UBC Theses and Dissertations

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UBC Theses and Dissertations

Generation of fast electrons in a CO₂ laser plasma interaction McIntosh, Grant W. J. 1983

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G e n e r a t i o n of F a s t E l e c t r o n s i n a C 0 2 L a s e r Plasma I n t e r a c t i o n by Grant W.J. M c i n t o s h B.Sc.(Hons.) U n i v e r s i t y of Manitoba 1981 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS PLASMA PHYSICS GROUP 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 tl>e> r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA December ] SS3 (g) 1983 Grant W.J. M c i n t o s h In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head of my department o r by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of *PAys/cr  The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date DE-6 (3/81) ABSTRACT F a s t e l e c t r o n s have been o b s e r v e d i n the +kL d i r e c t i o n f o r a C0 2 l a s e r plasma i n t e r a c t i o n . The t h r e s h o l d f o r f a s t e l e c t r o n p r o d u c t i o n was found t o be 1 0 1 3 Wcnr 2 . The number of e l e c t r o n s peaks and d e c r e a s e s f o r i n t e n s i t i e s g r e a t e r than 6 1 0 1 3 W c m - 2 . Some p o s s i b l e t h e o r i e s a r e suggested f o r t h i s b e h a v i o r . When a M a x w e l l i a n f i t t o the e l e c t r o n energy d i s t r i b u t i o n was used , a temperature of 121 keV was o b t a i n e d . The t h r e s h o l d and temperature a r e c o n s i s t e n t w i t h g e n e r a t i o n by S t i m u l a t e d Raman S c a t t e r i n g . The number of f a s t e l e c t r o n s i s a l s o shown t o i n c r e a s e d r a m a t i c a l l y as the amount of plasma near .25 n ^ i s i n c r e a s e d . A computer program was a l s o d e v e l o p e d f o r the i n t e r p r e t a t i o n of i n t e r f e r o g r a m s . i i i TABLE OF CONTENTS TITLE PAGE i ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES v LIST OF FIGURES v i ACKNOWLEDGEMENTS v i i CHAPTER 1 INTRODUCTION 1 CHAPTER 2 THEORY 3 P a r a m e t r i c P r o c e s s e s 3 Other Mechanisms 11 CHAPTER 3 EXPERIMENTAL SETUP 13 COt L a s e r System 13 Gas J e t Tar g e t 17 S p e c t r o m e t e r s 21 CHAPTER 4 EXPERIMENTAL RESULTS 31 P r e s s u r e Dependence 31 I n t e n s i t y Dependence 31 D i s t r i b u t i o n F u n c t i o n 38 CHAPTER 5 INTERFEROMETRY 41 G e n e r a l R e s u l t s 42 L a r g e S c a l e R e s u l t s 48 F i n e S c a l e R e s u l t s 51 CHAPTER 6 DISCUSSION AND CONCLUSIONS 56 P r e s s u r e Dependence 56 I n t e n s i t y Dependence 56 Energy D i s t r i b u t i o n F u n c t i o n 64 APPENDIX 1 INTERFEROGRAMS ANALYSIS 66 A P P E N D I X 2 R A M A N S C A T T E R I N G C A L C U L A T I O N S R E F E R E N C E S V LIST OF TABLES I S t i m u l a t e d Raman S c a t t e r i n g .Formulas 5 II Two Plasmon Decay .Formulas 10 LIST OF FIGURES I I - 1 Phase Energies for Stimulated Raman Electrons 6 111 -1 UBC C0 2 Laser System 15 I I I - 2 UBC Gas Jet Target and Mach Lines 19 III-3 MKII Electron Spectrometer 23 III-4 Calculated Electron Energy Range Relations 24 III-5 Calculated Magnetic Fields in Spectrometers 27 I I I - 6 MKIII Electron Spectrometer 29 IV- 1 Pressure Dependence in +ku Direction 33 IV-2 Pressure Dependence in -kL Direction 34 IV-3 Intensity Dependence of the Electron Signal (4 Torr).35 IV-4 Intensity Dependence of the Electron Signal (5 Torr).36 IV-5 Fraction of Shots where Electrons not seen 39 IV- 6 Electron Energy D i s t r i b u t i o n 41 V- 1 Sequence of Interferograms (2 Torr ).... 43 V-2 Sequence of Interferograms (3.5 Torr,low power) 44 V-3 Sequence of Interferograms (3.5 Torr,high power) 45 V-4 Sequence of Interferograms (5 Torr,low power) 46 V-5 Sequence of Interferograms (5 Torr,high power) 47 V-6 Gross Features of the Plasma : N and Veloci t y 50 V-7 Electron Density D i s t r i b u t i o n 52 V-8 E f f e c t s of Refraction 53 V- 9 Scale Lengths in the Plasma vs. Time 54 VI- 1 Calculated E f f e c t s of Refraction 63 A-1 Abel Inversion Geometry and Eff e c t s of Subroutines.... 68 ACKNOWLEDGEMENTS I would l i k e t o thank my s u p e r v i s o r Jochen Meyer f o r h i s p a t i e n c e and h i s h e l p i n the c o u r s e of my work. I would a l s o l i k e t o thank Dr. B r i a n H i l k o , Roman P o p i l and Hubert Houtman f o r t h e i r numerous comments and s u g g e s t i o n s . A s p e c i a l thanks i s e x t e n t e d t o John B e r n a r d f o r w r i t i n g the p l o t t i n g and ray t r a c i n g r o u t i n e s f o r the i n t e r f e r o g r a m s . F i n a l l y I would l i k e t o thank R i c h a r d K e e l e r and A l Cheuck f o r t h e i r i n v a l u a b l e t e c h n i c a l s u p p o r t . 1 CHAPTER 1 INTRODUCTION The study of e l e c t r o n s produced i n l a s e r plasma i n t e r a c t i o n s i s an i m p o r t a n t f i e l d of r e s e a r c h today. L a s e r f u s i o n r e a c t o r s must be d e s i g n e d so t h a t the so c a l l e d f a s t e l e c t r o n s produced by the i n t e r a c t i o n s do not i n h i b i t the f u s i o n p r o c e s s . T h i s t h e s i s i s a p r e s e n t a t i o n of e x p e r i m e n t a l r e s u l t s on the i n t e r a c t i o n of a C 0 2 l a s e r w i t h a gas j e t t a r g e t . In l a s e r f u s i o n t a r g e t s one wants t o a b l a t e the o u t e r l a y e r s c r e a t i n g an i m p l o s i o n which w i l l compress the 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 p r e s s u r e s a t low t e m p e r a t u r e s . At the s e extreme c o n d i t i o n s f u s i o n r e a c t i o n s can e a s i l y o c c u r . The i d e a l t e m p e r a t u r e would.be near 100 keV ; p r a c t i c a l c o n s i d e r a t i o n s lower the tem p e r a t u r e t o 10 keV. To a c h i e v e t h i s t e mperature the f u e l must s t i l l be h i g h l y compressed. The b e s t c o m p r e s s i o n i s o b t a i n e d . i f the f u e l i s kept c o l d u n t i l t he shock from the a b l a t e d l a y e r reaches i t . I f the f u e l has been warmed a much more p o w e r f u l l a s e r must be used . P o w e r f u l l a s e r s have pushed the f r o n t i e r s of r e s e a r c h away from the c l a s s i c a l r e g i o n s of EM - ma t t e r i n t e r a c t i o n s i n t o the n o n - l i n e a r r e g imes. New phenomena have been p r e d i c t e d and d i s c o v e r e d . Some of the s e phenomena have s e r i o u s i m p l i c a t i o n s f o r l a s e r f u s i o n schemes. In p a r t i c u l a r two d e t r i m e n t a l e f f e c t s can o c c u r : e l e c t r o n s of v e r y h i g h energy can be g e n e r a t e d and l a s e r l i g h t can be s c a t t e r e d away from the t a r g e t . These e f f e c t s o c c u r i n the underdense b l o w - o f f plasma of the t a r g e t . These h i g h energy e l e c t r o n s ( a l s o c a l l e d s u p r a t h e r m a l , f a s t or hot e l e c t r o n s ) can p r e h e a t the c o l d f u e l l e a d i n g t o reduced 2 c o m p r e s s i o n . The s c a t t e r e d l a s e r l i g h t d o e s n o t i n t e r a c t w i t h t h e a b l a t i n g l a y e r a n d t h e r e f o r e c o n t r i b u t e s n o t h i n g t o t h e a b l a t i o n p r e s s u r e w h i c h i s t o c o m p r e s s t h e f u e l . The o b j e c t i v e o f t h i s t h e s i s i s t o s t u d y t h e s e f a s t e l e c t r o n s i d e n t i f y i n g t h e i r o r i g i n , t h e number g e n e r a t e d , a n d t h e i r e n e r g y d i s t r i b u t i o n . I n c h a p t e r 2 a summary o f t h e t h e o r i e s w h i c h p r e d i c t f a s t e l e c t r o n s w i l l be p r e s e n t e d w i t h e m p h a s i s on S t i m u l a t e d Raman S c a t t e r i n g a n d t h e two p l a s m o n d e c a y i n s t a b i l i t y . C h a p t e r 3 w i l l o u t l i n e t h e a p p a r a t u s u s e d t o p r o d u c e a n d d e t e c t f a s t e l e c t r o n s . The r e s u l t s a n d some m i n o r d i s c u s s i o n o f t h e e l e c t r o n s e a r c h w i l l be e x p o u n d e d i n c h a p t e r 4. I n c h a p t e r 5 t h e r e s u l t s o f an a n a l y s i s o f i n t e r f e r o g r a m s w i l l be d i s c u s s e d . The r e s u l t s o f c h a p t e r s 4 a n d 5 w i l l be j o i n t l y d i s c u s s e d a n d t h e c o n c l u s i o n s w i l l be r e v i e w e d i n c h a p t e r 6. 3 CHAPTER 2 THEORY F a s t e l e c t r o n s can be g e n e r a t e d by q u i t e a number of p r o c e s s e s . . For t h i s t h e s i s the p a r a m e t r i c p r o c e s s e s a re t h e most i m p o r t a n t . P a r a m e t r i c P r o c e s s e s P a r a m e t r i c p r o c e s s e s have been p r e d i c t e d f o r a number of y e a r s . Indeed L o r d R a y l e i g h d i d the i n i t i a l c a l c u l a t i o n s i n the l a t e 19th c e n t u r y . There are f o u r t y p e s of p a r a m e t r i c decay i n s t a b i l i t i e s of i n t e r e s t t o plasma p h y s i c i s t s . I n each case , an i n c i d e n t e l e c t r o m a g n e t i c wave (EM wave) decays i n t o two o t h e r waves , s u b j e c t t o the c o n d i t i o n s of c o n s e r v a t i o n of fr e q u e n c y and c o n s e r v a t i o n of wave v e c t o r . The waves must a l s o obey the a p p r o p r i a t e d i s p e r s i o n r e l a t i o n s . In the p a r a m e t r i c i n s t a b i l i t y the decay p r o d u c t s a re an i o n a c o u s t i c wave (IA) and an e l e c t r o n plasma wave (EPW). In 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 (SBS) the decay p r o d u c t s a r e an IA wave and a s c a t t e r e d EM wave. In two plasmon decay the decay p r o d u c t s a r e two EPWs. In S t i m u l a t e d Raman S c a t t e r i n g ( S R S ) the decay p r o d u c t s a r e an EPW and a s c a t t e r e d EM wave. The t h e o r y b e h i n d a l l t h e s e i n s t a b i l i t i e s i s o u t l i n e d i n many papers (Dubois,1974; F o r s l u n d , K i n d e l and Lindmann,1975; J o r n a , l 9 7 4 ) . Only the r e s u l t s w i l l be quoted h e r e . SRS and two plasmon decay a r e the most i m p o r t a n t f o r the work c o n s i d e r e d h e r e . The r e s u l t s f o r the one d i m e n s i o n a l SRS can be gl e a n e d from a number of p a p e r s . There i s a summary of e x p r e s s i o n s f o r the t h r e s h o l d and growth r a t e s f o r homogeneous or inhomogeneous 4 plasmas i n T a b l e I . VQ 2 i s the q u i v e r v e l o c i t y d e f i n e d by e 2 E 0 2 /o 0 2/m 2 and i s found by 6 5 7 X 2 ( M ) K Wcm" 2 ). a£i i s the 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 1,5 10~ 5 / T 3 / 2 . yL i s the Landau damping c o n s t a n t =-.22 i/Trw, *exp( 1 / ( - 2 k 2 X P 2 ) ) / ( k 3 V e 3 ) o) p i s the plasma f r e q u e n c y and e q u a l s /(4ffne 2/m) . i s the fr e q u e n c y of t h e EPW and e q u a l s /(w p 2 + 3 K T k 2/m). L i s the d e n s i t y s c a l e l e n g t h d e f i n e d f o r d e n s i t y n as (1/n d n / d x ) - 1 . The f o r m u l a s a r e quoted from Chen (1974). . Upon s u b s t i t u t i o n of t y p i c a l numbers one f i n d s t h a t the t h r e s h o l d i n t e n s i t y i s near 1 0 1 3 Wcm~ 2 and t h a t the growth c o n s t a n t ( f o r e x p ( 7 t ) ) i s near 1 0 1 2 s~ 1 . From the fr e q u e n c y and wave v e c t o r matching c o n d i t i o n s one can d e r i v e much i n f o r m a t i o n about the phase v e l o c i t i e s and the d e n s i t i e s a t which SRS can o c c u r . The c a l c u l a t i o n s (see appendix) y i e l d t he f o l l o w i n g s a l i e n t f a c t s . (1) SRS can o n l y o c c u r i f d e n s i t i e s l e s s than .25 n are p r e s e n t .(2) The phase v e l o c i t y d rops o f f as the d e n s i t y i s d e c r e a s e d f o r Raman b a c k s c a t t e r . N c f t i s d e n s i t y a t which the plasma f r e q u e n c y e q u a l s the i n c i d e n t l a s e r f r e q u e n c y . The l a s e r l i g h t cannot p e n e t r a t e i n t o r e g i o n s where the d e n s i t y i s g r e a t e r than n ^ . The phase v e l o c i t y i s found t o be V p A /c= / ( l / ( ( n c R /n-1 )r 2)+3«T/(mc 2)) and £ i s the s o l u t i o n t o E 2-2£ + (2a/( 1+3«T/(mc 2) £ 2 ( a 2 - 1 ))-1-3/cT/(mc 2) £ 2 ( a 2 - 1 )) / ( a 2 - 1 ) = 0 where a=/(n c R /n) S i n c e «T/(mc 2) i s much l e s s than 1 the c o l d plasma a p p r o x i m a t i o n can be used : T=0. I f one d e f i n e s a temperature by K T M O T =mVpK 2/2 one can f i n d K T W o T as a f u n c t i o n of d e n s i t y . 5 T a b l e I S t i m u l a t e d Raman S c a t t e r i n g r F o r m u l a s P a p e r T h r e s h o l d I n t e n s i t y Growth C o n s t a n t F o r s l u n d K i n d e l Lindmann Y = k V 0 u p c 2 2 c 2 k 2 w 2 w # w ^ - t o j * Lee Kaw V e 3 / 2 . 5 2 3/2 c 3 / 2 > k ( ) L Y * V 0 u p 2c Jo r n a V g > 1 6 y r u*yK k 2 ( a ) 0 - u p Y = ^  wo w p ) 2 c Chen V2, 2 c 2 >:k^l Y = V 0 ( u a U p ) c Dubo i s > 1 0 9 W cm" 2 Y = E{ju)p 16rr nto 0 ) f T(<Vu^ K r u e r Y 5 1 0 1 7 L U / 3 x 2 / 3 > > F i g u r e 1 1 - 1 7 T h i s has been done and the r e s u l t s a r e p l o t t e d ( F i g . 11 — 1 ). Both f o r w a r d Raman s c a t t e r i n g ( h i g h energy) and backward Raman s c a t t e r i n g (low energy) a r e p l o t t e d . S t r i c t l y s p e a k i n g one s h o u l d c o n s i d e r r e l a t i v i s t i c e n e r g i e s . The e x p r e s s i o n f o r K T i n t h i s case w i l l be /cT H d T =mc2 ((1 +V ? h 2 / c 2 / / d - V f k 2 / c 2 ) ) - 1 ) For VpK a t . 