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Interaction of CO2 laser light with a dense Z-pinch plasma Albrecht, Georg F. 1979

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INTERACTION OF C0 2 LASER LIGHT WITH A DENSE Z-PINCH PLASMA by GEORG F. ALBRECHT D i p l . Phys, U. of S t u t t g a r t , 1974 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Department of Phys i c s v We accept t h i s t h e s i s as conforming to the re q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1979 0 Georg F. A l b r e c h t , 1979 I n 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 t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e 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 a n d s t u d y . I f u r t h e r a g r e e 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 c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r 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 n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V6T 1W5 B P 75-51 1 E Research Supervisor: J . Meyer Ab s t r a c t The i n t e r a c t i o n of a 250 MW CO^ l a s e r pulse w i t h a Z-pinch plasma has been observed. Heating by inverse bremsstrahlung and s t i m u l a t e d ^ . • rr i c . i n i 7 e l e c t r o n s B r i l l o u m s c a t t e r i n g o f f a plasma w i t h a few times 1 0 w 5 . cnw T^ °u 150 eV and temperature s c a l e lengths of a few mm i s shown to occur. The observed angular dependence of the B r i l l o u i n backscattered l i g h t i s i n good agreement w i t h - c u r r e n t t h e o r i e s . Some of the backscattered l i g h t shows i n t e n s i t y modulations at the e l e c t r o n gyro frequency. F i n a l l y , the s u c c e s s f u l nano second gat i n g of an O p t i c a l M u l t i c h a n n e l Analyser i s described and a p p l i e d to s p e c t r o s c o p i c a l d e n s i t y measurements of the plasma. I I TABLE OF CONTENTS Page ABSTRACT i i i LIST OF SYMBOLS i v INTRODUCTION 1 CHAPTER I : Theory of parametric decay i n an i n f i n i t e homogeneous^.plasma^and: i t s a p p l i c a t i o n to sti 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 3 1.1 I n t r o d u c t i o n 3 1.2 Ou t l i n e of the theory 5 1.3 D e r i v a t i o n of the general d i s p e r s i o n r e l a t i o n 6 1.4 S p e c i a l i z a t i o n f o r st i m u l a t e d b a c k s c a t t e r i n g 15 1.5 Stimulated B r i l l o u i n S c a t t e r i n g 19 1.6 The cj - k diagram 23 1.7 L i m i t a t i o n s of the t h e o r e t i c a l model 25 1.8 Numerical values 26 CHAPTER I I : Experimental i n v e s t i g a t i o n of the backscattered and t r a n s m i t t e d CO2 l a s e r l i g h t 30 2.1 I n t r o d u c t i o n 30 2.2 The Z-pinch plasma, measurements of r a d i u s , temperature and d e n s i t y and the CO2 l a s e r used f o r the laser-plasma i n t e r a c t i o n s t u d i e s 32 2.3 Experimental p r o v i s i o n s 35 2.4 S p e c t r a l l y i n t e g r a t e d backscattered C0£ l a s e r l i g h t as a f u n c t i o n of pinch-time 39 2.5 S p e c t r a l l y i n t e g r a t e d t r a n s m i t t e d CO2 l a s e r l i g h t as a f u n c t i o n of pinch-time 41 2.6 S p e c t r a l l y r e s o l v e d backscattered CO2 l a s e r l i g h t at pinch time t = 0 ± 25 nsec and the angular dependence of the backscattered l i g h t 45 I I I TABLE OF CONTENTS Page CHAPTER I I I : E v a l u a t i o n of the experimental r e s u l t s 49 3.1 The t r a n s m i t t e d C0 2 l a s e r l i g h t 49 3.2 The backscattered CO2 l a s e r l i g h t 55 3.21 Enhancement of the backscattered CO2 l a s e r l i g h t above thermal l e v e l s and the r e s u l t i n g i o n wave amplitudes i n the plasma 56 3.22 D i s c u s s i o n of the observed wavelength s h i f t ': of the backscattered CO2 l a s e r l i g h t 62 3.23 The angular dependence of the backscattered C0 2 l a s e r l i g h t and comparison with theory 67 3.24 The wavelength dependence of the l i g h t b ackscattered through the " s m a l l mask" 70 CONCLUSIONS 76 CHAPTER IV: The f a s t g a t i n g of an OMA and a c o n t r i b u t i o n to the d i a g n o s t i c s of the Z-pinch plasma 78 4.1 I n t r o d u c t i o n 78 4.2 Nanosecond g a t i n g of an OMA 82 4.3 Spectroscopic measurements of plasma d e n s i t y and temperature using the 4686S 1 ine of He I I 91 SPECIFICATIONS 94 REFERENCES 101 IV wave vec t o r s of the plasma d e n s i t y f l u c t u a t i o n s , the i n c i d e n t and the s c a t t e r e d l i g h t waves frequencies of the plasma d e n s i t y f l u c t u a t i o n , the i n c i d e n t and the s c a t t e r e d l i g h t waves e l e c t r i c f i e l d s of the i n c i d e n t and s c a t t e r e d electromagnetic waves currents i n the plasma at the s c a t t e r e d frequencies c o n d u c t i v i t y of the plasma at the s c a t t e r e d frequencies amplitude of the enhanced d e n s i t y f l u c t u a t i o n e l e c t r i c u n i t charge of e i t h e r i o n or e l e c t r o n e l e c t r i c u n i t charge of e l e c t r o n quiver v e l o c i t y of an e l e c t r o n i n the i n c i d e n t electromagnetic wave at frequencies ± u> mass of the e l e c t r o n mass of the atom (He) d i e l e c t r i c constant at w+ e l e c t r o n i c , i o n i c s u s c e p t i b i l i t y speed of l i g h t average e l e c t r o n d e n s i t y plasma frequencies of e l e c t r o n s / i o n s ponderomotive p o t e n t i a l see (1-12) high frequency coordinate of e l e c t r o n s o s c i l l a t i n g i r i an electromagnetic f i e l d c oordinate along which the e l e c t r i c f i e l d v a r i e s d i s t r i b u t i o n f u n c t i o n f o r i o n s , e l e c t r o n s Maxwellian e q u i l i b r i u m d i s t r i b u t i o n f u n c t i o n s e l f - c o n s i s t e n t p o t e n t i a l w i t h i n plasma damping of electromagnetic wave and ion wave (Landau and c o l l i s i o n a l ) Debye length angle between k and V q (Ch. I) a l s o angular coordinate (3.21) growth r a t e f o r a parametric decay i n s t a b i l i t y Shape f a c t o r of the Thomson s c a t t e r e d i o n feat u r e i o n , e l e c t r o n temperature inverse bremsstrahlung absorption length thermal c o n d u c t i v i t y Boltzmann' constant Coulomb log a r i t h m , f« 10 l a s e r l i g h t i n t e n s i t y i n c i d e n t on the plasma backscattered l a s e r l i g h t i n t e n s i t y (thermal, enhanced, denoted by s u p e r s c r i p t s ) angular divergence of backscattered l i g h t f o r d i f f e r e n t p h y s i c a l models length of i n t e r a c t i o n volume; s c a l e lengths inverse gain length of l i g h t a m p l i f i e d by SBS ion-acous t i c speed laser-plasma i n t e r a c t i o n volume VI to my f a t h e r , f o r h i s i n f i n i t e l o v e , t r u s t and support, to M a r i l y n , f o r a l l she taught me. to the pleasure of a l o v i n g woman's touch, i t i s the d i f f e r e n c e between l i v i n g and v egetating. VII 1. I n t r o d u c t i o n The current e f f o r t s i n c o n t r o l l e d thermonuclear f u s i o n research have brought about a great i n t e r e s t i n l a s e r .plasma-interaction s t u d i e s . The " i n e r t i a l confinement" type of experiment attempts to achieve c o n t r o l l e d f u s i o n by f o c u s s i n g enormous l a s e r powers (.5 peta watts) on small DT p e l l e t s (some lOOy diameter) and thus heat and compress the created plasma to temperatures and d e n s i t i e s at which thermonuclear 1 2 r e a c t i o n s y i e l d more energy than was put i n i n form of l a s e r energy. ' The key questions that a r i s e i n t h i s approach concern the c o u p l i n g of l a s e r l i g h t to the plasma, a problem of surmounting complexity. ; -The.type.of experiments i n which the plasma i s m a g n e t i c a l l y confined attempts to achieve c o n t r o l l e d f u s i o n not so much by reducing the mean fr e e path between r e a c t i o n s by plasma compression but by i n c r e a s i n g the confinement time to an extent that a s u f f i c i e n t amount of thermonuclear energy can be released before the plasma decays. One of the most important problems i n t h i s approach i s the h e a t i n g of the plasma to 5 6 s u f f i c i e n t temperatures. Dawson et a l . suggested that h i g h power l a s e r s be used to achieve s i g n i f i c a n t heating of m a g n e t i c a l l y confined plasmas because these types of plasmas are i n a d e n s i t y regime where the i n v e r s e bremsstrahlung absorption f o r CO2 l a s e r l i g h t becomes s i g n i f i c a n t . A number of experiments have been performed along these n. 5,7,8,9,28 lxnes. -In e i t h e r case i t . i s of great importance to i n v e s t i g a t e l a s e r plasma i n t e r a c t i o n s under d i f f e r e n t c o n d i t i o n s i n order to l e a r n about the p o s s i b l e p h y s i c a l processes i n v o l v e d and to a i d t h e o r i e s that t r y to p r e d i c t such processes. 2. The experiment described i n t h i s r e p o r t was set up to i n v e s t i g a t e which l a s e r plasma i n t e r a c t i o n processes can be studi e d w i t h the means c u r r e n t l y a v a i l a b l e i n t h i s l a b o r a t o r y . Chapter I describes the ba s i c s of the theory of one of the most important types of processes happening i n high power l a s e r plasma i n t e r s a c t i o n s , namely the parametric decay of a l i g h t wave i n plasma waves and s c a t t e r e d l i g h t waves. The case of sti 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 i s t r e a t e d i n some d e t a i l . Chapter I I describes the experiments performed w i t h a 250 MW CO2 l a s e r and a high d e n s i t y Z-pinch plasma and the r e s u l t s that were obtained. Chapter I I I discusses these r e s u l t s and conclusions are drawn about the occurrence of sti 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 , the angular divergence of i t s backscattered l i g h t and the i n t e n s i t y modulation of some of the backscattered l i g h t w i t h the gyro frequency of e l e c t r o n s . The abso r p t i o n of CO2 l a s e r l i g h t by inv e r s e bremsstrahlung i n some experiments i s v e r i f i e d . Chapter IV describes a c o n t r i b u t i o n to the d i a g n o s t i c s of the Z-pinch plasma. A new technique f o r s a t i s f a c t o r y nsec gat i n g of an O p t i c a l M u l t i c h a n n e l Analyser i s a p p l i e d to s p e c t r o s c o p i c a l s t u d i e s of the plasma d e n s i t y using the 4686S l i n e of He I I . 3. CHAPTER I Theory of parametric decay i n an i n f i n i t e , homogeneous plasma  and i t s a p p l i c a t i o n to 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 . 1.1 I n t r o d u c t i o n This chapter attempts to provide an e a s i l y readable p r e s e n t a t i o n of the theory of parametric decay i n s t a b i l i t i e s i n an i n f i n i t e homogeneous plasma. As i t i s n e i t h e r supposed to be a comprehensive review nor a cumbersome reproduction of work already done,"^ the p r e s e n t a t i o n i s l i m i t e d to the process most r e l e v a n t f o r the experiment to be described l a t e r , e.g. the process of lowest t h r e s h o l d and high growthrate namely stimulated B r i l l o u i n s c a t t e r i n g . Furthermore, the emphasis w i l l be on p h y s i c a l i n t e r p r e t a t i o n r a t h e r than the mathematical apparatus. F i r s t , a simple p i c t u r e of a parametric process s h a l l be given. Imagine an electromagnetic wave (& Q, wo) i n c i d e n t on a plasma with thermal d e n s i t y f l u c t u a t i o n s . Let us assume that the electromagnetic wave s c a t t e r s o f f a f o u r i e r component (1c, OJ) of these d e n s i t y f l u c t u a t i o n s thus c r e a t i n g a s c a t t e r e d electromagnetic wave at k = k^ — k (conservation of wave momentum) and frequency OJ = a)Q - oo (conservation of energy). In the case of s m a l l i n c i d e n t i n t e n s i t i e s t h i s s c a t t e r e d wave w i l l leave the plasma without p e r t u r b i n g i t any f u r t h e r . With i n c r e a s i n g i n c i d e n t i n t e n s i t i e s however, a s i t u a t i o n w i l l a r i s e i n which the i n t e n s i t y of the s c a t t e r e d l i g h t at" (u_, k*_) w i l l indeed be high enough to i t s e l f a f f e c t the plasma. Now we have the new s i t u a t i o n of two electromagnetic waves, namely one at (&Q> aj Q) and one at (1c , a)_) being simultaneously present i n the plasma. 4. I f both waves i n t e r a c t l i n e a r l y w i t h the plasma, nothing s p e c t a c u l a r w i l l happen and e l e c t r o n s w i l l o s c i l l a t e at frequencies w and o> . I f , however, the two waves couple to each other v i a the plasma, not only the frequencies at OJ and to w i l l occur i n the plasma, but a l s o the sum and d i f f e r e n c e frequencies to + to and to - to . I t w i l l be seen, t h a t ! t h i s o - o -non l i n e a r coupling indeed takes place v i a the ponderomotive f o r c e , to be explained l a t e r . Of these two frequencies, to + to and O) q - t o _ , the low frequency at m - oi w i l l couple much stronger to the plasma as the amplitude of an e l e c t r o n o s c i l l a t i n g i n an electromagnetic f i e l d i s p r o p o r t i o n a l to -^r. Therefore, the simultaneous presence of the two electromagnetic waves at (k » w ) and (k , to ) w i l l set up a plasma wave - > - - > - » - - > - - > - - > -a t k - k = k - (k - k ) = k and to - t o = 0 0 - (w - to) = t o . 0 - 0 0 0 - 0 0 In other words, the beat of the i n c i d e n t and s c a t t e r e d electromagnetic wave i n the plasma w i l l enhance e x a c t l y that d e n s i t y f l u c t u a t i o n which i n i t i a l l y s c a t t e r e d the i n c i d e n t electromagnetic wave. I f the plasma wave at (k, to) i s b u i l t up to an extent that t h e r m a l i z a t i o n cannot destroy i t anymore, more f l i g h t yet w i l l be s c a t t e r e d at the now enhanced d e n s i t y f l u c t u a t i o n , which i n t u r n w i l l beat again w i t h the i n c i d e n t l i g h t wave to enhance the d e n s i t y f l u c t u a t i o n even more and thus a s c a t t e r i n g i n s t a b i l i t y or "s t i m u l a t e d s c a t t e r i n g " w i l l take p l a c e . The t h r e s h o l d w i l l , as i n d i c a t e d , depend on how e f f e c t i v e the beat wave i s imprinted onto the plasma by the electromagnetic f i e l d s against the randomizing e f f e c t s of c o l l i s i o n a l and Landau damping. With t h i s simple p i c t u r e i n mind, the t h e o r e t i c a l model d e s c r i b i n g these parametric processes can e a s i l y be followed. 5. 