AN EXPERIMENTAL STUDY 0F7N0NL~INEAR PHENOMENA m A RESONANTLY. SUSTAINED WCROWAVE PLASMA ‘ 7 ‘ Thesis fortheDegree of Ph 0;. . MLCHEGAN STATE UNIVERSITY QUONG HON LEE 1970 ' LI B RA R '1" l‘xiichigan Stat-e: Unix-cram}! '. ."g-oyv'I-I P -! ' This is to certify that the thesis entitled AN EXPERIMENTAL STUDY OF NONLINEAR PHENOMENA IN A RESONANTLY SUSTAINED MICROWAVE PLASMA presented by QU ONG HON LEE has been accepted towards fulfillment of the requirements for Ph. D. degree in Electrical Engineering And System Science & Major professo\v Date “MK R0) \qu 0-169 ‘_ ABSTRACT AN EXPERIMENTAL STUDY OF NONLINEAR PHENOMENA IN A RESONANTLY SUSTAINED IVIICROWAVE PLASMA By Quong Hon Lee The nonlinear resonance phenomena exhibited in a resonantly sustained high frequency discharge and a dc mercury vapor discharge irradiated by high microwave power were experimentally studied. TWO experimental systems were employed; a coaxial system and a ridge waveguide system. The transition between the linear resonances of a bounded plasma and the resonantly sustained plasma was experimentally investigated. The linear resonances were found to become distorted as the incident microwave power was increased. Stable and unstable rf operating regions appeared in the resonance curve. By approximating the plasma system with an equivalent transmission circuit, these phenomena were qualitatively interpreted by applying transmission line theory. That is, criteria for coupling microwave power into the plasma was developed by plotting the equivalent plasma impedance on a Smith chart. When the incident pump power (f0) of a resonantly sustained plasma was above a certain threshold level, high frequency (f otfi ) Quong Hon Lee ‘ and low frequency (fi) oscillations were excited in the plasma. The ‘ low frequency oscillation appears to be caused by a standing ion- «' acoustic wave. By reducing the incident power to just below the threshold, a separately applied cw signal was able to be ampli- fied when the signal frequency is in the vicinity of f0 i fi' Since amplification existed above and below fo, the instability observed is a four frequency parametric interaction. At the occurrence of the parametric instability, the low frequency, fi, was observed to shift as the incident power, or the plasma size or a dc biased voltage across the discharge was varied. The sheath effects for a bounded plasma appear to account for this phenomenon. Also, when the plasma system is resonant simultaneously at £0, 3fo and fi’ the coupling between the low frequency oscillation, ii, and the third harmonic resonance was studied. The low frequency resonance causes the zeroth-order density profile to vary at fi, which in turn causes the third harmonic output to be frequency modulated. AN EXPERIMENTAL STUDY OF NONLINEAR PHENOMENA IN A RESONANTLY SUSTAINED MICROWAVE PLASMA BY Quong Hon Lee A THESIS Submitted to Michigan State University partial fulfillment of the requirements for the degree of E- DOC TOR OF PHILOSOPHY Department of Electrical Engineering and Systems Science 1970 "'I"f_lm‘ -’ 7- ” c—pwvn-P . wmnfif‘. vwfi'r 'v‘.’ l. Irv)" 1: ic- r , a; u' . O...‘ I» :m :J‘t . a,” . [Klan "we," .n-w yratuudc ;0 MI labia! ”.rw; . 3n!£nunu:t -leidtnct_. e ..‘ : :i vitrehrCh. Sincere my “Mums ,. a n -. ‘~ Him: and Dr. H. To my parents . . ., ..., mm w.- ”. Cote-sock Mr.& M11? So, M.- L9. 1,‘ " ' . " . FimnY. ‘ j. , . - « 'l—Z‘pfits‘fi‘j in ACKNOWLEDGMENTS The author wish to express his deep gratitude to his major professor, Dr. J. Asznussen, for providing continuous guidance and encouragement throughout the course of this research. Sincere appreciation is also due to the other members of the guidance committee, Dr. K. M. Chen, Dr. B. Ho, Dr. N. Hills and Dr. H. Hedges for their time and interests in this work; and to P. Colestock for providing the experimental curve shown in figure 4.6. Finally, it is gratefully acknowledged that this research was supported in part by the National Science Foundation under grant GK-2952. swm—s-w‘ r- Mm?“ TABLE OF CONTENTS Page ABSTRACT ACKNOWLEDGEMENTS................................ iii LISTOF FIGURES..................................... vi 1. INTRODUCTION.................................... 1 Z. A GENERAL REVIEW OF MICROWAVE-PLASMA INTERACTIONOIIIIIIUOOCOOOOUCOOUIOI....IOIOOOUOOU 5 2.1 Coupling of electromagnetic and electroacoustic waves. 5 .2 Resonances in bounded plasmas................. 12 . 3 Re sonantly sustained high frequency discharge . . . 18 4 Nonlinear effects in a resonantly sustained highfrequency plasma......................... 22 NNN 3. THE PLASMA CAPACITOR SYSTEM... ... .. ... ..... . 27 1 Theory.......0I...0..........IICOICIIIUCOOOIO 27 .2 Resonance at high frequency (03>wp >> wp+)...... 34 A. The variation of electric field as a function of plasmadensity........................... 37 B. The variation of electric field as a function of position.................................. 43 C. Impedance and equivalent circuit (a) > a) ) . . . 46 D. Magnitude of the admittance for the - main and temperature resonances . . . . . . . . . . 49 3.3 Resonance at low frequency (u) top. In the low frequency regime, the model again indicates the presence of standing longitudinal waves; namely, the presence of ion-acoustic wave. A transmission equivalent circuit for both high and low frequencies is developed. The effects of the plasma sheath are clearly indicated in this model and plots of the E field for cold plasma, temperature and ion-acoustic resonances are made. The nonlinear resonance effects arising from nonlinear micro- wave-plasma interactions are experimentally studied with a ridge waveguide system and with a coaxial type of microwave -vacuum system. Both of these structures can be approximated as a plasma- filled capacitor. A detailed description of the experimental apparatus is given in Chapter 4. Experimental results are presented in Chapter 5. The transi- tion between the linear resonance of a plasma column and a completely resonantly sustained plasma is presented. A qualitative description of this transition is developed using a Smith Chart representation of the plasma capacitor impedance. Stable and unstable regions are clearly indicated. A plasma instability in a resonantly sutained plasma is also investigated. Experimental results indicate that a four frequency parametric interaction is responsible for the instability. It is be- lieved that a nonlinear coupling between microwave frequency resonances and low frequency ion resonances is involved. In fact, the plasma capacitor theory for the ion-acoustic wave with the proper sheath, electron temperature, etc. , is able to explain certain experimentally observed low frequency variations. Nonlinear coupling between low frequency and high frequency resonances is also studied. Experimental and theoretical results indicate that low frequency oscillations (due to standing ion-acoustic waves) cause the zeroth-order density profile to oscillate at the low frequency. This variation of the density profile causes the third- harmonic resonance to vary resulting a frequency modulated third harmonic output. Experimental results describing the operation of a completely resonantly sustained plasma are also presented. Chapter 6 summarizes the work presented in this thesis. Also, certain problem areas which need additional attention in future research are pointed out. CHAPTER 2 GENERAL REVIEW OF MICROWAVE-PLASMA INTERACTION This chapter is devoted to the general review of literature pertaining to the nonlinear resonance effects in a resonantly sustained high-frequency discharge. The physical interpretation of these non- linear resonance phenomena can be perceived through the study of the coupling between electromagnetic and electroacoustic waves and the nature of linear resonances in a bounded plasma. As a start, the coupling of electromagnetic and electroacoustic waves in an infinite, non-uniform plasma is demonstrated by deriving a fourth-order differential equation for plasma potential. Then, the experimental and theoretical interpretation of the linear Tonks-Dattner resonances exhibited in plasma scattering and plasma capacitor are reviewed. Finally, the physical mechanism of a resonantly sustained discharge is described. Nonlinear resonance effects which bear direct relationship to the experimental investigation here are empha- sized. 2. l Coupling of electromagnetic and electroacoustic waves A warm plasma can support propagation of transverse electro- magnetic (EM) and longitudinal electroacoustic (EA) waves if the signal frequency a) is greater than the plasma frequency 00p. These EM and EA waves are uncoupled in an isotropic, homogeneous plasma of infinite extent. Field6 formulated the coupling problem quantitatively for an infinite plasma on the basis of a hydrodynamic model of the plasma. He concluded that coupling exists under any one of these three conditions: (1) plasma inhomogeneities, (2) static magnetic field and (3) density discontinuity, such as that at a vacuum-plasma boundary (really this is a Special case of (1)). Since the investigation to be reported involves no static magnetic field, only the coupling introduced by a non-uniform plasma will be considered below. The basic equations used are the first two linearized, hydrodynamic equations and the Maxwell's equations. In this analysis, the heavy ion and neutral molecule distri- butions are assumed unperturbed. A linearized theory for small perturbation of electrons is followed in deriving the coupling equations. The plasma variable E, n and p are written as the sum of a dc term (subscript zero) with no time variation and a perturbation term Jwt, while H and i; are per- (subscript 1) with a time dependence of e turbed quantities. If the perturbation of the waves causes adiabatic changes in the plasma, the steady state and perturbed pressures are directly proportional to their correSponding densities a»-.. - . --.... 7 peo = KTneo”) pe1 : YKTnel where 'y is a constant depending on the number of degrees of freedom of compression involved. Since the plasma density varies with posi- tion, neo(r) (denoted by neo from here on) is a function of ii. From the linearized hydrodynamic equations, two first-order and one zeroth—order equations are obtained.7 The zeroth-order equation is of the form ... mUe2 Vneo E = - —— (2.1) 0 7e 11 ea The first-order equations are anel — -V - (neovel) (2.2) 2 -eneo -> e -> Ue Jwvel _ m n Eo- m El- vne1 (2'3) e e1 e co where 2 'YKTe U : e m e Equation (2.1) shows that the static electric field, E0, created by the gradient of the steady state density, neo' is in the direction opposite to the maximum change of density. From the Maxwell's equations, the following first-order equations are obtained 8 w -e \7 "E1 = E— l’le:l (2.4) 0 Vle = 30) eoEl-neoe vel (2.5) vxvxfl = -ja)roxT-Il (2.6) Equations (2.1) to (2.6) can be combined to yield 2 2 -> e vneo —> 2—» Ue —> VXVXEI +——Z T (V-El)-keEl- —2V(V'El)=0 ’Yc eo c (2 7) where 2 2 (1)2 a) _(r) ke : ~2- (1' 2 c 2 1 C : “060 n e2 (1)2(r) : {:0 p eEo According to the vector theory8, a vector can be decomposed into the sum of an irrotational part and a solenoidal part. The former has a vanishing curl, while the latter has a zero divergence E = E + E. I s 1rr i where V - Es = 0 V x E - 0 1rr In terms of ES and firr’ equation (2. 7) can be written as 2 2 2 —> Ue 2 2 Ue vneo -> (kae)Es+ (:z—v +1 2 (v 'yv f '7 f kae 97 Bin" 2 W Eirr c c 2 (A) _ -> = L Vf- E (2.10) c2 5 If the effects due to the static electric fields are excluded equations (2. 9) - (2. 10) are the coupled equations derived by Field. The fields Eirr and Es are coupled through Vf(r); i. e. , the non-vanishing plasma non-uniformity. Thus, no coupling exists in a homogeneous plasma where f is unity. Furthermore, equation (2.10) shows that Es must have a component in the direction of Vf(r) in order for coupling to take place. This is important to consider when experimentally exciting EA waves by EM waves. a, , In consideration of only plane wave propagation in an infinite plasma, the operatorv can be replaced by {12. The zero divergence of the solenoidal part implies that Es corresponds to the transverse electromagnetic part of the field, while the vanishing curl implies that Eirr corresponds to the longitudinal electroacoustic part of the field. Since V x Eirr : 0, the electroacoustic part of the field, Eirr’ can be expressed in terms of the scalar, plasma potential op; i.e., E].er = -chp (2.11) Similarly, the electromagnetic part of the field, Es, is related to the vector potential, A, and the scalar electric potential, ¢e’ through ES 2 -ja)|.10A-V¢e (2.12) '13 = vxK (2.13) 1 The electric field in the plasma then has the form if =jqu-V¢ -V¢p (2.14) From the zero divergence of E8, the vector potential A can be directly related to the electric potential (be v2.1 v 0 K : —. e (2’ 15) 1w so Also, equation (2. 3) to (2. 5) can be combined to yield a single coupled equation of potentials . 2 2 U 2.. 2.. _ - ‘0 - e _vf 2 V A +keA - (1)—L110 icz 4»er +7—C2 (v —f )v ¢p+ kezv¢p (2.16) Equation (2.16) is obtained under the assumption of the Lorentz con- _p dition which relates the vector potential A to the electric potential 4’ e k2 e _ q; (2.17) JON-Lo e >1 V . The divergence of equation (2.16) is identical to equation (2.10) when expressed in potential form; that is 2 vzvq, +1 _1_ (_‘U _f)_v.37_f _V_f. V2,), _l_vf.v¢ p 'y x 2 2 f f p X 2 p D 01p_ '1’ D = '—12Vf-(jmp.A+V¢) (2.18) o e 'YX where XDZ = KTZ is the Debye's wavelength mmp_ Obviously, the electromagnetic potentials, A and the, are coupled to the plasma potential, 41p, through inhomogeneities in the plasma. By applying the quasi-static approximation to the last equation, one arrives at the very well known equation for the Tonks—Dattner resonances in a bounded plasma17 (the coupling of the electromag- netic part of the field is also included; i.e., E = -V¢P- V¢el 2 -2 IE. 2 1mg; :73 2 1 VZV (hp-ylgf V)V ¢p+7 xz(w2-f)-V. f]V (hp-112 D p- D -1 Vf - V¢p = —2- Vf-V¢e (Z. 19) 'YXD 2. 