2 5 n C R one f i n d s K T h o T =115 keV. In b o t h f o r w a r d and back s c a t t e r i n g the EPW g e n e r a t e d p r o p a g a t e s i n the d i r e c t i o n of the l a s e r l i g h t wave v e c t o r (+k^). Once the phase v e l o c i t y (or the k of the EPW) i s known one can b e g i n t o e s t i m a t e the number of e l e c t r o n s which may be i n i t i a l l y t r a p p e d by the wave. One assumes t h a t the background plasma can be d e s c r i b e d by a M a x w e l l i a n d i s t r i b u t i o n at temperature K T . The e l e c t r o n s which a r e t r a p p e d have e n e r g i e s between . 5mVpK 2-e<f> and . SmV^ 2+e<p . E 4> was d e t e r m i n e d u s i n g P o i s s o n ' s e q u a t i o n and i s e q u a l t o 4 7 r e S n / k 2 . These e n e r g i e s c o r r e s p o n d t o a range of v e l o c i t i e s and thus upon a s l i g h t change of v a r i a b l e s x 2 =. 5mv2//cT we f i n d t h a t n 7 « « P =n e ( e r f (x^ ) - e r f ( x t ) ) x u=avV(m/UT) )+cjp/k/( (Sn/na )m/UT)) x,. =w/k/(m/UT) -wf / k / ( U n / n n )m/UT) i The a b s o l u t e number of e l e c t r o n s g e n e r a t e d has a l s o been e s t i m a t e d by computer s i m u l a t i o n ( H i o b and B a r n a r d , 1983). For ( V 0 / c ) 2 = . 0 6 2 5 , 3 % of the e l e c t r o n s i n the plasma are c o n v e r t e d 8 E x p o n e n t i a l growth cannot c o n t i n u e i n d e f i n i t e l y . S i n c e i t i s the a m p l i t u d e of the e l e c t r o s t a t i c wave which i s growing one s h o u l d examine the mechanisms which p r e v e n t s f u r t h e r growth. T r a p p i n g of e l e c t r o n s i n l a r g e a m p l i t u d e waves (Ichimaru,1973) has been proposed as a mechanism which p r e v e n t s f u r t h e r growth. In t h i s case f u r t h e r energy i n p u t goes i n t o e l e c t r o n k i n e t i c energy ( f a s t e l e c t r o n s ) and not i n t o the growth of the wave a m p l i t u d e . M o d i f i c a t i o n of the s c a l e l e n g t h s and c o u p l i n g i n t o i o n a c o u s t i c waves i n the plasma has a l s o been proposed as a s a t u r a t i o n mechanism . I f the waves grow t o o l a r g e , pondermotive f o r c e s of the decay EPWs c o u p l e and cause the s c a l e l e n g t h t o s h o r t e n (Langdon and L a s i n s k i , 1 9 7 6 ; E s t a b r o o k and K r u e r , l 9 8 3 ) . T h i s means u n l e s s the l a s e r i n t e n s i t y can i n c r e a s e f a s t enough t o compensate f o r t h i s d e c r e a s e i n s c a l e l e n g t h the t h r e s h o l d c o n d i t i o n w i l l soon be exceeded and the growth s t o p s . The wave w i l l then damp away. The shape of the d i s t r i b u t i o n f u n c t i o n i s a l s o i m p o r t a n t . From the s i m p l e t h e o r y of t r a p p i n g one might expect a d i s t r i b u t i o n c e n t e r e d on the phase v e l o c i t y and d e c r e a s i n g l o c a l l y away from t h i s c e n t e r . T h i s would assume t h a t t h e r e a r e no r a n d o m i z i n g c o l l i s i o n s . From a d i s c u s s i o n of the r e l a x a t i o n t i m e s ( S p i t z e r , 1 9 6 2 ) one can e s t i m a t e the time needed t o t h e r m a l i z e a d i s t r i b u t i o n of e l e c t r o n s . F or T=1 keV and n= .25n C R one f i n d s the time n e c e s s a r y i s about 5 ns. T h i s means the background plasma has time t o t h e r m a l i z e . For T=80 keV the 9 time n e c e s s a r y i s g r e a t e r than 400 ns. T h i s i s much l o n g e r than the l a s e r p u l s e and hence we would not expect a M a x w e l l i a n d i s t r i b u t i o n . I t i s i n t e r e s t i n g t o note ,however, t h a t computer s i m u l a t i o n s p r e d i c t t h a t a M a x w e l l i a n d i s t r i b u t i o n r e s u l t s ( E s t a b r o o k , K r u e r and L a s i n s k i , 1 9 8 0 ) a t a hot temp e r a t u r e e q u a l t o the phase v e l o c i t y energy. The mechanism b e h i n d t h i s i s not u n d e r s t o o d . The time e v o l u t i o n of the e l e c t r o n s i g n a l has been f o l l o w e d i n computer s i m u l a t i o n s (Hiob and B a r n a r d ,1983). The r e s u l t s i n d i c a t e t h a t the f a s t e l e c t r o n s appear r a p i d l y (co t about 50) a f t e r the t h r e s h o l d i n t e n s i t y has been reached i n the p u l s e and remain c o n s t a n t t h e r e a f t e r . The o t h e r p a r a m e t r i c i n s t a b i l i t y of importance i s t h e two plasmon decay. The t h e o r y of t h i s decay has been c o v e r e d by a number of r e s e a r c h e r s (Jackson,1967; L i u and Rosenbluth,1972; Langdon and L a s i n s k i , 1 9 7 8 ; S i m o n , S h o r t , W i l l i a m s and Dewardre). The r e s u l t s of t h e s e and o t h e r s a r e condensed i n t a b l e I I . S u b s t i t u t i o n of numbers y i e l d s t h r e s h o l d s near 1 0 1 2 W cm" 2 and growth r a t e s near 1 0 1 2 s " 1 . The e l e c t r o n plasma waves g e n e r a t e d have a maximum growth r a t e near 45 degrees w i t h r e s p e c t t o the l a s e r beam a x i s i n the p l a n e of p o l a r i z a t i o n . The ks of the two plasmon EPWs a r e u s u a l l y i n i t i a l l y much g r e a t e r than k 0. T h i s would suggest t h a t the mean energy of the g e n e r a t e d f a s t e l e c t r o n s ( K T ) would be much l e s s than .5m(co/k e) 2=85 keV. For k=2k 0 t h i s would imply a K T about 20 keV 10 T a b l e I I Two Plasmon Decay t F o r m u l a s P a p e r T h r e s h o l d I n t e n s i t y Growth C o n s t a n t L i u R o s e n b l u t h V2, 3 V e 2 > k 0 L ys^oVa_ wP 2 k, 2L J a c k s o n E 0 > 16ojp2mY^ ko u p •» -» J» -A y = ek "E 0k *ko 8mk 2 up R o s e n b l u t h I(W c m _ 2 ) > T ( K e V ) 600L(mm) Y ^ C e 2 * ^ (2m 2 c5) J ' 2 Simon et a l . V? 3 K r u e r I(W c m " 2 ) > T ( e V ) I01 3 Duboi s > 1 0 9 W cm" 2 Y = Ef}k - k 0 k • E 0 OJf 32irntcTk 2k jE 0 11 assuming that .5 mVpH 2 = K T . However soon a f t e r s u r p a s s i n g t h r e s h o l d there appears a r a p i d decay i n t o s h o r t e r k waves ( B a l d i s and Walsh,1983). T h i s i m p l i e s that a higher K T a c t u a l l y observed e x p e r i m e n t a l l y i s not unreasonable. The frequency of these waves i s about t j p = c o 0 / 2 . The waves can s c a t t e r the i n c i d e n t l a s e r at a frequency = co„/2 + C J Q =3/2 u>0. T h i s s c a t t e r e d r a d i a t i o n can be d e t e c t e d and i s u s u a l l y c o n s i d e r e d a s i g n that two plasmon decay i s o c c u r r i n g . Other Mechanisms Other mechanisms have been proposed to generate f a s t e l e c t r o n s . These however u s u a l l y p r e d i c t much lower temperatures than observed h e r e i n . For completeness, these methods w i l l be b r i e f l y mentioned. F i l a m e n t a t i o n can generate f a s t e l e c t r o n s i n underdense plasmas. The i n c i d e n t l a s e r can be f u r t h e r focused by the plasma i t c r e a t e s . I t i s t h i s much higher l o c a l i n t e n s i t y which generates the f a s t e l e c t r o n s v i a parametric i n s t a b i l i t i e s (Cohen and Max ,1979; Ng et a l , l 9 7 9 ; Herbst et a l , l 9 8 l ). Both ion a c o u s t i c and e l e c t r o n plasma wave tu r b u l e n c e have been proposed . Turbulence i n t h i s context d e s c r i b e s the randomization of wave v e c t o r s from an i n i t i a l l y n e a r l y ordered s t a t e . EPW t u r b u l e n c e has been p r e d i c t e d to generate a Maxwellian d i s t r i b u t i o n at 20-30 keV ( S i l i n and Tikhonchuk ,1981). Ion a c o u s t i c t u r b u l e n c e has been shown i n computer s i m u l a t i o n s to generate a d i s t r i b u t i o n c h a r a c t e r i s e d by a 12 T H 0 T =1 ,5/( 1-n/n C R ) * T C O L D ( E s t a b r o o k , 1 981). For our c o n d i t i o n s t h i s would imply T„ O T =4 keV. Resonance a b s o r p t i o n i s the p r o c e s s i n which l a s e r l i g h t p e n e t r a t e s , t o the c r i t i c a l d e n s i t y l a y e r of the plasma a t an o b l i q u e a n g l e w i t h an e l e c t r i c f i e l d component p a r a l l e l t o the d e n s i t y g r a d i e n t and d r i v e s a p u r e l y e x p o n e n t i a l l y d e c a y i n g wave i n t o the plasma. There a r e numerous papers which d e s c r i b e resonance a b s o r p t i o n ( E s t a b r o o k and K r u e r , l 9 7 8 ; F o r s l u n d , K i n d e l and Lee,1977; Ko l o d n e r and Y a b l o n o v i t c h , 1 9 7 6 ) . These papers t h e o r e t i c a l l y p r e d i c t or r e p o r t measurements of 10-20 keV M a x w e l l i a n d i s t r i b u t i o n s f o r the f a s t e l e c t r o n s . T h i s i s not an i m p o r t a n t p r o c e s s i n our case s i n c e we do not r e a c h c r i t i c a l d e n s i t y i n our plasma. 13 CHAPTER 3 EXPERIMENTAL SETUP The i n v e s t i g a t i o n of SRS i n the l a s e r plasma i n t e r a c t i o n d e s c r i b e d i n t h i s r e p o r t uses 3 b a s i c components : a C0 2 l a s e r , the gas j e t t a r g e t and the d i a g n o s t i c s . These w i l l now be d i s c u s s e d i n d e t a i l . To study p a r a m e t r i c p r o c e s s e s i n l a s e r plasma i n t e r a c t i o n one must f i r s t f i n d an a p p r o p r i a t e l a s e r . From the t h e o r y p r e v i o u s l y d i s c u s s e d one b a s i c parameter i s the q u i v e r v e l o c i t y of an e l e c t r o n i n the l a s e r ' s e l e c t r i c f i e l d g i v e n by V 0 2 = e 2 E 0 2/(a> 0 2m 2) or p r o p o r t i o n a l t o I X 2 . We a l s o know t h a t t h i s term must be r e l a t i v e l y l a r g e (about 1 0 1 5 jum2 Wcm" 2). A C0 2 l a s e r i s i d e a l s i n c e i t s l o n g w a v e l e n g t h a t 10.6 nm means t h a t the r e q u i r e d i n t e n s i t y i s about 1 0 1 3 W cm" 2 which i s much more e a s i l y reached than the 1 0 1 5 Wcm*2 r e q u i r e d i f one was t o use 1 Mm l i g h t . The C0 2 l a s e r used a t UBC i s c a p a b l e of p r o d u c i n g the i n t e n s i t i e s r e q u i r e d t o stu d y SRS and o t h e r p a r a m e t r i c p r o c e s s e s . The COa, L a s e r System The C 0 2 l a s e r system i s o u t l i n e d on F i g 111 — 1 . I t was s e t up by J . B e r n a r d , J.Cherwoniak, C.J.Walsh, H.Houtman, R . P o p i l and J.Meyer . D i s t a n c e s between m i r r o r s ( i n cm) a r e i n d i c a t e d . The system can be d e s c r i b e d as f o l l o w s . The l o n g i t u d i n a l mode and the t r a n v e r s e mode a r e f i x e d by the CW HP ( c o l l e c t i v e l y known the h y b r i d l a s e r ) s e c t i o n . The HP st a n d s f o r h i g h p r e s s u r e and CW st a n d s f o r c o n t i n u o u s wave l a s e r . T h i s c o m b i n a t i o n i s used t o c r e a t e a s i n g l e mode 100 ns g a i n s w i t c h e d 14 p u l s e g e n e r a t e d when an e l e c t r i c a l d i s c h a r g e i n HP c r e a t e s a p o p u l a t i o n i n v e r s i o n i n t h a t s e c t i o n and thus changes the g a i n . The p o l a r i z a t i o n i s f i x e d by Br e w s t e r windows w i t h i n the c a v i t y and by t h e germanium p o l a r i z e r s GP l o c a t e d o u t s i d e the c a v i t y . The wa v e l e n g t h i s f i x e d by a temperature c o n t r o l l e d germanium e t a l o n used as the output c o u p l e r a t the e x i t of the CW s e c t i o n . The P o c k e l ' s c e l l i s a r r a n g e d so t h a t w i t h no v o l t a g e a p p l i e d t o i t t he p u l s e w i l l pass t h r o u g h the germanium f l a t G and be r e f l e c t e d i n t o the O p t i c a l E n g i n e e r i n g Spectrum A n a l y z e r SA. A 2 ns p u l s e of h i g h v o l t a g e i s a p p l i e d t o the PC . T h i s causes the p l a n e of p o l a r i z a t i o n t o r o t a t e by 90 degrees . S i n c e G i s se t a t the Br e w s t e r a n g l e t h i s new p o l a r i z a t i o n w i l l be r e f l e c t e d o f f G and i n t o t h e r e s t of the system. T h i s 2 ns HV p u l s e i s a p p l i e d near the peak of the g a i n s w i t c h e d p u l s e . T h i s p u l s e a l s o p r o v i d e s a c o n v e n i e n t t i m i n g p u l s e t o be used t o t r i g g e r d i a g n o s t i c s l a t e r i n the system. The 2 ns p u l s e p a s s e s t h r o u g h l e n s L1 which f o c u s e s the beam onto a p o l y e t h y l e n e sheet PS. There i s no breakdown a t the sheet and the p u l s e c o n t i n u e s on t h r o u g h l e n s L2. The l e n s c o m b i n a t i o n L1-L2 has a f o c u s a t F1 where a s p a t i a l f i l t e r i s l o c a t e d . F1 has two purposes : f i r s t t o c l e a n up the beam g o i n g i n t o the K103 p r e a m p l i f i e r and second t o p r o v i d e a breakdown p o i n t f o r t h e b a c k s c a t t e r e d beam from the l a s e r t a r g e t . The p u l s e i s a m p l i f i e d by the K103 p r e a m p l i f i e r . I t passes t h r o u g h L3 which f o c u s e s the p u l s e i n t o a second s p a t i a l f i l t e r F2. F2 a l s o has p r o v i s i o n f o r spark breakdown timed so t h a t the i n c i d e n t p u l s e can pass t h r o u g h unimpeded. The b a c k s c a t t e r e d 15 UBC CO a LASER LUMONICS K103 PREAMP LUMONICS TEA 601A AMP CWI GAS JET F i g u r e 111- 1 1 6 p u l s e w i l l f i n d a dense plasma when i t a t t e m p t s t o pass t h r o u g h t h i s p o i n t and hence w i l l be a t t e n u a t e d by r e f r a c t i o n and a b s o r p t i o n . The p u l s e i s a m p l i f i e d by the t h r e e s