UBC Theses and Dissertations

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UBC Theses and Dissertations

Investigation of optical mixing Albach, Gary George 1975

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INVESTIGATION OPTICAL  OF  MIXING  by GARY  S c . ,  GEORGE  ALBACH  B . S c , U n i v e r s i t y o f Waterloo, 1970 The U n i v e r s i t y of B r i t i s h Columbia,  A  THESIS THE  SUBMITTED  IN  REQUIREMENTS DOCTOR  in  PARTIAL FOR  THE  1972  FULFILMENT DEGREE  OF  PHILOSOPHY  t h e  Department  OF  OF  of PHYSICS  We  accept  required  THE  this  t h e s i s  as  conforming  to  standard  UNIVERSITY  OF  June,  BRITISH  1975  COLUMBIA  the  In  presenting  this  thesis  an advanced degree at the L i b r a r y s h a l l I f u r t h e r agree for scholarly by h i s of  this  written  make  it  It  Columbia,  gain s h a l l  permission.  Depa rtment Columbia  I agree  r e f e r e n c e and this  that  not  copying or  for  that  study. thesis  by the Head of my Department  is understood  for financial  for  the requirements  f o r e x t e n s i v e copying o f  purposes may be granted  The U n i v e r s i t y o f B r i t i s h  British  freely available  that permission  2075 Wesbrook Place Vancouver, Canada V6T 1W5  fulfilment of  the U n i v e r s i t y of  representatives. thesis  in p a r t i a l  or  publication  be allowed without my  ABSTRACT  The in  a  plasma  l i g h t are  and  shown  to  with  i n c o h e r e n t observed  by  the  gases  the  mixed  beams  a  the  power.  of  the  of  l a r g e  of  the  100  the  d e n s i t y  d e n s i t y  a  the  simple  t h r e s h o l d  for  i n c r e a s i n g  gas  of  l a s e r  on  the  i s  been  s c a t t e r e d  by  s c a t t e r i n g As  medium.  power  i n d i c a t e d  model.  and  of  in  For the  i n v e s t i g a t e d  d e n s i t y  p r e s s u r e  power  With  has  v a l u e .  the  been  o b s e r v a t i o n  beams. beam  that  t h e o r e t i c a l  magnitude  of  e l e c t r o m a g n e t i c  in  amplitude  means  enhanced  mixed  beams  modulations  t h i r d  R a y l e i g h  has  by  mixed  times  wave  pressure  i n c r e a s i n g  a  S i m i l a r  normal  l a s e r  d e n s i t y two  from  over  the  gas  to  that  both  d i o x i d e .  f i e l d  ruby  i n v e s t i g a t e d  c o h e r e n t l y  over  compared  with  been  d i s t r i b u t i o n ,  on  A  i n t e n s e  media,  s t a t i o n a r y  and  with  modulations carbon  in  gases  i n d i c a t e  s a t u r a t e s  both  dependence  r e s u l t s  r e s u l t s  In  two  f l u c t u a t i o n s .  wavevector produce  has  i n t e n s i t i e s  in  waves  mixing  gas  s c a t t e r e d  thermal  is  of  a  e x i s t  l i g h t  observed  l a s e r  in  s c a t t e r i n g .  plasma,  the  e f f e c t  the  a l s o  and  The modulations  i n c r e a s i n g d e n s i t y noted  in  TABLE OF CONTENTS  Page  ABSTRACT LIST TABLE  OF OF  . FIGURES. SELECTED  .  .  .  .  .  .  .  .  .  .  .  i i  .  .  .  .  .  .  .  .  .  .  .  v i  NOTATION  v i i i  ACKNOWLEDGMENTS  x i  Chapter 1  INTRODUCTION  2  PRINCIPLES  OF  1 OPTICAL  2.1  I n t r o d u c t i o n  2.2  S i n g l e  MIXING.  .  .  .  .  .  E l e c t r o n  V a l i d i t y 2.3.1  2 . 3 . 3 Vlasov  i n  a  Ideal  .  .  6  Wave .  .  .  .  .  C o n s i d e r a t i o n .  .  7  .  10  P r o c e d u r e s . . Breakdown of  .  .  .  .  .  .  .  .  .  10  L i n e a r i z a t i o n  .  .  .  .  .  .  .  .  .  12  E f f e c t s  Plasma  i n  of a  C o l l i s i o n s  Wave  .  .  .  .  .  14  M i x i n g  F i e l d 2.5  .  L i n e a r i z a t i o n  2 . 3 . 2  2.4  .  6  M i x i n g F i e l d 2.3  .  15 Gas  i n  a  F i e l d  Wave  M i x i n g 19  i i i  Chapter  3  Page  THE  PLASMA  EXPERIMENTS  24  3.1  I n t r o d u c t i o n .  .  3.2  F e a s i b i l i t y of F l u c t u a t i o n s  4  5  THE  .  24  Measuring  Induced 25  3.2.1  Method  3 . 2 . 2  Stray  3 . 2 . 3  S c a t t e r i n g From F l u c t u a t i o n s  3 . 2 . 4 3.3  .  of  .  .  .  .  L i g h t  Enhanced  Experimental  D e t e c t i o n  27 Thermal  S c a t t e r i n g  Apparatus  3.3.1  The  Plasma:  3 . 3 . 2  The  O p t i c a l  3 . 3 . 3  Second  25  28 .  .  .  .  30 36 36  System  37  Harmonic  G e n e r a t i o n  40  3 . 3 . 4  The  41  3 . 3 . 5  Timing  Ruby  Laser  Sequence  45  3.4  R e s u l t s  48  3.5  D i s c u s s i o n  51  3.6  Other  53  GAS  Experiments  EXPERIMENTS  55  4.1  I n t r o d u c t i o n  4.2  Experimental  4 . 3  R e s u l t s  58  4.4  D i s c u s s i o n  64  CONCLUSION  .  55 Procedure  55  74  i v  Page  REFERENCES  77  APPENDICES I II III  D e t a i l s Proof  of  of  the  the  Plasma  Ergodic  C a l c u l a t i o n of F l u c t u a t i o n s  the  80 Theorem.  Induced  .  99  D e n s i t y 104  v  LIST OF FIGURES  F i g u r e  1  2  Page  R e l a t i v e number d e n s i t y of trapped p a r t i c l e s v . s . 8 M i x i n g  f o r c e ,  dens i ty 3  v . s .  free  p o t e n t i a l ,  and  v e l o c i t y  diagrams  s c a t t e r i n g  geometries  ,  f o r  mixing  .  .  .  .  .  .  .  .  .  .  21  .  22  s  p o s i t i o n .  Wavevector  .  .  .  and 26  4  Spectrum  of  thermal  f l u c t u a t i o n s  .  .  .  .  .  .  .  .  .  31  5  Spectrum  of  induced  f l u c t u a t i o n s  .  .  .  .  .  .  .  .  .  32  6  O p t i c a l  system  f o r  mixing  7  O p t i c a l  system  for  measuring  e x p e r i m e n t s .  .  l a s e r  l i n e w i d t h .  .  8  F a b r y - P e r o t  9  E l e c t r i c a l  p a t t e r n  of  l a s e r  o u t p u t .  .  .  .  .  ,  44  sequence  11  Enhanced  12  mixing beams' power. . . . . . . O s c i l l o s c o p e t r a c e of s c a t t e r e d and  47  s c a t t e r i n g  l a s e r  Enhancement  v . s .  43  46  Timing  13  ,  .  system  10  pulse  38  from  monitors p r e s s u r e  vi  plasma  .  . of  .  v . s .  .  . . . l i g h t .  .  .  hydrogen.  .  .  .  .  .  .  49  .  .  .  .  .  .  56  .  .  .  59  .  .  .  Enhancement  v . s .  p r e s s u r e  of  n i t r o g e n .  Enhancement  v . s .  p r e s s u r e  of  argon  Enhancement  v . s .  P r e s s u r e  of  carbon  Normalized  enhancement  Enhancement T o r r  C0  v . s .  Enhancement  v . s . H2  T o r r  .  .  .  d i o x i d e  p r e s s u r e  ,  mixing  power  f o r  100  mixing  power  f o r  50  .  .  .  Plasma  v e s s e l  Plasma  c u r r e n t  Plasma  d i s c h a r g e  Streak  photograph  H o r i z o n t a l  Streak  .  .  2  and  25  v . s .  .  assembly  .  .  .  .  .  p u l s e  .  .  c i r c u i t  .  of  A r c :  of  a r c :  .  .  .  .  .  .  .  .  .  .  .  .  ,  .  .  s l i t  photograph  V e r t i c a l  .  s l i t  P r o f i l e  of  H„  l i n e  at  t  =  200  ysec  .  .  .  .  .  .  P  O p t i c a l  system  f o r  d i a g n o s t i c  s c a t t e r i n g  Measured f i t t e d  Spectrum anomaly  spectrum  to  of  acomputed  of at  s c a t t e r e d p r o f i l e .  s c a t t e r e d  l i g h t  u)n  vi i  l i g h t .  .  .  showing  .  .  TABLE OF SELECTED NOTATION  a  p a r t i c l e  e  charge  f  d i s t r i b u t i o n  f  0  f  v e l o c i t y  on  an  unperturbed  perturbed  (taken  f u n c t i o n  i n  d i s t r i b u t i o n  d i s t r i b u t i o n  a s y m p t o t i c  /=T  k  wavevector  m  mass  n  p a r t i c l e  p  p o l a r i z a b i 1 i t y  an  p o s i t i o n  t  time  v  x , y , z  as  d e n s i t y  of  a  f u n c t i o n  medium  v a r i a b l e  v e l o c i t y  p o s i t i o n  space  f u n c t i o n  ( e l e c t r o n s  v e c t o r  a  phase  i n  a  time  e l e c t r o n  ->-  r  p o s i t i v e )  l i m i t  i  of  e l e c t r o n  c o o r d i n a t e s v i i i  o r  m o l e c u l e s )  cn  i  m  j  u  D ( K )  =  EC<»4>)  e l l i p t i c  i n t e g r a l  of  the  second  F ( K , < J > )  e l l i p t i c  i n t e g r a l  of  the  f i r s t  F  plasma  . e, 1  ( 1 / K  du  ) [ K - E ]  2  d i s p e r s i o n  I  i n t e n s i t y  K ( K )  complete  M  mass  N  number  P  power  of  of  a  of  wave  i n t e g r a l  of  the  f i r s t  kind  ion  p a r t i c l e s  S(k,u))  Thompson  T  c h a r a c t e r i s t i c  U  p o t e n t i a l  V  kind  f u n c t i o n  e l l i p t i c  an  kind  s c a t t e r i n g  form  time  f a c t o r  s c a l e  energy  volume  W  t o t a l  a  s c a t t e r i n g  r a t i o  energy  of  k i n e t i c  or  sum  of  l i n e w i d t h s  of  mixing  beams  parameter  p o t e n t i a l energy  of  energy a  of  p a r t i c l e  l x  a  w e l l  to  the  thermal  6  D i r a c  e  p e r m i t t i v i t y  n  index  9  K  ^  X  y  v  d e l t a - f u n c t i o n  of  of  a  r e f r a c t i o n  medium  or  p o s i t i o n  m i x i n g a n g l e  v e l o c i t y  t r a n s f o r m  wavelength  p e r m e a b i l i t y  of  a  medium  frequency  £  p o s i t i o n  a  c o l l i s i o n  to  a n g u l a r  t r a n s f o r m  c r o s s - s e c t i o n  frequency  x  t r a n s f o r m  ACKNOWLEDGMENTS  I for  h i s  work. Dr.  help In  R.  Mark  would  s t a f f ,  the  extended  to  whose  a l s o  past  S i e b e r g a l l  the  B a l d i s .  Dave An  to and  and  through  the  and  to  course  s u g g e s t i o n s  leave  a r e  Dr.  J .  of  Meyer  t h i s  given  by  g r a t e f u l l y  acknowledge  the  p r e s e n t  p a r t i c u l a r l y  J i m  members and  Camm,  Jon  and  work  Aazam-Zanganeh. of  help  e x c e l l e n t  the  were  UBC  P r e s t o n ,  t y p i n g  the  Warm  Plasma  i n v a l u a b l e ,  of  thanks  P h y s i c s i n  Rob  M o r r i s  j o b  has  been  the  N a t i o n a l  p a r t i c u l a r and  done  by  H a l l e r . F i n a n c i a l and  UBC  This Energy  thanks  Meyer's  l i k e  f r i e n d s h i p  C h u c h l a n d ,  Council  s i n c e r e  support  Dr.  both  Doug  Sharon  encouragement  d u r i n g  Bosma,  Hector  my  a t e d .  t e c h n i c a l  group  express  a d d i t i o n ,  I  are  to  and  Nodwell  appreci  Jack  wish  C o n t r o l  a s s i s t a n c e  f e l l o w s h i p s work Board  i s of  from  has  supported Canada.  xi  been by  g r a t e f u l l y a  grant  from  Research r e c e i v e d . the  Atomic  Chapter  1  INTRODUCTION  In i n i t i a t e d in  the  recent  f u s i o n  f i e l d  s t a n d i n g s  of  a r e  of the  l a r g e  r a d i a t i o n  years  i n  f o r c e s  t h i s  e f f e c t  l e d  t h a t i s  i n c i d e n t  l a s e r may  on  been  which  beams  a  [ 1 , 2 , 4 - 1 0 ] .  One  formed  i t  [3]:  f r e q u e n c i e s , d e n s i t y  experimental  when  mixed  f l u c t u a t i o n  c o i n c i d e s  with  A  e f f e c t  s i m i l a r  e x c i t e  showed  the  that i n  may  be  on  a  n o n l i n e a r  and  extended.  beams  l a s e r  plasma, i f  the  plasma  expected  1  i f  a  by  a  s i t u a t i o n  been  of  i n  per-  d i f f e r e n t  enhance  d i f f e r e n c e  resonance  number  resonance  has  beams  the  the  t h e o r e t i c a l  plasma  can  plasma,  For  of  the  n a t u r a l  two a  i n  caused  v e r i f i c a t i o n  i n t e n s i t y e l e c t r o n  intense  s u b j e c t a t  under-  when  the aimed  a c t i v i t y  Better  e l e c t r o n s .  p a r t i c u l a r l y  mixed  be  l a s e r  i n c r e a s e d  the  i n v e s t i g a t i o n s , the  produce  developed  d e n s i t y  has  to  occur  i n t e n s e  a c t i n g  to  i n t e r a c t i o n s .  being  two  p e r t u r b a t i o n s  l o n g i t u d i n a l of  p r o c e s s e s  mixing  quest  has  l a s e r - p l a s m a  c o n t i n u a l l y By  the  r e a c t i o n s  e l e c t r o m a g n e t i c medium  years  the frequency  f r e q u e n c y .  d i f f e r e n c e  frequency  2  c o i n c i d e s here  with  shows  mixed  t h a t  l a s e r  d e n s i t y  the  even  beams  takes  ion  resonance.  for  a  zero  strong  place  and  a  The  work  frequency  be  d e s c r i b e d  d i f f e r e n c e  m o d i f i c a t i o n s t a t i o n a r y  to  of  the  d e n s i t y  of  the  e q u i l i b r i u m  wave  i s  set  up. From i t  was  r e c o g n i z e d  a l s o  be  t h i s  case  If  the  induced the  in  gases  ments on  to  the  l i g h t be  which  of  the  with of  can  gases  under  mixing  in  f l u c t u a t i o n s  by  s i m i l a r  study  are  a  plasma  c o u l d  means.  bound  In  to  n u c l e i i .  t i m e - i n d e p e n d e n t ,  whole  d i s p l a c e m e n t s ,  d e s p i t e  on  s t i m u l a t e d  c o n c e n t r a t e d a  s i n g l e two  of  gas,  the  l a s e r  intense or  d e n s i t y  i n f l u e n c e  on  R a y l e i g h a n a l y s i n g  beam. l a s e r  In beams  at  180°  g r a d i e n t  i s  shown  s c a t t e r i n g  the the  plasma,  l i g h t  s c a t t e r i n g backe x p e r i -  are to to  i n c i d e n t one be  set  p r o p e r t i e s  medium.  two free  several  ments,  from  p e r i o d i c  The which  d e n s i t y  s u i t a b l y  work  has  volume  A  up  o p t i c a l  a p p r e c i a b l e  p r e s e n t e d ,  same  another.  of  masses.  [ 2 2 , 2 3 ]  s c a t t e r e d  i s  achieve  Previous in  l a r g e  neutral  f o r c e  can  l a r g e  that  e l e c t r o n s  mixing  m o l e c u l e s t h e i r  c o n s i d e r a t i o n s  experimental  a n t i p a r a l l e l or  bound  d e s c r i b e d  e l e c t r o m a g n e t i c  e l e c t r o n s ,  important  i n c i d e n t  c o n d i t i o n  i s  s i t u a t i o n s .  s p e c u l a r l y  seen  waves to  be  here,  in  i n t e r a c t an  exact  In  l a s e r  f u s i o n  r e f l e c t e d  l a s e r  waves  analogy  e x p e r i mix  in  3  the  atmosphere  t i o n s The  range  from  gas  of  o c c u r s  i o n i z e d ,  the  As  three  in  a  and  for  equation  and  the  [28]  i s  f o l l o w e d .  the  d e t a i l s .  gas.  t h a t  induced  i n f l u e n c e  the  l i g h t .  s i m i l a r  A of  f a r  back-  high  from  may  f l u c t u a t i o n s r e s u l t s reader The  i s  of  the  s i t u a -  power  being  f u l l y  i n f l u e n c e  the  p r o p e r t i e s  other  for  to  of  of  used  to  deal  gas.  For  w a v e - p a r t i c l e aim  i s  by  the  mixed  to  to  the  two  mixing  case  of  For  to  of  the a  l a s e r  o p t i c a l  are f o r  a  Vlasov  gas  the [17]  c a l c u l a t e  l i t e r a t u r e  d e s c r i b e  models  with  c a l c u l a t i o n s the  Chapter  i n t e r a c t i o n s  the  plasma  plasma  B a s i c a l l y ,  a p p l i c a b l e .  induced  s p l i t  the  the  l i n e a r i z a t i o n "  be  cases  the  o u t l i n e s  one a  been  t h e o r y ,  p r o c e s s .  used;  "Landau  has  r e s p e c t i v e l y .  M i x i n g , "  r e f e r r e d  model  t e x t  experiments  are  both  the  with  mixing  shown  In  and  gas  s o - c a l l e d  is  the  c o n d i -  n e u t r a l  s c a t t e r i n g  of  d e a l i n g  o p t i c a l  theory  Only  where  i n d i c a t e  c a v i t i e s  l i g h t  O p t i c a l  " q u a s i 1 i n e a r "  d e n s i t y  4  to  f l u c t u a t i o n s  and  approaches  the  and  t y p i c a l l y  p o r t i o n  the of  the  plasma  plasma  3  l a s e r  i s  d e n s i t y  major  " P r i n c i p l e s  d i f f e r e n t  plasma  resonant  gas  c h a p t e r s ,  experiments  formed  the  [ 2 4 - 2 7 ] ,  medium. The  into  t a r g e t  s t r o n g l y  i n c i d e n t  d i s c h a r g e  l a s i n g  a  i o n i z e d  may  the  induced  e l e c t r i c a l  of  Chapters  i n s i d e  l a s e r s .  the  of  modulations  s c a t t e r i n g t i o n  f r o n t h i g h l y  o b s e r v a t i o n s  d e n s i t y  2,  in  the  beams. given the  mixing  in  a  4  gas is  has  been  presented  S e c t i o n  2.4  developed  almost  here  h i s  and  with  Appendix  Chapter t h a t  were  in  plasma.  a  performed  apparatus  of  such  are  in  t h a t The  in  a  to  that  w i t h  work  gas  are of  in  to  and  a  presented 3  to  the  The  f i n a l  and  b r i e f l y  c a l l y  o p t i c a l  and  i n c l u d e d t h a t  the  o p e r a t i o n  are  were  II  c a l c u l a t i o n s In b a r s ,  and of the  in  III  to  plasma  Meyer  and  i n c l u d e s  are  the  source  in  the  a n a l y s e  model  of  the a  provide  a  are  2.  d e n s i t y i s  s i m i l a r  d i s c u s s e d  of  of  the  the  major  c o n s t r u c t e d  s p e c i f i -  d e t a i l s  Appendix  I.  of  Also  d i a g n o s t i c  plasma. for  i t s  t e s t s  F i n a l l y , some  of  the  2.  diagrams  of  experimental  n o t e d ,  r e f e r  to  wave  model.  r e s u l t s  d e t a i l s  Chapter  summary  was  a  format  r e s u l t s of  by  induce The  and  experiments  f o l l o w e d  to  wave  e x p e r i m e n t a l  the  v a r i o u s the  d e n s i t y  procedure  e x p e r i m e n t s ,  given of  a  the  4.  the  experiments  the  of  p r o v i d e s  mixing  when  then  the  Chapter  the  to  s e c t i o n ,  reviews  Chapter  except  and  p r e d i c t i o n s  to  J .  work  d e t e c t  given  again  r e s u l t s  performed  Appendices  e r r o r  the  d e t a i l  performed  c h a p t e r  the  Dr.  This  r e s u l t s  them  and  As  design  The  r e l a t e s  Chapter  i s  p r o j e c t  separate  e x p e r i m e n t s . for  induce  experiments  r e f e r e n c e  done  d e s c r i b e s  a  by  III.  o u t l i n e d .  presented  d i s c u s s i o n  c o n s e n t .  C o n s i d e r a t i o n  f e a s i b i l i t y  are  3  e n t i r e l y  r e s u l t s , the  root  the  mean  5  square  of  d i c t i o n s  the or  l e a s t  and  d e s c r i b e d  are  shown  as  d e v i a t i o n s  from  squares  a c c o r d i n g l y . s o l i d  l i n e s  f i t s  the are  mean. drawn  Trends i n t e r r u p t e d  in  T h e o r e t i c a l as  s o l i d  experimental at  the  data  p r e -  l i n e s r e s u l t s p o i n t s .  Chapter  2  PRINCIPLES OF OPTICAL MIXING  I n t r o d u c t i o n  2.1  Since the  o p t i c a l  i n t e n s i v e has  mixing  c o n c e n t r a t e d medium  c o n s i d e r e d other  the  media.  