AN HPEREMENTALMD THERMAL". 5:," ”j?" STUDY oragsommsusmm_f V PLASMA m ~tMlGRGWA‘VE CAVITIES _‘ _ -.1' 7 Thesis for the Degree ht Rh. 1).: j _ _ _ rMICHIGANSTATEjUNWERSITY‘ 9._\ ROBERT MlCHA-EL FREDERECKS . _ ' L I l}- 1? '5; K. Y Michigan Stats Universi‘Y IWWHHl“UllllllllllllllW\Llllllellll 3 1293 105336 This is to certify that the thesis entitled An Experimental and Theoretical Study of a Resonantly Sustained Plasma in Microwave Cavities presented by Robert Michael Fredericks has been accepted towards fulfillment of the requirements for Ph.D. degree in Electrical Engineering Major professr‘ Date W “3 VSAV‘CU 0-7639 (”:0 I 6 1795. W/ \t . :H “.1 “M £0. :‘v-~ , ~~M“ I - :1‘.H~ 5,“: ‘ I " 'f‘ v- .‘ “~-'q ‘ ABSTRACT AN EXPERIMENTAL AND THEORETICAL STUDY OF A RESONANTLY SUSTAINED PLASMA IN MICROWAVE CAVITLE‘S BY Robert Michael Fredericks An experimental and theoretical investigation of reentrant and cylindrical microwave cavity-plasma systems is performed. Using the linearized warm electron and cold ion plasma equations, along with Maxwell's equations, the axially propagating modes in two cylindrical waveguiding structures are analyzed. Both low frequency (‘9 "’ wi) and high frequency (to ~ we) solutions are presented. This bounded plasma theory is employed to interpret some low frequency Parametric instabilities observed in cylindrical plasma resonators. For the reentrant cavity-plasma system when operating at low incident power levels (5 30 mw). the electromagnetic cavity resonance is perturbed by the presence of the plasma. Additionally, new cold and warm plasma resonances are observed in the neighbor- hood of this cavity resonance. These new modes are interpreted as the eigewnodes of the cylindrical plasma resonator in the cavity. At mode ' ° rate 1nc1dent power levels (~ 100 mw), nonlinear hysteresis flush mdcfl ‘ v- up; 3:1“ .u. Twrw .‘l""."" I5 run u a s Robe rt Michael Fredericks effects are observed in the total density and power absorbed curves as the dc plasma density is modulated. A three dimensional graphical analysis is developed to eXplain these observations. For an incident power level of approximately 20 watts, a new variable length retuning technique is introduced to excite a dense resonantly sustained plasma. Radio frequency instabilities and hysteresis effects are analyzed. Furthermore, two equivalent circuits explain the decoupling eXperienced at large plasma densities. For the cylindrical cavity-plasma system, the linear resonance characteristics of the lower order dipole modes are investigated. The warm plasma theoretical predictions are in good qualitative agreement with the experimental results. An experimentalplot of input impedance Shows that the higher order Tonks -Dattner resonances are weakly ex- cited because they are impedance mismatched to the microwave driving circu' 1t. Also, a dense resonantly sustained plasma is excited in the * TM . . 101 resonance. For this h1gh density mode of operation, it is found ' BXperimentally and theoretically that detuning caused by increases i n the plasma density has been minimized. AN EXPERIMENTAL AND THEORETICAL STUDY OF A RESONANTLY SUSTAINED PLASMA IN MICROWAVE CAVITIES BY Robert Michael Fredericks A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOC TOR OF PHILOSOPHY Department of Electrical Engineering and Systems Science 1971 To Cheryl ~:Va' D ""1“ ' l » . T ;F}. W "‘.'0A. . ..,;" M A “'I- 4.. ACKNOWLEDGMENTS The author wishes to express his sincere appreciation to his major professor, Dr. J. Asmussen, Jr., for providing guidance, wisdom and encouragement during the course of this investigation. Also, thanks are due to the other members of the guidance committee, Drs. D. P. Nyquist, K. M. Chen, G. Pollack and G. Kemeny for their time and interest in this work. A particular note of thanks is extended to Dr. B. Ho for his construction of the plasma tubes and to the National Science Foundation, Grant (SK-5617, and the National Aeronautics and Space Administration for financing in part this research. Finally, the author thanks his wife, Cheryl, for her understand- ing and encouragement throughout his graduate career, as well as for proofreading the manuscript. ii TABLE OF CONTENTS Page ABSTRACT ACKNOWLEDGEMENTS .................... ii LIST OF TABLES ....................... v LIST OF FIGURES ....................... vi I. INTRODUCTION .................... 1 11. REVIEW OF MICROWAVE PLASMA INTERACTION . 7 2.1 Introduction. . ....... . .......... 7 Z. 2 Linearized Warm Plasma Model ......... 10 2. 3 Resonances in Bounded Plasmas ......... l9 2. 4 Re sonantly Sustained Discharges ........ 25 2. 5 Plasmas in Microwave Cavities . ........ 33 III. ROTATIONALLY SYMMETRIC PROPAGATING MODES ON A PLASMA CYLINDER .......... 38 3.1 Dielectric Rod Waveguides. . . . . ..... 38 3.2 Model and Characteristic Equations. . . . . . . 41 3. 3 Discussion of the Dispersion Diagram ...... 47 3.4 Dispersion Diagram . . . . . . . . ...... . 57 IV. PROPAGATING MODES ON A CONCENTRIC WARM PLASMA ROD INSIDE A CYLINDRICAL METAL WAVEGUIDE I O O O O O O O O O ........ O O O O 61 4. 1 Theoretical Model, Characteristic Equation and Discussion .................. 61 4. 2 Diaper sion Characteristics of the Dipole Modes. 68 4 3 Dipole Modes of a Plasma Rod Inside a Cylindrical Cavity ................ 73 iii to v o a V s.- \b 'r' Fig uw- 2.. I. sw. :5. cc. “M4 1.5. Page 4.4 A Comparision of the Low Frequency Eigen- modes of a Cylindrical Plasma with Experi- mental, Parametrically Excited Plasmas. . . . 83 V. EXPERIMENTAL SYSTEMS .............. 93 5. 1 Experimental System .............. 93 5. Z Reentrant Cavity. . ............... 96 5. 3 Cylindrical Cavity ................ 101 VI. EXPERIMENTAL RESULTS-.LINEAR OPERATION. 107 6.1 Reentrant Cavity ................. 107 A. Classification and Interpretation of the Resonances .............. 107 B. Cavity Equivalent Circuits and Input Impedance ............... 115 6. 2 Cylindrical Cavity ..... . .......... 128 VII. EXPERIMENTAL RESULTS--NONLINEAR OPERATION . . .................... 140 7.1 Reentrant Cavity. . . . . . . .......... 140 A. Resonantly Enhanced Discharge ....... 140 B. Re sonantly Sustained Discharge ....... 153 7.2 Cylindrical Cavity. . . . . ........... 166 VIII. SUMMARY AND CONCLUSIONS ............ 180 REFERENCES ......................... 186 APPENDIX A ......................... 192 APPENDIX B . . . . . .................... 199 APPENDIX C . . . ...................... 204 APPENDIX D ..... . ................... 217 APPENDIX E ......................... 224 iv LIST OF TABLES Table Page 4.1 Theoretical and Experiment Low Frequency Eigenfrequencies for a Plasma Cylinder. a=4mm, Te: 2,000°K, we: (21rxl.9Ghz). . . . 88 7.1 Power absorbed by the reentrant cavity-plasma system versus short length. Pin = 20 watts. . . . . 164 a- -. _ -~.~. ..;.‘. El. 1" 4 . p—V . p—a Figure 2. 2.2 3. 3. 4. l .3 l 3 1 LIST OF FIGURES Page Dispersion diagram for electromagnetic (EM), electron plasma (EP) and ion acoustic (IA) waves in an infinite, warm electron, cold ion plasma (not to scale) .......... . .......... 18 Parallel plate plasma capacitor. The driving voltage, V(t), has a frequency w. The plasma has width L-S, and each sheath has width 8/ 2. The plasma frequency is we . . . . . ........ 28 Power absorbed and power loss by the plasma versus density. The four power absorbed curves correspond to different values for the driving > > > . voltage, Va Vb VC Vd ............. 29 An infinitely long plasma cylinder surrounded by free space. The radius of the cylinder is a. The plasma is assumed to be lossless, uniform and isotropic. . ................... 42 Skeletal dispersion diagram. The shaded areas denote regions where Landau damping is strong . . 49 Dispersion diagram for rotationally symmetric modes on a plasma cylinder. a = 0. 3 cm, T = 30, 0000K, m. = 3. 34 x 10'25 kg (mercury plaetsma), w = 211' x 3 G 2. The solutions are invalid in the sfiaded regions where Landau damping is strong . . 58 A uniform, lossless, isotropic, warm electron, cold ion plasma rod located concentrically inside a cylindrical metal waveguide. The radii of the plasma and the metal cylinder are a and b respectively........... .......... 62 vi «(J (Q. ‘ ‘I I... -!d 4.. .hv 4’ I.“ 75 I .‘v 1%. ! 3‘4 Figure 4.6 4.8 5.1 5.2 5.3 5.4 Diaper sion diagram for dipole modes. cue/217 : 105, b = 5.08 cm, a = 0.3 cm. . . . ........ Dispersion diagram for dipole modes. 0) /21T = 3 Ghz, b = 5.08 cm, a = .3 cm, T = 3o,%00K. The coupling of the TE* mode with the TDl mode is shown in the insert ......... I. . . . Dispersion bdiagram for dipole modes. (”e/2" = Ghz,b - _5.08 cm, a: .3 cm ......... Resonant frequency versus electron plasma frequency for cylindrical cavity. L8 = 6 cm, b = 5.08 cm, a = .3 cm, Te 2 30, 000° K. The TM010 is also shown for reference . ........ Resonant length, Ls’ versus electron plasma frequency. w = 217 x 3.03 Ghz, b : 5. 08 cm, Te = 30, 0000K, a = o. 3 cm ......... Experimental setup to observe Tonks-Dattner resonances and parametric coupling to low frequency oscillations ................ Cross Section (a) and detail (b) of the coaxial discharge structure of Asmussen and Lee4 Experimental set up and coaxial microwave system 0 O I O O O O OOOOOOOOOOOOOOOOO a. Cross section of the reentrant cavity b. Equivalent electrical schematic . ........ Cross section of the cylindrical cavity and microwave probe circuit .............. Resonant frequency versus cavity size ....... Oscilloscope trace of power absorbed by the cavity-plasma system. Horizontal axis displays tube discharge current, 24 ma cm'l. Vertical axis is relative power absorbed. Input frequency is 2 Ghz and gap length, Lp is 0. 1 cm ....... vii 3,,48 49 70 74 77 79 84 90 94 97 102 104 fl. . Figure Page 6. 2 Power absorbed versus discharge current. The cavity size, Ls, has been adjusted to cause the perturbed electromagnetic cavity resonance to interact with the plasma resonances. Same epxerimental parameters as Figure 6.1. . . . . . . 111 6. 3 EXperimental and theoretical variation of cold and warm plasma resonances versus plasma density .......... A .............. 114 6.4 Input impedance versus short length, LS, for the reentrant cavity. Data points are dots ....... 116 6. 5 Resonant length versus plasma frequency for perturbed electromagnetic cavity resonance. Input frequency is 3. 03 Ghz . . ........... 119 6. 6 Equivalent circuit of the reentrant cavity con- sisting of two transmission lines .......... 120 6. 7 Cavity impedance versus discharge current (noted in milliamps) . . . . . . . . . . ....... 123 6. 8a Oscillogram of power absorbed versus discharge current for four values of Ls' Input frequency is 1. 5 Ghz ...................... 124 6. 8b Envelope of power absorbed versus plasma density, in critical densities, for increasing cavity size . . . . . ................. 124 6. 9a Equivalent circuit of quarter wave, open circuit transmission line resonance ......... 126 6. 9b Equivalent circuit of quarter wave, short circuit transmission line antiresonance ....... 126 6. 10 Input impedance versus cavity length without a plasma for the TElll cavity mode. The numbers refer to the short length, L3, in cm. Input frequency is 3. 03 Ghz ........ . ....... 129 6.11 Oscillogram of power absorbed versus discharge current. The four photographs have Ls = 5. 85, 6. 05, 6.15 and 6. 30 cm reapectively. Top trace is the zero line reference . . ......... 131 viii o.- Figure 7.2 7.3 7.4 7.5 7.6 7.7 7. 8a 7.8b Input impedance versus discharge current in ma for the cylindrical cavity-plasma system. L8=6.10cm.. ...... Resonant length versus plasma frequency for the dipole modes. The input frequency is 3.03Ghz..... ............. Resonant frequency versus the square root of discharge current (on ) for the cylindrical cavity plasma system dipole modes. Ls : 6.0 cm Luminosity (top trace) and power absorbed (bottom trace) versus discharge current for increasing levels of incident power. (Reentrant cavity), Horizontal axis, 25 ma/ cm Input impedance versus discharge current (in ma) for two levels of incident power. . . . . Luminosity (top trace) and power absorbed (bottom trace) versus discharge current for increasing levels of incident power. (Reentrant cavity). Horizontal axis, 25 ma/cm . . . . . . a. Power absorbed versus total plasma density. b. Total plasma density versus discharge current. c. Power absorbed versus discharge current. . Plasma density versus incident power. L8 = 1.80 cm. . . . a. Power absorbed and power loss versus density. b. Density versus short length. c. Power absorbed versus short length. . . . . Luminosity versus short length. The luminosity has been calibrated in critical densities . . . . Input impedance versus short length for reentrant cavity-plasma system . Detail of 7.8a. Short lengths are noted in cm . ix Page 132 135 137 141 144 146 147 154 156 160 162 163 2) .‘a via his Figure Page 7. 9 Luminosity (in microamps) versus plasma density (measured by TM010 frequency shift). . . . 167 7.10 Luminosity (top trace) and power absorbed (bottom trace) versus discharge current for increasing levels of incident power (cylindrical cavity). Horizontal axis, 40 ma/ cm ........ 168 7.11 Plasma density versus short length for cylindrical cavity Pin : 23 watts .......... 170 7. 12 Impedance versus short length for cylindrical cavity. Points a, b, . . . , p correSpond to points a, b, . . . , p in Figure 7.11 ....... . . 171 7. 13 Plasma density versus incident power for a resonantly sustained plasma in TMIOI mode . . . . 174 7.14 TM* impedance versus incident power. The data points are labeled (P /N ), where P is incident power in watts an1 N018 plasma diensity inlOl l/cm3. . .. ................. 176 :1: 7.15 TM impedance versus short length. Data points are labeled (L so/N ) where L8 is lcaviézy length and No is plasma density in 1801 1/cm . . . . 177 El Cut off frequency versus electron plasma frequency for TMOI waveguide mode a :2 0. 3 cm, b = 1. 27 cm . .................... 225 "g\ a to.“ “I III II! 3‘ - § .1 r“- \ 5“~ CHAPTER I IN TR OD UC TION In this thesis, emphasis is placed on efficiently generating a dense resonantly sustained plasma in a microwave cavity. Such rnicrowave-plasma systems have the potential for efficiently creating, maintaining and confining a plasma by radio frequency power. Thus this subject has application for both heating and confining a plasma when attempting to sustain a controlled thermonuclear reaction. In addition, this type of plasma is becoming increasingly important for processing chemicals and in certain other experimental situations where a dense, stable plasma is required. However, there are several poorly understood phenomena associated with the generation of plasmas in microwave cavities. The shift of resonant frequency versus plasma density, the appearance of new resonances, the existance of radio frequency plasma instabili- ties and the nonlinear interaction of the cavity-plasma system with the driving microwave circuit all must be understood in order to efficiently generate a dense, hot plasma in a cavity. C“€‘ eh. . , on .u t“ A 5v U 128 1 .1- _ “ac: .‘ f‘ fr: ' “a...” 1 ‘ ~ p. ‘I': During the course of this investigation it was found that the problem of efficiently creating and maintaining a dense, resonantly sustained plasma in a microwave cavity has several facets. First/ the eigenvalue equation of a warm plasma in a cavity must be solved to determine how the cavity resonances are modified by the presence of the plasma, and to find new resonances which do not exist in the empty cavity. This theory must be eXperimentally verified for the low power (linear) operation of the cavity. Secondly, a retuning method must be develOped to overcome the detuning caused by the radio frequency sustained plasma in the cavity. A fixed frequency retuning technique is desirable. Thirdly, the problem of coupling from the microwave system (generator, coaxial feed lines, etc.) to the cavity-plasma system must also be analyzed. A choice among probe, loop or aperture coupling must be made based on the field configuration inside the cavity with the plasma present. Finally, the choice of which cavity-plasma system resonance to excite will depend upon: a. ability to couple to the resonance, b. ability to retune this resonance by the new re- tuning technique, c. ability of this resonance to concentrate the electromagnetic fields in the region of the cavity where the plasma exists. Conditions (a) and (c) will generally be in opposition with each other, and a trade-off may be necessary to improve efficiency. Each of :9?" .e. A. n a ‘n “at: A... r-m, , - ‘Ju ... “u [\I r» , 4" these aspects has been considered in the design, eXperimental Operation and theoretical treatment presented in this the sis. Two types of microwave cavities have been designed, con- structed] and experimentally and theoretically investigated. The first is a short gap or reentrant type. This cavity is basically a short circuited coaxial transmission line with the center conductcr slightly separated from one end forming a gap region. A plasma tube is placed inside the hollow center conductor and the plasma interacts with the fields of the cavity only in the gap region. The other cavity is a basic variable length, cylindrical cavity with a cylindrical plasma rod placed concentrically inside the cavity. In Chapter II the basic equations which will be used to mathe- matically describe the plasma are introduced. The dispersion characteristics of the electromagnetic, electron plasma and ion acoustic waves are discussed for an infinite warm plasma medium. Also a review of the pertinent literature is presented. Resonances in bounded plasmas, resonantly sustained plasmas and plasmas in microwave cavities are all discussed. The theory of rotationally symmetric, axially propagating Waves on a plasma cylinder is developed in Chapter III. This theory can easily be applied to eXperimental results presented in Chapter VI. Both high microwave frequency electron effects and low frequency ion effects are studied. The chapter begins with a general 13 discussion of dielectric rod waveguides; and necessary conditions if? .4” F’- Una tn. arax vie .- ,. LC? L)“ 'l) CE are stated for waveguiding modes to exist. The warm electron, cold ion plasma equations and Maxwell's equations are solved yielding a characteristic equation for the modes on a cylindrical plasma wave- guide. The qualitative aSpects of these modes are discussed and an exact numerical solution is presented. In Chapter IV, the boundary value problem of a warm electron, cold ion plasma rod located inside a cylindrical waveguide is solved. The case of the dipole modes propagating on this structure is discussed in detail and diaper sion plots are presented. Extending this waveguide theory further, curves of the resonant frequency and resonant length versus plasma density are presented for a cylindrical cavity with a concentric plasma rod. In the last section of Chapter IV, this bounded plasma theory is used to interpret some previOusly observed, para- metric instabilities in eXperimentally excited cylindrical plasmas. The experimental apparatus is presented in Chapter V. The driving microwave circuit and detecting devices are discussed. Also, the design criteria and description of the reentrant cavity and the cylindrical cavity are presented. The experimental linear response of the microwave cavity plasma systems is presented in Chapter VI. In this low incident power operation, three types of resonances are classified. They are the perturbed electromagnetic cavity resonances, cold plasma resonances and warm plasma resonances. These resonances are observed as both changes in absorbed power and changes in the input D" a- L. -. ~- :1- . . -c. I#. '1 r ”I n fl ‘ impedance of the cavity. The linear results are eXplained by the theoretical predictions of Chapters Ill and IV. The coupling of the warm and cold plasma resonances is observed and the effects on the coupling of the plasma-cavity system to the microwave circuit are highlighted. Using two equivalent circuits for the reentrant cavity, the decoupling of the cavity-plasma system is eXplained as the inter- action Of a low density resonance and a high density antiresonance. In Chapter VII, the eXperimental nonlinear response of the cavity-plasma systems is presented. The transition from the linear resonances to a completely resonantly sustained plasma is presented. A graphical analysis is developed to eXplain the observed hysteresis effects. For the reentrant cavity-plasma system a new retuning technique, using the variable length of the cavity, is introduced. With an incident power of twenty watts, a resonantly sustained plasma of over thirteen critical densities is excited in the reentrant cavity. For the cylindrical cavity at high incident power levels, the plasma can be resonantly sustained in the perturbed cavity, cold plasma or warm plasma resonances. However, eXperimentally it is found that a large volume of dense (over ten critical densities) resonantly sus- >2 tained plasma can be excited in the TM 101, cold plasma resonance, by properly adjusting the cavity size. In this condition, over 75% of the available microwave power is being absorbed by the cavity- plasma system. It. 9'- y'- ’7‘ In - -4. u. u..- 'i "T1 (II '2. he». a . J.: . I. V “FF! ‘- I“ Three manifestations of nonlinearities will be discussed. First, the response of the cavity-plasma systems displays hysteresis effects as a system parameter is varied. That is, the state of the system can be either one of two values, for the same set of system parameters, depending on the history of the state of the system. Another closely related manifestation of nonlinearities is that the response of the cavity-plasma system is not proportional to the level of the input signal. In Chapter VII it is shown eXperimentally that large increases in the incident power do not necessarily produce correSponding increases in the plasma density. Finally, the third manifestation of nonlinear action is the appearance of new para- metrically and harmonically excited frequencies when the cavity- plasma system is driven at one input frequency. Chapter VIII summarizes the work presented in this the sis and points out possible areas of fruitful future research. Pa “‘1 l a!» a... n a... at... ’1 1 v ‘s .Q. d CHAPTER II REVIEW OF MICROWAVE-PLASMA. INTERACTICN 2.1. Introduction This chapter will emphasize the background which will be needed to understand the problem of resonantly sustained plasmas in microwave cavities. The mathematical models to be used to describe the plasma and the electromagnetic fields are presented. Also, the pertinent literature discussing low power plasma resonance, resonantly sustained discharges, and the effects of plasmas in cavities is reviewed. Section 2. 2 defines the notation to be used. The warm electron and warm ion plasma equations are stated and, along with Maxwell's equations for the electromagnetic fields, are linearized. The cold ion plasma and the cold electron plasma models are shown to be Special cases of this warm plasma theory. Two types of waves can propagate in the warm plasma media; the transverse electromagnetic waves and the longitudinal electron plasma and ion acoustic waves. The diSpersion characteristics [\l pa .1 ‘1' gm . p ‘a ‘1! (7') for these waves in an infinite media are presented. Also, the criteria for the coupling of these two waves are stated. Next, the history of resonances in bounded plasmas is pre- sented. It was noted by Tonks6 that when an electric field is applied perpendicular to the axis of a cylindrical plasma, resonances are observed. Latter Dattnerll experimentally determined that the resonances were dipolar in nature and that they were due to oscilla- tion of charge. Parker, Nickel and Gould16 obtained excellent agreement between experiment and theory by applying the non- uniform, warm plasma equations to the prOper bounded geometry and using an "adjusted" value of electron temperature. Later, Baldwin21 solved the Tonks-Dattner resonance problem using a more sophisticated kinetic theory formulation. Although this method is more theoretically exact than Parker, Nickel and Gould's, it had quite poor agreement with experiment in a recent testzs. As in the theory of Parker, Nickel and Gould, Baldwin's results could be adjusted to agree with experiments if the electron temperature is chosen for best fit. In addition to these resonances at high frequencies, low frequency modes on plasma cylinders have been observed35. These modes are related to the ion acoustic wave. The third section presents a discussion of radio frequency sustained plasmas. Depending on the background pressure and driving frequency, these radio frequency discharges are multi- pacting plasmas, resonantly sustained plasmas or a diffusion \v Us :01 controlled plasma. The resonantly sustained plasma is characterized by its sharp luminous boundaries and the fact that it can be main- tained by a relatively low power driving source. Taillet4l, through a series of experiments and employing sheathed, cold plasma capacitor model, developed a qualitative explanation for these resonantly sustained plasmas. From an equivalent circuit point of view, energy exchange between the capactive sheath and inductive. overdense plasma creates a resonance condition. Taillet developed a stability criterion which determined when a resonantly sustained plasma can be maintained. Finally, Taillet noted that for a resonantly sustained plasma, a large increase in incident power generally causes only a small increase in plasma density. This last result is a funda- mental problem when trying to generate a dense, resonantly sus- tained plasma. The last section of this chapter discusses previous work describing plasmas in cavities. The eigenvalue problem, which determines the resonant frequency as a function of the plasma density, has received a great deal of attention to date. The plasma has been modeled using both cold and warm plasma theory. New re sonancea which do not exist in an empty cavity, have been found. The fact that the resonant frequency of the plasma-cavity system shifts as the plasma density varies makes it difficult to excite a high density, resonantly sustained plasma using a fixed frequency source. Recently a variable frequency retuning lO technique has been developed to overcome the detuning effect. 2. 