t a g e a m p l i f i e r . T h i s i s a l a b - b u i l t 3 module a m p l i f i e r d e s c r i b e d i n UBC Plasma P h y s i c s Report No.79 (Houtman and W a l s h ) . The beam i s a m p l i f i e d a g a i n by a commercial Lumonics a m p l i f i e r module and the a m p l i f i e d beam i s sent t o the gas j e t t a r g e t . The i m p e r f e c t c o n t r a s t r a t i o of the P o c k e l ' s c e l l has been compensated f o r i n the system. A problem a r i s e s because not a l l of the 100 ns p u l s e i s p e r f e c t l y r e j e c t e d when the 2 ns p u l s e i s s w i t c h e d o u t . T h i s p r o v i d e s wings on the l a s e r p u l s e which a r e of low i n t e n s i t y , but c o n t a i n a s u b s t a n t i a l amount of energy. The problem i s known t o be e l i m i n a t e d when no energy i s d e t e c t e d a f t e r the l a s e r i s f i r e d w i t h no 2 ns p u l s e a p p l i e d t o the PC. T h i s i s a c c o m p l i s h e d by f i l l i n g gas c e l l C1 w i t h 4-15 T o r r of S F 6 and 760 T o r r of He i n an approximate d i s t a n c e of 2 cm. He. I t i s a l s o i m p o r t a n t t o p r e v e n t s e l f - l a s i n g i n the l a s e r by making the r e a r m i r r o r of the t h r e e stage i n v i s i b l e t o low i n t e n s i t y C 0 2 r a d i a t i o n . T h i s i s done by f i l l i n g the gas c e l l C2 w i t h 1-1.5 T o r r S F 6 , 20 T o r r e t h a n o l , 100 T o r r Freon and 640 T o r r of He. Both c e l l s work on the p r i n c i p l e t h a t S F 6 i s a s a t u r a b l e a b s o r b e r . T h i s means t h a t low i n t e n s i t y r a d i a t i o n i s absorbed, but h i g h i n t e n s i t i e s cause the SF* t o b l e a c h and p e r m i t the p a s s i n g of the r a d i a t i o n . The p r o p e r t i e s of SF 6 have been measured by o t h e r r e s e a r c h e r s (Burak e t a l , 1 9 6 9 ) . C r o s s s e c t i o n a l q u a l i t y of the beam t o be f o c u s e d onto the gas j e t i s not p e r f e c t . A l t h o u g h the beam i s assumed t o be 17 G a u s s i a n , f o r the c a l c u l a t i o n of the f o c a l spot r a d i u s , e v i d e n c e from t h e t h e r m a l paper i n d i c a t e s t h a t t h e r e a r e a few i r r e g u l a r i t i e s i n the beam. These r e s u l t s must be i n t e r p r e t e d c a r e f u l l y s i n c e the response of the t h e r m a l paper i s d e c i d e l y n o n - l i n e a r . The s t r u c t u r e i s due t o the n o n - u n i f o r m i t y of the d i s c h a r g e i n the a m p l i f i e r s . The s t r u c t u r e however i s not i m p o r t a n t f o r the s e l o n g w a v e l e n g t h s . The measurement of the f o c a l spot s i z e u s i n g a two d i m e n s i o n a l g r a t i n g t e c h n i q u e ( see B e r n a r d Ph.D. T h e s i s 1984) i n d i c a t e s the t y p i c a l f o c a l spot s i z e i s 46 w a i s t r a d i u s . The l a s e r i s thus a b l e t o p r o v i d e a p u l s e of C0 2 r a d i a t i o n a t wavelength 10.6 um w i t h e n e r g i e s i n the range 0-15 J o u l e s and i n a time t y p i c a l l y 2 ns fwhm. Wi t h the f/5 f o c u s i n g l e n s i n t e n s i t i e s a p p r o a c h i n g 1 0 1 0 W cm" 2 may be reached. The parameters d e s c r i b i n g the C0 2 p u l s e were d e t e r m i n e d as f o l l o w s . The wavelength was noted f o r eve r y shot on the spectrum a n a l y z e r ; the i n c i d e n t energy was measured u s i n g a photon drag IR d e t e c t o r ( c a l i b r a t e d e v e r y e x p e r i m e n t a l s e s s i o n ) ; the t r a n s m i t t e d energy was noted u s i n g an A p o l l o energy meter ; and the p u l s e l e n g t h and shape were noted by d i s p l a y i n g the photon drag s i g n a l on a T e k t r o n i x 7104 o s c i l l o s c o p e . W i t h the l a s e r p u l s e we now can proceed t o st u d y l a s e r plasma i n t e r a c t i o n s . The plasma can be formed by a l a s e r p u l s e s t r i k i n g a t a r g e t or i t can be preformed as i n a z p i n c h . The Gas J e t T a r g e t 18 The gas j e t t a r g e t i s based on a d e s i g n from the N a t i o n a l R e s e a r c h C o u n c i l i n Ottawa and the U n i v e r s i t y of A l b e r t a . I t was s e t up a t UBC by R . P o p i l and A.Ng. I t uses a L a v a l n o z z l e t o produce a l a m i n a r j e t of n i t r o g e n gas. The Mach l i n e c a l c u l a t i o n s based upon G i l e s (Ph.D,1983) and S h a p i r o (1953) i n d i c a t e t h a t we s h o u l d have a l a m i n a r j e t of u n i f o r m m o l e c u l a r d e n s i t y ( w i t h i n 10 %) (see F i g 111-2) . A Mach l i n e i s a boundary between d i f f e r e n t f l o w r e g i o n s i n t h e gas j e t . The c a l c u l a t i o n s of the Mach l i n e s t h e m s e l v e s depend o n l y on the i n i t i a l Mach number (at the e x i t ) and the p r e s s u r e r a t i o between the background and the r e s e r v o i r . The t h r o a t w i d t h A* and the e x i t w i d t h A of the gas j e t n o z z l e are i n d i c a t e d i n the d i a gram. For t h e s e d i m e n s i o n s a Mach number M of 4.51 can be r e ached a t the e x i t . T h i s assumes t h a t t h e r e i s i s e n t r o p i c f l o w . T h i s Mach number can be c a l c u l a t e d from the f o r m u l a ( A 2 / A * 2 ) = ( 2 / ( 7 - 1 ) ( 1 + M 2 ( 7 - 1 ) / 2 ) ) * * ( ( 7 + 1 ) / ( 7 ~ 1 ) ) / M 2 (from C h o r l t o n , 1 9 6 7 ) . 7 i s the r a t i o of the s p e c i f i c h eats and i s u s u a l l y t a k e n as 1.4 . The d e n s i t y n a t the e x i t ( r e g i o n 1) can a l s o be c a l c u l a t e d u s i n g a n o t h e r f o r m u l a from C h o r l t o n . n 0 / n = ( 1 + M 2 ( 7 - 1 ) / 2 ) * * 1 / ( 7 - 1 ) ( n o = r e s e r v o i r d e n s i t y ) S u b s t i t u t i n g numbers one f i n d s n/n c R = ( 2 1 ^ )2.67/57.82 f o r STP c o n d i t i o n s i n the r e s e r v o i r . Assuming the n i t r o g e n i s f u l l y i o n i z e d the 5 T o r r j e t s h o u l d have n=1.43 n t R . T h i s i s a f t e r c o n v e r t i n g from STP c o n d i t i o n s t o ambient c o n d i t i o n s i n the r e s e r v o i r . In r e g i o n 2 the Mach number can be c a l c u l a t e d i f one r e a l i z e s t h a t the p r e s s u r e i n t h i s r e g i o n i s the same as the ambient p r e s s u r e P 8 of the background He. Assuming the p r e s s u r e i n r e g i o n 1 i s the same as i n the r e s e r v o i r P R , t h e i s e n t r o p i c UBC GAS JET F i g u r e 111 - 2 20 p r e s s u r e r e l a t i o n P„/P B =(1 + .5M2 ( 7 - 1 ) ) * * ( 7 / ( 7 - 1 ) ) w i l l y i e l d the the Mach number. S u b s t i t u t i n g the p r e s s u r e r a t i o used i n the experiment (P R/P B=314) we f i n d t h a t M=4.56 i e the same Mach number as i n r e g i o n 1. By the n 0/n f o r m u l a above t h i s i m p l i e s t h a t t h e r e i s almost the same d e n s i t y i n r e g i o n s 1 and 2. For d e n s i t i e s f u r t h e r up i n the j e t ( r e g i o n 3 and beyond) the method o u t l i n e d i n S h a p i r o can be f o l l o w e d . The d e n s i t y i n r e g i o n 3 s h o u l d be 1.29 n C R . The d e n s i t i e s c a l c u l a t e d a r e above the measured d e n s i t i e s s i n c e hydrodynamic motion and i n c o m p l e t e i o n i z a t i o n can o c c u r . These w i l l t e n d t o lower the d e n s i t y . The r e l a t i o n s h i p s i n d i c a t e t h a t the m o l e c u l a r d e n s i t y i n the j e t i s l i n e a r l y r e l a t e d t o the d e n s i t y i n the r e s e r v o i r and assuming the i d e a l gas law t o the p r e s s u r e i n the r e s e r v o i r . Thus we have a c o n v e n i e n t way of c o n t r o l l i n g the maximum d e n s i t y the l a s e r can see. S i n c e a l s o the background p r e s s u r e i s l i n e a r l y r e l a t e d t o the r e s e r v o i r p r e s s u r e f o r s t a b l e gas j e t c o n d i t i o n s one can l a b e l the d e n s i t y r e g i o n t o be e x p l o r e d by s p e c i f y i n g the background p r e s s u r e . In our case , t h i s i s v a l i d as l o n g the n o z z l e remains the same. S i n c e .4 n C R i s reached w i t h a 5 T o r r background 3/5 x .4 n C R w i l l be r e ached by a 3 T o r r background j e t i e .25 n C R . T h i s assumes t h a t t h e average Z remains th e same and t h a t the same hydrodynamic motion o c c u r s f o r b o t h j e t s . One t h i n g i n the diagram of the gas j e t not i n d i c a t e d e x p l i c i t l y i s the t h r o a t which c o n n e c t s the r e s e r v o i r t o the n o z z l e . T h i s i s the a c t u a l ' r e s e r v o i r ' f i l l e d by opening the 21 r e s e r v o i r v a l v e . The p r e s s u r e t o which t h i s i s f i l l e d depends upon the amount of gas r e l e a s e d from the r e s e r v o i r . T h i s i n t u r n depends on how l o n g the r e s e r v o i r v a l v e has been opened. S i n c e a manual v a l v e push b u t t o n i s used t h e r e i s some v a r i a t i o n i n t he time t h e v a l v e i s open. T h i s time i s r e p r o d u c i b l e f o r an i n d i v i d u a l but not from i n d i v i d u a l t o i n d i v i d u a l . The e f f e c t m a n i f e s t s i t s e l f d i r e c t l y i n the f i n a l r e s e r v o i r p r e s s u r e :the l o n g e r the v a l v e i s open the lower the r e s e r v o i r p r e s s u r e . The i m p l i c a t i o n i s t h a t the j e t c o n d i t i o n s a r e a f f e c t e d by the l e n g t h of time the v a l v e i s open. T h i s has been ob s e r v e d f o r x r a y e m i s s i o n under d i f f e r e n t c o n d i t i o n s . The b a s i c plasma parameters a r e as f o l l o w s : t e m p e r a t u r e s around 2 keV and 300 eV ( P o p i l Ph.D. T h e s i s 1984) de t e r m i n e d by x r a y a b s o r p t i o n f o i l t e c h n i q u e s ( P o p i l and Meyer,1981), d e n s i t i e s near .25 n C R r e a c h i n g .4 n C R maximum and t y p i c a l s c a l e l e n g t h s of 300 M m . The l a s t two parameters were 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 u s i n g a Jamin i n t e r f e r o m e t e r . The S p e c t r o m e t e r s The t h i r d element r e q u i r e d t o st u d y SRS i s d i a g n o s t i c equipment. In t h i s case two e l e c t r o n s p e c t r o m e t e r s were b u i l t , MK I I a s i n g l e c h a n n e l ( v a r i a b l e energy) and MK I I I a f o u r c h a n n e l system. MK I I i s the s i n g l e c h a n n e l s p e c t r o m e t e r b u i l t t o study the f a s t e l e c t r o n s produced by the l a s e r i n t e r a c t i o n w i t h the gas j e t a t UBC. I t i s a b a s i c a l l y s i m p l e d e v i c e . I t c o n s i s t s of an e l e c t r o m a g n e t ( H e l m h o l t z c o n f i g u r a t i o n ) ,an e n t r a n c e a p e r t u r e 22 and a d e t e c t o r s i t u a t e d a t 90 d e g r e e s . The s i z e and o t h e r d e t a i l s of the c o n s t r u c t i o n a r e shown i n F i g I I I - 3 the t h e o r y of i t s o p e r a t i o n i s q u i t e s i m p l e ( a t l e a s t t o f i r s t o r d e r ) . One assumes a u n i f o r m magnetic f i e l d w i t h i n the r a d i u s of the c o i l s and z e r o magnetic f i e l d e x t e r i o r t o the c o i l s . T h i s i s a good a p p r o x i m a t i o n ( L i v i n g o o d ) p r o v i d e d one uses an e f f e c t i v e r a d i u s . For the H e l m h o l t z c o i l the e f f e c t i v e r a d i u s i s near the p h y s i c a l r a d i u s . The c a l c u l a t e d f i e l d i n the median p l a n e and the measured f i e l d i n the same p l a n e a r e shown i n F i g I I I - 5 a . The c a l c u l a t e d f i e l d was found by merely i n t e g r a t i n g the the e x p r e s s i o n and the f o l l o w i n g evB=vp/r we can s o l v e f o r p=eBr and f i n d the r e l a t i v i s t i c k i n e t i c energy where 1= c u r r e n t i n magnet e=charge on e l e c t r o n m= mass of the e l e c t r o n r = r a d i u s of t h e o r b i t (same as p h y s i c a l r a d i u s i f e n t r a n c e and e x i t a r e a t 90 d e g r e e s ) . The e x p r e s s i o n f o r B as a f u n c t i o n of I was checked and found t o be as p r e d i c t e d ( w i t h i n 10 %) The f o c u s i n g p r o p e r t i e s of the magnet a r e h a r d t o judge e x p e r i m e n t a l l y . However L i v i n g o o d does g i v e good e x p r e s s i o n s f o r the t h e o r e t i c a l p r o p e r t i e s . T r a j e c t o r i e s f o r e l e c t r o n s of the same energy e n t e r i n g the a p e r t u r e a r e i n d i c a t e d i n the diagram. The energy r e s o l u t i o n i s a p p r o x i m a t e l y +/- dR/R. T h i s B= W i t h the magnetic f i e l d g i v e n by the e x p r e s s i o n B= 8 U 0 N I / ( / 1 2 5 r ) ( a t c e n t e r o f . m i d p l a n e ) E=mc 2(/(1 + (eBr/(mc)) 2 ) ~ 1 ) 23 M K . I I E L E C T R O N S P E C T R O M E T E R C U SHIELD. HELMHOLTZ COILS 50 TURNS .0403 IN. DIA. CU WIRE EA . .64 DIA. BRASS HOUSING RUBBER RING 5 MICRON AL NE 102 SCINTILLATOR (.25 THICK; F IBER OPTIC CABLE (.20* ID) PMT IRCA31034 -1846 V •+TO S C O P E DIMENSIONS IN C M UNLESS INDICATED OTHERWISE F i g u r e I I I - 3 E L E C T R O N E N E R G Y R A N G E P L O T o -r-l " LO ~ C O -o — LO — co — o — COLO -cr: -S = m — UJ (J^CO -o — LO — ro — o — in -¥ CD 1—1 10 1 1 1 3 3 1 i 1 ni i | i | i 111| 5 7 30« 3 5 1 10s ENERGY (EVJ i i i i 11111 3 5 710 s F i g u r e 111 - 4 25 corresponds to +/- 15%. The s o l i d angle subtended i s fixed by the exit aperture area A and the distance D from the gas jet to the exit aperture. The s o l i d angle i s A/D2. With A=.32 cm2 and D=9.74 cm the s o l i d angle i s 3.37 10" 3 sr. The detector used was a simple f o i l - s c i n t i l l a t o r - o p t i c a l fiber-photo m u l t i p l i e r combination . A 5 Mm aluminum f o i l (obtained from Goodfellow Metals, Cambridge,England) was placed in front of NE 102 p l a s t i c s c i n t i l l a t o r and the s c i n t i l l a t o r l i g h t was guided by a 60. cm o p t i c a l fib e r to an RCA 31034 photomultiplier. The signal was sent to a Faraday cage screened room and displayed on an oscilloscope. More elaborate spectrometers based upon a Thomson parabola were considered, but had to be rejected since, for the energies and dimensions required, the e l e c t r i c f i e l d needed was large enough to cause glow discharges in the 5 Torr background helium. The 5 Mm f o i l does present a problem as not a l l the electrons w i l l penetrate i t . Calculations based on the integration of the energy loss formula (Segre,l977) dE/dx = 27re',nln(E/Ion)/E (Ion = 9. 