1.2 O u t l i n e of the theory Maxwell's equations describe the generation of electromagnetic waves at frequencies U J + = u> ± w due to source terms at the same frequencies. The forces i n the plasma a r i s i n g from the beat of the i n c i d e n t and the s c a t t e r e d l i g h t wave are best described by i n t r o d u c i n g the ponderomotive p o t e n t i a l . Together with the s e l f - c o n s i s t e n t p o t e n t i a l , c a l c u l a t e d from Poisson's equation, i t w i l l provide the f o r c e terms i n the Vlasov equation. The l a t t e r then allows d e n s i t y f l u c t u a t i o n s at a), it i n the plasma due to the beat of two electromagnetic waves to be c a l c u l a t e d . The Vlasov equation f o r the d e n s i t y f l u c t u a t i o n s <5n and the equations f o r it form a set of coupled equations f o r fin and the s o l u t i o n of t h i s system i s a d i s p e r s i o n r e l a t i o n d e s c r i b i n g the propagation of electromagnetic waves as a f u n c t i o n of the propagation of plasma waves under various c o n d i t i o n s . This r e l a t i o n makes i t p o s s i b l e to c a l c u l a t e thresholds and growthrates f o r d i f f e r e n t types of processes. = E ( O J ± CJ ) 1.3 D e r i v a t i o n of the general d i s p e r s i o n r e l a t i o n As pointed out i n the o u t l i n e of the theory, we s t a r t out w i t h Maxwell's equations to describe the generation of electromagnetic waves at the sc a t t e r e d frequencies to , = to ± to . ± o E l i m i n a t i n g H + out of (*) 1 ' 3 B+ - 4TT ~ 1 3 g+ V ' = - - — — and V x H = — j . + -rr=-t ± c J ± c 8t V x E c 8 where B = y H and u = 1 one gets V x (V x E +) = and w i t h 8^E , 4-TT ' 8 J ± 1 : 2 at c 2 a t 2 V x (v x E ) = (grad d i v - A) E + ( I - D (1-2) (1-3) (1-4) t h i s becomes 1 9 2 f i + 4TT 3 J ± —75- — grad d i v E^ c 2 3t 6 ± (1-5) The R.H.S. of equation (1-5) means that the electromagnetic wave E + has source terms due to currents j f l o w i n g i n the plasma and due to s e l f - c o n s i s t e n t f i e l d s which w i l l have to be c a l c u l a t e d using Poisson's equation. F o u r i e r transforming the equation (1-5) using E (x, t) = 1 4TT2 >. i ( k x - w t ) 3 E(k, a)) e d 3 kdtj (1-6) a l l w a l l k K ' E + = E ( u ) + ) . . E - = E ( c o ) 7. we f ind c z- ± i -k +:.k + 4iri ( * ) (1-7) In order to a r r i v e at a d i s p e r s i o n r e l a t i o n , we have to express j i n terms of the e l e c t r i c f i e l d E,. This current i , a r i s e s due to the l i n e a r response of the e l e c t r o n s to the o s c i l l a t i n g f i e l d E + and the beat of the i n c i d e n t wave E o± wi t h d e n s i t y f l u c t u a t i o n s 6n at k, o>. Thus we get (1-8) where v o± xm a) o± e o of the d i e l e c t r i c constant 'e through E ,. The c o n d u c t i v i t y a can be described i n terms r>+ J 1 1U± -k (£± = 1 } (1-9), I n s e r t i n g the expression i n (1-7) we f i n d 2 \ i i. —£ ,2 e +] I - k + - k + a) 2 6n (£, a))Eo±(±ujo) (1-10) o The f i r s t term i n (1-8) modifies the vacuum pa r t of equation (1-7) i n t o one f o r a medium by s e t t i n g e + ^ 1. The second term of eq. (1-8) as i t appears i n equation (1-10) describes d i r e c t l y the generation of f i e l d s at E,(ti),) due to the beat of the i n c i d e n t f i e l d E ,'(±ci) ) w i t h the d e n s i t y ± ± o± o f l u c t u a t i o n s 6n(to) . I = E , = E (+w ) + E (-0) ) *** 0 ± 0 0 0 0 This i s a general r e l a t i o n s h i p f o r electromagnetic waves i n conducting media and f o l l o w s from Maxwell's equations. 8. I t w i l l l a t e r be of mathematical convenience to solve f o r E + by i n v e r t i n g (1-10). The r e s u l t i s Sn 2 ! ^ k ± k ± \ 1 k ± k ± E . (1-11) o± ± p n o There, D + = k + 2 c 2 - t o + 2 e + (1-12) w i t h k^ = k ± k , us = to ± to , ± o ± o 2 0) e = 1 - ^ o - and k 2 c 2 - o> 2 + o> 2 = 0 + ~ x M 2 to +^ o o p This form w i l l be used l a t e r . A f t e r the equation "for the electromagnetic f i e l d s (1-11) "has been e s t a b l i s h e d , we now proceed to c a l c u l a t e the d e n s i t y f l u c t u a t i o n s S n ^ k , to) a r i s i n g from the beat of the i n c i d e n t and the s c a t t e r e d lightwave E , (±to ) and o± o E +(co +) by s o l v i n g the Vlasov equation. To do t h i s , the ponderomotive force concept f i r s t has to be introduced. We wish to s o l v e , at l e a s t approximately, the equation of motion f o r a s i n g l e e l e c t r o n i n an electromagnetic f i e l d which i s a slow f u n c t i o n of p o s i t i o n i n the sense that i t s s t r e n g t h v a r i e s only s l i g h t l y w i t h i n the range of amplitude of the e l e c t r o n due to the o s c i l l a t i o n i n t h i s f i e l d . In zero order approximation, the e l e c t r o n w i l l o s c i l l a t e i n the a p p l i e d f i e l d according to ? e + , £ £ , itot , -itot,. / T 1 7 s E, = o- E where E = E (e + e ) (1-14) m to^ o e A l l other symbols have the u s u a l obvious meaning. The e l e c t r o n w i l l , however, a d d i t i o n a l l y experience a slow movement due to the f a c t that the e l e c t r i c f i e l d 1? i s not constant along the coordinates of o s c i l l a t i o n . 9. We th e r e f o r e expand E = E (R) + (,6R V) E (1-15) Choosing 6R = £ means to ask: how much does E change i f the e l e c t r o n p o s i t i o n i s changed during the o s c i l l a t i o n . An e f f e c t of equal order a r i s e s because the e l e c t r o n not only o s c i l l a t e s i n a pure e l e c t r i c f i e l d but i n the electromagnetic f i e l d of the l i g h t wave. The complete equation of motion can approximately be w r i t t e n as m(R + 1) = - e E (R) + (|-V) E + | x B (1-16) ~t e ~* e -> w i t h K = - E , 5 = -=7- E (1-17) lwm moj^ -y and |5- = - V x E hence B = - 5Z x E (1-18) 3t X03 We see that the second and t h i r d terms of the R.H.S. of equation (1-16) r e s u l t i n motion at the frequencies 0 and 2OJV'- r e s p e c t i v e l y . As high frequencies (2co) are of no i n t e r e s t here, we can f i n a l l y w r i t e the low_ frequency p a r t of the equation (1-16) as m R = - t (E'V)E + E x (V x E) ] (1-19) The f i r s t term a r i s e s from the s p a t i a l non-uniformity of the e l e c t r i c f i e l d and i s a d r i f t . t e r m analogous to the (vV)v term i n f l u i d equations; thei.second..term i s duetto the i n f l u e n c e of the magnetic part of the electromagnetic f i e l d on the e l e c t r o n and i s a f o r c e p o i n t i n g along the vect o r of propagation of the electromagnetic wave. Rew r i t i n g E x 07, x E) = V E 2 - (EV)E (1-4) 10. we see that both f o r c e s combine to form a ponderomotive f o r c e m R which can be derived from a ponderomotive p o t e n t i a l ip given by m * = " 7 * = ~ 2m^2 V ^ ( I " 2 0 ) I t must be understood that t h i s f o r c e v a r i e s slowly w i t h time, e.g. t » R and thus at frequences at which ions can respond v i a s e l f -c o n s i s t e n t f i e l d s . I f the described c a l c u l a t i o n i s c a r r i e d out f o r more than one electromagnetic wave the ponderomotive p o t e n t i a l i s found to b e ^ 2 E 2 * = f - Re | I (1-21) 2m 1 f1 lco 1 A A In the case considered here, there are three f i e l d s present, E (±to"), o o E +(to ) and E_(o)_) . An e x p l i c i t e v a l u a t i o n of (1-21) f o r these three f i e l d s shows that the ponderomotive p o t e n t i a l f o r the slow frequency of the plasma wave at to i s given by i|> = 0 ( E E + E E ) (1-22) to 2mto z o+ - o- + o We now t u r n to the problem of c a l c u l a t i n g the induced d e n s i t y f l u c t u a -t i o n s w i t h the Vlasov equation. I t should; be understood that an e l e c t r o n i n a plasma experiencing a strong electromagnetic f i e l d i s subject to three forces at two types of frequencies: The d i r e c t f o r c e of the e l e c t r i c p a r t of the i n c i d e n t electromagnetic f i e l d E (to ) which makes the e l e c t r o n o s c i l l a t e ' ' - at: to , a f o r c e due to o o c o' the s e l f - c o n s i s t e n t f i e l d between ions and e l e c t r o n s a r i s i n g from l o c a l d e n s i t y p e r t u r b a t i o n s , and a f o r c e derived from the ponderomotive p o t e n t i a l . The i o n s , on the other hand, are considered only to experience a f o r c e due to the s e l f - c o n s i s t e n t f i e l d s , s i n c e the d i r e c t l i -ef f e e t on them v i a the i n c i d e n t electromagnetic f i e l d and the ponderomotive f o r c e i s smaller than that f o r e l e c t r o n s by a f a c t o r of q./ra. * 1 1 -^ 7 • q /m e e To c a l c u l a t e a d i s t r i b u t i o n f u n c t i o n f. f o r ions and f f o r e l e c t r o n s l e we th e r e f o r e w r i t e 9f. • , 3f. 1 + v. V f. + - (Z elV $) T - 2 1 = 0 (1-23) )t i i m v " " 3v. 9 f i 3 f — v V f - - [ e V ( $ + - ) ] — — = 0 (1-24) 3t e e m e 1 8v e As the ponderomotive p o t e n t i a l appears only as a m o d i f i c a t i o n of the s e l f - c o n s i s t e n t p o t e n t i a l both equations can be solved by an, expansion f = f + f 1 around the e q u i l i b r i u m d i s t r i b u t i o n f u n c t i o n f as i s shown o n o i n any textbook on plasma p h y s i c s " ^ and the r e s u l t f o r the d e n s i t y f l u c t u a t i o n s i s Sn = y— (* + -) X (1-25) e 4ire e e k 2 + 6n. = -f— $:-v. (1-26) i 4ire A i The t h i r d equation f o r the three unknowns 6n ... <5n. and $ i s Poisson's e' i equation. - k z$ = 4ir (e 6n - e 6n g) (1-27) E l i m i n a t i n g 6n^ out of (1-26) and (1-27) and then e l i m i n a t i n g $ out of the remaining two equations y i e l d s * t charge-to-mass r a t i o f o r e l e c t r o n s / i o n s . X »X- s e e P- 14 ** e I c|> i s the s e l f - c o n s i s t e n t p o t e n t i a l i n the plasma 12. r 2 Sn = - (1 + x.) x T ^ T - ( I _ 2 8> e — I e 4Trez E 1" i s the ponderomotive p o t e n t i a l due to the beat of the i n c i d e n t and the sca t t e r e d electromagnetic wave and i s given by (1-22). With equation (1-28) the second goal i s achieved, namely the d e s c r i p t i o n of how den s i t y f l u c t u a t i o n s are set up due to the beat of two e l e c t r o -magnetic waves i n a plasma. I t remains to e x t r a c t a d i s p e r s i o n r e l a t i o n out of the two equations (1-28) and (-1-11). Equations (1-11) and (1-28) are a set of coupled equations f o r E + and 6n. Combining (1-28) and (1-22) and r e p l a c i n g i n the r e s u l t i n g expression E + and E_ from (1-11) one can cancel out fin^. The r e s u l t i n g equation i s the wanted d i s p e r s i o n r e l a t i o n (1 + X,)X„ V 2 w 2 I = 1 e k p e 4iTmcoi2 n o o •> ->• - -'r?- •+ + + * (\*+ 1 k k 1 + ) (1-29)-In order to b r i n g t h i s expression i n t o the more f a m i l i a r form appearing i n Ref. 14 note that 1 ^ 1 , + — and ( i + x ± ) x e x e i + x i E [(k*k) E ] = |E.£| 2 = E 2 k 2 s i n 2 H E , ^ ) o o 1 o 1 o o so that Eo. ( Y - | ^ ) E 1 = t 2 k 2 [ l - s i n 2 ( * E , k)] = | k x E L• kz oJ o o c 13. 4im e 2 _^  S u b s t i t u t i n g u 2 = and remembering that — : E was the 0 order & p m imu) o v o quiver v e l o c i t y of the e l e c t r o n i n the i n c i d e n t f i e l d E q one a r r i v e s at the f i n a l form of the d i s p e r s i o n r e l a t i o n k .xv K k x v z k, • v r 1 k «v i + o 1 . •1 — o 1 1 + a ' -+ 2 o D k 2 D k 2 k 2 U 2 e k 2 U 2 e + + . - - .-.+ +'+ -(1-30) From the d e r i v a t i o n of (1-30) i t f o l l o w s that the k«v terms i n the o R.H.S. of (1-30) a r i s e from the grad d i v FJ term i n the i n i t i a l wave equation (1-5), e.g. from the e l e c t r o s t a t i c components at the sideband -*--»- 1 3 2 frequencies u>, and w . The k x v terms a r i s e from the -AH -r—b n + - o c 9 t z term i n the wave equation (1-5), e.g. from the electromagnetic components at the sideband frequencies. The reason f o r both of these modes depending on each other i s that the term - ^ 2 ^+3 + ( e < l - I _ 7 ) in v o l v e s a co u p l i n g v i a the ponderomotive force of the e l e c t r o s t a t i c components. To get a f e e l i n g f o r how t h i s d i s p e r s i o n r e l a t i o n d e s c r ibes the propagation of a d e n s i t y wave and electromagnetic sideband modes as a f u n c t i o n of each other, consider the f o l l o w i n g s i m p l i f i e d case: imagine some kind of process where only the second term i n (1-30) on the R.H.S. i s of importance, e.g. D_ -> 0, and where the ions can be neglected. Then, (1-30) reduces to (— + 1) D = l k x v I 2 t 1 " 3 1 ) X k 2 1 - o 1 Ae I f there would be no ponderomotive forc e term, then the R.H.S. of equation (1-31) would be zero and the equation would reduce to (— + 1) • D = 0 (1-32) *e 14. Then one s o l u t i o n i s + 1 = 0 and, hence, to2 = to 2 describes an x e P undisturbed plasma o s c i l l a t i o n at to ;:;or D = 0 hence, k 2 c 2 + to 2 = to 2  p p' - ' - P -describes the undisturbed propagation of an electromagnetic wave at frequency to through that plasma. The R.H.S. of (1-31) t h e r e f o r e , a r i s i n g due to the ponderomotive f o r c e , i s the cou p l i n g c o e f f i c i e n t between the electromagnetic wave at to_ and the plasma o s c i l l a t i o n at to^. The st r e n g t h of t h i s c o u p l i n g i s c l e a r l y p r o p o r t i o n a l to k 2, i . e . the le n g t h of the wave v e c t o r of the d e n s i t y f l u c t u a t i o n squared. This means that short wavelength f l u c t u a t i o n s w i l l couple stronger to the electromagnetic sideband modes than long wave-length f l u c t u a t i o n s , and the cou p l i n g i s p r o p o r t i o n a l to the i n t e n s i t y of the i n c i d e n t electromagnetic wave at to . b o The s u s c e p t i b i l i t i e s x^ a n c^ X e a r e given through the Vlasov equation 14-19 i n terms of the e q u i l i b r i u m d i s t r i b u t i o n f u n c t i o n f by 3f 2 oe, i X X e , i k2 d 3 v 5Z (1-33) -k.v For f being Maxwellian, X e a n d X-^  h a v e t h e f o l l o w i n g approximate forms which w i l l be used throughout the c a l c u l a t i o n s : — tr a/ = — — f o r — » v X~ 9 k e Ae o)2 7~ « v X = 1 ? ^  9 (! + I ) k e Ae k2X„2 e to to _ P 1 v. ^  T 7 V = — De„ (1-34) T » v. x- = ~ o k l A x to2 to k « v i x ± = k s f r ( 1 •+ I 1 ) Di Here, w i t h v^ << v g , 1^ & are the imaginary p a r t s of the s u s c e p t i b i l i t i e s . Note that waves w i t h l a r g e k (short wavelength) are s t r o n g l y damped, waves with s m a l l k (long wavelengths) are weakly damped. 15. 1.4 S p e c i a l i z a t i o n f o r s t i m u l a t e d b a c k s c a t t e r l n g Consider the case where an i n c i d e n t electromagnetic wave decays i n t o a plasma wave and a s c a t t e r e d electromagnetic wave. For the case of a reasonably underdense plasma (co2 >> w 2) the electromagnetic wave w i l l propagate according to D « 0, t h e r e f o r e the k x V q terms i n (1-30) w i l l dominate. At moderate i n c i d e n t powers E q 2 , predominantly downconversion (generation of electromagnetic sideband modes at in - a r a t h e r than 0) + w^) w i l l occur, so that the D + term can be neglected s i n c e i t i s non resonant. This reduces (1-30) to The R.H.S. of t h i s equation s h a l l now be s i m p l i f i e d using p h y s i c a l arguments. For underdense plasmas, the d i s p e r s i o n r e l a t i o n f o r the l i g h t and plasma waves show that - OJ < < O) q and u)_, hence |k Q| a> |lc_| . |k| however need by no means be s m a l l and because the coupling c o e f f i c i e n t i n (1-35) i s p r o p o r t i o n a l to k 2, the i n s t a b i l i t y w i l l occur at l a r g e r a t h e r than at small k. These c o n s i d e r a t i o n s lead to the f o l l o w i n g p o s s i b i l i t i e s to arrange k , It and Ic: o 16. For a l l cases the diagram shows that |k |»2|k Q|cos 0 which i s as approximate as |k_| » |k Q| and means that the i n s t a b i l i t y w i l l predom-i n a n t l y occur at k_«*. -k-0> that i s the electromagnetic sideband mode at -> co w i l l be backscattered and k w i l l be about 2k . This means that the - o d e n s i t y f l u c t u a t i o n w i l l propagate p a r a l l e l to the i n c i d e n t l i g h t wave vec t o r . |k x v | 2 — o Therefore, j-—2 can be w r i t t e n as \t x v | 2 - o' 2 • 2 O w i t h <f> the angle between k and V q being near 90 . We now t r y to s i m p l i f y D_, a l s o using p h y s i c a l arguments. Introdu c i n g damping f o r the l i g h t wave means that D = k 2 c 2 - co 2 + co 2 i s complex because co = co + co - - P * • o -where co i s now w r i t t e n as co + i T . o o o Using k 2 c 2 - o o 2 + c o 2 = 0, the expression f o r D reduces to o o p r , 2 2 k'k c D = 2co (co + — + .1 T ) - o 2co co o In c l u d i n g a l l these approximations i n (1-35), one obtains i i k 2 v 2sin 2d> 1 , 1 o X„ + l + X, " . 2 . 2 t^Tc2 (1-36) e 1 o / , k z c z o , . „ . 