2 Resonances in bounded plasmas As early as 1931, Tonks9 observed the phenomenon of resonance oscillation in a bounded uniform plasma at a frequency (.0 less than the plasma frequency 03p. His obsergation was explained with an equivalent permittivity, e = 60( l - (10% ), in a uniform, ex- ternal field. This is now called the cold plafma approximation. About twenty years later, when performing some microwave scat- te ring experiments, Romelllo observed the main resonance of Tanks and an additional series of weaker resonances. These weaker resonance were not predicted with the uniform, cold plasma model of Tonks. Through extensive experimental investigations, Dattner11 concluded that the observed resonances are dipolar in nature arising from oscillations of separated charges. These additional dipolar resonances have often been referred to as "Tonks-Dattner" or "T-D" resonances. Attempts were first made to explain these multiple resonances with a non-uniform, cold plasma. Treating the plasma as a dielectric cylinder with an abrupt change in density at the edge, Herlofson12 found that the resonance was damped by the density gradient. Apprximating a non-uniform plasma cylinder with a discrete number of uniform, cylindrical V 13 dielectric shells, Kaiser and Close13 showed that there exists a new resonance for each discontinuity in density. Although a continuous spectrum of resonances were able to be obtained from such a non- uniform dielectric model, the theoretical predictions failed to com- pare satisfactorily with experimental results. Including the thermal motion of electron in a uniform plasma, Gouldl4 observed a main resonance similar to that predicted by the cold plasma approximation and a spectrum of temperature resonances at higher frequencies, wN’ which clustered near the plasma frequency, i. e. , 2 2 2 2 (UN =00p_(1-37\DkN) N:1,2,3,... where kN is an eigenvalue of the bounded plasma problem. These temperature resonances were derived from a hydrodynamic des- cription of the electron motions under adiabatic changes in a collision- 1ess plasma similar to the one discussed in 2.1. The distribution of the temperature reSOnance was, however, too c105ely Spaced to account for the correct experimental results. Better quantitative agreement with the experiments was possible when a non-uniform warm plasma was considered. Physically, as the electron plasma waves propagate up the density gradient, the wave becomes cut-off, and is reflected when it reaches a point where the applied frequency, (1), equals the local plasma frequency (11p . Consequently, only evanescent waves exist in its. 7. the high density plasma core at a radius rc where (”p-ire) = a), while standing longitudinal electron plasma waves of the type described by Bohm and Gross15 are trapped in the low density region bounded by the plasma core and the glass wall. In this model, both the elec- tron temperature and the electron density profile contribute to the resonance conditions. Using a perturbed tensorial pressure and a parabolic electron density profile, Vandenplas and Gould16 obtained a sixth-order linear differential equation for a non-uniform, warm plasma cylinder. However, the solution proved to be too difficult to obtain. In 1964, Parker, Nickel and Gould17 simplified the problem They by postulating a scalar perturbed pressure, p61: 'YK Tnel. obtained a fourth-order linear differential equation for plasma potential on the basis of linearized analysis for a small perturbation of electrons similar to that outlined in 2.1. The differential equation they derived for the longitudinal electron plasma wave is similar to equation (2. 19) that is 2 2 l 1 V_f 1 VZV¢-—(— -ZPV)V¢ +,17 (-‘°—-r)-v vzp¢-—— p ’Y Z Z 2 X a) 771 D P' p Vf ° V = 0 <1>13 Numerical solutions, through numerical integration of the above equation, were obtained with a density profile correSponding to the Tonks-Langmuirl8 model. The calculated resonances showed excellent quantitative agreement with experiments for the main and the first two temperature resonances. _ In this theory the electron temperature, Te had to be chosen for the best fit to the experimental data. 17 Re— cently, while experimenting with rare gases, Hart and Oleson19 observed that the temperature resonances are fewer and broader than those for a mercury vapor discharge. Gould's hydrodynamic model of a collisionless plasma did not include Landau damping which occurs in the low density region near the tube wall. Furthermore, the use of the boundary condition that the electron velocity vanishes at the wall was without good theoretical justification. Recently, W. M. Leavens20 and D. E. Baldwin have developed a kinetic model for the temperature resonances. In both cases the Landau damping that is present near the tube wall is in- cluded in the analysis. Experimental confirmation of these theories has been made without choosing the electron temperature for the "best fit. ” The resonance phenomena exhibited in microwave scattering from a plasma column appear also in the circuit characteristics of a plasma—filled capacitor system shown in Figure 2-1. The hydro- dynamic theory for the resonance behavior of a one-dimensional plasma slab-condenser system was examined theoretically by Vandenplas and Gould. 22 The plasma resonator which consists of two infinite parallel plates separated by a distance L is excited by an externally applied rf voltage source V(t). A uniform plasma slab of thickness L-s is placed symmetrically within the plates between ::::::: mun rntousucv x 11111111 ccccccccc “£53? ) two vacuum sheaths of total thickness 3. In the analysis, the capci- tor impedance was determined by applying a quasi-static approximation and the following boundary conditions (1) the continuity of total current, (2) the perfect relfection of electrons (i. e., 3e: 0) at the plasma- vacuum boundary, and (3) the voltage across the plates V(t) = g de. For the case of cold plasma (T20), the plasma condenser system was found to possess a single anti-resonance at the plasma frequency (1) , and a single "geometrical resonance" at a characteristic fre- p— 1 2 quency (n : (11p (s/L) / . The plasma-condenser system can be thought of as a circuit consisting of two condensers in series. That is, the vacuum sheath and the plasma slab constitute the two condensers . . . . 2 2 hav1ng respective capac1t1es cr : 60/8 and cp = 60(1 -u)p /00 )/(L-s). At resonance where o.) is less than (op, the plasma slab behaves like an inductive medium since [1 - ((1) 2/(1))] < 0. Thus, the resonance r can be interpreted as a series resonance between the capacitance of the sheath region and the inductance of the over dense plasma. In the warm plasma case (Te ;{ 0), the main resonance was 1/2 . . . found to occur at (1:~ UP-(S/ L) for ND << 3. In addition, a discrete temperature resonance spectrum appears at higher frequency (”N > u)p_ where 2 2 2 2 2 2 mN :(1)p_[1+(2N+1)‘YnKD/s]N:1,2,3,... The temporal effect also introduced a discrete anti-resonance spec- trum adjacent to (1) The collisions in the plasma were found to have N. .-.)- ._ -_‘.— - 18 relatively small influence on the main resonance, but significant damp- ing on the temperature resonances. For the case considered by Vandenplas, the temperature resonances were completely obscured by the collisional damping for lye/(1) = . 1. An experimental investi- gation by Mes sian and Vandenplas23 showed that the main anti- resonance and the main resonance agree remarkably well with theory. In addition to providing a physical insight into the nature of resonances in a bounded plasma, this simple plasma capacitor system may also predict the physical behavior of a resonantly sustained high- frequency plasma. An equivalent circuit of this warm plasma will be developed and discussed in Chapter 3. 2. 3 Resonantly sustained high frequency discharge The peculiar phenomena exhibited by a resonantly sustained high frequency discharge or high frequency "plasmoid” was first described by Wood1 in approximately 1930. The rf discharge was created in glass vacuum vessels by applying an electric field of 10 to 100 MHz to some external metal electrodes. Wood observed that the discharge was concentrated in sharply defined, stable, lumi- nous balls, spindles and pear-shaped bodies. That is, instead of appearing as a diffuse glow completely filling the tube, the plasma appeared to "float" and was surrounded entirely by a dark sheath region. Also, the discharge could be sustained with a low exciting voltage. Wood suggested that some electron oscillation similar to those described by Tonks and LangmuirZ4 could be sustained with a low exciting voltage. Wood suggested that some electron oscilla- tion similar to those described by Tonks and Langmuir24 could explain this phenomenon. There was no further progress on the resonantly sustained plasma until 1942. At that time, theoretical studies of the high fre- quency discharge by W. O. Shumannzs; and Allis, Brown and Everhard 26 showed that when a high frequency field is applied to a discharge, the field tends to concentrate in the resonance region of the discharge whe re the ionization rate is maximum. Years later (1958-61), E. R. Harrison27 and A. J. Hatchzs' 29 independently performed extensive experimental studies with rf discharges. They obtained conclusive evidence which supported Wood's hypothesis of an oscillatory phenomenon in the plasmoid. From probe measurements of rf potential in high frequency plasmas at low pressures, Hatch29 observed that the rf field in the plasma is 1800 out of phase with respect to the sheath field. This is contrary to what occurs in a diffusion controlled plasma. The phase reversal indicates the existence of electron oscillation in the plasmoid. Also, the dark sheath around the plasmoid can be associa— ted with the points of zero high frequency fields where phase reversals occur. In spite of all these experimental and theoretical efforts, no good physical explanation of the plasmoid phenomenon had been found until a few years ago (~ 1963). At that time, .1. Taillet3o combined 20 experimental observation of the electric field in the plasma and the role of the negative ions together with a straightforward application of the linear theory of bounded plasma resonance to produce a qualitative explanation of the plasmoid phenomenon. Taillet’s physical interpretation of the resonantly sustained radio frequency discharge was based on the linear theory of Vandenplas and Gould22 for a plasma-filled capacitor shown in Figure 2.1 and the balance of rf power in the plasma. To explain the resonances in the rf discharge, he formulated the plasma capacitor problem by calculating the rf electric fields in the vacuum sheath as well as in the uniform plasma slab. The electric fields of this cold, collision- less model predict Hatch's observation of a phase reversal and oscillation phenomena in a rf plasmoid. Taillet also verified these theoretical results experimentally with a slab discharge which was excited by applying a rf electric field of 15-50 MHz between two parallel and horizontal metallic discs. The rf electric field in the discharge was estimated from the vertical deviations of sharply focused electron beams which traverse the discharge horizontally in the absence of a field. Physically, a resonantly sustained rf discharge differs from the well-knoWn diffusion-controlled plasma in many reSpects. A resonantly sustained rf discharge is a discharge which is always in a resonant state. It occurs in a pressure regime lower than that of a diffusion plasma. Furthermore, it is characterized by a low 21 rf sustaining voltage. Due to the nature of resonance in the plasma, the rf field inside the plasma is 1800 out of phase with respect to the sheath field. Also, this resonance effect allows the E field inside the plasma and the sheath to be much greater than the field when there is no plasma present. In a resonantly sustained rf plasma, the ions are lost to the wall not by an ordinary diffusion process involving collisions against the neutral particles, but by direct fall. That is, in such a low pressure regime, the mean-free-path of the elastic ion—neutral collision is larger than the dimension of the confining vessel. The electron plasma frequency of a resonantly sustained dis- charge is of the order of magnitude or greater than the driving fre- quency. Whenever the generator frequency or the pressure is varied, the plasma reacts to such a change by adjusting its parameter (1. e., electron density, sheath thickness, etc. ), so that an eigenfrequency of the bounded plasma system remains approximately equal to the generator frequency or a harmonic of the generator frequency. A typical resonance-sustained high frequency discharge is pictured in Figure 2.1. Note that the plasma is completely surrounded by sheaths, and thus makes no contact with the capacitor plates. The plasma is created and maintained at low pressure by a high frequency power. Although much progress has been made toward a better under- standing of the radio frequency plasmoid, but the physical process of 22 the nonlinear phenomena associated with such a plasma is still poorly known. This is particular true in a microwave (GHz) dis- charge since most experiments in the past were performed in the MHz frequency range. In this investigation, the nonlinear phenomena of a resonantly sustained plasma were studied at frequencies of 3.03 and 9. 09 GHz. 2.4 Nonlinear effects in a resonantly sustained hiLh frequency plasma In the forgoing discussion of resonance behavior in plasma, only linear effects have been considered, and a linearized theory has been adopted to explain exPerimental results. However, as the inci- dent microwave power increased, nonlinear effects become important and the plasma will finally become resonantly sustained by the micro- wave power. These nonlinear properties have been the subject of experimental and theoretical research for a number of years. In the following, the research on harmonic and sub-harminic generation, parametric interaction and the nonlinear behavior of temperature resonances at high power is briefly reviewed. The harmonic generation in a plasma has been investigated by numerous researchers in the past. Recently, Asmussen and Beyer33 extended this investigation, both experimentally and theoretically, in a coaxial discharge structure. Their experimental results showed that maximum third harmonic power was generated when the plasma is resonant at the third harmonic as well as the Z3 fundamental frequency. Their results were qualitatively inter- preted with a nonlinear plasma-filled capacitor. In this kind of harmonic generation, the dominant nonlinear microwave source of harmonic power comes from the spatial and time varying electro- magnetic field as well as inhomogeneities in a bounded plasma. 34 In a recent experimental investigation employing a cylindrical ionized column of mercury, vapor inserted across a waveguide, en- hanced radiation at 1/3 the applied signal was observed by Demokan, Hsuan and Lonngren. 35 The proper resonance condition was for the bounded plasma to be resonant at the fundamental and the subharmonic frequency. The strong coupling between the incident microwave sig- nal and the fundamental longitudinal plasma wave at resonance generates a longitudinal field of large amplitude which acts as a driving term for the sub-harmonic. The resonance phenomena exhibited in microwave scattering by a plasma column are linear at low power. However, as the inci- dent microwave power is increased Hsuan, Ajmera and Longren2 observed that the temperature resonances exhibit a strong nonlinear behavior. They interpreted these nonlinear resonance effects at high power as due to the alteration of the zeroth-order density distribution of a plasma with an external electromagnetic field. On the basis of this assumption, they derived a nonlinear differential equation that includes the spatial inhomogeneity and temperature effect of the macroscoptic electric field. The solution predicts the following “- Z4 nonlinear phenomena: a hysteresis behavior of the resonance, a shift in the resonance frequency and an abrupt jump in the magnitude of the reflected and transmitted signal. These nonlinear resonance effects were believed to result from one of the three physical mecha- nisms: (l) heating, (2) ionization and (3) spatial inhomogeneith of the electric field. Similar nonlinear resonance effects Were noted by Hsuan, who formulated the problem of interaction between a plasma and radiation in a self-consistent manner. Concurrently, Massiaen and Vandenplas4 investigated the nonlinear resonance effects at high power with a plasma tube—waveguide system. A photo-electric device was installed to measure the plasma luminosity, and hence the average plasma density. In addition to the observation of hysteresis and defor- mation of resonance peaks, the plasma was seen to show preferential absorption of high-frequency power at resonance and a tendency to remain in a resonance state. In Chapter 5 this is discussed further. For sufficiently high incident microwave power, the plasma can be self-sustained by the high-frequency energy in a discrete set of resonance states. These resonances correspond to the linear tem- perature and cold plasma resonances at lower power. Then the plasma is resonance-sustained, the absorbed power and the corres- ponding plasma density varies only slightly with incident power. The parametric excitation of the modes of an infinite plasma has been studied theoretically by several investigators36-39. Experi- mentally, the parametric coupling between electron plasma and ion- 7 ,__..