a l s o  in  The  e l e c t r o n second,  the  the  K r o l l , beams  Ron has  i n v e s t i g a t i o n . s p e c i f i c  case  O u t s i d e  of  m i x i n g  work  i s i s  purpose  of  and  Rostoker  been  the  Much  of  of  plasma  t h i s ,  a few  i n t e n s e  p r e s e n t e d  equation f i e l d  then  At of  of  made  to  a  f o r  [1],  s u b j e c t t h i s as  the  have  beams  n e u t r a l  of  e f f o r t  people  l a s e r  h e r e ,  motion two  part  nucleus  c h a p t e r an  f i r s t  of in  i s  to  in  gases  p r e s e n t  u n d e r s t a n d i n g a  of  l i g h t  s p e c i f i c  c o n s i d e r e d  bound  t h i s  needed  p r o c e s s .  mixing  problem  l a s e r  [2-10].  models  mixing  the  on  of  c o n s i d e r e d .  t h e o r e t i c a l  t a k e n .  of  problem In  The  l i g h t  work  t h e o r e t i c a l  n o n l i n e a r  are  the  general a  i s  " t e s t " found.  f o r  two  a  f u l l y  i o n i z e d  an  ideal  gas.  6  the  approach  s i n g l e  beams  of  the  media.  i s e l e c t r o n The  F i r s t ,  In  plasma both  the and  7  cases  the  problem  is  c o n f i n e d  caused  by  the  v a r i a t i o n s  Only of  a  gas i s  plasma, in  a  wave  2.2  has  not  S i n g l e  m  bound  by  r e s t o r i n g to  chromatic  as  be  to  a  f i e l d  a  a  a  is  given  well  d i f f e r e n t as  i t s  the  A l l In  i s  Ze  of and  presence  waves  the  case For  used. to  a It  t h i s  F i e l d  charge  -e  mass  at  r e l a t i v i s t i c  the  e l e c t r o m a g n e t i c  for  l i t e r a t u r e .  e l e c t r o n  charge  -mw0x.  d e n s i t y  documented. model  Mixing  s i n g l e of  the  a p p l i c a t i o n  in  Wave  n e g l i g i b l e .  plane  i s  appeared  nucleus  f o r c e  o u t l i n e  d e t a i l  in  c a l c u l a t i n g m i x i n g .  a n a l y s i s  yet  c o n s i d e r  mass  assumed  more  E l e c t r o n  We  a  the  mixing i n  l i g h t  b r i e f e s t  s i n c e  presented  problem  the  to  i t s  M  e f f e c t s  of  two  and x  =  0  are  mono-  equation  of  motion  becomes:  To Far is  from given  f i r s t  resonance  order such  we t h a t  n e g l e c t (w  2 0  -  the w) 2  term >>  2yw  v  x  B.  the  s o l u t i o n  by:  E . c o s ( (JO . t  -  k .  •  r)  8  The  f i r s t  order  x  (  1  )  X  where  To  aiw)  second  v e l o c i t y  -  vT  -  V  1  -  o  I  i=  order  we  (t)  mx  1  the  K . y-p i  =  '(2)  ev  x I  0  recognized  .the  J  -e  x I  -  as  ?  £„.  gives  the  •  the  l ( t  This Meyer  and  2  e x p r e s s i o n  S t a n s f i e l d  s i n  is  [4]  2  ( w  s i n  2  t  seen for  p o l a r i z a b i 1 i t y ,  2  order  (a) j t  r)  2  (1)  r)  w2  2  r)  )  second  + a  B.(t)  a k" E  ->-  •  term  0 )  sin(a>.t  E.  - k.  sir\{u.t  1  c r o s s - p r o d u c t s  +  of  x  =  form  a^.E.  cal c u l a t e  + (2 '  Forming  the  is  =  mxx  where  has  an  -  to  t  2  be  •  force?  it,  sin  r)  r)  (2)  e q u i v a l e n t  unbound  (to2t  p a r t i c l e  to (w  that  0  =  0),  9  In the  the  d i f f e r e n c e  higher ment  f o l l o w i n g  frequency  frequency  of  e i t h e r  terms  a  free  a n a l y s i s  ( u i -  can  O J  not  only  )w i l l  2  cause  e l e c t r o n  o r ,  be  an  for  terms  i n v o l v i n g  r e t a i n e d  s i n c e  a p p r e c i a b l e a  g a s ,  the  d i s p l a c e -  whole  m o l e c u l e .  r e f e r e n c e  Using  t h i s  frame  moving  equation  (2)  where  =  k  can  ki In  produced  by  frequency frames  -  k2  a l l the  i s  be  s i m p l i f i c a t i o n w i t h  w r i t t e n  i s  the  these same  zero  wavevector  experiments  the  Under  This  t h i s  This  is  the  wave  mixing  f i n a l  molecule  r e s u l t  the  case  Despite to  - a  of  to  the  a wave,  the  the  two  means  w  and  c o n d i t i o n  induced mixing ~  x  beams the  co , 2  wave  wave. are  beat  r e f e r e n c e  equation  ( 3 )  becomes  -*  '  l  for  E  an  2  It  sin(k"  e l e c t r o n  •  r)  in  (4)  a  s t a t i o n a r y  f i e l d .  In m o l e c u l e .  =  E  v e l o c i t y  of  l a b o r a t o r y  -»•  mx  phase  t r a n s f o r m i n g  as  l a s e r .  and  c o i n c i d e .  the  and  move  a  i t s  of  a  gas  l a r g e  t h i s  mass,  s u b s t a n t i a l  f o r c e there  amount  a c t s  on  a  whole the  i s  time  f o r  s i n c e  t h i s  f o r c e  i s  10  t i m e - i n d e p e n d e n t .  As  v a r i a t i o n s  use  placement a  we  w i l l  of  f r e e  We  can  we  are  equation  e l e c t r o n s  (a)  now  in  a  to  s p a t i a l  c a l c u l a t e  plasma  and  d e n s i t y  the  d i s -  m o l e c u l e s  in  In  a  (b)  c o n s i d e r  a  =  In  f u l l y  i o n i z e d of  — mw^  a  the  where  gas  at  /  to  express  a  r\.  Using  the  a  =  the  two  =  the is  1  e  —  2.3.1  -  f r e q u e n c i e s .  and  co0  ~OJ.  For  a  gas  and  i t  i s  c o n v e n i e n t  3  u>L) terms  of  the  r e g i o n  the P  index  n.2  of  e/e >  =  index =  of  NaE,  r e f r a c t i o n  the  r e l a t i o n  r e f r a c t i o n t  n  e  0  in  pol ar i zabi 1 i  ty  by  =  ^ f ( n  2  -  1)  C o n s i d e r a t i o n s  L i n e a r i z a t i o n We  0  l j 2  p o l a r i z a t i o n  a  V a l i d i t y  =  2»  in  r ot  o p t i c a l given  w  —  2  P +  =  w0  resonance  1  (w0  e — Co  the  e l e c t r o n  o p t i c a l  e — m  and  ensemble  s p e c i f i c a l l y  plasma  w  2  S e c t i o n  more  media.  frequency  by  (4)  in  gas.  experimental  2.3  i n t e r e s t e d  must  2.2  i f  d e s c r i b e d  Procedures  now  c o n s i d e r  the  t e s t  by  a  the  consequences  p a r t i c l e  d i s t r i b u t i o n  i s  part  of  f u n c t i o n  imposed a  c l a s s i c a l  f ( r , v , t ) .  11  E s s e n t i a l l y  the  p a r t s ;  a  main  d i t i o n  a)  ~  v e l o c i t y  part •  t  and  d i s t r i b u t i o n  a  resonant  can  part  be  s p l i t  into  s a t i s f y i n g  the  two c o n -  v.  main  part  resonant  Since a  constant  in  in  the  resonant  r e l a t i v e  to  the  wave,  phase  p e r i o d i c  p o t e n t i a l  The d e s c r i b e d  p a r t i c l e s  by  w e l l s  a s s o c i a t e d  o n e - d i m e n s i o n a l the  Boltzmann  response  region  m a i n t a i n  t r a p p i n g  can  with  wave.  of  the  the  r e g i o n  occur  medium  i s  e q u a t i o n :  3f 3t  The  (a)  n o n l i n e a r  t h i r d  Hold  V  3x  term  3f/3v  +  a  3t  3v  can  be  c o n s t a n t .  c o l l  l i n e a r i z e d  By  p e r t u r b a t i o n s  from  expands  =  <<  3f •^7  f  0  +  f i  ( f i  in  two  c o n s i d e r i n g  amplitude f  .  fo)  small  e q u i l i b r i u m and  ways:  one  r e p l a c e s  3 f by  .  With  some  a p p r o p r i a t e  s i mpl i f i c t i on  1 2  of  c o l l i s i o n  equation  f o r  treatment  s i t u a t i o n  t h e  can  then  s o l v e  p e r t u r b a t i o n v a l i d  as  This  f i .  long  the  as  3f o av  Hold  " a "  have  a  2 . 3 . 2  one  remains  3fi 9v  (b)  term  constant  constant  Breakdown  It  i s  i n  terms  by  the  wave  to  a m p l i t u d e .  of  L i n e a r i z a t i o n  convenient of  c o n s i d e r i n g  at  t h i s  p o i n t  c h a r a c t e r i s t i c  to  time  a n a l y s e  s c a l e s .  the Important  are  2TT t r  2m  which  k  in  T0  =  relaxation in  the  Davidson the  main  r e g i o n , z a t i o n be the  part where  i n  v a l i d wave  (a) f o r i s  of  [11]  the  shows  f o r  t  > T  changing  for  t  an  oscillation electron  potential  trapped  well.  fluctuations  that  d i s t r i b u t i o n  growth  <<  time  a  the  of  medium  i t s  T0  is  period  T  8 f i / 8 v but  e v e n t u a l l y  t r and  f a s t e r  He  shows  u s e f u l than  as  s c a l e s t  i n  2  d e s t r o y s t h i s  when 3f/3v.  the  as  the  the  t  i n  resonant  l i n e a r i -  l i n e a r i z a t i o n a m p l i t u d e  of  to  However, procedure of  the  be  and  time  p o t e n t i a l  be  w e l l  o s c i l l a t i o n s . trapped  and  and  the  T  than  T  i s  problem  not  can  the  becomes  to  a s s u r e  over  many  exact  by  form  f.  The  important that  the  trapped  p a r t i c l e s  handled  second  f o r  >  t r u e ,  be  the  d i r e c t l y  c 0 n  t r  If  see  there  are  are  the  never  Landau  two  l i m i t i n g  cases  f o r  a n a l y s i s :  °  l i g h t  s o l v i n g  c o n s t a n t  t h i s  true  p r e s c r i b i n g  remains  T>  T  be  by  is  .  We  can  o p p o s i t e  l a s e r s ,  l a r g e r  the  1 i n e a r i z a t i o n  l i n e a r  f i e l d of  must  the  f o l l o w e d  e l e c t r i c  coherence here  can  i f  The  second  The  two  shown  to  mixing  d i a m e t e r ,  E  ~  10  9  t r  experimental  of  K  i n e q u a l i t y  s a t i s f y  beams  T  these  150  v o l t s / m .  MW  <  T  c o h  being  true  f o r  media  (plasma  l i m i t s .  For  e a c h ,  f o c u s s e d  a l l  and  ruby to  c a s e s .  gas) l a s e r  a  300y  14  i  aE  2  n " 2  10  2e  14.  ^  v o l t s  and T  The a  coherence  l a s e r  time  For 2  x  10  cm"  1 6  rad/sec  and  a  T0  ~  in  the  to  mixing  p e r t u r b a t i o n  . coh  a  ~  IO  IO"  sec  1 1  l i n e w i d t h  of  sec  T.  - 9  >>  .03  A  and  t h e r e f o r e  . tr  with  e l e c t r o n l/co  ~  next  T0  e l e c t r o n plasma  10"  c o n d i t i o n  t h e r e f o r e wave  has  plasma  the  3  The  a  T  ~  t r  i s  a p p r o x i m a t i o n  frequency  n  g  w  = ~ 8  x  10  1 2  s e c .  1 3  <<  T  s e c t i o n  f i e l d  d e n s i t y  i s  t r  the  f u l f i l l e d  response  c a l c u l a t e d i n . t h e  u s i n g  of  and a  the  plasma small  col 1 i s i o n ! e s s  Boltzmann  e q u a t i o n .  sound  a  t i o n  T0  to  gas  a  ~ >>  For  a  353  m/sec  T  and  t  atom  amplitude  2 . 3 . 3  gas  much  w i l l for  be the  and the  T0  ~  at 10"  9  a n a l y s i s  ( S e c t i o n  2.5)  i s  300°K  the  s e c .  This  for  a  done  t e s t  in  the  v e l o c i t y i s  of  the  c o n d i -  e l e c t r o n  bound  c o n s t a n t  a p p r o x i m a t i o n .  E f f e c t  Unless is  ( n i t r o g e n )  l o n g e r knocked  of  the  than out  d i s t r i b u t i o n  C o l l i  mean the of  s i o n s  time  between  o s c i l l a t i o n the  w e l l s  f u n c t i o n  w i l l  c o l l i s i o n s ,  p e r i o d  and  the  never  T^  ,  T  ] - | »  p a r t i c l e s  a s y m p t o t i c be  c o  r e a c h e d .  l i m i t  1 5  We  can  e l e c t r o n - i o n  e s t i m a t e  c o l l i s i o n s  v  where  the  e i  For I)  v  be  only  ei  .  ~  10 one  expected  the  to  order  at  300°K  and  the  T o r r  I  m  2 w  IKT)  frequency  f o r  o"  p  _  G  s  e  1  c  1.  used T  in ~  collthan  s i z e  gases  [  J  ~  l a r g e r  the 100  and  1  reduce  For  G  plasma  s e c "  1 0  me o  f a c t o r  c o l l i s i o n  3 / 2  [Ze2]  9  Gaunt  t y p i c a l  [12]  i  1 '3'  a  the  1 0 T^  of  used  s e c .  1 0  and  the  in  p r e s s u r e  experiments  (see  This  thus  i s  Appendix  seen  c o l l i s i o n s  to are  e f f e c t .  Chapter the  5  ( e . g .  c o l l i s i o n  n i t r o g e n )  frequency  between  molecules  v = ap  Thus  T  l i t t l e  2.4  ^  ~  IO  e f f e c t  Vlasov  - 9  on  two  i d e a l i z e d  which the  the  on  problem  K  mixing  a  in  (4)  a  c o n s i s t s  of  .  10*  s e c -  M i x i n g  S e c t i o n  be  [  1  3  ]  expected  to  have  We  f u l l y  F i e l d  2.2  e l e c t r o n  beams. a  *  p r o c e s s .  Wave  of  in  l  col 1 i s i o n s c a n  s i n g l e  mix  T  Trm  the  l a s e r  beams  6  and  Plasma  f o r c e  1  sec  Equation t u d i n a l  f  d e s c r i b e s  i n  the  the  mixing  f i e l d  c o n s i d e r  now  i o n i z e d ,  c o l l i s i o n l e s s  d e t e r m i n i n g  the  the  plasma  l o n g i of  s i t u a t i o n  response  i n  p l a s m a ; to  16  the  l o n g i t u d i n a l  c u l a t i n g two  the  e l e c t r o n  e l e c t r o m a g n e t i c The  the  can  be  The equation from 0  in  Only  found  treatment e v a l u a t e  to  and  v e l o c i t i e s .  This  of  the  d e n s i t y  Hagfor.s for t i o n  a  plasma f u n c t i o n  w i t h  c a l -  the  >  the  =  [16] a  the  i  by  i s  found  f i e l d  of  observes  the  coherent  [15]  one  can  to:  f o l l o w i n g  e q u a t i o n  f  to  d e n s i t y i  over  s p e c t r a l  the  has  a r i s i n g  a l l  i n t e r e s t .  S i n c e  i n d  the  harmonic  the  i n t e g r a t i n g  q u a n t i t y  < | n  as  d i s t r i b u t i o n  P o i s s o n ' s  e l e c t r i c  one  the  Vlasov  f i e l d s  e q u i l i b r i u m  Using  and  w r i t e  d e n s i t y  i n c o h e r e n t  the  spectrum  a s :  ( k , c o ) |  shown  M a x w e l l i a n  reduces  i n t o  f  found  of  [ 2 , 4 ] .  the  [14].  in  p r e s e n t e d  for  f l u c t u a t i o n s  2  i s  s o l u t i o n  a d d i t i v e  < | n ( k , w ) |  o u t l i n e  p e r t u r b a t i o n s  p a r t i c u l a r ,  are  by  e x p l a i n e d  the  are  i s  as  and  f l u c t u a t i o n s .  s c a t t e r i n g s  means  l i n e a r i z a t i o n  s e l f - c o n s i s t e n t  s e l f - c o n s i s t e n t  In  t h i s  induced  Landau  i n t r o d u c e d  With  the  n(k,w)  these  i s  any  B e r n s t e i n  the  the  equation  M a x w e l l i a n  t i m e ,  of  i s  b r i e f  f o r c e  with  be  f l u c t u a t i o n s  elsewhere  mixing  f l u c t u a t i o n s  of  a  s e p a r a t i o n .  assumed space  used  Boltzmann  together  charge  d e n s i t y  S p e c i f i c a l l y  waves.  s e c t i o n .  d e t a i l s  f o r c e s .  approach  col 1 i s i o n l e s s  previous  f  mixing  2  >  t h a t  +  < | n  the  e l e c t r o n  t  h  ( k ,to) |  thermal  v e l o c i t y  2  >  spectrum d i s t r i b u -  the  17  <  lnth(k»w)l  7TW ™  >  11 + aHF  + ZF^I'  e  :  (6)  f  where  F  e ^  x  ^  ^  =  ~  ""  +2  x  e1"  x e  dt  -  i i T  2  x  e  ^0  f o r  zero  magnetic  r e p l a c i n g  x  by  w h e r e  x  . £  f i e l d  .  beams  (Gaussian  f i e l d  amplitude)  f o r  plane as  wave  w e l l  l i n e w i d t h  the  of  o s c i l l a t o r s  Meyer  [2]  has  extended  the  that  the  spectrum  of  becomes:  found  with  f i n i t e  from  diameter  a p p r o x i m a t i o n  as  c l a s s i c a l  show  i s  t h i s  by  . 2£l  C o r r e c t i n g  non-zero  F.(x)  1  a  and  and  the  the  random  l a s e r  f o r  of the  temporal  e m i s s i o n  frequency  r e s u l t s  p r e v i o u s  the  induced  l a s e r  e l e c t r i c mode  (sum  p a r t i c u l a r of  the  behaviour  over  p r o b a b i l i t i e s ) , work  f l u c t u a t i o n s  [4]  to  18  F 1  Xn  =  Debye  Ii , 2 =  —  exp  sum  b  e f f e c t i v e  used to the  in  i s  Chapter  p r e d i c t  the  experiment.  of  the  3, s i z e  of  +  +  2  F . ) ZF.)  1  (a) -  co  s i n  expected  l i g h t  of  f i n a l  the  (6),  m i x i n g  mixing  beam  Comparison equation  a  ( F „ e  l i n e w i d t h s  S e c t i o n  f l u c t u a t i o n s , in  Z a  1  of  W =  This  2  (iW)'  power  frequency  =  a  +  1 0  +  5/2  2  t o  +  2  0  J  z  (7)  k;  l e n g t h  i n c i d e n t  =  ( 1 ) 1 0 , 2 0  +  w  "2T  where  (l  e  beams  of  mixing  beams  r a d i u s  r e s u l t 2  beams  where  induced  f o r  the  It  c a l c u l a t i o n s  are  f l u c t u a t i o n s  expected  is  made  to  the  to  p r e d i c t  the  s c a t t e r i n g  plasma.  e x p e r i m e n t .  s i z e  of  i s  done  the  enhancement  i n thermal  19  2.5  Ideal  Gas  As of  two  shown  (2).  can  f o r c e  reduces  cause  As problem  in  to  S e c t i o n  of  d e s c r i b e d  phase  f u n c t i o n d e n s i t y  i s  assumption  is  by  plasma  the  d e t a i l e d here  only  Rather  plasma,  the  i n i t i a l  M a x w e l l i a n ,  the  amplitude  be  the  the  f i e l d  given  low  of  by  frequency  m o l e c u l e s ,  t h i s  (4).  of  s e c t i o n ,  the  than  motion  gas  an  the  p e r t u r b a t i o n  the  to  the  l i n e a r i z e of  and  the  p a r t i c l e s  asymptotic  to  c r e a t e d  the  by  are  " f r e e . "  the  p o t e n t i a l  be  A  and  l a s e r - p a r t i c l e  A  thus  the  in  d i s t r i b u t i o n in  of  a  i s  w e l l  serves  i n t e r a c t i o n .  as  of  p a r t i c l e  model. to  r e s t as  By  f l u c u a -  p a r t i c l e s  of the  k i n e t i c  r e l a t i v e  III  be  p o t e n t i a l  the  the  Appendix  d e n s i t y the  d e f i n e d to  a  the  in  f u n c t i o n  p e r i o d i c w h i l e  6 ,  of  induced  f r a c t i o n  beams,  parameter, energy  r e s u l t s  of  "trapped"  mixing  appear  d i s t r i b u t i o n  c a l c u l a t e d .  c o n s i d e r e d  p a r t i c l e  only  p r e c e d i n g  c a l c u l a t i o n s  the  a  (4).  f o r c e  equation  response  in  found.  present  can  the  t h i s  c o n s i d e r i n g  t i o n s  in  c o n s i d e r e d From  a  that  d e s c r i b e d  the  e l e c t r o n  d i s p l a c e m e n t s  f i n d i n g  for  an  e x p e r i e n c e s  equation  d e r i v e d .  The we  by  done  space  the  F i e l d  2 . 2 ,  waves  the  t h a t  with  as  Mixing  a p p r e c i a b l e  c o n s i s t s  e q u a t i o n ,  and  Wave  Under  terms  in  a  e l e c t r o m a g n e t i c  equation  f o r c e  in  are  w e l l s the  p a r t i c l e s  r a t i o  of  energy  of  measure  of  the  Boltzmann  20  In t i o n s from  F i g u r e  i s  p l o t t e d  as  the  trapped  and  I I I - 1 2 b )  a r e  shown  apparent  f e a t u r e  1  the  amplitude  a  f u n c t i o n untrapped  together  i s  the  f a c t  always  c o n t r i b u t e  amplitude  of  induced  of  t h i s  The  f a c t  f o r c e  p o t e n t i a l are  a l l  mixing  U,  r e p r e s e n t a t i v e  shown  F i g u r e  f r e e  r e g i o n s  time  to  x  between  the  between  2  =  ir/k,  and  t h e  (equation  p a r t i c l e p l o t t e d  The  most  to  t h e  i n t e r p r e t a t i o n  help  4 ) ,  o f  the  v e l o c i t i e s  a g a i n s t  and  trapped  phases  p h y s i c a l  two  by  " f r e e "  i s  3ir/k, i n  i n  the  the  F i g u r e  2.  