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The growth and saturation of stimulated brillouin scattering in a CO2 laser-produced plasma Bernard, John Edward 1985

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T H E G R O W T H A N D S A T U R A T I O N OF S T I M U L A T E D B R T X L O U I N S C A T T E R I N G IN A C 0 2 L A S E R - P R O D U C E D P L A S M A By J O H N E. B E R N A R D B.Sc, University of Victoria, 1977 M . S c , University of British Columbia, 1979 A THESIS S U B M I T T E D IN PARTIAL F U L F I L L M E N T O F T H E R E Q U I R E M E N T S FOR T H E D E G R E E O F D O C T O R O F PHILOSOPHY in T H E F A C U L T Y O F G R A D U A T E STUDIES ( D E P A R T M E N T O F PHYSICS) We accept this thesis as conforming to the required standard. T H E UNIVERSITY O F BRITISH C O L U M B I A May 1985 © John E . Bernard, 1985 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of P h y s i c s  The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 John E . Bernard 11 A B S T R A C T The growth and saturation characteristics of stimulated Brillouin scattering (SBS) in the interaction of an intense (/ < 1013 W/cm 2 ) CO2 laser beam with an underdense plasma are investigated experimentally. The plasma is produced by focussing a short (2 ns F W H M ) , CO2 laser pulse onto a stabilized nitrogen gas jet which flows from a Laval nozzle into low pressure helium. The resulting SBS interaction is studied through observations of the intensity and spectral behavior of the backscattered light as well as through temporally resolved ruby laser Thomson scattering measurements of the spatial and spectral behavior of the SBS generated ion acoustic waves. SBS occurs primarily in the long scale length, low density plasma located in the background gas in front of the jet. Initially, the instability grows absolutely at a rate within a factor of two of the predicted temporal growth rate. The SBS reflectivity is observed to saturate at less than 10%. This low reflectivity is a result of two processes. First, the SBS interaction region and the associated ion acoustic waves are broken up into several smaller regions, hence limiting the coherence length of the waves, and second, the ion acoustic fluctuation amplitude saturates at less than 20%. The latter saturation is attributed to trapping of ions within the potential troughs of the ion acoustic waves. The observed occurrence of the first harmonic in the ion acoustic wave spectrum as well as temporal modulations in the wave amplitude and sidebands in the spectrum of the backscattered light can be explained as consequences of the ion trapping. i i i T A B L E OF C O N T E N T S A B S T R A C T ii T A B L E O F C O N T E N T S iii LIST O F T A B L E S vii LIST O F FIGURES viii A C K N O W L E D G E M E N T S xi C H A P T E R 1 I N T R O D U C T I O N 1 C H A P T E R 2 T H E O R Y 6 2.1 Introduction 6 2.2 General Features of Stimulated Scattering 6 2.3 Parametric Excitation 11 2.4 The Plasma Dispersion Relation 12 2.4a The Ponderomotive Force 16 2.4b The Plasma Fluctuations 18 2.5 Solution of the Plasma Dispersion Relation 22 2.6 Absolute and Convective Growth 27 2.7 The SBS Reflectivity 32 2.8 Saturation Mechanisms 33 2.8a Pump Depletion 34 2.8b Ion Heating 35 2.8c Ion Trapping 37 2.8d Harmonic Generation and Wave Breaking 40 2.8e Profile Modification 43 C H A P T E R 3 T H E C 0 2 L A S E R A N D T H E GAS J E T T A R G E T 45 3.1 Introduction 45 3.2 C 0 2 Lasers 45 Iv 3.3 The Hybrid Oscillator 47 3.3a Longitudinal and Transverse Mode Selection 49 3.3b Line Selection 51 3.3c Short Pulse Generation 51 3.4 The Amplifier Chain 53 3.4a The Amplifiers 53 3.4b The Optical System 59 3.4c Optics Protection 60 3.4d Absorption Cells 61 3.5 The Predicted Output of the Laser System 62 3.6 Observed Laser Performance 64 3.7 The Gas Jet Target 65 C H A P T E R 4 DIAGNOSTICS 69 4.1 Introduction 69 4.2 Interferometry 69 4.2a Theory 69 4.2b Experimental 71 4.2c Data Analysis 73 4.2d The Plasma Density 74 4.3 Temperature Measurements 79 4.4 The Infrared Diagnostics 82 4.4a Intensity Measurements 82 4.4b Spectral Measurements 85 4.4c Sidescatter Measurements 85 4.5 Thomson Scattering 86 4.5a Theory 86 4.5b Coherent Fluctuations 89 4.5c Incoherent Fluctuations 92 V 4.5d The Thomson Scattering Experiments 96 4.5e Intensity Calibration 99 4.5f Temporal Calibration 100 4.6 The Streak Camera Intensity Response 101 C H A P T E R 5 RESULTS 103 5.1 Introduction 103 5.2 The Backscatter Reflectivity 103 5.3 The Backscatter Spectrum 107 5.4 The Sidescatter Spectrum I l l 5.5 Thomson Scattering 117 5.5a Spatial and Temporal Behaviour 117 5.5b Spatial and Temporal Modulations 121 5.5c The Fluctuation Amplitude 125 5.5d Late Scatter 129 5.5e The k-spectrum of the SBS Driven Fluctuations 131 C H A P T E R 6 DISCUSSION O F T H E E X P E R I M E N T A L R E S U L T S 136 6.1 Introduction 136 6.2 The Ion Temperature 136 6.3 Threshold Behavior and Temporal Growth Rates 140 6.3a The SBS Threshold 140 6.3b The Temporal Growth Rate of the SBS Instability 142 6.4 The Spatial Behavior of the SBS Instability 148 6.4a The SBS Reflectivity 148 6.4b Estimates of the SBS Backscattered Reflectivity 150 6.4c The SBS Interaction Length 155 6.5 Saturation 157 6.5a Ion Trapping 158 6.5b The Observed Effects of Ion Trapping 164 v i 6.5c Pump Depletion 166 6.5d Ion Heating 166 6.5e Harmonic Generation 167 6.5f Profile Modification and Self Focussing 168 6.6 The Early and Late Scattering 170 6.6a Early Scattering 171 6.6b Late Scattering 174 C H A P T E R 7 CONCLUSIONS A N D SUGGESTIONS F O R F U R T H E R W O R K 179 B I B L I O G R A P H Y 185 A P P E N D I X 1 M E A S U R E M E N T O F T H E F O C A L S P O T SIZE 193 A P P E N D I X 2 R A Y T R A C I N G 201 A P P E N D I X 3 T H E I M A G E DISSECTOR 204 vii L I S T O F T A B L E S III—I The gas jet background and reservoir pressures 68 V - I The predicted sidescatter spectral shifts 114 V I - I The Predicted Ion Temperature 138 VI-II Table of the Infinite Homogeneous Growth Rates 143 VI-III Table of the Measured and Calculated Interaction Lengths 156 L I S T OF F I G U R E S 2-1 Plots of oj vs k for electromagnetic, electron plasma, and ion acoustic waves. 10 2-2 Ion wave growth rate and amplitude vs frequency. 24 2-3 Normalized frequency of the driven ion acoustic wave vs Te for several laser intensities 26 2-4 A numerical simulation of absolute instability 29 2-5 Numerical simulation of convective instability 31 2- 6 Ion trapping and its effect on the ion distribution function 39 3- 1 The hybrid oscillator and short pulse gating circuit 48 3-2 The gain-switched pulse from the hybrid oscillator 50 3-3 The CO2 laser amplifier chain 54 3-4 The gain as a function of time for the K103 amplifier 56 3-5 The gain of the 3-stage amplifier as a function of charging voltage and gas pressure 57 3-6 The gain as a function of time for the 3-stage amplifier 58 3-7 The gain as a function of time for the TEA-600A amplifier 59 3-8 Burnmark of the partly focussed laser output 66 3- 9 The gas jet target 67 4- 1 The Mach-Zender interferometer 72 4-2 A typical interferogram 74 4-3 Contour plots of the plasma density for a laser energy of 6.7 Joules 76 4-3 continued 77 4-4 The experimental arrangement for measuring backscatter 83 4-5 Thomson scattering from a coherent wave 88 4-6 A typical Thomson scattering experiment 90 4-7 The Thomson scattering arrangement used for studying the temporal and spatial behaviour of the scattering region 97 ix 4-8 The Thomson scattering arrangement used for studying the temporal behaviour of the ion acoustic k-spectrum 98 4- 9 The intensity response of the streak camera system 102 5- 1 The backscatter reflectivity as a function of incident laser energy 104 5-2 The temporal behaviour of the backscatter reflectivity 106 5-3 Examples of the backscattered spectrum 108 5-4 Histogram of the position of the peaks and prominent shoulders in the backscat-tered spectrum 109 5-5 The geometry for sidescatter from a blowoff plasma 113 5-6 Refraction of the incident laser beam in an expanding plasma 116 5-7 Streak record of the spatially resolved Thomson scattered light 119 5-8 Streak record of the spatially resolved Thomson scattered light and behaviour of the plasma density in the scattering regions 120 5-9 The axial plasma density as function of position and time 122 5-10 The temporal modulation period as function of the incident laser energy. 124 5-11 Peak ion acoustic fluctuation levels as function of laser energy. 127 5-12 Peak spatially averaged ion acoustic fluctuation levels as function of laser energy 128 5-13 A streak record showing late scatter 130 5-14 A streak record of the k-spectrum of the ion acoustic fluctuations 132 5- 15 Spectral width of the feature at 2ka as function of laser energy 133 6- 1 The Landau damping rate for ion acoustic waves as a function of Ti/ZTe. 139 6-2 Plot of the measured Thomson scattered intensity as a function of time. . 146 6-3 The absolute growth rate of the SBS instability as a function of incident laser energy 147 6-4 The calculated backscatter reflectivity vs the measured reflectivity. 152 6-5 The fraction of the total number of ions which are trapped vs the fluctuation amplitude 159 6-6 The ion acoustic wave amplitide as function of time 161 6-7 Calculated energy loss for an ion acoustic wave due to ion trapping as a function of 6n/n 162 6-8 The k-vector matching for the 2wp-decay process 171 6-9 The k-vector matching for generating ion acoustic waves from the decay of the 2uydecay EPW's 173 6-10 Electron and ion distribution functions which lead to the ion acoustic instabil-ity 176 Al-1 The experimental arrangement for measuring the intensity distribution at the focal spot 194 Al-2 An enlargement of a typical burnmark 198 Al -3 Contour map of the focal spot intensity distribution 199 Al-4 Graph of ln(Calculated relative intensity) vs (Measured radius)2 200 A2-1 The geometry assumed in the ray tracing 202 A2-2 A comparison of the ray paths calculated theoretically (dashed curve)and with the ray tracing program (solid curve) 203 A3-1 The image dissector 204 A3-2 The measured intensity response of the image dissector as a function of channel number 206 A3-3 A scope trace of the output of the image dissector showing two temporally separated backscatter spectra 207 xi A C K N O W L E D G E M E N T S I would like to thank Dr. J . Meyer for suggesting this project and for the supervision he provided during the course of the work. I would also like to thank him for the many useful discussions and his words of encouragement. This exper-iment was part of a larger group project involving several people. I would like to thank Roman Popil, Grant Mcintosh, Hubert Houtman, Brian Hilko, and Chris Walsh for the many useful discussions and the assistance I received in obtaining and analysing the data. Special thanks are extended to Hubert Houtman for op-erating and maintaining the ruby lasers and to A l Cheuck for his many electronic wonders and especially his willingness to repair broken equipment on short notice. In adition, I would like to thank Dr. A.J . Barnard for his help with the theoretical details and his collaboration in solving the plasma dispersion relation and also to Luiz DaSilva for his useful computer programs and programming advice. Finally, I would like to acknowledge the financial support I received throughout the course of this work from N S E R C , the H.R. MacMillan Family Fellowship, and the Plasma Physics Group. CHAPTER 1: INTRODUCTION 1 C H A P T E R 1 I N T R O D U C T I O N Soon after the invention of the laser, experiments involving the interaction of an intense laser beam with a plasma were performed. The high intensities available with lasers made it possible to study interactions between plasmas and electro-magnetic (e.m.) waves at intensity levels unavailable with other sources. At such intensities, the interaction can lead to significant changes in the plasma character-istics which in turn can affect the propagation of the laser beam. Besides heating the plasma, an intense laser beam can also drive large plasma fluctuations. These fluctuations can result either from the local deposition of energy in the plasma which tends to drive a turbulent spectrum of waves, or through a coupling of the incident electromagnetic wave to the plasma waves. The most important form of coupling is through 3-wave parametric processes1 in which the incident e.m. wave decays into either two plasma waves or a plasma wave and another e.m. wave. At critical density, an e.m. wave can decay via parametric decay into an electron plasma wave and an ion acoustic wave. (The critical density for a plasma is that density at which the electron plasma frequency is equal to the frequency of the incident electromagnetic wave.) At a density of | critical, the two plasmon decay process in which an e.m. wave decays into two electron plasma waves can occur. If the coupling results in the generation of a scattered e.m. wave, the process is known as either stimulated Raman scattering (SRS) or stimulated Brillouin scattering (SBS) depending on whether the scattering wave is an electron CHAPTER 1: INTRODUCTION 2 plasma wave or an ion acoustic wave. SRS can occur at densities equal to or less than | critical while SBS can occur at any density less than critical. The major reason for studying these parametric processes besides their basic scientific interest is that such processes are likely to have a major effect on the coupling of the laser light to the target in proposed laser fusion schemes.2 Current plans for laser fusion call for the illumination of a spherical shell target, several millimeters in diameter and containing deuterium and tritium, with a laser pulse of several nanosecond duration and ~ 1 M J total energy. The laser pulse ionizes the outer surface of the target and the back reaction from the resulting expansion of the heated plasma into the surrounding vacuum drives the inner part of the shell inward, compressing an eventually heating the deuterium and tritium mixture to fusion conditions. The majority of the laser light absorption occurs via classical absorption near the plasma critical density surface. The parametric decay and two-plasmon decay instabilities can also contribute to the absorption since the plasma waves generated in these proceses are damped and therefore give up their energy to the plasma particles. The SRS and SBS instabilities, however, can scatter much of the incident laser light out of the plasma before it reaches the critical density layer and hence these processes lead to a decrease in absorption. Since the length of the underdense plasma generated in a fusion experiment can be comparable or even larger than the pellet diameter, SRS and SBS are both expected to occur and lead to significant scattering losses. Because of its low threshold and ability to reflect almost 100% of the incident light, SBS is probably one of the most serious threats to laser fusion. It is therefore important to have a good understanding of this process. In the present experiment, SBS is studied in the interaction of an intense CO>2 laser beam with a subcritical gas jet plasma. While it is unlikely that laser fusion will use CO2 lasers, the long wavelength of the laser means that the plasma densities at which the SBS process occurs are low, and therefore optical diagnostics of the interaction region can be used. The strictly subcritical plasma used in this CHAPTER 1: INTRODUCTION 3 study also means that complicating effects, such as reflections from a critical density surface, are absent. The aim of this study is to obtain quantitative information about the SBS interaction in a well diagnosed experiment. The interaction is studied by examining the temporal and spectral features of the backscattered light as well as by measuring the spatial and spectral behavior of the SBS driven ion acoustic waves. For the latter measurements, time resolved ruby laser Thomson scattering is used. The important questions to be answered in this thesis are: 1. How does the SBS instability grow? Does it grow as an ablsolute or convective instability? How do the growth rates compare to those predicted by theory? 2. Where and when does the instability occur? What plasma conditions are favourable to its growth? 3. How is the reflectivity related to the fluctuation level of the ion acoustic waves? and 4. What causes the instability to saturate? The final question is particularly important since experiments have generally found that the SBS reflectivity saturates at a level well below the predicted maximum of almost 100%. Although the results obtained are, in some cases, specific to this experiment, they can be compared to theoretical predictions. Therefore, the results of the theory can be checked and verified. This thesis is organized as follows. The theory of the SBS process, particularly as it applies to this experiment, is presented in Chapter 2. The general formalism for 3-wave parametric effects is briefly presented and the dispersion relation for ion acoustic waves in an infinite homogeneous plasma, in the presence of a large amplitude e.m. wave, is derived. This dispersion relation is solved numerically for the experimental plasma conditions in order to determine the growth rate and frequency of the excited ion acoustic waves. The conditions and characteristics of absolute growth in a finite plasma are discussed. Finally, the various saturation CHAPTER 1: INTRODUCTION 4 mechanisms are explained and the predicted effects of each process on the instability are discussed. The CO2 laser and gas jet target are discussed in Chapter 3. Since a major part of this study involved the setting up and debugging of the laser system, considerable detail is presented on the CO2 laser. The diagnostics used in this work are discussed in Chapter 4. These include short pulse ruby laser interferometry which was used to determine the plasma den-sity, and the infrared diagnostics through which the backscattered reflectivity and spectra were determined. In addition, the temperture diagnostics (which were not part of this thesis) are briefly discussed and the results presented. The final part of the chapter is devoted to presenting the theory of ruby laser Thomson scattering, in particular, how it applies to the present experiment in which scattering from en-hanced ion acoustic waves is studied. Finally, the experimental Thomson scattering arrangement is described and the methods of data analysis presented. The results of the experimental measurements are presented in Chapter 5, where attention is given to the plasma conditions in the interaction region as de-termined from the backscatter, Thomson scattering, and interferometric measure-ments. The experimental results are discussed and compared to theory in Chapter 6. Here, the temporal and spatial growth rates of the instability are determined from the time resolved Thomson scattering and backscatter measurements. Details of the spatial, spectral, and temporal behavior of the backscattered light and ion acoustic wave amplitude are analyzed and as well a saturation mechanism is proposed and shown to be consistent with the experimental results. Finally, some effects of the two-plasmon decay instability on the observed ion acoustic waves are presented and discussed. CHAPTER 1: INTRODUCTION 5 The results are summarized and the important conclusions and original con-tributions of this work are pointed out in Chapter 7. Several suggestions for future work are proposed in this chapter. CHAPTER 2: THEORY 6 C H A P T E R 2 T H E O R Y 2.1 Introduction As was pointed out in the previous chapter, stimulated Brillouin scattering is potentially one of the most serious problems affecting proposed laser fusion schemes. It is therefore important to understand the SBS process and the effect of the plasma conditions on the SBS level. Stimulated Brillouin scattering as well as many of the other wave-wave inter-actions that occur at sufficiently high laser intensities can be understood in terms of parametric amplification. The theory of parametric excitation of coupled waves is discussed in this chapter and a general dispersion relation for the SBS process is derived. The dependence of the growth rate on various plasma parameters can be obtained from a study of this dispersion relation. Modifications to the theoretical growth rates as well as the convective and absolute nature of the instability are discussed for the experimental case of a limited interaction region. Finally, various mechanisms that can limit and saturate SBS are discussed. 2.2 General Features of Stimulated Scattering When laser light or any form of electromagnetic radiation passes through a plasma, the free electrons oscillate in the electric field of the e.m. wave and, because of their accelerated motion, reemit or scatter radiation. The ions in the CHAPTER 2: THEORY 7 plasma also oscillate, but because of their much larger mass, the acceleration is small and the amount of scatter is negligible compared to that from the electrons. If the electrons are randomly distributed in a typical laboratory size plasma the total scattered intensity is small and very little of the incident energy is reflected from the interaction region. However, because of the long range Coulomb forces, a plasma is capable of supporting collective oscillations which can scatter radiation coherently in much the same manner as the crystal planes in crystalized solids Bragg-scatter x-rays (This is discussed in some detail in the discussion of Thomson scattering of Chapter 4). The amplitude of these oscillations is small if the plasma is thermal and may be calculated. 