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UBC Theses and Dissertations

A study of stimulated Raman scattering in a CO₂ laser produced plasma McIntosh, Grant William John 1987

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A S T U D Y O F S T I M U L A T E D R A M A N S C A T T E R I N G I N A C 0 2 L A S E R P R O D U C E D P L A S M A b y G R A N T W.J. M C I N T O S H B.Sc.(Hons.) U n i v e r s i t y o f M a n i t o b a 1981 M.Sc. U n i v e r s i t y of B r i t i s h C o l u m b i a 1984 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y i n T H E F A C U L T Y O F G R A D U A T E S T U D I E S ( D E P A R T M E N T O F P H Y S I C S ) We ac c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A J a n u a r y 1987 © G r a n t W.J. M c i n t o s h 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by . his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of P h y s i c s The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date J a n u a r y 2 6 , 1987 D E - 6 G / 8 1 ) ii A B S T R A C T S t i m u l a t e d R a m a n s c a t t e r i n g ( S R S ) has b e e n s t u d i e d e x p e r i m e n t a l l y i n a CO2 l a s e r p l a s m a i n t e r a c t i o n . A N2 gas j e t t a r g e t was i r r a d i a t e d w i t h CO2 l a s e r l i g h t ( w a v e l e n g t h , A = 10.6/zm a n d f r e q u e n c y , w 0) at i n t e n s i t i e s u p t o 1 0 1 4 W c m - 2 . S R S o c c u r s at i n t e n s i t i e s g r e a t e r t h a n 3 x 1 0 1 3 W c m - 2 , w h i c h is f a i r agreement w i t h t h r e s h o l d p r e d i c t i o n s f o r a p l a s m a w i t h a n e l e c t r o n t e m p e r a t u r e of 3 0 0 e V a n d a d e n s i t y s c a l e l e n g t h o f 300>m. A f t e r t h i s i n t e n s i t y is reached, t h e p l a s m a waves g r o w e x p o n e n t i a l l y i n t i m e w i t h a g r o w t h r a t e o f 6 x 10~ 3w 0. T h i s was m e a s u r e d w i t h ps r e s o l u t i o n r u b y laser T h o m s o n s c a t t e r i n g . S p a t i a l g r o w t h was also o bserved. S c a t t e r e d i n f r a r e d l i g h t at 2 A was f o u n d . W e a k I R l i g h t i n a b r o a d b a n d n e a r 15/xm was a l s o f o u n d . T h e s a t u r a t e d p l a s m a wave l e v e l a n d t h e n u m b e r o f h o t e l e c t r o n s a t 150 k e V are w e l l c o r r e l a t e d , w h i c h i n d i c a t e s t h a t t r a p p i n g is r e s p o n s i b l e f o r t h e e l e c t r o n s . However, t r a p p i n g is n o t r e s p o n s i b l e f o r t h e s a t u r a t i o n of t h e i n s t a b i l i t y . F r e q u e n c y r e s o l v e d T h o m s o n s c a t t e r i n g r e v e a l e d t h a t i o n a c o u s t i c waves s t a r t at t h e p e a k of t h e S R S f l u c t u a t i o n s . O n c e t h e i o n a c o u s t i c waves grow t o a large a m p l i t u d e , S R S is q u e n c h e d a n d does n o t r e a p p e a r . i i i T A B L E O F C O N T E N T S TITLE PAGE i ABSTRACT ii T A B L E OF CONTENTS iii LIST OF TABLES v LIST OF FIGURES vi ACKNOWLEDGMENTS via LIST OF SYMBOLS ix C H A P T E R 1 Introduction 1 1.1 Preliminary Remarks 1 1.2 Motivation for the Present Work 3 1.3 Organization of the Thesis 4 C H A P T E R 2 Theory of stimulated Raman scattering 5 2.1 Introduction 5 2.2 Overview of SRS 5 2.3 Details of SRS and Two-plasmon Decay 10 2.4 Thresholds, Growth Rates and Fast Electrons 18 2.5 Connection between the Infrared Reflectivity and the Fluctuation Levels 28 2.6 Computer Simulations 30 C H A P T E R 3 Experimental Details 34 3.1 Laser and Target Characteristics 34 3.2 Thomson Scattering 38 3.3 Infrared Diagnostics 57 3.4 Electron Diagnostics 61 C H A P T E R 4 Results 65 4.1 General Observations 65 iv 4.2 Results from Spatially Resolved Thomson Scattering 66 4.3 Results from the Wavevector Thomson Scattering 70 4.4 Results from Frequency resolved Thomson Scattering 77 4.5 Scattered Infrared Light 79 4.6 Simultaneous Observations of Electrons and Plasma Waves 82 C H A P T E R 5 Discussion 89 5.1 Preliminary Numbers 89 5.2 Threshold Considerations 89 5.3 Spatial Details 92 5.3 Temporal Growth and Behaviour 95 5.4 Density Behaviour 100 5.5 Scattered Infrared Light 103 5.6 Saturation and Competition ^ 112 5.7 Connection between SRS and high energy electrons 118 CHAPTER 6 Summary and Conclusions 121 6.1 The SRS Process 121 6.2 Original Contributions 122 6.3 Suggestions for Further Work 123 REFERENCES 124 LIST OF TABLES 4-1 P a r a m e t e r s t o find a b s o l u t e fluctuation l e v e l s vi LIST OF FIGURES 2-1 Wavevector matching conditions for SRS 8 2-2 Frequency of scattered Raman light from eqn 2-1 9 2-3 Wavevectors of Raman fluctuations from eqn 2-2 10 2-4 Expected fast electron temperatures from eqn 2-3 11 2- 5 Density scale length decrease at 0.25n c r 33 3- 1 Layout of the CO2 laser system 36 3-2 Gas jet target drawn to scale 37 3-3 Laminar structure in the He jet 39 3-4 Two dimensional representation of scattering from a finite volume 41 3-5 Uncertainty in k due to a finite Au ; s 43 3-6 Projection effects on spatial Thomson scattering 45 3-7 Narrow angle k resolved Thomson scattering 47 3-8 Wide angle k resolved Thomson scattering 48 3-9 Fresnel diffraction geometry 49 3-10 Simultaneous wavevector matching for SRS and Thomson scattering 54 3-11 Frequency resolved Thomson scattering 56 3-12 IR diagnostics 60 3- 13 Mk.III electron spectrometer 62 4- 1 Threshold behaviour in the scattered IR light 66 4-2 Threshold behaviour in the number of fast electrons 67 4-3 Position of the scattering region relative to the plasma 69 4-4 Spatial extent of SRS scattering 70 4-5 More examples of spatially resolved streaks of SRS 71 4-6 Example of a wavevector resolved streak record 72 4-7 Further examples of wavevector spectra of SRS 73 vii 4-8 Reappearance of SRS in an exceptional case 74 4-9 Average k vector resolved spectrum of SRS 75 4-10 Typical streak records from the wide angle Thomson scattering 77 4-11 Example of wide angle Thomson scattering streak taken at a higher sweep speed 76 4-12 Typical frequency resolved Thomson scattering streak record 79 4-13 Frequency resolved Thomson scattering showing ion and electron features. 80 4-14 Spectrum of scattered IR near 2\0 82 4-15 Correlation of electron number with the fluctuation level 85 4-16 Correlation of electron number with the integrated IR level 86 4-17 Simultaneous electron and wavevector spectra 87 4- 18 The fitted fast electron temperature compared to that expected from the fluc-tuation spectra 88 5- 1 Evidence for spatial growth of SRS 94 5-2 Evidence for temporal growth 96 5-3 Experimentally measured growth rates 97 5-4 Absorption of SRS IR in a isothermal density ramp I l l 5-5 Summary of wide angle Thomson scattering 112 5-6 Ion acoustic waves from TPD saturation 115 A C K N O W L E D G M E N T S I a m d e e p l y i n d e b t e d t o m y s u p e r v i s o r , D r . J o c h e n M e y e r , f o r h i s c o n s t a n t i n t e r e s t a n d e n c o u r a g e m e n t i n t h e course o f t h i s w o r k , a n d f o r m a n y s t i m u l a t i n g d i s c u s s i o n s a n d suggestions. S p e c i a l t h a n k s are e x t e n d e d t o Z h a n g Y a z h o u f o r h i s t i r e l e s s a i d i n t h e i n f r a r e d d i a g n o s t i c s . H u b e r t H outman's s k i l f u l o p e r a t i o n o f t h e r u b y laser is a l s o m u c h a p p r e c i a t e d . I w o u l d also l i k e t o e x t e n d a t h a n k - y o u t o D r . A . J . B a r n a r d w h o a t t e m p t e d a g a i n s t t r e m e n d o u s o d d s t o i m p r o v e t h e E n g l i s h o f t h e t h e s i s . ( A n y r e m n a n t s o f a w k w a r d g r a m m a r are my o w n r e s p o n s i b i l i t y . ) F i n a l l y , I w o u l d l i k e t h a n k s t o D r . J o h n B e r n a r d , D r . R o m a n P o p i l , G r e g S t u a r t , K e l l y M a h a n d J i m B o o t h f o r m a n y h o u r s of m o r a l s u p p o r t , c o l o u r f u l d i s c u s s i o n s , a n d l a s t , b u t n o t l e a s t , t h e F r i d a y a f t e r n o o n beer g a r d e n s w h i c h m a d e g r a d u a t e l i f e b e a r a b l e ( B E E R a b l e ? ) . ix L I S T O F S Y M B O L S • A - t h e a t o m i c w e i g h t of ions i n t h e p l a s m a • JB, B - m a g n e t i c field • cs - i o n a c o u s t i c wave speed • c - v a c u u m speed of l i g h t • E , E - e l e c t r i c f i e l d ( s ometimes w i t h a s u b s c r i p t ) • e - t h e c h a r g e o n t h e e l e c t r o n • E M W , E M , e m - e l e c t r o m a g n e t i c wave • E P W , epw - e l e c t r o n p l a s m a wave • I - i n t e n s i t y ( u s u a l l y o f t h e i n c i d e n t l a s e r l i g h t ) • IA, i a - i o n a c o u s t i c • J - c u r r e n t d e n s i t y • kx - wave v e c t o r f o r o s c i l l a t i o n x • kj, - D e b y e w a v e v e c t o r • k0 - w a v e v e c t o r o f CO2 l i g h t i n t h e p l a s m a • kgT - t e m p e r a t u r e i n energy u n i t s ( u s u a l l y e V ) • L - d e n s i t y scale l e n g t h • m e, m - mass of t h e e l e c t r o n • N - n u m b e r d e n s i t y ( sometimes w i t h a s u b s c r i p t ) • n - e l e c t r o n n u m b e r d e n s i t y ( s o m e t i m e s w i t h s u b s c r i p t ) • n c r - t h e c r i t i c a l e l e c t r o n n u m b e r d e n s i t y • R - r e f l e c t i v i t y • re - c l a s s i c a l e l e c t r o n r a d i u s • S B S - s t i m u l a t e d B r i l l o u i n s c a t t e r i n g • S R S - s t i m u l a t e d R a m a n s c a t t e r i n g • T P D - t w o - p l a s m o n decay i n s t a b i l i t y • U - e n e r g y d e n s i t y ( u s u a l l y w i t h a s u b s c r i p t ) • V- - g r o u p v e l o c i t y of s c a t t e r e d e m wave X • vepw - g r o u p v e l o c i t y o f epw • vf>h->vp ~ phase v e l o c i t y of epw • v0 - t h e q u i v e r v e l o c i t y o f a n e l e c t r o n i n a n e l e c t r i c field • ve,vth - e l e c t r o n t h e r m a l speed • Z - average i o n i c c h a rge i n t h e p l a s m a • ux - a n g u l a r f r e q u e n c y f o r o s c i l l a t i o n x • Up - e l e c t r o n p l a s m a f r e q u e n c y • w 0 - a n g u l a r f r e q u e n c y of i n c i d e n t CO2 laser • <f>- e l e c t r o s t a t i c wave p o t e n t i a l • vti - t h e e l e c t r o n - i o n c o l l i s i o n f r e q u e n c y • A - t h e C o u l o m b l o g a r i t h m • A - w a v e l e n g t h ( s o m e t i m e s w i t h a s u b s c r i p t ) • \d - D e b y e l e n g t h • 6n - e l e c t r o n n u m b e r d e n s i t y fluctuation a m p l i t u d e • 7 - g r o w t h o r d a m p i n g r a t e (sometimes w i t h a s u b s c r i p t ) • k' - g r a d i e n t o f wave v e c t o r m i s m a t c h • dU, AU - d i f f e r e n t i a l s o l i d angle • fic - e l e c t r o n c y c l o t r o n f r e q u e n c y • 6 - a n g l e b e t w e e n i n c i d e n t a n d s c a t t e r e d r u b y l i g h t • Xe ~ s u s c e p t i b i l i t y • t - s h o r t p e r i o d o f t i m e • ^ - a n g l e b e t w e e n i n c i d e n t a n d s c a t t e r d CO2 l i g h t CHAPTER 1: Introduction 1 CHAPTER 1 Introduction 1.1 Preliminary Remarks E a r l y i n t h e t w e n t i e t h c e n t u r y , t h e i n t e r a c t i o n of l i g h t w i t h m a t t e r was i n -v e s t i g a t e d o n m a n y f r o n t s . O n e o b s e r v a t i o n of p a r t i c u l a r i n t e r e s t t o t h i s w o r k was made by R a m a n w h o n o t e d t h a t l i g h t s c a t t e r e d f r o m m o l e c u l e s was s h i f t e d b y fre-quencies c h a r a c t e r i s t i c of v i b r a t i o n s i n t h e m o l e c u l e . S u b s e q u e n t w o r k showed t h a t t h i s p h e n o m e n o n was u n i v e r s a l - l i g h t s c a t t e r e d f r o m s o l i d s , l i q u i d s , a n d gases c o u l d s how c h a r a c t e r i s t i c s h i f t s . A pl a s m a , t o o , has i t s c h a r a c t e r i s t i c f r e q u e n c i e s a n d l i g h t s c a t t e r e d f r o m i t does show a fr e q u e n c y s h i f t . T h e f r e q u e n c y d i f f e r e n c e w h i c h is of m o s t i n t e r e s t i n t h i s w o r k is a p p r o x i m a t e l y t h e e l e c t r o n p l a s m a frequency. ( T h i s p l a s m a f r e q u e n c y is p r o p o r t i o n a l t o t h e s quare r o o t of t h e e l e c t r o n n u m b e r density. T h e d e n s i t y at w h i c h t h e i n c i d e n t laser f r e q u e n c y e q u a l s t h e p l a s m a f r e q u e n c y is c a l l e d t h e c r i t i c a l d e n s i t y , ncr. ) L i g h t w h i c h leaves t h e p l a s m a w i t h t h i s f r e q u e n c y s h i f t is s a i d t o be R a m a n s c a t t e r e d . I n t h e l a t e 1960's, t h e t h e o r y of p a r a m e t r i c os-c i l l a t o r s ( o r i g i n a l l y d e r i v e d f o r e l e c t r i c a l a p p l i c a t i o n s ) was a p p l i e d t o a la s e r - p l a s m a i n t e r a c t i o n . A n i n t e r e s t i n g r e s u l t was f o u n d - t h e R a m a n s c a t t e r i n g was, f o r h i g h l i g h t i n t e n s i t i e s , a n i n s t a b i l i t y a n d w o u l d grow e x p o n e n t i a l l y . T h e n a m e for t h i s i n s t a b i l i t y is s t i m u l a t e d R a m a n s c a t t e r i n g ( S R S ) . T h e i n v e n t i o n of t h e laser m a de i t p o s s i b l e t o s t u d y t h e i n t e r a c t i o n o f e l e c t r o -m a g n e t i c r a d i a t i o n w i t h m a t t e r at p r e v i o u s l y u n a t t a i n a b l e i n t e n s i t i e s . I n p a r t i c u l a r , t h e power level s were s u c h t h a t t h e p a r a m e t r i c i n s t a b i l i t i e s c o u l d be e x c i t e d . T h e s e CHAPTER 1: Introduction 2 i n s t a b i l i t i e s were i n t e r e s t i n g f o r b a s i c s c i e n t i f i c reasons alone. H o w e v e r the same h i g h i n t e n s i t i e s a l s o m a d e laser f u s i o n a p o s s i b i l i t y . F u s i o n schemes p r o p o s e d t h e c o m p r e s s i o n of s m a l l p e l l e t s of d e u t e r i u m - t r i t i u m f u e l t o h i g h d e n s i t i e s a n d t o h i g h t e m p e r a t u r e s . A t these h i g h d e n s i t i e s a n d t e m p e r a t u r e s t h e f u s i o n r e a c t i o n c a n p r o c e d e at i t s o p t i m u m r a t e . T h e p r o p o s e d m e t h o d i n v o l v e d t h e i r r a d i a t i o n of t h e o u t e r s u r f a c e of t h e p e l l e t w i t h h i g h i n t e n s i t y laser beams. T h e o u t e r l a y e r w o u l d b l o w of f a n d t h e i n n e r core w o u l d be c o m p r e s s e d by t h e i m p l o s i o n t h a t is a con-sequence of t h e c o n s e r v a t i o n o f m o m e n t u m . T h e m a t e r i a l b l o w n off w o u l d be a p l a s m a a n d hence, t h e s t u d y of t h e laser i n t e r a c t i o n w i t h t h e p l a s m a became of p r a c t i c a l i m p o r t a n c e . C o m p u t e r s i m u l a t i o n s s h o w e d t h a t t h e i n t e r a c t i o n has some s e r i o u s a n d un-e x p e c t e d consequences. T h e r e was, of course, s c a t t e r e d l i g h t . I f t h e i n c i d e n t l i g h t is less e f f i c i e n t l y a b s o r b e d t h e p o s s i b l e c o m p r e s s i o n of t h e t a r g e t is r e d u c e d . M o r e im-p o r t a n t l y , a s s o c i a t e d w i t h S R S was t h e p r o d u c t i o n of h i g h energy e l e c t r o n s . T h e s e e l e c t r o n s w o u l d s t r e a m f o r w a r d f r o m t h e b l o w o f f p l a s m a a n d e n t e r t h e f u e l p e l l e t w h e r e t h e y w o u l d lose t h e i r energy a n d heat u p t h e p e l l e t , m a k i n g t h e c o m p r e s s i o n e v e n m o r e d i f f i c u l t . T h e a n a l y t i c a l a n d c o m p u t e r m o d e l s of t h e i n t e r a c t i o n of l i g h t w i t h t h e t a r g e t h a d t o be c o n f i r m e d e x p e r i m e n t a l l y . T h e first o b s e r v a t i o n o f S R S was made by B o b i n i n 1973 a n d t h i s r e m a i n e d t h e o n l y o b s e r v a t i o n u n t i l 1978. S i n c e t h e n , e x p e r i m e n t s have s h o w n t h a t S R S is i n d e e d a t r o u b l e s o m e p r o b l e m . T h e e n h a n c e d l e v e l s o f s c a t t e r e d l i g h t have been measured. T h e f a s t e l e c t r o n s p r e d i c t e d b y t h e s i m u l a t i o n s have been observed. Y e t , as m o r e a n d m o r e e x p e r i m e n t s were c o n d u c t e d a t h i g h e r a n d h i g h e r i n t e n s i t i e s , some u n e x p e c t e d r e s u l t s were o b t a i n e d . T h e m o st p u z z l i n g r e s u l t was a gap i n t h e f r e q u e n c y s p e c t r a of t h e s c a t t e r e d l a s e r l i g h t . 1 A gap i n t h e s p e c t r a c a n be t h o u g h t o f as a gap i n t h e d e n s i t i e s at w h i c h t h e s c a t t e r i n g is t h o u g h t t o be o c c u r r i n g . T h e o b s e r v e d gap c o r r e s p o n d e d t o d e n s i t i e s between 0.1 a n d 0.24 ncr. CHAPTER 1: Introduction 3 S i n c e e a c h s c a t t e r e d l i g h t wave is p r o d u c e d by a p a r t i c u l a r p l a s m a wave, one w o u l d also e x p e c t a g a p i n t h e p l a s m a wave s p e c t r u m . It is t h e r e f o r e w o r t h w h i l e t o s t u d y t h e p l a s m a waves by o t h e r m e t h o d s . If n o gap i n t h e p l a s m a wave s p e c t r u m is o b s e r v e d a n d i f a g a p is o b s e r v e d i n t h e I R s p e c t r u m , t h e n t h e m o d e l of S R S may h a v e t o be m o d i f i e d . T h e s t a n d a r d m o d e l of S R S is d e s c r i b e d i n d e t a i l i n c h a p t e r 2. It is a l s o i m p o r t a n t t o m e asure t h e p r o p e r t i e s of t h e h i g h energy electrons. If e i t h e r t h e i r n u m b e r o r t h e i r energies disagree w i t h t h e c o m p u t e r s i m u l a t i o n s , t h e n t h e p l a n s f o r l a s e r f u s i o n m a y have t o b e changed. T h e b e s t w a y t o check t h e s i m u l a t i o n s w o u l d be t o c o m p a r e t h e p r o p e r t i e s of t h e o b s e r v e d p l a s m a waves w i t h t h e e l e c t r o n s . A n i n s t a b i l i t y w h i c h grows e x p o n e n t i a l l y c a n n o t c o n t i n u e t o do so forever; t h e r e m u s t be some m e c h a n i s m w h i c h s a t u r a t e s a n d quenches t h e g r o w t h . Simu-l a t i o n s h ave s h o w n t h a t t h e presence of fast e l e c t r o n s c a n cause s a t u r a t i o n . O t h e r s i m u l a t i o n s have s h o w n t h a t c o u p l i n g b etween t h e e l e c t r o n p l a s m a waves a n d o t h e r waves i n t h e p l a s m a c a n cause s a t u r a t i o n . O n c e a g a i n e x p e r i m e n t s c a n be set u p w h i c h c a n d e t e r m i n e i f e i t h e r of these m e c h a n i s m s is r e s p o n s i b l e . 1.2 Motivation for the Present Work W i t h t h e b r i e f b a c k g r o u n d p r o v i d e d , t h e m o t i v a t i o n f o r t h e present work b e comes c l e a r e r . I n a p l a s m a g e n e r a t e d b y a CO2 laser ( w a v e l e n g t h A 0=10.6 rtm), a l l t h e a s p e c t s of S R S are o b s e r v a b l e w i t h s t a n d a r d techniques. T h e d e n s i t y fluctu-a t i o n s c a n be m e a s u r e d w i t h T h o m s o n s c a t t e r i n g , t h e s c a t t e r e d E M waves c a n be d e t e c t e d w i t h s e n s i t i v e i n f r a r e d d e t e c t o r s , a n d t h e e l e c t r o n s c a n be a n a l y z e d w i t h a s i m p l e e l e c t r o n s p e c t r o m e t e r a n d s c i n t i l l a t o r d e t e c t o r s . T h e m a j o r q u e s t i o n s t o be a n s w e r e d by t h i s w o r k are: • I n a p l a s m a of a k n o w n de n s i t y , are t h e r e e n h a n c e d fluctuations present cor-r e s p o n d i n g t o t h a t d e n s i t y ? CHAPTER 1: Introduction 4 • Is t h e r e s c a t t e r e d l i g h t c o r r e s p o n d i n g t o t h e o b s e r v e d fluctuations? • H o w are t h e n u m b e r a n d energy d i s t r i b u t i o n o f t h e h i g h energy e l e c t r o n s r e l a t e d t o t h e e n h a n c e d fluctuations? • W h a t causes S R S t o s a t u r a t e a n d decay? 1.3 Organization of the Thesis T h e t h e o r y a n d t h e o b s e r v a b l e consequences of S R S r e l e v a n t t o t h i s w o r k are p r e s e n t e d i n c h a p t e r 2. I n c h a p t e r 3, t h e e x p e r i m e n t a l d e t a i l s a n d t h e m e t h o d s of a n a l y s i s u s e d o n t h e d a t a are g i v e n . C h a p t e r 4 c o n t a i n s t h e r e s u l t s of my e x p e r i -m e n t s w h i l e t h e i m p l i c a t i o n s a n d c o m p a r i s o n s w i t h t h e o r y a n d o t h e r e x p e r i m e n t s a r e f o u n d i n C h a p t e r 5. I n C h a p t e r 6 a c o n s i s t e n t a n d concise s u m m a r y of t h e S R S p r o c e s s is f o u n d . T h e o r i g i n a l c o n t r i b u t i o n s of t h i s w o r k a n d suggestions f o r f u r t h e r w o r k e n d t h a t c h a p t e r . T h r o u g h o u t t h i s t h e s i s t h e C G S G a u s s i a n s y s t e m of u n i t s is used. O n l y e q u a t i o n s w h i c h are q u o t e d elsewhere i n t h e t h e s i s are numbered. CHAPTER 2: Theory of stimulated Raman scattering 5 CHAPTER 2 Theory of stimulated Raman scattering 2.1 Introduction A p l a s m a c o n s i s t s of a free e l e c t r o n gas i n t e r s p e r s e d i n a free i o n gas. A s such, a n e l e c t r o m a g n e t i c wave i n t e r a c t i n g w i t h i t c a n be h a n d l e d r e a d i l y . W o r k of t h i s n a t u r e h a s b e e n u n d e r t a k e n s i n c e t h e 1960's a n d has s h o w n t h a t m a n y n o n l i n e a r o p t i c a l p h e n o m e n a s h o u l d o c c u r i n a p l a s m a i f t h e i n c i d e n t E M f i e l d is l a r g e enough. In d e e d , t h e n o n l i n e a r a s p e c t s are h a r d l y a v o i d a b l e . I n t h i s c h a p t e r , t h e p h y s i c a l o r i g i n of s u c h n o n l i n e a r p h e n o m e n a is e x p l a i n e d . T h e d i s c u s s i o n b e g i n s w i t h a n o v e r v i e w of p a r a m e t r i c i n s t a b i l i t i e s w i t h e m p h a s i s o n S t i m u l a t e d R a m a n S c a t t e r i n g ( S R S ) . T h i s is f o l l o w e d b y a d e r i v a t i o n o f t h e b a s i c g r o w t h r a t e f o r S R S a n d t h e r e l a t e d t w o - p l a s m o n decay ( T P D ) i n s t a b i l i t y f o r a homogeneous p l a s m a . E x p r e s s i o n s f o r t he t h r e s h o l d i n t e n s i t y a n d t h e g r o w t h r a t e u n d e r m o r e r e a l i s t i c a s s u m p t i o n s are t h e n r e v i e w e d f o r c o m p a r i s o n w i t h e x p e r i m e n -t a l r e s u l t s . T h e c h a p t e r ends w i t h a d i s c u s s i o n of c o m p u t e r s i m u l a t i o n s . 2.2 Overview of SRS I n a p l a s m a , i n t h e absence of a n e x t e r n a l m a g n e t i c f i e l d , t h r e e t y p e s of waves c a n b e s u p p o r t e d : a h i g h f r e q u e n c y l o n g i t u d i n a l e l e c t r o n p l a s m a wave (EPW), a low f r e q u e n c y l o n g i t u d i n a l i o n a c o u s t i c wave (IAW), a n d a h i g h f r e q u e n c y t r a n s -v e r s e e l e c t r o m a g n e t i c wave (EMW). E a c h wave has a frequency, u; a n d a wave CHAPTER 2: Theory of stimulated Raman scattering 6 v e c t o r , k. T h e s e waves obey d i s p e r s i o n r e l a t i o n s w h i c h c a n be w r i t t e n i n t h e f o l -l o w i n g s i m p l i f i e d f o r m s 1 2: ^EPW = w p + 3kBTk2EPW/me, uiaw - cskiaw, a n d 2 2 2; 2 UEMW ~ up + c kEMW i w h e r e t h e p l a s m a f r e q u e n c y '4nnee2 a n d i o n a c o u s t i c speed cs = VZksT/mi have been i n t r o d u c e d . In a p a r a m e t r i c i n s t a b i l i t y , one of these waves decays i n t o t w o o t h e r s , s u b j e c t t o t h e s c a t t e r i n g c o n d i t i o n s k0 = ki + k2 a n d U}0 = U! + U2. T h e s e c o n d i t i o n s are s o m e t i m e s r e f e r e d t o as c o n s e r v a t i o n o f m o m e n t u m a n d energy, w h i c h is n o t c o r r e c t ; t h e y are phase m a t c h i n g c o n d i t i o n s . I n t h e decay of a n i n c i d e n t E M wave, f o u r decay modes are p o s s i b l e : EMW -• EMW + I AW ( S t i m u l a t e d B r i l l o u i n S c a t t e r i n g ) CHAPTER 2: Theory of stimulated Raman scattering 7 EMW -* EMW + EPW ( S t i m u l a t e d R a m a n S c a t t e r i n g ) EMW -> EPW + I AW ( P a r a m e t r i c D e c a y I n s t a b i l i t y ) a n d EMW -» EPW + EPW ( T w o - p l a s m o n D e c a y I n s t a b i l i t y ) . O t h e r d e c a y s (e.g. EMW —> EMW + EMW) are f o r b i d d e n s i n c e t h e d i s p e r s i o n r e l a t i o n s a n d t h e s c a t t e r i n g c o n d i t i o n s c a n n o t be s a t i s f i e d s i m u l t a n e o u s l y . If o n l y t h e d i s p e r s i o n r e l a t i o n s at t e m p e r a t u r e T=0 a n d t h e s c a t t e r i n g c o n d i -t i o n s a re c o n s i d e r e d f o r S R S t h e f o l l o w i n g c a n be f o u n d : (1) f o r t h e s c a t t e r e d E M wave, o f f r e q u e n c y u)s, i * = i - ^ ' 2-1 a n d (2) f o r t h e e l e c t r o n p l a s m a wave, (kepw) = ^[(1 - 4) + (1 - ^E ) - 2(1 - 4)1/2(1 - ^)*/W] 1" 2 - 2 w h e r e xp is t h e angle b e t w e e n t h e i n c i d e n t ( f r e q u e n c y u>0) a n d s c a t t e r e d E M waves. T h e d e n s i t y at w h i c h u0 = up is c a l l e d t h e c r i t i c a l d e n s i t y n c r so t h a t ufy/u:2 — n/ncr. O n e d i m e n s i o n a l c o m p u t e r s i m u l a t i o n s 2's have s h o w n t h a t h i g h e n e r g y elec-t r o n s c a n b e g e n e r a t e d by S R S a n d t h a t t h e e l e c t r o n s have, a t h i g h energies, a d i s t r i b u t i o n of v e l o c i t i e s t h a t decreases e x p o n e n t i a l l y . S u c h a decrease is c h a r a c t e r -i s t i c o f a M a x w e l l i a n d i s t r i b u t i o n . T h e s i m u l a t i o n s i n d i c a t e t h a t a n a p p r o x i m a t e t e m p e r a t u r e fcBr = m e c 2 ( [ l - ( - ^ ) 2 ] - 1 / 2 - l ) 2 - 3 c a n b e fitted. T h e p h y s i c a l e x p l a n a t i o n f o r t h i s b e h a v i o u r is n o t known. CHAPTER 2: Theory of stimulated Raman scattering 8 Figure 2-1 Wavevector matching conditions for SRS. The drawing is to scale. k0 7^ w 0/c because of dispersion in the plasma. The amount of scattered energy can be estimated. If we imagine n E M quanta decaying to n scattered E M quanta and n EPW quanta, we get the trivial relation-ships nhu0 nku!s nhu}epW ' — — — — 2 4 W 0 tt>s U>epw which are the Manley-Rowe conditions 4 that relate the total energy in a particular mode {nhu) to that in any other. The ratio of the respective frequencies indicates the relative amounts of energy in each mode. (These relations were originally de-rived for classical oscillators in which the energy in a particular mode is E 2/(87r) where E is the electric field associated with that mode.) The energy in a given CHAPTER 2: Theory of stimulated Raman scattering 9 m o d e does n o t have t o r e m a i n i n t h a t mode. T h e s c a t t e r e d E M r a d i a t i o n c o u l d be a b s o r b e d b y t h e p l a s m a t h r o u g h o t h e r processes, s u c h as in v e r s e b r e m s s t r a h l u n g ab-s o r p t i o n w h i l e t h e p l a s m a wave energy c o u l d be p a r t i a l l y o r c o m p l e t e l y t r a n s f e r r e d t o t h e e l e c t r o n s . T o s u m u p S R S has t h e o b s e r v a b l e consequences o f a s c a t t e r e d E M wave, a n e n h a n c e d e l e c t r o n p l a s m a wave f l u c t u a t i o n , a n d h i g h energy e l e c t r o n s . T h e r e l a t i v e a m o u n t of e n e r g y i n each o f these s y s t e m s s h o u l d be r e l a t e d t o each o t h e r . F o r V > = 1 8 0 ° , i.e. f o r d i r e c t e l e c t r o m a g n e t i c b a c k s c a t t e r , t h e values f o r these o b s e r v a b l e s have been p l o t t e d i n figu r e s 2 - 2 , 2 -3 a n d 2-4 as a f u n c t i o n o f p l a s m a density. <—r 1 1 — i 1 — * i N / N c r F i g u r e 2 - 2 F r e q u e n c y of s c a t t e r e d R a m a n l i g h t f r o m eqn 2 - 1 . CHAPTER 2: Theory of stimulated Raman scattering 10 r N/Ncr Figure 2-3 Wavevectors of Raman fluctuations from eqn 2-2. 