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Stimulated scattering in a plasma filling an optical cavity Myra, James Richard 1976

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STIMULATED SCATTERING IN A PLASMA FILLING AN OPTICAL CAVITY by OAMES RICHARD MYRA B . S c , Queen's Un i ve rs i t y , 1975 THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Physics) We accept th is thes is as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November, 1976 © J a m e s Richard Myra, 1976 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced deg ree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t ha t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment o f PVx^yics The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date p "Decgv^W >«376 ABSTRACT Stimulated scat te r ing processes in a homogeneous plasma ins ide an opt ica l cav i ty are studied t h e o r e t i c a l l y . In p a r t i c u l a r , at tent ion is focussed on the Raman and B r i l l o u i n i n s t a b i l i t i e s . The coupled equations for the wave amplitudes are solved subject to opt i ca l cav i ty boundary condit ions and i t is shown that for a wide range of plasma lengths and cav i ty mirror r e f l e c t i v i t i e s , the threshold values for i n s t a b i l i t y are approximately those for the temporal problem (waves uniform in space, growth or decay in t ime) . Sample resu l ts are ca lcu la ted numerical ly for the Raman i n s t a b i l i t y in a typ ica l laboratory plasma. The threshold values of inc ident laser i n tens i t y are e a s i l y exceeded with present ly a v a i l a b l e , high powered (200 MW) C0 2 l a s e r s . F i n a l l y , the physical s i g n i f i c a n c e of these resu l ts i s d iscussed , and a genera l i za t ion to the case of an inhomogeneous plasma is suggested. i i TABLE OF CONTENTS Page ABSTRACT i i LIST OF FIGURES . v TABLE OF SELECTED NOTATION vi ACKNOWLEDGEMENTS . ix Chapter 1 INTRODUCTION 1 1.1 Nonlinear Plasma Theory 1 1.2 Stimulated Scatter ing 2 1.3 Optical Cav i t i es 3 1.4 Moti vat ion -'. 5 2 BASIC EQUATIONS AND RELATION TO PREVIOUS WORK 6 2.1 The Plasma Equations 6 2.2 The Van der Pol Method 10 2.3 The Purely Growing B r i l l o u i n I nstabi 1 i ty 15 2.4 The Spat ia l Problem 18 3 THE OPTICAL CAVITY PROBLEM 22 3.1 Equations and Solut ion 22 3.2 Sample Results 30 3.3 Discussion ." 37 4 SUMMARY AND CONCLUSIONS 41 i i i Page BIBLIOGRAPHY _ 4 3 tv LIST OF FIGURES Figure Page 3.1 Marginal S t a b i l i t y Curves . 26 3.2 Marginal S t a b i l i t y Curve 28 3.3 Raman I n s t a b i l i t y Growth Rates 33 3.4 Raman I n s t a b i l i t y Thresholds 34 v TABLE OF SELECTED NOTATION time coordinate spat ia l coordinate general vector potent ia l , pump wave, backscattered wave density of j th species j = e lectrons or ions ra t io of s p e c i f i c heats v e l o c i t y of j th species charge of j th species ^i = e ' % = " e v e l o c i t y of l i g h t sca lar potent ia l equ i l ib r ium density of e lectrons and ions current density phenomenological damping rates of l i g h t wave (./sec) phenomenological damping rates of ion acoust ic waves or e lect ron plasma waves (./ sec) plasma frequency e lect ron plasma frequency ion plasma frequency e l e c t r o n , ion mass temperature of j th species in energy units e j = k B o l t T j density perturbat ion of j th species v i thermal v e l o c i t y v j = V m j quiver v e l o c i t y of electrons in pump e l e c t r i c f i e l d pump frequency, wavenumber backscattered frequency, wavenumber e l e c t r o s t a t i c frequency, wavenumber amplitudes of si n u s o i d a l l y varying quantities (superscript denotes d i r e c t i o n of propogation) group v e l o c i t i e s of plasma and l i g h t waves temporal growth rate, far above threshold ( s p e c i f i e s pump intensity) dimensionless form of AJ dimensionless form of n^ e ra t i o of -group v e l o c i t i e s dimensionless form of y dimensionless form of Y P r a t i o of s p e c i f i c heats for ions r i s e time of laser pulse c h a r a c t e r i s t i c time for purely growing B r i l l o u i n i n s t a b i l i t y ion acoustic frequency; u>s = kc s ion sound speed dimensionless growth rate piasma 1ength = ( 6 f + 8) + e ( 8 C + 6) vi i z = (6 f + 5) - e (0 C +6) A = (w2 - 4)* a = (4 - vi2 ) h ? = La/2 E e l e c t r i c f i e l d B magnetic f i e l d r*i , r 2 voltage r e f l e c t i o n c o e f f i c i e n t s R system r e f l e c t i v i t y c o e f f i c i e n t = (r, r 2 ) * r damping c o e f f i c i e n t describing mirror losses D _ 3 8 Dt 3t V c , f 3x Cp coupling c o e f f i c i e n t s S_, Sp energy fluxes of l i g h t , plasma waves cfp energy densities in l i g h t , plasma waves • » * v m ACKNOWLEDGEMENTS I wish to thank Dr. J . Meyer for his s u p e r v i s i o n , encouragement, and in te res t during the course of th is work. The ass istance and advice of Mr. D. Pawluk in the computer programming and several helpful d iscuss ions with Dr. B. Hewitt during the i n i t i a l stages of th is pro ject are .1 remembered and apprec iated. I am indebted to Mr. M.S. Johnson for his ass is tance in the preparation of th is document and to Dr. R.E. Burgess for his careful reading of the manuscript . v. I wish to express specia l thanks to my t y p i s t , Ms. Marj McDougall , for undertaking th is job under short no t i ce , ' and for her f ine work. Thanks are also due to many members of the Plasma Physics group for the i r c o n t r i b u t i o n s , d i r e c t or i n d i r e c t , to th is p ro ject . -F inanc ia l ass istance from N.R.C. in the form of a scholarship is g r a t e f u l l y acknowledged. This work was also supported by a grant from N.R.C. Chapter 1 INTRODUCTION 1.1 Nonlinear Plasma Theory One major area in modern plasma physics is that of nonl inear phenomena. In p a r t i c u l a r , plasma waves and i n s t a b i l i t i e s have been the subject of much research , both theore t i ca l and experimental . While a general theory of ': large amplitude plasma waves and turbulence has yet to be developed, there has been much progress in the area of weakly nonl inear processes ( K r a l l , 1973). These processes can be studied by extending the l i n e a r perturbat ion theory where one assumes small departures from e q u i l i b r i u m . Even though such an approach has obvious inherent l i m i t a t i o n s , the theory is s t i l l quite compl icated, and to gain useful information one must make fur ther r e s t r i c t i o n s . In the case of plasma waves, one can s ing le out a few waves and consider the nonl inear in te rac t ions occurr ing among them. A l t e r n a t i v e l y , one may wish to study changes in the bulk plasma propert ies owing to the presence of a large number of waves randomly superimposed on each other . These theor ies are known respec t i ve l y as the "theory of weak coherent waves", and the "theory of weak turbulence" . The work which fo l lows i s concerned with the former. 