8' 4' 5 If the frequency and wavevector of the incident, scattered, and plasma waves are (OJ0, k 0), (w,,k,), and (u;,k) respectively, then in any scattering process the relationships: must be satisfied. The (+) sign in Eqs. 2-1 is for the case in which the scattered electromagnetic wave is blue-shifted while the (—) sign is for the case where the scattered wave is red shifted. In the case of stimulated scattering, we are interested in the latter situation, that is, the case in which the scattered wave is redshifted and the amplitude of the plasma oscillation grows at the expense of the e.m. wave. Eqs. 2-1 can therefore be written as (2 - la) and k« — k 0 i k, (2-16) u0 = u9 + u (2 - 2a) and k 0 = k,+k. (2 - 26) CHAPTER 2: THEORY 8 These are the basic frequency and wavevector matching conditions that must be satisfied in any stimulated scattering process as well as in the decay of an e.m. wave into two plasma oscillations. The stimulated scattering process begins with an electromagnetic wave prop-agating through a plasma that contains a thermal level of plasma fluctuations. (If the plasma has just been formed or there are shock waves or other disturbances present, the plasma fluctuations may be above the thermal level.) The e.m. wave is scattered by the fluctuations in such a manner that the scattering relations, Eqs. 2.2, are satisfied. The incident and scattered waves can then beat together and, by a mechanism to be discussed later, drive the original plasma fluctuation at the difference frequency. If the frequency, w, and the wavevector, k, of the plasma fluctuation satisfy the dispersion relation of a resonant oscillation the wave is driven resonantly and can grow and lead to even higher scatter. An instability therefore de-velops which can lead to very high levels of scattered radiation and large amplitude plasma oscillations. Two kinds of stimulated scattering can occur in an unmagnetized plasma. Stimulated scatter from the high frequency electron plasma waves is called stim-ulated Raman scatter (SRS) while the stimulated scatter from the low frequency ion acoustic waves is called stimulated Brillouin scatter (SBS). The electron plasma waves and ion acoustic waves have the respective approximate dispersion relations, ( 2 - 3 ) and ( 2 - 4 ) Here kg is Boltzmann's constant, Te(Ti) is the electron (ion) temperature, CHAPTER 2: THEORY 0 and uve = and —e, the charge on an electron. The dispersion relation for an electromagnetic wave in a plasma can be written as If we plot each of these dispersion relation on a a; vs. k plot, the relations 2-2 can be used to determine the frequencies and wavevectors involved in the stimulated Brillouin and stimulated Raman processes. An example is shown in Fig. 2-1. (These oj — k diagrams are for 1-D processes only, i.e. forward scatter or backscatter. The general sidescatter problem would require a 4-dimensional u — k diagram.) , .2 _ , .2 _i_ -2 j.2 ( 2 - 5 ) CHAPTER 2: THEORY 10 E.M. Wave p me Electron Plasma Wave Figure 2-1 Plots of u vs k for electromagnetic, electron plasma, and ion acoustic waves. The parallelogram construction illustrates the case of SBS backscatter. The stimulated scattering process cannot occur in just any scattering event that satisfies Eqs. 2-2 and the relevant dispersion relations but only in those events for which the growth rate of the instability exceeds the damping rate. Plasma waves may be either collisionally or Landau damped while the incident and scattered e.m. waves are typically damped by collisions (i.e. inverse Bremsstrahlung). The pump intensity must be very high to exceed the instability thresholds imposed by damping. Stimulated scatter is therefore usually seen only in high power laser-plasma experiments. The threshold and growth of a stimulated scattering instability can be determined from the general theory of coupled parametric oscillations. This theory is summarized in the following section. CHAPTER 2: THEORY 11 2.3 Parametric Exci tat ion Parametric excitation of coupled waves occurs in a plasma when two waves, electromagnetic or electrostatic, are coupled due to the presence of a third. The word parametric refers to the process by which parameters governing the natural oscillations, such as the electrostatic restoring forces of ion acoustic waves, are modulated at a frequency close to a harmonic of the natural frequency, resulting in amplification. The effect of the waves on one another can usually be approximated by a coupling which depends linearly on the wave amplitudes. This approach was first used by Nishikawa6 in developing a general formalism for describing parametric wave-wave interactions. For a three wave process we have the three nonlinearly coupled equations: KX(t, x) = | + 2 7 i + u\ - | X(t, x) = XY(t, x)Z(t, x), (2 - 6a) ( d 32*) MZ(t, x) - j ^ + 2 7 0 - ^ + "0 ~ j Z(t>x) = x)Ytt> *). (2 - 6c) where 0^1,2,0 are the natural frequencies, 71,2,0 are the temporal damping rates, and ci,2,o are the phase velocities of the unperturbed waves X, Y, and Z. Setting the coupling constants A, fi, and v on the RHS of Eqs. 2-6 equal to zero then results in three equations describing three independent damped oscillations. With non zero coupling the RHS's represent force terms. In the solution of these equations, the pump, Z, is usually assumed to have a constant amplitude and Eqs. 2-6a and 2-6b are Fourier transformed leading to a set of three coupled equations describing the modes X(UJ, k) and Y(u ± OJ0, k ± k 0). Assuming a solution of the form u = w +1'7 CHAPTER 2: THEORY 12 where u represents the real frequency of X and 7 is its growth rate, Nishikawa has solved these 3 equations for the case of SBS: |wi| |W2| = Kl-The results are briefly summarized. An oscillatory growing solution is possible only if u0 > 0J2. In addition, the frequency, ui, of the fastest growing mode can differ greatly from u>i (i.e. for SBS, the ion acoustic oscillation frequency may be shifted far from the natural frequency, Wia). The threshold intensity of the pump required for growth of the instability is found to be proportional to the product of the damping constants, 7172- Finally, far above threshold, the maximum growth rate and the shift in u from u;,0 are both expected to increase with the cube root of the pump intensity. In the following section a set of three equations similar to Eqs. 2-6 will be used to derive a dispersion relation for the ion acoustic waves driven by SBS. 2.4 The Plasma Dispersion Relation In this section the dispersion relation for ion acoustic waves in the presence of an intense electromagnetic wave is derived. This dispersion relation can be used to determine the growth rate of the SBS instability and the effect of various parameters on the growth rate. In particular, the effect of plasma density, electron and ion temperatures, and laser intensity are studied. The frequency and k-vector range of the growth region as well as the centre frequency of the excited ion acoustic wave is also determined. Knowledge of these parameters is important since the shift and width of the SBS backscattered spectrum are often used for determining the electron and ion temperatures. The dispersion relation is generalized for the case of a plasma made up of two ion species as well as electrons. This is particularly relevant in the present experiment since the plasma is formed in a mixture of nitrogen and helium. In addition, ions in the plasma are likely to be in several ionization states. It was CHAPTER S: THEORY IS expected that the presence of two ion species would lead to multiple SBS driven ion acoustic modes which could be observed experimentally by their effect on the temporal and spectral behaviour of the scattered e.m. wave. Indeed, it has been known for some time that the presence of several ion species can lead to multiple ion acoustic modes with comparable damping in field free (i.e. no e.m. waves) plasmas. This was first predicted by Fried et. a l . 7 who studied the field free ion acoustic dispersion relation generalized for two ion species. The first experimental observation of multiple ion acoustic modes was by Nakamura et. a l . 8 who observed two grid excited ion acoustic waves with different phase velocities in an argon-helium plasma. The possibility of two sound velocities in neutral gas mixtures containing two types of molecules has also been suggested9. The work to be presented in this chapter, however, is the first study of the possibility of multiple SBS driven ion acoustic modes in a plasma containing more than one ion species. In determining the plasma dispersion relation describing SBS, Forslund et. a l . 1 0 have closely followed the general parametric coupling formalism of Nishikawa. By choosing to represent the incident and scattered e.m. waves by their vector potentials, A , and by picking the Coulomb gauge, i.e. V • A = 0, they are able to write the equations describing the incident and scattered waves and the electron and ion fluctuations in a form very similar to Eqs. 2-6. Here, however, we will closely follow the approach of Drake et. a l . 1 1 and use a kinetic approach and E -vectors to describe the electromagnetic waves. Our method will involve deriving: 1. an expression for the scattered electric field in terms of the pump electric field and electron density fluctuations, and 2. an expression for the fluctuating electron density in terms of the incident and scattered electric fields. To obtain the former, we start with the Maxwell equations: V x E ± + ^ = 0 ( ( 2 - 7 ) CHAPTER £: THEORY 14 and 1 <9E± V x B ±" ? l f ( 2 _ 8 ) and derive the wave equation for scattered electromagnetic waves which have fre-quency u) ± u0 and wavevector k ± k 0 . Taking the curl of Eq. 2-7, substituting Eq. 2-8 and using the identity, V x (V x E ± ) = V ( V • E ± ) - V 2 E ± , we obtain, V > E ± - V ( V . E ± ) - ! ^ = ^ . ( , - » ) This wave equation is of a form similar to Eqs. 2-6. The force term is due to the currents J ± flowing in the plasma. The term, V ( V • E ± ) , which represents a self consistent feedback in the wave, disappears in the A-vector representation with the Coulomb gauge. Since we are interested in the scattered field at frequency U±UJ0 and wavevec-tor k ± k O J Eq. 2-9 must be Fourier transformed in time and space. Using /oo />oo dr / d<E±(r,0e,k±r-,w±', -oo J —oo O O V  OO and • oo />oo /oo poo dr / <ftJ±(r>0c,"k±-r-'h'=t<, -oo J —oo where k ± = k ± k 0 and ui± = u ± ua, the result is ( 4 - 4 ) i - k±k= E ± = -ifJt0u±3±, (2 - 10) 1 0 0 \ where I = j 0 1 0 1. The currents J ± arise from the linear response of the 0 0 l) electrons to the field E ± and also from the coupling between the oscillation velocity CHAPTER 2: THEORY 15 of the electrons in the E field of the pump and the electron density fluctuation produced by the electrostatic plasma oscillation: J± = <r± • E± - e6ne(u,}i)v0(uj0,ko), (2 - 11) where a± is the linear plasma conductivity, 6ne(w,k) is the amplitude of the longitudinal plasma oscillations at (u>,k), and v 0((j 0 , k 0) = e E ^ ° ^ is the oscillation or quiver velocity amplitude of the electrons in the field of the pump at (u/ 0,k 0). The first term in Eq. 2-11 is just Ohm's law for an isotropic plasma. For UJ0 S> upe, we can write 1 ' a± = iu±{e± - e0I), (2 - 12) where e± = e±I is the linear dielectric constant at frequency w ± u)0. The contri-bution at (w±,k±) to J± from the second term in Eq. 2-11 can be found from the Fourier component of e6nev at (u>±,k±). For a pump of the form E 0 = E0e0 cos(k • r — w0t) we have: dJ (2) „2 e dt 2m 6neE0e0 [e-'(*°'-«'°0 + e + , ( k 0 r - u , 0 t ) j ^ ^ _ ^ and /oo />oo at(2) 2 IT< * roc /»oo -oo ./-oo 2m e J_00 ; - o o e t (k±k 0 )T- . - («± W o )« e - ' ( k° r - u ; o ' ) - r - e + ' ( k o r - u , 0 ^ ] 5 n e ( a ; , k ) . (2-14) Assuming that c*/0 » u and neglecting terms containing e,(k±2ko)'r-,(w±2u'<>)' as off resonant, we find for the current due to the beating of the pump and the plasma oscillation: . 2 J i 2 ) = T — f i n e ( W , k ) E 0 T ( u , 0 , k 0 ) , (2 - 15) meu)0 where E 0 + = ±E0e0e+ilk°r~u°t) and E„_ = fore be written as, CHAPTER 2: THEORY i ^ e o e - ' ^ " - " 0 ' ) . Eq. 2-11 can there-3± = iuj±(e± - c0)E± =F te25ne(w,k) (2 - 16) and Eq. 2-10 becomes, c 2k±k± 2 ^ ( u ^ k ) 'pe (2 - 17) Inverting the final expression, we find, E+ = -UJ 2 6ne(w,k) 1 pe I —k±k± E o=F> (2 - 18) where, D± = kic2-ul£-±, (2 - 19) This is the desired result for the scattered electric field at (u>±,k±) in terms of the pump electric field at (±(*; 0,±k 0) and the plasma density fluctuations at (w,k). In order to complete the positive feedback loop in the SBS process, the pump and scattered electromagnetic waves must drive the plasma density fluctuation to larger amplitude. The driving force is provided by the ponderomotive force which is briefly discussed in the next section. 2.4a The Ponderomotive Force The equation of motion of an electron in the field of an electromagnetic wave is, m e ^ = -e [E ( r )+vxB( r ) ] , (2 - 20) CHAPTER 2: THEORY 17 where E(r) and B(r) are the instantaneous electric and magnetic fields due to the e.m. wave at position r. To first order, the electric field is assumed to be uniform in space and the v x B ( r ) term is small so the resulting electron velocity is sinusoidal in time and lies in a direction parallel to E . The ponderomotive force arises because of two second order effects. The first is caused by spatial non-uniformities in the e.m. wave's electric field which produce a net drift in the electron motion since the electron moves further in the half cycle when it is moving from a strong field region to a weak field region than vice versa. The net effect is that the electrons move from regions of high intensity to regions of low intensity. For example, if a laser beam has a Gaussian crossection, this process can produce a plasma density profile which will focus the e.m. beam. This self focussing process can also cause a laser beam possessing spatial intensity modulations to break up into several beams or filaments. The second source of the ponderomotive force is the v x B term in Eq. 2-20, i.e. the force due to the interaction of the electron's velocity as a result of the — eE force, and the e.m. wave's B-field. The ponderomotive force can be found by averaging the effects of the nonuniform E-field and the v x B force over many cycles of the e.m. wave. The result i s 1 8 r" = - j £ ?v^ ' ( 2-2 i ) where E and u are the e.m. wave's E-field amplitude and frequency. It would appear that there is a net electron drift only if E2 is non-uniform. However, the v x B force causes the electron to drift with a small (constant average) velocity even if the e.m. field is uniform. For the present problem, there exists in the plasma the e.m. pump, E 0 , as well as two sidebands, E + and E _ , which can be electromagnetic and/or electrostatic in nature. The resulting low frequency force (i.e. neglecting terms with frequency CHAPTER 2: THEORY 18 ~ 2u)0) can be derived by considering the cross terms. The result is, ?NL = -e2 4m ew 2 (2 - 22) + { v ( E0E+) + V ( E0E _ ) } + {v(E+-E_)} Mw,k) I > (2w,2k) The first three terms are the d.c. drifts caused by each of the three waves inde-pendently. The fourth and fifth terms occur at frequency w and wavevector k and are the terms responsible for driving the original plasma fluctuations. The last term, which occurs at (2w, 2k) is negligible since one of E+ or E_ is small for SBS. Therefore, considering only the components that occur at (w,k), the ponderomotive potential $ (fyj, = —V\P) becomes, *(a,,k) = + ~ ^ | [(E. • B+) + ( B a • E_) ] . (2 - 23) 2.4b The Plasma Fluctuations We are now in a position to calculate the magnitude of the electron density fluctuations. For a collisionless plasma, the electron and ion distribution functions can be calculated from the collisionless Boltzmann equation (Vlasov equation): where is the normalized electron (ion) distribution function, mej is the electron (ion) mass, and F is the force acting on the electrons (ions). The low frequency forces acting on the electrons are the ponderomotive force (—W) and the electro-static restoring force (+eV$, where $ is the self consistent electrostatic potential). The ponderomotive force on the ions is smaller than that on the electrons by a CHAPTER £: THEORY factor of Zme/mi and can be neglected. The Vlasov equations for the electrons and ions therefore become: £ £ + v . V / , + i - [ e V * - V * ] ^ = 0 , (2 - 25) and dfi Ze dfi at m,- aw (2 - 26) By a procedure similar to that used in the derivation of the ordinary plasma dis-persion relat ion, 1 3 ' 1 8 these equations are linearized and Fourier transformed since we require / e i i t l (u ; ,k ) . The fluctuating electron and ion densities can then be de-termined from: 6n e t t(w,k) = ne,t- j d v / e i ) t l ( w , k ) . (2 - 27) J—oo The results are: 6n a ( W | k) = ^ ($(w,k) - i ^ ) X e ( a ; , k ) , (2 - 28) and eak2 £n t-(<j,k) = —^-$(w,k)x,-(w,k), (2 - 29) where M)=^jy^h^iM, (2_30) is the usual electron (ion) susceptibility. Eqs. 2-28 and 2-29 can be used along with the Fourier transformed Poisson equation to eliminate $(a>,k) and 5n,(o;,k). The resulting expression for the electron density fluctuation is, 5ne(u;,k) = ,k2 X e (o ; ,k)( l + x,-(aj,k)) 1 +Xe(w,k) +x,(u>,k) * ( " , k ) . (2 - 31) CHAPTER S: THEORY 20 Finally substituting Eq. 2-23 into Eq. 2-31, we obtain the desired expression for the amplitude of the electron density fluctuations in terms of the pump and scattered electric fields: , n e ( W j k ) = ^ X e ( u ; , k ) ( l + X t ( . , k ) ) _ Eqs. 2-18 and 2-32 can be combined to obtain the desired dispersion relation. After a little algebra, the result is: 1 1 + — ; — T T = k * e (o ; ,k ) 1+ x,(w,k) | k _ x v 0 | 2 (k_-v 0 ) s + k2_D- kiu2_€-/€0 (2 - 33) k + X V . l2 ( k + V 0 ) 2 ' k\D+ k\u\e+lt0 where we have introduced v 0 , the quiver velocity amplitude of the electrons in the electromagnetic field of the pump wave. Eq. 2-33 is the dispersion relation for the plasma oscillations at (u>, k ) modified by the presence of the pump e.m. wave. It describes the parametric coupling of the low frequency electrostatic wave at (w,k) with the high frequency mixed electrostatic-electromagnetic side bands at ( u / ± , k ± ) . For a wave with frequency, U>± ~ U J p C ) e±( ^ , k±)B 1_ c j .s o > ( 2_3 4 ) and the k ± • v 0 terms dominate Eq. 2-33. In these cases, the high frequency side-bands are predominantly electrostatic. Eq. 2-33 then describes the parametric excitation of either two electron plasma waves (the two-plasmon decay instabil-ity) or an electron plasma wave and an ion acoustic wave (the parametric decay instability). For \u±\ » |wpe|, e ± ( w ± , k ± ) / e 0 ^ 1, the sidebands are primarily elec-tromagnetic, and D± = 0. The k ± x v terms then dominate Eq. 2-33 and we have CHAPTER 2: THEORY 21 the case of stimulated scatter where the pump excites an electrostatic wave and shifted electromagnetic waves (i.e. SRS and SBS). Since we are interested in SBS the k± • v 0 terms in Eq. 2-33 are ignored. In addition, only the (w_,k_) (Stokes) sideband satisfies the frequency and wavevec-tor matching conditions (Eqs. 2-2), so the (w+,k+) (anti-Stokes) sideband can be neglected and Eq. 2-33 becomes: * 2|k_ x v 0 2 Xe(w,k) l + x.-(w,k) klD. For e_(u/_) = 1 — u ; 2 / u / 2 . , D- can be written as, ~*~ TTT7777TT= LTTJ * (2 " 3 5) D- = 2uj0 U) + Wo 2U)0 (2 - 36) D- = 0 then requires k • k 0 = A^:2 or k = 2>cocos0 where 5 is the angle between k and k 0. Therefore the dispersion relation for the ion acoustic wave generated in the SBS process becomes, 1 1 2fc2v2 cos2 G s in 2 <f> Xe 1 + Xi w0(w - Aw) where <p is the angle between E 0 and the scattered wavevector k_ and Au> = c 2k • k o / ^ o — C2AJ2/2W0. It should be noted that for zero pump intensity the R.H.S. of Eq. 2-37 is zero and we are left with the dispersion relation for plasma fluctuations in a field free plasma. The R.H.S. of Eq. 2-38 is the coupling term for SBS which maximizes for 9 = 0 and <f> = ir/2, i.e. for 180° backscatter. Eq . 2-37 is Eq. 16 of Drake et. a l . 1 1 It has been solved analytically by several authors for both weak and strong pump intensities (see for example Refs. 10 and 11). The threshold intensity required for the onset of growth as well as the temporal growth rates may be calculated for given plasma conditions. CHAPTER 2: THEORY 22 2.5 Solution of the Plasma Dispersion Relation It was desired to calculate the growth rate and frequency of the SBS driven ion acoustic waves for plasma conditions similar to those encountered in the experiment, including the effect of two ion species. Since the laser intensities varied over a considerable range, it was necessary to solve Eq. 2-37 numerically. Only the problem of SBS backscatter was studied for which the following relationships apply: k = 2k 0, k„ = —kO J and w_ « u0 » u. (2 - 38) Assuming then that the zero order electron and ion distribution functions can be represented by 1-D Maxwellians characterized by the temperatures Te and T,-, the electron and ion susceptibilities (Eq. 2-30) can be written as: AV(w,k) = k2 De,Di k2 1 + " z( u ) y/2kveti \\/2kvej) (2 - 39) where nee' kBTee0 1/2 <Vc>e and kj){ = Z 2 n,e2 ni / 2 kBTi€0 AD,' (2 - 40) are the inverse Debye lengths, m 1/2 (2-41) is the electron (ion) thermal velocity, and i r°° e~x* CHAPTER 2: THEORY is the plasma dispersion function. Z(() is defined in terms of the complex error function and can be calculated numerically from standard computer subroutines. Eq. 2-37 was modified for the case of two ion species, 1 and 2, by simply replacing \ i by Xii + Xi2- The resulting expression was written as, 1 + v + v + v Xe{l+Xii+Xi2)2k20vl 1 + Xe + Xa + Xi2 ; r — ^ — 0, (2 - 42) and solved by the Newton method in the complex plane. 1 4 In general both k and u are complex. The temporal problem can be studied by choosing k real and calculating the corresponding frequency, u>, and temporal growth rate, 7<, (u; = u + i'70). Plasma conditions were chosen to be similar to those encountered in the experiment: a helium (Z = 2) and nitrogen (Z = 7) plasma with a concentration ratio of H e : N ~ 1.7, Te = 300 eV, and ne ~ 0.1n c r = 101 8 c m - 3 . Fig. 2-2a shows the calculated growth rate, 7 0, as a function of the (driven) ion acoustic wave, frequency, u. Both u; and 7 0 are normalized by dividing by 2fccc, — the frequency of an undriven ion acoustic wave with wavevector 2ka. Here, c, is taken in a form appropriate for large Te/T, and a mixture of helium and nitrogen ions: Cg=fz}+^zW\ kBT 2_ Maximum growth always occurred at approximately 2ka. For intensities above ~ 5 x l 0 n W / c m 2 , lomax w a s found to increase with (intensity)1/3 as predicted 6 ' 1 0 , 1 1 (see section 2.3). The frequency of the driven ion acoustic waves was also seen to increase with pump intensity. At high intensities the frequency differed significantly from the field free value of 2k0ct. These highly perturbed modes are often referred to as quasi-modes. 