2.3 Details of SRS and Two-plasmon Decay The mechanisms which determine the presence and importance of SRS can-not be found by considerations of the previous section alone. The details of the interaction must be studied. In this section a derivation of the growth rate, 7 0 , of SRS in an infinite, homogeneous plasma is given. There are several assumptions made and these are clearer if explicitly pointed out as they are made. The homoge-neous growth rate is the maximum rate at a given density and defines a timescale for SRS. The lower growth rates in a finite or inhomogeneous plasma usually involve a knowledge of 7 0 . The following derivation is based upon a number of sources with the important ones being Jackson B , Shen 6 , Tsytovich 7 , Forslund et al.8, Drake CHAPTER 2: Theory of stimulated Raman scattering 11 Forward Scatter V = i — t -xepw .1 .15 .2 .25 n/rtr Figure 2-4 Expected fast electron temperatures from eqn 2-3. et al.9 and Langdon et al.10. It is not mathematically rigorous but the physical mechanisms should be clear. The starting point is Maxwell's equations: „ - 1 dE 4?r V x B = - — + — J c at c V B = 0 CHAPTER 2: Theory of stimulated Raman scattering V.E = 4nY/{NJq3) 2 - 5 . i I n t h i s s e c t i o n , t h e f o l l o w i n g s y m b o l s are used: E f o r e l e c t r i c f i e l d , B f o r m a g n e t i c field, c f o r t h e speed of l i g h t , Nj f o r n u m b e r d e n s i t y o f species j w i t h c h arge qj, e f o r t h e e l e c t r o n charge, v f o r t h e p l a s m a fluid v e l o c i t y , w for a n g u l a r f r e q u e n c y of a wave, a n d k f o r t h e w a v e v e c t o r o f a wave. T h e wave e q u a t i o n f o r E is w h i c h u p o n F o u r i e r t r a n s f o r m a t i o n gives -kik • E) + k2E - —rE = J . P h y s i c a l l y , one i m a g i n e s E, t h e e l e c t r i c field, as b e i n g g e n e r a t e d by c u r r e n t s , J . J m a y be g e n e r a t e d by a n o n l i n e a r r e s p o n s e i n a p l a s m a . If we ne g l e c t i o n c u r r e n t s we get J ( r , r ) = eN{r,t)v{r,t) 2 - 6 a n d J{k,u) = ejj N{k',u')v{k",oj")S{k -k'- k")6(u, - u'- u>")dzk'dzk"duj'duj". I n a n i n f i n i t e p l a s m a t h e i n t e g r a l s are r e p l a c e d by s u m s over a l l P, k" w h i c h s a t i s f y t h e m a t c h i n g c o n d i t i o n s . N ( f , t) a n d v ( r , t) are d e s c r i b e d i n t h e fluid m o d e l f o r t h e p l a s m a , t h e c o n s e r v a t i o n o f m o m e n t u m : ^ + V . V t = J - { E + * x B ) - y £ v N - 2 - 7 at me c rneN0 CHAPTER 2: Theory of stimulated Raman scattering 13 a n d t h e c o n t i n u i t y e q u a t i o n : dN + V • {Nv) = 0 dt 2 - 8 . T h e t e r m i n v o l v i n g V 7 Y i n E q n . 2-7 represents t h e p r e s s u r e g r a d i e n t force i n a n i d e a l gas. J(k, u>) m u s t be c a l c u l a t e d u s i n g r e a l i s t i c a s s u m p t i o n s . S i n c e t h r e e wave o f m o s t i n t e r e s t , t h i s w i l l l e a d t o a n a t u r a l c h o i c e of c o u p l i n g s between t h e t h r e e waves i n w h i c h t h e differences i n w a v e v e c t o r a n d f r e q u e n c y of t w o o f t h e waves are e q u a l t o k0 a n d w0. O n l y lowest o r d e r c o u p l i n g s w i l l be c o n s i d e r e d . T h i s set of e q u a t i o n s ( E q n . 2-5 t o 2-8) c a n be s o l v e d n u m e r i c a l l y w i t h a com-p u t e r . I n i t i a l a n a l y t i c b e h a v i o u r c a n be f o u n d b y u s i n g e x p a n s i o n t e c h n i q u e s . T h e d e n s i t y N is s e p a r a t e d i n t o t w o p a r t s , a c o n s t a n t average d e n s i t y N0 a n d a f l u c t u a t -i n g p a r t n. F i r s t , t h e l i n e a r r e s p o n s e of t h e p l a s m a t o a n E M W is f o u n d . T h e l i n e a r d i s p e r s i o n r e l a t i o n s f o r an E M W a n d a n E P W are d e r i v e d . T h e d e n s i t y fluctuation l e v e l w h i c h is e n h a n c e d by t h e n o n l i n e a r response of t h e p l a s m a is c a l c u l a t e d n e x t . T h e n o n l i n e a r d i s p e r s i o n r e l a t i o n f o r t h e s c a t t e r e d wave is g i v e n a n d t h e g r o w t h r a t e f o r S R S is d e r i v e d f r o m i t . I n t h e f o l l o w i n g t h e s u b s c r i p t s o n E, k, a n d u refer t o p a r t i c u l a r waves. T h e l i n e a r response of a p l a s m a t o a n e l e c t r i c f i e l d J(ki,u}\) is easy t o c a l c u -l a t e . W e need t o find i n t e r a c t i o n s i n t h e presence of a s t r o n g E M wave ( i d e n t i f i e d b y t h e s u b s c r i p t o) are 4tt dJ 4neN0 dv c 2 dt c 2 dt' We s u b s t i t u t e f o r t h e d e r i v a t i v e of v n e g l e c t i n g t h e v x B , v • V a n d d(nv)/dt t e r m s f r o m eqn 2-7. U p o n F o u r i e r t r a n s f o r m i n g , we get SksTAneikin mec2 CHAPTER 2: Theory of stimulated Raman scattering T o find n we neglect t h e V • (nv) t e r m i n t h e c o n t i n u i t y e q u a t i o n (eqn 2-8) a n d F o u r i e r t r a n s f o r m t h e r e s u l t t o get , u p o n r e a r r a n g e m e n t , n(fci,u;i) = T o find v , we F o u r i e r t r a n s f o r m t h e e q u a t i o n d e s c r i b i n g t h e v a r i a t i o n of v a n d n e g l e c t a l l t e r m s e x c e p t t h a t i n v o l v i n g E\. R e a r r a n g i n g , we find it E\ v meoj\ S u b s t i t u t i n g v i n t o t h e e x p r e s s i o n f o r n a n d n i n t o t h e e x p r e s s i o n f o r J , we o b t a i n iu1J(ki,oji) ul -, 3kBTu%ki(ki • E\) 2 = " T ^ 1 " I T~i S u b s t i t u t i n g t h i s i n t o t h e wave e q u a t i o n f o r E\ a n d r e g r o u p i n g t h e t e r m s we see t h a t [,_,++(*? - 4+1)/>i • A=4"'u,i/Nf(t"""i) bj\mtcl cl cl cl w h e r e J J V . L. c o n t a i n s t h e n o n - l i n e a r t e r m s a n d // is t h e u n i t d y a d i c tensor. W h e n t h e n o n l i n e a r t e r m is n e g l e c t e d , t h e d i s p e r s i o n r e l a t i o n s f o r t r a n s v e r s e ( k\ • E\ = 0) a n d l o n g i t u d i n a l waves are o b t a i n e d , namely, w i — up + (Tran s v e r s e ) a n d 2 2 , "lMBTk\ u{ = bjv H — — 5 ( L o n g i t u d i n a l ) . CHAPTER 2: Theory of stimulated Raman scattering 15 ( R e m e m b e r t h a t ojp « U\ .) T h e t e r m Jn.L. m u s t now be c a l c u l a t e d . I t is g e n e r a t e d by t h e d e n s i t y p e r t u r -—* —* b a t i o n n(fc2,u;2) b e a t i n g w i t h t h e v e l o c i t y g e n e r a t e d by t h e i n c i d e n t E M W ( w 0 , f c 0 ) w h e r e £2 = ko — k\ a n d u>2 = uj0 — u\. H i g h e r o r d e r c o u p l i n g s a n d t h e a n t i - S t o k e s c o u p l i n g are ne g l e c t e d . I n t h i s a p p r o x i m a t i o n we find iugJN.L. _ iuitn(k2,u)2)v(k0,u0) c 2 c 2 T h e q u i v e r v e l o c i t y v(k0,w0) is g i v e n b y v[k0,u)0) = . mew0 ( H e r e a f t e r , v0 a c t u a l l y refers t o v(k0,u0).) n(fc2,u;2) m u s t be c a l c u l a t e d f r o m t h e wave e q u a t i o n w h i c h is f o u n d b y com-b i n i n g t he c o n t i n u i t y a n d m o m e n t u m e q u a t i o n s f o r t h e fluid. T h e r e s u l t i n g e q u a t i o n f o r n, n e g l e c t i n g VtT • V u , is *d2n ZkBTV2n N0eV • E _ eJV 0 V • (v x B) dt2 A s u s u a l t h i s is F o u r i e r t r a n s f o r m e d t o get 2 , %kBTkl 2 N0k2 - w 2 + + u n = me v c J J v1 x{kx ^-Y'Hh - k' - k")dk'dk". T h e R H S gives r i s e t o a t e r m of t h e f o r m klN0e2E0 • Ei rn2eu}0ui\ CHAPTER 2: Theory of stimulated Raman scattering 16 Thus we find k\E0 • E\u\ n(k2,u2) v (w| — w 2 — ZkBTk2l/me)4nme(jj0uji n is generated by the beating of E0 and E\ and is enhanced because of the v x B force. Using this n and the v(k0,uj0) we find the expression for Jn.l. which can be substituted into the wave equation for E\. The result is uftkBT r r 2 ( 2 . r a - u}p1u)ln(k2,u)2)Eo This is readily inverted to give the scattered field # = 2 ^ " f e ^ a ) [ f f - r M i x 1 M i i £ 1 p u 0 N0 K k\ J ( f c 2 C 2 - W l 2 + a ; 2 ) fc2(u;2-u;2-u;23A;Brfc2/u;2me)J °' Upon substituting for n(A :2,W2) , taking the dot product of J50 with both sides, cancelling E0 • E\ and recognizing that (k\ x E0)2 — k2E2 — (fci • E0)2, one arrives at the dispersion relation (w 2 - w 2 - ZkBTkllme)mluy k\{k\c2 - u\ + w 2) (ki • E0)< * 2(w 2 - wj - 3fcj3Tfc2u;2/a;2me) 2 - 9 This expression is equivalent to that given by Drake 9 provided (a) ions are ne-glected; (b) thermal effects are included; (c) the fluid susceptibilities -w 2 Xe(w,fc)- P uj2 — Zv2k2 CHAPTER 2: Theory of stimulated Raman scattering 17 are u s e d i n p l a c e of t h e k i n e t i c ones; a n d (d) t h e a n t i - S t o k e s c o u p l i n g is neglected. T h e i m p o r t a n c e of t h e t e r m s i n t h e { } i n E q n . 2-9 is f o u n d by e x a m i n i n g the r e s p e c t i v e d e n o m i n a t o r s . If t h e d e n o m i n a t o r is s m a l l (near zero) t h e t e r m is q u i t e i m p o r t a n t . T h e first t e r m a p p l i e s t o S R S a n d t h e second t e r m a p p l i e s t o t h e T P D i n s t a b i l i t y . T h e c r o s s a n d s c a l a r p r o d u c t s d e t e r m i n e t h e i m p o r t a n t s c a t t e r i n g d i -r e c t i o n s . F o r S R S , t h e cross p r o d u c t i m p l i e s t h a t t h e r e s h o u l d be no s c a t t e r e d E M wave i n d i r e c t i o n s p a r a l l e l t o t h e i n c i d e n t , l i n e a r l y p o l a r i z e d E0 a n d t h a t E M scat-—* t e r i n g s h o u l d b e s t r o n g e s t i n t h e p l a n e p e r p e n d i c u l a r t o E0. F o r T P D , t h e s c a l a r p r o d u c t m eans t h e reverse s i t u a t i o n is t r u e i.e. t h e b e s t p l a n e s f o r v i e w i n g E P W are t h o s e p l a n e s i n w h i c h E0 l i e s . I f we neglect t h e s e c o n d t e r m i n t h e brackets i n E q n . 2-9 a n d r e a r r a n g e t h e r e s u l t we find (u\ - u\ - ZkBTk\lmt)(u\ - w 2 - c2k\) = w 2 f c | ( v 0 s i n ( 7 r / 2 - V ) ) 2 2 - 1 0 wher e 7r/2 — \p is t h e angle b e t w e e n k\ a n d v0. L e t u>2 = w 2 r + »7, = u\T — 17, 7/wi -C 1 (where 7 is t h e g r o w t h r a t e ) , u\T = Up-rSksTkl/rne a n d u\T = w 2 + c 2 f c 2 . W i t h the s u b s t i t u t i o n f o r u>2r <*>p a n d uj\r = w 0 — u%r i n t o E q n . 2-10 becomes 4 7 2w 2 ru;i r = u; 2 f c 2 t ; 2 s i n 2 ( 7 r / 2 -ib). T h e final r e s u l t ( w i t h m i n o r changes i n n o t a t i o n ) 8 is 0 r/g e p t t,t; 0a;psin(7r/2-r/>) 7o — u.o / . . • * 11 \/UJp(U0 - wp) T h i s 7 0 is t h e g r o w t h r a t e f o r S R S f o r a n i n f i n i t e , h omogeneous p l a s m a i n t h e r e g i m e w h e r e d a m p i n g (so f a r neglected) is n o t i m p o r t a n t a n d v0 is s m a l l enough t h a t t h e a p p r o x i m a t i o n 7/w <C 1 is v a l i d . 7 0 reaches a m a x i m u m w h e n xp = ir ( d i r e c t b a c k s c a t t e r ) . It s h o u l d be r e m e m b e r e d t h a t kepw d e p e n d s u p o n ip. CHAPTER 2: Theory of stimulated Raman scattering 18 When n = 0.25n.cr the growth rate for TPD that is derived from Eqn. 2-9 is incorrect. At that density k\ = 0, and both terms in the dispersion relation must be retained. Langdon 1 0 has considered this problem carefully and derived a slightly different dispersion relation for the TPD instability; the term corresponding to SRS remains unchanged. The modified dispersion relation reads (^2 - Up - 3kBTkf2/me) = —-f 9 l 2 , , 2 + 2{lj\ - wj - c2k\)k\ 2(u\ - w| - 3kBTk2/me)k2k2' Considering only the TPD instability component we find the growth rate for TPD to be • r ((kx-k0/2)2-(kl+kl/4)Yrl/2 \ 2Uk, - kJ2) J J koVo l2w„ - —r-This growth rate is a maximum if (kx - k0/2)2 = (k2 + k20/4) 2 - 13 which for kx,ky 3> k0 reduces to kx = iiky. For ky = 0, kx = k0 is the condition for maximum growth. When both terms are considered, the resulting wave is a mixed transverse-longitudinal mode. This is sometimes called the SRS-TPD hybrid mode. (The matching wave is purely longitudinal.) To summarize: SRS grows at a rate "y0 given by Eqn. 2-11 and for 0.25ncr the effects of SRS and the TPD instability cannot be separated. 2.4 T h r e s h o l d s , G r o w t h R a t e s a n d F a s t E l e c t r o n s The previous theory considered the case of undamped waves, a constant u 0, an infinite plasma, and no gradients in density, temperature or plasma velocity. These assumptions will be relaxed and the implications discussed. CHAPTER 2: Theory of stimulated Raman scattering The various waves generated can be damped by several means. Plasma waves can be damped by either collisionless Landau damping 1 1 , at a rate given by ll = WP(TT/8) l'2(kd/k)3exp(-0.5(kd/k)2 2 - 14 or by collisional damping Ic = Vci/2 or both, where ue{ is the electron-ion collision frequency 12 1.5 x 10~ 6Zn e(cm T 3 / 2 (eV) A is the Coulomb logarithm, ln(l27rn eA| )), with XD (cm)= 6.9(T( 0/i')/ne(cm~ 3)) 1/ 2, the Debye length. (For the plasma conditions encountered in the present experiment the value of A can be set to 10.) Electromagnetic waves are mainly damped by collisional damping (inverse bremsstrahlung) for which the decay rate is The finite damping introduced by these rates requires that a threshold growth rate It given by must be reached before SRS occurs and once threshold is exceeded, the plasma waves grow but at a lower rate than 7„ . (Well above threshold, growth at 7„ is recovered.) One obvious saturation mechanism is the decline in the incident laser light intensity as the interaction proceeds. Since the plasma waves and the backscattered waves are growing, some energy must be supplied. The incident EMW acts as the supply but its energy is not infinite. When energy is transfered to the waves and Tern — i/ei-w2/2w2 IT = ill + 1e)li 'em 2 - 15 CHAPTER 2: Theory of stimulated Raman scattering 20 t h e waves s u b s e q u e n t l y are d a m p e d , t h e energy c a n n o t be r e t u r n e d t o t h e p u m p wave. A st e a d y s t a t e s i t u a t i o n is s o o n reached. A t t h i s stage p u m p d e p l e t i o n has set i n a n d t h e waves no longer grow. T h i s process has b e e n s t u d i e d r e c e n t l y b y 1 4 ). M c K i n s t r i e showed t h a t t h e i n t e n s i t y r e f l e c t i v i t y R s a t u r a t e s i n a n i n f i n i t e , c o l l i s i o n a l p l a s m a a t w h e r e m = i0/lTi n o t e t h a t m > 1. T h i s suggests a l o w e r r e f l e c t i v i t y t h a n t h a t e x p e c t e d f r o m t h e M a n l e y - R o w e r e l a t i o n s . R is a s e n s i t i v e f u n c t i o n of t h e t h r e s h o l d g r o w t h r a t e It-I n a n y r e a l e x p e r i m e n t a l s i t u a t i o n , a p l a s m a is n e i t h e r i n f i n i t e n o r homoge-neous. T h e finite l e n g t h of t h e i n t e r a c t i o n r e g i o n has some consequences o n t h e n a t u r e of t h e i n s t a b i l i t y . Is i t a c o n v e c t i v e o r a n a b s o l u t e i n s t a b i l i t y ? T h e t e r m s c o n v e c t i v e a n d a b s o l u t e refer t o t h e p a t t e r n of g r o w t h i n space a n d t i m e . C o n -v e c t i v e g r o w t h is s u c h t h a t a f t e r a s h o r t t i m e r = L/v~ t h e a m p l i t u d e of a wave grows e x p o n e n t i a l l y i n space only. A b s o l u t e g r o w t h refers t o g r o w t h w h i c h o c c u r s e v e r y w h e r e , e x p o n e n t i a l l y i n t i m e . I n a finite b u t homogeneous p l a s m a of l e n g t h L , i t seems r e a s o n a b l e t o e x p e c t a m i n i m u m l e n g t h f o r c o n v e c t i v e g r o w t h of S R S t o o c c u r a n d a m a x i m u m l e n g t h above w h i c h t h e i n f i n i t e homogeneous l i m i t o f ab-s o l u t e g r o w t h is recovered. K r o l l 1 5 showed i f L < ( v ~ v e v w ) l l 2 l~i0, o n l y c o n v e c t i v e g r o w t h w i l l o c c u r , i f { v - v e p w ) l l 2 < L < v-/i0, a b s o l u t e g r o w t h o c c u r s b u t at a r e d u c e d r a t e a n d i f L > g r o w t h at i0 w i l l be r e c o v e r e d f o r a t i m e r , f o l l o w e d by c o n t i n u e d g r o w t h at t h e r e d u c e d r a t e . (v~ a n d vepw are t h e g r o u p v e l o c i t i e s of the s c a t t e r e d E M a n d p l a s m a wave respectively.) F o r s l u n d et a l . 8 showed t h a t t h e a b s o l u t e M c K i n s t r i e et a l . 1 8 (whose r e s u l t s agreed w i t h s i m u l a t i o n s of H i o b a n d B a r n a r d 2 - 1 6 2 - 1 7 CHAPTER 2: Theory of stimulated Raman scattering 21 i n s t a b i l i t y i n a finite p l a s m a leads t o b o t h t h e r e d u c e d t e m p o r a l g r o w t h ( E q n . 2-17) a n d a s p a t i a l g r o w t h (for w e a k l y d a m p e d waves) Imifc = l o 2 - 18. y/VepwV— T h e p h y s i c a l r e a s o n f o r t h i s b e h a v i o u r is c l e a r e r i f one r e a l i z e s t h a t t h e E M waves c a n escape f r o m t h e p l a s m a a n d also t h a t t h e d i s t u r b a n c e t a k e s a finite t i m e t o p r o p a g a t e across t h e p l a s m a . A t t h e edge of t h e p l a s m a f a c i n g t h e i n c i d e n t p u m p wave, t h e waves w i l l h ave a h e a d s t a r t i n g r o w t h c o m p a r e d t o t h o s e a t t h e f a r edge. T h e c o n v e c t i v e g r o w t h t i m e r is t h e finite t i m e f o r t h e effect a t t h e f a r edge t o reach t h e f r o n t edge of t h e p l a s m a . I n a n i n h o m o g e n e o u s p l a s m a t h e s t u d y of t h e i n s t a b i l i t y becomes m u c h more c o m p l i c a t e d , p r i m a r i l y b e c a u s e t h e w a v e v e c t o r m a t c h i n g c o n d i t i o n c a n n o t be sat i s -f i e d at a l l p o i n t s i n t h e p l a s m a . T h i s m i s m a t c h is best q u a n t i f i e d b y t h e d e r i v a t i v e , k — — (/c0(i) kgpfjj(x) /c$(x)) ax w h i c h is zero f o r a homogeneous p l a s m a . If t h e d e n s i t y v a r i e s l i n e a r l y w i t h x n(x) = n0(l - x/L), t h e n u2p(x) = u2p0(l-x/L). I t is easy t o show, f o r p e r f e c t f r e q u e n c y m a t c h i n g , t h a t a k m i s m a t c h k' = { u l l 2 c 2 L ) ( k - 1 - k;1 - c2me/kepw3kbT) CHAPTER 2: Theory of stimulated Raman scattering 22 exists which is approximately (for ks ^ 0) 6LkbTk The absolute or convective nature of SRS is important since the thresholds which the various theories predict for the instability depend upon the assumed nature of the instability. These threshold conditions are quite different for the different theories. Also, for comparison to later experimental results, the spatial distribution of the fluctuations is important. For example, Rosenbluth 1 6 showed that in an infinite, inhomogeneous plasma only convective growth could be expected with a maximum intensity growth This differs from the usual convective growth in that this level of growth is reached at all points where the instability occurs i.e. the fluctuations will be of uniform amplitude over some region of space. Rosenbluth's condition for this growth is that the term in the exponent divided by 2w should be greater than 1. On the other hand, Pesme 1 7 showed this convective growth only occurs for a plasma of length L > " y 0 / K , ' ( v e p w V - ) 1 / 2 and absolute growth (at the reduced temporal rate) occurs if (Uep^V-) 1/ 2 < L < 7 0 / 'K^VepvjV-) 1 / 2 ' ' . Dubois 1 8 states that Rosenbluth and Pesme are both wrong and that an absolute instability occurs in any size of plasma provided 7 2 / n ' v e p w v - > 1. In this case, the growth rate will be given by I/I0 = exp(27T72//c't;ep«;V_) 2-19 . 2-Yo ( l - ( K V, epw )2/37r-l/2) _ ( T L + 22^)] epw 2-20 . \fvepwv Vepw l>_ — V, epw CHAPTER 2: Theory of stimulated Raman scattering T h i s e x p r e s s i o n also i n c l u d e s a new effect. I n a finite, homogeneous p l a s m a , t h e r e is a t h r e s h o l d g r o w t h r a t e set b y c o n v e c t i v e d a m p i n g , namely, IT > epw 2 epw ) 2 - 2 1 . W h e n t h e e x p r e s s i o n s f o r i0 , k' a n d th e c o n d i t i o n f o r a b s o l u t e g r o w t h t h a t D u b o i s uses are c o m b i n e d , t h e t h r e s h o l d f o r a b s o l u t e g r o w t h i n an i n h o m o g e n e o u s b u t finite p l a s m a becomes C h a m b e r s s o l v e d t h e c o u p l e d m o d e e q u a t i o n s d e s c r i b i n g S R S n u m e r i c a l l y a n d p o i n t e d o u t t h a t t h e n a t u r e o f a n i n s t a b i l i t y i s a f u n c t i o n of b o t h t h e p l a s m a s i z e a n d t h e a s s u m e d b o u n d a r i e s . H e also showed t h a t t h e e l e c t r o s t a t i c m o d e energy is n o t u n i f o r m l y d i s t r i b u t e d over the i n t e r a c t i o n r e g i o n b u t tends t o be l a r g e r near t h e b o u n d a r y . T h e e x a c t p o i n t of m a x i m u m fluctuations depends u p o n t h e d e n s i t y g r a d i e n t a n d t h e r a t i o of t h e g r o u p v e l o c i t i e s of t h e waves generated. W i t h s e v e r a l d i f f e r e n t t h e o r i e s t o choose f r o m , t h e best t h i n g one c a n do is t o w a i t u n t i l some e x p e r i m e n t s have been m ade a n d t h e n d e c i d e w h i c h t h e o r y is c o r r e c t . A t 0.25n c r, t h e t h r e s h o l d f o r m u l a (eqn 2-22) is n o t v a l i d s i n c e ks = 0 a n d th e r e f o r e k' is i n f i n i t e . D r a k e a n d L e e 2 0 have s h o w n t h a t at t h i s d e n s i t y t h e t h r e s h o l d s h o u l d b e A ca v e a t s h o u l d b e p l a c e d on any p r e d i c t i o n of final fluctuation l e v e l v e r s u s p o s i t i o n . N o n l i n e a r effects, w h i c h are r e s p o n s i b l e f o r t h e s a t u r a t i o n o f t h e i n s t a b i l i t y , c a n affect t h e levels. T h e e x t e n t or i m p o r t a n c e of these effects is t h e s u b j e c t o f o n g o i n g i n v e s t i g a t i o n s . 2 - 2 2 . 0.52 (k0L)*/2 2 - 2 3 . c CHAPTER 2: Theory of stimulated Raman scattering So f a r o n l y a t h r e e wave i n t e r a c t i o n has been c o n s i d e r e d a n d t h e p l a s m a f l u i d p a r a m e t e r s have been v a r i e d . T h e r e are c o m p e t i n g wave-wave i n t e r a c t i o n s w h i c h c a n have a n effect o n S R S . I n t e r a c t i o n s l i k e S B S a n d t w o - p l a s m o n decay a n d o t h e r processes i n w h i c h t h e waves p r o d u c e d by S R S decay cascade i n f u r t h e r decays are c o n s i d e r e d t o e s t i m a t e how t h e y effect S R S . T h e effects of m o b i l e ions h ave so f a r b e en neglected. S t i m u l a t e d B r i l l o u i n S c a t t e r i n g ( S B S ) , w h i c h has a lower t h r e s h o l d a n d a lower g r o w t h r a t e t h a n S R S does, c a n d r i v e l a r g e a m p l i t u d e i o n a c o u s t i c waves i n t h e p l a s m a . T h e s e waves c o u l d d i s r u p t S R S i f t h e E P W g e n e r a t e d b y S R S s c a t t e r s off these i o n a c o u s t i c waves. A l t h o u g h t h e energy t r a n s f e r e d is s m a l l ( w t a <C u)epw), t h e E P W c o u l d be s c a t t e r e d i n any d i r e c t i o n . T h e s e s c a t t e r e d E P W c a n n o t phase m a t c h p r o p e r l y w i t h t h e p u m p wave a n d S R S is quenched. T h e two d i m e n s i o n a l n a t u r e of t h i s effect has n e v e r b een f u l l y d i s c u s s e d a l t h o u g h r e s u l t s f r o m one d i m e n s i o n a l s t u d i e s have s h o w n i t c o u l d be q u i t e i m p o r t a n t . ( T h e 2 D n a t u r e r e s u l t s f r o m t h e wave v e c t o r m a t c h i n g c o n d i t i o n . ) D a m p i n g of an E P W b y a n I A W has b e en c o n s i d e r e d b y D a w s o n 2 1, Sagdeev 2 2, a n d i n a n e a r l y (1970) c o m p u t e r s i m u l a t i o n 2 S. Th e effects of t h e T P D i n s t a b i l i t y 2 4 o n S R S are now d i s c u s s e d . T P D has a s l i g h t l y lower t h r e s h o l d t h a n S R S does a n d has a s i m i l a r g r o w t h r a t e . Its effect is t w o f o l d . It c a n d r i v e l a r g e a m p l i t u d e I A waves 2 5 w h i c h c o u l d a c t as t h o s e of S B S do. It c a n a l s o m o d i f y t h e l o c a l d e n s i t y p r o f i l e n e a r q u a r t e r c r i t i c a l d e n s i t y 2 5- 2 6 5 d e c r e a s i n g t h e scale l e n g t h L. A s L decreases, t h e t h r e s h o l d f o r S R S increases. A t l o w e r d e n s i t i e s , L increases a n d p e r h a p s t h e t h r e s h o l d i n t e n s i t y r e q u i r e d at t h o s e d e n s i t i e s decreases. T h i s p r o f i l e m o d i f i c a t i o n c o u l d be r e s p o n s i b l e f o r a g a p i n t h e s c a t t e r e d E M s p e c t r u m . S e c o n d a r y decay processes, i n w h i c h t h e s c a t t e r e d E M W o r E P W decay, have a l s o b e e n p r o p o s e d as s a t u r a t i o n m e c h a n s i m s f o r S R S . I n p a r t i c u l a r , t h e e l e c t r o -s t a t i c w a v e c o u l d decay i n t o a n o t h e r E P W a n d a n I A W . T h i s process was c o n s i d e r e d CHAPTER 2: Theory of stimulated Raman scattering 25 by Karttunen 27>28>29. For Up/u2, < 0.1 ,the damping rate due to this decay is Idecay/lo = { ^ ) { ^ ^ ) E e p w / E0 2 - 24. e p For T=300eV, n/ncr = 0.1 and Z=4, one finds for a CO2 laser plasma that idecay/lo = 0.5Eepu,/E0. During the nonlinear evolution of SRS, there can be instances where Eepw > E0 and thus, this effect might be quite important for the saturation of SRS. Related to the mode coupling effects are what I call convective effects. Koch and Williams 8 0 have considered coupling between forward and backscattered E M waves in a density gradient. The forward scattered waves are reflected from their critical surface and propagate down the density gradient, triggering the instability at lower densities by increasing the E M noise level. A similar mechanism may be applicable to the plasma waves generated near quarter critical density although away from 0.25ncr the E M waves travel much faster than the EPW do. The plasma waves generated by SRS will propagate forward up the density gradient and be reflected at their critical surface. The waves will subsequently propagate down the graident. This suggests that SRS should start at lower densities as time progreses. The rate at which it does so may be estimated by noting that dn/dt » (dn/dx)(dx/dt) 2 - 2 5 where dn/dx is related to the density scale length and dx/dt is vepw. This effect should be kept in mind when the experimental data are examined. The fluid model uses an average electron velocity. There are instances where the actual distribution of electron velocities is important. For example, Landau damping is sensitive to df /dv evaluated at the phase velocity of the wave. The quoted formula, eqn 2-14, is derived under the assumption that the distribution of electron velocities was Maxwellian. Since fast high energy electrons are generated CHAPTER 2: Theory of stimulated Raman scattering 26 w i t h v « vpn, t h e d i s t r i b u t i o n f u n c t i o n at vph m a y be m o d i f i e d . E s t a b r o o k 2 has s h o w n t h a t m o r e p l a s m a wave d a m p i n g does o c c u r . If t h e h o t e l e c t r o n s have a M a x w e l l i a n d i s t r i b u t i o n , t h e t o t a l r a t e is JL_ = ^ ^ ^ ( ^ ( ^ e x p i - M — ) 2 } LJepw V ve ^ k ' V F 1 V ve 1 1 + ( T ) 3 / 2 ^ e x p { _ 0 5 ( ^ ) 2 ( T •iHot n-T Ve 21 Hot 2 - 2 6 w h e r e nc is n u m b e r of c o l d e l e c t r o n s , njjot is t h e n u m b e r of fa s t e l e c t r o n s , = nc + n-Hot a n d Tjfot is t h e f a s t e l e c t r o n t e m p e r a t u r e . If Tjj0t is t o o l a r g e , a r e l a t i v i s t i c p a r t i c l e d i s t r i b u t i o n s h o u l d be u s e d i n p l a c e of t h e o r d i n a r y M a x w e l l i a n d i s t r i b u t i o n s e m p l o y e d t o d e r i v e t h e f o r m u l a q u o t e d . T h e effects o f a n y s t a t i c ( o r q u a s i s t a t i c o n 7" 1 t i m e scales) m a g n e t i c fields o n S R S h ave been n e g l e c t e d i n t h e d i s c u s s i o n so f a r . T h e effects of s u c h a m a g n e t i c field o n S R S have b e e n c o n s i d e r e d 35>36>37>S8.39. j n t h e s i m p l e s t case, S R S generates s c a t t e r e d l i g h t at t h e f r e q u e n c i e s us = u0J2 ± Uc/4 2-27 a n d i t s g r o w t h r a t e changes t o 7 = 7o(l±3nc/u;0) 2 - 2 8 whe r e 70 <C Uc a n d n = 0.25n c r have been assumed. nc is t h e e l e c t r o n c y c l o t r o n f r e q u e n c y . T h e s c a t t e r e d l i g h t is c i r c u l a r l y p o l a r i z e d . M a n y m e c h a n i s m s c a n generate large m a g n e t i c fields i n a laser- p r o d u c e d p l a s m a , t h e m o s t c o m m o n b e i n g t h e m a c r o s c o p i c V n e x VTe source. T h e e l e c t r i c f o r c e i n a p l a s m a is b a l a n c e d by a e l e c t r o n p r e s s u r e g r a d i e n t i.e. neeE = ( V p ) . If t h i s e l e c t r i c field is s u b s t i t u t e d i n t o Faraday's l a w a n d i f t h e p l a s m a behaves l i k e a n CHAPTER 2: Theory of stimulated Raman scattering 27 i d e a l gas, t h e V n e x V T t m a g n e t i c field r e s u l t s . A s i m p l e e s t i m a t e of t h e s a t u r a t e d field d u e t o t h i s m e c h a n i s m was m a d e by M a x et al. 3 4 w h o s howed = 3 6 ( r ( * e V ) ) - / 2 _ J _ y ( _ A _ ) 1 / 2 . F o r T = 3 0 0 e V , L (the d e n s i t y scale l e n g t h ) = 300 pm, A (the a t o m i c w e i g h t ) = 14 a n d Z ( t h e average charge) = 4, a field of 100 k G m i g h t be g e n e r a t e d . T h e m a g n e t i c field w h i c h is a r e s u l t of t h i s m e c h a n i s m w i l l be t o r o d i a l i n s h a p e w i t h t h e a x i s of t h e t o r u s a l o n g t h e laser a x i s . ( A p u l s e of f a s t e l e c t r o n s is e q u i v a l e n t t o a c u r r e n t a n d c o u l d a l s o generate a large m a g n e t i c field.) P a r t i c l e t r a p p i n g 3 1> S 2> S S i n t h e e l e c t r o s t a t i c wave p o t e n t i a l c a n g e n e r a t e h i g h e n e r g y e l e c t r o n s . T r a p p i n g o c c u r s f o r e l e c t r o n s w h ose v e l o c i t i e s v are s u c h t h a t 0.5met;^ — t(p < 0.5met;2 < 0.5mev2h + e<j> w h e r e t h e d e n s i t y fluctuation l e v e l (6n) is r e l a t e d t o t h e e l e c t r o s t a t i c p o t e n t i a l a m p l i t u d e , <f>, b y t h e F o u r i e r t r a n s f o r m e d P o i s s o n e q u a t i o n 6n/n0 = <f>k2 /4wn0e. ( T h i s c o n d i t i o n is n o t t h e u s u a l c o n d i t i o n f o r t r a p p i n g w h i c h is 0.5m e(v — vvh)2 < e<j>. If t h i s is u s e d a l i n e a r d e p e n d e n c e of N o n 6M 1/ 2 i n s t e a d o f 6n is obtained.) I n a M a x w e l l i a n p l a s m a t h e r e w i l l a l w a y s be some t r a p p i n g . We c a n m a k e a s i m p l e m o d e l f o r e s t i m a t i n g t h e n u m b e r of hot e l e c t r o n s g e n e r a t e d b y t r a p p i n g . We a ssume t h a t a c o n s t a n t f r a c t i o n , /, of t h e e l e c t r o n s w i t h energies b e t w e e n 0.5m(w/Ar) 2 + e<$> a n d 0.5m(w/fc) 2 — e<j> are t r a p p e d by the wave p o t e n t i a l , c o n v e r t e d t o h i g h energy e l e c t r o n s , a n d s u b s e q u e n t l y d e t e c t e d . If t h e b a c k g r o u n d p l a s m a has a M a x w e l l i a n CHAPTER 2: Theory of stimulated Raman scattering 28 d i s t r i b u t i o n of v e l o c i t i e s , t h e n u m b e r of fa s t e l e c t r o n s p r o d u c e d w i l l be Ne — / n 0 [ e r f ( x + ) — e r f ( x _ ) ] w h e r e x2±=0.5m{uj/k)2/kBT±e(f>/kBT. We c a n let x+ go t o i n f i n i t y , p r o v i d e d e<j>/kBT 1. B e c a u s e x _ 3> 1 , we c a n use t h e a s y m p t o t i c e x p a n s i o n f o r t h e r e s u l t i n g erfc x _ t o a r r i v e at Ne = /n 0exp(-x 2 _ ) / ( v / 7 r x _ ) o r lnNe = \n{fn0/{yfirx-))-a?