1 2 .. 1 .2 Stimulated Scat ter ing - . There are a number of nonl inear plasma phenomena which f a l l in the category of 3-wave i n t e r a c t i o n s . In genera l , one considers the e f f e c t s of coupl ing between waves which would ex is t independently in the plasma in a l i n e a r theory. Several 3-wave in te rac t ions are discussed by C.N. Lashmore-Davies (Lashmore-Davies, 1975). In many s i tua t ions of i n t e r e s t , one considers an i n i -t i a l wave of large amplitude, and two other small amplitude waves which perhaps grow up out of the background rad ia t ion or thermal noise of the plasma. In th is case, i t is appropr i -ate to study the i n i t i a l stages of the process by t reat ing the large amplitude, or pump wave, as a given quant i t y , and so lv ing the plasma equations for the space-t ime development of the exci ted waves. Stimulated B r i l l o u i n scat te r ing and st imulated Raman scat te r ing are examples of 3-wave i n t e r a c t i o n s . The pump wave is a transverse electromagnetic ( l i g h t ) wave and the exci ted waves are , r e s p e c t i v e l y , a l i g h t wave and an ion wave; and, a l i g h t wave and an e lect ron plasma wave. Stimulated scat te r ing may be viewed as the product of two simpler physical processes, namely, wave mixing and normal s c a t t e r i n g . In wave mix ing, one considers two l i g h t waves as g iven , and solves the plasma equations for the induced density wave. In normal s c a t t e r i n g , one l i g h t wave and a spectrum of densi ty waves ( a r i s i n g from thermal c o r r e l a t i o n s ) are g iven , and one solves for the scattered s i g n a l . Both of these processes can 3 occur simultaneously and in fact feed on each other. A scattered l i g h t wave can wave mix with the inc ident beam to induce (or enhance) a density wave. The density wave can, in tu rn , increase the amount of s c a t t e r i n g . It is th is type of feedback which gives r i s e to the backscatter ing i n s t a b i l i t i e s . From such a p i c t u r e , one might expect st imulated scat te r ing to dominate many physical systems. In a c t u a l i t y , the inc lus ion of the e f f e c t s of wave damping and other energy losses shows that there ex is t threshold values in the i n t e n -s i t y of the inc ident beam, below which the feedback mechanism is i n s u f f i c i e n t to cause i n s t a b i l i t y . Part of the aim of th is study is to derive the threshold condit ions for st imulated scat te r ing for a plasma within an opt ica l c a v i t y . 1.3 Opt ical Cav i t ies The in te rac t ions occurr ing in plasmas p a r t i a l l y f i l l -ing microwave or opt ica l c a v i t i e s have been the subject of inves t iga t ion in the past (Heckenberg, 1973; T a i t , 1973; Lochte-Holtgreven , 1 968). Diagnostic techniques employing interferometry and the mode structure in microwave c a v i t i e s (Shohet, 1 970) have been useful tools is pla;sma phys ics . The theory of such systems is well understood. In the fo l lowing we consider the st imulated scat te r ing process for a plasma within an opt ica l c a v i t y . The large amplitude pump wave is resonant within the c a v i t y . This s i t u a t i o n could be achieved in pract ice by using p a r t i a l l y r e f l e c t i n g mi r ro rs , or by a c t u a l l y inser t ing the plasma into the resonant cav i ty of a laser (Figure 1.1) 4 1 mirror mirror ( r 9 ) F i g . 1.1. Schematic of the Physical System Simi la r problems have already been considered for the case of SRS (Stimulated Raman Scatter ing) in substances other than plasmas (Lugovoi, 1969). In these cases , the medium couples pump and backscattered rad ia t ion through molecular t r a n s i t i o n s at the Raman frequency, whereas in a plasma th is is accomplished by a long i tud ina l density wave. It w i l l be seen that the nonzero group v e l o c i t y of the density wave gives r i s e to e f fec ts which d i s t i n g u i s h the behaviours of the two systems. For many values of the system parameters, the e f f e c t of convection of the density wave is n e g l i g i b l e . The conse-quence of th is for plasma experiments w i l l be d iscussed. 1.4 .Moti vat ion Experimental work has been done on wave mixing in plasmas and unionized gases in opt i ca l c a v i t i e s (Meyer, 1976). The behaviour of gases can resemble that of plasmas since atoms and molecules can be given induced d ipole moments. The p o l a r i z a b i 1 i t y of a gas i s thus re lated to the charge density in a piasma. There i s experimental evidence to support the existence of an i n s t a b i l i t y in gases analagous to the B r i l l o u i n ins tab -i l i t y . For th is case, and for the experiments with plasmas, the threshold inc ident laser i n t e n s i t y seems to be considerably lower than expected. These experimental resu l t s provide motivation for obtaining a better understanding of the e f f e c t s of op t i ca l cav i ty boundary condit ions on st imulated s c a t t e r i n g . Chapter 2 BASIC EQUATIONS AND RELATION TO PREVIOUS WORK 2.1 The Plasma Equations We consider a f i n i t e , homogeneous plasma in the f l u i d l i m i t , completely fi1 1ing an optical cavity of length I. The standard f l u i d and Maxwell equations are: 3v. q 3A ! ^ + Yj ; ?Yj = i f <-Y* - iw> + iVc" Yj x (V x A) - j ^ J - VN (2.1a) 3N, T t + ? ' tNjV.) = 0 (2.1b) ^ - c ^ . A - ^ F J- + c ^ * ( 2 . l c ) V 2* = - ^ ^ q N (2 .Id) Here, A i s the vector potential in the Coulomb gauge (V -A = 0), N. is the density of the j t h species (j = e or i ) , v. the f l u i d v e l o c i t y , $ the e l e c t r i c p o t e n t i a l , J the p a r t i c l e cur-rent density, and 9. the temperature in energy units (9 = k B o l t T ) . Thus, a Maxwellian plasma is assumed with each species being characterized by i t s own temperature, y i s defined from the equation of state r e l a t i n g pressure and densi ty . As usua l , we take p = N Y x constant . For the usual case of waves of s u f f i c i e n t l y high frequency, the ad iabat ic assumption is va l id and y is the ra t io of s p e c i f i c heats. An exception to th is case w i l l be seen to occur in sect ion 2.3 where