1 1 However, it should be pointed out that the quasi-modes are not oscillations in addition to that at 2A;0c,, but rather that there is a smooth increase in the frequency of the excited ion acoustic waves as the intensity is increased. CHAPTER 2: THEORY i i i i 1 1 1 • i 0 200 400 600 800 L\(A) Figure 2-2 Ion wave growth rate and amplitude vs frequency, a) Normalized growth rate vs. normalized frequency for a range of fc's. b) Unsaturated ion acoustic wave amplitude vs. frequency. The wavelength scale of the backscattered spectrum is also shown. CHAPTER £: THEORY 25 Therefore it is important to take this shift into account when calculating the electron temperature from the shifted backscatter spectrum. This has not been done in most previously reported experiments. During the period before saturation, the ion acoustic wave amplitude is pro-portional to e 10. This value is plotted in Fig. 2-2b. The spectrum of the backscat-tered light is proportional to the square of the ion wave amplitude. It can clearly be seen from Figs. 2-2a and 2-2b that the ion temperature has very little effect on either the growth rate or the ion wave spectrum. Even at the lowest intensities encountered in this experiment (~ 101 2 W/cm 2 ) the growth rate is reduced by only 6% and the F W H M of the backscattered spectrum by only 4% when the ion tem-perature is changed from 3 eV to 300 eV. Therefore the backscatter spectral shape serves as a very imprecise ion temperature diagnostic. Previous experiments which have used the F W H M of the backscattered spectrum as an ion temperature diag-nostic have usually treated the ion waves as undriven and calculated the spectral width from the Landau d a m p i n g . 1 5 , 1 8 ' 1 7 ' 1 8 For SBS, however, the coupling term on the RHS of Eq. 2-37 is usually dominant and the ion wave spectrum is determined primarily by the laser intensity and not by the Landau damping. The effect of various plasma conditions on u and ^ a have also been investi-gated. For fixed temperature and intensity the growth rate is a weak function of the plasma density. Maximum growth occurred at ~ 0.5 n c r . Also of interest is the electron temperature dependence of the frequency at maximum growth. This is shown in Fig. 2-3. The effect of the laser intensity on the frequency shift is clearly evident. In addition, the dependence of the frequency shift on the electron temperature can be seen. This latter effect is probably related to the fact that the electron Debye length, \j)e, approaches the fluctuation wavelength at lower values of Te, and single particle effects become important. A thorough search was made for multiple SBS driven modes, expected to be present in a multi-ion plasma. A fully ionized He-N plasma, however, is expected CHAPTER 2: THEORY 10 8 w u o _x rsj X o E 3 * 1 1 — i i i i i i Te/Tj =100 — —-T./T, =1 ^ V I O 1 4 W/cm2 1013 W/cm2 1011 W/cm2 04 100 200 300 400 Te (eV) 600 800 1000 Figure 2-3 Normalized frequency of the driven ion acoustic wave vs Te for several laser intensities. to have only a single SBS mode since helium and nitrogen have identical values of Z/rrii. Therefore a hydrogen-carbon plasma was studied. 1 9 Although multiple modes with similar damping were seen in the field free case, only one growing mode was observed when the pump was present. The other modes remained damped. A secondary peak, much smaller than the main peak was noted in the growth rate but this peak was present even for plasmas with a single ion species and its presence would not lead to any experimentally observed effects. Therefore it is concluded that the pump term in Eq. 2-37 is dominant in determining the growing modes. CHAPTER 2: THEORY 2.6 Absolute and Convective Growth The dispersion relation derived in the previous sections is for the case of an infinite homogeneous plasma. In this section, we will indicate how a finite interaction region and plasma inhomogeneities are expected to affect the instability threshold and growth rate. We begin by considering the effect of a finite interaction region. This problem has been treated by a number of authors . 1 0 ' 2 0 ' 3 1 Instead of dealing with coupled equations specific to SBS, we consider the more general equations for weak coupling in space and time between the two waves A;a and A- (The pump is assumed to have constant amplitude): 3 3 1-A- + + V- = loAia exp[»' f Ac(z')d"z'], (2 - 44) HaMa + + = loA- exp [-t J K(x')dx^ , where 7-,7,a are the damping rates and V _ , V t a are the group velocities of the coupled waves, 70 is the infinite homogeneous growth rate, and *c(x) = kQ(x) — kia(x)—k-(x) is the mismatch in wavenumber due to plasma inhomogeneities. These equations were solved numerically for the case of a uniform plasma (K(X) = 0) and no damping (7,,,, 7- = 0). The plasma region was represented by a 1-D, 200 step grid. At time T = 1 the ion acoustic and backscattered wave amplitudes were initialized to a spatially uniform "noise" level. The ion acoustic and backscattered "waves* had respective velocities of Via = 1*$^ (to the right) and V_ = - 4 ^ | | | | H (to the left) and Eqs. 2-44 were solved at each grid position for every time step. The plasma boundaries were simulated by fixing both the ion acoustic amplitude at position 0 and the backscatter amplitude at position 200 to "noise" at every time step. Figs. 2-4a and 2-4b show the results. It can clearly be seen that initially both waves grow at the same rate over the entire plasma. This is the period of absolute growth at rate 7C predicted for an infinite homogeneous plasma. Growth at 7,, at CHAPTER 2: THEORY a given position continues only until the effect of the plasma edges has propagated to that position. This occurs for position 150 at T = 13, when the backscattered disturbance which started at the right boundary reaches this point. Subsquently both waves grow at a reduced rate o f , 1 0 ' 3 0 (Because of the weak coupling there is some delay for both waves to reach the new growth rate.) It can be seen from the results of the above model that for a plasma with an initial uniformly distributed noise source, absolute growth appears to occur. Absolute growth usually refers to the case where the amplitude at a given position continually increases in time. However, for SBS the disturbances grow only by propagating which suggests that the instabilities are convective. In fact, if only a local disturbance was present at time zero, this disturbance would propagate and grow but the amplitude at any given point would eventually reach zero. It is only because of the fact that we start with a distributed noise source and continually feed in noise at the boundaries that the instability appears to be absolute. There is a critical length for absolute instability even in the case of a uniform plasma and zero damping. This is because of the fact that while growing in both space and time, the excited waves can escape from the system. The critical length, L. required for absolute instability is given by Pesme et. a l . a o as T = 27o (2 - 45) \via-v.\-L > 27o (2 - 46) Since the growth rates were calculated for conditions well above threshold, the effects of the damping of the backscattered e.m. wave were ignored in the derivation of the plasma dispersion relation. Drake et. a l . 1 1 have modified the CHAPTER 2: THEORY Incident *ave a ) 80 120 Position o m 5 5-1 I o A-a> •o 3 t 3 E o q 2-1 Temporal growth rates Ion acoustic Wave time step 80 b) Figure 2-4 A numerical simulation of absolute instability, a). Spatial profiles of the ion acoustic and backscattered wave amplitudes as a function of position, b). The wave amplitudes at position 150 as a function of time. CHAPTER 2: THEORY SO dispersion relation (Eq. 2-37) by adding damping terms for both the ion acoustic and e.m. waves. The resulting expression for the threshold intensity for SBS 1 8 0 ° backscatter is: 7, a7-ce 0m 2a; 3c,  / = e^k0 * (2"4?) In addition, damping can affect the nature of the instability. F o r 2 0 7o » 7c » TT, (2 - 48) where ~tT = (7 ta7- ) 1 / 2 and 7C = + ^\)\ViaV-\lt2, damping is negligible and absolute growth should occur provided 2-46 is satisfied. If IT < 1o < 7o (2 - 49) then there is only spatial amplification. To illustrate spatial or convective growth, Eqs. 2-44 were again solved numer-ically for conditions similar to those used in Fig. 2-4, but this time 7, 0 = 0.15 and 7_ = 0.02 (7 0 = 0.1). The calculated amplitude profiles of the ion acoustic and backscattered waves at various times are illustrated in Fig. 2-5. It can clearly be seen that both wave amplitudes eventually stop growing in time and exhibit only spatial growth. In the case of the ion acoustic wave, the spatial growth is very rapid but is followed by a long spatial decay. For the case of strong damping (but still satisfying relation 2-48) the spatial growth rate of the backscattered wave is given b y 1 0 Im(fc) * -4r- (2 - 50) 7«a y — Plasma nonuniformities can reduce the SBS growth rate by limiting the region over which the frequency and wavevector matching conditions (Eqs. 2-2) are satis-fied. Gradients in the plasma density lead to changes in the incident and scattered CHAPTER S: THEORY 0 AO 80 120 160 200 Position Figure 2-5 Numerical simulation of convective instability. Conditions were identi-cal to those in Fig. 2-4 except here, 7,a = 0.15 and 7_ = 0.02. wavevectors while temperature gradients affect mainly the ion acoustic wavevector. Velocity gradients can affect all three wavevectors by changing the local phase ve-locities. The net effect is to change the value of K(X) (see Eq. 2-44) from zero (its value for perfect wavevector matching). DuBois et. al .* 1 show that for a plasma of length satisfying Eq. 2-46, absolute instability should occur provided that (2 - 51) CHAPTER £: THEORY 82 where K' is the spatial derivative of K(X). The overall effect of each of: a finite interaction region, damping of the excited waves, and plasma inhomogeneities, is to reduce the SBS growth rate from its infinite homogeneous value, % . The resulting temporal growth rate for conditions satisfying those required for absolute growth is given by DuBois et. a l . 2 1 as: 7 = \ViaV-\W\ y/t\W) \Via + \V-\) K a l K " 1 (2 - 52) Vi. + |V-|' In the following section we will discuss briefly the effects of the absolute or convective nature of the SBS instability on the scattered light. 2.7 The S B S Reflectivity It can be seen from the previous section that for absolute growth, the SBS backscattered reflectivity is expected to increase exponentially with time: P- = Pne* where P_ is the backscattered power, P/vr is the power backscattered from the initial noise level, and 7 is the appropriate growth rate. Saturation mechanisms can impose a limit on the amplitude of the ion acoustic waves (see the following sections) and hence lead to a maximum value for the reflectivity. In the case where a ion wave fluctuation is saturated at a level of 6ne/ne and extends coherently over an interaction region of length, L, the power reflectivity can be calculated in a manner similar to the calculation of Bragg reflection from crystal planes. The result i s 2 6 , 2 7 R = tanh 2 TT L 2 A o He Tic, V Tier / J where XQ is the vacuum wavelength of the incident laser beam. If the damping of the ion acoustic and scattered waves is sufficient to allow only convective growth, CHAPTER 2: THEORY the fluctuation amplitude across the interaction region is not uniform but varies exponentially in space as illustrated in Fig. 2-5. Using the spatial growth rate for the scattered waves given by Eq. 2-50, we find that the scattered power at the front edge of the plasma is, P- = Pn exp[2lm(/:)L] = Pn exp where Pn is the power scattered from the noise at the back of the interaction region. As discussed in Sec. 2.6, plasma inhomogeneities can also limit the growth to the convective regime. Rosenbluth 2 2 has calculated the SBS reflectivity for an infinite non-uniform plasma with negligible damping. The result is, (2 - 55) where Pn is again the scattered power at the back edge of the interaction region. As illustrated by Eqs. 2-53, 2-54, and 2-55, the behaviour of the scattered light in an SBS experiment can be an important diagnostic in the study of both the plasma conditions and the SBS instability itself. 2.8 Saturation Mechanisms So far, we have only considered situations where the SBS instability grows at a constant rate determined solely by the initial plasma conditions. However, the instability cannot grow at the rate predicted by the linear theory forever. Eventu-ally, the SBS process begins to affect both the plasma and scattering conditions and can have the effect of decreasing or even stopping any further growth. This nonlin-ear behavior can saturate the plasma fluctuation amplitude and lead to scattered reflectivities significantly reduced from the predicted maximum of 100%. Many processes have been proposed as possible saturation mechanisms for SBS. These in-clude: 1. processes which reduce the SBS growth through a reduction in the incident (2 - 54) P- = Pn exp LIV-V,-.*'!! CHAPTER 2: THEORY intensity (pump depletion), 2. processes which increase the damping and therefore reduce the growth of the driven ion acoustic waves (ion heating), 3. processes which directly affect or limit the maximum possible amplitude of the ion acoustic waves change the plasma profile and hence limit the region over which the SBS instability occurs (profile modification and filamentation). Each of these saturation mecha-nisms is considered in the following sections. 2.8a Primp Depletion Exponential growth of both the ion acoustic and scattered wave amplitudes cannot continue indefinitely since this would lead to reflectivities of greater than 100%. If the SBS reflectivity approaches significant levels (> 10% or so), the pump intensity reaching parts of the plasma can be lowered resulting in smaller growth rates and reflectivities than those calculated for a uniform pump. The maximum backscattered amplitude can be determined from the Manley-Rowe relationships with the result: For SBS oj- « u0, and so reflectivities of almost 100% are possible. The Manley-Rowe relations can also be used to determine the maximum ion acoustic fluctuation level. The result i s 1 0 These results apply only in the case of an infinite homogeneous plasma. In most experimental situations the reflectivity is limited to values far below 100% by plasma inhomogeneities as well as saturation mechanisms affecting the ion acoustic, wave. The scattered and ion acoustic waves, once having reached their maximum amplitudes permitted by the Manley-Rowe relationships, begin to transfer their (ion trapping and harmonic and subharmonic generation), and 4. processes which E l _ E l (2 - 56) CHAPTER S: THEORY energy to the pump wave and decrease in amplitude. This leads to an oscillatory behaviour in the amplitudes with a period corresponding to several linear growth times (i.e. 1/7© or I/7). As the laser intensity is increased the modulation frequency is expected to increase until, in the limit of strong coupling, it approaches the ion acoustic frequency. 1 0 ' 3 8 Eventually, due to damping of the three waves, the modulations disappear. These modulations have been seen in simulations. 1 0 , 3 4 ' 3 6 Forslund et. a l . 1 0 have also found in their simulations that the ion acoustic wave grows exponentially until the pump depletion becomes significant after which the growth is algebraic (6ne/ne or. r 2). Even after the reflectivity has saturated, the pondermotive force can continue to drive the ion acoustic waves to higher amplitudes until eventually the waves break. 2.8b Ion Heating In an SBS scattering event, ion acoustic waves can be driven to very large amplitudes (6ne/ne > 10%). A significant amount of energy is therefore contained in the waves which are subsequently Landau and collisionally damped. This damping eventually leads to ion heating, either whole body heating (i.e. the temperature of the entire ion population increases) or the formation of a heated tail (only those electrons with thermal velocities close to the wave's phase velocity are heated). The increase in the ion temperature leads to an increase in the ion Landau damping and hence an even further increase in the ion temperature. This enhanced damping can lower the SBS growth rate and cause the instability to become convective. Ion heating can therefore saturate stimulated Brillouin scattering. For the simple case of convective growth in a uniform, finite plasma, the resulting reflectivity is given by Eq. 2-54. This result is valid only if the reflectivity is small. If pump depletion becomes significant, then the reflectivity, for cases of strong ion heating, can be found f r o m , 3 7 ' 3 8 R(l -R) = 5{exp[x(l - R)} - R}, (2 - 57) CHAPTER S: THEORY where and B is the noise level of the backscattered wave relative to the incident intensity. In order to close this description of damping, we must estimate the ion heating. This has been done by Phillion et. a l . 3 8 by balancing the energy flux deposited in the ion waves with that carried away by the heated ions. The approximate energy flux deposited in the ion waves can be estimated from the Manley-Rowe relations where Ii is the laser intensity and R is the reflectivity. If we assume that all the ions are heated and that they carry away the energy at their thermal velocity, v,-(i.e. they free stream), then the resulting ion temperature can be found f r o m , 3 7 ' 3 8 Another estimate of the ion temperature can be made if the fluctuation level is known. The energy density of an ion acoustic wave with a fluctuation level of 6ne/ne i s 3 5 If the ion acoustic wave is maintained at a level of 6ne/ne by the SBS process while damping at a rate of 7ta (i.e. the Landau,damping rate) then in time t the energy damped away is approximately, as R I L ^ = w0 Z (2 - 58) 2EliatV, CHAPTER £: THEORY where V is the volume of the interaction region. Therefore in time t the ions in the interaction volume are heated to a temperature of kBTi S 2-ZkBTeliat(^)2. (2 - 59) Both Eqs. 2-58 and 2-59 are for cases where the interaction time is longer than that required for the ions to reach equilibrium at the higher temperature. Since the equilibration time for the massive ions is substantial (~ 1 ns for our experimental conditions), the actual ion temperatures can be expected to be lower than those predicted above. The reflectivity predicted by Eq. 2-57 is for the case of a uniform plasma of length, L. If instead, the plasma is nonuniform, the ion temperature is not expected to have such a strong effect on the reflectivity. This is because the width of the driven ion wave spectrum increases with the increased damping leading to an increase in the length of the region over which the wave matching conditions are satisfied. This effect has been studied in the simulations of Manheimer and Colombant 2 9 who show that for a nonuniform plasma the effect of a change in the ion temperature on the reflectivity is much smaller than for a homogeneous plasma. 2.8c Ion Trapping In the previous section the ion acoustic waves were assumed to be damped primarily by linear Landau damping — collisionless damping which leaves the ion distribution essentually unchanged. However, if the ion acoustic wave has a large amplitude, it is possible for a significant number of ions to become trapped in the wave potential and so the ion distribution function can change considerably. Damping of this kind is known as nonlinear Landau damping or ion trapping. Ion trapping is illustrated in Fig. 2-6. It is assumed that the ion wave grows absolutely or convectively in such a manner that it traps ions as it grows. If the ion CHAPTER £: THEORY acoustic wave has a potential amplitude of <f> and propagates with a phase velocity of = u>/k, then ions with velocities, w, in the range given by, -m,(u - t^)2 < 2Ze<f>, (2 - 60) will be trapped in the wave potential. Their phase space orbits in the frame moving at the wave phase velocity are shown in Fig. 2-6a. Those ions with velocities much larger or smaller than v^, are not trapped but still contribute to the linear damping. 1 8 For the trapped ions, those with v < are accelerated and take energy from the wave while those with v > are decelerated and therefore give up energy to the wave. For a Maxwellian distribution, there are more slow than fast ions and so the net effect is a damping of the ion wave. The effect of the ion trapping on the ion distribution function is shown in Fig. 2-6b. A large number of slow ions are accelerated by the ion wave resulting in the formation of a "heated tail". Then, due to the increased number of ions with velocities in the vicinity of the wave phase velocity, the linear Landau damping is increased. In addition, much slower ions eventually fill the gaps in the ion distribution function left by those ions that are trapped and accelerated by the wave (i.e. the ion temperature equilibrates). This leads to whole body heating and enhanced linear damping and ion trapping. Kruer 2 7 has estimated the time required for strong ion heating (!F,- ~ T e/3) via the production of a heated tail as, where L is the interaction length and R is the reflectivity. Since this time is often comparable to the laser pulse length, ion heating by trapping likely does not result in reflectivities as low as those predicted from Eq. 2-57. t (2-61) CHAPTER 2: THEORY Figure 2-6 Ion trapping and its effect on the ion distribution function, a) A phase space plot in the frame of the ion wave showing the orbits of the trapped and untrapped ions; b) The effect of trapping on the ion distribution function. The ion wave fluctuation level required for significant ion trapping has been estimated from a water bag model by Dawson et. al .* 9 For long wavelength fluctu-ations {kXue ^ 1) the wave potential is given by, kfe-~^- ( 2 " 6 2 ) If the ion distribution function is assumed to be constant for v < y/Zv; and zero elsewhere then trapping begins when, 6n \mi{v4 - \fZvi) 2 = Ze<p = ZkBTe— It ft CHAPTER S: THEORY 40 or ^[(-nr-(it)T If the ions have a Maxwellian distribution then the trapping will occur at lower 6ne/ne than predicted by Eq. 2-63. However, the trapping will not have a strong effect on the wave until a large number of ions are trapped. That occurs when ions with velocities of ~ 2v,- become trapped. The choice of \fZv; as the cut on for trapping is in good agreement with 1-D simulations.2 9 Because of the exponential dependence between the number of ions and the (velocity)2 for a Maxwellian distribution, the number of trapped ions increases very rapidly with increases in the wave fluctuation amplitude. Therefore ion trapping can have the effect of essentially clamping the fluctuation amplitude at the saturation level across the entire interaction region. The SBS backscatter reflectivity in such cases is given by Eq. 2-53. It is also possible for ion trapping to lead to the collapse of the original waveform. This effect is considered in the following section. 