcs Lco Hbe Description Emission Region Length Emission Region Width Sole Length Length between the Cathode and Sole Cathode Offset Length Height of beam electrode VSim Grid: dX, dY , dZ, NLW Nd Cell length in X, Y, and Z Number of Cells per Line Width Number of Cells per Dielectric Other Parameters: Ibeam Prf Electron Beam Current RF input Power Pgf ea GFEA Drive Power Hc Height of the cathode from y = 0 edge for divergence free region Length between emitter to emitter of segmented cathode Approximated current density used in cathode approximation 3 Le2e Jy∗ Spatial Emission Profile: φx φt φof f set Jp Itot fDC Je 5.3.1.1 Spatial phase shift used for the sine wave emission profile Time phase shift used for the sine wave emission profile Phase shift offset from the RF accelerating region for the sine wave Profile Peak current density of the emission profile Total emitted current Fraction of the current density that is uniform Emission current density Grid The grid can be uniform or non-uniform. The cell sizes (dX, dY , dZ) of the uniform grid remain constant throughout the entire domain; whereas the cell sizes of the 100 non-uniform grid can vary in the same direction. The cell size throughout the domain is determined by the resolution requirement of the smallest geometry in the domain. Because of this limitation, the number of cells in a uniform grid can be quite large. The non-uniform implementation can have small cells located where they are needed and increased cells sizes where the resolution requirements are less stringent. A detailed description of the non-uniform and uniform grid can be found in sections 5.3.3 and 5.3.4, respectively. The minimum requirements on the x and z axis are determined by the slow wave circuit line thickness. The geometric dimensions for the top view (normal to the y-axis) on a uniform grid is shown in Fig 5.4. Fig. 5.4 does not show the actual dimensions of any slow wave circuits used in this work but is used for a good pictorial representation. Note that in this case, the line width is two cells wide (NLW = 2), which is the same for dX and dZ. This resolution is the chosen resolution for the CFA models used in this work. It would be more desirable to increase this resolution, but each increase in number of cells per line width, NLW , increases the number of cells in both the x and z axis. Studies on the effects of the NLW show minimal gains to accuracy for NLW > 2. Also note that due to this coarse resolution, the inner conductor of the coaxial port is square rather than circular. 101 Figure 5.4: View of the dimensions of the slow wave circuit and ports from the top view, normal to the y-axis. The green meander line comes down (in the y-direction) through the outer conductor of the coaxial cable, and then meanders above the dielectric (not shown) and ground plane shown in red on the x − z plane. The minimum requirement in y is determined by the dielectric thickness. The geometric dimensions on a uniform grid for the side view (normal to the z-axis) is shown in Fig 5.5. These dimensions are the actual dimensions used for SW3, the circuit used in the NU CFA. In this case the number of cells per dielectric, Nd , is two. These studies are also not presented here for brevity. This resolution is chosen for the simulations used in this work. Once again, more cells would be desirable but would increase the model size greatly. 102 Figure 5.5: View of the VSim model, showing the dimensions of the slow wave circuit and ports from the side view, normal to the Z-axis for the NU CFA study. 5.3.2 Create the geometry There are two methods to create a geometry in VSim, and this section focuses on the first method. The first method uses spatial coordinates to define the geometry rather than cell coordinates. These spatial coordinates are then translated to the grid coordinates for implementation of the finite difference method. This method allows for implementation of complex geometries. The slow wave circuit, the coaxial input and output ports, and all the exterior of the domain are defined using this method. The other method creates boundary conditions using cellular coordinates but is limited to cubic geometries. This method is defined and discussed in the electrostatic and electromagnetic sections. Many of the electrostatic boundary conditions are defined in this way. Also, the dielectric of the slow wave circuit is defined in a similar way. The geometry is created using a superposition of mathematical functions using spatial coordinates rather than cell coordinates. VSim supplies two main types of 103 functions to create the geometry: fill and void. These functions, used in conjunction with mathematical descriptions of the shapes, create the geometry. The fill