the isothermal approximation is more appropriate and Y = 1. Following Fors lund, K inde l , and Lindman (Fors lund, 1975) we may obtain coupled wave equations for the densi ty waves and electromagnetic waves in the plasma for the one dimensional problem (s lab geometry). t 2 A^ Jt + S " C 2 V 2 ) - = ~wp - ( 2 ' 2 a > 3 2 n e '., 3n 3 2 n T P " + 2 y p Tt" ~ y v e ~dx7~ + wpe ( n e " ni> = e2N 2 l P r £ V 2 (A ; A] (2.2b) e 3 2 n - 3 2 n . T P 1 - 3 v i + Vi ( n i ' n e ) = ° ( 2 * 2 c ) Here and are phenomenological damping r a t e s , vj = e j / m j » u p j = ^ i r N 0 e 2 ^ m j ' N o 1 S t h e e c ? u ^ ^ ibr-ium density of e lect rons and ions , and n. is the density per turbat ion . In th is one J dimensional s i t u a t i o n , A is perpendicular to £ , the d i r e c t i o n of propagation of the waves. Thus the tota l e l e c t r i c f i e l d - -** - C J t -8 i s conveniently s p l i t up into the e l e c t r o s t a t i c f i e l d -V4> in 1 3 the x d i r e c t i o n , and the electromagnetic f i e l d -— -^r- A in the C a t — y-2 plane. In these equat ions, A describes the tota l e l e c t r o -magnetic s i g n a l , namely, both the pump and excited l i g h t waves. The l e f t hand sides of equations (2.2) are the usual l i n e a r i z e d wave equations while the r ight hand sides of (2.2a) and (2.2b) contain nonl inear terms which can couple the waves. We shal l see that the r ight hand side of (2.2a) descr ibes the scat te r ing of a l i g h t wave of f of a macroscopic density wave in the plasma. The r ight hand side of (2.2b) ar ises from inc lus ion of the v x B force on the e lectrons and descr ibes the mixing of l i g h t waves in the plasma (Recall v a A , B = V x A, and V - { A x (V x A)> a V2 (A • A) ). These equations are to be supplemented with boundary condit ions which describe tota l absorption of the outgoing -e l e c t r o s t a t i c waves at plasma boundaries and r e f l e c t i o n of the outgoing electromagnetic waves at mirrors near the plasma boundaries. The problem at hand involves six waves since one has pump, backscattered, and e l e c t r o s t a t i c waves t r a v e l l i n g in both d i r e c t i o n s . We focus our at tent ion on three processes which can be character ized by the nature of the excited e l e c t r o s t a t i c wave. These are the Raman i n s t a b i l i t y corresponding to an e lectron plasma wave, the B r i l l o u i n i n s t a b i l i t y corresponding to an ion acoust ic wave, and the purely growing (Re<D = 0 , I m i o < 0 ) i n s t a b i l i t y resu l t ing from the mixing of the pump at (•a), k Q) with the pump at ( w Q , - k Q ) corresponding to a standing 9 density modulat ion. . In the theory which fo l lows , where we neglect pump d e p l e t i o n , these processes are uncoupled and can be treated independently. One could proceed by solv ing equations (2.2) d i r e c t l y . The so lut ion would ind icate the existence of modes within the cav i t y whose c h a r a c t e r i s t i c frequencies and wavenumbers would be determined by the cav i ty length and plasma parameters. For the case we cons ider , where the cav i ty is long compared to the wavelengths invo lved , the intermode spacing becomes smal l . Thus we may consider w and k as continuous v a r i a b l e s , with the understanding that , what is r e a l l y meant, is the nearest cav i ty mode (a) . k ) where w - w . k Z k . In what fo l l ows , J q q' q q we shal l only be interested in the amplitudes of the waves as a funct ion of pos i t ion and time. This allows a great s i m p l i f i -cat ion of the equations at the expense of los ing phase informa-t i o n . In th is s p i r i t we introduce the slow space-t ime amplitude and phase modulations appropriate for weak coupl ing ( i . e . , growth rate << frequency) n^ ( x , t ) , n^ ' ( x , t ) , ( x , t ) , A^ ' • • where the index j = 1 or 2 to descr ibe propagation in both d i r e c t i o n s . Assuming coherent, monochromatic, plane waves, l e t n. = n*.(x,t) cos(ut - kx) + n?(x , t ) cos(wt + kx) (2.3a) ne = R e ^ x , t ^ c o s ( w t " k x) + R e ^ x , t ^ c o s ^ a ) t + k x) '(2.3b.); 10 A = A> cos(w 0 t - k Qx) + A 2 cos(co0t + k Qx) + A*(x,t) cos(w_t - k_x) + AMx . t ) cos(co_t + k_x) (2.3c) where u) = to - 10 , k = k - k and we consider w, k r e a l . o o The equations for the slowly varying amplitudes are obtained by plugging (2.3) into (2.2) and looking at the Fourier compon ents of the resu l t ing equations for a s p e c i f i c value of OJ and This is equivalent to the Van der Pol method of space-t ime averaging over the fast o s c i l l a t i o n s (Tsytov ich , 1970). For maximum growth, one chooses (w, k) such that the wavenumber and frequency matching condit ions as well as the uncoupled d ispers ion r e l a t i o n s for l i g h t and plasma waves are s imu l -taneously s a t i s f i e d . 2.2 The Van der Pol Method The Raman I n s t a b i l i t y For th is case, the ions may be taken to be s tat ionary ( i . e . , n.. - 0 ) , y = 3, and we consider w ~ t o p e . Neglecting forc ing terms far o f f resonance, equations (2.2) y i e l d in complex notation ± k x) 0 j e i ( tot ± kx) = 11 Throwing away the nonresonant terms uncouples the j = 1 and j = 2 modes ins ide the plasma.(They are of course coupled through the boundary c o n d i t i o n s ) . The e lectron waves exci ted by the standing pump are thus merely a superposit ion of those excited independently by two t r a v e l l i n g pumps propagating in opposite d i r e c t i o n s . Carrying out the d i f f e r e n t i a t i o n in equations (2.4) and taking into account the slowly varying nature of the amplitudes, i . e . , 3 2n 3n • _ e « . k _ _ £ < < k * ( 2 .5a ) 3 2 PS 3n ~dtT~ < K 1 0 T T < < J 0 , 2 (2.5b) (with s i m i l a r i n e q u a l i t i e s for A_) , one obtains to order u>':, to2 the uncoupled d ispers ion r e l a t i o n s o)2 = w 2 + 3v^k 2 (2.6a) a , 2 » ca2 + c 2 k 2 (2.6b) P 3n 3fl and to order w-^-, W^ -T^ JF- the desired equations for the ampl i -tudes where the group v e l o c i t i e s Vp and v_ are given by 12 3u) 3v2k " ¥ e 3k 10 3d) TIT = V -c 2 k 0) (2.8) and the ± sign is determined according as the wave is t ravel l i n g to the r ight or l e f t . In terms of the dimensionless var iab les X = ( v p v j % (2.9) where kv 2(2)** [co(co0 - co)]' v , ' A o ' 2 ; 2 v b - 2m*c* and the renormalized amplitudes /k_yh A; (2.10a) CO. o c ( k Q k ) 2 N 0 e (2.10b) equations (2.7) become (2.11a) o f * + £ * + f * < c r B f j j ~ T j C j (2.11b) where 13 3T 3X A few comments may be made concerning equations (2 .11) . Instead of descr