a s s o c i a t e d and  d i s t a n c e  d e n s i t i e s i n  have  The  f r e e  r e g i o n s  This  i s  the  i n  i s  the  lower  understood  w e l l s .  P a r t i c l e s  a l l  time  the  bottom  p a r t i c l e s they  at  x  =  number seen  to  of  f r e e  d e n s i t y  v e l o c i t y thus spend  i n  t h e  n/k of  p a r t i c l e s  be  l a r g e r  p a r t i c l e s  diagram.  of by  r e g i o n  minimum  s i n c e  drawn The  t h i s  w i l l  i n d i c a t e d  t r a j e c t o r i e s .  c o n t r i b u t i o n  a t  x - d i r e c t i o n  w i l l  r e c t a n g l e  p o t e n t i a l t h e i r  the  e t c . those  Consequently  shown  i n  e l s e w h e r e .  t r a j e c t o r i e s  hand,  moving  c o n s e q u e n t l y  d e n s i t y  diagram  The other  and  than  e l s e w h e r e . as  F  the  there  space  v a r i e s  2  p a r t i c l e  f o r c e  c o n t r i b u t e  than  i n  f r e e  with  I I I - 1 2 a  a m p l i t u d e .  o p p o s i t e The  p a r t i c l e  (equations  volume.  the  phase  s i n g l e  with  f l u c t u a -  c o n t r i b u t i o n s  t o t a l  the  understood  a  e x p e r i e n c e  more  e a s i l y  t h a t  induced  The  the  wave.  on  A  in  i s  3.  the  p a r t i c l e s  with  p a r t i c l e s  the  of  of  trapped  p a r t i c l e s ,  c o n s i d e r i n g with of  l i t t l e a  w e l l  on  the  the  p e r i o d i c  energy  spend  and  thus  v i r t u a l l y  c o n t r i b u t e  F i g u r e  1:  R e l a t i v e trapped  number  d e n s i t y  p a r t i c l e s  v . s .  of 3.  f r e e  and  gure  2:  Mixing v . s .  f o r c e ,  pos i  t i o n .  p o t e n t i a l ,  v e l o c i t y ,  and  densi  23  to  the  d e n s i t y  is  again  between the  c e n t r e of  of  a  an  Figure  a m p l i t u d e s .  s t r o n g l y  f r e e  p a r t i c l e s .  50%  of  and  i n d i c a t e s  p a r t i c l e s well  in  weakly  to  In  trapped  This  the i s  w i l l  done  i n we  and  model With  at  in  the  wave's =  3  i t  very  f a r  comments  on  Figure  r e t u r n  e v a l u a t i o n  in  more  o b s e r v a t i o n s  to  and the  l i g h t  in  than  seem  the  1  a  much  do  the  that  over  r e a l i s t i c the  a  trapped  p o t e n t i a l  p a r t i c l e s  w i l l  w e l l .  Those  i m p o r t a n t . are  have  the  model of  c o n t r i b u t e  of  of  1.  c o n c e r n i n g  f o r  bottom  measure  Figure  accounted  few  as  a b s o l u t e  not  r e a l i t y  the  i l l u -  from  p i c t u r e  of  only  is  the  at  of  would  This  maximum  d e n s i t y  a m p l i t u d e  1  has  a  t h i s  p a r t i c l e s  p o s i t i o n  an  to  space  of  p o i n t  appear  be  4  the  as  second  l i k e l y  Chapter  i s  found  t r a p p e d .  that  This  not  are  the  at  are  e x p e r i m e n t a l  D i s c u s s i o n ) c r i t i c a l  f a c t ,  number  c o n s i d e r e d  p a r t i c l e s  the  trapped  be  d e n s i t y  see  F u r t h e r a f t e r  to  s t r o n g l y . we  the  phase  r e p r e s e n t i n g  a m p l i t u d e s  the  that  too  be  curves  us  In  r e p r e s e n t s  s i t u a t i o n  p a r t i c l e s  mind,  a c t u a l l y  which  trapped  more  e t c .  c o n s i d e r i n g  should  b r i n g s  The  4rr/ k ,  t r a j e c t o r i e s .  2  The  This  the  by  The  p h y s i c a l  1.  2TI/ k ,  "eye"  w e l l .  in  s t r a t i n g  Figure  =  " t r a p p e d "  a  p a r t i c l e s  of  x  a n t i c i p a t e d two  bottom  at  l a s t  r e s e r v e d  been  r e p o r t e d .  s e c t i o n  p r e s e n t e d  u n t i l  here  e x p e r i m e n t a l  (4.4 f o r  a  r e s u l t s .  Chapter  3  THE PLASMA EXPERIMENTS  3.1  I n t r o d u c t i o n  The r e t i c a l  p r e v i o u s  p r i n c i p l e s  d i n a l  d e n s i t y  wave  l a s e r  beams.  Two  chosen  as  experiments Chapter  by  the  s p e c i f i c This  t h a t  with  were  i n  the  the  mixing  of  media,  plasma  and  c h a p t e r  gases  general  g e n e r a t i o n  o p t i c a l  c a r r i e d  n e u t r a l  b r i e f  f i r s t ,  o u t l i n e  f o l l o w e d  and  cussed  l i g h t  of  the  mention  i s  made  i n  F i n a l l y p o s s i b l e  o p t i c a l  by  apparatus  the  d e s c r i b e d  now out  a r e  of  two  a  l o n g i t u -  i n t e n s e  g a s ,  p r e s e n t s f o r  a  were  the  plasma.  p r e s e n t e d  theo-  e x p e r i m e n t a l The  s e p a r a t e l y  i n  4. A  given  i n v o l v e d  examples.  i n v e s t i g a t i o n s  c h a p t e r  plasma  k-spectrum  the  a  experimental  d e t a i l e d  system.  The  p r e d i c t i o n s  h e a t i n g of  of  of  s e v e r a l  e f f e c t s  induced  and wave.  24  f e a s i b i l i t y  d e s c r i p t i o n r e s u l t s of  the  a r e  of then  p r e c e d i n g  experiments p o s s i b l e  done  i s  the d i s c h a p t e r . to  harmonics  d e t e c t i n  25  3.2  Feasibility 3.2.1  of  Measuring  Method  of  Assuming a  plasma  (or  detected. from in  the  The  principle  method  scattering  to  a spectrum  induced  wave and  must  £i  light  scattering  In  all  beams of  wavevector  k  is  scattered  off  In to  wavevector  must  two  scattered  the  high  plasma  this (a) of  be  power  signal  is  in  be  of  light  [18], may  all  be  decomposed  directions,  relation  further  can  equivalent  technique  the  It = Iti - l t , 2  Also,  in  a  restriction  it. a n d it r e f e r i s  to  or  is  geometry  satisfy here,  a gas  A third,  one  it  in  scattering  This  beams.  mixing  presented  case  equivalent  with  3 shows  the  the  where  configurations  in  how  light  the  mixing  induced  to  that  the  inci-  light.  experiments mix  to  fluctuations  satisfy  the  be m e t ,  Figure scattering  to  be  fluctuations  experiment,  scattered  of  propagating  primarily  2  that  as  diagnostic  thermal  waves  can  modulations.  thermal  jt r e f e r  It = It. - It m u s t i s and  the  wave  arises  is  standard  of  where  dent  chosen  random  Whereas into  question  density  the  from  a density  the  periodic  Fluctuations  Detection  that  gas),  Induced  to  the  two  and  above  two  requirements.  antiparallel  induce  possible  ruby  a density  diagnostic  laser  wave  beam,  wave. t  i  the  = lt , L  i.e.  mixing  exactly mixing  180° beams  impossible.  the  beams  diagnostic  and  thus  backscattered. detection  of  the  beam  is  scattered  Unfortunately a small  back-  26  k  1 =  k  k =- k 2  L  k = 2k  F i g u r e  3:  Wavevector geometri  e s .  diagrams  for  L  L  mixing  and  s c a t t e r i n g  27  Case the  frequency  the  wavevector  then is  be  the  (b) of  the  of  observed geometry  u s e f u l  might w i l l  to  come  f l e c t e d from  from  from  3 . 2 . 2  that  as  a  r e s u l t  l e n g t h .  from  induced and  U n f o r t u n a t e l y , As  (a  photon  can  of  n  1 6  10  the  t r a v e l c m  - 3  i s  in  to  l i g h t  w i l l  s c a t t e r e d  i t  l i g h t  i n t o  (a)  This  d e t a i l  s c a t t e r e d  one  the  s t r a y  i n c o h e r e n t  now  60°  e x p e r i m e n t s .  experiment of  double  d i r e c t i o n .  these  coherent  d e t e c t o r  l i g h t  r e -  s c a t t e r i n g  s c a t t e r i n g  c o n s i d e r e d  s c a t t e r i n g  p a r a m e t e r s ,  i n c i d e n t  w a v e l e n g t h .  =  l i g h t  have  observed  amount the  these  u s u a l l y  the  work  f o r  (b)  (c)  at  S c a t t e r e d  s o u r c e s :  s u r f a c e s ,  w i t h  i n c i d e n t  i n c i d e n t  amount L i g h t  beam  in  from  the  t u r n .  L i g h t  plasma  C o n s e q u e n t l y ,  the  the  three  of  t y p i c a l  f l u c t u a t i o n s  t h i s  of  Each  determine  in  one  Stray  s i g n i f i c a n t  the  f l u c t u a t i o n s ,  In to  d e s c r i b i n g  and  wave.  adopted  o b s e r v e .  wave.  beams  from  was  v a r i o u s  thermal  induced  60°  that  to  d i a g n o s t i c  induced  c o n s i d e r  expect  the  mixing  the at  Before is  shows  the  before  high  phase  i n c i d e n t  wave  has  s t r a y  l i g h t  being  i s  phase  i s  16  a l s o  km  s c a t t e r e d )  d e n s i t y  s h i f t e d  by  the  very  through great  and  same  u n s h i f t e d i s  a  However,  v e l o c i t y  have  c r o s s - s e c t i o n  average  thermal  w a v e l e n g t h .  l i g h t  performed  v e l o c i t i e s .  l i g h t  zero  s c a t t e r e d  the  observes  s c a t t e r e d  s c a t t e r i n g on  one  experiments  a  care  wavein small  plasma had  28  to  be  taken  to  d e t e c t o r . is  v i r t u a l l y  The  d e s c r i b e d  3 . 2 . 3  p l a c i n g  in  the  s c a t t e r i t the  i s  useful  to  thermal  r e l a t i v i s t i c in  frequency  an  the  Thermal  l a s e r s i z e  Oj  =  r  0  (l  -  dco  plane  have  s i n  2  9  c o s  done  a  to  p l a s m a ,  expected  from  wave, i n t o  the  n e g l e c t i n g  s o l i d  angle  form:  S(k,uO ^  e  cf>)  2  i s  the  dfi  (20)  d i f f e r e n t i a l  c r o s s - s e c t i o n  is  the  c l a s s i c a l  6  is  the  angle  {t.,  <j>  is  the  angle  (E.  is  the  i n c i d e n t  V  is  the  s c a t t e r i n g  g  is  the  e l e c t r o n  S(k,a))  i s  the  s p e c t r a l  n and  V n  T  2  P^/A  been  from  s c a t t e r e d  w i l l  o- U)  Thompson r  dumps  .  = -j-  2  have  s i g n a l  P.  where  l i g h t  the  F l u c t u a t i o n s  l i g h t  of  i n c i d e n t  power  i n t e r v a l  Ps  at  a l o n e .  for  e f f e c t s ,  and  experiments  the  f l u c t u a t i o n s  l i g h t  a p p a r a t u s ,  ruby  c a l c u l a t e  s t r a y  stops  Random  any,  doubled  general  on  From  i f  a l l  b a f f l e s ,  s e c t i o n  few,  frequency  In  dft  of  S c a t t e r i n g  Since  e l i m i n a t e  e l e c t r o n  t ,  r a d i u s  ) Es)  i n t e n s i t y volume d e n s i t y d e n s i t y  of  the  thermal  f l u c t u a t i o n s  29  If of  N.  frequency  and  (20)  I  v  i s  , s in  a  of  i n c i d e n t / s c a t t e r e d  pulse  of  d u r a t i o n  l a s e r  s  V  NT  The f o c a l  d e t e c t i o n S e c t i o n  number  T  then  photons  P  =  ^y^-  becomes  N  s e c t i n g  the  e  n  ~A~ °J  =  s c a t t e r i n g  volume  r e g i o n s  the  o p t i c s .  3 . 3 . 2 )  V  of  With ~  8  x  the 10"  S p e c i f i c a l l y , Appendix  I)  S ( k  and  a  du  27 "  w )  V  d  i s  d e f i n e d  by  l a s e r  beam  i n d i c e n t p a r t i c u l a r m  1 2  f o r  forward  '  o p t i c s  the  i n t e r -  and  used  the (see  .  3  the  experimental  s c a t t e r i n g  plasma  (see  angle  of  6 0 ° ,  we  ( f u l l  w i d t h - h a l f  f i n d  o  S  -  3  x  I O  -  1  3  ,  averaged  over  a  0.3  A  t r a n s o  m i s s i o n )  t r i a n g u l a r For  The  instrument  and  <J> =  instrument  c o l l e c t i o n p r o f i l e  To r a t i o  these  f 2 0 , to  dfi  dw  -  -  at  2  4.7  3471.5  A.  x  10  - 3  s r .  x  10  1 1  s e c  -  1  c o n d i t i o n s :  estimate  c o n s i d e r  doubled  ruby  1.4  1 7  10  of  corresponds  ^  x  o p t i c s  c e n t r e d  0. With  ing  p r o f i l e  l a s e r  i n c i d e n t  the  *  the  7  x  s i g n a l  i n c i d e n t  pulse  (80  photons  mj, and  IO"  1 6  produced l i g h t 20 thus  to  from be  n s e c ) . Ng  -  a  t h i s 4  MW  There 100  s c a t t e r frequency are  photons.  If  30  these  are  detected  e f f i c i e n c y  and  a  by  a  gain  of  9558QB  photomul t i pi i er  the  nsec  20  pulse  To that  produced  a c t u a l a  be  the  by  the the  thermal  and  R a y l e i g h  from  The  r a d i a t i o n  only  from  the  be  the  thermal  must  be  F i g u r e s  in  than  In  dft)  about  was  the  produced  150  mv  at  from  the  as  was  the o b s e r v a b l e  c a l c u l a t e d  from  3 . 5 ) .  of  i n t e r e s t  the  l e a s t  as  l a r g e  as  spectrum  of  each  at  4  any  and  5  show  i s  these  induced the  given  by  s p e c t r a  F i g u r e  t r a n s m i s s i o n  p r o f i l e  the  monocromator.  of  the  induced  f l u c t u a t i o n s  the  thermal  spectrum. x  107.  At  i s w  seen =  0  to the  4  a l s o  be  d e n s i t y  thermal  and  8  (dw,  mv.  l a r g e r  s c a t t e r e d  c o n d i t i o n s  about  be  l o a d ,  40  r a d i a t i o n .  experimental  by  anode  50Q.  f l u c t u a t i o n s  ( S e c t i o n  EMI  i n c o n v e n i e n c e  f l u c t u a t i o n s  data  an  d e t e c t a b l e .  an  enhanced  quantum  about  must  l i g h t  not  a  of  a v e r a g i n g  the  proved  to  s i g n a l  s i g n a l  s i g n a l  for  with  background  was  be  p r e d i c t e d  (7).  t h i s  plasma  23%  E n h a n c e d S c a t t e r i n g  To t i o n s  v o l t s )  peak  T h e r e f o r e ,  s c a t t e r i n g  3 . 2 . 4  a  of  ( t y p i c a l  f l u c t u a t i o n s  s c a t t e r e d  t h a t  106  1 500  plasma  background  This l i g h t  at  x  o b s e r v a b l e  o s c i l l o s c o p e .  random  5  produces  experiment  f l u c t u a t i n g  p h o t o m u l t i p i i e r  f l u c t u a t i o n s .  e q u a t i o n s  p l o t t e d  f o r  i n d i c a t e s The  f l u c t u a -  the the  spectrum  much  narrower  induced  spectrum  and  (6)  of  than i s  l a r g e r  33  The t u a t i o n s appear  (when  t h i s  d e t e c t i n g bandpass ment  enhancement  l a r g e  but  both  OJ  s c a t t e r e d  to  l i g h t  l i g h t  k" w e in  may  the  i n t e g r a l s  d e t e c t o r . t h a t  a)  f a c t o r  0,  both  c o n t r i b u t e .  The  the  =  The  i n f l u e n c e  system  on  bandpass  r a t i o  the of  the  dw  t n  the  measurement.  in  i s  k - v e c t o r  frequency  s i n c e  waves g(k)  in  the has  a  enhance-  g(k)  l e s s  the  than  now  of  (21)  bandpass  of  s p e c i f i c  d i r e c t i o n  response  We  of  observed  the  ( k , o » ) 12>  ^  ±[ti  u n i t y the  c o n s i d e r  Response  of the  In  4  the  as  F i g u r e well  t h i s  to  spectrum  thermal  t h a t  f a c t o r  not  d e t e c t o r  dw  a r i s e s  induced  the  (k,w)|2>  Frequency  Comparing  the  2  over  bandpass  that  w r i t e  w i l l  f l u c -  the case  -  t) 2  and  d e s c r i b e s  d e t e c t i o n  each  d e t e c t o r  s e p a r a t e l y .  d e t e c t o r  induced  of  the  are  the  these  form  ri  < | n  the  on  R e c o g n i z i n g and  off  s c a t t e r i n g )  depend  <In.  where  s c a t t e r e d  thermal  w i l l  i n s t r u m e n t . for  of  compared  of  of  area  the  Figure i s  spectrum  r a t i o the  as  we  see  spectrum 5  we  d e t e c t e d i s  see but  seen.  To  frequency of  the  that that  in  under  induced  response  thermal  only  equation  a for  (21)  spectrum  of  the  f l u c t u a t i o n s .  v i r t u a l l y  account  i n t e g r a l s the  Detector  a l l  p o r t i o n t h i s  we  i s  simply  in  Aw  to  the of note the the  34  area  under  the  thermal  spectrum  in  Aco, w h e r e o  bandwidth  of  the  monochromator  Forming  t h i s  Wavevector The because  we  frequency 6  and  but  none  of  v e c t o r  f u n c t i o n s  a l s o  and  frequency,  we  acceptance  f u n c t i o n  the  and  x  made  forms  l i g h t  these  was  simple  f i r s t  the  a c t u a l known  u n i t we  step  s t a r t  c a l c u l a t e  s c a t t e r e d  from  Then,  as  f u n c t i o n s  of  the  with  the  wave-  both done  with  i n t e g r a t e  r a t i o  (equations  the  (a  e a s i e r  the  only  k-spectrum  d e t e c t o r ,  g(k).  for  r a d i a t i o n  not  must  106)  c o n s i d e r a b l y  that  We  form  2(3  Detector  f l u c t u a t i o n s .  of  A).  with  the  the  +10  combined  in  m u l t i p l y  wavevectors  the f o r  the  over  a l l  r e s u l t  to  g(k). The  w i l l  must  =  was  m a t h e m a t i c a l l y  i n c o h e r e n t  E  s c a t t e r e d  response  of  to  f u n c t i o n a l  were  advantages.  d i s t r i b u t i o n s  p o s s i b l e  the  U n f o r t u n a t e l y ,  these  coherent  get  of  frequency was  f u n c t i o n ) .  known  the  o  A  of the  a n a l y s i s  with  dependence  i n s t r u m e n t a l e x a c t l y  preceding  These  gives  Response  s t a r t e d  7).  r a t i o  (-10  i s  A to  have  the  wavevector  d i s t r i b u t i o n s  of  s c a t t e r e d  l i g h t  form  l  s  ( t  s  )  «  dt  1 ^ )  < | n ( k ) |  2  >  (22)  35  where  =  kg  -  r e s p e c t i v e l y or  and  i n c o h e r e n t  the  and  the  and  d i s t r i b u t i o n s a  accepted  power,  f r a c t i o n  given  where  i s  for of  a x i s  k  the  g i v e s  coherent  powers them  to  by  I  C  have  w a i s t  h  the  e i t h e r  d e s c r i b e d  plasma  for  (22)  both  t h a t by  wave  d e n s i t y ,  for  powers  coherent  assumptions  be  the  s c a t t e r e d  Gaussian  i s one  the  can coherent  power. a  Gaussian  the  d i s t r i b u t i o n  system  sees  d e t e c t i o n  M u l t i p l y i n g and  only  power  1^  cone  t h i s  i n t e g r a t i n g  i n c o h e r e n t  and  the  of  Z.  t o t a l  and Q  form  to  can  t h a t  d e t e c t i o n  power  the  the  assuming  along  s c a t t e r e d  waves  and  a  f o r s p a t i a l  by  the  i s  Under  d i s t r i b u t i o n  of  the  i n c i d e n t r e f e r s  n  and  s c a t t e r e d  A g a i n ,  t i o n  power,  e x p r e s s i o n s  0  the  d e n s i t y  s i m i l a r  i n c o h e r e n t  u  are  s c a t t e r e d  f o r  by  e v a l u a t e  I.j  f l u c t u a t i o n s .  i n c i d e n t  d e s c r i b e d  k,  s c a t t e r e d  over  f l u c t u a t i o n s .  