2 . Linearized Warm Plasma Model One quite general mathematical description of a plasma is to use the Boltzman equation1 for each Species of particles present (electron, ions and neutrals). If these equations are used to describe the plasma, and hiaxwell's equations to describe the associated electromagnetic fields, a coupled set of equations is found which, in principle, could be solved to give the response of the plasma and the electromagnetic fields. In practice, unfor- tunately, these equations usually are not tractable and do not represent a realistic approach to many plasma problems. An alternative model is the warm plasma approximation for the plasma and Maxwell's equations for the electromagnetic fields. The warm plasma equations1 represent the first two velocity moments of the Boltzman equation. The infinite set of coupled moment equations is terminated after the first two by assuming the off diagonal terms in the pressure tensor are zero. These equations can be written (rationalized MKS units), BNe —> at + V- (Neve) = 0 2.1 d —> —> -» -> kTe Inca—Eve = -e( +VexB)- N VNe 2.2 e aNi —. ll d -» -> kTi mid-{vi = e(E+Vex B)- Ni VNi 2.4 where Ne = electron number density Ni = ion number density Ve = electron velocity Vi = ion velocity me = electron mass m1 = ion mass Te 1' electron temperature Ti '-' ion temperature E : electric field B = magnetic field e = electronic charge k = Boltzman's constant. Equations 2. 1 to 2.4 express the conservation of mass for elec- trons, conservation of momentum for electrons, conservation of mass for ions and conservation of momentum for ions reapectively. The plasma has been assumed to be lossless. Along with Max- we 11' s e quations 2, V-D = e(N.-N) 2.5 l e .. a” VXE: -—alt3 2.6 12 vx'fi e(N.V. - N 17 ) + —— 2.7 1 1 e e V-B 2.8 II C the warm plasma equations represent a reasonable, tractable set of equations for many problems of interest. Further assumptions will be used to simplify the above equations. Letting the media be homogeneous and expressing the variables as the sum of a dc term and a perturbed ac term, the variable 3 be come, N = N + n ert 2.9 e o e N. = N + n ert 2.10 1 o e v8 = 36cm - no dc drift velocity 2.11 —> -> ' 1; Vi : vier - no dc drift velocity 2.12 -> -> joot . . . E = E e - no static electric field 2. 13 -> -> jmt . . . B — Be - no static magnet1c field 2.14 Because the region is homogeneous, the dc number density for the electrons arrl ions is the same. Also, assume a free Space medium with free electrons and ions so that, Beef 2.15 O B = [lo-1:1. 2.16 When equations 2. 9-16 are substituted into equations 2.1-8, the resulting equations can be separated into a dc term, a fundamental . . ° t . . . ac term which varies as er , and a harmonic ac term which varies 13 ' t ' t . as eJPw . The @1200 terms are the nonlinear terms and are assumed to be second order small with respect to the first order terms (clot). The dc terms describe the static field and are not of interest here. The fundamental part of equations 2.1-2. 8(emt understood) 1 are , jwn +NOV°3 = 0 217 y kT 'm; =-eE--e eVn 218 JO) N e o jwn.+NV-:. = 0 219 1 o —> VikTi = _ 2. 2 jmmivi eE N Vni O o v E e 3— (n — n ) 2 21 E e 0 V xE = -j(i)p.o—H 2.22 .23 <1 N m u H. 8 m 0 m "U I N v - ii = o 2.24 The Ye and Yi have been introduced in equations 2.18 and 2. 20, 3 respectively, by assuming adiabatic change in the ac pressure . Also, the following notation has been introduced, 2 2 e =1-_i_-we 2.25 2 2 14 2 Noe2 me = 6 m - angular electron plasma 2.26 o e frequency 2 Noe2 a). = - angular ion plasma frequency. 2.27 1 Eomi Equations 2. 17-20 are the linearized warm plasma equa- tions for electrons and ions. If the electrons and ions are assumed to be "cold, " Ti: Te : 0, the warm plasma model is reduced to the "cold plasma model. " This model is equivalent to considering the plasma to be a media which has the relative dielectric constant Ep’ as in equation 2. 25. In this media, the fields are the usual electromagnetic fieldsl. That is, in an infinite media the only plane wave solution has a wave vector I: which is perpendicular to both E and H. By way of contrast, when the warm plasma model is used, it is known that, in addition to the electromagnetic waves, electron plasma and ion acoustic waves can propagatel. The electromagnetic fields are the same as in the cold plasma model. Because Ti << Te for the experimental plasmas of this the sis the ions are assumed to be cold (Ti 2 0). Then by taking the divergence of equation 2. 20, and substituting equations 2. l9 and 2. 21 into the result gives, n. = --——-—- 2.28 15 where 2.29 Equation 2. 28 shows that the motion of the ions is related to the motion of the electrons for warm electron, cold ion plasma theory. In order to gain insight into the nature of the waves which can exist in an infinite plasma media, the electric field is separated . . . . . . 4 into 1ts linearly independent solenOIdal and irrotational parts , E = E +15 2 3O sol 1rr where by definition v x E. s 0 2.31 1rr V - E E o 2.32 sol The H field is purely solenoidal (see equation 2. 24), '13 = 1'1 . 2.33 sol Taking the curl of equation 2. 22, substituting equation 2. 23, and separating the solenoidal terms and the irrotational components, gives for the solenoidal electric field, 32.. IE v + Ik e o 2. 34 sol e sol where _, l k = 91 E 2 R 2.35 e c p ’12 - a unit vector in the direction of propagation. .1. 2 c : l/(poeo) - speed of light 2.36 Similarly, : O, 2.37 Equation 2. 34 and 2. 37 show that the solenoidal part fields propa- gate as if they were in a media with relative dielectric constant 6 . The waves are identified as the electromagnetic fields and P the E and H vectors are transverse to the wave vector Re, the same as in cold plasma theoryl. By taking the divergence of equation 2. 18, and substituting equations 2.17, 2. 21 and 2. 28 gives, v23: . 1: I21? 2 o 2.38 Irr p Irr where 1 -> 6 EA kp = 5)- (ER) k 2.39 e + l: - a unit vector in the direction of prOpagation 1 v RT 2 U = ( _e___e_) - electron thermal 2.40 e m . e veloc1ty The waves represented by equation 2. 38 are longitudinal, that is I1) is parallel to Eirr' This longitudinal electric field is the field . . . . 1 as soc1ated With the electron plasma and 1on acoustic waves . The ion acoustic wave does not propagate for (D > (bi, because Ti: 01. .5 Also, since V x E. = 0, 1rr E. = - :74} o 2.41 1rr p Using equation 2. 41 in 2. 38 gives, 2 2 k -_- o 2.42 V (WP) + p (WP) 17 The vector identity4 V x V x_A = V(V - A) - V2; applied to the first term of 2. 42 gives 2 2 V(V ¢p+ kp up) = 0 or VZLIJ +k ZLIJ = constant 2.43 P P P The constant in equation 2.43 is related to the inhomogeneous solution for LPP. Because only the gradient of dip is of interest (from equation 2.41), this constant is not important and can be set equal to zero. Thus Eirr can be derived from a scalar poten- tial, and this scalar potential satisfies the wave equation 2. 43. The solenoidal components of the field will also be derived from scalar potentials in Chapter 3. For an infinite media, the solenoidal and irrotational fields have the dispersion characteristics (plots of equations 2. 35 and 2. 39) shown in Figure 2.1. The electromagnetic wave is cut off for the overdense region ((1) < we), and asymptotically approaches the light line for high frequencies. In the electron domain ((1) Nwe), the irrotational field is an electron plasma wave. This wave is also cut off for a) < me, but it asymptotically approaches the electron thermal line for high frequencies. In the ion domain (a) ~mi) the irrotational field is an ion acoustic wave. This wave does not propagate for (i) > mi, and asymptotically approaches the plasma sound velocity, 18 A .Aofimom on “05 manna now 300 50.3036 Esme? .oficflcfi an E mo>m3 Add oflmdoom so“ new Anmm: mEmma cofioofio .szv ofiocmmEofiooHo MOM Empmflp :owmpoamwQ E. H: x \ 11 \x no .oan \ \ \fl/ meadow mesmflm . \ D .034 38.838 28.50on \ O .225 Emfi //;&\ 4 .N 23E Hcomm n3 19 .{ W r-l Nit- U = (”e—e) 2.44 for low frequencies. The above discussion of the waves in a plasma assumed an infinite, homogeneous, isotropic (no static magnetic field) media. In such a media, the solenoidal and irrotational parts of the fields are uncoupled. However, it has been shown by Field5 that for warm plasma theory, the two components of the field are coupled when: a) the plasma is inhomogeneous, b) a static magnetic field is present, c) plasma density discontinuities exist. Condition (c) is really a special case of condition (a). All inve sti- gation in this thesis will be carried out without a magnetic field. Thus, only plasma inhomogeneity or boundaries will lead to coupling between the solenoidal and irrotational fields, within the context of warm pla sma theory. 2. 3. Resonances in Bounded Plasmas In the late 1920's and early 1930's, several investigators studied interaction of radio waves with a cylindrical, mercury vapor plasma. Tonks6 reported observing resonances, and attempted to explain them by using the cold plasma model, with an equivalent permittivity of 6 : €o(1--—%—), 2.45 20 >1: The resonances occurred when oo~ me. However, he could not successfully explain other resonances he observed. Later, this problem again be came important when resonances were observed in radar waves reflected from imized meteor trails. Many workers attempted to explain these resonances using cold 7’ 8’ 9’ 10. Experimentally, Dattner11 determined plasma theory that these resonances were dipolar and that they were due to oscil- . 2 lations of charge. Gould1 showed that the use of the Bohm- 13 . . . . . Gross dispersmn rel ation for a warm plasma gave qual1tat1ve . . 14, 15 agreement With experimental results. Others used a non- uniform density profile with warm plasma theory to improve the qualitative agreement. Employing a qua si-static approximation with the nonuniform, 16’ 17 obtained excellent warm plasma model, Parker, Nickel and Gould agreement between experiment and theory for the Tonks-Dattner reso- nances. According to their numerical solutions, the Tonks -Dattner resonances are caused by radially standing electron plasma waves. These electron plasma waves are assumed to be perfectly reflected by the dielectric tube which surrounds the plasma, and by the overdense (a) < we) center portion of the plasma cylinder; Be cause the plasma density is not uniform in experimental plasmas, the density (hence electron plasma frequency) used in equation 2.45 is an average plasma density. This average value is a function of the plasma density profile. 21 that is, the electron plasma waves exist in the outer, underdense ((1) > we) portion of the nonuniform plasma density profile. In the overdense center portion, the electron plasma waves are cut off (see Figure 2.1). There are standing wave resonances only for certain characteristic values of m and me (see footnote on page 20). The success of Parker, Nickel and Gould's warm, nonuniform plasma theory was judged on its ability to predict the spacing between observed scattering resonances. However, it was necessary to use an “adjusted" value of electron temperature, which was different than the experimentally measured value. Furthermore, this warm plasma theory is inconsistent with the fact that Landau damping should be very strong in the low density sheath region. Landau damping, which has been eXperimentally observed for electron plasma waves , is not, of course, predicted by warm plasma theory, but only from a careful solution to Boltzman's equation]. To overcome the latter weakness of the warm plasma theory, Leavens20 and Baldwin21 have solved the Tonks-Dattner resonance problem using the conductivity kernal method of Drummond, Gerwin and SpringerZI. This method involves eXpressing the current (electron plasma wave) in the plasma as a volume integral of a kernal times the electric fields. The kernel is found from a solution of the Boltzman equation. Using a numerical solution to this kinetic theory formulation, 2 Leavens O solved the Tonks-Dattner resonance problem without 22 having to assume a boundary condition for the electron plasma wave at the dielectric tube-plasma boundary. His solution included Landau damping as well as a description of the coupling of the exciting electromagnetic fields to the electron plasma waves. Leavens' theory predicted a resonance Spectrum similar to the eXperimental results; but, because he did not use a realistic density profile, he could not make a quantitative comparison between experiment and theory. Leavens also concluded that there was no difference of kind between the first and higher order resonances, which agrees with Figure 9 of Parker, Nickel and Gouldlé. Unfor- tunately, Leavens' numerical results do not give any understand- ing of the physics involved in the kinetic theory solution. Baldwin21 has recently solved the kinetic theory formulation analytically. Through his solution, the physics of the Tonks-Dattner resonances has been illuminated. From kinetic theory, Baldwin found (also see Baldwin and Ignat23) that the incident electromagnetic fields couple to the plasma waves at the point on the density profile where a) = me. This coupling excites a plasma wave which propagates toward the dielectric tube-plasma boundary, since all propagation toward the overdense center of the plasma is cut off. As the electron plasma wave propagates toward the dielectric tube -plasma boundary, the plasma density decreases. In the low density region, a) >> (be. eie< sec ele the ’ 9 ’3' '1 23 the phase velocity of the electron plasma wave approaches the electron thermal velocity. This condition, V 1 Ue' is the ph necessary condition for Landau damping to be strong3, and the electron plasma wave is absorbed by Landau damping. However, the energy of the wave is not thermalized due to collisions 4, but rather creates a ”bump" on the background Maxwellian distribution for electrons. The electrons in this "bump" stream toward the dielectric tube-plasma boundary at approximately the electron thermal velocity. The streaming electrons in the "bump" are then reflected by the negative potential of the dielectric tube (or the sheath). After reflection, the electrons in the "bMp” stream back toward the center of the plasma. When this returning stream enters the region where its velocity (approximately the electron thermal velocity) is near the local phase velocity of the electron plasma wave, the electron plasma wave is re-excited by absorbing the energy of the streaming electrons. This re-excitation process is similar to a beam plasma amplifierl. This intricate reflection mechanism allows standing electron plasma waves to exist for characteristic values of frequency and results in the observed resonances in the scattered fields. Although Baldwin's theory gives an excellent physical des- cription of the Tonks-Dattner resonances, it had quite poor agree- . . . 25 ment With eXperiment 1n a recent test by Ignat . No reasons for 9;" c “-5- OZ 6): 3 L, t €12 24 this poor agreement have been forwarded, but Ignat noted that the discrepency could be erased by assuming an unrealistic value for the electron temperature. Also, the question of the cold plasma resonances was clouded by Baldwin's results, because his theory indicates that different phenomena cause the lowest resonance than cause the higher order resonances. . It is only these higher order resonances which are described by Baldwin's theoryZI' 23' 25. Research on the Tonks-Dattner resonances continues to be of interest26' 27' 28' 29. Although nineteen resonances have been experimentally observed30, coupling to the high order resonances is poor. The coupling problem has been confused with the damping experienced by the re sonanceSZB' 30. Also, many types of non- linearities have been observed for moderate incident power levels. A problem closely related to the transverse Tonks-Dattner resonances on a plasma cylinder is axial propagation along a plasma cylinder located inside a metal waveguide31 or in free space . This phenomenon was explained by applying warm electron plasma theory and Maxwell's equations to an infinitely long, plasma cylinder In addition to these electron plasma wave phenomena, workers have also reported observation of ion acoustic waves on plasma cylinders. In 1961, Crawford34 observed low frequency ((1) ~ (oi) fluctuations on a plasma cylinder, and attributed them to radially . . . . 35 . standing 1on acoustic waves. Little and Jones observed axial propagating waves on a plasma cylinder in a similar frequency 25 range, and this effect was also related to ion acoustic waves. In 1966 Anderson and WeiSSglas36 made a theoretical study of the propagating modes of a warm electron, cold ion plasma cylinder located inside a metal wave guide. For the rotationally symmetric mode, they predicted an ion surface wave mode and a series of ion standing wave modes. Their results qualitatively explained the previously observed propagating modes35 and point out the value of considering the boundedness of realistic plasma problems. Later Klevans and Mitchell37 performed a similar theoretical study of a plasma column in free Space. 2.4. Re sonantly Sustained Discharges While Tonks6 was investigating plasma resonances, in 1931 Wood38 noted that he could maintain a plasma with only radio fre- quency power. This plasma,which he called a "plasmoid, " concen- trated itself into clearly defined shapes, rather than filling the entire vacuum discharge vessel. These plasma bodies were surrounded by a dark sheath. Once these plasmoids were excited, they could be maintained by low exciting power, although continuous and discon- tinuous changes were observed. Wood suggested that these effects were related to the plasma resonances reported by Tonks6. 9’ 40 and Taillet4l, 39 Wood's observations are now qualitatively understood. Hatch 3 Through a series of experiments by Hatch experimentally characterized three types of radio frequency plasmas, depending on the background pressure and driving frequency. Using 3T 26 a 15 Mhz excitation source, collisions are the dominant mechanism for a pressure of approximately .1 Torr. This type of plasma is called a diffusion controlled discharge. At lower pressures, approxi- mately 10.3 Torr, the rf plasma has the characteristics of a plasmoid, and is called a resonantly sustained discharge. For very low pressures, approximately 2 x 10.5 Torr, the plasma is a multipacting discharge. Using Langmuir probes in a resonantly sustained discharge, Hatch40 observed that the electric field in the plasma is opposite in phase to the electric field in the sheath. This phase reversal does not occur in a diffusion controlled discharge, and Hatch suggested that it is the phase reversal which is responsible for the observed sharp boundary in a resonantly sustained plasma. The background pressure in a resonantly sustained plasma is low enough that the mean-free- path for ion-neutral collisions is larger than the dimensions of the vacuum discharge vessel. Thus the positive ions are lost to the walls by direct fall down the potential well. In Taillet's41 experiment, an electron beam was used to probe a resonantly sustained plasma created between two parallel metal plates inside a vacuum vessel. The plasma is excited by applying a 15 to 50 Mhz signal to the plates. Taillet confirmed the phase reversal observed by Hatch40, and he also observed that the electric field in the plasma is much larger than the field without a plasma present. This demonstrates that the discharge is always in a resonant state: thus the title, resonantly sustained plasma. 27 Taillet developed a theory to qualitatively eXplain these observations based on a sheathed, uniform, cold plasma capacitor, as shown in Figure 2. 2. For this plasma capacitor4z’ 43 it is assumed that the dimension L is much less than the free space wave length for the driving frequency. The width of the Sheath, as shown in Figure 2. 2, is S/Z. Suppose for the moment, that a uniform plasma, with plasma frequency we and an effective elastic collision frequency v, exists in the plasma capacitor. This plasma will absorb power from the dr1v1ng source, due to elast1c collISIons, accordIng to the equation 9 2 2 u e v (—) P = w . 2.46 abs 2 2 2 (1) 2m 2 em -L 2) L) 2 2 (1) 0) 0) This power absorbed is plotted as a function of plasma density in Figure 2. 3. A family of curves is presented correSponding to different values of the driving voltage. It is easily seen that the power absorbed by the plasma is large in the neighborhood of the resonant density, Nres' This value of density is determined from equation 2.46 to be, (1) L Eome Nres = S 2 2'47 e or 1 L a __. - 2.4 we (1) ( S ) . 8 28 {\J W) 1 S / 2 1 I High Frequency Plasma . Generator / Sheath Figure 2. 2. Parallel plate plasma capacitor. The driving voltage, V(t), has a frequency (1). The plasma has width L—S, and each sheath has a width S/2. The plasma frequency is we. 29 A Power) relat1ve — —— Power Loss units | Power Absorbed I V >V >V >V l a a b c d ' I l I I 1 I l l l l l I l l I P (abs)a _,__-_________- -y / (Pabs)b --- ................. I l _, I M‘ l 1 > N N N N res b a Figure 2. 3. Power absorbed and power loss by the plasma versus density. The four power absorbed curves correspond to different values for the driving voltage, V > V > a b Vc > Vd. 30 When the plasma capacitor is operating near the resonant density Nres’ power absorbed by the plasma is a maximum. Additionally, this plasma capacitor model predicts that the electric field in the plasma is opposite in phase to the field in the sheath, similar to experiments. The resonance is caused (from an equivalent circuit point of view) by the capacitive nature of the sheath and the inductive nature of the overdense plasma. Also shown in Figure 2. 3 is a power loss line. This repre- sents the power lost in the plasma due to inelastic ionization and excitation collisions and wall losses (see equation 2 of Taillet4l). For a typical exciting voltage level b, there are two points where the power absorbed by the plasma is equal to the power lost by the plasma. If the plasma in the capacitor is being resonantly sustained by the driving voltage, the power absorbed is, in fact, equal to the power lost by the plasma. Thus in steady state operation for there are two equilibrium points, M and M shown in voltage V 1 2 b! Figure 2. 3. However, only one of these two points, M2, is a stable equilibrium; the other, M , is unstable. This is easily seen by l considering what happens to the plasma if it is slightly perturbed from an equilibrium condition. At point M , any reduction in the 2 plasma density has the tendency to increase the power absorbed by the plasma. This in turn causes an increase in ionization, hence increases the plasma density. Likewise, an increase in the plasma 31 density at point M has the tendency to reduce the power absorbed 2 by the plasma, which in turn causes a reduction in the plasma den- sity. Thus, equilibrium point M2 is stable. Using a similar argument, the other equilibrium point, M1, is unstable. Starting from M an increase (or decrease) in the 1. plasma density causes an increase (or decrease) in power absorbed. Thus the plasma jumps to the stable operating point M2 (or it abruptly extinguishes itself). Thus the plasma can never be resonantly sustained for densities less than Nres If the driving voltage is greatly increased from level b to level a, it is seen from Figure 2. 3 that the density of the resonantly sustained plasma increases only a small amount, from N to Na' b As shown in Figure 2. 3, this is caused by the fact that, despite the large increase in driving voltage, there is only the small increase in power absorbed. Alternatively, if the driving voltage is reduced to level c, the power loss line is tangent to the power absorbed curve at the density Nres' This condition is unstable, Since any slight decrease in the plasma density reduces the power absorbed to a level less than the power lost by the plasma. Thus the plasma will abruptly disappear. For a driving voltage of level (1, there is no equilibrium point where power absorbed equals power lost. Thus, it is impossible to resonantly sustain a plasma at this value of driving voltage. 32 The fact that plasma density does not increase proportionally to increases in driving voltage is a fundamental problem encountered in resonantly sustained plasma. Taillet's eXperimental results indi- cate that we mu) for the resonantly sustained plasmas generated in the capacitor geometry. As seen from Figure 2. 3, increases in driving voltage shift the stable operating point away from the region where the plasma is efficiently absorbing the available power. Thus, in order to obtain a dense, resonantly sustained plasma some retuning method must be used to enable the plasma to absorb a significant amount of the available power. One further remark is in order regarding Figure 2. 3. It is noted that the resonantly sustained plasma capacitor is always in a near resonant condition. This results in large values of electric field in the plasma, and it is these large fields which generate non- linearities in the plasma (e. g. , ionization). Although Taillet's work was performed for incident frequencies in the Mhz range, more recent workers have studied the Ghz range where warm plasma (e. g., Tonks -Dattner) resonances, as well as cold plasma resonances, are important. Hsuan, Ajmera and Lonngren44 observed that for moderate levels in incident power, the Tonks-Dattner resonances become "distorted" with hysteresis effects as the dc density is varied. They used anharmonic oscillator theory to explain these results. Later, it was shown that ionization is the dominant nonlinear 29, 43 mechanism , and nonlinear sheath effects are also possible. 33 At these Ghz incident frequencies, it is possible to resonantly sus- tain a plasma in a temperature resonance, as well as a cold plasma 45 resonance . In addition to hysteresis effects in absorbed power, other nonlinear phenomena occur in Tonks-Dattner resonances at moderate 43, 46 incident power levels. Harmonic generation , sub-harmonic . 47 . . . . . generation , and parametric excitation of low frequency osc1llations have been observedzg’ 43’ 48' 49' 50' 51. Several physical mechanisms 50’ 52’ 53’ 54 are used to explain these parametric oscillations. It will be shown in Section 4. 4 that coupling to the low frequency eigen- modes of a plasma cylinder is a possible explanation for some of these low frequency oscillations. 2. 