ib ing growth/decay of the waves by the slowly varying amplitudes fi and A , one could have used the formalism of complex co and k. Equations (2.2) would then have y ie lded a complex d ispers ion r e l a t i o n . In th is formalism, the assump-t ion of weak coupling (formerly expressed by equation (2.5) ) takes the form Im(co) << Re(to) Im(k) << Re(k) The imaginary part of the d ispers ion re la t ions would then y i e l d an equation equivalent to the pai r (2 .11) . One advantage of using (2.11) instead of the complex d ispers ion r e l a t i o n is that one el iminates the ' v a r i a b l e s ' Re(co) , Re(k), which for weak coupling are assumed to be f i xed anyhow by solv ing the wavenumber and frequency matching cond i -t ions together with the uncoupled d ispers ion r e l a t i o n s . A l s o , having separate equations for each wave instead of a s ing le d ispers ion r e l a t i o n for the system permits easy genera l i za t ion to include e f fec ts which may act on one wave and not the other ( e . g . , r e f l e c t i o n at a boundary). Equations (2.11) in various more general forms have 14 been used by many authors (Fors lund, 1975; K r o l l , 1965; Fuchs; Dubois, 1973) to study the behaviour of parametric processes. A more complete der ivat ion of them is given by Tsytovich (Tsytov ich , 1970). These equations w i l l be taken as the s ta r t ing point for the present study. The B r i l l o u i n I n s t a b i l i t y In th is case one considers a low frequency ion acous-t i c wave at (o>, k) . Take y = 1 and s p e c i a l i z e to the region of the ion wave where 3 v 2 k 2 << w2 << k 2 v 2 << u 2 . Then pro -l e pe ceeding as before by neglect ing the offresonant fo rc ing terms in (2.2a) and (2 .2b) , the j = 1 and j = 2 modes again become uncoupled. (Neglecting offresonant forc ing terms for the B r i l l o u i n i n s t a b i l i t y where w << to . i s a much more ser ious J o. approximation than in the Raman case, since offresonant terms are now o;.ly s l i g h t l y mismatched. The approximations i n t r o -duced here should be v a l i d as long as we assume perfect match-ing of the resonant wave, since then the stronger resonant process w i l l dominate. Lashmore-Davies gives equations fo r imperfect matching, when apparently our j = 1 and j = 2 modes can be coupled. In a moment, we w i l l consider a specia l case of the matched wave being nonresonant, namely the case of a) = 0. Otherwise, the thresholds for the imperfect ly matched case are larger than those for perfect matching.) With the j = 1 and j = 2 modes uncoupled, the ana l ys i s of the previous sect ion goes through with s l i g h t m o d i f i c a t i o n s , since now one must a d d i t i o n a l l y consider the ions . It can be shown that the desired equations again take the form of (2.11) with the same renormalized amplitudes as in 15 equations (2 .10) , except that now one must use the yQt v p , v appropriate to the ion acoust ic wave (Fors lund, 1975) / e « \ J 2 kv t o . In neglect ing the offresonant forc ing terms in (2.2b) we threw away a term proport ional to A* A 2 cos(2k Q x) in both the Raman and B r i l l o u i n cases. Since the d ispers ion r e l a t i o n for the e lectron plasma wave exhib i ts a cutof f at co„„ and th is r pe forc ing term is at to = 0, the e f fec ts on the electron plasma wave are smal l . This is not the case for the low frequency ion wave however. Furthermore, while the resonant fo rc ing term is proport ional to A Q , th is term is much la rge r , being proport ional to | A Q | 2 . The next sect ion deals with the pro -cess resu l t ing from th is term, which can be viewed as a spe-c i a l case of the B r i l l o u i n i n s t a b i l i t y where co = 0 hence the j = 1 and j = 2 terms in equations (2) coalesce as do the A Q and A waves. 2.3 The Purely Growing B r i l l o u i n I n s t a b i l i t y Let n.. = n . (x , t ) cos kx (2.13a) ne = n e C x » t ) c o s kx . , (2.13b) A = A Q cos(co Qt - k 0x) + cos(co Qt + k Qx) (2.13c) In th is s e c t i o n , we relax the condi t ion that n ( x , t ) , A_(x,t ) be slowly vary ing. Equation (2.2a) descr ibes the process 16 whereby an incoming wave A 0 c o s(a ) Q t - k Qx) can in teract with the density wave n g cos kx to produce a backscattered wave A o c o s ( u 0 t + k Qx) for k = 2 k Q . In l i n e a r theory, where one considers a nondepleted pump, th is equation is of no fur ther i n t e r e s t . Equations (2 .2b) , (2.2c) give kx . 2 fx x \_ i k x + V ( n e " n i ) e [ & + % Jt ' v e & ] V1" = _ ! ^ o ^ l f j [ 2 e 2 i k x] (2.14a) 2m*c' 3x^ [ o 0 J \ & - V ! F * -p*,] ",«UX - -J,V 1 k X - 0 (2.14b) Since the forc ing terms are independent of p o s i t i o n , and the density waves have zero group v e l o c i t y , hence do not propagate out of the in te rac t ion reg ion , i t is reasonable to look for so lut ions independent of pos i t ion ( i . e . , = 0) . \ & + % J t * v e k ' ] H e + "PVV Ri> - " N o k 2 v o <2-15*> \ & + Y1vtk2]Bt + Bpi(iii " V = ° <2-15b> The above system possesses a steady state so lut ion n | s , n| s given by " e S v o ( 3 v i k 2 + MJl> TT~ = _ u'.v* + y.u* v? (2.16a) o pi e ' i pe i 17 T y p i c a l l y y^v^k 2 << to2^ in which case the ion and e lect ron perturbat ions are equal and V = ( 2 - 1 7 a ) which can also be written as ss e 2 E 2 f f s s _ e t o o o e e 11 l (2.17b) where E Q i s the maximum value of the e l e c t r i c pump f i e l d . For such a steady state s o l u t i o n , i t is appropriate to take the ions to be isothermal ( i . e . , y^ = 1) . One thus recovers a special case of Meyer and S t a n s f i e l d ' s resu l t for enhanced density f luc tuat ions due to opt ica l mixing (Meyer, 1971). More genera l l y , one can consider the nonsteady s o l u -t ions to equations (2.15) which may a r i se due to the pump, v 2 , r i s i n g from an i n i t a l va lue , zero , to i t s maximum value. If the r i s e time I*S.T r and we l e t T * = < 2 - 1 8 ) then the appropriate so lut ion is determined by the r a t i o T R / T ^ . If T^<<T r the q u a s i s t a t i c approximation is v a l i d and equation (2.17) holds with time dependent v 2 . On the other hand, for T^>>T^, the impulse approximation is v a l i d . For th is case, the general (nonsteady) so lu t ion to equation (2.15) for y . j V 2 k 2 << to2., with i n i t i a l condit ions n(.o) = ^ 1^ - = 0 (2.19a) 18 and with 0 t < 0 (2.19b) constant t > 0 i s n p cos oo (2.20a) where 2 = i_(e + y . 