2co (co + -= + i r ) o 2co co o Using the d i s p e r s i o n r e l a t i o n s f o r the i n c i d e n t and the s c a t t e r e d l i g h t wave together w i t h the approximations (to 2 - co 2 ) «* 2to (co — co ) and o - o o -(k 2 - k 2 ) » -2 k k + k 2 o - o 17. I t f o l l o w s that o = Aco (1-37) CO - CO o o o Aco i s the d i f f e r e n c e between the i n c i d e n t and the s c a t t e r e d l i g h t wave which has to be equal to the frequency of the plasma wave. The d i s p e r s i o n r e l a t i o n t h e r e f o r e takes the f o l l o w i n g form: I t should be remembered that, according to the approximations made, t h i s d i s p e r s i o n r e l a t i o n describes the decay of an i n c i d e n t e l e c t r o -magnetic wave i n t o a plasma wave ( i o n or el e c t r o n ) and a backscattered electromagnetic wave i n a plasma that i s underdense enough so that D_, * 0. Three d i s t i n c t cases of i n s t a b i l i t i e s e x i s t : (1) The s c a t t e r i n g plasma wave i s l a r g e l y undamped and hence c l o s e to an eigenmode. I f the s c a t t e r i n g occurs o f f e l e c t r o n waves, the process i s c a l l e d "Stimulated Raman S c a t t e r i n g " ; i f i t occurs o f f i o n waves, i t i s c a l l e d "Stimulated B r i l l o u i n S c a t t e r i n g " henceforth abbreviated SBS. (2) I f e i t h e r mode i s s t r o n g l y damped, the processes are c a l l e d " s c a t t e r i n g o f f r e s i s t i v e quasi modes". (3) F i n a l l y , there i s a more e x o t i c form of t h i s type of i n s t a b i l i t y 14 which occurs at too high powers to concern us any f u r t h e r . This i s termed " s c a t t e r i n g o f f r e a c t i v e quasi modes". I t r e f e r s to the case where the frequency s h i f t <^ o-co_ i s l a r g e r than the eigen-frequency of the plasma o s c i l l a t i o n . k 2 v 2 sin2<j>. o 2co (co-Aco + i f ) (1-38) o o As the Landau damping of the plasma/wave depends c r i t i c a l l y on the parameter kA.^, w i t h A Q being the Debye s h i e l d i n g d i s t a n c e , the d i s t i n c -t i o n between the f i r s t two cases w i l l be formulated mathematically by kX < 1 or kA >> 1 r e s p e c t i v e l y . 19. 1.5 Stimulated B r i l l o u i n S c a t t e r i n g As stimulated s c a t t e r i n g o f f i o n modes was observed i n the experiment described l a t e r , t h i s case s h a l l now be"treated i n some d e t a i l . F i r s t , consider the case where the i o n wave i s only weakly damped. Then, the s u s c e p t i b i l i t i e s can be w r i t t e n as (see 1-34) 1 CO .2 D l 1 1 k 2 A D 2 ( u ) 2 - u T . 2 ) + co2 hence, . h -—; = = P 1 9  X 1 + X- w2 - c o . 2 e l p i co . 2 k 2 A 2 2 2 CO — CO . p i 2 - 2^!vL w i t h co^ = l + k 2 x 2 a n d i n c l u d i n g weak Landau damping, e.g. co.2 -> co.2 - 2 i r co. (1-38) can be w r i t t e n as: 1 1 a I k 2 v 2sin 2(i>(to 2 - co . 2) (U2 - U.2 + 21 Ta,.H. - Aco + i r Q ) = 2 ° ( 1 + k 2 , 2) P 1 (1-39) o D Thereby Aco i s , according to i t s d e f i n i t i o n , about equal to U K . I n order to f i n d regions of i n s t a b i l i t y , one sets co = co^  + iy> m u l t i p l i e s out the L.H.S., and separates r e a l and imaginary p a r t s . The r e a l p a r t describes the a c t u a l d i s p e r s i o n of the waves and the imaginary p a r t , i f negative, the damping; i f p o s i t i v e , thew growth '-.of the -longwave as na f u n c t i o n of time. S e t t i n g Y = 0 means to ask f o r the c o n d i t i o n s under which the wave n e i t h e r grows nor decays w i t h time which describes the t h r e s h o l d f o r the 20. i n s t a b i l i t y . The r e s u l t f o r the growthrate y i s Y = " j k 2 v 2(sin2<j>)to ..2 r + r - J (r - r ) 2 + : — ° . . . 2. 2 ^ o a V o a co co.(l + k zA z ) o i D (1-40) Note that the damping r a t e s r , are passive plasma parameters. The d r i v i n g term that determines when th r e s h o l d i s obtained and how la r g e the growthrate at threshold w i l l be i s k 2 v 2co . 2sin 2cf> o p i co u).(l + k 2 A 2 ) o l D When t h i s term increases (e.g. w i t h i n c r e a s i n g input power) the value of the root increases u n t i l i t i s l a r g e r than the sum of the two damping r a t e s , at which po i n t the i n s t a b i l i t y grows. The threshold c o n d i t i o n f o r weak damping ( k 2 A Q 2 << 1) and b a c k s c a t t e r i n g (* = 90") i s c a l c u l a t e d from (1-40): k 2 V 2C0 . 2 4 r r = ° p l s i n 2 * (1-41) o a co co. O 1 Note that the threshold decreases as the plasma d e n s i t y decreases and the temperature increases and that i t goes to zero with the damping of e i t h e r t!he l i g h t wave or the i o n a c o u s t i c wave going to zero. To f i n d the growthrate j u s t above threshold f o r r » T we expand the a o root i n t o a Taylor s e r i e s around T to get cl k 2V Q 2co _^2sin2cf> Y a t threshold = 4co co. ( l + k 2 A 2 ) r (1-^2) o l D a see a l s o page 29 21-. Large growthrates are obtained when the d r i v i n g term becomes dominant as compared to the damping terms. Then the growthrate i s , k v co . sin<f> , k v co . v = i ° _ E ± I o P 1 ( I _ 4 3 ) Tmax 2 • •> U v/co co. (1 + k z A 2 ) v\/o) co. V o i v D V o i f o r k z X D z « 1 and <j> 90 w. Next we consider the case where the i o n wave i s h e a v i l y damped ( k 2 A D 2 > 1). This means f o r the s u s c e p t i b i l i t i e s (see 1-34) 1 1 + I . . X E = ¥ ^ 7 ; x ± = ^ d 7 f o r T = T. i t i s L . 2 = Z L 2 , I:'.. i s the imaginary part of Y-e I Dx De l ° r l I n s e r t i n g t h i s i n t o equation (1-38) , s e t t i n g co = U K + i y but making no f u r t h e r assumptions on k 2X^ 2 <;< or >> 1 we o b t a i n v 2sin 2cJ) [to. - Aw + i ( T + Y ) ] ( Z k 2 A 2 + I . + 1 + Z) = ° . — ( Z k 2 A 2 + 1. + 1 ) (1-44) l o J D l 2co A „ ^ D l o D s o l v i n g the r e a l p a r t of t h i s equation f o r to and assuming 1\ i s s m a l l one obtains v 2sin 2cf 1 .Zk2A„2. + 1 . . o D co = flco + 7 — 7 — * „, ')\—2 1 1—7~^ ~ 2co Zk^ A Z + 1 + Z o D D f o r strong Landau damping ( k 2 A ^ 2 >> 1) t h i s reduced to v 2sin 2<() co = Aw + ° , 2 (1-45) 2co A ^ o D To see what t h i s expression means p h y s i c a l l y , we apply the arguments presented on p. 16,17 i n reverse and f i n d that the second term of eq. (1-45) can approximately be w r i t t e n as ,22. V o 2 V o 2 1 2 ^ 7 ~ ( w o " k 2 T 7 ( I " 4 6 ) o D D Hence, t h i s term represents a frequency imprinted by the l a s e r on the plasma that depends on the i n c i d e n t l a s e r power and the amount of Landau damping. S o l v i n g the imaginary p a r t :of (1-44) f o r the growthrate y one obtains v 2sin2<j> i y = — p + -5 — — — where I = -=- I . (1-47) Y o 2co A„ 2 (Zk^X * + 1 + Z ) 2 i i o D D This expression has various l i m i t s f o r k 2A 2 << 1, » 1 or« 1 For strong'Landau -damping of the i o n wave ( k 2 ^ 2 « 1) and Z = 2 the growthrate reduces to: , v 2sin2<|> y = - r 4 ° i 2 1 ( I _ 4 8 ) o 25 co A ^  o D and the thr e s h o l d becomes v Z 25r co A 2 o o D c 2 " I c 2 (1-49) one can see ; . t h a t J t h i s i n s t a b i l i t y d e r i v e s i t s e x i s t e n c e from the magnitude of the imaginary -part of the i o n i c s u s c e p t i b i l i t y . This term being s m a l l means that the t h r e s h o l d i s hi g h , the growthrate s m a l l : Note that f o r k 2A^ 2 >> 1 the r e s u l t f o r th r e s h o l d and growthrate depends much stronger on the magnitude of k 2A^ 2. 23. 1.6 The co - k diagram The s e c t i o n on s t i m u l a t e d B r i l l o u i n s c a t t e r i n g s h a l l be concluded by f u r t h e r i n t e r p r e t i n g stimulated b a c k s c a t t e r i n g on the s o - c a l l e d c o - k diagram. In the c o - k diagram co and k are the axes of a Carthesian coordinate system and the curves represent the d i s p e r s i o n r e l a t i o n s f o r l i g h t - , i o n - and e l e c t r o n waves. Figure (1-1): c o - k diagram. Curves are d i s p e r s i o n r e l a t i o n s f o r (a) l i g h t , (b) e l e c t r o n , and (c) i o n waves. Shown i s the decay of a l i g h t wave i n t o an i o n a c o u s t i c wave and a s c a t t e r e d l i g h t wave. Note that because v. << v << c, the d i s p e r s i o n curves f o r e l e c t r o n and 13. G i o n waves are b a s i c a l l y h o r i z o n t a l l i n e s compared to the parabola re p r e s e n t i n g the lightwave d i s p e r s i o n f o r s m a l l k. \ 24. Vectors ending on the l i g h t d i s p e r s i o n curve represent l i g h t waves.and those ending on the e l e c t r o n wave d i s p e r s i o n curve represent e l e c t r o n waves, e t c . Therefore, f o r a parametric decay process to be p o s s i b l e , i t s to-k vectors have to add up. Consider, as an example, the decay of a l i g h t wave i n t o an i o n a c o u s t i c wave and another l i g h t wave as shown i n F i g . (1-1). The p r o j e c t i o n s on the frequency a x i s describe the conservation of energy: co^  = co + co_. Because the frequency of the i o n a c o u s t i c wave (co) i s s m a l l compared to the frequency of the l i g h t wave ( Q^)» the frequency of the s c a t t e r e d l i g h t wave (W q ) w i l l be only s l i g h t l y l e s s than the frequency of.the i n c i d e n t l i g h t wave. This i s a general f e a t u r e of B r i l l o u i n s c a t t e r i n g . From the diagram, i t a l s o f o l l o w s that the wave vecto r of the s c a t t e r e d l i g h t wave w i l l be -k , w i t h k being the wave ° o o v e c t o r of the i n c i d e n t l i g h t wave. Hence, k, the wave vect o r of the i o n a c o u s t i c wave, w i l l be k « ~^ k Q ( s e e P«16 ). Furthermore the co - k diagram shows that 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 can only take place i n underdense plasmas. For the decay of e.g. a l i g h t wave i n t o two - e l e c t r o n waves^, i t i s obvious that the process can only take place where cbQ £" 2co^ whereas the decay of a lightwave i n an e l e c t r o n and an i o n a c o u s t i c wave"''"' can only occur at CO I . .CO . o p These are j u s t some p o i n t s which by no means exhaust the usefulness of *15 the diagram. Questions about processes l i k e cascading, parametric decay i n moving plasmas e t c . , can a l s o be answered by t h i s simple p i c t u r e . The B r i l l o u i n s c a t t e r e d l i g h t wave i s intense enough to i t s e l f give r i s e to a higher order B r i l l o u i n s c a t t e r e d l i g h t wave. 25. 1.7 L i m i t a t i o n s of the t h e o r e t i c a l model The t h e o r e t i c a l treatment of the described parametric i n s t a b i l i t y s h a l l be concluded by p o i n t i n g out the l i m i t a t i o n s of the model used and i t s relevance to a c t u a l experiments. Despite the use of Maxwell's and the Vlasov equations which are, f o r wave phenomena, q u i t e g e n e r a l , the presented d e s c r i p t i o n i s of very l i m i t e d v a l i d i t y . F i r s t l y , only i n f i n i t e , homogeneous plasmas were t r e a t e d ; secondly, a li n e a r i s e d > t h e o r y only can describe the onset of i n s t a b i l i t i e s ; and t h i r d l y , even i n such s i m p l i f i e d cases many approximations have to be made i n order to a r r i v e at a n a l y t i c a l l y s o l v a b l e equations. The l i n e a r theory f o r f i n i t e , inhomogeneous plasmas i s con s i d e r a b l y more complex and can 13 14 be t r e a t e d a n a l y t i c a l l y only f o r very simple d e n s i t y p r o f i l e s . ' A p r e s e n t a t i o n of that theory i s considered w e l l outside the scope considered here. However, a t h r e s h o l d expression f o r sti 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 from t h i s theory w i l l be used l a t e r ; i t s p h y s i c a l i n t e r p r e t a t i o n i s given on p. 29 . A complete theory d e s c r i b i n g the onset, growth and eventual s a t u r a t i o n of parametric i n s t a b i l i t i e s does not e x i s t . At present, p r e d i c t i o n s that i n v o l v e more complicated and d e t a i l e d models are made with the a i d of computer s i m u l a t i o n s . 2 ' ^ ' ^ " ' ^ ' ' Therefore, a l l that can be expected of the simple theory presented here, i s to give an i n i t i a l i n s i g h t i n t o the physics of parametric decay i n plasmas and produce numerical values that provide order of magnitude estimates f o r thresholds and growthrates. 26. 1.8 Numerical values f o r threshold and growthrate of 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 i n homogeneous and inhbmogeneous plasmas In order to evaluate the previous r e s u l t s n u m e r i c a l l y , the f o l l o w i n g l i s t of formulas i s handy: Here, i n t e n s i t i e s are i n W ^ 2 ~ S » temperatures i n eV. As the cgs system i s used, the dimensions of the r e s u l t s w i l l be i n cm, sec, esu, e t c . cm (1) quiver v e l o c i t y V q ( s e c ) °^ a n e l e c t r o n i n an e l e c t r i c f i e l d of i n t e n s i t y I ( W a t» S) f o r C0„ l a s e r wavelength cm^ z v = 265 / i o (2) Debye length A^ (cm) A,, = 740,/- : n i n cm"3 , T i n eV D V n (3) plasma frequency (~~j") co = 5.6 x 101* Vn~ ; n i n cm - 3 pe (4) c o l l i s i o n frequency between e l e c t r o n s v & e (—^  ) 2.8 x 10~ 5 - ^ V T O ; n.incm 3 , T i n eV v = Z.tf x 1U J 5-7-. ee m3/2 T (5) c o l l i s i o n a l damping of a l i g h t wave i n a plasma r (—^-) o sec. 2 T = 1.4 x 10 2 k - A — ; n i n c m - 3 , T i n eV o T3/2 ' cm (6) i o n a c o u s t i c speed v. (— ) 1a . sec v. = 6.9 x 10 5 / f ~ ~ ; Tv.lneV 1a e cm (7) e l e c t r o n thermal speed v ( ) e sec v = 6 x 10 7 JT~ : T i n eV e e i n c i d e n t l a s e r l i g h t frequency i s that of l a s e r l i g h t (8) e l e c t r i c a l c o n d u c t i v i t y a(esu) T3/2 a = 1.7. x 1 0 1 3 ( f o r He, Z = 2 ) , T i n e V The power of the CO^ l a s e r was ^ 250 MW, focussed to about 1 mm2. Hence the quiver v e l o c i t y of an e l e c t r o n i n t h i s f i e l d i s 7 cm * v = 4.2 x 10 ^ 2 -o sec v 2 ° r - f - = 1-9 x 10-6 Using the formulae p. 26 , the thresholds and growthrates reduce to co. 2 the f o l l o w i n g expressions ( v a l i d only f o r —^r << 1) CO o A) Stimulated s c a t t e r i n g o f f quasi r e s i s t i v e i o n modes Growthrate Y = - 1 . 4 x l 0 ~ • ' + 1.8 x 10 — Threshold V o , A „ -, n-24 n L + 2 x 1 Q 1 3 T 3/2 ' " T i 9 v x +-2;x 10 2 i — n = L 6 x l O : " - \ l i + 2 x l O " ^ B) Stimulated s c a t t e r i n g o f f i o n modes Growthrate y = 1.65 x 10 2 (T n) max e Threshold f o r a homogeneous plasma Z - l . 4.3 x M-S. 15 ** Threshold f o r an inhomogeneous plasma v 2 o , „ „„9 T_ = 6.8 x 10-where L^ , i s the s c a l e length of temperature gradient i n cm. - Note^that t h i s i s about the e l e c t r o n thermal speed i n a plasma of some. eV. -* * : - '."' r — - -•-"' = • " ; . ' „• " " - ' . „ - ' • * • • ' - * ' ' In .this expression, the Landau damping r a t e was neglected compared to the c o l l i s i o n a l damping r a t e . Numerical values A) s c a t t e r i n g o f f quasi r e s i s t i v e modes n 1 0 1 5 1 0 1 6 1 0 1 7 1 0 1 8 Y T . = 1 eV 1.8 x 10 9 1.8 x 1 0 1 0 1.7 x 1 0 1 1 4 x 1 0 1 1 e T = 10 eV 1.6 x 10 8 1.8 x 10 9 1.7 x 1 0 1 0 1.4 x 1 0 1 1 e T = 100 eV 1 x 10 7 1.6 x 10 8 1.8 x 10 9 1.7 x 1 0 1 0 e v 2 c e e T 1 0 1 5 1 0 1 6 1 0 1 7 1 0 1 8 T = 1 eV 1.6 x 10 9 1.6 x 10 8 l.OOx 10 7 1.6 x 10 6 T = 10 eV 5.5 x 1 0 " 1 0 5 x 10~ 9 5 x 1 0 - 8 5 x 10~ 7 T = 100 eV 2.8 x 10 1 0 1.8 x 10 9 1.6 x 10 8 1.6 x 10~ 7 e B) s c a t t e r i n g o f f i o n modes v n 1 0 1 5 1 0 1 6 1 0 1 7 1 0 1 8 Y T = 1 eV 5.2 x 10 5 1.6 x 10 6 5.2 x 10 6 1.6 x 1 0 7 e T = 10 eV 1.6 x 10 6 5.2 x 10 s 1.6 x 10 7 5.2 x 10 7 e T = 100 eV 5.2 x 10 6 1.6 x 10 7 5.2 x 1 0 7 1.6 x 10 8 e thresholds f o r a homogeneous plasma n v 2 c 2 1 0 1 5 1 0 1 6 1 0 1 7 1 0 1 8 T = 1 eV 4.3 x 10" 9 4.3^x 10~ 7 4.3 x 10" 5 4.3 x 1 0 - 3 e T = 10 eV 1.4 x 1 0 " 1 1 1.4 x 10" 9 1.4 x 10~ 7 1.4 x 10- 5 e T = 100 eV 4.3 x 1 0 " ^ 4.3 x 10"' 1 2 4.3 x 1 0 ~ 1 0 4.3 x 1 0 _ 8 e 29. B) thresholds f o r an inhomogeneous plasma L = 1 mm 2\ n -2- \ 1 0 1 5 1 0 1 6 1 0 1 7 10 ] T = 1 eV 6.9 x 10" 6 6.9 x 10" 7 6.9 x 10~ 8 6.9 x 10" 9 T = 10 eV 6.9 x 10" 5 6.9 x 1 0 - 5 6.9 x 10~ 7 6.9 x 10~ 8 T = 100 eV 6.9 x 10 _ t t 6.9 x 10~ 5 6.9 x 10" 6 6.9 x 10" 7 I t may be s u r p r i s i n g to see that the th r e s h o l d f o r a homogeneous plasma increases with i n c r e a s i n g d e n s i t y , however, the t h r e s h o l d f o r an inhomogeneous plasma decreases with i n c r e a s i n g d e n s i t y . The reason i s that i n d e r i v i n g the th r e s h o l d expression f o r a homogeneous plasma the assumed model was that d e n s i t y f l u c t u a t i o n s have t o be induced against the randomizing e f f e c t of damping (see p.4 ). Hence, v 2 the t h r e s h o l d o -^T r , i s p r o p o r t i o n a l to the product of the damping o 2 r a t e s f o r i o n and l i g h t wave. In d e r i v i n g the th r e s h o l d expression f o r an inhomogeneous p l a s m a " ^ ' h o w e v e r , the assumed model i s that the induced d e n s i t y waves t r a v e l out of the region where the wave vector matching c o n d i t i o n f o r sti m u l a t e d s c a t t e r i n g i s s a t i s f i e d and that compared to t h i s i n s t a b i l i t y suppressing e f f e c t , damping i s n e g l i g i b l e . Hence the th r e s h o l d derived f o r that case depends on Y~, LT the i n v e r s e temperature s c a l e length of the plasma. 