—-‘ . , '- 25 acoustic oscillations was investigated by Stern and Tzoar4o. In the experiment, a microwave signal of frequency 000 was incident trans- versely on a cylindrical dc plasma column. The signal frequency is approximately equal to a temperature or cold plasma resonance of the column. When the incident "pump" power was increased above a certain power threshold, radiation at two additional frequencies, moi wi’ was observed. This observation was interpreted as being produced by a four -frequency parametric interaction. That is, an interaction of a low frequency resonance at mi with high frequency resonances at (no: mi and (no. Above the threhold, the frequency, mi, compares closely to the value of the ion-acoustic oscillation which had a wavelength of the same order of magnitude as the inside diameter of the tube. The excitation of the ion oscillation at 001 was detected above the threshold by probe measurements. According to their experimental results, the pump field ”threshold" is con- siderably lower than that predicted by DuBois and Goldman4l, who formulated the problem on the basis of a Green's function perturbation analysis for an infinite plasma. DuBois and Goldman suggested that the unreSolved discrepancy might be expected to result from an effect other than the pure parametric excitation in a plasma. Namely, they postulated that the low frequency fi was excited by another type of instability (such as tWO stream instability) and the high frequency Spectrum resulted from a simple frequency mixing. ‘ ' ‘ ‘ 26 :-~' ‘3. ' i ‘ I In an attempt to explain the discrepancy between the theoretical ff'g'Z‘ i and experimental threshold condition, Amano and Okamoto42 pointed : 1' out that the density gradient of a plasma, which always exists in " laboratory experimental condition, may reduce the pOWer threshold somewhat for the experimental conditions of Stern and Tzoar. Taking higher-order mode coupling into account, R. Goldman43 suggested ‘- that a full treatment of the experimental phenomena would require an analysis of the neturals, fluctuation effects, heating effects and finite geometry effects. To this date, the theoretical interpretation of this experiment is unresolved. In Chapter 5, experimental results for a similar radiation induced instability in a resonantly sustained plasma are presented. It is. shown experimentally that the instability is caused by a four frequency parametric interaction. F*_ CHAPTER 3 THE PLASMA CAPACITOR SYSTEM 3.1 Theory In this chapter, a unified formulation is made for the resonance behavior of a warm plasma capacitor which approximates the experi- mental system used here. Such an analysis will provide physical insight into the experimental results to be presented later. As shOWn in Figure 2. 1, the one-dimensional plasma capacitor consists of two infinite parallel, metallic plates separated by a dis- tance L. The separation of the plates is much smaller than the electromagnetic wavelength. Thus, one can apply the quasi-static approximation. A slab of uniform, warm plasma is located symme- trically around the x: 0 axis between two vacuum sheaths of thickness s/Z between the plasma and the plate. The plasma capacitor system is excited externally with a rf voltage V(t). The formulation of the resonance problem is based on the small signal perturbation of a hydrodynamic plasma. The funda- mental equations used are Maxwell's equations and the linearized 27 28 hydrodynamic equations for an interpenetrating electron gas and ion gas. For time-harmonic variation, these equations are Jrnmével : -eE - Vnel - ”emevel (3.1) O J”"mi"11 ‘ 6E ' vnil ' ”mivil (3'2) 0 \ 0.. ' ~ : o 3.3 N07 vel+Jonel ( ) V 7’ ' = 0 3.4 No/ V11 ”‘0“11 ( ) "’ _ 8. _ J » E _ G (ni1 hell (3. 5) O \7 x E : -jwp.O—I:I (3.6) \/ x H : Noe(vi1- vel) + Jme‘oE (3.7) where the subscript, e, denotes electron and, i, ion; n and v are the perturbed macroscopic particle density and velocity. Since hydrodynamic equations are used, Landau damping is not included. Consequently, the result becomes erroneous in a certain low density region. Also, by assuming a uniform plasma, the steady state plasma density thus does not vary with position. Such an assumption introduce inaccuracy to the resonant frequency. Since the intention of this analysis is to study the general nature of resonance in a bounded plasma, and not to produce exact theoretical verification of experimental results, the non-uniformity of the plasma is neglected. However, it is understood that the exact resonance positions of a non-uniform plasma differ from the results obtained here. Z9 Maxwell's equations are used to derive the differential equa- tion for the electric field in a plasma medium. From equation (3.6) and (3. 7), it can be shown that Vxinf 2 175— '» Ne? V) (38) 0) H060 _ J(”Lo 0 Vi] e1 ' using equations (3.1) and (3. 2), equation (3.8) is reduced to Z —> —-> 2 Ui Uez VxVxE-QE=wp.oe (————vnfl- . Vnel) 00 'Jmui (D 'JQ/e (3.9) where 2 co 2 (n + s2—‘i’3 1- 2?“ .2J (3.10) C 00 +J®Ve 0) +Jwvi Z 2 2 Noe 2 Noe (l) _ — m ; (.0 + E m 60 e p o 1 KT KT 2 U : ; U.2 = e m 1 m e 1 Equation (3. l), (3. 3) and (3. 5) can be combined to produce 2 2 2 2 2 _ .. ' : 0 3. (m o)p_ Jwye + Ue V )nel +oop_ni1 ( 11) In the case of plane wave propagation, V = jk', equation (3.11) be- comes 2 2 2 Z Z - - ' - U + = 0 3.12 (on mp- Jwye e k )nel (op-mil ( ) Equation (3. 12) and the divergence of equation (3. 5) are then com- bined to give 3O 2 , 2 2 - k(k . E) = —— -—-———— E 2 o u) p- (jknel) (3.13) From equation (3. 9), (3. 12) and (3. 13), a differential equation for the electric field in plasma is obtained -ExKxE+sRE-E)-QE: o Bum where 2 2 2 . 22 U.(o)-(o -Jw,,-Uk) (3: a) 1 p- e e 2 2.2 22 2. c ((0 -Jco ve- Uek) a) -que Uzwz + e_E; OJM w ~J> Ti in the experimental plasma. At high frequency (a) Z (op_), the phase velocity of the ion- acoustic wave. is equal to the ion thermal speed (i. e. , vph: KTi/mi)' The cold ion approximation assumes that the ion-acoustic wave does not exist at high frequency. This is a good assumption since a more accurate analysis using particle distribution function, f, shows that the ion acoustic mode is heavily Landau damped for a) >> ij Te >> Ti' Thus, physically, it can not exist as a wave at these high frequencies. As a result, in the model developed here, the ions 33 form a stationary background in the plasma when a) > top . At low frequency (a) < 0013+), a look at the ion-acoustic mode shows that the cold ion approximation is not good for a) > a) , but p+ . . . 44 it IS valid for (n < a) . p+ This cold ion approximation reduces the diSpersion relation (3. 21) to a quadratic equation in k. That is 2 2 (0 (o p- + p: :1 -‘ -U2k w-‘w w que e Jui 01' (wz in) )(w2 w 2 jw ) w Z(oz joo ) 1/2 = 1’e p+ ”i J- Ill (3 26) 2 2 2 , . Ue (w - wp+ - JON/i) From equation (3. 25), the electric field in the plasma becomes E = Alcoskx+Azsinkx+Ao (3.27) If collisions in the plasma are neglected (ye = "i = 0), the phase velocity obtained from equation (3. 26) is 2 2 1 - mp+/u) 1/2 vphz Ue ( 2 2) (3.28) pm 2 Z 1 -wp+/oo - w _ The high and low frequency modes of the cold ion approximation are shOWn in the plot of phase velocity versus frequency below. 34 "n 1 | u.. ‘ 1 1 1 45 _ l | 2 ll, 1 ( cap-MOO «2+ 1 1 "e 2 6110 III/Sec. o J, 1 "p z 3.71103m/sec. 9 “9+ ’r “L on In the investigation below, a discussion for the high and low frequency modes of the plasma capacitor are given. 3. 2 Resonance at high frequency (a) > (op >> mp+) At high frequency, the ions are assumed fixed (i. e., Ui: 0, Vi = 0, v11 = 0 and nil : 0). The diaperson equation (3. 26) can then be written as 2 1/2 (1) (D _ ye k : U— (l-‘Lz -J: (3.29) e O) The electric field in the plasma slab, Ep’ is given by equation (3. 27) E : Acoskx+Asinkx+A (3.30) p l 2 o In the vacuum sheath, the quasi-static electric field, E0, satisfies Laplace ' 5 equation. Consequently 35 (1E0 ‘5;- = 0 01' E : B (3.31) O O The constants, B0 and A0. . . A2, are determined from boundary conditions. As often done, one boundary condition is derived from the assumption that electrons are perfectly reflected at the plasma- air boundary7. Thus, the normal component of the electron velocity, :el’ must vanish at x : i(L-s)/2. From equation (3.1), this assump- tion implies that there exists an electric field, 1?: = -(KTe/ Noe)Vnel, at the plasma-sheath interface physically, the electrons are not allowed to build up a charge layer at the interface when Te ;/ 0. Other boundary conditions are the continuity of total current and V(t) : SE ' d?. The latter is obtained from the quasi-static approximation which relates the externally applied rf voltage, V(t), to the electric fields in the plasma capacitor. To summarize, the boundary condition used are: (i) x ° v81: 0 at x: i(L-s)/2 " ' r. '15 =° "13’: " f ' th 1 (11) Jmko o ngo p(x) + Jp(x) or any x in e p asma -(L-s)/2 (L-s)/2 L/Z (iii) V(t) : S. E dx +5 E dx +5 Eodx ‘-L/Z ° (L-s)/2 p L-s)/2 Form the continuity of total current(ii), the boundary condition (1) and equations (3. 30) - (3. 31) the following relations hold at x z :(L-s)/2 36 O 13 A0 + Al cos k(L-s)/2 (3. 32) A2 = 0 (3.33) After applying the first boundary condition (i) to equation (3.1), the result is combined with equations (3. 5), (3. 30) and (3. 33) to give U 2k2 A0 = - (1+ 8 2 )Alcos k(L-s)/2 (3.34) (I) p... Using equations (3. 32) to (3. 34), the electric fields can be simpli- fied to the following forms Uezk2 k(L 3) E0 - - 2 A1 cos 2 (3.35) (i) p- U Zkz k(L-s) E =A coskx- (1+ e )cos—-—-— (3.36) p l 2 2 0.) p— Because of the symmetrical geometry, the electric field is an even function of position. The third boundary condition (iii) along with equation (3. 35) to (3. 36) yields A : “V1” 1 U 2k2 e k(L-s) E . k(L-s) L> mp_, ye), the phase velocity of the electron plasma wave approaches the electron thermal velocity Ue; i. e. , w/k ~ Ue. Consequently, X = Ue/f. A decrease in electron tem- perature decreases the wavelength X and thus allows more resonant states to exist in the plasma capacitor. (Note that the curves in Figure 3. 1 to 3. 5 are plotted for a constant (1)/Ue ratio. Thus, 1. is limited at high frequency. ) Collisionless Landau damping becomes important when kxD > 1.45 From equation (3. 29), this corresponds to (mz/wpf- 1)l/2 > 71/2. Thus, the Landau damp- ing of the resonances is important when wp_/(n :. 7. The Landau damped region has been indicated in the figures. The temperature effect has relatively small influence on the resonance frequency of the main resonance, while drastically changes the resonant fre- quencies of the temperature resonances. The resonant frequency for the main resonance of a warm plasma is slightly lower than / 2 mp 18 / L)1 which is the resonant frequency obtained from cold plasma theory. lepuw 42 1100) 900) | I --.... .Q... 11 --2x10 1 ' 1 oo 7 '-"'_ %=3Xlo4 O 500 300 2001 1501' 50 ::.=== . -— o‘ . 0 London .7 I don‘ping ’—1 0 In P... Figure 3. 3 Normalized amplitudes of the high frequency plasma field, 1LE /V1a VB- (1) _/(1) for different electron temperature, Te, (L=l.6mm, 2.6mm, Ve/w=.oo1,x=o, (1)/U =2x10 3 x10 ) e A 43 (5) The influence of the sheath on the resonances of a col- lisional, warm plasma is shown in Figure 3.4. As the sheath, 8, is increased, the temperature resonances are found to occur at lower plasma densities. In the case of the main resonance, increasing the sheath from . 2 mm to . 6 mm results in a change of (op-AD from 2. S7 to 1. 53 respectively. The cold plasma theory (Te: 0) explains 46 this resonance quite well. (B) The variation of electric field as a function of position. The Spatial variation of the real part of electric field is shown in Figure 3. 5(a) for the main resonance, and in Figure 3. 5 (b)-(c) for the temperature resonance. These resonances correspond to the case of Figure 3. 1. Each resonance is shown at a time when the electric field has reached its maximum value. As shown in Figure 3. 5(a), there is no wave phenomenon associated with the main resonance (wp_/u) = l. 526). It is clear that the physical origin of this resonance comes from a resonance between the inductive effect of the plasma and the capacitive effect of the sheath. For the temperature resonances, standing waves exist in the uniform plasma region. The occurrence of the tempera- ture resonance requires that (L-s) ~ (N + % )k, where N = l, 2, 3. . . . These results can be also obtained from the impedance approach as done by Vandenplas. 46 44 AEEo.meEEN.nmm.oux . unouomflv new 8\ 8 .m> Tm. \ 10 * O O—xbnfls: w. v A 3.353. .3 III 'l"'lIl " I I, I I o o ome u D\3 .So. u3\ s .88 c M11 303 gunman cacaosvoum swan on: no opsuflagm @0338qu w .m ondmfim 4 u 1: .m .mmosxowfi gunman -_---—---—----- _ — -- -~ ~ ‘ 9.350? m. 301:5.— O EEO.” W 1.1! EEN.Hm III. A IAI'I°3| O— m— -...) X:.8Iu /%% / %/ I ............. 46 (C) Impedance and equivalent circuit The impedance of the plasma capacitor system is defined as z :- V(t)/ I where I = jwe SE 0 o S is the surface area of the capacitor plate. In terms of E0, the impedance, Z, is of the form : ' 3 . 41 z V(t)/JUDEOSEO ( 1 Similarly, the admittance, Y, can be written as Y = JwEOSEo/Wt) Hence, the amplitude of the admittance, Y, is directly proportional to the amplitude of the electric field in the vacuum sheath; i. e., the admittance variation differs only by a constant from figures 3.1, 3. 3 and 3.4. Substitution equation (3. 39) in equation (3. 41) yields 2 2 (0 y (.0 _E.'.'._£‘. _._e_ -Z__E. a k -s 2 1 2 8(11w1+k8 Ztn(L)/. 7,__. ‘0 m (3.42) JmC 2 00 v (1__R:_j_?.) Z a) where C = 6 GS/s is the total capacitance of the two vacuum sheaths. If the collisions in the plasma is negelected, equation (3. 42) becomes 47 2 co .. Z .. .. 7.. I .1 1+ 1 L(s +_ E tank(L s)/2 (3.43) JmC 2 2 ks 2 2 2 (1) /oo -1 co (1) /(1) -1 P' P' where 2 1.» w 2 1/ k : fi- (1 -—P—‘Z ) e 0.) Equation (3. 43) can be represented by an equivalent circuit which consists of a los‘sless transmission line terminated in a short where C(l moi/(1)2) p: (Ls-1) “*9ij L = (Lia-l) < p (co wp_) 2 2 C(01) - w ) p— X is Cp or Lp depending on whether (.1) is greater or less than a) Z. = jZ tan 0 k(L-s) in Z (3.44) From equation (3. 43), the input impedance to the shorted trans- mission line is also given by 2 (1) z : j_2__ 2' tank(L-s)/2 (3.45) c (1)3 k(l-wPf/wz) 48 According to the transmission line theory, 2.0 : (7.'/Y')l/2 (3.46) j k (z' Y')1/2 (3.47) By defining the series impedance per unit length of the line, 2', and the shunt admittance, Y', as follows: Z' = ju) Yl . Z . 1w: +w e/Jw p- Equations (3.46) - (3.47) can be reduced to the following forms 2 2 1 2 k = (011160 -oop_/> ye . Thus the increase a) in height of the main resonance results from a shift of resonance position to a higher density. 50 —— $2.2mm ---- =;6mm \ ~Q- 1 - I6 20 K 1102) ' “s Figure 3. 6 Graphical solution of the high frequency capacitor impedance for different sheath thickness, 3, (L = l. 6 mm, (1)/U6: 2x104, s = .meand .6 mm) 51 3001 200 ' 100.; 30- 0 2 0 4'0 To 1'20 1'50 200 K (10 ) 4—u 9 Figure 3. 7 Graphical solution of the high frequency capacitor impedance for different electron temperature, ”I; , (L =1.6 mm, s = .6 mm, oo/Ue = 2x 104 and 3x 104) 52 Unlike the main resonance, the temperature resonance de- crease in height (i. e., the admittance decrease) for the above mentioned changes of Te and 3. Physical insight into this phenomenon can be obtained from the graphical solution of equation (3. 50). In Figures 3.6 and 3. 7, the two sides of equation (3. 