1 Z/( 1 + 1 .9Z**-.667) (eV) ) indicate that for 50 keV electrons most of the electrons make i t through the f o i l with less than 10 % energy loss (see Fig I I I -4 ) . The f o i l has two eff e c t s on the electrons. It attenuates the the number of electrons passing through i t by an amount exp(t/r(E)) . It also reduces the mean energy of the electrons passing through the f o i l . A further complication i s that xrays are generated as an electron loses energy in the f o i l ; these can subsequently be detected by the s c i n t i l l a t o r . Any losses due to 26 x r a y e m i s s i o n a r e n e g l e c t e d i n the a n a l y s i s . The response of the NE102 s c i n t i l l a t o r has been measured f o r low energy ( l e s s than 10 keV) e l e c t r o n s and was found t o be l i n e a r (von.Schmeling,1960). The response t o low energy x r a y s has a l s o been measured and was a l s o l i n e a r w i t h energy ( M e y e r o t t , 1 9 6 4 ) . T h i s has a l s o been c o n f i r m e d by measurements f o r t h i s work. I t i s assumed t h a t the response i s l i n e a r t o h i g h e r e n e r g i e s . T h i s assumption i s t r u e p r o v i d e d dE/dx i s s m a l l . The c o r r e c t e x p r e s s i o n i s dS/dx=adE/dx/(1+kb dE/dx) where kb = .01 g c m - 2 Mev-1 and i s a p p r o x i m a t e l y adE/dx i f kb dE/dx much l e s s than 1. S i s the s i g n a l due t o f l u o r e s c e n c e ( P r e s c o t t e t a l , 1 9 6 l ) The o p t i c a l f i b e r i s a good q u a l i t y f i b e r o b t a i n e d from Welch A l l y n , S k a n e a t e l e s F a l l s N.Y . The a t t e n u a t i o n c o e f f i c i e n t measured f o r t h e s c i n t i l l a t i o n l i g h t ( w avelength of maximum out p u t 423 nm) i s .032 cm" 1. T h i s r e s u l t s i n an 85% s i g n a l l o s s i n the f i b e r . Some e f f o r t s were made t o e l i m i n a t e n o i s e from the PMT s i g n a l . These i n c l u d e d u s i n g double s h i e l d e d c o a x i a l c a b l e f o r the s i g n a l and HV l i n e s . The PMT was encased i n a copper m e t a l box and i s o l a t e d from the box by M y l a r p l a s t i c . The box was mounted on a P l e x i g l a s s s t a n d , i s o l a t e d from the j e t h o u s i n g . The HV s u p p l y however was kept o u t s i d e the s c r e e n e d room. In s p i t e of t h e s e e f f o r t s the n o i s e l e v e l was about 100 mV or l e s s . T h i s means t h a t s m a l l s i g n a l s c o u l d not be unambiguously s o r t e d out and a r e t h e r e f o r e not a n a l y z e d . MAGNETIC FIELDS • EXP. THEORY 1. R / R 0 1.5 a. MK II 28 MK I I I i s the 4 c h a n n e l s p e c t r o m e t e r . I t i s an i r o n c o r e e l e c t r o m a g n e t based s p e c t r o m e t e r d e s i g n e d t o produce a l a r g e f i e l d and examine h i g h energy e l e c t r o n s (up t o 300 keV). The f o u r c h a n n e l a s p e c t i s i m p o r t a n t s i n c e t h e r e a r e l a r g e shot t o shot v a r i a t i o n s i n the e l e c t r o n s i g n a l . T h i s makes the a n a l y s i s of the s i n g l e shot s i n g l e c h a n n e l d a t a v e r y d i f f i c u l t and one cannot d e r i v e w i t h an e f f e c t i v e e l e c t r o n d i s t r i b u t i o n . The s i d e view i s shown i n F i g I I I - 6 . T r a j e c t o r i e s of e l e c t r o n s i n an assumed u n i f o r m magnetic f i e l d a r e shown. The l o c a t i o n of the s c i n t i l l a t o r s was chosen so t h a t the b e s t f o c u s was o b t a i n e d . The c a l c u l a t e d magnetic f i e l d a t the midplane f o r o n l y the c o i l s i s g i v e n i n F i g I I I - 5 b . The c e n t e r c o n t o u r l i n e i s a 110 Gauss l i n e and the c o n t o u r i n t e r v a l i s 30 Gauss. The f i e l d was c a l c u l a t e d f o r a 10 Ampere c u r r e n t i n the c o i l s . The a c t u a l f i e l d i s smoother and s l i g h t l y (x1.2) s t r o n g e r than p r e d i c t e d . The i r o n e x t e r i o r c o n s t r u c t i o n was o r i g i n a l l y i n t e n d e d t o p r o v i d e a more u n i f o r m f i e l d . As i n the s i n g l e c h a n n e l s p e c t r o m e t e r the d e t e c t o r c o n s i s t s of a f o i l - s c i n t i l l a t o r c o m b i n a t i o n . The f o i l i s 5 Mm t h i c k and the s c i n t i l l a t o r i s NE102 p l a s t i c s c i n t i l l a t o r . The c h a n n e l s a r e c o n n e c t e d v i a o p t i c a l f i b e r s t o the head of an O p t i c a l M u l t i c h a n n e l Analyzer(OMA) Model 12051 manufactured by P r i n c e t o n A p p l i e d Research C o r p o r a t i o n . The r e l a t i v e response was measured by u s i n g a t t e n u a t e d room l i g h t . The OMA r e q u i r e s some t i m i n g c o n s i d e r a t i o n s as i t reads the c h a n n e l s s e q u e n t i a l l y . To s y n c h r o n i z e the l a s e r l i g h t p u l s e , the gas j e t and the OMA r e a d c y c l e p r o p e r l y a two c h a n n e l d e l a y u n i t was used. The t i m i n g p u l s e from the OMA t r i g g e r e d b oth the 29 MK.III ELECTRON SPECTROMETER 4.0 EXIT FE BACK 5 MICRON AL - NE 102-OPTICAL FIBER - OMA UNITS:CM. FE SIDE PLATE 40 TURNS CU WIRE .040 IN. DIA. AL PLATE 2 CM. DIA. ENTRANCE FE FRONT FE CORE 1.0 THICK FE BOTTOM F i g u r e I I I - 6 30 j e t and t h e l a s e r i n d e p e n d e n t l y ( a f t e r b o t h s i g n a l s have gone t h r o u g h the d e l a y u n i t s ) . The j i t t e r of the j e t w i t h r e s p e c t t o the l a s e r i s l e s s than 1 ms which i s a c c e p t a b l e s i n c e the j e t i s s t a b l e f o r s e v e r a l t e n s of ms. The t r i g g e r t o the j e t i s a l s o a c c u r a t e l y r e p r o d u c i b l e ( l e n g t h 150 ms.) hence e l i m i n a t i n g the p o t e n t i a l problem of the manual p u s h b o t t o n f i r i n g of v a r i a b l e time l e n g t h . 31 CHAPTER 4 EXPERIMENTAL RESULTS The e x p e r i m e n t a l r e s u l t s can now be p r e s e n t e d . They w i l l be p r e s e n t e d i n the f o l l o w i n g o r d e r : the s i g n a l l e v e l as a f u n c t i o n of p r e s s u r e , the s i g n a l as a f u n c t i o n of l a s e r i n t e n s i t y and the s i g n a l as a f u n c t i o n of e l e c t r o n energy sampled. A l l e r r o r s b a r s a r e f o r s t a n d a r d d e v i a t i o n s of the mean u n l e s s o t h e r w i s e n o t e d . P r e s s u r e Dependence The d i s c u s s i o n of the v a r i a t i o n of the number of e l e c t r o n s as the r e s e r v o i r p r e s s u r e i s changed w i l l be p r e s e n t e d f i r s t . As can be seen from the graphs ( F i g IV-1,2) the number of e l e c t r o n s grows v e r y r a p i d l y as the background p r e s s u r e i n c r e a s e s above 3 T o r r . T h i s i s t r u e both f o r e l e c t r o n s i n the backward and. f o r w a r d d i r e c t i o n s . S i n c e i t i s known t h a t the maximum m o l e c u l a r d e n s i t y i s p r o p o r t i o n a l t o the r e s e r v o i r p r e s s u r e the p r e s s u r e a x i s c o u l d be r e l a b e l l e d as the average d e n s i t y a x i s . In t h i s c o n t e x t the r e s u l t s t a k e on a c l e a r e r meaning. The number of e l e c t r o n s seen i n a g i v e n energy range i n c r e a s e s d r a m a t i c a l l y as the amount of plasma near .25 n c r i s i n c r e a s e d . I n t e n s i t y Dependence The background p r e s s u r e can now be f i x e d and e l e c t r o n p r o d u c t i o n as a f u n c t i o n of l a s e r i n t e n s i t y can be ob s e r v e d . The r e s u l t s a r e p l o t t e d f o r the 4 T o r r and the 5 T o r r j e t s ( F i g I V - 3 , 4 ) . The s c a l e s a r e r e l a t i v e ; 10 u n i t s f o r the 4 T o r r 32 r e s u l t s c o r r e s p o n d s t o 1 u n i t f o r the 5 T o r r r e s u l t s . The e r r o r b a r s i n the energy i n d i c a t e the s i z e of the b i n s used when a v e r a g i n g the d a t a . The p l o t s a r e s i m i l i a r . The s i g n a l s t a r t s o f f s m a l l a t .5 1 0 1 3 Wcrn - 2 , i n c r e a s e s t o a maximum s i g n a l near 4 1 0 1 3 Wcm"2 and d e c r e a s e s t o almost z e r o beyond 7 1 0 1 3 Wcrn"2. The d e c r e a s e i s u nexpected as one would n a i v e l y e x pect the number of e l e c t r o n s t o s a t u r a t e , but not d e c r e a s e . I f one assumes t h a t a Boltzmann d i s t r i b u t i o n i s v a l i d , an attempt a t f i n d i n g the so c a l l e d hot e l e c t r o n t e m p e r a t u r e can be made by comparing' s i g n a l s a t d i f f e r e n t e l e c t r o n e n e r g i e s . A c o u p l e of c a v e a t s must be made . There i s no a p r i o r i reason t h a t a M a x w e l l i a n d i s t r i b u t i o n s h o u l d be e x p e c t e d . S i n c e the t e m p e r a t u r e c a l c u l a t e d depends c r i t i c a l l y on the r e l a t i v e s i z e s of the two e l e c t r o n energy s i g n a l s , s m a l l e r r o r s w i l l have d r a m a t i c r e s u l t s on the t e m p e r a t u r e c a l c u l a t e d . By n o r m a l i z i n g t h e s i g n a l s ( i e d i v i d i n g by the energy examined n=S/£) one g e t s the r e l a t i v e number of e l e c t r o n s a t a p a r t i c u l a r energy. S i n c e f o r a Boltzmann d i s t r i b u t i o n n(£)= no/Fexp( - £ / K T ) two n o r m a l i z e d s i g n a l s can be used t o f i n d K T = ( ^ 1 - ^ 2 ) / l n ( n l / | 2 / ( n 2 v / J T ) ) . The e x p e r i m e n t a l r e s u l t s i n d i c a t e t h a t n l / n 2 = 1.27+-.67 f o r the 5 T o r r j e t a t 4 1 0 1 3 W cm" 2. T h i s means t h a t , s i n c e £1-£2 = 50 keV, «T w i l l be 85+-220 keV. The l a r g e e r r o r i s due t o the l o g a r i t h m i n the c a l c u l a t i o n of K T . One q u a l i t a t i v e o b s e r v a t i o n t o be made i s t h a t , f o r the 4 33 pb ( Torr) F i g u r e I V - 1 34 P b (Torr ) F i g u r e I V - 2 35 18 » 4 Torr INTENSITY +KL DEPENDENCE • 50 KeV o 100 KeV i II » I ^ I i i 1 I i i 1 ' * — 2 3 4 5 6 7 AVERAGE INTENSITY (1013W cm"2 ) F i g u r e IV-3 5 Torr INTENSITY +KL DEPENDENCE • 50 KeV a 100 KeV t——41 1 I ti H < I n 1 2 3 4 5,„ 6 AVERAGE INTENSITY (1013 W cm"2 ) F i g u r e IV-4 37 Torr j e t , the 100 keV s i g n a l i s on average s m a l l e r than the 50 keV s i g n a l whereas i n the 5 T o r r j e t the 100 keV s i g n a l i s 1.5 to 2 times as b i g as the 50 keV s i g n a l . T h i s would seem to imply t h a t f o r the 4 T o r r j e t there i s a lower K T From the s i z e of the s i g n a l one can attempt to estimate the numbers of f a s t e l e c t r o n s d e t e c t e d . To do t h i s one must s t a r t with the s i g n a l and work backwards to the s c i n t i l l a t o r . C o n v e r t i n g the v o l t a g e s i g n a l (100 mV) to a c u r r e n t s i g n a l by d i v i d i n g by 50 ohms and i n t e g r a t i n g the c u r r e n t s i g n a l over time (10 ns fwhm) the t o t a l charge (2 10" 1 1 C) generated i s found. C o n v e r t i n g the charge to the number of e l e c t r o n s i n t o the scope through d i v i s i o n by the gain of p h o t o m u l t i p l i e r (4.8 10 s) the number of p h o t o e l e c t r o n s e m i t t e d (250) i s e s t i m a t e d . D i v i d i n g by the quantum e f f i c i e n c y of the photocathode (.2) the number of photons i n c i d e n t (1250) i s found. F u r t h e r c o r r e c t i n g f o r l o s s e s at i n t e r f a c e s ( 7 % / i n t e r f a c e x 3) and i n the o p t i c a l f i b e r (85%) and f o r l i g h t emission i n s c i n t i l l a t o r which does not get i n t o o p t i c a l f i b e r s (47r/acceptance angle=5.5) , m u l t i p l y i n g by the r a t i o of the energy of t y p i c a l s c i n t i l l a t o r photon (3eV) to the energy of i n c i d e n t e l e c t r o n (50 keV) and f i n a l l y d i v i d i n g by the f r a c t i o n of the i n c i d e n t energy u l t i m a t e l y emitted as v i s i b l e l i g h t by the s c i n t i l l a t o r (assumed 1) one a r r i v e s at the number of e l e c t r o n s i n c i d e n t on the s c i n t i l l a t o r . For a 100 mV s i g n a l and 50 keV e l e c t r o n s i n c i d e n t the number a r r i v e d at i s small , approximately 1 to 10 e l e c t r o n s . Since some s i g n a l s are the e q u i v a l e n t (through N.D. F i l t e r s and n o n - l i n e a r response of p h o t o m u l t i p l i e r ) to 70 v o l t s t h i s means approximately 1000 e l e c t r o