5. Plasmas in Microwave Cavities The placing of a plasma inside a microwave cavity has the potential for efficiently creating, maintaining and radio frequency confining a radio frequency discharge. However, associated with putting a plasma in a cavity, there are several poorly understood phenomena. The shift in resonant frequencies, the appearance of new resonances, the existence of radio frequency instabilities,“ and the interaction of the plasma with the microwave circuit all must be understood to efficiently generate a dense, hot plasma in a cavity. The answer to these questions has application for both heating and confining a plasma when attempting to sustain a controlled theromonuclear re action70. Also, this type of plasma is becoming 34 increasingly important for processing chemicals in electrical dis- 0 . . . . charges7 and in other experimental Situations where a dense, stable plasma is requiredn’ 72. Although Brown and Rose55 discussed the effects of plasmas in microwave cavities in the early 1950's, it was not until 1961 that Agdur and Enander56 theoretically solved the boundary value problem of a cold plasma cylinder located concentrically in a cylin- drical cavity. These results were later extended to include plasma losse857. Agdur and Enander characterized three types of resonances for this cylindrical cavity: a. TM with n=0 and/or p20, TE with n=0; m> o. nmp nmp These modes are pure TM or TE modes and the resonances are not greatly effected by the presence of the plasmas. * * b. TE and TM with n, m, p ,l o nmp nmp These modes are not pure TE or TM, but rather hybrid mode 84. The * is used as a reminder that these modes are greatly changed by the presence of the plasma and that the TE or TM designation is not strictly valid with a plasma in the cavity. They are labeled according to the TE or TM mode into which they degenerate when the plasma density goes to zero. * c. TM 11 > 0 nop -- These hybrid modes do not exist in the empty cavity. They have the characteristic that the resonant frequency goes to zero as the plasma density becomes small. For the n20 mode, the TE and TM modes are uncoupled and the mode is a pure TM mode, H H - TMOOI' The TE001 mode does not ex1st. 9“ '1‘ 17.3 n) 0 Fr. (1" 35 Shohet and Hatch58 numerically solved for the resonant fre- quencies of a plasma filled cavity. A step-wise nonuniform approxi- mation was used for the cold plasma density profile. Also, Kent59' 60’ 61 has studied a plasma filled cavity with an axial, static magnetic field at high frequencies (~9 Ghz). However, he has not achieved a dense resonantly sustained plasma in his cavities. The properties of a warm plasma in a cavity have also been investigated. For a Sphere of plasma inside of a spherical cavity, 64 . . Boyen, et.a1. classified the resonances into three groups: a. Empty cavity modes, which are perturbed by the presence of the plasma . b. Cold plasma modes, which do not exist in the empty cavity and are described by cold plasma theory.» c. Warm plasma modes, which also do not exist in the empty cavity, but can only be described by warm plasma theory. A WKB approximation was used to theoretically describe these warm plasma modes. Experimentally it was found that the warm plasma resonances could be excited only if they occur in the neighborhood of an empty cavity mode (a) or a cold plasma mode (b). Leprince65 has experimentally studied a cylindrical plasma located inside a fixed size cylindrical cavity. He observed three types of resonances (perturbed electromagnetic resonances, cold plasma resonances and warm plasma resonances) and displays eXperimental curves of resonant frequency versus density for these resonances. Additionally, he extensively studied some observed 36 parametric instabilities in the plasma. He noted that it was possible to sustain the plasma inside the cavity using the radio frequency driving source. Using a cold plasma model in cylindrical cavity, Vlachos and 62 . 63 . Hsuan and Simultaneously Barkley and Sexton reported usmg frequency Shift data from two suitably chosen cavity resonances to measure the radial density profile on a dc plasma column. They used two cavities, one for each mode, in their experiments. Although this increase in the cavity resonant frequency is . . 62, 63 . . . useful for plasma diagnostics , this shift detunes the cav1ty when a fixed frequency source is used. Arguments similar to those used to explain Figure 2. 3 Show that one can not generate a dense resonantly sustained plasma merely by increasing the incident micro- 66 . . wave power . Moreover, the use of tuning stubs to overcome this . . 59, 66 problem have been shown to be very ineffective . 66 . Halverson and Hatch have used a variable frequency, constant power output source to retune a cavity containing a resonantly sustained plasma. This allows them to overcome the detuning effects of the resonantly sustained plasma. Although this retuning technique has . . . . 67 . . . been verified for a spherical cav1ty , it requires a generating source which has a level output over a frequency shift range of about Af/f = 10%. This large band width requirement makes it desirable to develop alter- nate retuning techniques which can use a constant frequency source. At lower frequencies, Hatch and Heuckroth69 were able to retune a 37 resonantly sustained discharge by varying an external circuit element. Another method used to overcome these detuning effects is . 69 . . . to use an overmoded cav1ty . For such a cav1ty, the shift in resonant frequency caused by the plasma will always bring another cavity mode into a resonant condition. However, it is difficult to couple energy into the plasma in these overmoded cavities when me>>oo, that is when the plasma is overdense SIS! ! f \ 6 CHAPTER III ROTATIONALLY SYMMETRIC PROPAGATING MODES ON A PLASMA CYLINDER This chapter begins with a discussion of cylindrical dielectric rod waveguides. Then using the warm electron, cold ion plasma theory along with waveguide theory, the characteristic equation is derived for the rotationally symmetric modes on a cylindrical plasma wave guide. The qualitative features of this equation are discussed and a typical numerical solution is given. Both high frequency ((1) ~me) and low frequency ((1) ~wi) modes are presented. 3.1. Dielectric Rod Waveguides Before developing the theory of a cylindrical plasma wave- guide, it is instructive to briefly discuss cylindrical dielectric rod . 4, 73, 74 waveguides . An infinitely long dielectric rod of radius a is assumed to be surrounded by free Space. The dielectric constant of the rod is greater than the dielectric constant of the surrounding media. For the more familiar type of waveguides, that is metal pipes of various cross sections, the transverse components of the 38 39 electromagnetic fields are reflected by the conducting walls. The only possible direction of energy flow is in the axial direction. Given the proper excitation frequency, the dielectric rod also meets this requirement of purely axial energy flow. However, the trans- verse fields are not reflected by metal walls, but rather the fields are radially evanescent in the exterior region. Thus the fields are very small everywhere except inside the dielectric rod and in the exterior region immediately surrounding the rod. A heuristic visualization of the dielectric rod waveguide is obtained by considering the wave to undergo total internal reflection from the optically less dense surrounding region”. From this point of view, the wave propagates in the axial direction by bouncing back and forth inside the dielectric rod. This argument, of course, ignores the evanescent fields which exist exterior to the dielectric rod. The complete solution to the cylindrical dielectric rod wave- guide73 shows that, in general, the waves are not pure TE or TM, but rather they are hybrid in nature. That is, both the TE and TM modes are needed to satisfy the boundary conditions. For the special case of rotationally symmetric modes, the TE and TM modes un- couple and pure TE or TM modes exist by themselves. Furthermore, for these rotationally symmetric modes on the dielectric rod, there is a cut off frequency. That is, for frequencies less than the cut off 4O frequency, purely axial energy flow does not occur. Instead, energy flows in the radial as well as the axial direction. In this situation, the dielectric rod is acting as a dielectric antenna instead of a wave- guide. For nonrotationally symmetric modes, however, there is no cut off frequency and the dielectric rod is a waveguide for all fre- quencies. For the waveguiding modes on a dielectric rod, the axial phase velocity is less than Speed of light in the exterior free space media; but it is greater than the speed of light in the dielectric73 Since the phase velocity is always less than the Speed of light in free space, dielectric rod waveguides are called slow wave structures. For low frequencies the phase velocity is near the Speed of light in the exterior region, and the external fields are loosely bound (weakly radially evanescent) to the rod; and for high frequencies the phase velocity is near the speed of light in the dielectric and the external fields are tightly bound (strongly radially evanescent) to the dielectric rod. As was discussed in Chapter II, section 2, a cold plasma can be considered to be a Special type of dielectric. The relative dielectric constant of this cold plasma, as given by equation 2. 25, is di; 50' O 'x' ”'7 41 The use of waveguide techniques applied to a cold plasma cylinder will give fields which are radially evanescent and allow axial energy flow. That is, a cold plasma cylinder can be a waveguide. Using a quasistatic approximation (assume c —> 00) for a cold plasma cylin- der , Trivelpiece31 has solved for the rotationally symmetric and dipole modes on a plasma waveguide. Trivelpie ce's theory predicts for the rotationally symmetric modes that axial propagation occurs for all frequencies such that ep<-l. 3.1 Thus the rotationally symmetric waveguide mode exists only on overdense (we > (1)) plasma cylinders. In the quasistatic approxi- mation, the dipole mode has a cut off frequency, and propagation cannot occur at frequencies less than this cut off value. Thus even with the approximate theory of Trivelpiece, it is seen that a plasma cylinder has waveguiding modes. The remainder of this chapter will theoretically investigate the rotationally symmetric modes on a warm electron, cold ion plasma cylinder. New modes are predicted by this warm plasma theory which are not predicted by cold plasma theory. In Chapter IV these results are extended for a warm plasma cylinder placed concentrically inside a cylindrical metal wave guide. 3. 2. Model and Characteristic Equations Consider the infinitely long plasma cylinder shown in Figure 3. l. The cylinder has a radius a, and is surrounded by 42 Figure 3. 1. An infinitely long plasma cylinder surrounded by free space. The radius of the cylinder is a. The plasma is assumed to be lossless, uniform and isotrOpic. 43 free space. The plasma is assumed to be isotropic (no static magnetic field), uniform and lossless. As outlined in section 2. 2, the plasma is modeled with the warm electron, cold ion equations, 2. 17-24. Maxwell's equations are used both inside the plasma and in the exterior, free Space region. This boundary value problem will be solved using waveguide theory for axial propagating wave 54. Outside of the plasma, the fields are purely electromagnetic, solenoidal fields. Maxwell's equations for this region are (ejwt understood) V . if" = o 3 2 V1550 = -jmuoH° 3.3 VXHO = waOEO 3.4 V ' 3:10 = 0 3.5 The superscripto will be used to denote the exterior, free Space region. These fields can be derived from Hertz potentials TI; and .1180. The relations are4, +0 . +0 *0 E - -J(1)|J-0er&1 +VxVx1'Ie 3.6 H0 = VxVxIZ‘O +jc116 Vic-.110. 3.7 o e O The Hertz potentials fih and II: correspond to the TE and TM fields, reapectively. For cylindrical geometry, they can be expressed as -> o o -jkzz IIh = 21ph(r,¢)e 3.8 " O O -ijz IIe :: z the (r, ¢)e 3.9 whe re As 44 2 o 2 o 5' : Vt 1th + 41h o 3.10 2 o 2 o 6' = . Vt we + 418 o 3 11 2 2 2 . 6' : (1) p060 - kz = -5 ; radial wave number 3.12 V: - transverse Laplacian (.10, 60 - permeability and permittivity of free space A . . . . . z - unit vector in the ax1al direction. stated in Chapter II, inside of the plasma the electric field is expressed as the sum of a solenoidial part and an irrotational part, while the magnetic field is purely solenoidal. The solenoidal fields are described by Hertz potentials, ->l . -> 1 -> 1 E801 — -J(1)P-0Vx Ilh +VxVxIIe 3.13 "151' — vxvx‘fi' +111 6 vx‘fil 314 sol _ h J Eo p e ° ' Again, the Hertz potentials are eXpressed as —jk z "’ l A 1 Z IIh .. zwh (r,¢)e . 3.15 -jk z “*1 _ A 1 z 116 _ zqae (r,¢)e 3.16 -jkz where (e understood) 2 1 ,2 1 _ Vt 41h +71 41h _ O 3.17 2 1 ,2 l Vt LPe +71 lie — 0 3.18 2 2 2 2 ' = - = - . . 3 71 (1) 1106061) kz 71 , radial wave number .19 45 It has been shown in section 2. 2 that the irrotational field in the plasma can be derived from a scalar plasma potential, 41 . That is, in the plasma, let 1 -jkzz l‘1" : K1" (1', )e 0 3' 20 P P 4) -ijZ Equation 2.43 can then be written (e understood) 2 2 V ¢1+7'¢1=0 3.21 t p p p where 2 E 2 Z 2 7' = “9“ “R - k = -7 ; radial plasma 3.22 p 2 E z Ue + wavenumber The irrotational part of the electric field inside the plasma is El = -vq11. 3.23 p From the solution of equations 3.10, 3.11, 3.17, 3.18 and 3. 21 the electric and magnetic fields inside and outside the plasma can be determined. This "potential" formalism is commonly used to solve for the waveguiding modes along cylindrical structure 54. The coupling between the fields inside the plasma and the fields outside the plasma occursat the plasma-free space boundary, r=a. In particular, the irrotational fields and the solenoidal fields couple at this discontinuity, as stated in section 2. 2. The boundary conditions used are the continuity of tangential E and H fields, and the normal component of electron velocity equals zero30 at r=a. The cold ion assumption, Ti: 0, eliminates the need to assume a boundary condition for the ions. 46 The complete solution to this problem is detailed in Appen- dix A. The characteristic equation for symmetric TM modes on the plasma cylinder shown in Figure 3.1 is (equation A. 33) _ .15:fo 6.€.%)2(£;)2 l - + 3. 24 7 11(7 a) 7 11(71a) 6 Kl(6a) p Iowpa) l IO(—yla) K0(6a) 2 for the irrotational surface wave modes (7p > O). For the irrota- tional standing wave modes (ypz< 0), the characteristic equation is (equation A. 34) sis-:fépz 631551.32 '1 — + 3 25 J 'a - I a K 5 ' 7' 1(7' ) 7 1(71 ) 5 1(5a) p Jowpa) 1 Iota/1a) KO( a) The definitions for the notation used in equations 3. 24 and 3. 25 is given in Appendix A. The characteristic equations 3. 24 and 3. 25 have been written to segregate the terms which are contributed by the various com- ponents of the fields. That is, the LHS contains the contribution of the irrotational fields inside the plasma cylinder. The first term on the RHS is due to the electromagnetic fields inside of the plasma and the second term on the RHS is due to the electromagnetic fields outside of the plasma. If the cold electron plasma approximation is assumed, the LHS of the characteristic equations is zero. The RHS then reduces 47 to the cold plasma result of Allis, Buchsbaum and Bers78, equation 10. 47. The following classification method, TMfinn, will now be introduced to denote the modes characterized by equations 3. 24 and 3. 25. The m and n will correspond to the usual scheme for the electromagnetic waves on a cylindrical wave guide4. The m refers to the number of angular variations of the fields, and the n refers to the number of radial variations of the electromagnetic fields. The 1 superscript will be used to denote the number of radial variations of the irrotational fields. For the rotationally symmetric modes, m=0. Since the electromagnetic fields are described by radially evanescent modified Bessel functions, n20. For the irrotational surface wave modes, when the irrotational fields also vary as modified Bessel function, 1:0. The irrotational standing wave modes are described by regular Bessel functions of the first kind. Thus 1 = l, 2, 3, . . . depending on the number of half wave length standing waves which exist between the center and the edge of the plasma cylinder. . 50, according to this method, equation 3. 24 characterizes TM:o modes, while equation 3.25 characterizes TMio, l = l, 2, 3, . . . modes. 3. 3. Discussion of the Dispersion Diagram Before presenting a complete numerical solution for the characteristic equations 3. 24 and 3. 25, a qualitative discussion of 48 the se equations is instructive. Not only will it lend considerable insight into the nature of the fields, which may be overlooked in computer results, but it will give confidence in the numerical results when they agree with this insight. A skeletal diSpersion diagram is presented in Figure 3. 2. A log w-log kz scale is used so that several orders of magnitude of w and kz can be conveniently shown on the same figure. The phase velocity is Vph : w/kz , 3. 26 and the free Space light line, V = C, is shown for reference. ph The region to the left of the light line is the fast wave region, Vph> C; and the region to the right of the light line is the slow wave region, V < C. The group velocity is ph 8(logw) p 8(10gkz) . 3.27 Thus the group velocity is the product of the phase velocity and the slope, 8(logw)/3(log kz), of the dispersion curve on a log-log plot. The light line is also determined by plotting 62: 0, from equation 3.12. This divides the diSpersion diagram into a 52> 0 region or slow wave region with 6 a real number, and 62 < 0 region, i.e. , fast wave region with 6 an imaginary number. The solution of Helmoltz equation 3.11 which satisfies the condition for wave- 49 y 7 of ,. ./ ..,// / ./ / a o A“... .mcob- 3 anagrams sensed 0.22.3 mcowmo» ouocop noose pounce 2:. .Esumfip sofiasonn; gages—m .~ .m ouswmh E .\n< No. q o>o>> 335m 9.9.303 \ \ 059.0% :0. \ / o . M. 26) £32m 8:83 o v M» \ \ \ \ \ \ / x 2: 2a: 9» P08 n l 50 guiding action is the modified Bessel function Kn(5r)75, because for large real arguments 1T -X Kn(x) _ If; e . 3.28 This clearly Shows that Kn(6r) is the desired radially evanescent solution. In the fast wave region, there are no solutions for <1) and kz which are purely real numbers. The complex kz type solutions will give radially traveling waves and the energy flow will not be purely axial. Thus only slow wave solutions to the characteristic equation 3. 24 and 3. 25 will be presented in this chapter. However, in Chapter IV, fast wave solutions for a plasma cylinder located concentrically inside a metal waveguide will be discussed in detail. In the slow wave region, 712 > O. This is easily Shown by using equations 3.19 and 2. 25, 72 - k2 (1)2 E “ '7 1 z C p 2 (1)2 002 wez (0,2 1 : k —— + -— +— 2 2 2 2 C C (to (1) ) 2 2 2 we +mi = a + (_.__). 3.29 C2 Since 62> 0, equation 3. 29 shows that 712 is positive and so 71 is real. From equation 3.19, 71 is the radial wave number for the electromagnetic component of the fields inside the plasma cylinder. Thu EVE. 6 1'3 are the C01 51 Thus the electromagnetic fields inside the plasma are always radially evanescent and are represented by modified Bessel functions Ik(-ylr). Summarizing the above, the electromagnetic fields are radially evanescent with a maximum value at the plasma boundary. From analogy with a dielectric rod waveguide, when 6 is small (near the light line) the fields are weakly radially evanescent, and when 6 is large (far to the right of the light line) the electromagnetic fields are strongly radially evanescent or tightly bound to the surface of the plasma cylinder. Also shown in Figure 3. 2 is a plot of 7:: 0. These curves are equivalent to the infinite media dispersion relations for the electron plasma and ion acoustic waves shown in Figure 2. l . From Figure 3. 2, the 7: = 0 curves divide the slow wave region into four subregions, two with YPZ > 0 and two with 792 < 0. In the electron domain (to ~ we) the ypz < 0 (or 71.32 > 0) region is described by equation 3. 25. The oscillatory nature of the regular Bessel function Jk('yl')a) implies that there will be an infinite number of solutions to 1e equation 3. 25 in this region. These modes are denoted as TMoo where 1 = l, 2, 3, . . . . The irrotational fields will have 1 radial variations between the center of the plasma and the plasma-free space boundary. The additional superscript e is used to denote the electron domain where these modes exist. Since there are no cold plasma modes in this region78 (i.e. , there are no solutions to the cold plasma dispersion relation), the poles of the LHS of equation 3.25 52 will closely approximate the total solution to equation 3. 25. This implies that the TM:: modes are approximated by (see equation A16c of Appendix A) or 'Yp'a = Xi); ' 3.31 where X' is the 1th zero of J '(X). 01 o 2 The 7p > 0 region in the electron domain is described by equation 3. 24. The irrotational components of the fields in the plasma (as described by the LHS of equation 3. 24) are radially evanescent waves with functional form Ikwpr). Be cause of the behavior of modified Bessel functions, 1km ~ ex Kk(x) ~ e "x there can be only one solution to equation 3. 24 in the region. This solution is denoted as the TM22 mode, as described in the last section. Again the additional e superscript is used to imply that (1) ~ (1) . e In the ion domain ((1) 5, mi), there are again two types of 2 regions. For 7p > 0 (the lower right hand corner of Figure 3. 2), there can be only one solution to equation 3. 24. The irrotational field is a surface wave and the mode is denoted TMEL, where the ii E6 53 i is used to imply (1) ~ (111. This mode exists for all frequencies (1) such that, w < (oi/«f2 . 3. 31 Because there are no cold plasma modes in this region78, the RHS of equation 3. 24 is a large, relatively constant number. Thus the pole of the LHS of 3. 24, 7p = 0, will yield an approximate solution for the TM:: mode. The 7: < 0 (yp'2> 0) region in the ion domain lies to the left of the 7:: 0 curve and below the mi line. The irrotational fields in the plasma are radially standing waves and these modes are denoted as TMffO , 1 = 1, 2, 3, . . . . The poles of the LHS will yield an approximate solution to equation 3. 25, except where the RHS is small. From Trivelpiece's re sults3l, it is eXpected that the RHS will be small only near the light line. Thus, near the light line (when RHS and LHS of 3. 25 are both small) the electromagnetic field will interact with the irrotational fields. Also by analogy with Trivelpiece, a solution to 3. 25 is anticipated which will extend to the origin of the dispersion diagram near the light line. Finally, a short discussion on the validity of the results obtained from the plasma model is in order. The warm plasma model for the electrons does not predict Landau damping. This will be important in the electron domain when V 3kTi ion domain when V Fe . ph m. 1 ph as Ue' or in the These regions are shaded on 54 Figure 3. 2, and predictions of warm plasma equations are not valid in these regions. Although the ion temperature has been assumed to be zero for this theory, an ion velocity line for a typical eXperi- mental temperature of 3000K is shown in Figure 3. 2. The results obtained from cold ion theory used in this chapter are not valid near this ion thermal velocity line." The use of an eXperimentally more realistic, nonuniform density profile would be expected to accentuate the spacing between the TM:: modes. However, because the TM:: modes exist in the overdense center region of the plasma cylinder, they are less sensi- tive to a nonuniform density profile. Thus, this theory should be quite accurate for the TMZ: modes. It has been noted several times in this chapter that the poles of the LHS of the characteristic equation 3. 25 play a very important role in nature of the standing wave modes. By analyzing the deriva- tion of Appendix A, it is found that boundary condition 3e - ’1: = 0 at r=a determines the form of this term. In the electron domain, the perfect reflection of electrons from the boundary is justified by the fact that a negative sheath forms in a plasma which is in contact with the surrounding glass tubel. But in the ion domain, where both ions and electrons contribute to the irrotational waves, this boundary condition is less easily justified. However, perfect reflection of electrons is used here for lack of a better boundary condition. 55 Another boundary condition, if it could be found, would only slightly alter the quantitative aspects of these TMS‘) modes. Finally, in any eXperimental system, a glass tube surrounds the plasma. This additional dielectric layer greatly complicates the boundary value problem solved in Appendix A. However, the effects of this glass tube may be accounted for by considering the media outside the plasma r > a, to consist of a fictitious homogeneous l6, 17, 31, 32 dielectric . The relative dielectric constant of the effec- tive media, 6 eff’ will fall in the range < E < . 1 eff 6g 3 32 where 6g is the dielectric constant of the glass tube. When the complete boundary value problem is solved, includ- ing the glass tube of outer radius b, only the second term on the RHS of equations 3. 24 or 3. 25 is changed. This term becomes, (1) 2 l 2 €+Ep 5.) (F) r Kl(5b) 6C1 K T—(f’b' (1;: )C2 1 (5b) 1.66} g3)c +:—-—(——:6b)(;5 324)C 1 3.33 whe re 0 II -II (kga)K1 (kgb) + K1(kga)Il(kgb) 0 II 10(kgb)Kl(kga) + Ko (kgb)ll (kga) 56 = b k k b C3 10(kga)Kl(kg )+Ko( ga)Il( g ) 4 k b - k k b C4 10( ga)K0(kg ) Ko( g:1)IO( g ) 2 k 2 = k 2 - E 3.34 Term 3. 33, which includes the effect of the glass layer, may be rewritten as (1) Z 70 2 6+61) (V) (IT) e Z 3.35 KIW a) e o Koh/oa) where 2 2 2 (1) ‘YO : kz - 25 Eeff 3 36 Thus term 3. 33 has been simplified to 3. 35. By equating these two expressions, one obtains an equation for Eeff in terms of Eg' a, b, o.) and kz. This effective dielectric constant will be used when comparison between theoretical predictions of this chapter and experi- mental observations are made in Chapter VI. Far to the right of the light line, the external fields are strongly evanescent, and in an experimental system the fields will be negligible outside of the glass tube. For this case, Eeff w 6g, Alternatively, near the line, Eeff ..~.. 1 because the fields are loosely bound to the cylinder and the glass tube has only a small effect on the nature of the dispersion relation. 57 3. 4. Dispersion Diagram A complete numerical solution to the characteristic equa- tions 3. 24 and 3. 25 is shown in Figure 3. 3. A description of the numerical method is given in Appendix B. The parameters used in Figure 3. 3 have been chosen to match experimental parameters which will be introduced in section'S. 2. The appearance of the dispersion curves verifies the argu- ments given in the previous section. The TMf): modes, electron standing waves, are observed in the underdense (we< w) region on the dispersion diagram. The group velocity, vg, of these waves is very small, except for large kz where they asymptotically approach the electron thermal velocity. The TMf): modes are closely Spaced on the frequency axis (see the insert on Figure-3. 3). A more experimentally realistic plasma model, with a nonuniform density profile, would be eXpected to increase the frequency spacing between these modes The TMS: mode, electron surface wave, approaches the electron thermal line for high frequencies and the light line for low frequencies. When w > (oi, the effect of the ions on the TMZZ and TM:: modes is negligible. At lower frequencies (w i wi), the motion of the ions greatly effects the modes. The TME: mode, ion surface wave mode, is found to the right of the plasma sound line, As kz becomes large, m approaches the (oi/«[2 asymptote, and as kz becomes small so 58 6.3.5. 3. means—av «5%an 80:3 2830.. vow-z. on» 3 3:2: 3.: 2.0330. 2;. 0.»:0n u p~ u 3 .Aafiafia E388. ax 3-2 a 3 .n n .E .x 8o .2 n a .Eo n .o u a .3323 «End:— u no 259: 3305:;- Nzaoflflg you Enuwfiv comnuoa-mfl .m .m enough on .33 7635. coauuofim L 32 L 03 u 83 :33 L :2 /\ in the wt". is t A c rig] ele: the ions incr 59 approaches zero along the plasma sound line asymptote. The first five TMlo; modes, ion standing waves, are shown in Figure 3. 