_. T h e first t e r m o n t h e R H S is e f f e c t i v e l y c o n s t a n t . S u b s t i t u t i n g t h e d e f i n i t i o n of x2_ a n d u s i n g <f> f r o m t h e F o u r i e r t r a n s f o r m e d P o i s s o n e q u a t i o n we find ln(JVe) = (4-ne2n0/k2kBT){6n/n0) + C 2-29 w h e r e C is a co n s t a n t . T h i s r e l a t i o n s h i p c a n be c o n f i r m e d i f the f l u c t u a t i o n l e v e l a n d t h e n u m b e r of f a s t e l e c t r o n s are s i m u l t a n e o u s l y m e a s u r e d . 2.5 Connection between the Infrared Reflectivity and the Fluctuation Levels A n e l e c t r o n d e n s i t y p e r t u r b a t i o n o f d e p t h Sn/n c a n s c a t t e r a n e l e c t r o m a g -n e t i c wave. T h e le v e l of t h e s c a t t e r e d l i g h t c a n be e s t i m a t e d w i t h s e v e r a l a p p r o x i -m a t e f o r m u l a s . F o r t h e sake of d i s c u s s i o n , suppose t h a t a fluctuation l e v e l has been e s t a b l i s h e d a n d has been s a t u r a t e d b y some u n s p e c i f i e d m e c h a n i s m . If t h e s c a t t e r -i n g is s u f f i c i e n t l y weak, t h e a m o u n t of s c a t t e r e d l i g h t c a n b e f o u n d u s i n g n o r m a l s c a t t e r i n g t h e o r i e s . I n t h i s case, t h e l e v e l is c o n s i d e r e d due t o T h o m s o n s c a t t e r i n g . CHAPTER 2: Theory of stimulated Raman scattering ( T h i s is t h e s a me s c a t t e r i n g process t h a t is used f o r d i a g n o s t i c purposes. Here, i t is t h e i n c i d e n t CO2 laser l i g h t a n d n o t t h e p r o b e r u b y l a s e r l i g h t w h i c h is scattered.) T h e s c a t t e r e d power, Ps, i n t o s o l i d a n g l e d f i i n f r e q u e n c y range dw is g i v e n b y 1 dP* =nDS{k,u)r2J{27r) Pi dUduo w h e r e S{k,u>)= 1 VT 6n(k, u)n 2 n D is t h e l e n g t h of t h e p l a s m a , T is t h e t i m e of o b s e r v a t i o n , a n d V is t h e v o l u m e c o n t a i n i n g t h e p l a s m a waves. T h i s reduces t o a r e f l e c t i v i t y (for a si n g l e epw) of 4 0 R = r2eX20D2{Sn/n)2/4 2-30 w h e r e t h e s c a t t e r e d p o w e r is r a d i a t e d back i n t o t h e s o l i d angle s u b t e n d e d by t h e i n c i d e n t l a s e r beam. r e = e 2 / ( m e c 2 ) is the c l a s s i c a l e l e c t r o n r a d i u s . If t h e scat-t e r i n g is s u f f i c i e n t l y s t r o n g , however, t h i s f o r m u l a m u s t bre a k down. (T o see t h i s , c o n s i d e r t h e D —• 0 0 l i m i t . ) I n t h i s case, the c o u p l i n g o f t h e i n c i d e n t laser w i t h t h e p l a s m a m u s t be c o n s i d e r e d m o r e c a r e f u l l y . V i n o g r a d o v 4 1 has done so a n d f o u n d a n o n - l i n e a r r e f l e c t i v i t y f r o m a p l a s m a . K r u e r 4 2 has d e r i v e d t h e same r e s u l t a n d p r o v i d e d m o r e d e t a i l o n how i t was f o u n d . T h e r e s u l t f o r waves p r o p a g a t i n g i n one s p a t i a l d i r e c t i o n is R=t&nh2 (aD(6n/n)) w h e r e a is t h e c o u p l i n g c o n s t a n t between t h e v e c t o r p o t e n t i a l of t h e i n c i d e n t E M wave a n d t h e p l a s m a fluctuation, Sn/n. S u b s t i t u t i n g t h e e x p r e s s i o n f o r a , we get J2 = t a n h 2 1 2-31. 2 n X0 ncr V ncr ) T h i s r e s u l t has t h e c o r r e c t p h y s i c a l l i m i t s R —• 0 as D —> 0 a n d R —* 1 as D —> 0 0 . I t a l s o r e d u c e s t o t h e n o r m a l T h o m s o n s c a t t e r i n g f o r m u l a f o r weak s c a t t e r i n g (when CHAPTER 2: Theory of stimulated Raman scattering SO the approximation tanh(x) = x is valid). There is also a correction for the refractive index of the medium which is usually ignored in the normal Thomson scattering formula. If the beam is fully coupled into the medium, the Manley-Rowe relations can be used to deduce a reflectivity. This is the largest limit since there cannot be 100% reflection from a density fluctuation if, for no other reason, conservation of action density (nh) must be attained. (See the original justification of these relations in section 2-2.) The Manley-Rowe method uses an estimate of how much energy, U e p t v , is contained in the plasma waves. This energy is just U e p w = E2V/{8x) where E is the electric field associated with a electron density fluctuation, E = 4ire26n/(kn0) and V is the volume containing the plasma waves. The amount of scattered infrared light is then 2 - 3 2 . The reflectivity R = Uscat/Uinc where ? 7 t n c is the amount of incident energy in the laser pulse. 2.6 Computer Simulations SRS is straightforward only if many simplifying (and sometimes unrealistic) assumptions are made. Computer simulations which follow the temporal evolution of SRS are necessary, sometimes providing insight before analytic models are avail-able. This section briefly describes how simulations are performed. Some results pertaining to this work are highlighted. CHAPTER 2: Theory of stimulated Raman scattering 31 A collection of electrons and ions is placed in some region of space and the dynamics of these two species are followed by solving Maxwell's equations. (If there is no initial electromagnetic wave, the problem is called electrostatic.) If the number of particles becomes too large, computer solutions cannot be used since the number of interactions goes as N 2 . Simplifying assumptions must be made. Two types of approach are common, the particle in cell (PIC) codes 4 8 and the Vlasov codes. Most simulations have been made using the PIC method. This method divides the particles into cells by a mesh. The fields on the mesh are found by summing the fields due all the individual particles. Each particle within a cell is moved by the field at its position found by interpolation of the fields at the mesh boundaries of the cell. The procedure is repeated and the simulation progresses. Typical simulations would have 60000 particles and 60 cells. The second approach involves the solution of Vlasov's equation and Maxwell's equations. In this method, it is the particle distribution which is followed. An ap-proximation, the waterbag model, in which the boundaries of a uniform distribution are followed, make this way potentially much faster. Common to both approaches are problems associated with infinite (periodic) versus finite plasmas and one versus two dimensional effects. In addition, relativistic efffects must be included as the velocities of some of the particles can be large. Early simulations 4 4 in infinite, homogeneous plasmas showed that the thresh-old and growth rates analytically calculated are quite good. Other simulations 4 5 have shown that density scale lengths, background temperature, plasma noise lev-els and damping of waves play important roles in determining thresholds for SRS. When experiments are compared to the simulations, one should remain aware of the fact that simulations are only an approximation to reality. Several of the aforemen-tioned effects could be present in an experiment, whereas simulations are usually restricted to examining the effect of one or two of them. CHAPTER 2: Theory of stimulated Raman scattering 32 S i m u l a t i o n s ( u s u a l 1-D) 3 2 > 4 4 i d e n t i f i e d t w o s a t u r a t i o n m e c h a n i s m s , p u m p d e p l e t i o n ( d i s c u s s e d e a r l i e r ) a n d f a s t e l e c t r o n g e n e r a t i o n . T h e g e n e r a t i o n of fast e l e c t r o n s is a d a m p i n g m e c h a n i s m as t h e energy is r e m o v e d f r o m t h e E P W a n d t r a n s f e r r e d t o k i n e t i c e nergy i n t h e e l e c t r o n s . T h e fast e l e c t r o n s h ave a v e l o c i t y d i s t r i b u t i o n w h i c h a p p e a r s M a x w e l l i a n w i t h a t e m p e r a t u r e g i v e n a p p r o x i m a t e l y b y kf,T = Q.5mev2h. vpn depends p r i m a r i l y o n t h e p l a s m a d e n s i t y b u t t h e r e is a te m p e r a t u r e dependence t h a t s i m u l a t i o n s 1 show is i m p o r t a n t f o r T > I k e V . U p to 2% of t h e e l e c t r o n s i n t h e p l a s m a c o u l d be c o n v e r t e d t o f a s t ones. O t h e r s i m u l a t i o n s 2 5 > 2 6 have i d e n t i f i e d f u r t h e r s a t u r a t i o n m e c h a n i s m s . T w o d i m e n s i o n a l s i m u l a t i o n s showed p r o f i l e m o d i f i c a t i o n due t o T P D (or S R S ) a n d m o d e c o u p l i n g t o i o n a c o u s t i c waves are i m p o r t a n t . B o t h m e c h a n i s m s have a c o m m o n o r i g i n . I n T P D , t w o l a r g e a m p l i t u d e epw are gene r a t e d , E \ j 2 = E \ 0 y 2 o c o s ( ^ i , 2 * ?~ uit2t). T h e t o t a l e n e r g y d e n s i t y , g i v e n b y t h e square o f t h e t o t a l e l e c t r i c field, is gi v e n by E 2 = £ 2 0 + £ f 0 - £ l t ) - £ 2 o [ c o s ^ T h e p o n d e r m o t i v e f o r c e is p r o p o r t i o n a l t o VE2. T h e t e r m V(E20 + E20) is r e s p o n s i -b l e f o r m o d i f y i n g t h e p l a s m a d e n s i t y n e a r q u a r t e r c r i t i c a l . T h i s p r o f i l e m o d i f i c a t i o n is c l e a r l y seen i n figure 2-5 w h i c h is r e p r o d u c e d f r o m L a n g d o n 2 6 . T h e s e c o n d t e r m , V ( J E I • £ 2 ) 5 w m t i m e average t o z e r o unless u>i — u2 « 0 o r a t least is m u c h less t h a n Up. I o n a c o u s t i c waves, whose frequencies are m u c h less t h a n up, c o u l d b e e x c i t e d b y t h i s means a t fcta = k\ — k2. T h e s e i o n waves were seen i n t h e s ame s i m u l a t i o n s w h i c h f o u n d t h e p r o f i l e m o d i f i c a t i o n . A self-generated m a g n e t i c field f o u n d i n y e t a n o t h e r s i m u l a t i o n 2 affects S R S by d i s r u p t i n g t h e t r a n s p o r t of hot el e c t r o n s . A s E s t a b r o o k p o i n t s o u t , r e d u c i n g t h e h o t e l e c t r o n t r a n s p o r t increases t h e l o c a l h o t e l e c t r o n density. T h e i n c r e a s e d d e n s i t y leads t o in c r e a s e d d a m p i n g (see eqn 2-26). CHAPTER 2: Theory of stimulated Raman scattering 33 F i g u r e 2-5 D e n s i t y scale l e n g t h decrease at 0.25n c r. N o t i c e t h e s h e l f a t lower d e n s i t i e s . O n e o b s e r v a b l e effect suggested by s i m u l a t i o n s 4 4 i s , f o r l i g h t s c a t t e r e d f r o m 0.25n c r, t h a t t h e E M s p e c t r u m s h o u l d show h a r m o n i c s u>0j2 + m w „ w h e r e m is a n i n t e g e r . ( S i m i l a r b e h a v i o u r is also e x p e c t e d as a n i n d i r e c t r e s u l t of t h e T P D i n s t a b i l i t y . ) T h e s e h a r m o n i c s were s u p p r e s s e d i n t h e e a r l i e r t h e o r y w h e n t h e h i g h e r o r d e r c o u p l i n g s were ign o r e d . T h e s i m u l a t i o n s do not have t h i s r e s t r i c t i o n a n d show t h a t t h e s e c o u p l i n g s are present. O n e i n t e r e s t i n g s i m u l a t i o n 4 6 was done i n 2-D. I n i t , S R S b a c k s c a t t e r is seen t o e v o l v e i n t o S R S s i d e s c a t t e r . T h i s o c c u r s b ecause , as t h e p l a s m a h e a t s u p , t h e E P W w i t h t h e longest w a v e v e c t o r s become m o r e damped. S i n c e t h e E P W w i t h t h e l o n g e s t w a v e v e c t o r is t h e one r e s p o n s i b l e f o r d i r e c t b a c k s c a t t e r , t h e s h o r t e r ones are o b v i o u s l y t h o s e r e s p o n s i b l e f o r s i d e ( a n d e v e n t u a l l y f o r w a r d ) s c a t t e r . T h i s effect is n e g l i g i b l e f o r o u r p l a s m a since t h e s i m u l a t i o n s howed i t is n o t i m p o r t a n t i n c o o l ( < 1 k e V ) p l a s m a s . CHAPTER S: Experimental Details CHAPTER 3 Experimental Details In this chapter, the facilities used to generate a plasma and the diagnostics used to measure SRS-generated effects are described. These diagnostics include Thomson scattering to examine the plasma waves, infrared spectroscopy and an electron spectrometer. In addition, the methods of analysis of the results are out-lined. 3.1 L a s e r a n d T a r g e t C h a r a c t e r i s t i c s The study of parametric instabilities requires a system capable of generating a large v0. (Remember that v0 is the velocity at which an electron oscillates in the electric field of the incident laser light.) Since i ; 2 is proportional to IA 2, either a long wavelength or a high intensity must be used. A CO2 laser with A = 10.6/xm is ideal since a large v0 can be generated at modest intensities. In addition, ncr for 10.6/xm radiation is low so that visible light diagnostics (primarily Thomson scattering and interferometry) can be used without any absorption or refraction effects on the probe light. (Remember that ncr is the electron number density at which the plasma frequency equals the incident laser frequency.) The CO2 laser has been described elsewhere 4 7>4 8>4 9 and is only outlined here. The physical layout of the system and essential operating parameters are shown in figure 3-1. The system consists of a hybrid oscillator, from which a 2 nanosecond pulse is electro-optically switched out using a Pockels cell, and 3 amplifiers which CHAPTER S: Experimental Details 35 i n c r e a s e t h e p u l s e energy t o ne a r 10 J o u l e s . T h e s w i t c h o u t a s p e c t is i m p o r t a n t since t h e e l e c t r i c a l p u l s e used f o r t h i s also t r i g g e r s low j i t t e r d i a g n o s t i c s s u c h as a r u b y l a s e r 5 0 > 5 1 , a H a m a m a t s u T e m p o r a l D i s p e r s e r M o d e l C1370-01 (streak c a m e r a ) , a n d a T e k t r o n i x 7104 o s c i l l o s c o p e . O t h e r t h a n r o u t i n e m a i n t e n a n c e changes, t h e s y s t e m has r e m a i n e d i d e n t i c a l t o t h a t d e s c r i b e d b y B e r n a r d 4 8 e x c e p t t h a t p r o v i s i o n f o r c h a n g i n g t h e p l a n e of p o l a r i z a t i o n ( d e f i n e d by t h e p l a n e c o n t a i n i n g E a n d k) has been i n t r o d u c e d . P r e v i o u s l y , t h e p l a n e of p o l a r i z a t i o n was v e r t i c a l at t h e t a r g e t , b u t u s i n g a t w o m i r r o r s y s t e m i t was c h a n g e d t o h o r i z o n t a l . T h e s t a t e of p o l a r i z a t i o n was c o n v e n i e n t l y c h a n g e d b y t h e i n s e r t i o n o f a q u a r t e r w a v e C d S p l a t e (to gi v e c i r c u l a r p o l a r i z a t i o n ) o r a h a l f w a v e p l a t e ( t o change l i n e a r l y v e r t i c a l t o l i n e a r l y h o r i z o n t a l p o l a r i z a t i o n ) a f t e r t h e f i n a l G e f l a t o n t h e h y b r i d t a b l e . F o r t h e s p a t i a l l y r e s o l v e d T h o m s o n s c a t t e r i n g a n d t h e n a r r o w r a n g e (4°) w a v e v e c t o r (k) r e s o l v e d T h o m s o n s c a t t e r i n g , t h e p l a n e o f p o l a r i z a t i o n was v e r t i c a l . F o r t h e w i d e angle (18°) k r e s o l v e d T h o m s o n s c a t t e r i n g , t h e f r e q u e n c y r e s o l v e d T h o m s o n s c a t t e r i n g , a n d t h e I R m e a s u r e m e n t s , t h e p l a n e was h o r i z o n t a l . C i r c u l a r p o l a r i z a t i o n , a l t h o u g h a n o p t i o n , was never used. T h e t a r g e t u s e d was s i m i l a r t o t h a t set u p b y P o p i l 5 2. It is a p u l s e d n i t r o g e n gas j e t s t a b i l i z e d t o l a m i n a r f l o w b y a h e l i u m b a c k g r o u n d 5 8. T h e p l a s m a w h i c h f o r m s is s u b c r i t i c a l ( n m a x « 0.4n c r) a n d has a t i m e a v e r a g e d t e m p e r a t u r e of 300 eV. T h e t a r g e t c h a m b e r s u r r o u n d i n g t h e j e t was r e p l a c e d b y one w i t h w h i c h d i a g -n o s t i c s c o u l d be m o r e e a s i l y p e r f o r m e d . A new f o c u s s i n g lens was used. P r e v i o u s l y , a s y m m e t r i c b i c o n v e x K C 1 lens was used, whereas now, a K C 1 lens d esigned f o r m i n i m u m s p h e r i c a l a b e r r a t i o n is used. T h e f i r s t lens was use d f o r t h e s p a t i a l l y re-s o l v e d T h o m s o n s c a t t e r i n g only. A sc h e m a t i c o u t l i n e o f t h e t a r g e t ( g i v e n i n f i g u r e 3-2) shows t h e size of t h e j e t r e l a t i v e t o t h e i n c i d e n t laser b e a m a n d subsequent p l a s m a . T h e p l a s m a f o r m s i n t w o r e g i o n s w h i c h c o r r e s p o n d t o t h e edges of t h e j e t . T h e h e i g h t at w h i c h t h e laser s t r i k e s t h e j e t v a r i e s b y ± 2 mm. T h e n e u t r a l gas CHAPTER S: Experimental Details F i g u r e 3-1 L a y o u t of t h e CO2 laser system. T y p i c a l p e r f o r m a n c e p a r a m e t e r s are s h o w n CHAPTER S: Experimental Details 37 F i g u r e 3-2 G a s j e t t a r g e t d r a w n t o scale. j e t d e n s i t y a n d s t r u c t u r e is u n i f o r m over s u c h a v a r i a t i o n i n h e i g h t a n d t h u s , t h i s v a r i a t i o n is t h o u g h t t o be u n i m p o r t a n t . T e m p o r a l l y a n d s p a t i a l l y i n t e g r a t e d o p t i c a l s p e c t r a of t h e p l a s m a l i g h t i n -d i c a t e t h e presence of n u m e r o u s N I a n d H e I lines. N I li n e s are e x p e c t e d , b u t H e I l i n e s i n d i c a t e t h a t t h e gas o u t s i d e t h e j e t is b r e a k i n g down. I n t h i s o u t s i d e b a c k g r o u n d p l a s m a m o s t of t h e S B S o b s e r v e d b y B e r n a r d 5 4 o c c u r s . A d i f f e r e n t t y p e o f gas j e t was b r i e f l y e x a m i n e d . T h i s j e t , a He j e t i n a He b a c k g r o u n d , was e x p e c t e d t o have s i m i l a r d e n s i t i e s t o those of t h e N 2 j e t ( c o n f i r m e d b y i n t e r f e r o m e t r y ) . O f course, t h e r e w i l l be c o n s i d e r a b l e d i f f e r e n c e i n i o n mass a n d i t was e x p l a i n e d i n c h a p t e r 2 t h a t t h e ions are i m p o r t a n t i n d e t e r m i n i n g t he CHAPTER S: Experimental Details 38 s a t u r a t i o n o f th e T P D i n s t a b i l i t y . T h e s t r u c t u r e of the N2 j e t was d e t e r m i n e d o p t i c a l l y b u t t h e s t r u c t u r e of t h e new j e t c a n n o t be s t u d i e d b y t h a t m e t h o d as the i n d e x of r e f r a c t i o n f o r H e is m u c h s m a l l e r t h a n t h a t of N2. A dif f e r e n t m e t h o d was used. A glow d i s c h a r g e b e t w e e n t w o s m a l l ( l c m by 2.5 cm) p a r a l l e l s t e e l e l e c t r o d e s s e p a r a t e d b y 3 c m was e s t a b l i s h e d i n t h e b a c k g r o u n d gas . T h e e l e c t r o d e s were o r i e n t e d p a r a l l e l t o t h e gas j e t . T h e gas j e t was f i r e d a n d th e r e s u l t i n g glow i n t h e j e t was i m a g e d t h r o u g h a n imag e i n t e n s i f i e r a n d p h o t o g r a p h e d o n P o l a r o i d film. A sh u t t e r was used t o c o n t r o l t h e t i m e of e x p o s u r e r e l a t i v e t o t h e firing o f t h e j e t a n d al s o t o c o n t r o l t h e l e n g t h of t h e exposure. I n figure 3-3, t h e gas j e t s t r u c t u r e c a n e a s i l y be r e c o g n i z e d a n d a p p e a r s t o be l a m i n a r . 3.2 Thomson Scattering T h o m s o n s c a t t e r i n g was used t o e x a m i n e t h e E P W f l u c t u a t i o n s d i r e c t l y . S i n c e t h e b u l k o f t h e i m p o r t a n t r e s u l t s t o b e pr e s e n t e d were d e r i v e d f r o m t h i s d i a g n o s t i c , a s h o r t d i s c u s s i o n o f th e b a c k g r o u n d o f th e t e c h n i q u e is g i v e n . T h e o u t l i n e g i v e n c l o s e l y f o l l o w s t h e i n t r o d u c t o r y c h a p t e r s of S h e f f i e l d 5 6. W h e n a n i n d i v i d u a l e l e c t r o n is a c c e l e r a t e d i n t h e field of a n e l e c t r o m a g n e t i c field E0, i t r a d i a t e s i n t h e f a r field a d i p o l e p a t t e r n . T h e field a t a n ob s e r v e r (Es) is p r o p o r t i o n a l t o i x (5 x E0) R w h e r e s = R/r a n d R j o i n s t h e o b s e r v a t i o n p o i n t t o t h e source o r i g i n . W h e n t h e r e are m a n y e l e c t r o n s , t h e f a r f i e l d p a t t e r n is a s u p e r p o s i t i o n o f th e effects of e a c h e l e c t r o n w i t h c o r r e c t i o n s f o r t h e r e t a r d e d t i m e differences a n d s o u r c e phase di f f e r e n c e s at t h e i n d i v i d u a l e l e c t r o n s i n p l a s m a . T h i s gives r i s e t o a power p a t t e r n - c d f i dws r fT e 2 • -Ps(R,us)dfldu)s = —r l i m — - / dt ,s x (s x Eio) 4tt' T->oo T \-J-t rrieC^ f • - - - I2 / drne{r, t) exp(— iuj s ( t — s • f/c + R/c)) cos(/c, • r — w,f) JVol CHAPTER 8: Experimental Details 39 Figure 3-3 Laminar structure in the He jet. The bright glow to the left is the cathode glow in front of one of the electrodes. The direction and position of gas flow is indicated by the arrow. ne is the source weighting, the exponential term contains the retarded time factor and the cos term is the phase difference at the source. ne is given by /die f — - J dume(k,uj)exj>{-i{k-r-u)t)). There are two terms ( + or — ) in the exponentials which reduce to t[(w - (u>s ± u>i))t - (k — (uss/c ± ki)) • r — usR/c}. CHAPTER S: Experimental Details If a n e x p e r i m e n t is set u p a n d a s i g n a l is o b s e r v e d at f r e q u e n c y ios a n d at a n a n g l e 0 t o t h e i n c i d e n t p r o b e beam, i t does n o t f o l l o w t h a t t h e k is t h a t c a l c u l a t e d f r o m k = ks — k{. T h e response t o a g i v e n wave is n o t a ^ - f u n c t i o n unless a n i n f i n i t e l e n g t h of t h e wave is p r o b e d . If a finite segment is p r o b e d u s i n g T h o m s o n s c a t t e r i n g (or i n d e e d , a n y s c a t t e r i n g ) t h e r e w i l l be t y p i c a l l y a s i n e 2 t y p e response. A m o n o c h r o m a t i c wave c a n give r i s e t o a finite cone of r a y s . T o see t h i s c o n s i d e r t h e i n t e g r a t i o n s w h i c h have t o be p e r f o r m e d t o d e r i v e a 6 f u n c t i o n response. T h e i n t e g r a t i o n over t i m e r educes t o 6(u — (uis ± u;,)). A t t h i s p o i n t a n i n t e g r a t i o n over v o l u m e is u s u a l l y p e r f o r m e d t o give 6(k — (ks ± ki)). If we are d e a l i n g w i t h a finite v o l u m e , t h i s is n o t v a l i d a n d t h e r e s u l t i n g s c a t t e r e d l i g h t w i l l have some sp r e a d . C o n s i d e r i n g o n l y k i n t e g r a t i o n we get P(R) oc / / dkdfne(k, u) exp(i(k — (us/c ± fc,)) • f). Jk JVol A s s u m e t h a t ne(k,u) is a s l o w l y v a r y i n g f u n c t i o n of k. T h e n P(R) oc nes'mc2(kzLz)s'mc2(kxLx)sinc2(kyLy), w h e r e kx,y,z — {k — (u>si/c ± ki))x,y,z, a n d a r e c t a n g u l a r v o l u m e of p l a s m a was assumed. I t is o n l y t h e finite s c a t t e r i n g v o l u m e w h i c h gives r i s e t o t h i s d i f f r a c t i o n effect. F i g u r e 3-4 shows a t w o d i m e n s i o n a l r e p r e s e n t a t i o n of t h e s c a t t e r i n g f r o m a finite s e gment of p l a s m a . C o n t o u r s i n k space ( a r b i t r a r i l y d r a w n ) show t h e l o e i i o f d i r e c t i o n s of e q u a l s c a t t e r e d i n t e n s i t y . T h e m a x i m u m i n t e n s i t y o c c u r s w h e n ks — k{ + k. ( O n l y t h e + case is shown.) T h e e x a c t a n g u l a r e x t e n t d epends u p o n CHAPTER S: Experimental Details 41 F i g u r e 3-4 T w o d i m e n s i o n a l r e p r e s e n t a t i o n of s c a t t e r i n g f r o m a finite v o l u m e . T h e c o n t o u r s are a r b i t r a r y . t h e scale l e n g t h s i n t h e x a n d z d i r e c t i o n s , t h e a n g l e o f i n c i d e n c e , a n d l e n g t h o f k. T h e m a x i m u m a n g u l a r e x t e n t w i l l be ki L w h e r e Skx>z = —!