r e s p e c t i v e l y ,  th " -T  Io  Meyer  <M*>r>  the  a l l  into  forms  T  by  and  the  d e t e c -  e x p r e s s i o n s  p o s s i b l e  the  v a l u e s  d e t e c t o r  by  Denoting  these  [21]  shown  has  36  j  4rr2  a  c  0  +  [uoV  h  w  2  ( 2 u  +  2  r  2  ) ] *  [ u  x  where  w  mixing  ,  0  r  r e f e r  beams  to  the  a  =  10  the  r a t i o  From 3  m  _ 1 ,  x  the 10  and of  =  9 the  and  6  the  60°  thus  The  As by  a  f i l l i n g mixing by  means  cate  of  that  were  b u t i o n  1 6  the  d e t e c t i o n value  e x p r e s s i o n s  us  the  6/2  2  = Z^oiL  „  beam  +  r  2  ] *  =  and  < | n ( k) | 2 >  and  a n g l e .  w0  expected  observed  r  =  3uo»  forming enhancement.  i  n c  j/< I n ( k ) |  2  >  t  enhancement  300  th  cm"  f u n c t i o n .  in  3  Appendix  I  the  s t a b i l i z e d  pulsed  20  The  T o r r .  i n v e s t i g a t e d  plasma  10  x  i s  an  l a s e r  n  2  i n c i d e n t  p r e d i c t  of  temperature  =  s i n  2  the  found  w a l l  e l e c t r o n g  ( 2 u  Plasma  ruby  the  2  Apparatus  pressure volume  w  we  d e t a i l e d  p a r t i a l l y  +  of  these  A  3.3.1  a  response  we  Experimental  9  g i v e s  E  3.3  r  2  experimental  into  r e s u l t s  frequency  and  w0  n2(k)  w a i s t s  r e s p e c t i v e l y  S u b s t i t u t i n g  I0  2  l i g h t  s t a b l e  Tg  a  was  hydrogen  arc  plasma  and  4 2 , 0 0 0 ° K , M a x w e l l i a n  at  The  r e s u l t s  e l e c t r o n  e l e c t r o n  a in  the  experiments  r e p r o d u c i b l e an  produced  parameters  separate  s c a t t e r i n g .  i s  and  in  plasma  with  i n d i an  d e n s i t y  v e l o c i t y  d i s t r i -  n  37  3 . 3 . 2  The  O p t i c a l  Figure f o r  the  into  a  m i x i n g  vacuum L  4  ( - 2 0 0  mm)  along  This  p r o v i d e s The  m i x i n g  beams'  components  and  the  and  the MW)  Because  the  In  ruby  v a r i a t i o n changing  35%  r e f l e c t i o n  dye beam  the  through  beam  .centre 3  beam  in  ( 8 5  the  mm)  to  the and  r e f l e c t i o n  m i r r o r  M i .  i n t e r a c t i o n  t r a n s m i t t e d  l o s s e s  of  before  the  photodiode  s p u r i o u s  through  monitor  from  the  the  the o p t i c a l  plasma  to  about  c o n s i s t s  of  an  50%  that  l a s e r  s a t u r a b l e p l a c e d  of  an  the  i n  dye  and  c e l l in  c e l l  the  a c t i v e  ruby  i n t e n s i t y  dye  l a s e r  of  the  c a v i t y  beam. c o n s i s t i n g  o s c i l l a t o r  a l l o w s  the  ( c r y p t o c y a n i n e  path  the  o s c i l l a t o r -  m i x i n g  a  f r o n t  c o n t r o l l e d volume  by  s i m p l y  c o n c e n t r a t i o n . s p l i t t e r  d i r e c t l y of  a  l i g h t  in  a m p l i f i e r  l i g h t  the  p l a c e d  mately  f o r m a t i o n  of  A is  methanol)  a d d i t i o n ,  L  f o c u s s e d  .  i s  m i r r o r .  by  of  power  in  the  the  Lenses  m i x i n g  d i s s o l v e d  M i ,  to  i s  remove  d i e l e c t r i c  a  m i r r o r  to  path  c o m b i n a t i o n ,  of  f . L . )  R e f l e c t i o n  (200  prevents  mm  arrangement  beam  the  a m p l i f i e r  This  l a s e r  ruby  mm).  c a l i b r a t e d  reduce  The  c o l l i m a t e  amount  power.  o p t i c a l  f o c u s s e d  L.2(85  second  a  complete  L i ( 1 2 7  o r i g i n a l  on  l a s e r  the  then  l e n s  small  f a l l s  the  by  i t s  m i r r o r  of  l e n s  c o l l e c t  back  volume.  by  modes,  v e s s e l  shows  e x p e r i m e n t s .  p i n h o l e  t r a n s v e r s e  6  System  a f t e r  c o n s i s t i n g the  l i g h t  into  two  prisms  l a s e r . a  of  a  This  beam  g l a s s  r e f l e c t s  d i a g n o s t i c t h i s  t h i n  i s  beam. passed  p l a t e a p p r o x i -  A f t e r through  RUBY LASER  CO  F i g u r e  6.  Optical,  system  f o r  mixing  e x p e r i m e n t s .  39  an  i n v e r t i n g  and  L  6  (7 5  rubidium  t e l e s c o p e  mm)  to  c o n s i s t i n g  i n c r e a s e  dihydrogen  i t s  phosphate  of  l e n s e s  i n t e n s i t y . (RDP)  L A  5  (150  mm)  c r y s t a l  frequency  of  doubles  the  o  l i g h t of  to  3471.5  copper  A.  sulphate  fundamental  ruby  harmonic.  F i f t y  t u a l l y  the  the  L7  a l l  the (169  a  i s  onto The  L8  the s l i t  Since the  plasma t h a t the mm  of  a  1P28  mm)  s e t  Ls  i s  and  plasma  c h o i c e  the  of  observed  two  beams  diameter.  w h i l e  of  s e t  of  the  second blocked  t r a n s m i t t i n g  u l t r a v i o l e t system  MW  beam  two  i s  A f t e r dense  v i r -  85%  then  q u a r t z  of  f o c u s s e d  l e n s e s  p a s s i n g  f i l t e r s  wavevector  the  angle  r e l a t i o n s h i p ,  of  60°  with  beam  and  the  f i r s t  L9  (89  mm)  image  of  a  beam  30u  and  has  these  masked  the  as  This  beams  focussed  beam.  mixing  volume  1800).  h e i g h t  of  m a g n i f i c a t i o n  of  about  a  c r o s s - s e c t i o n  of  1OOu  was  based  small  as  f o c a l  volume  and to  a  both  mixing  (Spex to  to  a  dimensions  volume  s c a t t e r e d  r e s p e c t  monochromator  provide  l i g h t .  a r e  s o l u t i o n  a  volume  mixing  block  of  t h i s  4  s o l u t i o n  p h o t o m u l t i p l i e r .  to  L9  to  the  beam  a  aqueous  p a s s i n g  and  an  and  an used  mm f i x e d ) .  the  at  i s  400  u l t r a v i o l e t  background  l a s e r i n  window  width  the  by +  s l i t  The  keep  volume  entrance  observed  140uto  The  s a t i s f y  (300  lenses  w h i l e  l i g h t .  c o l l e c t e d  i n c i d e n t  Lenses  by  g m / l i t r e )  m i l l i m e t e r s  q u a r t z  To  the  l i g h t  c o n t a i n i n g  ruby  mixing  monitored  l i g h t  (140  mm a d j u s t a b l e  through is  c e l l  remnant  u l t r a v i o l e t  into  A  the a  p o s s i b l e  on  the  to  4 0 u . 3 . 4 , x need  reduce  i n t e r s e c t s  d i a g n o s t i c  beam.  c r o s s - s e c t i o n  about  A l l 0 . 3  40  The  mono-chroma t o r  has  a  1 200  line/mm  g r a t i n g  o  blazed e x i t  at  3500  s l i t  A  which  width  of  2  is  mm  used  the  in  f i r s t  instrument  o r d e r .  With  an  t r a n s m i s s i o n  p r o f i l e  o  was  n e a r l y  a  p e r f e c t  r e c t a n g l e  9558QB  p h o t o m u l t i p i i e r  behind  the  e x i t  s l i t  Tubes from  each  s t r a y  l i g h t  t i o n s , the  of  the at  as  a  between t i o n  used.  These  d e t e c t o r from  and  that  3 . 3 . 3  Second  for L9  of  (10  c l a i m s  a  e f f i c i e n c y  t i o n a l  r e f l e c t s  a r e a .  g e n e r a t i o n i n t e n s i t y . to  an  The  in  a  a  70  MW  T h e r e f o r e ,  i n t e n s i t y  of  power mm  into  470  achieved  with  x  i s 10  reduce  s t a b i l i z e a  R a y l e i g h  o p t i c s .  f-number  r e f l e c -  An  of  i r i s  the  generated  d e t e c -  a n g l e -  mm)  for  t h r e s h o l d  of  300  MW/cm2  In  Figure  beamof  0.6  e f f i c i e n c y  by  6  an  the  cm  beam  which and  beam  c r o s s - s e c -  2  for  l i n e a r l y  fundamental  MW/cm2  an  15  a  mm  in  x  i n c r e a s e s the  to  unwanted that  the  25%.  c o n v e r s i o n  c r y s t a l  serve  v e s s e l  Generation  damage of  vacuum  f u n n e l s  d e t e c t i o n  used  monocromator.  Harmonic  RDP  i s  matches  harmonic  of  g l a s s  the  the  manufacturer  s p l i t t e r  the  EMI  l i g h t .  the  e l i m i n a t i n g  r e d u c t i o n  dump  L8  by  An  was  s c a t t e r e d  being  to  c o n v e r s i o n  the  v o l t s  windows  viewing  c r y s t a l  1700  w i d t h .  into  F u r t h e r  Second  d e t e c t  f u l l  extend  o p t i c s  tuned  to  at  A  nylon  the  l e n s e s  operated  20  black  p a r t i c u l a r l y  plasma.  horn  of  of  harmonic with i s  i n v e r t i n g  input i n c r e a s e d t e l e s c o p e .  the  41  The a  a c t u a l  f a c t o r  t i o n the  i n t e n s i t y  of  l o s s e s  but  because  from  each  in  l i q u i d . by  Angle  micrometer  no  i s  tuning  maximum  power  r e q u i r e d  power  r e s u l t s the  t a i n e d  t h i s  4%  r e f l e c -  This  the  by  puts  recommended  damage. a  c e l l  f i t t e d  with  an  i s  i s  with  a n t i -  index  matching  a c c o m p l i s h e d  e f f e c t i v e l y Tuning  a n g u l a r  dependence in  1.2  j  at  u l t r a v i o l e t MW)  a  f o r  a  r o t a t e  the  c r i t i c a l  adjustments  the  time  of  the  of  and  about  This  r e q u i r e s  p o s s i b l e .  f l u c t u a t i o n s  a  2,  5  x  pulse  in  a  time 60  mj  e f f i c i e n c y  of  that  the of  30  in  a  on  i s  ruby nsec time  input  s h o r t e r pulse (40 of  15  c o n MW). nsec  10%.  Laser  S e c t i o n  beams  l a s e r  However, of  e f f i c i e n c y  T y p i c a l l y  c o n t a i n e d  Ruby  mixing  harmonic  c r y s t a l  pulse  Chapter  c o n v e r s i o n  p u l s e .  c o n v e r s i o n  The  of  second  fundamental  3 . 3 . 4  as  above  matching)  axes.  the  s u r f a c e .  f i l l e d  which  than  r a d . The  (4  three  l e s s  n e g l e c t e d lens  in  and  adjustments  i s  s l i g h t l y  (phase  i t s  The  and  held  windows  about  than  have  n o t i c e a b l e  c r y s t a l  10_lt  we  c r y s t a l  c r y s t a l  coated  c r y s t a l  prism  the  caused  The r e f l e c t i o n  the  0.72  i n t e n s i t y  maximum  at  2.3  must  with the  a p p r e c i a b l e  as  be  showed as  long  narrow  power  that  a  must  a m p l i t u d e .  as  the  p o s s i b l e .  s p e c t r a l be  kept This  coherence  e m i s s i o n  high  to  combined  induce  42  requirement the to  of  standard an  high  technique  a u t h o r ' s  inch is  x  1/2  pulses  the  inch by  c o u p l i n g  used t h e s i s  a  The  [19]. ruby  A  F a b r y - P e r o t  " e k a l o n " )  and  low  a c h i e v e d  power  by  o s c i l l a t o r  with  in  was  30  e t a l o n as  of  the  mode  66%  c o n t r o l .  l a s e r  r e p l a c i n g  m i r r o r  Figure  of  7.  This  in  the  a l l o w e d  experimental  with  C e l l  x of  in  a  4  Q-switch  1/2  inch  producing  6  the  r e f l e c t i v i t y  o s c i l l a t o r  of  M2  inch  capable  Measurements  a c t u a l  the  P o c k e l s 6  of  Figure  reduced  i s  d e s c r i b e d  n s e c .  used  l o n g i t u d i n a l  a  that  o s c i l l a t o r  a  number  the  i s  to  An  c o m b i n a t i o n  j o u l e s  c a v i t y  s i m i l a r  a m p l i f i e r  6  provide  i s  diameter an  r o d .  of  O p t i c s  l a s e r M.Sc.  f o l l o w e d  diameter  to  of  b r i g h t n e s s  a m p l i f i e r . The  the  s p e c t r a l  f r o n t  A p e r t u r e s  t r a n s v e r s e  by  l a s e r  c o n d i t i o n s .  be  The  p l a c e d  were  o p t i c a l  to  m i r r o r in  modes.  l i n e w i d t h the  (Laser  done  by  arrangement  monitored  under  F a b r y - P e r o t  e t a l o n  o in  Figure  had  7  separated  by  p l a t e s  of  95%  a p p r o x i m a t e l y  r e f l e c t i v i t y  26.386  mm.  at  This  6943  gave  a  A, f r e e  o s p e c t r a l  range  of  the  ruby  0.095  A  and  a  r e s o l v i n g  l i m i t  of  1.6  x  o IO"3  A  at  L i g h t plane  of  which  covered  and  a  f i l t e r  through  n e u t r a l  were  w a v e l e n g t h . the  d e n s i t y  h a l f then  the  e t a l o n f i l t e r  was of  i n t e r f e r e n c e  photographed  on  imaged  50%  onto  t r a n s m i s s i o n  p a t t e r n . T R l - x  the  f i l m  The  p a t t e r n  using  a  D i a g n o s t i c  b e a m  R U B Y  L A S E R  M i x i n g optics  1 2 7  C a m e r a  m m  A l u m i n u m plate G r o u n d glass screen  F i g u r e  7.  O p t i c a l  system  E t a l o n  Filter;  3 6 0 m m  d e n s i t y • 0 . 3  for  measuring  l a s e r  Filter; W r a t t e n n o  - 9 2  l i n e w i d t h .  CO  44  #92 of  Wratten using  ments no  a  of  H-D  f i l t e r  to  n e u t r a l  the  reduce  d e n s i t y  h a l f - w i d t h s  unwanted  f i l t e r  of  the  curve  was  r e q u i r e d  d e n s i t o m e t e r  the  h a l f - i n t e n s i t y  r i n g of and  were  the  found  s i m p l y  p a t t e r n .  the  r e s u l t s The  This were  for  by  g r e a t l y  r i n g s  the  p o i n t s  c o n t r i b u t e d  r e s u l t s  of  be  shots  The  technique  s i m p l i f i e d the  on  a,  a  because  m i c r o -  p a r t i c u l a r  the  major  other  source  a c c u r a t e showed  measure-  p a t t e r n  Using  with  the  to  many  in  f i l m .  comparison  e s t i m a t e d  l i g h t .  to  the  h a l f of  e r r o r  about  l a s e r  15%. beam  o  to  have  a  s p e c t r a l  at  f u l l  power  output  l i m i t s  appeared  in  l a s e r  the  width  to  mode  of  be  v a r y i n g 200  MW.  random  s t r u c t u r e .  and  between  .01  V a r i a t i o n s are  Figure  and  .03  between  a t t r i b u t e d 8  shows  the  a  l i n e w i d t h  to  A these  changes Fabryo  Perot  p a t t e r n  for  a  s i n g l e  shot  with  of  .02  A.  o  Figure  8:  Fabry-Perot power = 200  p a t t e r n MW).  of  l a s e r  output  (AX, = i  .02  A,  45  As measure was  an  the  a s i d e ,  a  l i n e w i d t h  " d e c o u p l e d "  from  s e r i e s  of  the  the  of  measurements  l a s e r  resonant  a l o n e ,  were  i . e .  o p t i c a l  the  system  done  to  l a s e r  of  F i g u r e  o  6.  By  i t s e l f ,  narrower be  the  s p e c t r a l  q u a l i t a t i v e l y  dye  c e l l  in  3 . 3 . 5  the  T e k t r o n i x H o r i z o n t a l  B  for  650  usee  lamps  and  the  t h i s a  time  pickup  and  B  i s  from  d i s p l a y e d  s t a r t  and  100  the  can s a t u r a b l e  the  and  The  t r i g g e r e d  i s  7B91  the  give  3  together  to  form  one  h o r i z o n t a l  A  plasma  v o l t a g e  then  i s  to  a  the  d i s a b l e s by  the  timebase  the  p u l s e .  t r i g g e r e d  pulse  t i m i n g  when  generated c u r r e n t  i s  the  p l u g - i n s .  e x t e r n a l l y  noise  It  shape  o s c i l l o s c o p e  f i r e .  the  f l a s h A f t e r  e x t e r n a l l y P o c k e l s  by  C e l l  n s e c / d i v i s i o n .  channels  to  beams'  of  shows  A  of  7A12  mixing  presence  10  and  high  V e r t i c a l  c o n f i g u r a t i o n  of  t r i g g e r a b l e .  at  The  o p e r a t i o n  7B90  r e j e c t  the  A.  b a s i c  F i g u r e  f l a s h l a m p s  to  the  .07  the  p u l s e s .  timebase  o s c i l l a t o r  to  mixing  of  beams.  and  with  l a s e r  l i n e w i d t h  the  o u t l i n e s  r e l e v a n t  7704  a  Sequence  9  system  most  for  mixing  Timing  e l e c t r i c a l the  width  had  a t t r i b u t e d  F i g u r e  of  l a s e r  L  independent t r a c e .  i n t e n s i t y  and  and  R  c o n t a i n  channels S c a t t e r e d  frequency  p l u g - i n s  which l i g h t  are  7A16  added  i n t e n s i t y ,  doubled  i n t e n s i t y  o  are  d i s p l a y e d  a g a i n s t  the  s e q u e n t i a l l y  other  through  on  t h i s  c o a x i a l  t r a c e delay  by  d e l a y i n g  c a b l e s .  one  !•  P u l s e  O  P u s n button  L a s e r  o s c .  f  lashlamps  W  Generator  D i s a b l eo s c . c h a r g i n g u n i t  T o  L a s e r  pulse  I O O / i s e c delay  a m p .  f  p i c k - u p  am  7 7 0 4  5 0 0  p s e c  delay  few  P l a s m a trigger  F l a s h l a m p s  a m p . lashlamps  to  Pockels Cell  t o  P l a s m a  b a n k gap  F i g u r e  9.  E l e c t r i  cal  sys  tern.  s p a r k  47  A m p l i f i e r  P l a s m a current  i  i  I  l  L  i  »  •  1000  500  F i g u r e  10:  Timing  i_  sequence.  t ( / t M C  .,  48  3.4  R e s u l t s The  r e s u l t s  beat  frequency  ment  i s  sents  d e f i n e d  the  c a l l y  the  Each  r e g i o n .  The of  of  v e r t i c a l the  The  l i g h t  the  made  i t  random  was  c a l c u l a t e d and  the  a p e r t u r e s thermal w i l l  the  the at  same  / I  h  into  t  the  11.  the  Enhance-  where  Is  d e t e c t o r  r e p r e -  and by  1 ^ the  p l o t t e d  v e r t i -  data  taken  of  the  i n d i c a t e t h a t  the  standard  d i f f e r e n t  data  the  were  at  in  i t s  a b s o l u t e a v e r a g e d . d e v i a t i o n  mean. I  .  was  =  0  open to  r e q u i r e s to  2mm,  d e t e c t  R a y l e i g h  d e t e c t i o n  some  the  background  l i g h t  s c a t t e r e d  C o n s e q u e n t l y ,  s c a t t e r i n g  data  t h i s  using  from value  50  T o r r  system.  i n c i d e n t  i n t e n s i t i e s  the  r a t i o  of  s c a t t e r e d  l i g h t  f l u c t u a t i o n s  to  and  into  R a y l e i g h  gas  form:  plasma gas  e x p l a n a t i o n .  plasma  equal  r  when  beams.  p o i n t s  f l u c t u a t i o n s .  I  ,  i s  For  the  plasma  d e t e c t o r  value  bars  h  the  mixing  a  F i g u r e  r a t i o  i n d i c a t e  i m p o s s i b l e  from  t  combined  the  in  in  into  the  e r r o r  bars  I  average  for  s l i t  thermal  plasma  have  is  p o i n t  e x i t  the  H2  from  -  This  c o n t a i n s  power  e r r o r  p o i n t s  Because  in  h o r i z o n t a l  l a s e r  Ig  s c a t t e r e d  power  p o i n t  The  r a t i o  a l o n e .  graph  mixing  presented  s c a t t e r e d  f l u c t u a t i o n s  The  l i m i t s  the  power  o p t i c a l  are  power  the  a g a i n s t  t i m e s .  zero  as  t o t a l  r e p r e s e n t s thermal  i s  of  "e "  q  N  T  S  °R  (  k  )  _  c o l l e c t i o n a  d e t e c t o r  s c a t t e r i n g  by  49  ENHANCEMENT 150 J — O  0  i  1!  1  100  50  •mix  50 Fi gure  11.  Enhanced beams'  s c a t t e r i n g power.  (  M  W  )  100 from  plasma  v . s .  mixing  50  where  n  and  e  N  are  the  m o l e c u l e s  r e s p e c t i v e l y .  bandwidth  of  the  The given been  by  c o n d i t i o n s .  in  over  S ( k , w ) .  