50) are plotted as a function of k for (1) > mp_. The points of intersection correspond to the electroacoustic (or temperature) resonances, while the asymptotic lines of the tangent function locate the anti-resonances (i.e., Z 2 00). The resonance and anti-resonance are Spaced very close to each other. The Spacing decreases with increasing order of electroacoustic resonance. One important effect of varying the temperature or sheath in a plasma is manifested in a shift of resonance position and a change in the number of resonances as discussed earlier. An increase in Te or a decrease in 5 results in a greater separation of the electroacoustic resonance and anti- resonance, and a subsequent increase of the resonance peak as shown by the graphical solutions. The experimental observation of electroacoustic resonance depends on the coupling of the external circuit to the electroacoustic resonance in a plasma. This of course is a matter of impedance matching. That is, matching the plasma capacitor impedance to the external microwave system impedance. One can observe from the graphical data that an impedance match between a resonance plasma and a given microwave system usually does not occur. This 53 is particularly true for the temperature resonances. In fact, it has been pointed out that the microwave coupling to the temperature resonances is poor“). The large impedance mismatch, of course, explains this. An analogy between the coupling of a metal cavity to a micro- wave system and the coupling of a resonant plasma to a microwave system can be made. In a metal cavity, one must adjust the coupling loop position, size, etc., or aperture-size, etc., for maximum coupling (i. e., to a critical coupled condition). With a plasma resonator, there are no such metallic loops available. In order to improve coupling, one must adjust (in the case discussed here) L, s and Te for maximum coupling. This is not always eXperi- mentally possible; thus many of the resonances are not observed at all, or are observed as very weak interactions. This is parti- cularly true for the higher order temperature resonances of a plasma column. The coupling to the plasma resonances can be improved by building metal impedance transforming networks (i. e. , slide screw tuners, microwave cavity, sliding short, etc.) around the plasma. 32 33 This has been done by those working with harmonic generation ' . 3. 3 Resonance at low frequency ((1) < wp’r) At low frequency, the thermal motions of the ions are neglected (U1: 0). The electric field in the plasma is given by equation (3. 27). 54 E = C coskx+C sinkx+C (3.51) p l 2 o where k is given by the diSpersion relation (3. 26). The electric field in the vacuum sheath is obtained again from Laplace's equation. E = G = constant (3.52) o 0 Since the ions are assumed cold (Ti : 0), equation (3. Z) is then reduced to _) esp . = . 3. 53 V11 (30) + Vi)mi ( ) The ion acoustic wave at the sheath has been seen to possess both reflective and absorptive characteristics depending on the nature of the sheath. The reflection of the ion-acoustic wave occurs at an electron rich sheath (positive bias on reflecting surface) or an ion rich sheath (negative bias on reflecting surface) of a “perturbed” 47 . . . . . plasma , while the absorption of the ion-acoustic wave is resulted . . 48 ' . . . . from an ion rich sheath . Thus, no definite boundary condition for the ion-acoustic wave at the sheath has been established. As a result, the same boundary conditions for the high frequency electro- acoustic wave are applied to the low frequency ion wave here. These boundary conditions are us ed for lack of better boundary conditions. (i) 3'5- 361 = o atx:i(L-s)/2 (ii) jweE = jme E +J o o o p p -(L-s)/2 (L-s1/2 L/Z (iii) V(t) : S1 E dx +5 E dx +3 Eodx ‘ L/Z O -L-s)/2 p L-s)/2 55 In the absence of the ion pressure (i.e., KTiVnel/Noe : 0), the ions may form a rf charge layer at the plasma-sheath interface. The continuity of total current (ii) is then of the form ' E = 7 ' E 3. 3006o o Noevil JrJOOEo p ( 54) Equations (3. 51) to (3. 54) are combined to produce the following results at x : i(L-s)/2. 2 CL) ._ ___£:r__ Ellis). GO - (1 2. )(CO+C1cos 2 ) (3.55) 0.) -qui c2 = o (3.56) From equations (3. 2), (3.4) and (3. 5), the following equation is obtained. 00 2 - 12+ 3 n11~ 2 2 . \ nel ( .57) (D cop+ ‘1in Applying the boundary condition (i) to equation(3. 1) yields KT (jkn ) (3.58) p Noe e1 By combining equations (3.13), (3. 57) and (3. 58) with the diSpersion relation (3. 26), the following result is obtained at x : i (L-s)/2. 2 2 . . (w +-61 +Jwvin'Jve) C = P ‘r C cos (I) pr-Jg) 1 k(L-s) (3.59) From the boundary condition (iii) and equations (3. 51) - (3. 52), it can be shown that 56 V(t) : 360 + (L-s) c0 - (3.60) k 2 Now, the constants are solved from equation (3. 55), (3. 59) and (3.60) in terms of V(t). After substituting the known constants into equations (3. 51) and (3. 52), the electric fields in the plasma and in the sheath have the following form 2 . 00 _(03 H Vi) V(t) . + coskx - w-Jue 2 . cos k(L-s)/2 (A) ‘0) 'JCDV. p-I' 1 Ep: 2 2 2 Swp_(m-in)+[S(Dp+-[(D(OJ-in)1((1)-J ye) - E (co-J Vi)wp- tanML'S) 00(0) - jVi) k (0) Z-wz+'w ) 2 . p+ J ‘1 (3.61) , Z Z . Z . - [(w-JueHw «013+- qui) -wp_(w-J v11] V(t) E : 0 (OD-.11) 120) 20) 2 _. 2 . . _ i p- k(L-s) swp_[(n-in)+[scop+-L00(0)-Jui)] ((1)-Jpe)-k ( 2 2+. ) tan 2 cop+ (D qui (3.62) (A) The variation of electric field as a function of plasma density. The electric fields at the center (x:0) of the plasma capacitor are calculated as a function of mp+/m for the following fixed para- meters; L 21.6 mm, s = .6 mm, m/Ue = 25, ve/(n: .1 and Vi/w : O. 001. Figure 3. 8 shows the variation of the normalized electric fields versus 00p+/00. The results obtained fromthis study are discussed below. (1) At low frequency, there exists a series of discrete resonances for (1) <0) +. The Spacing between these resonances increase FIELD 600 300 ' 100 mm fi'I—fifi 50 25* 57 __ IEPl/Vl ---- IEol/V| ~— 4-0 I 11 11 II II II 11 '1 I I1 5 ’ .\ J' ’ \ I \_‘I *4 Figure 3. 8 Normalized amplitude of the low frequency electric fields, (LEO/y'and1LEO/v1, vs. 0) 1/(1) (L: 1.6 mm, s = .6 mm, (1)/Ue = 25, x: 0, u_ (1) =.1, u+/co= .001) 58 with the increase of mp+/w. As a result, a higher concentra- tion of closely spaced resonances are found near wp+/(0 = 1. The explanation to this phenomenon is provided by the v vs. ph 00 diagram. At a frequency near the ion plasma frequency, a cluster of resonances is resulted since a small change in den- sity causes a big change in wavelength. However, at low fre- quency, there is only a small change in wavelength, or no change at all, for a big change in density. Thus, the separation between adjacent resonances becomes larger. (2) At resonance the electric field in the sheath, E0, and the electric field in the plasma, Ep, are much larger than the electric field in an empty capacitor. (3) The number of discrete resonances at low frequency also de- pends strongly on the electron temperature, Te. Resonances can occur at higher densities for a smaller electron temperature. Thus, decreasing the electron temperature increases the number of resonances in a collisional, warm plasma. To illustrate, by keeping a) constant while increasing oo/ Ue from 25 to 50, the number of plasma resonances increases from six to ten. The temperature effects can be eXplained with the dispersion relation (3. 26). If the ion collisions are neglected at low frequency ((0 << (0er << 00 ), equation (3. 26) reduces to 59 (1) (0 U0) ep+ The phase velocity of the plasma wave at low-frequency is then directly proportional to the electron temperature. That is K T e (0 k m. i For a constant (.1, decreasing the electron temperature, Te’ results in a corresponding decrease in the wavelength. Thus, higher resonance states are allowed to exist in the plasma. (B) The variation of electric field as a function of position The spatial variation of the real parts of the electric fields for the first three resonances are shown in Figure 3. 9. The real part of the field is calculated with a time phase which maximizes the amplitudes. Clearly, all the resonances result from standing acoustic wave propagations for (n < 00p+. The electric field in the sheath and in the plasma are discontinuous at the plasma-sheath interface as a result of charge accumulation. This change layer can be determined from the boundary condition 11' ( o-Dp) = 08 01‘ 600:0- Ep) : 0 Using equation (3. 53) and (3. 54), the surface charge induced on the surface of the plasma-sheath interface is of the form 60 —I¢(Epl/VI --.-111(101/11) + 1 0 :8. u 3\ s .H. 13\ s ....N u D? a .98: o. .1. m .85 o4 u 1: 00:0:0m0u an .x .noflwmoa .m> .>\ MA .303 0930.3 >onoswonm 33 posflmguoc 05 we used Hmom o .m 0.5th / 2, // // \ss . \ 3 h _ r. _ _ _ _ \\ x \\ \\\\\\\\: . \\\\\WM§ §§xx v.i \\\\\\\\\\\\\\\\\\\\ 600) :I.I7I U/U I'+ — o o o o o 2 all 200 . up In: :l.049 + a 1022 I'1- U/ = 61 's 2 , CL) "3031/1 where Ep is the electric field in the plasma given by equation (3.61). As shown in Figure 3. 9, the surface charge, p3, is pro- portional to the difference of the normalized electric field ampli- tudes at the plasma-sheath interface. Note that since the electric field is a function of time, the surface charge is also a function of time. That is is a rf surface char e densit . p s g Y (C) Impedance and equivalent circuit (a) < oop+) According to equation (3. 41), the impedance of the plasma capacitor is of the form Z : V t . SE ( )/JOO€O 0 If a collisionless plasma is considered, the electric field, E0, of equation (3.62) reduces to Z 2 2. Z Z .. (w _ (D + - a) )(m +- (0 HM” E I p B- P (3.63) 0 [SO 2 + x Z) L 2](\ Z 2) ‘2' 002 Ztan k(L-s) -Dp- up+ (‘0 ('L’p+ U) k wp- 2 where 2 Z 2 1/2 (1.) - (D " (L) k _, _93_ ( 3+ t) ‘ Ue Z 2 (D top+ From equation (3.63), the impedance, Z, becomes 62 ( 2+ 2 L (2” 2 2) tan ( -S) / - _ ) ‘ '— 2 — l m _ 013+ k wp+ w ks u) _a) 2 .. I! C 2 Z J 0 ((0 _. a) - (fl ) ((1) - (D ) (3.64) At low frequency, the ion plasma frequency w +' and hence a), are generally much smaller than the electron plasma frequency 0) Therefore, equation (3. 64) can be written as 1 me (L/s -1)(r)Z 2 tan k(L-s)/Z z = . 1- -— ) (3.65) 303C mi 2 ks Z/ 2 1 03p+ CDP+ (L) - where m. 1/2 2 -1/2 (I) 1 (L) \ k : .— (__) ( - , U m 1 2) (3 66) e e m p+ Similar to the high-frequency case, equation (3. 65) can be repre- sented by the following equivalent circuit n IL * fir " /// /, c lp //////% ; l.//,///4/// / ,//// r é/w/Z/é : . ”C" 4 7 /////////%¢//A |-—— (l—s)/2 __.‘ where L ‘ :3 (L/s-Zl) p mi C a) + In view of equation (3.65), the input impedance to the transmission line is of the form r . 2 e me tank(L-s)/Z /- = J— (a?) (3.67) (ep+ i (l-mz/mp:)l/Z 63 Also, from the transmission line theory, k(L-s) Zz'Zt 'JoanZ 1n where the characteristic impedance, Z0, and the propagation con- stant, k, are given by z : Qv/le/Z 0 jk = (Z'le/Z If the series impedance per unit length of the line, 2', and the shunt admittance are chosen as 7,. -j(1_‘}1- 2 w -1 Y! _ (flwtffl) C. JCDE The characteristic impedance, Z , and the prOpagation constant, k, 0 then have the form 2 -1/2 _ m 1/2 w k '5‘” (H6) (1-———Z) (3.68) P+ wp+ 1/2 2 1/2 - E L 7o”w+( ) (1' 2) (3'69) 9 E w p+ (0 2 m, = (421) .3. HG: U m e e 3 2 m 02+ sC i 2 E = *r-'—- (w -<0) 2 Z m U e ‘"‘f:::! 64 __z__( 2 2)-1 H -- SC (I)p+-(A) ()n the basis of the above information, the following equivalent cir- cuit can be drawn for the shorted transmission line. iqu I ’6??— I i 2 1“” V | “NJ. 1 | 1096 I l l I l __._ m J. l I it» 1 ‘ I l l I Since (I) <0) + is assumed, the propagation constant, k, is always positive real. At resonance (2:0), equation (3.65) is simplified to 2 5k _ 0‘ (L-s) : tanhfkr‘i) (3.70) 2 2 2k 2 Zme Ue Ue 2(k -- —— -1) m. 2 The resonance condition can be obtained from graphical solution of equation (3. 70). Figure 3. 10 shows the graphical solution of the first three resonances in the high plasma density region correspond- ing to the case plotted in Figure 3. 8. Note that the first resonance can be a half wavelength or a whole wavelength depending on the value of (n/ Ue ratio. For a large (n/ Ue ratio, only the high order of resonance can be excited. IOW A 65 A 5 Figure 3. 10 Graphical solution of low frequency capacitor impedance in is 20 (L: 1.6 mm, s = .6 mm, co/Ue : 25) 25 ‘K (103) CHAPTER 4 EXPERIMENTAL SYSTEMS 4.1 Introduction Nonlinear resonance effects at higher power have been in- vestigated with two different experimental systems: namely, the coaxial type system and the ridge waveguide system. In the coaxial type system, the plasma is created in a small break (> 1.6 mm) in the inner conductor by a high-frequency electric field of 3. 036 Hz. The nonlinear resonance effect at both the fundamental and the third harmonic frequencies have been analyzed. One unique feature of this system is its ability to vary the plasma parameters: The input power, the gap length and the pressure in the system can be adjusted independently for various resonance states. Furthermore, a small microwave signal or a dc bias can be externally applied to the system to study the sheath and parametric amplification effects. Although only dry air was used, different gases, such as A, Kr and Xe’ can be introduced into the system. A picture of this system is shown in Figure 4. l. The structurally simpler ridge waveguide system is similar to an exPeriniental system that has been used by other investigators. 66 Figure 4. 1 Experimental system: (a) overall view of the coaxial system; (b) close view of the coaxial discharge structure. 68 This system consists of a positive column of a mercury vapor dis- charge which is inserted across a rectangular ridge waveguide perpendicular to an externally applied electric field. Either a dc or a resonantly sustained high-frequency discharge can be created. In fact a combination of the two types of discharges can occur simul- taneously producing a dense plasma in the ridge waveguide system. In the following, the structural design and the assembly of microwave equipment pertaining to the two plasma systems described above will be discussed. 4. 2 Coaxial type system A block diagram of eXperimental apparatus for the coaxial type system is shown in Figure 4. Z. The system can be considered as a combination of three sub-system; namely, the microwave system, the plasma—vacuum system and the detecting system. Each of these sub-systems is discussed separately below. (A) The microwave system. The microwave signal of frequency 3. ()3 CH7. is provided by an external cavity klystron oscillator (ZKZ8) which is connected to a HP Z650A oscillator synchronizer to form a phase-lock loop. The synchronizer phase locks the klystron oscillator to a harmonic of an internal Spectrally pure crystal oscillator to produce a frequency stable, microwave souce. A frequency stable source is important here, because the plasma [Tledillnl is generally dis persive in nature, and the microwave circuits new are frequency sensitive. 69 . III t oscillator : power“ Muchromzer SUDD I II N IWT power / supp‘y external canty klystron coaxial . 3 Mil attenuator input _ TWT (Varian 6156) --* IWT —* l H dc block 0"t9"t capacitor coaxial wave l quarter-wave isolator meter stub / harmonic line stretchers\ titer incident ———--1 power \_\,/ to Power / ’i \ poets; of directional I9 re coup er Specimen? \ reflected -<’ ana y: r Dower N (i matched coaxial A ‘ XBand load atten ator (II’IV””,_/I"'IJ\ sliding short III I. NW \ / \ 1 22—6)“ i513; 'rII'I%L'_F - ‘JTIEIL’ :31: mwrge Photo-diode tuner ‘ :ggldiglr'n H'P'43] coaxial sliding Y Power Short meter ii to low freq, spectrum analyzer Figure 4. 