n s have reached the s c i n t i l l a t o r . T h i s corresponds to 38 roughly 3 10s electrons s r " 1 at 50 keV when divided by the s o l i d angle. Another piece of data analyzed was the percentage of shots where no (or very few electrons ) were observed. This confirmed both the pressure and the intensity observations (see Fig IV-5): the lower the background pressure the less l i k e l y electrons w i l l be observed and a dip at 4 10 1 3 Wcrn"2 confirming that i t w i l l be most l i k e l y to see electrons at th i s i n t e n s i t y . In the process of c a l i b r a t i n g the fibers for the OMA ,a few shots were taken with the f o i l - s c i n t i l l a t o r . combination placed in the plane perpendicular to E0 of the laser. The shots indicated that the t o t a l flux of xrays and electrons was peaked in the backwards d i r e c t i o n . The s c i n t i l l a t o r s were located at 153,140,128 and 117 degrees to +ku. This suggests the electron production i s not i s o t r o p i c , but peaked in the backwards (and presumably) forwards d i r e c t i o n s . The detailed angular variation was not a c t i v e l y pursued in th i s work, but has been reported elsewhere (Meyer et a l . , 1983). Di s t r i b u t i o n Funct ion The four channel spectrometer confirms that a Maxwellian d i s t r i b u t i o n f i t i s acceptable in the range 40 - 300 keV . The K T f i t t e d i s 121 keV (see Fig IV-6). The error in K T i s estimated at 25%. The signals from the OMA were corrected for r e l a t i v e channel response,energy loss in the f o i l and electron number F i g u r e I V - 5 4 0 a t t e n u a t i o n i n t h e f o i l a n d w e r e n o r m a l i z e d f o r d i f f e r e n t i n c i d e n t e n e r g i e s . T h e r e s u l t s w e r e f i t t e d b y a f u n c t i o n o f t h e f o r m f U ) = A / ( T ) e x p ( - U ) A T ) ( F i t 1 ) K T w a s f o u n d t o b e 1 2 1 k e V . T h e 4 c h a n n e l s y s t e m w a s l e s s s e n s i t i v e t h a n t h e o n e c h a n n e l s y s t e m . T h i s i s d u e t o p o o r e r q u a l i t y o p t i c a l f i b e r , s m a l l e r s c i n t i l l a t o r a r e a ( a n d s m a l l e r o p t i c a l f i b e r a r e a ) a n d a p o s s i b l y l e s s t i g h t l y f o c u s e d s p e c t r o m e t e r . A f e w w o r d s o f c a u t i o n m u s t b e e x p r e s s e d a b o u t t h e i n t e r p r e t a t i o n o f t h e f i t a b o v e . T h e f i t w a s m a d e s i n c e o t h e r e x p e r i m e n t s ( J o s h i e t a l . , 1 9 8 1 ; E b r a h i m e t a l . , 1 9 8 1 ; B e r g e r e t a l . , 1 9 8 3 ) h a v e s h o w n t h a t s u c h a f i t w i l l w o r k . N o u n d e r l y i n g a s s u m p t i o n s a b o u t t h e r e a s o n f o r t h e f i t m u s t b e m a d e . T o f u r t h e r e m p h a s i z e t h e p o i n t a n o t h e r f i t f U ) - A e x p ( - B ( / £ V C ) (/t-/c) ) ( F i t 2 ) w a s t r i e d . C w a s 8 2 k e V a n d B " 1 w a s 2 6 k e V . T h e f i t i s a l s o g o o d . M a n y m o r e d a t a p o i n t s o v e r a m u c h w i d e r e n e r g y r a n g e a n d w i t h m u c h s m a l l e r e r r o r b a r s m u s t b e u s e d t o r e s o l v e t h e d i f f e r e n c e . T h e c o n c l u s i o n s a r e t h a t S R S e l e c t r o n s a r e p r o d u c e d q u i t e e f f i c i e n t l y p r o d u c e d i n t h e . 2 5 n C R r e g i o n o f t h e p l a s m a a n d t h e e x p e c t e d t e m p e r a t u r e b a s e d u p o n t h e s i m p l e S R S t h e o r y o f t h e a b s o l u t e i n s t a b i l i t y a t t h e . 2 5 n C R r e g i o n i s i n g o o d a g r e e m e n t w i t h t h e e x p e r i m e n t a l r e s u l t . E L E C T R O N E N E R G Y D I S T R I B U T I O N 41 o o ENERGY (KEVJ I X W ) F i g u r e I V - 6 42 CHAPTER 5 INTERFEROMETRY The r e s u l t s of c h a p t e r 4 r a i s e d some i n t e r e s t i n g p o i n t s w hich r e q u i r e f u r t h e r s t u d y . The p r e s s u r e dependence of the number of e l e c t r o n s may be due t o the plasma d e n s i t y c h a n g i n g w i t h c h a n g i n g background p r e s s u r e . The i n t e n s i t y dependence may be due t o s m a l l s c a l e changes i n the d e n s i t y . These s m a l l s c a l e changes i n c l u d e m o d i f i c a t i o n of the s c a l e l e n g t h a t .25 n C R and w i l l a f f e c t beam r e f r a c t i o n i n the plasma. I t seems t h a t a d e t a i l e d i n v e s t i g a t i o n of the plasma d e n s i t y as a f u n c t i o n of time ,space and i n c i d e n t l a s e r energy must be u n d e r t a k e n . I n t e r f e r o m e t r y w i t h a s h o r t p u l s e d l a s e r would be the most e f f i c i e n t way of a c c o m p l i s h i n g t h i s o b j e c t i v e . From the r e s u l t i n g i n t e r f e r o g r a m s a g r e a t d e a l of i n f o r m a t i o n can be e x t r a c t e d . The f o l l o w i n g can be g i v e n as f u n c t i o n s of tim e , s p a c e and i n c i d e n t l a s e r energy : plasma v o l u m e , t o t a l e l e c t r o n number, d e n s i t y , s c a l e l e n g t h and r e f r a c t i o n e f f e c t s . The probe used f o r i n t e r f e r o m e t r y was a 80 ps ruby l a s e r ( w a v e l e n g t h 694.3 nm). A Jamin i n t e r f e r o m e t e r (Born and Wolf) was s e t up by B . H i l k o and H.Houtman. And the plasma was probed. The i n t e r f e r e n c e f r i n g e s were r e c o r d e d on P o l a r o i d f i l m and s e l e c t e d ones were p h o t o g r a p h i c a l l y e n l a r g e d and d i g i t i z e d . The d a t a was then a n a l y z e d by computer (see Appendix 1) and the r e s u l t s were p l o t t e d ( see f i g u r e s V—1 t o V - 5 ) . The .25 n C R l i n e i s emphasized. G e n e r a l R e s u l t s The f e a t u r e which i s most s t r i k i n g i s the e x i s t e n c e of 43 ce: on 2 T o r r 2 . 9 J 3.1 n s i n : 5.0 l 4.0 2 I. (nm.) 5.0 6.0 Ct: o: 6.7 J 1.7 n s 3.0 ii ft , y ; <L_j. •, i 4.0 6 . 0 mm. CO Cc on ro . _ o o o . X 3.0 4.0 Z (mm) i i j i _ 5.0 6.7 J 3 . 2 n s F i g u r e V - l 3 . 5 T o r r 3.7 J 1.0 n s 3.1 J 1.5 n s 2 . 9 J 2 . 4 n s , u k . • \ - ! 1_ .'-i 5.0 I frirn) F i g u r e V - 2 I (1 -t e.o 3 . 5 Torr 45 Z [mm) F i g u r e V-3 46 5 Torr 47 F i g u r e 48 t h r e e i s l a n d s o f p l a s m a . I n i t i a l l y t h e i s l a n d l o c a t e d a t t h e r e a r edge o f t h e g a s j e t i s f o r m e d . The n e x t i s l a n d t o a p p e a r i s t h e m i d d l e one l o c a t e d a t t h e f r o n t edge o f t h e j e t . T h i s i s l a n d a p p e a r s l e s s t h a n 200 ps a f t e r t h e f i r s t . The f i n a l i s l a n d a p p e a r s i n f r o n t o f t h e j e t a n d c o u l d be due t o b r e a k d o w n i n t h e 5 T o r r b a c k g r o u n d g a s . T h i s i s s u r p r i s i n g s i n c e t h e c a l c u l a t e d maximum d e n s i t y i s 3.5 1 0 1 7 cm" 3. C o n t a m i n a t i o n by n i t r o g e n f r o m t h e g a s j e t i s p o s s i b l e s i n c e t h e j e t w i l l h ave b e e n on f o r 7 ms b e f o r e t h e l a s e r p u l s e r e a c h e s i t . I t i s more p r o b a b l e t h a t e x p a n s i o n o r b l o w - o f f o f p l a s m a f r o m . t h e m i d d l e i s l a n d i s t h e c a u s e o f t h e f r o n t i s l a n d . I t i s i n t e r e s t i n g t o n o t e t h a t w h i l e t h e d e n s i t y i n t h e f r o n t p l a s m a i s h i g h i t i s n o t a s h i g h a s i n t h e c l u m p s l o c a t e d a t t h e g a s j e t p r o p e r . T h e s e i s l a n d s e x p a n d w i t h t i m e a n d e v e n t u a l l y m e r g e . A f t e r a few ns t h e p l a s m a a p p e a r s t o r o i d a l w i t h t h e r e g i o n o f maximum d e n s i t y m o v i n g away f r o m t h e a x i s . L a r g e S c a l e R e s u l t s The a b s o l u t e number o f e l e c t r o n s p r e s e n t i n t h e e n t i r e p l a s m a was c a l c u l a t e d by m e r e l y i n t e g r a t i n g ( v i a c o m p u t e r ) t h e d e n s i t y o v e r t h e v o l u m e : N= SJ n ( r ) 2 7 r r d r dz The number c a l c u l a t e d c a n be r o u g h l y e s t i m a t e d by u s i n g a p l a s m a v o l u m e = 7 r r 2l m u l t i p l i e d by t h e p r o d u c t o f t h e m o l e c u l a r d e n s i t y a n d t h e Z o f n i t r o g e n . T h i s y i e l d s an e s t i m a t e o f o r d e r 1 0 1 5 t o 1 0 1 6 w h i c h a g r e e s w i t h t h e numbers c o m p u t e d . 49 The expansion v e l o c i t y of the plasma was also examined. This i s done by following a contour l i n e as the plasma expands. The t o t a l plasma volume i t s e l f was also calculated. The results of the gross analysis of the plasma are in Fig V - 6 . The expansion v e l o c i t y i s e a s i l y determined by finding the slope of the curves. What was plotted was the contour position at a fixed Z vs time. The general result indicates that the plasma expands at about few times 10 7 cm s" 1 . The ve l o c i t y measured depends upon several factors. The incident energy i s important because as more laser energy is applied the plasma expands faster. The expansion rate also depends upon which contour one picks and which part of the plasma the contour i s found. The bulk plasma electron density d i s t r i b u t i o n was also calculated. This d i s t r i b u t i o n indicates the density regimes where the electrons in the plasma are to be found. Further the average density was also found by dividing the t o t a l number of electrons by the t o t a l plasma volume. This d i s t r i b u t i o n i s a convenient way of showing that at early times the density regime where the maximum number of electrons appear i s lower for the lower Torr j e t s . At later times the d i s t r i b u t i o n changes. This happens because the electrons are being moved into higher density regimes. Since the number of electrons i s presumably fixed t h i s means that electrons are being moved from lower density regimes . The d i s t r i b u t i o n thus gains at the high end V= Gross Features • 2 J • 4 J • 7 J E lectron Numbers 1.5 f.O a. T(ns) Expansion Rates D 4 J * 7 J .5 10 b. T (ns) 1.5 F i g u r e V - 6 51 t h a t t h i s i s happening. The g r o s s f e a t u r e s r e v e a l some not unexpected r e s u l t s . As more l a s e r energy i s a p p l i e d t o the plasma the g r e a t e r i s the r a t e a t which the number of e l e c t r o n s grows . As more l a s e r energy i s a p p l i e d the e a r l i e r the plasma appears . At any g i v e n time ( e a r l y i n the l a s e r p u l s e ) t h e r e a r e more e l e c t r o n s a t the h i g h e r e n e r g i e s . There a l s o appears t o be a ' s a t u r a t i o n ' as the number of e l e c t r o n s appears t o l e v e l o f f a t l a t e r t i m e s . F i n e S c a l e R e s u l t s On a f i n e r s c a l e more d e t a i l can be measured. C a u t i o n must be used s i n c e s m a l l d e t a i l s a r e o b s c u r e d by e r r o r s i n d i g i t i z i n g and c o m p u t a t i o n a l a r t i f a c t s g e n e r a t e d by. imposing a 'mesh' ( r e c t a n g u l a r c o o r d i n a t e system) onto the i n t e r f e r o g r a m s . Some of t h i s i s compensated f o r by u s i n g a smoothing r o u t i n e i n the a n a l y s i s . The main assumption of c y l i n d r i c a l symmetry ( i n o r d e r t o do the A b e l i n v e r s i o n ) i s a l s o v i o l a t e d i n some of the l a t e r i n t e r f e r o g r a m s . Ray t r a c i n g was performed by J . E . B e r n a r d . The r e s u l t s o b t a i n e d (eg F i g V-8) i n d i c a t e t h a t the plasma r e f r a c t s the l a s e r away from the r e a r plasma. A l s o i n the p r o c e s s of r e f r a c t i o n the l a s e r spreads o u t , l o w e r i n g the i n t e n s i t y i n the plasma. One can e s t i m a t e the d e n s i t y s c a l e l e n g t h i n the plasma. In p a r t i c u l a r L can be found near the .25 n C R l a y e r . L i s d e f i n e d as 52 Bulk Plasma Electron Density Distribution 3.8 J 1.5 ns 5 Torr N».IOIN, 'cr 2.7 J ' 3.3 ns 5 Torr N=099Nb co CM Z9J ' 2.85 ns 3.5 Torr N*.087Nc .4 .2 .3 ,4 F i g u r e V - 7 53 F i g u r e V - 8 5 4 Scale Length (.25 N c r ) Vs. Time .8 • • • A .6 A L (mm) .4 • • A .2 A • 5 Torr • 4 J A 7 J 1 . 2 3 4 TIME (ns) F i g u r e V - 9 55 L = 1 / ( l / n dn/dx) L i s p l o t t e d i n F i g V - 9 as a f u n c t i o n of time and energy. No d i s t i n c t e v i d e n c e i s seen f o r s c a l e l e n g t h m o d i f i c a t i o n but the tendency i s f o r the h i g h power s c a l e l e n g t h s t o grow s l i g h t l y s l o w e r and t o not r e a c h the maximum low power s c a l e l e n g t h s . T h i s c o u l d p o s s i b l y be a s s i g n e d t o pondermotive f o r c e s which d e c r e a s e the hydrodynamic e x p a n s i o n t h a t i s c a u s i n g the s c a l e l e n g t h t o i n c r e a s e . 