# THEORY AND COMPUTER SIMULATION OF A NEW TYPE OF PLASMA CHERENKOV MASER (ELECTRON BEAM, MICROWAVE GENERATOR)

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THEORY AND COMPUTER SIMULATION OF A NEW TYPE OF PLASMA CHERENKOV MASER Timothy David Pointon B.S. (University of Nevada, Reno) 1980 M.S. (University of Nevada, Reno) 1982 DISSERTATION Submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in Applied Science in the GRADUATE DIVISION of the UNIVERSITY OF CALIFORNIA DAVIS Approved: Committee in Charge Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENTS I wish to thank my parents, father, for their continued throughout my graduate studies. friends, whose work easier. members who support and in support particular and my motivation 1 am also grateful to my and willingness to help made my Thanks are inspired me due also to those faculty to put in the extra effort to learn new physics. A special DeGroot for acknowledgement is due to Dr. John his unique role as my major professor. His enthusiasm and wealth of ideas almost always presented me with new approaches to seemingly intractable problems I encountered in my research at Davis. Finally I wish to give special thanks to Adam Bridge and Robert Walraven provide unlimited resources and use of Multiware, of their the company's technical development of the CYLTMP code. Inc. They kindly computational assistance This was during invaluable che timely completion of this dissertation. - 11 - Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to TABLE OF CONTENTS Acknowledgements ii List of symbols v List of figures x Abstract xii CHAPTER 1. INTRODUCTION 1 Background 1 Problem Statement and Outline of Dissertation 8 Instability Mechanism 11 Mathematical Model 15 CHAPTER 2. LINEAR THEORY 20 Introduction 20 Dispersion Relation and Field Structure Equations 21 Normal Modes without the Beam 24 The Effect of Adding the Beam 34 Discussion 40 CHAPTER 3. THE CYLTMP PARTICLE SIMULATION CODE 41 Introduction 41 Field Solver 44 Particle Pusher and Particle/Grid Interpolation 46 The Full Computational Cycle 53 Diagnostics 56 iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4. CYLTMP SIMULATIONS OF THE PLASMA CHERENKOV MASER 61 Introduction 61 Choosing Input Parameters 62 Simulation Results 65 Saturation Mechanism 79 CHAPTER 5. SUMMARY AND FUTUREDIRECTIONS APPENDIX A. RELATION ANALYTIC TREATMENT OF THE DISPERSION APPENDIX B. ORTHOGONALITY RELATIONS 93 98 REFERENCES 105 110 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF SYMBOLS A A/ Vector potential Cross-sectional area of the beam Cross-sectional area of the cells centered on the ith radial row of the 2-D r-z grid Cross-sectional area of the plasma B A/ Pn Magnetic flux density. a & component — In thiswork, B<£ at the (ifj)'th point of the 2-D r-z in CYLTMP, at timelevel n+1/2 B has only ^ grid c Speed of light in vacuum e Magnitude of electronic charge (positive) E A Electric field. In this work E has only r and z components — Er and Efi ~ E2sj'EAj E a and Er at the (i,j)'th point of the 2-D r-z grid used in CYLTMP, at time level n Linear driver field array for g'th group at time level n Total field energy in a CYLTMP simulation system Electrostatic component of field energy in CYLTMP Electromagnetic component of field energy in CYLTMP Total kinetic energy of 'th species in CYLTMP Ezk onfr"') Rad;*-al profile of Ea for the TM6a mode with axial wavenumber ka foi One particle distribution function for species t< g Index for simulation particle groups g'th group of simulation particles for species c< i Index in the r-direction on the 2-D r-z grid used in CYLTMP v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. , Minimum and maximum indices of cadial rows ’ °ver which particles of group g extend Ift Alfven Current — Ik Beam current I0 (I^) Modified Bessel functions of the first kind of order zero (one) j Index in the z-direction on the 2-D r-z grid used in CYLTMP J0 (J^) Bessel function of order zero (one) J Current density. In this work, J has only a z-component — Ja J- i\ J z component of J at the (i,j)'th grid point at time level n+1/2 in CYLTMP K0(K^) Modified Bessel functions of the second kind of order zero (one) kB Boltzmann's constant k4 axial wavenumber ka(u n'th discrete axial wavenumber or "mode" in a CYLTMP system — a&k-^. kjjuft. Effective perpendicular wavenumber of the TM an'th mode L Axial length of interaction region and the r-z grid used by CYLTMP m Electron rest mass N 0(Na) Neumann function of order zero (one) N-. No. of groups of simulation particles for species N^ No. of simulation particles in group g NR No. of points in the r-direction for the 2-D grid NZ No. of points in the z-direction for the 2-D grid nL Unperturbed beam density vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. np Unperturbed plasma density n!L»'v Axial linear particle density of g'th group of particles at time level n n^J Particle density of species at the (ifj)'th point of the 2-D grid at time level n pa Electron's axial momentum component Pr *■ Axial momentum of the r'th simulation particle at time level n+1/2 R Radius of waveguide and radial extent of the r-z grid used by CYLTMP rt Inner radius of annular plasma r^ Outer radius of plasma r^ Radius of Beam r^ Radius of i'th radial row of the r-z grid on which E ^ is defined r£ fr^ S. Radii of the closest radial rows to r^ — fi±A£ z. Linear axial weighting functionfor gridparticle interpolation in CYLTMP tgfr{ Time at saturation of the instability i.e. when field energy is at its first maximum V0 Unperturbed beam velocity va Electron's axial velocity component vpv* Wave's axial phase velocity VjT4^* Axial velocity of r'th simulation particle at time level n+1/2 w* ; 3 Radial weights for grid-particle interpolation in CYLTMP Zj Axial location of the jth column of the 2-D r-z grid on which ^ is defined Zp Axial position of r'th particle at time level n oCon> n'th zero of J0 vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Normalized beam velocity -- V0/c ■^0 Normalized wave phase velocity — v ^ / c X “^2. Normalized beam energy — (1- ^ ) ** £ Linear growth rate Quantum unit of wavenumber in the finite CYLTMP system — Thickness of plasma — ra - r4 Dimensions of unit cell in the 2-D r-z grid Finite time step used in time integration of field and particle equations in CYLTMP Thermal velocity spread of beam Thermal velocity spread of plasma £ 1-D dielectric function £ a^. z-z component of the 3-D dielectric tensor V B e a m to microwave energy conversion efficiency ©b Beam temperature (in energy units) 0e Plasma temperature (in energy units) Charge density (TV w <T Charge density at the (i,j)'th point of the 2-D r-z grid at time level n Plasma conductivity Electrostatic potential Electrostatic potential at the (i,j)'th point of the 2-D r-z grid at time level n 3C Perpendicular wavenumber in vacuum i.e. radial profile of E % i n vacuum is a linear combination of I0(ocr) and K0(yr) Perpendicular wavenumber in plasma y.± Perpendicular wavenumber in the beam Angular frequency of the TM6A'th mode viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (Ote Electron cyclotron frequency — Plasma frequency of plasma electrons — OJpt» Plasma frequency of beam electrons _ S. Radial Laplacian operator — w J4.tr TV~» ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES £aga Figure 1.1 The experimental plasma Cherenkov maser 6 1.2 Radial profile of the interaction region of the experimental plasma Cherenkov maser 9 2.1 Plot of dispersion function versus M for fixed k^. 27 2.2 Radial profile of Ea for selected TMOA> modes 29 2.3 Radial profile of E s and Er for a T M ^ mode 31 2.4 2.5 2.6 3.1 4.1 4.2 w versus k4diagram for the TM qo branch of the dispersion relation Effect of the beam on the TM00 branch Growth rate of the TM of plasma density 35 38 modes as a function 39 2-D r-z grid used for the field solver in CYLTMP 43 Time history of EM field energy and beam and plasma kinetic energies for a CYLTMP simula tion of the device at n. =2x10' cm , © e = lOeV and © b= 100ev 68 4.5 Time history of most unstable TM0o mode, and comparison of growth rates from the code with linear theory for n~=2x1013 cm-3, ©e.= lOeV and @ t= lOOeV -*3 As in Figure 4.1, but with np = SxlO1* cm -•3 As in Figure 4.2, but with nP " 5x10 13 cm As in Figure 4.1, but with np = 8xl013 cm"3 4.6 As in Figure 4.2, but with nP = 8X10*”5 cnf^ 75 4.7 As in Figure 4.1, but with e b = lOkeV 76 4.3 4.4 X Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 72 73 74 4.8 As in Figure 4.2, but with 0^ = lOkeV 77 4.9 Beam to microwave energy conversion efficiency from CYLTMP simulations 78 Qualitative features of growth rate versus k and time history of electrostatic field energy for the 1-D two-stream instability 81 Phase space diagram of plasma electrons at saturation from the CYLTMP simulation of Figures 4.1 and 4.2 85 Phase space diagram of beam electrons at saturation from the CYLTMP simulation of Figures 4.1 and 4.2, together with axial profile of Ea on axis 86 4.10 4.11 4.12 xi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT Theory and computer simulation of a new experimental high-power microwave generator is presented. In this device, a circular waveguide is partially filled dense annular plasma. When electron beam pulse passes region, microwaves (;£ 20% ). modes in are an intense through emitted the with waveguide. The high radiation efficiency mechanism is waves by The linear theory is analyzed first. A dispersion relation and field structure the vacuum v ^ < c ) TM stimulated Cherenkov emission of these slow TM the beam electrons. a relativistic central The plasma creates slow (i.e. the with equations are derived for azimuthally symmetric TM modes of this system. Numerical solutions demonstrate the existence of the slow TM waves without the beam, and confirm that some are unstable in the presence of the beam. To analyze the non-linear theory a new particle simulation code has been developed. of This code is described in detail, and results simulations of the experimental device are presented. In these simulations, the system initially evolves in good quantitative agreement with linear theory, while the non-linear saturation experimentally amplitudes observed are efficiencies. consistent with Saturation linear instability is shown to be due to trapping of of the beam electrons, and the saturation amplitudes agree quite well with a simple trapping model. xii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 CHAPTER 1. INTRODUCTION Background In recent years, great effort has the generation been devoted to high-power electromagnetic radiation 4" 5 using intense relativistic electron beams . Quite apart from of the extremely high-power already been achieved, interest in from the fact that conventional these shorter microwave in than tube particular the gyrotron ( electron operating schemes stems they provide efficient, high-power sources at wavelengths far with levels that have is attainable technology. cyclotron In maser ), the millimeter to submillimeter range, and at even shorter wavelength, the free electron laser, have both proved to be practical efficient sources of tunable, high-power radiation. An alternative concept that has received less A .O attention is the so-called Cherenkov maser . device, a relativistic electron through waveguide beam passes lined with a layer of dielectric. alters the dispersion characteristics of and for judiciously chosen in vacuum. These a This lining the waveguide, parameters, creates normal modes with phase velocities lower than c - the light In this modes speed of can then be excited by Cherenkov emission from relativistic electrons satisfying Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. V > V fV\. . have Experiments shown that and efficient computer operation simulation" is possible at centimeter and millimeter wavelengths. One of the principal limitations on the power levels attainable current. charge in such devices is the If a beam is propagated in of the beam limit on the beam vacuum? this space itself creates a potential barrier that the beam electrons must overcome. that the It is easy to see provides a limit on the beam current. linearity of Maxwell's equations? this potential By the barrier is directly proportional to the beam density - -e^>Boc Thus for any fixed accelerating ultimately voltage VQ ? a beam density such that -e there is > -eV0 . This limitation can be overcome by injecting the beam dense plasma (n^ >> n b ). . into a As the beam propagates through the plasma? the plasma electrons are driven out of the beam volume? effectively neutralizing the space charge of the beam (The plasma electrons are preferentially out in since the driven they are less massive than the beam electrons lab frame). For a beam propagating in a conducting medium? the characteristic neutralization time for the radial ejection of the excess charge is = 1/4tt(T . However? for a typical laboratory plasma? 