2.8d Harmonic Generation and Wave Breaking Harmonic generation may saturate SBS in either of two ways. The first is that the production of harmonics can take away energy from the SBS gener-ated ion acoustic wave and so lower its amplitude as well as the resulting SBS reflectivity. 8 0 , 8 1 , 8 2 The second way is that the harmonic generation may become severe enough to lead to the formation of severely distorted wave profiles and the eventual escape of the ions from the waves. 1 0 , 2 9 This second process can lead to the collapse of the ion acoustic wave and is often referred to as wavebreaking. Since its propagation velocity is a function of its amplitude, an ion acoustic wave can steepen through the production of harmonics. If the wavelengths are all long (kXr)e -C 1) the dispersion relation is linear (w = c$k) and it is possible, provided all the waves are travelling in the same direction, for both the frequencies CHAPTER 2: THEORY 41 and wavevectors of the fundamental and harmonics to be matched over long regions of plasma. For such conditions the harmonics can be driven resonantly. Kruer and Estabrook 8 0 have computed the growth of the first harmonic by linearizing the two fluid equations and obtain, 6ne(2k) l/gne(fc)x ne -2\~n~rr^ (2_64) where t is the time. If the effects of finite k\j)e are included, the first harmonic is found to propagate at a different speed than the fundamental and gradually slips out of phase with it. The requirement that the time required for the first harmonic to grow to the level of the fundamental before a slip of A/4 takes place gives a limit of , 3 9 ^ > *2A L (2 - 65) below which steepening will not occur. Karttunen and Salomaa 8 1 have estimated the SBS reflectivity limited by first harmonic production for the case of small linear Landau damping. Their result, written in terms of the plasma length, L, and e.m. noise level c is L _ ( V o \ = 4s/2 l-ne/ncr t/4R\^l-y/R~ X0\vJ IT ( n e / n c r ) 3 / 2(i - R) [\ e J 1 + y/R Strong linear damping of the fundamental or first harmonic will lessen the effect of harmonic production on the reflectivity. However, it is difficult to distinguish experimentally between harmonic generation and linear damping since both effect the SBS reflectivity in similar ways. 8 1 ' 8 3 In addition to the production of harmonics, ion acoustic waves can decay into daughter waves of lower frequency (i.e. k —» ki + k.2 and u —* w\ + u>2). Again, efficient coupling and phase matching requires that the ion acoustic waves have long (2 - 66) CHAPTER 2: THEORY 42 wavelengths (kXj)e <C 1). For the field free case, ion acoustic decay will occur if the threshold condition 8 8 6J±>(K!lY/\ (2 -67) is satisfied. Here, 71 and 72 are the damping rates of the daughter waves of fre-quencies ui and u>2- The maximum growth rate is found to occur for k i = k2 = i k 38,34,35 Karttunen et. a l . 8 8 have calculated the SBS reflectivity limited by ion wave decay for the case of a uniform plasma and negligible linear damping. Their result: ncr 1 ln(2 + R/e) ne 4ln[(l + V ^ ) / ( l - v / 5 ) ] ' ] where e is the ion acoustic noise level, indicates there should be strong saturation for densities < 0.5n c r. For example, R < 20% for ne < 0.3n c r. The above analyses treat the original ion acoustic wave as though it was undriven. The pondermotive force arising from the SBS process, however, has the effect of reinforcing the original waveform. Therefore, there is an additional damping of the harmonics and subharmonics which could influence their growth. In addition, it should be pointed out that the limits for harmonic production and ion wave decay set by Eqs. 2-65 and 2-67 are likely not strong enough. The ion acoustic waves driven by SBS satisfy the dispersion relation, u) = cBk, only for very low intensities. At high intensities the waves are driven far off the field free resonance (see Sec. 2.5) and so harmonic generation and ion wave decay, at least by the methods outlined above, are strongly reduced. Wavebreaking is closely related to ion trapping. The bunching and bouncing back and forth of the ions trapped in the wave potential can have a strong effect on the shape of the wave leading to wave steepening and the production of harmonics. If enough ions are trapped, the wave potential and fluctuation amplitude can also be greatly reduced. Eventually the trapped ions modify the wave to such a degree CHAPTER 2: THEORY that it is possible for them to spill out of the waveform, destroying the original wave. Forslund et. a l . 1 0 have observed wavebreaking of this form in their particle simu-lations of SBS. They observed that even after the SBS reflectivity had saturated, the fluctuation level continued to grow because of the ponderomotive force due to the incident and backscattered waves. Eventually the plasma pressure caused the trapped ions to leak out of the wave potential at the regions where the pondero-motive pressure was lowest. They called this x-type wavebreaking because of the characteristic x-shape of the ion distribution in the phase space plots. Their sim-ulations, however, were for strong coupling with v\fv\ > 1 and produced values of 6ne/ne > 1. These fluctuation levels are much higher than those typically observed in experiments. Simulations with weak coupling did not show x-type breaking, but rather showed ions which leaked out and formed non-Maxwellian tails. The latter situation is more relevent to the present experiment. 2.8e Profi le Modif icat ion The SBS damping and saturation mechanisms considered up to this point have acted on the ion acoustic wave. It is also possible for the SBS interaction to influence the plasma conditions in the interaction region. In addition to ion heating and tail formation which have already been discussed, the deposition of energy and momentum in the plasma by the damped wave can change the plasma density profile. This process is known as profile modification. The plasma density profile is changed as a result of the ion acoustic wave damping which transfers momentum from the wave to the plasma particles. The plasma then piles up leading to a shortening of the interaction region and a reduction in the resulting scatter. This effect is strongest i f , 8 6 7,a > c./L, (2 - 69) where 7 t 0 is the damping rate of the ion acoustic wave and L is the interaction length, since then the ion acoustic wave will deposit most of its momentum in the CHAPTER 2: THEORY interaction region. High plasma pressure (i.e. high electron and ion temperatures) will tend to cause the plasma to resist this clumping and therefore reduces the net profile steepening. In addition to profile steepening by SBS, stimulated Raman scattering8 7 and two-plasmon decay 8 8 can strongly modify the plasma density profile in the vicinity of 0.25nc r, while resonance absorption 8 9 can lead to steepening at densities > n c r . In the present experiment SBS occurred at densities < 0.211^ so only the effect of SBS on the density profile should be relevant. This is considered in greater detail in Chapter 6 where the results of the experiments are discussed. The above list of SBS saturation mechanisms, while not exhaustive, does include most of those processes that have been found to be operating in previous experiments. The relative importance of each in the present experiment will be discussed further in Chapter 6. It should, however, be pointed out that once the instability has grown to a level for which the saturation is large, it may be difficult to disentangle all the non-linear saturation mechanisms from each other. CHAPTER S: THE C02 LASER AND THE GAS JET TARGET 45 C H A P T E R 3 T H E C 0 2 L A S E R A N D T H E G A S J E T T A R G E T 3.1 Introduction This chapter describes the construction details of the CO2 laser used to pro-duce and interact with the plasma. Much of the laser system was built in this laboratory and so considerable time was spent determining the operating condi-tions and the component configuration which gave the highest quality beam as well as the most reliable and trouble free performance. In hindsight, many of the prob-lems encountered in the construction of the laser chain could have been avoided. However, it must be pointed out that many of the features in this laser system were new to this laboratory and comparison to systems elsewhere was not always possible. Therefore experience and understanding had to be acquired in an often time consuming experimental approach. For these reasons I will try to stress the importance of the features and components in the laser system that are crucial to reliability. In addition, the purpose and theory of the Laval nozzle gas jet target is discussed. 3.2 CO2 Lasers The desired properties of the CO2 laser system are determined by the kind of experiments that are performed. The aim is to study parametric effects in the in-teraction of an intense laser beam with a plasma. The primary driving force behind CHAPTER S: THE C02 LASER AND THE GAS JET TARGET 46 these parametric effects is the ratio of the ponderomotive pressure to the thermal pressure which is related to the ratio of the quiver velocity of the electrons in the electromagnetic field of the laser to the electron thermal velocity vosc/ve. Since the quiver velocity is proportional to / A 2 , it is necessary to have high laser intensi-ties, Jo, and long laser wavelengths, Ao, in order to strongly drive the parametric interactions. Therefore the laser system is designed to produce high power pulses of good spatial quality which can be focussed to high intensities. A CO2 laser is chosen since its long wavelength output (A 0 = 10.6 fim) means that the paramet-ric processes can be studied at laser powers a factor of almost 100 less than those required for visible or 1 fim lasers. In addition, the long wavelength means that the critical density is low (ncr = 101 9 c m - 3 ) and so the plasma contitions and the parametric processes themselves can be studied with visible light (e.g. ruby laser interferometry and Thomson scattering). In order to maximize the power that can be delivered without component damage and to permit less time for heating of the plasma, a short laser pulse is chosen. The laser transition in the CO2 molecule is between rotational sublevels (quan-tum number, J) of two vibrational states. At 300 K the transition with the highest gain is between the J = 19 and the J = 20 rotation sublevels of the first asymmet-ric stretching vibrational state and the first symmetric stretching vibrational state respectively. This transition is referred to as the P(20) transition of the 10.6 fim band and has a wavelength of 10.591 fim and energy of 0.117 eV. The pumping of the inversion is achieved primarily by electron-molecule collisions in an electric discharge. Due to the long lifetime of the upper laser transition against sponta-neous decay and the rapid depopulation of the lower laser levels through collisions, a population inversion can be produced. Other gases are usually added to the CO2 to help in the production of the population inversion. Nitrogen is added to aid in the pumping of the upper laser level while helium is added both to improve the CHAPTER S: THE C02 LASER AND THE GAS JET TARGET discharge characteristics and to help depopulate the lower laser level. For further details on the properties of the excited states of CO2 molecule see Ref. 40. 3.3 The Hybr id Oscillator The purpose of the oscillator in the present experiment is to produce a short, temporally smooth laser pulse of moderate power which can be amplified in the amplifier chain. A hybrid oscillator4 1 is chosen for this purpose because it can be operated on a single longitudinal and a single transverse mode and so can produce temporally smooth Gaussian output pulses. (For more complete details on the modes in laser oscillators see for example Yariv's book). 4 3 The hybrid oscillator used in this experiment is shown in Fig. 3-1. It consists of a single folded laser cavity made up of both a high pressure (1 atmosphere) pulsed section and a low pressure (~ 26 Torr) continuous wave (CW) section; hence the name hybrid is used. The cavity is formed between a gold coated concave mirror (R = 8 m) and an uncoated germanium flat etalon separated by 263 cm. The high pressure section consists of two chambers, each with a pair of Ro-gowski electrodes 35 cm long, 2.4cm across and separated by 12 mm. The discharge is formed between these electrodes after ultraviolet preionization produced by a travelling spark between washers located along the sides of the Rogowski electrodes. The main discharge is fired at the end of the first quarter cycle of the ringing preion-ization circuit. The total energy dumped into the gas mix is 180J/1 provided by a 0.08/iF capacitor bank charged to 30kV. The gas mix consists of helium, CO2, and N2 in the percentage concentrations (He : CO2 : N 2 ) = (76 : 13 : 11)% with a total flow rate of 2.6 1/minute. Typical output energy is 1/4 to 1/2 Joule in a pulse ~ 100 ns (FWHM) long. The low pressure section is operated with a flowing 26 Torr mix of (He : CO2 : N2) = (84 : 8 : 8)%. The discharge is formed between two brass ring electrodes in w H ti-ro tr" cr >—' • o o o •1 p Cu CO tr o •-i <rt-X) 5T n (TO ert-»— • P W o • 3 LOW PRESSURE C.W. SECTION FROM WATER TEMPERATURE MONITOR / Ge ETALON COPPER MIRROR Ge BREWSTER ANGLE POLARIZERS SPECTRUM ANALYZER HIGH PRESSURE PULSED SECTION TO AMPLIFIERS 1 i O to § to S3 2 to £ •"3 oo CHAPTER 3: THE C02 LASER AND THE GAS JET TARGET a water cooled glass tube 75 cm long and 12 mm in diameter. The applied voltage is 12 kV at 8 mA. The continuous output is ~ 1.5 W. 3.3a Longitudinal and Transverse Mode Selection Each possible laser transition or line can be broadened so that it is possible for several longitudinal modes (standing wave patterns) to have comparable gain and oscillate simultaneously. This leads to beating between the modes and large temporal fluctuations in the output power. Some form of mode selection is therefore required. At atmospheric pressure, the width of the P(20) transition is typically of the order of 3.5 G H z . 4 8 The major broadening mechanism is homogenous pressure broadening. The natural width, that is, the homogeneous width of the transition, including the effects of the radiative lifetime as well as collisional relaxation, is only > 10 MHz and the Doppler width at 300 K is 40 - 50 M H z . 4 4 The frequency spacing of the longitudinal modes in a cavity of length, /, is c/2/. For our oscillator, / = 263 cm, which results in Aw = 60 MHz and so it is possible for approximately 50 modes to oscillate simultaneously. The hybrid laser design forces the oscillator to run on only one longitudinal mode by adding a C W low pressure section to the oscillator cavity. The homoge-neous width at half maximum of the P(20) line is given b y 4 6 Au = 7.58(FCo2 + 0.73FN a + 0 . 6 4 F H e ) P T o r r ( 3 0 0 K/r ) 1 / 2 M H z , (3 - 1) where PQQ2, ^ N 2 ' a n < * ^He a r e fracti 0 1 1 8 °f CO2, N2, and He in the gas mix. For our case (He : C 0 2 : N 2 ) = (84 : 8 : 8)%, P ~ 20Torr and T = 300 K and so the F W H M of the P(20) line in the C W section was only ~ 100MHz. In addition, since the transitions are homogeneously broadened, the molecules do not act independently but can influence one another. Therefore, in the steady state, once the mode with the highest gain has reached threshold (the gain equals the CHAPTER 5: THE C02 LASER AND THE GAS JET TARGET 50 losses for a cavity round trip), the modes with lower gain are held below threshold and do not oscillate. Therefore, on its own, the C W section should oscillate on only one longitudinal mode. When the high pressure section is fired, the longitudinal mode due to the C W section has such a head start on the other modes that the net output of the oscillator is almost purely single longitudinal mode. The output pulse shape, as measured by a photon drag detector-scope combination with a risetime of < Ins, is shown in Figs. 3-2. In Fig. 3-2a the C W section was not running and the output exhibits severe mode beating. With the C W section on, the output pulse appears earlier and is smoothed considerably as seen in Fig. 3-2b. Even if the C W section is somewhat below threshold4 1 and hence not lasing on its own when the high pressure section is fired, the differences between the initial (noise) photon densities is such as still to force the pulsed section to oscillate on only one mode. Figure 3-2 The gain-switched pulse from the hybrid oscillator, a). C W section off. b). C W section on. CHAPTER S: THE C02 LASER AND THE GAS JET TARGET 51 The transverse mode structure of a CO2 laser oscillator can usually be con-trolled by the insertion of apertures into the laser cavity. This is not necessary in our case probably because of the large length to diameter ratio of the CW discharge tube. The oscillator was observed always to operate in the TEMQQ mode. 3.3b Line Selection Although the gain of the P(20) transition at 10.6 /zm is usually larger than that of other transitions, wavelength sensitive loss mechanisms can often force the CO2 laser to lase on some other line. For this reason some sort of wavelength tuning is usually necessary. Initially, dielectric coated germanium mirrors were used on the output end of the hybrid oscillator. However, the laser output was found to damage these mirrors and so they were replaced by uncoated Ge flats. These fiats are found to act as etalons having a wavelength dependent reflectivity. By changing the optical length of the etalon, it is possible to select the laser output wavelength by maximizing the Q of the laser cavity for that particular wavelength. The refractive index of Ge at 10.6 /xm is 4.0, giving a single surface reflectivity of 36%. For a flat etalon the reflectivity therefore varies from 0 to nearly 80%. By varying the temperature, the refractive index (dn/dT = (3.81 ± 0.03) x 10 - 4 K) and the length (\/L(dL/dT) = 5.7 x 10 - 6 K) of the etalons can be changed. For our 3/16" thick Ge flats this results in a scan of one a free spectral range for a temperature change of 2.7 K allowing easy tuning at room temperature. The temperature of the Ge flat is controlled by placing it in contact with a water cooled copper flange. The water temperature is controlled typically to 0.1 K which results in accurate control of the laser wavelength. 3.3c Short Pnlse Generation The laser pulse generated by the hybrid laser has a FWHM of approximately CHAPTER S: THE C02 LASER AND THE GAS JET TARGET 100 ns. A Pockels' cell electro-optic switch is used to gate a 2 ns pulse from this output. The gating system is shown in Fig. 3-1. The laser pulse generated by the hybrid oscillator is almost completely polar-ized due to the two intracavity KC1 Brewster windows. Three Ge flats set at the Brewster angle further polarize the laser output before it enters an anti-reflection coated GaAs crystal which is the first stage in the electro-optic switch. With no voltage applied to the crystal, the polarization of the laser beam is unchanged ex-cept for a slight effect due to residual birefringence in the GaAs crystal. The beam therefore passes through the Ge Brewster flat, F4, with very little reflection and is dumped in the spectrum analyser (Optical Engineering CO2 Spectrum Analyser) which is used to monitor the laser wavelength before and during the laser pulse. When a voltage is applied across the GaAs crystal, a birefringence is produced which changes the polarization of the light entering F4 from linear to elliptical. The net reflectivity from F4 is given by (Yariv Ch. 9): 4 3 where V(t) is the time varying voltage applied across the GaAs crystal and V\f2 is the half wave voltage given by where, Ao is the vacuum wavelength (10.6/im), n is the index of refraction (3.3), r\\ is the electro-optic coefficient of the crystal (1.6 x 1 0 _ 1 2 m / V ) d is the crystal thickness between the electrodes (6mm), and / is the crystal length (39 mm). For our crystal V1/2 = 14.2 kV. CHAPTER S: THE C02 LASER AND THE GAS JET TARGET 53 The 2 ns pulse at approximately 14 kV used for gating the GaAs crystal is produced by discharging a 20 cm length of 50 ft cable charged to 28 kV. Initially a laser triggered spark gap was used to switch the cable discharge. However, this was unreliable because shot to shot variations in the oscillator output led to unacceptable temporal jitter in the gating pulse. (Laser triggered spark gaps are more suited to triggering by short laser pulses.) The laser triggered spark gap was therefore replaced by a u.v. triggered spark gap with a 5-10 ns jitter that is timed so as to produce a gating pulse at the GaAs crystal at the time of the peak in the oscillator output. The resulting pulse that emerges from the gating system is symmetric with a 2.0 ns F W H M and is polarized in the vertical direction. The contrast ratio (signal at 14 kV to signal at 0 kV) is estimated from d.c. measurements to be at least 200:1. Typical output powers and energies are 70 kW and 0.15 mJ respectively. 3.4 The Amplifier Chain The amplifier chain used in the experiments is shown in Fig. 3-3. The chain consists of three double passed amplifiers in succession along with the necessary spatial filters, absorber cells, beam expanders and damage prevention optics. These features are discussed in the following sections. 3.4a The Amplifiers The first amplifier is a Lumonics model K103 which is operated in the small signal regime (see Sec. 3.5) and acts as a preamplifier for the saturated power amplifiers that follow it. The gas mix for the K103 is the same as for the high pressure section of the hybrid oscillator (He:C02:N 2=(76:13:ll)% at 2.6 1/minute and 1 atmosphere) flowing first through the K103 and then to the oscillator. This gas mix is chosen to maximize the output of the oscillator-Kl03 combination while still maintaining reliability. Higher concentrations of C 0 2 and N 2 increase the output power but unfortunately result in occasional arcing in the K103. CHAPTER S: THE C02 LASER AND THE GAS JET TARGET 55 The K103 consists of 3 identical discharge sections. Each section is u.v. preionized by sparks between a series of balls on the ends of pins and a screen cathode. After preionization, the main discharge occurs between the screen cath-ode and the anode. The electrodes are specially shaped to keep the electric field as uniform as possible. The preionization must be uniform and the circuit fast enough to deliver all the stored energy to the gas via a glow discharge. Arcs concentrate the discharge energy to narrow channels and lead to ionization of the gas molecules and almost zero pumping. Unfortunately the K103 preionization system depends critically on the gap distance between the preionizer pins and the cathode. Uneven sparking results in severe arcing. Much time was spent adjusting the preionizer pins and carefully cleaning the electrodes but arcing remained a problem. It was finally eliminated with the addition of Tri-n-propylamine ( C H a C B ^ C B ^ N to the gas m i x . 