function fills the defined area with metal, and the void function removes metal from the defined area. Adding many functions like this together, complex geometries can be created. Note that the order in which the fill and void functions are defined affect the end result. To define different shapes mathematically, a few notable functions are used. The Heaviside function is used to define every shape. By defining mathematical functions which are greater than zero in parts where the desired shape is located, any geometry can be generated. Another important function which speeds up the geometry evaluation process is the modulus function. When creating a periodic geometry, such as the slow wave circuit, each period is defined in the same way but with an offset. One inefficient way to define the periodic structure is to define each segment with its own equation, but with a larger number of periods, evaluating this large set of equations takes time (>1 hour). By using the modulus function, one set of equations which define one period can be used to define the whole periodic structure. This approach reduces the geometry evaluation time to mere seconds. 5.3.2.1 Geometric Translation to Grid The finite difference method uses a cellular grid to perform all calculations; therefore the spatially defined geometry needs translation to the grid. There are two ways in which the geometry is converted to the grid: using a Dey-Mittra cut-cell approach [74, 75] or the stair step approach [73]. Each cell in the stair step approach can only be either conductor or vacuum; whereas the cut-cell approach can “weight” each cell as both conductor and vacuum. The model used in this work is all cubic architecture 104 except for the coaxial port. This model does not use the Dey-Mittra cut-cell approach, so the discussion is focused on the stair step. In order to actually implement the correct geometry, the geometry needs to align with the grid properly. Fig. 5.6 shows (a) the geometry alignment with the grid and (b) the corresponding y-component of the electric field of a generic run. In Fig. 5.6(b), the y-component of the electric field is defined at the nodes (where the grid lines intersect). Green shaded areas correspond to where the y-component of the electric field is zero, which corresponds to conducting regions since parallel electric fields are zero at the boundary of a conductor. Shown in Fig. 5.6(a), the geometry engulfs three nodal points in the width of the circuit. This translates to a conducting region of three nodes, shown in Fig. 5.6 as three nodes of Ey = 0. This corresponds to a line width of two cells which translates to 2dX, which is the desired line width. (a) (b) Figure 5.6: Top down view (normal to y-axis) of (a) the meander line geometry with a good alignment with the grid and (b) the corresponding Ey field of a generic run . The green section denotes locations where Ey = 0 which corresponds to conductor, and the blue part is vacuum. Also note that the geometry width is two cells wide plus a small offset in order to ensure that the nodes are actually engulfed. If numerical errors actually make 105 the width slightly less than two cells wide, the edge nodes will not be engulfed, and the node will be translated to vacuum erroneously. Fig. 5.7 shows a poor geometry grid alignment and the corresponding Y component of the electric field of a generic run. The geometry appears to be two cells wide but only engulfs two nodes. This translates to only two nodes being conductor which means the actual geometry is one cell wide. (a) (b) Figure 5.7: Top down view (normal to y-axis) of the meander line geometry with a poor alignment with the grid (a) and the corresponding Ey field of a generic run (b). The green section denotes locations where Ey = 0 which corresponds to conductor, and the blue part is vacuum. The geometry to grid translation also has an effect on the coaxial port. Fig. 5.8 shows the coaxial cable and the corresponding Ey field. Denoted by the blue nodes in Fig. 5.8(b), the vacuum portion has two nodes between the inner and outer conductor for most of the region and has only one node on the upper left and lower right corners. It would be desirable to have more nodes in this region to properly resolve the fields, but it is impractical due to computational time constraints. Also, proper modeling of the fields in this region is not that important so long as the power is conserved since 106 the main goal of the research is to improve the gain of the device. Also, the length of the coaxial cable from the meander line to the port boundary condition is very small, which minimizes any reflections caused by an impedance mismatch caused by the coarse resolution. Resolution studies were performed on the co