14. -3 say n p = 10 cm and Te =10eV? this time is -17 ^3x10 sec. Thus the conducting fluid model must by modified to include electron inertia? leading instead Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to 3 the characteristic ^b is the neutralization time TT*/ ^ &b/c, where thickness establishment of a of the beam-plasma beam. Since the equilibrium in a drift tube of length L requires a time of at least since typically L >> and , any such equilibrium must have almost complete neutralization of the space charge of the beam. Having neutralized the appears. The attractive limitation space charge, a new magnetic force between the beam electrons is now unbalanced, and so beam shown pinches itself 12. long agothat distribution down. the For a slab geometry, it was independent of the radial of the current density, the maximum current that can propagate in a space charge neutralized beam is the so-called Alfven current: IR = where -(DkPDpv Cm* ^ = v/c, and ^ = ( 1 - ) The maximum current in a drift tube is modified geometric factors neutralized of order unity. beam can by applying self-pinching magnetic field. only by The space charge be stabilized from this a sufficiently strong axial However, it should be pointed out that in applying the Alfven limit to a beam-plasma system, one must use the total current, including any currents in the plasma. In fact, large reverse currents are induced in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the plasma, that almost completely current behind neutralize the head of the beam. the beam An observer fixed in the plasma as the beam head passes by sees a rapidly rising magnetic fi^ld due to the azimuthal magnetic field of the beam passing by with the beam. The electric field associated with this time-varying magnetic field drives a current in the These plasma opposite to the beam current. induced plasma currents persist behind the head of the beam current for a decay distance characteristic in the plasma. of resistive In practice this distance is so large that essentially all of the beam except a thin layer at the head is fully current neutralized. With the foregoing considerations, it would appear that unlimited beam currents in plasmas are possible. practice, beam currents Since these are limited instabilities by result In instabilities. in growth of electromagnetic waves, the possibility of harnessing them for the generation of electromagnetic radiation presents itself. The obvious. possible Since neutralized beam high Thus power study relativistic the radiation currents, levels of advantage is of comes scaling to schemes is from essentially extraordinarily possible, at least in principle. electromagnetic electron such beam-plasma instabilities in systems has possible application in high-power radiation generation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 The pioneering work in this field has been the Soviet Union instabilities has (work in the focussed U.S. mainly done in on beam-plasma on electrostatic instabilities and applications to plasma heating ). In a series of papers in the late 1970's and early 1980's, the basic theory of what they 13*— 1 1 ? electronics" was presented culminated in efficient the first plasma constructed, the termed microwave . This work has recently experimental microwave basic "plasma device generator features . has An been of which are shown in Figure 1.1. In this device, the dense annular plasma created pulsing the plasma cathode first. by is Then, the beam diode is fired and the intense relativistic electron beam passes through the plasma. High-power microwave emission is observed to emerge from waveguide. This device maser described above. dielectric lining is from output coaxial analogous to the Cherenkov The annular plasma replaces the as theslow wave structure creating electromagnetic modes with Radiation the the phase device is velocities the Vpt\ < c. result of coherent Cherenkov emission of these waves by the beam electrons. Thus we call this device the plasma Cherenkov maser. In their experiments, they ran parameters inner and radius fixed: outer rb=0.55cm, Waveguide radii beam the following radius R = 1.45cm, plasma r,=0.67cm energy with and r;L=0.73cm, beam E^=480keV, beam current Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CoiIs f or ia I S7I FTI IS ?7 ! rv7 |JE x te rn A A A ! A ! B* fieId Cathode Anode HIgh-densIty pIasma , / \ Col Iector v/////y////y/y/7777///y//77s ^/Relativistic e l e c t r o n b e a m y/y V ///////////////////////////////2 /, Cathode inner coax i a conductor Circular waveguide N , __________ REB Diode Interaction region Figure 1.1. O u t p u t c o a x ia I w a v e g u i de The experimental plasma Cherenkov maser. Ib=900A, and a pulse length of 50ns. output power the We will focus here on results of varying the plasma density when the other parameters were held 22.5kG, fixed resonator ( External length 24cm, scatter of the beam) . The over the range wavelength 3cm varied np«&2xl0 ± from . corresponding to & 1.8cm. output Peak power of Since the beam power is 430MW, this is an efficiency of 21%. Power output was observed to below n p «#10 slowly for n p > 2x10 13 cm 13 -3 cm drop , while it dropped off -3 Despite the success of the device, there published angular 20) MW was observed at a plasma density cm rapidly minimum field density was varied |3 »4 -3 n p ~ 10 - 10 cm The output of (90 and magnetic plasma ( n ^ / n p 0.02 - 0.002 ). off of was observed for various configurations of the other experimental parameters. the Dependence is little theory on it ( compounded by the the fact that apparently some refences are unavailable outside the USSR ). The theory that is available is mainly analytic treatment of the linear theory, based on approximations. For the conditions for approximations limiting value the parameters used in the device, the validity are violated. this system is required for of the most of the Thus numerical analysis of a truly pertinent theory. This is the problem addressed by this dissertation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 Broblem Statement and Outline of Dissertation In this physics dissertation we will of the interaction region. address of the This will be done by considering the simplified problem of problem only the initial-value an infinitely long waveguide with the radial structure of the interaction region, in which the beam and plasma densities and velocities are initially uniform everywhere. Several This system is depicted in Figure 1.2. important questions about the actual device will remain unanswered. address First, with this model we cannot the problem of charge and current neutralization when the beam and plasma are physically seperated. is a transient established. net This process that depends on how the beam is We simply assume here that the plasma has a positive charge density to fully neutralize the beam charge, and that the plasma has a return current equal to the beam current. Secondly, oscillator ( rather than a the actual device is an single-pass amplifier), so that in analyzing the actual device, the axial boundaries cannot be ignored. simplified Nevertheless, the physics of this problem is still interesting, non-trivial and directly applicable to the most fundamental aspect of the experimental device - the generation of electromagnetic waves from beam-plasma instabilities in region. In what simplified model the interaction follows, we will loosely refer to the of Figure 1.2 alone as the plasma Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of 9 region 3 D> Q> > Q_ JZ O) X -Q 04 Figure 1.2. Radial profile of the the plasma Cherenkov maser. interaction O Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 Cherenkov maser or PCM. Any references to the full experimental device will be explicitly noted. The outline of this dissertation is as follows. In the remainder of this chapter, a brief description of the basic instability mechanism simplified and justification mathematical model used is given. 2, the linear theory of the device is the dispersion relation is for In Chapter presented. derived. Then First numerical solutions of the dispersion relation and resulting structure of the normal modes are given. we describe the new investigate the particle non-linear first present results of simulation problem. simulations compare the results with linear theory. by a discussion of the non-linear effects results model. are field In Chapter 3, code In of the used to Chapter 4 we the PCMf and This is followed in which the interpreted with a simple semi-quantitative Finally/ in Chapter 5, we summarize the results/ and discuss future directions for this project. This problem has dissertation/ we many have free focussed used in the experimental device. results in this work/ parameters. In this only on the parameters Thusf in all numerical the following parameters always have the constant values listed below: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 R r r r nb p* = = = = = = 1.45cm 0.658cm 0.718cm 0.538cm 2.3x10!* cm V6/c = 0.857 These parameters experiments. are Since close to those used in the the particle simulation code uses a finite radial grid size Ar, this is as close to the experiments as we can get. Instability. Mechanism Before plunging useful to develop ahead a simple beam-plasma interaction. that although with Cherenkov effects analysis, it physical picture of the emission is a single particle in densities at beam are dominant. Although the nature of Cherenkov emission is obscured with this terminology, the basic solution which Thus we talk of "beam waves" rather than single beam electrons. underlying is First, it should be pointed out effect, we are interested collective the of the result equations in remains the - detailed collective regime predict instability only if the phase velocity of these unstable waves is less than the beam velocity. The linear electromagnetic instability to generates the waves in the plasma Cherenkov maser is a two-stream instability. similar that the simple In fact, the physicsis one-dimensional very electrostatic Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. two-stream instability. geometic Since the latter has none of the complications of the PCM, let us first focus on it for the moment. In the 1-D electrostatic case, we consider the problem of an infinite homogeneous plasma of density np penetrated everywhere by with and density plasma dispersion homogeneous nb and velocity Va = electrons and fixed relation for the E s can can be be . beam electron For cold beam plasma ions, the electrostatic waves a O 1.1 io (fca-kVoY and Where beam written Etw.k') = 1 - "E - and a are plasma frequencies the plasma respectively. Without the beam term, the solution of equation 1.1 gives the normal modes of the plasma = iWpjf independent of k ), while dropping the plasma term gives the normal modes of the beam DO - k Vo st With both terms equation in CO included, for equation fixed k. are with part unstable positive mode, imaginary which gives is a quartic For sufficiently small k ( k^cOpt/Va ), a pair of roots the 1.1 rise complex. The corresponds to the root to an two-stream instability. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 We can understand this instability as follows. total energy ( kinetic energy of the The oscillating electrons plus electrostatic energy of the field ) of the beam waves is Thus for &>> 0, the slow beam wave ( negative sign ) is a "negative energy amplitude E0, the electromagnetic wave. wave". total is, energy for a certain wave of the beam relationship this happens between perturbations. In are the the 180 because of e slow out the density and beam of the total velocity is low, and vice versa. velocity wave, is the electrostatic seperation, resulting in a energy of the wave. net these high field lowering where The resulting drop in the kinetic energy exceeds the positive in phase phase. Thus the total density ( equilibrium plus perturbation) energy and field is actually lower than without the Physically perturbations That due of definite to charge the total By energy conservation, if such a wave is excited, the lost energy must go into the source that established the wave. The two-stream instability results from coupling of the plasma waves with the resonant slow beam waves ( i.e. those beam waves with , which correspond Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to 14 wavenumbers k for L6p\,« COpe,).The plasma wave excites the beam wave which reduces beam. the energy the This lost energy goes into the plasma wave, whose amplitude therefore increases, increasing of of the amplitude the beam wave, which drops the beam energy even more, and so on in a positive feedback loop. The waves grow together, transferring energy from the beam to the plasma waves. In the PCM, the interaction is beam wave still a is similar. The slow no longer purely electrostatic, but it is negative energy wave. The plasma replaced by an electromagnetic wave with an E s wave is component to drive the beam electrons into exciting the beam wave. Unlike the 1-D case however, the electromagnetic wave has transverse components coupled to Maxwell's equations. same rate as G 2 compared to PCM, get we electrostatic the oscillation conversion and Eg, component of growth rate frequency ). beam electromagnetic energy field plasma kinetic energy ) from an essentially interaction of by Consequently they too grow at the ( assuming the its is small Thus in the into both energy ( and longitudinal the E g. component of the electromagnetic wave with collective oscillations of the beam electrons. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 Mathemat i c.al...Model In general, the analytic treatment of electromagnetic instabilities in relativistic beam-plasma systems is extremely complicated. the plasma However analysis Cherenkov maser is greatly simplified by the fact that the transverse velocity components of the electrons of do not play beam a key role in the interaction ( unlike, for example, the gyrotron or FEL ). In fact, large transverse velocity components degrade performance, and in a well designed PCM they are reasonable of neglecting altogether. l&ce./ condition plasma electrons. the transverse Physically, realized with a strong axial the Thus as a approximation, we can make the very important simplification components small. this magnetic &<* ^pc< velocity condition is field satisfying >> 1 for both the beam and Since transverse velocity components are neglected, it follows that only azimuthally symmetric perturbations can be excited. of Maxwell's equations to Thus we restrict this case. In general, the normal modes of the electromagnetic field in a inhomogeneously filled with a analysis waveguide dielectric have all six field components, and cannot be seperated into purely and TE modes 20 . . However, the azimuthally symmetric modes of an inhomogeneously filled circular waveguide which the dielectric special case in which function such a TM depends seperation in only on r is a is possible. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 Since the interaction electric field, we retain electromagnetic field. have only Ep Ea , requires an E a component of the only the TM Azimuthally and modes of the symmetric TM modes components. With these assumptions, Maxwell's equations reduce to 5E r _ Si 3 E* = 3 x -3 Sr | ^Er H T W — j£ . . 