4 9 ' 6 0 Tri-n-propylamine has a low ionization potential (7.23 eV) and is eas-ily ionized by the u.v. preionization. It therefore contributes electrons to the early stages of the glow discharge which rapidly grows by an avalanche process. Part of the gas mix is flowed over liquid tri-n-propylamine (vapour pres-sure=6 Torr at 20°C) contained in a flask. The resulting partial pressure of tri-n-propylamine in the gas mix is < 1 Torr. Greater concentrations, while still sup-pressing arcs, lead to strong absorption of the laser pulse. P-xylene was also tried as a gas additive instead of tri-n-propylamine but unfortunately the partial pressures required to suppress arcs usually led to strong absorption of the pulse. Since the K103 is operated in the small signal regime, it is,possible to measure its gain using the 2 ns pulse coming from the hybrid oscillator. Typically, this pulse has a total energy of ~ 0.15 mJ. After the K103 amplifier, the maximum energy is ~ 42 mJ for a measured gain of 280 (double passed). This results in a small signal gain of ~ 2%/cm for the active double passed length of 282 cm. The optimum timing for the firing of the K103 discharge was determined by measuring the output pulse CHAPTER 8: THE CO2 LASER AND THE GAS JET TARGET 56 Figure 3-4 The gain as a function of time for the K103 amplifier. size as a function of the timing. The results are shown in Fig. 3-4. It can be seen that the gain in the K103 amplifier persists for at least 1-2 /*s. The second amplifier, which is homebuilt, is operated near the saturated regime. Details on its construction can be found in the report by Houtman and Walsh. 6 1 This amplifier consists of 3 stages, each with side-on u.v. preionization via a series of sparks between washers. The main discharge is between pairs of Rogowski shaped electrodes, 60 cm long, 8 cm wide, and separated by 5 cm. The electrical energy for each section is supplied by a 3-stage Marx-bank charged to 27 kV and supplying 140 J/1 to the gas mix. This amplifier is operated at 1 atm with a gas mix of He:C02:N2=(63:25:12)% at a total flow rate of 8.5 1/minute. The same gas mix subsequently flows through the final amplifier. Since the 3-stage amplifier is observed to arc occasionally, tri-n-propylamine is also added to its gas mix. Besides CHAPTER S: THE C02 LASER AND THE GAS JET TARGET preventing arcs themselves, the presence of tri-n-propylamine is observed to lower the number of streamers (thin channels of either enhanced or decreased gain). Since the 3-stage amplifier is homebuilt, its gain characteristics were measured in great detail. The small signal gain was determined by measuring the amplification of a ~ 1 W C W laser beam. The results for different charging voltages and pressures are shown in Fig. 3-5. The typical small signal gain at 27 kV and 1 atm. for a single passed active length of 180 cm is ~ 3%/cm. Using the amplified 2 ns pulse emerging from the K103 amplifier as input, the timing of the 3-stage amplifier was determined. The results are shown in Fig. 3-6. Significant gain lasts for at least 1 /is. 80T-7.0 -$760 Torr $ 1030 Torr J 1300 Torr n 3.0"" zo -1.0 36 4 0 20 24 28 32 Charging Voltage (KV) Figure 3-5 The gain of the 3-stage amplifier as a function of charging voltage and gas pressure. GAIN (arb. unitsj CHAPTER S: THE CO2 LASER AND THE GAS JET TARGET T TIME (JJLS) Figure 3-6 The gain as a function of time for the 3-stage amplifier. The final amplifier is a Lumonics type TEA-600A. Preionization is accom-plished by a flash board located behind a screen cathode. The main discharge occurs between Rogowski shaped electrodes 50 cm long and 8.9 cm wide separated by 8.9 cm. The discharge energy is provided by a 3 stage Marx bank charged to 48 kV supplying 190 J/1 to the gas mix. This amplifier is operated in the saturated regime (see Sec. 3.5). The small signal gain of the TEA-600A was never measured. However, since the applied electric field is identical in the 3-stage amplifier and the TEA-600A and the discharge energy/liter is comparable, it is likely that the small signal gain coefficient is about the same in the two amplifiers, ie. ~ 3%/cm. The amplified 2 ns pulse was again used to determine the TEA-600A timing. The results are shown in Fig. 3-7. 10 CHAPTER S: THE C02 LASER AND THE GAS JET TARGET 1 1 1 1 1 ~ l 8-6i GAIN (arb. units) A-2-0 2.0 TIME (jis) A Y 6.0 Figure 3-7 The gain as a function of time for the TEA-600A amplifier. 3.4b T h e Opt ica l System The optical system is designed with the objective of extracting the maximum amount of energy from the amplifiers in a uniform beam. For this reason, the amplifiers are double passed and the beam is kept as wide as possible as it passes through the amplifiers. The optical arrangement and the beam path are shown in Fig. 3-3. The beam from the hybrid oscillator is focussed by the lens system, L l , through the first spatial filter (SF #1) which is used primarily for optics protection CHAPTER S: THE C02 LASER AND THE GAS JET TARGET (see Sec. 3.4c). From there, the beam expands as it passes through the K103 ampli-fier and is collimated by the copper mirror, M4. The beam then passes above the first spatial filter and is focussed by L2 through the second spatial filter (SF #2) which removes the hot spots and band structure from the beam that were caused by reflections from the electrodes in the K103 amplifier. The beam then expands as it passes through the 3-stage amplifier and is collimated by the copper mirror, M6. Baffles are inserted between each of the sections of the 3-stage amplifier to help prevent reflections from the electrodes. After passing above the second pinhole, the beam is directed onto the convex mirror, M10, and expands during its first pass of the TEA-600A amplifier. Finally, the beam is collimated by M i l and, after passing above M10, is directed by a 6" plane copper mirror to the target chamber. The entire system is aligned with the aid of the beam from the C W section of the hybrid laser. A quarter wave plate placed between the GaAs crystal and F4 (Fig. 3-1) is used to switch part of the beam towards the amplifiers. 3.4c Optics Protect ion Up to 10% of the laser pulse energy can be reflected from the target in the backward direction through stimulated Brillouin scattering. This backscattered light is often shifted less than 10 A (2.7 GHz) from the incident wavelength and therefore falls within the 3.5 GHz bandwidth of the P(20) transition. Therefore the backscattered light can be amplified to very large energies as it makes its way back through the amplifier chain and so lead to damage of the various optical components. This damage is prevented by 3 passive optical shutters; spatial filters #1 and #2 and the sharp focus between the two lenses which make up L l . Backscattered light, amplified by double passes through the TEA-600A and 3-stage amplifiers, is focussed by M6 onto the second spatial filter. This produces an airspark which rapidly extends back towards M6 for a distance of 20-30 cm. CHAPTER S: THE C02 LASER AND THE GAS JET TARGET 61 Although most of the amplified backscattered light is absorbed, reflected and re-fracted by this airspark, a small amount is transmitted through the spatial filter and is amplified by the K103 amplifier. This produces a second airspark at the first spatial filter. Again, most of the energy is blocked, but sufficient intensity is transmitted to damage the anti-reflection coating of the GaAs Pockels cell. This problem was solved by replacing the single lens that focussed the hybrid output into SF #1 with the two lens combination, L l . Any backscattered beam that has suffi-cient energy to damage the GaAs crystal produces an air spark at the sharp focus between the two lenses and is attenuated. With the aforementioned additions to the laser chain, damage to the optical components is almost completely eliminated except for occasional damage to the window on the second absorption cell and to mirror, M6. 3.4d Absorpt ion Cells Due to the high gain and long length of the CO2 laser amplifiers, it is possible for a random noise signal inside the amplifier chain to grow and lead to amplifier self-lasing. An absorption cell, located at the back of the 3-stage amplifier and containing 1.5-2.0 Torr S F 6 , 20 Torr ethanol, 100 Torr freon-502 and 640 Ton-helium is used to stop self-lasing in the amplifier chain. Self-lasing in the vicinity of 10.6 fim is prevented by the SF6 which strongly attenuates any small noise signals but is bleached by the much stronger main pulse. 6 2 The other gases, freon-502 and ethanol, are added to the absorption cell to prevent self-lasing over the region from 9 to 11 fim. These two gases absorb strongly in the 9-10.3 fim region but are only weak absorbers of 10.6 fim radiation. 6 8 Finally, helium is added to the cell to aid in the recovery of the bleached SF6. 1 The first absorption cell is not needed for amplifier isolation but is instead used to enhance the contrast ratio of the main pulse. As mentioned earlier, the power contrast ratio of the pulse emerging from the Pockels cell is ~ 200 : 1. CHAPTER S: THE C02 LASER AND THE GAS JET TARGET 62 However, since the duration of the gain switched pulse from the hybrid oscillator is of the order of 100 times that of the gated signal, comparable energy is contained in the main pulse and the feed through. S F 6 pressures of 10 Torr buffered with 1 atmosphere of helium in the first absorption cell are found sufficient to eliminate the effects of the feed through in the final amplified pulse. 3.5 The Predicted Output of the Laser System In this section, the theory of CO2 laser amplifiers as it applies to this exper-iment is briefly presented. The predicted output of the entire laser chain is then calculated from the measured gain characteristics of the individual amplifiers. The result is compared to the measured output in the following section. The gain characteristics of a CO2 amplifier depend on the power and temporal behaviour of the input pulse. In calculating the gain, it is important to distinguish between short and long laser pulses. By short pulses we refer to those with a duration, rp, shorter than or comparable to the thermal relaxation time, TR, of the rotational sublevels within the vibrational states, TR is typically ~ 0.15 ns for an amplifier operated at atmospheric pressure.4 6 The thermalization is usually via collisions and is rapid because the typical energy difference between adjacent rotational levels (2 x 1 0 - 4 eV)is small compared to the thermal energy (0.026 eV) at 300 K . 4 7 Long pulses are those for which rp » TR but are still short enough (< 10 ns) that vibrational relaxation and the effects of pumping during the pulse can be ignored. For these long pulses the output energy density, Eout, is given by: Eout = E, ln{l + exp(a0L)[exp(Ein/Et) - 1]} J / cm 2 , (3 - 3) where is the input energy density, ao is the small signal gain, L is the length of the gain medium, and Eg = 2 g £ ( j 0 ) is the total vibrational level saturation energy. Here, w is the laser angular frequency, a is the cross-section for stimulated emission and K(JQ) is the fraction of the molecules in the upper vibrational state that are CHAPTER S: THE C02 LASER AND THE GAS JET TARGET in the Jo'th rotational sublevel. Two regimes are commonly encountered: At low input powers (i? t n <3C Eg), Eont = Ein exp(arn£)j ie- the energy density increases exponentially with length. This is called the small signal or unsaturated regime. For high input powers (i? t n » Et), E^t = Ein + EgCtoL J / cm 2 ; the output energy density increases linearly with length. This is called the saturated regime. The most efficient use of laser amplifiers is obtained in this regime since the maximum amount of energy available in the inversion is extracted. For small values of 2?,n, the small signal gain for short input pulses differs little from the value for long pulses. However, for short input pulses with large values of E{n there is insufficient time for the rotational sublevels to transfer their energy to the one being depleted by the stimulated emission and so the saturated gain is less than for longer pulses. This reduction in gain is important even in amplifiers where the pulse period is longer than the rotational relaxation period. In these cases the output can be written as, 4 6 Eont « Ein + LEtaQ{\ - exp[-TpK(J0)/TR}} (3 - 4) For the upper J = 19 sublevel 4 8 /c(19) = 0.068 so for the 2 ns F W H M pulse used in the experiments, the saturated growth rate is reduced by a factor of 0.60. This strong reduction in the gain for short laser pulses is the reason why the amplifiers are double passed in the present experiment. The time between the first and second passes allows for thermalization of the rotational sublevels and hence a repopulation of the upper laser level and a depopulation of the lower laser level. We are now in a position to calculate the theoretical output of the laser chain. For this purpose, we require the saturated energy densities and hence the cross sections for stimulated emission, a. These have been calculated in Ref. 46. For our gas mixes: a = 1.47 x 10~ 1 8 cm 2 , E,=94 mJ/cm 2 (3-stage and TEA-600A) a = 1.58 x 1 0 _ 1 8 c m 2 , E,=87 mJ/cm 2 (K103). CHAPTER S: THE C02 LASER AND THE GAS JET TARGET The pulse energy after the second pass of the K103 is 42 mJ in a beam with an approximate diameter of 2.2 cm. The resulting energy density of 11 mJ/cm 2 is much less than the K103 saturated energy density of 87 mJ/cm 2 , therefore justifying the small signal approximation used in the measurement of the gain in Sec. 3.4a. During the first pass of the 3-stage amplifier, the beam expands from a diameter of 0.7 cm to a final diameter of 3.5 cm at which it remains for the second pass. The energy density in the pulse is always greater than the saturated energy density of 97 mJ/cm 2 and so Eq. 3-4 is used to determine the gain. The resulting output from the 3-stage amplifier is 4.0 J or 420 mJ/cm 2 . The beam again expands during the first pass of the TEA-600A amplifier; this time from 6.7 cm to a final diameter of 7.0 cm. The output at the end of the second pass is collimated in a beam of ~ 7.0 cm diameter with an energy density of 270 mJ/cm 2 . The predicted total output energy is therefore ~ 10 J . In the above calculations several simplifications have been made. The reflec-tion loss at each salt window of ~ 7% and the absorption loss at the copper mirrors (< 2%) have been ignored as have the losses in the SFe absorption cells. In ad-dition, the beam was treated as being uniform over its diameter when in fact its actual profile was roughly Gaussian with some spatial modulations. Finally, it has been assumed that both the 3-stage and TEA-600A amplifiers were saturated at all times. This assumption may have resulted in an underestimation of the actual gain since the beam energy density at the beginning of the first pass in both amplifiers was barely greater than the saturation energy density. 3.6 Observed Laser Performance The laser system produces pulses of roughly triangular shape with a 1.2 ns rise and a 2.8 ns fall. The steepening is likely due to both the SF6 in the absorption cells and the saturation effects of the amplifiers. Typical output energies range from 6 to 12 Joules with the occasional shot up to 15 Joules. This agrees well with CHAPTER S: THE C02 LASER AND THE GAS JET TARGET the predicted output of ~ 10 J. Complete misfires occur on ~ 5% of the shots. The variation in the output energy is mainly due to changes in the hybrid output and the gain characteristics of the K103 amplifier. A Rofin photon drag detector sampling the reflection off the front window of the first SFe cell is used to monitor the K103 output during most experimental runs. A good correlation is observed between the K103 and system outputs. The effects of saturation are clearly seen at higher energies. Spatial intensity modulations are present in the structure of the beam. These consist mainly of horizontal and vertical bands; the vertical bands due to reflections off the amplifier electrodes and the horizontal bands believed to be due to the effect of streamers. Much of the high frequency structure diffracts away in the 12 m from the TEA-600A amplifier to the target chamber. A burnmark made at the target chamber by partially focussing the laser output onto a piece of calculator thermal paper is shown in Fig. 3-8. Due to the nonlinear behaviour of the paper, the band structure is enhanced. The focal spot is roughly Gaussian in shape with an intensity waist of ~ 50 fim and a depth of focus of ± 2 mm (see Appendix 1 for details). A 10 J pulse therefore results in a peak focal spot intensity of 6.4 X 101 3 W / c m 2 . In operation, the laser can be fired once every 2-3 minutes. The spectrum analyser is used to monitor the laser line on each shot. 3.7 The Gas Jet Target For the interaction experiments it was desirable to have a strictly underdense plasma with long scale lengths and little bulk plasma motion. A gas jet target was chosen as experience at N R C and the University of Alberta 5 4 had shown this target to be suitable for these studies. For our experiments the gas jet target consisted of a laminar nitrogen jet flowing out of a planar Laval nozzle into low pressure helium. Nitrogen was chosen as the target gas because it would result in the required plasma CHAPTER 3: THE C02 LASER AND THE GAS JET TARGET Figure 3-8 Burnmark of the partly focussed laser output. density when ionized. For the background gas, helium had the desired property that it is usually difficult to produce a laser spark in this gas. The target chamber was an aluminum cylinder 25 cm in diameter and 55 cm long lying on its side. Plexiglass extensions on both ends supported the KC1 focussing and transmission lenses. Side ports were used for optical diagnostics (interferometry etc.) and a top port was used for x-ray detectors. The chamber was evacuated through the bottom port by a rotary vacuum pump. Filling and evacuation of the target chamber and high pressure reservoir were automatically controlled. For further details, see Ref. 55. The jet nozzle (see Fig. 3-9) was made of plexiglass with stainless steel jaws 7 mm long. The nozzle throat was 70 /xm across and the mouth opened to a final width of 1.2 mm. Nitrogen for the jet was stored in a small reservoir at a pressure of 1550 Torr. Upon firing, a solenoid valve opened, permitting the nitrogen to flow CHAPTER S: THE C02 LASER AND THE GAS JET TARGET 67 From Ruby Laser> (Thomsoij^Scattering Interferometry) 5 Torr Helium Background From C 0 2 .aser Figure 3-9 The gas jet target. The general layout of the target chamber is shown in Fig. 4-1. to the nozzle where the pressure change was detected by a piezo detector. After a period of ~ 10 ms the jet stabilized and a delayed pulse was sent to the delay unit controlling the CO2 laser system. An estimate of the conditions in the jet will now be made. Assuming adiabatic and isentropic flow in a perfect gas, the pressure, p, and area, o, at a point inside the nozzle can be written as. 6 6 Po \ 2 J ! \ -7/(7-1) and /p\ - ( 7 + l ) / 2 7 / ° = VPO / M / ^1 P O P O CHAPTER S: THE C02 LASER AND THE GAS JET TARGET 68 where po and po are the pressure and density on the high pressure side of the nozzle, 7 is the ratio of specific heats (7 = 1.4 for N2), Af is the local Mach number, and m is the mass flow rate which is assumed to be constant inside the nozzle. For a Laval nozzle, the Mach number is fixed at a value of 1 at the throat. Substituting M = 1 at the throat (athroat = 0.51mm2) the design pressure ratio, pmo«th/po, can be determined. The results for 5 Torr, 3.5 Torr and 2 Torr background pressures are listed in Table III-I. Table IH- I Background Design Experimental Plasma Density Press. Entrance Press. Entrance Press. (Completely Ion.) (Torr) (Torr) (Torr) (cm" 3) 5 1430 l & O 1.2 x 101 9 3.5 1000 1100 8.09 x 101 8 2 572 631 4.64 x 101 8 The discrepancy between the design entrance pressure and the pressure used in the experiment is due to the fact that the experimental parameters were determined for a nozzle with slightly different dimensions. The nitrogen leaving the nozzle was cold (~ 58 K) and of sufficient density (for the 5 Torr jet) to produce a supercritical plasma if complete ionization was achieved. Further details on the performance of the gas jet as well as the characteristics of the resulting laser produced plasma are reported in subsequent chapters. CHAPTER 4: DIAGNOSTICS C H A P T E R 4 D I A G N O S T I C S 4.1 Introduction An understanding of the processes involved in a laser-plasma interaction experiment is possible only if the plasma conditions are well known. Therefore it is important to diagnose carefully the plasma density and temperature as well as the plasma's dynamic behaviour. This chapter describes the techniques used for determining the plasma conditions in the present experiment. The infrared techniques used in the study of the scattered CO2 laser radiation as well as the Thomson scattering measurements used in the study of the ion acoustic waves driven by stimulated Brillouin scattering are also outlined. 