0 . ) s mi V De ' r r (2.20b) Since we have n. - n = h" e l e e i kx , equation (2.20) shows that the t rans ient introduced by turning the pump on suddenly at t = 0 cons is ts of the term which is to say that as well as set t ing up the off resonant a) = 0, k = 2k Q density per tu rbat ion , the pump launches two resonant, large amplitude ion acoust ic waves. These waves may seed the normal B r i l l o u i n i n s t a b i l i t y or produce observ-able density f l u c t u a t i o n s even when one is below the power threshold for i n s t a b i l i t y . l a s e r , t g ~ 30 ps. so that one is in the q u a s i s t a t i c regime. 2.4 The Spat ia l Problem Before proceeding with the B r i l l o u i n and Raman For a typ ica l z -p inch plasma ( 6 e 10 ev.) and CO 19 problems for a plasma within an opt ica l c a v i t y , i t may be be.st to review the so lut ion for a t r a v e l l i n g pump inc ident on a f i n i t e plasma (the spat ia l problem) in the present nota -t i o n . This work was done by Krol l ( K r o l l , 1965) for the case of a s o l i d , and is quoted and fur ther discussed by Forslund et a l . for the case of a plasma. For the pump inc ident from the l e f t , the equations are e(B c c + c) - c' = f* (2.21a) 3 f f* + f* - f*' = c (2.21b) Assuming so lut ions of the form e q x + 5 t [e(B c + 6) - q]c = f* (2.22a) [ 3 f + 6 + q ] f* = c (2.22b) The d ispers ion r e l a t i o n is [e(3 c + 6 ) - q] [e f + 6 + q] = 1 (2.23) Approximate so lut ions may be obtained by neglect ing e which is always smal l . We shal l write 3 for 3^ in the remainder of th is s e c t i o n . -q (3 + 6 - q) = 1 (2.24) Define qi and q 2 by 20 qi 2 -h{& + 6 ± [(3 + 6 ) 2 - 4p| (2.25) The so lu t ion s a t i s f y i n g the boundary condit ions f ( 0 , t ) = 0 c ( L , t ) = c 6 e 6 t (2.26) i s c e 5 t ^ ~ C q j / q , ) ^ ^ ( 2 . 2 7 a ) 0 e q > L - ( q i / q 2 ) e q 2 L f * - n r e q i X - e q z X , 9 9 7 . x f - - q i C o e -qTL ; , . q 2 L (2.27b) e H 1 - ( q i / q 2 ) e H 2 ' -Normally, the opt ica l input into the plasma is due to no ise , thvs 6 = 0 and c Q is a small constant. The a m p l i f i c a -t ion of the noise on propagating through the plasma may be ca lcu lated from (2 .27) . This behaviour is termed convective i n s t a b i l i t y and is predominant for 8 > 2. For 3 < 2, the d i f f e r e n t i a l equations can also admit a nont r i v ia l so lut ion s a t i s f y i n g boundary condit ions c o r r e s -ponding to zero opt ica l input , f*(0, t ) = c ( L , t ) = 0. This so lut ion is c = A e 6 t £ q i X - (qi/q2)eq2XJ (2.28a) f * = _ q i A e 6 t [ e q i X - eq2XJ (2.28b) where 6, q x , q 2 are chosen to s a t i s f y 21 9LL e<*2l = 0 (2.29) and the d ispers ion r e l a t i o n (2 .24) . These can be combined into the s ing le equation When (2.30) possesses a so lut ion with 6 > 0, the plasma wave is abso lute ly unstable. It can be shown that such a so lu t ion with p o s i t i v e 6 can ex is t only when The f i r s t condi t ion is one on the power density of the pump and the damping ra tes , the second states that the plasma must be longer than some c r i t i c a l length which is at least ir/2. (Here L is in the dimensionless units of equation (2.9) ). S imi la r phenomena w i l l be seen to occur in the next sect ion where we consider a plasma within an opt i ca l cav i t y . (2.30) ( i ) e < 2 ( i i ) L > L c (3 ) > 7r/2 Chapter 3 THE OPTICAL CAVITY PROBLEM 3.1 Equations and Solut ion In accordance with the preceding sec t ion , we consider the fo l lowing equations for l i g h t waves described by C i and c 2 and plasma waves described by f t and f t e ( 6 c C ! + c j - ci = f t (3.1a) B f f t + ft + ft' = cj (3.1b) e ( 3 c c 2 + c 2 ) + c i = f t (3.1c) B f f t + f t - ft' = c 2 (3. Id) subject to the boundary condit ions r i C i ( 0 , t ) = c 2 ( 0 , t ) (3.2a) c z ( L , t ) = r 2 c 2 ( L , t ) (3.2b) f f ( O . t ) = 0 (3.2c) f t ( L , t ) = 0 (3.2d) In the above, L is the length of the plasma (measured in 22 23 dimensionless units) which is also taken to be the length of the opt ica l c a v i t y . 1*1 and r 2 are the r e f l e c t i o n c o e f f i c i e n t s of the cav i t y mir rors . The density gradients near the ends of the plasma are assumed to be gent le , i . e . , V n | « k. so that we neglect r e f l e c t i o n s of both l i g h t and plasma waves at the plasma boundaries. It can be shown that the threshold condit ions for the case where the plasma does not completely f i l l the cav i ty and damping of the l i g h t wave outside the plasma is n e g l i g i b l e are iden t i ca l to those derived below. The pa i r (3 .1a ) , (3.1b) possess exponential so lu t ions g qx + 6t^ T h e r e s u ; ] t i n g d ispers ion r e l a t i o n is |V(Bc + <5) - qjj|3 f + 6 + q j = 1 (3.3) Let the so lut ions of th is quadratic be qi and q 2 . qi2 = ~h^f + 6 - e (3 c + 6) ± ( [ 3 f + 6 + e (8 c + 6 ) ] 2 - 4 j^( 3.4) Then the so lut ions s a t i s f y i n g the boundary condit ions are of the form ci = (Ae^ 1* + B e q 2 X ) e 6 t (3.5a) f f = ( D e q i X + E e q 2 X ) e 6 t (3.5b) c 2 = ( F e " q i X + G e " q 2 X ) e 6 t (3.5c) 24 ft = ( H e " q i X + J e " q 2 X ) e 6 t (3.5d) Making use of the set ( 3 . 1 ) , (3.2) to e l iminate the amplitudes A, B . . . J , one obtains the condit ion determining 6, the growth rate of the system. « < » c +"> - *" - ( r . r , ) 1 * ± . « ' • • ( . As usual (Fors lund, 1975; K r o l l , 1965; Fuchs) the so lut ion ex is ts in two d i s t i n c t regimes according to whether (3 f + eBQ) i s greater than or less than 2. A clue to th is is given by equation (3.4) since in the l a t t e r case, there ex is ts the pos-s i b i l i t y of the roots qi and q 2 coalesc ing or going complex for non-negative 6. It is convenient to make the s u b s t i t u -t ions w = (3 f + 6) + e(3 c +6) (3.7a) z = (3 f + 6) - e(3 c + 6) (3.7b) R = C r i r 2 ) % (3.7c) Af ter some rearrangement, equation (3.6) may be expressed in the form z = £ an | ( c o s h ^ + ^ sinh^ j l | (3.8a) where A = (w2 - 4)^ . This resu l t is s t i l l va l id for w < 2. Equ iva lent l y , for th is case one obtains 25 i *n|(cos If + J s in ^ l| (3.8b) 2 = 1-where a = (4 - w ) 2 . These forms are convenient for numerical computations and also for deducing some a n a l y t i c a l p roper t ies . The curves of marginal s t a b i l i t y (6 = 0) are shown in Figure 3.1 for the case R = 1 . For a given plasma (y_» Y p > v _ » v p ) and given l i g h t wave beam parameters (wQ, v ) one can ca lcu la te the quant i t ies L, e8 c > 3f. The pai r (e3 c, 8 f) determine a point on the graph and L determines a p a r t i c u l a r marginal s t a b i l i t y curve. Insta -b i l i t y is indicated when the point l i e s below the curve. There is a regime