1 ‘4 1.3 where Ea, Ep, To complete particles. waves and the Since are all functions of r, z and t. model, we are U> *'10^ » we treat neutralizing must dealing the background. we the with high frequency plasma For describe the ions as a fixed beam and the plasma electrons, collisional effects are completely negligible, but kinetic non-linear present electron effects behaviour are in critical the strong as the instability grows. species f * (r,z,pa,t). with a 1-D in determining the electric fields Thus we describe each distribution function Since the transverse momentum components are zero, the distribution function satisfies the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1-D 17 Vlasov equation where va = pg/^ m, and ^ J 1 + p^/m**c* The electron current is 9* J«(r,z,t) = -ef 2. -foi ( 1.7 o. Equations 1.3-1.7 are the basic equations used the mathematical model of the PCM. here for These equations can only be handled numerically. In the early stages of amplitudes are unimportant. small, the and instability, kinetic the wave effects are In this case, it is appropriate to use linearized fluid electrons. As a further assumption, we will assume the velocity equations the beam and plasma that spread of the beam and plasma electrons is negligible ( J^v^ << V0 and beam for the velocity). &v^ << ^ , where is the The linearized fluid equations for the density and velocity perturbations of the plasma are = o 1.8 = - eEa at Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 while the linearized fluid equations for the beam perturbations are ^ + O W + Vo = 0 1.9 ^W In + these Vo ^5 V?tb equations, equilibrium radius = it quantities ( where they has been - e £ -a assumed n^, n p and are that the are independent of non-zero). The linearized current density is 1.10 Now assume fields have that the perturbation the form f (r) e quantities and . g0 iving for the perturbation current as a function of E ^ , we can rewrite Maxwell's equations as i£L = i V , E r -|I*- Er l.u « 1-12 1.13 where is the z-z component of the linear dielectric tensor, defined by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 r f £„(Wrk*,r) =< "z* 4 o«r$rk - 1 1 rVa< r < r l ] .14 - n ^ r ^ f 2. V. 1 Equations 1.11-1.14 are the starting point for the linear theory. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 CHAPTER 2. LINEAR THEORY In this chapter? we analyze the linear theory of the We first derive the dispersion relation relating to PCM. and k2 ? and the equations for the field structure of modes. Analysis stages. First? we find the solution relation these equations is to the That is we look for the normal infinitely plasma. long waveguide containing For brevity? we call waveguide”. beam in two dispersion this modes the only the annular system ( recall fluid that we are the using "plasma starting point the These modes are for the analysis of the system with the beam? both for the initial-value problem axial cold model for the plasma and the beam)? both 03 and ka are real for these modes. without of Since there are no losses in the absence of collisionless the done and resulting field structure in the absence of the beam. the of the boundaries)? and the ( with or steady-state boundary-value problem. However as mentioned in 1? the initial-value problem of the we focus only on infinitely long waveguide. real? independent Thus variable? we and consider use the chapter k3 roots the effect of adding the beam. u) (k%) with positive imaginary a dispersion relation to find the allowed frequencies co(k9 ). consider as Then we We find complex parts which Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 correspond to unstable waves. Finally the results of linear theory are summarized. Dispersion Relation and Field Stjcucture eauations From Maxwell's equations - equations 1.11-1.13, we easily derive an equation for E & 2.1 where ■=* The transverse field components are easily obtained from E « using Maxwell's equations ELr “ Ca£ c-z.— 2.2 \JZ' 2.3 Writing out equation 2.1 for each region explicitly, have vacuum: plasma: Vr beam Vr* : + 0Cp « O 2.4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. we 22 where the constants , OCp and are given by -i) 2.5 ^ i f c s s * - 0 We choose the signs this way because we are interested in the and frequency ye? are range to< ck2 , £»>< The equations for E s are both real. equations of order zero. y. In this range Bessel Since E s must be finite at the origin, we can write the general form of the solution as ~ AJ0 (orkr) BId (xr) E a (r) = + CKft(xr) rt < r < r, < 2.6 rt4 DJ0 (a^r) + ENtt(«pr) r * r^ V^, FI0 (9Cr) + GKa (jcr) Ja and N s are Bessel and Neumann functions of order zeror while zero. I e and The unknown K0 are modified Bessel functions of order radial boundary conditions determine The requirement quantities. E ^ (r=R) = 0. Secondly, since E% component at interfaces, itmust interfaces. component of first the be Finally, D, E r beam/vacuum by the that field plasma/vacuum continuous across and hence across the interfaces. is is a tangential and the these continuity of the normal must be continuous Using the relations Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 J o (z) = N*<z> = “» ±(*) K*<z> = “K^ 2) the boundary conditions lead to a set of homogeneous linear equations for the coefficients AJ0 (*jrt ) = B I d (3crb ) + C K 0 ( a r fc) BIe (7cr, ) + CK„(oc r, ) = DJ0 (5Cfr, ) + E N 0 (3C|lr|) D J o t e pr * .) + E N ^ C X p r^ ) = F l^ jc r^ ) + GK 0 ( ^ c r z PI0 (JCR) + GK 0 (3CR) =0 l*brh* ) 2.7 = 7C(”BI^ (# rfc) + CK^jcr^)) 3C(-BIA(5cr| ) + CK4 (*r4 )) = ) + ENA (^r4 )) + ENA (s^r2 )) = 3C(-FI4 (*ra ) + G K ^ r ^ ) ) We get non-trivial solutions if the coefficients is zero. This determinant leads to of the the dispersion relation Q. 2.8 1o(3cr,>) [ l 5(3cr,') + Q K 0 Cxni] In this equation, Dp - yt-n^fotp VTNafapri} is the dispersion function without the 29 X d(7fr0 beam, while Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the 24 other undefined quantities are "I* — -t PJftQfpO) +■ PNdC^ffaT) f % ( « 4- l « ( y R ^ Ka.(^r^\ \ I » (X *i) K atxR ) — lofjCR?} Ko C^CHt) / Q 2 .10 3C IjLtyQ 4-S lo txq,^ s -Sk,(xO £ ss This X b *3a.(Xh^ Jo^b^ form of thedispersion relation 21.In deriving this is given in Ref. form, we have used the relation W(R 0 (z), IQ (z)) = 1/z, where W is the Wronskian operator. We by begin the analysis of equation 2.8 first considering solutions without the beam. Normal Modes without the Beam Without the beam, the general form for E , equation 2 .6 , simplifies to Ea =i BIa(*r) 0 ^ r < r, ° J o ^ p r^ + EN0 (jCjj) r, < r< PIe(5Cr) + GKB(xr) r2 < r ^ r^ 2.11 R While the dispersion relation - equation 2.8, becomes Dp = 0 2.12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 Analytic treatment of relation can be done even this only simplified in certain limiting cases. Rather than tackle this immediately, look at dispersion it is clearer to the numerical solutions to this equation first. Plots of the numerical solutions provide a qualitative understanding of the nature of the normal modes, and help to motivate the analytic approximations. Two programs have been developed for solution of the linear problem. the numerical ANALYZE_DSPFN solves the dispersion relation, while GET_FIELDS used these to find the corresponding field results structure. Both ANALYZE_DSPFN and GET_FIELDS contain a switch to turn the beam on or off, allowing these programs to investigate either the plasma waveguide alone, or the full linear PCM problem. ANALYZ EJDSPFN uses iteratively find the complex roots from the dispersion relation. Muller's to (k4 ) ANALYZ E__DSPFN is an for 99 real to ka As with all iterative root solvers, it performs only as well as the Thus method initial interactive requires some careful input from the user. guess. program that It allows the user to graphically locate the neighborhood of a zero and reset convergence parameters before actually calling root solver. the By contrast the calculations performed in GET_FIELDS are straightforward, and can be done without any user supervision. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 Before solving the dispersion relation for the plasma waveguider it useful to recall the results for the azimuthally symmetric TM waveguide. modes of the empty circular These modes satisfy the dispersion relation 2.13 and have radial structure 2.14 r where n = 1, 2, ...). being Note the nth zero at Thus the high on In the plasma the dispersion characteristics. frequency roots to equation 2.12 must be given approximately by equation 2.14. At frequencies the plasma will have a strong effect. k 2 and go Dp over , Dp a is also real. range of Figure 2.1 shows a frequency for the parameters at a plasma density of n p = 2 x10 wavelength of ( frequencies t*> >>00p«. , the plasma has high very little effect Ja that for any GO and k^, the phase velocity of these waves is greater than c. waveguide of X» = 3cm. This density For real plot of experimental -3 cm and a is roughly the density at which the efficiency of the device is maximumf while this wavelength is that of the most unstable wave in the presence of the beam. We see that there are roots to equation 2.12 corresponding to modes with Vp^ < c. In fact there are an infinite number of low frequency roots. However the very low frequency roots are not physical. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 100 50 CL o -50 -100 -2 1 o «/«pe Figure 2.1. The dispersion function without the beam at plasma density np = 2 x10 '* cm , plotted over a range of frequency for fixed k^= .25wfft/c ( wavelength 3cm). = .25 is the freqency at which vpi^ = c. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 They arise from using the cold fluid model for the plasma beyond its range of validity. In reality, waves with phase velocities comparable to the the plasma are strongly thermal Landau damped. plasma model is valid only for W >>kay ^ . since v^ << velocity of Thus the cold Nevertheless, c, some of the higher phase velocity slow waves are physically significant. When the field structure corresponding to the of the dispersion relation these modes becomes clear. profile of Ea for roots is computed, the nature of Figure 2.2 shows the radial the first three modes with Vpy* > c (a-c), and the first three modes ( in order of decreasing frequency) with essentially the slightly by < c (d-f). vacuum waveguide the plasma. TMon modes, for n = 1, 2, exist only Modes with Vp^ > c are TM modes, perturbed We therefore label these modes ... . Modes with v^ < c because of the plasma. It is natural to label these modes in order of decreasing frequency, counting downwards. Thus we call the highest frequency slow wave the TM QO mode, the next highest frequency slow wave the ™o,-i or™ o T mode an<* so on* Higher TM0^ modes have successively more wiggles in the plasma while the field structure in the vacuum region is almost the same. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 .5 1 0 0 1 0.0 0 1 (d) (a) 1 2 0 0 1 0 1 2 0 1 (e) 1 i 0 1 Vl! !> 1I I I}! I <f 0 k r /R 1 r /R 1 (c) Figure 2.2. Radial profile of E g that E a(r=0) = 1) corresponding roots in Figure 2.1: (a) to = .366 (b) (0 = .520 (c) to = .798 ( normalized so to the following (d) w = .191 (e) tO = .039 (f) to = .020 The plasma occupies the shaded region. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 The TMoo mode is of particular interest in the Not only does it satisfy the condition for Cherenkov emissionr v ^ < c, but it has the simplest the structure to excite by a relativistic electron beam. The full structure of T M ^ wave from Figure 2.2 is shown in Figure 2.3. large Note outside that the the transverse plasma. Thus components the plasma. outside It is this characteristic that allows good coupling of the TMW wave with a TEM wave in waveguide are electromagnetic energy of the TM M mode is predominantly located the in of all the slow waves, making it the easiest plasma field PCM. of the experimental device, the while component is largest on axis to maximize the output the Ea interaction with the beam. The question of completeness of the TMork modes is technical problem beyond simply assume here field at represented time by t=0 a the scope of this work. that any azimuthally in the plasma linear symmetric waveguide combination of the can J-Oo We TM be plasma waveguide TMen modes Ea(r,a.It-o>)A <Jk2e *^2 . a 2.15 Ws-fc Subsequent time evolution of the field is then Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 0.0 0.2 r/R (a) 4 3 2 1 0 1 «-* 0.0 0.2 0.4 0.6 0.8 1 .0 r/R (b) Figure 2.3. Radial profile of (a) Ea and (b) Er, for the TMoo wave with k*= .25u)f£/c at n^ = 2x10 cm"3 . The fields are normalized so that E% (r=0) =1. The plasma occupies the shaded region. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 2.16 _i. ^ot\( To find the expansion orthogonality coefficients, we need an condition for inodes with the same k8. The simple orthogonality condition for TM modes of the vacuum waveguide comes from Sturm-Liouiville theory. is not a Sturm-Liouiville However equation 2.1 equation, because is a function of both CO and r ( the plasma is dispersive, and only partially fills the waveguide). It is clear just from looking at the radial profile of the T M e^ and modes that they do not satisfy jrAc Nevertheless, there exists a more condition that rather than general comes special relation general * orthogonality directly from Maxwell's equations mathematical structure ' . The and its derivation are in Appendix B. For the special case here in which we consider two m TMeo modes and <\. having the same k4' the orthogonality condition can be written r~iv\ 2.18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 Physically, this condition ensures that the total in the field energy given by equation 2.17 and its associated transverse components, is equal to the sum of the energy in its component modes. Finally in this section, we consider the analytic treatment of the dispersion relation, equation 2.12. For the TMon modes with n ^ 1, equations 2.13 are reasonable approximations. However, principal interest is to derive and for analytic the PCM, the approximations for the slow TM modes, especially the T M m mode. done in ref. A. The 21, and this work is contained in principal result 2.14 This is Appendix is an approximate dispersion relation for the slow TM modes 2.19 where n = 0 , "1 , "2 , ... and r n = 0 n < 0 Where A p = r^ - r^. Equation A f<<C4 , 2.19 and assumes In that £5 1. JCR « 1, >:> These conditions can only be satisfied for a low frequency wave and a very dense plasma. Figure 2.4 compares the TM 00 thin branch of the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 dispersion relation computed 1*3 -3 2.19 at n p = 2x10 cm . ra *+4, equation 2.19 numerically Even is though still approximation of the TM ^ branch. with a equation 9CR 2 fairly and good The field structure of the TM o0 mode/ is given to lowest order by 0 ^ r ^ r- Ea =< 2.20 \xS^ 'A ker Er = ra>< r ^ R 0 ^ r£ r < 2.21 r- < r < R sc 8* % r This concludes the discussion of the normal modes of the plasma waveguide. We now consider the effect of the slow adding the beam. Ths ..Effsct .Adding the.