3. For large kz, these modes asymptotically approach the (pi line. Near the light line, the coupling of the TM:: modes with the low frequency extension of the TM22 mode is clearly seen. This coupling is caused by the interaction between the solenoidal and irrotational fields in this region. As kz becomes small, the TM:: mode reappears and extends to the origin of the dispersion diagram near the light line. As mentioned, the parameters used to obtain Figure 3. 3 were chosen to match experimental apparatus which will be intro- duced later. Although this result is a special case of the general solution to the characteristic equation, a few remarks will develop the insight needed to visualize the effects of varying these parameters. The most important manifestation of the electron temperature is the electron thermal velocity line and the plasma sound speed line. A cooler (hotter) electron temperature will shift these lines to the right (left) on the dispersion diagram. Also, a cooler(hotter) electron temperature will decrease (increase) the spacing between the TM:: and TM:: modes. An increase (decrease) in plasma density will increase (decrease) we and mi and will shift all modes upward (downward) on the dispersion diagram. The mass of the ions appears in two parameters, mi and U8. Lighter ions will increase (oi, relative to we, and will shift the plasma sound line to tl‘ f? 9—.4 t] In 60 the left. From equation 3. 31 the radius of the plasma cylinder is very important in the determination of the spacing of the TM:: and TM:: modes. A larger (smaller) radius will decrease (increase) the spacing between the irrotational standing wave modes. Any experimental plasma cylinder will be enclosed in a glass tube. There are no fundamentally new phenomena generated by 32, 78 this additional dielectric layer For weakly evanescent modes, Vphas C, the electromagnetic fields extend far outside of the glass layer and the effective dielectric constant, Eeff of the outer region is approximately 1. For tightly bound waves, Vph<< C, the external fields have decayed to a negligible value for radii greater than the radius of the glass. The only mode which is effected by this is the electron surface wave. The other modes are primarily irrotational modes (LHS of characteristic equation is most important) and the electromagnetic parts of the fields make a very small contribution to the total solution. The effect of the glass layer on the electron surface wave is to create a backward wave region near the light line for 78 N w < we . This is caused by the transition between the strongly evanescent region and the weakly evanescent region. the of ¢ de 1: 1159 OfF 4.1 CHAPTER IV PROPAGATING MODES ON A CONCENTRIC WARM PLASMA ROD INSIDE A CYLINDRICAL METAL WAVEGUIDE This chapter presents the analysis of the boundary value problem of a plasma rod located concentrically inside a cylindrical waveguide. It is found that the existing modes of the waveguide are modified by the presence of the plasma. Additionally, new cold plasma modes and a new series of warm plasma modes appear. Typical dispersion curves for these modes are presented. This theory is then applied to a plasma inside a cylindrical cavity. Curves of cavity resonant frequencies and resonant lengths versus the plasma density are presented for the dipole modes. Finally, the theory is used to interpret previously reported, experimental observations of parametrically excited oscillations at low frequencies (m~wi). 4.1. Theoretical Model, Characteristic Equation and Discussion Consider the infinitely long waveguiding structure shown in Figure 4.1. The metal circular cylinder of radius b is assumed to be perfectly conducting. Located concentrically in this metal cylin- der is a plasma rod. As in Chapter III, the plasma is modeled as 61 62 A uniform , lossless, isotropic, warm electron, cold ion plasma rod located concentrically inside a cylindri- cal metal waveguide. The radii of the plasma and the metal cylinder are a and b respectively. Figure 4. l. 63 lossless, uniform and isotropic. The linearized warm electron, cold ion equations, 2. 17-. 24, are used to describe the plasma. Maxwell's equations are valid both inside the plasma and in the exterior region between the plasma and the metal cylinder. This structure will be analyzed using waveguide theory to determine the axially propagating modes. In contrast to the previous chapter, the transverse field components are perfectly reflected from the metal walls of the outer cylinder. Thus nonradiating fast wave solutions, V > C, exist in ph this waveguide. In addition, slow wave solutions, V < C, also ph prOpagate because of the presence of the coaxially located plasma rod. In general, both TE and TM fields will be needed to satisfy . the boundary conditions. That is, hybrid modes are present. For the rotationally symmetric modes, the TM and TE modes uncouple and each one can exist separately. As in Chapter III irrotational components of the fields in the plasma couple to the solenoidal fields at the plasma boundary, r = a. Before solving this problem, two special cases are considered. First, when the plasma density is zero, the waveguide in Figure 4.1 is the usual circular waveguide81 . There are only fast wave solu- tions for this type of waveguide, and the modes are pure TE and TM modes. Second, when the plasma density becomes very large, the fields cannot penetrate into the plasma. In this case, the plasma cylinder behaves similarly to a metal rod. Thus, the wave guide in 64 Figure 4.1 is similar to a coaxial metal line. It is known that TE and TM modes as well as TEM solutions exist on a coaxial struc- ture81. The TM and TE modes are uncoupled and propagate as fast waves, while the TEM transmission line mode propagates at the speed of light, Vph = C. Of course, experimentally generated plasmas fall between these two limiting conditions of the zero density or the infinite den- sity plasma. For these plasma densities the modes must be hybrid to satisfy the boundary conditions, and will have properties which will fall between the two limits stated above. Using the general methods developed in section 3. 2, the characteristic equation is derived in Appendix C. The fields are assumed to vary as exp(jcot - jkzz + jn¢). Theangular dependence ' is thus exp(jn¢), n = 0, i1, i2, i3, . . . . From equation C45, the characteristic equation is where the four by four matrices Mc’ U and V are defined in Appen- dix C. The terms in equation 4. l have been separated into the irrotational contribution (LHS) and the solenoidal contribution (RHS). Although equation 4. l is a very complicated expression, the contribution due to the warm and cold plasma modes is easily found. If Te -e 0, equation C40 of Appendix C shows that Pn -> 00. In this 65 case the LHS of equation 4.1 is zero, and the total characteristic equation reduces to |M|=o. 4.2 30, 56 This is equivalent to the cold plasma approximation . New modes which are predicted by equation 4. 2 are called cold plasma modes. Also, the modes which exist on the cylindrical waveguide, but are modified by the presence of the plasma, are predicted by equation 4. Z. The terms in the denominator of the RHS of equation 4. l are the coupling terms between the ir rotational and solenoidal components of the fields. The LHS of equation 4.1 is very small with respect to the RHS except for a), kz and we where, P -> 0. 4.3 n From equation C37 , this is equivalent to, y 'a = X. . 4.4 The modes predicted by equation 4. 4 are called the warm plasma modes. These modes are caused by radially standing irrotational fields in the plasma (e. g. , Tonks-Dattner modes). By inserting experimental numerical values for the para- meters in equation 4. 1, it can be shown that this characteristic equation is generally closely approximated by the cold plasma result (equation 4. Z) or the warm plasma result (equation 4. 4). 66 However, this approximation is not valid in those regions where RHS m LHS in equation 4.1. Physically this occurs only when there is strong interaction between the irrotational and solenoidal fields. Several special cases of the characteristic equation 4.1 are discussed in Appendix C. For rotationally symmetric modes, n = 0, it is shown that the TE and TM fields uncouple and only the TM modes couple to the irrotational fields. For Vph < C, n = O and b -> 00, the characteristic equation 4.1 is shown to be equivalent to equation 3. 24. Additionally, it is shown that for an arbitrary angular variation (n = 0, i1, i 2, . . . ), equation 4.1 reduces to the surface modes on a cylindrical plasma waveguide32’ 33 when b -> 00. Because the modes on the waveguide shown in Figure 4.1 are in general hybrid in nature, the usual waveguide identification scheme is not applicable8l. In addition, there are new cold plasma modes (primarily determined by the RHS of 4.1) and warm plasma modes (primarily determined by the LHS of 4.1). Thus it is con- 56, 64 venient to classify three types of modes . They are: l. Perturbed cylindrical waveguide modes * >l< a. TE orTM ;n>0,m>O nm nm In reality, the modes are not TE or TM, but rather they are hybrids. They are labeled with respect to the TE or TM modes on a circular wave guide into which they degenerate when the plasma density equals zero. The asterisk is used to denote these hybrid modes. It is 67 found that the presence of the plasma greatly effects these modes. b. TE or TM ; m > O om om These modes are pure TE or TM modes. The presence of the plasma does not greatly effect these modes, which are also labeled with res- pect to the TE or TM modes on a circular wave- guide into which they degenerate when me = 0. Cold plasma modes * TM and TM ; n > 0 00 no The TMoo mode is a pure TM mode. The solenoidal fields of this mode are surface waves (V < C). Also, ph for me >> 0) this mode becomes essentially the TEM * coaxial transmission line mode (V = C). The TMno modes are hybrids, and may be eitlhler slow waves, or fast waves depending on the plasma density. These modes do not exist in the empty waveguide, and are only affected by the finite electron temperature when Vph N Ue andw > we. Thus, these modes are called cold plasma modes. Warm plasma modes TD“! ; n _>_'_ O, 1 Z 1 These TD (Tonks-Dattner) modes primarily occur because of radially standing irrotational fields inside the plasma. The 1 subscript is used to denote the number of radial variations in the irrotational field. These modes do not exist in the empty waveguide and are only described by warm plasma theory. gl 68 This classification scheme allows one to identify the pre- dominant characteristics associated with the modes of the waveguide. Also, it will be used to describe the resonant modes in a cavity formed from a section of the waveguide shown in Figure 4.1. 4. 2. DiSpersion Characteristics of the Dipole Modes In this section the dispersion curves for the dipole modes of the waveguide shown in Figure 4.1 are presented. That is, equa- tion 4.1 is numerically solved for a) versus kz for n = l and we a fixed value. These dispersion curves will develop intuition which will be valuable when the dipole resonances of a cylindrical cavity are theoretically studied in the next section and experimentally studied in sections 6. 2 and 7. Z. In particular, the diSpersion * * . characteristics for the TEM, TM10 and TDM modes are presented. The dipole modes have several features which are of interest later in experimental work. The TE11 mode has the lowest cut off frequency in both the cylindrical waveguide and the coaxial wave- guide. An analysis of the field pattern of the TE):1 mode shows that it concentrates the electric field in the center of the waveguide where the plasma exists 56. The TD” modes are the usual dipolar Tonks-Dattner modes. The waveguide shown in Figure 4.1 is a geometry which can easily be theoretically and experimentally in- vestigated to study these resonances. Finally, it is found experi- mentally that a high density plasma can be resonantly sustained a): 101 mode (see section 7. 2). inside a cavity in the TM 69 The dispersion characteristics (00 vs. kz) have been found for the dipole modes for three values of electron plasma frequency, we. The choice of parameters a, b, Te’ etc., is based on experi- mental apparatus to be introduced in the next chapter. Only solutions in the slow wave region near the light line and in the fast wave region have been found. That is, V > C/5 for frequencies in the range ph 1 Ghz < 20—"- < 4 Ghz. Also, the effects of the ions have been ignored in this section ((1)1 = 0 ). A description of the numerical methods used to solve the characteristic equation 4. l is given in Appendix D. Briefly, the total solution to equation 4. 1 has been approximated by the cold plasma result, equation 4. 2, and the warm plasma result, equation 4.4. This approximation is very accurate everywhere except where the RHS m LHS in equation 4.1. In this region the background developed in section 3.4 for a full solution to a characteristic equa- tion has been used to intuitively complete the diSpersion curves. In Figure 4. 2 the dispersion characteristics of the dipole modes with (De/Zn = 105 are presented. For this value of electron plasma frequency and for the range of frequencies of interest, the ratio (De/co is nearly zero. Thus the curves in Figure 4. 2 are essen- tially the same as the empty waveguide (me = 0) results. That is, the a): a: TEll and TM11 modes are the usual cylindrical waveguide modes“. * The new cold and warm plasma modes, TM10 and TD”, on this waveguide occur for frequencies far less than 1 Ghz. Therefore, they are not shown. L” . (”U o kfi \ 0 {J3 2w Ghz 4. 1L * TM 70 Light Line / O 10 20 30 4O 50 60 7O 80 O 100 k , rn"1 z we Figure 4. 2. Dispersion diagram for dipole modes. 3— = 10 1T b: 5.08 cm, a=0.3 cm. mo an: the ETC TE the me C01; int 71 In Figure 4. 3 the dispersion characteristics for the dipole modes are shown when me/ 211 : 3. Ghz, i.e. , a) ”we. Several very interesting aSpects of these modes are observed in this diagram. * The cut off frequency for the TEll mode has been shifted from 1. 72 Ghz in Figure 4. 2 to 2.14 Ghz in Figure 4. 3. Frequencies a): less than this cut off value can not prOpagate in the TEll mode. * The TM10 mode, which did not exist on the empty wave guide is also seen in Figure 4. 3. In the slow wave region of interest, this mode is essentially determined by the relation (D = we/NI—Z . The group velocity of this mode in this slow wave region is very small. For lower frequencies, this mode crosses the light line and becomes a fast wave with a cut off frequency of 1.69 Ghz. Thus * * the TM10 mode has a lower cut off frequency than the TE11 mode. The first five TD]! modes are shown in Figure 4. 3. These modes are essentially the warm plasma modes on the plasma cylinder, and they are very closely Spaced in the underdense (a) > we) region of the dispersion diagram. These TD modes have essentially zero 11 group velocity except in the region where they interact with the * TEll mode (see insert on Figure 4. 3). In this interaction region the coupling to the TD modes should be strong and presents a 1! means to study these Tonks -Dattner modes. In other regions, the coupling to these modes will be difficult because the solenoidal fields in the waveguide are small. 4- “HI ‘00-. Ghz 11 72 TE* / 11 /\ / Light Line TD 11 / l I L f l M /, TE* / 11 / 3.15 , TD15 J TM* / TD 10 / 3.104 14 0) — TD / 2" 3.05 13 / [TDIZ 3.00 / TDll / A’“ 2.95 1,1 4. t / 40 45 50 k Z % : t : .L i : + t r 0 10 20 30 40 50 60 70 80 90 100 k , m"1 z 00 Figure 4. 3. Dispersion diagram for dipole modes. -£- = 3 Ghz, b = 5.08 gm, a = . 3 cm, Te = 30, 000°K.“The coupling of the TEM mode with the TD” mode is shown in the insert. Ghz. essen frequ< the I? mode The T is out soluti since onyx diSpe- ShOWI mode bEQU' 4.3. deveL Cavity Calcuj with a are ac 73 (0 Next, Figure 4.4 shows the dispersion curves for -Z—:- : 10 1.1 Ghz. Because the plasma is very dense (me/0) "' 100), the modes are essentially the dipole modes for a coaxial waveguide81. The cut off :1: ll modes is 3.62 Ghz; i.e., the same as that of frequency of the TE >1: the TM11 coaxial mode. Likewise the cut off frequency of the TM10 mode is l. 71 Ghz, or the same as that of the TEll coaxial mode. The TD” modes exist in the underdense part of Figure 4.4 which is out of the frequency range of interest. There are no slow wave solutions to equation 4. 1 in the frequency range shown in Figure 4.4, since the TM?0 mode crosses the light line and becomes a slow wave only when 00 as me/NI’Z. From these three example solutions to equation 4.1, the dispersion characteristics of the dipole modes on the waveguide shown in Figure 4.1 are understood. The modification of existing modes and the appearance of new cold and warm plasma modes has been highlighted . 4. 3. Dipole Modes of a Plasma Rod Inside a Cylindrical Cavity. The dispersion curves of the previous section are useful to develop understanding of the dipole resonances of a cylindrical cavity. In this section the characteristic equation is employed to calculate resonance curves of a cylindrical cavity partially filled with a plasma. The dimensions of the cavity and the plasma rod are set equal to parameters of an experimental cavity-plasma system 74 Light Line ) ' , Ghz IT 0 E . t 4 4. : : t .L e 4 0 10 20 3O 4O 50 6O 7O 8O 90 100 k , In"1 2 w 11 Figure 4.4. Dispersion diagram for dipole modes. EST; = 10 Ghz, b: 5.08am, a=.3cm. 75 to be introduced in the next chapter. Hence the theoretical pre- dictions of this section will be compared with experimental results in section 6. 2 and section 7. 2. Physically this cavity is formed from a section of the wave- guide shown in Figure 4.1 by placing perfectly conducting metal plates over the ends. The boundary conditions at the metal end plates80 then imply that for each mode, where L8 is the length of the cavity. Thus for a given value of p and Ls’ equation 4. 5 gives the axial wave number. This, in turn, is fixed in the characteristic equation 4.1, which is then solved for the resonant frequency, ‘00, versus electron plasma frequency, we. Alternatively, if the input frequency to the cavity is assumed to be fixed, the characteristic equation 4. 1 can be solved for the resonant length, Ls’ versus electron plasma frequency, me, by using equa- tion 4. 5. Such curves will be useful when experimentally inve sti- gating a variable length cavity. As in the last section, the effects of the ions are ignored by assuming mi = 0. Also only slow waves near the light line (cg/kz < C/5) and fast wavesare considered in the frequency range 1 Ghz < m/Zw < 4. 4 Ghz. The details of the numerical methods, which are the same as in the previous section, are presented in Appendix D. ef re 01' de: she 853 ”:0 76 In Figure 4. 5, the numerical solution ofw vs. we is shown when L8 = 6 cm. The third subscript l on the modes is used to de- note one axial variation in the fields. That is, p = 1 in equation 4. 5 so, Tr L : )‘g/Z: —, 4.6 8 kz where Kg is the guided wavelength in the waveguide81. In addition to the dipole modes, the TM mode is also shown on Figure 4. 5 010 for contrast. It is easily seen that the TM010 mode is much less >1: effected by the presence of the plasma than is the TElll mode. It is this property which makes the TM010 such a valuable cavity . . 62, 63, 73 resonance for diagnostic purposes . >1: The effect of the plasma density on the TElll mode is very :1: dramatically shown in Figure 4. 5. In the region 0) m we. the TEI 11 mode strongly couples to the TD modes, i.e., the warm plasma 111 modes (see insert on Figure 4. 5). As the plasma becomes over- :1: dense (we > 00), the resonant frequency of the TE mode rises 111 sharply along the line 00 = we /\/—2. Then the curve levels off and asymptotically approaches a frequency correSponding to the TM111 . . 81 . mode on a coax1al cav1ty . In the region where 0) zme/JZ, the magnetic fields in the cavity are very weak56. The energy which is stored in the cavity oscillates be tween the electric fields of the over- dense plasma and the electric fields in the free space neighborhood immediately surrounding the plasma. 77 I ‘ 0) / TElll _ e (1) - 2/ 4. db / IA 111 / 1 Ghz 3. 1n ' TM>lE / 101 TMOIO MN) 3 10% 131 D T 151 D D111 ii 14 Z. .. TE* 121 111 3.05) 3.00 fl 4 Z 9 3.0 3.1 0.) e 0 $ l l L 1 - 422?; n i L fin 0 l 2 3 5 6 7 8 9 10 we '21-;- , Ghz Figure 4. 5. T 010 Resonant frequency versus electron plasma frequency for cylindrical cavity. L e = 30, 0000K. The TM8 6 cm, b: 5.08 cm, a: .3 cm, is also shown for reference. 78 The presence of the plasma in the cavity causes the new T mode to exist. This mode does not exist in the empty cavity. M* 101 :1: The TM101 mode is a standing surface mode (Kg < ch/m) for fre- quencies less than 0) = 21rc/ 2LS : 211 x 2. 5 Ghz. For frequencies >1: above this value, the TM . . x > 101 is a standing fast wave mode ( g * 21rc/00). With a very dense plasma in the cavity, the TM101 mode's resonant frequency approaches the value determined by the TElll resonant frequency in a coaxial cavity. Thus for electron plasma frequencies greater than we : ZTI' x 5 Ghz, the solenoidal fields of :1: the TM101 mode will be very similar to the TE111 mode's fields in a coaxial cavity resonator. Finally, a series of TD modes exists in the cavity. The 11 l electromagnetic fields of these modes externalto the plasma are very weak except in the region shown in the insert of Figure 4. 5. Thus these modes will be difficult to excite except in the region 3): where they couple to the TE mode. The TD modes are essen- 111 ll 1 tially radially standing electron plasma waves on the plasma cylinder. That is, the dipole Tonks-Dattner resonances. In addition to tuning a cavity by adjusting the input frequency, it is also possible to use a variable length cavity with a fixed input frequency to tune a cavity. Although these two methods appear to be equivalent, there are some important differences which can be seen in Figure 4. 6. 79 6.10 / 6.09.. /(/ D T 111 1 7 .L l LS I 3 6.08- 11 TDlll LS, cm *— TD121 6.07 , g. L, >:< 2.80 2. 3. 3. TE 90 w 00 10 111 __e 6 7' Z‘IT M):< T 101 5-1- 03 =~f200 e 0 L 1 I l 3 l 0 :2 3 4 5 6 i i? <3 10 we 21: , Ghz Figure 4.6. Resonant length, L , versus electron plasma frequency. w = 2n x 3.03 Ghz, ‘5 = 5. 08 cm, Te = 30, 0000K, a: 0. 3 cm. 80 In Figure 4.6, the resonant length of the cavity is plotted as a function of the electron plasma frequency, we. The input fre- quency has been fixed at w : 2w x 3. 03 Ghz. The resonant length of :1: the TElll mode increases with increasing we. The coupling of the >1: TD”1 modes to the TE111 (shown in the insert of Figure 4.6) occurs in the neighborhood of we z w. The re sonant length of this mode begins to increase rapidly in the neighborhood of we > (ox/’2. However, the >1: TElll mode approaches a vertical asymptote in Figure 4.6. Thus :g: the cavity cannot be tuned to resonance in the TE111 mode for we > 211 x 4. 4 Ghz by adjusting the cavity length when the input fre- quency is w = 21r x 3. 03 Ghz. If the input frequency to the cavity were greater than the cut off frequency for the TM mode on a coaxial waveguide (217 x 3.62 11 :1: Ghz from Figure 4.4), then a plot of Ls versus we for the TIE?111 mode would approach a horizontal asymptote for large we. This can easily be seen by observing the shift in the cut off frequency of :1: . the TF.‘ll mode in Figures 4. 2, 4. 3 and 4.4 as we increases. Using these Figures and noting from equation 4. 5 that kzoc l/Ls, it is observed that w must be greater than the cut off frequency of the :1: ' d 111cav1ty mo e) TM11 coaxial waveguide mode, if L8 (for the TE is to have an upper bound as we increases. Thus the resonant length, Ls’ of the cavity may have a vertical asymptote (as in Figure 4.6) or a horizontal asymptote (if w is greater than the cut off frequency of the TM11 coaxial waveguide mode) as we increases. This behavior 81 is in contrast with the variable frequency tuning method where the * TE111 mode always approaches a horizontal asymptote for large (1) . e >l< The resonant length of the TM101 mode increa ses sharply (D for 2% z 4. 5 Ghz. However, it approaches an asymptotic length determined by the TE mode on a coaxial waveguide. It is seen 111 that the resonant length L8 is very insensitive to changes in plasma 0) density for i > 8 Ghz. This effect will be very important in 2Tr section 7. 2 for increasing the plasma density of a radio frequency generated plasma inside an eXperimental cavity. A few comments are in order here on the validity of the numerical results of this chapter. It is known from section 2. 3 that the nonuniform plasma density profile, which exists in experimental plasmas, has a very pronounced effect on the Spacing and position of the electron plasma standing wave modes. Because a uniform plasma was assumed in this chapter, only the qualitative properties of the TD modes will be given by Figure 4. 5 and 4. 6. These problems 111 will also effect the manner in which the TD”1 modes interact with the TElll mode. Thus the warm plasma modes predicted by this chapter can only be expected to have qualitative agreement with eXperiment. Also, it was stated on page 20 that the value of we used in the theory is a function of the plasma density profile. Thus we of this chapter must be interpreted as an average value. 82 It has been noted several times in this chapter that the plasma density we = wNI—Z plays an important role in the nature of the dipole resonances. This value is related to the scattering resonance of a 6,17, 30 cold plasma cylinder When the plasma has a realistic density profile and the surrounding glass tube is theoretically taken into account, it is known that this relation is modified to w = wx/k, e where k > 2 is a constant. Thus it is seen that the theoretical predictions of this chapter describe the qualitative features of the plasma cylinder located inside a cylindrical waveguide or cavity. In order to obtain quantitatively correct results, the problem must be analyzed using a realistic density profile and including the effects of the glass tube. Finally, one other aspect of the resonances in this section is important. The existence of curves on a graph like Figure 4.6 does not necessarily imply that the resonances can be eXperimentally excited; rather excitation of resonances is a matter of impedance matching between the resonance and the driving microwave network 28' 80. Thus, just because many resonances are theoretically pre- dicted, only those which are impedance matched to the driving network will be eXperimentally observed. Additionally, in order to excite a resonance of the cavity- plasma system, some type of exciter (probe, loop or aperture) must be introduced into the cavity. This exciter must be placed in a region 83 of the cavity where the field pattern allows coupling of the exciter 80 . . to the resonant mode . Thus impedance matching and proper ex- citing techniques are necessary to eXperimentally observe these theoretically predicted re sonance S. 4.4. A Comparison of the Low Frequency Eigenmodes of a Cylindrical Plasma with Experimental, Parametrically Excited Plasmas ‘ The bounded plasma theory for the waveguide modes on a plasma cylinder will now be used to calculate the low frequency eigenmodes of some parametrically excited plasma cylinders. There has been some discussion on the nature of eXperimentally observed, low frequency, parametric instabilities. It will be shown in this section that some of these parametric instabilities can be explained by the bounded plasma theory developed in Chapters III and IV. These phenomena have been reported in two types of experiments: the 2 ”usual" Tonks-Dattner experiment 9’ 50' 51 and the "plasma capacitor” eXperi mental setup43’ 48’ 49. The common eXperimental setup used to observe the Tonks- Dattner resonances is shown in Figure 4. 