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LBL-29720 Lawrence Berkeley Laboratory UNIVERSITY OF CALIFORNIA Accelerator & Fusion Research Division Theory of the Ion-Channel Laser DJEi Whittum (Ph.D. Thesis) September 1990 HeceWed by OSW FEB191991 DISTRIBUTION OF THIS DOCUMENT IS UNLIMITEC Prepared for the VS. Department of Energy under Contract Number DE-AC03-76SF00098. DISCLAIMER This document was prepared as an account of work sponsored by the United Stales Government. Neither the United Stales Government nor any agency thereof, nor The Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe; privately owned rights. 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DISTRIBUTION OF THIS DOCUMENT IS UNLIMITED Theory of the Ion-Channel Laser Copyright 01090 'David H. Whittum ' The United States Department of Energy has the right to use this thesis for any purpose whatsoever including the right to reproduce all or any part thereof Theory of the Ion-Channel Laser by David H. Whittum Abstract A relativistic electron beam propagating through a plasma in the ion- focussed regime exhibits an electromagnetic instability with peak growth rate near a resonant frequency a>~2"fi o>p, where yis the Lorentz factor and cop is the betatron frequency. The physical basis for this instability is that an ensemble of relativistic simple harmonic oscillators, weakly driven by an electromagnetic wave, will lose energy to the wave through axial bunching. This "bunching" corresponds to the development of an rf component in the beam current, and a coherent centroid oscillation. The subject of this thesis is the theory of a laser capitalizing on this electromagnetic instability. In Chapter 1 a historical perspective is offered. In Chapter 2, the basic features of relativistic electron beam propagation in the ion-focussed regime are reviewed. In Chapter 3, the ion-channel laser (ICL) instability is explored theoretically through an eikonal formalism, analgous to the "KMR" formalism for the free-electron laser (FEL). The dispersion relation is derived, and the dependence of growth rate on three key parameters (detuning S, Pierce parameter p, and betatron parameter a$ is explored. Finite temperature effects are assessed. From this work it is found that the typical gain length for amplification is longer than the Rayleigh length and we go on to consider three mechanisms which will tend to guide the radiation. First, we consider the effect of the ion channel as a dielectric waveguide. We consider next the use of a conducting waveguide, appropriate for a microwave amplifier. Finally, we examine a form of "optical guiding" analgous to that found in the FEL. In Chapter 4, the eikonal formalism is used to model numerically the instability through and beyond saturation. Results are compared with the numerical simulation of the full equations of motion, and with the analytic scalings. The analytical requirement on detuning spread is confirmed. In Chapter 5 results are summarized and prospects for future work are considered. Table of Contents page Index of Figures vii Index of Tables x Acknowledgments xi Dedication xv Chapter 1: Introduction 1 A. Overview of Beam-Plasma Physics 2 1. Early work on beam-plasma physics 3 2. Induction linear accelerators and beam-plasma physics 4 3. Beam-plasma theory and experiment 5 4. Novel beam-plasma concepts 7 B. Overview of Free-Electron Radiation Devices 8 C. Summary 11 References 13 Chapter 2: Short Pulse Propagation in the IFR 21 A. The Ion-Focussed Regime 22 B. Equilibrium Propagation 26 1. Steady-state plasma electron flow 27 a. MHD equations 27 b. Large skin-depth limit (V«l) 33 c. General solution 39 2. Ion-Collapse 46 3. Beam equilibrium 50 ii a. Single-particle Hamiltoniai 50 b. Longitudinal wake 52 c. Equations of motion 53 d. Continuous plasma focus 56 G Beam Break-Up Instabilities in the IFR. 60 1. Beam break-up equation 61 2. Electron-hose instability. 65 3. Ion-hose instability 73 D. Other Plasma Effects in IFR propagation 76 1. Scattering 77 2. Radiation 79 3. Ionization 80 4. Streaming and other instabilities 82 5. Erosion 84 E. Summary 85 References 87 Chapter 3: Theory of the Ion Channel Laser 93 A. Concept 95 B. Eikonal Formalism 98 1. Particle equations 99 2. Particle motion in a prescribed field 108 a. Bounce motion 109 b. Ponderomotive force 115 c. Invariants 127 d. 1-D equations 132 iii page 3. Maxwell's equations 133 a. Eikonal equation 135 b. Dispersion relation 136 c. Cubic regime 142 d. Quadratic regime 147 4. Finite temperature effects 158 a. Cubic regime 159 b. Quadratic regime 161 C. Maxwell-Vlasov Dispersion Relation 163 2. Phase-space integrals 163 2. Method of characteristics 166 3. Step-radial profile. 