STIMULATED SCATTERING IN A PLASMA FILLING AN OPTICAL CAVITY by OAMES RICHARD MYRA B.Sc, THESIS Queen's U n i v e r s i t y , 1975 SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of We accept t h i s Physics) t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA November, 1976 ©James Richard Myra, 1976 In presenting this thesis an a d v a n c e d d e g r e e a t the I Library further for agree scholarly by h i s of shall this written the U n i v e r s i t y make it freely that permission for It gain PVx^yics of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 p "Decgv^W of of Columbia, British >«376 for extensive the requirements reference copying of I agree and this shall that not copying or for that study. thesis by t h e Head o f my D e p a r t m e n t is understood financial of The U n i v e r s i t y Date for permission. Department fulfilment available p u r p o s e s may be g r a n t e d representatives. thesis in p a r t i a l or publication be a l l o w e d w i t h o u t my ABSTRACT Stimulated s c a t t e r i n g processes in a homogeneous plasma i n s i d e an o p t i c a l cavity In p a r t i c u l a r , is Brillouin attention instabilities. lengths for and i t and c a v i t y instability is The coupled equations f o r to o p t i c a l shown that mirror instability are approximately in a t y p i c a l incident presently available, plasma i s laser the p h y s i c a l discussed, cavity boundary the t h r e s h o l d those f o r intensity plasma. The are e a s i l y threshold exceeded with 2 lasers. s i g n i f i c a n c e of these r e s u l t s suggested. ii time). the Raman high powered (200 MW) C 0 and a g e n e r a l i z a t i o n values the temporal growth or decay in laboratory the wave a wide range of plasma are c a l c u l a t e d n u m e r i c a l l y f o r values of Finally, for reflectivities, problem (waves uniform in space, Sample r e s u l t s theoretically. focussed on the Raman and amplitudes are solved s u b j e c t conditions are s t u d i e d is to the case of an inhomogeneous TABLE OF CONTENTS Page ABSTRACT i i LIST OF FIGURES . v TABLE OF SELECTED NOTATION vi ACKNOWLEDGEMENTS . ix Chapter 1 2 INTRODUCTION 1 1.1 Nonlinear Plasma Theory 1.2 Stimulated S c a t t e r i n g 2 1.3 Optical 3 1.4 Moti vat ion Cavities -'. 5 BASIC EQUATIONS AND RELATION TO PREVIOUS WORK 6 2.1 The Plasma Equations 2.2 The Van der Pol Method 2.3 The Purely Growing 2.4 4 The S p a t i a l THE OPTICAL 6 10 Brillouin I n s t a b i 1 i ty 3 1 15 Problem 18 CAVITY PROBLEM 22 3.1 Equations and S o l u t i o n 22 3.2 Sample R e s u l t s 30 3.3 Discussion SUMMARY AND CONCLUSIONS iii ." 37 41 Page BIBLIOGRAPHY _ tv 4 3 LIST OF FIGURES Figure Page 3.1 Marginal S t a b i l i t y Curves 3.2 Marginal S t a b i l i t y Curve 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 . 26 28 TABLE OF SELECTED NOTATION time c o o r d i n a t e spatial coordinate general v e c t o r p o t e n t i a l , pump wave, b a c k s c a t t e r e d wave d e n s i t y of j t h s p e c i e s j = e l e c t r o n s or ions ratio of s p e c i f i c heats velocity of j t h charge of j t h ^i = e ' % velocity scalar = of " species species e light potential e q u i l i b r i u m d e n s i t y of e l e c t r o n s current and ions density phenomenological wave (./sec) damping r a t e s of light phenomenological damping r a t e s of ion a c o u s t i c waves or e l e c t r o n plasma waves (./ sec) plasma frequency e l e c t r o n plasma frequency ion plasma frequency electron, ion mass temperature of j t h j Bolt j e = k density s p e c i e s in energy T perturbation vi of j t h species units thermal v j = V velocity m j q u i v e r v e l o c i t y of e l e c t r o n s i n pump electricfield pump frequency, wavenumber backscattered frequency, wavenumber electrostatic frequency, wavenumber amplitudes of s i n u s o i d a l l y v a r y i n g q u a n t i t i e s ( s u p e r s c r i p t denotes d i r e c t i o n of p r o p o g a t i o n ) group v e l o c i t i e s of plasma and l i g h t waves temporal growth r a t e , f a r above t h r e s h o l d ( s p e c i f i e s pump i n t e n s i t y ) dimensionless form d i m e n s i o n l e s s form ratio of A J of n^ e 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 rise heats time o f l a s e r f o r ions pulse c h a r a c t e r i s t i c time f o r p u r e l y growing Brillouin instability ion a c o u s t i c frequency; u> = k c s ion sound speed dimensionless growth r a t e piasma 1ength = ( 6 + 8) + e ( 8 + 6) f C vi i s z = (6 A = (w a = (4 - ? = La/2 E electric field B magnetic field r*i , r 2 R + 5) - e ( 0 f 2 C +6) - 4)* vi 2 voltage ) h reflection coefficients 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 d e s c r i b i n g D Dt _ Cp S_, Sp cf p 3 3t V mirror losses 8 c , f 3x coupling coefficients energy f l u x e s of l i g h t , plasma energy d e n s i t i e s • » * v m waves i n l i g h t , plasma waves ACKNOWLEDGEMENTS I wish to thank Dr. encouragement, and i n t e r e s t J. Meyer f o r during his the course of The a s s i s t a n c e and advice of Mr. helpful Dr. stages of Hewitt during the i n i t i a l .1 remembered and a p p r e c i a t e d . for R.E. Burgess I wish to express McDougall , f o r for her f i n e for his special undertaking discussions this careful v.. of t h i s job Ms. short Johnson manuscript. Marj work. P h y s i c s group f o r project. Financial scholarship are n o t i c e , ' and Thanks are a l s o due to many members of the this with M.S. reading of the under the document and thanks to my t y p i s t , this work. project I am indebted to Mr. h i s a s s i s t a n c e in the p r e p a r a t i o n to Dr. this D. Pawluk in computer programming and s e v e r a l B. supervision, their contributions, direct or Plasma indirect, a s s i s t a n c e from N . R . C . is g r a t e f u l l y i n the form of a acknowledged. T h i s work was a l s o supported by a grant from N.R.C. to Chapter 1 INTRODUCTION 1.1 N o n l i n e a r Plasma Theory One major area in modern plasma p h y s i c s n o n l i n e a r phenomena. instabilities theoretical In p a r t i c u l a r , that of much r e s e a r c h , both While a general theory l a r g e amplitude plasma waves and t u r b u l e n c e has y e t developed, there has been much progress n o n l i n e a r processes ( K r a l l , s t u d i e d by extending 1973). the l i n e a r of plasma waves and have been the subject and e x p e r i m e n t a l . is of ': to be in the area of weakly These processes can be perturbation 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 limitations, gain u s e f u l In the theory is still q u i t e c o m p l i c a t e d , and to i n f o r m a t i o n one must make f u r t h e r restrictions. the case of plasma waves, one can s i n g l e out a few waves and c o n s i d e r the n o n l i n e a r Alternatively, interactions occurring among them. one may wish to study changes in the bulk plasma p r o p e r t i e s owing to the presence of a l a r g e number of waves randomly superimposed on each o t h e r . are known r e s p e c t i v e l y as the "theory and the " t h e o r y of weak t u r b u l e n c e " . is inherent concerned with the former. 