56 CHAPTER 6 DISCUSSION AND CONCLUSIONS The r e s u l t s of c h a p t e r 4 and c h a p t e r 5 w i l l now be d i s c u s s e d . The c o n c l u s i o n s w i l l a l s o be r e v i e w e d . P r e s s u r e Dependence In c h a p t e r 4 i t was noted t h a t the e l e c t r o n s i g n a l i n c r e a s e s d r a m a t i c a l l y as the background p r e s s u r e was i n c r e a s e d . S i n c e i t i s c a l c u l a t e d t h a t the maximum m o l e c u l a r d e n s i t y i s p r o p o r t i o n a l t o the r e s e r v o i r p r e s s u r e i t i s r e a s o n a b l e t o expect t h a t the s p a t i a l l y and t e m p o r a l l y averaged plasma d e n s i t y w i l l be lower f o r the lower T o r r j e t s . A comparison of the 2 T o r r and 5 T o r r j e t s c l e a r l y i n d i c a t e t h i s and i s c o n f i r m e d by the b u l k plasma e l e c t r o n d i s t r i b u t i o n f u n c t i o n . S i n c e a t lower d e n s i t i e s the growth r a t e and the phase v e l o c i t y a r e much lower one s h o u l d expect the number of e l e c t r o n s t o be lower f o r a f i x e d energy. The l o w e r i n g of the phase energy w i l l lower the 100 keV:50keV s i g n a l r a t i o . For example i f the average d e n s i t y i s .2n C R the phase energy i s 39 keV. Assuming t h a t the f a s t e l e c t r o n d i s t r i b u t i o n i s M a x w e l l i a n , the s i g n a l r a t i o s h o u l d be 1 00/50 V ( 1 00/50) exp (-50/39) = .8 whereas a t .25n C R the r a t i o s h o u l d be 100/50*/(100/50)exp(-50/115)=1.8 . T h i s i s what i s o b s e r v e d f o r the 4 T o r r and 5 T o r r r e s u l t s . I t s h o u l d be noted t h a t f o r lower d e n s i t i e s the k of the EPW i s l o n g e r . T h i s w i l l i n c r e a s e the Landau damping and th u s the waves w i l l decay sooner. A l s o SRS i s c o n v e c t i v e f o r lower d e n s i t i e s and the growth r a t e i s lower than i n the a b s o l u t e c a s e . I n t e n s i t y Dependence 57 The a b s o l u t e number of f a s t e l e c t r o n s g e n e r a t e d by SRS i s an i m p o r t a n t number. I f the t o t a l number i s v e r y s m a l l then t h e i r r e l a t i v e importance i n p r e h e a t i n g a l a s e r f u s i o n t a r g e t becomes n e g l i g i b l e . I n c h a p t e r 4 i t was shown one c o u l d use an e l e c t r o n energy d i s t r i b u t i o n f i t t e d by where /cT= 121 keV. In c h a p t e r 3 the energy r e s o l u t i o n of MK I I was noted a t +-15%. An e s t i m a t e of the t o t a l f r a c t i o n of the d i s t r i b u t i o n e n t e r i n g the s c i n t i l l a t o r can be made by i n t e g r a t i n g over the d i s t r i b u t i o n At lOOkeV we f i n d N/N 0=.13. The l a r g e s t e l e c t r o n s i g n a l a t t h i s energy c o r r e s p o n d s t o a t l e a s t 1000 e l e c t r o n s . The t o t a l number of e l e c t r o n s i s thus g r e a t e r than 10000. T h i s number i s a lower bound on the t o t a l number of f a s t e l e c t r o n s . T h i s i s due , i n p a r t , t o the c o n s e r v a t i v e e s t i m a t e of the a b s o l u t e number of e l e c t r o n s done i n c h a p t e r 3. No i n t e g r a t i o n over the s o l i d a n g l e i n t o which the e l e c t r o n s were e m i t t e d was performed ( o t h e r than the o b v i o u s one of the s p e c t r o m e t e r ) . One must a l s o r e a l i z e t h a t t he e l e c t r o n s t h a t have reached the d e t e c t o r c o r r e s p o n d t o the most e n e r g e t i c e l e c t r o n s c r e a t e d . The e l e c t r o n s which have escaped from the plasma must have overcome l a r g e p o t e n t i a l b a r r i e r s u which c o u l d be up t o 200 keV. The f a s t e l e c t r o n s escape the plasma; the more massive p o s i t i v e i o n s a r e l e f t b e h i n d g i v i n g t he plasma a p o s i t i v e charge and hence the plasma o b t a i n s a p o t e n t i a l . The e l e c t r o n s t h a t we see a t N ( £ ) = C / | e x p ( - £ / K T ) ( C i s f o r n o r m a l i z a t i o n ) dx x 2 = . 5 m v 2 / U T ) 58 energy i- must have had energy £+U i n t h e plasma. The o v e r a l l e f f e c t of the p o t e n t i a l i s t o reduce the number of e l e c t r o n seen by a f a c t o r exp(-u/»cT). Some i d e a of the t o t a l energy c a r r i e d by hot e l e c t r o n s can be d e r i v e d from the Manley-Rowe r e l a t i o n s . These r e l a t i o n s can be thought of as f o l l o w s . For n g i v e n i n c i d e n t p h o t o n s , n decay s c a t t e r e d photons and n decay plasmons can be g e n e r a t e d . Each photon or plasmon has energy h/{2n)u. The t o t a l energy i n a p a r t i c u l a r form i s nh/(27r)co. Hence the f o l l o w i n g r e l a t i o n s h i p h o l d s : Thus the t o t a l p o s s i b l e energy a v a i l a b l e f o r hot e l e c t r o n s , i s J p=u p^ s/u s. At .25 n C R .a maximum energy of .5£ 0 c o u l d appear as f a s t e l e c t r o n s . T h i s i s an o v e r e s t i m a t e as i t n e g l e c t s the k i n e t i c energy of the background plasma which w i l l be r a i s e d by SRS. Computer s i m u l a t i o n s (Hiob and Barnard) i n d i c a t e t h a t a p p r o x i m a t e l y 25% of the i n c i d e n t energy s h o u l d appear as f a s t e l e c t r o n s . At l e a s t .05% of the i n c i d e n t energy appears as f a s t e l e c t r o n s i n our case . When one c o n s i d e r s t h a t a p p r o x i m a t e l y 70%+ of the i n c i d e n t energy i s not t r a p p e d i n the plasma ( i e i s t r a n s m i t t e d , B r i l l o u i n b a c k s c a t t e r e d and r e f r a c t e d ) ( P o p i l Ph.D T h e s i s 1984 ; B e r n a r d Ph.D T h e s i s 1984) q u i t e a c o n s i d e r a b l e f r a c t i o n of the absorbed energy has shown up as f a s t e l e c t r o n s . F a s t e l e c t r o n s a r e c o n s e q u e n t l y q u i t e i m p o r t a n t and must be c o n s i d e r e d when d e s i g n i n g l a s e r f u s i o n r e a c t o r s . The t h r e s h o l d f o r SRS i s a l s o i m p o r t a n t . The i d e a l l a s e r f u s i o n l a s e r w a velength must be chosen so t h a t t h e r e i s v e r y 59 good a b s o r p t i o n ( l e a d i n g t o h i g h a b l a t i o n p r e s s u r e s and t o good compression) w i t h no f a s t e l e c t r o n p r o d u c t i o n . ( The advantage of s h o r t e r wavelengths i s t h a t i n v e r s e b r e m s s t r a h l u n g a b s o r p t i o n , a major h e a t i n g mechanism,is p r o p o r t i o n a l t o X" 2 ). S i n c e the t h r e s h o l d s o b s e r v e d i n t h i s work ( 1 0 1 3 Wcrn"2) ar e i n good agreement w i t h the t h e o r y the t h e o r i e s s h o u l d work w e l l i n c a l c u l a t i n g the s h o r t e r w avelength t h r e s h o l d s . Hence, f o r .53 Mm l i g h t i n t e n s i t i e s s h o u l d be kept below <4 1 0 1 5 Wcrn"2 t o p r e v e n t hot e l e c t r o n p r o d u c t i o n . S i n c e i t has been shown t h a t f a s t e l e c t r o n s a r e produced one s h o u l d ask o n e s e l f whether t h e r e i s a l i m i t t o the number t h a t can be produced. An extreme upper l i m i t i s s e t by the Manley- Rowe r e l a t i o n s ( above). F o r t u n a t e l y t h e r e a r e o t h e r mechanisms t h a t s a t u r a t e and quench SRS. S a t u r a t i o n i s o b s e r v e d when the s c a t t e r e d SRS l i g h t l e v e l s t o p s r i s i n g e x p o n e n t i a l l y (as i s s h o u l d near t h r e s h o l d ) and l e v e l s o f f a t i n c i d e n t i n t e n s i t i e s w e l l above t h r e s h o l d . S i n c e the number of f a s t e l e c t r o n s d e t e c t e d i s r e l a t e d t o the a m p l i t u d e of the u n d e r l y i n g SRS plasma wave , a s a t u r a t i o n and quenching of the e l e c t r o n s i g n a l i n d i c a t e s a c o r r e s p o n d i n g s a t u r a t i o n and q u e n c h i n g i n SRS wave a m p l i t u d e . I t i s t h e r e f o r e i n s t r u c t i v e t o examine the mechanisms which a r e r e s p o n s i b l e f o r the s a t u r a t i o n of SRS. One s a t u r a t i o n mechanism i s p a r t i c l e t r a p p i n g (Hiob and Barnard,1983) . The u n d e r l y i n g plasma wave g e n e r a t e d i n the SRS i n t e r a c t i o n grows t o such a l a r g e a m p l i t u d e t h a t an a p p r e c i a b l e f r a c t i o n of the e l e c t r o n s i n t h e plasma a r e t r a p p e d i n the e l e c t r o s t a t i c p o t e n t i a l of the wave. The d e t e c t i o n of f a s t 60 e l e c t r o n s i s a n e c e s s a r y but not s u f f i c i e n t p r o o f of t h i s s a t u r a t i o n mechanism. N o n - l i n e a r Landau damping of any l a r g e a m p l i t u d e plasma wave w i l l a l s o g e n e r a t e f a s t e l e c t r o n s . Other mechanisms can s a t u r a t e SRS b e f o r e the i n s t a b i l i t y has grown l a r g e enough t o g e n e r a t e f a s t e l e c t r o n s . The e l e c t r o s t a t i c wave g e n e r a t e d can s u b s e q u e n t l y decay i n t o a daughter EPW and IA wave. Indeed K a r t t u n e n ,1980 c l a i m s ' . . . t h a t the Langmuir decay s t a r t s t o dominate the s t i m u l a t e d Raman s c a t t e r i n g even a t v e r y low l e v e l s of the Langmuir wave a m p l i t u d e i n d i c a t i n g a s t r o n g s a t u r a t i o n e f f e c t on SRS except a t e x t r e m e l y h i g h t e m p e r a t u r e s ' . By comparing the growth r a t e s f o r the decay p r o c e s s and SRS we f i n d f o r our case (/cT=1keV, n=.25n C R ) t h a t 7 l / 7 s r s =215 E E P W / E 0 . Thus i t i s p o s s i b l e t h a t t h i s i s a mechanism f o r the s a t u r a t i o n of SRS a t h i g h i n t e n s i t i e s . Another mechanism which can s a t u r a t e SRS i s s c a l e l e n g t h m o d i f i c a t i o n (Chen and L i u ) . The EPW g e n e r a t e d by Raman s c a t t e r i n g (and more i m p o r t a n t l y , by the two plasmon decay i n s t a b i l i t y ) cause a g r a d i e n t i n E 2 . T h i s g r a d i e n t r e s u l t s i n a f o r c e ( t h e pondermotive f o r c e ) which tends t o move i o n s and e l e c t r o n s out of the r e g i o n where the waves a r e b e i n g g e n e r a t e d namely near '25n C R . The r e s u l t of t h i s i s t o change an i n i t i a l l y l i n e a r d e n s i t y ramp i n t o a s t e e p g r a d i e n t a t ,25n C R and a p l a t e a u a t a lower d e n s i t y . The SRS i s c o n v e c t i v e (growing i n space o n l y ) f o r d e n s i t i e s below .25 n C R and i s a b s o l u t e (growing i n space and time) o n l y near .25n C R . Thus 61 f a s t e l e c t r o n s w i l l not be seen as SRS w i l l shut down b e f o r e the wave can t r a p a l a r g e number of e l e c t r o n s . Other t h e o r i e s which p r e d i c t s a t u r a t i o n i n c l u d e the decay of the b a c k s c a t t e r e d EM wave ( K a r t t u n n e n and Salomaa,1979) and r e f l e c t i o n f r o n t p r o p a g a t i o n ( K r u e r e t a l , 1 9 7 5 ) . These w i l l not be d i s c u s s e d h e r e . There can be o t h e r p r o c e s s e s independent of the s a t u r a t i o n mechanisms d i s c u s s e d above which can e x p l a i n the d i s a p p e a r a n c e of the e l e c t r o n s a t h i g h i n t e n s i t i e s . These must be d i s c u s s e d as our r e s u l t s a r e not p r e d i c t e d by any t h e o r y nor seen i n o t h e r e x p e r i m e n t s . Two such p r o c e s s e s can o c c u r i n our e x p e r i m e n t . E a r l y r e f r a c t i o n causes the l a s e r i n t e n s i t y seen by the plasma a t .25 n C R l a y e r d e c r e a s e e a r l i e r f o r the h i g h power s h o t s . T h i s happens s i n c e the plasma i n f r o n t of the gas j e t due t o the i o n i z a t i o n of the He background gas forms e a r l i e r f o r the h i g h power s h o t s and thus r e f r a c t i o n of the l a s e r can s t a r t e a r l i e r and the i n t e n s i t y drops below the t h r e s h o l d i n t e n s i t y e a r l y i n the l a s e r p u l s e . A l t h o u g h not shown on a l l the i n t e r f e r o g r a m s p l o t t e d the o r i g i n a l i n t e r f e r o g r a m s show some e v i d e n c e (eg b l u r r i n g of f r i n g e s or v e r y s m a l l f r i n g e s h i f t s ) of breakdown i n the background Helium gas. W h i l e the d e n s i t y i s low ( l e s s than .1 n C R ) the l o n g d i s t a n c e s ( g r e a t e r than 3 mm) compensate and the net e f f e c t c o u l d be c o n s i d e r a b l e r e f r a c t i o n . A ray t r a c i n g c a l c u l a t i o n shows t h a t t h i s i s a r e a s o n a b l e a s s u m p t i o n (see F i g u r e VI-1) . The d e n s i t y form assumed i s shown i n the f i g u r e and i s independent of the Z c o o r d i n a t e . E a r l y r e f r a c t i o n i s a l s o s u p p o r t e d by t e m p o r a l l y r e s o l v e d 62 t r a n s m i s s i o n measurements ( B e r n a r d , Ph.D T h e s i s 1984). These show t h a t as the i n c i d e n t l a s e r i n t e n s i t y i s i n c r e a s e d the t r a n s m i t t e d power th r o u g h the f/5 e x i t l e n s d e c r e a s e s t o z e r o e a r l i e r . In f a c t f o r the h i g h e s t powers the t r a n s m i s s i o n i s over b e f o r e the peak of the l a s e r p u l s e has a r r i v e d . I t i s d i f f i c u l t t o e s t i m a t e the i n s t a n t a n e o u s power w i t h r e f r a c t i o n t a ken i n t o a c c o u n t . A rough e s t i m a t e can be o b t a i n e d from the ray t r a c i n g where the r a d i u s c o n t a i n i n g the r a y s squared can used as a measure of the r e l a t i v e i n t e n s i t i e s . A modest 50% i n c r e a s e i n r a d i u s w i l l reduce the i n t e n s i t y by more than a f a c t o r of 2. .Secondly, r a p i d e x p a n s i o n c o u l d e x p l a i n the d i s a p p e a r a n c e of f a s t e l e c t r o n s . The plasma expands expands r a p i d l y l e a v i n g a low d e n s i t y h o l e . T h i s h o l e appears e a r l i e r f o r the h i g h power s h o t s and thus the b u l k of the l a s e r p u l s e w i l l see an e f f e c t i v e l y lower d e n s i t y plasma. Thus sa m p l