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Experimental investigation of the mixing of two optical frequency EM waves in a plasma Godfrey, Lawrence Allan 1977

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AN EXPERIMENTAL INVESTIGATION OF THE MIXING OF TWO OPTICAL FREQUENCY EM WAVES IN A PLASMA by LAWRENCE ALLAN GODFREY B.Sc., University of B r i t i s h Columbia, 1971 M.Sc., University of B r i t i s h Columbia, 1973 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Physics) We accept this thesis as conforming to the required standards THE UNIVERSITY OF BRITISH COLUMBIA September, 1977 ©Lawrence Allan Godfrey, 1977 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb ia , I a g ree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e 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 purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that 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 not be a l l o w e d w i thout my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Co lumbia 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 Date <>£PT. /9<7 7 i i ABSTRACT Q The effect of o p t i c a l mixing of two tunable dye lasers at frequencies near the plasma frequency has been experimentally investigated i n a helium plasma 1et„ It has been shown that the wave mixing produces longitudinal plasma o s c i l l a t i o n s at the frequency and wave vector of the mixing force. The driven waves were detected by scattering a t h i r d diagnostic l i g h t wave from th e i r density f l u c t u a t i o n . The scattering signals increased to as much as seven times the sig n a l observed when scattering from the thermal fluctuations alone. The spectrum of the spectral density function of the induced fluctuations has been measured, as well as the dependence of i t s amplitude on the power of the mixing l i g h t beams. These results agree well with t h e o r e t i c a l c a l c u l a t i o n s based on a simple model of the mixing effect of a single electron in the f i e l d of two electromagnetic waves. The response of the plasma to o p t i c a l mixing a t ! d i f f e r e n t frequencies has also been measured. This spectrum agrees i n part with theoreti c a l predictions, but has features not explained by the simple model. i i i TABLE OF COHTEHTS ABSTRACT. ,, i i TABLE OF CONTENTS i i i LIST OF FIGURES V LIST OF TABLES v i i ACKNOWLEDGEMENTS. , v i i i CHAPTER I INTRODUCTION ,. 1 CHAPTER II THEORY AND EXPERIMENTAL CONSIDERATIONS. . 6 A. Physical Model 6 B. Theory. 1 2 C. Methods Of Detection 23 D. Signal Calculations 25 E. Experimental Design. ................... 32 F. V a l i d i t y Considerations. ................ 36 CHAPTER III EXPERIMENTAL APPARATUS 15 A. The Plasma Jet 47 B. The Ruby Laser 51 C. The Tunable Dye Lasers. 52 D. The Complete Experimental System And Diagnostics. 57 E. Alignment Procedure. .................... 63 F. E l e c t r i c a l System And Timing. 67 i v CHAPTER IV DESCRIPTIONS AND DISCUSSION OF EXPERIMENTS 73 A. Thomson Scattering Measurements of Electron Temperature and Density. ....... 73 B. General Experimental Condtions And Procedure. 75 C. Data Reduction. •• 78 D. Dependence Of Mixing Signal On Mixing Pover. 81 E. Spectrum Of The Induced Fluctuations. ... 88 F. The Response Function Of The Plasma. .... 93 CHAPTER V CONCLUSIONS AND SUGGESTIONS 107 BIBLIOGRAPHY 112 APPENDIX A DOUBLE CONVOLUTION OF A LORENTZIAN WITH TWO ARBITRARY LINE PROFILES. 115 APPENDIX B DATA REDUCTION .123 A. Oscillogram Analysis. .........123 B. Evaluation Of Relative Scattering Cross Sections ..134 APPENDIX C PLASMA JET COLLISION RATES , 143 APPENDIX D DISCUSSION OF THE LOCATION OF THE DIPS IN THE MEASURED RESPONSE FUNCTION 145 V LIST OF FIGORES FIGURE II-1 Electron In The F i e l d Of Two EH laves. . 8 FIGURE II-2 A Typical Induced And Thermal Fluctuation Spectrum. .................. 20 FIGURE II-3 Theoretical Response Of The Plasma To Different Driving Frequencies. ......... 21 FIGURE II-4 Have Vector Hatching 30 FIGURE III-1 The Complete Optical System. 46 FIGURE III-2 The Plasma Jet , 48 FIGURE III-3 The Plasma Jet Power Supply C i r c u i t . ... 50 FIGURE III-4 Dye Laser System. 53 FIGURE III-5 Dye Laser Beam Spectral Line P r o f i l e s . . 55 FIGURE III-6 E l e c t r i c a l System 68 FIGURE III-7 Logic Seguence And Timing Pulses. 70 FIGURE 17-1 Thomson Scattering Spectrum To Determine Electron Temperature And Density. ...... 74 FIGURE IV-2 Typical Osscillograms. 79 FIGURE IV-3 Total PH Signal Compared To The F i t t e d Values. .. ». ..84 FIGURE IV-4 Photomultiplier Signal Due To Light Mixing Alone As A Function Of The Product Of The Laser Powers 87 FIGURE IV-5 Spectrum Of The Induced Have 90 FIGURE IV-6 Spectral Line P r o f i l e s (Average Of Several Shots) 92 v i FIGURE IV-7 Thomson Scattering Spectrum For The Mixing Experiment 95 FIGURE IV - 8 Measured Response Of The Plasma To Different Driving Freguencies . 97 FIGURE IV-9 Dye Laser Power Before And After The Mixing Region: The Fixed Frequency Laser. 102 FIGURE IV-10 Dye Laser Power Before And After The Mixing Region: The Variable Frequency Laser. 103 FIGURE B-1 Thomson Scattering Photomultiplier Signal As A Function Of Incident Ruby Power. ...126 FIGURE D-1 Position Of The Nth Valley As A Function Of N 146 v i i LIST OF TABLES TABLE II-1 Possible Laser Systems 34 TABLE IV-1 Parameters Of The Experiment Held Constant 76 v i i i ACKNOWLEDGEMENTS I wish to sincerely thank Roy Nodwell for i n v i t i n g me to join the D.B.C. Plasma Physics Group under his supervision, and for suggesting the o p t i c a l wave mixing experiment for my Ph.D. project. I would also l i k e to thank Frank Curzon for his expert assistance in the preparation of t h i s thesis during Roy Nodwells leave of absence. My thanks goes to a l l at the Group for stimulating discussions, experimental assistance, and comradeship. Brian Hilko and my good f r e i n d , Charles Ashley T u l l y , I I , agreed to help me take data, even u n t i l many hours past midnight. Marj K i l l i a n , Daryl Pawluk, and Hannas Barnard have helped me prepare the computer programs reguired for the data analysis, as well as providing programs of their own. Our past and present electronic technicians, Doug Sieberg and Alan Cheuck, have always responded quickly to repair sparking power supplies and burning r e s i s t o r s . Peter Haas, Tony Knop, Beat Meyer and Ole Christiansen have made well f i n i s h e d and precise components for key parts of t h i s experiment. I would especially l i k e to thank my wife, Laurie. During the years involved with t h i s experiment, Laurie has assisted i n taking data, typed, proof read, labeled drawings, and given me moral support. Wives of graduate students receive far too l i t t l e recognition of ix their contributions. I g r a t e f u l l y acknowledge f i n a n c i a l support from the H. R. HacMillan and the University of B r i t i s h Columbia scholarship funds, as well as the Plasma Physics Group. This work has been supported by grants from the Atomic Energy Control Board of Canada and the National Research Council. 1 Chapter I INTRODUCTION Wave mixing was f i r s t proposed by K r o l l , Ron and Rostocker 1 i n 1964 as a means of increasing the usually very small cross-section for scattering of laser l i g h t for plasma diagnostics. When two intense l i g h t sources are incident on a plasma, the electrons motion has a mixing effect which r e s u l t s a longitudinal driving force. The frequency of t h i s driving force i s equal to the freguency difference of the two incident EM waves. Matching t h i s freguency difference to a plasma resonance allows one to drive well-defined plasma waves. Besides being used as a diagnostic a i d , o p t i c a l wave mixing can be a possible t o o l to help f i l l the gap between th e o r e t i c a l and experimental studies of nonlinear wave-plasma interactions. Wave mixing can be classed as a nonlinear wave-plasma interaction in which d i f f e r e n t electromagnetic, electron, or ion wave motions cannot be considered as independent modes of the plasma. There has been a large increase in interest i n nonlinear wave-plasma interactions during recent years, qenerated mainly by the quest for controlled nuclear fusion, but also by experiments to study and control the ionosphere with high power microwaves. Both these areas of research reguire detailed knowledge of plasma wave phenomena. 2 To date the experimental studies 2 have been limited mostly to plasmas of density less than 10»* cm-3 because of the necessity of using microwave power sources. Driving waves i n a plasma generally requires that the frequencies of the driver be close to plasma normal mode resonance frequencies. Presently, the sources of hiqh power EH radiation are unavailable between the ranqe of a few GHz from microwaves and a few thousands of GHz from C0 2 lasers. It has been impossible to do controlled studies of plasmas with densities of 10*• to 10 1 9 cm - 3 u n t i l S tansfield, Nodwell and Meyer 3 shoved that plasma waves can be driven by mixing two intense, high frequency l i g h t sources. Besides t h e i r l i m i t a t i o n in densities, microwave experiments have the drawback that the smallest wavelength of the radiation i s of the order of a millimeter. This i s comparable to the sheath thickness of the plasma boundaries, i n which the electron density goes to zero. The plasma boundary can not be considered sharp, a circumstance which complicates the interpretation of experimental r e s u l t s . It i s also d i f f i c u l t to create plasmas which are uniform over the area of a microwave beam, a further complication to data interpretation. The perturbation caused by mechanical probes placed i n the plasma vessels must also be considered. Stable uniform plasmas with densities approaching of 10 l» cm - 3 have only recently become available*)s for plasma wave studies using C0 9 lasers. 3 Wave studies at t h i s density are s t i l l in the i r infancy. Of course, i r r a d i a t i n g DT p e l l e t s with Terawatt laser l i g h t beans can be considered as plasma waves studies, but the processes of these experiments are almost too complex to unravel. Wave mixing then appears to be a very desirable diagnostic t o o l . With the beating of two laser beams, i t should be possible to produce plasma waves of controlled amplitude, spectrum and wave vector d i s t r i b u t i o n . The freguency of the waves can match the resonant modes of plasmas of any density, including the previously inaccessible 10 1* to 10** cm - 3 density range. The mixing experiments themselves t e s t our interpretation of the physics of the interaction of the mixing l i g h t beams with the plasma, and the generated waves, with t h e i r well defined freguency and wave vectors, can be used for further wave-plasma studies. The mixing l i g h t beams are transparent to the plasma, and there are no mechanical probes to cause perturbations. The well defined interaction region i s small enough that plasma density gradients usually need not be considered. Optical wave mixing, once i t s experimental f e a s i b i l i t y and pr i n c i p l e s have been established, w i l l allow a systematic study of wave-wave and wave-particle interactions i n a wide variety of plasmas. This thesis presents the experimental r e s u l t s of wave mixing of two tunable dye la s e r s near the electron resonance in a plasma with a density of 2x10* 6 cmr 3. I t 4 i s both a v e r i f i c a t i o n and extension of Stansfield's work* i n which i t was shown that o p t i c a l wave mixing has a measurable e f f e c t on the plasma. The experiments presented here were chosen to test the theore t i c a l results of Meyer 7 which take into account the s p a t i a l and spectral content of the mixing beams. Have mixing has been reported by other authors, but mainly i n the microwave region. Stern and Tzoar 8 mixed two high frequency (^ 30 GHz) microwave beams i n a 0 . 8 cm diameter mercury discharqe and observed an increase in the naturally occurrinq radiation from the plasma at i t s resonance frequency {^ 3 GHz). Kuhn, Leheny and Marshal 9 did microwave mixing using a neon afterglow i n a square X-band wave-guide. They obtained good quantitative agreement between their experimental re s u l t s and theory modified to account for their f i n i t e geometry. The induced density f l u c t u a t i o n s were observed by probes and scattering of microwaves. Other a u t h o r s 1 0 have reported 'wave mixing*, but their work i s fundamentally d i f f e r e n t from the wave mixing reported here. These authors excited normal modes of the plasma with microwaves. I t i s these electron waves which interact, rather then the i n i t i a l mixing beams. Since t h i s thesis i s an experimental report, emphasis has been placed on the physics of the wave mixing as well as the techniques of the actual experiments. The main part of the thesis i s therefore divided i n t o four sections. 5 Chapter I I presents a simple physical model of wave mixing i n a plasma. The theory of Meyer i s outlined. Experimental considerations are presented including checks on application of the theory to the experimental conditions. Chapter III presents the experimental system. Included i s a description of the plasma apparatus, the wave mixing l i g h t sources, and the o p t i c a l and e l e c t r i c a l arrangements. The experiments themselves are presented i n Chapter IV. The results are compared with theory. The method of data reduction has been placed in Appendix II to make t h i s chapter more readable. F i n a l l y a summary and conclusions are given i n Chapter V. This chapter d e t a i l s the o r i g i n a l contributions of the author. Suggestions for improvements of the present experimental arrangement, as well as for further possible studies are presented. 6 Chapter II THEORY AND EXPERIMENTAL.CONSIDERATIONS In t h i s chapter the physics of wave mixing i s presented. The model of a test electron in the f i e l d of two high freguency electromagnetic waves i s described. The r e s u l t s of t h i s model are incorporated into the c o l l i s i o n l e s s Vlasov eguation so that the perturbations caused by the wave mixing can be evaluated. Two possible methods of detecting these perturbations are discussed. The scattering cross-section of an electromagnetic wave by the induced plasma waves i s then evaluated, including the expected r e l a t i v e signal amplitudes for laser l i g h t scattering. Then the considerations i n choosing the experimental arrangement and eguipment used in the thesis are presented. F i n a l l y , the assumptions made in the theoretic a l derivation are evaluated for the actual experimental s i t u a t i o n to check their v a l i d i t y . A. Physical Model An electron i n the f i e l d of a single high freguency electromagnetic (EM) wave w i l l not experience a detectable perturbation i n i t s o r b i t . This assumes that the frequency of the l i g h t wave i s so much higher than any 7 natural freguency of the plasma, that the electron, even though i t w i l l obtain a considerable velocity, does not have time to move from i t s o r b i t . When there are two high frequency EN waves present, t h i s s i t u a t i o n changes. The test p a r t i c l e •mixes 1 the forces from the two waves, res u l t i n g i n a low freguency driving force which can perturb the electron o r b i t . The mixing e f f e c t can be understood using the following description(Figure II-1). The electron i s accelerated by the e l e c t r i c f i e l d (E 1) of the f i r s t EH wave. I t then experiences a Lorentz force from the magnetic f i e l d (B 2) of the second wave in the d i r e c t i o n J 1 X£ 2. Simultaneously the e l e c t r i c f i e l d of the second wave imparts a v e l o c i t y to the electron so that i t also experiences a Lorentz force in the d i r e c t i o n E^B^. when these forces are added together and the sums performed, one contribution comes from a term whose frequency i s the frequency difference of the two EH waves. This small acceleration, i f the mixing frequencies are judiciously chosen to match a natural plasma resonance, can measurably perturb the electron from i t s natural o r b i t . The magnitude and d i r e c t i o n of the force at the frequency difference i s calculated below, assuming plane monochromatic mixing waves. A more detailed c a l c u l a t i o n i s outlined i n section B. The force applied to the electron by the two EM waves i s FIGURE I I - l Electron in the f i e l d of two EM waves. 9 mV = e /\ (E. + V x B.) i = l ~ - 1 . . . ( D where mr e and V are the mass, charge and velocity of the electron, and Ej and Bj are the e l e c t r i c and magnetic f i e l d s of the i t h EM wave with freguency a>j and wave vector k j . we have for an EM wave at time t and position r: E. = E. Cos(6_,) — i —10 i ... (2) Since the phase ve l o c i t y of the mixing waves ( Wj /kj ) i s c, the speed of l i g h t , to f i r s t order we may neglect the magnetic f i e l d term i n (1). Integrating the f i r s t order acceleration V<1>: ' C 1 ) - ,/» £ V v ' = e m 7. E. 1 = 1 ^  (3) we get the f i r s t order ve l o c i t y V<1>: 2 E V ( 1 ) - e/m X) — Sin(9 ) + V ( o ) - i = l u i 1 " ... d) We may a r b i t r a r i l y choose the o r i g i n a l velocity v<°> to be 10 zero for oar test p a r t i c l e . The electron has obtained a vel o c i t y given by (4) . The maximum excursion B m a x that the electron w i l l make from i t s position R<°> i s found by integrating (4) again: R = - e/m J2 — +R ( 0 > i = l (u>,)2 2 E. R —max ... (5) If the e l e c t r i c f i e l d s are strong enough that V approaches c (requiring >90,000 MH focused to 100 microns), we s t i l l have for v i s i b l e l i g h t that R m a x < 1 0 - 7 m. This displacement i s much too small to be measured by the technigues i n t h i s experiment. The velocity i t does obtain , however, interacts with the magnetic f i e l d to produce a second order acceleration v<2> of the test electron given by: 2 V(2> - - e/B V(1> X 2 i, j = l 2 ... (6) Using (2) and (3) we have 2 V = o) ± 1 J S l n ( e i ) S l n ( 9 j ) ... (7) He now make the simplifying assumption that the mixing beams are polarized perpendicular to the plane defined by 11 {k.jfkg). Then we have 2 E E V ( 2 ) = e2/m2 Y. [ s l n ( e • + e ) .+ Sin(9 - 8 )] k . . . ( 8 ) There are now a t o t a l of 8 terns for the second order veloci t y . The terms which o s c i l l a t e at the sum frequencies can immediately be neglected, as we have shown above. We are l e f t with •(2) -e 2 E 1 0 E 2 0 — 2m2c oi^ co2 — v — — 1 T 2 . . . ( 9 ) H h S r e Ak = k 2 - kx ...(10) Ak and Aw w i l l be refered to as the mixing wave vector and freguency. Note the important feature that the test electron experiences a longitudinal force since the direction of propagation i s p a r a l l e l to the wave vector Ak. The wave mixing should then drive longitudinal plasma electron waves at freguency Aw and wave vector Ak. When one considers the s p a t i a l and spectral structure of the mixing waves, then one would expect a spectrum of plasma waves with a spread i n frequency and wave vector. The calculation of these spectra i s outlined i n the followinq section. 12 B. Theory The e f f e c t of wave nixing on the plasma has been calculated by several authors. These include the o r i g i n a l approximate c a l c u l a t i o n of the scattering cross s e c t i o n , 1 calculations f o r magnetic f i e l d s * 1 , for density g r a d i e n t s 1 2 ? 1 3 > 1 • , numerical s i m u l a t i o n s 1 9 > 1 * , plasma h e a t i n g 1 7 and for the e f f e c t on the pump wave 1 8? 