7a. A plasma discharge tube is inserted through the narrow side of a rectangular wave guide shown in Figure 4. 7b. This waveguide is excited in the dominant TElo mode; thus the axis of the plasma tube is perpendicular to both the electric field vector, E, and the axial wave vector, 1:, of the rectangular waveguide. This E field has only a vertical com- ponent as in Figure 4. 7b, and its intensity varies Spatially as shown 84 (a) (b) (C) X ’ 0 w Figure 4. 7. Experimental setup to observe Tonks -Dattner resonances and parametric coupling to low frequency 0 s cillations . in ar 85 in Figure 4. 7c. At low incident power levels, when the plasma density in the discharge tube is varied, the common Tonks-Dattner resonances are observed in the scattered fields of the plasma tubell. These resonances are primarily dipolar in nature, due to the vertical . . . . 27, 76 component of the exc1ting electric field , The usual interpretation of the Tonks-Dattner resonances has been to assume that the exciting field is uniform and that only . . l6 "transverse" dipolar resonances are exc1ted . As seen from Figure 4. 7c, the exciting field is not uniform, but rather varies spatially in a manner similar to a half wavelength standing wave. Thus for the analysis of this section, it is assumed that Tonks- Dattner resonances are caused by a resonant halfwave section of . 76 . . . plasma waveguide . That is, a cylindrical plasma resonator. The fields on the cylindrical resonator are assumed to be reflected by the metal side walls of the rectangular waveguide. Thus the axial wave number on the plasma cylinder can be found from, 217 kz-EW' 4.7 where W is the width of the rectangular wave guide. Additionally, it is assumed that the standing wave modes on the plasma cylinder are not appreciably effected by the top and bottom walls of the metal waveguide. Given the dimensions of the rectangular waveguide shown in Figure 4. 7, this assumption is especially valid for radially stand- ing wave modes on the plasma waveguide because the external fields are tightly bound to the plasma cylinder76 31'? C51 in t the. as 2 mm a 0} Watt Ilatt has metl 'Ihes infin tionz are I more Seeti. Predi 86 Since the dipole modes of the cylindrical plasma resonator are primarily excited in the Tonks-Dattner experiment, equation C51 of Appendix C with n = 1 can be used to determine the resonance spectrum. Because a uniform plasma density profile has been used in this theory, only qualitative agreement can be eXpected between theory and experiment in the electron domain (w N we). However, as noted in section 3. 3, this theory should be quite accurate for modes in the ion domain (w ~ wi). In addition, all of the qualitative 1i discussion in section 3. 3 for the TM 00 modes (i.e., rotationally symmetric ion standing wave modes) is valid for dipole modes on a cylindrical plasma waveguide. When the incident power is increased to approximately one watt, and the plasma density in the tube is adjusted to a Tonks- Dattner resonance value, parametric coupling to low frequencies has been observed in the experimental setup of Figure 4. 729' 50' 51. For incident frequencies in the 2 to 4 Ghz range, the observed para- metric instabilities have a typical frequency of the order of 100 Khz. These low frequency oscillations have been interpreted by using the infinite plasma diaper sion relation for ion acoustic waves (see equa- tionZ. 44 and Figure 2. l) and assuming that these ion acoustic waves are reflected by the glass tube-plasma bounchry. However, using the more realistic cylindrical plasma resonator interpretation of this section, these low frequencies can be related to the eigenfrequency predictions of equation C54. f1 fl: th. In in ~ Th. as,- date 87 Using the experimental parameters (plasma density, tube radius, etc. ) from references 29, 50 and 51, it is noted that only the dipolar ion acoustic standing wave modes fall within the eXperimental frequency range. These eigenfrequencies are very closely approxi- mated by the poles of the LHS of equation C 54. That is, I I 2 4.8 J1 (yp a) 0 OT ' 2 '° = 23... 4. Ypa X1191 1,9 ’ 9 where X11 is the 1th zero of the derivative of the regular Bessel 75 function of the first kind . Inserting the eXpression for yp' (equation 3. 22) into 4. 9 and noting that 2 2 ' >> Yp kz and 2 2 2 40.) U y' i e p 2 (<1, 2 2 ,2 (we +Ue YP ) the eigenfrequencies are i- 2 w. (x' /a) i 11 fl : -2— 2 2 2 . 4.10 1r x: (ll/a) +we /Ue In any experiment we, hence wi, can be determined by observing in which Tonks-Dattner resonance the parametric coupling occurs. The resonance frequencies can be calculated from equation 4.10 by assuming a temperature which produces a be st fit to experimental data of reference 51 and are displayed in Table 4.1. as 1'6 1'6 the tel fit ref Ca) (3&1 for 88 Table 4. 1 Theoretical and Experimental Low Frequency Eigenfrequencies for a Plasma Cylinder a = 4mm, Te = 2, 0000K and me = (211' x 1.9 Ghz) Bounded Plasma Theory EXperiment-Ref. 51 m fl (Khz) fl /f3 , fn(Khz) fn/f3 l 21 0. 215 40 O. 4 2 61 O. 62 6O 0. 6 3 98 l. 0 100 1. O 4 133 l. 36 130 l. 3 As was done in reference 51, the raw data is also presented as frequency ratios, normalized to the third, most easily excited, resonance. These ratios are in good agreement with eXperimental results, but this appears to be due to the properties of the zeros of the Bessel functions and are approximately the same for any electron temperature chosen. The electron temperature which gives the best fit in Table 4. l is Z, 0000K, or about ten times lower than experimental electron temperatureSSZ. An alternate, but more realistic approach, is to assume the most easily excited instability, the third reported by reference 51, is the lowest order dipolar resonance. If this is done, calculations yield an electron temperature of 45, 0000K. Likewise, calculations of electron temperature from equation 4. 10 yield 65, 0000K for the data of reference 50 and 117, 0000K for the data of reference 29. Th of F1; 43, ra: ge< Be I'Ot to] pla pla isti Unc the exP shO' aPp: 89 These electron temperatures are within an order of magnitude of experimental values, but are slightly high. In the experimental setup of Lee and Asmus sen shown in Figure 4. 8, parametric coupling to low frequencies is also observed 43' 48' 49. A thin plasma cylinder (length is much less than the radius) is resonantly sustained by the incident 3. O3 Ghz signal. This geometry is similar to the plasma capacitor shown in Figure 2. 2. Because the incident fields from the coaxial transmission line are rotationally symmetric, only the rotationally symmetric modes are assumed to be excited in the plasma. The parametrically excited oscillations which have been observed in this experiment are found to be a function of the length of the plasma, L. The plasma cylinder is again assumed to be a section of plasma waveguide. This waveguide is terminated by the metal end plates of the inner conductor as in Figure 4. 8a and thus the character- istic equation 3. 24 can be used to describe the low frequency modes. oi Under these conditions it is found that only the TM001 resonance, the ion surface wave mode, will give eigenfrequencies close to experimentally observed values. For this TMBBI mode, it was shown in section 3. 3 that the total characteristic equation is closely approximated by the infinite plasma re sult. That is, Y = o. 4.11 90 (Outer Conductor J Coaxial . . Inner Sliding Conductok; .———L _ Short I————J Incident Signal ‘__7' r 1 (b) Quartz Tube Inner Conductor Plasma Cylinder Figure 4. 8. Cross Section (a) and detail (b) of the coaxial discharge structure of Asmussen and Lee ’ 48’ 49. 91 This explains how the use of the infinite plasma diSpersion relation by Lee and Asmus sen49 gave valid results for the eigenfrequencies on the cylindrical plasma resonator. For these TM?01 resonances the fields external to the plasma are tightly bound to the plasma cylinder. Thus the outer conductor of the coaxial structure in Figure 4. 8a has a negligible effect on the resonances. Equation 4.11 yields an electron temperature cf 13, 0000K. In conclusion, a bounded plasma theory has been derived and used to calculate low frequency eigenfrequencies observed in cylindrical plasma resonators. If parametric coupling to "ion acoustic modes” is assumed, then the instabilities occurring in two experimental plasmas are caused by coupling to two different resonant modes: for example, dipolar ion standing modes in the Tonks—Dattner waveguide experiment and rotationally symmetric ion surface modes in the "plasma capacitor" experiment. Reasonable agreement between eXperiment and theory is obtained when the theory is compared to the most strongly excited low frequency oscillations. Slightly lower and more experimentally reasonable electron temperatures can be obtained if wave reflection from a "perturbed sheath"84 is assumed. This assumption decreases the effective radius, a, in equation 4.10 and thus causes T8 to decrease. Hence the explanation of parametric coupling to ion acoustic modes can not be ruled out as a possible mechanism in the eXperiments. However, other low frequency oscillations, such as 92 relaxation oscillationsgs, sheath oscillations may also be excited. 43, 52, 53, 54 , etc CHAPTER V EXPERIMENTAL S YSTEMS Two microwave cavities were designed and fabricated to eXpe rimentally study the linear and nonlinear behavior of bounded plasmas. The first is a short gap or reentrant type and the second is a cylindrical cavity. Both of the cavities have a sliding short at one end to give a new variable for the tuning problem of plasmas in cavities. The plasmas were created with a hot cathode, mercury vapor discharge tube and can be maintained by dc power, microwave power or a combination of these two sources. 5. 1. Experimental System A line diagram of the experimental system is presented in Figure 5.1. The microwave signal frequency of 3. O3 Ghz is generated by an external cavity klystron which is connected to an oscillator synchronizer to form a phase-lock loop. The synchronizer locks the klystron oscillator to a harmonic of an internal crystal oscillator to produce a frequency stable microwave source. Such a frequency stable source is important, because the plasma media is diSpersive and the 93 94 oscillator klystron synchronizer power supply A , dc , TWT l: volta e loop power supply rf ’ feedback / isolator external cavity variable klystron attenuator to incident power meter / \ directional to reflected power meter, couplers crystal detector, / Spectrum analyzer h sliding . probe coaxial to spectrum analyzer % < , —~$—L—a slotted , line calibrated variable , attenuator —"_"l —"W“— _ photodiod filament 1 supply h] plasma tube H. V . supply W 10K 52 fl 0 discharge 60 Hz current modulation Figure 5.1. EXperimental set up and coaxial microwave system. 95 microwave circuits are frequency sensitive. The stabilized microwave power is delivered to a variable attenuator which regulates the input power to a traveling wave tube amplifier (TWT). By adjusting the variable attenuator, the ampli- fied output power from the TWT can be varied from 0 to 25 watts. An isolator protects the TWT's output helix from being damaged by reflected microwave power. This microwave generating source was used in mo st experimental work; however, a variable frequency source was employed when a number of frequencies were required. A pair of directional couplers are used to measure incident and reflected power from the cavity. The reflected signal could also be rectified by a crystal detector and diSplayed on an oscilloscope as a parameter of the system is varied (e.g., discharge current). In addition, the Spectrum of the reflected signal may be analyzed by attaching a Spectrum analyzer to the reflected signal directional coupler. The section of coaxial slotted line is used to measure the input impedance of the cavities. A reference plane is established on the slotted line for a short circuit load. The phase of the cavity impedance is measured by noting the difference of the null in the SWR pattern and the reference planeso. Because large SWR's are often encountered, a special "double detector” system79 was needed to measure accurately the amplitude of the impedance. This technique uses a calibrated variable attenuator and a Spectrum analyzer (see Figure 5.1). After dete Spec in t} the obse cou; info an rnea Cha Sust Cath the , Part COul 5,2. basi. 96 determining a reference level at a null in the SWR pattern on the spectrum analyzer, the probe in the slotted line is moved to a peak in the SWR pattern. The attenuator is increased to lower the input to the spectrum analyzer to the previously determined reference level. The SWR is then found directly from the difference of the two settings on the attenuator. This Special technique accurately measures very high SWR's. These impedance measurements are more accurate than observing reflected power because the directivity of the directional couplers is rather poor (E 20 db) for very high SWR's. Also phase information is available from impedance measurements which is lost by merely observing rectified reflected power. The average luminosity emitted by the plasma is used as a measure of average plasma densityzg. It is experimentally shown in Chapter VII that this technique is valid for both dc and radio frequency sustained plasmas. The plasmas were obtained in a sealed, mercury vapor, hot cathode, dc plasma discharge tube. By varying the discharge current, the average plasma density could be varied from 1. 5 x 1010 to 20. x 1010 particles per cm3. As shown in Figure 5.1, a 60 Hz sweep circuit could also be used to vary the discharge current for some eXperiments. 5. 2. Reentrant Cavity The reentrant cavity is shown in Figure 5. 2a. This cavity is basically a coaxial transmission line, which is short circuited at one 97 ufimEozom 33.3036 «cofimkfiswm .n >fi>mo unmpunoon on» we coufioom mmOuU .d N .m ops—mam L 1‘ m \r ‘IH ‘l hf L Hug .1 I . E 3IU 1m .mmgw mo 332.80 3.503on 9,329“ .Q.O Eu w. .0; 80 o. 66.3 Macadam 3020: 3533500 nos“: :9: MERE pouosmfioo .830 II/ / .. // Eu vm .N a mlvgggg/gg/Zggggggg/ZZ/ sowz‘ é a _ . w. . .Qf _ _ 1 A _ x _ fl mwoflUOuosflF Q03 mcfiadoo ”.59: |\ \ S .m «Ema 82:02: 1388 23355 A3 98 end and has a gap at the other. That is, the inner conductor is separated from one end to form a "gap region” as shown in Figure 5. 2a. The ratio of the radii of the inner and outer conductors was chosen to give this coaxial transmission line a characteristic . 8O . . . . impedance of 509 . Only the TEM transmissmn line mode is excited on the coaxial section of the cavity, be cause the cut off frequency of the lowest order waveguide mode on the coaxial wave- guide, the TE mode, is 5.4 Ghz. 11 The mercury vapor discharge tube was placed inside of the hollow inner conductor. Thus a plasma cylinder, of height Lp' is present in the gap region inside the cavity. This plasma interacts with the fields in the gap. A close fit of the inner conductor and the end plate of the cavity on the glass tube is obtained by flexible metal fingers which grip the plasma tube. The length of the gap, LP, and the length of the short, Ls' are adjustable. For small gap lengths, the plasma geometry is similar to the plasma capacitor shown in Figure 2. 2. By adjusting the sliding short, the cavity's resonant frequency can be tuned from less than 600 Mhz to over 4 Ghz for either a low or high density plasma in the cavity. Be cause of the small size of the reentrant cavity (approximately a quarter wavelength long, and a quarter wavelength in diameter for the frequencies used) it has a large power density for a given input power level. Additionally, the field strength in the gap is very strong for small gap sizes. A typical 99 gap length used in the experiments is 1 mm. A 100p formed from the inner conductor of a 5052 miniature coaxial transmission line is located on the sliding short. This line can be rotated so that the plane of the 100p is normal to the$ direction. In this position, the cavity is critically coupled80 to the H4) component of the TEM magnetic field when the plasma density is zero. However, as shown in section 6. l the cavity-plasma system becomes decoupled (i.e. , not critically coupled) as the plasma density increases. In order to measure the average luminosity emitted by the plasma, a small hole (diameter m 1 mm) was drilled in the cavity wall near the gap region and a photodiode is inserted into this hole. As mentioned in the previous section, this luminosity is proportional to the average plasma density in the gap region. Because the fields of the TEM mode are rotationally symmetric, only rotationally symmetric fields are excited in the gap. Viewing the gap as a section of cylindrical waveguide, with diameter equal to the diameter of the outer conductor, only the TMom and TEom modes may be excited by the incident TEM field. However, the TEom modes will not be excited because their field components are normal to the field components of the incident TEM fieldsgo. Thus only the TMom modes are excited in the section of waveguide forming the gap. These modes are cut off in the frequency range used here. In the cut off condition, the TMom modes have a larger amount of energy stored in electric field than in the magnetic field; thus their lOO . . . ., 8 . impedance 18 capac1t1ve 0. This argument clearly demonstrates that the impedance of the gap region is capacitive. A heuristic understanding of this cavity is obtained by model- . . 81 . . ing the gap as a capac1tor . In order to resonate, the capac1t1ve reactance of the gap must be cancelled by the inductive reactance of 73, 80 the short circuited transmission line This can be written 'X + °X . = O, 5.1 J gap J line where 'X - ca acitive reactance of the a , J gap P 8 P jxline - reactance of the short circuited transmission line. Because the capacitance of the gap region is small, the reactance of the gap region, J'Xa =-———-.wé . 5.2 gp J gap will be very large. Equation 5.1 then implies that the reactance of the short circuited transmission line , J'Xline = j50tan(BoLs) . 5.3 where (30 = m/C. 5.4 must be very large and inductive. This condition holds for cavity lengths, Ls’ which satisfy, 217/ (30 4 L < 8% 101 Thus this simple model of the reentrant cavity shows that the resonant length, Ls’ is less than, but approximately equal to, a quarter wave- length. Figure 5. Zb shows an electrical schematic of the reentrant cavity. The coupling loop is modeled as an ideal transformer with an n: 1 turns ratio and the impedance jXL represents the phase shift due to the stored energy in the evanescent fields near the loop80. A length of 5052 transmission line extends to the gap region. The gap is modeled as a capacitor and a resistor, which will depend on the parameters of the plasma. More detailed equivalent circuits for the reentrant cavity will be developed in section 6. 1. The reentrant cavity has one additional feature which makes it of interest for plasma research. This short gap cavity measures only localized plasma properties because only a small section of the plasma cylinder interacts with the fields of the cavity. This differs from the TM010 cylindrical cavity (discussed in the next section) which measures properties of the plasma of a large section of the plasma cylinder. This characteristic of the short gap cavity is valuable for some plasma diagnostics 5. 3. Cylindrical Cavity The cylindrical cavity is shownin Figure 5. 3. This variable length cavity was designed to excite several resonances in the 2 to 4 Ghz frequency range. For the TElll mode, it was desired to have 102 .fidofio 00.0.3 05.03958: was birds Hmufiucfigo 23 mo cowuoom macho : .m wudmfih Sent 2.5.— L1 Asimov 6.25358 Com _fl .m .m upsmE 103 the ratio of the resonant length to radius to be approximately one for a 3. 03 Ghz incident signal. A cavity radius of two inches (5. 08 cm) was determined to meet the aforementioned design criteria. Several of the resonant modes of this cylindrical cavity are shown on Figure 5. 4. In this Figure, the resonant frequency is shown as a function of cavity length. It is seen that the resonant length for the TE111 mode at 3. 03 Ghz is approximately two inches. Also shown on Figure 5. 4 is the resonant frequency versus length for a few selected, coaxial waveguide modes These modes exist in the cavity if a small metal center conductor is placed concentrically inside the cavity. These modes are of interest because a very dense cylindrical plasma has electrical properties similar to a metal rod inside the cavity (see. Figure 4. 4). As in the reentrant cavity, a photodiode is placed in a small hole in the cavity wall to measure the luminosity (density) of the plasma. Also shown on Figure 5. 3 is a backward wave sweep oscillator (BWO). The 3. 03 Ghz source is used to drive the radio frequency plasma in the cavity; while the sweep oscillator is used to probe the plasma by observing shifts in the resonant frequencies of other mode 362' 63. The sweep oscillator is protected from coupling to the higher power 3. 03 Ghz source by a circulator and a stub filter, which presents open circuit impedance to the 3. 03 Ghz signal. The cylindrical cavity is driven by the microwave system shown in Figure 5.1 by the radial probe. This coupling probe excites the 104 .muwm >fi>mo mdmpo> >200:ng unmcomwom Jim 0.53m .80.m ._. 1H om . 2 3 E .2 . o.“ wb’ h b ‘ I U358 003 030382005 #30300 I I l I Omocoa 3500 #30300 II II J; m oboe 3300 30:902.? / o v N. o d P b dl 11 db ‘1 -(r O NnO .m 105 radial electric fields in the cavity. The coupling to the cavity can be adjusted by varying the length of the probe. It was absolutely necessary to use probe coupling, because the magnetic field in the cavity is very small for certain plasma densitie856. Thus, loop coupling is not efficient. Aperture coupling was not feasible because the microwave drive system is a coaxial transmission line. In addition to the above primary design criteria, the cylindrical cavity has many other features which may be used in future research. The electric fields of the TM011 mode are axial quadrupole modes, which are capable of exerting radio frequency forces on a plasma for confinement purposes83. Likewise the TEle mode is a transverse quadrupole mode . Thus further investigation of these modes is justi- fied by the continued quest for a controlled thermonuclear reaction. As mentioned in section 2. 5, it is possible to measure the radial plasma density profile of a dc plasma cylinder by comparing the 7 62 63 shifts of resonant frequency of two suitably chosen cavity modes ’ . This result can theoretically be extended by using the TM d 010 an TM modes in the cylindrical cavity to measure the radial profile 110 :1: of a resonantly sustained plasma driven in the TE111 mode. Also, the axial density profile can be measured by comparing the frequency shifts of the TM010 and TM011 modes. These profile measurements could also be performed on resonantly enhanced plasmas41 to deter- . . . . . . . 45 mine‘ if profile changes contribute to the nonlinearities observed . 106 Finally, for many purposes the cylindrical cavity is a more logical microwave device to study Tonks-Dattner resonances than the usual waveguide method (see Figure 5. 7). When driven in the TElll mode, the fields in the cavity are overwhelmingly dipolar; and thus the monopole and quadrupole fields,which are excited in the 27, 76 usual scattering experiments, should, not be present Also, the quadrupole Tonks-Dattner resonances could be studied by exciting the TE:11 mode in the cavity16'17. CHAPTER VI EXPERINIENTAL RES ULTS- - LINEAR OPERATION A necessary step in the development of understanding of the nonlinear operation of the cavity-plasma systems is the investigation of the linear resonances. Thus this chapter will present the experi- mental results of linear (i. e. , low incident power) operation of the reentr ant and cylindrical cavity-plasma systems. The observed resonances are classified, and the theoretical results of Chapters III and IV are used to explain the eXperimental observations. The power absorbed and input impedance of the cavities are measured as the cavity size, incident frequency or dc plasma density are varied. In this linear operation, the plasma in the cavity is created and maintained by the dc discharge only. That is, the steady state plasma behavior is effected very little by the absorbed microwave power. 6.1. Reentrant Cavity A. Classification and Interpretation of the Resonances The reentrant cavity shown in Figure 5. 2 is driven by the microwave circuit shown in Figure 5.1. The plasma in the gap 107 108 region is created by the dc discharge in the plasma tube. When the incident power is less than 30 mw, the response of the cavity-plasma system is linear. The resonances are observed by monitoring the reflected power or measuring the input impedance of the cavity- plasma system. Figure 6.1 shows a typical experimental oscilloscope trace of absorbed power versus the discharge current in the plasma tube. The resonances of the cavity-plasma system are observed as in- creases in absorbed power. This absorbed power is sensed by the directional coupler shown in Figure 5. l, rectified by a crystal detector and displayed on the vertical axis of an oscilloscope. The discharge current is measured by the voltage across the small resistor in the plasma tube circuit. From the theoretical predictions of Chapter III and experi- mental observations, the resonances of the cavity-plasma system are classified into three groups64' 65: a. electromagnetic cavity resonances, which are perturbed by the presence of the plasma, b. cold plasma resonances, which do not exist in the empty cavity and are predicted by cold plasma theory, c. warm plasma resonances, which do not exist in the empty cavity and are only predicted by warm plasma theory. In Figure 6.1, the large dip on the left is the perturbed electromagnetic cavity resonance (a). The dip on the right, in the 109 a .50 is ,n. . 4 Eric... as... new A m. 24m 7.03.70.» . Eu 0:. VN Janina: 41:9: unfit i2:— z.»