1 These theories of weak coherent The work which waves", follows 2 .. 1 .2 Stimulated Scattering - There are a number of n o n l i n e a r fall in the category of 3-wave considers exist independently 3-wave i n t e r a c t i o n s (Lashmore-Davies, In g e n e r a l , one between waves which in the plasma in a l i n e a r would theory. Several are d i s c u s s e d by C.N. Lashmore-Davies 1975). In many s i t u a t i o n s tial plasma phenomena which interactions. the e f f e c t s of c o u p l i n g . of interest, one c o n s i d e r s an wave of l a r g e a m p l i t u d e , and two other ini- small amplitude waves which perhaps grow up out of the background radiation or is thermal noise of the plasma. ate to study the i n i t i a l In the plasma equations f o r of the e x c i t e d are examples of is a transverse waves a r e , and (light) wave and the Stimulated of two s i m p l e r p h y s i c a l excited scattering processes, scattering. one c o n s i d e r s two l i g h t waves as g i v e n , and s o l v e s the plasma equations f o r In normal s c a t t e r i n g , The pump wave a l i g h t wave and an ion wave; and, a namely, wave mixing and normal In wave m i x i n g , and s t i m u l a t e d Raman 3-wave i n t e r a c t i o n s . may be viewed as the product solves for treating the s p a c e - t i m e development scattering electromagnetic respectively, (arising appropri- as a given q u a n t i t y , l i g h t wave and an e l e c t r o n plasma wave. waves it waves. Stimulated B r i l l o u i n scattering case, stages of the process by the l a r g e a m p l i t u d e , or pump wave, solving this the induced d e n s i t y one l i g h t wave and a spectrum of from thermal c o r r e l a t i o n s ) the s c a t t e r e d s i g n a l . are g i v e n , wave. density and one Both of these processes can 3 occur s i m u l t a n e o u s l y and in f a c t feed on each o t h e r . scattered induce turn, l i g h t wave can wave mix with the i n c i d e n t (or enhance) a d e n s i t y i n c r e a s e the amount of feedback which gives rise the is the is for Optical the t h r e s h o l d interferometry cavities in the past instabilities. In actuality, in the energy inten- for this stimulated cavity. in plasmas p a r t i a l l y have been the s u b j e c t (Heckenberg, Diagnostic and the mode s t r u c t u r e of such systems i s well the f o l l o w i n g tools the resonant c a v i t y in microwave is cavities pla sma p h y s i c s . The ; understood. we c o n s i d e r the s t i m u l a t e d cavity. The resonant w i t h i n the c a v i t y . or by a c t u a l l y of a l a s e r of techniques employing could be achieved in p r a c t i c e mirrors, fill- 1973; T a i t , 1973; a plasma w i t h i n an o p t i c a l amplitude pump wave i s reflecting of Part of the aim of conditions occurring 1 970) have been u s e f u l situation type the feedback mechanism a plasma w i t h i n an o p t i c a l L o c h t e - H o l t g r e v e n , 1 968). process f o r this in Cavities investigation In wave c a n , stimulated values beam, below which ing microwave or o p t i c a l theory beam to of wave damping and other exist threshold The i n t e r a c t i o n s (Shohet, is systems. to cause i n s t a b i l i t y . to d e r i v e scattering 1.3 incident insufficient study there It one might expect of the e f f e c t s shows that s i t y of scattering. to dominate many p h y s i c a l inclusion losses The d e n s i t y to the b a c k s c a t t e r i n g From such a p i c t u r e , scattering wave. A by using inserting (Figure 1.1) scattering large This partially the plasma into 4 1 mirror mirror (r ) 9 Fig. 1.1. Schematic of the P h y s i c a l System S i m i l a r problems have a l r e a d y been c o n s i d e r e d f o r case of SRS (Stimulated than plasmas ( L u g o v o i , Raman S c a t t e r i n g ) 1969). In is at the Raman f r e q u e n c y , accomplished by a l o n g i t u d i n a l to e f f e c t s which d i s t i n g u i s h systems. through m o l e c u l a r whereas in a plasma t h i s d e n s i t y wave. seen t h a t the nonzero group v e l o c i t y rise other these c a s e s , the medium couples pump and b a c k s c a t t e r e d r a d i a t i o n transitions in substances It will quence of t h i s the d e n s i t y the behaviours for wave i s be of the d e n s i t y wave of negligible. plasma experiments w i l l gives the two For many values of the system parameters, the of c o n v e c t i o n of the effect The c o n s e - be d i s c u s s e d . 1.4 .Moti vat ion Experimental work has been done on wave mixing plasmas and u n i o n i z e d gases i n o p t i c a l The behaviour of gases can resemble that of atoms and molecules can be given polarizabi1ity cavities of a gas i s in (Meyer, 1976). plasmas s i n c e induced d i p o l e moments. thus r e l a t e d to the charge The density in a piasma. There i s experimental of an i n s t a b i l i t y ility. For t h i s the t h r e s h o l d case, lower than expected. for of o p t i c a l cavity the in gases analagous to the B r i l l o u i n incident motivation evidence to support and f o r laser instab- the experiments with plasmas, intensity seems to be c o n s i d e r a b l y These experimental obtaining existence a better results provide understanding of the boundary c o n d i t i o n s on s t i m u l a t e d effects scattering. Chapter 2 BASIC EQUATIONS AND RELATION TO PREVIOUS WORK 2.1 The Plasma Equations We c o n s i d e r limit, completely standard fluid a finite, f i 1 1 i n g an o p t i c a l i n the f l u i d c a v i t y of l e n g t h I. The and Maxwell equations a r e : 3v. !^ homogeneous plasma q + Yj ?Yj iVc" ; = 3A i f <-Y* - i w > + (2.1a) Yj x (V x A) - j ^ J - VN 3N, Tt + ? ' tNjV.) = 0 ^ - c ^ . A - ^ F J V* 2 + (2.1b) c ^ * . ( 2 l c ) (2.Id) = - ^ ^ q N Here, A i s the v e c t o r p o t e n t i a l i n the Coulomb gauge (V - A = 0), N. i s the d e n s i t y o f the j t h s p e c i e s fluid v e l o c i t y , $ the e l e c t r i c ( j = e o r i ) , v. the p o t e n t i a l , J the p a r t i c l e rent d e n s i t y , and 9. the temperature i n energy u n i t s (9 = k B o l t T). species Thus, a Maxwellian plasma i s assumed with being c h a r a c t e r i z e d by i t s own temperature, each y is cur- d e f i n e d from the equation of s t a t e r e l a t i n g density. As u s u a l , we take p = N case of waves of s u f f i c i e n t l y assumption i s v a l i d exception to t h i s x constant. Y high f r e q u e n c y , and y i s the r a t i o case w i l l pressure and For the usual the a d i a b a t i c of s p e c i f i c h e a t s . An be seen to occur in s e c t i o n 2.3 where the isothermal approximation i s more a p p r o p r i a t e and Y = 1. Following 1975) F o r s l u n d , K i n d e l , and Lindman ( F o r s l u n d , we may o b t a i n coupled wave equations f o r the waves and e l e c t r o m a g n e t i c