1 9. A l l of the above treatments assume plane monochromatic mixing waves, except for the l a t t e r which allows for f i n i t e beam diameters. The c a l c u l a t i o n which i s most d i r e c t l y applicable to an experiment has been done by Meyer, 7 who takes i n t o account the s p a t i a l and spectral structure of the mixing waves. An outline of t h i s derivation i s given here. Let us f i r s t anticipate some general r e s u l t s of this derivation. We have calculated i n the above model the driving force on a free electron due to the mixing of two high frequency EM waves. Electrons i n a plasma, however, are not free, but are influenced by the presence of other charged p a r t i c l e s . Wave mixing can be thought of in terms of a variable freguency longitudinal driving force acting on the resonant system of ions and electrons that make up the plasma. The c a l c u l a t i o n of the e f f e c t of mixing EM waves on the plasma must f i r s t r e f l e c t the c h a r a c t e r i s t i c s of the mixing forces. Their frequency and wave vector spectra w i l l r e s u l t i n s i n i l a r spectra for the driven density waves. The amplitude of the waves for different d r i v i n g frequencies w i l l depend on the resonant 13 response structure of the plasma system. This a b i l i t y of the plasma to sustain the driven waves i n turn w i l l depend strongly on the wavelength of the driven waves compared to the plasma scale length. The c o r r e l a t i o n parameter, °c = k d/Ak, i s a measure of t h i s r a t i o , where kd=ezB0/t() *Te i s the inverse Debye length. The plasma responds to induced density fluctuations in such a manner that the fluctuations are reduced and charge ne u t r a l i t y i s maintained. In practice, however, the charge density i s not zero over scale lengths smaller than the Debye length: variations i n electron density within a Debye sphere does not strongly influence the neighbouring plasma because of the shielding e f f e c t of the ions (Debye shielding). For small values of alpha, the wavelength of the driven wave i s much smaller than the Debye length. Therefore, the plasma should respond to the driving force as i f i t were a system of independent electrons. I t should be possible to drive plasma waves over a large freguency range. The plasma i s very e f f i c i e n t at maintaining charge neutrality on scale lengths larger than the Debye length. For large values of alpha, the wavelength of the induced wave i s much larger than the Debye length, so that the driving force cannot produce plasma o s c i l l a t i o n s of appreciable amplitude. However, a uniform plasma does have natural o s c i l l a t i o n resonances, one of which i s near the plasma frequency. I t should therefore be possible to drive large amplitude waves for large alpha, as long as 14 the driving freguency matches a plasma resonance. The c a l c u l a t i o n of the effect of the mixing on the plasma should then r e f l e c t the sharp resonance for large alpha and the broad spectra for small alpha, as well as the freguency and wave vector structure of the driving force. Let us now proceed with the outline of the derivation. The e f f e c t of wave mixing on the ions i s neglected because of t h e i r large mass compared to the electrons. The second order acceleration of the electron, analogous to eguation (9), i s calculated assuming that the electromagnetic waves are generated by c l a s s i c a l damped o s c i l l a t o r s . The damping frequency Y i s much smaller than the central frequencies •( co10 , M2O ) °f t n e t w o waves. The second order acceleration i s inserted i n t o the c o l l i s i o n l e s s Vlasov equation for the electrons. The usual Vlasov equation i s used for the ions, i n l i n e with neglecting the wave mixinq on the heavier p a r t i c l e s . Perturbation theory i s then evoked to l i n e a r i z e the Vlasov equations. There i s no external e l e c t r i c f i e l d , but a constant maqnetic f i e l d i s allowed. The equilibrium v e l o c i t y d i s t r i b u t i o n i s assumed to be Haxwellian. Fourier transforms in space and Laplace transforms in time are performed. B e r n s t e i n ' s 2 0 and Hagfors*s 2 i calculations aid in the evaluation of the density fluctuation. Realizinq that the measurable quantity i s not the density fluctuation Ne (Ak,Aa>) , but i s i t s spectral density 15 S(A J C , A < J)O C <| N E (Ak,Aw) | 2>, < I N E (Ak,Aco) I *> i s now calculated. To perform t h i s derivation, i t i s assumed that the observation time T i s much larger than the damping rate r . Assumptions about the mixing beams must also be made. These are: -the l i g h t waves are composed of a great number of modes during the time T, -each l i g h t beam i s considered to be a stationary random process with random st a r t i n g times and phases for the modes, -each l i g h t beam i s assumed to be a plane Gaussian wave with e f f e c t i v e diameter b. Se f i n a l l y get <|Ne(Ak,Au>) | 2 > = <|N h ( A k , A u ) | 2 > + < | N i n d ( A k , A i o ) | 2> ... (11) where N V Im(-F) 1 + Z a 2 F j 2 + Zc^ImC-Fj F 0 < N , (Ak Aco) 2 > = -2 e ' 1 i 2_ th 1 TTAO) h + a 2 ( F + Z F } 2 1 e i ... (12) 16 CO 2±_ ff P 1 ( M 1 ) d a ) l d h > 4 y 2 + (Ao) - a) + a) ) F (1 + Za 2F J) e v i 1 + a z(F + ZF.) U10W20 x — exp r M + ( 4 k ) 2 \ the variables in t h i s expressions are: . (13) -Ze i s the charge on the ions, -oC- kd/|Ak| (correlation parameter), -k(j = e*N0/«0/cT (inverse Debye length), - «o i s the permittivity of free space, - K i s Boltzman's constant, -F e w Fj i f© r n o external magnetic f i e l d , reduce to: -x 2 F = F (x) = 1 - xe e e x = Aw/(Aka) a 2 = 2KT /m e 'MT \ h f h -x2 e dt - 1 TT xe F. = F i e r / M T E \ * i ... <1<0 -Te , Tj are the electron and ion temperatures, -M i s the mass of the ion, -I., ,1, are the mixing beam powers, 17 - /3 i s the angle between the e l e c t r i c f i e l d vectors of the mixing beams, - Y i s the damping rate, - w. i s the l i n e centre of the spectral io r d i s t r i b u t i o n s (Pj (a>j)) of the i t h mixing beam. Pj i s normalized t o unit area, -b i s the radius of the e f f e c t i v e s i z e of the Gaussian plane waves, - 9 m j x i s the mixing angle included between k1 and k 2. -the z-axis i s taken as p a r a l l e l to Ak, and the y-axis i s perpendicular to the plane (k1 , k 2 ) . In the above expression f o r the density fluctuations, <|Mtn (£k ,Aw)| 2 > i s immediately recognized as the contribution from the thermal fluctuations. These exist with or without wave mixing. The second term, <jN i n d ( A k , A o ) | 2 > , i s the r e s u l t of the wave mixing. It can be rewritten i n the form A N i n d ( £ M u O |2> = I 1 X 2 C(.N e,T e) W(Aio) K(Ak) RCAto/Akj N , T ) ...(15) where -C i s a constant for a pa r t i c u l a r plasma: C (8irmc)2 . .. (16) 18 •Ii and I2 are the powers of the incident beams, •H i s the ef f e c t of the f i n i t e frequency width of the mixing beams w = 2x, 2 2 (lit 10 20 00 ff P l ( a ) l ) P 2 ( a ) 2 ) d a > l JJ 4y 2 + CAo) - ^  + ( d t 02 ... (17) This function i s peaked at A<O= O>10 - « 2 0 • •K i s the eff e c t of the s p a t i a l structure of the mixing beams 9 b 2 K = Cos /3 -r— exp (Ak) 2 25 + ( A k ) 2 Sin2(%9 . ) * mix . .. (18) This function, unlike the thermal spectrum, i s sharply peaked i n the di r e c t i o n of Ak. -R describes the response of the plasma to mixing at Ak,AU> R = F (1.+ Za2F.) e 1 1 + a 2(F + ZF.) e 1 ... (19) The experiments reported i n t h i s thesis were designed to t e s t some of the predictions of t h i s formula (15). F i r s t , the induced fluctuations were measured as a function of the mixing i n t e n s i t y to test the I 1 x l 2 19 dependence. Second, the spectrum of the induced wave was measured, keeping a l l parameters of the mixing constant. This tests the function 9. F i n a l l y , the response function of the plasma was measured at dif f e r e n t driving frequencies. This i s the most important aspect of the thesis. Measurement of the intensity and frequency dependence T^x^xH i s necessary, but these are functions of the con t r o l l a b l e mixing beams. A measurement of the response function tests our understanding of the physical processes involved i n the interaction of the mixing l i g h t beams with the plasma. Figure II-2 i s a schematic of a possible spectrum with wave vector equal to 4k, and scattering parameter alpha=1. The height of the spectrum due to the induced component i s •arbitrary* because of i t s dependence on the powers of the mixing waves. These powers have been chosen to give egual contribution from the thermal component and the induced component of the density fluctuation spectrum. The width of the induced spike i s determined by the width of the spectral probability functions Pj (60j), as long as the mixing beams* spectral l i n e p r o f i l e s are narrower than the width of the response function R. If the freguency difference between the two mixing waves i s changed, then the position of the induced contribution w i l l s h i f t , and i t s peak height w i l l change according to the value of the response function. The response of the plasma for d i f f e r e n t values of alpha are i l l u s t r a t e d in Figure IIr3. The electron 20 i — r FIGURE II -2 A typical induced and thermal fluctuation spectrum. 21 F I G U R E II-3 Theoretical response of the plasma to different driving frequencies. 22 temperature and density have been fixed at No = 2x10**c«fJand Te=19000 °K, so to change alpha, we must change the wave vector of the induced wave. The amplitude of the response functions have been normalized to unity. The actual values of the maximum response for alpha equal to 1, 2 and 4 are 0.35, 0.47 and 156, respectively. The curve for alpha equal to 10 i s too narrow to resolve without sp e c i a l programing techniques. To i l l u s t r a t e a curve f o r very small alpha, parameters different than f o r our plasma must be used. The smallest possible alpha that i s obtainable in t h i s experiment i s 0.97 for e m j x=180°. The brocken l i n e in Fiqure II-3 has been drawn for conditions exactly the same as for the alpha=1 case, except that the temperature i s 190,000 °K. This alpha=0.1 curve has a maximum value of'u0.98. The resonance of the plasma, described by the response function R, has the q u a l i t a t i v e dependence on alpha as described i n the introduction to t h i s section. For small values of alpha, the wavelength of the wave beinq driven i s much smaller than the Debye lenqth. This means that the electrons tend to move independently of one another, and one would expect that the response of the plasma should have a broad spectrum. For large values of alpha, the wavelength of the driven wave i s much larger than the Debye length. Now c o l l e c t i v e e f f e c t s are important, and the waves are strongly damped unless they s a t i s f y the Bohm-Gross dispersion r e l a t i o n . The graphs show that the resonant 23 response does approach the plasma frequency from the high frequency side as alpha i s increased. The intermediate case i s guite i n t e r e s t i n g , since the driven waves are governed both by the random motion of the p a r t i c l e s and the longer range c o l l e c t i v e effects. He then have the broad resonances as for alpha=2. In the next section, possible methods of observing the induced fluctuations are presented. C. Methods Of Detection Since K r o l l , Ron, and Rostoker* f i r s t proposed wave mixing as a means of increasing the l i g h t scattering cross section of a plasma, scattering i s an obvious possible choice to examine the eff e c t of wave mixing. The scattering cross section i s d i r e c t l y proportional to <|Ne <Ak,&Cj) | 2 > = <|Hth (Ak,AG>) |*> • <|Njnd (Ak,AO ) | 2 > . The scattering signal w i l l be composed of two parts: the thermal fluctuation s i g n a l , and the induced fl u c t u a t i o n s i g n a l . If the mixing scattering signal as a function of frequency and wave vector i s experimentally determined, then by subtracting the normal scatterinq s i g n a l component we are l e f t with the induced scattering s i g n a l . This gives a r e l a t i v e measure of <|Njnd (Ak,A<o) |*> as a function of frequency and wave vector. Another method of determining the e f f e c t of the mixing i s to look at the intensity of one the of the 24 mixing beams, as suggested by Weibel 1 8 in 1976. I t has been shown by many a u t h o r s 2 2 that during the interaction of the three waves, energy i s transferred from the higher freguency EM wave to the plasma wave and the lower frequency EM wave. As Weibel argues, i f the frequency of one of the mixing beams i s changed rapidly so that the mixing freguency i s swept through a resonance of the plasma (a resonance of R), then t h i s energy transfer w i l l be modulated. He suggests that, under favourable conditions, a power of the order of 10 MW i n one of the mixing beams would be s u f f i c i e n t to produce a 1% change i n the power of the other beam. He does not suggest a means of sweeping the freguency of one of the mixing beams on the time scale reguired for short pulse lasers (MO 8 nm/sec). An experiment to test t h i s concept would be to look at the intensity of the modulated beam with the mixing beams tuned to match a resonance of R. The intensity of the higher freguency beam should vary i n the presence of the lower freguency beam. Measurements of the in t e n s i t y of the higher frequency beam before and afte r the interaction region would indicate any change in in t e n s i t y due to the mixing. signal error considerations suggest that the the minimum useful change in power would be of the order of 10%, requiring a 100 MW tunable l i g h t source. Such a high power reguirement for a dye laser makes t h i s method of observing the presence of the induced density fluctuation unattractive. 25 We are l e f t with scattering diagnostics to detect the fluctuations. The experiments presented here, of course, were designed f o r scattering since Veibel's recent suggestions were not available when t h i s research program was begun. D. Signal Calculations For the waves driven by the wave mixing to be detectable by laser scattering, the amplitude of the corresponding density fluctuation mast be at l e a s t the order of magnitude of the thermal fluctuations at t h e i r maximum. Since thermal fluctuations are re a d i l y observable, the increase in scattering s i g n a l due to the wave mixing should also be detectable. The r e l a t i v e amplitude of the induced scattering s i g n a l as compared to the thermal scattering s i g n a l i s calculated below. The d i f f e r e n t i a l scattering cross section per unit freguency d o / 2 T , per unit s o l i d angle dfi, per unit incident f l u x , f o r each electron i s d 2 0 = r o 2 ( l - Sin2eCos2<J>) S(Ak,Aui) du dfi 2n . . . ( 2 0 ) Here - r 0 i s the c l a s s i c a l electron radius, -9 i s the scattering angle, -4> i s the angle between the pol a r i z a t i o n vector of the incident l i g h t and the 26 scattering plane, -S i s the spectral density function given by S = NH<lNe(^Au>l2> o ..•(21) -N0 i s the average electron density, -V i s the scattering volume. The amount of power scattered I s c a t i s then /2TT<|N.|2>\ W - V T A V A N v ) T o 2 a - Sin 2 0 C o s 2 * ) dco d« o 2* ...<22) where I d i s the diagnostic power through the area k. One actually measures a range of Co and k, so that I 0b s , the sig n a l received i s Iobs^obs'^obs ) = fdu T k ^ - * W V A < ° - *» o b 8) I 8 c a t ..•(23) where T k and T w are the wave vector and freguency transmission functions of the detection system. The wave vector transmission function i s centred at the wave vector -cT-obs ' * n e r e JSd a n ^ k Q b s are the wave vectors of the incident and scattered l i g h t , respectively. The frequency transmission function i s centred at A c o 0 b S , the frequency s h i f t of the detector from the diagnostic frequency. 27 These transmission functions should take into account the losses of the detection system, but f o r our purposes we may set them to unity at t h e i r maximums. The s o l i d angle integral can e a s i l y be approximated for the thermal and the induced fluctuation components to the scattered l i g h t . The thermal fluctuations do not have a strong angle dependence over the usually small s o l i d angle of the detection system. The integral can then be replaced by the observation s o l i d angle d f i 0 D S times <| Mth (£k,Ac?) | 2> calculated at extreme (see eguations 13, 18). The response function of the plasma R can be considered constant over the range of the observed Ak, but the wave vector function K cannot. Since bk»1 i n the exponential i n K, the induced fluctuations scatter l i g h t i n a very narrow cone i n the dir e c t i o n k^-kg. The i n t e g r a l over the s o l i d angle of the observation system gives: ' 4* = A k o b s . The induced fluctuations are i n the other s c a t .. . (24) and the observed signal i s given by 28 ^bs = T r o 2 ( 1 " s^ecos 2*) x dn [dnobs /*> V l N t h ( ^ o b s ' A u ) l 2 > I . I , C S i n 2 ^ . f 1 2iS fdu>T R(Ato/Ak.,_ ) W(Au>) (Ak v ) 2Sin 2 J s e y u obs J + e - / WAU ' 2 U i f obs scat This can be rewritten: I , = S,I. + S 0I.I-I 0 obs I d 2 d 1 2 Sl (^bbs' A Uobs ) = r o 2 ( 1 - Sin 2 e C o 3 2 ( ( , ) dP. x y d c o V l N ^ C ^ . A a , ) ! ^ S 2 ( A kobs' A U )obs ) = E r o 2 ( 1 * Sin 2 e C os 2*) C Sin23s9 ' ' /• X (Ak . Vii** / d u Tu> R ( A ^ A k o b s ) W obs scat J . . . (25) ...(26) The quantities I d , I 1 , I 2 and I^g are Measured in an actual experiment; The data then qive values f o r S1 and S2* which are d i r e c t l y related to the spectral density functions for the thermal and the induced fluctuations. Mote that to maximize the scattered l i g h t , a l l the l a s e r s should be polarized perpendicular to the scattering plane. The scattering volume V i s determined by the the 29 intersections of the laser beans, and the viewing area of the detection system. The scattering volume f o r the thermal fluctuations are usually larger than those for the mixing, since the observation area i s purposely made larger than the laser beam diameters to c o l l e c t a l l the scattered l i g h t . The difference in volumes i s re f l e c t e d by the e in equation (25). « i s the r a t i o of the mixing volume to the scattering volume. Since the wave vector function K i s so peaked i n the d i r e c t i o n of the mixing wave vector, i t i s e s s e n t i a l that the detection system be arranged so that i t s transmission i s a maximum for Ak. This can e a s i l y be done as long as Ak = k x - k 2 - kj - k Q b s ..,(27) where k 1, k 2 are the wave vectors of the mixing beams, and k d, k Q b s are the wave vectors of the incident diagnostic beam and the scattered l i g h t (Figure Since k 1 = k 2 and k d = k O D S , the mixing and scattering angles are related by Sin-gQ^^ |k^jl ^ n ^ s c a t ...(28) I f the spectral d i s t r i b u t i o n s P, (a,) i n »(Aw)(10) are Lorentzian l i n e p r o f i l e s of f u l l width half maximum Fw, : % 1 V 1 0^ 2TT ( i g F w . ) 2 + K - a,, ) 2 i i io ...(29) FIGURE II - 4 Wave vector matching. 31 then B reduces to w tAto) = ~ — ^ " f 7 T,—To ; — T o -2<i)10 So (l5Pw) + (A" " "10 + ...(30) where Fw= 4Y • F W 1 * F W 2 . Generally, however, the di s t r i b u t i o n s P, are not Lorentzians. The function W can be calculated for experimentally determined l i n e p r o f i l e s as described in Appendix A. Be can now calculate the r e l a t i v e scattering signals using values of the plasma parameters and experimental conditions for th i s experiment, and assuming Lorentzian l i n e p r o f i l e s for the mixing beams: 6J 2 f J=2.30x10»s sec-» (820.0 nm) , 6J1Q=2.31x10»« sec-* (816.1 nm) , t J 1 0 - c c 2 0=1. 09x10*3 see- 1 (31.0 nm) , Fv 2 =3.7x10*2 sec-' (1.3 nm) , Fw1 =2.3x10** sec-* (2.3 nm) , r 0 = 2.82x10-»s. m, V n o r m=2.5x10"»* a.*, vmix = (2/3) xV n o r m, N0 = 2.0x10** cm-3, Te=19000 O K , 6 m i x =150.70, cG = 1. 0, I-,=I2=7 MB, d f i O D S =5.0x-3 Sr (f/12.5). 