2 system. Experimental apparatus block diagram for the coaxial 70 As shown in Figure 4. Z, the frequency stable microwave power is fed into an isolator. The principle function of the isolator is to block the reflected microwave power from returning to the stabilized klystron source. A short tuning stub may be inserted to match the system for maximum power transfer. Under matched conditions, the maximum attainable power from the klystron source is approxi- mately 6 mw. This stabilized microwave power is then delivered to a variable attenuator which regulates the input power to a Varian 615 G traveling wave tube amplifier (TWT). By adjusting the variable attenuator, the amplified output power from the TWT can vary from 0 to 25 watts. The output helix of the TWT is protected from being damaged by the reflected microwave power with another isolator. Harmonic power can be generated by the klystron source as well as the TWT whenever they are over-driven. When analyzing the output spectrum of nonlinear plasnia-microwave interactions, the harmonics generated by other than the plasma are generally undesirable. To minimize the transfer of the source harmonic power to the plasma- vacuum system, a low-pass Microlab LA-40N filter is placed between two line stretchers which provide translational freedom for the low- pass filter without alterating the length of the coaxial line. A proper adjustment of the line stretchers resonates the third harmonic power in the coaxial line, as well as reducing the third harmonic power entering the plasma-vacuum system. 71 In general, only part of the microwave power delivered to the plasma-vacuum system is absorbed by the plasma, while the rest is reflected back to the microwave source. Directional couplers were used to sense the incident and reflected power in the system. Also, in the study of parametric amplification in a resonantly sus- tained plasma, a directional coupler was used to couple an externally r applied signal to the high—frequency discharge. in As shown in Figure 4. Z, the third harmonic power generated by plasma nonlinearities was coupled out from the coaxial discharge ' structure with a x-band waveguide system. By adjusting the x-band sliding short and the slide screw tuner for best match, a significant amount of the third harmonic power can be cotipled to the x-band load. (B) The plasma-vacuum system The cross section of the coaxial device used in this experi- ment is shown in Figure 4. 3. The discharge structure is made of non-magnetic stainless steel which has a relatively low vapor pres- sure. A teflon insert which provides structural supports for the anode electrode is designed as a quarter-wave matching transformer. Thus, the input characteristic impedance of the coaxial system is matched to the plasma-vacuum system. In order to hold a high vacuum inside the discharge structure, O—rings lightly coated with thin vacuum grease are placed around the teflon insert. TranSparent mylar sheets are. used to seal the waveguide windows. The use of 722 .opazosbnm owpmgomfiofimwxooo ozumo.sQEOom mmOpU m.¢.ou5wfih paahao o_zo2mq= “mom; 82;; 35:35 a: mao=ocs sucsgom_u «mica Ill 3:: bag:— aop¢d=mz_ 29.5”— :mm 2.59: E: dice-.3 5:»: E2.” 3:: .8 5:58. 2 2: 2.; 53-325 225:...“ ”Eggs. 73 mylar sheets not only permits the coupling of the third harmonic power to the x—band system, but also allows one to observe the dis- charge frono outside. The cylindrical interior of the coaxial vacuum device can be held at a reduced, variable pressure by means of high vacuum valves and a rotary mechanical pump. With the mechanical pump alone, the lowest attainable pressure in the system is about 10 microns. Further decrease in pressure requires the use of a Vac-Ion pump which operates efficiently only with a low, initial pressure. When using the Vac-Ion pump, the pressure in the system can be reduced to the order of 10“5 torr (mm Hg). A thermocouple vacuum gauge and a calibrated Mcloed gauge. are used to read the pressure in the system. Pressure readings are taken after the system reaches an equilibrium state. The accuracy of the experimental results depends strongly on how well the system can hold its vacuum. This is parti- cularly true when experimenting with pure gas. As shown in Figure 4. 4, the reflection of the high-frequency input power at the coaxial sliding short sets up a standing wave inside the coaxial discharge structure. Any variation of the position of the coaxial sliding short will cause a corresponding translation of the standing wave. To produce a breakdown, a microwave power of about 20 watts at 3. 03 GHz is applied to the system. The pressure in the system is adjusted to approximately the breakdown value which depends on the gas. In the case of air, the breakdown pressure l L I 74 I, J 7 l'd' iilor'l‘g W/////////// /////////////////////////7////[//MA ~ ‘ voltage standing \II’II Hamill \ /////// /////7///// ////// fl/fl/fl/ // /[//////// 124/ WWW/A low freq, t t be P Probe \ quar z u \lasma W mova le catho e Figure 4.4 Voltage standing wave pattern for breakdown by high E field. direc ional coup er Photo-d iocle / microwave Input(3.o3 6H2) ridge fibvoltfitfi. rectangular to ridge waveguide transition '— movable short circuit Figure 4. 5 Ridge waveguide system. varies from 2-4 mm Hg. With the inner conductors shorted together, the current maximum is located at the conductor break. Then, the nmvable electrode is pulled out slowly to form a gap of length l. 5 nun. The high electric field in the gap breaks down the gas in creating a plasma. The physical mechanisms involved in the high-frequency breakdown are well known. 49 Clearly, the high—frequency break- down depends not only on the input power, gas type and pressure in the system, but also on the gap length and the position of the sliding short. The gap length can be varied by adjusting the movable elec-- trode which is connected to the grounded, external shell of the coaxial discharge structure. A dial micrometer indicator which is fastened to the movable electrode provides precise, continuous measurements of the gap lengths. Since the gap length is much smaller than the electromagnetic wavelength, the coaxial structure can be approxi- mated as a plasrna-filled capacitor. In the low pressure, high field regime, the plasma originally located inside the coaxial conductor gap is ejected to a position be- tween the inner and outer conductors. In order to keep the plasma confined to the gap at all pressures, a quartz tube is placed around the center conductor at the gap as shown in Figure 4-3. The size of the tube has an important effect on the output of the third harmonic 34 . . . . power. To detect the low-frequency ion osc1llations 1n the plasma, a thin wire probe is inserted into the plasma through the hollow cathode electrode as shown in Figure 4—3. Whenever a parametric instability occurs in the plasma, strong low-frequency ion oscillation at fi and Zfi have been detected through a low-pass filter with a Tektronix 3141.0 Spectrum a analyzer. As shown in Figure 4, 3, a dc bias can be applied externally across the conductor gap through a. quarter—wave stub. To provide dc isolation of the coaxial center conductor, the end plate of the quarter-wave stub is isolated from the grounded, outer conductor by a ring of teflono The width of the teflon ring is chosen according to the radial waveguide theory, so that at the inner radius, the teflon surface behaves electrically like a short circuit. To prevent the plasma from being shorted out, a dc belocking capacitor is inserted between the quarter-wave stub and the line stretchers. With no dc bias applied to the system, the dc voltage induced across the conductor gap by the resonantly sustained discharge can be measured by con- necting a digital voltmeter to the quarter-wave stub. Also, the low- frequency ion oscillations can be sensed through the quarter—wave stub with a Tektronix oscilloscope. (C) The detecting system. The Tektronix 1L40 Spectrum analyzer and the HP 431C power meter were used to sense the incident and reflected microwave power through directional couplers at 3, 03 CH7. as well as the third harmonic 77 power at 9. 0‘) (3117.. The display of the relative power as a function of frequency on the Spectrum analyz r leads to the observations of parametric amplification, frequency shift and frequency modulation in the resonantly sustained high-frequency plasma. Care must be taken not to over-drive the spectrum. analyzer. Otherwise, Spurious response generated within the Spectrum analyzer may produce an erroneous display of signal. The average luminosity emitted from the plasma has been 4 . - shown by others to be proportional to the relative average plasma density . In this eXperiment, the average luminosity from the plasma was measured by recording the lmninosity current in a simple dc photo—diode circuit. The photo--diode which was placed adjacent to the mylar window of the discharge structure senses the luminosity and the plasma volume changes. Experimentally, a X-Y plotter was used to record the incident power, pin’ the absorbed power, Pa} ‘, the third harlnonic power, >s P d' and the rectified dc. voltage across the gap versus the relative sr ' ‘ average plasma density < N>. The voltage input to the X-Y plotter was taken directly from the dc calibration jack of the power meter. These recordings diSplay the hysteresis phenomena in plasma. 4. 3 The ridge waveguide system. A block diagram of the ridge waveguide system is shown in Figure 4. 5. Clearly, the basic difference between the coaxial type system and the ridge—waveguide systeln lies in the geometry of the 78 discharge structure and how the plasma is created. In the former case, the plasma is created and maintained solely by the microwave power in a small conductor gap; whereas in the latter case, the plasma is created and maintained by either dc or microwave power in a cylindrical plaszna column. This column is inserted across a ridge waveguide perpendicular to an applied high-frequency field. A similar system, without the ridge in the waveguide, has been commonly IRA... used by others to investigate the linear T-D resonances exhibited in *‘V-nu‘ £31.}; I ‘1 the microwave scattering at low powerlY, and recently the nonlinear resonance effects of a resonantly sustained plasma at higher power ’4. After the plasma is created with a dc voltage, a high—frequency electric field can be applied to the s-band ridge waveguide system. For sufficiently high microwave power, the plasma can be sustained by microwave energy even if the dc voltage is removed. The same microwave system (klystron, TWT, etc.) described above provides the necessary microwave power. Structurally, rectangular ridges are fastened along the center of the top and bottom waveguide walls. In order to produce a gradual discontinuity at the joint between the ridge and the waveguide wall, the ridge are tapered. The perturbation of the ridge lowers the cutoff frequency of the dominant mode, but raises the cutoff frequency of the next higher mode. As a result, the introduction of the ridge into the waveguide causes a greater separation between single-niode . 50 fl . operation . b urthermore, the ridge are used here to concentrate a strong high-«frequency electric field across the narrow gap where the. plasma column is located. Hence, a more efficient use of micro— wave power is possible. Since the ridge waveguide is terminated in a movable short circuit, the standing wave inside the waveguide can be translated to a position where the maximum electric field strength falls directly on the plasma column. As shown in Figure 4, 5, a photo—diode. is inserted into a small hole (diameter : 2mm) through the ridge. on the top waveguide wall in order to measure the average luminosity emitted from the plasma. Unlike the coaxial case where the diode is placed relatively far away fron‘i the small plasma gap, this diode is embedded in a tiny hold Ineasuring only the density of a small fraction of the plasma. That is, it sees only a small portion of the plasma, and thus does not sense plasma volume changes, With this experimental set-up, the luminosity of a resonantly sustained plasma which is proportional to the luminosity current, 199 can be directly related to the average plasma density. This was done by measuring: the luminosity current, I and the corresponding electron plasma density, n , in a dc dis- l” e charge. The electron plasma density was determined by the cavity perturbation method with a cylindrical cavity placed adjacent to the ridge waveguide. By varying the dc discharge current, a curve of 119 versus II was drawn. As shown in Figure ~‘l-6, the electron plasma density, n , increases with the increase otf luminosmy current. The e 17 3 mafia /m 50» 40* 30’ 20' 10* ) 80 Figure 4.6 i0 4‘0 60 I, .Lurninosity vs. plasrna density. e— —6 100(lflamp) 81 nonlinear characteristics of the photo-diode attribute to the nonlinear relation between ne and If A Tektronix 1L4O spectrum analyzer was used to display the reflected power through a directional coupler. Also, by sweeping the dc discharge current, the temperature and cold plasma resonances at high and low powers were observed with a Tektronix oscilloscope through a crystal detector. CHAPTER 5 EXPERIMENTAL RESULTS The nonlinear resonance effects to be reported here arise from two different types of discharge: a dc discharge and a resonance-sustained high frequency discharge. The nonlinear phenomena for the temperature and cold plasma resonances were observed in the microwave scattering from a dc plasma column which was irradiated by a high power microwave signal. Such a study leads to the physical understanding of the more complex non- linear phenomena of a resonance-sustained high frequency plasma which is created and maintained entirely by microwave power. In the following discussion, the nonlinear resonance effects of the high frequency discharge, including parametric amplification, frequency shift, frequency modulation, rectification and hysteresis phenomenon, are described separately. The physical interpretation of the eXperi- mental results are also considered. 82 83 5. 1 Nonlinear effects of the temperature resonances at high power: the. transition from a dc sustained to a rf resonantly sustained plasma When a low power microwave signal (< 50 mw) is incident on a bounded plasma, a series of linear temperature (Tonks -Dattner) resonances can be observed as the plasma density is varied. If the incident microwave power is increased, the linear resonances be- come"distorted, " and finally when the rf power is sufficiently high, the plasma becomes completely sustained by the rf power; i.e., it becomes what is commonly called a resonantly sustained plasma. The experimental observation of the transition from low rf power, linear resonances to the completely resonantly sustained plasma is presented here. Using these observations, a qualitative theory is developed which eXplains the behavior of the plasma microwave system. These nonlinear resonance effects were studied experimentally with the ridge waveguide system which is discussed in Chapter 4 and shown in Figure 4. 4. Instead of the usual matched load termination, this waveguide system is terminated in a sliding short. The position of the sliding short is adjusted so that the waveguide impedance at the plasma is zero (i. 6., open circuit). Thus, maximum electric field strength impinges upon the dc discharge column located be- tween the ridges. The incident microwave power in the waveguide system is partly absorbed by the dc plasma, while the rest is reflected. The power absorbed by the plasma is then equal to 84 the difference of. the. incident and reflected power; that is pabs : Pin - pref In this microwave system pin : constant for each experimental observation. Thus the resonances are observed as ”dip" in re- flected power as the dc discharge current is variedl7. Typical experimental results are shown in the oscilloscope pictures of Figures 5. l (a) to (e) in which the plasma luminosity (top curve) and the relative reflected power (bottom curve) correspond to the ordinate, and the dc discharge current corresponds to the abscissa. The horizontal center line is the zero reference with the luminosity increasing upward and the reflected power increasing downward. As mentioned in Chapter 4, the luminosity is proportional to the relative plasma density. At low microwave power, the usual linear temperature resonances are observed. These are displayed in‘Figure 5.1(a). 