i n g f i x e d e n e r g i e s w i l l y i e l d the same r e s u l t s as l o w e r i n g the background p r e s s u r e . S i m u l a t i o n s done w i t h CASTOR (which i s a 2-D hydrodynamic code f o r l a s e r plasma i n t e r a c t i o n s ) i n d i c a t e t h a t the blow-out does s t a r t v e r y e a r l y i n t o the p u l s e . T h i s e x p l a n a t i o n i s p l a u s i b l e o n l y i f the r e f r a c t i o n h y p o t h e s i s p r e s e n t e d i s wrong as the e x p a n s i o n h y p o t h e s i s r e q u i r e s the l a s e r t o pass t h r o u g h the same f o c a l volume f o r the low and h i g h power s h o t s . The Raman s c a t t e r i n g must a l s o o c c u r l a t e i n the p u l s e i e when the dense plasma has a c t u a l l y blown out of the f o c a l volume. The above d i s c u s s i o n i s f i n e f o r g e n e r a l e x p l a n a t i o n of the i n t e n s i t y dependence, but d e t a i l e d n u m e r i c a l a n a l y s i s i s 63 C a l c u l a t e d E f f e c t s of Re f rac t ion Due to Breakdown P lasma in F r o n t of Ta rget 1 2 3 Z ( mm) F i g u r e V I - 1 64 d i f f i c u l t the t h e o r e t i c a l t h r e s h o l d s a r e not u n i q u e , but r a t h e r a r e o r d e r of magnitude e s t i m a t e s and v a r i a t i o n s from t h e o r y t o t h e o r y of two or t h r e e a r e q u i t e p o s s i b l e . For example f o r L=300 um and n=.25n O R K r u e r ' s t h e o r y p r e d i c t s I-r=5 1 0 1 3 Wcrn"2 whereas Chen p r e d i c t s I T=1.58 10 1" Wcrn" 2. S i n c e the e x p e r i m e n t s were performed v e r y c l o s e t o the c a l c u l a t e d t h r e s h o l d s s u b t l e v a r i a t i o n s i n L or I c o u l d have d r a m a t i c e f f e c t s on the e l e c t r o n s i g n a l . I t i s , however, v e r y d i f f i c u l t t o prove any of t h e s e p o s s i b i l i t i e s w i t h o u t more e x p e r i m e n t s . To see s c a l e l e n g t h m o d i f i c a t i o n the d e n s i t y v a r i a t i o n s must be known q u i t e a c c u r a t e l y w i t h s m a l l e r r o r b a r s and f o r e a r l y t i m e s when the r a d i u s of the plasma i s l e s s than 200 M m . W h i l e i n t e r f e r o m e t r y i s c a p a b l e of d o i n g t h i s ( i n p r i n c i p l e ) the amount of d a t a a n a l y s i s can become r a t h e r arduous. T h i s , i s why the d i g i t i z i n g was done and the computer program was w r i t t e n . The g a i n i n the ease of a n a l y s i s i s b a l a n c e d by a ( p r a c t i c a l ) d e c r e a s e i n a c c u r a c y . R e f r a c t i o n c a l c u l a t i o n s depend on the q u a l i t y of the i n t e r f e r o m e t r i c a n a l y s i s as the ray t r a c i n g i s e x t r e m e l y s e n s i t i v e t o s m a l l r a p i d changes i n the d e n s i t y . Only the most g e n e r a l t r e n d s can be e x t r a c t e d from the ray t r a c i n g . Energy D i s t r i b u t i o n F u n c t i o n The energy d i s t r i b u t i o n f u n c t i o n f o r the f a s t e l e c t r o n s w i l l now be d i s c u s s e d . One f u n c t i o n which f i t s the d a t a w e l l i s the one would e x p e c t i f a M a x w e l l i a n d i s t r i b u t i o n r e s u l t e d . As noted e a r l i e r however the d a t a a r e i n s u f f i c i e n t t o c l a i m t h a t a M a x w e l l i a n i s the b e s t f i t . I f i t assumed t h a t i t i s a K T of 121 keV i s o b t a i n e d . T h i s i s v e r y c l o s e t o the phase energy f o r 6 5 an Raman EPW g e n e r a t e d a t d e n s i t i e s near .25 n c r (=115 keV ,Chapter 2 ) . A M a x w e l l i a n d i s t r i b u t i o n i s the d i s t r i b u t i o n e x p e c t e d i f the f a s t e l e c t r o n s have come t o e q u i l i b r i u m among them s e l v e s ( t h e r m a l i z e d ) . The time f o r t h e r m a l i z a t i o n has been c a l c u l a t e d and we found the time n e c e s s a r y i s much l o n g e r t h a n the l a s e r p u l s e t i m e . I t does not seem p o s s i b l e t o f i n d a p h y s i c a l mechanism f o r t h e r m a l i z a t i o n . As noted e a r l i e r o t h e r f i t s a r e p o s s i b l e . For example i n c h a p t e r 4 F i t 2 was t r i e d . The p h y s i c a l b a s i s f o r a t t e m p t i n g t h i s f i t was the assumption t h a t we had a beam of e l e c t r o n s w i t h some d r i f t v e l o c i t y and some G a u s s i a n shaped sp r e a d about t h i s d r i f t . v e l o c i t y . The e s s e n t i a l r e s u l t s of t h i s work a r e as f o l l o w s . F a s t e l e c t r o n s a r e g e n e r a t e d q u i t e e f f i c i e n t l y by SRS whenever a s i g n i f i c a n t amount of .25 n C R plasma i s p r e s e n t . The f l u x e s g e n e r a t e d a r e g r e a t e r than 3 10 6 s r " 1 from the t o t a l number of e l e c t r o n s i n the f o c a l volume of 1 0 1 2 . The e n e r g i e s of t h e s e e l e c t r o n s a r e c o n s i s t a n t w i t h a M a x w e l l i a n type d i s t r i b u t i o n a t a KT=121 keV. The most p r o b a b l e energy i s i n good agreement w i t h t h a t e x p e c t e d from the s i m p l e t h e o r y of SRS. The e f f e c t s of r e f r a c t i o n a r e i m p o r t a n t f o r the p r o d u c t i o n of e l e c t r o n s when the i n c i d e n t i n t e n s i t y i s near the t h r e s h o l d i n t e n s i t y . F u r t h e r work must be done t o c o m p l e t e l y u n d e r s t a n d the p h y s i c s of the time e v o l u t i o n of t e m p e r a t u r e , d e n s i t y and l o c a l i n t e n s i t y i n the l a s e r plasma i n t e r a c t i o n s t u d i e d i n t h i s work and t h e i r e f f e c t on t h e number of f a s t e l e c t r o n s . 66 APPENDIX 1 INTERPRETATION OF INTERFEROGRAMS:COMPUTER PROGRAM A computer program was w r i t t e n which w i l l i n t e r p r e t d i g i t i z e d i n t e r f e r o g r a m s . A l i s t i n g of the program f o l l o w s . In o r d e r t o u n d e r s t a n d the computer program some background i n f o r m a t i o n on i n t e r f e r o m e t r y must be p r e s e n t e d . I n t e r f e r o m e t r y i s based upon the i n t e r f e r e n c e of two beams E1 ( t h e r e f e r e n c e beam) and E2 ( t h e probe beam). S i n c e b oth beams i n our case have a common source (a ruby l a s e r ) the i n t e r f e r e n c e p a t t e r n w i l l depend o n l y on the phase d i f f e r e n c e between them. In the absence of plasma the phase d i f f e r e n c e i s caused by d i f f e r e n t p a t h l e n g t h s . T h i s s i t u a t i o n i s c h a r a c t e r i s e d by e v e n l y spaced s t r a i g h t f r i n g e s . I f a plasma i s p r e s e n t the phase d i f f e r e n c e w i l l depend on the o p t i c a l p a t h l e n g t h i n t h e plasma as the plasma has an index of r e f r a c t i o n The phase d i f f e r e n c e i n a c y l i n d r i c a l symmetric plasma i s g i v e n by where M 0 i s the vacuum index of r e f r a c t i o n . F o l l o w i n g Fan and S q u i r e one makes a change of v a r i a b l e s t o get A b e l i n v e r s i o n i s performed t o f i n d M*(r) from A ( y ) ( f o r the geometry see the f i g u r e ) . The computer program does the a n a l y s i s of the i n t e r f e r o g r a m s . The e f f e c t s of t h e v a r i o u s s u b r o u t i n e s a r e dx) 67 o u t l i n e d s c h e m a t i c a l l y i n the f i g u r e . The c e n t e r of the f r i n g e i s c a l c u l a t e d u s i n g the f i r s t moment of the f r i n g e s h i f t . ^ C E N T E R " £ ( A ( i ) * y ( i ) ) / £ ( A ( i ) ) The r a d i u s i s c a l c u l a t e d by moving out from the c e n t e r u n t i l t he f r i n g e s h i f t e q u i d i s t a n t from the c e n t e r i s below the e r r o r amount. The phase s h i f t a t r a d i u s r i s found by a v e r a g i n g the phase s h i f t s a t r above and r below the c e n t e r . The phase s h i f t as a f u n c t i o n of y i s then A b e l u n f o l d e d t o p r o v i d e n ( r ) u s i n g the computer program of Fan and S q u i r e . A b a s i c a p p r o x i m a t i o n used i s t h a t M*=/( 1-n/n C R )-1 i s about n / ( 2 * n C R ) The program i s s t r a i g h t f o r w a r d t o use. One d i g i t i z e s a photograph so t h a t an u n d e v i a t e d f r i n g e f o l l o w s a v e r t i c a l l i n e . A l l f r i n g e s s h o u l d s t a r t e v e n l y and end e v e n l y w i t h a margin around t h e f r i n g e s . The b e g i n n i n g and end of a f r i n g e s h o u l d l i e on t h e u n d e v i a t e d p a r t of the f r i n g e . At the b e g i n n i n g of the d a t a f i l e c e r t a i n p a rameters s h o u l d be s p e c i f i e d . These a r e : (1) photo number (2) p o s i t i o n of minimum and maximum y (chosen so t h a t t h e r e i s a s m a l l margin around the f r i n g e s ) (3) number of e v e n l y spaced s l i c e s between ymin and ymax (up t o 150) , the number of f r i n g e s (up t o 125) and the amount of smoothing n (averages 2n+1 p o i n t s ) (4) the s p a c i n g between f r i n g e s on the photograph and the a c t u a l s p a c i n g of f r i n g e s ( m a g n i f i c a t i o n = photo s p a c i n g / a c t u a l s p a c i n g ) (5) e r r o r ( t o l e r a n c e ) i n r a d i a n s :below t h i s l e v e l phase s h i f t s a r e made z e r o and s i g n (+1 or -1) of the dominant f r i n g e s h i f t ( s h i f t s of o p p o s i t e s i g n a r e made z e r o ) . B e f o r e the da t a f o r 6 8 i I m i Digitized a iv Dlint (*A - -A -A - A Abel Inversion Geometry Y Plasma u(r) Probg a. Subroutine Effects ii Readin —»• D •*— v Abdata iii Dlact -i-a -rA - r A •A +A •+A +A +4 vi Abel A Phase Shift n Density b. F i g u r e A - 1 69 each f r i n g e the number of the f r i n g e and the number of p o i n t s on the f r i n g e should be s p e c i f i e d . The p o i n t s along the f r i n g e do not have to be evenly spaced; the program does the i n t e r p o l a t i o n to even s p a c i n g . The amount of memory r e q u i r e d to run program depends upon the s i z e of the a r r a y s . The program as l i s t e d r e q u i r e d about 20K f o r the o b j e c t code compiled under FORTRAN IV-H and the a r r a y s r e q u i r e about 300K. The e x e c u t i o n time r e q u i r e d i s g e n e r a l l y under 5s CPU time on an AMDAHL 470 V-8 f o r an i n t e r f e r o g r a m of 100 f r i n g e s and 90 s l i c e s per f r i n g e . 70 10 NY, NFR, D) DELINT, NY, NFR, ERROR, D, SIGN, NFR, DELRAD, IRADS, FLAG, SIGN, DIMENSION RAW(150,125), DELACT(150,125), DELINT(150,125), 1 DELRAD(150,125), IRADS(1000) DIMENSION ROUT(1000), DENSO000), XOUT(2500), YOUT(2500), 1 ZOUT(2500) DIMENSION XT(2500), YT(2500), ZT(2500) LOGICAL FLAG(1000) DO 10 I = 1, 1000 FLAG(I) = .TRUE. CALL READIN(RAW, NY, NFR, D, DELY, FRACT, SIGN, ERROR, 1 YMAX, YO, NSMTH) WRITE (6,120) CALL DLACT(RAW, DELACT, WRITE (6,120) CALL DLINT(RAW, DELACT, 1 NSMTH) WRITE (6,120) CALL ABDATA(DELINT, NY, 1 ERROR) NFRMAX = NFR - 1 C COMPUTE NUMBER OF DATA POINTS TO BE USED IN PLOTTING PROGRAMS NOUT = 2 1 NP = NFRMAX * NOUT WRITE (8,130) NP N = 0 DO 50 IFRNO = 1, NFRMAX C DO ABEL INVERSION WHERE POSSIBLE IF ( .NOT. FLAG(IFRNO)) GO TO 30 NDS = IRADS(IFRNO) + 1 WRITE (6,140) IFRNO CALL ABEL(DELRAD, NDS, IFRNO, 1 DELY, FRACT) C CREATE DATA ARRAYS FOR 3-D PLOTS C XOUT = Z POSITION (DIRECTION LASER IS C YOUT = RADIUS C ZOUT = DENSITY DO 20 I I = 1, NOUT N = N + 1 = FLOAT(IFRNO) * FRACT = ROUT(II) DENS(I I ) ROUT, DENS, NOUT, D, INCREASING FRINGE NO.) WITH ZEROS * FRACT / D XOUT(N) YOUT(N) 20 ZOUT(N) = GO TO 50 C IF NO FRINGE SHIFT ,PAD 30 DO 40 J = 1, NOUT N = N + 1 STEP = (YMAX - YO) / 2. / FLOAT(NOUT) XOUT(N) = FLOAT(IFRNO) * FRACT YOUT(N) = (FLOAT(J) - 1.) * STEP ZOUT(N) = 0.000 40 CONTINUE 50 CONTINUE Q******************************************** C REWRITE DATA IN 'PROPER' ORDER IE REVERSE THE Z DIRECTION C THIS SECTION REMOVABLE IF FRINGES DIGITIZED CORRECTLY C CHANGE Z COORDINATE N = 1 DO 70 I = 1, NFRMAX 71 DO 60 J = 1, NOUT XOUT(N) = FL0AT(119 - I ) * FRACT N = N + 1 60 CONTINUE 70 CONTINUE C REARRANGE N = 1 DO 90 I = 1, NFRMAX DO 80 J = 1, NOUT XT(N) = XOUT(NP - I*NOUT + J) YT(N) = YOUT(NP - I*NOUT + J) ZT(N) = ZOUT(NP - I*NOUT + J) N = N + 1 80 CONTINUE 90 CONTINUE C REWRITE DO 100 I = 1, NP XOUT(I) = X T ( I ) YOUT(I) = Y T ( I ) ZOUT(I) = ZT(I) 100 CONTINUE r j * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C WRITE 3-D PLOT DATA ON UNIT 8 (= SOME FILE) WRITE (8,150) (XOUT(I),YOUT(I),ZOUT(I),I=1,NP) 110 STOP 120 FORMAT ( 1 ', ' ' ) 130 FORMAT (14) 140 FORMAT (' ', ' RESULT FOR FRINGE NO', 14) 150 FORMAT ((6(G10.4,2X))) END C=========================================================== SUBROUTINE READIN(RAW, NY, NFR, D, DELY, FRACT, SIGN, 1 ERROR, YMAX, YOO, NSMTH) C THIS PROGRAM IS TO DO INTERPOLATION ON DIGITIZER DATA DIMENSION Z O 0 0 0 ) , Y ( 1 0 0 0 ) , RAW( 1 50 , 1 25) , I Z(lOOO), 1 I Y O 0 0 0 ) C READIN ALL DATA RELEVANT TO THE PROGRAMS EXECUTION WRITE (8,100) READ (5,110) I PHOTO -WRITE (8,110) I PHOTO WRITE (6,230) I PHOTO READ (5,120) YO, YMAX WRITE (6,130) YO, YMAX READ (5,140) NY, NFR, NSMTH WRITE (6,150) NY, NFR, NSMTH READ (5,160) D r FRACT WRITE (6,170) D, FRACT READ (5,180) ERROR, SIGN WRITE (6,190) ERROR, SIGN C I PHOTO = PHOTO NUMBER C YO= MIN.Y VALUE YMAX= MAX Y VALUE C NY= NO. OF Y SLICES USED NFR= NO. OF FRINGES NSMTH= NO. OF C POINTS TO BE AVERAGED IN SMOOTHING C NY=NO. OF EVENLY SPACED YSLICES (Y IS ALONG FRINGE) C NFR=NO OF FRINGES MAX.125 (MAX NO OF Y =150 -BOTH LIMITED C BY THE DIMENSION STATEMENTS) 72 C D= FRINGE SEP. ON PHOTO FRACT= ACTUAL FRINGE SEP. AT JET C ERROR= TOLERANCE ON FRINGE SHIFT IN RADIANS SIGN= + OR - 1 DELY = (YMAX - YO) / FLOAT(NY) YOO = YO C DO 80 IFRNO = 1, NFR C SET UP ARRAYS FOR ONE FRINGE (FRINGE NO=IFRNO) READ (5,200) I F , N READ (5,210) ( I Z ( I ) , I Y ( I ) , 1 = 1 , N ) C N=1 IMPLIES A STRAIGHT FRINGE IF (N .NE. 1) GO TO 10 N = 3 I Y ( 1 ) = IFIX(YMAX*1000.) I Y ( 2 ) = IFIX((YMAX + YOO)/2.