32 T w i s a symmetric function, unit height, 0.2 nm and 0.6 nm vide at the top and the base, respectively. Ve get for the thermal term: I t h / I d = 5x10-1* and for the mixing term: Imix /!d - 20x10 -»5. The t o t a l signal ve observe during the mixing should be 5 times the signal observed from the thermal fluctuations alone. The actual pover varies between 5 and 8 megavatts, so the expected mixing s i g n a l v i l l be betveen 2 and 4 times the thermal s i g n a l . The fact that the experimental l i n e p r o f i l e s are not true Loreutzians v i l l change the value for the induced s i g n a l only a small amount, since the freguency transmission function i s much wider than the l i n e p r o f i l e s ( i . e . the shape i s not important, just the t o t a l integrated intensity) . E. Expermimental Design The cal c u l a t i o n s of the scattering signals i n section C indicate that high pover l i g h t sources are necessary to study vave mixing successfully. The mixing lasers must have sim i l a r frequencies, and at least one should be tunable to a l l o v tuning of t h e i r frequency difference to a plasma resonance. The diagnostic laser should produce at least 10 WW at a frequency which matches photomultiplier responses i n order to give reasonable s i z e 33 scattering signals with our experimental conditions. The diagnostic laser should also have a freguency which i s s i g n i f i c a n t l y d i f f e r e n t from the frequencies of the mixing lasers. If the wavelengths are too close, then to s a t i s f y the wave vector matching conditions, the detection system w i l l be colinear with one of the mixing beams. Stray l i g h t would then be a severe problem. Table II-1 i l l u s t r a t e s the most reasonable choices of laser systems which s a t i s f y the above conditions. System 2 has the advantage that one of the mixing beams can be a very high power s o l i d state l a s e r . The added cost (a second s o l i d state laser) and complexity (frequency doubling) are large disadvantages. System 3 also has the advantage of mixing with a high power l a s e r . The diagnostic dye lase r can be chosen to lase at freguency which matches the maximum response of photomultipliers. This wavelength i s much di f f e r e n t than the mixing wavelengths. Unfortunately, commercial flashtube pumped dye lasers have only recently been able to produce the megawatts of power required f o r the diagnostics, and even then an o s c i l l a t o r amplifier system i s required. f The f i r s t system then was chosen f o r t h i s experiment. I f two tunable dye lasers are used for the wave mixing, then they are automatically i n the same spectral region. The dye lasers can either be flashtube pumped or laser pumped. Laser pumping has the advantage 34 MIXING BEAM MIXING BEAM DIAGNOSTIC BEAM SYSTEM 1 Dye Laser Dye Laser Ruby Laser SYSTEM 2 Dye Laser Buby Laser Frequency Doubled Glass Laser SYSTEM 3 Dye Laser Ruby Laser Dye Laser TABLE II-1 Possible l a s e r systems. 35 that the laser required for scattering diagnostics could also supply the pump energy. Laser pumped dye lasers have higher e f f i c i e n c i e s than flashtube units. The operation freguency of a laser pumped dye laser i s always lower than the pump (diagnostic) beam. Thus the mixing and diagnostic lasers are s p e c t r a l l y separated, and the inverse frequency dependence of the mixing can be exploited. Also the timing of the lasers i s automatic since a laser pumped dye laser lases almost coincident with the pump. The choice of a plasma for a new diagnostic experiment i s governed by several c r i t e r i a . I t should be r e l i a b l e , e a s i l y available, preferably well studied, and s a t i s f y the conditions of the theory to be tested. The open a i r D.C. plasma jet with which our laboratory i s fam i l i a r i s a good candidate. Electrode design improvements now allow i t s operation at several hundred amperes i n d e f i n i t e l y . I t has been studied by several authors with s p e c t r o s c o p i c 2 3 and laser diagnostics 2*> 2*> 2*. However, the plasma produced by the plasma jet i s not i n l o c a l thermodynamic e q u i l i b r i u m , 2 7 and i s not c o l l i s i o n l e s s . These aspects of the jet are considered i n section E. The actual experimental arrangement must f a c i l i t a t e examining the induced density fluctuations as a function of freguency and wave vector. The range of the driving frequencies which produce a measurable response i n the plasma, a function of the mixing wave vector, must not 36 be too narrow to resolve. Also the ever present thermal fluctuations must be considered. For these reasons, a diagnostic scattering angle of 110° with corresponding mixing angle of 150.7° were chosen to give a thermal scattering spectrum which i s e s s e n t i a l l y f l a t out to 3 nm from the ruby l i n e (alpha=1.0). The angular separation of the laser beams i s adeguate to allow a co-planar system, unlike i n Stansfield's® work. The frequency dependence of the scattered l i g h t can be observed by tuning a dye laser and/or the detector to d i f f e r e n t frequencies. observing the wave vector dependence i s much more d i f f i c u l t , since a choice of scattering and mixing angles fi x e s the observation wave vector. The entire o p t i c a l assembly must be reorganized to change 4*obs* Because of t h i s d i f f i c u l t y , the present experiment was performed as a single observation wave vector. F. V a l i d i t y Considerations In deriving the expressions (11-13) fo r the density fluctuations, many assumptions were made, both about the wave mixing l i g h t sources and the plasma. These assumptions w i l l now be examined for v a l i d i t y , beginning with the mixing beams. The l i g h t sources are considered as being generated by c l a s s i c a l damped o s c i l l a t o r s . The number of modes present i s large enough, and t h e i r phases and 37 starting times uncorrelated so that each beam can be considered as a stationary random process. He can only say that t h i s i s a plausible description of a laser system. Assumptions about the damping rate Y , however, are well s a t i s f i e d . I t i s assumed that 1/T « y « a> 1 ( ), U) 2 Q ..,(31) where the damping rate can approximated by the time f o r a single pass through the o s c i l l a t o r ruby rod. The damping rate i s then 1x10* s e c - * , much smaller than the central frequency of the dye lasers, 2.6x10*s s e c - 1 . The inverse observation time (1/T) i s 4x10* sec -*, completing the inequality. The estimate of the damping rate i s very crude, since i t depends on the unknown d e t a i l s of the laser system. The laser outputs are assumed to be plane Gaussian beams. This assumption i s not l i k e l y to be s a t i s f i e d for our lasers. Observation of the near f i e l d output using foot-print paper indicates that i t i s •spotty'. That i s , the smooth variation of burn expected for a Gaussian beam i s not present. This uneven output w i l l have several consequences. F i r s t , one must now question the form of the function K (Ak) and the mixing s i n t e n s i t y dependence of the induced fluctuations as given i n (15). I f the spiking i s random both i n space and time, then the average value of the i n t e g r a l s i n k space reguired to obtain (15) would be expected to s t i l l give the I x i dependence, but on a longer time scale than 38 before. Since some structure can be seen i n the burn patterns of the lasers, the time scale for the spiking i s not s u f f i c i e n t l y short to average out the beam i n t e n s i t y during one l a s e r pulse. This suggests that only on the average of several shots w i l l we get the I 1 x l 2 dependence. This structure can also be considered as a source of error f o r the mixing experiment. The laser monitors measure an average power of the dye lasers, and do not t r u l y represent the product of the e l e c t r i c f i e l d s in the mixing region. K (Ak) should s t i l l be sharply peaked around 4,k, but may be reduced i n amplitude. Since the detection system e s s e n t i a l l y integrates over a l l the wave vectors present i n the induced f l u c t u a t i o n s , i t s exact form i s unimportant for our considerations., There are three main assumptions about the plasma. F i r s t , the plasma i s supposed to be c o l l i s i o n l e s s . Since the waves i n a plasma are Landau damped, t h i s c r i t e r i o n r e a l l y states that the waves must be damped before c o l l i s i o n s disrupt the wave motion. The Landau damping rate for waves with phase velocity near the thermal v e l o c i t y of the electrons cannot be calculated using the approximate solution f o r small wave vectors. E c k e r , 2 8 using a more exact c a l c u l a t i o n , gives a value for the Landau damping rate CJ^ roughly egual to electron plasma freguency Cdr? u>„ = 0.85(D £ p 39 The plasma frequency i s 8x1O 1 2 rad/sec. The c o l l i s i o n a l damping rates are dependent on the choice of the vel o c i t y to be used for the electrons. An electron i n the mixing region experiences the induced o s c i l l a t i o n , but s t i l l retains i t s thermal v e l o c i t y . Since i t s thermal velocity i s so much higher than the velocity which i t obtains through the mixing process, the average thermal v e l o c i t y should be used. The c o l l i s i o n rates )Jea between electrons (e) and species (a) can then be taken d i r e c t l y form Appendix C: ui = 0.05 u)„ e n x. ue l = 0.005 <i>r w = 0.07 o)0 ee * The subscripts n and i refer to the neutrals and the ions. These c o l l i s i o n rates are at least ten times smaller than the Landau damping rate. Since the waves are damped 10 e-foldings before being interrupted, c o l l i s i o n s can be expected to play only a small r o l e i n the damping. The c o l l i s i o n l e s s requirement i s approximately s a t i s f i e d . The fact that c o l l i s i o n s have l i t t l e e f f e c t on the spectrum of the density fluctuations for t h i s choice of k i s backed by the excellent agreement between scattering theory for c o l l i s i o n l e s s plasmas and experimental spectra for t h i s plasma source. C o l l i s i o n s uo v i l l have a lesser e f f e c t on the very vide spectrum for alpha=1 i n t h i s report, so one should consider the f i t s for spectrums with higher alpha (smaller Ak), such as i n It i s also assumed that the plasma i s i n l o c a l thermodynamic equilibrium (LTE). That i s , the ions and electrons should have Naxwellian v e l o c i t y d i s t r i b u t i o n s characterized by the same temperature. The plasma from the plasma jet i s not i n L T E , " since the ion temperature i s lower than the electron's. However, laser scattering experiments show that at least the electron v e l o c i t y d i s t r i b u t i o n i s Haxwellian**. Different temperatures for the ions and electrons w i l l have no e f f e c t on the r e s u l t s of t h i s experiment, since there i s no change to the thermal fluctuations in the spectral region of i n t e r e s t . This i s because the phase v e l o c i t y of the waves being examined are so much larger than the thermal speed of the ions that there are v i r t u a l l y no ions with s u f f i c i e n t v elocity to inter a c t with the wave. I t i s only when Tj /T e = (H/m)*/« that the ion d i s t r i b u t i o n makes a contribution to the regions examined. This reasoning also applies to the high phase velocity induced waves, with the added feature that the ions are considered stationary in the f i e l d s of the mixing beams because of t h e i r higher mass. The Vlasov equations were l i n e a r i z e d when the density fluctuations were calculated. This requires that the fluctuations be much smaller than the average density. 41 The smallness of the perturbation can be v e r i f i e d i n two ways. F i r s t , we know that the thermal fluctuations are such smaller than the average density. The induced fluctuation spectrum measured i n these experiments are only a few times the thermal l e v e l , so they too can be expected to be much smaller than the thermal l e v e l . Secondly, S t a n s f i e l d * , using plane monochromatic wave approximations, has shown that the amplitude of the induced wave i s fN , where H i s the mean electron density, and f i s given by the expression 2 E,_ E__ 1 - e. f ~ 4 10 20 ; A a m w 1 0 " 2 0 eSL p h .,,(32) Here ^ i s the e l e c t r o n i c d i e l e c t r i c function and V p h i s the phase velocity of the wave. Using the value of the f i e l d of 3x10* V/m (10 HW focused to 0.2 mm diameter spot) and a maximum value of (1-«t)/ej=5 at Vph equal to the mean thermal v e l o c i t y , we get f = 5x10-*. Thus, the amplitude of the driven waves are much smaller than the average density. One great advantage that las e r scattering claims as a diagnostic t o o l i s that the plasma i s not perturbed by the diagnostic beam. This i s only true i f the energy absorbed from the diagnostic beam i s small compared to the thermal energy. Kunze 2» gives for the energy absorbed by the electrons through inverse bremsstrahlung as: 42 AT N x r 1 1 A 5.32 x 10~7 - ^ — g / , l - e x p ( - f £ ) -f AT (KT )• / Z L e J A . (33) Unfortunately, t h i s expression for the amount of energy absorbed i s maximum for low temperature, high density plasmas such as ours. The units f o r t h i s equation are cm-3 for density, ev for kT, cm f o r wavelength, watts cm - 2 for laser i n t e n s i t y , and seconds for the pulse length AT. Using 40 flW focused to 150 micron diameter gives I d/A =2.3x10** MW cm-2 for the 20 nanosecond pulse. This formulae then gives the energy absorbed as AT e/ Te=2. This assumes that the relaxation time amongst the electrons i s much smaller than the length of the laser pulse, and that the energy stays with the electrons. The f i r s t reguirement i s s a t i s f i e d since the energy exchange time between electrons i s approximately 3x10-** sec (Appendix C). The assumption that the energy remains e n t i r e l y with the electrons i s not v a l i d . He are not interested i n the temperature relaxation time between the electrons and the ions and neutrals, which i s long because of the reduced energy transfer as temperatures approach their LTB values. He are interested i n the energy exchange rate at the present temperature difference. S p i t z e r 3 0 gives t h i s energy loss rate as: 43 dT T e e dt eq t .27 (V 3/2 sec eq N £nA o ... (34) where A i s 9 times the number of electrons i n the Oebye sphere, T e and Tj are in °K, and N0 in cm-'. The value for the rate of change of temperature f o r the temperature difference i n the interaction region i s : Thus, the energy deposited to the electrons w i l l quickly be carried to the ions. As discussed in Appendix C, the neutrals and ions are close to equilibrium between themselves, so that the neutrals w i l l also absorb some of the energy. The neutrals, because of the increasing population of higher atomic energy l e v e l s and radiation losses, can absorb energy without r a i s i n g t h e i r k i n e t i c energy an equivalent amount., Therefore, the enerqy deposited to the electrons w i l l r e s u l t in a raised neutral and ion temperature, and possibly a s l i g h t l y increased electron density. There i s some evidence that the diagnostic laser (the most intense of the three lasers) i s not perturbing the plasma an appreciable amount. I f the diagnostic power i s increased, one would expect the perturbation to increase. The change in plasma parameters would change d(T e)/dt = 1 ev/(4x10-* 2 sec). 44 the scattering cross section, which would show up as a change i n slope of a scattered i n t e n s i t y versus incident intensity graph. Such a graph (Figure B-1) shows no s i g n i f i c a n t deviation from a straight l i n e , i n d i c a t i n g that the c a l c u l a t i o n of the energy absorbed i s an over-estimate. The calculation of the induced fluctuations i s * qua s i - s t a t i c ' since i t i s assumed that the mixing beams are present f o r a l l time. Actual changes i n the laser intensity w i l l be followed c l o s e l y by the induced fluctuations so long as the time scale of the changes are longer than the response time of the plasma. The dye lasers used sometimes show a tendency to mode-lock, but the modulations are a few nanoseconds wide. The plasma can respond to changes in mixing in t e n s i t y on the order of the inverse plasma frequency {~10 - 1^ sec). Be can therefor assume that the mixing l i g h t waves produce plasma waves with no 'phase lag*. In summary then, the assumptions used i n the derivation of the spectrum of the induced density fluctuations hold for our experiment, except f o r plane Gaussian mixing beams. The e f f e c t of non-Gaussian beams i s expected to give some departure from the simple dependence I-, x I 2 x K (Ak) , but only in the form of possibly reduced signals with more shot to shot v a r i a t i o n . U5 Chapter III EXPERIMENTAL APP&RATOS This chapter i s devoted to a description of the experimental apparatus and e l e c t r o n i c s . The o p t i c a l system i s i l l u s t r a t e d in Figure III-1. I t s main components include a ruby laser which provides the power for the diagnostic scattering as well as the pumping of the dye l a s e r o s c i l l a t o r - a m p l i f i e r s . The diagnostic beam and dye laser mixing beams are directed by mirrors and prisms toward the plasma je t where they are monitored and focused into the centre of the plasma. The scattered l i g h t i s collected by another set of lenses and analyzed using a monochromator and photomultiplier. The l a s e r monitors and photomultiplier signals are delayed with respect to each other and recorded on oscillograms. The spectra of the dye lasers are also monitored by a second monochromator, and an o p t i c a l multichannel analyzer after they pass through the interaction region. An e l e c t r o n i c l o g i c system controls the f i r i n g of the ruby laser, since i t must lase only when the remaining diagnostic equipment i s •ready*. A more detailed description of the experimental system follows. 46 FIGURE I I I - l The complete optical system. Legend: D, Brewster angle dump; F, glass funnel and optical fibre; GP, thin glass plate; H, Rayleigh horn light dump; I, i r i s ; L, lens; M, mirror; MON, Brewster angle laser monitor (shown rotated 90° about the laser optical axis). «7 A. The Plasma Jet A plasma jet produces a steady state, open a i r plasma by passing a D.C. current through a gas which flows between two electrodes. Morris 2* has shown that i n the ra d i a l d i r e c t i o n , the electron temperature and density of this plasma i s constant to 5% within a range of 1 mm. Design improvements of the plasma jets used by Chan and o t h e r s 2 3 - 2 7 3 1 have made t h i s plasma source r e l i a b l e for extended periods of operation. These c h a r a c t e r i s t i c s make th i s simple device an excellent experimental plasma source: r e l a t i v e l y inexpensive, reproducible and reasonably uniform, long l i v e d , and free of surrounding windows which can r e s t r i c t the diagnostic geometry and increase stray l i g h t in scattering experiments. Figure III-2 i s a schematic of the plasma j e t . Helium gas flows between the tungsten cathode and the copper anode where i t i s p a r t i a l l y ionized, and then out of the anode o r i f i c e , creating a free standing plasma flame. The design improvements mainly consist of increased water cooling of the anode by direc t i n g the water symmetrically over the inside of the copper surface, with higher flow rates i n the regions of highest heat loading. Improvements were also made to the tungsten cathode design. Now copper i s f i r s t melted onto tungsten with a n i c k e l interface and then machined and soldered to a brass tube. The tungsten-nickel-copper-brass design gives a stronger join and better heat conduction compared to soldering. Also the cathode water flow was redesigned ANODE COPPER RETAINING RING 'O' RINGS He GAS NYLON mm OUT (1 Of 2). MATERIAL: BRASS (EXCEPT WHERE NOTED) TUNGSTEN CATHODE COPPER WATER DIRECTING PLUG CERAMIC IPOLY FLOW' FITTING IN ( l o t2) WATER OUT FIGURE III - 2 The plasma jet. 49 to eliminate stagnation points. The j e t i s i d l e d at low current, about 70 amperes, with a welding supply (Figure i n - 3 ) • This reduces the duty cycle of the battery as well as increases the l i f e t i m e of the electrodes. The DC welding supply has maximum ratings of 500 amperes, 60 volts and 10 KVA. When the lasers are ready to be f i r e d , the current source i s switched by relays to a 48 v o l t , 240 ampere-hour battery, and the current raised in six steps to 230 amperes by combining 1 ohm, 1000 watt r e s i s t o r s i n p a r a l l e l with the ball a s t r e s i s t o r . The large number of steps reduces the erosion of the anode, thereby increasing the rep r o d u c i b i l i t y of the jet. The current i s then trimmed to 230 amperes using a current regulated power supply. This power supply, connected in p a r a l l e l with the jet and isolated from the battery by a diode, acts as a variable current source of up to 30 amperes, regulated to better 0 than 1%. The jet i s allowed a few seconds to come to eguilibrium before an experimental shot i s made. The j e t current i s monitored by measuring the voltage drop across a 10-• ohm shunt with a d i g i t a l voltmeter which reads to the nearest tenth of a m i l l i v o l t . This gives a direct reading of the current: 0.1 mv = 1 A. The shunt i s accurate to 1% and the voltmeter to 0.1 mv, so that the absolute current accuracy i s ±3 A, and the current r e p r o d u c i b i l i t y i s ±1 A. The axis of the jet i s v e r t i c a l , and perpendicular to the monochromator and la s e r axis. The 1 PLASMA JET SHUNT -ih % - i n \ v 4 - i f i \ \ o i 4 - i n it (SIX UNITS) h 1 1N2055 1N3288 1 1N1164A T WELDING CHARGING 48 V - REGULATED SUPPLY UNIT DC -Z • SUPPLY X FIGURE III-3 The plasma jet power supply c i r c u i t . 