waves in the plasma f o r dimensional problem ( s l a b Jt t ^A 2 3 n 2 '., e TP" + 3n 2 y the one geometry). " S + density C 2 V 2 - ) = ~ p - w ( ' 2 > 2 a 3 n 2 p Tt" ~ e y v ~dx ~ 7 eN r£ e + w pe e " i> ( n n = 2 V 2 l P 3 n- Here u pj and 1 and = ^ i r N 0 e 2 (2.2b) 3 n. 2 TP (A ; A] 2 2 - 3 v i Vi + ( n i ' n e ) = ° ( are phenomenological damping r a t e s , vj ^ j' m N i o n s , and n. o is 1 S t h e ec ? ^ ^ ibr-ium d e n s i t y of u the d e n s i t y perturbation. In = * 2 e 2 c ) j/ j» m electrons t h i s one J dimensional s i t u a t i o n , A is p e r p e n d i c u l a r to £ , the of propagation of the waves. - -** Thus the t o t a l - C Jt - direction electric field 8 is conveniently split the x d i r e c t i o n , y-2 plane. In up i n t o the e l e c t r o s t a t i c field 1 and the e l e c t r o m a g n e t i c f i e l d 3 -— -^r- A in C a The l e f t hand s i d e s of equations linearized wave equations while and (2.2b) contain nonlinear see that scattering the r i g h t the r i g h t hand s i d e of and V - { A x V x A, (V (2.2a) (2.2b) arises x A)> a V (A 2 • A) the wave from and d e s c r i b e s in the plasma ( R e c a l l c o n d i t i o n s which d e s c r i b e t o t a l outgoing (2.2a) describes va A , B = ). These equations are to be supplemented with electrostatic waves. are the usual of a macroscopic d e n s i t y The r i g h t hand s i d e of l i g h t waves light hand sides of i n c l u s i o n of the v x B f o r c e on the e l e c t r o n s the mixing of electro- terms which can couple the waves. of a l i g h t wave o f f in the plasma. (2.2) the t — these e q u a t i o n s , A d e s c r i b e s the t o t a l magnetic s i g n a l , namely, both the pump and e x c i t e d We s h a l l -V4> in boundary a b s o r p t i o n of the outgoing waves at plasma boundaries and r e f l e c t i o n e l e c t r o m a g n e t i c waves at m i r r o r s of the near the plasma boundaries. The problem at hand i n v o l v e s six pump, b a c k s c a t t e r e d , and e l e c t r o s t a t i c both waves s i n c e one has waves t r a v e l l i n g directions. We focus our a t t e n t i o n on three processes which can be c h a r a c t e r i z e d by the nature of the e x c i t e d wave. These are the Raman i n s t a b i l i t y electron plasma wave, the B r i l l o u i n (•a), < 0) k ) Q instability resulting with the pump at ( w Q electrostatic corresponding instability to an ion a c o u s t i c wave, and the p u r e l y Imio in growing to an corresponding (Re<D = 0 , from the mixing of the pump at , -k ) Q corresponding to a standing 9 density modulation.. In the theory which f o l l o w s , n e g l e c t pump d e p l e t i o n , be treated these processes are uncoupled and can independently. One could proceed by s o l v i n g The s o l u t i o n would cavity equations f r e q u e n c i e s and wavenumbers be determined by the c a v i t y involved, the intermode spacing becomes s m a l l . what i s r e a l l y variables, we s h a l l only be i n t e r e s t e d a f u n c t i o n of p o s i t i o n with the meant, i s the nearest mode (a) . k ) where w - w . k Z k . q q' q q J For i s long compared to the Thus we may c o n s i d e r w and k as continuous understanding t h a t , would length and plasma parameters. the case we c o n s i d e r , where the c a v i t y wavelengths (2.2) d i r e c t l y . i n d i c a t e the e x i s t e n c e of modes w i t h i n the whose c h a r a c t e r i s t i c cavity where we In what follows, i n the amplitudes of the waves as and t i m e . This allows a great simplifi- c a t i o n of the equations at the expense of l o s i n g phase i n f o r m a tion. In t h i s spirit we i n t r o d u c e the slow s p a c e - t i m e amplitude and phase modulations a p p r o p r i a t e (i.e., f o r weak c o u p l i n g growth r a t e << frequency) n^ ( x , t ) , n^' ( x , t ) , (x,t), A^ ' •• where the index j = 1 or 2 to d e s c r i b e propagation directions. in both Assuming c o h e r e n t , monochromatic, plane waves, let n. n = n*.(x,t) = e R e^ x , t ^ c cos(ut o s ( w t - kx) + n ? ( x , t ) " ) k x + R e^ x , t ^ c o s ^ cos(wt + kx) a ) t + k x ) (2.3a) '(2.3b.) ; 10 A = A> c o s ( w t 0 A*(x,t) where u) cos(w_t - k x) k_x) = to - 10 , k o + A Q cos(co t 2 0 + AMx.t) = k - the slowly varying by plugging into ents of the r e s u l t i n g This is equivalent averaging over (2.2) taneously 2.2 amplitudes are and l o o k i n g equations for one chooses relations for (2.3c) (w, k) light obtained at the F o u r i e r a specific space-time (Tsytovich, such that as well compon value of OJ and to the Van der Pol method of and frequency matching c o n d i t i o n s dispersion cos(co_t + k_x) the f a s t o s c i l l a t i o n s maximum growth, + Q k and we c o n s i d e r w, k r e a l . o The equations f o r (2.3) + k x) 1970). For the wavenumber as the uncoupled and plasma waves are s i m u l - satisfied. The Van der Pol Method The Raman I n s t a b i l i t y For t h i s (i.e., n.. - case, the ions may be taken to be 0 ) , y = 3 , and we c o n s i d e r w ~ t o . p e forcing terms f a r o f f complex notation resonance, equations (2.2) ± k x) 0 j e i ( t o t ± kx) = stationary Neglecting yield in 11 Throwing away the nonresonant terms uncouples the j = 1 and j = 2 modes i n s i d e the plasma.(They are of course coupled through the boundary c o n d i t i o n s ) . The e l e c t r o n waves excited by the standing pump are thus merely a s u p e r p o s i t i o n of excited opposite independently by two t r a v e l l i n g and t a k i n g amplitudes, into out the d i f f e r e n t i a t i o n in equations account the slowly varying in nature of (2.4) the i.e., 3 n • _ e 2 « . 3 PS ~dt ~ k 3n _ _ £ < < 2 T < K 10 (with s i m i l a r i n e q u a l i t i e s 2 pumps propagating directions. Carrying to those 3n TT a, 3n = 2 2 w < < J * ( .5a) 2 (2.5b) 0 , 2 for A_), the uncoupled d i s p e r s i o n o) k one obtains to order u>', : relations 2 + 3v^k » ca + P 2 c 2 k (2.6a) 2 (2.6b) 2 3fl and to order w-^-, W^-T^JF- the d e s i r e d equations f o r tudes where the group v e l o c i t i e s Vp and v_ are given by the ampli- 12 3v k " e ¥ ling 2 TIT 10 and the ± sign c k 3d) 2 3u) 3k = V - 0) i s determined a c c o r d i n g as the wave i s to the r i g h t or l e f t . In (2.8) travel terms of the d i m e n s i o n l e s s variables X= (2.9) (v vj% p where kv 2(2)** v v and the renormalized [co(co - 0 , b - ' o' ; 2m*c* A 2 co)]' 2 amplitudes A; /k_yh (2.10a) CO. o equations (2.7) c(k k) N 2 Q e (2.10b) 0 become (2.11a) o B where f f* j + £* j + ~ T f * < c j C r j (2.11b) 13 3T 3X A few comments may be made concerning equations Instead of d e s c r i b i n g growth/decay varying amplitudes fi of the waves by the (2.11). slowly and A , one could have used the formalism of complex co and k. Equations a complex d i s p e r s i o n r e l a t i o n . t i o n of weak c o u p l i n g (2.2) would then have y i e l d e d In t h i s f o r m a l i s m , the assump- ( f o r m e r l y expressed by equation (2.5) ) takes the form Im(co) << Re(to) Im(k) << Re(k) The imaginary part of the d i s p e r s i o n r e l a t i o n s yield an equation e q u i v a l e n t to the p a i r One advantage of using dispersion relation Re(co), Re(k), together then (2.11). i n s t e a d of the complex that one e l i m i n a t e s the 'variables' which f o r weak c o u p l i n g are assumed to be anyhow by s o l v i n g tions is (2.11) would the wavenumber and frequency matching c o n d i - with the uncoupled d i s p e r s i o n r e l a t i o n s . having separate equations f o r dispersion relation for reflection Equations Also, each wave i n s t e a d of a s i n g l e the system permits easy generalization to i n c l u d e e f f e c t s which may act on one wave and not the (e.g., fixed other at a boundary). (2.11) in v a r i o u s more general forms have 14 been used by many authors Dubois, 1973) to study (Forslund, starting 1970). point The B r i l l o u i n In tic of for of them i s given These equations w i l l the present (o>, k ) . 