177 4. Arbitrary profile 179 a. Modified dispersion relation 180 b. Cubic regime 184 c. Quadratic regime 185 d. Example: parabolic profile 186 e. Example: Gaussian profile. 187 D. Radiation-Guiding 188 1. Ion-channel dielectric guiding 190 a. Weak guiding 194 b. Moderate guiding 195 c. Strong guiding 198 2. Effect of conducting waveguide 199 a. Rectangular waveguide 199 b. Circular waveguide 200 3. Optical guiding 201 a. Cubic regime 206 b. Quadratic regime 209 iv page E. Examples 212 F. Overdense Regime 216 G. Summary 221 References 223 Chapter 4: Numerical Simulations 226 A. The Codes 227 l.Eikonal equation solver (ECL) 229 a. Inputs 229 b. Initialization 230 c. Numerical requirements 232 2.Full-equation solver (FULLCL) 233 a. Inputs 233 b. Initialization 233 c. Numerical requirements 234 B. Microwave examples 234 1. Ion-channel guiding example 236 a. Summary of ECL results 237 b. Comparison with FULLCL results 240 c. Particle motion 243 d. Effect of detuning spread 253 2. Effect of optical guiding 260 a. Comparison of ECL, FULLCL results 260 b. Effect of detuning spread 261 3. Waveguide example 266 a. FULLCL results 266 b. Error in plasma density 266 4. Discussion 268 v Page C. Sub-millimeter examples 270 1. High-gain example. 270 2. Low-gain example , 273 D. Infrared example 274 E. X-Ray example 278 F. Summary , 279 References 281 Chapter 5: Conclusions 282 A. Summary 283 B. Prospects 285 References 287 vi Figures P»g« Figure 2.1: Magnetically s^lf-focussed regime 22 Figure 2.2: Ion-focussed regime 23 Figure 2.3: Ion-channel 25 Figure 2.4: Ion-channel equilibrium (I~l kA) 36 Figure 2.5: Ion-channel equilibrium (I~4 kA) 38 Figure 2.6: Ion-channel equilibrium (J~16 kA) 40 Figure 2.7: Variation of channel radius with current 42 Figure 2.8: Beam equilibrium 43 Figure 2.9: Continuous plasma focus concept 56 Figure 2.10: Beam break-up 61 Figure 2.11: Electron-hose instability 66 Figure 2.12: Ion-hose instability 74 Figure 3.1: Ion-channel laser concept 96 Figure 3.2: Bounce frequency 114 Figure 3.3a: Pierce parameter (1</?,,<1.05; l«fe<5) , 140 Figure 3.3b: Pierce parameter (1</JV<1.05; l«fc<5) 141 Figure 3.4a: Gain in the cubic regime 143 Figure 3.4b: Phase in the cubic regime 144 Figure 3.5: Gain threshold versus S/n 149 Figure 3.6a: Growth rate, lm(Q, versus 6 and n 150 Figure 3.6b: RefQ versus 5 and jU. 151 Figure 3.7: Growth rate versus 5 and /i (larger scale) 153 Figure 3.8 : Maximum growth rate versus pJp 154 vii page Figure 3.9: Detuning corresponding to maximum growth 155 Figure 3.10: HEu mode area vs. /, 0£l£A kA 193 Figure 3.11: HEn mode area vs. J, lkA£J£4 kA 194 Figure 3.12: HEn mode area vs. J, 4kA«£fcl00 kA 196 Figure 3.13: HEn radial wavenumbers vs V 197 Figure 3.14a: Fiber parameter (cubic regime) 207 Figure 3.14b: Overlap integral (cubic regime) 208 Figure 3.15a: Fiber parameter (quadratic regime) 209 Figure 3.15b: Overlap integral (quadratic regime) 210 Figure 4.1: ECL results for first microwave example 238 Figure 4.2: More ECL results for first microwave example 239 Figure 4.3: More ECL results for first microwave example 239 Figure 4.4: Comparison of ECL, FULLCL, analytic 241 Figure 4.5: More comparison of ECL, FULLCL 241 Figure 4.6: Beam centroid oscillations 242 Figure 4.7: Emittance variation 242 Figure 4.8: Motion in y<Jz plane 243 Figure 4.9: Detail of motion in \/r-qz plane 247 Figure 4.10: Detail of qz variation 250 Figure 4.11: Effect of spread in axial momena 254 Figure 4.12: Power versus detuning spread 255 Figure 4.13: Effect of spread in transverse energy 257 Figure 4.14: Effect of optical guiding 259 Figure 4.15: Comparison of ECL/FULLCL, with optical guiding 259 Figure 4.16: Beam centroid oscillations, with optical guiding 261 Figure 4.17: Effect of spread in axial momena, with optical guiding 262 viii Figure 4.18: Power versus detuning spread with optical guiding 263 Figure 4.19: Power versus detuning spread, for both examples 264 Figure 4.20: Effect of spread in transverse energy, with optical guiding..