-Sg.am When the beam is added, become can unstable, due to collective Cherenkov emission by the beam electrons. the T M on modes TM0q mode We focus only on the instability with a of beam of the appropriate energy, since this is the mode of interest in the PCM. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 0.4 0.3 0.2 0. 1 0. 1 0.2 0.3 0.4 0.5 ^ ^ z/^pe Figure 2.4. The T M 0o branch of the dispersion relation at n. = 2 x 1 0 cm"3 . Solid curve is the numerical solution and the dashed curve is equation 2.19. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 The standard approach to the analytic beam-plasma weak ( n ^ the interactions « n^ is treatment of to assume that the beam is ), and treat it as a perturbation normal modes of the beamless system. of The details of this calculation for the PCM can be found in Appendix The A. principal results are that the frequency of the most unstable TM qq wave is given by W** « Wpe, — 2.22 where the cutoff frequency is defined by COe. 2.23 85 The wavenumber of this wave is given by W = kaV0 , while its growth rate is \ a Equation 2.24 is implies a \ 2 i V y 6? valid minimum 2.24 if << 1. Equation 2.22 plasma density for which instability occurs. In fact this is observed in the experiments. As 1$ -3 the plasma density is reduced below 10 cm , the power output drops off sharply. However, substituting the experimental parameters into equation 2.23, the predicted cutoff density is n ^ 2 .8 x10 cm . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 The analytic results, equations 2.22 - 2.24 fact good are in approximations only for beam densities so low as to be of little investigate the practical linear interest. effects To carefully of the beam, numerical solution of the full dispersion relation, equation 2.8 is required. The numerical solution of equation 2.8 follows the same procedure as that described for the beamless case. The effect of adding the beam used in the experiments to the TM00 branch of the dispersion relation 13 _3 at a plasma density of n p « 2x10 cm is shown in Figure 2.5. The unstable over the rate plot range shows 04 r U that tv- ^ the 0.35. TM^ mode is The growth is maximum at k-g.® .25tOp(6/c. Since ii ..a. 2.5x10 sec , the maximum growth rate is -a 3.5x10 sec . Thus, this wave increases by an order of magnitude in 0.66nsec! to This is extremely fast compared the 50nsec pulse length used in the experiments. growth rate of the TM ^ branch with plasma density parameter is shown in Figure 2.6. which as a This plot is in CGS units to compare the "absolute" growth that The We find IZ -3 the TMO0 wave is unstable only for np1^ 6x10 cm , is in good agreement with rates. the experimentally observed values. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 0. 3 3 © QC 0.0 0.3 0.4 0. 5 (a) 0.016 0.0 1 2 © Q. 3 3 0.008 v_/ E 0. 00 4 0.000 0.4 0.0 (b) Figure 2.5. (a) Real and (b) imaginary parts of the TMfto branch of the dispersion relation with the beam at n p = 2x10 cm"3 . The dashed curve in (a) is the (purely real) TMm branch without the beam. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 xl 0® 6 5 4 3 2 1 0 0 2 4 6 8 10 1 .6 2.0 k z (c m " 1) (a) x10s 16 12 8 *o 4 0.4 0.8 1 .2 k z (c m " 1) (b) Figure 2.6. Growth rate of the TMoo branch of the dispersion relation with plasma density as a parameter. Each curve is labelled with its plasma density in units of 10*^ cm'3 . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 D iscussion The TM QO mode central of the role in the PCM. waveguide plays a In the presence of a beam with the appropriate energy, it is rapidly plasma unstable and compared to typical pulse lengths. it couples very efficiently with the TEM grows very Furthermore, mode of the for the output waveguide in the experimental device. The linear lower limit for experiments. exponential theory successfully operation However linear of accounts the theory PCM predicts growth of the unstable waves. that slow and ultimately terminate the waves attainable determining non-linear non-linear wave growth theory. problem We is unbounded of these Since the critical in the efficiency of any actual device, we need assumptions, and even handle. amplitude in The processes are outside the realm of linear theory. maximum a observed Analytic must then therefore turn treatment make is to some extremely of the simplifying difficult to computer simulation to investigate the non-linear effects. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 CHAPTER 3. THE CYLTMP PARTICLE SIMULATION CODE Introduction Particle analyze simulation methods are widely used to . Of non-linear problems in plasma physics particular interest to plasma microwave multi-dimensional (2-D, 2-1/2D fully electromagnetic codes 2i’2TJ electronics are and 3-D), relativistic, . These codes retain the most physics, and require tremendous computing power. Since our computational resources are VAX limited to computers (VAX 780, MicroVAX II ), we cannot perform such calculations in a realistic amount of time. the small time used steps required Lorentz in to force. such codes is spent on the intricate push the particles with for -eE^. the electrons is of the Since with of the essential called CYLTMP TM waves and Plasma) to analyze this model. CYLTMP can be used not only system equation PCM is contained in these equations, we have developed a particle simulation code (CYLindrical full the much simpler There is no magnetic force. physics the However, if we restrict ourselves to the model described by equations 1.3 - 1.7, the motion Most of azimuthal for the PCM, but on any symmetry in a circular waveguide with a strong magnetic field constraining the to this general in computer only move axially. applicability, and the Despite tremendous savings particles Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 time compared with a full 2-D relativistic electromagnetic code, CYLTMP appears to be the first code of its kind. We therefore discuss it in detail here. CYLTMP self-consistently advances the value of forward in time the electromagnetic field components on a two-dimensional r-z grid, and the the one-dimensional (z) positions and momenta of simulation representing the plasma and beam. are advanced using The field densities interpolated positions and velocities. turn advanced using components finite-difference approximations of Maxwells equations (equations 1.3-1.5), with current particles from The particle charge the and particle momenta are in electric fields interpolated fromm the grid, and the particle resulting new momenta. positions updated with the The field solver is essentially a simplified version of the one used in the NRL code CYLRAD 9.1 as described by Boris interpolation between the novel features of CYLTMP. . The grid particle and pusher particles and are the In this chapter, we consider in detail these two halves of the calculation, how they are coupled together in the complete computational cycle, and finally some of the diagnostics used in CYLTMP. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 z=0 Z= L -O — S3— O — S3 o — S3— O — S3 I I I I NR D — • — I O— I S3— I I I I I I I I I I I I I I r=R -O — S3— O — S3 O— » 1 1 1 ! NR-1 □— •— □— 6 -□— © — □ — 9 I 1 B I h I zl 2 O — S3— O — S3— 7K 2 □- I O — S3 I • □I — ©I AAr -©— I 1 O1— S3— 1, O — S3— O 1 □ - © — □ — ©1 1 2 2 NZ II 0 - S3 I ©I I I $3— O — S3 - r=0 NZ+1 Jz m i □ - " EI f S3 — I □— © - 0 - 9 - - p, § Figure 3.1. The 2-D r-z grid used for the field solver in CYLTMP. The "computational box" 0 « z < L , 0 < r £ R is shown with the thick outline. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 F ie ld Sply.sx The two dimensional r-z grid on which components are defined is shown in Figure 3.1. covers a computational box O ^ z ^ L , O ^ r ^ R . row of the field This grid The extra points in each direction is included to simplify the boundary conditions and particle pusher. quantities are The field defined at the following grid points and times Eaij = E*((i-YUr.(i-V)Aa,a*) Ertj = Ec( tt-OAr, -1)A*.,C&) {U-£> Af, 0 - V)A5L H D $) = &t) 3-1 .<i-i) ae , I'-frt') Note that these quantities are staggered in both time and space. The electromagnetic field components are advanced with the following finite difference approximations to equations 1.3-1.5 3.2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 where T\ = fo- V) &r = fi d: and c^- x Because of the staggering of the SL field definitions/ equations 3.2-3.4 are second order in both space and time ( i.e. finite difference truncation errors are Equations 3.2-3.4 hold in the interior mesh/ but compute the conditions. boundary points, we need specific boundary boundary CYLTMP currently conditions - is runs periodic conducting boundary at r=R. r=0 to The handled by symmetry. in with the z, and simplest a perfectly computational boundary Explicitly/ these boundary conditions are: 1. Periodic in z. For all quantities/ the j=NZ+l column is equal to the j=l column. 2. Symmetry at r=0 3.5 n-4*1 3. | i Conductor at r=R 3.6 Equations 3.2-3.6 form a complete set of equations/ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. once 46 we have a prescription for computing Particle Pusher and Particle/Grid Interpolation Since the particle motion the force law is is one-dimensional, simply £, - qE^e%, the basic particle handling facilities in CYLTMP are essentially those in a one-dimensional review the 1-D and electrostatic code. non-relativistic Let us first electrostatic case clarify the discussion and define some notation. rest of this dissertation "particle" mean "simulation particle". will used be to For the taken to Actual plasma particles will be referred to as "real particles". In a 1-D electrostatic code, the considered to be particles can infinite slabs of charge with uniform charge density normal to the direction of inhomogeneity. We will take the z-axis to this direction. The particles move in a computational box ( here a line ) from z=L, on which a grid Z*^ = j A z is defined. step, a charge density based on the particle self-consistent With this field we particle, repeated and z=0 to At each time is assigned to the grid points positions. Then difference approximation to Gauss' law is the be used a finite to find electrostatic field Ej on the grid. compute the force acting advance it forward in time. continuously, simulating the on each The cycle is self-consistent collective behaviour of this multi-particle system. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 Central to the operation of the code is function S(Zj-zr) that to or from a particle at z r species c& at density 3.S. density a force particle of q^vrS(Zj-zr)/£z q,«,S(Zj-zr}/Azr and to grid point Zj . turn, the field at grid point Zj acts with A with velocity vr contributes particle density S(Z^-zr)/&z, charge current weight interpolates a quantity at grid point zr the q^EjS(Zj-zr). on this In particle The most common choice of weighting function is linear weighting j * jo j = je + 1 st 3.7 j = j© »jo + 1 where Z? £ z_ < Z. . , and 4® r Ja-HL With this function a &z = ftr - Z; 1 j particle contributes particle density only to the two grid points it lies between. relative contribution to directly proportional to the distance between the these points and the particle. simple This two grid weighting points scheme is has a physical interpretation in terms of "finite-size" particles. their these The Since the grid points neighbors by a distance Az, considered to be at the center of a are seperated from each grid point can be cell of length A z extending outAz/2 from the grid point in each direction. If we now imagine the particles to have a finite length Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 equal to the grid spacing bzt S^(Zj-zr) fraction of the particle in the jth this linear weighting in the is simply the cell. CYLTMP uses axial directionr so its particles can be considered to have a length of Az in the z-direction. We can summarize the computational cycle electrostatic code as follows. ft (umh and velocities z r and vf at of a 1-D Given particle positions time step n , we first compute a charge density on the grid by " A a i r l ’* ? S l a i -a?') 3.8 Next solve the field equations to get the electric Ej* on the grid. field Then advance the particle positions and velocities to the new time step ( correct to by the staggering in time between vr, zr and Er ) » -eg?** ^ 3.9 where the electric field at particle r is Er" - 2 EjS(zr a nr ) 3.10 3 With the particle positions and velocities at time step n+1, we can then repeat this cycle. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 In CYLTMP, although one-dimensional, the the particle particles assign points on one or more radial rows. given species o< are divided motion charge to grid The particles up into Particles from group for to by the user at t=0. the for radial rows t0 imax,ou^ ' these radial boundaries being specified a groups G ^ , contribute charge and current densities of species is inputs Physically each particle in group G*^ can be visualized as an annulus of thickness Az and inner and outer radii r7Wtoi oj and rtIfllWX.O*} respectively, representing a collection of real particles in this volume, all moving with the same velocity. reality, since E % varies with r, the real the eventually volume of the particle different velocities, and However on a grid with the will annulus in CYLTMP is particles in the weighted that particles will in acquire break many radial rows, slowly from one row to the next. In up. can vary The approximation made we use the same Es for all the real annulus, this E a being a carefully average of E^ at the "near" grid points. Given that there is no radial motion of the real particles, they all maintain if the same axial velocity, the annulus moves as a single rigid super-particle. Furthermore, if the real particles in the annulus start with a prescribed radial density profile, this profile is also preserved in time. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 Although several the radial particles rows, we assign can their model arbitrary profiles by radially weighting the charge different rows spanned by the particle. group have the particle density same on density the charge to radial assignment to All particles in profile. Thus, the grid between rows i a n d i-wvm.oi^ can be written in the factored form 3.11 where ^ = 3‘12 is the axial linear particle density density of particles in group G ^. Note that CYLTMP uses linear weighting in the axial direction, and that *s dimensionless. It is convenient to put the 1/length3 dimension in nQ - the "background" density that defines the electron plasma frequency used as the unit of time. Particle densities for each species are then defined relative to w«*cy, are the radial weights for group above discussion, they remain uniquely defined constant in na . The From the time, being by the initial radial density profile. At t=0, the unperturbed density profile is 1 U (f\ ^ * YUo ( O Vtfte) 3-13 where g^ and h ^ are dimensionless functions specified the user. The axial positions of by the particles are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 defined so that 3*14 where n <,*^o = N ^ / N g density of group group ). is the average linear particle ( N,*^ is the number of particles in Comparison of equations 3.11, 3.12 and 3.14 show that the radial weights are defined by U o / w^ l Thus n u .0 = ^ 3-15 in CYLTMP, the full charge density on the2-D grid from all the particles is given by ^ “ HoZ. ^ 3.16 Similarly, the current density on the grid is ° 3-17 where (?VO u ^ * 21 V r Si( 5 ^ - Z.^) 3.18 This completes the particle to grid interpolation scheme. We also need to interpolate particles for the particle picture E g from the grid to the pusher. From our physical of the simulation particle, the average force on the real particles within its annular volume is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 Thus in CYLTMP we compute the following radially weighted average driver field for group /71. where A^ = TT ( r£ radial cell. - r^ ) i s the endface area of the ith Note that the radial weight factors for the driver field can also be computed once t=0. 3.19 and We then move the particles in group a 1-D electrostatic code. for all at just as in The field at the rth particle in group G«u^ is 3.20 We update the particle momentum with this field tv+K. n ^ ”* ^ = ^ E r At 3.21 With the new momentum, we then update the particle energy and position * 3.22 n * 'i z ^ - z " , jv At 3-23 ;i ? v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 The F.ull.Computation Cyglfi With the above definitions, we can now full computational cycle used in CYLTMP. examine the Starting with ri a the quantities at time level n - Eayj , fk A**% zc , - w e perform the following steps: » 1. Compute B^.> 2. Advance the particle positions and momenta and pr from equation 3.2. . For each driver field using particle using in the equation o+l zr . group, we compute the linear equation 3.19. group 3.21 and to Then for each we obtain the new momentum the new position using equation 3.23. t\*k from 3. Compute J2 - calculation is equations 3.17 actually and 3.18. This performed as the particles are being moved. 4. Advance and E rij t0 time level n+1 using equations 3.3 and 3.4. nfrl 0+1 We now have Ea-j '2 , Eri^ , , o+l zr . o+V , and pjand can , repeat the cycle. As emphasized above, all the radial weights can be computed once and stored for use at each later time step. Thus the particle mover is almost as fast as a electrostatic mover. The trick of pure 1-D spreading the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 particles out over more than one radial row results in tremendous saving a in the number of particlesrequired ( and hence CPU time per run ). theexperimental In PCM, the plasma thickness is about l/25th of the radius of the guide, while the beam radius is about 9 times the thickness. plasma Typical CYLTMP simulations require about 4096 particles ( 256 grid points and 16 particles/cell thermal velocity distribution). scheme in which the radial particles If were we tied a used a simple to a single row, the minimal calculation would need 25 radial rows, with one row for the plasma We would be forced to used (l+9)x4096 andnine forthe beam. = 40 960 particles. However, examining the radial profile of the Ea for TMd0 wave, is slowly varying in the beam region, while E r and vary rapidly simulation near we the plasma. use spread over 2 rows particles a typical CYLTMP 50 radial rows, 1 group of particles for the spread In over 6 plasma, rows for and 3 groups the beam. double the radial resolution of the above of Thus we example while using only 16 384 particles. The time required to do the grid calculations on a 256x50 grid is about the the time required to push this same as number of particles. Using the simple scheme would halve the time spent on the grid but increase factor of 2.5. better results! the time spent on the particles by a CYLTMP runs 33% faster and produces The radial weighting and particle group radial row assignments are run-time user inputs. Thus if Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 a series of runs are matter to first find radial row to be performed, it is a simple optimum assignments particle by simply thicknesses trying and different combinations on a single run. Having shown that CYLTMP is an efficient solution to its restricted model, we compare relativistic electromagnetic code. require it with a full 2-D Such a code would far more particles than CYLTMP - at least 50 000 particles for a 256x50 grid. Furthermore each particle must go through a complex electromagnetic particle pusher to advance its position and momentum at each Finally, a 2-D "divergence continuity At code cleaning" is must step preserved go to to through an ensure that used in CYLTMP step. elaborate charge second order in &T,A% and It is easy to show that the charge densities time - and current equations 3.16 and 3.17 - already satisfy the continuity equation to second order, and so CYLTMP does not have to do this step. With these conservative considerations, estimate it also probably a that CYLTMP uses only 2-5% of the CPU time required by a full 2-D code. must is Furthermore, one consider that the added complexity of the 2-D code makes it more difficult to maintain and use. The price paid for these savings is that CYLTMP is restricted to the simplified model given by equations 1.3-1.7, while this is just a special case for a full 2-D code. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56 Diagnostics Since we will CYLTMP be simulations, history energy. of the results of the it is appropriate to define some of the diagnostics used. time discussing the The principal diagnostics are components of the the total system The total field energy is 3.24 where tVv= tT(r*^-r^) and ^ “ TTfr^ This energy is the sum of the electrostatic energy due to charge seperation ( plasma waves ) and the energy in purely electromagnetic waves (V*E = 0 ). We make this distinction by writing the electric field as S, * Ee« + Ses 3-25 Only the purely electromagnetic component of the energy can be radiated out of the system as microwaves, so it is important to make this distinction quantitative. easily This is done by using the scalar and vector potentials in the Coulomb gauge 3.26 Eei - - V § We then define 3.27 the electrostatic and electromagnetic components of the field energy in the system as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 £es = ^EtA * £esc,v)^| 3.28 + 3.29 Note that at a given point E •E = E-_ + E— 2E •E E5 ElA ^ES v a* a * and that in our geometry Egs*E5M ^ 0. ^F£ = ^6S + Thus + 2 J < W Eps * With our choice of boundary conditions and gauge to condition on integral vanishes. A, it is easy the Coulomb show that the Thus definitions 3.24 and 3.25 lead to the consistent condition Efe - ^es In CYLTMP, Once +^'E<v\ is 3.30 computed directly from the fields. we have the charge density from the particle mover, we can then solve for the scalar potential using the finite difference approximation to Poisson's equation r? S iu t,\ - Z<-i 1 + <~i 5 i . - i , S be'*' Si.yH - 3.31 * - 4nrf ii This is a set of linear equations that can be efficient matrix zLtH methods that exploit the solved by sparseness of the . The electrostatic energy is then computed from Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We then compute ^ SMfrom Bpg and This using equation 3.30. division of 18re consumes considerably more CPU time than the computation of itself, because the solution to equation 3.31 is non-trivial. The total kinetic energy for species = 2 1 Z I ft'iVWv 2- is ■ 3.33 where we have used the identity V .!)««■ - T a j * * ' - This is useful to avoid the roundoff computing error inherent ^ - 1 for non-relativistic particles. In addition to these energies, CYLTMP can also the time save history of Fourier modes of 1-D quantities and Fourier-Bessel modes of 2-D quantities. is in periodic Because the grid in the z-direction and has N 2 grid points, there are N a distinct axial wavenumbers in the system !>«■ " A k * * + 3.35 where A k a = 2Tt N-a.ks 3.36 For a function defined on the grid points f^ = f (Zj), define the finite Fourier transform r . 1 T £ £» _ s ; r * » e 3 - 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. we 59 Its inverse is ° e n 3.38 The most useful FFT mode histories saved by CYLTMP are for the linear particle density of a specified group \ ^ £ ^0 These are used to observe the behaviour of longitudinal waves in the selected species. To analyze the electromagnetic field CYLTMP can save Fourier-Bessel modes on the grid, of the magnetic field %ij 3.40 Where NWPk is an arbitrary normalization constant, and <Aoin is the mth zero of JQ . These mode amplitudes are directly proportional to the amplitude of the with the same ka , larger amplitude than TM mode once the latter has grown to much all the other plasma waveguide modes with the same ka . In addition to time histories, CYLTMP also saves "snapshots” of selected quantities on the grid at chosen time steps. 1-D quantities and their FFT's versus axial grid point Z These quantities include CYLTMP can also or wavenumber k n ^ , are plotted respectively. ' and • generate contour maps of B^.* , and the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 electrostatic potential of . , together with vector maps Finally, to display maximum information about the particles, CYLTMP can generate snapshots of z-v^ and z-p^ phase space for arbitrarily chosen particle species. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 CHAPTER 4 CYLTMP SIMULATIONS OF THE PLASMA CHERENKOV MASER injttpflwgEifln In this simulation chapter we apply code to the PCM. the CYLTMP We begin with a description of how the principal parameters in the are determined. of the PCM parameters are presented. corresponding i3 These to -3 cm density are used varying use in the over the . Results of these simulations are in In addition, the able to carefully examine the effect of beam temperature on the efficiency. linear simulations simulations those good agreement with these experiments. simulations code Next, results of a series of simulations experiments, with the plasma range 2-8x10 particle Saturation of the instability is found to be due to trapping of the beam electrons in the Enfield of the most unstable wave. We present substantiate simulation this results assertion. to Finally, qualitatively we find fair quantitative agreement between the maximum energy loss of the beam in the simulations and those predicted from simple semi-empirical trapping model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a 62 Choosing Input Parameters Since some care is needed parameters for procedure used CYLTMP, for the we PCM to determine begin by the input outlining simulations. the All PCM simulations used the following parameters: grid size: NR = 50 NZ = 256 radial parameters: /^r = 0.0299cm rb = 18 A r = 0.538cm r4 = 22 & r = 0.658cm r^ = 24 Ar = 0.718cm The beam was split into three groups, each particle being 6 Ar wide, while a single group was used for the plasma with particles 2 Ar thick. While Ar value of and the radial parameters were A z simulated. From linear there technical want as are point many the used depended on the plasmadensitybeing theory, number of the most unstable TM Az, fixed, two is factors also we computed wave - kao. to consider discussed below ). points/wavelength as requires A z to be as small as possible. the wave In choosing ( a third First, we possible. This Secondly, since we are trying to model a continuous range of wavenumbers, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 we want & to be as small as possible. From equation 3.36, we see that this is accomplished by as many points as possible, making possible, and making k ^ one of the latter two requirements requirements arbitrary for spatial we ka&= 10 A k % . points/wavelength ran The the an with the maximum = 256, and This for modes. resolution. As always number of axial grid points setting higher as large as are completely opposite to the maximum compromise, Az using Az fixed by gives Ak^/k^ = 0.1 and 25 most unstable wave. This choice appears to have worked quite well. Once Ar, Az and n^ cannot be arbitrarily are determined the chosen. The time step finite difference equations 3.2 - 3.4 are numerically unstable too large. limit can befound by In principle this upper deriving the modifications that finite Az, make to Ar At and known first appear in frequencies to be stable in the continuous limit. is a very difficult calculation, and all we need At is the dispersion relation, and finding the At at which imaginary parts modes if A t to be less than this upper bound. of This is for Thus in practice, we simply use the limit for a homogeneous plasma in 2-D Cartesian geometry At* X AX Ay% multiplied by a safety factor, 4.1 4 typically in the range Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 .7-.8. This has been perfectly satisfactory. The particle momenta of species ot were to approximate initialized a Maxwellian distribution in the species rest frame « N where o = kgT^, and N ^ such that fa(p'a') C is a — 4.')j 4*2 normalization constant = 1 ( where is the modified Bessel function of the second kind order one). of The particle momenta were then transformed into the lab frame Pa = where the > ( ^'2- +■ 4,3 |&0 = Vd /c is the drift velocity of lab frame. Numerical instabilities can result if either the beam or plasma temperatures In practice, since the cold plasma is plasma species o*. in are too low n^ , only the instability with a problem. The for stability temperature condition in on the the 1-D non-relativistic electrostatic case is ^ where the * P X p= < \ 3 | * st^ie electron Debye plasma.Fortunately, this length non-physical instability is self-stabilizing