in which the character of the margi -nal s t a b i l i t y curve changes d ramat ica l l y . A clue to in te rpre t th is behaviour is found in equation (3 .8b) , in the s i n g u l a r i t y of the logarithm func t ion . Thus i m p l i c i t l y we consider w < 2. The condit ion for the argument to vanish can be expressed as tan c = ^ r (3.9) 2 /I 2 \*S 0 - C 2 /L 2 ) L a where C = — and 0 <; C < L. Equation (3.9) has a so lut ion when L > T T / 2 . For a long plasma, there are many s o l u t i o n s , which correspond to values of w denoted by wm. The s i m i l a r i t y between the present s i tua t ion and that discussed in sect ion 2.4 should be noted. In the l a t t e r case we had a s i m i l a r c r i t i c a l length appearing, below which there could be no absolute i n s t a b i l i t y . While the opt ica l cav i ty plasma considered here has some new features and provides d i f -ferent condit ions for i n s t a b i l i t y , some of the old features 26 2 Fig. 3.1. Marginal S t a b i l i t y Curves Parameters are R=1.0 and L=10,1,.1,.01 as indicated. The L=10 and L=°° curves are coincident to drawing accuracy. 27 s t i l l remain. P h y s i c a l l y , th is could be expected since the l i m i t of small mirror r e f l e c t i v i t i e s must return us to the solut ions of the spat ia l problem. Returning to (3 .8b) , we see that in the neighbourhood of a s ingular po int , wm, the marginal s t a b i l i t y curve is well described by the equations w = constant = wm m Z -*• oo or equiva lent ly $ f + e3 c = constant = wm (3.10) Thus for a long plasma, the marginal s t a b i l i t y curve can be mult ivalued ind ica t ing the presence of more than one unstable mode. The modes spoken of here are not to be confused with the cav i ty modes corresponding to d i f f e r e n t f requencies . As the r e f l e c t i v i t y approaches zero , the marginal s t a b i l i t y so lut ions approach the s i n g u l a r i t y and one recovers the spat ia l thresholds as quoted by Forslund et a l . Figure 3.2 d isp lays the curve for R = 0 . 1 . As R i s decreased f u r t h e r , the pattern moves up and to the l e f t , unt i l f i n a l l y the s t ra ight l i n e so lut ions occupy the region of usual physical i n t e r e s t . It is in te res t ing to look at some specia l cases analy -t i c a l l y . In the l i m i t of large L, we can look for a so lu t ion with w >> 2. Then (3.8) gives immediately 0.0 0.5 1.0 1.5 Pig. 3.2. Marginal Stability Curve L=10 R=0.10 29 Z = A ( B f - e B c ) 2 = (B f .+ e B c ) 2 - 4 (3.11) e B f B c = 1 The l a s t equation is the usual threshold condi t ion for the temporal problem. Reverting to dimensional var iab les th is takes the form Y* - Y . Y p (3.12) The so lut ion is v a l i d for w >> 2 which together with (3.11) means 8^ >> e B c - This condit ion is almost always s a t i s f i e d owing to the small values of e, the ra t io of the group v e l o c i -t i e s of the plasma and l i g h t waves. To have recovered the threshold for the temporal prob-lem is a b i t s u r p r i s i n g , since the temporal problem corresponds to the case where both the l i g h t and plasma waves are r e f l e c t e d back into the in te rac t ion reg ion . Furthermore, i t is known that the threshold and growth rate for the f i n i t e plasma in the spat ia l problem (where no waves are re f lec ted) do not ap-proach the temporal values in the large L l i m i t . Exper imental ly , th is may be s i g n i f i c a n t since the temp-oral thresholds are e a s i l y exceeded, and the condit ion L >> 1 is considerably less st r ingent than the requirement for s i g n i -f i c a n t growth in the spat ia l problem, L >> 8^; and, of course, also considerably less st r ingent than the condi t ion for absolute i n s t a b i l i t y in the spat ia l problem, B f + c8 < 2. 30 In the l i m i t of small L, the threshold condit ion becomes independent of the damping of the plasma wave. For a rb i t ra ry w, one obtains on expanding (3.8a) or (3.8b) to order La , e3 c = J$L. (3.13) P h y s i c a l l y , the damping of the plasma wave is now unimportant because the main source of d i s s i p a t i o n of plasma waves is the convection of energy into the plasma boundaries, governed by the parameters v p and L. In dimensional v a r i a b l e s , the cond i -t io n i s j u s t Y* = Y. ^ (3'.14) which is one of the same form as (3 .12) . 3. 2 Sample Results Solut ions in terms of the dimensionless var iab les are d i f f i c u l t to apply d i r e c t l y to a given experimental s i t u a t i o n since the time and length scales are themselves funct ions of the inc ident laser power. In order to obtain the growth r a t e , 6, as an e x p l i c i t funct ion of inc ident laser power, i t is neces-sary to resort to numerical techniques. The computations have been done for the Raman I n s t a b i l i t y in a typ ica l plasma: N = 1 0 1 7 c m - 3 , 6 = 10 eV. o e The values of the various parameters used in th i s sample c a l c u l a t i o n are l i s t e d in Table 3 . 1 . Some comments are in order regarding the damping c o e f f i c i e n t s of the e l e c t r o -31 magnetic and e l e c t r o s t a t i c waves. While i t is poss ib le to obtain a good theoret ica l estimate of the Landau damping of the e lectron plasma wave, the same cannot be said for c o l l i -s ional damping. Lacking any better est imate, the c o l l i s i o n a l damping c o e f f i c i e n t s for the plasma wave and l i g h t wave have been obtained from the equations = * ve1 IT Y p = ^ e i where v . i s the e f f e c t i v e e l e c t r o n - i o n c o l l i s i o n frequency given by irN eHnA 3 ' e e * 1 e e o The v a l i d i t y of the concept of an e l e c t r o n - i o n c o l l i -s ion frequency for a f u l l y ionized plasma may well be ques-t ioned , and the resu l ts quoted should be viewed with some skept ic ism. However, i t is not our aim to predict accurate numerical r e s u l t s , but merely to compare the features of the unstable plasma in an opt ica l cav i ty with s i tuat ions previously considered. This point shal l be returned to in the conclusions of Chapter 4. For the sample plasma invest igated here, i t w i l l be noted that Landau damping is completely neg l i g ib le at the wavelength of the excited e lect ron plasma waves (since kAp << 1) . For many s i tuat ions of in te res t th is w i l l not be the case and the cont r ibut ion of Landau damping to the tota l damp-ing c o e f f i c i e n t must be inc luded. 