~3m_r 32.0 12:92:02 £753.; wEmflu 21; 1 f >0ccavcc. 0:9; 4.32.2330 .7322.— 6.2.3,: -3300 .E. .3 7.5.2520 naked as was: cacouozzuao ._ .s iii SON A u v— 110 high plasma density region, is a cold plasma resonance (b), and the two small dips in the middle are the warm plasma resonances (c). These cold and warm plasma resonances can be excited only when they occur in the neighborhood of the perturbed cavity resonance. That is, the short length, Ls’ in Figure 5. Z is adjusted to cause the perturbed electromagnetic cavity resonance to occur in the neighbor- hood of the plasma resonances. The perturbed electromagnetic cavity resonance allows a sizeable electric field to exist in the gap and this excites the resonances in the plasma. The electric field in the gap is nearly axial and thus it is perpendicular to the flat faces of the plasma cylinder in the gap. The coupling to the cold and warm plasma resonances can be improved over that shown in Figure 6.1 by adjusting the cavity size to Cause the electromagnetic cavity resonance to overlap with the plasma resonances, as in Figure 6. 2. This coupling of the perturbed electromagnetic cavity resonance and the warm and cold plasma resonances is due to the strong interaction of the evanescent solenoidal fields in the gap and irrotationaly fields in the plasma resonator. To theoretically analyze the cold and warm plasma resonances, the plasma cylinder in the gap is considered to be a section of plasma waveguide. This plasma waveguide is terminated at each end by the coaxial cavity to form a half wavelength cylindrical plasma resonator of length LP, the length of the gap. The observed cold and warm lll . _ .o FEEL ma {33:30:15:02.: :7... ~r..~ 41.10:..W.;~L arr—L4J3 fur—u Lu—o} ~.vf«L(«~:_ ..E41 ,1...:u5:::0:._.. UQLEZLLQ .~ L:>...3 .56.:2 . Lubriizc 1597;121:147. .7052; ..£.H .N ... 111...: 90 llZ plasma resonances are essentially the eigenfrequencies of this cylindrical plasma resonator. Because the fields incident on the gap region from the coaxial section of the reentrant cavity are rotationally symmetric, the wave- guide theory developed in Chapter III can be used to analyze the cylindrical plasma resonator in the gap region. For the small gap lengths reported here, the axial wave number, kz, is large, k : 211' z 2L P It was shown in Chapter III that for large values of kz, far to the right of the light line in Figure 3. 2, the fields outside the plasma cylinder are strongly radially evanescent. Thus the outer conductor of the coaxial cavity has a negligible effect on the plasma resonances. The theoretical nature of the cold and warm plasma resonances is now clearly understood. The cold plasma re sonance in Figure 6.1 08 001 resonance on the (in the high plasma density region) is the TM cylindrical plasma resonator in the gap region. As stated in Chapter III, this mode occurs in the overdense (ypz > 0) region where the irrotational fields are surface waves. That is the irrotational fields vary as modified Bessel functions, Ikwpr). Because this “4331 resonance is also predicted by cold plasma theory, Te = 0, it is called a cold plasma resonance (b). 18 The warm plasma resonances in Figure 6.1 are the TMOOI' 1 = l, 2, 3, resonances on the cylindrical plasma resonator. These 113 resonances occur in the underdense (ypz < 0) region where the irrotational fields are radially standing waves. That is, the irro- tational fields vary as a regular Bessel function, thlp'r). These resonances are not predicted by cold plasma theory, rather only from warm plasma theory, Te > 0. Thus they are called the warm plasma resonances (c). The experimental variation of the resonant frequency versus oe plasma density is shown in Figure 6. 3. The experimental TM001 1e and TM001 , 1 = l, 2, 3, 4 resonances have been observed. Special care was tax en when the data for Figure 6. 3 was recorded to adjust the perturbed electromagnetic cavity resonance to the same relative position with respect to the cold and warm plasma resonances. This technique was necessary to avoid the interaction of the resonances shown in Figure 6. 2 and thus to obtain valid data points. The dc plasma density in the tube was measured by the TM 62, 63, 73 . d . . 010 cy11n rical cav1ty, frequency shift method Extension of the eXperimental curves in Figure 6. 3 for larger plasma densities was not possible due to current limitations of the dc discharge tube. Also shown on Figure 6. 3 is the theoretical prediction for the TMggl resonance given by equation 3. 24. As discussed in section 3. 3, the third term of this characteristic equation3. 24 has been modified to the term given in 3. 35, which includes the effects of the glass tube surrounding the cylindrical plasma resonator. Equation A1 of . 32 . . . O'Brien was used to calculate the effective dielectric constant, 114 Simeon «Emma mamuo> moocmCOump mEmmHa Sum? was Boo mo coSmCQ/ 2330.865 cam amusoewpoaxm m .o ouswwh m-EuoHoH x >fimGoQ mamgnw 0N ma 0H m o a d J a O “WWEH .mmm «EmmHnH 8.235-.. .mxm + l a HAMMER .mom mammfinm 300-.. .mxm .IIOIII HOOEH .mom «Emmfinw UfiooulfioonH 60 I N 115 6 eff’ of the surrounding media. This equation yields, Eeff : 5' O forE = 5.1, k = Zn/ZL , L = 0.1 cm. 8 Z P P Numerical methods similar to those described in Appendix B were used to solve the characteristic equation with an assumed electron temperature of 30, 0000K. The theoretical prediction, shown in Figure 6. 3, is in excellent agreement for the eXperimental TMggl resonance. The theory of Chapter III, which used a uniform density profile, is not capable of quantitative predictions for the warm plasma, TMégl, resonances (see section 2. 3 for a complete discussion of this concept). However, an improved theory, similar to Parker, Nickel and Gould's“), using a nonuniform density profile 1e 001 modes on the should have quantitative agreement with the TM cylindrical plasma resonator. B. Cavity Equivalent Circuits and Input Impedance The perturbed electromagnetic cavity resonance, (a), shown in Figure 6. l and 6. 2 is the primary interest of this section because this resonance will be used to resonantly sustain the plasma in Chapter VII. The basic nature of the electromagnetic cavity resonance Without a plasma present was described in section 5. 2. The experimental input impedance of the reentrant cavity Versus the cavity length is shown as the solid curve on the Smith CZhart8o in Figure 6.4. The impedance was measured using the 116 i‘: E... »#.W . .... . «sausage». 4 ............... I.“ 6 off... in . ‘6éfi on... $0.... . £¢¢bfil~4 o g...» .hhnflwnu- 9 owe»... . aura»... ...............w~. . as...» «assent»; ...1........ sass. i. .flmJ a... ‘ so u so. In“. 0 .u f II ' .. .V O O. ‘9 O O. O 000 a «guovfiouoo . for the Input impedance venue short length. L reentrant cavity. Data points are data. Figure 6. 4. 117 "double detector, ” minimum shift, SWR technique (see section 5. l) on the slotted line shown in Figure 5.1. There is no plasma in the cavity for this plot and the input frequency is 3. O3 Ghz. Several features of the cavity can quickly be determined from . . . . 80, 86 , a Smith Chart plot of its input impedance . The magnitude of the reflection coefficient, lI‘l, for a given impedance is proportional to the distance from the center of the Smith Chart to the location of the impedance point. Also, the power absorbed by the cavity is, z Pab = Pin(l - Ir‘l ) 6.2 where Pin is the incident power. Thus for a given value of incident power, the power absorbed will be large for impedance points near the center of the Smith Chart; or the power absorbed will be small for impedance points near the edge of the Smith Chart. In Figure 6. 4 the input impedance of the cavity forms a circle on the Smith Chart as the cavity length is varied. This cir- cular nature of the resonance is typical of Smith Chart plots of cavity input impedance80. Because the circle passes through the center of the Smith Chart, the cavity is critically coupled80 and the empty cavity resonant length, Lso' is l. '78 cm. Notice that for cavity lengths slightly different than Lso’ the cavity is greatly detuned. That is, II‘ I xi 1 and from equation 6. 2, Pa 0. Alternatively, b W when operating in the critically coupled position, L8 = L80, IF] = 0, so all of the incident signal is absorbed by the cavity. 118 The impedance jXL shown in the equivalent circuit 5. 2b, can be determined from Figure 6.4. This impedance is caused by the evanes- cent fields which exist in the neighborhood of the coupling 100p shown on Figure 5. Za80. Subtracting the inductive impedance, jXL, from the input impedance of the cavity rotates the impedance plot on the Smith Chart to the normal80 position shown as the dashed curve on Figure 6. 4. All input impedance plots for the reentrant cavity given in this section and in Chapter VII will have the jXL impedance subtracted out. This will allow a clearer view of the true nature of the cavity- plasma system input impedance. When a dc plasma is present in the gap region of the cavity and the input frequency is fixed at 3. O3 Ghz, the resonant length of the perturbed electromagnetic cavity resonance increases as the plasma density increases. This behavior of the reentrant cavity is similar to the effect observed in Figure 4. 6 for the cylindrical cavity. That is, for a fixed input frequency, the cavity size increases for larger plasma densi- ties. Figure 6. 5 shows the eXperimental shift in L8 as a function of electron plasma frequency. In order to gain an understanding of this effect, the reentrant cavity has been modeled as two transmission lines as shown in Figure 6.6. The transmission line on the right, (b), is the trans- mission line equivalent of the TM waveguide mode80 described by 01 eQuation C44 of Appendix C. This line has length LP, the length of the gap region, and is short circuited by the end of the cavity. As 119 4 - / / / / / / / / 3 -. / we // 2" , Ghz 2 db W Experiment l .. — —- — Theory 0 r A, 1.75 1.80 1.85 1.90 L , cm s Figure 6. 5. Resonant length versus plasma frequency for perturbed electromagnetic cavity resonance. Input frequency is 3. O3 Ghz. 120 (a) (b) o. fla’ z'ca 'X b short J cb circuit #WA+ 44 Figure 6.6. Equivalent circuit of the reentrant cavity consisting of two transmission lines. . shown in section 5. 2, only TMom’ m = l, 2, 3, . . . modes are excited in the gap region, and a single mode expansion4 will be used here to model this segment of the reentrant cavity. The characteristic impedance of this line is jXCb and its attenu- ation constant is ob. The characteristic impedance and the "propaga- tion constant" are imaginary because for the dimensions of the reentrant cavity, the TM01 waveguide mode is cut off. With a cold plasma cylin- der located concentrically inside the waveguide, the cut off frequency of the TM01 mode is a function of plasma density (see Appendix E for details). Thus the characteristic impedance and the attenuation con- stant, which are functions of the cut off frequencyao, will depend on the plasma density. The transmission line on the left, (a), in Figure 6. 6 repre- sents the segment of the reentrant cavity formed by the 50:2 trans- mission line. The length of this line is L8 and its propagation constant is Ba : w/c. The impedance jXL which terminates this line is the impedance due to the evanescent fields in the neighborhood of the cOupling loop. This impedance is known from Figure 6.4. 121 For the reentrant cavity to be in resonance, the reactance of the gap region added to the reactance of the coaxial section must sum to zero, (see equation 5.1). For the transmission line representation given in Figure 6. 6, this is written, JXL+ 350 tan([3aL8) 50 -jX »tanh(o L )=0, 6.3 50 - XLtanUBaLs) cb b p where, .1. 2 fc2 ch ’ *1 T ‘1 6 4 f L 2 2 f 21rf c CL1:) T _2' '1 6'5 f T] : IZOWQ ' 6.6 fc = cut off frequency for the TM waveguide 01 f = 3.03 Ghz, input frequency. As stated, the cut off frequency, fc, is a function of plasma density. Using Figure E1 of Appendix E for fc versus we, equation 6. 3 has been solved for L!3 versus we. The results, shown in Figure 6. 5, are in fair agreement with the experimental observations. In a more complete description of the reentrant cavity, the gap would be modeled using all the TMom' m : O, l, 2, 3, . . . , modes of the waveguide shown in Figure 4. 1. In addition, the mom’ m = l, 2, 3, . . . , modes on a coaxial waveguide are excited by the gap discontinuity in the coaxial section of the cavity. The full use of these infinite sets of 122 modes would be expected to give better quantitative agreement with the experimental results of Figure 6. 5. The increase in resonant length of the cavity as the plasma density increases can also be observed in the input impedance of the cavity. For L8 = 1.84 cm, the impedance of the cavity is shown in Figure 6. 7. This length is larger than the empty cavity resonant length, Lso = l. 78 cm, and so the reactance is inductive for low values of plasma density. Increasing the plasma density critically couples the cavity, and a further increase in the plasma density causes the impedance to become capacitive. The impedance then approaches an antiresonant condition for very large values of plasma density. Thus for increasing cavity lengths, the plasma density must be increased to resonate the cavity. For a further increase in the cavity length, L8 = l. 92 cm shown in Figure 6. 7, a very large plasma density is needed to re sonate the cavity. However, the cavity is no longer critically coupled for this value of length and plasma density; rather it is over coupledgo. Thus, in this resonant condition, only part of the incident signal can be absorbed by the cavity-plasma system; the rest is reflected. This reduction in the coupling to the cavity-plasma system is dramatically shown in oscilloscope trace Figure 6.83. This photo- graph shows the power absorbed by the cavity versus discharge cur- rent for four values of Ls' all exposed on the same picture. The incident frequency was reduced to l. 5 Ghz in order that the dc plasma 123 1.84 cm 1.92 cm Figure 6. 7. Cavity impedance versus discharge current (noted in milliamps). 124 £ . twilixiaiiwiiipleilgwu. a.-. . . . .4. ea“.-. . H. d L .x l ‘* Figure 6. 83. Oscillogram of power absorbed versus discharge current for four values of Ls. Input frequency is l. 5 Ghz. l abs P. in 0 I n 1 J o i 2' 3 4 5 N , cm- c Figure 6. 8b. Envelope of power absorbed versus plasma density, in critical densities, for increasing cavity size. 125 density attainable in the plasma tube would be several critical densities*. The envelope of the traces in Figure 6. 8a is shown in Figure 6. 8b. The discharge current has been normalized in terms of critical densities for the incident 1. 5 Ghz signal. This envelOpe shows that for low density Operation the cavity-plasma system is critically coupled (complete absorption of the incident signal); but there is nearly com- plete reflection of the incident signal for densities greater than five critical densities. The explanation of this unanticipated impedance mismatch between the microwave exciting system and the cavity-plasma system is based on the equivalent circuits, Figures 6. 9a and 6. 9b. With only a low density plasma in the gap, the reentrant cavity is very similar to an open circuited, quarter wave transmission line resonator. The appropriate ecpivalent circuit for such a resonator is the RLC series representation shown in Figure 6. 9a86. However, for large plasma densities (several critical densities) the electrical properties of a plasma are similar to a metal rod. Thus a dense plasma short circuits the gap; and the cavity is then operating in a manner very *A critical density, N , is defined as that density for whichu) = (i). That is, for a frequeiicy a), e a) meEo N = -—-—- . c 2 e 126 _. jX n:l Figure 6. 9a. Equivalent circuit of quarter wave, open circuit transmission line resonance. JXL . Rb __ c , ’15 b L1:. n: 1 Figure 6. 9b. Equivalent circuit of quarter wave, short circuit transmission line antiresonance. 127 similar to a short circuited, quarter wave antiresonance. The appro- priate equivalent circuit is the RLC parallel representation shown in Figure 6. 9b86. The values of the R's, L's and C's used in Figures 6. 9a and 6. 9b are functions of Ls’ Lp, (s, me, etc. The turns ratios on the transformers and the phase shift jXL are the same as in equivalent circuit 5. 2b. The detuning of the cavity-plasma system illustrated in Figures 6. 7 and 6. 8 is understood by considering the circuits 6. 9a and 6. 9b. For low plasma densities, the cavity is critically coupled and the appro- priate equivalent circuit is 6. 9a. However, with a cavity size of approximately a quarter wavelength, as the plasma density increases the cavity is operating in an antiresonant ( '2‘ -> oo) condition. This condition is described by equivalent circuit 6. 9b. Thus it is the transi- tion from low density resonance to the high density antiresonance which decouples the cavity for the eXperimentally available, moderate plasma densities. This decoupling effect for high density plasmas will limit the ability of the reentrant cavity to generate a dense, resonantly sus- tained plasma in Chapter VII. In conclusion, the reentrant cavity-plasma system has been experimentally studied. New warm and cold plasma resonances, which do not exist in the empty cavity, have been explained by the bounded plasma theory develOped in Chapter 111. These resonances can only be excited when they exist in the neighborhood of a perturbed cavity resonance. 128 Also, two equivalent circuits for the perturbed electromagnetic cavity resonance have been developed. These apply to the experimentally observed low density, quarter wave resonance, and the high density, quarter wave antiresonance. With these two equivalent circuits, the decoupling eXperienced by the cavity-plasma system for large plasma densitie s is unde rstood . 6. 2. Cylindrical Cavity In this section the low power linear characteristics of the lowest order dipole modes of the cylindrical cavity-plasma system are pre- sented. This cavity, shown in Figure 5. 3, consists of a plasma rod located concentrically inside a metal cylinder. For low incident microwave power levels, the plasma is created and maintained by a dc discharge. The dipole resonances of this cylindrical cavity-plasma system have been theoretically analyzed in section 4. 3. Figure 6.10 shows the input impedance of the TE 111 mode versus cavity length. The incident frequency is 3. O3 Ghz and the plasma density is zero. As in the previous section for the reentrant cavity, the impedance plot forms a circle on the Smith Chart when L8 is varied from 5. 80 to 6. 20 cm. It can be seen from Figure 6.10, that the TE111 resonance of the cylindrical cavity is under coupledgo, because the normalized resistance is less than one at resonance (resonance is defined at the point on the circle where If] is minimum). In addition, the reactance associated with the energy stored in the evanescent fields 129 u 607 a... Figure 6. 10. Input impedance versus cavity length without a plasma for the TE 11 cavity mode. The numbers refer to the short fength, Ls’ in cm. Input fre- quency is 3. 03 Ghz. 130 in the neighborhood of the driving probe is determined from Figure 6. 10. In all future impedance plots for the cylindrical cavity, this reactance, jXC, will be subtracted to eliminate this capacitive phase shift. Figure 6. llc shows a typical power absorbed versus discharge current oscillogram for the cylindrical cavity-plasma system. The resonance in the high density region (almost off the photograph) is the TETII resonance. In addition, the first six TDlll resonances are also observed. It can be seen from this oscillogram that the higher order TD resonances (in the low density region) are weakly 111 excited. (Figure 6. 11 will be discussed in more detail later in this section. ) The input impedance of the cavity-plasma system for increasing discharge current (plasma density) is shown in Figure 6.12. Because the cavity's length has been adjusted to be longer than the empty cavity resonant length, the reactance of the input impedance is inductive for low plasma densities. Also, the cavity is poorly coupled. However, as the plasma density increases, the impedance follows a series of swirls. Each loop in Figure 6.12 corresponds to a TDlzl resonance. That is, the cavity is locally absorbing a maximum amount of power at each resonance, because the magnitude of the reflection coefficient has a minimum. The four swirls in Figure 6. 12 correspond to the TD” 1, 1 = l, 2, 3, 4 resonances observed in Figure 6.11c. 131 L,,s.as 6.05 6.15 6.30 I, Invurn‘ Ir. (Nxxlluuramul pu\\l'r :lll:n"|x"1\,lr~~\l- uh“ hay-u- r urrvnt. Tht- (our plinh»;rar)hs hil\1' 1.. ‘n “-. 1,31%, (1. 1S and M. H) (”I V‘(':[)('\Ll\'1'l\. HIM) [Luv is lhv .wrn lino rvl'vrt-m r~_ 132 Figure 6.12. Input impedance versus discharge current in ma. for the cylindrical cavity-plasma system. LS = 6.10 cm. 133 From Figure 6.12, it is seen that the lower order TDlll modes are better coupled to the driving microwave system. In fact, the TD resonance is critically coupled (i.e., it passes through 111 the center of the Smith Chart). The excitation of higher order TD”l resonances is prevented by the impedance mismatch between the driving microwave‘system and the cavity-plasma system. Thus, it is expected that a decrease in the characteristic impedance of the driving microwave system would give better coupling to the higher order TDMl modes (i.e. , Tonks-Dattner resonances). Figure 6.11 displays a sequence of oscillograms which show the power absorbed by the cavity-plasma system versus discharge current (plasma density). The top trace in each photograph is the zero line for complete absorption of the incident signal. This sequence of oscillograms was taken for increasing values of cavity length, Ls' In Figure 6. 11a, the length of the cavity has been adjusted to be less than the empty cavity resonant length. It is seen that the TD , 1 = l, 2, 3, resonances are weakly excited in this trace, with 111 the TDlll most strongly excited. In Figure 6. 11b, the short length has been adjusted to a value slightly larger than the empty cavity resonant length for the TElll a: mode. Thus the TE111 resonance appears as the broad resonance in the low density end of Figure 6. 11b. In addition, the existance of the * TE111 mode allows a sizeable solenoidal field to exist in the cavity. It is this electromagnetic field which excites the irrotational fields of 134 the TD111 modes. Thus, the first seven TDMl modes can be observed. an: In addition, the coupling of the TD”1 modes and the TE111 mode has shifted the TD111 mode to a higher density. This shift is qualitatively predicted by Figure 4. 6. A further increase in the cavity length gives the result shown in Figure 6. 11¢. This resonant spectrum is similar to the coupling region shown in the insert on Figure 4.6. The resonance on the far >'.< right is now the TE111 resonance (only half of it appears on Figure 6. 11c). The TDul resonances are still strongly excited, but com- parison of Figures 6.11b and 6. llc shows that they are shifting into the higher density region. Finally, for a cavity size much larger than the empty cavity resonant length, Figure 6. 11d, the coupling to the TD resonances has greatly decreased. The position of the TD 111 111 mode is approaching the value it had in Figure 6. 11a (see insert in Figure 4.6). The TE):11 resonance has moved off the picture into the high density region. The eXperimental results of resonant length versus plasma frequency* has been summarized in Figure 6. 13. It is seen that the theoretical predictions of Figure 4.6 are in excellent qualitative agree- >5: ment with the experimental observations. Excitation of the TM101 *In Figure 6.13 and 6.14, the electron plasma frequency has been approximated by the square root of the discharge current. This is justified by the fact that the discharge current is approximately proportional to the average plasma density. 135 *r 6-4? TD141 TD131 TD121 L,cm s 6.0 .flzmcow manna H308 .Q Since“. mamma H33 m5mp0> “contamnm meonH .m v .N. oudmfm .m» hoe/om on ---------A E r— neuron \ m $2m..: 3V :3 148 the cavity will correspond to the equilibrium points in Figure 7.4a where power absorbed matches power loss. (The power loss line in Figure 7. 4a applies only to the resonantly sustained condition. ) Although two such equilibrium points exist, only g on the high density side of the power absorbed curve is stable. Also, as pointed out in section 2.4, the power absorbed curve must have a stable inter- section with the power loss line to resonantly sustain the plasma. Alternatively, if only a dc plasma exists in the cavity, all states in Figure 7. 4a are stable. This only happens when the inci- dent power is low enough that there is no radio frequency ionization of the plasma. Using Figure 7.4, it is possible to understand the intermediate case of moderate incident power levels where both dc and radio frequency power contribute to create the total plasma * density . Consider first incident power level Pin in Figure 7.4. For 1 this low power level, there is no radio frequency ionization of the plasma. Thus the total density follows the continuous path AD in Figure 7. 4b as the dc plasma density (discharge current) is varied. Thus the power absorbed versus total density follows the path ad in Figure 7. 4a on curve 1. These two curves are combined to give * . . Concurrent with this research, Leprince, Matthieus sent and AllisB'7 have developed an alternative analysis of this effect. 149 path a'b' in the power absorbed versus discharge current in Figure 7.4c. Thus for low incident power (linear operation), Figure 7.4 predicts the linear response of the cavity—plasma system observed in Figure 7. la. For the incident power level Pinz in 7. 4, the dc density of the plasma can be resonantly enhanced by radio frequency ionization in the neighborhood of the cavity-plasma system resonance. That is, some of the incident power absorbed by the plasma ionizes the plasma. This moderate incident power operation of the cavity-plasma system, shown in Figure 7. 1c, is also qualitatively explained by Figure 7. 4. Starting from point A, the total plasma density rises linearly with the discharge current until point B. The state of the system in Figure 7. 4a follows the path ab as the cavity be comes progressively better tuned by the increasing dc plasma density. However, at point b, the cavity-plasma system is absorbing enough radio frequency power to cause radio frequency ionization in the plasma. This new ionization causes an increase in the total density N, without an in- crease in the dc density. From Figure 7. 4a, the increase in the total density better tunes the cavity-plasma system, resulting in more power absorbed and more radio frequency ionization. This type process (similar to the stability discussion for Figure 2. 3) results in the jump from b to c. State c is stable because an increase (or decrease) in the total density results in a decrease (or increase) in the power absorbed. El {.17. .11. if." '11! 