2 and ( 2 . 2 b ) , uncoupled. Brillouin study. << w 2 2 instability J << k v 2 = 1 and j approximation than in the Raman c a s e , process w i l l s l i g h t l y mismatched. dominate. case are l a r g e r of the previous the t h r e s h o l d s = 1 and j the form of terms f o r since offresonant terms intro- our j for perfect can be shown that resonant equations = 1 and j for = 2 modes c o n s i d e r a s p e c i a l case namely the case of the i m p e r f e c t l y matched matching.) s e c t i o n goes through with s l i g h t (2.11) the a much more s e r i o u s = 2 modes uncoupled, the s i n c e now one must a d d i t i o n a l l y It terms = 2 modes again become Lashmore-Davies gives than those f o r With the j pro- forcing s i n c e then the stronger In a moment, we w i l l Otherwise, Then as long as we assume p e r f e c t match- of the matched wave being nonresonant, a) = 0. . The approximations imperfect m a t c h i n g , when a p p a r e n t l y can be c o u p l e d . 2 region pe forcing where w << to . i s o. ing of the resonant wave, << u the o f f r e s o n a n t offresonant duced here should be v a l i d 2 e the j (Neglecting are now o;.ly ion a c o u s - Take y = 1 and s p e c i a l i z e to the the ion wave where 3 v k l (2.2a) be taken as the case one c o n s i d e r s a low frequency ceeding as before by n e g l e c t i n g in by T s y t o v i c h Instability this wave at 1965; F u c h s ; the behaviour of parametric p r o c e s s e s . A more complete d e r i v a t i o n (Tsytovich, 1975; K r o l l , c o n s i d e r the analysis modifications, ions. the d e s i r e d equations again take with the same renormalized amplitudes as in 15 equations v (2.10), appropriate except that now one must use the y e kv J In n e g l e c t i n g to. the o f f r e s o n a n t we threw away a term p r o p o r t i o n a l the Raman and B r i l l o u i n cases. forcing to A* A 2 terms in cos(2k x) a cutoff r This ion wave however. Furthermore, while and A 2.3 forcing being The next s e c t i o n deals with the p r o - 2 from t h i s term, which can be viewed as a s p e - case of the B r i l l o u i n j = 1 and j the resonant term i s much l a r g e r , Q to | A | . cess r e s u l t i n g cial to A , t h i s Q plasma i s not the case f o r the low frequency term i s p r o p o r t i o n a l proportional relation at co„„ and t h i s pe term i s at to = 0 , the e f f e c t s on the e l e c t r o n wave are s m a l l . (2.2b) in both Q Since the d i s p e r s i o n the e l e c t r o n plasma wave e x h i b i t s forcing p to the ion a c o u s t i c wave ( F o r s l u n d , 1975) / «\ 2 for v , Qt instability = 2 terms i n equations where co = 0 hence the (2) c o a l e s c e as do the A Q waves. The Purely Growing Brillouin Instability Let n.. = n . ( x , t ) n A = A In t h i s Q e = n C »t) cos(co t Q s e c t i o n , we r e l a x be slowly varying. e x cos kx c o s (2.13a) kx . , k x ) + cos(co t + k x ) 0 Q Q the c o n d i t i o n that n ( x , t ) , Equation (2.2a) (2.13b) (2.13c) A_(x,t) d e s c r i b e s the process 16 whereby an incoming wave A c o s ( a ) t 0 the d e n s i t y A cos(u t o 0 wave n c o s for Q k = 2k . In Q [ & Equations % + Jt ' (2.2b), 1 _!^o = kx \& - ! F Since the f o r c i n g density + \ & + Y o n \_ i k x i ) e (2.14a) J group v e l o c i t y , region, it v e '] e k H 1t ]t v k2 "PVV + + B B is 1kX -0 (2.14b) hence do not propagate for = 0). i> - " o N k 2 v o pi i " V ° (ii and the reasonable to look (i.e., R further x] 0 < - *> 2 15 <- > = The above system possesses a steady s t a t e given i s of no x " (n e independent of p o s i t i o n % J t * where one terms are independent of p o s i t i o n , out of the i n t e r a c t i o n \ & 2 fx V e -p*,] ",« - -J,V * waves have zero solutions . + UX V with give ^lfj[2 2ik 2m*c' 3x^ [ theory, equation (2.2c) V" e & ] v can i n t e r a c t Q linear c o n s i d e r s a nondepleted pump, t h i s interest. k x) kx to produce a b a c k s c a t t e r e d wave g + k x) - Q 2 15b solution n| , s n| s by "e TT~ o S o u'.v* pi v = _ ( 3 v i Jl> M k 2 + + y.u* v? e ' i pe i (2.16a) 17 Typically y^v^k perturbations << to ^ 2 in which case the ion and e l e c t r o n 2 are equal and V = (2 - 17a) which can a l s o be w r i t t e n as ss ff _ e E o 2 ss e o where E is Q 2 t (2.17b) o e e the maximum value of such a steady s t a t e s o l u t i o n , ions to be isothermal (i.e., it 1 1 l the e l e c t r i c is appropriate y^ = 1 ) . case of Meyer and S t a n s f i e l d ' s r e s u l t density fluctuations due to o p t i c a l tions to equations v , rising If the r i s e 2 from an i n i t a l * T If R T^<<T equation (2.17) r case, for y.jV k 2 2 (Meyer, 1971). which may a r i s e due to the pump, value, zero, to i t s maximum v a l u e . and we l e t < solution i s determined by the the q u a s i s t a t i c approximation holds with time dependent v . 2 hand, f o r T^>>T^, this mixing f o r enhanced = then the a p p r o p r i a t e T /T^. r to take the one can c o n s i d e r the nonsteady s o l u - (2.15) time I*S.T For One thus recovers a special More g e n e r a l l y , pump f i e l d . the impulse approximation the general (nonsteady) << to ., with i n i t i a l 2 n(.o) = ^1^- solution 2 - 1 8 ) ratio i s v a l i d and On the other is v a l i d . to equation For (2.15) conditions = 0 (2.19a) 18 and with 0 t < 0 (2.19b) t > 0 constant is n p cos (2.20a) oo where 2 s = i_( m V D i e e (2.20b) + y.0.) ' r r Since we have n. - n = h" e i kx , equation (2.20) shows that the l e e t r a n s i e n t i n t r o d u c e d by t u r n i n g the pump on suddenly at t = 0 c o n s i s t s of the which i s term to say that as well a) = 0 , k = 2 k resonant, Q density perturbation, able density threshold for fluctuations 2.4 g offresonant the pump launches two instability These waves or produce even when one i s below the observpower instability. For a t y p i c a l t up the l a r g e amplitude ion a c o u s t i c waves. may seed the normal B r i l l o u i n laser, as s e t t