51 helium gas flow rate i s 8x10 - 3 1 sec -* (25 standard cubic feet per hour). In order to maintain the charge on the battery during an experiment, a charging unit i s connected to the battery. When the j e t i s using the welder current source, a relay switches power to the charger. B. The Ruby Laser A ruby laser system capable of high output power i s required because the ruby laser provides both the pumping power for the dye lasers and the incident radiation for the scattering diagnostics. A Q-switched o s c i l l a t o r - a m p l i f i e r combination was constructed for t h i s experiment, similar to that designed by A l b a c h 3 2 . Both the o s c i l l a t o r and amplifier employ 6 inch long by 1/2 inch diameter Brewster cut ruby rods. The o p t i c a l pumping of each rod i s provided by two l i n e a r flashlamps i n double e l l i p t i c a l c a v i t i e s . Op to 4 ki l o j o u l e s of energy i n a 1 millisecond pulse i s provided by a separate capacitor bank for each flashlamp. The ruby rods and flashlamps are enclosed i n glass jackets to permit water cooling. Refer to Albach and Churchland 3 3 for construction d e t a i l s . The o s c i l l a t o r laser cavity has a Laser Optics •ekalon 1 Fabry-perot etalon of 66% r e f l e c t i v i t y (R) as the output mirror, and a d i e l e c t r i c coated, >99% R rear r e f l e c t o r . Q-switching i s done using a Glan-air prism 52 polarizer and Pockels c e l l placed between the ruby rod and rear r e f l e c t o r . The etalon aids in lo n g i t u d i n a l aode control, and 8 mm diameter i r i s e s in the cavity help to control transverse modes. The prism polarizes the ruby l i g h t perpendicular to the scattering plane. The laser system, with fresh o p t i c a l components, i s capable of delivering over 500 megawatts peak power i n a 20 nanosecond pulse, but i t i s regularly operated at less than 200 megawatts to preserve components. The spectral l i n e width of t h i s laser system can be expected to be less than 0.07 nanometers as indicated by measurements on l a s e r s of s i m i l a r c o n s t r u c t i o n 3 * * 3 * . To try to maintain r e p r o d u c i b i l i t y , the ruby laser was f i r e d no more than three times every ten minutes. However, during the course of the hundreds of shots required for an experiment, the pulse c h a r a c t e r i s t i c s from shot to shot vary more than 10X, and the output power could drop 50% by the end of a run. C. The Tunable Dye Lasers Two tunable, o s c i l l a t o r - a m p l i f i e r dye lasers, i l l u s t r a t e d by Figure were constructed f o r t h i s experiment. Several d i f f e r e n t dye lase r configurations were examined i n the search for high power, narrow spectral l i n e output c h a r a c t e r i s t i c s . I t i s not within the scope of t h i s thesis to describe these studies, so only information on the actual system used i s presented. FIGURE III - 4 Dye laser system. 54 The dye lasers are capable of producing 5 to 10 megawatts of power i n a 15 nanosecond f u l l width half maximum (FWHM) pulse. The spectral l i n e shape i s variable from l e s s than 0.1 to more than 0.3 nm FflHtt by varying the diameter of the i r i s e s . Figure 17-5 compares t y p i c a l output p r o f i l e s as measured by an o p t i c a l multichannel analyzer. The dye lasers can be tuned through more than 15 nm centred at 810 nm. The o s c i l l a t o r output r e f l e c t o r i s a 30% R plane mirror. The plane grating rear r e f l e c t o r provides tuning. This 1200 l i n e s per mm grating, manufactured by Bausch and Lomb, i s blazed f o r 750 nm in f i r s t order. The 32 mm wide ruled area has l i n e s 30 mm long. I t i s Littrow mounted with the l i n e s perpendicular to the scattering plane. The grating mount has two opposing, pre-stressed angle bearings for smooth ro t a t i o n a l adjustment. This tuning i s done by a large drum micrometer on a lever arm approximately 10 cm long. The lever arm can be adjusted so that a 0.01 mm micrometer movements gives 0.1 nm tuning (1 micron = 0.01 nm). The maximum of the output can be tuned to a desired wavelength to within ±0.02 nm, and i s stable to the same figure. The dye c e l l windows, f l a t to X/4, are mounted at the Brewster angle. The o p t i c a l path length through the dye i s approximately 10 mm. & flow system with a 300 ml reservoir i s used to prolong the operating time f o r the dye lasers. The dye used i s 3,3*-diethyl-2,2'-FIGURE III - 5 Dye laser beam spectral line profiles. 56 thiatricarbocyanine iodide (DTTC iodide) dissolved i n dimethyl sulphoxide (DMSO) at a concentration of 2x10 _ s Molar for the o s c i l l a t o r and 4x10-* for the amplifier. The dye solution i s replaced during an experiment whenever the output power begins to drop, usually a f t e r about 75 laser f i r i n g s . The Brewster angle dye c e l l s and the natural tendency for dye lasers to lase with the same polarization as the pumping laser ensure that the dye laser outputs are polarized perpendicular to the scattering plane. I r i s e s and a Galilean telescope placed i n the cavity help to control the spectral l i n e shape. The 1 cm maximum diameter i r i s e s reduce the angular divergence of the output, and therefore reduce the spread i n angle of incidence at the grating. The telescope, with a n t i r e f l e c t i o n coated lenses of -4 cm and 16 cm f o c a l lengths, expand the dye laser beam so that more l i n e s of the grating are illuminated. This improves the resolving power of the grating and also has the extremely favourable effect of reducing the power flux on the grating. The longer l i f e t i m e of the grating with the telescope present makes the beam expander almost mandatory. With the telescope, the grating l a s t s i n d e f i n i t e l y , but can be damaged i n a single shot without i t . Each o s c i l l a t o r i s pumped with 15 megawatts of the ruby lase r output, and each amplifier with about 65 megawatts. The gain of the amplifier i s 25, which means, for 10 megawatts output power, only 45S of the pump energy 57 i s converted to o s c i l l a t o r output energy.. This i s ouch below the 2035 reported possible for t h i s dye 3*. The 20SS figure, however, refers to output with l i t t l e wavelength selection, which can account for a factor of two or sore in the difference in e f f i c i e n c i e s . Care was not taken to maintain high purity of the dye and solvent. This, combined with the use of only medium quality optics, makes the H% e f f i c i e n c y not unreasonable, although below expectations. The spectral l i n e shape of the output i s very sensitive to the position of the o s c i l l a t o r pumping beam. If the pump i s off-centre, then the output p r o f i l e i s skewed. & 1/4 mm displacement of the 10 mm diameter pumping beam produces changes in the spectral output which are read i l y seen using the ORA monitor. D. The Complete Experimental System And Diagnostics Pigure III-1 on page 46 i s a schematic of the op t i c a l system. The output from the ruby laser i s divided into three beams by d i e l e c t r i c coated mirrors (M1,2) of selected r e f l e c t i v i t y . Forty percent of the power i s directed to pump one of the dye laser systems, an equal amount i s directed to the second dye laser, and the remaining 20% i s transmitted through the mirrors f o r laser scattering diagnostics. The r e f l e c t i v i t i e s quoted for a l l the d i e l e c t r i c mirrors are the suppliers' figures for the angles of incidence i n the experiment. 58 The diagnostic beam goes through an i r i s which makes the focused spot more symmetric. I t i s then directed to the plasma by two mirrors of >99SS B (M3,4) and focused by a 10 cm f o c a l length lens (L1). After passing through the plasma, the diagnostic beam i s trapped i n a glass Rayleigh horn. The dye laser pumping beams are themselves divided into two parts. The glass plates (GP1,2) r e f l e c t approximately 20% into the dye c e l l s of the o s c i l l a t o r s . The remaining pumping beams are redirected by prisms (PI,2) into the amplifier c e l l s . After being mostly absorbed by the dye solutions, the pumping beams then enter Brewster angle dumps. The dye laser mixing beams are directed towards the plasma by prisms (P3,4), and then focused into the interaction region by 10 cm f o c a l length lenses (L2,3). After passing though the plasma, the dye laser beams are collected by glass funnels. Optical f i b r e s at the base of the funnels dir e c t some of the co l l e c t e d l i g h t to the entrance s l i t of a monochromator-optical multichannel analyzer combination (henceforth refered to as the OMA). The Princeton Applied Research Model 1205 OHA i s a vidicon device with an image i n t e n s i f i e r f i r s t stage. It i s placed at the exit port of a monochromator so that the image plane of the image i n t e n s i f i e r i s coincident with the exit f o c a l plane of the monochromator. This produces an image of the exit plane of the monochromator on the vidicon surface. Light on the vidicon surface 59 produces a charge proportional to the l i g h t i n t e n s i t y . The charge i s measured by an electron beam which scans across the vidicon surface i n 500 l i n e segments, or channels. The s i g n a l from each channel i s then displayed d i g i t a l l y and on an oscilloscope. & single count represents two photons incident on the vidicon at maximum gain for the image i n t e n s i f i e r at i t s best s p r e c t r a l response. The OHA i s oriented so that the channels are p a r a l l e l to the entrance s l i t of the monochromator. Thus each narrow v e r t i c a l channel plays the role of a monochromator exit s l i t and a photomultiplier. & complete spectrum of 500 points can be obtained on a single shot basis. As any experimenter who has p a i n f u l l y processed photographic plates or point by point spectra can well imagine, the 500 channel OHA i s an extremely valuable t o o l . A frosted glass plate and neutral density f i l t e r s placed between the entrance s l i t and the ends of the glass f i b r e s serve to control the in t e n s i t y of l i g h t reaching the OHA. The monochromator has a dispersion of 10 A/mm. This, combined with the 2,5 microns channel spacing, gives a combined dispersion of 0.025 nm per channel. Due to the cross t a l k between channels, the resolution of the OHA i s 1.5 channels FHHM. For the 10 micron entrance s l i t used, the system resolution i s 0.04 nm, guite s u f f i c i e n t to resolve the 0.1 to 0.3 nm FWHH dye laser s p e c t r a l l i n e s . Since the wavelength spread of the 500 channels of the OHA i s greater than the 60 wavelength difference of the dye lasers, both lasers can be displayed simultaneously. For every f i r i n g of the lasers, the spectral l i n e s are plotted by an x-y chart recorder. Also the t o t a l integrated i n t e n s i t y , and the maximum signal and i t s channel p o s i t i o n i s recorded using d i g i t a l information supplied by the OMA. Since the OMA has a free running read and display cycle which cannot be controlled externally, a timing system i s required. This system i s described i n section F. A lens combination (L4,5) c o l l e c t s scattered l i g h t and focuses i t onto the entrance s l i t of the second monochromator. The baffl e s and i r i s e s i n the c o l l e c t i o n system, set for f/12.5, are very important for stray l i g h t reduction. These aids, combined with the laser and Rayleigh horn viewing dump, make the stray l i g h t l e v e l s neg l i g i b l e i n the regions of the scattering spectrum of interest. The i r i s e s are indicated i n Figure III-1, but the b a f f l e s , since t h e i r position i s a unique function of the exact o p t i c a l arranqement, are not. The entrance s l i t of the monochromator i s masked in the v e r t i c a l d i r e c t i o n and has i t s width set so that i t forms a 300 micron square viewing hole. This helps to define the scat t e r i n g volume i n the plasma, since any l i g h t which does not go through the image of the entrance s l i t i n the plasma w i l l not go through the entrance s l i t i t s e l f . An i r i s placed i n s i d e the monochromator r e s t r i c t s the f-number from i t s f u l l f/6.3 to f/12.5. This i s to ensure good stray l i g h t r e j e c t i o n , but at the same time 61 maintain s u f f i c i e n t normal scattering signals. The mixing signals, because of t h e i r small s o l i d angle, t r i l l be unaffected by t h i s change i n f-number. The inverse dispersion of the monochromator i s 1 nm/mm, with a maximum resolution of 0.01 nm. The e x i t s l i t i s set to 300 microns or vider, depending on the experiment being performed, to give a 0.3 nm PWHH or wider transmission function. The scattered l i g h t i s detected by a room temperature, GaAs photocathode photomultiplier (RCA—C31034B) with the standard dynode r e s i s t o r chain suggested by RCA. The photomultiplier (PH) i s r e s t r i c t e d to very low l i g h t l e v e l s indicated by a maximum 30 second average anode current rating of 10 microamps at 1500 volts. To reduce the average current to t h i s l e v e l when the PH i s exposed to the continuum radiation of the plasma jet, i t i s necessary to i n s e r t a chopping wheel i n the detection system. To allow a maximum reduction in the plasma l i g h t l e v e l s , the chopping wheel i s placed as close to the entrance s l i t as possible, with an aperture just s u f f i c i e n t to l e t the converging l i g h t through. The 3 mm wide aperture i s at 9 . 8 cm radius on a 25 cm diameter aluminium disk. The free running chopping wheel, rotated b y a 3600 RPH synchronous motor, must also be considered in the timing sequence. The chopping wheel reduces the average l i g h t l e v e l by 1/100, giving a PH anode current of 1 microamp at 1500 v o l t s , well below the stated maximum. Besides monitoring the spe c t r a l output of the 62 dye l a s e r s , the diagnostic and nixing laser beans are monitored by pin photodiodes. Brewster angle glass plates (GP 3,4,5) placed just before the focusing lenses r e f l e c t some of the laser l i g h t onto frosted glass plates. Some of the dispersed l i g h t reaches the photodiodes through neutral density f i l t e r s which reduce the intensity to the linear operation l e v e l for the diodes. The signals from these diodes, as well as the photomultiplier s i g n a l , are transmitted by 50 ohm coaxial cables to delay cables and a Tektronix 7904 oscilloscope. The cables are terminated with 50 ohms at the oscilloscope, but not at the diodes or the PM. The delay cables combined with two dual plug-in units allow a l l four signals to be displayed on the oscilloscope simultaneously. The ruby laser s i g n a l i s displayed immediately, the dye lasers with 100 and 200 nsec delay, and the PH with 400 nsec delay. The delay cables are lengths of very low loss 50 ohm cable. The r i s e time f o r the dye laser monitor system i s about 1 nsec (0.8 nsec f o r the oscilloscope and l e s s than 1.0 nsec for the diodes). The ruby laser monitor system r i s e time i s r e s t r i c t e d to 1.5 nsec by the slower plug-in unit at the oscilloscope, and the PH system to 3 nsec by the slow response of the PM. The signal-to-noise for the displayed signals i s 20 to 1. This large figure i s possible because of the very low D.C. PM current from the low continuum radiation of a Helium plasma. The noise pickup from the f a s t f a l l i n g high voltage Pockels c e l l driver i s reduced to a 63 neg l i g i b l e l e v e l by sheathing the si g n a l cables i n extra grounding braid, and housing the PH assembly i n a brass case. E. Alignment Procedure There are over 50 o p t i c a l components in t h i s experiment, so that careful alignment and r e l i a b l e mounts are necessary. The table on which most of the components are mounted consists of 1/2 inch thick aluminium plates, tapped with 1/1-20 holes on 2 inch centres. This home-made o p t i c a l table i s f a r superior to triangular o p t i c a l r a i l s , but does not perform as well as commercial units would be expected to. Alignment i s accomplished with the aid of four separate HeNe CW las e r s , one for each o p t i c a l axis. These lasers reguire a d d i t i o n a l steering mirrors and pin holes, but greatly reduce the set-up time for a run. One HeNe beam enters the ruby l a s e r through the rear r e f l e c t o r . This i s possible since t h i s mirror has greater than 99% H at 694 nm, but less than 50% B at 633 nm. The other beams cross above the centre of the plasma j e t toward the dye lasers and the monochromator. The beams are co-planar and p a r a l l e l to the table top. The anode of the plasma j e t i s replaced by an alignment template which has four pairs of pin holes, each pair 35 cm apart. These pin holes have been made to define accurately the angles for the four o p t i c a l axis 64 above the plasma j e t . . The HeHe beams are then directed by mirrors (M1,2 f o r the ruby axis, extra mirrors f o r the others) through the pin holes. The j e t mount allows height and t i l t adjustments to make the template coplanar with the alignment beams. The consistency of the angles i s better than 10-? radians (0.3 mm i n 30 cm), as indicated by observing the beam positions at several meters distance after d i f f e r e n t alignments. The accuracy i s estimated at the same value. The anode i s replaced and the o p t i c a l components for the dye lasers are centred, s t a r t i n g at the plasma jet. The focusing lenses L2,3 are inserted at a l a t e r time. The standard technique of observing the r e f l e c t i o n s from the laser mirrors i s used to a l i g n the laser c a v i t i e s . The grating must then be rotated to r e f l e c t 820 nm rather than 633. Because the operating wavelength for the dye lasers i s outside the v i s i b l e region, focusing the telescopes by conventional means i s d i f f i c u l t . However, the extremely high gain of the dye lasers can be used to an advantage, since the o s c i l l a t o r s w i l l lase even i f the telescopes are poorly aligned. By monitoring the spectral outputs with the OMA while the telescopes are adjusted, focusing i s e a s i l y and accurately done. The scattered l i g h t c o l l e c t i o n system i s aligned next. Mirror M5, lenses 1.4,5, and the chopping wheel are on a separate o p t i c a l bench with vernier distance scales on the o p t i c a l mounts. The chopping wheel aperture was 65 « cat to be i n the scattering plane vhen i t reaches i t s highest position. Mirror M5 i s adjusted so that the alignment beam i s r e f l e c t e d p a r a l l e l to the axis of the bench. The monochromator i s centred so that the HeNe beam enters the entrance square and i s centred on the f i r s t mirror. Lenses L4,5 are centred. To aid i n f i x i n g the distance between the plasma jet and the focusing and c o l l e c t i o n lens, a pin mounted in a plug which f i t s i nto the anode aperture i s used. The pin defines the centre of the v e r t i c a l axis of the j e t , and i t s t i p positions the interaction region. The l i g h t between the two c o l l e c t i n