30. CHAPTER 2 Experimental i n v e s t i g a t i o n of the backscattered and  transmitted CO2 l a s e r l i g h t 2.1 I n t r o d u c t i o n The purpose of the CO2 l a s e r l i g h t s c a t t e r i n g experiments was to i n v e s t i g a t e which i n s t a b i l i t i e s could be detected w i t h the CO2 l a s e r power a v a i l a b l e and the plasma parameters given by the Z-pinch. As the theory shows, st i m u l a t e d s c a t t e r i n g should manifest i t s e l f i n l a r g e amounts of backscattered l i g h t . We t h e r e f o r e measured the amount of backscattered CO2 l a s e r l i g h t under d i f f e r e n t plasma c o n d i t i o n s (e.g. as a f u n c t i o n of the c o l l a p s e phase of the p i n c h ) . As the s p e c t r a l decomposition of the backscattered CO2 l a s e r l i g h t w i l l show which st i m u l a t e d b a c k s c a t t e r i n g process takes p l a c e , we searched the backscattered l i g h t f o r s i g n a l s at the f o l l o w i n g frequencies (C O q being the frequency of the i n c i d e n t CO2 l a s e r l i g h t ) : near C O q, to detect 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 and p o s s i b l y , . 1 5 cascading 3 -t 1 14,15,20,23 near T CO , to detect two plasmon decay 2 o near c o ^ , to detect s t i m u l a t e d Raman s c a t t e r i n g at the c r i t i c a l . 14,15,22 de n s i t y The transmitted or forward s c a t t e r e d l i g h t was searched near C O q to look f o r - the normal tran s m i s s i o n of CO2 l a s e r l i g h t through an underdense plasma . *14,15,24,25 - f l l a m e n t a t i o n ,. • *15 - cascading see footnote p. 24 A d d i t i o n a l l y , the angular dependence of the CC^ l a s e r l i g h t backscattered due to 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 was i n v e s t i g a t e d as f a r as the setup allowed. The r e s u l t s can be summarized as f o l l o w s : The only parametric i n s t a b i l i t y that was detected w i t h absolute c e r t a i n t y was stimulated B r i l l o u i n s c a t t e r i n g . Due to the low t h r e s h o l d and high growthrate t h i s i n s t a b i l i t y i s a l s o of great importance f o r l a s e r f u s i o n 20 experiments. Fi l a i r i e h t a t i o n was not present to a degree that i t could have been detected. This i s i n agreement w i t h simple t h e o r e t i c a l models about 24 25 t h i s i n s t a b i l i t y . ' No other i n s t a b i l i t y was found, most c e r t a i n l y because the power a v a i l a b l e d i d not a l l o w i t to exceed any other thresholds. The angular dependence of the backscattered l a s e r l i g h t shows a somewhat s u r p r i s i n g r e s u l t . f o r which a p l a u s i b l e e x p l a n a t i o n w i l l be presented. F i l a m e n t a t i o n i s the c e r a t i o n of low d e n s i t y channels by the l a s e r l i g h t along i t s d i r e c t i o n of propagation. 32. 2.2 The Z-pinch plasma, measurements of r a d i u s , temperature and d e n s i t y  and the CO^ l a s e r used f o r the laser-plasma i n t e r a c t i o n s t u d i e s . A d e t a i l e d p r e s e n t a t i o n of the design parameters and the discharge bank of the Z-pinch i s given i n Ref. 26. The d e s c r i p t i o n given here i s only intended to provide the necessary background f o r the experiments described subsequently. The Z-pinch c o n s i s t s of a pyrex glass v e s s e l with h o l l o w copper e l e c t r o d e s i n s e r t e d i n each end. 5.6 k j of e l e c t r i c a l energy stored i n a 84 uF c a p a c i t o r bank are discharged i n t o 1.2 Torr He to produce a pinch plasma which reaches maximum compression and temperature about 2 usee a f t e r the i n i t i a l breakdown. End on framing photography w i t h a TRW image converter camera gave the 26 f i r s t photographs of the c o l l a p s i n g plasma. . As t h i s geometry d i d not a l l o w a d e t a i l e d observation of the phase of maximum compression, e.g. minimum r a d i u s , s i d e on streak photography was used to s p a t i a l l y and t -temporally r e s o l v e t h i s f i n a l phase of c o l l a p s e . '„ These measurements revealed a minimum luminous ra d i u s of the plasma of 2 mm. The f i r s t measurements of plasma temperature and d e n s i t y were done 2 6 s p e c t r o s c o p i c a l l y u s i n g the 4686$ l i n e of He I I . .. For the means a v a i l a b l e at the time, these measurements y i e l d e d good estimates of a 18 6 maximum d e n s i t y of 8 x 10 — and a maximum temperature of 30 eV to 40 eV. For a c t u a l dimensions see s p e c i f i c a t i o n s at end of re p o r t . A* see F i g . 5-2 t see footnote p. 33 33. Next, a Ruby l a s e r Thomson s c a t t e r i n g system was set up to v e r i f y the 26 ** measurements. The attempt to use an O p t i c a l M u l t i c h a n n e l Analyser (OMA) to record the e l e c t r o n f e a t u r e of the Thomson s c a t t e r e d spectrum r e s u l t e d i n the development of a technique that allowed the f a s t g a t i n g of the OMA without d i s t o r t i o n s . A d e s c r i p t i o n of t h i s technique and i t s a p p l i c a t i o n to improved s p e c t r o s c o p i c a l d e n s i t y measurements i s presented i n Chapter IV. Before the signal-to-bremsstrahlung r a t i o i n the Thomson s c a t t e r e d e l e c t r o n f e a t u r e could be improved to y i e l d a t r u l y s a t i s f a c t o r y spectrum however, the observation of the Thomson s c a t t e r e d i o n f e a t u r e permitted the making of very good measurements of plasma temperature and d e n s i t y . ^ These r e s u l t s are shown i n F i g . (2-3) and w i l l be used throughout t h i s r e p o r t . The CO2 l a s e r used i n the laser-plasma i n t e r a c t i o n s t u d i e s was a Lumonix T600 module i n unstable resonator c o n f i g u r a t i o n . In t h i s l a s e r , an e l e c t r i c a l discharge transverse to the o p t i c a l a x i s i n v e r t s the v i b r a t i o n a l l e v e l s of CO2 i n a He.Ne.CO2 m i x t u r e a t atmospheric pressure. The l a r g e gain of the i n v e r t e d medium allows the use of an unstable resonator f o r c o u p l i n g .out' the l a s e r l i g h t . The T600 module employed a c o n f o c a l unstable resonator c o n f i g u r a t i o n as shown on the f o l l o w i n g page. see Ch. IV, F i g . 4-1. see s p e c i f i c a t i o n s t I am indebted to B r i a n H i l k o f o r l e t t i n g me use the r e s u l t s of h i s experimental work. ***see Ch. IV. 34. k * 1 m Figure (2-1): The unstable resonator of the Lumonix T600 TEA C0 2 l a s e r . Both m i r r o r s have F as a common f o c a l spot. Due to t h i s arrangement, the wave l e a v i n g the c a v i t y i s a plane wave and the output aperture of the CO, l a s e r has an annular shape as shown i n F i g . (2-2) below. Figure (2-2): Annular output of the CO^ l a s e r due to the use of an unstable resonator c a v i t y . The height i s 10.5 cm, the width 8 cm and equal to the separation of the e l e c t r o d e s between which the discharge pumping the l a s e r t r a n s i t i o n takes place. This output aperture shape allowed the measurements which are described i n 2.6 and evaluated i n 3.23. 35. 2.3 Experimental p r o v i s i o n s Before the CC^ l a s e r could s u c c e s s f u l l y be focussed i n t o the plasma, the pinch v e s s e l had to be modified and new plasma parameters, now changed by these m o d i f i c a t i o n s , had to be measured. The sketch below shows the p r i n c i p a l setup to focus the CC^ l a s e r i n t o the plasma. C O 2 in E l e c t r o d e s i ^ Salt lens Pinch vessel I t i s obvious that the l a s e r l i g h t had to be protected from the plasma to a c t u a l l y be able to form a f o c a l spot i n the center of the v e s s e l . Otherwise defocussing of the l a s e r beam by the plasma would l i m i t the power f l u x of the i n c i d e n t l i g h t to too low values. The problem was solved by using a quartz funnel as i n d i c a t e d i n the next sketch. f r — y = O uartz funnel Quartz must be used as glass q u i c k l y s u f f e r s from a phenomenon c a l l e d " c r a z i n g " which c o n s i s t s of myriads of very f i n e cracks, a r i s i n g from * the temperature shock due to absorbed UV r a d i a t i o n . The f u n n e l must be embedded i n s o f t e r m a t e r i a l s , e.g. nylon, otherwise the mechanical shock of the pi n c h i n g plasma w i l l destroy i t w i t h i n a very few shots. I am indebted to Ray E l t o n of N.R.L. f o r t h i s i n f o r m a t i o n . 36. In order to measure the t r a n s m i s s i o n of the CC^ l a s e r l i g h t , a second funnel has to be i n s t a l l e d a c c o r d i n g l y . I t must be expected that these m o d i f i c a t i o n s change the plasma parameters. Therefore, the plasma de n s i t y and temperature were measured a l s o w i t h both funnels i n s t a l l e d . The time of maximum compression, e.g. minimum rad i u s was measured w i t h no, one and two funnels i n s t a l l e d , using streak and shadow photography. These measurements were c a r r i e d out w i t h i n the program of B r i a n H i l k o ' s Ph. D. work, hence, d e t a i l e d comments and r e s u l t s w i l l appear i n h i s t h e s i s . The r e s u l t s , as f a r as they are r e l e v a n t f o r the experiments to be d e s c r i b e d , are shown i n the F i g . (2-3) o v e r l e a f . The top t r a c e i n each p i c t u r e shows the d e n s i t y ; the bottom tra c e shows the temperature of the pinch plasma as a f u n c t i o n of time. Time t = 0 i s chosen as the time d l ** when -jT^ - = 0 , 1 being the t o t a l current i n the pinch as measured w i t h a 26 Rogowski c o i l . The ,top and bottom p i c t u r e (no and two funnels) are experimental data obtained through i o n f e a t u r e Thomson s c a t t e r i n g . The e r r o r bars i n d i c a t e the spread of i n d i v i d u a l data and the u n c e r t a i n t y w i t h which temperature and d e n s i t y can be deduced from the experimental r e s u l t s . The middle p i c t u r e (one funnel) i s i n f e r r e d from the other two. The shaded area i n d i c a t e s an estimated e r r o r . The time axes of a l l three p i c t u r e s represent the true time lags from p i c t u r e to p i c t u r e . About the dashed l i n e i n the bottom p i c t u r e , see s e c t i o n 3.1. The a d d i t i o n a l data p o i n t s i n the d e n s i t y t r a c e of the top p i c t u r e are measurements from the Stark broadening of the 4686$ l i n e of He I , described i n 4.3. see footnote p. 33. ** This time reference w i l l be used e x c l u s i v e l y throughout the r e p o r t . 37. 1 4 1 1 1 • 1 i r — i 1 < 1 < 1 1 1 200 100 0 100 200 300 400 t[nsj 100 0 100 200 300 400 Figure (2-3) 38. The j i t t e r of the pinch discharge was reduced from about 100 nsec to fr e q u e n t l y l e s s than 10 nsec by f i r i n g a p r e i o n i z a t i o n discharge i n the v e s s e l p r i o r to the main pinch discharge. The a c t u a l c i r c u i t m o d i f i c a -t i o n of the Z-pinch discharge bank i s described i n the appendix. This r e d u c t i o n i n j i t t e r n a t u r a l l y was of great importance i n the s p e c t r a l shot to shot scanning of the backscattered (X^ l a s e r l i g h t (see 2.5). An expensive problem ( i n terms of money) f i n a l l y a r i s e s due to the f a c t that the plasma as i t pinches i s not confined a x i a l l y . P a r t of i t , t h e r e f o r e , i s e j e c t e d w i t h high speed through the hole i n the cathode towards the s a l t lens (see e.g. F i g . 2.5). The r e s u l t i n g impact i s s u f f i c i e n t to i n f l i c t v i s i b l e mechanical damage p a r t i c u l a r l y i n the c e n t r a l r e g i o n of the s a l t l ens a f t e r 10 shots. The problem was solved p a r t i a l l y by p l a c i n g an o b s t a c l e i n the plasma beam which i s sm a l l enough to not hinder the incoming C0„ l a s e r l i g h t . 39. 2.4 S p e c t r a l l y Integrated backscattered CO^ l a s e r l i g h t as a f u n c t i o n  of pinch-time. The experimental setup i s shown i n F i g . (2-5) on the next page. L i g h t s c a t t e r e d back from the plasma t r a v e l s back out through the s a l t lens and onto the e x i t s a l t window of the CG^ l a s e r . This window i s t i l t e d and due to F r e s n e l r e f l e c t i o n , ^ 4% i s r e f l e c t e d towards the Gen Tech energy meter. The r e s u l t i s shown i n F i g . (2-4) below. The background, as l a t e r experiments showed, a r i s e s from the f a c t that the plasma i t s e l f r a d i a t e s i n the i n f r a r e d . I [% of i n c i d e n t energy] .6-- 2 0 0 -100 0 100 200 300 t [ns] Figure (2-4): S p e c t r a l l y i n t e g r a t e d backscattered l a s e r l i g h t as a f u n c t i o n - o f .time. E r r o r bars are standard e r r o r s of the mean. A see s p e c i f i c a t i o n s at end of r e p o r t . Figure (2-5): Setup,for observing the - spect r a l l y - ; i n t e g r a t e d backscattered CC^ l a s e r l i g h t ( s e c t i o n 2.4). 41. 2.5 S p e c t r a l l y Integrated transmitted CO^ l a s e r l i g h t as a f u n c t i o n  of pinch-time. The experimental setup i s shown i n F i g . (2-7) on the next page. For high A t r a n s m i t t e d energies, a c a l o r i m e t e r was used as i n d i c a t e d . For lower tra n s m i t t e d energies a background again a r i s e s from the i n f r a r e d emission of the plasma i t s e l f . These low energies were th e r e f o r e measured w i t h a gold doped germanium detector , which time resolv e d the s i g n a l and thus allowed to d i s c r i m i n a t e against the i n f r a r e d emission from the plasma. A t y p i c a l o s c i l l o s c o p e trace i s shown below. 20 mV 100 ns Figure (2-6): Transmitted C0 2 l a s e r l i g h t with i n f r a r e d emission from the pinch. Au Ge d e t e c t o r . The r e s u l t s of the experiment are shown i n F i g . (2-8) and F i g . (2-9). * A p o l l o energy meter, see s p e c i f i c a t i o n s at end of report ** see s p e c i f i c a t i o n s Figure (2-7): Setup to measure the t r a n s m i t t e d CC^ l a s e r l i g h t ( s e c t i o n 2.5). Figure (2-8): % of transmitted l i g h t as a f u n c t i o n of time, A p o l l o energy meter. E r r o r bars denote the standard e r r o r of the mean. 44. I [% of incident energy] 0 t[ns] ure (2-9): % of transmitted l i g h t as a f u n c t i o n of time, Au Ge de t e c t o r . E r r o r bars denote the standard e r r o r of the mean; squares are s i n g l e measurements. 45. 2.6 S p e c t r a l l y resolved backscattered CO^ l a s e r l i g h t at pinch time t = 0 ± 25 nsec and the angular dependence of the backscattered l i g h t . The experimental arrangement i s shown i n F i g . (2-10) on the next page. This setup was chosen f o r the f o l l o w i n g reasons: S p h e r i c a l m i r r o r s , i f used o f f a x i s , produce astigmatism. I f they are a d d i t i o n a l l y t i l t e d -out of the xy plane, t h i s astigmatism appears r o t a t e d . The. second plane m i r r o r . i n Fig.