150 The jump b - c (a - will be used to denote jMps) in Figure 7.4a is observed as the jump B - C in the total density versus dc density in Figure 7.4b. Also, the power absorbed versus discharge current will follow the path a'b' - 'c' for increasing discharge current. From points c, C or c', a further increase in discharge current will detune the cavity, following the paths cd, CD or c'd' reapectively in Figures 7.4a, b and c. Starting at the high level of discharge current (point D), de- creasing the dc plasma density will cause the cavity plasma system to absorb radio frequency power. However, the high density side of the power absorbed curve in Figure 7. 4a is stable; thus there are no jumps until point e is approached. As the state of the system approaches e, the plasma is resonantly enhanced. That is, the total plasma den- sity is the sum of a dc part (supplied by the discharge current) and a radio frequency part (supplied by radio frequency ionization of the plasma). At point e the cavity-plasma system is absorbing all of the incident power; and the radio frequency plasma density can not increase to compenstate for a further decrease in dc plasma density. This results in the state of the system jumping from e to f. This jump appears as E - F in the total density versus discharge current plot, 7.41), and in the jump e' .- f' in the power absorbed versus dis- charge current curve, Figure 7. 4c. Further decreases of the dis- charge current will follow the linear curves fa, FA or f'a'. 151 It is clearly seen that the graphical analysis in Figure 7. 4 explains the observed hysteresis effects in Figure 7.1c. The state of the system follows the discontinuous path ab - cde - fa. This results in the hysteresis loop a'b' - c'd'e’ - f'a' in the experimentally observed power absorbed versus discharge current. Also, the total density versus discharge current will follow the hysteresis loop AB - CDE - FA. Finally, Figure 7.4 shows that radio frequency ionization of the plasma is the dominant mechanism which causes the nonlinear effects observed in the power absorbed curves in Figure 7.1c. There has previously been speculation as to whether these hysteresis effects were caused by nonuniform E force 344, nonlinear shifts in the resonant frequency“, or ionization. For the incident power level P1113 in Figure 7.4, the hysteresis 100ps in the power absorbed and total plasma density versus discharge current are now more pronounced due to the higher level of incident power. This result is similar to the experimental observation of Figure 7.1e. Finally in Figure 7.4, the incident power, Pin4’ has been increased to a level greater than the power loss in the cavity-plasma system (Ploss in Figure 7. 4a). That is, power absorbed by the cavity-plasma system is greater than the threshold necessary to resonantly sustain the plasma. In this case, the hysteresis effects in Figures 7.4b and 7.4c have disappeared. Instead, as the discharge current is varied, Figures 7. 4b and 7. 4c follow the continuous paths 152 DC and d'g' respectively. For high values of discharge current, the state of the system in Figure 7. 4a is at point d. When the dc plasma density decreases, the state of the system approaches point g. If the dc plasma is removed, the cavity-plasma system will be resonantly sustained at point g. The jump in the input impedance observed in Figure 7. 2 is similar to effects observed in other eXperimental microwave- 28, 45 plasma systems . It is recognized that the cavity-plasma system can develop an instability when alrl > O. . BN 7 1 That is, if [1"] decreases for increasing plasma density, that por- tion of the Smith Chart plot may be unstable- A threshold condition for this instability has recently been published87. Similarly, when ELF—[<0 7.2 8N that portion of a Smith Chart plot is stable and jump discontinuities do not occur. Thus the graphical analysis of Figure 7. 4 explains the non- linear effects observed in the resonances of Figure 7.1. Further- more, arguments similar to these could easily be developed to explain the hysteresis effects observed in Figure 7. 3. The only difference would be that the curves representing the state of the system would have more than one resonance, as in Figure 7. 3a. 153 B. Re sonantly Sustained Discharge It was observed in Figure 7. If and 7. 3f that it is possible to resonantly sustain the plasma in the reentrant cavity with an incident power level of approximately two watts. However, as demonstrated in Figure 2. 3, it is not possible to generate a dense resonantly sus- tained plasma by merely increasing the incident power level. This fact has been eXperimentally verified for the reentrant cavity-plasma system. In Figure 7. 5 the luminosity (plasma density) has been plotted as a function of incident power. The incident frequency is 3. O3 Ghz and the cavity length is l. 80 cm. For zero incident power, a small discharge current gives a reading for plasma density of approximately 0.1 critical densities. For an incident power of 4 watts, the plasma density rises sharply due to radio frequency ionization. However, further increases in the incident power up to 20 watts do not appreciably increase the plasma density. As was shown in section 2. 4, the presence of the resonantly sustained plasma detunes the plasma system (see Figure 2. 3). Thus large increases in incident power result in only a very small increase in absorbed power and hence a small increase in plasma density. Thus, a retuning technique is necessary to overcome the detuning effects of the resonantly sustained plasma. One method which can be used to retune the cavity-plasma 41, 66, 67 system is the frequency shift technique To test this method, 154 .Eo ow; n m4 3330a “commas“ mamuo> Kinsmen mEmmHnH .m .N. 95me .n 333 .:.m m a o v N o u a u U z .3 a: 155 the frequency stabilized source in Figure 5.1 was disconnected and a variable frequency generator was substituted to feed the TWT. Experimentally, it was very difficult to maintain a constant output power from the TWT, because the output of the generator varied greatly with frequency. Continual adjustments were necessary to keep from overdriving the TWT and yet hold the output at a constant 20 watts. With the short length set at 2. 2 cm, a resonantly sustained plasma of over eleven critical densities was excited by increasing the input frequency from 2. 70 to 3. 04 Ghz, a shift of about 10%. Although this experiment verified the frequency shift retuning tech- nique, it was difficult to use because of the nonlevel output of the available variable frequency source. Another very effective, easily implemented retuning method for the reentrant cavity-plasma system is to increase the cavity size, by increasing the short length Ls, when the plasma is resonantly sustained by the 3. 03 Ghz incident signal. A plot of experimental density versus short length, for an incident power of twenty watts, is shown in Figure 7. 6b. A typical experimental run proceeds as follows. At A in Figure 7. 6b, the cavity size is such that a nonresonant condition eXists for the driving frequency 3. 03 Ghz. Thus, the plasma is entirely sustained by the dc discharge. As the cavity length increases from A to B the low density cavity-plasma resonant length is approached, 156 cums“: ”Iona mamuo> ponHOmnm .330m .0 cameo.“ tone 26pm; Knudsen .n Simeon 36.3., mmoH hospoa was vengeanm uoBOnH .m 0.5 0.53m .839“ . .0 ADV A 5 $5.95 on n .nA Am m m A A . . . A v A A m A .832” ' U .0 .N HQWH 30 Ann I ll magmas“ I III Amy 157 and beyond B the cavity, although somewhat detuned, starts absorbing radio frequency power. This, in turn, causes additional radio fre- quency ionization resulting in the observed increase in luminosity (density). As the length L8 is increased further, the density increases until C is reached. At C the plasma density undergoes an abrupt decrease to D where the plasma is again maintained entirely by the dc discharge. This transition is irreversible and is labeled a "dynamic instability" or jump in keeping with similar phenomena observed in lower frequency discharges As the short length is decreased the plasma density remains at the dc level until E. Here there is a sharp retuning of the cavity- plasma system and the plasma density jumps to F on the original curve, since the plasma again becomes primarily radio frequency sustained. A further decrease in L8 decreases the density down the original curve to B where the plasma is sustained by dc energy. The graphical analysis presented in Figure 7. 6 explains the large increase in the resonantly sustained plasma density and the hysteresis effect. This analysis is fundamentally the same as Figure 7.4. That is, Figure 7.6 interconnects the three parameters of the cavity-plasma system: power absorbed, plasma denstiy and short length. In Figure 7. 6a a family of power absorbed curves is shown for increasing values of short length. That is, Figure 7.6a displays the experimental fact (established in Figure 6. 8a) that cavity resonant 158 length increases for increasing plasma density. Furthermore, the coupling to the resonance decreases as the plasma density and short length increase. This decoupling is shown in Figure 7.6a as the de- creasing envelope of the power absorbed curves, similar to Figure 6. 8b. It was shown in section 6. 1B that the transition from the low density resonance to the high density antiresonance decouples the cavity for moderate densities. Also shown in Figure 7.6a is a power loss line. This represents the sum of the power lost due to ohmic heating in the metal cavity, plus the power lost in the plasma due to inelastic ionization and excitation collisions and wall losses. In a resonantly sustained plasma, the power lost matches the power ab- sorbed4l. Figure 7. 6b displays an experimental curve of luminosity (plasma density) versus cavity length. The density traces a hysteresis loop ABC - DE - FBA as the cavity length is fir st increased and then decreased. Finally, Figure 7.6c displays the power absorbed (equal to power loss in Figure 7.6a) versus cavity length. This curve also traces a hysteresis loop a'b'c' - d'e' - f'b'a' as the cavity length is increased and then decreased. As the cavity length is increased from point A, it is seen in Figure 7. 6a that the stable, high density size of the power absorbed curve approaches the dc plasma density level. This allows the cavity to absorb some of the incident microwave power (see Figure 7.6c). This in turn causes radio frequency ionization of the plasma. 159 Further increases in the- short length shift the power absorbed curves in Figure 7. 6a to the higher density region, and thus the power ab- sorbed increases and the resonantly sustained plasma density increases. However, point c in Figure 7.6a is unstable because the power loss is equal to the power absorbed. This causes the jump in the density (C - D) and in the power absorbed (c' - d'). Thus, operation for lengths larger than C are not possible, since there is no inter- section of the power loss line and the power absorbed curve. As the short length is decreased from 2. 2 cm, it is seen from Figure 7. 6a that the unstable, low density side of the power absorbed curve approaches the dc density point, e. Thus, the state of the system jumps to the stable point f. (or F in Figure 7.6b and f' in Figure 7. 6c). Further decreases in short length detune the cavity and the state of the system returns to point a in Figure 7.6a. Thus the graphical analysis of Figure 7. 6 explains the observed hysteresis effect in Figure 7. 6a. The data of Figures 7. 6b shows that an increase in the short length of 5 mm, or 1/20 of a wavelength, increases the average plasma density to thirteen times the critical density, or to approximately 1. 5 x 1012 cm-3. Thus, varying the cavity size is an easily imple- mented retuning technique which can greatly increase the density of a resonantly sustained plasma. In Figure 7. 7, the variable length retuning technique is demon- strated for different incident powers. It is observed that doubling of 160 .moflfimcov 30320 CA @395th some. men 33058.3 och. .fwsoH toga weapon, >fimocfihdA .N. .s «.3th m 80. A m.~ N.N EN o.N 04 wA NA ” .r n . .r a o .flvd .IN Arm .r¢ lfm L10 .15 n. u m 2 4o .9: +3 #2 .1: x. 161 the incident power from ten to twenty watts, does not result in a doubling of the plasma density. In fact, because of the detuning of the cavity-plasma system observed in Figure 6. 8, increasing the incident power to levels above twenty watts will not greatly increase the density of the resonantly sustained plasma in the cavity. Figure 7. 8a and 7. 8b display the input impedance of the re- entrant cavity-plasma system when the increasing length retuning technique is performed. That is, Figure 7.8a and b shows how the input impedance varies as the plasma density is increased by in- creasing the short length. For an incident power level of four watts, increasing the short length clearly shows that the magnitude of the reflection coefficient is decreasing. Thus from equation 6. 2, the power absorbed is increasing (causing the rise in plasma density). However, when the incident power is increased to ten watts, there is very little change in the input impedance; although from Figure 7. 7 there is a large increase in plasma density. Finally in Figure 7. 8 with an incident power of twenty watts, there is almost no change in the input impedance of the cavity-plasma system, deSpite an increase in the plasma density to thirteen critical densities. In this high density condition, the reentrant cavity plasma system is operating in the antiresonant condition modeled by the equivalent circuit 6. 9b. Table 7.1 shows the magnitude of the reflection coefficient and power absorbed by the reentrant cavity-plasma system as the 162 P. in o 4 watts D 5 watts A 10 watts O 20 watts Figure 7. 8a. Input impedance versus short length for reentrant cavity-plasma system. r”.- “'_. “It“ n~ 163 Po 1nC O 4watts n Swatts A 10 watts O 20 watts Figure 7. 8b. Detail of 7. 8a. Short lengths are noted in cm. '2! 2. 9 O u 20 18 164 short length is increased. In the completely detuned position, Ls = 1. 7 cm, 3.6 watts are being lost to ohmic heating of the cavity. The power absorbed increases, as the short length increases, until 5. 0 watts are being absorbed. The difference of only 1. 4 watts is being used to resonantly sustain the plasma. Thus the variable length retuning technique for the reentrant cavity can create a dense resonantly sustained plasma; although it does not use efficiently the available microwave power. A Table 7. 1 Power absorbed by the reentrant cavity-plasma system versus short length. Pin = 20 watts. Ls’ cm II‘I . Pab' watts .91 .88 .87 .87 .87 .87 NHOOCDQ mmmmwsw OOOOO‘O‘ It appears that for some applications changing the cavity size can be a practical method of increasing the density of a resonantly sustained plasma in a cavity. This method satisfies the goal of obtaining a retuning technique for a fixed input frequency. Unfortunately, the drastic reduction in coupling (Figures 6. 8 and 7. 8b) eXperienced by the reentrant cavity at high densities made it impossible to use efficiently the available microwave power to drive 165 the plasma. Thus, for the reentrant cavity, it appears unlikely that densities greater than those obtained here are possible due to the decoupling effects. This problem is overcome in the next sec- tion for the cylindrical cavity, because there is more than one type of resonance to excite. In the reentrant cavity, only the perturbed electromagnetic cavity resonance is capable of exciting a dense plasma, since for the cold and warm plasma resonances on the plasma cylinder in the gap, we “ 0.). ”mm“ m One other type of experiment has been performed. By using an air filled plasma tube, it was possible to vary the background pressure in the plasma. As in the mercury plasma, large increases in luminosity were observed by using the variable length retuning technique; but this luminosity was not calibrated to measure plasma density. Also, it was observed that luminosity of the resonantly sustained plasma varied as the background pressure was changed. More work to determine the relationship between plasma density, cavity size, background pressure, and type of gas would be valuable to understand these phenomena. In conclusion, it has been shown that the graphical analysis of Figure 7. 4 ani 7.6 was useful to understand the nature of the observed hysteresis effects. Similar graphs could easily be con- structed to explain experimental results reported by others. The only modification would be that the horizontal axis would be the experi- . . 66 . . 68 mental variable, either frequency or an external Circuit element . 166 7. 2 Cylindrical Cavity Before discus sing the nonlinear Operation of the cylindrical cavity, the experimental result verifying that luminosity is pro- portional to plasma density will be presented. The luminosity is measured by the photodiode in Figure 5. 3 and the plasma density is * measured by the frequency shift of the TM cavity mode . The 010 dc plasma is. generated by the discharge tube and the resonantly sustained plasma is excited in another resonance of the cylindrical cavity by the 3. 03 Ghz source. The experimental curve, Figure 7. 9, clearly shows that the luminosity is proportional to plasma density for either a resonantly sustained plasma or a dc plasma; and the proportionality constant is the same for each type of plasma. The higher luminosity readings for the larger values of the dc plasma is caused by the change in background pressure in the tube due to ohmic heating. Thus Figure 7. 9 displays the fact that luminosity is proportional to plasma density for the same background pressure. The transition from the linear to nonlinear operation of the cylindrical cavity-plasma system is shownin Figure 7.10 for increasing *The exact solution (not perturbation theory) has been used to relate the observed frequency shift to the plasma density (see Figure 4. 5). Although a uniform density profile has been assumed, Figure 10 of Agdur and Enander shows that the use of a non- uniform profile does not effect the accuracy of this mode for the range of plasma densities excited here. 167 a rd. flan! . .GmEm >ocodvoum voHZH .3. pesdmmegv 33sec manna mam po> AdemoquE at .339»ng O .o .n «Ema Eu . Am- :o: z L w m w m N m o O \ O \ U .\ N \ K ¢ \ O r o o\ manna up O o\ \ w mEmeAm cocwmumsm. handsome." O \ O ofi \ K \ NA 1 m 0V .83 \ a; \ . A: \o o? %\ a R Vx . ON \o ,\>o A NN ,120mw , 300 mw 168 Figure 7.10. Luminosity (top trace) and pnwrr absorbed (liulluln trace) versus dim liargr current for int reusing lru-ls of ini'idvm puwrr (tylindrical (avityl. Hori/unlal axis, 40 Ina/(m 169 values of incident power. As in Figure 7. l for the reentrant cavity, this sequence of oscillograms for the TD modes displays the 111 luminosity (top trace) and power absorbed by the cavity (bottom trace) as the discharge current (dc plasma density) is modulated. The experimental set up is the same as described for the reentrant cavity in section 7. 1A. The experimental results are similar as those observed in 3 section 7.1A for the reentrant cavity. The phenomena are the same, and the graphical analysis of Figure 7. 4 can be applied to explain the hysteresis effects observed in Figure 7.10. Also, these results are very similar to hysteresis effects observed in Tonks-Dattner resonances observed in waveguide experiments (see Figure 4. 7) 28, 29, 43, 44, 45 It is observed in Figure 7.10d that it is possible to resonantly sustain the plasma in the cylindrical cavity in either the TD 111 or the TEIII resonances. With an incident power of 23 watts, Figure 7.11 shows the density versus cavity length, as the length is first increased and then decreased. It is observed that the density does not rise monotonically with length as it did for the reentrant cavity. However, the state of the system is still a function of its history (i.e. , nonlinear effects). In Figure 7. 12, the input impedance of the cavity is shown as the short length is increased and then decreased. The data for Figure 7.11 and 7.12 were taken on the same experi- mental run, and the corresponding points are labeled. The impedance 170 C .muums? MN H v.0 me am >fi>mo Hmoflupswfi>u ecu sumsoA tone mamum; crimson mEmflnH Eu .mA N \0 As one o.m m.m 1L. 1» db 4 .2 c. 23E 171 Figure 7.12. Impedance versus short length for cylindrical cavity. Points a, b, . . . , p correspond to points a, b, . . . , p in Figure 7.11. ”WM“?— 172 is seen to follow an erratic path in Figure 7. 12. However, there are two very important results which can be observed from Figures 7.11 and 7.12. First, when the resonantly sustained plasma density is large (points d, 1 or m in Figure 7.11), the impedame of the cavity-plasma system is in a resonant condition (points d, 1 or m in Figure 7. 12). This is different than the anti- 7 resonant condition of the input impedance of the reentrant cavity when the plasma is in a dense resonantly sustained condition. Second, from Figure 7.11 it is observed that when operating in the high density mode, the cavity length is approximately 5. 96 cm. This value is less than the 6. 07 cm empty cavity resonant length of the TE111 mode (see Figure 6.10). Thus the experimental high * 111 resonance. ’Also, the TD density mode can not be the TE 111 resonances are not high density modes because we m w for these resonances (see Figure 4.6). Thus it is concluded that when the plasma is excited in the high density condition, it is being resonantly * sustained in the TM101 mode. This result is predicted by the theoretical results of Chapter IV shown in Figure 4.6. That is, in :3: Figure 4. 6 the resonant length of the TM101 mode for large densities is approximately 5. 95 cm, which is less than the empty cavity resonant length of the TElll mode. Thus the theory has good agree- ment with experiment. Further, as was stated in section 4. 3, in a: this high density operation, the fields of the TM101 mode are essen- tially the same as the TE resonance of a coaxial cavity. 111 173 a: The TM101 mode has one other very important feature. It is seen from Figure 4.6 that for densities greater than four critical * densities (we : 2w x 6 Ghz), the resonant length of the TM101 resonance changes only slightly for increasing values of plasma density. Thus, for a fixed short length in the neighborhood of this * high density resonance, the TM101 mode will suffer very little detuning despite large increases in plasma density. Hence, the magnitude of the reflection coefficient will increase only slightly for large increases in plasma density. Using equation 6. 2, it is seen that increases in incident power should give nearly proportional * increases in absorbed power (hence plasma density) for the TM101 mode. This prediction has been experimentally verified. Figure 7.13 shows the plasma density as a function of incident power. It is clearly seen that the plasma density is proportional to incident power for densities above five critical densities when the cavity- >9: plasma system is operating in the TM101 mode. Because the maximum output power of the TWT was 28 watts, it is not known how far the results of Figure 7. 13 can be extrapolated. However, the concept of the horizontal asymptote >11: in the theoretical Figure 4. 6 shows that, once excited in the TM101 mode, the cavity-plasma system will suffer minimal detuning for large increases in the plasma density. 174 .358 2:29 E mEmma omcwmumsm 3398?: m new wok/om ”Bogus: mango.) banner Mamflna .2 .n oudwfim E 3nt . .m mm om «um NN om m: A: “I - S w o w N 0 «1p a 4F “ 4. a r T .7 + a J. .1 x 0 .1 H H: cofimummo @608 08/ O .1 N IO‘Q‘O! 0 II M .. v .I m 8H 0 r. o £03m» oao oboe 2H .0\ \ o \0. Eu . * /h\ f. )2: z \0 \\ \o 41 n \ \ o o\o\ \ .\O\O‘ #% co 175 * Figure 7.14 displays the input impedance of the TM101 resonance for increasing incident power (hence increasing plasma density). This plot shows that, not only does the plasma density increase with increasing incident power, but the cavity coupling remains approximately the same (actually improves slightly) for these higher densities. That is, the (same proportion cf the available incident power is being absorbed by the cavity-plasma system as the incident power increases. Although this increasing coupling phenomena must saturate for further increases in incident power, the important concept is that there is minimal detuning of the cavity- plasma system for large increases in plasma density. By carefully adjusting the length of the exciting probe in Figure 5. 3 and the length of the cavity for an incident power of 28 watts, the maximum resonantly sustained plasma density which could be achieved was 1. 29 x 1012 particles/ cm3, or over eleven critical densities. For this case, the standing wave ratio on the slotted line in Figure 5.1 was 9 db. Thus II‘I = .48 and Pab : .77 Pin; or 77% of the incident power is being absorbed by the cylindrical cavity- plasma system. This represents a great improvement in efficiency over the results of the reentrant cavity for creating a dense resonantly sustained plasma. Finally Figure 7.15 shows an experimental curve of input impedance versus short length when the cavity-plasma system is 3:: Operating in the dense TM101 mode. The plasma density at each 176 27 * Figure 7. 14. TM impedance versus incident power. The data points are labeled(P, /N ), where P, is incident . fin . o it}: . 3 power in watts and o 18 plasma densi y in 10 /cm . 177 * Figure 7.15. TM101 impedance versus short length. Data points are labeled (L /N ) whlef'e L83is cavity length and No is plasma density i’n 10 cm . 178 point is also shown on this curve for an incident power of 20 watts. * This plot shows that the TM101 mode is undercoupled, although it is less undercoupled than is the TE empty mode (see Figure 111 6.10). Thus, a comparison of Figure 7.15 and 6.10 experimentally shows that the presence of a plasma in a cavity tends to over couple the cavity. Although some improvement of the coupling can still be obtained for the TM):01 mode, the cavity-uplasma system can never be critically coupled, because the condition of power absorbed equal to incident power is unstable (see Figure 2. 3). In summary, a large volume (a cylinder of O. 8 cm diameter and 6 cm long) of dense (over eleven critical densities) plasma has been obtained in the cylindrical cavity with an incident power of 28 >5: watts. It was eXperimentally shown that the TM10 1, cold plasma, resonance is excited for this high density operation. Also, once this mode is resonantly sustained the cavity-plasma system is only minimally detuned by large increases in plasma density. In fact, the plasma density increases linearly with increases in the incident power, up to the limits of the available microwave source. Many future research areas are suggested by the results of this section. The increase in coupling for increases in incident power in Figure 7. l4 and the absence of a "dynamic instability” when the short length is changed in Figure 7. 