'I‘he dip in the trace indicates that the microwave power is absorbed at resonance. Note that at this low incident power, the resonance is critically coupled to the waveguide system. As the incident power is increased, Figures 5.1(b) to (e) show that the temperature resonances, which appear in the reflected power, are distorted. The luminosity levels off and decreases slightly when the plasma is in resonance. This indicates that when the microwave power is absorbed by the plasma, this energy ionizes the plasma and tends scale cal. (. 05v/cm) (. 05v/cm) (. lv/cm) (. lv/cm) (. lv/ cm) (C) (. Zvv/cm) (d) (. lv/cm) (. Zv/ cm) (e) (. lv/cm) pin a- 5 watts b (. 5v/ cm) Figure 5.1 Nonlinear effects of the Tonks-Dattner resonances: (i) luminosity vs. dc discharge current, and (ii) reflected power vs. dc discharge current. 86 to keep the plasma. density approximately constant. In Figure 5. 1 (e), the. luminosity levels for a large variation in dc discharge current. In fact, IdC 0 when the plasma drops out of the resonant state. In this case, the incident power level is almost sufficient to sustain the plasnzil completely in a resonant state. Another important fact is that the coupling of microwave power into the plasma depends on the incident microwave power. At high microwave power, the main resonance is no longer critically coupled to the waveguide system. in o ruer to understand the nonlinear behavior of the plasma, the physical behavior of the ridge waveguide system can be approxi- mated by the plasma capacitor system of Figure 2. 1. The equivalence of these two systems is illustrated in the diagrams below. The v...'aveguide system can be replaced by its equivalent ll‘étnSIY‘llSSlon system; thus, the plasma capacitor is the load in such a waveguide transmission system as shown below. 87 —.J_. P. load m m —r- P 'ef I~— m—~+ The load impedance of the plasma capacitor varies with the plasma density. Such a variation can be plotted on the Smith Chart. For illustration, the behavior of the main or cold plasma resonance (the largest resonance in Figure 5.1) is discussed. However, this concept applies to the temperature resonances as well. The capacitor admittance can be obtained from the boundary conditions given in Chapter 3. That is, V(t) = (sgr - s + L)I~:p For the cold collisional plasma model, the relative equivalent permittivity, Er’ is of the form 2 2 (0_ y_ 0.)- 6 Z (1_————L—)-J—_ ——R—— r 2 Z a) Z 2 Clo-y (1)-y Solving the above equation, the capacitor admittance can be written as 88 a 2 I.- 2 o m - v - (op (l + Ju_/m) Y. i (l;(: 2 Z 2 L/s(m -y_ )- mp_(l + jV_/(n) where C is the total capacitance of the vacuum sheath as indicated in Chapter 3, The amplitude and phase of Y versus (mp_/w) are plotted in Figure 5. Z for s/L : .l and y-/m = .1. Figure 5. 3 shows the plot of the impedance Z, the reciprocal of Y, for different plasma densi- ties on a Smith Chart. Note that the impedance curve is asymmetri- cal about the real axis because of collisional damping in plasma. For the convenience of illustration, a typical plot of the impedance 7. for a constant sheath thickness, and various a) is sketched below for u << (n (i.e, for small collisional loss). unstable region Nun—2) AMPLITUDE OF ADMIYTANCE Y O ‘0 89 I, \ Phase Of Y I up! w is is 5 Figure 5. 2 Admittance of a cold plasma capacitor model (s/L: .1, ye/(o: .1) 9O 1 a; c. O ’ g I. C v I . 0.. it li\\ w“ r w 0 on r; . 7.6 t ..m , K . .3. u .u n v ‘ , Q m r .. _ .s (5 J l) / MC \ 4 , .. ex. i n. \(\( ”H/ 13 ax i , L . . amba- ., ...MM t... /\ I . .i . a 12 . t a a. a 3. u it. «A» oo . ... $ r av... s t. i s 0“. .. 10¢ =.1) = .1, ye/(D dance (s/ L igure 5. 3 Smith Chart representation of the cold plasma capacitor lmpe ()1 The stability of an rf discharge can be developed from the Smith Chart impedance. A dc plasma is always stable; however, there exist stable and unstable regions when the plasma is partly or totally sustained by rf power. The stable region of a high frequency plasma corresponds to the portion of the resonance curve where increasing the plasma density decreases the microwave power absorbed by the plasma; whereas the unstable region corresponds to where increasing the density increases the absorbed power. These stable and unstable regions can be identified on the Smith Chart. Since the microwave power absorbed by the plasma is related to the incident power and the reflection coefficient '1", through 2 pabs Z Pin” ‘ II"! ) the stable region corresponds to the capacitive portion of the chart where d (1“)/dmp_ > O; that is, increasing the plasma density increases H"|, and hence decreases pabs' The unstable region corresponds to the inductive portion of the chart where d (Fl/dog _ < 0. Thus, in- creasing the plasma density decreases, IF', and increases pabs' As shown in the above sketch, the point, r0, corresponds to the resonance peak where the plasma is pure resistance. Any per- turbation in density tends to shift the plasma off resonance. In fact, decreasing the plasma density from ro to rl results in a de- crease in the absorbed power (or an increase in 'F'). Thus, the plasma becomes unstable, and tends to jump to a lower resonance 92 state. However, increasing the plasma density from ro to r2 decreases the power absorbed by the plasma. Such a decrease in absorbed power tends to pull the plasma back into a resonant state. It is to be emphasized that a large collisional damping has a stabilizing effect on the plasma. As shown in Figure 5. 3, the plasma is stable in the inductive region near the real axis, because for a small re- gion of inductive impedances decreasing the plasma density results in an increase in the absorbed power. The absorbed power, , versus the average plasma den- pabs sity < N>, is plotted below in the vicinity of the main resonance for three different values of incident microwave power with pinl < < o - h f o o I pinz pi].13 T e phase 0 the impedance 18 also prov1ded to correlate the stable and unstable regions on the resonance curves. ” P) '1... l PHASE COPOCi'iV9( s'Ob'e). 0° Phase J inductive (unstable) |l I! ii iii ()3 This figure can be thought of as a superposition of oscillo- scope traces. As shown by the luminosity curve of Figure 5.1, the density varies only slightly when in a resonance. Thus, there is a slight shift in the resonant frequency as I is decreased. Also for dc low incident power, p, is approximately equal to the maximum inl absorbed power, i. e., critical coupling occurs. However, pabs; at higher incident power, pabs is less than pinZ and pin3 which results from a decrease in the efficiency of coupling microwave power into the plasma system. Physically, the coupling prOpe rty of the ridge waveguide sys- tem can be directly related to the reflection coefficient, F, of its equivalent transmission system. The condition for critical coupling (i.e., pabs: pin) is the vanishing of '1"). Thus, the coupling problem can be approached from the impedance point of view. The effects of the sheath thickness, 5, and the incident power, pin’ on the coupling of microwave power into the plasma are demonstrated in the impedance curves of constant 5 and constant pin for the main resonance. A family of impedance curves is sketched below for different, constant sheath thickness (51 > 32 > 33). Also, for constant incident micro- wave. power, the impedance of the plasma system is experimentally measured as the dc current, Idc’ in the plasma is varied. A family of impedance curve for different, constant incident power (pin < l < . . . pinZ pin3) is shown in the diagram below. ()4 sl inc-905171? critical.-~ -..... w P. l l / \ dec sing \ in] As shown in the sketch above, a criteria for efficient coupling is to reduce. the reflection coefficient II‘I which corresponds to the distance between the point of unity and the impedance curve. Note that for the case of critical coupling at low incident power pinl' the impedance curve passes through the real axis at the point of unity which corresponds to a zero reflection coefficient. For higher incident powers pinZ and pin3’ the decrease in coupling is demon- strated on the Smith Chart as an increase in II"I . Inefficient coupling can also be resulted from a small sheath. It is indicated in the above sketch that increasing 3 simultaneously decreases II" I. As discussed in Chapter 3, increasing the sheath thickness, 5, dis- places the resonance position to a lower density. Thus, at resonance, the conductivity of the plasma becomes smaller, or the resistivity becomes larger. As a result, the reflection coefficient, IFI, 95 decreases with an increase in s. This decrease in II‘I increases the efficiency of coupling microwave power into the plasma. If one follows a curve of constant pin' it is noted that as Idc is decreased, more rf power is absorbed, and thus the sheath is constantly changed. It changes from a small sheath to a large sheath as Idc is decreased. This change in sheath explains the distortion of I. this! “I.” .m; the experimental resonance curves at high microwave power (see Figure 5. 1). This alteration in s (or alteration of the density pro- file in a more accurate model) has also been proposed by others to I“ explain the change in resonances at moderate incident microwave powerz. In summary, the coupling of microwave power into the plasma can be improved by either adjusting the plasma parameters or the characteristic impedance of the transmission system. 5. 2 Parametric amplification in a resonantly sustained high fre- quency plasma In the following, the eXperimental results of a radiation induced parametric instability in a completely microwave sustained plasma are discussed. It is believed that this phenomenon can be interpreted as a nonlinear coupling of the electron plasma modes, ion-acoustic modes and the pump field. The result is a four fre- quency (f0, fi’ fO _-%_ fi) parametric interaction. By reducing the pump power just below the power threshold required for the instability, a separately applied cw input signal can be amplified if the signal is approximately equal to the sum frequency (fo + fi) or the difference 96 frequency (fo- fi). The eXperimental results indicate that the para- metric instability depends not only on the incident power threshold, but also depends on the geometry of the system, the density gradient (or sheath in the plasma capacitor model) and dc bias across the dis- charge. The coaxial system discussed in Chapter 4 and shown in Figure 4. 2 was used in this investigation. The discharge was created 3 and resonantly sustained inside the center conductor gap by a fre- quency stabilized high power microwave source which is the 'pump' for the parametric instability. In all the experiments, the gap spacing is much smaller than the center conductor diameter; thus, the plasma geometry can be approximated by a plasma filled capaci- tor. The theoretical results of Chapter 3 can be applied here also. As indicated in Chapter 3, high frequency (electron plasma) reson- ances at, fl, (center frequency of temperature or cold plasma resonances), and low frequency (ion-acoustic) resonances at, fi’ exist in such a model. As a result of plasma nonlinearities, har- monic resonances also exist. This resonantly sustained high fre- quency plasma capacitor has demonstrated a resonance nature at both the fundamental (3. 03 GHz) and harmonic frequencies. These resonances are the nonlinear high power temperature and cold plasma resonances of the capacitor. They are similar to the high poer resonances in the plasma column which was just studied in the previous section. ll ‘ .I. III ...II’ 97 Experimentally, a parametric instability was observed when the pump frequency, f0, approximately equal to f was 3. 03 GHz 1' and fi varies from several hundred KHz to 4 W2. The relative positions of the high and low frequency resonances with respect to the pump frequency are sketched below. fi f1 ‘0 I '0". ‘o+ '. The eXperimental plasma (fl >> fi) is completely resonantly sus- tained and hence in accordinance with the discussion of the previous section, is operating in a stable region on the resonance curve. Such a position is indicated in the figure. From the theory of Chapter 3, one can observe that when the plasma is in resonance, the 1? field strengths are very large (many times larger than that of an empty capacitor). These large B fields cause nonlinear phenomena to occur. In particular, the high frequency resonance couples nonlinearly to the low frequency resonance fi' Since the system is also resonant at fo i fi’ a four frequency parametric interaction results. As noted in Chapter 3, the plasma geometry has a number of resonances, and the instability can occur in many similar resonant was! 98 states. However, the power required for the instability is quite different for each resonance. For example, the incident power threshold can vary from approximately 12 watts up to 25 watts (5 watts - 15 watts absorbed power) depending on the resonance and collisional damping of the plasma. Here, the discussion is restricted to the behavior of the instability while the capacitor remains in a single resonance state. When the incident power is above a certain threshold, and the pressure is reduced to a value where collisional damping can be neglected (approximately 400 microns in dry air) coherent sidebands at £0 _+_ fi always appear in the reflected power. A typical reflected spectrum and an oscilloscope trace of a reflected spectrum are shown in Figures 5. 6 and 5. 4 respectively. Just above the instability power threshold, only f0 _;-_f1 components are present. However, as the input power is slightly increased, other discrete frequency com- ponents, f0 i Zfi’ f0 i 3fi’ etc., gradually appear. If the incident power is increased continuously, the side frequency components shift in position and then disappear. The shifts in frequency will be discussed in the next section. At the very onset of the sidebands, low frequency oscillations were detected with a low frequency probe. As shown in Figure 4. 3, a coaxial probe, which was connected to a low frequency spectrum analyzer, was inserted into the plasma through the hollow inner conductor. This probe detected the existence of strong low 99 Figure 5. 4 Reflected power spectrum at fundamental frequency (3. 03 GHz) (dispersion = 1 MHz/cm, pressure : 400 micron) Figure 5. 5 Low frequency oscillations detected by low frequency probe (f, = Z. 4MHz, pressure: 400 micron) 1 amplitude power power amplitude [1,,1 100 —-I-— —-——pump signal 3 ZOdb _L__ l.i I 2f, 3r, Tr,- at, fo-Zf, g- f, Figure 5.6 Diagram of the reflected power spectrum amplification regions \ I..\ \ ‘t \ "\ \ ’ ‘ a \ o’ ‘aw/ \ A. " '.:~- " A L \- .....‘ f“ f0. f: fo'. 2% g+3fl ‘ freq.f I _ ,pump signal ‘ ......... ._-_ f; 3r, g-zr, w, Figure 5. 7 Diagram of parametric amplification region. f.+2f, f.+3f, freq. f 101 frequency oscillations at f1 and Zfi whenever the high frequency side- bands appear in the reflected power spectrum. Thus, the existence of a low frequency oscillations in the plasma was experimentally proved. Figure 5. 5 shows a picture of such a low frequency oscil- lation. In the experiment, a dc biased voltage was externally applied across the conductor gap through the quarter-wave stub as shown in Figure 4. 