i n g ~ 30 ps. The S p a t i a l z-pinch plasma ( 6 so t h a t one i s e 10 e v . ) and CO in the q u a s i s t a t i c regime. Problem Before proceeding with the B r i l l o u i n and Raman 19 problems f o r a plasma w i t h i n an o p t i c a l be.st to review on a f i n i t e tion. the s o l u t i o n plasma (the for a travelling spatial problem) This work was done by K r o l l of a s o l i d , et a l . for cavity, may be pump i n c i d e n t in the present (Kroll, and i s quoted and f u r t h e r it 1965) f o r nota- the case d i s c u s s e d by Forslund the case of a plasma. For the pump i n c i d e n t from the l e f t , the equations are e ( B c + c) c - 3 f * + f* f Assuming s o l u t i o n s this small. = c (2.21b) q x + 5 c [3 + 6 + q ] f* = c c +6) Approximate s o l u t i o n s always (2.21a) [e(B f + 6) - = f* f*' of the form e The d i s p e r s i o n r e l a t i o n [e(3 c' t q]c = f* (2.22a) (2.22b) is - q] [e f + 6 + q] = 1 may be obtained by n e g l e c t i n g We s h a l l w r i t e 3 f o r (2.23) e which is 3^ in the remainder of section. -q(3 + 6 - Define qi and q 2 by q) = 1 (2.24) 20 -h{& + 6 ± [ ( 3 + 6 ) qi 2 - 4p| (2.25) 2 The s o l u t i o n satisfying the boundary f(0,t) c(L,t) conditions =0 = c e (2.26) 6 t 6 is c e 5 ^ t f* q - n r - o f q i ~ e > 0 C Cqj/q,)^^ - L ( e -qTL e - q i X e H 1 Normally, noise, the o p t i c a l thvs 6 = 0 and c /q ) 2 e - e ; , . q L (qi/q )e '- c a l c u l a t e d from ( 2 . 2 7 ) . This 9 2 into 9 7 the plasma i s due to through behaviour 2 7 a ) H 2 2 i s a small c o n s t a n t . Q . , . x (2.27b) q z X input t i o n of the noise on propagating instability q i ( 2 q 2 L The a m p l i f i c a - the plasma may be i s termed convective and i s predominant f o r 8 > 2. For 3 < 2, the d i f f e r e n t i a l a nontrivial solution satisfying ponding to zero o p t i c a l input, equations can a l s o admit boundary c o n d i t i o n s f*(0,t) = c(L,t) = 0. corresThis solution is = A e c f where 6 , q , q x 2 * 6 t = _ £ q i q i - X Ae 6 t [e (qi/q )e J 2 q i X - q2X eJ are chosen to s a t i s f y q 2 X (2.28a) (2.28b) 21 9LL <*2l e and the d i s p e r s i o n r e l a t i o n into the s i n g l e = 0 (2.24). (2.29) These can be combined equation (2.30) When (2.30) possesses a s o l u t i o n with 6 > 0, the plasma wave is absolutely with p o s i t i v e The f i r s t unstable. (Here L i s can be shown that such a s o l u t i o n 6 can e x i s t only when (i) e < (ii) L > L (3) condition 2 c > 7r/2 i s one on the power d e n s i t y and the damping r a t e s , be longer It the second s t a t e s that than some c r i t i c a l the plasma must length which i s at l e a s t i n the dimensionless u n i t s S i m i l a r phenomena w i l l of the pump of equation (2.9) be seen to occur in the next where we c o n s i d e r a plasma w i t h i n an o p t i c a l ir/2. cavity. ). section Chapter 3 THE OPTICAL CAVITY PROBLEM 3.1 Equations and S o l u t i o n In accordance with the preceding s e c t i o n , we c o n s i d e r the f o l l o w i n g c 2 equations for l i g h t waves d e s c r i b e d by C i and and plasma waves d e s c r i b e d by f t e(6 C! c B ft + ft f e(3 c c 2 B ft f subject + cj to the boundary + c ) 2 + ft - the above, L i s ft = ft (3.1a) = cj (3.1b) + ci = ft (3.1c) ft' (3.Id) = c 2 conditions c (L,t) In ci + ft' riCi(0,t) z - and = c (0,t) 2 (3.2a) = r c (L,t) (3.2b) 2 2 ff(O.t) = 0 (3.2c) ft(L,t) =0 (3.2d) the length of the plasma (measured 22 in 23 dimensionless units) which i s the o p t i c a l 1*1 and r cavity. of the c a v i t y of mirrors. a l s o taken to be the l e n g t h are the r e f l e c t i o n 2 The d e n s i t y the plasma are assumed to be g e n t l e , V n | « at the plasma b o u n d a r i e s . conditions fill for the c a v i t y T h It = ~h^f can be shown that e r e s u ;]ti g the threshold + 6 - e(3 Then the s o l u t i o n s c to those d e r i v e d qjj|3 f + 6) ± ( [ 3 satisfying solutions is + 6 + qj = 1 q u a d r a t i c be qi (3.3) and q . + 6 + e(8 f the below. possess exponential dispersion relation n + <5) - c (3.1b) Let the s o l u t i o n s of t h i s 2 of both l i g h t and plasma waves are i d e n t i c a l (3.1a), |V(B qi i.e., and damping of the l i g h t wave o u t s i d e The p a i r + 6t^ near the ends the case where the plasma does not completely plasma i s n e g l i g i b l e qx coefficients k. so that we n e g l e c t r e f l e c t i o n s g gradients of 2 c + 6)] 2 - 4^j( 3.4) the boundary c o n d i t i o n s are of the form c ci = (Ae^ * + B e q 2 X ) e 6 t (3.5a) ff = (De q i X + Ee q 2 X ) e 6 t (3.5b) = (Fe" q i X + Ge" 2 1 q 2 X ) e 6 t (3.5c) 24 ft = (He" + Je" q i X Making use of the set ( 3 . 1 ) , A, B . . . J , q 2 X ) e (3.5d) 6 t ( 3 . 2 ) to e l i m i n a t e the amplitudes one o b t a i n s the c o n d i t i o n determining 6, the growth r a t e of the system. « < » c +"> - *" - As usual exists (Forslund, 1 1975; K r o l l , in two d i s t i n c t i s greater (r.r,) * ± .«'•• 1965; Fuchs) A c l u e to t h i s equation ( 3 . 4 ) s i n c e in the l a t t e r sibility of the r o o t s for non-negative 6. the s o l u t i o n regimes a c c o r d i n g to whether than or l e s s than 2. qi and q It case, . ( ( 3 + eB ) Q f i s given by there e x i s t s the pos- c o a l e s c i n g or going complex 2 i s convenient to make the s u b s t i t u - tions w = ( 3 + 6) + e ( 3 + 6 ) (3.7a) z = ( 3 + 6) - e ( 3 + 6) (3.7b) f c f c R = Crir ) (3.7c) % 2 After some rearrangement, equation ( 3 . 6 ) may be expressed i n the form z = £ an | ( c o s h ^ + where A = (w Equivalently, 2 - 4)^. for this This result ^sinh^j l | is s t i l l case one o b t a i n s valid (3.8a) f o r w < 2. 25 2 = i1- *n|(cos If where a = (4 - w ) . 2 The curves of marginal for l| (3.8b) These forms are convenient computations and a l s o f o r 3.1 + J sin ^ for deducing some a n a l y t i c a l stability numerical properties. ( 6 = 0) are shown in the case R = 1 . For a given plasma (y_» Y > v_» v ) p p and given wave beam parameters (w , v ) one can c a l c u l a t e the e8 c > 3f. The p a i r ( e 3 , 8 ) determine a point on the graph c f and L determines a p a r t i c u l a r bility is light quantities Q L, Figure marginal s t a b i l i t y i n d i c a t e d when the point lies curve. below the Insta- curve. There i s a regime in which the c h a r a c t e r of the m a r g i nal this stability curve changes d r a m a t i c a l l y . behaviour i s found in equation ( 3 . 8 b ) , of the l o g a r i t h m f u n c t i o n . The c o n d i t i o n f o r Thus i m p l i c i t l y A c l u e to in the interpret singularity we c o n s i d e r w < 2. the argument to vanish can be expressed as tan c = ^ 0 r (3.9) 22 /I 22 \*S - C /L ) La where C = — when L > T T / 2 . and 0 <; C < L. Equation (3.9) has a s o l u t i o n For a long plasma, there are many s o l u t i o n s , which correspond to values of w denoted by w . m The s i m i l a r i t y between the present s i t u a t i o n and that d i s c u s s e d in s e c t i o n 2.4 should be noted. we had a s i m i l a r c r i t i c a l In the l a t t e r length a p p e a r i n g , below which could be no absolute i n s t a b i l i t y . While the o p t i c a l case there cavity plasma considered here has some new f e a t u r e s and provides ferent conditions for instability, some of the old dif- features 26 2 F i g . 