15 imply that the properties of * the plasma are changing when excited in the high density TM101 mode. Changes in the electron distribution from Maxwellian, the electron or 179 ion temperatures, background pressure and effective collision fre- quency probably occur and should be studied. In the high density mode, the plasma is becoming significantly ionized and the theoretical plasma models used here are not necessarily valid. However, the results of this section show that it is possible to efficiently generate a dense, resonantly sustained plasma in a microwave cavity, and that the detuning effect of increases in plasma density has been minimized. CHAPTER VIII SUMMARY AND CONCLUSIONS Chapter II develops the linearized warm and cold plasma models. The dispersion relations for electromagnetic, electron plasma and ion acoustic waves are discussed; and the pertinent literature is presented for resonances in bounded plasmas, resonantly sustained plasmas and plasmas in microwave cavities. In Chapter III the warm electron, cold ion plasma equations, along with Maxwell's equations, are employed to find the rotationally symmetric modes on a plasma cylinder. A characteristic equation is derived and a qualitative discussion of the dispersion diagram is pre- sented. One complete numerical solution of the characteristic equa- tion is presented for both high frequency (w m we) and low frequency (00 Mei) axially propagating modes. Chapter IV presents a solution for the characteristic equation of the modes on a plasma rod located concentrically inside a cylindri- cal waveguide. In addition to the existing cylindrical wave guide modes, new cold plasma modes and new warm plasma modes can propagate in this waveguide structure. Dispersion diagrams are presented for the 180 lowest orde' resonantle modes. TI patterns a‘ section 4. hfierpret stabilitie micr owe de scrib1 several For the a new I are ob Plasm; hood c Predic Rance: Te Sona Cavity . f0)- inCr equiVale: 181 lowest order dipole modes. Also, curves of resonant frequency and resonant length versus plasma density are calculated for these cavity modes. The coupling of the warm and cold plasma modes, the field patterns and the excitation of the dipole modes are discussed. In section 4.4, the bounded warm electron, cold ion theory is used to interpret some previously reported low frequency parametric in- stabilitie s . The experimental apparatus is presented in Chapter V. The microwave system, reentrant cavity and cylindrical cavity are all described. Some special eXperimental techniques are explained and several possibilities for future research are suggested. The linear experimental results are reported in Chapter VI. For the reentrant cavity, in addition to the perturbed cavity resonance, a new cold plasma resonance and a series of warm plasma resonances are observed. Experimentally, it is found that the new cold and warm plasma resonances can only be excited when they occur in the neighbor- hood of the perturbed electromagnetic cavity resonance. The theoretical predictions of Chapter III qualitatively explain the warm plasma re so- nances and are in excellent quantitative agreement with the cold plasma resonance. It is experimentally observed that the perturbed electromagnetic cavity resonance for the reentrant cavity-plasma system can be tuned for increasing plasma densities by increasing the short length. An equivalent circuit, based on two transmission lines and the bounded 182 plasma theory of Chapter IV, explains this result. However, it is found that the presence of the plasma decouples (over couples) the cavity. This result is observed in both the power absorbed and the input impedance of the cavity-plasma system. Two equivalent circuits for the cavity explain this decoupling as the transition from a low density resonance to a high density antiresonance. The linear response of the dipole resonances of the cylindri- cal plasma-cavity system is also investigated in Chapter VI. It is noted that the theoretical predictions of section 4. 3 are in excellent qualitative agreement with the experimental results. The Tonks- Dattner resonances can only be observed when they occur in the neigh- borhood of a perturbed cavity resonance. A Smith Chart plot of the input impedance of the cylindrical cavity demonstrates that the ability to observe the Tonks -Dattner resonances is a matter of impedance matching between the cavity and the driving microwave system. In Chapter VII, the nonlinear response of the cavity-plasma systems is presented. At moderate incident power levels (~ 1 watt) the resonances become distorted with hysteresis effects as the dis- charge current is modulated. A graphical analysis was developed to explain these results. It was found that radio frequency ionization of the plasma is the dominant nonlinear effect. For the reentrant cavity, the plasma can be resonantly sus- tained with an incident power of approximately two watts. To demon- strate the need for a retuning technique, it was observed that the 183 density of a resonantly sustained plasma increases only a small amount despite large increases in incident power. By using the variable length retuning technique, a resonantly sustained plasma of over thirteen critical densities was excited in the reentrant cavity. A grq3hical analysis was developed to eXplain this large increase in plasma density as well as hysteresis effects as the cavity length is first increased, then decreased. The nonlinear response of the cylindrical cavity-plasma system is described in Chapter VII. For an incident power level of over twenty * watts, a dense resonantly sustained plasma was excited in the TM101 mode by initially varying the cavity length. However, the theory of Chapter IV predicts that once excited, the detuning of this resonance will be very small for large increases in plasma density. This is experimentally verified by observing that the plasma density rises linearly with increases in the incident power (up to the 28 watts avail- * able). Thus for a high density TM101 resonance, the detuning effects of the resonantly sustained plasma have been minimized. >1: When operating in the dense resonantly sustained TM101 mode, the cavity is under coupled. Experimental results presented here indicate that 77% of the incident power is absorbed by the cavity-plasma * system in the TM101 resonantly sustained mode. Thus more efficient coupling could be obtained by changing the exciting loops and probes of the coaxial input; although some reflected power is necessary for stability . 184 In addition to those stated at the end of Chapter V, there are many possibilities for future research. First, the results of Figure 7. 13 should be extended for higher values of incident power. The limits of the linearity between incident power and plasma density should be determined. Also, other cold plasma modes which have a horizontal asymptote on a resonant length versus plasma density plot (see :1: Figure 4.6) should be investigated. For instance, the TE resonance 111 will have this property if the incident frequency is over 3. 75 Ghz. Also, the theory of Chapter IV should be extended to include the effects of a static B field78. The use of a variable magnetic field may give a new method for retuning a cavity-plasma systeng. The process of how the electromagnetic energy is transferred to heat and to ionize a plasma when we >> to is poorly understood. Such situations arise when the cylindrical cavity is operating in the TM101 mode and when the reentrant cavity is tuned for high plasma densities. The electromagnetic “skin depth" is small and it is expected that under the experimental conditions changes in electron temperature, ion temperature, effective collision frequency and distribution functions all occur. Thus experiments should be conducted to measure these effects. Finally, when the ”gap" length is increased to approximately 1 to 4 cm, it has been observed that a plasma, with density greater than the critical density, is excited in the reentrant cavity by the incident, high power microwave signal. Because the electromagnetic 185 waveguide modes are cut off and only cold and warm plasma surface waves can propagate, it appears that the plasma is being sustained by propagating surface waves. 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Vandenplas, Appl. Phys. Lett., 15, 30 (1969). A. J. Hatch and L. E. Heuckroth, J. Appl. Phys., 41, 1701 (1970). 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 85. 85. 86. 190 H. O. Eason, J. of Microwave Power, 4, 88 (1969). E. Okress, Microwave Power Engineering, Vol. 11, Academic Press, 1968. D. T. Tuma, Bull. Am. Phys. Soc., 15, 1468 (1970). A. M. Messiaen and P. E. Vandenplas, Phys. Flds., 13, 2950 (1970). C. C. Johnson, Field and Wave Electrodynamics, McGraw-Hill, 1965. J. D. Jackson, Classical Electrodynamics, John Wiley, 1966. M. Abromowitz and I. A. Segun, Handbook of Mathematical Functions, Dover, 1965. B. Kezar and P. Weissglas, J. Appl. Phys., 36, 2479 (1965). D. G. Moursund and C. S. Duris, Elementary Theory and Application of Numerical Analysis, Michigan State University Press, 1965. W. P. Allis, S. J. Buchsbaum, A. Bers, Waves in Anisotropic Plasmas, Massachusetts Institute of Technology Press, 1963. C. G. Montgomery, Technique of Microwave Measurements, Boston Technical Publishers, Inc. , 1965. S. Ramo, J. R. Whinnery, T. Van Duzer, Fields and Waves in Communication Electronics, John Wiley, 1965. N. Marcuvitz, Waveguide Handbook, Dover, 1951. A. C. Raptis and K. E. Lonngren, J. of Microwave Power, 4(3), 182 (1969). R. K. M. Landshoff, The Plasma in a Magnetic Field, Stanford University Press, 1958. S. Aksornkitti, H.C.S. Hsuan, K. E. Lonngren and I. Alexeff, Phys. Flds. 11, 1838 (1968). C. W. Mendel and R. A. Stern, Bull. Am. Phys. Soc. 13, 304 (1968). R. E. Collin, Foundations for Microwave Engineering, McGraw- Hill, 1966. 191 87. P. Leprince, G. Matthieussent and W. P. Allis, J. of Appl. Phys., 42, 412 (1971). APPENDIX A In this appendix, the details of the solution for the character- istic equation for the TM modes on the plasma cylinder, shown in Figure 3. l, are presented. It is desirable to express the boundary condition y; - ’1“ = 0 at r=a in terms of the electric fields. To do this, consider the conservation of momentum equation 2.18, which can be written, 2 -eE. eE U Vn —> 1rr sol e e Ve : 'oom - 'wm - ‘wN ’ Al J e J e J o and Gauss' Law, equation 2. 20, V-Ez-s-(n.-n). A2 60 1 e Dividing the E field into solenoidal and irrotational parts gives, v-E = v-(E. +E‘ ) = V- E because V-E = 0 A3 1rr sol and from equation 2. 41, Eirr : -v¢p’ 192 193 Equation A3 can be written, V.E=-vo. A4 n. = . A5 it’— 26+ 8 i Substituting equations A4 and A5 into A2 gives, -en 2 e -1 V 11: = (—-———- - 1) p 60 0,2 2 6+ 00. 1 which can be simplified to E E n = o + V24: e e p 2 -* -e + o.) -» ve - jwm [E801+ 2 6+ ICirr]° A7 e we Thus, ve - I1> : 0 at r=a is equivalent to 2 —> (,0 —> 5 (E801 + w 2 E+Eirr) . r e 0 at r=a. A8 For the symmetric case, 8/ 34> 0, only the TM and the irrotational fields will couple. The TE fields have only a (b-direction component of the electric field,and thus can not couple to the TM or irrotational fields, which have only 9- and Q-direction components 194 of electric field. Thus the TE modes are not of interest here because they do not couple to the irrotational fields. Furthermore, a careful analysis shows the rotationally symmetric TE modes do not exist-78. For r > a, equation 3.11 can be written, 20 20 Vt¢e-5¢=0 e and from equation 3. 12, 2 2 2 6 = - k 01 p.060 A9 2 . . 4 . . The solution 13 , (exp(Jcot - szz) understood) 0 : 6 6 the AIO( r) + BK0( r). A10 The choice of modified Bessel functions is justified in section 3. 3. For r< a, equations 3.18 and 3. 21 are 2 1 2 1 V 41e - Yl qJe — O 2 1 2 l vt 4J9 - Y1D the _ 0 where 2 2 2 v1 — kz -w 110606!) All 2 2 ”262 2 v : kz _ 2 : - Y. A12 e + The solutions are l LlJe — CIo(ylr) + DKo(ylr) A13a l L): = FI r +GK r Al3b p 0(Yp ) 0(vp ) 195 where A, B, C, D, F, G are constants. 75 . . . For large arguments , Io(x) —>oo as x «v 00. This implies that A = 0 in order that 4180 remain finite. Likewise, for small arguments, Ko(x) —> 00 as x —> 0. This implies that D = G : 0 in order that the] and kllpl remain finite. Thus, equations A10, A1 3a and Al 3b become : 6 4 the BKo( r) A1 a 411 - CI ( r) A14b e _ 0Y1 1111 - FI ( r) Al4c p 0 Y1) ' From equations 3.6, . 7, .13, .14, . 23, the fields associated with the potentials A14a, b, c are, r > a E0 = vxvx’z‘oe" O 841 . e A 2 O = - -6 (sz 3r )r ties A15a 11° = jwe Vx€¢° 0 e 34180 A = -° 5b r <1> 1 <92 1 4 -+ A_ .- (E3°l+w2 €+Eirr) r-0 atr—a Al7c e where the first term of each vector component of E1 is the solenoidal part and the second is the irrotational part. Substituting equations A16a, b, c, d into Al7a, b, c and expressing the resulting linear equations in matrix form gives, 197 _ 2 2 _ _ _ -6 6 -°k B Ko( 3) Y1 106118) J zlowpa) -6 ' 6 ' = . Ko( a) Epyllowla) 0 C 0 A18 2 o -jk 1'( a) - i”— e 1 '( a) F z'Yl 0 Y1 w 2 +Yp o Yp 1— e d L— a A necessary and sufficient condition for a nontrivial solution to equation A18 is that the determinant of the coefficients be zero. So, 2 2 -6 6 -‘k Ko( 3) Y1 lob/la) J zIona) -6 ' 6 ' = . Al Ko( a) Epyllowla) 0 0 9 (‘02 -' ' __ t 0 szvllo (v13) w 2 €+vplo(vpa) e Using the expression for the derivative of the modified Be 8 sel function s75, IO'(X) = 11(X) K '(X) = -K (X) - o l and expanding equation A19 gives, after some rearrangement, 615311-15)" e.e.z<§,->Zo l = + A20 Il(y a) 11(y1a) 5 K1(6a) Yp 10(ypa) Yl 10(y1a) Ko(6a) Equation A20 is the desired characteristic equation for the symmetric modes on a plasma cylinder. For the case where YPZ < 0, it is useful to write equation A20 as 198 w 2 Y1 2 w 2 6 2 5(a) (g) 6.6, g)(;;') -1 J (11 'a1 ‘ I (y a) + K 16a) ”“1 Y I _]'___E'._ Y _l.___l__ 5 _—l—6_ P Jowpa) 1 lowla) Ko( a) The two equations A20, A21 describe the irrotational surface modes and the irrotational standing wave modes, respectively, which propagate on a plasma cylinder. A thorough discussion of the nature of these modes is given in Chapter III. APPENDIX B This Appendix discusses the numerical method used to solve the characteristic equation derived in Appendix A. Also, a typical computer program is presented. The method used to solve the transcendental equations 3.24 and 3. 25 is the Newton-Raphson iterative technique77. Equation 3. 24 is rewritten a s stall—:02 e.e.<.—:-)2<£z—)Z -1 DIF = + + L Y 1161 a) v 116/1a) 6 K1(6a) p Iowpa) l Iowla) Ko(6a) B1 With the parameters mi, Te’ a, we fixed and for a given value of w, DIF will be zero when the value for kz is chosen which satisfies equation 3. 24. From an initial ”guess" for kz, a better choice is . 77 calculated from the relation , DIF(k ) k _ k - z z new z d(DIF(kz )) d(kz) 199 200 The axial wave number k is then set equal to k . A check is z 2 new made to see if IDIFI < 10'8 . B3 If the inequality B3 is true, the values of w and k2 are solutions to equation 3. 24. If the inequality B3 is not true, another kz new is calculated from B2 and the above procedure is repeated. If B3 is not satisfied after 50 iterations, the program self-aborts. The MSU Library Subroutine BES, 00000116, was used to evaluate the required Bessel functions. This subroutine evaluates Bessel functions of the fir st and second kind and modified Bessel functions of the first and second kind. The results are accurate to nine significant figures. The derivative dDIF/dkz shown in equation B2 was approxi- mated by using a finite difference approximation, d(DIF) ~ DIF(kz+ Akz) - DIF(kz) d(kz) Akz B4 where Akz is a small increment. The accuracy of the approximation B4 is not critical because the test for convergence is the condition B3. However, the approximation was quite good because seldom was more than three iterations needed for convergence. The following page is a reproduction of a typical program. This program was run on a CDC 3600 computer. al-II 1111 - mfltmmorwh1hnH©HMHsu M> 1*" mdeLHfiuH #\moOH-..>Y. 11 11 11111 11 1 111 1. -IW.W~.4.O HUM... 1.1111. ‘11- 1111" 1111 11 II. 1111 -'ll 0 z -- -11 - -1-1.¥-,;.i-1:i:111:1mJ1fiH1¢/Wm114- d.) 411.. ... (x. H CU... .1 It! v..v..:. 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I! -::1i1-11!111 --1. -awa3*wa3.\.3*zvu - : 11.1- -31111!1HfifimwMWHW1chm11:11-13111!11; 1 -1 1! 1 11. cauo so .II. 1'11 1.1.4.111 31.11 1.1 - 11!.) amouFonm_o .«uqum mu" 11:1 .11. 1.111111 0“ u 0... 00 11.1. 1 11 . _w1u.m14u.1._1--.111 - 1 - 11 --- APPENDIX C In this appendix, the details are presented of the solution for the characteristic equation for the modes on a plasma rod located concentrically inside a metal waveguide. The warm electron, cold ion model is used for the lossless, uniform and isotropic plasma. The plasma rod is assumed to have a radius equal to a. The metal cylindrical waveguide is assumed to be perfectly conducting and to have an inner radius equal to b (see Figure 4.1). As pointed out in Appendix A, for the rotationally symmetric modes the TM and TE components of the electromagnetic fields are uncoupled in a cylindrical geometry. However, in this appendix all modes of arbitrary angular dependence are desired, so both TM and TE fields are needed to satisfy the boundary conditions; that is, the modes are hybrid modes. In the region outside of the plasma, a < r < b, equations 3.10 and 3.11 become Vtattleo + 51241;) 20 .20 Vtxph +5 41h I 0 C1 C2 II 0 204 205 where 2 2 Z I _ _ 5 — w [1060 kz . C3 The solutions for equations Cl and C2 are (exp(j1.1t - jkzz + jn¢) understood), 0 _ I l the — AIJn(6 r) + AZNn(6 r) C4 0 - l I 41h _ BlJn(6 r) + BZNn(6 r), C5 where Jn and Nn are regular Bessel functions, and A1, 2, B1 and B2 are arbitrary constants. Applying the boundary conditions4 at r : b, it is found that equations C4 and C5 can be written in terms of a new pair of constants, A and B, as, -E- II A(Jn(6'r)Nn(6'b) - Nn(6'r)Jn(6'b)) C6 £- 5" ll B(Jn(6'r)Nn(6'b) - Nn(6'r)Jn(6'b)), C7 where the prime denotes differentiation with reapect to the argument of the Bessel functions. Inside the plasma, 0 < r < a, equations 3.17, 3.18 and 3. 21 are, Vtquel +y1'ZqJel = 0 C8 vt21phl +y1'z111h1 = 0 C9 Vtzwpl +yp'2111p1 = 0 C10 where 2 Z 2 I _ .. Y1 - w P-OE E kz C11 206 Z 2 ‘*’ E 2 v ' = —P- - k c12 p U 2 z e 6-l- The solutions are, 1 _ I I the — C Jn(\(1 r) + C1 Nn(\(1 r) C13 1 _. I I Lth ... D anl r) +Dl anl r) C14 1 I I = 5 41p F Jan r) + F1 anp r) Cl where again, eXp(jwt - jkzz + jn¢) is understood. Because Nn(x)» 00 as x-> 0, the constants C1’ D1 and F1 are all zero. Thus equa- tions C13, C14 and C15 become _ I the - Canl r) C16 1 _ , , ¢1=FJ(Y'r). C18 P n P The field in the region a < r < b can be found from equations 3.6 and 3. 7 to be, E0 = VxVquJeo-jwpon’z‘wno 84‘eo A nkz o A 2 o _ "_- __ I - [11sz at )+4>( r 418 )+z(+6 the )] mp- n aq. o +[?( . h ¢ho)+$(JwI-lo 3r )] (:19 207 *o_ . A o A o H — Jweonz¢e +Vx§7xzxph -nw€ 31p 0 = W ° ¢e°)+$(-jweo a: )1 34: o nk A . h " Z 0 A 2 o +[r(-sz 3r )+¢(—r— Lvh )+z(5'¢h )] czo and substituting equations C6 and C7 into C19 and C20 gives, "0 E = r[-jkza'A(Jn'(5'r)Nn(6'b) - Nn'(6'r)Jn(6'b)) wP-n + B(Jn(6'r)Nn' (6'13) - Nn(6'r)Jn'(6'b))] nk +331 TEAUnm'l-mnm'b) - an'mn‘é'r” + ijO5'B(Jn '(6'1‘)Nn'(6'b) - Nn'(6'r)Jn'(6'b))] " I2 I I I I + z[ +6 A(Jn(6 r)Nn(6 b) - Nn(6~ r)Jn(6 b))] C21 -*o A no060 _ I I I I H — r[ - A(Jn(5 r)Nn(6 b) - Nn(6 r)Jn(6 b)) ' jkzé'B(Jn'(6'r)Nn'(6'b) - Nn'(5'r)Jn'(5'b))] +$[ -jw606'A(Jn'(6'r)Nn(6'b) - Nn'(6'r)Jn(6'r)) nkB Z + (Jn(6'r)Nn'(6'b) - Nn(6'r)Jn(6'b))] A I2 I I I _ I I I + z[ +6 B(Jn(6 r)Nn (6 b) Nn(6 r)Jn (5 b)]. czz Now, inside the plasma, 0 < r < a, the equations for the fields are (equation 3.13, 3.14 and 3. 21), *1 A l . A l l E _ vaxque -pr.onz¢h -VLIJP C23 208 ->l , A l A l H : Jweoepszupe +Vxszleh . Using equations C16, C17 and C18 in C23 and C24 yields, +1 A ‘9“ n E : r[-jkz¥l'CJn'(Yl'r)+ r A nkz +¢[-r—- CJnWl'r) +jwl¢ovl +2[+Y 'ZCJ (v 'r)+jk FJ (V 'r)] l n 1 z n p 3.11 = ?[-:uf-9-€-P CJ (v 'r)-jk Y 'DJ '6! 'r)] r n 1 z 1 n l A nk + ¢ [-JwEoele +2[+y 'ZDJ (y 'r)] l n l The remaining boundary conditions are, Ezo-Ezlzo atr=a E° E1 o 4’ - 4) = atr=a Hz°-Hzlzo atr=a Ho Hl-O atr-a ¢ ¢ _ - 1 Z 1 —§ (0 —> [5 (E801 + 2 €+ Eirr) r - Oatr_a w e I I I ___E. I CJn (V1 r) + 1' DJn(Yl r)] C24 D Jn608. M14 : 0 -kz M15 : P 1'1 nkz M = 21 awEOGn 22 23 24 25 31 32 33 34 35 41 42 43 44 45 51 52 5"l 210 O 0" N :3 O 211 kz M53 : .006 o n M54 -aS n 2 (.1 M55 ’ ' 2 E+ w whe r e, [J '(6'a)N (6'b) — N '(6'a)J (5113)] 6' n I n I n I nI [Jn(6 a)Nn(6 b) — Nn(6 a)Jn(6 b)] Cl 11 z 115' >0, = 6 [In'(53)Kn(5b) - Kn'(53)1n(5b)] [In(6a)Kn(6b) - Kn(6a)In(6b)] 2 as >0, R _ 6' [Jn'(6'a)Nn'(6'b)-Nn'(6'a)Jn'(6'b)] ' [J (6'a)N '(6'b) - N (6'a)J '(6'b)] n n n n if 6'2 > o, [I '(6a)K '(6b) - K '(6a)1 '(6b)] = 5 n n n n [I (6a)K '(6b) - K (6a)I '(6b)] n n n n 2 if 6 >0, I I ‘ I n l anl a) I _ Y In (Yla) 1 Inwla) C31 C32 C33 C34 C35 C36 212 J '(v 'a) 2 P :YI n 2'-.. ifYI>o C37 n p J (Y a) p n l I '(v a) :y In( a) ify2>0. C38 P an P Relations C31 and C33 will be used for fast waves, Vph > C, while C32 and C34 will be used for slow waves, V < C. Relation ph C37 will be used when the irrotational waves are radially standing waves; while C38 will be used when the irrotational fields are radially evanescent surface waves. If the electron temperature is set equal to zero (Te -v 0), the characteristic equation reduces to the cold plasma approximation. In this approximation, 2 i we 2 v =1—L-k) p 2 2 U6 e + w JEB 7,-6- E asTe+0 . e + and thus I I 51. ’l a In'WEa) n(Ue E+ ) I (v a) 7 up “ii/Ea) + *1 as Te->0. C39 213 So l I V p Inwpa) Thus, in the cold plasma approximation, elements M15 : M25 ..— 0‘ and the characteristic equation can be written, where M .. = M..; i,j=1,2,3,4. c1] 13 Equation C42 can be expanded to give 2 2 I I 21.2. Yn 6:3. Y1 Ep G s R s n n n n 2 2 2 2 nkz w [GP-l] azchSRS nn nn C40 C41 C42 C43 C44 which describes the modes which exist on a cold plasma rod located concentrically inside a cylindrical waveguide. The equation C44 is equivalent to equation 3 of reference 56. If the Characteristic equation C30 is expanded, the re sult can be rearranged into the form, 214 IM l 2 1 = C n C45 .2)— P 'klel'; lvl 26+ n w e where the matrices U and V are U.. 3 M. . 9 isj: 1:22314 C46 13 1+l,J V.. =M.. , i=1;j=1,2,3,4 1J 1J and . oI i=293I4Ij:19233!4 C47 1+lIJ and Mc is defined in equation C43. Some special cases of the general characteristic equation C45 are of interest. For the rotationally symmetric modes, the TM and irrotational fields uncouple from the TE modes and give two separate results. For the TE modes, v' 2 1 5' .——-=0 C48 SO RE) where 50 and R0 are defined in equations C33 and C35 (or C36), for first wave solutions. Slow wave solutions for equation C48 do not exist. The TM and irrotational fields for the rotationally symmetric case are coupled. The characteristic equation is, w 2 ' 2 w 2 2 (3;) 6+ 6p (6 /kz) (23:) 6+ (Yl/kz) _ z - . 4 P G S C 9 o o o 215 If the radius of the metal cylinder is allowed to go to infinity, b —' an, equation C49 reduces to (for slow waves only, 62 > 0) 2 (534,61, (6/12 (5:) egg/1312 = + 1,1»! a) 6 Kl(6a) 11(6a) YP I (Y a) K (621) Y1 1 (53,) o p o 0 C50 which is the same as equation A24 for the rotationally symmetric modes on a plasma cylinder. Thus the result of Appendix A is the Special case of equation C29 when b -> co and n : 0. If in equation C45 b -> co, the resulting equation is equivalent to equation 24 of reference 33, except the ion mass has been assumed to be finite in these calculations. For the dipolar mode, n = l lM l l = —— C -— C51 2 l 2— P 'k IUI " - IUI 2 6+ 1 z a b -r 00 we n = 1 when b—> oo Kl'(6a) R1 = Cl = 5 W C52 and only slow wave solutions, 62 > 0, are possible. From the form of equation C51, it is clear that the poles of the LHS, 2 v =0 if v >0 C53 P P or J'( 'a):0 ify'>0 C54 l¥p P 216 determine the solution to C51 except when [MCI is small. That is C53 or C54 approximate the solution to C51, except where IMCI m 0. APPENDIX D This Appendix will discuss the approximations and numerical methods used to solve equation 4.1. Only the dipole (nzl) modes will be considered; and the ions are assumed to be infinitely heavy (Q11: 0). Equation 4.1 can be written l - IMCI D1 J ’(Y a) n (g y._i__s_ -kz|U|-;|V| me p lep'a) nzl As stated in section 4. l and AppendixC, the equation D1 has been separated into warm plasma (irrotational) terms on the LHS and the cold p1asma (solenoidal) terms on the RHS. For typical numerical values of w, kz and we’ the value of the LHS of equation D1 is usually small. Thus the total solution to equation D1 simplifies to the zero of the RHS, IMI=0. D2 This is the cold p1asma approximation; and modes predicted by equation D2 are called the cold p1asma modes. 217 218 Alternatively, it is found for typical numerical values of w, kz and we that the LHS has very sharply peaked poles. These poles are dete rmined by I I ._ J1 (yp a) _ 0 D3 or ' ZX'; = I2I39°°° D4 Ypa 11 1 1 Where X1; is the 1 th zero of the derivative of the regular Bessel function of the first kind. Thus where lMcl 51’ 0, the total solution to equation D1 is closely approximated by equation D4. These are the warm plasma, Tonks-Dattner modes. Thus the total solution to the characteristic equation D1 is closely approximated by the cold plasma result (equation D2) and the warm p1asma result (equation D4). The above approximations break down in two regions. The first is near the electron thermal line (Vph % Ue) where it is known from Figure 3. 3 that the electron temperature is very important in the total solution. However, in Chapter IV only the slow wave region near the light line and the fast wave region are investigated. Thus the in- fluence of the finite electron temperature on the cold p1asma modes can be safely ignored. The other region where this approximation breaks down is in the fast wave region where the RHSas LHS in equation D1. That is, a region where the solenoidal fields strongly interact with the irrotational fields. An analogous situation was encountered in Figure 3. 3 when the 219 electron surface wave mode coupled with the ion acoustic standing wave modes. In Figure 3. 3 the equations were solved exactly and the insight gained can be used to intuitively analyze equation D1. For the numerical results of Chapter IV, equation D2 is written, DIF= IMI, D5 C where MC is defined in Appendix C. The values of w. kz and we which make 8 D6 are then the solutions. The Newton-Raphson iterative technique77, ID1F|< 10' as described in Appendix B, was used to evaluate the transcendental equation D5. For Figures 4. 2, 4. 3 and 4.4, we was fixed and the equations were solved for w vs. kz. For Figure 4. 5, k2 = 17/ Ls was fixed and the equations were solved for (.1 vs. we; and finally for Figure 4.6, w = 211 x 3. 03 x 109 was fixed and the equations were solved for Ls vs. we. The solutions to equation D4 were superimposed on the cold p1asma results and the interaction regions were analyzed. 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