in the sense that if equation 4.4 not satisfied initially, of is the instability heats up the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 plasma until it is, at which point the instability off. turns In CYLTMP, a plasma simulation particle represents many more electrons than a beam particle, so keeping the plasma temperature low greatly reduces the noise level of the entire simulation. a low plasma Thus our simulations started with temperature ( ©e=10eV, and simply let the non-physical instability stabilize while the evidence of simulations, smaller physical the non-physical grew. One instability can see of the but it saturates much earlier and at a much amplitude Consequently instability itself it than the physical instability. was assumed to be harmless, and all the simulations were done with the lOeV plasma temperature. Simulation Results Two series of simulations were performed. series The first was designed to test CYLTMP and the linear theory codes ANALYZE_DSPFN and GET_FIELDS against one another. In these simulations, a single TM qq wave was excited in a system with extremely low beam and plasma temperatures even here the the problem of the cold plasma did not affect have a major effect on the physical The instability). fields on the grid and perturbations in position and momentum for each species were initialized from ( the components linear at codes. selected The using electromagnetic points, the output field appropriate Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 Fourier-Bessel mode of the electromagnetic field, and the appropriate linear particle density FFT observed performed, and in About every a simulations the oscillation case, theory. Further, total & 1% ) with energy was conserved to better than .5%, despite the violence of the - all dozen frequency and growth rate agreed very well ( linear were to oscillate sinusoidally with an exponentially growing amplitude envelope. were modes instability converting more than 30% of the beam energy into field energy. CYLTMP This was taken to be overwhelming evidence that ( and incidentally the linear theory codes ) were working well. Confident that CYLTMP was working well, we began the second series of simulations. This series focussed on the realistic case of excitation of the TM00 waves thermal noise in the system. Experimentally, it is not difficult to obtain a relatively low-temperature << c ) plasma, but it is much ( i.e. more difficult to produce a relativistic electron beam with a small in axial momentum. We from therefore spread focussed on the PCM efficiency as a function of both plasma density and temperature, keeping the plasma temperature at lOeV and other parameters equal to those used in the In these beam simulations, the plasma experiments. particles were initialized with a small negative drift velocity so that the net plasma current was equal and opposite to the beam Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 current. Furthermore a fixed ion background was placed in the same region as the plasma so that the net charge in the system was zero ( consistent with periodic boundary conditions ). Figures 4.1 and results of B^lOOeV. much a 4.2 show simulation some of principal 13 -3 = 2x10 cm and in which With these temperatures the W p and A,Vb are both less than the beam velocity V0 = .857c, so that the cold fluid theory should still be valid. configuration, 93.5% of the total In the system kinetic energy of the beam, 1.6% kinetic plasma, 2.8% is initial energy energy of magnetostatic currents. total field of Figure 4.1a shows field energy, over and component in beam and plasma drift time its history of electrostatic the and The total field energy rises 30% of the energy in the system. electromagnetic part the the and electromagnetic components. to the electrostatic energy due to the radial charge seperation, and 2.1% is electromagnetic energy the is overtakes the Note that the electrostatic rises to over 20% of the total energy. Figure 4.1b shows the time history of the kinetic energy of both the beam and the plasma. At saturation, the beam has lost over 40% of its initial kinetic energy. of this energy has Roughly 50% gone into electrostatic waves ( of which roughly half is in electrostatic field energy, the other half and is in kinetic energy of the oscillating Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 TOTAL ES EM 200 400 600 BOO 1000 1200 800 1000 1200 (a) 1. 0 BEAM 0.8 0.6 0 .4 0.2 PLASMA 0.0 200 400 600 Q 4I (b) Figure 4.1. CYLTMP time history of (a) total field energy and its electrostatic and electromagnetic components, and (b) kinetic energy of the beam and plasma. n f = 2xlOra cm and = lOOeV in this simulation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 0"* 0 200 400 600 800 1000 1200 «pef (a) 0.0 1 5 o a. 0.010 3 0 .005 0.000 0 .4 (b) Figure 4.2. Fourier-Begsel modes of the electromagnetic field - from the CYLTMP simulation at n„ = 2x1013 cm"3 and ©»,= lOOeV. (a) Time history of mode 9 ( k^= .234wfe/c ) - the most unstable mode. (b) Linear growth rate of S&n, for n=7 to 12. The curve is the numerical solution of the TM«o branch with the beam at this plasma density. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 plasma electrons electromagnetic ). The waves. other We 50% define is the in purely efficiency of conversion of beam energy into microwaves as * = 4 .5 -do For this simulation, we get an efficiency of 18%. Figure 4.2 provides evidence that the instability is indeed the growth of TM 00 waves linear theory of Chapter 2. history of Figure 4.2a shows the time the most unstable Fourier-Bessel mode of the EM field { m=l, n=9 ). rather as predicted by the than Mode 9 was mode 10 because the the most unstable used to compute Az was calculated assuming the plasma is stationary, whereas the plasma is actually drifting opposite to the beam. After some erratic behaviour amplitude, this mode at grows early exponentially distinct growth rate until it saturates The growth m=l,n=7 to time m=l,n=12 Fourier-Bessel field theory drift velocity). the with a very quite are modes plotted in together with the growth rate of the TM linear small abruptly. rate obtained from the linear regime for the electromagnetic from and simulation of Figure the 4.2b, branch computed ( with compensation for the plasma The agreement between linear theory and is excellent. We will return to the non-linear behaviour later. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 Figures 4.3 to 4.6 show the effect plasma density. Figures of varying the 4.3 and 4.4 correspond to the results displayed in Figures 4.1 and 4.2, but with 1*3 -3 np = 5x10 cm , while Figures 4.5 and 4.6 are results at n r = 8x10 quite good, 1^ -3 cm but plasma density. and 4.8 density as from the to get poorer with increasing This is discussed the These in lOkeV. The agreement with linear theory is appears show temperature. . effect are Figures the 1 linear results and theory Figures of varying the using 4.7 plasma the same 2, but a temperature of At this temperature, cold below. significant deviations can be seen, although we still have an efficiency of 6%. The efficiencies obtained from the second series simulations are shown in Figure 4.9. seen to decrease as the plasma density is consistent efficiency increased with the of The efficiency is increases. experiments, in This which the was observed to increase rapidly as np 13 -3 above 10 cm , reach a maximum value of of *'**20% at np= 2x10 cm and then decrease as the plasma density is increased still further. Unfortunately, it is very difficult 13 -3 np«2xl0 cm . to do However the we simulations already below know from linear theory that the TM DO waves are stable below \z cm_x , which explains the low density cutoff on np ~ 6x10 the efficiency. Finally, note that the efficiency Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 0° TOTAL ,ES EM 200 400 600 800 1000 1200 800 1000 1200 ^pe ^ (a) BEAM 0 .4 PLASMA 200 400 600 (b) Figure 4.3. As n p = 5x10 ^ cm"3 . in Figure 4.1, but with Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 1(T 1-7 200 400 600 800 1000 1200 (a) 0.015 0.010 a. 3 0.005 0.000 (b) Figure 4.4. nf = 5x10 cm ka= .33w^,/c). As in Figure 4.2, but with . (a) Time history of mode 10 ( (b) Modes n=5 to 12 are shown. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 10"1 TOTAL ES___ EM 10"' 200 400 600 800 1000 1200 800 1000 1200 (a) BEAM 0.8 0 .4 0.2 PLASMA 0.0 200 400 600 6>pe* (b) Figure 4.5. As np = 8x10 13 cm”3 . in Figure 4.1, but with Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 200 400 600 800 1000 0.2 0 .4 1200 CJn.t (a) 0.010 0.0 0 8 Ol 0.006 4) 3 0.004 0.002 0.000 0.2 c k z/<ype (b) Figure 4.6. As in Figure 4.2, but with np = 8x10V3 cm"3 . (a) Time history of mode 10 ( kft= .SSw^/c ). (b) Modes n=5 to 12 are shown. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 10° TOTAL Ed 200 400 600 BOO 1000 1200 800 1000 1200 (a) BEAM 0.8 0.6 0 .4 0.2 PLASMA 0.0 200 400 600 (b) Figure 4.7. As in Figure 4.1, but with 0^= lOkeV. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 10"* 200 400 600 800 1000 1200 (a) 0.015 o a 3 0.010 0.005 0.000 (b) Figure 4.8. As in Figure 4.2, but with ©v» = lOkeV. (a) Time history of mode 10 ( k^= .26(0*, /c ). (b) Modes n=5 to 12 are shown. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 _ T n r . T.. 0.20 ▲ ^b«om ▲ 100aV ■ IksV ♦ lOkeV 0.15 ■ P ■ 0.10 - * A - m — ♦ - ♦ ▲ 0.05 ♦ 0.00 ■1 0 2 4 6 8 10 np ( x 1 0 13crrf3) Figure 4.9. CYLTMP simulation results of beam energy to microwave conversion efficiency as a function of plasma density with beam temperature as a parameter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 decreases with increasing beam temperature. This too is consistent with experiment, although they could not temperature directly. Instead, over the anode of the beam scatter, and the beam. they diode consequently to vary used a thin foil introduce angular an axial velocity spread, in It was observed that the efficiency dropped as the foil thickness increased. Saturation Mechanism As the linear instability evolves and waves grow, the wave amplitude the unstable spectrum becomes increasing peaked at the wavenumber of the most wave, since these waves Given a small initial amplitudes magnitude. can grow noise grow faster than the others. level, linearly unstable the. unstable by several wave orders In this case, by the time non-linear of effects are important, the wave amplitude spectrum is essentially a delta function at the k^of Thus the early non-linear dominated by the most saturates by the of the wave. wave. The instability the beam electrons in the Ez field of the most unstable TMm for unstable stage of the instability is unstable trapping most wave. The energy source instability is the kinetic energy of the freely streaming beam electrons. Once the TM 00 wave becomes electrons too large, the are troughs of the wave and no longer freely amplitude trapped in the stream through Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 the system. Thus the energy supply is turned off and the instability saturates. To clarify non-linear these behaviour ideas, of electrostatic two-stream we the first consider the non-relativistic 1-D instability. It has already been pointed out that the linear behaviour of this system and the PCM are qualitatively behaviour similar. The non-linear is also similar, and the simpler case has been z ° i - 3Z extensively studied . Figure qualitative features of the growth rate versus k , which is similar to Figure 2.5b. the growth Wf*/V6. rate We peaks consider dominated by this wave. In the 4.10a shows electrostatic k = k 0 , which is the non-linear effects the case, close to to be The general features of the time history of the electrostatic field energy are shown in Figure 4.10b. Consider an electron with velocity Vc; =V0-vpv» in the wave frame (i.e. in the lab the frame travelling with velocity v ^ frame) ? -S.sinking1, through where for an electrostatic now, Since the potential energy V = -e$ the electron energy is conserved. potential is constant in time. is constant in time, Clearly an electron is most easily trapped if it has this velocity at the bottom of a potential well. The energy equation for such an electron is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 6 (a) (b) Figure 4.10. Qualitative features of (a) growth rate versus k, and (b) time history of the electrostatic field energy, for the non-relativistic 1-D electrostatic two-stream instability. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 mv + eJ f'l. \\<W<> = - e. The electron is just trapped when its velocity is zero at the crest of a potential barrier. Thus the minimum potential amplitude for trapping is |L = 4.6 Z-e- For potentials oscillate larger back ) in this* the electron and forth in the potential well. trapped electrons minimum than ( those starting near a Deeply potential a strong potential well stray far from the potential minimum. will cannot In this case, the electrons oscillate in a roughly parabolic potential . 