32 Table 3.1 Summary of Parameters for Sample Ca lcu la t ion Raman Instabi1i ty E lectron Density 1 0 1 7 cm" 3 Electron Temperature 10 eV Laser Wavelength (in vacuum) 1 .06 X 10" 3 c m Laser Frequency 1 .78 X 101 ** r a d / s e c Laser Wavenumber ( in plasma) 5 .90 X 10 3 c m " 1 Backscattered Frequency 1 .60 X 101 1 1 r a d / s e c Backscattered Wavenumber (in plasma) 5 .29 X 10 3 c m " 1 E l e c t r o s t a t i c Frequency 1 .80 X 101 3 r a d / s e c E l e c t r o s t a t i c Wavenumber i i .12 X 10* c m " 1 Electron Plasma Frequency 1 .78 X 101 3 r a d / s e c Electron Debye Length 7 .44 X 10' 6 c m Landau Damping C o e f f i c i e n t 2 .02 X 10" l k s e c " 1 C o l l i s i o n a l Damping C o e f f i c i e n t 3 .02 X 101 0 s e c " 1 Total Damping C o e f f i c i e n t for E l e c t r o s t a t i c Wave 3 .02 X 101 0 s e c " 1 Total Damping C o e f f i c i e n t for Electromagnetic Wave 3 .76 X 10 8 s e c " 1 33 i to c to 10 L 8 ~ 6 1000 ,oo | 1 spatial I 1 problem 0.1 / r : / / / . \ . 8 9 10 11 log(l 0) (W cm"2) 12 13 F i g . 3.3- Raman I n s t a b i l i t y Growth Rates i n a Typ i c a l Plasma within an Optical Cavity. R=0.8 and plasma lengths, SL, i n cm are indicated f o r each curve. See table 3.1. f o r plasma parameters. 34 13 « spatial threshold 12 « 11 cm" M) 10 ' >^  o )6o| 9 —- jo^ \ 1000 8 7 • i • i i i i 0 .2 .4 .6 .8 1.0 R F i g . 3.4. Raman I n s t a b i l i t y Thresholds i n a Ty p i c a l Plasma within an Optical Cavity. Plasma lengths, A, i n cm are indicated. See Table 3.1. f o r plasma parameters. 35 In Figure 3.3 the growth rate of the Raman I n s t a b i l i t y is shown as a funct ion of inc ident laser power. The mirror r e f l e c t i v i t y c o e f f i c i e n t for th is c a l c u l a t i o n is R = 0 .80. Plasma lengths are I = 1000, 10, 1 and 0.1 cm. For the case of a perfect opt ica l cav i ty (R = 1 .00) , the curves for the four I values a l l co inc ide with the X, = °° curve in th is p l o t , to within pen accuracy. The rightmost curve is for the spa-t i a l problem (R = 0) . Again, the curves for the four I values are nearly co inc ident . The dependence of the threshold power leve ls on the mirror r e f l e c t i v i t y c o e f f i c i e n t , R, is given in Figure 3 .4 . One of the surpr i s ing features here is the dramatic drop of f of the thresholds as one moves away from R = 0. For a long plasma, the t r a n s i t i o n from the spat ia l to the temporal value is p a r t i c u l a r l y sudden. It is poss ib le to explain and reproduce these curves from a simple a n a l y t i c a l model. 36 F i r s t consider an i n f i n i t e plasma ins ide an opt i ca l cav i t y . As long as the inc ident laser power is s l i g h t l y above the temporal threshold , the waves w i l l grow in the i r own reference frame as they propagate. Thus for an i n f i n i t e plasma, the wave amplitudes w i l l become a r b i t r a r i l y large at i n f i n i t y . No matter how small the r e f l e c t i v i t y c o e f f i c i e n t i s , so long as i t is nonzero, there is always an i n f i n i t e amount of energy re f lec ted back into the central region of the plasma, hence one has absolute i n s t a b i l i t y . For a plasma of f i n i t e length , the processes of con-vect ive e - f o l d i n g s , and mirror losses compete. For uncoupled l i g h t waves in an opt ica l cav i ty ( i . e . , no in te rac t ion with density waves, e t c . ) the e f f e c t i v e damping c o e f f i c i e n t d e s c r i b -ing amplitude loss due to the mirrors is r = in 1 (3.15) The equation descr ib ing the gain of the l i g h t wave is rlt = -Y .A . + C_f«e (3.16) where - 2 . = J_ + v JL Dt 3t c 8x and c_ is the coupling c o e f f i c i e n t (see equation 2.7b) . At threshold , neglect ing convection of the density wave due to the smallness of v p , we take n g to be given from (see equation 2.7a). 37 0 = -Y nrl + r A ' p e p - (3.17) The gain c o e f f i c i e n t for the l i g h t wave is thus approximately -Y . + T 1 = 7 - Y . (3.18 An estimate of the threshold is then given by This simple formula describes the resu l ts e x c e l l e n t l y for R f 0. In the l i m i t of small R, the simple desc r ip t ion is i n v a l i d due to the breakdown of equation (3.15) g iv ing T. It is c lear p h y s i c a l l y , however, that the thresholds for th is problem can never exceed those of the spat ia l problem, and r i s e monotonically to them as R tends to zero. 3.3 Discussion Prev ious ly , various authors have derived the temporal and spat ia l thresholds for stimulated scat te r ing processes. For most experimental s i t u a t i o n s , the spat ia l th resho lds , which are several orders of magnitude l a r g e r , have been taken to be the relevant ones. In cont ras t , Figures 3 . 1 , 3 . 3 , and 3.4 i l l u s t r a t e the relevance of temporal thresholds to the onset of absolute i n s t a b i l i t y in a wide var ie ty of experimental s i tuat ions where a modest f r a c t i o n of backscattered r a d i a t i o n is re f lec ted back into the in te rac t ion region of the plasma. v P a r £ n i f + Y - ^ P (3.19) 38 From a theoret ica l point of view, the opt i ca l cav i ty problem provides a smooth l ink between the spat ia l and temporal prob-1 ems. It is not surpr is ing that r e f l e c t i o n of the exc i ted waves at the plasma boundaries could convert convective i n s t a b i l i t y into absolute i n s t a b i l i t y ; however, since st imu-lated scat ter ing generates two types of excited waves, namely, e l e c t r o s t a t i c and electromagnet ic , i t is not obvious what the e f f e c t of r e f l e c t i o n of the l i g h t waves alone would be. The resu l ts indicate that the temporal desc r ip t ion is a good ap-proximation for a moderately long plasma, and reasonable mir -ror r e f l e c t i v i t i e s . The mirrors e f f e c t i v e l y const ra in the l i g h t waves to be p r a c t i c a l l y uniform in space, hence any i n s t a b i l i t y must be temporal. It is useful to consider the energy transport by the excited waves. Again, we take the Raman i n s t a b i l i t y for i l l u s t r a t i o n . Using conservation of wave a c t i o n , we obtain for the ra t io of energy f luxes S_ £_v_ w_vc c 2 k_ c 2 S~ = £ L v ~ = loTT = 3VJk ~ 6vJ (3.20) P P P p e e where $ . is the energy dens i ty . Thus in a typ ica l s i t u a t i o n , 3 the energy transported from the in te rac t ion region by the density wave is n e g l i g i b l e . Containment of the exci ted l i g h t waves is p r a c t i c a l l y equivalent to per iod ic boundary condit ions for the system. It may be poss ib le to general ize these resu l t s for a homogeneous plasma, to the case