— . L<x,z F o r L = 200/zm, a n u n c e r t a i n t y o f 0.2°results ( F W ) . If L X i Z —> oo, t h e r e c o u l d still be a finite a n g u l a r r a n g e of s c a t t e r i n g i f e i t h e r k o r ki were n o t fixed. I n a r e a l e x p e r i m e n t t h e r e w i l l be a ran g e o f ki due t o the finite c o ne an g l e of t h e i n c i d e n t p r o b e beam. T h i s effect was m i n i m i z e d b y i m a g i n g t h e v e r t i c a l f o c a l p l a n e o f t h e lens o n t o t h e s l i t o f t h e s t r e a k c a m e r a . T h e d i s t r i b u t i o n o f l i g h t i n t h e f o c a l p l a n e of a lens is r e l a t e d t o t h e F o u r i e r t r a n s f o r m of t h e o b j e c t 5 7 w h i c h i n t h i s case is t h e p l a s m a . T o be pr e c i s e , i n t h e pr e s e n t case, t h i s is t r u e o n l y f o r d i r e c t i o n s p a r a l l e l t o t h e i n c i d e n t CO2 laser b e a m because of t h e a s t i g m a t i c CHAPTER 8: Experimental Details n a t u r e of t h e f o c u s s i n g used. O n e i m a g i n e s t h e l i g h t s c a t t e r e d f r o m t h e p l a s m a t o be a s u p e r p o s i t i o n of m a n y c y l i n d r i c a l waves each of w h i c h is c o n v e r g i n g t o a different l i n e i n t h e f o c a l p l a n e of t h e lens. If t h e r e were no p l a s m a present, t h e r e w o u l d be a s i n g l e c y l i n d r i c a l wave c o n v e r g i n g t o a l i n e . B e c a u s e of t h e f i n i t e s i z e o f the lens t h e wave does n o t converge t o a l i n e , b u t r a t h e r gives r i s e t o t h e u s u a l s i n g l e s l i t d i f f r a c t i o n p a t t e r n . A s i m i l a r p a t t e r n w i l l be f o u n d f o r each of t h e c o n v e r g i n g waves w h i c h m a k e u p t h e l i g h t s c a t t e r e d f r o m a p l a s m a . I n t h e present case, t h e a n g u l a r e r r o r due t o t h i s p a t t e r n f o r l i g h t w h i c h passes t h r o u g h a n f/100 lens at 694.3nm is a b o u t 0.01 r a d i a n s , a s m a l l a n g l e w h e n c o m p a r e d t o t h e s c a t t e r i n g angles a n d t h u s w i l l be i g n o r e d . I n a n y o t h e r p l a n e t h e d i f f r a c t i o n p a t t e r n s w i l l be s p r e a d out a n d m a y o v e r l a p t h e r e b y r e d u c i n g t h e a n g u l a r r e s o l u t i o n . T h e r e are a d d i t i o n a l effects w h i c h c a n give ris e t o a f i n i t e r ange of s c a t t e r i n g angles. T h e s e are b e s t u n d e r s t o o d i f we assume t h a t a n i n f i n i t e l e n g t h of t h e p l a s m a wave is p r o b e d ( t h e r e b y e l i m i n a t i n g t h e finite l e n g t h effect) a n d t h a t t h e i n c i d e n t p r o b e b e a m is a p l a n e wave ( t h e r e b y e l i m i n a t i n g t h e finite a ngle effect of t h i s beam). If we are c o n s i d e r i n g t h e p r o b i n g of a p l a s m a w i t h n o a priori a s s u m p t i o n s a b o u t t h e waves, t h e n t h e f r e q u e n c y of t h e s c a t t e r e d l i g h t at a g i v e n angle m u s t be w e l l k n o w n i n o r d e r t o deduce t h e k of t h e p l a s m a wave. However, t h e r e w i l l a lways be a finite f r e q u e n c y s p r e a d of t h e s c a t t e r e d l i g h t w h i c h w i l l r e s u l t i n a n u n c e r t a i n t y i n ks a l o n g t h e d i r e c t i o n of s c a t t e r of A u f / c . T h i s r e s u l t s i n a n u n c e r t a i n t y of the p l a s m a k d o i n g t h e s c a t t e r i n g . T h i s u n c e r t a i n t y is s h o w n i n figure 3-4 In p r i n c i p l e , i f b o t h t h e f r e q u e n c y a n d d i r e c t i o n o f t h e s c a t t e r e d p r o b e l i g h t a r e k n o w n , t h e p l a s m a wave p r o b e d is fixed a n d t h e r e s h o u l d be n o f u r t h e r c o m p l i -c a t i o n s . I n p r a c t i c e t h e r e r e m a i n s one p r o b l e m t h a t c a n give r i s e t o a finite a ngle of s c a t t e r e d l i g h t . T h e i n c i d e n t laser b e a m is e f f e c t i v e l y c o m p o s e d of p l a n e waves w i t h k = c o n s t a n t b u t p r o p a g a t i n g at a range of angles t o t h e b e a m a x i s . W i t h i n t h e cone a n g l e of a laser, a n a n g u l a r s p r e a d ± A ^ i n t h e d i r e c t i o n o f k is p o s s i b l e . CHAPTER S: Experimental Details 43 Figure 3-5 Uncertainty in k due to a finite Au>s. Simple trigonometry and calculus shows that this gives rise to an angular range in ks of k cos ^ ±A62 = - - A # 3 - 2 A;, cost? where \& is angle between the incident probe and the CO2 laser beam axis. This reduces to an FW uncertainty of 0.2°for $ = 124°and 0 = 3°. The total angular uncertainty shown in the data presented (e.g. Fig.4- 6) is the sum in quadrature of the finite volume effect (Eqn.3-1) and the finite cone angle effect (Eqn.3-2). Four different variations on Thomson scattering were performed: one spatially, two k-vector resolved and one frequency resolved. In each case, the frequency and wavevector matching conditions were chosen such that SRS driven EPW were observable. The frequency selection condition was obtained by using an interference filter which passes 670 nm light in an 11 nm FWHM passband. The frequency shift expected for light (blueshifted) from 0.25rccr would give the scattered light a wavelength of 672.3 nm which is well within the passband. Light scattered from lower densities would have longer wavelengths. SBS scattered light and stray ruby CHAPTER S: Experimental Details 44 light both have wavelengths near 694.3 nm which is well outside the passband. Two filters were used in most cases. A second filter, a 694.3 nm filter tilted to 26°, had wider passband but still effectively rejected unshifted probe light. This combination of filters provided a signal to noise ratio of more than 104. The k vector matching was obtained by a judicious choice of incident probe beam angle and scattered angles. The length of the incident probe wavevector is ujiubylc while that of the scattered light is (u)RUby + w p ) / c - The length of the EPW wavevector generated at 0.25n.cr is \/SOJ0/2C. Since these vectors sum to zero, elementary geometry yields cos 0 = (k2epw - kRuby - k2Scat)/(-2kRubykScat) and sin * = kScat sin 0/kepw where 8 is the angle between the incident and scattered probe light and ^ is the angle between the incident probe and the epw. ^ was set assuming the EPW generated at 0.25ncr was parallel to the incident CO2 laser beam. \I> was 124° while 0 for 0.25ncr was 2.6°. (Many previous Thomson scattering setups assume that kRuby — kscat, i-e-> no frequency shift, in which case cos0 = 1 - k2pw/2k2Ruby. This shift cannot be ignored in this case since the difference in length between the incident and scattered ruby wavevector lengths, although small, is comparable to the length of the plasma wavevector probed. This means the scattering geometry must be carefully setup.) CHAPTER S: Experimental Details 45 T h e s p a t i a l l y r e s o l v e d T h o m s o n s c a t t e r i n g was p e r f o r m e d t o d e t e r m i n e where S R S o c c u r r e d a n d t o o b t a i n some i d e a of t h e siz e of t h i s r e g i o n . I n t h i s e x p e r i -m e n t t h e p l a s m a was m a g n i f i e d by a f a c t o r of 8 a n d i m a g e d o n t o t h e s l i t of t h e s t r e a k camera. A mask was used t o l i m i t t h e a n g u l a r e x t e n t of t h e s c a t t e r e d p r o b e l i g h t i m a g e d t o 2.6-4.5°which c o r r e s p o n d s t o w a v e v e c t o r s e x p e c t e d t o b e present at d e n s i t i e s i n t h e ra n g e 0.15-0.25n c r. T h e f r e q u e n c y range was l i m i t e d by the a f o r e m e n t i o n e d i n t e r f e r e n c e filters. S i n c e we are o b s e r v i n g a 2-D r e g i o n f r o m an angle , t h e r e are p r o j e c t i o n effects w h i c h cause a d i s t a n c e p e r p e n d i c u l a r t o i n c i d e n t CO2 b e a m t o be o b s e r v e d as a le n g t h . C o n s i d e r figure 3-6 w h e r e t w o e x t r e m e cases are s h o w n w h i c h b o t h give rise t o t h e same o b s e r v e d w i d t h W. T o o b t a i n a l e n g t h I ( i n t h e d i r e c t i o n o f t h e CO2 l a s e r ) , a w i d t h w ( p e r p e n d i c u l a r t o t h e CO2 ) is needed. P h y s i c a l l y , w is l i m i t e d t o a m a x i m u m size b y t h e size o f t h e C 0 2 laser b e a m i.e. 2 r 0 a n d a m i n i m u m size o f zero. F o r a g i v e n W, / is l i m i t e d i n l e n g t h . I n t h e s t r e a k r e c o r d s p r e s e n t e d , w has been a ssumed t o be 0 a n d t h u s / is o v e r e s t i m a t e d . If w is set t o t h e m a x i m u m v a l u e , t h e o b s e r v e d / are g e n e r a l l y r e d u c e d b y a b o u t a t h i r d . F i g u r e 3-6 P r o j e c t i o n effects o n s p a t i a l T h o m s o n s c a t t e r i n g . CHAPTER S: Experimental Details 46 T w o w a v e v e c t o r (k) r e s o l v e d s c a t t e r i n g e x p e r i m e n t s were set u p w h i c h are o u t l i n e d i n figures 3-7 a n d 3-8. I n t h e first e x p e r i m e n t , o n l y a l i m i t e d a n g u l a r r a n g e (2-4.5°) was i m a g e d whereas i n t h e t h e second e x p e r i m e n t a m u c h w i d e r range (2-18°) was i m a g e d . T h e first case w i l l be d e s c r i b e d i n some d e t a i l as t h e b u l k of t h e r e s u l t s i n t h i s t h e s i s were o b t a i n e d f r o m t h i s e x p e r i m e n t . A n i n c i d e n t r u b y la s e r p u l s e o f 6ns d u r a t i o n was f o c u s s e d a s t i g m a t i c a l l y u s i n g a f/100 glass lens t o a h o r i z o n t a l f o c u s a t t h e gas j e t w h i c h is 3 m m w i d e a n d t o a v e r t i c a l f o c u s 30 c m b e y o n d t h e j e t , w h e r e i t is d u m p e d i n t o a s t a c k e d r a z o r b l a d e p i l e . T h e i n c i d e n t CO2 laser b e a m f o c a l v o l u m e a n d t h e r u b y laser b e a m were a l i g n e d b y f o r m i n g a pi n h o l e i n a n A l f o i l t a r g e t l o c a t e d at t h e j e t , c e n t e r i n g t h e r u b y laser a l i g n m e n t H e N e la s e r o n t o t h i s p i n h o l e a n d u s i n g t h e l i g h t d i f f r a c t e d t h r o u g h t h e p i n h o l e t o a l i g n t h e o p t i c s t o t h e s t r e a k c a m e r a . F o r r o u g h a l i g n m e n t , a s e c o n d H e N e laser w h i c h s i m u l a t e d t h e s c a t t e r e d p r o b e l i g h t w as used. T h e F o u r i e r t r a n s f o r m p l a n e , t h e p l a n e at w h i c h t h e v e r t i c a l f o c u s was l o c a t e d , was i m a g e d o n t o t h e s t r e a k c a m e r a s l i t . A mask l o c a t e d i n t h i s p l a n e l i m i t e d t h e a n g u l a r range. T h e mask a l s o l i m i t e d t h e w a v e v e c t o r s t o th o s e w i t h i n t h e cone a n g l e o f t h e CO2 laser i n t h e d i r e c t i o n p e r p e n d i c u l a r t o t h a t r e s o l v e d at t h e s l i t . T h e t w o i n t e r f e r e n c e filters were i n c l u d e d t o l i m i t t h e f r e q u e n c y of t h e s c a t t e r e d l i g h t a n d t o r e d u c e s t r a y p r o b e a n d p l a s m a l i g h t noise. A c y l i n d r i c a l lens was p o s i t i o n e d i n f r o n t o f t h e s t r e a k c a m e r a s l i t so t h a t t h e p l a s m a was i m a g e d o n t o t h e s l i t f o r d i r e c t i o n s p e r p e n d i c u l a r t o t h e s l i t ; t h i s lens w o u l d n o t affect any im a g e a l o n g t h e s l i t o t h e r t h a n t r a n s l a t i n g i t i n space. In figure 3-7, t h e u p p e r ins e t shows the r e l a t i v e sizes of t h e i n c i d e n t r u b y , CO2 a n d s l i t i m a g e a t t h e j e t . T h e a r r a n g e m e n t of t h e i r sizes p e r m i t t e d t h e m a x i m u m s i g n a l t o be g a t h e r e d . A fiducial s i g n a l was g u i d e d t o t h e t h e c a m e r a f r o m t h e r u b y laser t o p r o v i d e a r e c o r d of any fluctuations i n t h e r u b y laser i n t e n s i t y i n c i d e n t u p o n t h e j e t . T h e o p t i c a l p a t h l e n g t h s of t h e i n c i d e n t r u b y laser p u l s e a n d t h e fiducial p u l s e were a r r a n g e d so t h a t b o t h pulses were c o i n c i d e n t at t h e st r e a k c a m e r a s l i t . CHAPTER S: Experimental Details F i g u r e 3-7 N a r r o w angle k r e s o l v e d T h o m s o n s c a t t e r i n g . CHAPTER S: Experimental Details 48 Plane Mirror //Ruby • f/100 Collimating' Mirror f/2.5 Plane Imaged /' V ' !'//. onto Slit ' ' Light Mask [below scattered beam] —Imaging Lens ••Interference Filters ^Cylindrical Lens n-Streak Camera F i g u r e 3-8 W i d e angle k r e s o l v e d T h o m s o n s c a t t e r i n g . CHAPTER S: Experimental Details 49 T h e o p t i c a ] s y s t e m was c a l i b r a t e d a b s o l u t e l y u s i n g a F r e s n e l d i f f r a c t i o n t e c h-n i q u e . S i n c e t h i s m e t h o d is s o m e w h a t u n u s u a l , i t w i l l be d e s c r i b e d i n d e t a i l . T h e t e c h n i q u e r e l i e s u p o n F r e s n e l d i f f r a c t i o n f r o m a s t r a i g h t edge i n t o t h e g e o m e t r i c s h a d o w ( d a r k ) zone. A s t r a i g h t edge ( r a z o r blade) was p l a c e d v e r t i c a l l y a t t h e j e t . T h e l i g h t i m a g e d o n t o t h e s t r e a k c a m e r a was t h e l i g h t d i f f r a c t e d i n t o t h e d a r k zone by t h i s edge. T h e r u b y laser was a t t e n u a t e d u s i n g n e u t r a l d e n s i t y f i l t e r s b y f a c t o r s of 104 b e f o r e a n d 10 a f t e r t h e s t r a i g h t edge. T h e r e s u l t i n g d i f f r a c t i o n p a t t e r n was s t r e a k e d a n d a n a l y z e d as f o l l o w s . Straight Edge F i g u r e 3-9 F r e s n e l d i f f r a c t i o n geometry. D i f f r a c t i o n t h e o r y shows t h a t I(w) = [C(w)2 + (l-S(w))2}l'0 w h e r e C a n d S are t h e F r e s n e l c o s i n e a n d sine i n t e g r a l s a n d , = J ^ ) - + W - T CHAPTER S: Experimental Details 50 T h e t e r m s , r',s' a n d x are d e f i n e d i n figure 3-9. We are i n t e r e s t e d i n l a r g e w where a s y m p t o t i c e x p a n s i o n s c a n be u s e d t o y i e l d I{w) = 21'JTTW2. It is also u s u a l t o c a l c u l a t e I f o r t h e p l a n e of o b s e r v a t i o n . T h i s e n t a i l s m u l t i p l i c a t i o n by a scale f a c t o r (see K l e i n 5 9 ) since x/r' = x0/(r' + s'). T h e i n t e n s i t y we w a n t s h o u l d also be n o r m a l i z e d t o t h a t at t h e edge. S i n c e l'0{Po) = 1 Source I [r' -TS>) a n d ledge = ^Source! r •> i t f o l l o w s t h a t r0(P0) = r'ledge/(r' + s'). S u b s t i t u t i n g f o r I'0 a n d x i n t h e e x p r e s s i o n f o r I(tv) we find I(x0) = \Ieige(s'I x0f firs' w h i c h r e d u c e s t o I[8) I'hdge « A/Wtail 2 0. W h e n c o m p a r i n g the d e r i v a t i o n s u m m a r i z e d above t o a more c o m p l e t e d e r i v a t i o n f o u n d i n s t a n d a r d t e x t b o o k s , i t m i g h t be u s e f u l t o n o t e t h a t o u r source is t h e second ( v e r t i c a l ) f o c u s l o c a t e d o n t h e same side of t h e edge as t h e p l a n e of o b s e r v a t i o n . T h i s has no effect o n t h e r e s u l t s f o u n d above. CHAPTER S: Experimental Details 51 A t a p a r t i c u l a r a n g l e , 60, a n d i n a r a n g e 60 a n a m o u n t J(60)/ledge is d e t e c t e d b y t h e s t r e a k c a m e r a a n d gives a s i g n a l N i . C o r r e s p o n d i n g t o t h i s s i g n a l , t h e f i d u c i a l h a s a l e v e l N2. T h e n any s u b s e q u e n t s i g n a l o b s e r v e d over t h e same a n g u l a r range is r e l a t i v e l y c a l i b r a t e d s i n c e H e r e , N.D. s t a n d s f o r t h e n e u t r a l d e n s i t y of t h e f i l t e r s used. A t t h i s p o i n t we sti l l c a n n o t d e t e r m i n e t h e a c t u a l f l u c t u a t i o n l e v e l as t h i s r e q u i r e s a k n o w l e d g e of t h e s c a t t e r e d power (or p o w e r t i m e s St). T h e c o n v e r s i o n f r o m i n t e n s i t y t o power r e q u i r e s a c o m p l e t e k n o w l e d g e of a l l t h e areas i n v o l v e d . T h r e e come t o m i n d : (a) t h e a r e a of t h e r u b y laser r a d i a t i o n at t h e s c a t t e r i n g v o l u m e ; (b) t h e a c t u a l a r e a f r o m w h i c h s c a t t e r e d l i g h t is e m i t t e d ; a n d (c) t h e a r e a i n t h e o b s e r v a t i o n p l a n e w h i c h c o r r e s p o n d s t o a p i x e l i n t h e s t r e a k c a m e r a . I n a d d i t i o n , i n s c a t t e r i n g t h e o r i e s i t is u s u a l t o c a l c u l a t e t h e d i f f e r e n t i a l s c a t t e r i n g (power p e r a n g u l a r r a n g e ) . T h e r e s u l t i n g s c a t t e r e d power is T h e f i r s t t e r m i n square b r a c k e t s o n t h e r i g h t is t h e r a t i o of t h e d e t e c t e d a r e a i m a g e d o n t o t h e s t r e a k c a m e r a t o t h e a r e a t h e r u b y laser i l l u m i n a t e s at t h e j e t . H e r e / repr e s e n t s t h e f r a c t i o n o f t h e a r e a i n t h e y d i r e c t i o n i m a g e d a n d is e q u a l t o 1.0 i f t h e e n t i r e a r e a is imaged. T h e s e c o n d t e r m c o r r e c t s f o r t h e f a c t t h a t t h e a r e a of t h e T h o m s o n s c a t t e r i n g a is n o t n e c e s s a r i l y t h e same as t h e t o t a l a r e a of t h e r u b y laser r a d i a t i o n . T h e f i n a l t e r m c o n v e r t s t h e e n t i r e e x p r e s s i o n t o a d i f f e r e n t i a l s c a t t e r i n g e x p r e s s i o n . {Iscat/1edge) absolute — (AlO t-N.D. /ns' t a n 2 d0) ^signal N2 Nl Nfiducial dPs/<L6 _ I{0) \s'60Ayf] r A x A y i r 1 CHAPTER S: Experimental Details 52 T h e s c a t t e r i n g c r o s s - s e c t i o n has been c o m p u t e d i n t e r m s of t h e f l u c t u a t i o n l e v e l b y S l u s h e r a n d S u r k o 4 0 f o r a s i n g l e epw {Sn/n0)2 = 2dPs/d6/nyeL2\2Pi nc w h e r e L = l e n g t h o f t h e s c a t t e r i n g v o l u m e cros s e d , re is t h e c l a s s i c a l e l e c t r o n r a d i u s a n d A is t h e w a v e l e n g t h of t h e i n c i d e n t probe. A f a c t o r of 1/2 has b e en i n t r o d u c e d s i n c e t h e t h e o r e t i c a l p o w e r (above) i n c l u d e s c o n t r i b u t i o n s f r o m a l l f r equencies. T h e e x p e r i m e n t l o o k s a t p o w e r r a d i a t e d i n 1/2 of t h e f r e q u e n c y range ( o n l y t h e a n t i -S t o k e s c o m p o n e n t ) . S u b s t i t u t i n g f o r t h e s c a t t e r i n g c r o s s - s e c t i o n a n d s i m p l i f y i n g , one finds S n 2 = \2 10-NDN2Ayf] r N s i g n a l n o 7r 2r 2A t a n 2 d0Nx J \-n2,L2oNfiducial w h e r e t h e t e r m i n s i d e t h e first b r a c k e t is a c o n s t a n t f o r a g i v e n o p t i c a l s e t u p w h i l e t h e t e r m s i n s i d e t h e s e c o n d b r a c k e t c a n v a r y f r o m shot-to-shot. T h e w i d e an g l e k r e s o l v e d s y s t e m h a d t h e same i n c i d e n t o p t i c s as t h e n a r r o w a n g l e s y s t e m . T h e s c a t t e r e d l i g h t was c o l l i m a t e d b y a m i r r o r a n d d i r e c t e d o u t t h r o u g h a l a r g e w i n d o w . ( T h i s was necessary s i n c e t h e w i n d o w u s e d i n t h e first e x p e r i m e n t was t o o s m a l l t o p e r m i t a l a r g e a n g u l a r r a n g e t o be examined.) A na r r o w s t r i p of a r e a i n t h e F o u r i e r t r a n s f o r m p l a n e was d e m a g n i f i e d a n d i m a g e d o n t o t h e s l i t t h r o u g h i n t e r f e r e n c e filters a n d a c y l i n d r i c a l lens. T h e s c a t t e r e d l i g h t h a d t o be sent over t h e gas j e t a n d a m a s k was l o c a t e d s u c h t h a t n o p l a s m a l i g h t c o u l d r e a c h t h e s l i t d i r e c t l y . ( W i t h o u t t h e mask, t h e s c a t t e r e d s i g n a l was lost i n t h e p l a s m a lig h t . ) R e a s o n a b l e c a r e was t a k e n t o ensure t h a t t h e m a s k d i d not c u t o f f p a r t of t h e s c a t t e r e d l i g h t . C o m m o n t o b o t h k r e s o l v e d s y s t e m s was t h e m e t h o d of d e t e r m i n i n g t h e scat-t e r i n g angles. T h e a b s o l u t e angle (or p o s i t i o n of k n o w n 6) was d e t e r m i n e d u s i n g a H e N e la s e r w h i c h p a s s e d t h r o u g h t h e CO2 f o c a l v o l u m e . Its angle was f o u n d by CHAPTER S: Experimental Details 53 noting the perpendicular distance, di , between the light from this HeNe laser and the HeNe laser used to align the ruby laser at a measured distance, d 2 , from the jet. The angle was then arctan (di /d 2 ) . This angle was measured to ±0.2 °. The streak camera channel corresponding to this angle was recorded. The uncertainty in this angle was due primarily to an uncertainty in dx. Since dj was 20 ± 2mm and d 2 was 300 ± 2 mm, this means 60 - 2.6 ± .2°. The angular dispersion (degree / channel) was measured using a mask of equal dark and light regions of width w placed at the known distance d 2 . The mask was examined by the streak camera in focus mode (the continuous or nonstreak mode of the camera) either in transmission (first setup) or reflection (second setup). The dispersion was calculated by noting the channels corresponding to the light-dark transistion and dividing the difference in channels into the angle arctan(u; / d 2). For the narrow angle Thomson scattering system, the average dispersion was 0.00022 radians per pixel. To interpret the resulting k-resolved records, some assumptions about the frequency of the scattered light had to be made. Of course, the use of interfer-ence filters had restricted the range of w selectable, but the filters still had a finite bandpass. In the first k-resolved scattering, it was assumed that the waves which scattered the light were those generated by SRS (and thus have known frequen-cies.) Since two scattering conditions had to be satisfied, those of SRS and those of Thomson scattering, it was possible to assign to a given angle a unique density and a unique wavevector. This simultaneous wavevector matching is shown in figure 3-10. The k-resolved streak camera records were interpreted as follows. An inci-dent E M wavevector, k0, of length w 0 ( l - Wp/a)p) 1/ 2/c decays into a scattered E M wavevector, ks, of length w 0 (l - 2u>p/u;0)1/2/c and an EPW wavevector, kepw, whose length depends upon the angle \& at which the ks is directed. The usual k and CHAPTER S: Experimental Details 54 Figure 3-10 Simultaneous wavevector matching for SRS and Thomson scattering. u matching conditions apply to this SRS process. In Thomson scattering the in-cident probe light with wavevector, kr, of length ur/c, scatters off the E P W and exits the plasma with a wavevector kTS of length (w r + uepw)/c. For a fixed angle of incidence <p, 6 is fixed for a given density since the point C is fixed (see Fig.3-10). C is the intersection of a circle centered at A with radius kTS representing all the possible krs directions (the dashed arc) and a second circle centered at B with radius k£ respresenting the possible kepw directions (the solid circle). The solution for various densities at our given angle of incidence (<p = 56°) shows that we can observe SRS driven epw corresponding to E M scattering at ip — 180° (0.25ncr), at rp = 143° (0.2ncr), and at rp = 135° (0.15ncr). (Only the solution which gives plasma wavevectors longer than k0 was considered.) Since a unique frequency can be assigned to a given angle, it is possible to correct for the bandpass edge rolloff of the interference filter. The transmission of the filter was measured using a monochromator attached to an optical multichan-nel analyzer (OMA). A broadband, incandescent source (a flashlight) was used to illuminate the filter and the spectrum of the transmitted light was recorded. At a given 6 an urs is deduced . The signal at that angle is divided by the transmission CHAPTER S: Experimental Details 55 of the filter at the scattered frequency. This method is crude, but assures that the k spectra are correctly interpreted. In the second k experiment, it was assumed that all the waves probed were generated at 0.25n c r . This is not true in light of the above analysis, but for 6 > 4.5°, scattered light from SRS E P W would be shifted in frequency and would not be able to penetrate the interference filters. If scattered light is observed at these angles, it cannot be generated by SRS. It can also not be generated by SBS, as the scattered light frequency from that process is nearly the same as the probe and cannot penetrate the filters. (SBS light would have a maximum near 7° if there were no filters present.) The assumptions made about uTS are verifiable if the scattered light is col-lected and sent through a spectrograph. This was done in the w-resolved Thomson scattering. The plasma was imaged (demagnified) onto the entrance aperture of a spectrograph through a single interference filter. This filter was the 694.3nm filter tilted to 22°and thus permitted a wide wavelength range near 672nm to enter the spectrograph while still rejecting the unshifted ruby light by a factor of more than 100. The scattered light was collected over the same angular range that the narrow angle k resolved Thomson scattering examined. No entrance slit was used, since the image itself was small enough. The exit plane of the spectrograph was imaged onto the streak camera slit. The spectrograph permitted a range of 30 nm to be examined at a time. The relative dispersion was determined using several Ar I lines (696.5, 687.1, 675.2 nm) in the wavelength range of interest. Light from an argon Geisler tube was focussed over the gas jet through the same optics that were used for the Thomson scattering. These 3 lines were displayed across the slit of the streak camera and the positions of the lines were recorded by integrating the streak camera output using the internal trigger capabilities of the camera. From the recorded channel position of each of the lines the dispersion was calculated to be 0.172nm / channel. The streak camera was used in focus mode while these lines CHAPTER 8: Experimental Details 56 illuminated it. The channels corresponding to line center were used. The absolute wavelength was found by assuming the SBS line to be unshifted ruby laser light at 694.3 nm. The relative dispersion and the channel separation from the SBS line was used to find A at any channel. The experimental setup is shown in figure 3-11. Figure 3-11 Frequency resolved Thomson scattering. A ruby laser interference filtered tilted to 22°is located in front of the entrance of the monochromator to reduce stray ruby laser light. An OMA could not be used to record a time integrated scattered light spectrum since there are, as noted earlier, several optical spectral lines present including one N I line which occurs exactly where Thomson scattered light from quarter critical density should occur, at 672.3 nm. In time resolved measurements, these lines will appear much later (i.e.when the plasma has cooled) than the Thomson scattered light. The wavelength resolution of the system was measured by sending the well attenuated ruby laser light pulse through a pinhole at the jet into the monochroma-tor and streaking the resulting single line. Since the line width of any ruby laser is less than 0.1 nm, any width observed is due to instrument broadening. The FWHM observed corresponded to 1.38 nm. A few comments about the streak camera used are now made since it is being used to its limits. The linear dynamic range Smax/Snoise was measured by Bernard Monochromator co 2 Ruby CHAPTER S: Experimental Details 57 w h o m e a s u r e d a d y n a m i c r a n g e of 100. However, t h e u p p e r c o u n t n u m b e r is de-p e n d e n t u p o n how m a n y p e r s i s t e n c e i n t e g r a t i o n s were p e r f o r m e d . F o r a g r e a t e r n u m b e r , t h e u p p e r l i m i t increases. S i n c e t h e noise l e v e l a l s o increases, t h e d y n a m i c r a n g e s h o u l d n o t change. T h e t e m p o r a l r e s o l u t i o n m u s t a l s o be emp h a s i z e d . T h e m a n u f a c t u r e r ' s m e a s u r e d t e m p o r a l r e s o l u t i o n is 2 picoseconds. T h i s l i m i t is set b y se v e r a l f a c t o r s , one of w h i c h is t h e size of th e e n t r a n c e s l i t as i t i m a g e d o n t o t h e p h o t o c a t h o d e of t h e s t r e a k c a m e r a . F o r o p t i m u m t e m p o r a l r e s o l u t i o n , t h i s i m a g e m u s t b e m a d e as s m a l l as p o s s i b l e . T h i s c a n be e n s u r e d i f t h e e n t r a n c e o p t i c s of t h e s t r e a k c a m e r a are f u l l y i l l u m i n a t e d . I n each k case, t h e c y l i n d r i c a l lens was p l a c e d s u c h t h a t t h e e n t r a n c e o p t i c s of t h e s t r e a k c a m e r a were f u l l y i l l u m i n a t e d . T h e t e m p o r a l r e s o l u t i o n is l i m i t e d by two o t h e r f a c t o r s . T h e a c t u a l s t r e a k r a t e used is i m p o r t a n t : t h e p i x e l r e s o l u t i o n is t h e sweep t i m e d i v i d e d i n t o 256 channels. F o r t h e g r o w t h r a t e s m e a s u r e d , t h i s d i g i t i z i n g r e s o l u t i o n is 2.2 ps. T h e o t h e r f a c t o r is t h e i n t e n s i t y f a c t o r : t h e b r i g h t e r t h e s t r e a k e d l i g h t , t h e w i d e r t h e r e c o r d e d image is. A l t h o u g h t h i s effect is d i f f i c u l t t o q u a n t i f y , i t s h o u l d be n o t e d t h a t n o r e a l difference i n g r o w t h r a t e s was f o u n d b e t w e e n (a) t h o s e m e a s u r e d f o r i n t e n s i t i e s s i m i l a r t o t h e ones a t w h i c h t h e m a n u f a c t u r e r m e a s u r e d t h e t e m p o r a l r e s o l u t i o n a n d (b) those at h i g h e r i n t e n s i t i e s w h i c h were w i t h i n t h e d y n a m i c range o f t h e camera. W e l l s a t u -r a t e d s h o t s were n o t a n a l y z e d . T h e r e l a t i v e s e n s i t i v i t y a cross t h e s t r e a k t u b e was als o m e a s u r e d . T h e cent e r of t h e s t r e a k t u b e is a b o u t a t h i r d m o r e s e n s i t i v e t h a n t h e edges. T h i s was m e a s u r e d by u n i f o r m l y i l l u m i n a t i n g t h e s l i t b y a t t e n u a t e d r u b y l i g h t a n d s t r e a k i n g t h e r e s u l t . A q u a d r a t i c response across t h e t u b e was o b s e r v e d a n d u s e d t o c o r r e c t t h e k s p e c t r a . 