2 Thus, they execute simple harmonic motion at the so-called trapping frequency ( a). *c 4.7 iw With these considerations, we can explain the qualitative features of Figure 4.10b. As the two-stream grow, and close evolves, the waves we can soon neglect the effect of all but the most unstable wave. is instability to The phase velocity of this wave V9 , electrons travel electrons and so much are that more trapped v^ in the wave frame, the beam slowly first. than In the plasma fact, the wave Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 83 amplitude never comes close to being large enough to trap the plasma electrons, which behave linearly right up to saturation of the instability. the trapping amplitude As the wave , the fluctuations evolve non-linearly n ^ 10r*^ at the regions decelerates the electrons. end up in these bunches. in sinusoidal into dense above density bunches ( space where the Ea field Most of the beam electrons These electrons thus decelerate and supply energy to the growing wave. the grows In the lab frame, electrons continue to lose kinetic energy until they are completely turned around in the have wave velocity -V^ in the wave frame ). maximum field continue their the occurs. oscillation t their frequency hence amplitude in velocity the its ( i.e. This is when the the electrons potential in the lab frame, increases. draw energy from the wave and Half As frame wave well at frame, and Thus the electrons now amplitude decreases. a trapping period later, the electrons attain their maximum velocity and the wave amplitude is at a minimum. A short time later, the wave amplitude is increasing once again. as the wells. that Thus we get slow oscillations in the field energy electrons slosh back and forth in the potential However, equation 4.7 is an only an approximation works best for deeply trapped Furthermore equation 4.7 is strictly valid wave amplitude is constant. electrons. only if the These effects randomize the relative phases of the electrons in their oscillations in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 the wells, and potential distinct. the bunches become less Thus the amplitude of the slow oscillations in the field energy decays. This simple physical picture also accounts very well for the qualitative features of the non-linear behaviour of the field energy in the PCM simulations. Direct evidence that trapping of the beam electrons has occurred in the simulations is seen by examining the z-va phase space of theelectrons.Figure 4.11 shows the z-va phase space of theplasmaelectrons simulation after of Figures saturation electrons of exhibit at top*,t = 600 4.1 and 4.2. the The The plasma only mildly non-linear behaviour. electrons at By this vortices in phase space are a characteristic signature of trapping. velocity the This time is right instability. contrast Figure 4.12a shows the beam time. for axis at They are roughly centered on V = Vpv» ( asymmetric because the of relativistic effects)/ and the space axis where E^ssO and dE^/dz > 0 ( so that -eE^, is negative to the right of the center of a vortex and positive to its left). on the perimeter of a vortex are oscillating back and forth in theregion of space Recall are that there nine occupied by wavelengths unstable wave in this system/ and were lack Electrons it that vortex. of the most not for the of a vortex at z^«135, we would have nine vortices. The missing vortex is due to destructive interference Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of Figure 4.11. Phase space density of the plasma electrons ( every second particle shown) in the simulation of Figure 4.1 at uy,t = 600 (just after saturation). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 0.8 <n 0 .4 0.2 50 200 100 250 Z (a) 0 .0 4 o a. 0.02 3 o £N Ui ® -0 .0 4 150 100 200 250 Z (b) Figure 4.12. (a) As in Figure 4.11, but for the beam electrons, and (b) axial profile of E a along the cylinder axis at this time. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 the field of the mode 9 TM0ll wave with the other TM00 waves (from Figure 4.2b, their amplitudes are not totally negligible). Thus the vortices represent beam electrons trapped in the troughs of the most unstable wave. The qualitative features accounted for by trapping. in efficiency with interesting of that Figure 4.9 are also First consider the decrease increasing although plasma the density. linear It growth is rate determines how fast an unstable wave grows, it has almost no effect on the saturation amplitude. Instead, the trapping amplitude depends strongly on Ve- v ^ . the larger V ^ - v^ is, the larger the trapping amplitude is (we will make this quantitative later shows the density ). normalized phase velocity growth rate ^ for Clearly, for the most which the simulations Table 4.1 VPV\/C and the unstable waves were at each performed ( obtained from ANALYZE_DSPFN) nP (xlO® cm"3 ) .782 .795 .804 .824 2 3 5 8 .0127 .0137 .0141 .0094 Table 4.1. Phase velocity and growth rate of the most unstable TM00 wave as a function of plasma density. Recall that = .857. Thus monotonically Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 decreases with increasing np . The trapping model predicts that the efficiency should decrease with 13 (at least for np> 2x10 cm ). This is increasing n p consistent with experiment that and the simulations. Note there is no apparent correlation between the growth rate and saturation amplitude. The decrease temperature is easy to see. velocity in also efficiency with increasing consistent with trapping. With a velocity spread Av*,, difference parameter is not V8 - A v b-vpK. Furthermore, since vpV%atV0 , » so A v v>/V0 , the trapping chapter 2. This is the critical V6-vpVs, but A v b/(Ve -vpyj amplitude sensitive to velocity spread than the is linear far more theory of This is also seen in the simulation results. The oscillation frequencies and growth rates at are beam © b =ikeV almost identical to those at © b=100eV, both in good agreement with the "cold" linear efficiencies at A v fa, the However, the © b=lkeV are typically down 25-30% from the ©j^lOOeV values. spread theory. Also, amplitude with an initial thermal of the oscillations in the field energy after saturation should decrease because of the reduced coherence of the electron motion in the wave 13 -S troughs. At n f>= 2x10 cm we see oscillations at e ^ l O O e V as shown in Figure 4.1a, but they are absent at 0^=10keV in Figure 4.7a. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 Having seen that accounts for non-linear many trapping of the behaviour, we predictions. The beam the beam electrons qualitative features of the need simplest so-called rigid-rotor cold of 23 model . some trapping We assume f(z,pfi,t=0) = n ^ (p_g. “Po )• quantitative model an In is the initially the rigid rotor model we simply assume that at saturation, all the electrons The have turned distribution f (z,pa ,t=t around at ) = n^ (p-£ -p.f ). in the wave frame. saturation Because is of the strong bunching of the beam electrons at saturation, this is not as crude an approximation as one might at first think. This model works quite well in the non-relativistic case, 6 33 but must be modified for a relativistic beam ' Consider an electron with velocity frame. in the lab In the wave frame ( moving with velocity in the lab frame), the electron moves with velocity pi P° ~~ 4.8 Inverting ffpW -t 4.9 *0 If the electron turns around in the wave frame, its final _/ / velocity is yj-= - £»& • Thus in the lab frame, the final velocity is P° -f t, - 1 4.10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 The kinetic energy lost by the beam in the lab ( V - ' k Jmc2*, ’ &r= ( 1 " z )-'s.• where frame = ( 1 -^o ) * T^e initial electron energy is and in the wave frame is ^6 = ( 1 - $ 0f>fY^) -V i. where = (1 - 4.H f>^) Inverting this relation, we get "io m f® Pp*) 4.12 From this it follows trivially that 'fo * 4.13 Thus the energy lost in the lab frame can be written Uo'-fc We follow = the 4-i4 notation of Ref. 6 and introduce a normalized strength parameter S =a £ ^ A 4.15 where A = 1 The energy lost by the electron relative to its total (including rest mass ) energy is then given approximately by the dimensionless relation '>i» ^ --+ S 4.16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 This is the fractional energy loss of a as single electron measured in the lab frame when it turns around in the wave frame. If the rigid-rotor model were usedf equation 4.16 would therefore also give the fractional energy loss of the beam. contained It is shown therein, that in a Ref. 6, and references much better estimate of the fractional energy loss of the beam is - 1.5S ( 1 + 4.17 This a semi-empirical expression based on the foregoing analysis together with examination of computer simulation results. The factor increase in while of 1.5 comes from an effective S due to a decrease in ff* near saturation, -3/*_ additional (1+S) factor is due to a the relativistic effect in which only a fraction of the electrons rotate coherently in phase space. The results of applying equation simulations are shown in Table 4.17 to the 4.2 ( Note that while &E/^mc can be simply expressed solely in terms of the meaningful quantity is AE/(])o“l)mc more PCM S, and this is the quantity listed in the table ). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 (xlOftfcm~3 ) S .484 .400 .342 .213 2 3 5 8 theory simulation .38 .38 .36 .32 .41 .36 .29 .19 Table 4.2. Comparison of beam energy loss at saturation from theory and simulation. This model's prediction fair agreement of the beam with temperatures used loss is the simulations. agreement at higher density is beam energy in The poorer probably due to the finite in the simulations. Since V0-v^ is much smaller at the higher densities ( see Table 4.1), even running at lOOeV, the thermal spread of the beam electrons significantly reduces the efficiency the "cold" case. Despite the success of this model in predicting the maximum efficiency is exactly what microwaves. over still fraction energy unknown of the loss of the because we lost energy beam, do the not know goes into This question requires further study. gummaiy Computer simulations of the PCM have been using CYLTMP. These simulations have performed confirmed linear theory of the instability presented in chapter Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the 2. 93 More importantly, the saturation amplitudes predicted by the simulations are in good agreement with experimentally observed instability efficiencies. and subsequent Saturation non-linear of the linear behaviour is qualitatively explained by trapping of the beam electrons in the Es field of the most unstable TM00 wave. the magnitude Finally, of the beam energy lost at saturation can be explained fairly well with a simple trapping model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 CHAPTER 5. SUMMARY AND FUTURE DIRECTIONS In this dissertation, we have investigated the basic physics of the generation of electromagnetic radiation in the interaction region of the plasma Cherenkov maser. In chapter 1, a relatively simple mathematical model of this system is created by neglecting the components of theory this of principal the exact has In chapter 2, the linear been particle device the plasma of the particle waveguide, solved. The together with effect of the beam on these simulation code, with described the using this code are presented in chapter 4. field amplitude the linear theory. They electromagnetic is created by rapid growth of the slow T M ^ in good quantitative agreement with a in Simulations of the interaction region of have confirmed that the large numerical waves solutions The saturation amplitude of the field has been shown to be limited beam fully The non-linear theory has been addressed chapter 3. of velocity is the exact numerical solution of the calculations modes. new model result slow TM waves in electrons. transverse by trapping of the electrons in the troughs of the most unstable wave, in fair agreement with a simple trapping model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 Both the linear lead to results and non-linear consistent with calculations experiment. theory calculations have shown that the growth the TMoa waves have Linear rates of decrease significantly as the plasma 13 -3 density is reduced below n^ ~ 10 cm , and in fact these 12, .5 waves are stable for np^6xl0 cm . This is in good agreement with the sharp drop in power output observed in the experiments as the plasma density is dropped below 13 -X np~ 10 cm . More importantly, the efficiencies computed with experiment. CYLTMP are in fairly good agreement with In particular, the simulation results show a peak efficiency of about 20% when using a "cold" beam at 13 -3 n^= 2x10 cm , just as is seen in the experiments. However, despite these successes, much work remains to be done on the theory of this device. First, no attempt to been made. optimize the efficiency Clearly, to optimize the efficiency, we need to choose parameters that maximize Va - vpV> having an appreciable growth rate. experimental device. operation nf = 2x10 13 -3 cm For ( and these that even greater efficiency can other current parameters efficiency A 0 = 3cm. while still We have used only the radial parameters, beam energy and beam optimal has ea of the we find 20%) at However, it is possible be achieved by values of R, r4 ,rz , rt , Va and nb . Of course we can expect that considerable effort went using into Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the 96 choice of these parameters for the Soviet experiments. However, it is not clear to what degree the approximations, analytic they when to relied exact fully on numerical solutions are clearly required optimize the choice of parameters. This search for optimum parameters could be automated with a program incorporating parts ANALYZE_DSPFN. These optimum parameters can of then be tested with CYLTMP. Perhaps the single most important step to with this project is to conditions used in CYLTMP. actual device can be reflecting propagating to fill boundary at A before and a coaxial scheme to eliminate back into the system from the coaxial The simplest approach would the end of the coaxial guide region with an if reaches the end zsjn. would be injected at z=0r and those reinjected a z=0, artificial absorbing resistive current to wave boundary For the fields, we would have waveguide would be needed. be axial A more realistic model of the waveguide extending beyond z=L. waves the taken constructed with only a moderate increase in complexity. perfectly modify be at z=0. damp out the The beam particles "collected" at z=L Finally the plasma particles can be treated with self-consistent finite conductivity boundary walls at both ends CYLTMP simulations to axial boundaries. . These modifications would allow investigate Furthermore the we effects could of the also examine Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 transient effects as the beam is initially established such as current neutralization of the plasma. To complement this version of CYLTMP, a new of ANALYZE_DSPFN would be needed for the linear theory. Simple axial boundary conditions for are a version the linear theory perfectly conducting wall at z=0, and a partially |8,u reflecting boundary at z=L . While to and k^must still satisfy equation 2.8, the axial boundary conditions force both u> and k^to be complex. used to determine This modified program can be optimum parameters efficiency with the axial boundaries. to maximize the These parameters can then be tested with CYLTMP. Finally, in a real design of an actual device, we must examine the effects of not neglecting the transverse velocity components. The complicated, but still theory, need a we electromagnetic linear theory tractable. much more For the non-linear multi-dimensional particle code. is relativistic Although this code would be expensive to run, we could use it sparingly. For a given problem, we could use several runs of CYLTMP ( at a fraction of the cost of a single run of the big to code ), determine most of the "interesting" input parameters, thereby minimizing the number of runs needed with the big code. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 In conclusion, the plasma Cherenkov maser is a interesting addition electron generators efficiency and to of output the coherent power experimental device are most that growing family radiation. of the impressive. of free The high very It very is first hoped further work with this device will continue, to see if it really can achieve the great expectations these early results promise. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 APPENDIX A ANALYTIC TREATMENT OF THE DISPERSION RELATION In this appendix, solutions to the we derive dispersion approximate relation for the slow TM waves without the beam, and then derive an for the (complex) analytic approximation frequency shift for the TM00 wave in the presence of the beam. Given the complexity dispersion is that relation, it clear of the such analysis requires many simplifying assumptions, which is why it is left to the appendices. first presented in Ref. The major results below were 21. However, this article is extremely terse, and so it seems appropriate present this work here also, with additional steps to clarify the derivations. We start dispersion by relation » 3cR<< 1 and satisfied wave w << considering for a • without equation the beam. 2.12 - the We assume that 1. Physically, these conditions are very dense plasma and a low frequency Since ocpr^ >>1, it OCp rt >> 1, while JCR <<1 implies that follows « 1. that We use the asymptotic expansions for the Bessel functions A. 1 and the lowest order small argument expansions Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. for 100 |z| « 1 JoCa) ^ 1 3iW> %.[£)& \ — 1 KoU') ** ^ a. 2 -k. Keeping only the appropriate lowest order terms, we find the quantities in equations 2.9 and 2.10 simplify to 0CJ.C, ( "Tool? A A.3 \ ca>if + T si\i? / T Xt) fis y P suv ^<3* 4- Crt(u)-VX jl\ Xp 6efo \H •+■ Xu) — Psu*(ip+\i') A. 4 A. 5 where (J = 0fprA - ^ , \x= 3(p^ , and A p= r4 - r^. From these equations, it follows that equation 2.12 simplifies to X lW X j. = *•« Equation A. 6 has an infinite number of which implicitly defines U> for roots a fixed for Xj, ( k2 ). For A f<<^k and In ■Sfe*1, the right hand side of equation A.6 is very small. the zeroes of In this case, the roots must lie close to the tangent function. Expanding tangent function equation A.6 is approximately a -7 where n = 0, 1, 2, ... ( i.e. Xj_> 0 ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the 101 Since we assume that but neglect it we keep it for n = 0, for n > 0. Thus the roots are approximately l~A^ — j— T" V rk » O J / A. 8 OTT n>o In keeping with the nomenclature of Chapter 2, we define **■ k iof, C i ----- rrzr ^ * 0 -< *.» I (p£f n < ° Then, squaring both sides, equation A.8 can be written ^9Cp h^) ® IK-Io* ^p) or - o - which leads to the dispersion relation Won ® \ ** VCs■ l C* ----r— ; A. 10 It should be emphasized once again that equation A.10 derived under the assumptions .xR <<1, and In ^ ^ plasma. small. Ap<< rz 1. We next consider the field wave. 3Cj,r4 >>l, is structure of the TM 00 measures the change in phase of E s across the Clearly for the T M 00 wave this phase change is Thus the radial structure of this field component is approximately Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 Alo(xr') o Blofor^ + C K 0( ^ 0 Ea must vanish at r = R Using approximations Sr^Ct. and A.2, be and Aml1 continuous at r = r^. keeping only the lowest order terms o ^ r< E * — 1 InrM A. 12 Finally from equations 2.2 and A.12 ( noting that the next term in Es for r < r4 in the expansion of I0(5cr) is Mjxr) ), we find *< (^ r Er ~ o A. 13 We now consider the effect of the beam on wave. TM ^ To make progress, we assume that the beam density is low ( W^»\»<*w pe.), and treat the beam as a of the the T M 00 wave. perturbation As mentioned in the introduction we consider the interaction to be a resonant coupling of the the T M 00 wave the beam with the slow beam wave. density is so low that at We assume that frequencies of interest the dispersion relation of the slow beam wave is simply 60 = kaV0 satisfying (i.e. k2V0 >> Wpb this relation, ). For to and ka The condition for resonant coupling at wavenumber ka is that to = Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. an<^ 103 to = U)$o • The dispersion relation for the TMO0 wave is Applying the resonance condition, the resonant frequency is found to be A. 14 We see that resonant interaction cannot occur for densities so low that plasma U)f<< $ekj.ooV0 . To find the growth rate at the resonant frequency, we return to the full dispersion relation - equation 2.8. From equations A.3 - A.5, noting that for the TM wave, tan )sA esXi.» we have A.15 For the beam assumptions terms, that we already 3C rb <<1. know from Although previous C0» k ^ , we assume that the beam density is so low that In this case approximations 19Cb)a e.\oc\ A.2, the and beam so JXbr^j << 1. terms in Using equation 2.10 become A. 16 Q Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 104 Finally, to lowest order, equation 2.8 is simply Thus the Dp = Q. dispersion relation for the T M 00 wave with the beam is approximately 0* - J jOpb IT * r, v * ^ ( w -k,vy With the right hand side equal to zero this is dispersion equation A - 17 just the for the TM06 wave without the beam, which the resonant TM ^ wave satisfies with the frequency given approximately = oW . by equation A.14 and wavenumber Treating the beam term as a perturbation, we look for solutions to equation A.17 of the form 00 * Woo + where )£wl << After some lengthy algebra, we find 5 (2. 6O0o + A. 18 where 5 A. 19 o Then in the limit J « _Sw I «■.! *00 A. 20 The frequency shift is given simply by [iW-S* \ W 60/ tfs \ 2. Taking the cube root, the unstable root - i.e. A. 21 the with positive imaginary part is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. root From equation A. 22 the real shifted downward. part of the frequency is This ensures that the phase velocity of this wave is less than the beam velocity. The growth rate is Finally, plugging back the solution A.21 into condition A.19, we see that the frequency shift A.22 is valid if « k. A -24 This concludes the analytic treatment of relation. It is the dispersion clear that many assumptions have been made to derive these results. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 APPENDIX B ORTHOGONALITY RELATIONS In the analysis of the electromagnetic fields general cylindrical waveguide ( i.e. in arbitrary transverse profile, but uniform along the cylinder with a axis) the cylinder axis taken to be the z-axis, we find a discrete set of wave-like solutions to Maxwell's equations r 1 8,1 j3lx We are using the notation of Jackson is a general indices. E\ the *34» v , section 8.11. a mode index which may stand for one or two and are the transverse, and E^and longitudinal label indicates field components respectively. the direction in which the The (±) wave is travelling. In this appendix we focus only on the of two such modes X an<3 • We assume the waveguide to be filled with a non-permeable arbitrary dielectic both dispersive restriction and placed tensor a the medium is lossless. medium ( JV =1) £ (x,y,w,k») function on r%i £ orthogonality of with an which can be space. The only is that it be Hermitian, i.e. In this case, both 00 and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. k% 107 are real. If ^ is a scalar constant, the modes B.l break up into TM and TE modes. longitudinal component ), and it is easy to Each mode has a single ( E a (B^ ) for TM (TE) modes 2.0 show that these longitudinal components satisfy the simple orthogonality relation B.2 where S is the transverse surface of the waveguide. the general problem under consideration here, the modes cannot be seperated into purely equation B.2 no longer TM and applies. conservation. This complex conjugate of a and general tied to relation and its derivation 23 which only =ky. Maxwell's equations for mode Dotting equation B.3 for modes, closely are modifications of those given in Ref. considered modes in which TE However orthogonality relation exists which is energy For X* X take the form with Bp, equation B.3 and for adding the t+) 0 (+) with d x gives B.5 From similar manipulations on Hermiticity of £ equation B.4, using we obtain Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the 108 fcjr.vxar*§r-™r* = B 6 -i £ ^ * LW > £ " w ^|.lw **»W ) ] .g ^ Subtracting equation B.6 from equation B.5, and using the vector identity B «Vxk - ft-Vxg ^ V-£xB we get v-(sr»sr*6?*»s;v tr B. 7 (*> X We write V X where ^ as - & + **la is the transverse gradient operator. Using the explicit form of the fields - equation B.l, the left hand side of equation B.7 can be written 7.(s?*sT * s * &X ) jn-MfeV + * tav* *£?*§>) Note that in the last term, we only have fields the transverse because the longitudinal components cannot give a z-component in the cross product terms. Using divergence theorem Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the where C is the boundary curve of S and n is pointing normal on C. But vanishes. outward = nxE. *6 = 0 on the surface of a conductor. integral the Thus the and line It therefore follows from equations B.7 and B.8 that Equation B.9 is First# that note the if general we variable# and multiply both result we let sides are seeking. as a continuous of equation B.9 This is the simple statement that the energy of the flows at the group velocity. by wave For different modes in equation B.9 there are two cases of special interest. In the first case# we consider look to be fixed and at the orthogonality of modes with different ks(&). Physically, this applies to the boundary-value problem in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 110 which a localized source or boundary wall is being driven at a single frequency w*. away from allowed the source Then have wavenumbers ^(We). all waves frequency propagating but with the Assuming that is independent of k2 , equation B.9 reduces to + E ^ ° B.ll s Now in the derivation - (O b |n and _ to d p. ofB.ll, had we originallyreplaced e /-> with bj*and Oja the result wouldhave been For ifc follows that [ * & =° b-12 Equation B.12 states that the total power sum flowing in a of such waves is equal to the sum of the total power flowing in each wave. All overlap integrals of different modes in the integral for the total power vanish. In the second case, we consider the problem in which the modes have frequencies W (k4). initial-value a common ka, but with different This situation is applicable to problem studied in this the dissertation. Equation B.9 then reduces to KFor iS»0 •& -v the integral vanishes. From b .13 equation B.13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ill it follows that the total field energy in a sum of such waves is equal to the sum of the total energy in each individual wave. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 112 References 1. IEEE trans. Plasma Sci., Vol. PS-13, no. 6 (1985) (entire issue devoted to the subject) 2. A. Gover, P. Sprangle, IEEE J. vol. 3. Quantum Electronics, to Physics QE-17, 1196 (1981) R.B. Hiller, An Introduction the Intense Charged Particle beams, Ch. of 6, Plenum Press, NY 1982 4. Free electron Physics of Quantum S. Jacobs et. 5. Novel generators Sources of Coherent Electronics, vol. Radiation, 7, edited by al., Addison-Wesley 1980 of Coherent Quantum Electronics, vol. Radiation, Physics of 5, edited by S. Jacobs et. al., Addison-Wesley 1978 6. D.S. Lemons, L.E. Thode, Phys. Rev. Lett., 56, 2684 (1986) 7. M. Shoucri, Phys. Fluids., 26, 2271 (1983) 8. E.P. Garate, J.E. Walsh, IEEE trans. Plasma. Sci., vol PS-13, 524 (1985) 9. E. Garate, R. Cook, P. Heim, R. Layman, J. Walsh, Appl. Phys., 58, 627 (1985) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. J. 113 10 . A.N. Didenkor A.R. Borisov, Yu.G. Shtein, Sov. Tech. Phys. G.P.Fomenko, Lett. 11. A.T. Lin, W.W. Chang, Y.T. Yan, IEEE Sci.,Vol. 12. H. 13. 9, 26 (1983) Trans. Plasma PS-13, 531 (1985) Alfven, Phys. Rev., 55, 425 (1939) B.I. Aronov, L.S. Bogdankevich, A.A. Rukhadze, Plasma Phys., 18, 101 (1976) 14. L.S. Bogdankevich, A.A. Rukhadze, Sov. Phys. Tech. Phys., 22, 145 (1977) 15. L.S. Bogdankevich, M.V. Kuzelev, A.A. Rukhadze, J. 16. Plasma. Sov. Phys., 5, 51 (1979) M.V. Kuzelev, A.A. Rukhadze, Sov. Phys. Tech. Phys., 24, 654 (1979) 17. A.A. Ruhkadze, Modern Plasma Trieste Course 1979, Physics, International pp. 537-91, Atomic Energy Agency, Vienna 1981. 18. L.S. Bogdankevich, M.V. Kuzelev, A.A. Rukhadze, Phys. 19. Usp., 24, 1 (1981) M.V. Kuzelev, F.Kh. Mukhametzyanov, A.A. Rukhadze, Phys. Sov. M.S. Rabinovich, P.S. Strelkov, A.G. Shkvarunets, Sov. JETP, 56, 780 (1982); Sov. Phys. Dokl, 1030 (1982) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27, 114 20. R.E. Collin, Field Theory of Guided Waves, Ch. 6, McGraw-Hill, NY 1960 21. M.V. Kuzelev, F.Kh. Mukhametzyanov, A.G. Shkvarunets, Sov. 22. J. Plas. D.E. Muller, Phys., 9, 655, (1983) Mathematical Tables and Aides to Computation, 10, 208 (1956) 23. D. Marcuse, Light Transmission Optics, p. 322, Van Nostrand Reinhold, NY 1982 24. J.M. Dawson, Rev. 25. C.R. Birdsall, Mod. Phys.,55, 403 (1983) A.B. Langdon, Plasma Physics via Computer Simulation, McGraw-Hill, 1985 26. J.P. Boris, Proc. Simulation of 4th. Plasmas, Conf. p.3, R.A. Shanny, Naval Research on the Numerical Edited by J.P. Boris, Lab., Washington D.C., 1970 27. A.B. Langdon, B.F.Lasinski, Methods of Physics, vol. 16, p. Computational 327, Edited by B. Alder et. al.. Academic Press, 1976 28. R.W. Hockney, J.W. Eastwood, Computer Simulation using Particles, McGraw-Hill, 1981 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 115 29. W.E. Drummond, J.H. Malmberg, J.R. Thomson, Phys. 30. J.R. Thomson, Phys. 31. T.M. O'Neill, T.M. O'Neill, Fluids., 13, 2422 (1970) Fluids, 14, 1532 (1971) J.H. Winfrey, J.H. Malmberg, Phys. Fluids, 14, 1204 (1971) 32. S. Kainer, J. Dawson, R. Shanny, T. Coffey, Phys. Fluids., 15, 493 (1972) 33. L.E. Thode, R.N. Sudan, Phys. Rev. Lett., 30, 732 (1973) 34. J. .D. Jackson, Classical Electrodynamics, Wiley, 1975 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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