of a plasma with densi ty . iv. 39 gradients . Normally, st imulated scat ter ing is inh ib i ted by the presence of a density gradient due to the fact that the excited waves of a p a r t i c u l a r wavelength propagate out of the region where they are resonant for fur ther i n t e r a c t i o n . Looked at in another way, the excited frequency and wave-number i s a funct ion of pos i t ion in the plasma, thus the e f f e c t of propagation prevents s i g n i f i c a n t growth of any one wave (Chen, 1974). In the opt ica l cav i ty problem, we have seen that the l i g h t waves can dominate the behaviour of the system. In a plasma with density g rad ients , a l i g h t wave with a p a r t i c u l a r frequency and wavenumber gets to cross the region where i t is resonant for parametric in te rac t ion many times. Thus the density gradient scale length loses i t s s i g n i f i c a n c e . Under these circumstances, one again expects absolute i n s t a b i l i t y to set in near the homogeneous, temporal thresholds . The threshold condit ions in various regimes of in te res t a l l have simple physical explanations in terms of energy b a l -ance. The pump provides the input , d i s s i p a t i v e losses due to the damping of the l i g h t wave and plasma wave, and convec-t ion of energy out of the plasma due to propagation of the l i g h t and plasma waves provide the losses which must be counter balanced. These e f fec ts are character ized respect i ve ly by YQ, y_» Yp> and various combinations of the parameters v_, Vp, I, and R. In the temporal problem, the loss mechanisms are d i s s i p a t i v e , thus YQ = Y_Yp (3.21) 40 In the spat ia l problem, the main factors to be accounted for are the damping of the plasma wave and the fac t that the excited plasma and l i g h t waves propagate away from each other as they grow. Y 0 = %(v_/v p)* Y p ( 3 . 2 2 ) For an extremely short plasma within an opt ica l c a v i t y , the dominant e f fec ts are convection of the plasma wave and damp-ing of the l i g h t wave. F i n a l l y , for a plasma of more reasonable length (L >> 1) ins ide an opt ica l c a v i t y , the loss mechanisms are damping of the plasma wave, damping of the l i g h t wave, and mirror l o s s e s ; hence, the threshold is < • *-* P + TP x l n * <3-2 4> Chapter 4 SUMMARY AND CONCLUSIONS It has been shown that a plasma is considerably more unstable to st imulated scat ter ing processes when i t is placed ins ide an opt ica l cav i t y . Solut ions for the slowly varying wave amplitudes ind icate that the temporal thresholds are relevant to many such experimental s i t u a t i o n s , e s p e c i a l l y where the plasma is long or the cav i ty losses are smal l . Since the temporal thresholds are general ly several orders of magnitude below the spat ia l thresholds for absolute i n s t a b i l i t y , th is resu l t may be quite s i g n i f i c a n t . It would be rash to suggest that the numerical values ca lcu lated for the thresholds and growth rates are accurate in an absolute sense. Estimates of c o l l i s i o n a l and Landau damping are necessar i l y crude. In a d d i t i o n , there are many assumptions and r e s t r i c t i o n s s p e c i f i c to th is model, which may be d i f f i c u l t or impossible to achieve in p r a c t i c e . Per-haps the most serious of these are the assumptions of i s o -tropy in the y -z plane (perpendicular to the d i r e c t i o n of propagation of the waves) and homogeneity of the plasma. Such a one dimensional model cannot s t r i c t l y apply to schemes where the laser beam is converging, as is quite common in p r a c t i c e , to achieve high power dens i t ies in the focal spot. In a d d i t i o n , st imulated side scat ter ing at a rb i t ra ry angles to the pump wave is not accounted fo r . 41 42 Frequently , experiments involve lasers with an extremely short pulse. It might be noted that , for the above cav i ty analys is to be v a l i d , one must have the round t r i p t r a n s i t time for a photon very much smaller than the time for which the laser pulse e x i s t s . It is also obvious that for s i g n i f i c a n t growth, one must have the pulse e x i s t -ing for many e - f o l d i n g times of the system. The threshold in tens i t y leve ls for many other non-l i near processes in a plasma ( e . g . , s e l f focussing) l i e in roughly the same range as those for stimulated s c a t t e r i n g , though general ly are s l i g h t l y higher. If beam fi1amentation e x i s t s , so that local f lux dens i t ies exceed the average va lues , then these other nonlinear processes may modify the plasma, or couple d i r e c t l y to the e f f e c t being s tud ied . This creates fur ther d i f f i c u l t i e s which are awkward or impossible to handle t h e o r e t i c a l l y and may dominate the experimental r e s u l t s . While the exact numbers which come out of such a theory may thus be quite u n r e l i a b l e , the trends and basic conclusions concerning the e f f e c t of the opt ica l cav i ty should s t i l l stand. The experimental observation and study of nonl inear plasma phenomena is obviously c r u c i a l to our understanding of the f i e l d . The opt ica l cav i ty may well f a c i l i t a t e such work. BIBLIOGRAPHY Chen, F . F . , Laser Interact ion and Related Plasma Phenomena, Vo l . 3A, H. Schwarz and H. Hora Teds.), Plenum, New York, 1974, pp. 291-313. DuBois, D. , D.W. Forslund and E.A. Wi l l iams, Phys. Rev. L e t t . , 33 (1 973) , p. 1013. Fors lund, D.W. , J .M. Kindel and E.L. Lindman, Phys. F l u i d s , 18 (1975), p. 1002. Fuchs, V . , Phys. F l u i d s , (to be publ ished) . Heckenberg, N.R. , G.D. Ta i t and L.B. Whitbourn, J . Appl . Phys . , 44 (1973), p. 4522. K r a l l , N.A. and A.W. T r i v e l p i e c e , P r i n c i p l e s of Plasma  Phys ics , McGraw-Hi l l , 1973. Kro l l , N.M. , J . Appl . Phys . , 36 (1 965), p. 34, sec. I I IB2. Lashmore-Davies, C . N . , Plasma Phys . , 17 (1975), p. 281. Lochte -Hol tgreven, W. (ed.), Plasma D iagnost i cs , North-Hol land , 1968, p. 512-606 and re f . t h e r e i n . Lugovoi , V . N . , Soviet Physics JETP, 29 (1969), p. 374. Meyer, J . and G.G. Albach, Phys. Rev. A, 13 (1976), p. 1091. Meyer, J . and B. S t a n s f i e l d , Can. J . Phys . , 49 (1971), p. 2187. Shohet, J . L . , J . Appl . Phys . , 41 (1970), p. 2610. T a i t , G .D . , L .B. Whitbourn and L.C. Robinson, Phys. L e t t . , 46A (1 973), p. 239 . Tsy tov ich , V . , Nonlinear E f fec ts in Plasmas, Plenum, New York, 1970, p. 32. 43 

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