g lens must be as p a r a l l e l as possible i n order to keep f-numbers matched and to reduce l i g h t loses to a minimum. A telescope, with cross-hairs i n i t s f o c a l plane and focused on i n f i n i t y f or v i s i b l e l i g h t , i s used to do the i n i t i a l adjustment: when the lenses are one foc a l length from the pin or entrance square, the image i s in the plane of the cross hairs. To overcome possible error i n alignment of the telescope, an i t e r a t i v e process i s used to make fine adjustments. The entrance sguare i s illuminated from inside the monochromator, and the image i s checked by parallax for coincidence with the pin. One lens i s adjusted to make the image co-planar and the distance moved i s recorded. The lens i s then moved one-half the distance back and the second lens moved the same distance to compensate. The image i s checked f o r parallax again, and the procedure repeated. Usually only two i t e r a t i o n s 66 are required. This procedure fixes the position of the good quality achromat c o l l e c t i n g lenses to better than 0.5 ram. Note that a step f i l t e r which transmits above 620 nm i s used when viewing the pin and image. The distance between the plasma and the laser focusing lenses i s determined by t r i a l and err o r , since t h i s distance depends on the unknown divergence of the laser beams. The lasers are f i r e d with an unexposed, developed piece of negative f i l m i n the region of the fo c a l volume. When the lenses are the correct distance from the plasma, the smallest damage spot on the f i l m i s centred above the j e t . The depth of focus of the laser beam-lens system i s about 1 mm, so that t h i s adjustment can e a s i l y be done. The actual centring of a l l beam spots and c o l l e c t i o n volume i n the interaction region must be performed just before the experiment i t s e l f . The image of the entrance square, with the step f i l t e r i n the l i g h t path, i s centred on the pin. The focused spot of the ruby laser diagnostic beam i s checked to be coincident with the focused spot of i t s HeNe alignment beam, and centred at the t i p of the pin. The pin i s removed and the f i l m inserted so that the entrance square image and the ruby alignment beam spot are centred at the same point on the fil m . The l a s e r s are then f i r e d i n d i v i d u a l l y , and the focusing lenses adjusted so that the laser damage spots are i n the centre of the image of the square. The 100-200 micron diameter l a s e r spots are usually within one radius 67 of the f i n a l position before the fin e adjustments, which are accurate to about 25 microns. The i r i s e s and baffles are placed when the optics for that section have been aligned. The laser dumps and funnel c o l l e c t o r s are f i n a l l y inserted, stray l i g h t checked, and the experiment performed. F. E l e c t r i c a l System And Timing Figure IIX-6 i s a block diagram of the e l e c t r i c a l system. The l o g i c unit i s necessary since the ruby laser must be f i r e d only during the 10 microseconds that the chopping wheel window i s i n place, and the 760 microseconds that the OHA i s ready to accept l i g h t . The l o g i c unit receives pulses from the OHA and chopping wheel which occur at known times before these units are *ready*. The 0H& prepulse i s a standard part of the unit. The chopping wheel prepulse i s obtained when l i g h t from a l i g h t emitting diode i s r e f l e c t e d from a silv e r e d section of the ro t a t i n g shaft onto a matched photodiode. Rotation of the pickup on the shaft housing gives coarse adjustment of the prepulse delay time. The horizontal movement of the entire assembly provided by i t s o p t i c a l mount gives f i n e adjustment. This pulse i s processed by a discriminator and pulse shaping network before being sent to the l o g i c unit. When i t i s known that the OHA and chopping wheel window w i l l e n t i r e l y overlap, a pulse with variable delay MEMORY ENABLE PUSH BUTTON TRIGGER PULSE OPTICAL SHAPING SENSOR I CHOPPING WHEEL [DELAYED TRIGGER OUT HV TRIGGER TO -*> OSCILLATOR FLASHLAMPS DELAY UNIT HV TRIGGER TO -•> AMPLIFIER FLASHLAMPS DELAY UNIT POCKELS -H CELL DRIVER OSCILLO-SCOPE TO POCKELS CELL PICK-UP COIL EXTERNAL TRIGGER IN FIGURE III - 6 Electrical system. 69 i s sent to various components. This pulse goes d i r e c t l y to the high voltage t r i g g e r unit f o r the o s c i l l a t o r flashlamps, through a delay generator to the tri g g e r unit for the amplifier flashlamps, and through another delay generator to t r i g g e r the Pockels c e l l d r i v er. A c o i l of wire around the c o a x i a l cable to the Pockels c e l l picks up enough sign a l to trigger the oscilloscope when the Pockels c e l l voltage i s lowered. Figure III-7 gives more d e t a i l of the l o g i c seguence and timing. A represents the chopping wheel window, and B, i t s prepulse. This prepulse t r i g g e r s two pulses of variable width: C i s a timing window, and D i s the delayed out timing. If the leading edge of the prepulse E from the OMA i s within the timing window C, then the acceptance time for the OMA and the chopping wheel w i l l overlap, and the lasers can be f i r e d . When E and C overlap, a pulse i s sent to the OMA memory enable G to i n d icate that the OMA i s to store the signals from i t s vidicon during the next read cycles. For pulsed l i g h t signals i t i s best to scan the vidicon and add the information into memory for several read cycles. To enable the OMA console controls to determine the number of read cycles, the memory enable pulse i s more than one second long. Only when memory enable G i s high and the delay timing pulse D i s lowered does a trigger pulse H go the other e l e c t r o n i c s . The widths and delays of the various pulses were determined i n the following manner. To ensure a minimal CHOPPING WHEEL WINDOW CHOPPING WHEEL PREPULSE TIMING WINDOW DELAYED TRIGGER OUT TIMING OMA PREPULSE OMA WINDOW MEMORY ENABLE DELAYED TRIGGER OUT OSCILLATOR FLASHLAMPS AMPLIFIER FLASHLAMPS RUBY LASER OUTPUT PULSE POCKELS CELL VOLTAGE 4 t i JL 4L JL -H FIGURE III - 7 Logic sequence and timing pulses. 71 wait u n t i l the two units are synchronized, the timing window T3 should be as wide as possible. I t i s limited in width by the width of the ready time T6 of the OHA. The OMA prepulse delay T5 i s fixed by the i t s e l e c t r o n i c s , and the chopping wheel window T1 i s fixed by the speed of the motor. T7, the time between the f i r i n g of the o s c i l l a t o r flashlamps and the laser i t s e l f , are fixed by the requirements of maximum output power and the capacitor bank discharqe time. S i m i l a r i l y , the delays for the o s c i l l a t o r flashlamps and Pockels c e l l triqqerinq, T8 and T9, are fixed. So the following times are determined by the requirements of the experiment: T1=18 T3=T6=760 T5=832 T7=T9=800 T8=600. A l l times are in microseconds. If the OMA prepulse occurs at the beginning of the timing window, then the end of the OMA window i s required to overlap the chopping wheel window. This insures that as the prepulse E overlaps the timing window C at l a t e r times, the machine windows w i l l s t i l l overlap, though at e a r l i e r times within the OMA window: T5+T6=T2+Tl/2 T2=1591. The delay TU must be adjusted so that the o s c i l l a t o r flashlamps capacitor bank has enough time to discharge before the laser i s f i r e d : T4*T9=T2*T1/2 TU=800. Now a l l the pulse widths and delays T1-T9 are defined. 73 Chapter IV DESCRIPTIONS AND DISCUSSION OF EXPERIMENTS In t h i s chapter the experiments are described and the r e s u l t s discussed. F i r s t a description of the plasma as determined by Thomson scattering diagnostics i s given. Then the general experimental conditions and procedures are presented, and the method of data reduction i s outlined. F i n a l l y the d e t a i l s of the experiments are presented as s e l l as a discussion of the r e s u l t s . A. Thomson Scattering Measurements of Electron Temperature and Density A ruby laser l i g h t scattering experiment vas performed to determine the electron density and temperature of the plasma to be used in the main experiments. The experimental arrangement was exactly as described i n Chapter I I I , except that the mirrors for diverting ruby laser l i g h t for pumping the dye lasers were not present. Further d e t a i l s of t h i s standard diagnostic method'* are not given. Figure IV-1 shows a t y p i c a l scattering spectrum of the normalized scattered l i g h t s i g n a l as detected by the photomultiplier as a function of wavelength s h i f t from the ruby laser l i g h t wavelength. A^ o b s(ld 3sec 1) 1 0 •1.5 I 2 0 • EXPERIMENTAL POINTS 1.0 { • THEORETICAL LEAST SQUARES FIT: OC = 1.03 No= 2.l6xio16cm3 Te = 19,500 °K NORMALIZED PM SIGNAL (arbitrary units) - A A o b s ( n m ) — -10 2.0 30 I 9 I 4£ 50 _ l _ FIGURE IV-1 Thomson scattering spectrum to determine electron temperature and density. 75 The error bars are the standard deviations of the mean of usually 9 , but at least 6 measurements. The s o l i d curve i s a least squares f i t of the experimental data to the theoretic a l scattering spectrum as derived by S a l p e t e r 3 7 , assuming a Maxwellian ve l o c i t y d i s t r i b u t i o n for the electrons. The theoretical, spectrum was convoluted with the monochromator transmission function. The parameters that were varied for the f i t were r e l a t i v e amplitude, electron temperature, and the cor r e l a t i o n parameter alpha. The electron density was then inferred using the f i t t e d values for temperature and alpha. As can be seen from the quality of the f i t , the plasma i s very well described by a Maxwellian electron velocity d i s t r i b u t i o n with a temperature of 19,500 °K, and a density of 2.16x10** cm - 3. Ho unusual features can be seen on t h i s spectrum, nor on other Thomson scatterinq spectra obtained at t h i s time. The electron temperature and density of the plasma i s i n the range of values as measured on si m i l a r plasma jets using laser s c a t t e r i n g 2 3 * 2 S and spectroscopic technigues 2 3> 2*. The f i t t e d values are i d e n t i c a l to those obtained by the author in a previous laser scattering experiment with a s i m i l a r plasma j e t 2 * . B. General Experimental Condtions And Procedure Table IV-1 l i s t s the experimental conditions which were held constant during a l l experiments. In t h i s 76 1. Ruby Laser C h a r a c t e r i s t i c s wavelength linewidth t o t a l output energy pulse FWHM t o t a l average output power diagnostic beam power beam diameter focused beam diameter focused i n t e n s i t y focused beam f-number 694.27 nm 0.07 nm or l e s s 3.6 J 20 nsec 180 MW 40 MW 8 mm .15 mm 2x10** ' B/on1 f/12.5 Dye Laser Ch a r a c t e r i s t i c s wavelength linewidth t o t a l output energy pulse FWHM average output power beam diameter focused beam diameter focused intensity focused beam f-number 810-820 nm .1 to .3 nm 10 0 mJ 15 nsec 8 HI 6 mm .2 mm 2x10 1 0 w/cm1" f/17 Other Parameters diagnostic scattering angle mixing angle l i g h t c o l l e c t i o n f-number He flow rate anode-cathode distance anode-mixing region distance 110 degrees 150.7 degrees f/12.5 8x10- 3 1 sec-* 9. 5 mm 1.8 mm TABLE IV-1 Parameters of the experiment held constant. 77 table, the e f f e c t i v e size of the f o c a l spots are assumed egual to the area A of the damage spot on the negative film used i n the alignment. The power densities I are calculated using the time width T of the pulses, the t o t a l energy E, and the area: I = (E/TxA) The same general procedure i s used for a l l the the mixing experiments. The spectral lineshapes of the dye lasers are monitored on every shot, and the ruby laser pumping beams adjusted to maintain reasonably symmetric lineshapes. The wavelength and i n t e n s i t y at the l i n e centre, and the t o t a l integrated i n t e n s i t y are recorded, using the d i g i t a l information from the OHA. The ruby laser and dye laser output power i s monitored by photodiodes on every shot, and recorded by photographing the oscilloscope trace. The mixing experiments require that the scattered l i g h t signals be determined with and without the dye laser beams i n the int e r a c t i o n region. To accomplish these measurements, a pair of shots are made with and without the mixing beams. The parameter being investigated (for instance the mixing frequency) i s changed to a new value, and another pair of shots obtained. The ent i r e range of the parameter i s covered in th i s stepping manner. The parameter i s then reset to i t s o r i g i n a l value, and i t s range covered several more times. This procedure ensures that a l l values of the parameter are covered evenly, and that any changes i n experimental 78 conditions w i l l be e a s i l y noted. Each pair of shots, including t h e i r associated recording of OMA data and laser cooling time, require an average of 7 minutes. The e x i t s l i t of the monochromator was set between 300 and 600 microns wide, depending on the spectral resolution desired. The narrower s l i t gives a triangular instrument p r o f i l e with PHHM of 0.3 nm. The wider s l i t gives a trapezoidal instrument p r o f i l e with 0.6 nm PWHM. The oscillograms are analyzed as described i n the following section. Typical oscillograms for normal and mixing shots are i l l u s t r a t e d i n Figure IV-2. These oscillograms correspond to the experimental conditions for Figure IV, at a wavelength s h i f t of -3.05 nm. C. Data Reduction An outline of the measurement methods for the oscillograms and the evaluation of the numbers obtained i s presented i n t h i s section. A detailed description of the methods and the i r j u s t i f i c a t i o n are given i n Appendix B. ; The data a c g u i s i t i o n rate for the mixing experiments presented i n t h i s t h e s i s i s unfortunately rather low. For t h i s reason i t i s imperative that as much and as accurate information as possible be extracted from the oscillograms. Different analysis techniques were t r i e d using the data from the experiment described i n Section D of t h i s chapter. I t was concluded that the best 79 MONITORS: z> or uj or to UJ < CO CD LU or U CO < UJ c e UJ - I CL h -- j P o CL FIGURE IV-2 T y p i c a l Ossci l lograms. 80 analysis method i s to d i g i t i z e the oscillograms; that i s , to measure the height of the signals as a function of time (distance) along the oscillogram. numerical methods can then be used to evaluate the parameters of the mixing experiments, employing integration of the signals over time. i s related to the ruby laser power I d ( t ) and the dye laser powers I-| (t) and I 2 (t) through the parameters S1 and S 2 by the formula (see eguation (26)): S1 and S? can change as some parameter of the experiment cross section and I 1 x I 2 x S 2 i s the mixing scattering cross section.) Estimates for the parameters are made using the formula: The photomultiplier s i g n a l at t i n e t (I 0bs (*)) W* 1 = S l Td ( t ) + S2 Xd ( t ) X l ( t ) J 2 ^ (35) i s changed. (Recall that S i s the normal scattering s 1=1 1 N i = l ...(36) S 2 M I [ / l d l j l 2 d t ] 2 (37) 81 where there are N normal scattering shots and H mixing shots, and the superscripts re f e r to the shot number. These formula are least squares estimates of the parameters S1 and S 2 using egual weighting for a l l points, as described in Appendix B. To evaluate the i n t e g r a l of the t r i p l e product I dxl 1xl 2» i t i s necessary to determine the position on the oscillograms that correspond to the same time for the 3 different signals. The method used to determine these r e l a t i v e time o r i g i n s i s described in Appendix B. The standard deviation of the i n d i v i d u a l estimates of the parameter s 1 i s reduced by 2 as compared to the more usual method of measuring the heights of the signals alone. This means that i t requires <t times fewer shots to obtain the same accuracy for the mean value of S1 i f one d i g i t i z e s the oscillograms rather than measure the peaks. We therefore have compensated for the f a c t that we are only able to c o l l e c t an average of 8 shots for the mixing and normal scattering combined at each parameter setting. D. Dependence Of Mixing Signal On Mixing Power It i s obvious that the amount of l i g h t scattered due to l i g h t mixing i n the plasma w i l l depend on the diagnostic beam in t e n s i t y as well as the in t e n s i t y of the dye laser mixing beams. This experiment was performed to v e r i f y the r e l a t i o n s h i p between the laser i n t e n s i t i e s and 82 the l i g h t detected by the photomultiplier predicted by theory. Determining the scattered s i g n a l dependence on power i s i n fact e s s e n t i a l i f we are to proceed further. When other parameters of the experiment are investigated, we are required to compensate for the variat i o n s i n the detected si g n a l due to changes i n scattering and mixing powers. If the lasers gave more consistent output power, t h i s correction would not be necessary. The width of the exit s l i t of the monochromator was adjusted to 600 microns. The centre of i t s transmission function was set to correspond to a wavelength s h i f t of -2.87 nm from the ruby laser l i g h t wavelength. The dye laser whose o p t i c a l axis corresponds most closely to the ruby laser axis was set to 820 nm. To set the mixing frequency to the same value as the observation freguency, the second dye laser was tuned to a wavelength of 816.1 nm. An absorption c e l l was placed i n the path of the pumping beam for the o s c i l l a t o r of the second dye laser. The c e l l contained cryptocyanine dissolved i n d i s t i l l e d water. To vary the output power of t h i s second dye las e r , three d i f f e r e n t concentrations of cryptocyanine solution were used. Also the i r i s ^ of this laser were changes from 8 to 6 mm diameter. This varies the area of the output beam, and therefor the t o t a l output power. Changing the i r i s apertures also changed the linewidth of t h i s dye l a s e r , as i s recorded using the OHA monitor. A 83 series of oscillograms vere taken, alternating one normal scattering shot with two mixing shots. The normal scattering cross section, measured by S 1 , i s f i r s t evaluated using eguation (36). The l e a s t squares estimate of S 1 i s 1.53*. 03. The standard deviation of the points themselves i s approximately 10%. The mixing signals can now be evaluated. The PH signal i s a sum of the normal scattering signal and the mixing s i g n a l : I = S I, + S 1,1,1, obs 1 d s d 1 2 ... (38) To v e r i f y t h i s formula we f i r s t must subtract the normal signal from the t o t a l P H s i g n a l and compare t h i s with the t r i p l e product. This gives a value for S 2 of 1.01±.09. The data i s represented on two separate graphs. The f i r s t graph. Figure IV-3, i l l u s t r a t e s the t o t a l PM sig n a l for both the normal scattering and the mixing shots as a function of the calculated s i g n a l using equation (38) above and the f i t t e d values for and S 2 . The data should f i t the s o l i d curve whose slope i s unity and intercept i s the o r i q i n . This qraph has several important features. The f i r s t of course i s that the data f i t the equation above quite well. No trend away from the straight l i n e can be seen, although the fluctuations of the data are guite large. Another feature i s that the fluctuations i n the 8 4 FIGURE IV-3 Total PM signal compared to the fi t t e d values. 