3. The variation of the dc bias, and hence the plasma sheath, was seen to have an influence on the instability power thres- hold. In fact, by holding the incident power and the pressure constant, the parametric instability could be excited by slowly increasing the dc bias. However, too large a dc bias caused the side frequency components to shift in position, and then totally disappear. Physi- cally, the disappearance of the sidebands is caused by shifting the sheath voltage. Such a shift causes the plasma to be in or out of a resonance (i. e., ion-acoustic resonance for the plasma capacitor model). If the incident pump power was reduced to a level just below that required for the instability (i. e., to a value where there are no sidebands present), the plasma capacitor was able to amplify a separately applied input signal. With the pump power adjusted to a value slightly below the threshold, a separate, variable frequency signal of approximately . 06 mw was incident on the plasma capacitor. This signal was introduced into the coaxial system through the 102 directional coupler as shown in Figure 4. 2. The reflected power was sensed through another directional coupler with a spectrum analyzer. As the signal frequency, fs’ was varied manually, the magnitude of the reflected power was observed to vary significantly from frequency to frequency. For frequencies where f8 > fo + 3fi and f8 < fo- 3fi’ the reflected power at f5 was approximately equal to the input power indicating complete reflection from the plasma capacitor. However, when the signal was in the vicinity of foi fi and £01 Zfi’ a large increase in reflected power was observed indi- cating that amplification was occurring. The region of parametric amplification is shown in Figure 5. 7. In particular, the frequency fS was amplified by 13-24 db (depending on the pressure and resonance) when it was in the vicinity of f0- £1. The gain at fo+ fi is only a few db less. When the signal of fs 3 f0- fi (or f0+ fi) was being amplified, a signal at fszfo+ fi (or fo- fi) also appeared indicating the presence of a four frequency cw parametric interaction. A similar four frequency effect was observed when the input signal was in the vicinity of fo 3; 2fi' but the gain of the interaction was less than 8 db. A picture of the amplified signal is shown in Figure 5. 8. This picture was taken when the signal frequency, fs’ was slowly swept through the ampli- fication region in the vicinity of f() i fi' Note that the signal is too small to be observed outside the amplification region (the short, bright lines near the base line are caused by the Spectrum analyzer). 103 frequency Figure 5. 8 Parametric amplification of a separately applied microwave signal (Pin 2 35 watts, pressure 3400 microns, dispersion: Z MHz/cm, L : l. 25 mm). (a) (C) time Figure 5. 9 Low frequency oscillations detected by sensing recti- fied voltage across the inner conductor gap (pin 2 35 watts, pressure 3400 micron, sweep time = . 5 psec/cm, L r: l. 25 mm). 104 Another way of detecting the low frequency oscillations was to sense the voltage across the quarter-wave stub with an oscilloscope (see Figure 4. 2). The oscillations were displayed on the oscilloscope with a time sweep. A typical oscilloscope trace of the low frequency oscillation is shown in Figure 5. 9(a). In this figure, the system is in self-oscillation; i. e. , sidebands appear in the reflected power spectrum. The frequency fi is shown to be approximately equal to 2 MHz. As a result of low frequency harmonics at Zfi' etc. , the wave form is distorted from being a pure sine wave. If one reduces the microwave pump input to just below the instability threshold, Figure 5. 9(c) shows that the low frequency oscillations disappear (the small ripple is caused by noise). Now, if a small microwave signal of fS is applied to the system, low frequency amplification, as mentioned above, occurs and the entire system breaks into oscillation. These amplified low frequency oscillations are shown in Figure 5. 9(b). In summary, the experimental results indicate that the in- stability observed is a four frequency parametric amplification since amplification exists above and below the pump frequency. During this investigation, a similar parametric amplification was reported 1. by others The power threshold for parametric interaction is found to be of the same order of magnitude as that of Stern and 37, 41 4O . 'l'zoar . Thus, the theory of DuBois and Goldman does not account for the eXperimental results here. Also it is believed that 105 the low frequency oscillations observed in the parametric ampli- fication and in the related phenomena to be discussed in the next section correSpond to the ion-acoustic oscillations. The high fre- quency oscillations correSpond to the electron temperature and cold plasma resonances. Further development of the plasma capaci- tor model including nonlinear effects might help to explain the experi- mental results presented here. . .‘J‘ A?! "'I“., 5. 3 Frequency shifts of the low frequency fi It was noted in the last section that when the resonantly sus- tained plasma was in a parametric resonance, the low frequency, fi' varied as the incident power varied; as the plasma size changed; or as a small, separately applied dc voltage was increased or decreased. In an attempt to better understand the parametric instability, this change of f1 was studied. It was found that if the low frequency oscillation is assumed to correSpond to an ion-acoustic resonance, then the plasma sheath has an important influence on fi' Experimentally, the plasma size is changed by varying the conductor gap spacing, L. As the gap spacing, L, is varied, the light-intensity measurements indicate that the average plasma density remains constant. Thus, the system remains in the same resonance4. The frequency fi is measured by observing the shift of the side frequency component in the reflected power spectrum. Figure 5. 10 shows the variation of f1 versus L for a constant incident Linn»... 106 Figure 5. ll for different capacitor Spacing, L. A (cop- = 6n x109, s: .6 mm, Ue : 7x10 , L1: 1.4mm, L2: 1.6 mm, 50' fLCMHz) 4-2 3-2 2-0 l-3 m as .75 o-7 L(mmll-59 l-65 r72 l-78 P84 l°89 1-97 2-04 4.0. 3.0. pressure = 320 microns Q" (incident power)“ 30 watts 20L l°0’ x 4. O ;_A A A A A A A (g1 l-5 l-6 l°7 l8 l9 20 2| capacitor spacing, L .(mml Figure 5.10 Shift of low frequency, f., vs. capacitor spacing, L. (pin 2 30 watts, pressurie = 32. 0 microns) '20 l 1 I ‘1 IooI . l 80’ \ bor \ N 40) I\_\ ‘ L . \~ 20» . J - oo 2 4 6 a lO 12 . «(103) Graphical solution of low frequency capacitor impedance - 107 power. Clearly, the frequency fi is not related to the gap spacing, L, in a simple manner. In previous experiments with a mercury-vapor discharge, the frequency fi was identified as a standing ion-acoustic wave with 7. . . . . 4O . a wavelength given by the insme diameter of the tube . It is be- lieved that standing ion-acoustic waves also exist in the present . . 47 . experiment (a "perturbed plasma" as descr1bed elsewhere 18 assumed here). As discussed earlier in Chapter 3, the wavelength Xi and frequency fi of the ion-acoustic wave are related by 'y KTe 1/2 £91 3 (TI, ) (5.1) 1 when ZTl'fi << (ep+; Te >> Ti' These approximations hold for the high frequency discharge discussed here. In general the electron temperature, Te’ can be assumed to be approximately constant in this investigation. Thus, the frequency of the standing ion-acoustic wave is necessarily non-dispersive; that is, fi Xi : constant. If one assumes that the standing acoustic wave occupies the complete length L between the plates, then, as the distance between the plate is varied, the frequency fi should change in accordance with the equation above; i. e. , Lfi = constant. However, as shown in Figure 5.10, the normalized (with reSpect to the minimum L fi product) Lfi product is'far from constant. The curve indicates that as L decreased, the standing acoustic wave occupied a decreasing per- centage of the total L. That is, the region of allowed ion-acoustic 108 propagation decreased faster than L. Thus, it appears that the sheath appears to play an important role in reflecting the ion- acoustic wave. A more accurate interpretation of the behavior of fi vs. L requires a theoretical analysis of the plasma capacitor model dis- cussed in Chapter 3. Since the average plasma density and the electron temperature remain essentially constant when L is varied, the Debye length, and hence the sheath thickness 5, can be approxi- mated by a constant. The low-frequency, capacitor impedance given by equation (3. 65) is solved graphically for the following plasma parameters: 5 : .6 mm, mp_ : 6n x109, Te : 3 x lOSKO. The graphical solution corresponding to the lowest ionsacoustic mode frequency is shown in Figure 5.11 for different values of L (1.4 mm 5 L _<_ 2. 2 mm). Note that s is fixed in these calculations. As L is increased, the sheath region makes up a greater amount of the capacitor spacing for small L than for large L. As shown in Figure 5. 12, the theoretical curve is generally consistent with the experimental curve of Figure 5.10. Better agreement could be achieved with a choice of T8 or s which are closer to the experi- mental condition. When the incident power is increased, the effect of the sheath again exhibits itself as a decrease in fi. A typical change in fi vs. incident power for a constant L is shown in Figure 5. 13. As the incident power is increased, the plasma attempted to remain in the 109 llmm) 1.4 1,9 1.3 2.0 22 (i4; I {.5 i 2.5 MW") Figure 5.12 Shift of low frequency, fi' vs capacitor Spacing, L. (theorectical) (a) 9 (wall) IS 20 2| 22 23 24 2.5 3.5. In fi(MHz) 2-8 2-8 2-8 s 2 30 :- pressure = 400 microns 2.st L = 1°59 mm 0L:r I A A 1 A n O lO-O ll-O l2~O I30 I40 l5-O : 6nx109, 5: .6 mm, Us 2 7x105) (L. power absorbed (watts) Figure 5. l3 Shift of low frequency, f,, vs. the absorbed power 1. 59 mm, Pressure = 400 micron ) 110 resonance state by modifying the zero-order density profilez. This modification decreased the sheath and allowed the standing ion-acoustic wave to occupy a larger space between the capacitor plates. The eXperimental results could be compared with that of the plasma capacitor model if the variation of s with pin was available. The frequency shift of the standing ion-acoustic wave was ___ also observed when a dc biased voltage was applied across the capacitor plates. As the dc bias is increased, the low frequency ‘ fi increases. The increased dc voltage increases the size of the r sheath resulting a smaller plasma slab. Thus the frequency of resonance, fi, increases in accordance with the plasma capacitor model of Chapter 3. Figure 5.14 shows the variation of the fre- quency fi as a function of the dc voltage V Note that the curve dc' of ii vs. Vdc is asymmetrical. This lack of symmetry is caused by the asymmetrical density profile of the plasma which induces a dc voltage to appear across the center conductor gap. The profile is asymmetrical, because one probe is held at ground and the other floats. In conclusion the frequency fi usually can not be directly related to the external dimension L. However, if one assumes that Xi : 2L when the plasma is operating in the horizontal region of the curve in Figure 5. 13, then the value of Te obtained from the equa- tion (5.1) is close to experimentally measured values. For example, Te : 3. 7 x 104 K0 in Figure 5.10. This agreement between the fShlfi (MeHZ) l-3‘ " - - , - A - 1' - A _ - - - - -|4 -|2 -IO -8 -6 -4 -Z O 2 4 6 8 IO l2 l4 Vdc (volts) Figure 5. l4 Shift of low frequency, f,, vs. externally applied dc biased voltage. (p.n =-.1 32 watt, Pressure = 350 4 micron, L = 1.5 mm 112 "sheathed" plasma capacitor model and experimental results supports the conclusion that the low frequency resonances are caused by stand- ing ion -acoustic wave 8 . 5. 4 Frequency modulation of the third harmonic output Here a study of the coupling between a high frequency resonance and a low frequency resonance is discussed. The experimental plasma is again the resonantly sustained coaxial plasma capacitor discussed in the previous two sections and is shown in Figure 4. 2. This capacitor can be resonant simultaneously at the incident frequency fl, third harmonic frequency 3f and the low frequency fi' In particular, 1 emphasis is placed on the nonlinear coupling between the low fre- quency oscillation discussed in Section 3. 3 and the third harmonic resonance. Investigating the coupling between fi and 3fi has an experimental advantage over studying the coupling between it and f1 0 3 . . . ’ ) since there 18 no external dr1v1ng (as has been done by others4 electromagnetic radiation present to mask the observation of 3fi. The resonantly sustained high-frequency plasma is formed in the center conductor gap (L i 1.6 mm) of the coaxial system. The gas is dry air. The third harmonic (9.: 9. 09 GHz) is created within the plasma, and is coupled through an x-band waveguide system into a high frequency spectrum analyzer as shown in Figure 4. 2. It is found experimentally that when the plasma capacitor is simultaneously resonant at fi’ f1 and 3f1, the driving fundamental 113 power at f is coupled to f1 and 3f1. At high pressure, only a single 1 output third harmonic frequency is observed. However, as the pres- sure in the system is gradually reduced, the third harmonic output drops suddenly to a low value (5 2 mw) and is modulated as shown in Figure 5.15. In this low pressure regime, the electron-neutral collisional frequency is much smaller than the applied driving fre- quency. At the onset of the modulated harmonic output, low frequency oscillations at f1 and Zfi can be detected with the low frequency probe. The sidebands that appears in Figure 5.15 are related to the ion- acoustic oscillations; that is, they occur at 3fl ifi, 3f1 1: 2fi' 3fl i 3fi' etc. The sidebands are essentially equal in amplitude when they first appear. As the pressure in the system is further reduced, the sidebands may become very unequal in amplitude, while the carrier at 3fl may be strongly suppressed. In addition, the amplitude of the upper or lower sidebands may interchange. Such changes can be also achieved by variations of the incident power level or gap spacing. The frequency modulation of the third harmonic output can be qualitatively interpreted on the basis of low-frequency perturbation of the non-uniofrm zeroth-order density profile. When the plasma capacitor is resonant at the fundamental frequency fl and the low frequency fi, the fundamental energy is coupled by a poorly under- 4 . stood process to the low frequency oscillatlon. These low frequency oscillations are superimposed on the zeroth-order density Anouufle 0mm u oudmmopnm .88 o; u 1H 1 a a n I H Q 0 1H - c I.” Siamese - goers e2. emz me _ u M £0 so a - on 50 mo m - a «5350 0305.5: HEB» woudgvog cascade: 05 mo Ssh—comm hoe/om wouooaom ma .m oudmfim 4.1!! -1! Isl)! . profile resulting in a time varying density profile. That is, the low frequency density profile becomes n : n (x) + n cosmt + . .. (x is o 01 1 the position between the capacitor plates). Also the capacitor is resonant at 3fl causing an easily detectable amount of power at 3fl 33 . to be present . However, the temperature resonance of the third harmonic varies at m1, and the third-harmonic Spectrum is modulated as shown in Figure 5.12. The above physical description can be formulated theoretically by writing the equations which describe the behavior of a third har- ‘fi-dmi Iii—mil." W; monic signal while in a temperature resonance. By assuming the above mentioned time varying profile and neglecting the high fre- quency motion of the ions, the electron density has the following form. ne(x,t) : n (x)+n ((o t)+n t 5.2 e0 e1 1 01(mi ) “L “02(2031” ( ) where the unperturbed and perturbed quantities are designated by the subscripts (0 and l) reSpectively. It is assumed that the Spatial average plasma density is much greater than the high fre- e quency perturbation, n and the low frequency perturbations, el' n andn ;i.e., >>n ;>>n 01 02 e0 e1 e0 01 “02' BY ”pen" mental observation of the low frequency oscillation, n02 was noted to be less than 1101/2. Note that instead of assuming a uniform plasma slab and vacuum sheath as in the pr evious discussion, this model includes a spatially varying density profile, i. e. , neo(x), (u (x). P" 116 F rom the first two hydrodynamic equations for the electron fluid and the Poisson's equation, dnc _ . + V, . 