Oj  S e c t i o n  is  the  The is  given  a  hydrogen  a l l  of  i s  for  f a c t o r 4,  =  for Tg  -J1  a  to  the  and  on  gas  the  =  plasma S(k,oo)  the  =  found  parameter  Thompson  for  with  S(k,a))  0.8  s c a t t e r i n g  r a t i o  T..  is has  e x p e r i m e n t a l the  t o t a l  from  (23)  +  +  0.57  1/kX  D  R a y l e i g h  c r o s s - s e c t i o n s  by  For  x  10  The the  plasma  dw  l i g h t  hydrogen 1 8  c m -  3  f a c t o r  s c a t t e r e d  v  i  ^ 4T2"  mC  Or  ~ 1.8  e l e c t r o n s  dependent  form  Figure  f r e q u e n c i e s  r  N  f a c t o r  The  3 . 2 . 4 ,  =  a  a  plasma  c r o s s - s e c t i o n  s(k) =  where  i s  of  d e t e c t o r .  =  For  s c a t t e r i n g  F  s c a t t e r i n g  a ( k , w )  p l o t t e d  d e n s i t y  at  50  a n d Oj/o  F  i s  into  R  Torr  (n  -  -  1)  1)  ~  1.32  x  IO-",  r a t i o  of  ~ 4 . 4 .  found the  N2 (n  by  forming  bandwidth  of  the the  d e t e c t o r  51  to  the  t o t a l  s c a t t e r e d  over  a l l  wavelengths.  For  an  i n s t r u -  o  mental  width  of  20  A  c e n t r e d  the  l a s e r  l i n e  OA  + 1 F  on  dco S(k,co)/S(k) = 0 . 2 2  = o  OA Using  these  numbers  I  the  s c a t t e r i n g  Scat. ,  L  -  Scat.  /  i  / T  plasma  The m u l t i p l i e d H2-  This  p o i n t  by  the  from The  the  a c t u a l  F i g u r e  I  11  g  as  f u n c t i o n  orders  of  powers.  3.5  D i s c u s s i o n  This that  was  wave  of  of  the  to  0  x  10  c o r r e s p o n d s  to  t h i s  s c a t t e r e d  from  50  by  I ^  a  of  n  i  observed  t  '  a t  has  s  1  S  t  power  the  n  i n  e  r a t i o Torr  ( i n t e n s i t y  the  l i g h t  i s  i s  seen  p l o t t e d  d e t e c t  a  the  I by i n  beams. to  a v a i l a b l e  to  by  d e f i n e d  mixing  devoted  and  denoted  q u a n t i t y  h i g h e s t  been  produce  f r e q u e n c y .  i s  Enhancement  s c a t t e r e d  c h a p t e r  conducted zero  =  X  s i g n a l  T  magnitude  beam  -  i n t e n s i t y ) .  a  two  m i  i n t e n s i t y  ^ h ^ t h '  enhancement  -, .. - , n - 3  8.7  f l u c t u a t i o n s ) .  -  The  n  d e s i g n a t e d  thermal  s c a t t e r e d  r a t i o  I  i s  =  becomes:  gas  average  i n t e n s i t y  s c a t t e r e d  ( t o t a l  a t  _  —  r a t i o  be  about  mixing  experiment  plasma  d e n s i t y  52  D i s c u s s i o n s c e n t r e d be  p r i m a r i l y  e x c i t e d  were  the  to  not  the  In  any  c a s e ,  can  the of  the  of  l a s e r  11  a  induced  or  wave Of  the  11  in  r e s u l t s  could  primary  q u e s t i o n  indeed concern  as  d i s p e r s i o n  to  whether  r e l a t i o n .  that  we  Figure  see  to  be  a l s o  such  p r e d i c t s  do  Thus,  through  the  a  the  from  that  with  the  o r i g i n  c o n -  the power  the  the  s p e c t r a l  with  square  in  model  the  enhancement  c o n t r a d i c t  the  that  i n c r e a s e s  experimental  not  s e v e r a l  with  expected  i n c r e a s e  the  11  c l e a r l y  i n c r e a s e s  f l u c t u a t i o n s  to  Despite  curve  i s  power.  expected  Figure  shown  model  beams'  square  of of  the  u n c e r t a i n t i e s  the  t h i s  can  p r e d i c t i o n :  be  f i t t e d  to  d a t a . The  powers  of  induced  The  same the  l i g h t able  enhancement  over  the  in  a  unambiguously  wave  This  the  i s  of  such  plasma  F i r s t  induced  This  p a r a b o l i c  the  r e s u l t s  2.  power.  r e s u l t s  the show  drawn.  the  m i x i n g  Figure  the  beams.  Chapter  of  e x p e r i m e n t a l  a m p l i t u d e .  c o l l i s i o n s  r e s u l t s  be  of  mixing  d e n s i t y  the  e x c i t e d .  can  amplitude  to  whether  s a t i s f i e s  the  From c l u s i o n s  of  wave  be  around  d e t e c t a b l e  e f f e c t s  or  wave  a  p r i o r  100  d e n s i t y  s c a t t e r e d w h i l e  of  was  is  not.  has  a l s o  the  seen  mixing  severe  from  l i g h t  per  wave  c o n c l u s i o n  presence  t u a t i o n s  MW  i s  an  be  beam.  induced from  over  100  This  i n d i c a t e s  a p p r e c i a b l e  evidenced  plasma  s c a t t e r e d  to  by  wave the  was  random  that  a m p l i t u d e .  the  background  at  f a c t  t h a t ,  r a d i a t i o n ,  d i r e c t l y thermal  d e t e c t f l u c -  53  The powers  was  average  140,  This  agrees  ment  of  3 .6  Other  with  w i t h i n  60°  and  the  same  over  f a c t o r  of  2  from  the  a d d i t i o n  170  with  h i g h e s t being  the  two  In  mixing  beam  r e c o r d e d .  p r e d i c t e d  enhance-  was  s c a t t e r i n g  a p p l i e d the  at  w i t h to  p e r t u r b a t i o n o b s e r v e d . r e q u i r e s  60°  and  both  The concerned  the  s i m u l t a n e o u s l y  from  one  1 2 0 ° ,  the  one  with  ruby  at  60°  r e s u l t s  c l o s e l y  experiment  d e t e c t i n g  a  at  the  1 2 0 ° . the  spectrum agreement  l i g h t that  no  extreme was  of  weakly  p r e v i o u s  a l s o  s c a t t e r e d  f u n c t i o n  be  e l e c t r o n  independent  treatment  b a s i c a l l y  spectrum  simple  for  This  out  f u n c t i o n of  the  the  t o t a l  the  beams  around  d i s t r i b u t i o n  d i s t r i b u t i o n  in  spectrum  pointed  s i n c e  o r i g i n a t i n g  one  r e g i o n  the  d e t e c t o r  s c a t t e r e d  s i m u l t a n e o u s l y .  measuring  Ignoring  at  from  and  of  beam  l a s e r  the  r e s u l t i n g  r e a s s u r i n g  mixing  two  matched  e l e c t r o n  the  l i g h t  combined  of  i s  the  s p e c t r a l  The  the  we  plasma,  that  120°  beams  the  peak.  This that  that  The  experiments  ion  see  the  o b s e r v e d .  s e r i e s  6,  perturb  case  a  d o u b l i n g .  If  than  work  frequency  at  s p e c t r a ,  one  p r e v i o u s  Figure  volume.  d i f f e r e n t of  o b t a i n e d  in  other  s t r o n g l y  the  without  s c a t t e r e d  plasma  be  f e a t u r e  done  beam  l i g h t  together  was  of  s t a r t i n g  were  d i a g n o s t i c  observes  i n t o  values  the  Experiments  experiments  should  a  at  300.  Before  the  enhancement  Chapter  p e r t u r b e d . s e c t i o n s  s i n u s o i d a l  d e n s i t y  2  54  wave  of  there  the  appears  enough the  h a l f  to  a  2  the  may  n(k)  =  <<  a i .  The gated in  by  the  would the  ruby  been  s t u d i e d  t h i r d  Figure induced  ruby 6).  beam  d e n s i t y  mixing  in  i t s  l a s e r  t h i s  powers  wave  l a r g e  K-spectrum.  That  i s  kx  as  +  a  kx  cos  2  cos(kx/2)  i s  -f-  frequency  The  presence  of  an  enhanced  s c a t t e r e d  but,  as  d o u b l i n g t h i s  s c a t t e r i n g  e x p l a i n e d  above,  ]  0  e a s i l y  the  by  « •  i n v e s t i -  components sub-harmonic s i g n a l  at  a l s o  occurs  t h i s  had  d e t a i l .  r e s u l t s  We  high  d r i v i n g  U n f o r t u n a t e l y ,  beams  in  The  the  removing  detected  mixing  cos  &i  beam.  At  form:  v a r y i n g  f r e q u e n c y .  the  the  jl +  term  be  from  a  have  d i a g n o s t i c then  of  sub-harmonics  n o  simply  wavelength.  p o s s i b i l i t y  d e t e c t  d e n s i t y  where  ruby  showed  s c a t t e r e d  t h e r e f o r e  used.  d e t e c t a b l e  from  conclude  f l u c t u a t i o n s  powers  no  are  the  enhancement  mixing  that  in  v i r t u a l l y  volume  the  (as  plasma  s i n u s o i d a l  of in the for  Chapter  4  THE GAS EXPERIMENTS  4.1  I n t r o d u c t i  on  Chapter of  the  wave to  method  i n  a  up  to  and  150  s t u d i e d  the  mixing  as  a  d i s c u s s e d  4.2  Experimental  The in a  a  s i m i l a r  plasma.  with  the  i n  The  now  were  of  both  experimental  the  r e s u l t s of  the  at of  gas  The  d e s c r i p t i o n l o n g i t u d i n a l the  g a s e s .  used  f u n c t i o n  a  extends  v a r i o u s  s c a t t e r i n g  l i g h t  d e t a i l e d  d e t e c t  Enhanced  procedure  Hydrogen,  p r e s s u r e s a  and  t e c h n i q u e s  a r e  o b t a i n e d . theory  These  of  ranging  d i a g n o s t i c  p r e s s u r e  Chapter  a r g o n ,  beam  power  of  o u t l i n e d  r e s u l t s  a r e  2.  Procedure  experiments way  i n  a  and  chapter  d i o x i d e  by  then  produce  mixing  beams.  f o l l o w e d  p r e s e n t e d  This  carbon  T o r r .  has to  o p t i c a l  was  f i r s t ,  used  plasma.  i n c l u d e  n i t r o g e n  3  as  those  o p t i c a l  e x c e p t i o n s  i n  a l r e a d y  system  that  v a r i o u s  lens  was L8  55  gases  were  d e s c r i b e d the had  same a  f o r as  f o c a l  i n  conducted the  case  F i g u r e  l e n g t h  of  of 6  169  mm,  56  the  monochromator  entrance  s l i t  and  the  was  wide.  was  u s e d ,  e x i t  s l i t  however  no  200u  d i s c h a r g e  was  200y  The  was  wide  same  and  300u  vacuum  i n i t i a t e d  high  v e s s e l  between  the  e l e c t r o d e s . As  b e f o r e ,  s e q u e n t i a l l y no  plasma  was  on  of  100  s c a t t e r e d from  the  s i n g l e  background  e x c e l l e n t .  base  a  s i g n a l s  from  12  s i g n a l ;  u l t r a v i o l e t  and  a  The  the  the  t y p i c a l  f i r s t  second  ruby  t r a c e .  present  shows  n s e c / d i v i s i o n .  l i g h t  monitors  o s c i l l o s c o p e  r a d i a t i o n  F i g u r e  the  beam  d i s p l a y e d  However,  w i t h  s i g n a l / n o i s e t r a c e  p u l s e  and  were  with  a  r e p r e s e n t s  t h i r d  p u l s e s  monitors  r a t i o time the  are  r e s p e c t i v e l y .  i  J  F i g u r e  12:  O s c i l l o s c o p e l a s e r  For s c a t t e r e d  the  l i g h t  i n t e n s i t y  magnitude.  To  an  a p p r o p r i a t e  f i l t e r  t r a c e  monitors  v a r i o u s  of  -J  preserve was  (100  gases  at  v a r i e d the  of  by  s c a t t e r e d  and  n s e c / d i v )  d i f f e r e n t more  l i n e a r i t y  p l a c e d  l i g h t  between  p r e s s u r e s  than of  four  the  l e n s e s  the  orders  p h o t o m u l t i p i i e r L8  and  L9  57  in  Figure  6.  The  d e n s i t y  of  these  f i l t e r s  had  p r e v i o u s l y  o  been  c a l i b r a t e d In  held l e t  denote  d i a g n o s t i c 1^  denote  the  o f f  d e f i n e d  by  C 0  E  ,  u s i n g  the  the  i n t e n s i t y  when  the  -  (I5  ranging  a  second as  gas  p r e s s u r e .  Carbon  25  T o r r  measurements v a r i e d the  by  dye  no  when range  the  the  l i g h t  ) / I  of  set  R  .  10 1^ of  d i o x i d e  chosen  the  to  to  R  in  each  mixing 100  a l l o w  Power  in  power  enhancement  I  the  c o n c e n t r a t i o n  of  H2,  power  and  beams  are  i s  was  then  determined  A r ,  N2  and  c a s e .  the  T o r r the  the  enhancement  in  We  from  m i x i n g  T o r r  was  v a r i e d .  f u l l  the  The  150  at  was  have  experiments of  power  s c a t t e r e d  when  and  mixing  gas  beams  This  A.  enhancement at  and  c o n s t a n t  hydrogen  w i d e s t m i x i n g  range beams  at of  was  c r y p t o c y a n i n e  in  c e l l .  v i s i b l e was  R  3472.6  the  s c a t t e r e d  made.  changing  observed  shock  be  of  f u n c t i o n  were  to  In was  a  of  mixing  from  determined  at  s c a t t e r i n g ) . I  was  and  p r e s s u r e  i n t e n s i t y  =  l i n e  experiments  measurements In  50  neon  of  (Rayleigh  p r e s s u r e s 2  set  a  and  beam  blocked  at  one  c o n s t a n t Ij  using  a l l in  l i g h t heard  e x p e r i m e n t s , the  from and,  d e t e c t i o n  around  mixing  the.H  the in  evidence  volume.  V i s u a l  of  gas  breakdown  o b s e r v a t i o n  volume,  no  c h a r a c t e r i s t i c  hydrogen,  no  r a d i a t i o n  to  observe  system f t  no  l i n e .  was  set  was  the  showed  a c o u s t i c monitored  s p e c t r a l  58  4 . 3  R e s u l t s Figures  l i g h t  as  a  Power  in  the  each.  f u n c t i o n  We  c r e a s i n g d i o x i d e  see  a  of  the  with  of  induced are  response that  i t  of  i s  s t i l l  s e r i e s  of of  o p t i c s ,  prime  50  100  MW  the  beginning  of  5  at  50  focal s l i t  each.  was  l e n g t h , 30u  (Figure  the  Torr  wide 19).  i n  i s  hydrogen,  S e c t i o n  and By  This  changing  Lg  wide  =  89 40u,  mm an  4.2 is the f . L .  MW  i n carbon o b s e r v e d .  observes  c h a r a c t e r i s t i c s i n  compared  which to  and  that  that  induced  the  c o n c e r n ,  response  This  i s  with  frequency  provided  mixing  o p t i c a l used  shown  in  and  a  beam  and  the  20  a  d e t e c t i o n of  about at  enhancement 13  Figure  of  by  d e s c r i b e d  by 6  the  to  monochromator  r e d u c i n g  d e t e c t o r  powers  an  Figure in  the  d i f f e r e n t  set-up  was  o p t i c s  of  demonstrated  two  enhancement  e f f e c t i v e l y  with  one  thermal  major  100  commonly  the  means  used.  bandwidth.  at  the  were  wide  the  a  about  experiments  This  not  gases  hydrogen,  enhancement  gas  conducted  of  By  a  at  in  upon  s c a t t e r e d  the  maximum  200  was  wavevector  case  a  plasma  narrow.  one  to  dependent the  of  weakest  the  spectrum  In  each  of  c o n s t a n t  over  i m p o r t a n c e .  measured.  T o r r .  of  s u f f i c i e n t l y  experiments  in  i s  d e t e c t o r  However,  i s  before  spectrum,  the  for  n i t r o g e n ,  U n l i k e  comparably  of i s  and  thermal  enhancement  held  e f f e c t  experiment  i n c o h e r e n t  s p e c t r a  was  enhancements  d e t e c t o r .  the  the  mentioned  the  p r e s s u r e  beams  argon  where  m i x i n g  the  that  show  of  mixing  As in  13-16  was  a n g u l a r  L8  p o i n t =  300M  entrance o b t a i n e d acceptance  59  10 E =  8  •S-'R  I  6 4  P  50 F i g u r e  13.  Enhancement  i  I  v . s .  l I  100  J  p r e s s u r e  of  ~  0  l  (Torr) L  150  hydrogen.  100 N'  80h 0  0  60 40h  20  5  P (Torr) n  i  •  50 F i g u r e  14.  Enhancement  i  100 v . s .  p r e s s u r e  150 of  n i t r o g e n .  6 0  F i g u r e  15.  Enhancement  v . s .  p r e s s u r e  of  a r g o n .  61  CO,  ENHANCEMENT  200  100  L  P R E S S U R E (Torr)  • J  1  i  la  50  Figure  16.  Enhancement  150  100  v.s.  pressure  L  of  carbon  dioxide  62  of  the  d e t e c t o r  observed  r e d u c i n g  enhancement This  d i s t r i b u t i o n Over  and  i s  i s  understood  s c a t t e r e d  r e l a t i v e l y  l a r g e  angles  the  induced  r e l a t i v e l y  narrow  cone.  a  of  s i z e  f u n c t i o n  r e s u l t s  the  shown  d e t e c t i o n the  case  in  the  to  enhancement  i s  by  is  the In  If  each  of  (n  -  I )  108  the  p a r t i c u l a r  the  p o i n t s  are  seen  e n t i r e  of  to  to  Figures  of  be  n  The  p r e s s u r e .  s a t u r a t i o n  the  the to  the  i s  of  In  index best  at  of  theinduced  in  which  over  800.  d e n s i t y  attempt  wave's  f o r n i ( x )  i s  made  enhancement.  we  to  the  might  s i n c e low  n  expect 2  (k)  by and  e r r o r s see  <*  a  2  p r e s s u r e s . E  r e f r a c t i o n  f i t s  we  19),  r e s p o n s i b l e  d i o x i d e  cases  the  second  enhancement  e x p e r i m e n t a l a l l  be  same,  the  Carbon  the  4,  a  then  Figure  n o r m a l i z e d  o n l y  are  by  u s e d ,  the  i s  the  an  in  If  the  f o r  measured  gas  be  to  number  are  case  l i n e s  13-16.  w i t h i n  in  i s  o b v i o u s l y  a p e r t u r e .  used  s e c t i o n  v e r t i c a l l y  where  gas.  agree  range  i n d i c a t i n g  t h i s  p l o t t e d x  2  of  i s  would  2  waves.  c o n c e n t r a t e d  q u a n t i t y  13-16  enhancements  show  C0  a n g u l a r  s c a t t e r i n g  m u l t i p l i e d  from  the  r e f e r r e d  (and  Torr  Figures  p o l a r i z a b i 1 i t y  q u a n t i t y  be  next  q u a n t i t y  17  were  f l u c t u a t i o n the  i s  d e t e c t i o n  p h y s i c a l  t h i s  normalized  the  i n c o h e r e n t  R a y l e i g h  above  150  determine  Figure The  the  would at  the  enhancement  13-16  important  produced.  r e s p e c t i v e the  axes  enhancement  that  of  from  s c a t t e r i n g  d e s c r i b e d  v e r t i c a l  volume,  c o n s i d e r i n g  the  The  Figure  system  The the  by  l i g h t  w h i l e  d e t e c t i o n  i n c r e a s e d .  of  c o n s t a n t  a l l  the  eye  d i v i d e d of through  n i t r o g e n over  the  evidence  a m p l i t u d e  and  .  63  Pressure Figure  17.  Enhancement gas  n o r m a l i z e d  to  (Torr) p o l a r i z a b i 1 i t y  p r e s s u r e .  (index  of  r e f r a c t i o n  v . s .  0  taken  at  7000  A  and  STP)  64  an  a c t u a l  pursued  decrease  more A  measure  at  s i g n a l  v a r i o u s from  mixing  powers with  the  the  one  the of  h i g h e s t the  l i g h t  4.4  of  d e n s i t y  wave  mixing  of  presented can two  be  E  =  (I  -  g  I  (Figures  r e s u l t s i s  a  C0  R  0  in  i s  30  seen  ) / I  C0  t o t a l 50  T o r r  the i s  R  and  h i g h e s t  seen  are  2  the  to  16). e v i d e n t .  t h r e s h o l d  MW  of  to  l e v e l  mixing  i n d i c a t e s  and,  together the  at  r e p r e s e n t s  This  in  the  At  13  to  power  and  2  pronounced  around  at  s a t u r a t i o n  with  next  beam  off  a  of  the  s e c t i o n  in  e a r l i e r .  o r d e r s  of  r e s u l t s  induced  i n t e n s e  Enhancements two  Torr  at  d i s c u s s e d  developed  i s  D i s c u s s i o n  The  the  model  p o i n t  performed  show  IR.  used.  be  19  s c a t t e r i n g  amplitude  w i l l  and  =  the  This  mixing  Im^x  enhancement  wave's  of  100  p o i n t  r e g i o n  powers  e f f e c t , the  for  there  the  the  l a s e r  induced  t h r e s h o l d  that  18  r e s u l t s of  we're  f u n c t i o n  R a y l e i g h  i n  Second,  a  The  f e a t u r e s  hydrogen.  s e c t i o n .  