85 t o t a l PM signal for the mixing case follows the trend of the normal scattering signals. The dotted l i n e s encompass a l l the data points which correspond to normal scattering alone. The majority of the points for the mixing shots also f a l l within these l i n e s . The l a s t feature i s that the increase i n signal due to mixing i s quite s i g n i f i c a n t . The largest PM signals for mixing are more than twice as large as the signal for normal scattering alone, with approximately the same diagnostic beam i n t e n s i t y . This corresponds very well with the c a l c u l a t i o n of the r e l a t i v e magnitude of the mixing and normal scattering signals in Chapter I I . The calculations there indicate a r a t i o of mixing s i g n a l to normal sig n a l of 4:1. Making a correction for the wider instrument p r o f i l e used here gives a r a t i o of 3 :1, as compared to the actual value of 1:1. The s l i g h t l y lower experimental value may be due partly to the non-Gaussian laser beams, as discussed i n Chapter I I . However, anticipating the r e s u l t s of the measurement of the response of the plasma to d i f f e r e n t d r i v i n g freguencies, the main reason f o r the low experimental r a t i o i s the choice of mixing frequency. The s i g n a l at s l i g h t l y different mixing freguencies can be expected to be a factor of 3 to 4 higher than at the freguency f o r t h i s experiment. This would make the r a t i o of mixing signal to normal sig n a l approximately 4:1i Considering the possible error i n the mixing and scattering volumes used i n the ca c l u l a t i o n of the expected s i g n a l of 4031 or more, t h i s 86 agreement i s excellent. A second method of i l l u s t r a t i n g the data i s to plot the PH si g n a l due to mixing alone as a function of the t r i p l e product, i . e . I O D S - S1 x l d versus I^xI-jXl^. This graph, Figure IV-4, should stand on i t s own as a good f i t to a straight l i n e through the o r i g i n . The o r i g i n intercept can only be assumed i f the value of S 1 i s accurately known, as i s the case. However, unlike a graph of normal scattering signal versus ruby laser s i g n a l , some data points can (and do) become negative. The s o l i d curve i s a l e a s t sguares f i t through the o r i g i n , and the dashed curve i s a lea s t sguares f i t for a straight l i n e with an intercept. These two f i t t e d l i n e s are v i r t u a l l y i d e n t i c a l . This diagram graphically i l l u s t r a t e s how the variations in the s i g n a l due to mixing alone are compounded by the normal scattering s i g n a l variations. The standard deviation of the estimate for S 2 i s just under 9%, but the standard deviation of the points themselves i s approximately 50%. The v a l i d i t y discussion of Chapter I I indicated that the ef f e c t of the non-Gaussian character of the mixing laser beams would only r e s u l t i n possible reduction of mixing e f f e c t and increase in s i g n a l error. The r e l a t i v e s i g n a l s i z e s indicate very l i t t l e reduction, and the increase in the scatter of points for the mixing i s small. Apparently the d i f f e r e n t i n t e n s i t y structure of the beams has had l i t t l e e f f e c t . 88 E . Spectrum Of The Induced Fluctuations As predicted by equation (13), the spectrum of the density fluctuations driven by the l i g h t mixing should be of the form where W(Ao) i s a double convolution of the normalized frequency d i s t r i b u t i o n of the l i g h t sources driving the waves with a lorentzian of FWHH=4Y, and R(AO/Ak) i s the response function of the plasma. This experiment was performed to v e r i f y that the mixing spectrum i s determined by the dye laser spectra in t h i s manner. The dye lasers were fix e d at freguencies si m i l a r to those i n the previous experiment, corresponding to 820.75 and 816.15 nm. The expected maximum in the spectrum of the induced spectral density function i s at a wavelength s h i f t of -2.81 nm from the ruby laser l i n e . The scattering l i g h t signal was observed at d i f f e r e n t monochromator settings about t h i s freguency s h i f t (AX=-2.79 nm). The scattered l i g h t signal due to mixing ( I m j x ) then gives the spectrum of the induced wave, convoluted with the instrument transmission function T w, f o r these dye laser spectra (from equation ,(Ak,Au>)|2> a R(Aco/Ak)W(Aio) (39) (26)) : I mix T (Aio - Au> ) R(Aio/Ak) W(Aco)du) ,(40) 89 To obtain some resolution of the mixing spectrum, the width of the instrument transmission function should not be wider than the spectrum being observed. In general, the smaller the width i s , the higher the resolution and the smaller the s i g n a l . In our case, since the width of the entrance s l i t of the monochromator i s fixed to define the scattering volume, we can optimize resolution and s i g n a l strength by setting the exit s l i t to the same width. The transmission function then i s 0.36 nm wide, about egual to the expected width of the induced spectrum. The monochromator wavelength setting was changed in 0.2 nm increments from 690.77 to 692.27 nm. Between 4 and 8 shots, with an average of 6, were made at each setting for mixing, with s i m i l a r s t a t i s t i c s for normal scattering. The oscillograms were analyzed as previously described with the resulting spectrum shown in Figure IV-5. Since the normal scattering spectrum changes by le s s than 2% over t h i s wavelength region,the normal scattering cross section (S1 ) was assumed to be constant. An average value of S1 was used when determining the mixing s i g n a l . The scattering signals at different wavelengths had a standard deviation of ±7??, ind i c a t i n g that any r e a l change in S1 of a few percent i s n e g l i g i b l e . The standard deviation of the mean of S1 i s 2%. As usual the spectra of the dye lasers were recorded on every mixing shot. This allows a d i r e c t comparison of the obtained spectrum with the spectrum (J -I0 m i x (10 usec' ' )-1.0 1O.8 INTENSITY (arb i t rary units) EXPERIMENTAL POINTS THEORETICAL FIT 10.6 l0.4 I 1 INSTRUMENT WIDTH 10.2 A-Amix(nm) •0.4 -0.2 0.0 0.2 0.4 Nri.6 FIGURE IV-5 Spectrum of the induced wave. 91 predicted by eguation (40). The FWHN of the spectral l i n e s , for a single shot, were 0.19 to 0.22 and 0.10 to 0.12 nm for the two dye lasers. Single shot spectra are shown in Figure III-5. The centre maximum and the width of the dye laser p r o f i l e s change from shot to shot by ±1 OMA channels from their average values. This has the effect of increasing the e f f e c t i v e width of the p r o f i l e s for the entire run. Six representative spectra, approximately every 10th shot, were chosen as t y p i c a l p r o f i l e s for the run. These spectral l i n e s , with 0.13 and 0.23 nm FWHM are i l l u s t r a t e d in Figure IV-6. The Lorentzian and Gaussian l i n e p r o f i l e comparisons i n t h i s figure, least squares f i t s for amplitude and width, show that neither adequately describe the assymetric experimental p r o f i l e s . Using the re s u l t s of appendix A and the measured instrument transmission function, equation (40) was evaluated numerically to qive the s o l i d curve i n Fiqure IV-5. The amplitude of the t h e o r e t i c a l curve r e l a t i v e to the experimental points was determined using a least squares f i t with weights equal to the inverse of the variance of the experimental points. The instrument width i s also indicated on the diagram. We again have excellent agreement between theory and. experiment. Our instrument width i s not s u f f i c i e n t l y narrow to make out fin e structure of the induced spectrum, but the widths and general shape match very well. 92 FIGURE IV-6 Spectral line profiles (Average of several shots). 93 F. The Response Function Of The Plasma The most important aspect of the derivation of the induced fluctuation i s the function R(Ak#Aw), the response of the plasma to d i f f e r e n t d r i v i n g freguencies and wave vectors. The experiment described here i s designed to evaluate the response function. H which assumes Lorentzian p r o f i l e mixing beams, we have that the mixing s i g n a l received i s The detection system freguency transmission function T^> i s peaked at AO=ACd0bS , and the Lorentzian s t y l e denominator i s peaked at AQ = ^ 1 0 - w 2 0 « Thus to get a reasonable s i z e signal due to mixing, the detector must be tuned to &*bb^wio~°°20' T h e expected width of the response function of the plasma i s much larger than the width of the transmission function and the Lorentzian denominator. In this case, the mixing s i g n a l at A6J0bS=k)10-6()2o i s a measure of the response of the plasma to the d r i v i n g frequency AtoODS. I f a dye laser i s tuned to d i f f e r e n t frequencies, and i s 'followed* by the detection system, keeping ^^ibs^^lO -^20 * ¥ e have a d i r e c t measure of the response function f o r d i f f e r e n t d r i v i n g frequencies. The expected response function i s more than 4 nm wide, so we have qood resolution using a 0.3 nm wide transmission function. Recalling equation (26), and using the form for 94 The e x i t s l i t of the monochromator was set for the triangular instrument p r o f i l e with 0.3 nm FHHH. The frequency difference between the monochromator setting and the ruby l i n e , and between the dye lasers, were changed in increments of 1.17x10** see—*, from 2.74x10*2 to 2.52x10*3 sec—*. This corresponds to 0.3 nm steps from 693.55 to 687.85 nm. Between 3 and 7, with an average of 3.9, and between 3 and 8, with an average of 3.6 shots were recorded f o r the mixing and normal scattering, respectively. The spectra of the dye lasers were monitored on every mixing shot to ensure consistent lineshape and freguency. The dye and ruby laser output powers were recorded for normalization purposes as usual. The oscillograms were analyzed as described e a r l i e r , except that the f i t t e d values of the thermal signals rather than the experimental values were used when calc u l a t i n g the mixing sign a l s . The normal shots constitute a laser scattering diagnostic experiment by themselves. That i s , the spectrum of the scattering from the thermal fl u c t u a t i o n s as a function of frequency gives the electron temperature and density f o r t h i s p a r t i c u l a r run. The normal scattering data are presented in Figure IV-7. The s o l i d l i n e i s the t h e o r e t i c a l least squares f i t using the method described i n section A. There are two important features of t h i s spectrum. F i r s t , the l e a s t squares values for the 1.0 2*0 i1.5 EXPERIMENTAL POINTS FIGURE IV-7 Thomson scattering spectrum for mixing experiment. Dye laser beams blocked off. 96 temperature (19200 «>K) and density (2.13s 10** m-3) are exactly the same for the e a r l i e r diagnostic run in section A. This again i l l u s t r a t e s the r e p r o d u c i b i l i t y of the plasma. The second feature i s the few points that are more than one standard deviation away from the t h e o r e t i c a l curve. On a purely s t a t i s t i c a l basis one expects approximately 1/3 of the data points to l i e one standard deviation from the expected curve. However, the variations i n s i g n a l may be r e a l . This i s indicated by the smooth change from above the f i t t e d curve to below i t . Also, the frequencies of the sharp t r a n s i t i o n s at AX=-3.0 and -1.5 nm are harmonics of each other. The plasma freguency wavelength s h i f t has been indicated on the graph. When analyzing the data to obtain the response function of the plasma, i t i s necessary to subtract the thermal scattering signal from the t o t a l s i g n a l . The estimate for the thermal s i g n a l ( S ^ for a p a r t i c u l a r freguency can be found from the few shots at that one freguency, but a better estimate i s to choose the value of the t h e o r e t i c a l f i t for that frequency. In t h i s way the data from the entire run i s used, r e s u l t i n g i n a more accurate estimate of the mean value of the signal at any one frequency. The f i t t e d values for S-, were therefore used. The r e s u l t i n q spectrum of the response function of the plasma as a function of frequency i s displayed in Figure IV - 8 . This qraph obviously departs v i o l e n t l y from A 0) r t h c(10 1W 1) 0D8 2.0 E X P E R I M E N T A L POINTS LINE CONNECTING DATA POINTS THEORETICAL CURVE 0t=tO3 T e = 1 S £ 0 0 " K N„=2.13 10 1 6 cm i 1,0 4,0 FIGURE IV-8 Measured response of the plasma to different driving frequencies. 98 the smooth spectrum described i n Chapter I I . The broken l i n e i s drawn as a (suggestive) aid to the eye. The previous two experiments were done with the mixing freguency corresponding to the region of reduced s i g n a l around AX=-2.8 nm. There are certain trends to t h i s spectrum. The peak at AX=-3.15 nm has a signal approximately four times the maximum normal scattering s i g n a l . This corresponds closely to the expected s i g n a l due to the induced fluctuations, so one would expect that the valleys i n the spectrum are the anomalies rather than the peaks. This idea i s also backed by the fact that the theoreti c a l response function, calculated using the experimental values for the temperature and density, and normalized to the l a s t six data points, f i t s the envelope formed by the maximum values very well. This i s indicated by the s o l i d l i n e in Figure IV-8. A second feature has to do with the position of the valleys. There are d e f i n i t e valleys at AX=-2.85 and -«.35 nm, but the reduced signals at A\=-1.65 and -1.05 nm may not be s t a t i s t i c a l l y s i g n i f i c a n t . I f we assume a l l four are valley s , then t h e i r positions may be related i n a systematic way. Because of the speculative nature of these observations, further discussion i s r e s t r i c t e d to Appendix D. Another trend i n the spectrum of the response function i s the co r r e l a t i o n of the discontinuity around AX=-3.0 nm to the s i m i l a r one at the same position i n the 99 thermal scattering spectrum. This may indicate that the anomalous scattering s i g n a l i s enhanced by the e f f e c t of the wave mixing. The s i m i l a r discontinuity for the normal scattering spectrum at 1/2 the frequency s h i f t does not have such a c o r r e l a t i o n . It might be suggested that the large variations in the spectrum are at least partly due to using the f i t t e d thermal signals rather than the actual experimental signals. If the actual signals are subtracted from the t o t a l PH s i g n a l to the give the mixing signal, then we obtain a spectrum with the same features as before. The drop i n signal at ZiX=-1.65 nm i s increased, but the others remain approximately the same. The small change i s to be expected, since the mixing signal i s more than 30 times larger than the variations of the thermal signals from the f i t t e d curve. This spectrum i s not unique. For instance, an experimental run with a smaller anode-cathode distance produced a plasma with f i t t e d temperature and density of 21000 °K and 1.1x10»* cm - 3. The response spectrum of the plasma i s again sharply modulated. However, there i s no readi l y seen rela t i o n s h i p between the positions of the valleys as i s suggested i n Appendix D. There are two general reasons for the unexpected modulations i n the response function of the plasma. F i r s t , our physical model may be wrong or incomplete. Second, there i s possibly a phenomenon not related d i r e c t l y to the wave mixing which does not allow the waves 1 0 0 to develop to the expected amplitude. Before i t may be concluded that the model i s incorr e c t , a l l p o s s i b i l i t i e s which come under the second category must be exhausted. There are two possible methods through which the energy from the mixing beams w i l l not produce the proper amplitude waves. F i r s t , the mixing beams may be attenuated at freguencies which correspond to the valleys. Second, the mixing beams may produce the expected mixing waves, but they are damped by some process i n the plasma. If the mixing beams are attenuated at s p e c i f i c freguencies, the driv i n g force would be reduced to below the expected value. Then the calculated i n t e g r a l s of the t r i p l e product of the laser powers would be larger than their actual value. This lowers the estimate of S 2, creating a •valley*. However, data i s available which excludes the p o s s i b i l i t y of the dye lasers being attenuated at the valleys and not at the peaks. The dye laser photodiode monitors placed before the laser beams are focused into the plasma measure the dye laser power as a function of time, integrated over a l l freguencies. The OMA monitor measures the dye laser powers as a function of freguency, integrated over time. The OHA l i g h t c o l l e c t i n g funnels are placed after the dye lasers pass through the interaction region. He therefore have a measure of the dye laser powers before and a f t e r the i n t e r a c t i o n region simply by integrating the OMA spectra with respect to freguency, and the photodiode signals with respect to time. The integral of the l i n e 101 p r o f i l e s were recorded daring the actual run using b u i l t in OHA functions. The time i n t e g r a l s are done using the d i g i t i z e d oscillograms and the computer programs written for integrating normal scattering signals. The data i s presented i n Figure 17-9 and IV-10, where the s i g n a l a f t e r the interaction region i s plotted as a function of the signal before. Figure IV-9 compares the data for the f i x e d freguency dye laser f o r the three data points around AX=-2.55 nm to the data for the adjacent peaks (-19.5, -3.15, and -3.45 nm) . Figure IV-10 i l l u s t r a t e s the same information f o r the variable freguency dye laser. The error bars are the estimated error i n measuring the area under the oscillogram traces, determined i n Appendix B. The s o l i d curves are least sguare f i t s through the o r i g i n for the signals at the valleys, and the broken l i n e for the peaks. These graphs show that there i s no s t a t i s t i c a l l y s i g n i f i c a n t absorption of the dye laser beams as t h e i r freguency difference i s tuned from the peak to the valley. Be can exclude freguency dependent attenuation of the beams as a cause of the modulations. Possible damping of the driven waves cannot be so e a s i l y ruled out. The v a l i d i t y discussions i n Chapter II indicate that Landau damping i s the dominant damping mechanism, but there are other possible mechanisms not yet considered. An example i s strong coupling of the mixing waves to other natural modes of the plasma. If s u f f i c i e n t energy i s transferred to other waves, then the scattered 102 FIGURE IV-9 Dye laser power before and after the mixing region: the fixed frequency laser. 103 FIGURE IV-10 Dye laser power before and after the mixing region: the variable frequency laser. 10*1 l i g h t s i g n a l could be reduced. However, the following considerations suggest that nonlinear mode coupling i s not the explanation. For the mixing waves with freguencies and wave vector [A&fAk) to be coupled to other waves ( A ^ ,41^) , a thi r d wave {AoQt^Q) must already be present to s a t i s f y the momentum and energy conservation requirements: Aw. = Aw ± Aw 1 o ^ 1 = ^ * ^ 6 ...(42) (See for instance Sagdeev and G a l e e v 3 8 ) . The existing waves (Aci>Q,A]CQ) must be normal modes of the plasma, as must the waves produced (Aw^A^^. Because the driven waves have a large wave vector, either 4£-| or AkQ must also be large. Density waves with large wave vectors are not usually normal modes of the plasma, so the requirement that both the e x i s t i n q and coupled waves be normal modes i s not s a t i s f i e d . I f there i s only one wave (AO0,4kg) already exis t i n g , then a l l the produced waves w i l l have d i f f e r e n t frequencies, but i d e n t i c a l wave vectors. A cl a s s of waves with such a dispersion r e l a t i o n i s improbable, so that one must conclude that there i s a set of waves present for the driven waves to interact with. The amplitude dependent energy transfer must be from high frequency waves to lower frequencies. Therefore, f o r the enerqy transfer to be e f f i c i e n t , the frequency of the exis t i n g and produced waves must be lower 105 than the nixing waves, and the amplitude of the existing wave must be considerable. Laser scattering experiments performed at at l e a s t f i v e d i f f e r e n t wave vectors by di f f e r e n t authors have not observed large amplitude waves present. It i s concluded that neither attenuation of the mixing beams, nor nonlinear coupling of the driven waves to other waves can explain the modulations of the response function. However, we cannot choose between damping and an incomplete model as the cause of the unexpected modulations. Other damping mechanisms must be considered, and further t h e o r e t i c a l work i s required before we a make a d e f i n i t e conclusion. In the above analysis, only possible anomalous behavior r e s u l t i n q from the interaction of the mixing beams and the electrons and ions has been considered. I t i s also possible, however, for the laser beams to i n t e r a c t with the neutrals i n the plasma. This i n t e r t a c t i o n would be at d i s t i n c t frequencies because of the atomic energy l e v e l structure of the atoms. The small cross-sections for these interactions may be compensated by the very high density of the neutral atoms (~10 4 8 cm - 3). Such interactions cannot be predicted by the simple model which has been presented. However, one would expect that such interactions would produce modulations i n the scattered l i g h t s i g n a l which are not symmetric with respect to the diagnostic laser frequency. A measurement of the symmetry of the response function w i l l i n d i cate i f 106 laser-neutral interactions i s a possible cause of the response function modulations. 