4.2 Interferometry 4.2a Theory Interferometry uses a plasma's density dependent refractive index to deter-mine the plasma density. For an unmagnetized plasma of low enough density and high enough temperature that the effects of collisions can be ignored, the refractive index n is given by, 6 T /i = (1 - ne/ne)1'2, (4 - 1) CHAPTER 4: DIAGNOSTICS 70 where n e is the plasma electron density and ~ne is the critical density for the fre-quency of the probe beam. The refractivity of the plasma is mainly due to the free electrons. Eq. 4-1 is modified in cases where the probe wavelength is very close to a resonance line of an ion species present in the plasma. The effects of collisions become important in Eq. 4-1 only when the effective collision frequency, Vei, is comparable to or greater than the probe frequency. An estimate of is the electron-ion collision frequency for 90° total deflection of the electrons due to distant collisions with ions of charge, Z , and temperature, T,- « T V , 1 8 ' 5 8 where A = 12irne\ 3D/Z w 9.4, Z is the charge on the ions, and AD is the electron De-bye length. For the plasma in this experiment fcpTe ~ 300 eV and ne ~ 1018 c m - 3 , so Ud ~ 4.3 x 1010 s - 1 compared to a ruby laser (A = 6943 A) frequency of 4.3 x 1014 s - 1 . Therefore collisions are not important in Eq. 4-1. Interferometric measurements of a plasma's density are made by measuring a fringe shift. The fringe shift is calculated from the difference between the actual length of the probed region, L, and the optical length, (idl; where f.s. is the fringe shift measured in radians, dl is the length of a small region with refractive index /z, and Ap is the probe vacuum wavelength. If the plasma's refractive index profile is known or can be determined, Eqs. 4-1 and 4-3 can be used along with a measurement of the fringe shift to calculate the plasma density. (4-2) (4-3) CHAPTER 4: DIAGNOSTICS 71 4.2b Experimental In the present experiment it was necessary to know the plasma density at all locations in order to calculate density scale lengths as well as to understand better the interaction processes between the CO2 laser and the plasma. For this reason an interferometer was set up that allowed 2-D imaging of the plasma. The experimen-tal arrangement is shown in Fig. 4-1. The interferometer was of the Mach-Zender design with the reference beam passing behind the target chamber. (Earlier interfer-ograms were obtained by B. Hilko and H . Houtman using a Jamin interferometer, but the plasma was found to expand into the reference beam, complicating the interpretation of the interferograms.) An 80 ps Q-switched, mode-locked, and cavity dumped ruby laser pulse 6 9 was used as the probe beam. A HeNe beam, collinear with the ruby beam was used for alignment. The ruby laser timing was determined by a u.v. triggered spark gap which received a trigger pulse from the u.v. triggered spark gap that controlled the 2 ns gating system for the CO2 laser. This latter spark gap was modified so it was actually a dual, parallel, 50 ft system that sent signals to both the CO2 laser Pockels' cell and the ruby laser. The relative timing of the CO2 and ruby pulses was varied by changing cable lengths to the ruby laser trigger. The timing was monitored on a Tektronix 7104 oscilloscope. A spatial filter consisting of two 30 cm f.l. glass lenses and a 340 /xm pinhole was used to clean up the ruby beam before it entered the interferometer. Wedged optical flats were used in the interferometer as beam splitters and windows in order to prevent unwanted interference fringes. The reference and seeing beams were separated by several millimeters at the second wedge beamsplitter which was tilted so as to recombine the two beams at the film plane. In this way, a background or reference fringe pattern was produced on the film. The fringe spacing was normally 0.6 mm. Two glass lenses magnified the image of the plasma by a factor of ~ 9.2 giving a fringe spacing for the magnified image of 65 fim. CHAPTER 4: DIAGNOSTICS Figure 4-1 The Mach-Zender interferometer. CHAPTER I- DIAGNOSTICS 78 4.2c Data Analysis As shown by Eq. 4-3, the total fringe shift arises from the integrated total of the optical lengths of the infinitesimal regions of length dl. Since the plasma density was not constant in any plane perpendicular to the CO2 laser beam, it was neces-sary to unfold the contributions of the various plasma regions before the density profile could be determined. For this purpose, cylindrical symmetry about the axis of the CO2 laser beam was assumed (symmetry in the experimental interferograms justified this assumption) and the fringe shifts were Abel unfolded. Details of the Abel unfolding procedure can be found in the thesis of G . Mcintosh. 6 0 The Abel unfolding was performed on a computer after the interferograms were first photo-graphically enlarged (factor of 3) and the fringe positions digitized (resolution of 0.001 in) Every fringe profile in the regions containing plasma was digitized. After Abel unfolding, contours were fitted through the data producing a contour map of the plasma density. Examples are shown in Chapter 5. Because of the small fringe shifts (the fringe shift was typically less than one fringe spacing), digitizing errrors introduced sudden jumps in the values of the density determined from adjacent fringes. In order to smooth the resulting contour maps, it was found necessary to average fringe shifts in the adjacent fringes. Usually this involved replacing the fringe shift at a given radius by the unweighted average of the old fringe shift and the fringe shifts in the two fringes on each side of the fringe in question. This pro-cedure had the undesirable effect of smoothing the real density jumps somewhat, but since the fringe spacing corresponded to only 65 fim in the plasma, the problem was not serious for the SBS studies. Finally, geometrical ray tracing at A = 10.6 fim was performed in order to determine the effects of refraction on the laser inten-sity reaching various parts of the plasma. Details of the ray tracing are given in Appendix 2. CHAPTER 4: DIAGNOSTICS 74 h — Jet » Figure 4-2 A typical interferogram. 4.2d T h e Plasma Density A series of interferograms, taken over a range of times during the laser pulse were obtained at each of 2.8 J, 4.1 J, and 6.7 J average total laser energy. A sample interferogram obtained for the conditions, .Eraser = 2.9 J, time— 3.8 ns, is shown in Fig. 4-2. From each series of interferograms it was possible to study the temporal development of the plasma density profile. The series of contour plots resulting from the interferograms obtained for an average energy of 6.7 J is shown in Fig. 4-3. The position of the jet, the total laser energy, and the time at which the interferogram was obtained (with respect to the start of the laser pulse) is shown in each plot. In addition, the paths of 5 beamlets have been calculated from the ray tracing program and are plotted. (The beams are numbered 1 to 5 from the axis outward.) For many of the interferograms it was not possible to digitize accurately the positions of the CHAPTER 4: DIAGNOSTICS 75 fringes in the low density plasma located at the front of the main plasma region. The plasma density in this region was therefore estimated by manually measuring the maximum fringe shifts. The approximate density was then calculated with the assumption that the density profile could be represented by a step function in the radial direction. The resulting axial profiles are shown in the inserts of the plots of Fig. 4-3. The shot to shot reproducibility (same energy and time) of the plasma density profile was quite good. This is evident from Fig. 4-3 where it can be seen that the prominent features appear to change in time in an orderly fashion. There were, however, some slight shot to changes in the extent of the plasma into the region in front of the jet. Two things that are immedately obvious from a study of Fig. 4-3 are that the initial plasma breakdown did not occur at the centre of the jet, but rather at its edges, and that the breakdown was not confined to only the jet. Both features are a consequence of the plasma breakdown mechanism. Previous experience with gas jet targets at N R C 6 1 and the University of Alberta 6 4 has shown that for an oxygen jet in a low pressure helium background, the laser induced breakdown occured only in the jet. This is believed to be a con-sequence of the lower ionization potential of oxygen compared to helium, and the higher gas density and laser intensity in the region of the jet than in the back-ground gas. Therefore, it was believed that if the oxygen was replaced by nitrogen, the breakdown would be still confined to the jet. Interferometry, as well as side-view streak photographs of the plasma light revealed this not to be the case. While it is true that the threshold laser intensity required for multiphoton ionization of helium is higher than for nitrogen, once a free electron is produced, helium is expected to ionize more rapidly than nitrogen of similar density. 6 8 This latter rapid ionization occurs by an avalanche or cascade process and is a result of collisions between atoms (or molecules) and energetic electrons. A free electron oscillating in the electromagnetic field of the incident laser beam gains energy from CHAPTER 4: DIAGNOSTICS Energy: 6.5 J Time=0.9ns 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7. I Imm) K Jet *1 d' b) Energy=6.5 J Time=1.6ns o. 0 Energy=6.5J Time=2.4ns Figure 4-3 Contour plots of the plasma density for a laser energy of 6.7 Joules. The contour interval is 0.05ncr where = 101 9 c m - 3 . CHAPTER 4: DIAGNOSTICS Figure 4-3 continued. the electric field by making collisions with atoms or molecules in the gas (inverse Bremsstrahlung). In helium, the energy of the first electronically excited state is high (19.81 eV compared to an ionization energy of 24.58 eV) and so most of the collisions are elastic and the electrons can rapidly gain enough energy for ionization. The nitrogen molecule, however, has numerous low energy rotational, vibrational, and electronic states and so electrons may undergo many inelastic, non-ionizing collisions in nitrogen before they have sufficient energy for ionization. It is therefore believed that although the laser intensity was highest near the centre of the jet, the ionization process occurred at a faster rate at the edges CHAPTER 4: DIAGNOSTICS 78 where there was likely significant mixing of the nitrogen and helium. Subsequent rapid ionization of the helium background gas in front of the jet was probably enhanced both by free electrons escaping from the plasma and by u.v. preionization. Such processes should also occur in an oxygen jet. The fact that the breakdown was confined to the jet region in the previous oxygen jet experiments while there was considerable breakdown outside the jet in this experiment can probably be attributed mainly to different focussing geometries and laser intensities. It is apparent from Fig. 4-3 that whereas the breakdown plasma formed at the front of the jet expanded rapidly in both the radial and axial directions, the breakdown plasma formed at the back of the jet grew comparatively little. This is likely due to a substantial reduction in the intensity reaching the back plasma region because of absorption, reflection, and refraction of the incident laser beam in the front plasma region. Both the ray tracing and the power transmission measurements support this hypothesis. It was observed that a plasma was formed only after 0.8 to 1.0 Joule of (tem-porally integrated) energy had been delivered by the laser. If the total laser energy was substantially greater than 1.0 J, the initial expansion of the front plasma re-gion was very rapid; radial and axial expansion rates for the 0.1 n c r contour were measured to be 0.7 mm/ns and > 2.0 mm/ns respectively (approximately energy independent). Most of the axial expansion was in the direction of the focussing lens. For laser energies above 4 J, the breakdown extended up to 5 mm into the back-ground helium while for energies below 3 J the ionized region of helium typically extended less than 2 mm. In this region, which was later found to be the source of the SBS scatter, the plasma density varied from ~ 0.1n c r, far from the jet, to ~ 0.2n c r in the region close to the jet. In the radial direction, the plasma profile developed a doughnut shape which persisted for several nanoseconds after the end of the laser pulse. This structure suggests that there was significant plasma motion and expansion away from the CHAPTER 4: DIAGNOSTICS 79 interaction region. Plasma expansion is also indicated by the low peak densities measured in the jet region. For electron temperatures of approximately 300 eV and ion temperatures of the order of 50 eV, complete ionization of the nitrogen should have occurred (final ionization energy= 667 eV) and led to peak densities of approximately 1.2 ncr (see Chapter 3). This value is considerably larger than the observed peak density of 0.4-0.5 n c r . The densities in the plasma which extended into the helium, however, were considerably higher than' predicted. The plasma density in this region was observed to be ~ 0.1 ncr compared to a predicted density for complete ionization of the 5 Torr helium of only 0.035ncr. Therefore, there was likely some contamination of the background gas with nitrogen, probably a result of incomplete evacuation of the chamber between fills and mixing of the the jet gas with the background gas during the 10 ms between the opening of the jet solenoid valve and the firing of the laser. If the contaminating nitrogen molecules were completely ionized, the required density of nitrogen atoms was approximately 0.6 times the density of the helium atoms. This composition is used in Chapter 6 in the calculation of the SBS growth rates. 4.3 Temperature Measurements The plasma temperature measurements to be described in this section are part of the Ph.D. work of R. Popil. Details of the techniques used to measure the electron temperature as well as the results obtained can be found in Ref. 55. The major temperature diagnostic used in this experiment was the foil ab-sorber technique. Thin metal foils of various thicknesses were placed directly in front of plastic scintillators which emitted light, the intensity of which was directly proportional to the x-ray energy absorbed in the plastic. The light was transmitted to a photomultiplier tube via a light pipe. The experimental arrangement consisted of four scintillator-photomultiplier systems each with a different absorber foil. For CHAPTER 4: DIAGNOSTICS the x-rays emitted by a thermalized plasma of a given temperature, each foil thick-ness has its own transmission characteristics with the peak in the transmission spectrum occurring at higher and higher photon energies as the foil thickness is increased. Theoretical curves of the scintillator light output as a function of the absorber thickness can be constructed for various plasma electron temperatures and the experimental results compared to these curves to determine the experimental temperature. Two thermal temperatures were measured in this manner. Using aluminum and copper foils with aluminum equivalent thicknesses ranging from 5 to 900 /xm a temperature of 2 keV (ksTe) or 2.3 x 107 K was measured. However, using beryllium foils ranging in thickness from 15 to 90 fim, an electron temperature of 300 eV or 3.5 x 106 K was determined. Which of these two temperatures applies to the bulk of the plasma was determined from optical investigations of the plasma expansion. Using these optical measurements along with blast wave theory and total absorption measurements, the peak electron temperature was found to range from 300 eV to 600 eV. Therefore, the lower temperature of 300 eV as measured with the beryllium foils probably represents a reasonable time and space average for the bulk electron temperature. If it is assumed that the 2 keV component of the electron temperature exists during the entire CO2 laser pulse, it can be concluded that ~ 10% of the electrons were at 2 keV with the remainder at 300 eV. While the 300 eV temperature can be attributed to classical inverse bremmstrahlung absorption, no definite source has been found for the 2 keV electrons. Both temperatures were observed to be essentially independent of the incident laser intensity. This result is somewhat unexpected but can probably be attributed to increases in the size of the plasma and the number of heated electrons with an increase in laser intensity. The SBS process occurred mainly in the low density (~ 0.1ncr) plasma ex-tending in front of the jet, at least 1-2 mm from the regions of high density where the majority of the absorption and heating likely occurred. It is therefore possible CHAPTER 4: DIAGNOSTICS 81 that the electron temperature in the interaction region was different from the mea-sured temperature of 300 eV. The time required for energy deposited in the high density region to heat the interaction region can be estimated from the classical thermal conductivity. For a temperature gradient of dT/dx, the energy flux, Q, in the x direction is given by, Q = ~Kdx-> where K is the Spitzer6* conductivity, _ 28.3(fcTe)5/2 J  K ~ Z l n A s K m" Here, kTe is the electron temperature in eV, Z is the ion charge, and In A = ln[127rneA£,e] is the Coulomb logarithm. For our conditions (Te ~ 300 eV, ne ~ 1018 c m - 3 , and ^effective — 4) K = 9.7 x 10s J / sKm. Considering two plasma re-gions, one at Te — 300 eV and the other at Te -C 300 eV, separated by 2 mm (approximately the separation between the absorption region and the SBS active region) and assuming that the cold region has a radius of 0.4 mm and a length of 1 mm, the time required to heat the electrons to 300 eV through thermal conduc-tion from the hot region is ~ 21 ps. This result, however, is for the case where the temperature gradient is already established and should therefore be compared to the time required for 300 eV electrons to free-stream a distance of 2 mm. This time is ~ 0.3 ns. In either case, the time required to heat the interaction region to 300 eV is significantly shorter than the interaction period of at least 2 ns. Finally, the ion temperature is estimated. Since, primarily only the electrons are heated by the laser, we must consider the time required for the ions to thermalize with the electrons. This time is given by Spitzer6* as, 2Zme\' 2Zme „ 3.3 x i o u r e ° ' ' n . l n A CHAPTER 4: DIAGNOSTICS 82 where Te is measured in eV. The result is r^ - ~ 3 x 10 - 7 s compared to an electron thermalization time of ree ~ 1.5 x 1 0 - 1 0 s. Therefore, whereas the electrons should be thermalized at 300 eV for most of the laser pulse, the ion temperature should remain low. The ion temperature at times during the pulse can be estimated f r o m 6 8 dTj = T e - Tj dt Tgi Solving this equation for t = 2 ns (the time of the peak of the laser pulse), we find T{ ~ 2 eV. Therefore, there was negligible heating of the ions via collisions with the electrons. In Chapter 2 we discussed ion heating through damping of the SBS driven ion acoustic waves. It will be shown in Chapter 6 that ion temperatures comparable to the electron temperature can be produced in cases of strong nonlinear damping. 4.4 T h e Infrared Diagnostics This section describes the diagnostics used to measure the intensity and spectrum of the scattered infrared light. The time dependent intensity of both the backscattered and transmitted light were measured as well as the spectrum of the light backscattered at 180° and sidescattered at 135° and 90° . These infrared scattering measurements are complemented by the ruby laser Thomson scattering measurements that are described in the next section. 4.4a Intensity Measurements The development of the SBS instability as well as the level of the driven fluctuations can be studied through measurements of the scattered light intensity. The infrared diagnostics used in the experiment to measure the backscattered and transmitted light levels are shown in Fig. 4-4. Approximately 7% of the incident light was split off from the main beam by a 5 in diameter KC1 flat located just before the main focussing lens. This light was reflected off two first surface mirrors and sent ~ 6 m to a 8 in diameter, 163 cm f.l. CHAPTER 4: DIAGNOSTICS Figure 4-4 The experimental arrangement for measuring backscatter. CHAPTER 4: DIAGNOSTICS 84 concave mirror that focussed the beam onto a power meter (photon drag detector — Rofin model 7415) and an energy meter (GenTek LED-200-C). The backscattered light was collimated by the 5 in diameter KC1 lens and sent to the 8 in diameter mirror which focussed it onto the photon drag detector and onto a second GenTek energy meter. All mirrors were aluminized and capable of reflecting ~ 98% of the 10.6 fim radiation. The light transmitted or slightly refracted by the plasma was collected and partially collimated by two 5 in diameter, 50 cm f.l. KC1 lenses and a 2 in diameter, -10 cm f.l. KC1 lens. This light was sent ~ 7 m across the room where it was focussed onto the same photon drag detector used to monitor the incident and backscattered power. The optical lengths were adjusted so the incident, backscattered, and transmitted signals could be displayed on a scope with a 2 ns/div sweep. A Tektronix 7104 oscilloscope was used to monitor the photon drag detector output. The system risetime was ~500 ps, set mainly by the photon drag detector risetime and the cable dispersion. The oscilloscope traces were digitized to permit computer analysis. The true time dependent transmitted and backscattered signals were obtained by dividing the raw signals by the instantaneous incident signal. The outputs from the two GenTek energy meters were monitored on a Tektronix 468, two channel digitizing oscilloscope. Temporal and intensity calibration of the transmitted signal with respect to the incident signal was accomplished by simply noting the timing and relative sizes of the two signals when no plasma was produced. Backscatter calibration was accomplished by placing a first surface aluminized plane mirror above the jet and orienting it so it reflected the incident CO2 laser beam in the backward direction (~ 98% backscatter). No focussing lens was used for this calibration. Normally, only the incident and backscatter powers were monitored. The transmitted energy, however, was usually monitored with an energy meter (Apollo Lasers Digital Energy Meter and A L C Calorimeter) located at the back of the target chamber, directly behind the two 5 in diameter KC1 collection lenses. CHAPTER 4: DIAGNOSTICS Attenuators were made from 5 mil polyethylene sheet. Mylar attenuators were originally used but they had two serious drawbacks. First of all, mylar has strong infrared absorption bands in the region of 10.6 fim, making its attenuation strongly wavelength dependent, and secondly, there is a polarization effect in mylar so the attenuation depends on the orientation. Polyethylene has almost constant attenuation over the region 9.6-11.1 fim and the polarization effect is weaker than that in mylar. The attenuators were mounted in metal frames and calibrated in the same positions as used in the experiment. The calibration was accomplished using the 2 ns pulse reflected from the plane mirror that simulated 98% backscatter. 