(2-10) receiving;the backscattered l i g h t serves to e l i m i n a t e t h i s r o t a t i o n of astigmatism. The r e s t of * the o p t i c s i s set up to match the f number of the monochromator , to keep the astigmatism at a minimum by imaging as l i t t l e as p o s s i b l e o f f a x i s and, image as s t i g m a t i c a l l y as p o s s i b l e . F i n a l l y , the choice of components was very l i m i t e d . Another problem a r i s e s due to the surface i r r e g u l a r i t i e s of the CO2 l a s e r output window. As i t had to be used to r e f l e c t p a r t of the back-s c a t t e r e d l i g h t towards the d e t e c t i o n o p t i c s , the s a g i t t a l focus at the entrance s l i t was not very sharp and much i n t e n s i t y was l o s t there. To measure the angular dependence of the backscattered l i g h t , two types of vmasks were used on the m i r r o r i n d i c a t e d i n F i g . (2-10) >. .. . • -A Small Mask only r e f l e c t e d that l i g h t towards the d e t e c t i o n o p t i c s A* that came back through the inner p a r t of the CO2 l a s e r output annulus. A B i g Mask only r e f l e c t e d l i g h t towards the d e t e c t i o n o p t i c s that came d i r e c t l y back through the outer p a r t of the annulus. A see s p e c i f i c a t i o n s at end of report AA see 2.2 45a. Figure (2-10): Setup to s p e c t r a l l y decompose the. backscattered C0„ l a s e r l i g h t ( s e c t i o n 2.6). 46. B i g Mask Small Mask Figure (2-11): Backscattered l i g h t was tra n s m i t t e d through the. unshaded areas. The s p e c t r a l i n t e n s i t y d i s t r i b u t i o n of l i g h t s c a t t e r e d back through the b i g mask and of l i g h t s c a t t e r e d back through the s m a l l mask i s shown i n F i g s . (2-12) and (2-13). K a r b units) 1 0 . 5 8 0 1 2 3 4 5 6 7 8 9 . 5 9 0 1 2 A Figure (2-12) S p e c t r a l d i s t r i b u t i o n of l i g h t s c a t t e r e d back through the b i g mask. The dashed l i n e represents the u n s h i f t e d CC^ l a s e r l i n e . I t s true width i s s m a l l compared to the instrument p r o f i l e . 100 u s l i t s . E r r o r bars denote standard e r r o r of the mean; the shaded area shows the approximate noise l e v e l . Time i s 0 ± 25 nsec. 47. K a r b units) Figure (2-13): S p e c t r a l d i s t r i b u t i o n of l i g h t s c a t t e r e d back through the s m a l l mask. The dashed l i n e at 10.583urepresents again the u n s h i f t e d CO2 l a s e r l i n e . I t s true width i s s m a ll compared to the instrument p r o f i l e determined by 100 u monochromator s l i t s . E r r o r bars denote standard e r r o r of the mean. The shaded area shows the approximate noise l e v e l . The v e r t i c a l s c a l e i s 7.9 times that of F i g . (2-12). Time i s 0±25 nsec. F i g . (2-13) shows that l i g h t s c a t t e r e d back through the inner p a r t of the CO2 l a s e r output annulus i s not only s h i f t e d i n wavelength, but e x h i b i t s wings on both s i d e s of the c e n t r a l l i n e . A p o s s i b l e e x p l a n a t i o n f o r t h i s e f f e c t i s given i n 3.24. K a r b . u n i t s ) Figure (2-14): S p e c t r a l l y r e s o l v e d backscattered CC^ l a s e r l i g h t , no mask used. L e f t , the u n s h i f t e d CCv, l a s e r l i n e at 10.583 u. I t s width i s the instrument width. S l i t s 130 u, e r r o r bars denote standard e r r o r of the mean. Shaded area i n d i c a t e s the approximate noise l e v e l . These data were taken at a much e a r l i e r date than those shown i n F i g . (2-12) and (2-13). CHAPTER 3 49. E v a l u a t i o n of the experimental r e s u l t s 3.1 The transmitted CO., l a s e r l i g h t F i g . (3-1) shows the tran s m i t t e d and backscattered i n t e n s i t y as a f u n c t i o n of d e n s i t y as i t can be condensed out of F i g . (2-3) and Fig s , (2-4) and (2-9). I [%>of i n c i d e n t e n e r g y ] Backsc. Transm. •11 0.5-0.4} 0.3 + 0.2 + 0.1 f • • i • i i 4 5 6 7 891 16 X10 i • — i — . i > i—i—i—i—r-3 4 5 6 7 891 17 X10 ~I !—I—r-i—• | i—|—i—I 3 4 5 6 7 89 x1018 n e [ c m 3 ] Figure (3-1): Backscattered (b) [one funnel i n s t a l l e d ] and transmitted (a) [two funnels i n s t a l l e d ] energy as a f u n c t i o n of d e n s i t y of the plasma. V e r t i c a l e r r o r bars are e x p e r i -mental e r r o r s from F i g s . (2-4) and (2-9), h o r i z o n t a l e r r o r bars are due to den s i t y estimates from F i g . (2-3). 50. This measurement w i l l now be compared with the theory of l i g h t absorp-t i o n due to inverse bremsstrahlung (see a l s o p .1 ) . The two funnels i n s t a l l e d i n the pinch v e s s e l i n order to be able to make these l i g h t t r a n s m i s s i o n measurements (see F i g . (2-7)) l e f t 2 cm of plasma between them. T h i s , t h e r e f o r e , i s the length over which the CO2 l a s e r l i g h t i s absorbed. For the c l a s s i c a l theory of i n v e r s e bremsstrahlung to be v a l i d , the f o l l o w i n g requirements need to be s a t i s f i e d . (a) C O q >> ujp, e.g. the plasma must be q u i t e underdense which i s f u l f i l l e d (b) c V e2E 2 2mto 2 *B o < k T, e.g. the quiver energy of the e l e c t r o n i n the f i e l d of the l a s e r l i g h t must be small compared to the k i n e t i c energy due to thermal motion which i s f u l f i l l e d as w e l l . e 2E 2 (c) o * 9 7 •' 2 m a ] 2 < "ft (do > i . e . the quiver energy of the e l e c t r o n i n die o f i e l d of the l a s e r l i g h t must be so s m a l l that the energy of photons created by bremsstrahlung i s not comparable to the photon energy absorbed i n i n v e r s e bremsstrahlung. In our case, the two terms are comparable. M o d i f i c a t i o n s i n the c l a s s i c a l expression f o r i n v e r s e bremsstrahling absorption e 2 V 27 however, only become necessary i f 2 >>-n co0 > which i s not the case f o r the experiment described. In the Rayleigh Jeans l i m i t f o r Planck's. r a d i a t i o n law, the l i n e a r a bsorption c o e f f i c i e n t can be w r i t t e n as KB " A ' ^ y , */2 GOVO). ( I I I - l ) • 3/6ii c co 2(mkT) ' o 51. * u c o 2 F o r % = U C 0 2 ' 6 ~k^f~ » Z n e = n i a n d G ( T ' "CO^ * Z' 5> t h i s reduces to n 2 K = 7.06 x 10~3k - J J 2 ( I H - 2 ) T Having £ cm of plasma, the i n t e n s i t y I of the t r a n s m i t t e d l i g h t as a f u n c t i o n of £ i s given by = e " < B * ( I I I - 3 ) o In comparing the experiment w i t h the theory however, the f o l l o w i n g problem a r i s e s : 2 - C - f y £ 1(1) T3/2 As — — = e , the r e l a t i v e e r r o r i n the i n t e n s i t y o r a t i o i s » i ( A ) I n 2 2 c — £ K A ) " T3/2 I - A n 3 AT n 2 T ( I I I - 4 ) where and 4Jr are the r e l a t i v e e r r o r s i n d e n s i t y and temperature so n T . that f o r K £ > 1, a 10% e r r o r i n — or 4ir leads to an order of magnitude n T e r r o r f o r the i n t e n s i t y r a t i o . Taken however i n the form KB = - j r * ^ r 1 ( I I I " 5 ) o one can see from the reverse argument of ( I I I - 4 ) that a measurement of allows a good determination of K- and hence of n 2/T ^ 2 , or I a e o one of the q u a n t i t i e s i f the other q u a n t i t y i s known. 52. According to the d e r i v a t i o n of the above expression, T i s the temperature of the plasma before the a c t u a l absorption process heats the plasma. I(£) The f o l l o w i n g t a b l e shows the values of K F Tcomputed from — and the o values of T computed from K.and the d e n s i t i e s given from F i g . (2-3). E l e c t r o n d e n s i t y e cc 1 0 1 [ % ] o K [ — ] •LcmJ T [eV] e L J 8 x 1 0 1 6 10.3 1.14 2.5 9 x 1 0 1 6 8.9 1.21 2.8 1 x 1 0 1 7 7.6 1.29 3.1 1.5 x 1 0 1 7 3.3 1.71 4.4 2 x 1 0 1 7 1.5 2.10 5.7 2.5 x 1 0 1 7 .75 2.45 6.9 3 x 1 0 1 7 .30 2.90 7.8 3.5 x 1 0 1 7 ^.05 ^3.6 ^9 These c a l c u l a t e d values f o r T are a l s o shown as the dashed l i n e i n e F±g°. (2-3), bottom p i c t u r e , where one can see that they connect w e l l w i t h the temperature curve measured by Thomson s c a t t e r i n g . I t must be kept i n mind that not a l l the (X^ l a s e r l i g h t that was not transmitted need a c t u a l l y be absorbed by inverse- bremsstrahlung but can 28 w e l l be r e f r a c t e d out of the plasma. This would lower the absorption c o e f f i c i e n t K" and as a consequence increase the c a l c u l a t e d e l e c t r o n B temperature T. Evidence f o r the complexity of the i n t e r a c t i o n volume i s shown i n the p i c t u r e on the next page ( F i g . 3-2). Many thanks to B r i a n H i l k o f o r l e t t i n g me use t h i s p i c t u r e . 53. CC>2 l a s e r Figure (3-2): This photograph i s a shadowgram done i n Ruby l a s e r l i g h t w i t h the image plane *v» 1 cm away from the plasma, viewed s i d e on. The p i c t u r e i s magnified 3.4 times. The CC^ l a s e r i s i n c i d e n t from the r i g h t . The vacuum f o c a l spot i s i n the middle of the p i c t u r e . The two funnels (see F i g . (2-7)) are j u s t outside the p i c t u r e t o the l e f t and r i g h t ; time i s - 30 n s e c * ** I t i s evident that some plasma i s pushed away by the CO2 l a s e r , that the beam i s somewhat fanned out and bent away from i t s o r i g i n a l d i r e c t i o n and that i t l o s e s c o n s i d e r a b l y i n i n t e n s i t y as i t penetrates the plasma The t r a n s m i t t e d C0^ l a s e r l i g h t shows no s h i f t o f f the (X^ frequency of 10.583 y. The measured spectrum i s shown i n F i g . (3-3) on the f o l l o w i n g page. The r e s u l t , to some ex t e n t , r u l e s out modulational i n s t a b i l i t i e s " ^ ' " ^ '^^'^ r e s u l t i n g i n sidebands i n the foreward s c a t t e r e d l i g h t . At the given s p e c t r a l The sharp l i g h t l i n e s i n d i c a t e the region where the i n c i d e n t l a s e r l i g h t has decreased the plasma d e n s i t y . The w e l l - d e f i n e d annular shape of the f o c a l r e g i o n i s a l s o i n d i c a t e d This f a c t w i l l be of great importance f o r arguments presented i n S e c t i o n 3.24. 54. I [arb. unit^ 10.580 10590 Figure (3-3): The s p e c t r a l l y r e s o l v e d transmitted CC^ l a s e r l i g h t at pinch times < -20 nsec. C i r c l e s i n d i c a t e the wavelength r e g i o n scanned. The shaded area denotes the noise l e v e l . The setup used was analogous to the ones shown i n F i g . (2-7) and (2-10). r e s o l u t i o n the frequency s h i f t r e s u l t i n g from such i n s t a b i l i t i e s would have to be comparatively l a r g e to be observed. The occurrence of f i l a m e n t a t i o n can p r i n c i p a l l y not be r u l e d out; i t i s however, u n l i k e l y to occur as, f o r the re l e v a n t d e n s i t i e s , homogeneous plasma thresholds are only j u s t exceeded. This i s i n good agreement w i t h 24 25 estimates from other t h e o r i e s ' which p r e d i c t f o r our case a maximum den s i t y depression due to f i l a m e n t a t i o n of — ^ 5%. n F i n a l l y , there i s , with the setup used, no p o s s i b i l i t y of d i s t i n g u i s h i n g a minor f i l a m e n t a l e f f e c t from simple l i g h t t ransmission through an underdense plasma (see, however, p. 76 ). These arguments lead to the conc l u s i o n that the decrease i n transmitted C0 2 l a s e r l i g h t i n t e n s i t y - i s due to i n c r e a s i n g inverse bremsstrahlung absorption of a plasma of i n c r e a s i n g d e n s i t y . compare with F i g s . (2-12) - (2-14). •55. 3.2 The backscattered CO., l a s e r l i g h t The r e s u l t s of the experiments concerning the backscattered CC^ l a s e r l i g h t w i l l be evaluated i n the f o l l o w i n g s e c t i o n s : 3.21 Enhancement of the backscattered CC^ l a s e r light-above thermal l e v e l s and the r e s u l t i n g i o n wave amplitudes i n the plasma. 3.22 D i s c u s s i o n of the observed wavelength s h i f t of the backscattered l a s e r l i g h t . 3.23 Angular dependence.of the backscattered CO^ l a s e r l i g h t and comparison w i t h theory. 3.24 The wavelength dependence of the l i g h t backscattered through the -small mask . see p. 46 5.6. 3.21 Enhancement of the backscattered CO^ l a s e r l i g h t above thermal  l e v e l s and the r e s u l t i n g i o n wave amplitudes i n the plasma. In order to determine the enhancement of s c a t t e r e d l i g h t above thermal l e v e l s , the i n t e n s i t i e s s c a t t e r e d from thermal d e n s i t y f l u c t u a t i o n s (Thomson s c a t t e r i n g ) must be known f i r s t . The theory of Thomson s c a t t e r i n g from plasmas shows that the l i g h t i n t e n s i t y I as scattered.from thermal f l u c t u a t i o n i s given by I (a) = S(a) I . N. -4- ( l - s i n 2 e cos 2*) ( I I I - 6 ) s 8n T xnc e rz r Here, = .66 x 1 0 - 2 1 + cm 2 i s the Thomson s c a t t e r i n g c r o s s e c t i o n . Z a 4 S(a) = • - f o r the i o n f e a t u r e ( l + a 2 ) [ l + a 2 ( l + Z Y-)] 1 1 a = — — , where k i s the wave vect o r of the d e n s i t y f l u c t u a t i o n D which does the s c a t t e r i n g . i s the Debye s h i e l d i n g d i s t a n c e . I . = i n c i d e n t l a s e r l i g h t i n t e n s i t y xnc J = number of e l e c t r o n s present w i t h i n the s c a t t e r i n g volume. The l a s t term on the R.H.S. of equation ( I I I - 6 ) describes the d i p o l e f i e l d of the e l e c t r o n o s c i l l a t i n g i n the i n c i d e n t l a s e r f i e l d i n a coordinate system explained i n F i g . (3-4) on the f o l l o w i n g page. *41, 51 57. Figure (3-4): E x p l a i n i n g the coordinate system used i n the c a l c u l a t i o n s i n Secti o n 3.21. The d i p o l e f i e l d i s r o t a t i o n a l l y symmetric w i t h respect to the x a x i s . The * coordinate r o t a t e s around the Z a x i s . " B a c k s e a t t e r i n g " means 6 -> 180 . In order to f i n d the i n t e n s i t y s c a t t e r e d back through the pinch lens (see F i g . ( 2 - 5 ) ) , equation ( I I I - 6 ) w i l l have to be i n t e g r a t e d from <j> = 0 to 2ir and 9 = 180° - 4p- to 180°. There A6 corresponds to the angle of the "u f/5 pinch lens--, A0 f o r f-numbers >> 1. f-number Assuming that the s c a t t e r i n g occurs i n the underdense region of the plasma, |k| * 12k | (see p. 16 ) so that a = * 2k~\ ' F°r t*ie D o D d e n s i t i e s and temperatures in v o l v e d (see F i g . (2-3)) one can see that i n a l l cases a >> 1. With the plasma being a He-Plasma, S(a) then becomes 2 X 3" " ^e' t* i e n u m ^ e r °f e l e c t r o n s i n the s c a t t e r i n g volume i s given by N n e h where n i s the e l e c t r o n d e n s i t y , A^ the f o c a l area of the C0„ l a s e r , e F 2 being ^ 1 mm2 and I i s the length of the i n t e r a c t i o n r e g i o n . F i n a l l y , we m u l t i p l y eq. ( I I I - 6 ) by y because only the long wavelength si d e of the Thomson s c a t t e r e d i o n feat u r e was observed as being enhanced. Accounting f o r a l l these p o i n t s , one a r r i v e s at 58. therm = T x 8 . 1 3 x l 0 _ 3 0 n x £ ( I I I - 7 ) BS i n c e where now I,,,, i s i n J o u l e s , and I . i s i n Joules/cm 2. BS xnc With 27 Joules of i n c i d e n t CO^ l a s e r energy focussed to 1 mm2 t h i s reduces to therm = 2 2 ^ 1 Q _ 2 6 £ x q ( I H - 8 ) J J O e An "exact" value f o r I i s not known. Considering the geometry i n v o l v e d i t i s , however, reasonable to assume that i t i s of the order of a few mm. Supported by experimental evidence, ( F i g . (3^-2)), I ^ 1 cm was used. From formula ( I I I - 8 ) and F i g . (3-1), the enhancement V. T observed thermal , , , , „ , , , . Igg ' BS C a n c a l c u l a t e d . The r e s u l t s xs p l o t t e d xn F i g . (3-5) on the f o l l o w i n g page and shows that the i n t e n s i t y enhancement drops smoothly as the d e n s i t y of the plasma i n c r e a s e s . To t r u l y understand the behaviour shown i n F i g . (3-5) i n terms of Stimulated B r i l l o u i n s c a t t e r i n g i t would be necessary to have d e t a i l e d i n f o r m a t i o n about the temperature s c a l e length of the plasma as a f u n c t i o n of time. Figure (3-5): Showing the i n t e n s i t y enhancement above thermal l e v e l s as a f u n c t i o n of d e n s i t y i n the plasma. V e r t i c a l e r r o r bars are experimental e r r o r s from F i g . (2-4); h o r i z o n t a l e r r o r bars are due to the d e n s i t y estimates from F i g . (2-1). With t h i s information not being a v a i l a b l e at t h i s p o i n t , the q u a l i t a t i v e behaviour, of -I - . / I • .. versus n could enh therm e perhaps be understood-by the f a c t that i n c r e a s i n g d e n s i t y and temperature at decreasing dimensions (the plasma i s pinching) means decreasing s c a l e lengths, hence i n c r e a s i n g t h r e s h o l d s . see p.29 • 6 0 . To c a l c u l a t e the d e n s i t y f l u c t u a t i o n that gives r i s e to t h i s enhancement, one again needs to know the thermal f l u c t u a t i o n f i r s t . 