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 i n d i c a t e d . The L=10 and L=°° curves are coincident to drawing accuracy. 27 still remain. limit of small m i r r o r solutions Physically, of a s i n g u l a r to point, could be expected s i n c e the reflectivities of the s p a t i a l Returning this must r e t u r n us to the problem. (3.8b), w , we see that the marginal m in the stability neighbourhood curve is well d e s c r i b e d by the equations w = constant = w m m Z -*• oo or equivalently $ Thus f o r f + e3 a long plasma, multivalued i n d i c a t i n g mode. = constant = w c (3.10) m the marginal stability curve can be the presence of more than one unstable The modes spoken of here are not to be confused with the c a v i t y modes corresponding As the r e f l e c t i v i t y stability solutions the s p a t i a l approaches z e r o , approach the s i n g u l a r i t y thresholds 3.2 d i s p l a y s to d i f f e r e n t line the marginal and one as quoted by Forslund et a l . the curve f o r R = 0.1. solutions recovers Figure As R i s decreased the p a t t e r n moves up and to the l e f t , straight frequencies. until finally occupy the region of usual further, the physical interest. It tically. is interesting to look at some s p e c i a l cases In the l i m i t of l a r g e L, with w >> 2. Then (3.8) gives we can look f o r immediately analy- a solution 0.0 0.5 1.0 Pig. 3.2. Marginal Stability Curve L=10 R=0.10 1.5 29 Z = A (B f - eB ) c = (B .+ 2 f eB B f The l a s t equation i s temporal problem. c 2 - 4 (3.11) =1 c the usual Reverting eB ) threshold condition for to dimensional v a r i a b l e s the this takes the form Y* - The s o l u t i o n is v a l i d means 8^ >> e B - (3.12) p f o r w >> 2 which together This c Y.Y condition i s almost always owing to the small values of e, the r a t i o ties of the plasma and l i g h t To have recovered lem i s a b i t surprising, with (3.11) satisfied of the group veloci- waves. the t h r e s h o l d for s i n c e the temporal the temporal prob- problem corresponds to the case where both the l i g h t and plasma waves are reflected back into known that the i n t e r a c t i o n the t h r e s h o l d the s p a t i a l it the f i n i t e is plasma problem (where no waves are r e f l e c t e d ) Experimentally, thresholds is considerably ficant Furthermore, and growth r a t e f o r proach the temporal values oral region. may be s i g n i f i c a n t growth in the s p a t i a l L >> 1 than the requirement f o r problem, L >> 8^; and, of less stringent absolute i n s t a b i l i t y s i n c e the temp- exceeded, and the c o n d i t i o n less stringent also considerably do not ap- in the l a r g e L l i m i t . this are e a s i l y in than the c o n d i t i o n in the s p a t i a l problem, B f + c8 signicourse, for < 2. 30 In the l i m i t of small L, becomes independent of the t h r e s h o l d condition the damping of the plasma wave. a r b i t r a r y w, one o b t a i n s on expanding (3.8a) or For (3.8b) to order L , a e3 Physically, c = J$L. (3.13) the damping of the plasma wave i s now unimportant because the main source of d i s s i p a t i o n of plasma waves convection of energy the parameters v p into is the the plasma b o u n d a r i e s , governed by and L. In dimensional v a r i a b l e s , the c o n d i - tio n i s j u s t Y* = Y . ^ which i s one of the same form as 3. 2 (3'.14) (3.12). Sample R e s u l t s Solutions difficult in terms of the dimensionless v a r i a b l e s to apply d i r e c t l y s i n c e the time and length sary to r e s o r t been done f o r N = 10 o 1 7 cm In order f u n c t i o n of incident the Raman I n s t a b i l i t y l a s e r power, it rate, is neces- The computations have in a t y p i c a l plasma: , 6 = 10 eV. e The values of the v a r i o u s sample c a l c u l a t i o n are l i s t e d are in order to o b t a i n the growth to numerical t e c h n i q u e s . - 3 situation s c a l e s are themselves f u n c t i o n s of the i n c i d e n t l a s e r power. 6, as an e x p l i c i t to a given experimental are regarding parameters used in i n Table 3 . 1 . this Some comments the damping c o e f f i c i e n t s of the electro- 31 magnetic and e l e c t r o s t a t i c waves. While i t is possible to o b t a i n a good t h e o r e t i c a l estimate of the Landau damping of the e l e c t r o n plasma wave, the same cannot be s a i d f o r sional damping. Lacking any b e t t e r damping c o e f f i c i e n t s for e s t i m a t e , the colli- collisional the plasma wave and l i g h t wave have been obtained from the equations = where v . i s given * e1 v IT Y the e f f e c t i v e p ^ei = electron-ion collision frequency by irN e H n A e 3 ' 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 s i o n frequency tioned, for a fully and the r e s u l t s skepticism. numerical However, results, i o n i z e d plasma may well be ques- quoted should be viewed with some it is not our aim to p r e d i c t accurate but merely to compare the f e a t u r e s unstable plasma in an o p t i c a l cavity considered. be returned This point colli- shall with s i t u a t i o n s of the previously to in the c o n c l u s i o n s of Chapter 4. For the sample plasma i n v e s t i g a t e d here, it noted that Landau damping i s completely n e g l i g i b l e wavelength << 1 ) . of the e x c i t e d electron For many s i t u a t i o n s case and the c o n t r i b u t i o n of plasma waves interest this will at (since will be the kAp not be the of Landau damping to the t o t a l damp- ing c o e f f i c i e n t must be i n c l u d e d . 32 Table 3.1 Summary of Parameters f o r Sample C a l c u l a t i o n Raman I n s t a b i 1 i ty Electron Density 10 E l e c t r o n Temperature 10 eV Laser Wavelength Laser (in vacuum) Frequency Laser Wavenumber ( i n Backscattered plasma) Frequency 1 7 cm" 3 1 .06 X 10" 1 .78 X 10 1 5 .90 X 10 3 1 .60 X 10 1 Backscattered Wavenumber (in plasma) 5 .29 X 1 0 Electrostatic Frequency 1 .80 X 10 Electrostatic Wavenumber E l e c t r o n Plasma Frequency i i cm 3 ** c m " 3 c m " 1 1 rad/sec 3 c m " 1 1 rad/sec 11 .12 X 10* 1 .78 X 10 rad/sec 1 rad/sec 3 E l e c t r o n Debye Length 7 .44 X 1 0 ' Landau Damping C o e f f i c i e n t 2 .02 X 10" C o l l i s i o n a l Damping Coefficient 3 .02 X 10 1 0 s e c " 1 T o t a l Damping C o e f f i c i e n t f o r E l e c t r o s t a t i c Wave 3 .02 X 10 1 0 s e c " 1 Total Damping C o e f f i c i e n t f o r E l e c t r o m a g n e t i c Wave 3 .76 X 10 8 cm 6 l s e c " k s e c " 1 1 33 10 L 1000 8 ~ ,oo |1 spatial I1 6 i to c problem 0.1 /r to // : / 8 . \ 9 . 10 log(l ) 0 11 12 13 (W cm" ) 2 F i g . 