4.4b Spectral Measurements Information about the plasma electron and ion temperatures as well as the plasma motion can be obtained from the spectral behaviour of the backscattered light. In order to measure the spectrum of the backscattered light, the energy meter monitoring the backscattered energy (see Fig. 4-4) was replaced by a 0.5 m spectrometer. This spectrometer had a 160 fim entrance slit and a grating with ~ 150 lines/mm blazed at 50° for 10 fim. Calibration was accomplished using the many orders of the 6328A line from a HeNe laser. Preliminary spectra were obtained in a shot to shot manner but this was a slow process and subject to large errors due to shot to shot variations in the spectra. An image dissector was built to permit the measurement of spectra in a single shot. Details of the construction and calibration of the image dissector can be found in Appendix 3. For shots with several large temporal peaks in the backscatter, the image dissector permitted some time resolution in the spectra. However, in most cases, the spectra obtained were time integrated. 4.4c Sidescatter Measurements Sidescatter spectra were obtained for both 135° and 90° sidescatter. The small size of the target chamber as well as its somewhat impractical shape made CHAPTER 4: DIAGNOSTICS the placing and alignment of optics inside the chamber difficult. The sidescatter optics were supported by a space frame hanging from an aluminum plate placed in the top port of the target chamber. The sidescattered light originating in the plasma was reflected by a plane mirror and collimated by a 2 in diameter, 10 cm f.l. KC1 lens having a useful aperture of 4.0 cm and giving an effective collection f-ratio of //2.5. The collimated light was then reflected off another mirror and sent out of the target chamber through a KC1 window located above the main focussing lens. A mirror fastened to the 5 in diameter KC1 beamsplitter then directed the scattered light into the usual backscatter optics. Since the light was not quite collimated, an additional 2 in diameter, 10 cm f.l. lens was required to focus the scattered light into the spectrometer. Low signal levels also necessitated the replacement of the 160 fim entrance slit with one measuring 230 fim. 4.5 T h o m s o n Scattering The ion acoustic waves driven by stimulated Brillouin scattering were studied by ruby laser Thomson scattering. Reviews of this plasma diagnostic technique can be found in Refs. 64, 65, 66, and 67. 4.5a Theory A general development of the theory of Thomson scattering will not be at-tempted in this section. Rather, the emphasis will be on the theory as it applies to the present experiment. In Thomson scattering, free charges oscillate in the electromagnetic field of the incident laser beam and so radiate electromagnetic radiation as a result of their acceleration. For a single charged particle, the scattered power per unit solid angle (dPt/dCl) is given by, dPt dn ( 4 - 4 ) CHAPTER 4: DIAGNOSTICS 87 where q is the charge, § is the unit vector in the direction from the oscillating charge to the observer, and v is the acceleration of the charge due to the incident E . M . wave, calculated at the retarded time at which the radiation left the charge. The variation of dPt/dCl with direction has the doughnut shape characteristic of dipole radiation. Only the scattering from free electrons needs to be considered since the acceleration of the ions in the incident E . M . wave is less than that of the electrons by a factor of the electron:ion mass ratio. Scattering from a plasma made up of many particles is considerably more complicated than scattering from a single free charge. This is because radiation scattered from each of the electrons in the plasma interferes to give the total scat-tered power. Collective plasma motions such as electron plasma oscillations and ion-acoustic waves which are due to interparticle interactions (mainly Coulomb forces) can therefore largely determine the pattern and spectrum of the scattered radiation. The scatter in these cases can be treated in a manner similar to the treatment of Bragg reflection of E . M . radiation from crystal planes. 6 8 In the case of electron plasma oscillations, the scattering can come directly from the electrons in the density fluctuations. However, for ion-acoustic waves, the scattering comes from the electrons that are dragged along with the ions. The situation as it applies to scatter from an ion-acoustic wave is shown in Fig. 4-5. Incident light of frequency, w,-, and wavevector, k,-, is scattered by a plasma oscillation which propagates in the direction shown with phase velocity, v = u/k (k and uj are the plasma wave's wavevector and frequency). If v <S c, the magnitudes of the scattered and incident wavevectors are approximately equal and the angle of reflection will equal the angle of incidence. The Doppler shift in the scattered frequency is then given by: v» — uji = sin - = 2v|k,-| sin - . (4 — 5) CHAPTER 4: DIAGNOSTICS 88 Electron Density Figure 4-5 Thomson scattering from a coherent wave. Now, if |k,| « |k,-|, then | k , - k t | = 2|k,|sin0/2, ( 4 - 6 ) which is just Bragg's law for scatter from planes with wavevector, k = k, — k,-. Hence, Eq. 4-5 becomes u)t — w,- = — u. Therefore, if an experiment is set up so an incident light wave, (kt-, wt), scatters into a wave with wavevector, k,, and frequency, ut, the wavevector and frequency of the plasma fluctuation doing the scattering will be given by: ±k = k, - k,. Both thermal and driven plasma fluctuations can be studied by Thomson scattering. An important parameter in any scattering experiment is the scattering parameter, a, defined by: CHAPTER 4: DIAGNOSTICS 89 A;AD 4jrsin § y ^ I X , For a <IC 1, fluctuations are probed whose wavelength, A = 2n/k, is much less than the plasma Debye length. On this scale, Coulomb fields of the free charges are not effectively screened and the scattering electrons appear to have a random distri-bution. On the other hand, for a » 1, the fluctuations probed have wavelengths much longer than the plasma Debye length and the individual electrons are effec-tively shielded so only cooperative effects are apparent. Scattering with large values of a is called collective scattering. A scattering experiment is shown in Fig. 4-6. The light scattered in the volume V is detected by an observer located at point PT. Volume V is the scattering volume defined as that part of the plasma that is at the intersection of the incident laser beam and the region imaged onto the detector. We wish to calculate the scattering from both coherent and incoherent fluctuations located in the volume V and relate the intensity of the scattered light to the level of the fluctuations. Coherent fluctuations are those which have a single frequency and are in step so a single wave train fills the entire scattering volume. By incoherent, we refer to those fluctuations that have a single frequency but have an angular and spatial spread so the scattering volume appears to be filled with many waves having the same frequency and wavelength. Both cases apply to some extent in the present experiment. 4.5b Coherent FInctnations The electric field amplitude scattered from a single electron and observed at ( 4 - 7 ) CHAPTER 4: DIAGNOSTICS 90 Figure 4-6 A typical Thomson scattering experiment, point Pt at time t, is given b y , 6 4 ' 6 9 * W ) = ( 4 l r l ^ ^ (4~8) where E{0 is the time and space independent electric field amplitude of the incident light wave polarized in the direction of the unit vector, E ,„ . The geometry is defined in Fig. 4-6. The right hand side of Eq. 4-8 is evaluated at the retarded time, t'} to allow for the transit time of the scattered light from electron j to the observer. A single fluctuation is assumed to be present in the plasma so the electron density is given by: n(r',t') = no + 5ncos(k-r'-w<'), ( 4 - 9 ) CHAPTER 4: DIAGNOSTICS 91 where nQ is the background electron density, 6n is the fluctuation amplitude, and ±k = k, — k,. The total scattered electric field is found by summing (integrating) over the contributions from each of the electrons in the scattering volume, V] E^(R, t) = e2Eif * ( > Xf k + *ncos(k • r' - ut')} cos(k, • r' - u^dr'. v (4 - 10) Substituting t' = t - * + £s • r', for the retarded time, we find: + y c o s (kt- ± k - (w,- ± w) * ) r' + (oJi ± u) j - (wt- ± u)t J dr' c/ c ( 4 - H ) For V » I/Ac3 the large contributions in Eq. 4-11 occur for the cases when k, ± k = k« and a;,- ± u> = ut. The time averaged scattered power measured by a detector located at P, is therefore: dP. =\e0cR2\ETg\2 = l-e0cr2eE2i0 S x (S x E , J 2^V2d* = --{SnfPslLWjn, (4 - 12) where Pt- is the power in the incident laser beam, a0 is the radius and Lt is the length of the scattering volume in the direction of the incident beam, dCl is the scattered light solid angle, and re = e2/4ne0mec2 is the classical electron radius. It has been assumed that the incident light is polarized in the direction perpendicular to the scattering plane so |s x (i x E $ 0 )| 2 = 1. Therefore, after calibrating the detector response, it is possible to determine the fluctuation amplitude of a single monochromatic plasma oscillation from a measurement of the scattered power, Ps. This result has also been derived by Slusher and Surko. 7 0 CHAPTER 4: DIAGNOSTICS 92 It is worthwhile at this time to point out the origin of the angular spread in the scattered light, dCl. An infinite collimated light wave scattered by an infinite monochromatic series of Bragg planes would be expected to be scattered into a single direction. However, in any real experiment, only a finite region of plasma is illuminated by the incident laser beam and diffraction effects determine the spread in the scattered light. For an incident Gaussian beam with a 1/e waist of uQ, the scattered solid angle is Xf/nu 2,. 70 4.5c Incoherent Flnctnations As opposed to the previous derivation, no explicit-form of the density fluc-tuations is assumed in this section. Instead, the general form, n e(r', t'), is used and Eq. 4-10 becomes: e^(r' *>=e2EiL*JlxeRio)] I ne(r'>f,) co8(k«r' - tjit')dr'- (4 -i3) v The time averaged power scattered into the solid angle dCl is proportional to the time average of the square of the total scattered electric field; dP,(R,t) dn i r°° = e0cR2Km- l E ^ R , * ) ! 2 * , (4-14) 1 -*oo 1 J_oo or, dP,{R,t) dn e0cR2 lim 'Ut—duis/2 (u>,>0) dt (4 - 15) where, E f ( R , " . ) = B^R^e-^dt, J —OO (4 - 16) CHAPTER i: DIAGNOSTICS 9 3 is the temporal Fourier transform of E f (R, t). Parseval's theorem has been used in going from Eq. 4-14 to Eq. 4-15. It has been assumed that E , ( R , OJ) is significant only in the regions from — ut — dut/2 to — u, + dwt/2 and u, — dut/2 to ut + dwt/2 (a factor of 2 has appeared in Eq. 4-15 since only u, > 0 is considered). If we write n e(r', t') in terms of its Fourier transform: »«(*'.*') = / _ ^ 7 ^ 3 / _ ^ n e ( k , o ; ) e x p [ - t ( k r ' - u ; 0 ] , (4-17) and substitute t' = t — ^ + i s • p' for the retarded time, Eq. 4-15 becomes, d P , ( R , u , . ) _ e 4 E 2 J s x ( i x E J | 2 ^ u,+dws/2 dn im / -oo Tir J (4ne0)24irc3m2 Tijis—duis/2 (we>0) >v x i j e x p t j i (^-K-w,))t - (k - ( u — Ui c c + exp t (W - («. + Ui))t - (k - ( 5 i ± ^ i - k t ) ) r ' - } (4 - 18) Integrating over time t, we obtain the delta functions 2nS(u)—and Eq. 4-18 becomes: uis+dui,/2 dQ (4ire0)24irc2m2 r-oo Tir J J J.^ (2ir> u;»—rfu;»/2 V (w»>0) ^ | n e ( k ,u, - wi)exp-t (k - (k, - k , ) ) r ' + y i z j + n e(k, u>„ + w,-) exp - t (k-(k.+k,-)W + (4 - 19) CHAPTER 4: DIAGNOSTICS 94 If we let V —• oo and integrate over r' the delta functions, (2ir)35(k — (kt =f k,)), appear and Eq. 4-19 can be written as: dP.(Rtv.) _ Iirl H m 2ir T,V->oo ^ j dwe n e(k,+k , - ,£j , -f w,) + n e(k,-k , - , w , - a / t ) Us—dus/2 (ws>0) (4 - 20) where /,• is the incident intensity. In the present experiment, |s x (s x E, 0 )| 2 = 1. Finally, allowing u)g to be both negative and positive, Equation 4-20 can be written as: dn Us+dwa/2 IrL 1 f = t f r,vmoo f J D<JT NE(K* " K " U ' ~ U I ) hr\dut 2ir T,v^oo T (4-21) Equation 4-21 is often written in terms of the spectral density function defined b y : 6 4 1 <IMk,")|2> S^kjU/) = lim r,v-oo TV ne (4 - 22) where rieo is the average electron density. The angular brackets represent the ensem-ble average. It should be noted that S^k, of) is not dimensionless but has dimensions of time. The somewhat clumsy notation is a result of the Fourier transforms. For a stationary homogeneous plasma, we can write 5(k, u) simply as: 5(k, w) = lim 1 |ne(k,u>)|2 T,V-+oo TV n, (4-23) e o Using Eq. 4-21, we can write: (4 - 24) CHAPTER 4: DIAGNOSTICS 85 This result has been derived by Sheffield6 4 and Slusher and Surko 7 0 by somewhat different techniques. For scattering from ion-acoustic waves, the scattered light per unit frequency band is approximately constant for —w, a < u < Uia where <j,a is the ion-acoustic frequency. For this case we can integrate Eq. 4-24 over dw = dut and obtain: S,(k) = 2irdP,(R) Iir*neVdn (4 - 25) Finally, it is necessary to relate 5,(k) to the fluctuation amplitude. Consider the time and space average of |ne(r, t)/neo\2: ne(r,t) 1 f°° J ± f°° J ne(r,<)n!(r,<) =T^fvLdtLdr k • (4-26) Using Parseval's theorem and Eq. 4-23 and integrating over w, we have, neo \2JT/ T l e o J ^ (4 - 27) Assuming that ^ (k) is constant over the experimental range of k% we obtain the desired relation between the fluctuation amplitude and S^k): *M| =(±) ,mf f [A. neo \2irJ neo J J J (4 - 28) Exp. range of k's For the case of incoherent collective scattering, Eqs. 4-25 and 4-28 can be used to calculate the fluctuation amplitude from the measured scattered power. It is important to point out that the incoherent scattering result gives the space and time averaged fluctuation level. If a scattering experiment is set up so as to probe a single k vector in the plasma, the resulting fluctuation amplitude, as CHAPTER 4: DIAGNOSTICS determined by Eqs. 4-25 and 4-28, should be the same as that obtained with the assumption of coherent scattering (Eq. 4-12). 4.5d T h e T h o m s o n Scattering Experiments Thomson scattering was used in the present experiment in order to determine the location, amplitude, and wavelength (wavevector) of the ion-acoustic waves driven by stimulated Brillouin scattering. The optical setups used to measure the spatial structure and wavevector spectrum (k-spectrum) are shown in Figs. 4-7 and 4-8 respectively. The probe beam used in the experiments was a 6 ns, 6943A pulse produced by a Q-switched, mode-locked, and cavity dumped ruby laser system. 5 9 A horizontal line focus approximately 5 mm long and 200/im wide was formed at the location of the plasma by the combination of a plano-convex lens (f . l .~ 135 cm) and a cylindrical lens (f.l .~ 33 cm). A blackened baffle prevented the scatter off the entrance window from reaching the collection optics. Blackened cardboard located inside the chamber served as a viewing dump. The unscattered light was dumped into a Rayleigh horn located at the exit window. The scattered light passed through a plexiglass window and was collected by a 12.5 cm diameter, 30 cm focal length lens. Plasma light was prevented from reaching the streak camera by an interference filter (bandpass=110A F W H M centered at 6943A) located at the first image of the plasma. For spatial measurements, the plasma image was magnified by a second 5 in diameter, 30 cm lens and focussed onto the streak camera slit (25/xm). The streak camera (Hamamatsu Temporal Disperser C1370-01/LT.I—Analyzer System) produced a picture of the intensity falling on the slit as a function of time. The output from the streak camera was imaged onto a T V camera and the resulting picture was displayed on a T V monitor as well as digitized in a 256 x 256 array. The T V monitor could be photographed or the digitized picture transferred to tape for later computer analysis. CHAPTER 4: DIAGNOSTICS 97 ^Streak Carnea Figure 4-7 The Thomson scattering arrangement used for studying the temporal and spatial behaviour of the scattering region. As seen from Eq. 4-6, ruby laser (k = 9.05 x 106 m _ 1 ) scatter from ion acoustic waves driven by SBS backscatter (k = 2kco2 — 112 x 106 m - 1 at 0.1 nc) was expected to occur at 7.1°. Light was incident at an angle of ~ 3.3° with respect to the normal onto to the CO2 laser beam so scattered light was expected to be observed at ~ 3.8° . The k-vector of the probed plasma fluctuation therefore made an angle of 0.3° with respect to the CO2 laser beam. This is well within the 4.0° cone angle of the CO2 focussing optics. Various masks could be placed at the exit window to select the scattered k-vector. CHAPTER 4: DIAGNOSTICS Figure 4-8 The Thomson scattering arrangement used for studying the temporal behaviour of the ion acoustic k-spectrum. (The plasma fluctuation k-vectors are not drawn to the same scale as the ruby laser k-vectors.) For the k-spectrum measurements, the second 5 in diameter lens was moved so as to image a horizontal strip on the back surface of the first 5 in diameter lens onto the slit of the streak camera. The focus and magnification were determined with the aid of a grid placed in contact with the back surface of the lens. The range of scattering angles and hence ion-acoustic wave vectors that could be studied was limited by the size of the exit port on the vacuum chamber as well as by the size of the first collection lens, k-vectors up to k = 4k0 (at 0.1nc) were studied, corre-sponding to scattering angles of 14.3° from fluctuations with wave-vectors making an angle of 3.8° with respect to the incident CO2 laser beam. A monitor signal was split off from the main ruby beam and sent to the streak camera slit through a ~ 20 cm length of single mode optical fiber to produce CHAPTER i: DIAGNOSTICS 99 a fiducial on the streak record. Dispersion in the optical fiber was found to introduce little change in the beam's temporal structure; a 52 ps dye laser pulse was observed to have a period of 59 ps after passing through the fiber. Refraction of the probe beam into the direction of the expected Thomson scattered light can only occur in the presence of extremely steep density gradients in the plasma. The fact that both shadowgraphy and schlieren images of the plasma were obtained in ruby light demonstrated that refraction was occurring. To deter-mine the magnitude of this refraction, ray tracing calculations (see Appendix 2) at 6943A were performed on several density distributions that were obtained from interferometry. The results indicate that the maximum deflections due to refraction were of the order of 0.1° and hence did not obscure the Thomson scattering results. 4.5e Intensity Cal ibrat ion The fluctuation levels of the ion-acoustic waves could be determined from measurements of the scattered power only after the scattering optics were cali-brated. Rayleigh scattering from a neutral gas is often used for this purpose. 6 5 However, because of the small scattering angles in the present experiment, the Raleigh and Thomson scattering volumes were different, making this technique un-reliable. Therefore, another method of calibration was used. A partly diffusing mylar sheet located directly above the gas jet was used to scatter ruby laser light. After passing through a small aperture that was positioned so as to select scattering angles similar to those observed in the experiment, this scattered light was focussed in the usual manner onto the streak camera slit. The streak camera only measured the intensity of light scattered by that part of the mylar that was imaged into the slit. A Hamamatsu photodiode (model M103U-03-175) was then placed directly in front of the slit and the total scatter from the illuminated part of the mylar was measured. The same photodiode was used to measure the unscattered intensity with the mylar removed. The fiducial monitored CHAPTER 4: DIAGNOSTICS 100 the ruby laser intensity for all shots. Since each illuminated section of mylar is expected to scatter the same fraction of light through the aperture which selected the scattering angles, the photodiode and streak camera should detect the same ratio of scattered to incident light from the respective imaged sections of mylar. Therefore, from the photodiode measurements, the streak camera and focussing optics were calibrated for a known reflectivity. Besides a knowledge of the scattered power, a calculation of the coherent fluc-tuation level (Eq. 4-12) requires values for the scattering volume, V , and scattered solid angle, dQ. The scattered solid angle was measured in the k-spectrum exper-iment (Fig. 4-8) and was found to be (2.6 ± 0.4) x 10~4 sr (see Chapter 5). The scattering volume had a height and width of ~ 6 fim and ~ 20 /xm respectively (the imaged size of a single streak camera pixel) and a depth of <- 100/xm (the width of the focussed CO2 laser beam). However, due to the small solid angle of the scattered light, the effective aperture of the scatter collection lens was small and so diffraction effects limited the resolution. Therefore, the values for the height and width are almost assured to be underestimates. If the fluctuation level is calculated using Eqs. 4-25 and 4-28 (incoherent waves), then only the depth of the scattering volume, the scattered solid angle, and the experimental k-vector range are required. The range of the k-vectors was calculated assuming that their directions uniformly filled the cone angle of the CO2 focussing optics (half angle~ 4°) and that their magnitudes ranged from I.86&0 to 2.14fco (at 0.1nc). The magnitude range was determined from the observed scattered solid angle. 4.5f Temporal Cal ibrat ion The timing for the interferograms and streak photographs was determined from observations of the plasma density and associated plasma light. It was ex-perimentally observed that it took ~ 1 Joule of laser energy to break down the gas CHAPTER 4: DIAGNOSTICS 101 jet. Therefore, for a typical shot of 6-7 J , the first plasma appeared at ~ 0.3 ns into the laser pulse. This observation was checked by sharply focussing the trans-mitted CO2 laser beam onto a carbon block and imaging the light from both the jet plasma and the carbon plasma onto the streak camera slit. Provided that the carbon plasma was formed almost instantaneously (< 100 ps into the pulse), the two timing techniques gave consistent results. 