3.3 Infrared Diagnostics T h e s e c o n d aspect of S R S w h i c h was i n v e s t i g a t e d was t h e s c a t t e r e d E M r a d i -a t i o n . S p e c t r a l l y i n t e g r a t e d a n d s p e c t r a l l y r e s o l v e d m e a s u r e m e n t s were p e r f o r m e d . CHAPTER S: Experimental Details 58 Common to both measurements are the problems of absorption and dispersion in KC1 and rejection of SBS. The first problem manifests itself in that at 15/zm, KC1 starts to absorb IR with transmission per cm dropping from 90% (Fresnel losses only) to 30% at 21/zm. By the Kramer-Kronig relations, strong absorption is associated with strong dispersion. In KC1 , n=1.454 at 10.6/zm while n=1.39 at 21.2/zm. When imaging the plasma at various IR wavelengths, the resulting chromatic aberration must be considered. In our case, a single KC1 lens was used and its focal length was kept as short as possible. The lens subtended 0.016sr solid angle. The second problem concerns SBS rejection. This was important since SBS is very strong and could be expected in second order in the monochromator. Since SRS can generate a signal at 21.2 /zm, we had to be sure that no second order SBS light was detectable. Two 14//m redpass (cut-on) filters (OCL#L-13514-9) were used to provide a rejection of SBS of greater than 2.5 x 107. No SBS signal was seen through the two filters. This was checked by setting the monochromator to 10.6/xm; no signal was observed at this wavelength. (A weak signal was seen through a single filter.) The detector used in all SRS measurements was a Ge:Cu detector manufac-tured by the Santa Barbara Research Corp. It had a 5 mm diameter detector area behind a KRS-5 flat. (KRS-5 is a standard optical material for use in the far in-frared. It is made from pressed, polycrystalline thallium bromo-iodide.) To detect light at 21 microns, the detector must be liquid He cooled. The spectral response, according to the manufacturer's specifications, should be constant to within a factor of 2 over the range 15 — 21/xm. Two types of spectrally integrated measurements were carried out. The inte-grated signals were observed to get some idea of the energy scaling of the signal and more importantly, to see how large a signal could be expected. The optics were set up as in figure 3-12(a). A crude resolution of the spectrum was performed using a CHAPTER S: Experimental Details 59 single redpass filter and a 15 micron bandpass filter (FWHM = 2 microns). This filter was an OCL#W-15000-9A filter. The signal levels were sufficient that it was thought possible to resolve spec-trally the signal on a finer scale. A second IR arrangement was setup as in figure 3-12(b). In this arrangement, since a smaller signal was expected, chromatic aber-rations were considered more carefully. For this reason, a shorter focal length lens was used. The shorter focal length also allowed the plasma image to be reduced in size. This second consideration was important for lineup considerations. Although the plasma image itself was small (200^m), its actual position varied day-to-day by up to 2 mm. It is this change which had to be reduced so that the scattering volume was assured to be imaged into the entrance of the monochromator. The entrance aperture was positioned so that 20 micron light was perfectly imaged into it. The exit slit was imaged 1:1 onto the detector through the 2 redpass filters. The wavelength resolution, based upon the slit width imaged onto the detector was 0.2^m FW. The effects due the varying f# mismatch with wavelength were calculated by multiplying the solid angle subtended by the collection lens with the ratio of the area of the collimating mirror of the monochromator to the area over which the collected light would be spread at this mirror's position. The resulting effective solid angle is plotted in figure 3-12(c). Complications in the lineup also involved the HeNe laser light used to align the IR optics since it was affected to a greater degree by the chromatic aberration. The HeNe beam was used primarily on the optical axis of the system so that image position problems occurred only along the axis. There were no beam angle deflections. This ensured that the optical system would be centered for any wavelength; only the image positions along the optical axis would be uncertain. The system of alignment had one drawback. In order to minimize lineup errors, an f# mismatch had to occur for the signals into the monochromator. A better system would use two KRS-5 lenses which have much better dispersion .CO. CHAPTER S: Experimental Details 60 34.6cm " 0 — , 8 c m || Ge:Cu Detector f.l=20cm 2x1tym Redpass Filters 2 * K C l 1A/zm • 1°£/zm bandpass 18 34.6cm 1ft0 A I Monochromator — U . i?cm—r *4=16/im 60 lines/mm f.l.=50cm f/=10 Slit Width=6mm 2>KCl f.l.=10cm f.l=25cm GetCu Detector 2xU/zm Redpass Fi l ters i — r 19 21 F i g u r e 3-12 I R d i a g n o s t i c s , (a) S p e c t r a l l y i n t e g r a t e d (b) s p e c t r a l l y r e s o l v e d (c) E f f e c t i v e s o l i d angle d e t e c t e d CHAPTER S: Experimental Details 61 and transmission properties. The first lens would be used to collimate the scattered IR while the second could match the f# of the monochromator properly. Cost and time delays were the reasons this approach was not used. Complete use of only reflection optics would eliminate any chromatic aberration problems. However, severe spherical aberrations would have been introduced if mirrors were used to collect and direct scattered light from the target chamber since these mirrors would have to be used offaxis in order to avoid the jet nozzle on the way out. However, once the light has entered the monochromator, chromatic aberration is no longer a problem since only mirrors are used thereafter. 3.4 Electron Diagnostics The third aspect of SRS which was investigated was fast electron generation. For this, the spectrometer outlined in figure 3-13 was used. An electromagnet disperses and line focuses the electrons. These are detected by a NE102 plastic scintillator disk (behind 5 fim thick Al foil) connected to an OMA with optical fibers. The magnet current was kept fixed so that electrons at 50, 100 and 150 keV could be observed. (No signal was seen in the lowest energy channel which is indicated in figure 3-13 as low energy electrons cannot penetrate the A l foil.) Once the electron energy spectra are taken, there comes the question of in-terpretation. To fit a temperature (k#T), a Maxwellian distribution of velocities must be assumed although there is some question as to whether this is completely justified. Further, it is not clear whether a 1-D or 3-D distribution should be tried. In the following, I will attempt to explain the choice I made. Since the electron spectrometer is located far from the plasma, the electron distribution found is bi-ased. An effusion process is the appropriate model to apply. (For a simple model of an effusion process, imagine a pressurized sphere with a small hole which allows the gas to escape.) This implies f{vx,vy,vz) oc vzf(vz)f{vx)f{vy) CHAPTER S: Experimental Details 62 Fe Front-+—4.5-Gas Jet — 5jjm AI-NE102-Optical Fiber-OMA FeBack Units: cm R=1.17,2J02 ,2.89 3.55 in CO •6.3 B QQQ^Q •1-Fe Side (.125*) -40 x .040" Cu Wire -Al Plate Fe Core VJL. Fe Bottom F i g u r e 3-13 M k . I I I e l e c t r o n s p e c t r o m e t e r . A l l d i m e n s i o n s are i n c m u n l e s s other-w i s e i n d i c a t e d . CHAPTER S: Experimental Details 63 for the spectrometer centered on the z axis. The spectrometer is assumed to have an entrance aperture of radius r and is located a distance z0 from the plasma. If r <C z0, then vx, vy will both be <C v2. In fact, vx/vz = vy/vz = r/z0. The observed signal is proportional to the total energy deposited on the scintillator S oc J J j 0.5me(v2x + Vy + vl)f(vx,Vy,vz)dvxdVydvz. The integration is performed over the scintillator area. The spectrometer was designed to focus the electrons where focussing means that all electrons which leave a point at the plasma (the object) with a given speed converge at a line on the scintillator (the image). (Of course, only those electrons which pass through the entrance aperture of the spectrometer are focussed.) Elec-trons from the same object point but having different speeds will be focussed to different image lines. (Since the spectrometer uses a dipole magnetic field, there will only be focussing in the plane perpendicular to the B field.) This situation is analogous to chromatic aberration in ordinary optics. Light which leaves a object point passing through the entrance pupil of the system is focussed to an imaged point. Light of different frequencies (the analogy of different speeds) will be focussed to different points. The analogy continues if one considers the spectrometer as a cylindrical lens. The above argument justifies the following. Since vx + vy + v2 = v2 « v2 , we can approximate S as S oc 0 . 5 m e v 3 e x p ( - 0 . 5 m e v 2 / k B T ) ( i 3 t ; CHAPTER S: Experimental Details w h e r e a M a x w e l l i a n v e l o c i t y d i s t r i b u t i o n has been assumed. T h e t e r m d3v can be e v a l u a t e d ( a p p r o x i m a t e l y ) as (vd<t>)(dv)(vAr/z0) w h e r e d(f> = r/z 0 a n d dv = eBAr/mec . B is t h e m a g n e t i c field of t h e s p e c t r o m e t e r a n d A r is t h e r a d i u s o f t h e s c i n t i l l a t o r . ( T h i s c h o i c e of t h i s dzv presumes a 3-D d i s t r i b u t i o n o f v e l o c i t i e s , at least over a l i m i t e d s o l i d angle.) If we let E = 0.5m eu 2, we f i n d S/E oc £ 3 / 2 e x p ( - £ / k B T ) 3 - 4 w h i c h is t h e f u n c t i o n w h i c h was used t o find k # T . C o r r e c t i o n s f o r c h a n n e l - t o - c h a n n e l v a r i a t i o n s i n s e n s i t i v i t y have been made. T h e l i n e a r i t y o f t h e e l e c t r o n s i g n a l s d e t e c t e d was checked; t h e l a r g e s t s i g n a l was 8 0 % of t h e m a x i m u m l i n e a r range. A no n - l i n e a r least squares r o u t i n e ( U B C C o m p u t i n g C e n t r e N L 2 S N O ) was used t o f i t t h e f u n c t i o n above t o t h e o b s e r v e d energy s p e c t r a t o find k # T . CHAPTER 4: Results CHAPTER 4 Results In this chapter, the main results of my research are presented with a few, very brief comments made on them. A detailed discussion of their implications is given in the next chapter. 4.1 G e n e r a l O b s e r v a t i o n s Theory suggests that once a certain laser intensity is reached, SRS should occur. In experimental terms, this means a certain energy in a typical pulse time must be provided. Therefore, it is interesting to plot various signals versus incident energy. In figures 4-1 and 4-2, the relative 14 — 22/zm integrated IR scattered light energy and the relative number of electrons at 150 keV are shown as a function of pulse energy. The amount of each can be seen to grow nonlinearly. The vertical error bars are the standard errors of more than 4 data points while the horizontal errors represent the size of the energy bins used for averaging. A threshold energy upper limit of about 3 Joules, corresponding to an intensity of 1.9 x 1013 W c m - 2 , can be inferred from the electron numbers. This agrees with the upper limit on the threshold deducible from the integrated reflectivity measurements where the limit is set at the intensity where the signal to noise ratio equals one. For calculation purposes in the next chapter, an intensity slightly above this threshold of 3 x 1013 CHAPTER 4: Results 66 3 4 5 6 7 8 9 Incident Energy (J) F i g u r e 4-1 T h r e s h o l d b e h a v i o u r i n t h e s c a t t e r e d I R l i g h t . T h e l i g h t was i n t e g r a t e d f r o m 14 t o 22/zm. N o s i g n a l was f o u n d a t l o w energy. W c m 2 w i l l be used. T h i s c o r r e s p o n d s t o average i n t e n s i t y o n t a r g e t c o r r e s p o n d i n g t o 5 J . M o s t o f t h e r e s u l t s were o b t a i n e d at t h i s energy. 4.2 R e s u l t s f r o m S p a t i a l l y R e s o l v e d T h o m s o n S c a t t e r i n g I n a finite o r a n i n h o m o g e n e o u s p l a s m a , t h e r e is a c h a r a c t e r i s t i c l e n g t h L w h i c h is i m p o r t a n t i n d e t e r m i n i n g t h r e s h o l d s . It is a l s o i m p o r t a n t i n d e t e r m i n i n g t h e n a t u r e o f t h e i n s t a b i l i t y i.e. w h e t h e r i t is a n a b s o l u t e o r c o n v e c t i v e i n s t a b i l i t y . T h i s L c a n b e t h e l e n g t h of t h e p l a s m a , the d e n s i t y s c a l e l e n g t h , o r m o r e ge n e r a l l y , t h e l e n g t h o f t h e i n t e r a c t i o n r e g i o n . CHAPTER 4: Results 67 AO 30 <20 10 2 3 4 5 6 7 8 9 Incident Energy (J) F i g u r e 4-2 T h r e s h o l d b e h a v i o u r i n t h e n u m b e r of f a s t e l e c t r o n s . N o e l e c t r o n s were o b s e r v e d a t l o w energy. I n t h e p r e s e n t e x p e r i m e n t , t h e p l a s m a size a n d d e n s i t y ( a n d t h u s L ) were che c k e d i n d e p e n d e n t l y u s i n g i n t e r f e r o m e t r y a n d s p a t i a l l y r e s o l v e d T h o m s o n scat-t e r i n g . S i n c e a n e x t e n s i v e series of i n t e r f e r o g r a m s was a v a i l a b l e 6 0 , i t was u s e f u l t o c o r r e l a t e w h e r e S R S o c c u r r e d w i t h t h e p l a s m a d e n s i t y r e g i o n s f o u n d f r o m i n t e r f e r -o m e t r y . T h i s c a n be done by c o m p a r i n g t h e s p a t i a l l y r e s o l v e d T h o m s o n s c a t t e r i n g r e s u l t s w i t h t h e o b s e r v a t i o n s of t h e p l a s m a l i g h t t a k e n t h r o u g h t h e same o p t i c s . I n f i g u r e 4-3(a), a d e n s i t y c o n t o u r p l o t of t h e p l a s m a f o r c o n d i t i o n s at w h i c h S R S o c c u r s is s hown. T h e v i e w is f r o m 90°to th e CO2 l a s e r a x i s a n d t h e c o n t o u r s are c y l i n d r i c a l l y s y m m e t r i c a r o u n d t h e a x i s . T h e q u a r t e r c r i t i c a l d e n s i t y c o n t o u r is CHAPTER 4: Results 68 emphasized while the contour interval is 0.05ncr Early in the laser pulse, the inter-ferometry shows that the plasma is about 200 microns long. In figure 4-3(b) the position of plasma light and the region where the Thomson scattering occurred are illustrated in streak camera photographs. The Thomson scattering comes from the very beginning of the plasma in time. (The bright line across the plasma light is a streak camera digitizing error.) Although the two distinct plasma regions of figure 4-3(a) are not seen in the plasma light of figure 4-3(b) because the the plasma is viewed from 60°to the laser axis instead of 90°, one can still identify the plasma re-gion where the SRS occurs since the SRS occurs in the plasma region which forms first. The front plasma region of figure 4-3(a) formed first. This is clear since this front region has a well developed annular shape (the result of a shock wave) whereas in the rear plasma region, the plasma still has the characteristic initial spherical shape. In figure 4-4, a contour plot of the spatially resolved Thomson scattering is given. The intensity contours are in steps of 1 /4 of the maximum level. The size is 200 microns, although from shot to shot this varied from 190 to 410 microns (see the further examples of figure 4-5). The diffraction limited resolution of the spatially resolved Thomson scattering optical experiment was limited by the aperture used to define the wavevector range of the scattered probe light that was permitted to enter the optical setup. The resolution calculated was 36/mi which agrees with the observation that features 50 microns in size can be seen. A streak camera pixel corresponds to 16 microns. The axial distance has been calculated assuming no radial width to the scattering region (see section 3-2 for details). If the width is set to 2r„, the axial distance is reduced to 140 fim. The scattering appears to be coming from only one region in space; there are no multiple regions of scattering as was observed in the case of S B S 4 7 . A lower limit on the average density scale length is established by assuming one end of the scattering region is at nu density while the other end is set to nj, density. Since 0.15-0.25 critical density is the CHAPTER 4: Results 1 a> c Plasma Light Thomson Scattering Figure 4 -3 Position of the scattering region relative to the plasma, (a) Density contour plot (b) Plasma light and scattering region as observed through the same optics. Note that the Thomson scattering occurs at same temporal and spatial position as the beginning o/the plasma light. maximum density region expected for the given mask and interference filter used in the Thomson scattering, n# can be set to 0.25ncr and to 0.15nc r. From the formula dx tiff + n>L Ax L = n— ~ dn 2 njj — nj, a scale length of 400^m at 0.2 ncr can be deduced. This is a lower limit since nfj — ni may be smaller than assumed. Scattering from densities outside the experimentally permitted limited range will make no contribution to observed scattering. This CHAPTER 4: Results 70 s c a l e l e n g t h is o f t h e same o r d e r as t h e scale l e n g t h d e t e r m i n e d by i n t e r f e r o m e t r y . F o r c a l c u l a t i o n p u r p o s e s i n t h e n e x t c h a p t e r , a t y p i c a l c h a r a c t e r i s t i c s i z e of 300>m w i l l b e e n used. F i g u r e 4-4 S p a t i a l e x t e n t o f S R S s c a t t e r i n g . T h e i n t e n s i t y c o n t o u r s a r e i n steps of 1/4 o f t h e m a x i m u m l e v e l . 4.3 R e s u l t s f r o m t h e W a v e v e c t o r T h o m s o n S c a t t e r i n g B y i m a g i n g a n a r e a o f t h e v e r t i c a l f o c a l p l a n e o n t o t h e s t r e a k c a m e r a s l i t i n s t e a d o f t h e p l a s m a ( w h i c h is p o s i t i o n e d a t t h e h o r i z o n t a l f o c a l p l a n e of the a s t i g m a t i c lens - see figure 3-7), t h e w a v e v e c t o r s g e n e r a t e d i n t h e p l a s m a c o u l d be p r o b e d . I n p a r t i c u l a r , E P W w i t h w a v e v e c t o r s of l e n g t h k/k0 « 1 a n d fre q u e n c i e s CHAPTER 4: Results 71 0 100 200 400 500 Axial Distance (Aim) Axial Distance (/im) F i g u r e 4 -5 M o r e examples of spat ia l l y resolved streaks of S R S . CHAPTER 4: Results 72 U/LU0 « 0.5 were of particular interest as these are the waves which SRS is supposed to enhance to levels well above the noise. In figure 4-6, a typical streak record shows two bursts of SRS, each about 15ps long and separated by 15ps. The relative k width of each burst can vary from shot-to-shot (see the further examples of figure 4-7). The dashed line in figure 4-6 is explained in the next chapter. Figure 4-8 shows the fluctuation level, averaged over the observation window, plotted as a function of time. This is an exceptional case in which there is weak reappearance of SRS after 200ps. n/n, .25 c r .24 .23 —I h .21 + .19 1— 1. 1.16 121 1.28, 1.34 f k/k0 i . & CL 20 -40-60 Figure 4-6 Example of a wavevector resolved streak record. CHAPTER 4: Results 7 3 Scattering Angle (°) 1 1 1 1 1 P" 2.5 3.0 15 Scattering Angle (°) Figure 4-7 F u r t h e r e x a m p l e s of w a v e v e c t o r s p e c t r a o f S R S . T h e c o n v e r s i o n f r o m angle t o d e n s i t y is t h e same i n figure 4-5. T h e s e c o n d s t r e a k is s a t u r a t e d . CHAPTER 4: Results 74 0 200 400 600 Time (ps) Figure 4-8 Reappearance of SRS in an exceptional case. The k-resolved spectra, by inference (see section 3.5 for details), show that SRS occurs at all densities 0.15-0.25 nCT. This observed range is limited by the interference filters used. In figure 4-10, we see some evidence of scattering at angles less than 2.6°which means that the EPW wavevectors are shorter than k0. SRS can generate such wavevectors but only if the scattered IR light is scattered forward. Forward scattering was usually not present. In figure 4-9, a raw k spectrum averaged over 10 shots is presented (dashed curve). The maximum fluctuation level seems to be, on average, independent of density. The effects of the finite bandpass of the interference filter (the dotted line) which was discussed in chapter 3 and the streak camera non-uniformity of response have been deconvoluted to yield the solid CHAPTER 4: Results 75 c u r v e . I t is q u i t e s u g g e s t i v e of t h e f r e q u e n c y s h i f t i n t h e r u b y s c a t t e r e d l i g h t t h a t t h e e x p e c t e d r o l l o f f due t o t h e i n t e r f e r e n c e filter a c t u a l l y c a n b e seen t o o c c u r i n t h e d o t t e d c u r v e . n/n c r .25 .23 .21 .19 .17 F i g u r e 4-9 A v e r a g e k v e c t o r r e s o l v e d s p e c t r u m o f S R S . T h e d a s h e d c u r v e repre-sents t h e u n c o r r e c t e d s p e c t r u m , t h e d o t t e d c u r v e r e p r e s e n t s t h e i n t e r f e r e n c e filter b a n d p a s s a n d t h e s o l i d c u r v e r e p r e s e n t s t h e average fluctuation s p e c t r u m c o r r e c t e d f o r t h e b a n d p a s s r o l l o f f . W i d e a n g l e T h o m s o n s c a t t e r i n g was set u p t o s c a t t e r f r o m t h e e l e c t r o n p l a s m a waves g e n e r a t e d b y S R S , as w e l l as those waves g e n e r a t e d by t h e T P D i n s t a b i l i t y . S i n c e t h e E P W g e n e r a t e d by T P D are m o s t intense i n t h e p l a n e of p o l a r i z a t i o n o f t h e i n c i d e n t l a s e r , t h e T h o m s o n s c a t t e r i n g i n t h i s s e t u p was p e r f o r m e d i n t h i s p l a n e . F i g u r e 4 - 1 0 shows s e v e r a l e x a m p l e s of t h e s t r e a k r e c o r d s o b t a i n e d i n t h i s CHAPTER 4: Results experiment. It is interesting to note that the angular width varies considerably from shot-to-shot for nominally unchanged experimental conditions and that the angle at which the scattering originates also varies. On average, the scattering ranges from 3.4 ± 1.1° to 6.5 ± 1.3°. These angles correspond to k/k0 from 1.2 ± 0.3 to 2.1 ± 0.4 . Scattering at angles less than 4.5° is due to SRS-generated EPW whereas scattering at larger angles must be due to TPD. This average range (from 3.4°to 6.5°) is independent of incident laser energy. The maximum angle at which scattering was observed was about 9°, which corresponds to k/k0 = 2.9. In figure 4-11, it can be seen quite clearly that there is a progression in time from shorter to longer wavevectors. Scattering Angle (°) Figure 4-11 Example of wide angle Thomson scattering streak taken at a higher sweep speed. CHAPTER 4: Results 77 F i g u r e 4-10 T y p i c a l s t r e a k r e c o r d s f r o m t h e w i d e a n g l e T h o m s o n s c a t t e r i n g . T h e scales a r e t h e same f o r a l l t h e records. S c a t t e r i n g at angles less t h a n 1.5°was b l o c k e d . 4.4 R e s u l t s f r o m F r e q u e n c y r e s o l v e d T h o m s o n S c a t t e r i n g T h e f r e q u e n c y r e s o l v e d T h o m s o n s c a t t e r i n g was p e r f o r m e d t o c o n f i r m t h e w a v e l e n g t h s h i f t i m p l i e d by the w a v e v e c t o r r e s o l v e d r e s u l t s . T h e f r e q u e n c y s h i f t o f th e s c a t t e r e d l i g h t is j u s t t h e p l a s m a f r e q u e n c y of t h e E P W p r o b e d . I n f i g u r e 4-12, a t y p i c a l s p e c t r u m is p r e s e n t e d . T h e i n s t r u m e n t r e s o l u t i o n o~ins = 1.38nm F W H M is a p p r e c i a b l e b u t t h e o b s e r v e d w i d t h o0DS is m u c h w i d e r t h a n t h e i n s t r u m e n t w i d t h alone. A s i m p l e e s t i m a t e , o2 = a2bs — ofns suggests t h e a c t u a l w i d t h tr is 1.4 nm. T h i s i m p l i e s a s i m u l t a n e o u s a p p e a r a n c e of S R S over t h e d e n s i t y r a n g e 0.17-0.19 n c r ; t h i s is c o n s i s t e n t w i t h t h e n a r r o w k range observed. ( N o t e t h a t t h e s a m p l e k CHAPTER 4: Results 78 spectrum of figure 4-6 is much narrower than the averaged spectrum of figure 4-9.) The dashed line will be discussed in the next chapter. The cut-off in both the k and frequency spectra could be due to the finite aperture of the exit window of the Thomson scattering setup. The exit window cuts off scattering at angles greater than 4.5°which corresponds to density of 0.15nc r. Figure 4-9 seems to suggest that there are fluctuations present at even lower densities. This lower density regime was not further explored for two reasons. First, other experiments have seen SRS at those densities but not at higher densities. Second, substantial changes to the scattering geometry would have had to been made in order to detect such waves. The importance of ion acoustic waves in SRS can been seen in frequency resolved Thomson scattering which can show the ion feature (spectrally unshifted probe light) as well as the electron feature (probe light well shifted). An example of this is shown in figure 4-13, a full screen spectrum of the Thomson scattered light seen through the narrow angle geometry. The intensity contours are in steps of 1/4 of the maximum level. SRS starts earlier than the IA waves do, but once the IA waves become well established, SRS is saturated and quenched. The levels of the ion acoustic waves must be quite large since a single interference filter with a rejection ratio of more than 100 was used to reduce the levels of the Thomson scattered light which was scattered from the ion acoustic waves. The unshifted light is not stray ruby light. If it were, the signal should have started much sooner as the ruby laser was irradiating 2 to 3 ns before the start of the plasma. The ruby laser light also continues to irradiate the plasma for 3 ns after the start of the plasma whereas in the streak records, the unshifted light stops during the sweep. Further, an ion feature has been observed under identical experimental conditions 5 4 and this feature has the same temporal duration as the feature observed in this work. CHAPTER 4: Results 79 X (nm) Figure 4-12 Typical frequency resolved Thomson scattering streak record. The intensity contours are in steps of 1/4 of the maximum level. An interesting density-time effect can be seen in both figure 4-6 and 4-12. As time progresses, SRS seems to shift to lower densities. The typical rate is -0.001 nCT ps _ 1 . Explanations for this will be discussed in the next chapter. 