3.3- Raman I n s t a b i l i t y Growth Rates i n a T y p i c a l Plasma w i t h i n an O p t i c a l C a v i t y . R=0.8 and plasma l e n g t h s , SL, i n cm a r e i n d i c a t e d f o r each curve. See t a b l e 3.1. f o r plasma parameters. 34 13 « spatial threshold 12 11 cm" « M) 10 >^ ' o )6o| —-jo^ 9 \ 1000 8 7 • 0 .2 i • .4 i .6 i i .8 i 1.0 R F i g . 3.4. Raman I n s t a b i l i t y T h r e s h o l d s i n a T y p i c a l Plasma w i t h i n an O p t i c a l C a v i t y . Plasma l e n g t h s , A, i n cm are i n d i c a t e d . See Table 3.1. f o r plasma parameters. 35 In is Figure 3.3 the growth rate of the Raman I n s t a b i l i t y shown as a f u n c t i o n of reflectivity coefficient Plasma lengths I values a l l cavity problem (R = 0 ) . are nearly on the m i r r o r Figure 3.4. (R = 1 . 0 0 ) , The rightmost is reflectivity One of the s u r p r i s i n g of the t h r e s h o l d s particularly reproduce these curves The is R = 0.80. For the case the curves curve for mirror is for the in t h i s plot, for spa- the the four I The dependence of the t h r e s h o l d coefficient, features R, is here i s sudden. from the s p a t i a l It is values power given in the dramatic as one moves away from R = 0. a long plasma, the t r a n s i t i o n value calculation A g a i n , the curves coincident. levels drop o f f this power. c o i n c i d e with the X, = °° curve to w i t h i n pen a c c u r a c y . tial for laser are I = 1000, 10, 1 and 0.1 cm. of a p e r f e c t o p t i c a l four incident For to the temporal p o s s i b l e to e x p l a i n and from a simple a n a l y t i c a l model. 36 First cavity. c o n s i d e r an i n f i n i t e As long as the i n c i d e n t the temporal threshold, r e f e r e n c e frame as they plasma, is, amount of energy the plasma, ing propagate. is is nonzero, reflected Thus f o r an i s always an back i n t o hence one has a b s o l u t e the c e n t r a l e-foldings, l o s s e s compete. waves, etc.) cavity The equation d e s c r i b i n g infinite region For of uncoupled no i n t e r a c t i o n with describ- is in 1 (3.15) the gain of the l i g h t wave is = - Y . A . + C_f« rlt at coefficient damping c o e f f i c i e n t amplitude l o s s due to the m i r r o r s r = large the processes of c o n - (i.e., the e f f e c t i v e own instability. length, in an o p t i c a l above infinite become a r b i t r a r i l y there and m i r r o r slightly grow in t h e i r For a plasma of f i n i t e l i g h t waves density power No matter how small the r e f l e c t i v i t y so long as i t vective laser the waves w i l l the wave amplitudes w i l l infinity. plasma i n s i d e an o p t i c a l (3.16) e where -2. Dt and c_ i s threshold, + v 3t JL c 8x the c o u p l i n g c o e f f i c i e n t n e g l e c t i n g c o n v e c t i o n of the smallness of v , p 2.7a). = J_ we take n g (see equation 2 . 7 b ) . At the d e n s i t y wave due to to be given from (see equation 37 0 -Y rl + r A 'p e p - = The gain c o e f f i c i e n t f o r -Y. (3.17) n + the l i g h t wave i s T = 1 An estimate of the t h r e s h o l d approximately - Y. 7 is thus (3.18 then given by v P ar This for R f 0. is invalid It is £ n if + Y (3.19) -^P simple formula d e s c r i b e s the r e s u l t s In the l i m i t of small R, the simple d e s c r i p t i o n due to the breakdown of equation (3.15) clear physically, however, that giving the t h r e s h o l d s problem can never exceed those of the s p a t i a l r i s e m o n o t o n i c a l l y to them as R tends to 3.3 excellently for T. this problem, and zero. Discussion Previously, and s p a t i a l various thresholds For most experimental which are several to be the r e l e v a n t 3.4 i l l u s t r a t e for authors have d e r i v e d the temporal stimulated scattering processes. situations, the s p a t i a l orders of magnitude l a r g e r , ones. In c o n t r a s t , Figures thresholds, have been taken 3 . 1 , 3 . 3 , and the r e l e v a n c e of temporal t h r e s h o l d s onset of absolute i n s t a b i l i t y in a wide v a r i e t y of to the experimental s i t u a t i o n s where a modest f r a c t i o n of b a c k s c a t t e r e d r a d i a t i o n is r e f l e c t e d back into the i n t e r a c t i o n region of the plasma. 38 From a t h e o r e t i c a l provides point of view, the o p t i c a l a smooth l i n k between the s p a t i a l cavity problem and temporal prob- 1 ems. It i s not s u r p r i s i n g that r e f l e c t i o n waves at the plasma boundaries could convert instability into a b s o l u t e i n s t a b i l i t y ; of the excited convective however, since stimu- l a t e d s c a t t e r i n g generates two types of e x c i t e d waves, namely, e l e c t r o s t a t i c and e l e c t r o m a g n e t i c , 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. results i n d i c a t e that the temporal d e s c r i p t i o n proximation f o r ror The m i r r o r s l i g h t waves to be p r a c t i c a l l y It is useful the r a t i o hence any to c o n s i d e r the energy t r a n s p o r t of energy S~ P = £_v_ £Lv~ P P by the for fluxes w_v = c k_ 2 c loTT = p d e n s i t y wave i s 2 (3.20) e Thus in a t y p i c a l from the i n t e r a c t i o n negligible. waves i s p r a c t i c a l l y c 3VJk ~ 6vJ e the energy d e n s i t y . the energy t r a n s p o r t e d for uniform in s p a c e , the Using c o n s e r v a t i o n of wave a c t i o n , we o b t a i n S_ 3 constrain A g a i n , we take the Raman i n s t a b i l i t y illustration. where $ . i s effectively must be t e m p o r a l . e x c i t e d waves. for i s a good a p - a moderately long plasma, and reasonable m i r - reflectivities. instability The situation, region by the Containment of the e x c i t e d equivalent to p e r i o d i c boundary light conditions the system. It may be p o s s i b l e to g e n e r a l i z e these r e s u l t s for a homogeneous plasma, to the case of a plasma with d e n s i t y . iv. 39 gradients. Normally, stimulated scattering is inhibited by the presence of a d e n s i t y g r a d i e n t due to the f a c t that e x c i t e d waves of a p a r t i c u l a r wavelength the region where they are resonant f o r Looked at in another way, the e x c i t e d number i s a f u n c t i o n of p o s i t i o n e f f e c t of propagation prevents wave (Chen, In propagate out further the of interaction. frequency and wave- i n the plasma, thus significant the growth of any one 1974). the o p t i c a l cavity problem, we have seen that l i g h t waves can dominate the behaviour plasma with d e n s i t y of the system. the In a g r a d i e n t s , 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 resonant f o r parametric i n t e r a c t i o n many t i m e s . density gradient s c a l e length l o s e s i t s The t h r e s h o l d c o n d i t i o n s all have simple p h y s i c a l ance. The pump provides the i n p u t , Under instability thresholds. in v a r i o u s explanations the significance. these c i r c u m s t a n c e s , one again expects a b s o l u t e to set i n near the homogeneous, temporal Thus