Figures  previous  enhancement  power.  as  enhancement  n o t i c e s  in  experiments  measured  thermal the  next  of  powers.  Two F i r s t  set  I  e v i d e n t  the  p r e s s u r e .  s i g n a l  agree  in  enhancement  gas  s c a t t e r e d H2  f u l l y second  the  c o n s t a n t  being  magnitude  in  in  l a s e r  show a  g a s ,  as  a in  s t a t i o n a r y a  plasma,  by  beams.  s c a t t e r e d  were  that  l i g h t  measured.  i n t e n s i t y  This  of  over  i n d i c a t e s  that  65  1000 'scat (arb. units)  Figure  18.  Enhancement  v . s .  m i x i n g  power  f o r  100  Torr  C 0  2  .  F i g u r e  19.  S c a t t e r e d i n t e n s i t y and 25 T o r r H2.  v . s .  mixing  power  f o r  50  67  a p p r e c i a b l e m i x i n g  f l u c t u a t i o n s  depends  on  s t a t e d  the  experiments the  narrowed, d i r e c t  as  the  means  t i v e l y  the  the  for  in  amplitude  gas  p r e s s u r e  d e c r e a s i n g  p o t e n t i a l  produced  by  the  a s y m p t o t i c  Figure  17.  in  the  The  s a t u r a t i o n  f o r  the  the  e f f e c t g a s e s .  average  c o h e r e n t  to  the  the  the  that  the  k - v e c t o r  in  of  f o r  gas a c c e p t a n c e  p r e v i o u s  d e t e c t o r  i n c r e a s e d . the  s e c t i o n was  This  i s  k-spectrum  f l u c t u a t i o n s  wide  p r e v i o u s wave  spectrum  of  hydrogen,  of  i s  r e l a -  s c a t t e r e d  f u n c t i o n  then  time  about  the  seen  40  between  out  of Torr  of  i n  w e i g h t ,  at  a  the H2.  m o l e c u l a r  of the  time  Chapter  2.  curve  of  hydrogen  at  a  lower wave At  s t a r t s  probably  hydrogen  frequency  S a t u r a t i o n of  c o n s i d e r e d  that  i n c r e a s i n g  a c t u a l l y  p a r t i c l e s  atomic  c o l l i s i o n i s  w i t h  development  choose  lowest  i n d i c a t e s  C o l l i s i o n s  i n h i b i t  we  s e c t i o n  s a t u r a t e s  knocking  can  example  p r e s s u r e s  the  example  case  By  m o l e c u l a r  other  d e t e c t o r :  p r e s s u r e s .  they  Having  h i g h e s t  the  the  e f f e c t .  an  the  induced  h i g h e r  the  at  of  d i s t r i b u t i o n As  the  angle  enhancement  f l u c t u a t i o n s .  the  w e l l s ,  observed  statement  c o n t r a s t  and,  at  The  from  17  of  i nf 1 uence t h i s  273°K  be  the  enhancement  the  i n c o h e r e n t Figure  occurs  can  p a r t i c u l a r  acceptance  s c a t t e r e d  the  in  observed  narrow,  of  o p t i c s .  evidence  l i g h t  from  t h i s  that  e a r l i e r ,  p r o p e r t i e s  d e t e c t i o n  showed  the  d e n s i t y  p r o c e s s . As  of  i n  given  has p r e s s u r e .  p r e s s u r e a m p l i t u d e  40  Torr  c o l l i s i o n s  and i s  than  68  about  8  x  1 0 "  s e c  1 0  [ 2 0 ] .  c h a r a c t e r i s t i c  p e r i o d  p o t e n t i a l  w e l l  T  where  p o l a r i z a b i 1 i t y  the  =  t p  This  of  must  be  o s c i l l a t i o n  (2ir/k  E  a  0  =  compared  f o r  )(2m/a)2", 2e  0  (n  -  a  to  the  p a r t i c l e  (Chapter  l ) / p .  2,  For  i n  a  S e c t i o n  the  2 . 3 ) ,  c o n d i t i o n s  11  being  c o n s i d e r e d  bottom  of  a  T"tr  w e l l ,  a  between  we  that  c o m p l e t e d , At  the  p r e s s u r e s  of  atomic in  p o i n t  d i s t r i b u t i o n d e c r e a s e s  i s  being  behaviour  w i t h i n  the  same.  s a t u r a t e s w e i g h t .  amplitude  i n  At  f i r s t  the  a r g o n ,  a t  about  40  2,  t h i s  number  be  the  the  S e c t i o n  2 . 5 . 2 ,  were  r e a c h e d ,  c o l l i s i o n  i n d e e d ,  f r e -  f o r m a t i o n  i n h i b i t e d .  the  the  The  would  expect  the  other  gases  to i f  of  of  Figure e x p e r i m e n t a l  enhancements  p r e s s u r e s ,  hydrogen  One  gases  l i m i t s  n o r m a l i z e d  i n c r e a s e d in  t h a t ,  a c c o r d i n g l y .  c o n s i s t e n t  p r e s s u r e s ,  make  never  where,  of  low  of  hydrogen  e x c e p t i o n  v i r t u a l l y p l i t u d e  in  see  Chapter  would  the  At  can  In  l i m i t  the  thus  We  o s c i l l a t i o n s  to  With  e r r o r .  u n l e s s  i n c r e a s e d  a m p l i t u d e  show  c o l l i s i o n s .  Torr  a s y m p t o t i c  s e c .  p a r t i c l e  40  of  17  10"  over  has  wave  x  a s y m p t o t i c  quency the  2  trapped  o s c i l l a t i o n s e x p l a i n e d  ~  the  a r e  wave  because  of  observe  the  same  decrease  p r e s s u r e  were  r a i s e d  the  i t s  am-  lowest  s u f f i c i e n t l y h i g h . In c r e a s i n g  p r e s s u r e  dependence e x p l a n a t i o n obvious used.  a r g o n ,  on  the  but  wave  does  p o l a r i z a b i 1 i t y  f o r  t h i s  d i s t i n c t i o n  e f f e c t that  amplitude  not as i s  argon  appear do  the  o f f e r e d i s  the  s a t u r a t e s to  f o l l o w  other except only  with the  g a s e s . to  note  i n same No the  n o n - m o l e c u l a r  gas  69  From m i x i n g  power  we  amplitude  a l s o  there  a  ment  was was  the  form  o c c u r s  amplitude from  wave,  power  an  the and  has  observed  in  been  an  Figure  o b s e r v a b l e  Figure  18  to  c o r r e s p o n d s in 3  the  thermal  at  the  we  form  to  mixing  measures  the  be  an  the  r a t i o  energy  t h r e s h o l d ; the  and  of  a  (at  number  of  free  for  v a l u e s  of  3  are  trapped  and  and  l e s s t h e i r  1  wave  of  is  If  s o l i d  enhancethese  v a l i d i t y  and  the  r o l e  the  2,  of  that  the  l i n e ,  of  Chapter  composed  p a r t i c l e s  t h r e s h o l d  wave  in  MW.  With  100  f i e l d  a  r e c a l l c o n -  these  of  the  a m p l i t u d e  c o u l d  the  p o t e n t i a l  in  trapped  than  .04.  the  C0  have  the  a E  2  of  / 2  ~  energy  at  2 7 3 ° K  we  m i x i n g  power  3  a  c r e a t i o n  i s  2  o p t i c s  s t r e n g t h  p a r t i c l e ,  that  f o r  Torr  the  p o t e n t i a l  maximum  c o n c l u s i o n  the  domi na t e s .  30  of  the  a d d i t i o n ,  no  f o r  emphasized  the  the  e l e c t r i c  volume  which  Figure  phase.  In  wave's  18.  d e n s i t y about  to  over  by  E x p e r i m e n t a l l y , of  of  power,  of  versus  induced  e x p l a n a t i o n  induced  out  the  beneath  trapped  r e p r e s e n t e d  enhancement of  e v a l u a t i o n  Returning of  of  2  i n c r e a s i n g  p o s s i b l e  180°  p a r t i c l e s  C0  s a t u r a t i o n  in  A  free  are  i n  with  r e q u i r e s  c o n t r i b u t i o n s  the  that  model.  t r i b u t i o n s  of  saw  o b s e r v e d ,  t h e o r e t i c a l  trapped  r e s u l t s  t h r e s h o l d  o b s e r v a t i o n s  that  the  seen  used 4  x  10" of  a  f i n d ~  t h i s  10 3  from  v o l t s / m  8  ev.  S i n c e  w e l l  to  3  -  0 . 1 6 ) .  .04 Thus  diagram  such  as  Figure  1,  p a r t i c l e s  must  be  n e a r l y  equal  As  c o n t r i b u t i o n  3 to  i n c r e a s e s , the  wave  more  p a r t i c l e s  a m p l i t u d e  70  One n e g l e c t i n g F i g u r e  1  e n t i r e l y  the  that  a m p l i t u d e  the  with  free  p a r t i c l e s  at  =  @  trapped  carbon  of  show  no  d i o x i d e , At  t h r e s h o l d  with  c a s e s ,  observe  e n t i r e l y .  We  p a r t i c l e s  alone  a  s a t u r a t i o n  in  by  themselves  the  power.  However,  i n d i c a t i o n  25  power  f o r  power the  there but  of  in  from  d e t e c t o r  the by  n i ,  the  t o t a l  the  the  the  see  from  observed  t h r e s h o l d  r a t i o  of  =  a  Torr  i s  f o r  powers  -  i n v o l v i n g  10_1* the  m  and  w a i s t s  the of  H E  the  we  i n t o  gives  U n l i k e i s of  In we  2 }  a  I2^v.  mi x  i s An  the e s t i m a t e  the  presented  the  a  important  from  e x p r e s -  e x p e r i m e n t a l  f 1 u c u t a t i o n s . the  enhancement  (.18)  th  a  Torr  produced.  a  T o r r .  enhancement  i n c o h e r e n t  2ira  25  50  dependence  3 . 2 . 4  s c a t t e r e d  these  of  at  c a l c u l a t e d  S e c t i o n  H2.  i n d i c a t i o n  s e c t i o n s ,  i s  as  amplitude  e v i d e n t  the  be  and  25  wave's  p r e d i c t e d  now  In  and  e x c e p t i o n  can  =  50  p o s s i b l e  t h a t  coh  i n t e n s i t y  p r e v i o u s  power  I  where  by  s c a t t e r e d  the  d e n s i t y  c o h e r e n t  E  i s  r e s p o n s i b l e  number  of  nothing  in  enhancement.  for  the both  p o s s i b l e  mentioned  q u a n t i t y ,  observed  shown  Torr  q u a n t i t y  f l u c t u a t i o n  has  s a t u r a t i o n  d e p a r t u r e s  p h y s i c a l  Forming  no  with  As  t h i s  19  mixing  a p p a r e n t .  sions  p a r t i c l e s  show  a r i s e s  .04.  f u n c t i o n  of  p o s s i b i l i t y  i n c r e a s i n g  F i g u r e  a l l  f r e e  d i f f e r e n t  th  f a c t o r the  (.18)  m i x i n g  a r i s e s  beams,  from  i n c i d e n t  the beam  term and  71  d e t e c t i o n thermal sents and  o p t i c s .  spectrum  the  the  We as  can  w r i t e  < | n ( k ) |  e q u i l i b r i u m  2  >  t  the  ^  =  d e n s i t y ,  V  n  frequency  i s  the  f o r  a  (see  assuming  a  Gaussian  n(f)  performing  a  q u a n t i t y  wave's  a  i s  n  r e p r e -  0  e x p e r i m e n t a l  volume  =  n  0  +  F o u r i e r < | n(k)  s p e c t r a l  nio  e x p [ - l / r  t r a n s f o r m °ne  12>con  a m p l i t u d e  n  r  is  the  the  diameter  width  of  the  0  at  for  ( x  in  the  d e t e c t i o n  +  2  show  y  n\0  w a i s t  of  volume  23)  2  ) ]  of  cos  and  the  form  kz  then  t h a t  the  a  r*  plasma  d e n s i t y  space  has  TT 3  a  equation  can  (k)  =  2  gas  f o r  da)  d i s t r i b u t i o n  n „ ( k )  where  where  n e u t r a l  S ( k , t o )  the  g,  f a c t o r  1  By  V  0  i n t e g r a t e d  the  (24)  e v a l u a t i n g induced  form:  (25)  the and  m i x i n g n i  0  i s  beams, given  by  (24).  With  these  s u b s t i t u t i o n s ,  E  using  V  =  irr a. 2  =  .18  2lik  the  r  "  enhancement  f.2  n o *  n  * ° -  becomes  36 ^ J J L no  (26)  72  As by  an  a s i d e ,  c o n s i d e r i n g  the  t h i s  r a t i o  is  ^ o f /  1  seen  to  have  ^ *  Using  the  the  proper  form  r e l a t i o n s  t h a t  2  I  c  o  <*  n  n?  V2and  0  I  <*  t h  Returning volume  V  given  -  3  x  10"  to  2  .  n  0  i s  in  The  m  an  the  d e n s i t y  n  as  r a t i o  the  1.5  produced  x by  of 10  3  expect  (26),  induced  m  1 8  0  -  3  E  =  800.  as  - 3  mixing  can  x  1  0  for  E  n  an  change  * °  ]  ^  .  experimental  i n  d e n s i t y  i s  E  *  L oj n  c o n s i d e r x  perturbed  the  We  = n o  example  c o n s i d e r e d ,  ~  we  - 3  system  nio  fi/4ir  by  As C 0  V  equation  m3  1 1  ^  where  „  n  a l s o  IO21*  m  - 3  This  to  the  the  f l u c t u a t i o n  e s t i m a t e  the  plasma  e x p e r i m e n t s .  g  =  0 . 8 ,  n  =  x  m"  2  x  10"  and  6  m  0  -  4  2  2  x  ( n  for  150  the 1 0  / n  0  Torr  d e t e c t i o n )  d e n s i t y  ~  5  x  IO  and  in  number  of  the  d e n s i t y  p r o c e s s .  the  1 0  with  e q u i l i b r i u m  i n  2  and  gives  produced  0  r e s u l t s  10  3  1 6  and n r  E 3  = .  s i z e  S e t t i n g 100  g i v e s  the  f l u c t u a t i o n s f a c t o r  ( m o / n  0  )  ~  - 7  Chapter  5  CONCLUSION  The that to  was  developed  study  the  p a r t i c u l a r t i o n s in  p r e v i o u s  of  o p t i c a l  i t  has  plasma  the  that  a of  d e n s i t y .  This  a  t h i r d ,  a  to  of  F u r t h e r  be  in  the  that  were  of  two  x  10"  s t r o n g ,  be  i n  d e t e c t e d beam  over  the  and  by  f o r  100  beams,  those  i o n i z e d  produced  p r o c e s s  l i g h t  which  e l e c t r o n  s c a t t e r e d  measured  i n  f l u c t u a t i o n s , between  the by  e x p e r i m e n t a l l y the  t h e o r e t i c a l  p o s s i b l e  k-spectrum  of  the  wave  73  the  in  Agreement  i n v e s t i g a t e induced  from  comparison  to  for  an  enhancements  p r e d i c t e d  s i n u s o i d a l  plasma  with  experiments  v i r t u a l l y  In  f l u c t u a -  m i x i n g  e q u i l i b r i u m  the  were  found  f u l l y  was  the  thermal  was  a  wave  times  6  random two  l a s e r  by  model  performed  s t a t i o n a r y  induced  the  g a s e s .  l a s e r of  i n t e n s e  that  d e n s i t y  was  enhancements  harmonics wave  from  f a c t o r  measured model.  wave  i n t e n s i t y  s c a t t e r i n g  w i t h i n  2  d i a g n o s t i c  s c a t t e r e d to  about  experiments  conducted  s t a t i o n a r y  amplitude  d e s c r i b e d  can  n e u t r a l  Experiments show  have  shown  d e n s i t y  and  c h a p t e r s  mixing  been  p a r t i c l e  both  and  three  mixing  sub-  showed powers  the used.  74  A p p r e c i a b l e gas  atoms  were  s t a t i o n a r y beams from of  of a  the  two  d e t e c t o r been of  d e n s i t y same  t h i r d  over  experi  and  plasma of  for  hydrogen,  Evidence causing  has t h i s  mixing  t h r e s h o l d  p o s s i b l e  gas  produced  in  i s  shown  1000  two  by  using  enhancements be  n e u t r a l a  i n t e n s e  l i g h t  l a s e r  s c a t t e r i n g  l i g h t  o b s e r v e d .  apparent  that  c o u l d  by  of  A g a i n ,  s c a t t e r e d  were  the  d e n s i t y  m i x i n g .  d e t e c t e d  magnitude  in  a c t u a l l y been  The  i n t e n s i t y r o l e  enhancement  the  o p t i c a l  in  from  the  the has  system  s c a t t e r e d  expected  wave  r e q u i r e d  d e p a r t u r e s  With  f o r were  dependence these  f u t u r e  d i r e c t i o n  i n  l i g h t gas  to the  both  induce  to  to  wave  these from  the  h i g h e r  i n  done. power  was  a  has  In  s  i n c r e a s d e f i n i t e  one  With  plasma  cannot  By  ex-  p r e d i c t i o n s  mixing  mind  p r o v i d e d  most  s i m p l e  l a s e r s .  probably  d e t e c t a b l e  model  on  range.  d i o x i d e  w i t h  of  e f f e c t s .  enhancement  be  there  of  and,  are  carbon  observed  t h e o r e t i c a l  noted of  a  p r e s s u r e  e x p e r i m e n t a l  with  a l s o  a m p l i t u d e  c o l l i s i o n s  a d d i t i o n ,  experiments  work is  was  the  gas  the  that  experiments  and,  of  w i t h i n  show  amplitude  Examination  q u a d r a t i c  suggest  In  power  power  to  that  i n c r e a s i n g  decreases  b e h a v i o u r . of  i n d i c a t e  w i t h  presented  e x p l a n a t i o n s  p e r i m e n t s ,  obvious  experiments  s a t u r a t e s  beams'  a m p l i t u d e .  but  be and  e x p e r i m e n t s ,  wave  s a t u r a t i o n  a  o p t i c a l  d e t e r m i n i n g i t  the  ments.  induced  of  of  in  by  Enhancements  n e a r l y  the  ing  can  frequency  i n  The  a  wave  orders  p l a y s  i n t e n s i t y  induced  beam.  pursued  the  a l s o  f l u c t u a t i o n s  the  power. help most  d r i v i n g  the  75  d e n s i t y  wave  e x p l o r e  a  to  whole  i n t e r a c t i o n s P o s s i b l e The only  a  f i e l d  in  plasma  frequency  l a r g e r  the  mixing  beams.  This  use  of  a  can  e n v i s i o n  carbon  a  in  ( e . g .  b e t t e r  f u s i o n  d i o x i d e  phase The  which  the  v e l o c i t y )  would  have  g a s e s ,  waves in  the  the  a  f a l l  mixing l o n g e r  to  be  l a s e r .  an  ideal  d e n s i t y  low of  the  regime.  t h i s  r e a l m .  suggests f o r  s i t u a t i o n  E x t r a p o l a t i n g with  for  wave-  with  p r o c e s s e s  the  one  l o n g e r  p l a s m a s ,  not the  f u r t h e r  even  b a s i c  to  w a v e - p a r t i c l e  wave-length  HCN)  in  i n t o  process  done  been  f o c a l to  to  the i n v o l v e d  of  and  plasmas  the now  region  have  has  e l e c t r o n in  i s  the that  beams been  of  been  e x p e r i m e n t a l l y  plasma  resonance  r e g i o n  of  of  ion  the  s l i g h t l y  p o s t u l a t e d  zero  (high  f r e q u e n c y .  r e s o n a n c e ,  d i f f e r e n c e  but  to  date  f r e no  p u b l i s h e d .  the  experiments  the  the  ever  volume. d e t e c t  induced  in  mixing  i n c r e a s i n g  leads  c o n t i n u e d of  [3],  r e q u i r e  For in  a l s o  a l s o  and  t u r b u l e n c e "  experiments  r e g i o n  Experiments  r e s u l t s  the  mixing  i n t e r e s t i n g  quency.  wave-wave  f e a s i b l e  machines.  i n  l a s t  of  understanding  O p t i c a l v e r i f i e d  becomes  " s t r o n g  e f f e c t s  appears  mixing  aim  of  s o - c a l l e d  but  i t  n o n l i n e a r  dependence power  l a s e r s  of  h e a t i n g  h i g h e r  l e n g t h  a m p l i t u d e s ,  wave.  concerned  l a s e r  power  i n c r e a s i n g  However, p o s s i b l e  to  w i t h  o p t i c a l  induce  l a r g e  problem  f u r t h e r  of  gas  experiments  sub-harmonics  i n  the  mixing amplitude  breakdown can  be  k-spectrum  REFERENCES  1.  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W h i t e ,  697  (1 9 7 3 ) .  L a v a l ,  T . M .  O ' N e i l ,  M.N.  Sagdeev,  JETP  L e t t e r s  (USA)  G.  14,  J . 3J_,  Soures 1 184  Phys,  Rev.  Rosenbluth 17,  35  and  M . J .  29.  W u l f f ,  Phys.  L u b i n ,  Phys.  Rev.  (1 9 7 3 ) .  Sagdeev, R.Z. and A . A . G a l e e v , W.A. Benjamin (1969).  Z.  537  —  28.  H . ,  (1 9 6 9 ) .  (1 9 6 8 ) .  (1 9 7 3 ) .  27.  (1971).  ~ ~  M.N.  L e t t e r s  " E l e c t r o n i c  p u b l i s h e d ) .  (1974).  25.  Columbia  I l l , Oxford  be  Rev.  3 2 , ~ ~  B r i t i s h  G i l b o d y ,  V o l .  D i e t z  2 0 ,  of  H.B.  (to  Phys.  D.R.  Phys.  (1965).  Impact  R.M.  and  1_0,  Burhop  p r i v a t e  M.A.  K r u s k a l ,  E . H . S .  Ionic  J . ,  M.D.  K a t z e n s t e i n ,  M . S c .  H . S . W . ,  and  R e s .  F l u i d s  D.E. and J . 