4.5 Scattered Infrared Light If SRS is present, scattered IR light at frequencies shifted from the inci-dent CO2 frequency should be present. The results from the Thomson scattering, which examined the epw, suggested that IR associated with Raman scattering from densities 0.15-0.25 critical density should be present. X (nm) F i g u r e 4-13 F r e q u e n c y r e s o l v e d T h o m s o n s c a t t e r i n g s h o w i n g i o n a n d e l e c t r o n fea-t u r e s . A s y s t e m a t i c s e a r c h f o r t h e s c a t t e r e d I R l i g h t was u n d e r t a k e n at 162°to k0. ( O b s e r v a t i o n s o f d i r e c t b a c k s c a t t e r were n o t a t t e m p t e d as t h e t h i c k K C 1 lens w o u l d a b s o r b m o s t of t h e s c a t t e r e d li g h t . ) S p e c t r a l l y i n t e g r a t e d m e a s u r e m e n t s at a n a n g l e o f 144°to k0 were a l s o p e r f o r m e d . T h e s p e c t r a l l y i n t e g r a t e d m e a s u r e m e n t s were s u m m a r i z e d i n figure 4-1 w h e r e t h e energy d e p e n d e n c e of the s i g n a l is c l e a r l y seen. N o t e m p o r a l r e s o l u t i o n was p o s s i b l e due t o t h e l i m i t e d b a n d w i d t h of t h e d e t e c t o r ; t h e r e f o r e , a l l t h a t c a n be s a i d is t h a t t h e I R s c a t t e r i n g was less t h a n 1 ns i n d u r a t i o n since t h e t i m e r e s o l u t i o n of t h e Ge:Cu d e t e c t o r used was Ins. T h e s p e c t r a l l y i n t e g r a t e d r e f l e c t i v i t y c a n be deduced b y c o m p a r i n g t h e s i g n a l t h r o u g h one r e d p a s s filter S\ t o t h a t t h r o u g h t w o redpass filters S2. T h e f i l t e r p r o p e r t i e s CHAPTER 4: Results 81 are such that 70% of any SRS signal, SSRS, is transmitted while less than 0.05% of any SBS signal, SSBS, is transmitted. Since Si = 0.7SSRS +0.0005SSBS and 5 2 = (0.7)2 SSRS + (0.0005)2 SSBs, it follows, given the observed ratio S1/S2 = 2.4, that SSBS/$SRS = 1000. Since the SBS reflectivity is 10% 5 4 , the SRS reflectivity should be 0.01%. This method of estimating the SRS reflectivity assumes SRS and SBS have the same angular distribution. This method gives the reflectivity at 180° ,i.e., where the SBS level is 10%. Out of the cone angle of the CO2 laser, the SBS level drops rapidly. In fact, at 135°, the SBS reflectivity is only 0.2% and thus the SRS reflectivity at that angle will be correspondingly smaller. Thus, when one deduces absolute reflectivities from the signal size, this angular dependence of the signal strength should be remembered. The spectrally resolved scattered IR light was also examined with two meth-ods. One way used the 15 micron interference filter. The signal observed this manner was 40% of the total integrated signal. This indicates a large amount of SRS is occuring at densities near 0.1 critical density at an approximate reflectivity of 0.004%. However, IR scattered at these densities was not further explored since, as mentioned earlier, it was the scattering from higher densities (in the so called gap) that was of more interest. A second spectrally resolving system which used a monochromator was set up. The region from 16 to 22 microns was examined at 1 micron intervals. A signal was found only near 21 microns. At all other wavelengths, the scattered light must be at levels below 10% of that at 21 microns. The wavelength bandwidth of the monochromator, fixed by the exit aperture, is 0.2/xm FW. The region near 21 CHAPTER 4: Results 20.2 20.6 21. Wavelength (fim) Figure 4-14 Spectrum of scattered IR near 2A0 2U microns was examined at 0.2 micron steps and the results are shown in figure 4-14. The intensity seems to be split around exactly 2A 0. 4.6 Simultaneous Observations of Electrons and Plasma Waves In a CO2 laser experiment, all the observable aspects of the SRS process can be easily measured. This presents a unique opportunity to correlate these aspects. It is important to correlate the high energy electrons with the SRS as these electrons are one of the main problems in laser fusion schemes. The electrons can be correlated with SRS in two areas. The number of elec-trons can be compared with the amount of SRS measured through Thomson scatter-ing or through IR levels. The energy spectra of the electrons can be compared with CHAPTER 4: Results 83 the scattered I R spectra or with the the wavevector spectra from Thomson scatter-ing. Both comparisons are important for either large numbers or high energies or both are detrimental to laser fusion schemes. The simplest comparisons are made between the number of hot electrons at 150keV, N , and the saturated fluctuation level Sn or the amount of S R S integrated reflectivity R. In figures 4-15 and 4-16, the relative number of electrons at 150 keV, as measured by the electron spectrometer, are compared with the 6n/ne and R. A straight line relation is indicated between ln(N) and the fluctuation level or reflectivity. This is what was predicted by Eqn. 2-29. It is worthwhile noting that, in both cases, a change in the abscissa coordinate by a factor of 2 results in the same fractional increase in ln(N). The relative number of electrons can be compared directly because the same detector located at the same position was used to record the data. The straight lines are least squares fits whilst the error bars in figure 4-15 are estimates in the error for a single shot, due primarily to uncertainties in the signal/fiducial ratio in the Thomson scattering streakcamera data. Each datum represents a shot in which both the number of high energy electrons and either the fluctuation level or the spectrally integrated reflectivity were observed simultaneously. In order to correlate the number of electrons with the fluctuation level, the Thomson scattering data must be interpreted using the actual ( not relative) levels. In Table 4-1, the parameters used to find the fluctuation level via Eqn.3-3 are given. The maximum signal to reference count ratio NsigTiai/Nfiduciai measured was 12 which corresponds to a fluctuation level of 1%. The largest errors in this level are due to uncertainties in L and a. CHAPTER 4: Results 84 T a b l e 4-1 Parameters to find absolute fluctuation levels Parameter Value Ni 200 N 2 180 Ay 84 [im f 1 N.D. 5 e0 1.8° a 0.03mm2 L 115 t^m ne 0.25ncr 66 0.00022 rad (Here, A y is calculated for a diffraction limited focus.) The second correlation of high energy electrons with SRS can be seen through the energy spectra of the electrons compared with the wavevector spectra of the epw. (The prediction of computer simulations is described by Eqn. 2-3.) In this experiment, fluctuation spectra and electron energy spectra were recorded simulta-neously with the objective of testing the predictions. In figure 4-17, typical spectra are shown. Electron energy spectra (Fig.4-17(a)) and the corresponding wavevec-tor spectra (Fig.4- 17(b)) are given. As the number of electrons in the high energy channel increases, the time-averaged wavevector spectra shift to shorter k (higher phase velocity waves). This behaviour can be quantified. A temperature k#T is fitted, as described in chapter 3 (Eqn.3-4), to many spectra of the electron energy distribution. The wavevector spectra require a bit more analysis as there is a range of k vectors to choose from. The shorter k vector corresponding to one-half the square of the maximum fluctuation level was arbitrarily used to find the phase velocity, vp and in figure 4-18, the k#T versus 0.5meu2 is shown. The vertical error bars represent the standard errors of at least 4 data points while the horizontal error bars represent the size of the bins used for averaging. CHAPTER 4: Results 85 3-{ I 1 L I I I 0 .002 .004 .006 .008 .010 .012 6n/n Figure 4-15 Correlation of electron number with the fluctuation level. Eqn.2-29 predicted a straight line relationship on a semilog plot. The prediction was based upon a simple trapping model. There are convolution effects, since the electrons observed are time and space (density) integrated. There is also a range of EPW responsible for the generation of the electrons. These effects are minimized if averages over many shots, such as those in figure 4-18 are used. CHAPTER 4: Results 8 6 F i g u r e 4-16 C o r r e l a t i o n of e l e c t r o n n u m b e r w i t h t h e i n t e g r a t e d I R l e v e l . A s t r a i g h t l i n e r e l a t i o n s h i p is p r e d i c t e d by E q n . 2-29 i f one makes t h e r e a s o n a b l e a s s u m p t i o n t h a t R is p r o p o r t i o n a l t o 6n. CHAPTER 4: Results - 6 > a a> -8 V 1 i ' 1 — N S v > . . . - • -A ^ s - Data Rt KT(keV) O 104 • 56 V A N . A 50 i • • 0 50 100 150 Energy (keV) n/n c r •25 .23 1 h -.21 — 1 — .19 .17 — I 1 . Figure 4-17 Simultaneous electron and wavevector spectra. (a) The electron spectra (b) the k vector spectra . There are three different irradiations shown. As the fitted temperature increases, the wavevector spectrum shifts to higher phase velocity waves. CHAPTER 4: Results 88 F i g u r e 4-18 T h e f i t t e d f a s t e l e c t r o n t e m p e r a t u r e c o m p a r e d t o t h a t e x p e c t e d f r o m t h e f l u c t u a t i o n s p e c t r a . T h e s t r a i g h t l i n e is suggested by s i m u l a t i o n s . CHAPTER 5: Discussion 89 CHAPTER 5 Discussion In this chapter, the main results of my research are discussed and compared with the relevant theories and other experiments. A selfconsistent picture emerges. 5.1 Preliminary Numbers If one wishes to compare theory to experiment, specific quantities, such as T e, L, v0, ne,etc. all of which must be derived from experiment, must be used. In the following sections, typical values are used extensively to derive orders of magnitude for thresholds, growth rates, etc. In order to avoid repetition and to be consistent, the experimentally measured numbers (which do vary from shot to shot) used in the calculations (unless otherwise indicated) are: kBT=300 e V (from Popil 5 2 ), L=300 /xm (measured from spatial Thomson scattering), v0/c =0.05 (for a 5J incident laser pulse), ne =0.23 ncr (from interferometry 47>60)5 Sn/n0 = 0.01 (measured with Thomson scattering), and Z=4 (from Popil 5 2). 5.2 Threshold Considerations Theories of SRS predict that certain threshold intensities must be reached before the interaction proceeds. Many threshold estimates can be made. In this section, these estimates are made and a comparison to the observed threshold is CHAPTER 5: Discussion 90 made. Computer simulations44 have shown the estimates of thresholds, based upon analytic theory, are quite good. Various theories are available for comparison with the present experiment. The simplest model is the one for an infinite, homogeneous plasma in which the stimulated growth rate must be greater than the damping rate for the waves (Eqn. 2-15). To overcome collisional damping (Landau damp-ing being negligible), an intensity around 6.5 x 109 W c m - 2 is needed which, in this experiment, is exceeded by 4 orders of magnitude. In a finite, homogeneous plasma, there is the convective threshold (Eqn. 2-21) which must be exceeded. This threshold intensity is 9.6 x 107 W c m - 2 which again is exceeded by many orders of magnitude. However, if the laser intensity on our target is reduced by only a factor of 10, no scattering is seen. This behaviour is a direct result of the fact that our plasma is laser produced and is not preformed. Because of this, a certain amount of the incident energy must be used to form the plasma. Bernard 5 4 has estimated that this takes about 1 J. In such a plasma, large density gradients are expected and thus, the inhomogeneous plasma theories are more appropriate to use in finding thresholds. There are several regimes for these theories. At 0.25ncr, the threshold for absolute growth set by Drake and Lee (Eqn. 2-23) is 1=4 x 1012 W c m - 2 . At lower densities, a threshold for absolute growth (Eqn. 2-22) of I =6 x 10 1 3 W c m - 2 at 0.2 ncr is found. On the other hand, the convective threshold set by Rosenbluth (Eqn. 2-19) at 0.2 ncr requires a i0 of 6.3 x 10 1 2 s - 1 (or an intensity of 2.6 x 10 1 3 Wcm - 2 ) is required. It seems the observed threshold of 1.9 x 10 1 3 W c m - 2 is above the absolute thresholds at 0.25recr, but near the thresholds (absolute or convective) at 0.2nCf. Since SRS is observed at the lower densities, there is a discrepancy (less than a factor of 3) between theory and experiment. Other experimenters have noted such a discrepancy, with the theoretical thresholds usually being larger than those experimentally observed, although not all investigators agree on the size of the discrepancy. Elazer et a l . 6 1 , Villeneuve et a l . 6 2 , Walsh et a l . 6 3 and Turner et a l . 6 4 CHAPTER 5: Discussion 91 report thresholds close to those predicted while Seka et a l . 6 5 ' 6 6 , Shepard et al. 6 7 , Tanaka et al. 6 8 and Figueroa et a l . 6 9 report thresholds below the theoretical estimates by factors of 5 to two orders of magnitude. Several explanations have been proposed. The most plausible explanation is that experiments and theory are not being compared consistently. For instance, what intensity should be used - a peak, an average over 60% or 90% energy radius, the instantaneous intensity at the time of the interaction or something else? The theories usually do not consider these practical points. Theorists assume that a plane wave of uniform amplitude is interacting with the plasma. In an experiment, the focussed laser beam can be thought of as a superposition of many plane waves. Is this important? There are effects which can perturb an ideal plane wave so that its amplitude is not spatially uniform. For example, pondermotive filamentation can raise laser intensities locally, perhaps even well above thresholds. This process has a threshold intensity 7 0 of which is 10 1 2 Wcm - 2 in our case. Its growth rate is similar to that of SBS and should not be able to increase the intensity of the incident laser beam in the plasma on timescales on which SRS occurs (less than 50ps). A proper calculation of a threshold requires a knowledge of the plasma density profile. The reason is that the spatial gain in an inhomogeneous medium involves and this depends on exactly how k0,kepw and ks vary with x. All of the quoted formulas assumed a linear profile to simplify the calculations whereas real profiles are not necessarily of this form. Figueroa et al. 6 9 and Villeneuve et al. 6 2 have /((Wcm- 2) - [2.5 10 1 5 /^mA M m ] r e V n c /n CHAPTER 5: Discussion 92 shown that the form of the profile assumed (Gaussian, exponential, parabolic, linear, etc.) affects the threshold calculated (by as much as several orders of magnitude). Finally, if one uses a convective threshold, such as that of Rosenbluth, some level of growth must be assumed. Rosenbluth used e2jr « 535 as the level above which the threshold is said to be exceeded. Tanaka claims a more realistic estimate can be made if a growth of 50 is used which reduces the required intensity by 1.6. In addition, to apply correctly a convective threshold formula, some initial noise level should be measured or calculated. One possible noise source is random plasma fluctuations.11 A detector may see a signal (e.g. scattered light) but only if this signal exceeds the background levels can it be said SRS is occurring. The results from the Thomson scattering, where saturated fluctuation levels at 50 to 100 times the noise level are observed, suggest Rosenbluth's explanation for any threshold discrepancy is correct. The convective growth threshold calculated under this reduced growth requirement agrees with that observed although it is absolute growth which is observed. The IR signal observed at 2A0 could be a result of the TPD instability. There-fore, an estimate of its threshold intensity must be made. From Simon 2 4 , we find his equation (54) which reduces to /(Wcm-2) > {velc)2{k0L)-H.% x 1018/X2o{fim). For our conditions, this reduces to an intensity threshold of 1.4 1011 Wcm -2. Thus, the thresholds for both SRS and TPD are exceeded in my experiment and the interpretation of the other results is therefore complicated by this fact. 5.3 Spatial Details SRS can occur on a density gradient. However, the growth of the instability depends upon the magnitude of the gradient. The plasma length observed can yield some idea of whether absolute or convective growth should be expected. (The CHAPTER 5: Discussion 93 d e n s i t y p r o f i l e is also i m p o r t a n t . ) T h e lower l i m i t f o r (reduced) a b s o l u t e t e m p o r a l g r o w t h i n a finite, h omogeneous p l a s m a , (vepv)v-)1/2/")0, is 5.6 m i c r o n s w h i l e t h e u p p e r l i m i t , u _ / 7 0 , is 36 m i c r o n s . T h e c o n d i t i o n f o r a b s o l u t e g r o w t h , L m u c h l a r g e r t h a n 5.6 /xm, is w e l l s a t i s f i e d a n d t h e r e f o r e , t h i s t y p e o f g r o w t h m i g h t b e e x p e c t e d . S i n c e t h e o b s e r v e d d e n s i t y s c a l e l e n g t h is l o n g e r t h a n 36 //m, t h e r e s h o u l d be a sh o r t p e r i o d o f t e m p o r a l g r o w t h at ~i0, f o l l o w e d by g r o w t h a t 7 f (Eqn.2-17). T h i s s h o r t p e r i o d ( L / v _ ) , a b o u t 2 ps, is a t e n t h o f t h e o b s e r v e d d u r a t i o n o f S R S a n d , hence, t h e o b s e r v e d t e m p o r a l g r o w t h s h o u l d be a t ^ T u n l e s s n o n l i n e a r effects b e c o m e i m p o r t a n t . F u r t h e r , t h i s s h o r t p e r i o d is less t h a n t h e t e m p o r a l r e s o l u t i o n of t h e s t r e a k c a m e r a a n d t h e r e f o r e , i f t h e waves do grow at 7 0 , we c a n n o t observe t h i s g r o w t h . T h e s p a t i a l l y r e s o l v e d T h o m s o n s c a t t e r i n g gives some i d e a o f how t h e p l a s m a wave fluctuations v a r y w i t h p o s i t i o n . T h i s is i m p o r t a n t since the n a t u r e of t h e i n s t a b i l i t y d e t e r m i n e s how t h e fluctuations ( s o m e t i m e s c a l l e d n o r m a l m odes i n t h e l i t e r a t u r e ) w i l l be s p a t i a l l y d i s t r i b u t e d . I n p a r t i c u l a r , i n a finite b u t homogeneous p l a s m a , t h e r e s h o u l d be s p a t i a l g r o w t h . F o r t h e d e t e r m i n a t i o n o f t h e s p a t i a l d i s -t r i b u t i o n , t h i s finite, homogeneous m o d e l of t h e p l a s m a s h o u l d be v a l i d s ince t h e o b s e r v e d l e n g t h of t h e S R S s c a t t e r i n g r e g i o n is m u c h l a r g e r t h a n t h e u p p e r l i m i t f o r a b s o l u t e g r o w t h . T h i s means t h e d e n s i t y i n h o m o g e n e i t y is n o t i m p o r t a n t at least f o r t h i s a s p e c t of S R S . I n t h e s p a t i a l l y r e s o l v e d T h o m s o n s c a t t e r i n g , some e v i d e n c e f o r s p a t i a l g r o w t h was f o u n d . I n figure 5-1, e x p o n e n t i a l g r o w t h i n space c a n be seen w h i c h i n d i c a t e s t h a t s p a t i a l g r o w t h has o c c u r r e d . ( T h e ins e t i n t h e figure shows t h e c o m p l e t e s p a t i a l v a r i a t i o n o f t h e fluctuation level.) O v e r 10 shot s , a n average a m p l i t u d e I m ( k ) of 0.026 ± 0.004/xm - 1 was f o u n d . A s p a t i a l g r o w t h r a t e i n a m p l i t u d e c a n be c a l c u l a t e d f r o m E q n 2-18 w i t h t h e p a r a m e t e r s m e n t i o n e d i n s e c t i o n 5.1. T h e r e s u l t is 0.28//m _ 1 w h i c h is a f a c t o r of t e n l a r g e r t h a n observed. It m u s t be e m p h a s i z e d t h a t t h i s p r e d i c t i o n is f o r a finite, h o m o g e n e o u s p l a s m a . T h i s p r e d i c t i o n o f t h e s p a t i a l g r o w t h r a t e is t h e o n l y one CHAPTER 5: Discussion Axial Distance (fim) F i g u r e 5-1 E v i d e n c e f o r s p a t i a l g r o w t h of S R S . T h e i n t e n s i t y o f t h e T h o m s o n s c a t t e r e d l i g h t is p l o t t e d as a f u n c t i o n o f s p a t i a l p o s i t i o n . A s t r a i g h t l i n e o n t h e s e m i l o g i n d i c a t e s t h a t p o s s i b l e s p a t i a l g r o w t h is o c c u r i n g . T h e inset shows t h e f u l l s p a t i a l e x t e n t of t h e s c a t t e r i n g o n a l i n e a r p l o t . t h a t I a m aware of. T h e p o s s i b l e reasons f o r t h i s d i f f e r e n c e w i l l be d i s c u s s e d w i t h t h e t e m p o r a l g r o w t h ra t e s , a lso m e a s u r e d ( w h i c h show a s i m i l a r d i s c r e p a n c y ) . T h e s p a t i a l r e s u l t s o b t a i n e d s h o u l d n o t be c o m p a r e d d i r e c t l y w i t h o t h e r exper-i m e n t s . F o r c o m p a r i s o n p u r p o s e s , t h e differences i n t h e wavelengths of t h e i n c i d e n t l a s e r l i g h t m u s t be t a k e n i n t o a c c o u n t . T h e s i m p l e s t s c a l i n g p a r a m e t e r is L/A0. I n t h e present e x p e r i m e n t , t h i s r a t i o is a b o u t 20 t o 40. M o s t o t h e r e x p e r i m e n t s 6 1 , 6 4 , 6 5 , 6 6 , 6 8 , 7 6 , 8 1 a r e p e r f o r m e d a t s h o r t e r w a v e l e n g t h s (less t h a n one m i c r o n ) f o r s i m i l a r L. E f f e c t i v e l y , t h e o t h e r e x p e r i m e n t s have l o n g e r p l a s m a s a n d t h e r e f o r e , CHAPTER 5: Discussion 95 higher levels of SRS might be expected from them. Experiments which used CO2 lasers have been done with ratios of 30 (Villeneuve et a l . 6 2 ) and more than 10000 (Offenberger et a l . 7 2 , Watt et a l . 7 3 ' 7 4 ) although the latter two experiments were performed at much lower densities (less than 0.01n c r). A l l of the above experi-ments, except for the experiment of Villeneuve et al. 6 2 , report much higher levels of SRS than those observed in this work. The plasma size is probably the most important reason for this difference. 5.3 Temporal Growth and Behaviour In the section on spatial growth, it was decided that the instability should be absolute and should grow at the reduced temporal rate (Eqn. 2-17) in accordance with the theory of Forslund et al. 8 . In this section, this hypothesis is examined. Temporal growth is characterized by an exponential growth in time. In figure 5-2, the plasma wave amplitude can be seen to grow exponentially which indicates that there was absolute growth. The figure shows the increase of the scattered light intensity at 6 = 3°. In figure 5-3, growth rates from similar plots are shown as a function of wavevector. One notices that the growth rate does not vary within the standard deviations, at least over the range investigated, and, by inference, does not depend upon density dramatically. For comparison to the measured growth rates, various theoretical growth rates can be calculated. The theoretical growth rates for an infinite plasma (Eqn. 2-11) are about 30 times too large. Other theoretical growth rates are lower than the growth rate for an infinite, homogeneous plasma. Since the plasma is not infinite, the next level of complexity requires the rates to be calculated in a finite, homogeneous plasma (Eqn. 2-17). This was done and the results are shown in figure 5-3 where they are divided by 10 (the solid curve). The theoretical rates do not vary greatly with density, at least over the region of most interest. The measured rates are about 7 times lower than expected from this theory. CHAPTER 5: Discussion 96 Time (ps) F i g u r e 5-2 E v i d e n c e f o r t e m p o r a l g r o w t h . T h e p l o t s shows t h e i n t e n s i t y o f t h e T h o m s o n s c a t t e r e d l i g h t as a f u n c t i o n o f t i m e . T h e s t r a i g h t l i n e o n t h e s e m i l o g p l o t i n d i c a t e s t h a t t e m p o r a l g r o w t h has o c c u r e d . O t h e r c o m p l i c a t i o n s c a n f u r t h e r r e d u c e t h e t h e o r e t i c a l g r o w t h r a t e . F r o m t h e c o m p a r i s o n of t h e p l a s m a l i g h t t i m i n g w i t h t h e t i m i n g o f t h e s c a t t e r i n g , i t was f o u n d t h a t S R S o c c u r s e a r l y i n t h e laser pulse. S i n c e t h e e x a c t i n t e n s i t y at t h e t i m e of t h e s c a t t e r i n g was n o t k n o w n , an average laser i n t e n s i t y was u s e d i n t h e c a l c u l a t i o n . F u r t h e r , t h e m e a s u r e d g r o w t h r a t e s p r e s e n t e d were a v e r a g e d w i t h o u t a n y r e g a r d t o i n c i d e n t laser i n t e n s i t y . T h u s , i t d i d n o t seem r e a s o n a b l e t o c a l c u -l a t e t h e g r o w t h r a t e at i n t e n s i t i e s d i f f e r e n t f r o m t h e average i n t e n s i t y u s e d i n t h e e x p e r i m e n t . However, since t h e g r o w t h r a t e is p r o p o r t i o n a l t o y/l , a lower i n t e n -s i t y r e s u l t s i n a lower g r o w t h r a t e . T h i s c o u l d a c c o u n t f o r (at w o r s t ) a f a c t o r of CHAPTER 5: Discussion 97 n/n 8x10 f k -cr 23 .21 .19 .17 e(°) F i g u r e 5-3 Experimentally measured growth rates. These growth rates were de-termined from plots such as the one in Figure 5-3. 3, since I is constrained by threshold and plasma production considerations. An inhomogeneous plasma will also result in a lower growth rate. Again, since there are variations in the scale length, a simple calculation of the growth rates ignored any effects of inhomogeneities. If the full expression of DuBois (Eqn. 2-20) is used, the inclusion of a typical inhomogeneity observed in the experiment will reduce the rate to 25% of the rate in a homogeneous plasma. This term accounts for the wavevector mismatch mentioned earlier. Any damping will also reduce the growth rate, but Landau and collisional damping alone are too small to have any significant influence. In an infinite, homogeneous plasma, the scattered IR light can be generated into directions other than direct backscatter. The growth rate for this sidescattering is CHAPTER 5: Discussion l o w e r t h a n t h a t f o r d i r e c t b a c k s c a t t e r . I n t h e present e x p e r i m e n t , t h e T h o m s o n s c a t t e r i n g i n d i c a t e d t h a t some s i d e s c a t t e r i n g was o c c u r r i n g . T h e s e s i d e s c a t t e r ef-f e c t s w h i c h c hange t h e v a l u e of ^ 0 b y s i n ( 7r/2 — %p) (see E q n . 2 - l l ) have o n l y a s m a l l effect o n t h e final g r o w t h r a t e s i n c e t h e m a x i m u m %p e x p e c t e d is 135 °. I n c l u s i o n o f t h i s i n E q n . 2 -11 c a n reduce t h e t h e o r e t i c a l g r o w t h r a t e over t h e r a n g e of s c a t t e r i n g o b s e r v e d t o 7 0 % of t h e b a c k s c a t t e r r a t e (at m o s t ) . Some c o m b i n a t i o n of a l l these effects c a n r e a s o n a b l y reduce t h e e x p e c t e d g r o w t h r a t e s t o levels a t w h i c h t h e o r y a n d e x p e r i m e n t c a n be made t o agree a l t h o u g h t h e e x a c t m i x t u r e c a n n o t b e s t a t e d . E v e n g i v e n t h e u n c e r t a i n t i e s i n t h e e x p e r i m e n t a l p a r a m e t e r s w h i c h are needed t o c a l c u l a t e t h e g r o w t h r a t e s , the net r e s u l t is s t i l l t h a t t h e m e a s u r e d g r o w t h r a t e s are of t h e c o r r e c t o r d e r of m a g n i t u d e as t h o s e r a t e s p r e d i c t e d b y t h e o r y . A c o n s i s t e n c y between t h e t e m p o r a l a n d s p a t i a l g r o w t h r a t e s c a n be a t t a i n e d i f a s l i g h t l y h i g h e r e l e c t r o n t e m p e r a t u r e ( a p p r o x i m a t e l y 375 e V ) a n d a m u c h lower l a s e r i n t e n s i t y ( a p p r o x i m a t e l y 4.0 x 1 0 n W c m ~ 2 ) are p o s t u l a t e d . B y c h o o s i n g t h i s t e m p e r a t u r e , t h e r a t i o of t h e t h e o r e t i c a l t o t h e m e a s u r e d t e m p o r a l g r o w t h r a t e s is e q u a l t o t h e same r a t i o f o r t h e s p a t i a l r a t e s . B y c h o o s i n g t h i s i n t e n s i t y , t h e r a t i o c a n be r e d u c e d t o one. T h i s c o n s i s t e n c y also r e q u i r e s t h e p l a s m a t o be finite a n d homogeneous. M o r e t h e o r e t i c a l w o r k s h o u l d b e done o n t h e s p a t i a l g r o w t h r a t e i n a n i n h o m o g e n e o u s p l a s m a b e f o r e a r e a l i s t i c c o m p a r i s o n t o t h e present r e s u l t s is u n d e r t a k e n . T h e t e m p o r a l b e h a v i o u r of S R S is i n t e r e s t i n g f o r a n o t h e r reason. Some s i m -u l a t i o n s s h o w t h a t t h e r e s h o u l d be m o d u l a t i o n s i n the l e v e l of t h e s c a t t e r i n g . T h e s i m u l a t e d m o d u l a t i o n r a t e s are of t h e same m a g n i t u d e as c a n be r e s o l v e d b y t h e p r e s e n t e x p e r i m e n t a n d t h e r e f o r e t h e s t r e a k r e c o r d s s h o u l d be e x a m i n e d f o r e v i d e n c e of s u c h m o d u l a t i o n s . S h o r t p e r i o d ( a b o u t 15 ps ) m o d u l a t i o n s were f o u n d (see figure 4-6) . T h e r e a s o n f o r t h e s h o r t p e r i o d f l u c t u a t i o n s is n o t k nown. T h e r e a p p e a r a n c e i n t h e e x c e p t i o n a l case (figure 4-8) m a y be r e l a t e d t o s i m i l a r fluctuations o b s e r v e d i n t h e T P D i n s t a b i l i t y a n d S B S . CHAPTER 5: Discussion 9 9 Temporal behaviour of the SRS scattered light has been observed in other experiments, although growth rates of SRS in a laser produced plasma have never before been measured. SRS has always been subnanosecond in duration, with the shortest (instrument limited) pulses reported being 20ps long 7 5 . Modulation times of 50ps to 500ps have been reported 7 5 , 6 5 , 6 6 , 6 4 , 7 6 , 6 3 , 6 7 . j n a similar C 0 2 laser-plasma interaction experiment 6 2 , the pulse widths were lOOps and separated by 200-500ps. The reasons for the bursts of SRS are not at all clear. Two suggestions have been made in the literature. These are variations in the incident laser intensity 6 4 and localized, moving density plateaus 6 6 . The intensity of the laser beam used in this experiment is believed to be temporally smooth; no variations on the instru-ment limited scale of 500ps were seen (see figure 3-1). The spatially resolved Thom-son scattering did not reveal any evidence for moving, localized density plateaus. (These plateaus would be a result of the TPD profile modification process. The flattened regions would propagate away from 0.25ncr. Temporal behaviour would then be associated with the periodic saturation, quenching and reappearance of TPD.) Modulations have been observed in simulations in which periods of 100 laser periods (about 3 ps) were found but no detailed explanation for such behaviour was given. 