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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.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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