* 1000.) I Y ( 3 ) = IFIX(YOO*1000.) I Z ( 2 ) = I Z ( 1 ) I Z ( 3 ) = I Z ( 1 ) C FIRST SEE I F IY IS IN DESCENDING OR ASCENDING ORDER 10 IF ( I Y ( 2 ) - I Y ( 1 ) ) 20, 20, 40 C DESCENDING ORDER 20 DO 30 J J = 1, N Z ( J J ) = F L 0 A T ( I Z ( J J ) ) / 1000. Y ( J J ) = F L O A T ( I Y ( J J ) ) / 1000. 30 CONTINUE GO TO 60 C ASCENDING ORDER 40 DO 50 J J = 1 , N Z(N - J J + 1) = F L O A T ( l Z ( J J ) ) / 1000. Y(N - J J + 1) = F L O A T ( I Y ( J J ) ) / 1000. 50 CONTINUE 60 CONTINUE C CALCULATE THE RAW DATA C SLICE Y=1 MUST CROSS UNDEVIATED FRINGES (ALSO Y=NY) C RAW(A,B)=Z DIST. FROM Z ORIGIN FOR FRINGE B ON LINE A C DO SPLINE INTERPOLATION CALL SPLINA(Y, Z, N) YO = YOO DO 70 J = 1, NY CALL TERPA(YO, OUT) RAW(J,IFRNO) = OUT YO = YO + DELY 70 CONTINUE 80 CONTINUE DO 90 I = 1, NY 90 WRITE (6,220) (RAW(I,J),J=1,NFR) RETURN 1 00 FORMAT ( ' ' , ' ' ) 110 FORMAT (13) 120 FORMAT -(2F8.3) 130 FORMAT (' ', ' YO=', F8.3, ' YMAX=', F8.3) 140 FORMAT (313) 150 FORMAT (' ', ' YSLICES =', 14, ' FRINGES =', 14, 1 ' SMTH =', 14) 160 FORMAT (2F8.3) 170 FORMAT (' ', ' FRINGESPACEPHOTO=', F8.3, 73 1 ' FRINGESPACEJET=', F8.3) 180 FORMAT (2F8.4) 190 FORMAT (' ' ERROR RADIANS=', F8.4, ' SIGN USED', F8.4) 200 FORMAT (16, 15) 210 FORMAT ( 6 ( 1 X , 2 I 6 ) ) 220 FORMAT (' ', (14F7.3)) 230 FORMAT (' * , ' PHOTO * ' , 15) END C========================================================= SUBROUTINE DLACT(RAW, DELACT, NY, NFR, D) C CALCULATE THE ACTUAL PHASE SHIFT(ON A FRINGE) DIMENSION RAWO50, 125) , DELACT( 1 50 , 1 25) DO 20 IY = 1, NY DO 10 IFRNO = 1, NFR C USE AVERAGE OF END POINTS AS POSITION OF UNDEVIATED FRINGE DLAVE = (RAW(1,IFRNO) + RAW(NY,IFRNO)) / 2. C D=FRINGE SPACING ON SOURCE DELACT(IY,IFRNO) = (RAW(IY,IFRNO) - DLAVE) * 6.28 / D 10 CONTINUE WRITE (6,30) (DELACT(IY,J),J=1,NFR) 20 CONTINUE RETURN 30 FORMAT (' ', 20F6.2) END C========================================================== SUBROUTINE D.LI NT (RAW, DELACT, DELI NT, NY, NFR, ERROR, 1 D, SIGN, NSMTH) DIMENSION RAW(150,125), DELACT(150,125), DELINT(150,125) DIMENSION Z( 1 2 5 ) , DEL(125), Y(10000) C SPLINE INTERPOLATION USED C CALCULATE THE PHASE SHIFT ALONG LINE OF UNDEVIATED FRINGE NFRMAX = NFR - 1 DO 90 IY = 1, NY C FIRST SEE IF Z IS IN DESCENDING OR ASCENDING ORDER IF (RAW(1,2) - RAW(1,1)) 10, 10, 30 C DESCENDING ORDER 10 DO 20 IFRNO = 1, NFRMAX Z(IFRNO) = RAW(IY,IFRNO) DEL(IFRNO) = DELACT(IY,IFRNO) 20 CONTINUE GO TO 50 C ASCENDING ORDER 30 DO 40 IFRNO = 1, NFRMAX Z(NFRMAX - IFRNO + 1) = RAW(IY,IFRNO) DEL(NFRMAX - IFRNO + 1) = DELACT(IY,IFRNO) 40 CONTINUE 50 CONTINUE C DO THE ACTUAL INTERPOLATION CALL SPLINA(Z, DEL, NFRMAX) DO 60 IFRNO = 1, NFRMAX C USE AVERAGE OF END POINTS AS POSITION OF UNDEVIATED FRINGE ZO = (RAW(1,IFRNO) + RAW(NY,IFRNO)) / 2. CALL TERPA(ZO, OUT) DELINT(IY,IFRNO) = OUT IF (OUT*SIGN .LT. 0 .OR. ABS(OUT) .LT. ERROR) 1 DELINT(IY,IFRNO) = 0.0 74 C DUE TO ERRORS IN DIGITIZING,SMALL PHASE SHIFTS ARE MADE ZERO C ALSO IF FRINGE GOES 'WRONG' WAY PHASE SHIFT IS MADE ZERO 60 CONTINUE C SMOOTH THE DATA DO 70 KKK = 1, NFRMAX Y(KKK) = DELINT(IY,KKK) 70 CONTINUE CALL GSMTH(Y, NSMTH, NFRMAX) DO 80 KKK = 1, NFRMAX DELINT(IY,KKK) = Y(KKK) 80 CONTINUE WRITE (6,110) (DELINT(IY,J),J=1,NFRMAX) 90 CONTINUE 100 RETURN 110 FORMAT (* ', 20F6.2) END C========================================================= SUBROUTINE ABDATA(DELINT, NY, NFR, DELRAD, IRADS, FLAG, 1 SIGN, ERROR) C PREPARE DATA FOR ABEL INVERSION PROGRAM LOGICAL FLAG(1000) DIMENSION ICNTR(IOOO), IRADSO 000), DELRAD ( 1 50 , 1 25) , 1 DELINTO 50, 125) C CALCULATE CENTRE OF FRINGE PATTERN C CENTRE =SUM(IY*DELINT(IY,IFRNO))/SUM(DELINT(lY,IFRNO) NFRMAX = NFR - 1 DO 30 IFRNO = 1, NFRMAX SUM1 = O SUM2 = 0 DO 10 IY = 1 , NY SUM1 = SUM1 + FLOAT(IY) * DELINT(IY,IFRNO) SUM2 = SUM2 + DELINT(IY,IFRNO) 10 CONTINUE C I F TOTAL AMT. FRINGE SHIFT TOO SMALL REPORT NO FRINGE SHIFT IF (ABS(SUM2) .LT. ERROR) GO TO 20 ICNTR(IFRNO) = (SUM1/SUM2) + .5 WRITE (6,110) IFRNO, ICNTR(IFRNO) GO TO 30 20 FLAG(IFRNO) = .FALSE. WRITE (6,100) IFRNO 30 CONTINUE C NOW CALCULATE RADIUS OF FRINGE PATTERN C RADIUS = MIN. DIST. FROM CENTRE WHERE DEL=0 ON BOTH SIDES DO 70 IFRNO = 1, NFRMAX IF ( .NOT. FLAG(IFRNO)) GO TO 70 DO 40 I = 1 , NY IP = ICNTR(IFRNO) + I IM = ICNTR(IFRNO) - I C IF CANNOT FIND ZERO FRINGE SHIFT ON ENDS REPORT ERROR IF ( I P .GT. NY .OR. IM .LT. 1) GO TO 60 IF (ABS(DELINT(IP,IFRNO)) .LT. ERROR .AND. 1 ABS(DELINT(IM,IFRNO)) .LT. ERROR) GO TO 50 40 CONTINUE 50 WRITE (6,130) IFRNO, I IRADS(IFRNO) = I IF (IRADS(IFRNO) .GT. 2) GO TO 70 75 60 WRITE (6,120) IFRNO FLAG(IFRNO) = .FALSE. 70 CONTINUE C CALCULATE DEL AS A FUNCTION OF RADIUS C DELRAD= AVERAGE OF DEL AT PLUS/MINUS EQUAL DIST. FROM CENTRE DO 90 IFRNO = 1, NFRMAX IF ( .NOT. FLAG(IFRNO)) GO TO 90 JMAX = IRADS(IFRNO) + 1 DO 80 J = 1, JMAX JDIF = JMAX - J JP = ICNTR(IFRNO) + JDIF JM = ICNTR(IFRNO) - JDIF DELRAD(J,IFRNO) = (DELINT(JP,IFRNO) + DELINT(JM,IFRNO)) 1 SIGN C SIGN HERE BECAUSE OF WAY FRINGE SHIFT CALCULATED 80 CONTINUE 90 CONTINUE RETURN 100 FORMAT (' ', 'NO SHIFT FRINGE NO', 1 4 ) 110 FORMAT (' ', ' CENTRE OF FRINGE', 1 5 , ' I S ' , 1 5 ) 120 FORMAT (* ', ' FRINGE NO', 1 4 , ' IS TOO SKEWED/TOO LARGE /T 1L' ) 130 FORMAT (' ', ' FRINGE NO', 1 4 , ' HAS RADIUS', 1 4 ) END C================================================================ C C P C PROGRAM A B S C C ABEL INVERSION - DIRECT C============================================================= C WRITTEN BY L.S. FAN AND W. SQUIRE. C REF: COMP. PHYS. COMMUN. 10 (1975) 98 C XLO=CONSTANT LIMIT OF INTEGRAL C NA=NUMBER OF DATA POINTS ( WHICH DO C NOT HAVE TO BE EQUALLY SPACED) C NP=NUMBER OF DIVISIONS FOR INTEGRATION, C (LIMITED TO 300 BY DIMENSION STATEMENT) C NSS=NUMBER OF STEPS OMITTED NEAR SINGULARITY C NOUT=NUMBER OF OUTPUT POINTS C X=ARRAY WITH INDEPENDENT VARIABLE IN C DESCENDING ORDER C Y=ARRAY WITH CORRESPONDing VALUES OF DEPENDENT VARIABLE C========================================================= SUBROUTINE ABEL(DELRAD, NDS, IFRNO, ROUT, DENS, NOUT, D, 1 DELY, FRACT) C PROGRAM FOR SOLVING ABEL'S C EQUATION USING TABULATE DATA DIMENSION U( 3 0 0 ) , T ( 3 0 0 ) , X ( 3 0 0 ) , Y ( 3 0 0 ) , DELRAD(150,125), 1 IRADSO 000) DIMENSION ROUT(1000), DENSOOOO) C CREATE DATA ARRAYS YSCALE = DELY * FRACT / D C YSCALE=DISTANCE 1 UNIT IS IN REAL UNITS (MM) DO 10 I = 1, NDS X ( I ) = (NDS - I ) * YSCALE Y ( I ) = DELRAD(I,IFRNO) 10 CONTINUE C WRITE OUT DATA ARRAYS 76 DO 20 I = 1, NDS WRITE (6,60) X ( I ) , 7(1) 20 CONTINUE C SET UP SPECIFICATIONS FOR PROGRAM IOUT = 6 XLO = (FLOAT(NDS) - 1) * YSCALE NP = 50 NA = NDS NSS = 2 NOUT = 21 WLAMDA = 694.3E-06 / 2. / 3.1415926 C WLAMDA = WAVELENGTH OF PROBE LASER (RUBY) (MM)/2PI DSCALE = (10.6/.6943) ** 2 C DSCALE=NCRPROBE/NCRC02 WRITE (6,40) XLO, NA, NP, NSS, NOUT WRITE (IOUT,50) C WRITE OUT HEADINGS FOR OUTPUT WRITE (IOUT,70) C INVERSION OF ABEL'S INTEGRAL EQUATION C SMOOTHING DATA AND INTERPOLATION BY CALLING SPLINA C SET UP INTERPOLATION OF INPUT DATA CALL SPLINA(X, Y, NA) H = -XLO / FLOAT(NP) C INVERT ABEL INTEGRAL BY CALLING VLIGM NP = NP - NSS CALL VLIGM(XLO, H, NP, U, T) U( 1 ) = 0.0 T(1) = XLO C INTERPOLATION AND EXTRAPOLATION FOR OUTPUT NS = NP + 1 CALL SPLINA(T, U, NS) DO 30 I = 1, NOUT V = FLOAT(I - 1) / FLOAT(NOUT - 1) * XLO CALL TERPA (V, W ) VO = V WO = W * WLAMDA ROUT(I) = VO DENS (I ) = WO * 2. * DSCALE C IF DENS TOO LARGE DO NOT USE APPROXIMATIONS IF (DENS(I) .GT. .1) DENS(I) = (1. - (1. - WO)**2) * 1 DSCALE C IF DENSITY .LT. ZERO, MAKE IT ZERO (WILL FOUL UP RAY TRACING C OTHERWISE) IF (DENS(I) .LT. 0.0) DENS(I) = 0.0 WRITE (IOUT,80) ROU T ( l ) , DENS(I) 30 CONTINUE RETURN 40 FORMAT (' ', ' XLO=', F8.3, ' NA=', 14, ' NP=', 14, 1 ' NSS=, 14, ' NOUT=', 14) 50 FORMAT ('1') 60 FORMAT (2F10.4) 70 FORMAT (' 3X, 'RADIUS', 29X, 'N(CALCUL.)'/) 80 FORMAT (G10.4, 26X, G10.4, 8X, G10.4) END C= = = = = = = = = = = = = = = = = = = = = === = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = === = = = = SUBROUTINE SPLINA(X, Y, N) 77 C SUBROUTINE USING RAMAMOORTHY AND NARAYANA C SPLINE PROCEDURE FOR INTERPOLATION C SUBROUTINE HAS TWO ENTRIES SPLINA AND TERPA C ARGUMENTS FOR SPLINA ARE: C X=ARRAY WITH INDEPENDENT VARIABLE IN C DESCENDING ORDER C Y=ARRAY WITH CORRESPONDing VALUES OF DEPENDENT VARIABLE C N=NUMBER OF DATA POINTS C ARGUMENTS FOR TERPA ARE: C XV=VALUE OF INDEPENDENT VARIABLE C YV=LOCATION WHERE CORRESPONDING VALUE OF DEPENDENT C VARIABLE IS PUT DIMENSION W(300), Q(300), B ( 3 0 0 ) , A ( 3 0 0 ) , C ( 3 0 0 ) , S ( 3 0 0 ) , 1 Z(300) DIMENSION X ( N ) , Y(N) DATA A ( 1 ) , C(1) / ~ 1 . , 0.0/ C CHECK TO SEE IF DATA POINTS ARE TOO CLOSE (MAYBE EQUAL) DO 10 J = 2, N W(J) = X ( J ) - X ( J - 1) IF (ABS(W(J)) .LT. .005) WRITE (6,60) X ( J ) 10 CONTINUE NM = N - 1 DO 20 J = 2, NM WJ = W(J) WP = W(J + 1) WS = WJ + WP QJ = WJ / WS • Q(J) = QJ QA = 1. - . 5 * Q J * A ( J - 1) A ( J ) = .5 * ( 1 . - QJ) / QA B ( J ) = 3. * (WP*Y(J - 1) - WS*Y(J) + WJ*Y(J + 1)) / WP / 1 WJ / WS C ( J ) = (B ( J ) - .5*QJ*C(J - 1)) / QA 20 CONTINUE S(N) = C(NM) / ( 1 . + A(NM)) S(NM) = S(N) NMM = N - 2 DO 30 J J = 1, NMM J = NMM - J J + 1 S ( J ) = C ( J ) - A ( J ) * S ( J + 1) 30 CONTINUE RETURN C============================================================= ENTRY TERPA(XV,YV) C INTERPOLATION PROCEDURE DO 40 J J = 2, N J = J J IF (XV .LT. X ( J ) ) GO TO 40 GO TO 50 40 CONTINUE 50 WJ = W(J) D1 = (XV - X ( J - 1)) / WJ D2 = ( X ( J ) - XV) / WJ D3 = WJ * WJ / 6. YV = D1 * ( Y ( J ) + D3*(D1*D1 - 1 . ) * S ( J ) ) YV = YV + D2 * ( Y ( J - 1) + D3*(D2*D2 - 1 . ) * S ( J - 1)) RETURN 60 FORMAT (' G12.5, ' CHECK:TWO POINTS VERY CLOSE ') END C=========================================================== SUBROUTINE VLIGM(XLO, H, NP, U, T) C SUBROUTINE FOR SOLVING LINEAR VOLTERRA INTEGRAL C EQUATION OF THE FIRST KIND USING THE GENERALIZED C MIDPOINT RULE WITH THE KERNEL AS WEIGHT FUNCTION C XLO=LOWER LIMIT OF INTEGRAL C H=STEP SIZE C NP=NUMBER OF STEPS C KERNEL WITH RESPECT TO T C U=SINGLY SUBSCRIPTED ARRAY FOR SOLUTION C T=SINGLY SUBSCRIPTED ARRAY FOR INDEPENDENT VARIABLE DIMENSION U ( 1 ) , T(1) HXT(X,TM) = -2. * SQRT(TM*TM - X*X) S = 0.5 * H X = XLO + H T(2) = X - S HLO = HXT(X,XLO) HUP = HXT(X,X) CALL TERPA(X, FXX) U(2) = FXX / (HUP - HLO) DO 20 I = 2, NP X = H * FLOAT(I) + XLO CALL TERPA(X, FXX) T( I + 1) = X - S SUM = 0.0 HUP = HXT(X,XLO) NM = 1 - 1 DO 10 J = 1, NM HLO = HUP HUP = HXT(X,T(J + 1) + S) 10 SUM = SUM + U ( J + 1) * (HUP - HLO) U(I + 1) = (FXX - SUM) / (HXT(X,X) - HUP) 20 CONTINUE RETURN END C========================================================= SUBROUTINE GSMTH(Y, N, NP) C DOES GENERAL SMOOTHING USING 2N+1 POINT AVERAGING DIMENSION 7(100 0 0 ) , WY(10000) DO 30 I = 1, NP SUM = Y ( I ) NU = MIN0(I - 1,NP - I,N) IF (NU .EQ. 0) GO TO 20 DO 10 J = 1, NU SUM = SUM + Y ( I - J ) + Y ( I + J ) 10 CONTINUE 20 WY(I) = SUM / (2.*FLOAT(NU) + 1 . ) 30 CONTINUE DO 40 K = 1, NP Y(K) = WY(K) 40 CONTINUE RETURN END 7 9 APPENDIX 2 RAMAN ENERGY CALCULATIONS The c a l c u l a t i o n s t o f o l l o w w i l l show the d e n s i t y r e g i o n s where SRS can oc c u r and what a r e the phase v e l o c i t i e s t h a t can occur . Wave M a t c h i n g ( 1 ) CJq =us +o>K ( 2 ) k 0=k s+k D i s p e r s i o n R e l a t i o n s ( 3 ) C L > K 2 = C J p 2+3/cTk 2/m ( 4 ) o ) s 2 = c j p 2 + c 2 k s 2 ( 5 ) a > 0 2 = c j p 2 + c 2 k o 2 ( 1 ) and ( 2 ) i n t o ( 4 ) y i e l d s 6(a) ( w 0 - c j K ) * * 2 = w p 2 + c 2 ( k 0 - k ) * * 2 Expanding and ( 5 ) t o e l i m i n a t e C J 0 2 6(b) w K 2 - 2 o 0 w K = c 2 ( k 2 - 2 k k 0 ) = c 2 k 0 2 U 2 - 2 £ ) (k=$k Q) E l i m i n a t e c 2 k 0 2 by ( 5 ) ( 7 ) u K 2 - 2 c u 0 u K = (cu 0 2-a> p 2) U 2-2£) Rep l a c e C J k 2 by (3) and d i v i d e by C J P 2 ( 8 ) 1 + 3 K T £ 2 ( < J 0 2 / c j f 2 - 1 ) / ( m c 2 ) - 2 c J 0 / w p V / ( 1 + 3 K T ^ 2 ( c j 0 2 / c j p 2 - 1 )/(mc 2)) = K V " p 2 - 1 ) ( { 2 - 2 { ) Rearrange and c a l l cj 0/cjp=j/(n C R/n)=a ( 9 ) £ 2 - 2 £ + ( 2 a £ l + 3 K T £ 2 ( a 2 - 1 ) / ( m c 2 ) ) - 1 - 3 K T £ 2 ( a 2 - 1 ) / ( m c 2 ) ) / ( a 2 - 1 ) = 0 L e t T=0 ( 1 0 ) £ 2 - 2 £ + ( 2 a - 1 ) / ( a 2 - 1 ) = 0 T h i s q u a d r a t i c has r e a l r o o t s i f a>2 or n < n C R / 4 . The phase v e l o c i t y i s e a s i l y c a l c u l a t e d . ( 1 1 ) v p H /c=/(w P 2 + 3KTk 2/m)/(kc) ( 1 2 ) v / c = / ( o > p 2 / k 2 / c 2 + 3 K T / ( m c 2 ) ) L e t t i n g k=£k 0 and e l i m i n a t i n g k Q 2 c 2 by (5) (13) v ? H /c=/(o) f 2 / U 2 ( c o 0 2 - W p 2 ) ) + 3 K T / ( m c 2 ) ) . F i n a l l y we get (14) v p H / C = / ( 1 / U 2 ( a 2 - 1 ))+3*T/(mc 2)) A g a i n l e t T=0 (15) mv p H 2 / 2 = m c 2 / 2 / U 2 ( a 2 - 1 )) . T h i s i s the e x p r e s s i o n t h a t was p l o t t e d . 81 REFERENCES B a l d i s , H . A . , W a l s h , C . J . Phys. F l u i d s 26,1364 (1983) B e r g e r , R . G . , B r o o k s , R . D . , P i e t r z y k , Z . A . Phys. F l u i d s 26,353 (1983) B e r n a r d , J . E . Ph.D T h e s i s 1984 ( U n i v e r s i t y of B r i t i s h Columbia) Born,M.,Wolf,E. P r i n c i p l e s of O p t i c s (6 t h . Ed.) Pergammon P r e s s B u r a k , I . , S t e i n f e l d , J . I . , S u t t o n , D . G . J . Q u a n t . S p e c t r o s e . 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