2 Since the two explanations above do not seem to be applicable to this exper-iment, another reason must be found. One is that ion acoustic waves disrupt the SRS EPW and that modulations in the IA waves are reflected in SRS. The origin of these large amplitude ion acoustic waves is of some interest. Are they a result of the SRS-TPD processes or are they a result of SBS? Since SBS has a lower threshold intensity than SRS does, one might suspect that SBS starts before SRS does but grows at a lower rate for a longer time and might be the source of ion acoustic waves when the laser intensity reaches the threshold for SRS. However, there is a complication to this simple picture - it takes a certain amount of energy to produce a plasma in which these instabilities can occur. A typical laser rises from 0 to 5 x CHAPTER 5: Discussion 100 1 0 1 3 W c m ~ 2 i n 1.2ns. T h e r e q u i r e d a m o u n t of energy t o f o r m a p l a s m a , a p p r o x i -m a t e l y 0.8J, w i l l have been d e p o s i t e d i n t o t h e t a r g e t b y 0.7ns a t w h i c h t i m e , t h e l a s e r i n t e n s i t y w i l l b e 3 x 1 0 1 3 W c m - 2 . T h i s i n t e n s i t y is w e l l above t h e o b s e r v e d t h r e s h o l d s f o r S B S ( a b o u t 1 0 1 2 W c m - 2 ) a n d S R S ( a b o u t 1.9 x 1 0 1 3 W c m - 2 ) . S i n c e t h e i n t e r a c t i o n s c a n n o t p r o c e e d b e f o r e a p l a s m a has f o r m e d , a g o o d a p p r o x i m a t i o n t o t h e p h y s i c a l s i t u a t i o n i s t h a t S R S a n d S B S s t a r t s i m u l t a n e o u s l y at 0.7ns. I n th e n e x t 20ps, b o t h i n s t a b i l i t i e s grow i n t i m e at t h e m e a s u r e d r a t e s o f 1 0 1 0 s - 1 f o r S B S a n d 1 0 1 2 s - 1 f o r S R S . T h e r e l a t i v e e-foldings at t h e e n d of t h i s 20 ps w i l l b e 0.2 f o r S B S a n d 20 f o r S R S . T h u s , S B S w i l l b e r e l a t i v e l y u n i m p o r t a n t a n d any l a r g e a m p l i t u d e i o n a c o u s t i c waves o b s e r v e d i n t h i s e x p e r i m e n t at t h e t i m e of t h e S R S i n t e r a c t i o n m u s t have some o t h e r o r i g i n . I n p r e f o r m e d p l a s m a s , t h i s p l a s m a p r o d u c t i o n c o m p l i c a t i o n does n o t enter i n t o c o n s i d e r a t i o n . I n s u c h cases, i t is b e s t to e x a m i n e e a c h p r o b l e m i n d i v i d u a l l y . T w o s u c h e x p e r i m e n t s h a v e been p r e s e n t e d i n t h e l i t e r a t u r e . R o z m u s 7 7 a n d O f f e n b e r g e r 7 8 have s h o w n t h a t f o r a large, low d e n s i t y p l a s m a , S B S c a n o c c u r before S R S does whereas W a l s h 6 S showed exper-i m e n t a l l y t h a t , i n a p r e f o r m e d p l a s m a w h i c h has c o n d i t i o n s s i m i l a r t o t h e present e x p e r i m e n t , t h e presence of large a m p l i t u d e I A waves sto p s S R S . T h e r e s u l t s pre-s e n t e d here so f a r h i n t t h a t c o m p a r i s o n s w i t h t h i s l a t t e r case are a p p r o p r i a t e . T h e c o n n e c t i o n o f i o n a c o u s t i c waves t o S R S w i l l be d i s c u s s e d as a s a t u r a t i o n m e c h a n i s m l a t e r . I o n a c o u s t i c waves have been a s s o c i a t e d w i t h t h e s a t u r a t i o n o f T P D . 5.4 Density Behaviour O n e i m p o r t a n t p o i n t a b o u t S R S w h i c h has n o t been d i s c u s s e d yet: i n m a n y p r e v i o u s e x p e r i m e n t s 6 2 , 6 4 , 6 5 , 6 6 , 6 7 , 6 8 , 6 9 , 7 9 t h e r e is a gap i n t h e s c a t t e r e d E M spec-t r u m , c o r r e s p o n d i n g t o s c a t t e r i n g f r o m d e n s i t i e s a p p r o x i m a t e l y 0.1 t o 0.24 ncr. T h i s e x p e r i m e n t p r o b e d two aspects of S R S w h i c h s h o u l d be p r e s e n t , namely, scat-t e r e d I R l i g h t a n d e n h a n c e d e l e c t r o n p l a s m a waves. I n t h i s s e c t i o n , t h e d e n s i t y fluctuations f o u n d by wave v e c t o r a n d f r e q u e n c y r e s o l v e d T h o m s o n s c a t t e r i n g are CHAPTER 5: Discussion 101 examined to determine if the fluctuations are enhanced for densities in the region of interest. Upon examination of the typical streaks presented in chapter 4, it is imme-diately obvious that the EPW are enhanced at densities below 0.25raCf. This can be seen in both the wavevector and frequency resolved streaks. This means, in the present experiment, that the gap does not exist as far as the plasma wave fluc-tuations are concerned. In the next section, the levels of the IR that should be associated with these fluctuations are estimated and compared with the observed IR spectra. In Chapter 2, a hypothesis was put forward that the SRS should propagate to lower densities at a rate fixed by the plasma wave group velocity and the density gradient scale length. The present experimental data will be examined to see if there is any evidence for this. In both figures 4-6 and 4-12, SRS behaves qualitatively as predicted. On a quantitative level, the dashed line in these figures represents the rate at which the density should change if the convective model (Eqn. 2-25) is applied with the parameters L=400jzm, n — 0.2n c r, and T=300eV. In both cases, the observed rate is faster than the predicted rate of -0.0006ncr ps _ 1 which is a result of the shot-to-shot variation in the density scale length. In a plasma with a steeper density gradient, the rate is predicted to increase. Thus, the convective hypothesis seems to be quantitatively consistent with the data. The convective model is interesting in that it proposes that the EPW gen-erated at density ris increases the noise levels for EPW at lower densities. As was mentioned in chapter 2, the EPW initially propagate up the density and are reflected at their critical surface. The EPW then propagate down the density gra-dient. As the EPW propagates from the higher to lower densities, its wavevector must become longer since its frequency does not change. This means that the wave damping via Landau damping becomes greater. By a slight rearrangement of the p l a s m a wave d i s p e r s i o n r e l a t i o n , one finds t h a t CHAPTER 5: Discussion 102 n{x)/ns = 1 - 3(kvth/wepw)2 w h e r e uepw « up(ns)- G e n e r a l l y , L a n d a u d a m p i n g is i m p o r t a n t w h e n kv^/u^ is a b o u t 0.3 . A t t h i s p o i n t , t h e d a m p i n g becomes g r e a t e r t h a n t h e g r o w t h r a t e f o r S R S ( f o r t h e m o d e s t i n t e n s i t i e s used i n t h i s e x p e r i m e n t ) . T h i s l i m i t suggests t h a t t h e p l a s m a waves c a n p r o p a g a t e t o a m i n i m u m d e n s i t y o f 0.73ns- T h i s lower d e n s i t y c u t - o f f is o b s e r v a b l e i n some shots. F o r e x a m p l e , i n figure 4-6, t h e w a v e v e c t o r s s t a r t a t 0.25n c r. T h e l i m i t e x p e c t e d is 0.18n c r a n d t h e l i m i t o b s e r v e d is 0.19n c r. I n figure 4-12, t h e p l a s m a waves s t a r t at 0.19n c r a n d e n d at a b o u t 0.15n c r. T h e l i m i t e x p e c t e d is 0.14n c r. I n figure 4-12, t h i s c u t - o f f m a y be c o i n c i d e n t a l since t h e i n t e r f e r e n c e filter a n d t h e a n g u l a r l i m i t a t i o n s of t h e T h o m s o n s c a t t e r i n g also l i m i t t h e o b s e r v a b l e s p e c t r u m . I n figure 4-6, these effects s h o u l d n o t be r e s p o n s i b l e f o r t h e c u t - o f f s i n c e t h e c u t - o f f p o i n t i n t h a t figure is f a r f r o m t h e c u t - o f f set by t h e T h o m s o n s c a t t e r i n g s e t u p . T h i s f u r t h e r s u p p o r t s t h e c o n v e c t i v e m odel. T h e t e m p o r a l b e h a v i o u r of S R S has b e en o b s e r v e d i n o t h e r cases. I n each case, S R S a l w a y s p r o c e e d s t o s h o r t e r w a v e l e n g t h s , a l t h o u g h t h e e x p l a n a t i o n s f o r t h i s vary. S e k a et a l . 6 6 c l a i m s t h a t t h e t e m p e r a t u r e change i n t i m e is r e s p o n s i b l e . D i f f e r e n t i a t i n g t h e d i s p e r s i o n r e l a t i o n f o r a n E P W ( f r o m page 6 ) , one finds 1 duepw _ . k 2 3kg dTe up dt Up 2 m dt U s i n g t h e m e a s u r e d duepw/dt = — 0 . 2 2 p s - 2 we find a dT/dt = — lOOeV p s - 1 . T h i s r a t e is t o o l a r g e a n d t h e w r o n g d i r e c t i o n . A re a s o n a b l e e s t i m a t e w o u l d be a n i n c r e a s e of 3 0 0 e V i n 1.2ns. T h e r e f o r e it is u n l i k e l y t h a t a c h a n g i n g t e m p e r a t u r e is r e s p o n s i b l e f o r t h e o b s e r v e d effect. A s e c o n d effect was suggested b y F i g u e r o a et a l . 6 9, namely, t h e p l a s m a d e n s i t y m i g h t b e c h a n g i n g . R e s u l t s f r o m t h e i n t e r f e r o m e t r y i n d i c a t e t h a t t h e p l a s m a CHAPTER 5: Discussion 103 d e n s i t y v a r i e s v e r y l i t t l e a t t i m e s less t h a n one nanosecond. T h i s m e c h a n i s m does n o t seem t o be r e s p o n s i b l e f o r t h e present r e s u l t s . T h i s s h o c k wave s h o u l d n o t ch a n g e t h e d e n s i t y o n 20ps t i m e s c a l e s . 5.5 Scattered Infrared Light I n c h a p t e r 4, t h e r e s u l t s p r e s e n t e d s howed t h a t s c a t t e r e d I R l i g h t was f o u n d o n l y n e a r 2 1 ^ m a n d i n a b r o a d b a n d ne a r 15/xm. However, t h e T h o m s o n s c a t t e r i n g i n d i c a t e d t h a t t h e r e s h o u l d be m o r e I R l i g h t t h a n t h i s . I n t h i s s e c t i o n , a n a t t e m p t is m a d e t o r e c o n c i l e t h e T h o m s o n s c a t t e r i n g r e s u l t s w i t h t h e I R r e s u l t s . T h e ob-s e r v a t i o n s seem t o i m p l y t h a t e n h a n c e d E P W are d r i v e n at d e n s i t i e s b e l o w 0.25n c r y e t t h e r e is a lac k of d e t e c t a b l e s c a t t e r e d l i g h t c o r r e s p o n d i n g t o t h o s e d e n s i t i e s . I n a d d i t i o n , t h e r e are l a r g e s i g n a l s observed at 2 A P a n d at w a v e l e n g t h s c o r r e s p o n d i n g t o a p p r o x i m a t e l y 0.1n c r. T h e s e r e s u l t s are d i s c u s s e d i n t h e f o l l o w i n g order. F i r s t , a p l a u s i b l e o r i g i n f o r t h e s i g n a l at 2 A 0 is given. T h e n , t h e levels of S R S at lower d e n s i t i e s is e s t i m a t e d f r o m the r e s u l t s o f t h e T h o m s o n s c a t t e r i n g . F i n a l l y , these es-t i m a t e s are c o m p a r e d w i t h t h e I R d e t e c t o r t h r e s h o l d . It is s h o w n t h a t a d e t e c t a b l e s i g n a l s h o u l d have been generated. T h e gap is t o be e x p l a i n e d . T o t h i s end, t he c u r r e n t e x p l a n a t i o n s are o u t l i n e d a n d a p p l i e d t o t h i s e x p e r i m e n t . A v a r i a t i o n of one o f these e x p l a n a t i o n s is s h o w n t o be c o n s i s t e n t w i t h t h e r e s u l t s observed. T h e o r i g i n o f t h e 2 A 0 s i g n a l is p r o b a b l y n o t d u e t o S R S , b u t r a t h e r due t o l i n e a r m o d e c o n v e r s i o n f o r t h e T P D i n s t a b i l i t y epw. I n t h i s m e c h a n i s m , t h e E P W p r o p a g a t e s u p a d e n s i t y g r a d i e n t u n t i l i t reaches i t s t u r n i n g p o i n t a t w h i c h p o i n t t h e wave n o longe r p r o p a g a t e s . T h e e l e c t r o s t a t i c wave energy is t r a n s f e r e d the elec-t r o n s a n d t h e e l e c t r o n s r e r a d i a t e t h e energy as e l e c t r o m a g n e t i c waves. T h i s process is c a l l e d i n v e r s e r e s o n a n c e a b s o r p t i o n 9 2. O n l y frequencies present i n t h e E P W c a n r e p r o d u c e d i n t h e E M wave. T h e r e f o r e , i t is necessary t o l o o k o n l y at mech-a n i s m s w h i c h cause s p l i t t i n g i n t h e frequencies of t h e epw. I n o t h e r e x p e r i m e n t s 7 5 , 6 1 , 8 0 , 6 5 , 6 6 , 8 1 , 7 9 , 8 2 , 6 7 , 6 8 , 8 3 , 8 4 , 6 9 ^ f r e q u e n c y s p l i t t i n g has been o b s e r v e d near 2A 0. CHAPTER 5: Discussion 104 T h e e x p l a n a t i o n s g i v e n f o r s u c h e x p e r i m e n t s are a p p l i e d t o t h i s e x p e r i m e n t . T w o h y p o t h e s e s are u s u a l l y p u t f o r w a r d ; t h e p e a k t o p e a k s p l i t t i n g c a n be due t o e i t h e r a f i n i t e t e m p e r a t u r e 9 8 AXpp{fim) = 4.4 x 10~3\0(fim)T{keV) o r a f i n i t e m a g n e t i c f i e l d 9 8 AXpp{fim) = 1.87 10" 2 \ 2 0 { f im )B {MG) . T h e f i n i t e t e m p e r a t u r e effect is a re s u l t o f s m a l l c o r r e c t i o n s f o r t h e E P W d i s p e r -s i o n of t h e S R S E P W w h i c h were p r e v i o u s l y i g n o r e d i n t h e t h e s i s . A t e m p e r a t u r e o f 8 k e V o r a m a g n e t i c f i e l d of 200 k G is r e q u i r e d t o e x p l a i n t h e o b s e r v e d s p l i t -t i n g of a b o u t 0.4/zm. T h i s 8 k e V t e m p e r a t u r e is u n r e a s o n a b l e s i n c e the m e a s u r e d t e m p e r a t u r e is o n l y 300 eV. A l t h o u g h n o m a g n e t i c f i e l d was l o o k e d f o r i n t h i s ex-p e r i m e n t , o t h e r e x p e r i m e n t s 8 4 a have r e p o r t e d fie l d s i n excess of 2 M G a l t h o u g h t h o s e e x p e r i m e n t s were p e r f o r m e d u n d e r v e r y d i f f e r e n t c o n d i t i o n s . A field o f 2 0 0 k G is r e a s o n a b l e s i n c e a r o u g h e s t i m a t e m a d e i n c h a p t e r 2 showed t h a t , f o r o u r con-d i t i o n s , t h e t h e r m o e l e c t r i c effect c o u l d generate a field of o n t h e o r d e r o f lOOkG. A field of t h i s m a g n i t u d e w i l l c hange t h e t e m p o r a l g r o w t h r a t e b y ± 1 5 % . A t h i r d p o s s i b l e e x p l a n a t i o n c a n be p r o p o s e d . S i n c e t h e r e are l a r g e a m p l i t u d e I A W present, a c o u p l i n g of these waves w i t h t h e E P W w o u l d r e s u l t i n s i d e b a n d s b e i n g g e n e r a t e d a t up ± w t a. However, the o b s e r v e d f r e q u e n c y s p l i t t i n g is 10 t i m e s t h e s p l i t t i n g e x p e c t e d f o r I A W of A:,a = 4k0. S i d e b a n d s p l i t t i n g has been o b s e r v e d f o r l i g h t scat-t e r e d at 3/2u>0 i n a recent e x p e r i m e n t 8 5. A n o t h e r p o s s i b l e s o u r c e of t h e s p l i t t i n g is due t o e l e c t r o n t r a p p i n g . T h e s c a t t e r e d l i g h t w i l l be m o d u l a t e d at t h e b o u n c e f r e q u e n c y 1 1 of t h e e l e c t r o n s i n t h e wave i.e. ub = k0ve{6nln)1 CHAPTER 5: Discussion 105 T h i s leads t o a m o d u l a t i o n o f 4.3 x 1011 s _ 1 w h i c h is n a r r o w e r t h a n t h e observed s p l i t t i n g o f 1.7 x 10 1 2 s _ 1 . T h u s , we c o n c l u d e t h a t t h e o b s e r v e d s p l i t t i n g is c o n s i s t e n t o n l y w i t h t h e presence of a m a g n e t i c field. F u r t h e r e x p e r i m e n t s w o u l d be necessary t o p r o v e t h i s d e f i n i t e l y . T h e T h o m s o n s c a t t e r i n g r e s u l t s q u a l i t a t i v e l y i m p l y t h a t S R S s c a t t e r e d I R s h o u l d b e present. I n t h i s s e c t i o n , a q u a n t i t a t i v e e s t i m a t e o f t h e I R s i g n a l size is a t t e m p t e d . F u r t h e r , t h i s e s t i m a t e d p o w e r w i l l be c o m p a r e d t o t h e m i n i m u m power t h e d e t e c t o r c a n measure. T h e a m o u n t of s c a t t e r e d I R c a n be e s t i m a t e d since t h e T h o m s o n s c a t t e r i n g was a b s o l u t e l y c a l i b r a t e d . I t is i m p o r t a n t , however, t o keep t r a c k o f t h e d i f f e r e n c e b etween power r e f l e c t i v i t y a n d energy r e f l e c t i v i t y . T h e o r e t -i c a l f o r m u l a s p r e d i c t power e s t i m a t e s whereas e x p e r i m e n t s m e a s u r e energies. T h e c o n v e r s i o n f r o m p o w e r t o energy r e q u i r e s a t i m e e s t i m a t e . I n t h e f o l l o w i n g , S R S is a s s u m e d t o o c c u r f o r l O p s , w h i c h is a p p r o x i m a t e l y t h e m i n i m u m d u r a t i o n of t h e T h o m s o n s c a t t e r i n g . T h e first e s t i m a t e o f t h e a m o u n t of s c a t t e r e d l i g h t is made u n d e r t h e a s s u m p t i o n t h a t t h e I R l i g h t is s i m p l y T h o m s o n s c a t t e r e d f r o m t h e ob-s e r v e d fluctuations. I n s e c t i o n 2.5, a n e x p r e s s i o n was g i v e n w i t h w h i c h t h e power s c a t t e r e d b y a g i v e n fluctuation p e r s o l i d a n g l e a n d i n a g i v e n f r e q u e n c y range c a n be c a l c u l a t e d . T h i s c a l c u l a t i o n is i n d e p e n d e n t o f t h e w a v e l e n g t h o f t h e l i g h t w h i c h is t o b e s c a t t e r e d . I n s e c t i o n 3.3, t h i s f o r m u l a was a p p l i e d t o r u b y l a s e r l i g h t T h o m -s o n s c a t t e r i n g a n d a n e s t i m a t e o f t h e a b s o l u t e s c a t t e r e d power c o u l d be fou n d . I f t h e CO2 l a s e r is u s e d i n p l a c e o f t h e r u b y las e r , i t is s t r a i g h t f o r w a r d t o show t h a t w h e r e Aw is t h e b a n d w i d t h of t h e s c a t t e r e d l i g h t w h i c h c a n be d e t e c t e d , AD is t h e s o l i d angle i n t o w h i c h the d e t e c t e d s c a t t e r e d l i g h t was e m i t t e d , a n d L is t h e l e n g t h of t h e s c a t t e r i n g r e g i o n . T h e m a x i m u m (dPs / Pi) Ru\,y c a n be c a l c u l a t e d w i t h t h e m e t h o d o u t l i n e d i n c h a p t e r 3 a n d w i t h t h e p a r a m e t e r s of T a b l e 4.1. T h e r e s u l t is iPs\ AUJSCQ2 An Co 2 LCQ2 Pi / Ruby Au>sRui,y AtlRuby Ljiuby CHAPTER 5: Discussion 106 1.5 1 0 - 8 . T h e t e r m AD,co2 l s s e t D v t n e c o l l e c t i o n o p t i c s of I R d i a g n o s t i c s . T h e c a l c u l a t e d s o l i d a n g l e d e t e c t e d is g i v e n i n f i g u r e 3-12(c) as a f u n c t i o n o f w a v e l e n g t h . I n t h e c a l c u l a t i o n , a t y p i c a l s o l i d a ngle of 0.0016 s r is used. T h e s o l i d a ngle of t h e r u b y l a s e r s c a t t e r e d l i g h t w h i c h was d e t e c t e d is set by t h e a n g u l a r r a n g e of a si n g l e p i x e l (0.00022 r a d ) at a d i s t a n c e o f 30 c m f r o m t h e t a r g e t a n d by t h e h e i g h t of a m a s k o p e n i n g (1 cm) at t h e same d i s t a n c e . T h e A f 2 # t t i j , f o u n d t h i s w a y is (0.00022 x 1 c m ) / (30 cm) o r 7.3 1 0 - 6 sr. T h e f r e q u e n c y range a c c e p t e d b y t h e I R d i a g n o s t i c s was set b y t h e e x i t a p e r t u r e of t h e m o n o c h r o m a t o r . T h i s reduces t o 8.4 1 0 1 1 r a d s _ 1 . T h e f r e q u e n c y r a n g e f o r t h e r u b y laser T h o m s o n s c a t t e r i n g c a n be e s t i m a t e d t w o ways. T h e m o s t c o n s e r v a t i v e e s t i m a t e w o u l d set AWRu6J/ e q u a l t o t h e b a n d p a s s of t h e i n t e r f e r e n c e filter. T h i s is n o t u s e d s i n c e t h e f r e q u e n c y r e s o l v e d T h o m s o n s c a t t e r i n g (e.g. figure 4-12) i n d i c a t e d a m u c h n a r r o w e r range is m o r e a p p r o p r i a t e . A c c o r d i n g l y , t h e f r e q u e n c y w i n d o w used is 1.2 x 1 0 1 3 r a d s _ 1 w h i c h c o r r e s p o n d s t o a w a v e l e n g t h r a n g e i n t h e s c a t t e r e d r u b y l i g h t f r o m 6 7 5 n m t o 678nm. I also assume t h a t t h e s c a t t e r i n g l e n g t h s f o r t h e r u b y a n d c a r b o n d i o x i d e lasers are e q u a l . W h e n a l l t hese e s t i m a t e s are c o m b i n e d , one finds t h a t t h e t y p i c a l I R power r e f l e c t i v i t y is 2.2 x 1 0 - 7 . F o r a t y p i c a l i n c i d e n t p o w e r of 2.5GW, we find t h a t 5.5 n J of s c a t t e r e d I R energy s h o u l d have r e a c h e d t h e d e t e c t o r . T h i s s h o u l d be a lower l i m i t . A second e s t i m a t e of t h e a m o u n t of s c a t t e r e d l i g h t , b a s e d u p o n t h e M a n l e y - R o w e r e l a t i o n s (Eqn.2-32) i n d i c a t e s t h a t t h e r e s h o u l d be 60/zJ of s c a t t e r e d l i g h t b u t t h e d i r e c t i o n o f t h i s l i g h t c a n n o t be s p e c i f i e d . T h i s e s t i m a t e is based u p o n t h e a m o u n t of energy c o n t a i n e d i n t h e o b s e r v e d p l a s m a waves. T h e s e e s t i m a t e s are now c o m p a r e d w i t h t h e s e n s i t i v i t y of t h e G e : C u d e t e c t o r u s e d i n t h e e x p e r i m e n t . F o r s i g n a l s o f d u r a t i o n m u c h less t h a n t h e RC t i m e of t h e d e t e c t o r , t h e o u t p u t s i g n a l is p r o p o r t i o n a l t o t h e i n p u t energy, n o t the i n p u t power. F o r RC a b o u t I n s , a noise v o l t a g e of 5 m V, a n d a power s e n s i t i v i t y o f 0.2A/W i n t o 5 0 0 , we find t h e m i n i m u m d e t e c t a b l e energy o f UMIN = ( I n s ) ( 5 m V ) / ( 0 . 2 A / W 50f2) of l p J . E v e n g i v e n t h e r o u g h n e s s of t h i s e s t i m a t e , we c a n n o t e t h a t i t is s t i l l m a n y CHAPTER 5: Discussion 107 o r d e r s of m a g n i t u d e b e l o w t h e e s t i m a t e of t h e s c a t t e r e d energy. T h u s , I c o n c l u d e t h a t a s i g n a l s h o u l d have been det e c t e d . T h e l a c k of s c a t t e r e d l i g h t is c o n s i s t e n t w i t h m a n y e x p e r i m e n t s (e.g. Ref.65,66 a n d references t h e r e i n ) a l t h o u g h m o s t of these same e x p e r i m e n t s a l s o observe a s i g n a l at 2 A 0 (never b e f o r e i n a CO2 laser e x p e r i m e n t , however). T h e e x p l a n a t i o n s p u t f o r w a r d have v a r i e d w i d e l y , b u t c a n be r e d u c e d t o two. T h e m a j o r i t y of t h e i n v e s t i g a t o r s b e l i e v e p r o f i l e m o d i f i c a t i o n is i m p o r t a n t . T h i s m a y be i m p o r t a n t f o r q u e n c h i n g t h e i n s t a b i l i t y . However, i t is d i f f i c u l t t o see how t h i s m e c h a n i s m c a n e x p l a i n t h e gap i n t h i s e x p e r i m e n t . If the p r o f i l e has been m o d i f i e d t o s u c h a n e x t e n t t h a t S R S does n o t o c c u r at c e r t a i n d e n s i t i e s , i t s h o u l d n o t be p o s s i b l e t o observe E P W g e n e r a t e d at s a i d d e n s i t i e s . O t h e r i n v e s t i g a t o r s b e l i e v e t h a t t h e source of n o i s e f r o m w h i c h S R S m u s t grow is i m p o r t a n t . T h i s noise c a n be e l e c t r o m a g n e t i c o r i t c a n be e l e c t r o s t a t i c . F o r t h e source of t h e e l e c t r o m a g n e t i c noise, b l a c k b o d y r a d i a t i o n has b e e n suggested. S i n c e i n t h e l o n g w a v e l e n g t h r e g i o n t h i s noise is p r o p o r t i o n a l t o 1/A 4, t h e I R at longer w a v e l e n g t h s g e n e r a t e d b y S R S w i l l have t o grow f r o m a m u c h s m a l l e r noise l e v e l . T h e e l e c t r o s t a t i c noise is c a l c u l a t e d i n s t a n d a r d t e x t b o o k s 1 1 a n d is g i v e n b y (af t e r i n t e g r a t i o n over a l l f r e q u e n c i e s ) [ 6 n ) = N T T W o ? w h e r e N is n u m b e r of e l e c t r o n s i n t h e p l a s m a v o l u m e . A s k becomes l a r g e r , the n o i s e l e v e l becomes l a r g e r . S i n c e longer p l a s m a wav e v e c t o r s are needed f o r S R S p h a s e m a t c h i n g at lower d e n s i t i e s , S R S g r o w i n g f r o m e l e c t r o s t a t i c noise s h o u l d be m o r e p r e d o m i n a n t a t low d e n s i t i e s . I n e i t h e r case, t h e a c t u a l g a i n ( i n c l u d i n g losses due t o i n v e r s e b r e m s s t r a h l u n g ) at a g i v e n w a v e l e n g t h r e q u i r e s a n i n t e g r a t i o n over a c t u a l d e n s i t y a n d t e m p e r a t u r e p r o f i l e s . S eka et al. 6 5 > 6 6 were a b l e t o r e p r o d u c e t h e i r r e s u l t s q u i t e w e l l u s i n g h y d r o d y n a m i c s i m u l a t i o n r e s u l t s f o r t h e d e n s i t y a n d t e m p e r a t u r e p r o f i l e s a n d a s s u m i n g S R S grew f r o m b l a c k b o d y noise. CHAPTER 5: Discussion 108 A novel explanation was proposed by Simon and Short 8 6 > 8 7 who pointed out that a pulse of fast electrons moving in a cold plasma leads to an enhancement of plasma waves whose phase velocities lie on that part of the distribution where df /dv > 0. Spectra which are frequently identified as SRS are, by this theory, the result of ordinary Thomson scattering of the CO2 laser from these enhanced waves. They state enhanced scattering occurs if the denominator in their scattering crossection is zero or negative. The expression for this denominator is given by Vei/*1/2/uepw + (Ep/Tcf2 exp(-Ep/Tc) + fho?l2y2{y - 1) exp(-a(y - l) 2 ) where Ep = 0.bme{uepw/kepw)2, a = 3Thotk2x/2/Tck2epw, y2 = 2Epk2epw/(Zk2xThot), and kx is the component of kepw along the direction of a pulse of hot electrons. In 1-D, the problem becomes easier to examine since kx will then be just k and Ep has already been calculated (see figure 2-4). In the present experiment, the fraction of hot electrons, fh, is 0.01 at maximum 6 0. The observed hot electron temperature, Thot, is about lOOkeV. If various scattering angles and electron densities are tried in this formula, no region of enhanced scattering is found. The reason for this is that the collisional term in the denominator (the first term of the above expression) is totally dominant. For the denominator to be zero or negative, the final term must be large and negative. While there is no problem in making the term negative (i.e. making y less than 1), the exponential term makes the term small compared to the collisional term. This can be clearly seen by noticing that a is on the order of 500 for the temperatures specified earlier. The theory of Simon also predicts enhanced plasma waves. However, it is felt that the standard theory adequately explains all the observed phenomena (except for the lack of IR light from low densities). This does not preclude further more detailed experiments to further test the theory of Simon et al. It may be the correct explanation. CHAPTER 5: Discussion 109 T h e I R l i g h t m a y be s c a t t e r e d i n some u n e x p e c t e d d i r e c t i o n . S h e p a r d 6 7 has o b s e r v e d t h a t R a m a n s c a t t e r i n g is p r e f e r e n t i a l l y s c a t t e r e d w i t h i n t h e cone angle of t h e i n c i d e n t l a s e r l i g h t . A s m e n t i o n e d e a r l i e r , t h e severe a t t e n u a t i o n of K C 1 (and t h e p r o b l e m o f e l i m i n a t i n g S B S ) p r e v e n t e d e x a m i n a t i o n o f t h i s d i r e c t i o n . R e f r a c t i o n effects c o u l d r e d u c e t h i s p r e f e r e n t i a l s c a t t e r i n g . If t h e d e n s i t y g r a d i e n t s are p a r a l l e l t o t h e w a v e v e c t o r o f th e i n c i d e n t laser l i g h t (as t h e y are i n t h i s e x p e r i m e n t at t h e t i m e of t h e S R S i n t e r a c t i o n ) , r e f r a c t i o n effects w i l l be s m a l l . T h e r e is one a s p e c t of S R S w h i c h I b e l i e v e has n o t b e e n c o m p l e t e l y e x a m i n e d a l t h o u g h Seka et a l . 