107 Chapter ? CONCLUSIONS AND SUGGESTIONS The e f f e c t of o p t i c a l mixing of two tunable dye lasers at freguencies near the plasma freguency has been experimentally investigated. I t has been shown that the wave mixing produces longitudinal plasma o s c i l l a t i o n s at the freguency and wave vector of the mixing force. The driven waves were detected by scattering a t h i r d diagnostic l i g h t wave from th e i r density fluctuation. The wave mixing experiments presented i n t h i s thesis reguire a very r e l i a b l e plasma source because of the long operation times. With the improvements which I have made to the electrode design, as well as my addition of the current regulated power supply and battery charging unit, the plasma j e t can now be used for the length of time required. I have been able to produce an increase in scatterinq s i q n a l due to the o p t i c a l mixing an order of magnitude more than previously reported. The scattering signals increased to as much as seven times the s i g n a l observed when scattering from the thermal f l u c t u a t i o n s alone. I have made the f i r s t measurements of the dependence of the amplitude of the induced spectral density on the power of the o p t i c a l mixing beams. This 108 dependence agrees well with theory, even though the dye lasers were not plane Gaussian waves as assumed i n the derivation. The power dependence was used t o normalize the signals i n the subseguent experiments. The freguency spread of the mixing beams produces a spectrum of induced waves. The accuracy of my measurement of the spectrum of the induced spectral density function i s a factor of two better than previous measurements. The experimental spectrum agrees very well with the t h e o r e t i c a l p r o f i l e convoluted with the monochromator freguency transmission function. I have made the f i r s t measurements of the response of the plasma to different mixing freguencies. This spectrum departs r a d i c a l l y from the t h e o r e t i c a l predictions. The spectrum i s modulated, with changes in mixing s i g n a l by as much as a factor of four at adjacent measurement points. The envelope formed by the peaks of the spectrum f i t the t h e o r e t i c a l curve well. This good agreement, combined with the absolute value of the maximum signal being close to the expected value, indicates that the valleys i n the spectrum are anomalies rather than the peaks. I have shown that nonlinear mode coupling and freguency dependent attenuation of the mixing laser beams are probably not the cause of the modulations i n the response spectrum. They are perhaps due to an unknown damping mechanism, or perhaps to an incomplete physical model. No d e f i n i t e conclusion can be made at t h i s time. &s happens frequently, these experiments have 109 raised questions while answering others. I t has been shown that the wave mixing produces plasma waves of appreciable size and known s p e c t r a l content. This method of producing waves can be used to study wave-wave and wave-particle interactions, and perhaps wave mixing w i l l become a valuable t o o l for plasma physicists. However, further work must be done to understand completely the response of the plasma to different driving frequencies. There are some improvements to t h i s experiment which I strongly recommend. The f i r s t i s to improve the data acguisition rate. This can be accomplished by using higher r e p e t i t i o n rate l a s e r systems, and an 'automatic* multi-channel detection system. The techniques of usinq o p t i c a l multi-channel detectors with analoque to d i g i t a l conversion and recording of data d i r e c t l y onto magnetic tape are becoming quite standard, and would be invaluable to t h i s experiment. The second improvement would be to change the method of detection of the induced waves. Presently the amplitude of the waves i s small, so that the scattering cross section i s only a few times the thermal value. The signals can be d i f f i c u l t to detect and evaluate, and alignment i s c r i t i c a l . A possible d i f f e r e n t detection method i s to look at the coupling between allowed and forbidden components of spectral l i n e s . The e l e c t r i c f i e l d s associated with the driven electron waves should couple these spectral l i n e s through the A.C. Stark e f f e c t . Baranger and Mozer" have shown that plasma waves 110 should produce s a t e l l i t e s around forbidden coaponents, and Hicks* 0 that harmonics are also possible. Ringler** detected such s a t e l l i t e s caused by waves which appear naturally i n the plasma. The e l e c t r i c f i e l d strength of these waves were the same order of magnitude as waves driven here. If Thomson scattering i s s t i l l used, then recent developments i n commercially available dye lasers could be exploited. Flashtube pumped dye laser o s c i l l a t o r -amplifier systems are now available with ten's of megawatts output power, 0.1 nm linewidth, 500 nsec FHHH pulses. The wave mixing could be done using a ruby laser and a dye las e r , the diagnostics performed with a second dye laser (System 3 i n Table II-1). Such a system has many advantages. The guantua e f f i c i e n c y of the detectors at the wavelength of the diagnostic laser are high. The ruby laser could improve the mixing by orders of magnitude over the presently attained l e v e l . The normalization of the s i g n a l to the laser powers can be done easier, since the i n t e g r a l of the product of the l a s e r powers i s just the product of the dye laser powers times the i n t e g r a l of the ruby laser pulse. (The much longer dye l a s e r pulses can be considered constant as compared to the diagnostic pulse.) This l a s t i n t e g r a l and the multiplication can be done e l e c t r o n i c a l l y . There are many possible future experiments. These include the ones already mentioned, namely the investigation of the response function of the plasma and 111 the Baranger-Hozer e f f e c t . Other experiments include a dir e c t measurement of the Landau damping rate. I f the mixing i s done at s u f f i c i e n t l y small angles, the damping rate of the driven vaves w i l l be low enough that they can propagate an appreciable distance from the inte r a c t i o n region. The amplitude of the wave as a function of distance would give the damping rate. This experiment would require a plasma with much lower c o l l i s i o n rates than the plasma j e t . Of course, the present experiment could (and should) be performed using changed geometry to study the waves of di f f e r e n t wave vector. 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Modern Physics 5, 257 (1933). s 115 Appendix A DOOBLE CONVOLUTION OF A LORENTZIAN WITH TWO ARBITRARY LIME PROFILES We have, in the t h e o r e t i c a l derivation of the spectral density <|Njnd (4Jt,AD)|*> (equation (13)), the double i n t e g r a l I: CO Z 4Y 2 + (a) - 0)1 + «o2)S I(u) = 2y 't 4 Y 2 + (a) - O J 1 + «o2)2 ...(A1) 7 i s a constant and Pj (£jj) i s the spectral d i s t r i b u t i o n of the i t h nixing l i g h t source normalized to unity: oo / (A2) I f the p r o f i l e s can be described by Lorentzian with FWHM FWj and l i n e centres &)jo: w = — • 1  2tt (J5FW,)2 + (u> - m )2 I i io ...(A3) then the in t e g r a l I{U>) simply reduces to 116 IO) = f OsFw)2 + (to - cu 1 0 + o ) 2 0 ) 2 ...IkH) where FH=47 • Fw1+Fw2. In general, however, we do not have Lorentzian l i n e p r o f i l e s . I f the l i n e p r o f i l e s are measured as a function of freguency, then by l i n e a r l y i n t e r p o l a t i o n between the measured points we have a numerical description of the l i n e p r o f i l e s : W = 0 u i ± c i , i W = a i k u i + b i k c i , k i"± - ^ c i , k + i (u,) = 0 0), > CJ 1 1 i — i,n (»5) The a j k and b j k are determined by the measured values Pj <6?j) at 6>j = C j k , k=1,n: ik ci,k+r c i , k -,k • V " - ci, k> - a l f k C i ) 1 .. .<A6) Using t h i s l i n e a r a n a l y t i c a l expression for the Pj '{(0\) , i t i s possible to integrate I[o) ) numerically. However, we must eventually c a l c u l a t e the observed quantity which i s proportional to the convolution of t h i s double i n t e g r a l with the monochromator transmission function and the response function of the plasma: 117 00 Imix C A a )obs ) a I TCAco - Ato^) RCAto) I(CO) dto ...(A7) Performing t h i s t r i p l e i n t e g r a l numerically with standard integration routines available.to users of the IBM 360-67 computer at U.B.C. i s p r o h i b i t i v e l y expensive. Thus, the in t e g r a l I{OJ) should be evaluated a n a l y t i c a l l y . This i s done below. Substituting the li n e a r expressions for the l i n e p r o f i l e s (eg. 5) into l(co) gives : n-1 m-1 ~2,k+l I (to) = 2y r2,k+l ^l,k+l J J d u i °2,i X k j l ( a2,i"2 + b2.i ) ( al.k t°l + b l . k > 4y2 + (JO - o>1 + to 2 ) 2 . . . (A8) where there are m and n points describing the p r o f i l e s P-| and P 2 respectively. Osing the substitutions x i = X 2Y = a 2 , i a l , k u = (0 2Y Q i , k • = a 2 , i e i , k cx. . = S 2 Y a i , J Y k " = a l , k e 2 , i \i • = b i , 3 = e 2 , i P l , k sui -• C±,1 2Y .. . (A9) we have 118 n-1 m-1 i=l k=l 6 2,i+l 6l,k+l i,k . 2 X 1 L 1 + (u - x + x.)2 J U 0 4 1 1_ J- ^ 2 , i w l , k ... (A 10) Also define 1* i,k I R i,k ff d x l d x 2 X l X 2 = J J 1 + (u - X ; L + x 2) ff d X l d X 2 X 2 , " // 1 + (u - * 1 + x 2)' /T d x i d x 2 x i 2 JJ 1 + (u - X ; L + x 2) /if d x l d X 2 , (A 11) Then I i , k = V k ^ . k + ^.k^.k + ^.k^.k + S i , k I l , k ... (A12) The i n t e g r a l with respect to X| i s done f i r s t , using the transformation X1 = -Z • (u+X 2). The X1 i n t e g r a l range i s now Z k Q t o Z k l where The integration with respect to X 2 i s then performed using the transformation X 2 = Z-u + &|k+£» The i n t e g r a l range now 119 i s Z K° to Z'k^  where j. HI = n " 6l,k+A + 62,i4m (A 1*0 The d e t a i l s of these straight forward i n t e g r a l s are not given. Define A m - ^ ^ n i,o ( Z £ ) n l n [ l + (ZJE)2]dZ£ i,o Zk,£ . . . (A 15) Then I R £=o 1 0 = * £=o 1 1 j L k - £ ( - 1 ) £ + 1 [ ^ i , k , , + ' Al,k,£ " ( 2 6i,k+Jl " U ) Al,k,£ ~ 6l,k+£ ( 6l > k+Jl " U ) Al,k,£ ] ,k+£ " u ) Li,k,£ ... (A 16) The standard i n t e g r a l s A" k^ and are: 120 Li\k,Jl = ^ ^ ( - D ^ t l n C l + Z2) + Z 2ln(l + Z2) - Z2] m=o i > k > £ = ^ ("D [ a n d + Z2) - 2Z + 2 arctg(Z)] m=o Al,k,A = 3 ^  (-l) m + 1[Z 3arctg(z) - hZz + %l n ( l + Z 2)] m=o i,k,* = (-1) [arctg(Z) + Z2arctg(Z) - Z] m=o A ° _ / •, sm+1 i,k,£ = ^ (-D [Z arctg(Z) - %l n ( l + Z 2)] m=o ... (h 17) where the sub- and super-scripts of Z have been dropped. Combining terms, we f i n a l l y obtain: 121 «•> -2 E e s E <-«•** ^ » I ) K - \,k * - F l t k ] i=l k=l m=o £=o " * Z 2 p i , k + 1 5 a r C t g ( Z ) [ " Q i , k + R i , k " u P i , k ^ - Z arctg(Z)[(P l j k6 1 ) k + J l + Q ± > k) ( 6 ^ - u) + ^ ^ k - S ± > k] -%Z arctg(Z)[P i ) k(26 1 ) k + J l - u) + + R.^] - | Z 3arctg(Z)[P ± j k] + % l n ( l + Z2) [| P i > k + ( P i j k 6 1 > k + , + Q i , k ) ( « l f k + J l - u ) + * 1 , W . A , k - « l f k ] + H Z ln(l + Z2) [ P i > k ( 6 1 > k + £ - ") + R ± f k] + H Z 2ln(l + Z2) [P ] .. (A 18) where Z = Z ^ as above. This a n a l y t i c a l solution can be checked by using the Pj ( O j ) which describe Lorentzian l i n e p r o f i l e s as i n eguation A3. The re s u l t s of eguation A18 can then be compared d i r e c t l y to equation A4. Excellent agreement (1 part in 104) i s found between the two with Fw-, =5.66x10" sec-*, Fw2 =7.01x10*2 sec-*, y =10» sec-*. These corresponds to 0.20 and 0.25 nm FHBfi l i n e for the 122 narrow and wider p r o f i l e respectively, centred at 820 and 816.1 no. The P's were described by points every 0.02 na, over a range that was 8 and 10 times the PffHH f o r the p r o f i l e s . The large ranges are necessary because of the contribution to the double i n t e g r a l from the intense wings. 0 123 Appendix B DATA REDUCTION In t h i s appendix a detailed description of the data redaction i s given. F i r s t , tvo d i f f e r e n t methods of measuring the signals on the oscillograms are evaluated. Then the methods actually used for d i g i t i z i n g the oscillograms and reducing the data are presented. The errors in the measurement are analysed. F i n a l l y the di f f e r e n t methods of evaluating the scattering parameters are discussed and the actual method used i s j u s t i f i e d . A. Oscillogram Analysis As indicated i n Chapter IV, the data acq u i s i t i o n rate for the mixing experiments described i n t h i s t h e s i s i s unfortunately very low. For instance, the spectrum i n section F of Chapter IV reguire approximately 20 hours continuous work, including the " l a s t minute" alignment. However, there are only an average of four shots for each wavelength position for the mixing, and four shots f o r the normal scattering. Therefore, i t i s imperative that as much and as accurate information as possible be extracted from the oscillograms as possible. Consider a normal scattering experiment where 124 the photomultiplier s i g n a l I n o r m i s equal to a constant S1 times the ruby monitor sig n a l I d : I n o r m= S1 XIQ-. The constant v i l l change for d i f f e r e n t scattering angles and f-numbers, wavelengths, o p t i c a l transmission e f f i c i e n c y of the lenses and gratings, and plasma conditions. I t i s necessary to obtain an estimate of S1 from a series of measurements of I norm a n ( * I ( j while attempting to keep these parameters constant. The most simple method for estimating S1 i s to measure the maximum height of the ruby laser monitor and photomultiplier signals using a ruler or magnified s c a l e . An appropriately weighted average of the r a t i o of the measured heights I n o r m / I d now gives S 1 . This method makes two basic assumptions. I t i s f i r s t assumed that the peaks in the signals occur at the same instant in time. Secondly, i t i s assumed that the peak photomultiplier s i g n a l i s a l i n e a r function of the peak scattering l i g h t , even though the photomultiplier integrates the s i g n a l because of i t s slow risetime. These two assumptions are reasonable, and spectra obtained in t h i s manner can f i t the theory very well. However, t h i s method has a drawback i n that not a l l the information from oscillograms i s being used: the photomultiplier s i g n a l i s related to the laser s i g n a l at every instant i n time, not just at the maximum values. If we could determine the heights of the signals as a function of time then we could do an average of the r a t i o of heights for the two signals at the same instant i n time. A l t e r n a t i v e l y , we could 125 calculate the r a t i o of the area's of the signals* This l a t t e r method i s convenient since one does not have to determine accurately the r e l a t i v e time o r i g i n s of the tvo signals. Also the r a t i o of the areas i s probably more correct, for s t a t i s t i c a l reasons as indicated i n section B, and because the PH signal i s already s l i g h t l y integrated. The l a t t e r method of determining the constant S has the disadvantage that the measurement of the height of the sign a l s as a function of time i s an extremely tedious task. Therefore, a compromise i s usually made in vhich the peak heights of the sign a l s are measured, but a larger number of observations are taken. This i s not possible for the mixing experiments described i n t h i s t h e s i s . The improvement i n data reduction made by using the heights of the signals as a function of time as compared to measuring the peaks heights only i s i l l u s t r a t e d by analyzing the normal scattering signals of the experiment i n Chapter 17, Section D. When the 21 normal scattering oscillograms are analysed by measuring the peak heights of the signals alone, one obtains a value of 20% for the standard deviation of the estimates of S about the mean of S1 . When the the areas of the signals are determined, t h i s standard deviation i s reduced to 10%. Figure ft-1 i l l u s t r a t e s t h i s improvement. The d i f f e r e n t standard deviations indicate that, to obtain equally accurate estimates of the mean value of S1 , four times as many observations must be made when the peak heights of 126 FIGURE B-l Thomson scattering photomultiplier signal as a function of incident ruby power. 