29 6 N As i s w e l l known, the r e l a t i v e thermal d e n s i t y f l u c t u a t i o n therm N of N p a r t i c l e s w i t h i n a given c o n t r o l volume c o n t a i n i n g on the average N >>> 1 p a r t i c l e s , i s 6 N therm 6 n therm N n _1_ ( I I I - 9 ) where n denotes the corresponding p a r t i c l e d e n s i t i e s , therm The i n t e n s i t y I,,,, s c a t t e r e d from these thermal f l u c t u a t i o n i s D O T therm . therm ? I B S ^ < | 6 n |2> (111-10) Hence, the r e l a t i v e i n t e n s i t y enhancement above thermal l e v e l s i D o e n k / l T , r , t ' i e r m , f o l l o w s from the d e n s i t y f l u c t u a t i o n enhancement above do Jib , g n e n n thermal l e v e l s :— through - therm ° o n 6 n enh 6 n therm enh "BS therm BS ( I I I - l l ) T h i s , together wi t h ( I I I - 9 ) y i e l d s the absolute d e n s i t y f l u c t u a t i o n g i v i n g r i s e to the observed i n t e n s i t y enhancement as 6 n enh /nV x LP enh UBS L I B S therm (111-12) where V^p i s the i n t e r a c t i o n volume, ^enh Note that — depends only weakly on the i n t e r a c t i o n volume. As we discuss enhancements of s e v e r a l orders of magnitude, an ina c c u r a t e guess i n V^p by a f a c t o r 10 does not change the contents of a statement about 6 n 61. 6 n e n n * therm F i g . (3-6) shows the observed as w e l l as the t h e o r e t i c a l -^ -2 n n ^ n . The o v e r a l l enhancement above thermal f l u c t u a t i o n s i s about four orders of magnitude. 6 th ~ enh n on n n -8 -4 x10 . x10 20 10 9 8 7 6 5 2 H 1 .9 .8 .7 .6 •II •I . i : 1 i l l 1—I—i i ——T : 1 I I—i—«—r—r-i i :—> 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 x101? x l d 8 n j cm 3 ] Figure (3-6): Showing the absolute d e n s i t y f l u c t u a t i o n s g i v i n g r i s e to the enhanced backscattered l i g h t s i g n a l as a f u n c t i o n of plasma d e n s i t y . E r r o r bars l i k e i n F i g . (3-5). The s t r a i g h t l i n e represents r e l a t i v e thermal d e n s i t y f l u c t u a t i o n s . 62. 3.22 D i s c u s s i o n of the observed wavelength s h i f t of the backscattered  CO^ l a s e r l i g h t . The wavelength"shift of the backscattered CC^ l a s e r l i g h t was measured,:.in s e v e r a l independent experiments, to be (4.7 ± .4) x 1 0 - 3 um towards the red end of the spectrum. In p r i n c i p l e , t h i s i m p l i e s a net r e c e s s i o n v e l o c i t y v of the r e f l e c t i n g object of v = | f (111-13) r 2 A o which f o r the quoted Aa = 4.7 x 1 0 - 3 ym i s v = 6.6 cm/usec. The very nature of the b a c k s e a t t e r i n g experiment however does not a l l o w d i s t i n g u i s h i n g between a s h i f t a r i s i n g from bulk plasma motion and a s h i f t a r i s i n g from the s c a t t e r i n g o f f t r a v e l l i n g i o n a c o u s t i c waves. Therefore, the v e l o c i t y of the bulk plasma motion towards the i n c i d e n t CC>2 l a s e r s h a l l be estimated. Then, the h e a t i n g of the plasma by i n v e r s e bremsstrahlung under the given plasma parameters w i l l be considered. From t h a t , we t r y to conclude under .which, .circumstances:.-the:" enhanced backseat t e r i n g of C0 2 " l a s e r ? l i g h t occurs. To estimate the a x i a l escape v e l o c i t y , we proceed two ways. Momentum conservation f o r an i n f i n i t e s i m a l mass element of the r a d i a l l y c o l l a p s i n g plasma shows that the a x i a l escape v e l o c i t y must be of the order of the same f o r other experiments of t h i s type, e.g. 30,31. 63. r a d i a l c o l l a p s e v e l o c i t y . From s t r e a k photography t h i s was measured to be ^ 4 cm/usec. Considering, on the other hand, f r e e thermal expansion out of the ends of a plasma column ( r a d i a l l y confined by a magnetic f i e l d ) , one would estimate the excape v e l o c i t y to be l a r g e r or equal to the i o n thermal speed which, at T ^ 25 eV, i s about 3.5 cm/usec. With both estimates l e a d i n g to the same r e s u l t , we assume 4 cm/usec f o r the a x i a l escape v e l o c i t y v 1 esc Next, we combine the a x i a l escape v e l o c i t y of the plasma w i t h the measured wavelength s h i f t of the backscattered l i g h t to make a statement about the temperature of the region from which the enhanced b a c k s c a t t e r i n g occurs. S c a t t e r i n g o f f i o n a c o u s t i c waves d r i v e n by the i n c i d e n t CC^ l a s e r l i g h t w i l l r e s u l t i n a red s h i f t p r o p o r t i o n a l to the group v e l o c i t y v. of the i o n a c o u s t i c wave. I f t h i s i o n a c o u s t i c wave t r a v e l s i n a plasma that moves as a whole towards the i n c i d e n t CC^ l a s e r l i g h t w i t h v e l o c i t y v , the net wavelength s h i f t due to both motions w i l l be esc v. , - v = f T 1 (111-14) i a esc 2 A o e.g. i f Vj,a= v e s c , no s h i f t w i l l r e s u l t . With v £ s c estimated, as 4 cm/usec and an observed wavelength s h i f t of AA of 4.67 x 1 0 - 3 um, t h i s i s v. = 10.6 cm/usec, xa Using the d i s p e r s i o n r e l a t i o n f o r i o n a c o u s t i c waves at equal i o n and e l e c t r o n temperatures and t a k i n g k x D << 1 (see p. 5 7 ) , one obtains f o r the temperature of the plasma i n which the i o n wave t r a v e l s see footnote p. 33 64. T = 70 eV f o r v = 0 cm/usec esc T = 180 eV f o r v = 4 cm/ysec esc Next, we proceed to make a temperature estimate from i n v e r s e bremsstrahlung c o n s i d e r a t i o n s . The inverse bremsstrahlung absorption of l a s e r l i g h t by a plasma of n a 2 x 1 0 1 8 c m - 3 T » 25 eV, which are the parameters at the time e ) e of i n t e r e s t (see F i g . (2-3) mi d d l e ) , i s found to be K - 22 cm -l B Hence, over 2 mm, 99% of the i n c i d e n t r a d i a t i o n would be absorbed. Because of the f i n i t e temperature c o n d u c t i v i t y of the plasma i t i s unreasonable to assume such a l o c a l h eating of a 17 cm long plasma column and one has to take .temperature d i f f u s i o n i n t o account. Considering the l o s s of thermal energy out of a volume l i m i t e d by the C0 2 l a s e r f o c a l area and these 2 mm absorption length due to temperature 19 32 d i f f u s i o n only, using the thermal c o n d u c t i v i t y ' .5/2 ** 10k^ (2k T ) ° / Z KT = 3/2 h \ , < m - 1 5 > TT m ^ e 4 l n A o and t y p i c a l assumed temperature s c a l e lenths of the plasma, one a r r i v e s at a r e p r e s e n t a t i v e d i f f u s i o n time of n 2 x T = £ 2 x 4.9 x 10 2 2 - f y 9 (T. eV) (111-16) T T T5/2 i n For s c a l e lengths J c m <v mm, n » 2 x 1 0 1 8 c m - 3 and T > 20 eV, x m i s found T e 0/ T -5/2 to be < 10 nsec. As x m i s p r o p o r t i o n a l to T , higher temperatures r e s u l t i n much smaller times t ^ . *see ( I I I - l ) InA Sr' 10 i s the Coulomb lo g a r i t h m Hence, i t must be assumed that the i n t e r a c t i o n volume V , where the J_iir CC>2 l a s e r energy i s dumped i n t o the plasma, i s many times the 99% inverse bremsstrahlung absorption l e n g t h , e.g. the e n t i r e l e n g t h of the plasma column. The energy content E = N R^T of the plasma column at n g ^ 2 x'.AO18cm-3 and T ^ 25 eV i s about E = 2.0 J o u l e s . e Hence, absorbing a l l the i n c i d e n t 27 Joules of l a s e r l i g h t w i t h i n the 27 plasma leads to a f o l d increase i n temperature or to a heating to T = 350 eV. e In applying inverse bremsstrahlung c o n s i d e r a t i o n however, s e v e r a l assump-t i o n s were- made, which are i n r e a l i t y not f u l f i l l e d . The d e r i v a t i o n s l e a d i n g to I I I - l , I I I - 3 , and 111-15 assume that the heating i s i n f i n i t e s i m a l , e.g., that the temperature before and a f t e r the absorption process i s e s s e n t i a l l y the same, that k i s not a f u n c t i o n of distance and that K„ i s n e i t h e r a f u n c t i o n of temperature nor time. 1 i Attempts w i t h i n t h i s l a b o r a t o r y to solve the exact problem n u m e r i c a l l y are under way. Considering that the inverse bremsstrahlung absorption c o e f f i c i e n t decreases as the plasma temperature increases and that most l i k e l y some of the 28 i n c i d e n t CO^ l a s e r l i g h t i s r e f r a c t e d away from the plasma without being absorbed,cone w i l l have to conclude that the plasma i s heated to l e s s 9 than 350 eV by i n v e r s e bremsstrahlung absorption. An e a r l i e r experiment however . w i t h somewhat d i f f e r e n t geometry, i n d i c a t e d heating to 200 eV. There e x i s t s however a l s o the p o s s i b i l i t y that the region from where the b a c k s c a t t e r i n g occurs i s a c t u a l l y a region of lower d e n s i t y which l i e s Barnard, G u l i z i a , p r i v a t e communications. 66. outside the main plasma body and f o r which the inverse bremsstrahlung absorption c o e f f i c i e n t i s lower and t h e r e f o r e the temperature increase due to h e a t i n g by i n v e r s e bremsstrahlung absorption i s a l s o lower. The experiment discussed i n 3.24 seems to support t h i s view. S t r i c t l y speaking, there i s not yet enough experimental data to d e c i s i v e l y conclude on the circumstances under which the observed enhanced s c a t t e r i n g occurs but from the measurements and c o n s i d e r a t i o n s presented i t can be concluded that the enhanced s c a t t e r i n g o f f i o n a c o u s t i c waves (stimulated B r i l l o u i n s c a t t e r i n g ) occurs' i n a region i n f r o n t of the main core of the plasma where the e l e c t r o n d e n s i t y i s some a. 1 0 1 7 cm - 3, the temperature i s around 180 eV and the temperature s c a l e lengths are of the order of m i l l i m e t e r s . The enhancement i n backscattered i n t e n s i t y and i n induced d e n s i t y f l u c t u a t i o n s above thermal l e v e l s i s l i k e l y to be somewhat higher than i n d i c a t e d by F i g s . (3-5) and (3-6). 67. 3.23 The angular dependence of the backscattered CO., l a s e r l i g h t and  comparison w i t h theory. The angular dimensions of the setup f o r measuring the backscattered C0 9 l a s e r l i g h t were as f o l l o w s . i n t e r a c t i o n v o l u m e Figure (3-7): Angular dimensions of the CC^ l a s e r f o c u s s i n g o p t i c s (see a l s o p. 3 4 ) . The arrows m c i d a t e that the l a s e r l i g h t was i n c i d e n t through the annular r e g i o n ; backscattered l i g h t was observed s e p a r a t e l y through the annular region and through the c e n t r a l r e g i o n of the annulus. The spot i n the middle i n d i c a t e s the area obscurred by an ob s t a c l e put i n the way of the a x i a l l y escaping plasma to prevent severe mechanical damage to the KC1 lense. By using appropriate masks i n the b a c k s c a t t e r i n g o p t i c s (see F i g . (2-10) and (2-11)) and comparing the backscattered peak i n t e n s i t i e s i t was found that 7.9 times more l i g h t was s c a t t e r e d back through the b i g mask as compared to the sm a l l mask ( F i g s . (2-12) and (2-13)). As an estimate i t i s therefore reasonable to say that the divergence angle of the backscattered l i g h t i s about 3° to 5°. Comparison w i t h theory Omitting the ^  cos2<j> dependence p r e d i c t e d f o r the backscattered l i g h t by the theory f o r i n f i n i t e , homogeneous plasmas as too simple, we w i l l r e s t r i c t ourselves to the comparison with two more elaborate t h e o r i e s p r e d i c t i n g the t y p i c a l angular divergence f o r B r i l l o u i n backscattered l a s e r l i g h t . 68. 14 The f i r s t theory i s the treatment of st i m u l a t e d b a c k s c a t t e r i n g i n a f i n i t e , inhomogeneous plasma which p r e d i c t s a t y p i c a l angular spread Se__ of (quoted i n Ref. 33) DO 6 6 B S - 7Ti (III"17) There, 3£ i s the t o t a l gain.observed when a backscattered e l e c t r o -magnetic wave t r a v e l s through a medium of len g t h £ w i t h a gain of 3 per u n i t length. 33 , The second theory was forewarded r e c e n t l y by Lehmberg. Knowing that the f i r s t theory p r e d i c t s f o r t y p i c a l l a s e r f u s i o n experiments angular spreads which are f a r l a r g e r than observed, the second theory assumed not only that a s c a t t e r e d electromagnetic wave t r a v e l s through a medium wi t h gain, but a l s o , that i t has to s a t i s f y a Bragg c o n d i t i o n set up by the i n t e r f e r e n c e of d i f f e r e n t angular p a r t s of the i n c i d e n t l i g h t beam i n the plasma. From t h a t , an angular r e s o l u t i o n f o r the backscattered l i g h t of \, I, 9min = ( s T ) 4 ( f ) 2 i s d e r i v e d - < i r i- 1 8> o Here, k i s the wave ve c t o r of the i n c i d e n t l i g h t . ° The p r e d i c t i o n s of both t h e o r i e s f o r a t y p i c a l l a s e r f u s i o n experiment on one hand, and the experiment described i n t h i s r e p o r t on the other hand, w i l l now be compared. T y p i c a l l a s e r f u s i o n This experiment experiment-^ parameters k (Nd glas s ) = 6 x lOVcm k*.(C0o l a s e r ) = 6 x 10 3/cm o o 2 g£ 10 to 15 3£ 19. * £ 100 ym £ 5 mm 3 (1 to 1.5) x:103/cm 3 38/cm *see F i g . (3-5) T y p i c a l l a s e r f u s i o n experiment This experiment T h e o r e t i c a l P r e d i c t i o n s e D min BS ^ 1.2 or 60 ^ .14 or 8° o 6e„_ ^ .93 or 53° 0^ . ^ .063 or 3.6° D min Observed I t i s c l e a r that the f i r s t theory f a i l s i n e x p l a i n i n g both types of experiments. The experimental evidence suggests the correctness of Lehmberg's formula. The author wishes to emphasize that even though he claims to understand the p h y s i c a l p r i n c i p l e behind 111-18 as i t i s described i n Ref. 33, he would at t h i s p o i n t be unable to. r e d e r i v e 111-18 from f i r s t p r i n c i p l e s . The d e s c r i p t i o n of the r e s u l t s obtained i n Ref. 33 i s however e x p l i c i t enough that numerical values can be obtained from the presented formulae. 70. 