207 (1969).  21.  24.  (1 9 5 7 ) .  Phys.  and  Green  546  J .  s  J . M .  1_50,  61 4  " N o n l i n e a r  (1 9 5 8 ) .  Plasma  T h e o r y , "  78  30.  Huddl e s t o n e ,  R.H.  and  Techniques,"  E.E.,  AP  31.  S a l p e t e r ,  Phys.  32.  C h u r c h l a n d , M.T. and 655 (1974).  33.  B y r d ,  S.L. (1  Leonard,  "Plasma  D i a g n o s t i c  965).  Rev.  R.A.  120,  1528  N o d w e l l ,  (1960).  Can.  J.  Phys.  52,  P.F. and M.D. Friedman, "Handbook of E l l i p t i c I n t e g r a l s , " Lange, Maxwell and S p r i n g e r L t d . (1954).  i  79  APPENDIX  I  DETAILS OF THE PLASMA  1.1  Design  C o n s i d e r a t i o n s  When  the  mixing  was  on  c h a r a c t e r i s t i c s  the  source  f i r s t  experiment  p r o p o s e d , of  to  study  c e r t a i n the  high  power  r e s t r i c t i o n s  plasma.  I d e a l l y  o p t i c a l  were the  placed plasma  should (I  )  have as h i g h an e l e c t r o n d e n s i t y as p o s s i b l e ( > I 0 c m ) in o r d e r t o m a x i m i z e t h e number o f e l e c t r o n s i n t e r a c t i n g w i t h t h e l a s e r beams; 1 6  ( ii)  (iii)  ( iv)  (v) (vi )  - 3  be f u l l y i o n i z e d i n o r d e r t h a t t h e e f f e c t of n e u t r a l s c o u l d be d i s regarded ; have a M a x w e l l i a n v e I o c i t y distrib u t t o n f o r t h e i o n s and e l e c t r o n s in o r d e r to s i m p l i f y the c a l c u l a t i o n s o f t h e wave m i x i n g t h e o r y ; r a d i a t e a minimum amount o f b a c k g r o u n d l i g h t from f r e e - f r e e and f r e e - b o u n d t r a n s i t i o n s in order to a l l o w o b s e r v a t i o n s o f low i n t e n s i t y s c a t t e r e d light; have  reproducible  characteristics;  p r o v i d e c o n v e n i e n t a c c e s s f r o m many d i r e c t i o n s f o r the v a r i o u s o p t i c a l paths r e q u i r e d in the e x p e r i m e n t .  80  As a l l  these  new  plasma  ments.  no  c o n d i t i o n s source  A  pulsed  high  e l e c t r o n  with  a  helium  to  a  was  was  chosen  decided  of to  f o r  chosen and  to  keep  which  design  the  wave  because  of  d e n s i t i e s  equipment.  r e j e c t e d  a  vacuum  pulsed  and  s a t i s i f e d b u i l d  mixing the  t h a t  v i s i b l e  and  s t a b i l i z a t i o n  a  e x p e r i -  reasonably can  O p e r a t i o n  the  arc  s t a b i l i z a t i o n of  i t s  imposes  d e t a i l e d  p a r t i a l l y  s a t i s f i e d  a  because i t  As  confinement  i f  Magnetic  l i m i t a t i o n s  the  was  a v a i l a b l e  be  in  achieved  hydrogen  plasma  r a d i a t i o n  minimum.  a b l e .  i t  r e a d i l y  s p e c i f i c a l l y  amount  c o n s i d e r a t i o n s  was  i t  arc  Plasma  but  was  temperatures  minimum  or  plasma  w a l l  a l l  the  vessel  on in  i s was  be  o p t i c a l next  s t a b i l i z e d c r i t e r i a to  and  access  to  s e c t i o n ,  f o r  t h i s  Using  o u t l i n e d  design the  plasma. f i n a l  hydrogen  above  e x c e l l e n t  and  the the  important  r e p r o d u c e -  c o m p l e x i t y  a r c .  have  hot  c o n s i d e r e d  t e c h n i c a l  the  proved  to  are  and  s t r a y  in  d e s i g n gas a d d i t i o n  l i g h t  character!" s t i cs .  I.2  C o n s t r u c t i o n  1.2.1  a  g l a s s  the  O p e r a t i o n  C o n s t r u c t i o n  F i g u r e v e s s e l .  and  The tube  s i m i l a r  1-1  the  d e t a i l s  e l e c t r o d e s which  of  are  serves  " E i e r u h r "  type  Plasma  the short  to of  Vessel  c o n s t r u c t i o n aluminum  c o n f i n e d e v i c e  the [29],  of  the  plasma  c y l i n d e r s  i n s i d e  plasma. a  3  cm  U n l i k e s e c t i o n  i s  a n o d e a s s e m b l y 3  g l a s s  t u b e  c o p p e r ball w i n d o w a s s e m b l y  c a t h o d e | j * A A- ' V *  r  y  v  a s s e m b l y  ,  :  r  J  insulator  i  "Li J  ' 'g' ^ =5::5s ' g r o u n d to v a c u u m  F i g u r e  1-1.  s y s t e m  Plasma  Vessel  (approx.  1/3  Assembly s c a l e ) .  81  removed This  from  was  l a s e r  beams  metal  g l a s s was  to  a l l o w  d i r e c t  to  reduce  the  and  d i f f i c u l t  between  b a l l .  Eighteen  of  the  t h e s e .  In  lenses  or  These  from  the  This the  as  at  1-1 to  b a f f l e s  f o r  the  by  two  machined the  a i t  hemispheres,  c e n t r e  the  in  because  in  above  " b a l l "  i n h e r e n t  i s  l o c a t e d  in  of  the  the  b a l l ;  h o r i z o n t a l and  covered  below by  each  windows,  0 - r i n g s ,  black  w i t h i n  to  The  in  3  prevent  nylon  tubes  cm  the  of  extend plasma.  r e f l e c t i o n s  from  system.  (cathode) prevent b a l l  " f u n n e l s . "  e n c l o s e d  plasma  are  ,  e l e c t r o d e  the  plasma  are  into  in  45°  s e a l e d  high  of  g l a s s  p a r t i c u l a r  are  equator  the  to  and  ports  windows  in  of  ports  Figure  v o l t a g e  the  d i a m e t e r .  spun  E l e c t r i c a l l y  done  i n s i d e  and  in  two  problems  volume  the  d e t e c t i o n  e l e c t r o d e was  have  i n c l i n e d  dumps  l i g h t  l i g h t  the  o p e r a t i o n  shown  the  upper  lower  others  in  f u n n e l s  around  v a r i o u s  serve  e n t e r i n g  g l a s s  to  funnels  chosen  o b s e r v a t i o n  l i g h t As  inward  the  evenly  and  to  experimental  midway  plane  was  the cm  20  s t r a y  copper  forming  access  vacuum  chamber  l e a s t  spaced  a  tube,  o p t i c s .  copper  The  s i x  the  d e t e c t i o n  m a i n t a i n  v e s s e l  the  of  and  s p h e r i c a l  is  w a i s t  done  To a  the  b a l l  which the  i s  i n s u l a t e d  (anode) is  at  ground  d i s c h a r g e  i n s t e a d  of  to  but  from  the  not a l s o  only  from  from  the  p o t e n t i a l . a t t a c h i n g  cathode.  to  82  1.2.2  Operation  The  of  plasma  i s  a p p r o x i m a t e l y  300  measured  Rogowski  F i g u r e  by  1-2  F i g u r e  a and  ysec  i s  1-3  s t o r e s i s  e l i m i n a t e  by  any  i o n i z a t i o n  a  a  a  of  the  As  and  f e r r i t e  e l e c t r o d e s  b a l l a s t  p r o t e c t s  the  and  power  d i s c h a r g e .  I.3  Measurements  A f t e r s e r i e s  of  to the  main  in  in  6  kA  c u r r e n t  the  c u r r e n t  p u l s e  waveform, i s  d i s c h a r g e  the  energy  as  shown  c i r c u i t  c a p a c i t o r  of  20  standard  i n of  the  f o r m a t i o n  the  of  produce  i s  the  combined  c i r c u i t w i t h  a  the  of  kv.  The  a r c ,  of  F i g u r e  The in  c u r r e n t  spark To  a  p r e -  c o n t i n u o u s  kfi  connected  glow.  the  20  by  400  i n d u c t a n c e  from  at  t r a n s f o r m e r .  a c c o m p l i s h e d  i n  kj  bank  t h r e e - e l e c t r o d e  p u l s e  i n d u c t o r  1-3  a  wire-wound a c r o s s  r e s i s t o r the p u l s e  the serves  c i r c u i t of  the  Performance  c o n s t r u c t i o n  experiments  c h a r a c t e r i s t i c s .  l i n e ,  s e r i e s  supply  of  a  is  shown  r e s i s t o r  a  a  i n t e g r a t o r ,  v o l t a g e  plasma  supply  a  by  lag  power  a  by  maximum  s e p a r a t e  as  The and  delay  high  time  d i s c h a r g e .  plasma  c o i l  i n i t i a t e d  t r i g g e r e d  glow  l o n g .  by  C i r c u i t  1-3.  d i s c h a r g e gap  D i s c h a r g e  produced  produced  B a s i c a l l y F i g u r e  the  was  and done  i n i t i a l to  t e s t i n g  determine  the  was  c o m p l e t e ,  p l a s m a ' s  Plasma ( V e r t : Speed:  c u r r e n t 2  p u l s e  k a . / d i v . 50  u s e c / d i v .  + to  p u l s e x f m r  trigger unit 9 o  4 0 0k  kjUULr r  =  r T m 2L  8kv  ferrite  I  T T P — w \ — D ' o6 P l a s m a  o  L = 22  uH  = 1 0 UF/:2 0  k v d c  ! 0  Figure  s e c t i o n s  1-3.  Plasma  d i s c h a r g e  c i r c u i t .  CO  85  Plasma s t r e a k  photography;  broadening e l e c t r o n b u t i o n  of  the  of  f o l l o w i n g  gathered  in  taken  v e r t i c a l  arc  onto  entrance  of  0 . 5 .  A  h o r i z o n t a l system funnel  The  the  on  the  a  plasma  r o t a t i n g  s l i t  of  v e l o c i t y  d i s t r i -  s c a t t e r i n g . t e c h n i q u e s with  of  0.5  plasma  was  i s  the  p r e s e n t e d  i n f o r m a t i o n  g l a s s  camera mm  to  determined  m i r r o r  the  the  o b s e r v a t i o n  some  s l i t  the  u n i f o r m i t y  a  a  width  measurements  a  dove  s t r e a k  was  The  imaged  m a g n i f i c a t i o n  d e f i n e d  o b s e r v e d .  by  camera.  f u n n e l s with  then  be  smear  over  system then  f u n n e l s . of  the  helium  at  to  r o t a t e  d e f i n e d  a  1  The  l a r g e r  mm  high  o p t i c a l than  the  This  a  was  the  p r i s m plasma  v e r t i c a l done  to  was image  s l i c e measure  of the  i n s e r t e d by  9 0 ° .  plasma a x i a l  a r c .  Measurements and  Stark  r e p r o d u c i b i l i t y ,  l i g h t  t o g e t h e r  between  of  o p t i c a l  camera  between  by  o p e n i n g s . For  into  of  width  s l i c e  a l l o w e d  measured  by  Photography  column  s l i t  measured  c a s e .  S t a b i l i t y  the  were  e l e c t r o n  p a r t i c u l a r  s e c t i o n s ,  Streak  photographs  and  l a s e r  these  each  1.3.1  by  was  plasma  d e n s i t y  measured  confinement  d e n s i t y  l i n e s ;  t e m p e r a t u r e ,  were  and  e l e c t r o n  hydrogen  Each in  s t a b i l i t y  f i l l i n g  were  made  p r e s s u r e s  on from  d i s c h a r g e s 1  to  100  in Torr  hydrogen and  86  c u r r e n t s charges and  from were  3  to  in  S t a b l e ,  hydrogen  at  r e p r o d u c i b l e  the  higher  d i s -  p r e s s u r e s  c u r r e n t s .  h o r i z o n t a l  and  the  used is  k i l o a m p s .  produced  F i g u r e a  6  a l l  45.6  image.  l e n g t h  on  to  6  f i l m  of  c u r r e n t  flow  diameter  of  However,  a f t e r  diameter  of  lower  the  more  arc  than  about  the  are  about  shows  i s  u n s t a b l e  cm  160  f u n n e l s  (the  psec and  350  that  arc  burns  25  H2  c o n d i t i o n s Streak in  speed  the  to  in  the  t h i s  c u r r e n t  extends  on  Torr  y s e c .  when  the  f i r s t  to  a  p h o t o g r a p h ) .  c o n s t r i c t s  s t a b i l y  using  same  seen  and  a p e r t u r e  the  i s  upward  are  p o r t i o n  3  arc  c o r r e s p o n d s  f e a t u r e s  very  the  flows  s e c t i o n  i n t e r e s t i n g  the  e x p e r i m e n t s ,  time  p u l s e ,  of  p r e s s u r e  These  and  luminous  The  f i l l i n g  l a s e r - p l a s m a  the  the  photograph  kA.  e n t i r e  Several photograph.  a The  i s  subsequent  The  s t a r t s  shows  pulse  ysec/cm  f i g u r e . t o t a l  s l i t  c u r r e n t  in  1-4  u n t i l  to the  the c u r r e n t  s t o p s . The  e a r l y  Inside  the  g l a s s  in  gap  between  the  u n s t a b i l i z e d  f u n n e l s them  the one  behaviour  plasma would  i s  i s  to  be  e x p e c t e d .  w a l l - s t a b i 1 i z e d  expect  the  plasma  but  to  expand. The some by  time  Dr.  flow  B.  i s  a c t u a l s t i l l  Ahlborn  s t a b i l i z a t i o n  c o o l e r  o u t e r  reason  not and due  s e c t i o n s  t h a t  known. Dr. to of  J .  One  arc  c o n s t r i c t s  l i k e l y  Uhlenbusch  mass the  the  t r a n s p o r t  column.  a f t e r  e x p l a n a t i o n  is  the  p r o c e s s  r a d i a l l y  L i t t l e  from  o f f e r e d of the  i n v e s t i g a t i o n  Figure  1-4.  Streak  photograph  H o r i z o n t a l  Figure  1-5.  Streak V e r t i c a l  of  arc  of  arc  s l i t .  photograph s l i t .  88  was  done  in  d i r e c t i o n  t h i s  of  area  the  as  in  of  i t s  1-5  Figure  shows  the  plasma  under  1-4  but  now  w i t h  v e r t i c a l  c e n t r e d  on  the  a x i s  of  the  f u n n e l s .  seen  be  u n s t a b l e  i n  i t s  e a r l y  to  v i s i b l e  r a d i a t i o n  1.3.2  in  the  the  a  some  s e r i e s  broadening were  n e c e s s i t y  u n f o l d i n g  [30].  A x i a l  s i m p l e r  and  were  of  and  the  cromator. o r d e r , EMI  onto  a  by  hole  same  the  c o n d i -  s l i t  the  image  plasma  to  the  was  used  w i t h  have  i s  i n c r e a s e d  entrance  and  e x i t  s l i t s  done  abandoned  through the  an  arc  to  i t s  of  1200  line/mm  was  Spex  at  taken  the  i n v e r s i o n proved  new  upper  c e n t r e .  s l i t  set  to  Abel  through a  measure  due  a  d e n s i t y  r a d i a l  column  f a b r i c a t i n g  both  Data  e l e c t r o n  I n i t i a l l y  plasma  a  the  were  through  e n t r a n c e  p h o t o m u l t i p i i e r .  the  and  of  soon  through  This  9558  to  L i n e  l i n e .  r e s u l t s  imaged  the  stages  the  but  the  with  prisms  e l e c t r o d e  of  a c c o m p l i s h e d  assembly  Again  experiments  attempted  e l e c t r o d e l e n s e s  nature  c o n s t r i c t i o n .  the  measurements  a  i n d i c a t i o n  of  of  measurements of  i t s  Broadening  p r o v i d e  plasma  Stark  a f t e r  Stark  To  secondary  p r o j e c t .  Figure t i o n s  because  the  A  s e r i e s  hole  1700  mono-  g r a t i n g  50u  and  i n  i n  1st  an  s h o t - b y - s h o t  at  o i n t e r v a l s  of  5  A  P l o t t e d the  Ha  l i n e  at  a  around in time  the  Figure 200  l i n e 1-6  ysec  c e n t r e . i s  the  a f t e r  measured  the  s t a r t  p r o f i l e of  the  of  4816  4846  4876  4906 WAVELENGTH (X) CO  F i g u r e I - 6 .  P r o f i l e o f H  Q  l i n e a t t  =  2 ' 0 0 y s e c .  90  d i s c h a r g e . of  the  f i t  The  p o i n t s  curve  e r r o r  about  by  eye  the  to  Assuming and  f o l l o w i n g  [30] or  one  the  can  q u a r t e r  bars  of  The  the  standard  s o l i d  l i n e  i s  d e v i a t i o n a  best  d a t a . best  simple  estimate  widths  mean.  the the  r e p r e s e n t  f i t  i s  a  treatment  the the  i n  e l e c t r o n c u r v e .  L o r e n t z i a n Huddlestone  d e n s i t y  For  p r o f i l e  the  from  data  and the  of  Leonard h a l f  Figure  1-6  o  the to  f u l l an  width  at  e l e c t r o n  h a l f  d e n s i t y  maximum  i s  of  2  n  g  ~  about x  10  16  A,  c m -  1 6  3  c o r r e s p o n d i n g .  A  f u l l  width  o  at  q u a r t e r  maximum  i n d i c a t i n g  that  the  Despite t h i s  i s  an  numbers ments  the  1.3.3  next  Laser  ments,  a n a l y s i s  was  obvious  d e t e r m i n i n g e l e c t r o n volume  v e l o c i t y  curve  s i m p l e  L i g h t  of  of  t r u l y  data  along w e l l  n  w i t h  ~  2.5  x  10  c m  1 6  the  the  nature  plasma This  parameters and  d i s t r i b u t i o n  by  l i n e the  and of  more  an  the  the  f a c t  s i g h t ,  ruby  [18]  l a s e r  was in  that  measure-  the  mixing  l i g h t  to  the  of  measure  experimental of  e x p e r i -  s c a t t e r i n g  technique  used  i n d i c a t i o n  f u n c t i o n .  3  these  a c c u r a t e  o p t i c a l  now-standard  d e n s i t y  provide  of  -  L o r e n t z i a n .  h a n d l i n g the  g  S c a t t e r i n g  c h o i c e .  to  not  to  s e c t i o n .  plasma  a l s o  c o r r e s p o n d s i s  d e n s i t y  temperature  and  A  s u p r i s i n g l y  Because  an  27  the  average  agree  of  of  the plasma  e l e c t r o n  ,  91  R e f e r r i n g the  o p t i c a l  from  the  b a s i c  d i a g n o s t i c at  100  MW  and  the  forward  Figure  f o r  the  modes  plasma  into  from and  d i r e c t i o n  of  shown  focused  the  6  wave  arrangement  s c a t t e r i n g  t r a n s v e r s e i n t o  system  to  a  mixing  here a  to  The  l i g h t  1700  one  For  ruby  p i n h o l e  3  sees  experiments  1-7.  the  beam.  Spex  Chapter  Figure  s c a t t e r e d  by  of  operated  s p u r i o u s  was  d e t e c t e d  d e r i v e d  s i m p l e  was  remove  p i n h o l e  i s  the  l a s e r  that  at  monochromator  then 60°  imaged  from  w i t h  a  the  1200  o  line/mm  g r a t i n g  m u l t i p l i e r .  blazed  Entrance  at  7000  A  and  and  e x i t  s l i t s  an  RCA  were  C31034  both  photo-  s e t  at  300u  o  widths width  to at  give h a l f to  a  t r i a n g u l a r  t r a n s m i s s i o n .  was  masked  was  n e g l i g i b l e .  a  height  Figure the  s c a t t e r e d  a f t e r that  the a l l  e r r o r  mean.  to  the  data  using in  Several p l o t s This of  show i s  the  due  400y.  1-9  the  two  t y p i c a l  were  taken  c u r r e n t  were  a  l e a s t  theory  [31]  of  f u l l  e n t r a n c e  s l i t  at  3  the  d e t e c t o r  s p e c t r a 250 the  of  of  ysec same  performed.  curve  i s  A  at  p u l s e ,  d e v i a t i o n  time The  the  squares  f i t  and  p a r a -  the  diagrams.  f e a t u r e s  l a r g e l y  l i g h t  standard  S a l p e t e r ' s  the  the  S t r a y  show  Both  p r o f i l e  a d d i t i o n  experiments  s o l i d  s l i g h t l y  l e n s e s  In  plasma  mixing  The  shown  and  the  i n d i c a t e  the  of  l i g h t .  of  o p t i c a l  of  meters  1-8  ruby  s t a r t  bars  i n s t r u m e n t  should  d i f f e r e n t to  e x a c t l y  the on  be  temperatures  problem  the  noted.  plasma  of  F i r s t and  a l i g n i n g  a x i s .  a  the  two  d e n s i t i e s . the  Because  f o c i i the  photomultiplier ^ / '  m m  w i n d o w ^ ^ ^ 127 m m  R u b y  1 6 6 m m  laser  p i n h o l e 8 5  m m  l i g h t H e - N e  127  d u m p  l a s e r  F i g u r e  alignment mirrors  1-7.  O p t i c a l  system  for  d i a a n o s t i c  s c a t t e r i n g  93  F i g u r e  1-8.  Measured f i t t e d  to  spectrum a  of  computed  s c a t t e r e d p r o f i l e .  l i g h t  94  Figure  1-9.  Spectrum of s c a t t e r e d anomaly at w .  l i n h t  showinn  95  g l a s s  f u n n e l s  p o s i t i o n A l l  made  to  i s  experiments  were  performed  time  they  were  e r r o r ,  t o l e r a n c e s  to  in  the  about  t h i s  the  a c t u a l  m i l l i m e t e r .  