127 the signals are used instead of the areas. Obviously, for this experiment, the method of measuring the heights of the signals as a function of time i s necessary. By using th i s analysis method, the four shots at each value of experimental parameter i s egual to more than 10 shots using the simpler method. It i s appropriate at t h i s point to d e t a i l the method of determining the s i g n a l heights as a function of time. A commercial d i g i t i z e r i s used (Instronics Limited, •Gradicon 1)* which gives the position of a curser i n the form of x,y co-ordinates punched on regular computor data cards. The coordinates are given i n thousandths of an inch. The curser consists of a set of f i n e cross-hairs with a dot at t h e i r intersection, etched in c l e a r p l a s t i c . The diameter of the dot i s approximately 100 microns. The cross-hairs and oscillograms are viewed through a plano-convex lens temporarily mounted on the curser. The lens, with a magnification of over 10, makes positioning the centre of the curser on the 400 microns wide image of the oscilloscope trace on the oscillograms r e l a t i v e l y easy. The f i r s t oscillogram i s taped to the d i g i t i z e r table so that the horizontal gratic u l e l i n e s are p a r a l l e l to the x-axis of the table. This i s checked by moving the curser along the g r a t i c u l e l i n e and observing changes in the y-coordinate. A single computer card i s taped to the table so that the remaining oscillograms can be mounted p a r a l l e l to the x-axis by butting them against t h i s stop. To avoid parallax error, a small red dot was placed at the 128 centre of the lens. The f a i n t image of t h i s dot i s maintained above the curser centre and the point of measurement. The actual d i g i t i z i n g i s performed i n the following manner. The curser i s set to the i n t e r s e c t i o n of the main g r a t i c u l e l i n e s at the centre of the oscillogram. The (x,y) coordinates are set to (0,0). The coordinates of the 4 intersection points of the g r a t i c u l e lines 3 d i v i s i o n s up and down and 3 d i v i s i o n s l e f t and right of the o r i g i n are measured., These coordinates can be used to determine the r e l a t i v e magnification of t h i s set of oscillograms as compared to the c a l i b r a t i o n oscillograms described l a t e r . Hext, the curser i s set so that i t s horizontal l i n e i s centred on the baseline of the oscillogram trace, and i t s v e r t i c a l l i n e bisects the ruby laser monitor pulse. The position of the curser i s recorded so that the position of the ruby pulse on the oscillogram i s known. The (x,y) coordinates are reset to (0,0). How a l l measurements of height are r e l a t i v e to the baseline, and distance along the oscillogram are r e l a t i v e to the approximate centre of the ruby pulse. The curser i s f i n a l l y moved along the oscilloscope trace, and i t s position punched on cards. ' Approximately 25 (x,y) coordinates are punched for each s i g n a l . How the integration of the signals can be done numerically, l i n e a r l y i n t e r p o l a t i n g between the coordinates of the signals. The integration ranges are determined i n the following manner. F i r s t , the time 129 origins of the oscillograms i s s h i f t e d so that half the area of the ruby pulse i s located on each side of the new o r i g i n . The s n a i l s h i f t in o r i g i n i s only a fev percent of the t o t a l integration range of approxinately 1 cn, and i n f a c t makes l e s s than 1/4 % difference i n the value of the i n t e g r a l . The heights of the oscillograms as a function of distance are added together. This gives the heights as a function of distance for an "average oscillogram", and the points at which the signals go to zero can e a s i l y be determined. Equal lengths of integration ranges i s used for both the photomultiplier and the ruby signals. The estimate S1 i s now made in using the following two formulae: i=l *1 = "tifiLu^Xfiidt ( /^orm d t / i=l J ... (B1) where I n 0 rm a n ^ ^ a r e t n e signals for the i t h experimental shot of a t o t a l of N shots. These estimates are i n f a c t l e a s t squares estimates using d i f f e r e n t weighting methods, as discussed i n Section B. 130 Evaluating the mixing signals i s not as straight forward. The signals on the mixing oscillograms are the o r e t i c a l l y related in the following manner: obs I d 2 d 1 2 . . • t OZ| where I and I are the photomultiplier and ruby laser monitor signals, and I , and I are the dye laser monitor signals. The PM sign a l due to the mixing alone i s Tm±x TobS ~ S l I d " S 2 I d I l I 2 , 0 . % As t h i s equation indicates, to make an estimate of the mixing constant S 2 for t h i s p a r t i c u l a r parameter s e t t i n g , we must subtract the expected signal due to normal scattering. To work as accurately as possible we must again do the i n t e g r a l of these signals with respect to time. The evaluation of the t r i p l e product i n t e g r a l by numerical methods i s a simple task, once the sign a l s are d i g i t i z e d . However, the r e l a t i v e time o r i g i n s are now very important. The dye lasers begin to lase a f t e r the ruby laser, so that t h e i r lasing peaks do not coincide. I t i s not s u f f i c i e n t to do the integration s t a r t i n g at the r i s i n g edge of the signals, nor centred on the i r peaks.. The true r e l a t i v e time o r i g i n s were determined by connecting the cables which carry the signals from the photodiode monitors to a pulse generator. A 20 nsec wide pulse i s sent down a l l three cables and oscillograms made. 131 Measurements of the s h i f t s of the delayed signals r e l a t i v e to the di r e c t s i g n a l correspond to the s h i f t s of the origins of the dye l a s e r pulses to the ruby laser pulse. There i s no time s h i f t caused by the monitors, since they are i d e n t i c a l i n construction and equidistant from the scattering volume. This technique gives an absolute measurement of the or i g i n s h i f t s , independent of oscilloscope and camera l i n e a r i t y , cable lengths and lasing times. Nov that the time o r i g i n s are determined, the int e g r a l of the t r i p l e product can be evaluated. This integration i s performed using the trapezoid r u l e v i t h .001 inch increments. The ruby laser s i g n a l time o r i g i n i s redefined, and the heights of the signals are determined by l i n e a r i n t e r p o l a t i o n , as i s done i n the normal scattering case. The same integration range for the ruby pulse i s used. The value of the i n t e g r a l i s then given by: x / v l n—1 / W 2 d t" h E [ xi +i - xi] x [H(x. + 1) H( Xi+1 + d l ) H ( xi+1 + d l ) + H ( x i ) H ( x± + dl> H<xi + di>] ...<B4) where H(x) i s the height of the oscillogram trace at the distance x, d-| and d 2 are the s h i f t s of the time o r i g i n s , -Xj = .001 inches, and the ruby laser i s i n the region x1 to x n. 132 This experiment i s not being done on an absolute scale. That i s , the scattering i n t e n s i t i e s are being determined r e l a t i v e to each other. This allows us to multiply the calculated signals by constants to give convenient numbers to handle. The constant S-) i s of the order of unity without any adjustment, but the t r i p l e product i n t e g r a l i s multiplied by 10 3. This d i g i t i z i n g technique, and especially the Textronix 7900 series oscilloscope, proved to be suprisingly l i n e a r . In the analysis we have made the assumption that the oscillograms are l i n e a r i n time, that the constant r e l a t i n g distance along the oscillogram to real time does not change s i g n i f i c a n t l y along the trace. The s p e c i f i c a t i o n s for t h i s oscilloscope indicate that th i s constant should not change more than 3% across the entire face, and not more than Vt i n the middle half. Measurements of the l i n e a r i t y using a 50 MHz pulse generator indicate that t h i s oscilloscope i s well within s p e c i f i c a t i o n s . It i s l i n e a r to within less than 1% over any 150 nsec (3 cm) distance. Thus no correction for nonlinearity of the time scales i s necessary. Using the o r i g i n s h i f t c a l i b r a t i o n oscillograms, the s h i f t i n the signals varies only .01 inches when the f i r s t pulse s t a r t s at the gr a t i c u l e 1 cm i n from the edge of the face as compared to star t i n g 4 cm i n from the the edge (about 1/4 % ) . The changes in o r i g i n s h i f t due to di f f e r e n t s t a r t i n g positions of the ruby pulse are approximately t.002 inches, which also can be neglected. 133 (This i s a measure of the average l i n e a r i t y over 3 cm rather than the l i n e a r i t y within the 3 cm.) The oscillograms for the c a l i b r a t i o n pulses can be used to indicate the errors involved i n determining the areas and the t r i p l e product i n t e g r a l . The measured area of a c a l i b r a t i o n pulse whose width i s egual to that of the ruby laser pulse has a standard deviation of 134. The standard deviation of the t r i p l e product i n t e g r a l i s 2%. Another estimate of the measurement errors can be found by going through the d i g i t i z i n g sequence several times using the same experimental oscillogram. This gives a standard deviation of H% for the area, and 5% f o r the t r i p l e product i n t e g r a l . This larger error can be attributed to the fact that the rectangular c a l i b r a t i o n pulses are more regular and easier to see than the laser monitor pulses. The standard deviations of S1 and S 2 are much larger than 536, so that the measurement accuracy i s quite s u f f i c i e n t f o r our purposes. The measurement of the peak heights of the signals are surely also accurate to 5%. Why, therefore, i s the standard deviation of the estimates for S-| twice as large? Obviously there must be sources of error other than measurement alone which are handled better by the integration method. These sources include innacuracy in determining the baseline, and the variations in the width of the ruby laser pulse. Because of the fast risetime of the ruby monitor, a change i n t o t a l energy with the same r e l a t i v e change in width can maintain the same peak 134 height., However, due to the integrating e f f e c t of the PH, i t s s i g n a l peak height w i l l change. This can r e a d i l y be seen i f one considers the exaggerated case of a square pulse for the ruby laser, and an integration t i a e constant for the PH of the order of the width of the ruby pulse. The PH pulse would then be approximately twice the width of the ruby pulse, and i t s height proportional to the ruby pulse width. The fact that the scattering signal i s composed of mixing s i g n a l and normal Thomson scattering s i g n a l i s a disadvantage i n the data reduction. The s i g n a l due to mixing alone has i t s own large fluctuations which are compounded by those of the normal scattering. B. Evaluation of Relative Scattering Cross Sections Be now have numbers which represent the incident ruby diagnostic beam I d , the mixing product l d x l ^ x l 2 and the scattered l i g h t l 0 b S > These numbers must be used to evaluate S1 and S 2 for one particular set of experimental parameters, in the the formula /4s d t - sifTddt + hj^A^ ... (B5) S1 and S 2 x l 1 x l 2 are proportional to the normal and mixing scattering cross sections, respectively, and the superscripts i r e f e r to the i t h scattering measurent. 135 The in t e g r a l signs w i l l be dropped for s i m p l i c i t y . Let as r e s t r i c t ourselves to the normal scattering experiments f i r s t . Three of many possible estimates f o r S1 using the n measurements are: .. . (B6) It i s shown below that these three estimators are i n f a c t just l e a s t sguares estimates using d i f f e r e n t weighting methods. * 3 Consider the s i t u a t i o n of a random variable T which i s a known function of X: Y = F(X ; a 1 } a 2 , ... ) + e ... (B7) The CO. are unknown parameters of the function F, and c i s a random variable which represents the errors involved in a r e a l experiment. A seri e s of measurements (Xj,Yj) are made from which the "best*1 estimates oC, for the parameters oCj are to be made. The accepted d e f i n i t i o n of 136 "best" i s usually estimators which are l i n e a r i n the Yj, have an average value which i s egual to the actual parameters, and have the smallest variance. If the random variable c has zero expectation value and variance a 2 , then the best estimators are found using weighted least squares method. That i s , the choice of parameters which minimizes the sum N 2 S = W± [Y - F(X i;a 1,a 2 > ... )] W. = [o(e±)]~2 ... (B8) are the best estimates of the true parameters. Setting ^S/doOi = 0 f ° r t n e d i f f e r e n t oGj gives a set of equations which can be solved for the parameters i n terms of the measured quantities (Xj ,Yj ). A simple example i s a normal scattering experiment with a CH laser for the l i g h t source**. How Y = oCj X • « , where Y i s the number of scattered photons, X i s the length of time to obtain the counts, and OC-j is proportional to the scattering cross section. One would expect that the error € would have zero average value and some variance a 2 . The dependence of a on X w i l l be discussed presently. This experiment then q u a l i f i e s for the least squares method. Settinq s - ]T w i r Yi - aix±] ...(B9) 137 ne have *1 Mow consider d i f f e r e n t weighting schemes. For the simple case above l e t us assume that e i s governed by Poisson s t a t i s t i c s only. In t h i s case, the variance of e is proportional to the true number of counts i n the time X, which i n turn i s d i r e c t l y proportional to X. A Therefore, W oc X~», and 0C| i s ... (B11) This, of course, corresponds to the S 1 b estimator i n (B6). Consider next the case for constant percent error. This would correspond to our simple CM scattering experiment i f , whenever the counting rate i s reduced, a longer period of time i s used to obtain more counts and constant standard deviation. Mow N L W Y X 1=1 N E w- x i = i (B10) 138 a/x = c W i " c X i " 2 i = l X i E(1) i=i c = constant ... IB12) This corresponds to the S 1 a estimator i n equation (B6). F i n a l l y we have the case where e i s constant, independent of X. This corresponds to the si g n a l being buried i n noise, so that the fluctuations i n s i g n a l are due to the background alone. This gives W± = c N XY X i Y i 1 £ i=i (x ± ) 2 ... (B13) which corresponds to the S 1 c i n (B6). The weighting scheme which i s chosen i s dictated by the conditions of the experiment, and i s never as clear cut as in the above simple examples. Some general observations can be made however. F i r s t , in an actual experiment, the values of X can never be absolutely known. I f the variance of X cannot be neglected, the analysis i s s t i l l v a l i d , but now 139 e w i l l also take into account the variations i n X. Second, i f one i s able to maintain the experimental variable X nearly constant, then a l l estimators w i l l be exactly equivalent, independent of the form of the dependence of « on X.; Keeping the X*s constant may be convenient from the point of view of not having to determine the weighting scheme, but i t does does not always give good estimates of the parameters. Consider the case for Y = mX • b • « . Keeping X constant gives no estimate of m or b. It w i l l , however, give an estimate of the d i s t r i b u t i o n of *. Third, i f a large number of measurements are made, then the three estimators are e s s e n t i a l l y equivalent. This follows d i r e c t l y from the fact that the expectation values of a l l the estimators are equal to the true values i n the l i m i t of a large number of observations. Por example, consider the case of the normal scatterinq siqnals i n Chapter V, Section D. The standard deviation of the estimate of S1 , for 20 measurements, i s 3%, but the three estimators of S1 d i f f e r by only 2/3%. He thus see that the weiqhting scheme i s only important when one must change X, or when a small number of measurements are made. The present work has few measurements and large variations i n laser powers, so that the weights must be chosen. The weighting scheme v a l i d for this experiment i s between the weighting for Poisson s t a t i s t i c s and for mo constant error (S 1 b ,S 1 c , equation (B6)). The photomultiplier s i q n a l i s proportional to the number of photons scattered into the c o l l e c t i o n system. Since there are only of the order of 100's of photons i n the scattered l i q h t detected by the P M , s t a t i s t i c a l fluctuations are an important source of error. The measurement inaccuracy, which i s at lea s t partly independent of the area, contributes a s i m i l a r amount to the t o t a l error. He must choose between the two weighting schemes, but fortunately the estimates for S1 and S 2 using weighting schemes (b) and (c) give e s s e n t i a l l y the same r e s u l t s . Scheme (a), which assumes constant percent error, i s r a d i c a l l y d i f f e r e n t . As an example, the estimate f o r S 2 for the data of Chapter ?, Section D, using 38 points, has a standard deviation of the estimator of 10%. The three estimators d i f f e r by less than 4%. However, i f the centre 5 data points are analyzed separately, schemes (b) and (c) give a 10% smaller value f o r S 2, but are s t i l l within 2% of each other. Weighting scheme (a) gives a value that i s 47% low. For these experiments, the values for the parameters presented are those calculated using the weighting f o r constant error. The values using scheme (b) were calculated to look for large discrepancies, but are not presented. In a l l cases, schemes (b) and (c) were found to be equivalent. S-, i s determined from the normal scattering alone, as outlined above. S 9 w i l l be determined from the 141 mixing shots where the photomultiplier s i g n a l w i l l be the sum of the normal scattering signal and the mixing s i g n a l : 1 A " 1 u ~ S . I . + e* mix obs 1 d Se also have from the theory that ... (B14) ^ i x - ( + £ , , ) ... (B15) £' represents the errors involved in the measurement of the observed and diagnostic laser signals, and the calcu l a t i o n of S1 . £" represents the error involved i n the calculation of the t r i p l e product i n t e g r a l . This gives S . I . I - I . , = I , - S . I , + e' - e" 2 d 1 2 obs 1 d ... (B16) He now have the same s i t u a t i o n as described above with Y = I , - S . I , obs 1 d X - W 2 a l = S 2 ... (B17) Assuming that the random variable C1" s t i l l has average value zero and a new variance ( V ) , we can again use the least sguares analysis. The weighting scheme should l i e between Poisson s t a t i s t i c s and constant e r r o r . The W2 estimates for S 2 were calculated using weighting schemes (b) and (c). Since both schemes proved to be equivalent, only the res u l t s of the f i r s t are presented. 143 Appendix C PLASMA JET COLLISION RATES The t h e o r e t i c a l c a l c u l a t i o n of the induced density fluctuation i n Chapter II reguires that the dominant damping mechanism i s Landau damping. The c o l l i s i o n freguencies must be calculated to ensure that c o l l i s i o n a l damping i s not important. This appendix c o l l e c t s information from d i f f e r e n t sources on the c o l l i s i o n freguencies between the electrons and other species present i n the plasma. E c k e r 2 8 has calculated the c o l l i s i o n rates and energy exchange times for test electrons moving i n a f i e l d of Maxwellian electrons or protons. The average c o l l i s i o n rate between electrons f o r the temperature and density of our plasma i s : Vee = 1.25 2A<>> where ^ o ^ l x l O 1 * rad/sec. ( S p i t z e r ' s 3 0 formula give a value lower by a factor 2.) The energy exchange rate between electrons i s required for the c a l c u l a t i o n of the perturbation of the plasma caused by the l a s e r beams. Ecker gives a value the energy exchange rate T e e of: 144 T e e = 1.3/VCO). Ecker also gives a value for the electron-ion c o l l i s i o n rate V e - i . This c o l l i s i o n rate i s , correcting for the lover temperature of the ions: Pe-i= 0.09 y<o>. Calculation of c o l l i s i o n phenomena between charged p a r t i c l e s and neutrals depends strongly on the choice of the in t e r a c t i o n potential. I t i s thus appropriate to use measured values for the electron-neutral c o l l i s i o n rates.** Using the experimental r e s u l t s of Brode**, the electron-neutral c o l l i s i o n rates ^e-n are: i^en =0.8 V<©>. Ecker gives a value for the Landau damping rate for a plasma with electron temperature and density of the plasma j e t of: Thus we see that the c o l l i s i o n freguencies are an order of magnitude smaller than the damping rate. 145 Appendix D DISCUSSION OF THE LOCATION OF THE DIPS IN THE MEASURED RESPONSE PONCTION It has been indicated i n Chapter IV, Section F, that there i s a possible r e l a t i o n s h i p between the positions of the modulations i n the measured response function of the plasma. There are d e f i n i t e valleys at AX=-2.85 and -4.35 nm, and possible valleys at -1.65 and -1.05 nm. I f we assume that a l l four are v a l l e y s , and plot the positions Z\Xn of the nth v a l l e y as a function of n, we get the excellent straight l i n e shown in Figure IV-9 (A). In t h i s graph, the s t a r t i n g integer f o r the f i r s t * valley* was chosen as n=1 since i t produced a straight l i n e plot. If AX n i s plotted for d i f f e r e n t s t a r t i n g n, the curves deviate r a d i c a l l y from straight l i n e s . 2 However, note that i f one plots (AX^ rather than AXn versus n 2, s t a r t i n g now at n=0, a second straight l i n e f i t i s obtained, with curves for d i f f e r e n t choices of s t a r t i n g n>0 (Figure D-1 (B)). This dual dependence of AX non n i s related to the s l i g h t a r b i t r a r y nature on the i n i t i a l choice of n, and also to the mathematics of small numbers. If 2 2 2 2 ( A x n ) z = u0r + axy n ... (D1) 146 147 Then «. - \> [l + O "2] ...(D2) for Xi / % Q « 1 , where and are constants. For the case £1# i t i s possible to imagine that changing the index by one could again produce a reasonable straight l i n e f i t . Thus, i f there i s a d e f i n i t e r e l a t i o n s h i p amongst the valleys, we do not know i f i t i s of the form (D1) or (D2). The error bars in Figure D-1 <B) represent ± 1/2 FWHM of the monochromator transmission function, T w. The same error bars (±1.5 nm) are s l i g h t l y larger than the c i r c l e s in (A). 

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