3.24 The wavelength dependence of the l i g h t backscattered through the * sma l l mask The experimental r e s u l t ( F i g . (2-13)) shows that the spectrum of the CO^ l a s e r l i g h t s c a t t e r e d back through the small mask i s not simply a l i n e s h i f t e d by 4.7 x 1 0 - 3 ym The center of the l i n e i s s t i l l s h i f t e d by 4.7 x 1 0 _ 3 ym but a d d i t i o n a l l y wings are present, suggesting a modulation frequency co^ of coM % (3.5 to 2.9) x 1 0 1 0 M sec To my knowledge, the angular dependence of st 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 has not been measured before w i t h t h i s type of geometry (see F i g . (3-7)) and such a modulation has not yet been observed. Note that t h i s modulation frequency i s now an a b s o l u t e l y determined frequency that i s not obscured by any immeasurable s h i f t s due to bulk motions, :. '. .. et c . I n t r y i n g to provide a p h y s i c a l e x p l a n a t i o n i t was found that the modulation could best be explained through the g y r a t i o n of e l e c t r o n s i n the pinch magnetic f i e l d . The necessary reasoning s h a l l now be presented. For a s c a t t e r e d spectrum to be modulated, a n o n l i n e a r c o u p l i n g between the modulation o s c i l l a t i o n and the o s c i l l a t i o n of the s c a t t e r i n g e l e c t r o n i n the electromagnetic f i e l d s must e x i s t . In a s t a t i o n a r y magnetic f i e l d , the Lorentz f o r c e provides t h i s c o upling i n a very n a t u r a l way by simply modulating the v e l o c i t y of the e l e c t r o n w i t h the gyro frequency w where A see 2.6 71. to = — (111-19) g m For these modulations to be v i s i b l e however, the wave vec t o r "of the d e n s i t y f l u c t u a t i o n s c a t t e r i n g the i n c i d e n t electromagnetic wave must be very near perpendicular to the vector of the magnetic f i e l d 4 8 ' " ^ (e.g. <5°). Other-wise, the thermal motion of e l e c t r o n s moving p a r a l l e l to the magnetic f i e l d produces enough Doppler broadening to smear out the modulations. Consider-ing the setup i n F i g . (2-10) and the angular dimensions of the f o c u s s i n g and b a c k s c a t t e r i n g o p t i c s ( F i g . ( 3 - 7 ) ) , one can see .that t h i s requirement; i s f u l f i l l e d . Secondly, the spectrum of thermal e l e c t r o n d e n s i t y f l u c t u a t i o n s perpendicular to an a p p l i e d magnetic f i e l d already shows modulations at m u l t i p l e s of the e l e c t r o n c y c l o t r o n frequency. Thirdly.;,^ i t w i l l be seen that no other c h a r a c t e r i s t i c frequency i n a plasma of a d e n s i t y of 2 x 1 0 1 8 c m - 3 i s i n the v i c i n i t y of the observed modulation frequency. In order to support these statements, we w i l l f i r s t use (111-19) to c a l c u l a t e the magnitude of the magnetic f i e l d at the i n t e r a c t i o n volume. Then we w i l l estimate t h i s magnitude from plasma dynamic c o n s i d e r a t i o n s . The observed spectrum ( F i g . (2-13)) suggests a modulation frequency of co o , -,nlQ rads * O L, = (3.2 ± .6) x 10 M sec S e t t i n g t h i s equal to the gyro frequency to and using (111-19), t h i s suggests a magnetic f i e l d at the i n t e r a c t i o n volume of B 2000 Gauss The magnitude of t h i s magnetic f i e l d w i l l now a l s o be estimated from the current flow and the s i z e of the i n t e r a c t i o n volume i n the plasma, The lower l i m i t of t o ^ i s hard to estimate as the observed frequency modulation i s at the l i m i t of what can be s p e c t r a l l y r e s o l v e d . 72. applying the theory of the s k i n e f f e c t i n uniform c y l i n d r i c a l conductors. A As t h i s theory i s presented i n almost any t e s t book on electrodynamics , i t w i l l not be presented here to any extent, only i t s r e s u l t s w i l l be used. A A Current t r a c e s of the Z-pinch discharge show the magnitude of the current f l o w i n g through the plasma to vary on a time s c a l e of ^ 1 usee, e.g. w i t h a frequency of 0 i n c rads coT -'- 2TT x 10° I sec Assuming that the plasma i s a uniform conductor of r a d i u s r = 2 mm o • f ++ w i t h a temperature of 180 eV , i t s r e s i s t i v i t y a w i l l be a « 4 x 1 0 - 5 0, cm,; or that of a metal. From that one can c a l c u l a t e a .skin depth 6 = i C„ (c = speed of l i g h t ) (111-20) v2irt0ja •of <5 = .33 mm. As the r a d i u s of the plasma i s r Q = 2 mm, one can see that the m a j o r i t y of the current f l o w occurs at r a d i i l a r g e r than the r a d i u s of the back-s c a t t e r i n g i n t e r a c t i o n volume ( r » .5 mm). Applying the r e s u l t s of the s k i n e f f e c t theory f o r a uniform c y l i n d r i c a l conductor w i t h outer r a d i u s = 2mm and a current s k i n depth of 6 = .33 mm, one f i n d s that the current f l o w i n g through a c e n t r a l part A see p a r t i c u l a r l y 49,50 A A f see s p e c i f i c a t i o n . see p. 66 A A A ff see p.32 see p.26 73. w i t h r = .5 mm i s 6.7 x 1 0 - 3 of the t o t a l c u r r e n t . With the t o t a l current measured to be 110 k amps at the time the back-s c a t t e r i n g takes p l a c e , t h i s means that the current f l o w i n g through a c i r c u l a r c r o s s - s e c t i o n w i t h a r a d i u s equal to that of the f o c a l spot i s 740 amps. To c a l c u l a t e the magnetic f i e l d surrounding t h i s c u r r e n t , one uses Stokes theorem to evaluate Maxwell's equation V x H = j f o r the given geometry to o b t a i n B(r) = ~ - I (111-21) zirr For r = .5 mm and I = 740 amps t h i s r e s u l t s i n a magnetic f i e l d B of B = 3000 Gauss In view of the approximations made, t h i s compares w e l l w i t h the measured value (p. 71). The approximations made were the f o l l o w i n g : (1) The plasma was considered a uniform c y l i n d r i c a l conductor of r Q = 2 mm; i n r e a l i t y however i t has a r a d i a l temperature p r o f i l e , hence, a r a d i a l c o n d u c t i v i t y p r o f i l e . . . Fbr:.the assumption to h o l d , the temperature p r o f i l e must be reasonably f l a t w i t h i n the c e n t r a l r e g i o n of the plasma. For the convenience of measuring I i n amps, t h i s formula i s given i n MKSA u n i t s , y _o = 1 Q x 1 0-7 Y_sec 2ir Am 74. (2) The s c a t t e r i n g volume was considered a t h i n annular r i n g w i t h r = .5 mm. Considering that the aperture through which the CC^ l a s e r l i g h t enters the plasma i s an annulus already and that ray bending i n the r a d i a l d e n s i t y p r o f i l e of the plasma i s l i k e l y to change the aspect r a t i o of t h i s annulus to values c l o s e r to 1, the assumption made seems reasonable and i s as w e l l supported by evidence from shadow photographs ( F i g . (3-2 ,)). F i n a l l y , we s h a l l show that at the d e n s i t y of n g = 2 x 1 0 1 8 cm - 3, which i s the d e n s i t y at the pinch time of i n t e r e s t , no other c h a r a c t e r i s t i c frequency of the plasma i s near the observed modulation frequency of 10 rads co. .= 3 x 10 m sec (1) e l e c t r o n plasma frequency: co = 8 x 10 13 rads pe sec (2) i o n plasma frequency (He ions) co . = 1.8 x 1 0 1 2 r a c* s px sec 10 rads (3) e l e c t r o n gyro frequency co f o r 3000 Gauss ge co = 3.x 10. ge sec (4) .ion gyro frequency co . f o r 3000 Gauss g l i / in7 rads co . ' 1.4 x 10' g i sec (5) resonance frequency of e l e c t r o n waves i n a magnetic f i e l d , the upper h y b r i d frequency co,, = /co 2 + C0 2 > ; CO uh pe ge pe (6) resonance frequency of i o n waves i n a magnetic f i e l d co., = /k. 2v. "z + co . z ; co . 2 << k. 2v. 2 i h i l a g i g i l l a A These frequencies are explained i n any plasma physics book (16-19). 75. (6) continued Here, ^k^ 2v^ 2 i - s t n e i ° n a c o u s t i c frequency discussed i n S e c t i o n 3.22. Hence, U K ^ i s very c l o s e to the i o n a c o u s t i c frequency ( w i t h i n 10~2'''%) . (7) I f an e l e c t r o s t a t i c i o n wave t r a v e l s e x a c t l y perpendicular to a magnetic f i e l d , i t has a resonance at the lower h y b r i d frequency. I c , n 8 rads co... = vco to . = 5 x 10° l h ge g i sec I t needs to be emphasized that the arguments presented are not meant to be unique statements about the p h y s i c a l process g i v i n g r i s e to the observed s p e c t r a l i n t e n s i t y modulations. They ra t h e r are i n i t i a l suggestions based on the a v a i l a b l e experimental data. More experiments, e.g. b a c k s c a t t e r i n g of CC^ l a s e r l i g h t at d i f f e r e n t currents i n the pinch and a genuine t h e o r e t i c a l treatment would be needed to i n t e r p r e t the described data c o n c l u s i v e l y . W i t h i n the scope of the data and the reasoning presented however, i t can be suggested that the observed modulation i n the backscattered spectrum i s most l i k e l y due to the g y r a t i o n of e l e c t r o n s i n the magnetic f i e l d of the pinch. 76. CONCLUSIONS The experiments described i n Chapter I I and evaluated i n Chapter I I I showed which processes can be observed when a 250 MW C0 2 l a s e r i n t e r a c t s w i t h a plasma at d e n s i t i e s from 1 0 1 7 cm - 3 t o s o m e 1 0 1 8 cm - 3. At low d e n s i t i e s , the good agreement between theory and experiment shows that s i g n i f i c a n t absorption of CO^ l a s e r l i g h t by i n v e r s e bremmstrahlung takes place. At higher d e n s i t i e s , s t i m u l a t e d B r i l l o u i n s c a t t e r i n g i s 46 seen to occur, however, at l e v e l s w e l l below s a t u r a t i o n . The observed angular divergence of l i g h t backscattered/by 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 was seen to agree w e l l w i t h t h e o r e t i c a l p r e d i c t i o n s . ' F i n a l l y , i f has been observed that some of the B r i l l o u i n backscattered l i g h t i s modulated w i t h the gyro frequency of e l e c t r o n s i n the magnetic f i e l d i n the plasma, an e f f e c t that could be of great i n t e r e s t f o r the 35-37 49 i n v e s t i g a t i o n of magnetic f i e l d s i n l a s e r f u s i o n plasmas. ' As f a r as other i n s t a b i l i t i e s ' are concerned, we f e e l that the power a v a i l a b l e w i t h the C0 2 l a s e r was too low to exceed t h e i r t h r e s h o l d s . Higher C0 2 l a s e r powers would make i t consi d e r a b l y e a s i e r to measure how the observed i n t e n s i t y enhancement of the backscattered l i g h t v a r i e s as a f u n c t i o n of the i n c i d e n t C0 2 l a s e r power. I t would a l s o a l l o w the imaging of the b a c k s c a t t e r i n g r e g i o n on " f o o t p r i n t " paper and thus ob t a i n d i r e c t i n f o r m a t i o n about the geometry of the i n t e r a c t i o n volume. 24 25 From simple, experimentally v e r i f i e d t h e o r i e s about f i l a m e n t a t i o n ' i t can be p r e d i c t e d t h a t , w i t h the parameters described i n Chapter I I I , a d e n s i t y depression of ^ 3% should r e s u l t . With ten times higher C0 2 l a s e r power, t h i s d e n s i t y depression w i l l become s e v e r a l tens of percent, hence, should e a s i l y be observable. Higher CC^ l a s e r powers would a l s o provide a p o s s i b i l i t y of making a c o n t r i b u t i o n to the question of stim u l a t e d Raman s c a t t e r i n g . Much ,. _ ,10,11,12,14,15 , t . , 22 pre d i c t e d but perhaps only once observed, our experiment would provide i n t e r e s t i n g d e n s i t y s c a l e lengths f o r t e s t i n g the p r e d i c t i o n s . Appropriate changes i n the discharge bank of the Z-pinch would a l l o w the measuring of the spectrum.of the backscattered l i g h t at d i f f e r e n t plasma c u r r e n t s , e.g. d i f f e r e n t magnetic f i e l d • s t r e n g t h s w i t h i n the laser-plasma i n t e r a c t i o n r e g i o n . This would make i t p o s s i b l e to f u r t h e r t e s t i f the model assumed i n 3.24 i s c o r r e c t . F i n a l l y , a change i n geometry, e.g. fo c u s s i n g the l a s e r r a d i a l l y i n t o the plasma, w i l l have the advantage of a b e t t e r defined i n t e r a c t i o n region and w i l l a l l o w d i a g n o s t i c access to i n v e s t i g a t e two plasmon A 21>23 decay. 78. CHAPTER 4 The f a s t gating of an O p t i c a l M u l t i c h a n n e l Analyser (OMA)^  and a c o n t r i b u t i o n to the d i a g n o s t i c s of the Z-pinch plasma_ 4.1 I n t r o d u c t i o n The f i r s t measurements of e l e c t r o n d e n s i t y and temperature of the Z-pinch plasma were s p e c t r o s c o p i c a l measurements using the Stark broadening of the 4 6 8 6 $ l i n e of He I I . 2 6 In order to have these measurements supplemented by a second method, a Thomson s c a t t e r i n g system (see F i g . ( 4-1)) was set up. A 3 Joule Ruby l a s e r was focussed i n t o the plasma and the backscattered l i g h t observed under 173°. At the d e n s i t y and temperature i n d i c a t e d by the i n i t i a l s p e c t r o s c o p i c a l measurements, t h i s should have r e s u l t e d i n a Thomson-scattered spectrum c h a r a c t e r i z e d by • a. 1.2. 79. Ruby laser ! 1 Figure (4-1): The Ruby l a s e r Thomson s c a t t e r i n g system. 80. The spectrum, obtained on a shot-to-shot b a s i s and recorded w i t h a p h o t o m u l t i p l i e r , i s shown below and y i e l d s n « 1 0 1 8 cm - 3 , T = 40 eV e e I K a r b u n i t s ) ; o • Figure (4-2): Thomson s c a t t e r e d spectrum obtained w i t h setup i n F i g . (4-1), recorded w i t h a p h o t o m u l t i p l i e r . The next step was to use a 500 channel OMA i n order to detect t h i s e l e c t r o n f e a t u r e i n one s i n g l e shot. Despite much e f f o r t i nvested to a r r i v e at a t r u l y s a t i s f a c t o r y r e s u l t , t h i s was not achieved w i t h i n the work described i n t h i s r e p o r t . Part of the reason can be understood by examining Figure (4-3) which shows a Thomson s c a t t e r e d s i g n a l detected w i t h a p h o t o m u l t i p l i e r . 81. 100 ns /d i v Figure (4-3): A p h o t o m u l t i p l i e r t r a c e of the Thomson s c a t t e r e d s i g n a l . The f i r s t pulse i s the Ruby monitor. The missing piece i n d i c a t e s the place of a foreward s c a t t e r e d s i g n a l which.is w e l l o f f the screen and of no relevance here. The subsequent r i s e i s the delayed s i g n a l showing the plasma l i g h t w i t h the backscattered s i g n a l i n d i c a t e d . AX was lOoX; pinchtime was - 50 < t < 0 ns. The Figure shows c l e a r l y that the s i g n a l - t o - p i n c h l i g h t r a t i o i s about .2 or s m a l l e r . This a l s o gives r i s e to the l a r g e s t r a y of data p o i n t s i n F i g . (4-2). With the means a v a i l a b l e at the time, i t was found that the e v a l u a t i o n of the i o n feature of the Thomson s c a t t e r e d spectrum provided a very good measurement of plasma d e n s i t y and temperature, hence, the e l e c t r o n feature approach was not pursued any f u r t h e r . The work however d i d r e s u l t i n developing a technique f o r s u c c e s s f u l 40 nanosecond gati n g of the OMA. I n the f o l l o w i n g s e c t i o n t h i s technique s h a l l be described and a p p l i e d to improved s p e c t r o s c o p i c a l measurement of d e n s i t y and temperature of the plasma. see footnote p. 33 82. 4.2 Nanosecond g a t i n g of an o p t i c a l m ultichannel analyser (OMA) Many types of time-resolved measurements i n plasma physics r e q u i r e short time gat i n g of a d e t e c t i o n system during comparatively long d u r a t i o n high l i g h t l e v e l s . One such case i s the d e t e c t i o n of a Thomson s c a t t e r e d s i g n a l of nsec d u r a t i o n during the high bremsstrahlung emission of a dense plasma l a s t i n g many ysec. I f the detected s i g n a l s are weak, two requirements must be f u l f i l l e d : The c o n t r a s t r a t i o of the g a t i n g system must be high (e.g. > 10,000) and the g a t i n g process must not r e s u l t i n d i s t o r t i o n s of the recorded s i g n a l . How the OMA can be made to f u l f i l l both c o n d i t i o n s w i l l now be described. An OMA can be used i n two modes of operation. I n the continuous mode (also c a l l e d " r e a l time" mode), the 500 channels of the OMA are scanned every 32 msec w i t h an open time of 768 usee between scans. In t h i s mode, the OMA can s t o r e about 5,000 counts per channel, each count corresponding to ^ 20 v i s i b l e photons. In the gated mode, the 500 channels are s e n s i t i s e d during a d e s i r e d time i n t e r v a l , which must be s h o r t e r than and w i t h i n the 768 ysec open time. Gating i s achieved by h o l d i n g the photocathode of the image i n t e n s i f i e r at ^ 7 kV. A g a t i n g pulse of 1.1 kV to 1.4 kV b r i n g s the v o l t a g e of the photocathode up to the ^ 8.2 kV r e q u i r e d f o r f u l l s e n s i t i v i t y . The manufacturer claims that g a t i n g times as short as 10 nsec are p o s s i b l e , but i t w i l l be shown that t h i s i s only the case i f the s i g n a l r e c e i v e d w i t h i n the g a t i n g time i s s h o r t e r than 10 nsec. Otherwise, very severe see s p e c i f i c a t i o n s d i s t o r t i o n s w i l l r e s u l t . In order to t e s t the response of the OMA detector under various c o n d i t i o n s , the f o l l o w i n g experiment was set up "_~_~j!WMiiMiiiiiiiiitiimmuntHM Z P i n c h 7i—• Ground Glass S c r e e n j i Po la r i ze r s a n d PockelsCel l [ ( inrad 261-150) , , , , , , , Screened Room Extens ion Ga te Pulse 4 in IT • • O M A Spect rograph ( P A R 12501 ISIT ) Figure (4-4): Experimental setup to t e s t the response of the OMA i n gated mode. The spectrograph i s set to zero order to e l i m i n a t e s p e c t r a l dependences. The Z-pinch serves as a white l i g h t source of ysec d u r a t i o n . The detector i s mounted i n the f o c a l plane of a monochromator set to zero order to transmit white l i g h t . On the entrance s l i t plane a package c o n s i s t i n g of 10 s l i t s (25 thou wide, 25 thou apart) i s mounted. The white l i g h t image of t h i s s l i t package i s r e g i s t e r e d by the de t e c t o r . such a s l i t package was already used by (26) to optimize the gating voltage. 