2 dt \/ neve 0 (5'3) 82? 1 e. -—> -e e :2 . L . + —— T : 4 0t We VWe m L n m VP Veve (5 ) e e e \7 . E .- :9. 440 B ' th 1. d ' d d t f dV - 3 th y assuming a n01 an nOZ are 1n epen en 0 x an _ ax , e following equation for the third harmonic electric field in the one- dimensional plasma can be derived from equations (5. Z) to (5. 5) . 82 + E) + e2 + )+ 2 E (82 + a > 2 ye at rm 6 (n01 nOZ (Do 1— 2 1’8 at e o 3t 8t E + (nonlinear terms) (5.6) ext where '19 : phenomenological collisional frequency 2 . . . 54 m0 = th1rd harmonlc eigenfrequency for the TDR 2 z 82 2 3 =00 (X)-U ----+U —(fnn )— p- e 8x2 e 3x eo 8x The external electric field, E xt’ is related to the total current J e and the perturbed electric field, E through equations (5. 3) and 19 (5.5) ext Since no external microwave source at third harmonic frequency is applied, F can be set to zero. If one assumes that cos 309 t ext 1 is the normalized phenomenological forcing term at 3fl due to the nonlinear terms in equation (5.6), and (ext) r N cosmt 1 1 D01 01 Zrot) : N cos ?n).t, 1 1 noz( 02 equation (5.6) can be simplified to Z 2 Z N e. N e a a 01 Z __ + U -— + -——————- (COSLU.t + 9) ~'- COS(ZU).t + C1)+(D 2 e. 81; 1T1 p 1 « 1 o 3t e ‘o e o E : cos 303$ (5.7) 1 1 where 9 and C1 are the phase angles of the ion oscillations with reSpect to the forcing term. Equation (5. 7) can be simplified further with the following transformation .-Vet/2 El : y‘e (5.8) For 0) >> V /Z, the transformed equation has the form 0 e 82 2 at : +G (t)y : e cos 3mlt (5.9) 8t where. <(o > N N g - 01 p- 02 (J (t) ._ m [1) 2 COS(0)it +9)+ (n so (0 e0 0 o 118 Z 2 e 80 3 n) e o a - 2 ve/ Equation (5. 9) is a linear second order differential equation with time varying coefficients. Experimentally, for cold plasma and Z ”- temperature resonances, l/Z < <00 >Z/w < 4, but as assumed F — p- o - ; earlier N / << 1. Thus, 01 e0 <(~ Z>N <03 2>N " J) L. I; 01<<1,- "Bf—"OE «1. r (1) to 0 so 0 eo Under these assumptions, the low frequency perturbation causes only a small variation of the zeroth-order density profile. As a . . 55 . . result, the WKBJ approx1mation can be applied to find the comple- mentary and particular solution of equation (5. 9). (A) Complementary solution From WKBJ approximation, the complementary solution to the homogeneous differential equation of (5. 9) is of the form -1 Z . yC '2 C(t) / [Ascos ¢(t) + A6811] ¢(t)] (5.10) where A5 and A6 are arbitrary constant. From the binomial series eXpansion Z 2 < > < > C t) , [1 -+ -1- mp_ N01 c s( t+(3)+l (0 - N02 ’( 3' ‘00 Z 2 O ('01 2 2 (1)0 80 £00 90 cos (Zmit + 0)) 119 < > N <00 >2 N “MW ' “”2“ i (09' 01 c s( t+9) l P" 02 " 2““ ’4 z O ‘01 4 2 (1) e0 (1) e0 0 o cos(2(nit +d)] (5,11) en) - ) CHUdt < >2 \J \1 f "' 3t+1 ~0p_ [;QLCN t+9)+“‘l—QE—CS(Zwt+) F _ (10 2m. 2 O (01 O i a 1 (no eo eo . (5.12) ! Substituting equations (5. 8), (5.11) and (5.12) into equation (5.10) yields -at -1/2 Ec — e 000 [1 - mlcosmit +9)-m2cos(2mit+a)] (5.13) , 1 . . . . {Ascos [mot .k181n((nit + 9) +k281n(2wit + (1)] + A651n[wot +k151n (wit +0) + k Sin(?(l.\it +0)) 1 2 where < 3 > \J < \ >2 N n) g l (l. _ 1 01 n — l 0P” 01 ._ 4 Z ’ 12 ~ 4 Z (0 e0 (1) e0 0 o < x >2 < > N k 1 Qp- o1 , k _ 1 a)- 02 1' &n 2 ' 2 am 2 (no eo 1 mo eo Both amplitude and frequency modulation are exhibited in the com- plementary solution. Since m1 and m2 are much less than unity, the amplitude modulation produces only a negligible effect. The process of frequency modulation induces a series of side frequencies 120 which can be found by eXpanding the sine and cosine terms of equation (5. 13) in a series of Bessel functions of the first kind through the following identities: cos(k sinG) : Jo(k) + 2J2(k)cos 29 + 2J4(k)cos 49 + . . . (5.14) sin(k sine) -._- 2J1(k)sin0 + ZJ (k)sin39 + (5.15) 3 The amplitude of the side frequency components depend on the product of the Bessel functions of the first kind with arguments k, and k2. In the steady state (t _. 00), the complementary solution vanishes as a result of collisional damping in the plasma. (I3) Particular solution From WKBJ approximation and the method of variation of parameters the particular solution of the nonhomogenous differential equation (5. 9) is of the form yp : A(t)xl + B(t)x2 where x1 : C(t)'1/2cos¢(t) x2 : C(t)-l/Zsin¢(t) t x e cos 3mlt A(t) : - 2 . . (5.16) x x - x x 1 Z 1 Z - t . x1e cos 3mlt B“) Z \ x x -5: x 121 1/2 4) (t) and C(t)- are given by equations (5.11) and (5.12). Since the amplitude modulation factors ml and mZ are much less than unity, the amplitude modulation effects are neglected. As a result, - 2 x z: u) 1/ cos ¢(t) (5.17) l 0 -1/2 . x2 2 (130 Sin ¢(t) (5.18) 2m 4m2 xlx2 - xlngl + cos(wit+6) + cos(2wit+ o.) O m << 1; thus x 5: ~11 x :1. Then, A(t) Again noting that ml. 2 1 2 l 2 and B(t) can be written as A(t) g -(1)0-l/28.e-at cos (t) cos 3(nlt dt (5.19) B(t) a: (nod/2 Se-at sin ¢(t)cos 3(olt dt (5. 20) To simplify the calculations, the frequency modulation factor k1 is restricted to be less than 1. 8 (i. e., k 51.8). Thus, terms of 1 order J4(kl), J3(k2), J3(k1) J2(k2) or higher Will be neglected Since they are small for k _<_ l. 8. Furthermore, only the particular cases 1 in which the ion oscillations are in phase (i.e., 9 : a. = 0 or luv), and 1800 out of phase (i.e., 9 : 0. = IT) with the driving term will be dis- cussed. Experimentally, the sidebands appear only in the relatively low pressure regime where the collisional frequency, ye, are small. Here the limiting cases of ye << mi and "e >> mi are discussed. It 122 appears that the collisional frequency plays an important role in the process of frequency modulation. In finding the particular solution, the quantities A(t), B(t), x1 and x2 are calculated from equations (5.17) to (5.19). Through the use of identities given by equations (5. 12) to (5. 15), these quantities can be expressed in terms of the product of two infinite series of Bessel functions of the first kind and sine or cosine functions. Consequently, the general form of the particular solution involves the product of four infinite series of Bessel functions and the sine or cosine functions. The complexity involved in the calcula- tion of the side frequency components for the particular solution is obvious. In the following, the lengthy calculation is omitted. The side frequency components for the particular solutions in their final forms are given. To ease the writing, it is convenient to denote .IN(kl) by JN and JN(k2) by JN*° Since the system is assumed to be resonant at the third harmonic, (so is set equal to 3(01. Case (a): 9=G:O; (o =3o) o l >>1 Forye ”i sin3n) t . ~ 1 *2 Z 2 2 2 ‘ Ep.-. 6a
    3 ' =’.'< 3!: - " " coo . 3 - o + (J3J2 JIJo JlJl J3J1 + )s1n( (1)1 (1)1)t >3 :1: * >1: , +(J1Jo - JIJI - J3J1 - J3Jz + ...)sm(3w1+(oi)t >1: >1: :1: , +(J2JO+J2JZ-JOJ1+...)51n(3ool-2(oi)t F‘- >.‘: >1: 9':- . + (JZJo + .1212 + JOJl +. . . )81n(3(1)1+ 2min .. +JJ*JJ*JJ* '3w 303M: 11+12' 3o+°”)81n( 1" i a: :1: =1: , +(J1J1-J1JZ+J3J0+...)sm(3m1+3wi)t+...] f' The above results are plotted in Figure 5. l6(l)-(2) for N02: .4N01, Roz: .4k01. In the high pressure regime (ye >> (oi), the amplitudes of the sidebands are essentially zero possibly doe to collisional damping. However, in the low pressure regime (ye << (oi), the third harmonic electric field is frequency modulated with sidebands unequal in amplitud e . Case (b): Ozazn; (1) :30.) o 1 Forve>>toi sin3tot l *2 2 2 2 2 E ~-————-—-— +2] 2 2 +... p- bagel [(Jo) ”0 1+ J2+ J3 ) *2 2 Z Z Z 2 4 +(J1) ( Jo +4.11 +2.12 +4J3 + J1J3+ ) *2 2 2 2 2 +(JZ) (2Jo+4J1+1J2+4J3+...) :2: 3:: 2 , +(J0 J2)(4J2-8J1J +...) +...) 3 124 << u_ mi J J i" J E ~ ° ° [(.1 J "351nm t (5 23) p " (Jami o o 1 ° a}: 3:: * 31‘ , +(J1Jo -J1Jl - 13.11 - £73.12 + ... )Sln(30.)l- (Din: :2: >3 * * . +(JJ -JJO—JJ -.J'J +...)sm(3wl+wi)t 3 Z l 1 l 3 1 :1: :I: 3:: . + (JZJo + JZJZ + JOJ1 + .. .)sm(3wl- 2min =:< :1: >1: . + (JZJO + 12.12 - .1011 + . . . )81n(3(ol+ 2min * at: :1: , + (J3Jo - JlJZ 4 J1J1+ ...)51n(3(ol- 3wi)t + ( J J*+J J*- J J*+...)sin(3w +3to.)t +...] l l l 2 3 o 1 1 The results for this case are also pIotted in Figure 5.16(3)-(4). In comparison with the previous case, the difference in phase causes an interchange in amplitude between the lower frequency sideband and its corresponding upper frequency sideband. As shown in Figure 5. 16, in the low pressure regime, the sidebands are essentially equal in amplitude for small k How- 1. ever, for k1 = 1.8, the sidebands can become very unequal and the center frequency can be strongly suppressed. The physical process of the modulation phenomenon can be understood from the diagram below. 125 The low frequency ion-acoustic oscillations at (oi are superimposed on the zeroth-order density profile. This perturbation causes the density profile to vary with time at mi and 201i. The elctroacoustic resonances at m1 as well as at 31101 depend on the plasma density profile, and thus any change in the profile causes the resonant fre- quency at 3011 to change. The variation of the density profile results in a frequency modulated third harmonic output. This phenomenon is similar to a perturbation to the inductance or capacitance in a LC resonant circuit. The theoretical results agree with experimental observations. At high pressure ("e >> (oi), there is no detected modulation. As the pressure is lowered (ye (oi), modulation appears, and at still lower pressures (”e << (oi), the center frequency can be suppressed as shown in Figure 5.16. These results explain why this modulation does not play an important role in plasma harmonic generations at high harmonic output power56. Note that the modulation index, k1, is inversely proportional to (oi; that is, as (oi increase, k, decreases. Thus, when mi increases, the number and amplitude of the sidebands de- creases. However, for high harmonic power outputs at 9. 09 GHz, the length L is approximately 1. 0 - 1. 2 mm which corresponds to a low frequency oscillation greater than 15 MHz. Thus to observe a modulation, the density variation N must be 10 times greater than 01 M o A xv "NM v oz can m3! 3538, Ha non—30¢ cadunneavmgw n: .m oufimfim -_. _dl11. .11..__11.e .3vv: 2L 126 §AA we Amy J: .1;_ _ .._ 713v... Amy u3AA M a... :. v.7; 0.7.... ¢.O x 127 that at I. > 1. 5 mm (fi < 4 MHz). Apparently any low frequency variation present are not large enough to produce detectable modu- lation. ' .‘ CL".4 'VF 128 CHAP TE R 6 SUMMAR Y AND CONC LUSIONS A review of pertinent literature is present in Chapter 2. The history and mathematical formulation of electroacoustic reso- nances in bounded plasmas is presented. Relevant literature dis- cussing resonantly sustained plasma is also discussed. In Chapter 3, a model of a typical discharge structure is formulated. This model called a "plasma capacitor" is made up of a uniform, warm plasma region, sheath region and two parallel, infinite metallic plates. Equivalent circuits for the high frequency (0117.) and low frequency (MHz) regions are developed. At high frequency, this "circuit" predicts a cold plasma and a series of temperature (or electroacoustic resonances). The effects of plasma sheath, collisional damping. etc., on these resonances is studied. At low frequencies, more resonances appear due to the standing ion-acoustic waves. The theoretical data presented here are used in Chapter 5 to interpret some of the experimental results. 129 Two experimental discharge structures were constructed and put into operation. The description and some of the practical problems associated with the operation of these discharge structures and the associated microwave equipment are discussed in Chapter 4. A series of experiments were performed. The results and interpretation of these eXperiments are given in Chapter 5. The transition between the well known linear resonances of a bounded plasma and a completely resonantly sustained plasma was examined experimentally. It was found that the linear resonances become distorted, and stable and unstable rf operating regions appear. By plotting the equivalent impedance vs. mp on a Smith Chart, the distortion of resonances is qualitatively explained. In fact, stable and unstable regimes of rf power operation are clearly indicated by the Smith Chart formulation. In particular, it was found that an rf plasma could be maintained in a stable state on only one side of the resonance curve. Also the experimental results and theoretical interpretation indicate that the size of the plasma varies with a variation in absorbed power. That is, the sheath size changes in order to maintain the plasma in a resonant state. When the plasma is completely resonantly sustained at a frequency f0, it was observed that for incident power above a certain "threshold" level, oscillations at fi and f() i fi (fi << f0) would appear. ”When the incident power was adjusted to just below the threshold level, a separately applied signal was able to be amplified. Thus 130 it was eXperimentally proved that this instability results from a four frequency parametric interaction. When the instability is observed, the plasma is resonantly sustained in either temperature or cold plasma resonance. The low frequency oscillation at fi appear to be caused by a standing ion- acoustic wave. That is a correctly "sheathed" plasma capacitor r. from Chapter 3 is able to predict the proper order of magnitude of this frequency and to explain the frequency shifts associated with the instability. Thus, the physical origin of the instabilities appears to i be a mixing of four longitudinal resonances of f0, fi' fo 1; fi° Such four frequency mixing of longitudinal waves in a plasma has been studied theoretically by a number of investigators. However, the lowest predicted power threshold levels from these theories is several orders of magnitude higher than the threshold levels observed in the experiments. This discrepancy between theory and eXperiment has not yet been resolved. When the plasma capacitor is resonant simultaneously at £0, 3fo and fi the coupling between the low frequency oscillation fi and the third harmonic resonance is studied. The results of this study indicate that the low frequency resonance causes the zeroth- order density profile to vary at fi' This variation of the density profile causes the third-harminic output of the capacitor to be frequency modulated. 131 Future research is needed in a number of areas. A nonlinear capacitor model, which accounts for the coupling between the four resonances of the rf induced unstability should be developed. It may be possible to extend the present linear capacitor model by adding proper nonlinear phenomena. A goal of this study would be to pre- dict the proper incident power threshold for this instability. Precise impedance measurements of a resonantly sustained plasma should be made. By plotting these measurements on a Smith Chart, one should be able to describe the behavior of the plasma. 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