6 5 > 6 6 have i n c l u d e d t h e effect i n t h e i r c a l c u l a t i o n s . T h i s has t o do w i t h t h e p r o p a g a t i o n of t h e s c a t t e r e d I R t o t h e b o u n d a r i e s of t h e p l a s m a . It seems e v i d e n t t h a t one s o l u t i o n t o t h e i n c o n s i s t e n c y o f t h e E P W / I R r e s u l t s is t h e p o s s i b i l i t y t h a t t h e s c a t t e r e d l i g h t is b e i n g a b s o r b e d as i t t r a n s v e r s e s a l a r g e r e g i o n of p l a s m a bef o r e r e a c h i n g t h e d e t e c t o r . I n p a r t i c u l a r , i f t h e s c a t t e r e d l i g h t m u s t pass t h r o u g h a r e g i o n of c o l d , dense p l a s m a , inverse b r e m s s t r a h l u n g a b s o r p t i o n c a n r e d u c e t h e s i g n a l a p p r e c i a b l y . I n a p l a s m a , t h e inv e r s e b r e m s s t r a h l u n g c o e f f i c i e n t 8 7 a is g i v e n by , _ 7.8 x 1 0 - 9 Z n 2 ( c m - 3 ) A hib(cm ) = ryr — 7 $ V ( l - ^ 2 / ^ ) 1 / 2 w h i c h c a n be a f u n c t i o n of p o s i t i o n . I n p a r t i c u l a r , t h e s i g n a l w h i c h s t a r t s off at I 0 o n one side of a s l a b of p l a s m a of t h i c k n e s s L is r e d u c e d t o / = 7 0 e x p ( - / kib(x)dx). Jo I n a p l a s m a w i t h d e n s i t y , t h e r m a l a n d i o n i z a t i o n g r a d i e n t s , t h e a b s o r p t i o n coeffi-c i e n t is c e r t a i n l y a f u n c t i o n o f p o s i t i o n . T o see how i m p o r t a n t t h i s effect was, t h e a b s o r p t i o n was c a l c u l a t e d f o r S R S ge n e r a t e d o n an i s o t h e r m a l , l i n e a r d e n s i t y ramp. I n t h i s m o d e l t h e d e n s i t y v a r i e s f r o m q u a r t e r c r i t i c a l d e n s i t y t o 0 i n a dist a n c e L. T h e f r e q u e n c y of t h e s c a t t e r e d l i g h t was c a l c u l a t e d a t each p o i n t o n t h e r a mp, CHAPTER 5: Discussion 110 a s s u m i n g us — u0 — up(x) a n d t h a t t h e l i g h t t r a n s v e r s e d o n l y t h e d i s t a n c e f r o m i t s o r i g i n t o t h e p l a s m a b o u n d a r y w i t h t h e l i g h t s c a t t e r e d f r o m q u a r t e r c r i t i c a l d e n s i t y l a y e r t r a n s v e r s i n g t h e e n t i r e p l a s m a . F i g u r e 4-13 shows t h e e s c a p i n g f r a c t i o n of l i g h t g e n e r a t e d a t a g i v e n density. It is c l e a r t h a t l i g h t g e n e r a t e d at t h e h i g h e r d e n s i t i e s is s e v e r e l y a t t e n u a t e d . T h i s a t t e n u a t i o n is also a s e n s i t i v e f u n c t i o n o f t h e t e m p e r a t u r e of t h e p l a s m a . A t e m p e r a t u r e of 3 0 0 e V has been a s s u m e d p r e v i o u s l y b u t t h i s t e m p e r a t u r e is a c t u a l l y t e m p o r a l l y averaged. F u r t h e r a p p l i c a t i o n o f t h i s m o d e l r e q u i r e s t h e e x a c t t e m p e r a t u r e at t h e t i m e of t h e S R S w h i c h is n o t e a s i l y m e a s u r a b l e . C o m p u t e r s i m u l a t i o n s f o r t h e gas j e t show t h a t , e a r l y i n t o t h e laser p u l s e , t h e e l e c t r o n t e m p e r a t u r e is less t h a n lOOeV. I n a d d i t i o n , t h e h e a t i n g m o d e l u s e d b y P o p i l 5 2 a s s u m e d t h a t t h e e l e c t r o n t e m p e r a t u r e i n t h e b u l k of t h e p l a s m a g r e w l i n e a r l y w i t h t i m e d u r i n g t h e first 1.2ns, i.e. d u r i n g t h e rise t i m e of t h e i n -c i d e n t laser p u l s e . W h a t e v e r t h e t e m p e r a t u r e i s , t he p l a s m a s h o u l d be s p a t i a l l y i s o t h e r m a l as t h e t h e r m a l c o n d u c t i v i t y o f a h o t p l a s m a is large e n o u g h t o p e r m i t t h e r m a l i z a t i o n o n t i m e s less t h a n lOps. ( T h i s c a n be seen q u a l i t a t i v e l y b y n o t i n g t h e c o l l i s i o n p e r i o d f o r a 3 0 e V p l a s m a is a b o u t 3 1 0 ~ 1 3 s . T h e p l a s m a t h e r m a l i z e s a f t e r a few periods.) T h e r e is one o b v i o u s i n c o n s i s t e n c y w i t h t h i s m o d e l — a s i g n a l is o b s e r v e d w h i c h m u s t o r i g i n a t e n e a r t h e q u a r t e r c r i t i c a l l a yer. F r o m t h e averaged T h o m s o n s c a t t e r i n g r e s u l t s , S R S c a n be seen t o have s i m i l a r fluctuation levels a t a l l den-s i t i e s i n t h e r a n g e e x a m i n e d a n d hence, s i m i l a r levels of s c a t t e r e d i n f r a r e d l i g h t a r e e x p e c t e d over t h e c o r r e s p o n d i n g w a v e l e n g t h r e g i o n . However, t h e I R s i g n a l o b s e r v e d e x t e r n a l l y is t h e p r o d u c t of b o t h t h e a t t e n u a t i o n a n d t h e a c t u a l a m o u n t o f I R p resent at i t s o r i g i n deep i n s i d e t h e p l a s m a . A t 0.25re c r, a d d i t i o n a l l i g h t at oJo/2 c a n be g e n e r a t e d as a r e s u l t of t h e l i n e a r m o d e c o n v e r s i o n of t h e T P D p l a s m a waves. Stokes s c a t t e r i n g o f t h e i n c i d e n t C 0 2 l a s e r l i g h t f r o m the T P D p l a s m a waves c a n a l s o generate a d d i t i o n a l 2 A 0 l i g h t . I n e i t h e r case, t h e I R l i g h t at w 0/2 w i l l be CHAPTER 5: Discussion 111 Figure 5-4 Absorption of SRS IR in a isothermal density ramp. This was calculated for 10.6/xm light incident on a plasma which has a density ramp from 0 to 0.25ncr in 100/^ m and has an average Z—4. Light generated at a given density has a frequency UJS = U0 — Up. enhanced over the IR light at shorter wavelengths. The amount of electrostatic en-ergy in TPD waves of wavevector 4k0 with a fluctuation level of 10% and which are contained in a volume r0 in radius and 20/jm long is comparable with the estimate of the energy in the SRS EPW i.e. O(jxJ). This is the maximum amount of energy available for conversion from the electrostatic mode to electromagnetic mode. The parameters quoted for TPD were measured in a previous experiment 9 1 performed on this plasma. CHAPTER 5: Discussion 112 5.6 Saturation and Competition SRS was shown to grow exponentially, but eventually the plasma wave must saturate. There are many saturation mechanisms which will be discussed in the following. In an experiment, any combination of the mechanisms could be present and it may not be possible to separate their effects. One possible saturation mechanism is profile modification. Previous work at this university showed some indications of this mechanism, based upon an interfero-metric study of the plasma. Since the profile is modified by the pondermotive forces setup by the beating of the EPW generated by the TPD instability, an independent check of this mechanism could be performed. Wide angle Thomson scattering was set up to scatter of SRS epw, as well as those generated by the TPD instability. In figure 5-5, a summary of this experiment and the previous work is given. The range of observation is shown as a dashed line while the range where waves were actu-ally seen is plotted as a solid line. The contour for zero growth i.e. the threshold curve is shown as a dotted line. This line was calculated from the growth rate of Langdon (Eqn.2-12) for 1= 3.3 x 10 1 3Wcm~ 2 with Landau damping (Eqn.2-14) at &£T=700eV (from Ref. 91) and the inhomogeneous damping of Simon (Eqn.(51) of Ref. 24) for L=2G>m. (Simon's damping reduces to u0/(8kyL).) Thomson scattering at angles greater than 9° was never observed in over 100 shots. This indicates an absence of large TPD wavevectors in the present experi-ment. Threshold considerations do not seem to be an adequate explanation as there is a large region where the growth rate should certainly be positive yet no EPW were observed in this region. Of course, this assumes that all the contributions to the growth and decay of the TPD were included in the calculation of the threshold contour. When the previous work is compared with this one, an interesting obser-vation can be made: both cutoffs appear at the same kx value of « 5/2k0 . By wave vector matching for the TPD instability a wave with kx2 = —3/2k0 must be present. (The ky of these two waves are equal in size and opposite in direction.) CHAPTER 5: Discussion 1 1 3 F i g u r e 5-5 S u m m a r y o f w i d e a n g l e T h o m s o n s c a t t e r i n g . T h e u p p e r s o l i d - d a s h e d c u r v e is f r o m Ref.91; t h e lower s o l i d - d a s h e d c u r v e is f r o m t h e present work. T h e d o t - d a s h c u r v e represents t h e c o n t o u r of m a x i m u m g r o w t h f o r S R S - T P D i n a ho-m o geneous p l a s m a i n t h e absence of d a m p i n g (Eqn.2-13). T h e t h r e s h o l d f o r T P D is g i v e n as a d o t t e d l i n e . I n t h e lower s o l i d c u r v e t h e wavevectors of T P D a n d S R S h a v e n o t b e s e p a r a t e d . T h e u p p e r s o l i d c u r v e is e n t i r e l y T P D wavevectors. I o n a c o u s t i c ( I A ) waves of 4k0 p a r a l l e l t o k0 have been o b s e r v e d a n d a s s i g n e d t o t h e decay of t h e T P D i n s t a b i l i t y . T h i s is s h o w n i n f i g u r e 5 -6 w h i c h is r e p r o d u c e d f r o m B e r n a r d 4 7. S i n c e t h e difference between t h e t w o E P W w a v e v e c t o r s is a l s o 4k0, i t is b e l i e v e d t h a t I A waves are c l o s e l y c o n n e c t e d w i t h t h e s a t u r a t i o n of S R S as w e l l as T P D . T h e p r o p o s a l here is t h a t t h e E P W g e n e r a t e d at q u a r t e r c r i t i c a l d e n s i t y s t a r t at k — k0 a n d p r o c e e d t o longer w a v e v e c t o r s ( F i g u r e 4 -6 shows t h e s t a r t o f s u c h b e h a v i o u r ) b u t s t o p w h e n k\x — k2z « 4k0 a n d u>ta = Wi — u2 at w h i c h p o i n t , CHAPTER 5: Discussion 114 t h e E P W of S R S decay i n t o a n I A wave a n d a n E P W t r a v e l l i n g i n t h e o p p o s i t e d i r e c t i o n . T h i s p r e v e n t s f u r t h e r g r o w t h . T h e wave v e c t o r m a t c h i n g f o r t h i s process is s h o w n i n f i g u r e 5-6(b). T h e i m p o r t a n c e of i o n a c o u s t i c waves o n S R S is a l s o seen i n figure 4-13 w here l a r g e a m p l i t u d e I A waves were seen at the same t i m e as t h e S R S waves b u t once t h e I A waves became w e l l e s t a b l i s h e d , S R S was quenched. It t h u s seems t h a t t h e i o n a c o u s t i c waves are r e s p o n s i b l e f o r t h e s a t u r a t i o n a n d q u e n c h i n g of S R S . T h e e x a c t c o n n e c t i o n b e t w e e n t h e i o n a c o u s t i c waves a n d t h e s a t u r a t i o n of S R S is now discussed. T h e r e are at least two p o s s i b l e m e c h a n i s m s . R o z m u s 7 7 showed t h a t t h e presence of large a m p l i t u d e I A waves causes a f r e q u e n c y s h i f t i n t h e E P W where t h e r e s u l t i n g f r e q u e n c y m i s m a t c h c a n d r i v e S R S off reso-n a n c e (i.e. u>epw ^ u)0 — u)s ). S R S c a n still o c c u r a t off resonance c o n d i t i o n s b u t t h e g r o w t h r a t e is m u c h red u c e d . F a r f r o m resonance, t h e s t i m u l a t e d g r o w t h r a t e w i l l d r o p b e l o w t h e t h r e s h o l d r a t e a n d S R S stops. A secon d a p p r o a c h , d i s c u s s e d b y K a r t t u n e n 2 7 , 2 S, has t h e E P W of S R S d e c a y i n g i n t o a n o t h e r E P W a n d a n I A wave. T h e s c a t t e r i n g c o n d i t i o n f o r t h i s is kepvj\ —• kepW2 + k{a . I n a w e l l u n d e r d e n s e p l a s m a a n f o r b a c k s c a t t e r i n g , kepw\ = 2k0 a n d t h e f r e q u e n c y m a t c h i n g c o n d i t i o n f o r t h e p l a s m o n decay w i t h t h e h i g h e s t g r o w t h r a t e r e q u i r e s t h a t kepW2 — —kepwi a n d k{a = 2kepw\. A l t h o u g h l i t t l e e nergy is t r a n s f e r r e d t o t h e i o n waves, t h e s c a t t e r e d (or decay) E P W now has t h e w r o n g d i r e c t i o n t o c o n t i n u e i n t h e p r i m a r y S R S pro-cess. T h e K a r t t u n e n m o d e l p r e d i c t s a s a t u r a t i o n of t h e S R S r e f l e c t i v i t y R a t a val u e o f 7T2 kepw2 /kepwi\* (Ve\ 4 4 kja v ks ) V c ) w h i c h f o r t h e present e x p e r i m e n t is a b o u t 1 0 - 6 . T h e g r o w t h r a t e f o r t h i s decay (Eqn.2-22) is a l s o l a r g e e n o u g h f o r t h e decay t o o c c u r o n luIV2?iSftpscales. S i m u l a t i o n s have also s h o w n t h a t t h e presence of S B S , a source of s t r o n g I A waves, leads t o a r e d u c t i o n of S R S . A s i m p l e s i m u l a t i o n , p e r f o r m e d by B a r r 8 9 w h o CHAPTER 5: Discussion 115 t 1 1 Fiducial additional 200x attenuation F i g u r e 5-6 Ion acoustic waves from T P D saturation, (a) The wavevector record at the ion acoustic frequency, (b) The wavevector matching process involved. CHAPTER 5: Discussion 116 s u p e r i m p o s e d a s i n u s o i d a l i o n d e n s i t y r i p p l e o n a p l a s m a , s h o w e d t h a t a fluctuation l e v e l o f 1 5 % is s u f f i c i e n t t o q u e n c h c o m p l e t e l y S R S . S i n c e I A waves of t h i s l e v e l a r e e v e n t u a l l y p r e s e n t 4 7 , i t is r e a s o n a b l e t o e x p e c t S B S t o s a t u r a t e S R S . B a r r was a b l e t o s how t h a t t h e g r o w t h r a t e o f S R S i n t h e presence of a n i o n d e n s i t y r i p p l e o f d e p t h e — 6n/n0 reduces t o w h e r e q = e/(6Ar 2A 2) . H i s s i m u l a t i o n s also s howed t h a t t h e s c a t t e r e d S R S l i g h t w o u l d b e m o d u l a t e d i n f r e q u e n c y by t h e f r e q u e n c y o f t h e i o n a c o u s t i c waves. I n t h i s e x p e r i m e n t , m o d u l a t i o n of t h e s c a t t e r e d l i g h t a t 2 A 0 was observed. However, t h e i n f l u e n c e of i o n a c o u s t i c waves was n o t r e s p o n s i b l e f o r t h e o b s e r v e d m o d u l a t i o n . T h i s was s h o w n s e c t i o n 5.5. O t h e r s i m u l a t i o n s 2, i n w h i c h i o n s are n o t fixed, show r e d u c e d level s of S R S w h e n c o m p a r e d t o i d e n t i c a l s i m u l a t i o n s i n w h i c h t h e ions were f i x e d . F i n a l l y , y e t o t h e r e x p e r i m e n t s 9 0' 6 S show t h a t l a r g e a m p l i t u d e i o n a c o u s t i c waves c a n s a t u r a t e t h e epw. T h e s a t u r a t i o n m e c h a n i s m w h i c h is m o st l i k e l y f o r S R S i n t h e p r e s e n t case is t h e effect o f i o n a c o u s t i c waves b u t t h e r e are o t h e r m e c h a n i s m s w h i c h m u s t be d i s c u s s e d . S i n c e h i g h e n e r g y e l e c t r o n s are o b s e r v e d , t h e m e c h a n i s m s o f e l e c t r o n t r a p p i n g a n d e n h a n c e d L a n d a u d a m p i n g due t o t h e p o p u l a t i o n o f f a s t e l e c t r o n s s h o u l d be m e n t i o n e d . E l e c t r o n t r a p p i n g 8 2 c a n s a t u r a t e S R S because i t a c t s as a n ef f e c t i v e d a m p i n g o n t h e l a r g e a m p l i t u d e E P W . T h e energy of t h e waves is t r a n s f e r e d t o t h e k i n e t i c e n e r g y of t h e t r a p p e d e l e c t r o n s a n d d a m p i n g of a wave is d e f i n e d as a r e m o v a l of e n e r g y f r o m t h e wave. T h i s m e c h a n i s m is n o t i m p o r t a n t i n t h e present case as t h e m a x i m u m t r a p p i n g lev e l s (6n/n = 1 % , c o r r e s p o n d i n g t o a t r a p p i n g p o t e n t i a l o f I k e V ) , a r e t o o s m a l l t o c a p t u r e s i g n i f i c a n t n u m b e r s of el e c t r o n s . T h e ob s e r v e d n u m b e r of f a s t e l e c t r o n s is p u z z l i n g s i n c e t h e r e are v e r y few e l e c t r o n s i n the t a i l CHAPTER 5: Discussion 117 of a Maxwellian distribution which could be trapped to produce the hot electrons. A non-equilibrium population of velocities may be established because the slower, larger amplitude EPW from the TPD instability may move more electrons into the velocity regime where the EPW from SRS can trap and further accelerate them. This is, however, highly speculative. A significant population of high energy electrons, perhaps a result of the electron trapping in the TPD plasmons, can saturate SRS through a different route 2 . There is an additional damping of SRS if a hot electron tail is present. In the absence of hot electrons, an exact solution of the nonrelativistic dispersion relation (in the absence of a laser field ) shows that over the range k0 to 3k0 the frequency of the EPW is up (within 3%) while the damping is much less than 10~6wp. If a 1% population of fast electrons at lOOkeV is added, the situation changes. The frequency remains near up but the damping grows to levels between 6 x 10_3u;p at k0 to 4 x 10 _ 4 w p at 3fc0. (The approximation, Eqn.2-26, give similar results in this range.) The reason for this large effect is quite simple. Landau damping is resonant phenomena between waves at vph and particles at a mean speed ve which becomes large when vpn « ve. The added lOOkeV electron's velocity is nearly resonant with that of the EPW generated at 0.25ncr and even a small population can affect the damping greatly. Since up « w 0/2, the excess damping is approximately 3 x 10 _ 3u; 0. This rate is of the same order of magnitude as the observed growth rates. This damping may be partially responsible for the reduced growth, but is not large enough to quench SRS. When this damping and the growth are comparable, it is necessary to include the effect of the laser field. This would entail solving the dispersion relation of Drake 9 generalized to a 2 temperature electron distribution. The fraction of hot electrons used above is based upon the total number of fast electrons observed. This fraction is time integrated. However, the number of electrons associated with the TPD instability grows exponentially with the duration CHAPTER 5: Discussion 118 of the instability (up to 1 ns) 9 1 . Therefore, the number of hot electrons actually present during the short duration of SRS will be much less than 1% and the damping due to them will be much lower than calculated. Hot electron damping should not saturate SRS in this experiment. 5.7 Connection between SRS and high energy electrons We have seen that SRS grows at various densities and discussed the mech-anism which saturates and quenches the instability. In the section on saturation, electron trapping was dismissed as a saturation mechanism. This is not to be in-terpreted as a statement that electron trapping is unimportant. Indeed, electron trapping is responsible for the generation of high energy electrons. Both the number and energy distribution are well correlated with the electron plasma waves which produce the hot electrons. The first item to be discussed is how the number of electrons are related to the level of the density fluctuations. In chapter 2, a simple trapping model was presented which suggested there would be an exponential relationship between the electron number and the fluctuation level (Eqn.2-29). For the data in figure 4-15, the predicted slope (at 0.24 critical density) is 423. The slope found experimentally is 400 which is surprisingly good agreement. It should be pointed out that the pre-dicted slope is quite sensitive to the density chosen. The slope ranges from 570 at 0.25 critical density to 230 at 0.2 critical density. The same trapping model can be applied to figure 4-16, provided the reasonable assumption that the reflectivity R is proportional to the square of the fluctuation level is made. This assumption pre-sumes that the scattering level is small (see chapter 2.5). A conveniant comparison to figure 4-15 is allowed if the plot against y/R. The TPD decay instability can also trap and accelerate electrons by the same mechanisms as SRS. However, if the trapping model is applied, the predicted in-crease is much slower than for SRS. This is because the predicted slope is pro-CHAPTER 5: Discussion 119 p o r t i o n a l t o k~p2w. S i n c e t h e p l a s m a waves due t o th e T P D are 3k0 or l o n g e r 9 1 , at 0.25n c r, t h e s l o p e w i l l a t least 9 t i m e s s m a l l e r . T h i s does n o t agree w i t h t h e o b s e r v e d slope. T h e r e is a l t e r n a t e r e l a t i o n s h i p between t h e n u m b e r of h o t e l e c t r o n s o b s e r v e d a n d t h e r e f l e c t i v i t y w h i c h is a r e s u l t o f t h e M a n l e y - R o w e r e l a t i o n s . T h e s e r e l a t i o n s p r e d i c t t h a t t h e energy i n t h e E P W is p r o p o r t i o n a l t o t h e energy s c a t t e r e d . If a c o n s t a n t f r a c t i o n o f t h e energy i n t h e E P W is t r a n s f e r e d t o fast e l e c t r o n s , t h e n t h e e n e r g y c o n t a i n e d i n t h e h o t e l e c t r o n s w i l l be p r o p o r t i o n a l t o t h e energy s c a t t e r e d . F u r t h e r , i f t he energy d i s t r i b u t i o n of t h e h o t e l e c t r o n s is c o n s t a n t , t h e n t h e n u m b e r of h o t e l e c t r o n s at a g i v e n energy, N, w i l l be p r o p o r t i o n a l t o t h e r e f l e c t i v i t y , R. O t h e r e x p e r i m e n t s 8 4> 8 6> 6 6 have o b s e r v e d s u c h a l i n e a r r e l a t i o n s h i p over m o r e t h a n 3 o r d e r s of m a g n i t u d e of v a r i a t i o n i n R. O u r i n s t r u m e n t s do not have t h i s d y n a m i c r a n g e a n d t h e r e f o r e , i t c a n n o t be s t a t e d w h e t h e r t h i s l i m i t has been r e a c h e d . S i m u l a t i o n s have s h o w n t h a t t h e h o t e l e c t r o n energy d i s t r i b u t i o n is approx-i m a t e l y M a x w e l l i a n w i t h a t e m p e r a t u r e k # T = 0.5mev2h. I n t h i s e x p e r i m e n t , t h e e l e c t r o n p l a s m a waves a n d t h e e l e c t r o n energy d i s t r i b u t i o n were o b s e r v e d s i m u l a -t a n e o u s l y . I n figure 4-18, t h e r e s u l t s s h o w n i n d i c a t e t h a t t h e s i m u l a t i o n s give a re a s o n a b l e p r e d i c t i o n o f k # T . T h e c o m p a r i s o n of t h e e l e c t r o n energy s p e c t r u m t o the w a v e v e c t o r s p e c t r u m has n e v e r been done before b u t o t h e r e x p e r i m e n t s 7 3> 7 4> 8 8 t h e c o n n e c t i o n . I n some of these e x p e r i m e n t s 8 8, th e m a x i m u m d e n s i t y o f t h e p l a s m a c o u l d be c o n t r o l l e d a n d t h e e l e c t r o n s p e c t r a (or e q u i v a l e n t x-ray b r e m s s t r a h l u n g s p e c t r u m ) were observed. H i g h e r k ^ T were o b t a i n e d as t h e p l a s m a d e n s i t y a p p r o a c h e d q u a r t e r c r i t i c a l d ensity. I n d i f f e r e n t e x p e r i m e n t s 7 S > 7 4 , t h e s c a t t e r e d E M r a d i a t i o n was used t o i n f e r a k # T w h i c h was c o m p a r e d w i t h t h e k # T fitted t o t h e e l e c t r o n s p e c t r a o b s e r v e d at t h e same t i m e . T h e same b e h a v i o u r was seen. I n s u m m a r y , t h e s i m u l a t i o n p r e d i c t i o n s of k f l T have been c o n f i r m e d i n m a n y ways - t h r o u g h t h e v a r i a t i o n o f t h e p l a s m a CHAPTER 5: Discussion 120 d e n s i t y , t h r o u g h t h e s t u d y of t h e s c a t t e r e d E M r a d i a t i o n a n d , f r o m t h e present w o r k , t h r o u g h t h e s t u d y of t h e epw. CHAPTER 6: Summary and Conclusions 121 CHAPTER 6 Summary and Conclusions A s u m m a r y of t h e S R S p r o c e s s is g i v e n , as I b e l i e v e it t o be. T h e o r i g i n a l c o n t r i b u t i o n s a r e e x p l i c i t l y o u t l i n e d i n t h e second s e c t i o n . T h e t h e s i s ends w i t h s u g g e s t i o n s f o r f u r t h e r work, based u p o n q u e s t i o n s w h i c h arose i n t h e course of t h i s w ork. 6.1 T h e S R S P r o c e s s S t i m u l a t e d R a m a n s c a t t e r i n g is a process i n w h i c h a n e l e c t r o m a g n e t i c wave, i n c i d e n t u p o n a p l a s m a , decays i n t o a s c a t t e r e d E M wave a n d a n e l e c t r o n p l a s m a wave. T h e i n t e r a c t i o n i n v o l v e s a feedback w h i c h enhances the s c a t t e r i n g . S R S s t a r t s w h e n t h e i n c i d e n t E M i n t e n s i t y is near t h e t h r e s h o l d value. E s t i m a t e s of t h i s i n t e n s i t y c a n v a r y by as m u c h as a n o r d e r of m a g n i t u d e . A f t e r t h e t h r e s h o l d i n t e n s i t y is e x c e e d e d , t h e i n s t a b i l i t y grows e x p o n e n t i a l l y . T h e c o n v e c t i v e o r a b s o l u t e n a t u r e of t h e g r o w t h is d e t e r m i n e d b y t h e p l a s m a siz e a n d t h e i n c i d e n t i n t e n s i t y . If t h e p l a s m a is s u f f i c i e n t l y l o n g , t h e g r o w t h is a b s o l u t e a n d e x p o n e n t i a l g r o w t h i n b o t h space a n d t i m e c a n be e x p e c t e d . A s t h e waves grow i n a m p l i t u d e , o t h e r effects s a t u r a t e t h e i n s t a b i l i t y . T h e i n f l u e n c e of i o n waves is the p r i m a r y cause of s a t u r a t i o n . P u m p d e p l e t i o n , e l e c t r o n t r a p p i n g a n d e n h a n c e d L a n d a u d a m p i n g due t o h o t e l e c t r o n s a r e s e c o n d a r y processes. P r o f i l e m o d i f i c a t i o n , w h i c h h a p p e n s o n CHAPTER 6: Summary and Conclusions 122 a l o n g e r t i m e s c a l e , is r e s p o n s i b l e f o r t h e q u e n c h i n g i.e. t h e d i s a p p e a r a n c e of the i n s t a b i l i t y . A n e s s e n t i a l c o m p o n e n t of S R S is t h e s c a t t e r e d E M wave. I t is n o t e s s e n t i a l t h a t t h i s wave escape t h e p l a s m a ; i t need o n l y be present over t h e i n t e r a c t i o n r e g i o n . A s t h i s s c a t t e r e d wave p r o p a g a t e s t o the p l a s m a b o u n d a r i e s , i t is a b s o r b e d a n d m a y n o t escape f r o m t h e p l a s m a . T h i s gives r i s e t o t h e g a p o b s e r v e d i n t h e s p e c t r u m o f t h e s c a t t e r e d I R l i g h t . T h e weak s i g n a l a t 2 A 0 is due t o t h e t w o - p l a s m o n decay i n s t a b i l i t y . L i n e a r m o d e c o n v e r s i o n of t h e T P D E P W c o u l d generate a l a r g e a m o u n t of l i g h t a n d a s m a l l f r a c t i o n o f t h i s l i g h t m i g h t escape t h e p l a s m a . T h e a l t e r n a t e e x p l a n a t i o n f o r t h e r e s u l t s , t h e e n h a n c e d T h o m s o n s c a t t e r i n g t h e o r y of S i m o n et a l . , c o u l d p o s s i b l y e x p l a i n t h e r e s u l t s . It is f e l t , however, t h a t t h i s is n o t t h e case s i n c e t h e s t a n d a r d t h e o r y of S R S ( f r o m c h a p t e r 2) e x p l a i n e d a l l t h e o b s e r v e d d a t a ( t h r e s h o l d , g r o w t h ra t e s , f r e q u e n c y a n d w a v e v e c t o r of t h e p l a s m a waves o b s e r v e d , etc.) e x c e p t f o r t h e o b s e r v e d la c k o f I R l i g h t . T h i s e x p l a n a t i o n f o r t h e la c k o f I R l i g h t is u n k n o w n . Simon's theory is t h e first one t o e x p l a i n t h e gap. A s s o c i a t e d w i t h S R S is t h e g e n e r a t i o n of h i g h energy e l e c t r o n s . T h e s e elec-t r o n s are a r e s u l t o f t r a p p i n g a n d w a v e b r e a k i n g of t h e epw. T h e en e r g y d i s t r i b u t i o n o f these e l e c t r o n s is c o n s i s t e n t w i t h t h e s i m u l a t i o n s w h i c h suggest a M a x w e l l i a n d i s -t r i b u t i o n w i t h kf,T = 0.5mv2h. 6.2 Original Contributions S e v e r a l o f t h e s u m m a r y s t a t e m e n t s are b a s e d u p o n t h e w o r k p e r f o r m e d b y t h e a u t h o r . T h e me a s u r e m e n t s of t h e s p a t i a l a n d t e m p o r a l g r o w t h o f S R S E P W were first m a d e b y h i m . ( P h y s . Rev. L e t t . 57,337 (1986)) H e was al s o t h e first t o observe e l e c t r o n p l a s m a waves at d e n s i t i e s i n t h e range 0.15 — 0.25n c r. T h e s c a t t e r e d E M s i g n a l w i t h a w a v e l e n g t h o f 2 A 0 has never been o b s e r v e d p r e v i o u s l y i n a CO2 laser e x p e r i m e n t . T h e c o r r e l a t i o n o f t h e E P W fluctuation levels w i t h t h e n u m b e r of fast e l e c t r o n s has never been m a d e befo r e e i t h e r . T h e a u t h o r was also CHAPTER 6: Summary and Conclusions 123 the first investigator to show the simulations of the hot electron production were valid. This was done by measuring the EPW spectra and comparing them with the simultaneous electron velocity distributions (Phys. Fluids 29,3451 (1986)). 6.3 S u g g e s t i o n s f o r F u r t h e r W o r k In the course of this work, many questions concerning SRS were answered. Some questions were also raised. A spectrally and temporally integrated IR signal near 15/um was observed. The Thomson scattering geometry did not permit exam-ination of the EPW that should be associated with this IR. 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