84. F i g . (4-5) shows the response when the detector operates i n the r e a l time mode. The r e s o l u t i o n i s about 7 channels; the i n t e n s i t y response across the 500 channels i s reasonably f l a t . D e f i c i e n c i e s are due to pincushion d i s t o r t i o n . r e a l t i m e A*\ r \ n u u U u u u 1 0 0 2 0 0 3 0 0 4 0 0 c h a n n e l N o -5 0 0 Figure (4-5): Response of the OMA i n r e a l time. F i g . (4-6) shows the response i n gated mode at long gat i n g times. The gati n g was done e l e c t r o n i c a l l y only; the g a t i n g time was 1.2 ysec. I t i s seen that some r e s o l u t i o n i s l o s t and a s t r e t c h i n g of the spectrum across the channels occurs. These d i s t o r t i o n s , however, are minor and, as they are l a r g e l y i n t e n s i t y independent, can be accounted f o r . I n t e n s i t y 5 0 0 W U U c h a n n e l N o . 4 0 0 5 0 0 Figure (4-6): Response of the OMA when e l e c t r o n i c a l l y gated w i t h 1.2 ysec. I am indebted to W. Seka f o r i n f o r m a t i o n about work that has been done w i t h OMAs at the U. of Rochester Laboratory f o r Laser E n e r g e t i c s . 85. F i g . (4-7) and F i g . (4-8) show the response when the gatin g time i s 50 nsec and 6 nsec r e s p e c t i v e l y . , I n t e n s i t y 2 0 0 0 + c h a n n e l N o . 4 0 0 5 0 0 Figure (4-7): Response of the OMA. when e l e c t r o n i c a l l y gated with"50 nsecT . I n t e n s i t y 2 0 0 0 H 1 0 0 5 0 0 c h a n n e l N o . Figure (4-8): Response of the OMA when e l e c t r o n i c a l l y gated with 6 nsec. The d i s t o r t i o n s f o r 50 nsec gat i n g time are obviously severe and f o r 6 nsec g a t i n g time the detector becomes u s e l e s s . These d i s t o r t i o n s are presumably caused by an i n c o r r e c t and time-42-v a r y i n g voltage a p p l i e d to the photocathode during the switch<on and switch of f of the gatin g pulse. F i r s t i t was t r i e d to improve the operation of the OMA i n gated mode by improving the e l e c t r o n i c s of the gatin g c i r c u i t r y as suggested by i n (45) 86. A d d i t i o n a l l y , the cable c a r r y i n g the HV gati n g pulse to the photocathode of the f i r s t i n t e n s i f i e r was removed from i n s i d e the OMA and fed d i r e c t l y i n t o the OMA head te r m i n a t i o n (see 4-4). I t , however, soon was evident that the remedy could not r e a d i l y be achieved by e l e c t r o n i c a l means only. To gate the incoming l i g h t w i t h an e l e c t r o - o p t i c a l switch r a t h e r than ga t i n g the OMA e l e c t r o n i c a l l y e l i m i n a t e d the d i s t o r t i o n s i n the recorded spectrum, but c a r e f u l measurements showed that a con t r a s t r a t i o of about 400:1 was the best that could be achieved w i t h t h i s method. To combine the advantages of both methods, i . e . the d i s t o r t i o n f r e e ga t i n g by swi t c h i n g the incoming l i g h t and the good c o n t r a s t obtained by gatin g the OMA e l e c t r o n i c a l l y , the f o l l o w i n g experimental setup was used. In order to l e t the l i g h t s i g n a l reach the de t e c t o r only during the time when the e l e c t r o n i c a l g a t i n g pulse voltage i s constant, an o p t i c a l g a t i n g pulse (30 nsec) was f i t t e d temporally i n t o a 50 nsec e l e c t r o n i c gating pulse. The voltage of the e l e c t r o n i c . g a t i n g pulse was adjusted to give the l e a s t d i s t o r t i o n s across the whole spectrum at long gating times. The timing of the two pulses can be done with synchronized cable discharges. Figure (4-9): E l e c t r o n i c a l and o p t i c a l g a t i n g pulse temporally f i t t e d i n t o each other. Pulse height i s 1.2 kV. 87. F i g . (4—9) shows that during the time the o p t i c a l g a t i n g pulse i s switched, the voltage of the e l e c t r o n i c g a t i n g pulse i s q u i t e s t a b l e . F i g . (4-10) shows again the r e a l time response of the OMA, but the setup ( F i g . (4-4)) now in c l u d e s the Pockels c e l l . The i n t e n s i t y d i s t r i b u t i o n across the spectrum now a r i s e s mainly from the f a c t that the open aperture of the Pockels, c e l l (= 15 mm) was only s l i g h t l y l a r g e r than the s e n s i t i v e area of the OMA. F i g . (4-11) shows the performance of the OMA i n gated mode f o r both gat i n g pulses combined. Figures (4-10) and (4-11): Response of the OMA (a) i n r e a l time (b) f o r the i d e n t i c a l setup when e l e c t r o n i c a l l y gated w i t h 50 nsec and o p t i c a l l y gated w i t h 30 nsec. 88. The absence of any background such as i n F i g . (4-7) i s immediately obvious. The i n t e n s i t y d i s t r i b u t i o n of the r e a l time spectrum i s c l o s e l y r e p r o -duced. F i g . (4-12) shows that the r e s o l u t i o n i n t h i s double gat i n g mode i s improved as compared to the e l e c t r o n i c a l g a t i n g only. number of channels 15-10 5 100 200 300 400 500 channel number Figure (4-12): A comparison of the focusing p r o p e r t i e s f o r the i n v e s t i g a t e d cases. The number of channels on the r i s e and f a l l of the s l i t image i s shown as a f u n c t i o n of channel number f o r the three cases. The upper s o l i d Surve shows the e l e c t r o n i c a l l y gated mode, the lower s o l i d curves shows the r e a l time behaviour and the dashed one shows the r e s u l t s from the combined g a t i n g pulses. I t f i n a l l y was observed that i n the gated mode, the i n t e n s i t y s e n s i t i v i t y of the OMA i s only l i n e a r up to at most ^ 2000 counts/channel. This i s i l l u s t r a t e d i n F i g . (4-13). 89. c o u n t s a t t e n u a t i o n 3 1 6 1 0 0 31.6 1 0 3.16 f a c t o r Figure (4-13): Shown are the number of counts i n a given channel as a f u n c t i o n of i n t e n s i t y . The i n t e n s i t y was v a r i e d by i n s e r t i n g f i l t e r s of i n c r e a s i n g n e u t r a l d e n s i t y i n the l i g h t path between the source and the OMA (see F i g . (4-4)). From the measurements presented, the f o l l o w i n g conclusions can be drawn. The background observed i n the spectrum when the OMA i s gated e l e c t r o n -i c a l l y only ( F i g . (4-7)) a r i s e s from leakage of photocurrent due to l i g h t reaching the de t e c t o r when only 7 kV are a p p l i e d . The d i s t o r t i o n i n the i n t e n s i t y d i s t r i b u t i o n i s indeed due to the switchon and s w i t c h o f f process with l i g h t f a l l i n g on the OMA s e n s i t i v e elements as suggested by (42) The advantages of the double g a t i n g technique are, apart from removing the d i f f i c u l t i e s mentioned above, t h a t : The c o n t r a s t requirements f o r the o p t i c a l gate are not any more determined by the time the l i g h t source i s a c t i v e , but by the e l e c t r o n i c g a t i n g time. Hence, to achieve f a s t e r g a t i n g times, only the o p t i c a l g a t i n g pulse needs to be shortened. 90. The q u a l i t y requirements (e.g., squareness) f o r the e l e c t r o n i c . (as w e l l as o p t i c a l ) gating pulse are much r e l a x e d , as long as the voltage i s steady during the s w i t c h i n g time of the o p t i c a l gate. This completely e l i m i n a t e s the need f o r elaborate g a t i n g c i r c u i t s . 91. 4.3 Spectroscopic measurements of plasma d e n s i t y and temperature using  the 4686$ l i n e of He I I . The double gating technique described i n 4.2 w i l l now be a p p l i e d to spectroscopic temperature and d e n s i t y measurements using the 4686$ l i n e of He I I . Even though t h i s type of d i a g n o s t i c s had already been used, improved measurement appeared d e s i r a b l e . F i g . (4-14) shows, as an example, the 4686$ p r o f i l e at pinch time t = + 300 nsec, composed of f i v e separate measurements. -100A 0 • i p o A Figure (4-14): Data of e a r l i e r s p e c t r o s c o p i c a l measurements Composed l i n e p r o f i l e at 4686$ at t = + 300 nsec. Despite e x c e l l e n t experimental work, the means a v a i l a b l e at that time simply d i d not a l l o w b e t t e r data to be obtained. 26 92. 93. F i g . (4-15) shows a s e l e c t i o n of 4686$ p r o f i l e s obtained w i t h the double gating .technique (30 nsec o p t i c a l , 50 nsec e l e c t r i c a l ) and a monochromator 26 of l e s s d i s p e r s i o n than used p r e v i o u s l y . The experimental setup was e s s e n t i a l l y the same as shown i n F i g . (4-4), of course without ground glass screen, and the plasma was v i e w e d i s i d e on.— As a comparison, the r i g h t p i c t u r e i n F i g . (4-15) shows the r e s u l t obtained with 50 nsec e l e c t r o n i c a l gating only. I t must be emphasized 26 that the OMA used by Houtman had only one image i n t e n s i f i e r . The OMA employed i n the measurements described here had two image i n t e n s i f i e r s which r e s u l t e d i n more severe d i s t o r t i o n s when the OMA was gated e l e c t r o n i c a l l y only. E v a l u a t i n g the width of these l i n e s to o b t a i n the e l e c t r o n d e n s i t y and the t o t a l l i n e to continuum i n t e n s i t y r a t i o to o b t a i n the e l e c t r o n temperature J » ^ 4 of the plasma at a given time, one obtains the data shown i n F i g . (2-1) which agree well, w i t h the i o n fe a t u r e Thomson s c a t t e r i n g measurements, as f a r as the d e n s i t y i s concerned. As f o r the temperatures, the t h e o r i e s a l l o w i n g the determination of temperature from the l i n e to continuum r a t i o of i n t e n s i t i e s Opf) are too u n c e r t a i n f o r r a t i o s < 1. a. T h i s , and the disappearance of the 4686A5 l i n e at higher temperature due to a decrease i n the amount of H e l l i o n present, does not a l l o w f o r a good temperature measurement at pinch times of i n t e r e s t . f o r — «* 1, at •• d e n s i t i e s of ^ 5 x 1 0 1 8 , (46) p r e d i c t s about 40 eV, (47) p r e d i c t s about 5 eV. S p e c i f i c a t i o n s Z-pinch D e t a i l s of the Z-pinch are described i n Ref. 26 . Here, only the most important c h a r a c t e r i s t i c s s h a l l be described. He - f i l l i n g pressure inner r a d i u s v e s s e l outer r a d i u s v e s s e l m a t e r i a l of v e s s e l E l e c t r o d e s e p a r a t i o n (copper) Bank capacitance Bank inductance Bank energy at 11.5 kV charging v o l t a g e T y p i c a l maximum current of discharge Time of maximum d e n s i t y and temperature 1.2 Torr 5.08 cm 5.72 cm pyrex 35.6 cm 84 yF 33 n H 5.6 k J 150 k Amp 2.0 ys a f t e r i n i t i a l breakdown 95. Figure (5-1): Shows the c i r c u i t r y of the Z-pinch discharge bank. Note the added p r e i o n i z a t i o n discharge at the anode, mentioned i n 2.2. Crytron Trigger Unit 100 M Z - P i n c h 100 M - 300 k • W l ! Charge HV Supply 11-5 kV 5 k Dump 96. A t y p i c a l t r a c e as picked up by the Rogowsky c o i l i s shown i n the f i g u r e below. Maximum current I = 180 k amp. 5 V 5 0 0 n s , d I 1 ' d t Figure (5-2) : - j ^ - as a f u n c t i o n of time; Rogowsky c o i l s i g n a l . (2) C0 2 l a s e r Type: Lumonix T600 Ca v i t y : unstable resonator c o n f i g u r a t i o n Output geometry: see F i g s . (2-1) and (2-2). T y p i c a l output pulses at an energy of 27 Joules i s shown i n the s e c t i o n (3) Detectors. (3) Detectors Au Ge: Gold doped germanium semiconductor d e t e c t o r , l i q u i d n i t r o g e n cooled, s e n s i t i v e f o r X < 11 u. C a l i b r a t e d s e n s i t i v i t y at 10.6 u i s 6.7 V/mJ. S e n s i t i v e area 16 mm2. F i g . (5-3) shows the C0 2 l a s e r pulse as r e g i s t e r e d by the Au Ge d e t e c t o r . 1 0 0 m V 1 0 0 n s Figure (5-3): l a s e r p u l s e , Au Ge d e t e c t o r . Pyrd e l e c t r i c d e t e c t o r : Molectron Corp Model P3-00 50% of maximum s e n s i t i v i t y f o r X > 8 u. C a l i b r a t e d s e n s i t i v i t y at 10.6 u i s b e t t e r than 1.5 V/mJ. S e n s i t i v e area 1 mm2. F i g . (5-4) shows the CO2 l a s e r pulse as r e g i s t e r e d by the P y r o . e l e c t r i c d e t e c t o r . Figure (5-4): C0 0 l a s e r p u l s e , Pyro e l e c t r i c d e t e c t o r . 5 0 m V 1 0 0 n s Photon Drag Detector: Opticon Corp L t d . , Model 7425 This detector was used e x c l u s i v e l y as a timing monitor f o r the CO^ l a s e r , hence, a s e n s i t i v i t y c a l i b r a t i o n was unnecessary. A t y p i c a l t r a c e i s shown below. Figure (5-5): COv, l a s e r pulse, Photon Drag Detector. Note the beating of l o n g i t u d i n a l l a s e r modes. Gen Tech Energy Meter: Gen Tech Inc. Model LED-200-C Fast b a l l i s t i c energy meter, 5 msec response time. mV C a l i b r a t e d s e n s i t i v i t y at 10.6 ym i s 7.8 A t y p i c a l t r a c e i s shown below. mJ Figure (5-6): Response of Gen Tech energy meter to CO2 l a s e r pulse. A p o l l o Energy Meter: A p o l l o Lasers I nc., Model ACM-100 Range: 5 mJ to 2 k J , d i g i t a l readout. 99. (4) S a l t Optics A l l i n f r a r e d t r a n s m i t t i n g o p t i c s c o n s i s t e d of :,KCL., a l l m i r r o r s were aluminum-coated f r o n t surface m i r r o r s . The f i g u r e below shows the measured tran s m i s s i o n of a 6 mm KCL s a l t window (as used w i t h the Au Ge detector) and of a 12 mm KCL f l a t ( t h ickness of the s a l t lense) as a f u n c t i o n of wavelength. °/o T r a n s m i s s i o n 10oi 5 0 10 15 2 0 2 5 (5) Monochromator and Gratings" Monochromator: y m . J e r r a l Ash, 100 urn or 130 ym s l i t s G r a t i n g s : a) Yobin Ivon, 50 x 50 mm, 153 lines/mm Blazed at 10.6 ym D i s p e r s i o n 110 8/: mm. b) Bausch and Lomb, 2.7" x 2.7", 60 lines/mm Blazed at 16 ym D i s p e r s i o n 300 A*/ mm 100. (6) O p t i c a l M u l t i c h a n n e l Analyser Supplied by P r i n c e t o n A p p l i e d Research Corp, P r i n c e t o n N.J. The OMA used f o r the experiments described i n Chapter IV was type 12051. As t h i s device i s of considerable complexity, i t w i l l , - not be tr e a t e d f u r t h e r i n t h i s appendix. For f u r t h e r i n f o r m a t i o n r e f e r to the i n s t r u c t i o n manual, a v a i l a b l e from P.A.R. corp. (7) O s c i l l o s c o p e s T e k t r o n i x 7704 w i t h v e r t i c a l amps 7A16 and 7A12 Tektro n i x 466 f a s t storage o s c i l l o s c o p e . 101. References 1. Lubin, M., Goldman,. E., Soures., J . , Goldman,. L., Friedman, W., L e t z r i n g , S., A l b r i t t o n , J . , Koch, P., Jaakobi, B., Plasma Physics and C o n t r o l l e d Thermonuclear Fusion Research, V o l . 2, IAEA, Vienna, 1975. 2. N u c k o l l s , J.H.; Brueckner, K.A.; Kidder, R.E.; K a l i s k i , S.; Soures, J . , Goldman, L.M., Lubin, M., Laser I n t e r a c t i o n and Related Plasma Phenomena, V o l . 3B, ed. H.J. Schwarz and H. Hora, Plenum, N.Y., 1974. 3. Mead, W.C., Haas, R.A., Kruer,,W.L., P h i l l i o n , D.W., Kornblum, H.N., L i n d l , J.O., MacQuigg, D.R., Rupert, V.C., Phys. Rev. L e t t . 37, 489 (1976). 4. L i u , C.S., Rosenbluth, M.N., White, R.B.; Biskamp, D., Weber, H.; Kruer, W., Valeo, E., Estabrook, K., Thomson, J . J . , Langdon, B., L a s i n s k i , B., Plasma Physics and C o n t r o l l e d Thermonuclear Fusion Research, V o l . 2, IAEA, Vienna, 1975. 5. Workshop on l i n e a r magnetic confinement, S e a t t l e , Washington, 1977. 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