a x i a l  plasma  between  a  the  r e g i o n  and  parameters values  f e l l ,  p l o t t e d  in  f i g u r e s . This  s i z e  doubt  measured  experimental  these  in  c l o s e  a x i s  w i t h i n  of  plasma  the i s  i s  agree  i n f o r m a t i o n ,  e r r o r  b a r s ,  s u i t a b l y The  1-9  not  the  each  of  are  t h a t , w e l l  together  leads  to  w i t h  the  p o i n t  to  be  noted  c o m p a r i s o n ,  from  way  of  w i t h  the  value  o b t a i n e d  from  p o i n t  concerns  the  0  that  the  r e p r o d u c e a b l e .  second  of H  r e a s o n a b l e  c o n c l u s i o n  by  measurements  the  the  Figures  e l e c t r o n the  1-8  and  d e n s i t i e s  Stark  broadening  .  P  curves the  t h i r d  the  diagrms  e l e c t r o n  other The  i n  The  v e l o c i t y  d i s t r i b u t i o n  experimental i s  M a x w e l l i a n  e l e c t r o n The  1-9  that  c u r v e .  i s  made  f i n a l  more  f a c t  that  t h i s  l i e s  at  wavelength  frequency  tends  f i t s  the  the  f e a t u r e one t h i s  p a r t i c u l a r  to  can  s h i f t support  i s  curves  a  the  so  standard might p o i n t  be i s  these  the  has  p o i n t  a  the to of  i n  away  e x p e c t e d ;  f i n d i n g s  c u r v e s ,  f u n c t i o n .  d e v i a t i o n  above  that No  that  study  data  c o r r e s p o n d i n g the  s o l i d  assumption  w e l l  under  one  the  M a x w e l l i a n ,  d i s t r i b u t i o n i s  that  the  d u p l i c a t e  plasma  v e l o c i t y  than  under  d i s t r i b u t i o n  that  S t a t i s t i c a l l y  a  d e r i v e d  f u n c t i o n  data  c o n c l u s i o n  are  f a c t  Figure  from  however curve  the  the the  and  plasma  Churchland  and  96  Nodwell  [32]  No  f u r t h e r  no  e f f e c t  w i t h  i n v e s t i g a t i o n on  only  in  1.4  Summary  f u r t h e r  to of  anomolous t h i s  experiments  t i o n  and  used  to  p r e c e d i n g  o p e r a t i o n measure In  of  the  hydrogen  arc  a  ( i i )  and  was  i t  plasma  with  ±  the  i s  done  p r o f i l e s . as  i t  mentioned  0.3  x  43,000  These  ±  background  allowed  design  by  had  here  w a l l - s t a b i 1 i z e d  d e n s i t y  of  ;  3  temperature  °K; v e l o c i t y  f u n c t i o n ;  7000  A  of  in  the  smaI I low  s p e c t r a l enough  Ievei  to  s c a t t e r e d  l i g h t .  t o g e t h e r  of  o b t a i n e d  c h a r a c t e r i s t i c s :  e l e c t r o n  d e t e c t i o n  vacuum  c o n s t r u c -  r e s u l t s  p a r t i a l l y  c m "  4000  the  techniques  the  e l e c t r o n  permit  c o n s i d e r a t i o n s  and  r a d i a t i o g  l a s e r  the  e l e c t r o n  around  the  a  1 6  region  f e a t u r e s ,  s o u r c e ,  i s  IO  Maxwellian  ruby  o u t l i n e d  f o l l o w i n g  d i s t r i b u t i o n (iv)  have  parameters  rep roduceabIe  of a  plasma  rep roduceabIe  a  ( i i i )  the  the  2.0  the  anomaly  s e c t i o n s  plasma  b r i e f ,  (i)  access  s c a t t e r i n g  p a s s i n g .  The  pulsed  r e s p e c t  w i t h  v e s s e l S e c t i o n  are 1.  the seen  convenient to  meet  o p t i c a l a l l  APPENDIX  II  PROOF OF THE ERGODIC THEOREM  Appendix  Ergodio  III  makes  Theorem:  use  In  of  the  the  an  a s y m p t o t i c  phase  p a r t i c l e  Proof:  F o l l o w i n g  O ' N e i l  b u t i o n  p a r t i c l e s  between  of 2  [17]  space  t r a j e c t o r i e s  a  d e n s i t y  t r a j e c t o r y  c o n s i d e r i n  temporal  an  small  W and  a  c o n s t a n t .  d i s t r i -  space  r e g i o n  AW.  P a r t i c l e s  w i t h  s h o r t e r  energy  p e r i o d s  W than  those  of  energy  W +  AW,  which  i s  always  the  case  for  o s c i l l a t i o n  s i n u s o i d a l  As  time  patch form f i l l  p r o g r e s s e s  of  p a r t i c l e s  into  two  the  i n  a  p o t e n t i a l  e v e n t u a l l y  97  along  i n i t i a l  phase  W +  have  i s  l i m i t  a  s p i r a l , the  w e l l  i n i t i a l w i l l  s p i r a l  region  t r a j e c t o r i e s  d e -  and  between  w i l l the  98  D e f i n e :  f  =  " c o a r s e  g r a i n "  =  average  value  l a r g e  phase  c o n t a i n  By  the  d e n s i t y  L i o u v i l l e  rate  For  the  areas  strands  of  the  in  the  asymptotic  of  flow  of  phase  r a t e  of  flow  of  c o n s i d e r  flow  = =  no. -—'•  past  the  p a r t i c l e s ~ '—— u n i t 11me  =  ( p a r t i c l e  =  fvs  to  of  phase  space  p o s i t i o n  1  2  f l o w .  In  cC Ir o s s i n g  general  area  S  3  d e n s i t y )  phase  enough  s p i r a l ) .  past  space  ; —Hr—" Hr——:7  l a r g e  over  l i m i t :  p o s i t i o n  of  2 - d i m e n s i o n a l  Ax  d e n s i t y  phase  d i s t r i b u t i o n  ( i . e .  t h a t ,  of  Av'  space  a c t u a l  c o n s e r v a t i o n  now  case  the  the  rate  We  of  Theorem,  r e q u i r e s  =  many  d i s t r i b u t i o n  ( p a r t i c l e  space  as  v e l o c i t y )  used  (S)  here:  / p a r t i c l e s l i n e  S  =  S  (Ax2  flow  with  +  a c r o s s  v e l o c i t y  Av2)  v  99  V e l o c i t y  v  i s v  r e s o l v e d  plus  the  p o i n t  to  a n y t h i n g  i n t o  two  change  in  components; v e l o c i t y  p o i n t  on  the  phase  but  a  h o r i z o n t a l  v  =  v  c o n s t a n t ( i . e .  space  l i n e  to  move  diagram  r e q u i r e s  v e l o c i t y from  along  an  a c c e l e r a t i o n )  T h e r e f o r e  Returning  e q u a t i n g  to  the  is  chosen  +  2  such  t h a t  v  flow  past  1  =  f j  r a t e  of  flow  past  2  =  f  2  2  )  t r a j e c t o r i e s , =  x  of  these  v  p a r t i c l e  r a t e  f i  Now,  i n i t i a l  (y  Ax  Vi ( v  =  0:  Avi +  2  v 2z)*  (Ax  V !  A  V l  the  =  f  2  ( v  2  e n e r g i e s  +  v  2  of  ) *  (Ax  the  2  +  A v  p o s i t i o n  2  +  2  AW  AW  =  3W 3x  =  3W  AV;  3v  Ax  2  +  2  Av2;)*  ) *  t r a j e c t o r i e s  p o i n t :  1  p o s i t i o n  g i v e s :  c o n s i d e r i n g  p o s i t i o n  where  3W  3V  AV;  (1)  at  each  1  Using  the  Lagrange  f o r m a l i s m  aw  j ^ -  9v  and  equating  AW  at  the  d e f i n i t i o n ,  i s ,  in  the  Forming  the  cross  -v  Combining  2  Ax  (1),  2  mv  9W/9x  =  -mv  Avi  v e l o c i t y  d i a g r a m ,  =  two  V i  Be  9W/9v  p o s i t i o n s  =  v  -v  i s  vector  product  +  v  (2),  Av  2  (3)  (v,v)  (v2,  2  s i n c e  p o s i t i o n  2  was  Ax  +  2  =  J-  v2)  (v*  v  Av  2  (Ax,  *  +  v  1  aw 9x  g i v e s :  p e r p e n d i c u l a r  to  ) *  l i n e  S;  t h a t  Av).  (Ax2,  2  2  Av2)  (Ax  2  +  g i v e s  A v  2  ) *  g i v e s  f i  and  2  =  any  =  ft  a r b i t r a r y  p o i n t  along  the  As  t r a j e c t o r y stant  along  the the  d i s t r i b u t i o n phase  space  f u n c t i o n  f  t r a j e c t o r y .  i s  shown  to  be  c o n -  APPENDIX  III  CALCULATION OF THE INDUCED DENSITY FLUCTUATIONS  For f u n c t i o n s the  a  used  reader  i s  I I I . l  more i n  of  t h i s  Motion  x - a x i s  to  i n  It  p o l a r i z a b i 1 i t y  the a  (4)  an  of  the  throughout  e l l i p t i c  the  t e x t ,  [33].  Space  and  equation  i n  and  r e f e r e n c e  Phase  equation  along  d e s c r i p t i o n  appendix  r e f e r r e d  Using the  complete  s e t t i n g of  o p t i c a l  motion wave  E i * E f o r  =  2  a  mixing  E  and  m o l e c u l e  f i e l d  becomes  mx  To  x  s o l v e  x  =  2  m u l t i p l y  d t  (  x  both  =  -  at  s i d e s  by  k  s i n  kx  ( I I I - l )  x  and  i n t e g r a t e  )  —  dt|2  (x)'2~1 J  K  =  - ^ - f e l i d11_ 2  101  cos  kx _  u s i n g  102  Hence  1  •  2  where  and  W i s  the  the  F -  t o t a l  kx  =  W  (111-2)  kx  +  W  (111  cos  energy  v e l o c i t y  x  Now, p a r t i c l e s  mx2  we  to  aE  W  =  :  cos  d i s t i n g u i s h  between  f r e e  and  - 3)  trapped  d e f i n e  ,  _  2gE2 aE + 2W 2  In  a  p o t e n t i a l aE  depth has  Two  cases  Next,  to  a r e  p o s s i b l e ;  K  *  1  p a r t i c l e  i s  K  >  1  p a r t i c l e  i s  t r a n s f o r m  V i n t e g r a l  s e t  K  =  ^  equation y  .  (111  w e l l  of  2  -j-  the  t o t a l  p a r t i c l e  energy  W  untrapped trapped  - 3)  i n t o  i n  the  w e l l  form  of  an  e l l i p t i c  103  S u b s t i t u t i n g  C  s i n c e  £  i n t o  (111  -3)  gives  K  =  i s  always  K  z  r e a l ,  s i n  E,  2  make  the  =  2  s i n  phase  (111  -4)  f o r  n  K  K  (t  d e s c r i b e  the  motion  of  a  <  > 1 rapped)  t e s t  o r b i t s  shown and  Equations  (111-4)  f i r s t  can  terms  of  be  i n t e g r a t e d  f o r  ( I I I - 4 b )  p a r t i c l e  F  the  =  I dt  s o l u t i o n s  (1  a r e :  -  K  u s i n g  2  s i n  appear the  untrapped  the  k i n d :  F U , K )  In  >  i n  1 The  the  ( I I I - 4 a )  s p a c e .  K  of  1  rapped)  t r a n s f o r m a t i o n  n= k  Equations  <  (un t  2  t)  as  trapped p a r t i c l e s .  e l l i p t i c  i n t e g r a l s  1 04  21 i  aE: 2m  F ( H , 1 / K  I I I . 2  The  The space b u t i o n  a r e  2  )  -  F ( n o , 1 / K  D i s t r i b u t i o n  e q u a t i o n s  g i v e n  by  f u n c t i o n  i s  )  =  motion  f o r  of  Theorem.  In  the  This  means  t h a t  p r e v i o u s  phase  space  diagram  c o n s t a n t  d e n s i t y  can  be  the  p a r t i c l e  the  average  found  a v e r a g i n g  t r a j e c t o r i e s of  f  0  over  a l l  *  *  the  along  a  (111-5a )  K  >  1  C111-5b)  f ) .  phase the  c i i s t r i  phase  temporal  space  p a r t i c l e  d e n s i t y  t r a j e c t o r y  c o n s t a n t  be  in  Appendix  Assuming  i n i t i a l phase  drawn  i n t e r p r e t e d  f u n c t i o n  i n  i n  c a l c u l a t e  t r a j e c t o r i e s can  t h e  p a r t i c l e To  l i m i t  the  d i s t r i b u t i o n  by  ]  a s y m p t o t i c  (proved  a s y m p t o t i c  ^  1  an  i s  the  a  ( I I I - 4 ) .  made  ( c o n s t a n t  k  <  F u n c t i o n  e q u a t i o n s  use  Ergodio  of  2  K  f o r  a E  i s  2  t h e  l l )  i n as  t r a j e c t o r i e s  l i n e s  p a r t i c l e s ,  [ 1 7 ] ,  o f  c o n s t a n t ,  d i s t r i b u t i o n  space  t h e  over  Forming  f ,  105  tir/2  ' f  0  Av  d£  o f  ( I I I - 6 )  TnTz  =  K Av  e v a l u a t e  =  d i s t a n c e  t h i s  We  between  d i s t r i b u t i o n  e v a l u a t e  express  Av  the  Av  by  two  using  p a r t i c l e  mv2  , +  r  aE  two  2 z  d£  phase  steps  t r a j e c t o r y  •„2 s i n ^  c  £  K  (1  2  s i n  2  taken:  (111  -2)  to  at  2  =  - p -  E  m  5)  t r a j e c t o r i e s  as  m  2a  =  are  equation  sin  Av  space  -T2K  K  s i n  A(l  2  5) <  Expressing  f  than  asymptotic  v  the  0  as  a  f u n c t i o n  of  K  d i s t r i b u t i o n  r a t h e r becomes  1  106 rTT/2  d£ K<1  (1  f„(K,g)  -  K  s i n  2  untrapped  (111-7a )  t r a p p e d  ( I I I - 7 b )  £ )  2  TT/2 1 K>1  d  n  f  K ( 1 / K )  1  -  °l >") K  s i n  . n i *  2  TT/2 where  K(<)  =  F(K,TT/2)  dC  = J  is  t h e  f i r s t  This f o r  t h e  beams. and  i s  v e l o c i t y  d i s t r i b u t i o n  f u n c t i o n  a f t e r  t h e  i s  t h e  f  i s  o f  t h e  s i m p l y  sum over  f o r  t h e  p a r t i c u l a r  The  r a t h e r  c a n  than  a  o f  v  i n  d e f i n e  r a t i o  energy  t h e  a l l  t h e  some  mixed  t h e  time  l a s e r  sum o f  ( I I I - 7 a )  p a r t i c l e s .  o f a  a s y m p t o t i c  case  procedure  f o ( K >5 )  t h e  of  i n f l u e n c e  f u n c t i o n  e v a l u a t e  We  k i n e t i c  o f  now  s u b s t i t u t i n g  measures  £ ) *  2  i n t e g r a l  We  K  s i n  2  Maxwellian  d i s t r i b u t i o n . o f  K  e l l i p t i c  under  which  fo  terms  t h e  complete  ( I I I - 7 b )  -  k i n d  p a r t i c l e s The  f u n c t i o n  complete  (1  f o r  f  i s  0  a  c o n s i s t s  of  both  <  1  and  3  =  a E  K  M a x w e l l i a n  e x p r e s s i n g K  >  1  f  and  i n  0  then  ( 1 4 ) . a  parameter  p o t e n t i a l  m o l e c u l e .  m o l e c u l e .  t h a t  d i s t r i b u t i o n  energy Here,  a  2  / m a  of  a  i s  t h e  which  2  w e l l  to  t h e  thermal  107  In equation  order  to  express  f  2  =  Jj*  O i l  +  m  W  S i nc e  f i n d  2  This  d i s t r i b u t i o n  K  2  s i n  2  b u t i o n ,  terms  of  K  we  w r i t e  ( I I I - 2 )  V  we  i n  0  K  f  =  =  =  0  2  — — a /TT n..  -  we  e x p ) - +r --  ( I I I - 8 )  2  For  a  |4  n  1  s u b s t i t u t e  may  between  e x p | -  1  s  we  Thus  ——  ;  2  exp  d i s t i n g u i s h i n g  - ~ Z  aE  23a  e x p r e s s i o n  s i n  0  =  k ) (  m  (1  K  >  1  the  we  f o r  the  trapped  and  f r e e  -  2  K  s i n  2  2  2  M a x w e l l i a n  t r a n s f o r m  w r i t e  l• -- |i- - C cO oS s 2  into  i n i t i a l  5)}  T T "  d i s t r i  p a r t i c l e s :  ic <  1  K  1  >  a / ?  S u b s t i t u t i n g equations  (111-7)  gives  these the  e x p r e s s i o n s asymptotic  f o r  f  0  i n t o  d i s t r i b u t i o n  f o r :  ( I l l - 9 a )  ( I I I - 9 b )  108  Untrapped  particles  TT/2 e x p ( - 2 8 / K K<1  a / ?  Trapped  2  )  exp(2B  K ( K )  K  (1  2  s i n  2  g)  s i n  2  E,)  ( I I I - 1 0 a )  d£  particles  TT/2 e x p ( - 2 g S a/T?  where  S  =  I I I . 3  Induced  d e n s i t y  J  (1  mixed  g i v e n  S p a t i a l  l a s e r  f  0  = f K<1  i s  2  s i n  2  dn  the  beams  >r  V a r i a t i o n s  w i l l  induce  a  p e r t u r b a t i o n  average  =  n  00  (f  unperturbed  - f  0  )  dv  p a r t i c l e  d e n s i t y  + ff •  KK>1  We  (111-1  by  ni (x)  n  S  Density  +  where  -  n) n ) *  2  1/K  The in  K(S)  c o s  2  a r e  i n t e r e s t e d  i n  the  F o u r i e r  component  and  Ob)  109  n(k)  J  =  H  -n  where  x  =  t h e  c o s i n e  2 £ / k i s  Using s e c t i o n the  t h e  ( I I I - l l )  r e s u l t s  n  s i n c e  K  <  I  k  f  f  n ( k )  i s  ( I I I - l l )  k —  =  f  2 7 T / k  n ( x )  K  c o s  kx  dx  w i t h  even.  a s y m p t o t i c  c a n be  dv  'o  d i s t r i b u t i o n ^  i n t e g r a t e d .  As  of  shown  t h e  p r e v i o u s  next  a r e  dK K  ( k )  \>,< >  c o s 2£  o  t r a n s f o r m  used  d£  1  0  7T  K<1  Z  =  ^ a n  S  ( 3 ) *  0  ds  f  K  >  (K)  ]  [ K ( K )-  ( S )  K(S)  2D(K)J  -  2S  :  (111-1 2a )  D(s)] (111-12b)  1  Here:  (K  -  i n t e g r a l  o f  E q u a t i o n s v a r i t i o n  o f  p a r t i c u l a r  case  M a x w e l l i a n , a s y m p t o t i c  t h e  t h a t  and  t h e  a r e  i n i t i a l  e u q a t i o n s  e q u a t i o n  i s  d e n s i t y  t h e  d i s t r i b u t i o n  E  second  (111-12)  induced  these  Using becomes  E)  f  can  g i v e n ( I I I - 8 ) ,  t h e  complete  e l l i p t i c  kind  the  f i n a l  form  f l u c t u a t i o n s . d i s t r i b u t i o n be  e v a l u a t e d  by  e q u a t i o n s e q u a t i o n  f o r For  the t h e  f u n c t i o n u s i n g  t h e  (111-10). ( I I I - l l )  s p a t i a l  i s  no f .'I  TT  n(k) =  an 0 (e)*  d£  die  c o s 2 ?  **"  « 1  T  (1  -  K  2  s i n  2  ? ) *  l/sin£ d<  O  (1  K  Using over  f  t h e  d e f i n i t i o n  K  i s  <  1  k  s t i l l  \  =  K  2  [ K ( K )  ( 1 1 1 - 1 3 )  s i n  -  2  ?)  E ( K ) ]  t h e  i n t e g r a l  s t r a i g h t f o r w a r d  \<i< > •  This  D ( K )  -  l  l e a v e s  3£  f o r  a n o ( B )  t h e  1  die  t r a p p e d  K<1  JK(K)  p a r t i c l e s  -  2D(<jJ  (111-1  4 )  111  There  are  two  p o s s i b l e  orders  for  the  fir ei  i n t e g r a t i o n  K  1/sin  ther  d K  . oo  or  •TT  d<  e l s e  1  where  c() =  a r c s i n  Choosing  4/2 \ > l <  k  >  d£  +  i  d£  4>  1/K  the  second  form  dK K  an (3)' 0  g i v e s  1  2  T  dg K>1  (1  -  K  =  2  TT  cos  de (1  we  can  2  s i n  2  O  w r i t e  »(J)  'TT  d?  +  d(J>  d£  = <  IT -<J>  we  K  e = TT-C  where  Since  -  26  f i n d  -  d£  cos 2  s i n  2g 2  £)  112  a res i n  —  K  We  now  change  c o o r d i n a t e s , ,  d 5  K  (1  s i n  %  to  2?  =  n  and  =  1  -  K  2  2g s i n  s u b s t i t u t e  d n  -  .  -4-  J -  s i n  s  i  2  n  n  *  )  n  2  get  8/2  an  7T  0  where  get  2  1 dn (1  S e t t i n g  dK  (3)'  rTT/2  to  s i n  =1 (1  cos  cos  d£  ~rP  S  the  =  1/K  f i n a l  we  express  I  in  I  =  K(S)  form  for  the  f  K> 1  • 2 s m ^  n  -  ~r  s i n  terms  of  e l l i p t i c  -  2S  trapped  2  2  I  n ) *  D(S)  p a r t i c l e s  i n t e g r a l s :  2  £)  113  S  ds  f  K  >  r  ( S )  ]  ^K(S)  -  2 S  2  D(S) ( 1 1 1 -1 5 )  For t i o n  i s  Using  t h e  p a r t i c u l a r  M a x w e l l i a n ,  the  f  i s  case  given  by  m=0  d e f i n i t i o n  a s y m p t o t i c  C  d i s t r i b u -  (111-10).  « 1  a / ?  m  exp  (1  m,  "X  -  K  i n t o  a / ?  -  I  t h e  £ ) *  2  I m  1  S T  ( - 2 6 S  K(S)  equations  f o r  dg  s i n  2  becomes  -2$  1  form  m  K ( K )  « 1  S u b s t i t u t i n g  m!  x x  cos  d i s t r i b u t i o n s  1  p a r t i c u l a r  equations  J -  1  these  i n i t i a l  i  y  t h e  t h e  expansion  m  and  that  (111-14)  induced  and  d e n s i t y  JJJT ml  2  )  m  C  ( - 2 e ) m C 2?mm  (111-16b)  2m  (111-15)  ( I H - 1 6 a )  g i v e s  f l u c t u a t i o n s :  t h e  114  djc  \<l( >  K(K)  k  exp  -26  r  K T < T  k  -L - i  an (3)  S  2  0  ds  ( - 2 6 S  [ K ( K )-  Co  = K(K)  C  =  2  ±  [ E ( K )-  2m(2< C  2m+2  2  5  3  <  2  a n d ni ma  m!  (1  -  2  1)  -  i c  C  2  m  2  (111-17) =  )  -  (111-17a )  "2m  2 S  D ( s )  2  KTS7  m  (111-17b)  "2m  ) K(icj]  +  (2m -  2(2m +  x  K(S)  C  E(<r|  =  Equations 0.1  14M!  ml  m  D ( K )= ^  2  m  -!T7T  \<l< >  with  2D(  -  7T  5 .  were  1)  1)(1 K  2  -  K  2  )  C  a  computer  2 m  _  2  '  e v a l u a t e d  on  f o r  

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