is regimes of interest in terms of energy dissipative bal- l o s s e s due to the damping of the l i g h t wave and plasma wave, and convect i o n of energy out of the plasma due to propagation of l i g h t and plasma waves provide balanced. Q I, and v a r i o u s p and R. the l o s s e s which must be counter These e f f e c t s are c h a r a c t e r i z e d r e s p e c t i v e l y Y , y_» Y > In dissipative, the by combinations of the parameters v_, Vp, the temporal problem, the l o s s mechanisms are thus YQ = Y_Yp (3.21) 40 In the s p a t i a l problem, the main f a c t o r s to be accounted f o r are the damping of the plasma wave and the f a c t that excited plasma and l i g h t as they grow. Y For an extremely = waves propagate away from each other %(v_/v )* Y p 0 short the (3.22) p plasma w i t h i n an o p t i c a l cavity, the dominant e f f e c t s are c o n v e c t i o n of the plasma wave and damping of the l i g h t Finally, wave. f o r a plasma of more reasonable length i n s i d e an o p t i c a l cavity, the l o s s mechanisms are damping of the plasma wave, damping of hence, the t h r e s h o l d < (L >> 1) the l i g h t wave, and m i r r o r losses; is • *-* + P T P x l n * < - > 3 24 Chapter 4 SUMMARY AND CONCLUSIONS It has been shown that a plasma i s c o n s i d e r a b l y unstable to s t i m u l a t e d s c a t t e r i n g i n s i d e an o p t i c a l c a v i t y . wave amplitudes relevant indicate Solutions that Since the temporal It this thresholds result are g e n e r a l l y thresholds would be rash to suggest the t h r e s h o l d s in an absolute sense. assumptions and r e s t r i c t i o n s may be d i f f i c u l t or in the y - z of are In a d d i t i o n , to t h i s beam i s c o n v e r g i n g , to achieve In a d d i t i o n , model, which not accounted 41 for. of plasma. apply to schemes as i s q u i t e high power d e n s i t i e s Periso- to the d i r e c t i o n stimulated side s c a t t e r i n g to the pump wave i s are many in p r a c t i c e . Such a one dimensional model cannot s t r i c t l y practice, accurate these are the assumptions of plane ( p e r p e n d i c u l a r values and Landau there propagation of the waves) and homogeneity of the where the l a s e r orders the numerical i m p o s s i b l e to achieve haps the most s e r i o u s small. significant. that specific are absolute and growth rates crude. varying several for placed especially Estimates of c o l l i s i o n a l damping are n e c e s s a r i l y is thresholds l o s s e s are may be q u i t e calculated for the slowly situations, long or the c a v i t y of magnitude below the s p a t i a l instability, for the temporal to many such experimental where the plasma i s tropy processes when i t more common in in the f o c a l at a r b i t r a r y spot. angles 42 Frequently, extremely short experiments pulse. It i n v o l v e l a s e r s with an might be noted t h a t , the above c a v i t y analysis trip time f o r a photon very much s m a l l e r than transit to be v a l i d , for one must have the time f o r which the l a s e r pulse e x i s t s . that f o r significant growth, ing f o r many e - f o l d i n g The t h r e s h o l d linear exists, so that intensity then these other are s l i g h t l y local levels further theoretically focussing) stimulated higher. If in scattering, exceed the average being s t u d i e d . values, plasma, This creates i m p o s s i b l e to handle and may dominate the experimental theory may thus be q u i t e non- beam f i 1 a m e n t a t i o n which are awkward or While the exact results. numbers which come out of such a unreliable, the trends and b a s i c c o n c l u s i o n s concerning the e f f e c t of the o p t i c a l still exist- lie processes may modify the to the e f f e c t difficulties obvious f o r many other self flux densities nonlinear or couple d i r e c t l y the times of the system. the same range as those f o r though g e n e r a l l y is also one must have the pulse processes in a plasma ( e . g . , roughly It round cavity should stand. The experimental observation plasma phenomena i s o b v i o u s l y the f i e l d . The o p t i c a l cavity crucial and study of nonlinear to our understanding may well facilitate of such work. BIBLIOGRAPHY Chen, F . F . , Laser I n t e r a c t i o n and Related Plasma Phenomena, V o l . 3A, H. Schwarz and H. Hora Teds.), Plenum, New York, 1974, pp. 291-313. DuBois, D . , D.W. Forslund and E.A. L e t t . , 33 (1 973) , p. 1013. Williams, F o r s l u n d , D.W. , J . M . Kindel and E . L . F l u i d s , 18 (1975), p. 1002. Fuchs, V . , Phys. Fluids, (to be Kroll , N.M., J . Appl. Lashmore-Davies, C.N., Phys., published). Whitbourn, Principles 36 (1 965), p. Plasma P h y s . , J. V.N., Soviet Meyer, J . and G.G. p. 1091. Physics A l b a c h , Phys. Meyer, J . and B. p. 2187. Stansfield, Shohet, Appl. J.L., J. JETP, 34, sec.IIIB2. 17 (1975), Phys., Can. J . A, p. 281. T s y t o v i c h , V . , Nonlinear E f f e c t s New York, 1970, p. 32. North- p. 374. 13 (1976), Phys., 41 (1970), T a i t , G . D . , L . B . Whitbourn and L . C . L e t t . , 46A (1 973), p. 239 . 43 29 (1969), Rev. Appl. of Plasma L o c h t e - H o l t g r e v e n , W. (ed.), Plasma D i a g n o s t i c s , H o l l a n d , 1968, p. 512-606 and r e f . t h e r e i n . Lugovoi, Rev. Lindman, Phys. Heckenberg, N . R . , G.D. T a i t and L . B . P h y s . , 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 h y s i c s , M c G r a w - H i l l , 1973. Phys. 49 (1971), p. 2610. Robinson, Phys. in Plasmas, Plenum,
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Stimulated scattering in a plasma filling an optical cavity Myra, James Richard 1976
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Title | Stimulated scattering in a plasma filling an optical cavity |
Creator |
Myra, James Richard |
Date Issued | 1976 |
Description | Stimulated scattering processes in a homogeneous plasma inside an optical cavity are studied theoretically. In particular, attention is focussed on the Raman and Brillouin instabilities. The coupled equations for the wave amplitudes are solved subject to optical cavity boundary conditions and it is shown that for a wide range of plasma lengths and cavity mirror reflectivities, the threshold values for instability are approximately those for the temporal problem (waves uniform in space, growth or decay in time). Sample results are calculated numerically for the Raman instability in a typical laboratory plasma. The threshold values of incident laser intensity are easily exceeded with presently available, high powered (200 MW) CO₂ lasers. Finally, the physical significance of these results is discussed, and a generalization to the case of an inhomogeneous plasma is suggested. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-02-18 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085751 |
URI | http://hdl.handle.net/2429/20450 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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