4.6 The Streak Camera Intensity Response All Thomson scattering intensity measurements were made with the Hama-matsu streak camera. This detector is reported by the manufacturer to have a dy-namic range of ~ 80 but no further information is provided concerning the linearity of the intensity response of the particular unit used in this experiment. Therefore an experiment was performed to determine the linear range of the streak camera-T V camera-digitizer system for conditions similar to those used in this study. The streak camera slit was illuminated with ruby laser light by bringing a ruby beam to a line focus on a piece of partially diffusing mylar and then diffusely focussing the image of the mylar onto the slit. The resulting response to this illu-mination was measured and later used in the calibration of the intensity response measurements. A gradual decrease in the focus mode (d.c.) sensitivity of ~ 50% between the centre and the ends of the slit was observed by moving the defocussed image along the length of the slit. (A HeNe laser beam, collinear with the ruby beam, was used for this measurement.) Since the focus mode and streak mode out-puts had similar shapes, it is reasonable to assume that the streak mode sensitivity also rolled off by ~ 50% between the centre and the edges of the slit. The intensity calibration was performed by measuring the response to light which passed through a 6-step calibrated neutral density step wedge placed directly in front of the slit. The results are shown in Fig. 4-9 for two settings of the streak camera micro-channel plate (MCP) accelerating voltage. Several shots were made at CHAPTER 4: DIAGNOSTICS 102 1 5 10 50 100 500 1000 Intensity (arb. units) Figure 4-9 The intensity response of the streak camera system. Note that addi-tional attenuation was added to the ruby beam for the M C P = 4 measurements. each setting in order to determine the response over a wide range of intensities. The uncertainty of the signal levels, set mainly by nonuniformities in the illumination, was ± 1 0 % . As expected, the dark signal level increased when the micro-channel plate setting was changed from M C P = 3 (~ 5 counts) to M C P = 4 (~ 10 counts). For both settings, the response was approximately linear from the dark signal level to ~ 500 to 600 counts, resulting in a dynamic range of about 100. Above approxi-mately 800 counts the response became very nonlinear and eventually saturated at 1050-1100 counts. CHAPTER 5: RESULTS 103 C H A P T E R 5 R E S U L T S 5.1 Introduction The results of the stimulated Brillouin scattering experiments are presented in this chapter. The important plasma parameters, the density and temperature, have already been discussed in Chapter 4. Further details on the plasma conditions were obtained from the backscatter and Thomson scattering measurements and were used to construct a fairly detailed picture of the plasma conditions in the SBS interaction regions. Details of the backscatter and sidescatter measurements as well as the results from the spatially and k-vector resolved Thomson scattering experiments are presented in this chapter and compared to one another. Where appropriate, the results of the present experiment are compared to results from previous work in other laboratories. Detailed discussion of the results is left to Chapter 6. 5.2 T h e Backscatter Reflectivity This section reports on the results of the infrared backscatter reflectivity measurements. Radiation scattered back into the focussing lens is here referred to as backscatter. The backscatter reflectivity is then defined as the ratio of the amount of backscattered to incident radiation. Both the energy and power reflectivities were measured using the arrangement shown in Fig. 4-4 and are plotted as a function of CHAPTER 5: RESULTS 104 total laser energy in Fig. 5-1. The results from measurements spread over several days were averaged to construct this graph. The error bars represent the standard deviation of the mean of from 2 to 30 points in a given energy bin. 10-at ° 1-If) o o m 0.5-- I — I — r • > i r - i — i — i — i — r < > H f 1 o Power Reflectivity o Energy Reflectivity 0.11 i0 i — i — i — i — i — i — i — i — i — i — i — i i 0 2 U 6 8 10 12 M Laser Energy (J) Figure 5-1 The backscatter reflectivity as a function of incident laser energy. The majority of the backscatter is due to SBS. The timing of the backscatter signal as well as the fact that no backscatter was observed when the chamber was completely evacuated rules out reflection from the apparatus as a source of scatter. Since the plasma density was always subcritical, there was also no reflection from a critical surface. In addition, Fresnel reflection off sudden changes of refractive CHAPTER 5: RESULTS 105 index can be expected to be negligible since the density gradient scale length was long compared to the CO2 laser wavelength. For low laser energies (1-4 J), the SBS reflectivity was observed to be a strong function of the incident energy. The growth in this region was roughly exponential increasing at a rate of (4.4 ± 1.3) J - 1 . However, above 4 J incident energy, the reflectivity changed very little and appeared to saturate at 7-8% for energies above 6 J. Since no plasma was formed for laser energies below ~ 1 J, some of the energy dependence in the reflectivity was probably due to changing plasma conditions. The saturation behaviour, however, indicates that the plasma conditions were not the dominant parameters determining the reflectivity at the higher energies. The backscatter reflectivity as function of time, given by the instantaneous reflected to incident power ratio, is shown in Fig. 5-2 for various total laser energies. Each of the curves in Fig. 5-2 represents the average of several temporal profiles in the same energy bin. Also shown, by arrows, is the time at which a tempo-rally integrated total energy of approximately 0.8 J had been delivered by the laser. This time roughly corresponds to the time at which the plasma was first formed. The backscatter began approximately 200-300 ps later into the pulse. Observations of the temporal behavior of the transmitted light also revealed an interesting be-haviour. Close to 100% transmission occurred up to the time when the plasma began to form. The transmission then rapidly dropped to zero, probably because of absorption, reflection, and severe refraction of incident beam (see the results of the ray tracing shown in Figs. 4-3). The time corresponding to the end of transmission is shown in Fig. 5-2 by the second set of arrows. There was a clear temporal modulation in the backscatter reflectivity, espe-cially at the higher laser energies. The backscatter usually appeared as two peaks and occasionally as three. It can be seen from Fig. 5-1 that the energy reflectivity was generally less than the peak power reflectivity. This is to be expected if the CHAPTER 5: RESULTS 106 0 1 2 3 Time after peak of incident pulse (ns) Figure 5-2 The temporal behaviour of the backscatter reflectivity. reflectivity did indeed increase with the instantaneous intensity and is also a con-sequence of the reflectivity modulations. However, since the photon drag detector-scope combination had a risetime of ~0.5 ns, changes in the reflected power more rapid than those shown in Fig. 5-2 were filtered out. This implies that the true power reflectivities were likely higher than measured. We can obtain some idea of the actual duration of the backscatter by considering the results of previous work. In previous CO2 laser experiments with underdense gas targets, SBS backscat-ter reflectivities ranging from only a few percent 7 1 ' 7 * ' 7 8 ' 7 4 to greater than 50% 7 6 ' 7 8 ' 7 7 have been observed. The SBS backscatter was found to be very spikey in nature v CHAPTER 5: RESULTS 107 in all the long pulse exper iments 7 1 , 7 2 , 7 * ' 7 4 ' 7 6 ' 7 7 ('incident > 10 n s ) suggesting that some sort of saturation mechanism caused the instability to collapse after it had grown to a certain level. The vastly different measured peak reflectivities, ranging from less than 1% 7 1 ' 7 4 to greater than 50%, 7 6 ' 7 7 , however, suggest that possibly some other mechansim, such as changing plasma conditions, may have been re-sponsible for the spikey behaviour. Due to detector risetimes of > 1 ns, it was not possible to determine the duration of the spikes except that it was less than 2 ns. In other short pulse gas target experiments 7 5 ' 7 8 single backscatter pulses, considerably shorter than the incident pulse were observed, but again, no structure finer than several hundred picoseconds could be resolved. A recent C O 2 laser experiment,7 9 employing an optical Kerr cell in conjunction with a streak camera to permit time resolution of the order of < 9 ps, found that the SBS backscatter occurred mainly in a single pulse of length ~400 ps ('incident = * n s ) » f ° U ° w e d by a few smaller pulses. If we assume that in the present experiment, the backscatter also occurred in pulses of several hundred picoseconds duration and were instrument broadened (instru-ment F W H M ~ 500 ps) to the observed F W H M of ~ 800 ps, then the deconvolved pulse length was approximately 600 ps. This result indicates that the measured peak power reflectivities were low by a factor of approximately 0.75. Therefore, the power reflectivity likely saturated at approximately 10%. 5.3 T h e Backscatter Spect rum The backscatter spectrum was measured with the image dissector as described in Chapter 4 and Appendix 3. Previous SBS backscatter experiments performed in this laboratory have measured the backscatter spectrum in a shot to shot manner. 1 8 This technique, however, relies on the assumption that the spectrum at a given laser energy changes very little from shot to shot. This was not the case in the present experiment. CHAPTER 5: RESULTS 108 At/(GHz)-30 -2*0 -l'o 6 AO *20 LV (GHz) -30 -20 -10 0 •10 *20 Figure 5-3 Examples of the backscattered spectrum. Sample spectra which show all the observed features are shown in Figs. 5-3. The instrument profile, obtained by placing a mirror in front of the focussing lens to reflect the incident beam into the backscatter diagnostics, is shown for comparison. Typically, the instrument profile had a F W H M of 2-3 channels. The wavelength of the peak of the backscattered spectrum was observed to change considerably from shot to shot but was almost always found to lie at one of three positions which corresponded to red shifts from the P(20) CO2 line of ~ 7 A, 33 A, and 61 A. This is illustrated in Fig. 5-4 where the number of times a given spectral peak or prominent shoulder was measured is plotted versus the shift from the incident CO2 line. Data from two days are shown. Since the image dissector channels were separated by either 11 A (1st day) or 13 A (2nd day), the uncertainty in the positions of the peaks is approximately ± 5 A. In addition, the relative heights of the various peaks could vary considerably depending on the positions of the peaks relative to the CHAPTER 5: RESULTS centres of the channels. This probably explains why the central peak, so prominent on the 1st day, was missing on the 2nd day. 109 25T i 1 1 1 1 1 1 1 r J220-c Figure 5-4 Histogram of the position of the peaks and prominent shoulders in the backscattered spectrum. The central peak (AA ~ 33 A) was the most prominent feature on 47% of the shots while the peaks with shifts of ~ 7 A and 61 A were dominant on 44% and 9% of the shots respectively. At low energies {Eraser < J), the spectrum was usually narrow and consisted of only one peak, usually the central one. The average F W H M of this peak, after subtracting off the instrument profile (assumed to be Lorentzian), was ~ (10 ± 3) A. As the laser energy increased, the spectra usually broadened and the other peaks or prominent shoulders appeared. An additional feature became detectable at high laser energies in the form of at least two temporally separated spectra which often differed considerably in CHAPTER 5: RESULTS 110 appearance. It is believed that the appearance of these two temporally separated spectra is related to the fact that there were also two temporal peaks in the total backscatter reflectivity at high laser intensity. There was a trend for the peak of the second spectrum to have a larger red shift than the peak of the first spectrum (see Fig. 5-3b). In fact, if the two spectra were normalized to the same peak intensity, they often appeared to be mirror images of each other about a line located at a shift of ~ 33 A . The large temporal changes in the spectrum, indicate that considerable blurring of the single or double spectra likely took place. Therefore, only gross features of the spectra should be considered in any analysis. Temporal changes 7 9 ' 8 0 - 8 4 and multiple p e a k s 1 6 ' 7 6 ' 7 6 - 7 9 ' 8 0 ' 8 1 ' " ' 8 8 ' 8 4 have been seen in backscattered spectra in a number of previous experiments. Several expla-nations for these features have been proposed. Additional blue shifted peaks in the spectra have often been attributed to classical reflection from the moving critical density front which is present in solid target experiments. 8 0 ' 8 1 ' 8 ' In addition, SBS reflection from an expanding plasma can be blue shifted if the expansion velocity is supersonic.8 0 Other authors 7 6 ' 7 6 have attributed the presence of several spectral peaks, some blue shifted, to an anti-Stokes branch in the Brillouin process. This could come about if the backscattered wave generated in the primary SBS process is of sufficient intensity to cause SBS scattering itself. The resulting backward mov-ing ion acoustic waves could then scatter the incident laser beam, leading to a blue shift. The frequency splitting of the primary and secondary backscattered waves is then given b y 7 6 ' 7 9 Aw = 2uia = 4k0c„. (5 - 1) In addition, it has also been shown through simulations1 6 that multiple reflec-tions off the plasma boundaries can lead to multiple backscattered lines separated by the ion acoustic frequency. Finally, changes in the laser intensity during the pulse can lead to several lines in the backscattered spectrum since the frequency of CHAPTER 5: RESULTS 111 the SBS driven ion acoustic waves is intensity dependent 7 9 ' 8 8 (i.e. quasi-modes are generated). In the present experiment, the mean separation between the 3 spectral peaks is A A = (27 ± 5) A or Au = (45 ± 8) x 109 rad/s whereas the ion acoustic angular frequency of the fastest growing mode for our conditions (incident — (5 —"10) x 101 2 W / c m 2 , Te = 300 eV, ne = O-ln^) is approximately 2 x (2k0ct) = 270 x 109 rad/s (see Fig. 2-3). Therefore, it is unlikely that either multiple reflections from the plasma boundaries or anti-Stokes scattering can account for the observed spectra. The large discrepency between the observed red shifts and the predicted red shift of 160 A (w,a = 270 x 109 rad/s) is believed to be due plasma motions caused by a blowoff of the interaction plasma from the denser plasma located in the jet. One possible explanation for three shifted peaks is that the scatter arose from three regions of plasma, each with a different blowoff velocity. In order to test this hypothesis , the spectrum of the SBS sidescattered light was measured for scattering angles of both 135° and 90° . 5.4 T h e Sidescatter Spectrum Stimulated Brillouin sidescatter should occur under most conditions that are also favourable for the strong growth of SBS backscatter. The SBS growth rate is proportional to [cos 9} * / 2 sin <f> where 9 is the angle between k and k<> and <f> is the angle between E 0 and the scattered wavevector, k- . 1 1 Therefore, the growth rate for backscatter should be only slightly larger than that for large angle sidescatter. In cases where the incident k-vector, k„ , lies along the density gradient, the growth of backscatter can be greatly reduced because of phase mismatches. If the sidescat-tered k-vector lies in a direction perpendicular to the density gradient, however, it does not suffer from phase mismatches due to plasma inhomogeneity, and so under certain conditions the sidescatter growth rate can approach that for backscatter.85 The growth of the sidescatter instability is usually limited in practical experiments CHAPTER 5: RESULTS 112 by the size of the interaction region, which in turn is determined by the diameter of the incident laser beam. Experiments 7 8 ' 8 0 ' 8 8 ' 8 7 have usually found a high degree of ray retracing, sug-gesting a dominance of backscatter over sidescatter much stronger than predicted purely from the ratio of growth rates. It has been suggested88 that this ray retrac-ing is due to the interference between the nonparallel (focussed) components that make up the incident beam. This interference produces an identical interference pattern in the SBS gain coefficient which can act as a volume hologram. Therefore, any beam travelling in a direction opposite to any part of the incident beam will grow rapidly and Bragg diffract into those directions opposite to the other parts of the incident beam. This means that any backscattered beam will be diffracted so as to fill the cone angle of the incident pump beam. Since the growth rate for sidescatter is slightly less than for backscatter, sidescattered beams do not grow as rapidly and are diffracted less efficiently. Since both sidescatter and backscatter are due to Bragg reflection from driven ion acoustic waves, the resulting red shifts are expected to be different. For the present experiment, we wish to calculate the shift of the SBS sidescattered spectrum, taking into account any plasma blowoff. The situation is illustrated in Fig. 5-5. Incident light with wavevector, k 0 , is scattered at an angle, /9, by ion acoustic waves with wavevector, k , a , which move at a velocity of v , 0 . v t a results from two motions as shown in Fig. 5-5: the velocity of the ion acoustic waves with respect to the plasma, v f l u c t u a t i o n = v f l u c t u a t i o n c , i i r L , and the blowoff velocity of the plasma with respect to the lab, V b i o w o n r = M c , . Here, M is the blowoff Mach number with respect to the field free ion acoustic velocity and is assumed to lie in the direction anti-parallel to k 0 . (From symmetry considerations, this is likely the case in the present experiment.) Solving for « , 0 in terms of c,, M , and /? we find for CHAPTER 5: RESULTS 113 Figure 5-5 The geometry for sidescatter from a blowoff plasma, the wavelength shift, v where Ki is the calculated ratio of the SBS driven ion acoustic wave frequency to 2kacs (see Fig. 2-3). The predicted line shifts for sidescatter at 135° and 90° are listed in Table V - I . These shifts were calculated from the shifts of the observed lines in the backscatter spectrum using Eq. 5-2. Plasma conditions of Te = 300 eV and ne = 0.1 ncr were assumed and KL, the correction factor for the driven ion acoustic frequency, was taken to be equal to 2.0. Since the backscattered light was thought to originate in three separate scat-tering regions, the sidescatter optics were set up so as to allow scattered light to be collected from different parts of the interaction region. This was accomplished by 2\0KLc, ft n e V / 2 . 13 AA = I 1 J sin — c \ ncrJ 2 CHAPTER 5: RESULTS 114 Table V - I Backscatter Blowoff Mach Predicted Predicted Shift Velocity Number Shift Shift (A) (m/s) M at 135° at 90° (A) (A) 7 2.3 x 105 1.9 7 7 33 1.9 x 105 1.6 32 30 61 1.5 x 105 1.2 59 52 changing the angle of the first collection mirror. Unfortunately, at the time when the sidescatter data was obtained, it was believed that the backscatter was from the plasma at the front and back of the jet and so the sidescatter optics were set up to sample the scattered light from these regions. However, both refraction of the scattered light within the plasma and the method used to focus the scattered light onto the spectrometer slit, probably permitted some of the scatter from the blowoff plasma to be sampled. Approximately 0.2% of the incident light was sidescattered at 135° into the //2.5 collection optics. This sidescatter exhibited two prominent peaks in its spec-trum at redshifts of 0-20 A and 50-80 A. In addition, there often was a strong shoulder or peak at a red shift of 100-110 A. The relative strengths of the various features varied considerably from shot to shot, but were not found to depend on the laser energy or the region in the jet which was sampled. However, the shifts of the two main peaks at 0-20 A and 50-80 A were observed to increase further to the red as the laser energy was increased. This observation is in contrast to what was observed for backscatter for which no energy dependence was seen in the shifts of the three peaks. The sidescatter spectrum at 90° was considerably more complicated than that at 135°. The spectrum for 90° sidescatter was composed of at least two peaks; a very prominent one at a redshift of 60-80 A and another one with zero average shift. This latter peak sometimes appeared double with average shifts of 20 A to the blue CHAPTER 5: RESULTS 115 and 10 A to the red. The relative prominence of the peaks at shifts of 60-80 A and 0 A did not appear to depend on the incident laser energy or the region in the jet which was sampled. However, at the sampled position furthest toward the back of the jet, a spectral peak with a red shift of at least 110 A appeared. (Because of the complicated sidescatter collection system, the actual position was difficult to estimate.) This peak dominated the spectrum from this region of plasma at least for small incident laser energies. Finally, at the sampled position furthest toward the front of the jet, the spectrum was observed to extend far into the blue. A prominent peak with a blue shift of 80-120 A was observed on one shot. There is poor agreement between the predicted shifts and those observed. For the 135° scatter, the observed shifts of 0-20 A and 50-80 A agree well with the predicted shifts of 7 A and 59 A respectively. However, the predicted peak at a shift of 32 A, which corresponded to a very prominent peak in the backscatter spectrum, was absent from the sidescatter spectrum. In addition, the peak in the sidescatter data at 100-110 A was not predicted from the backscatter data. For the 90° scatter, only the observed shift of -20-10 A agrees well with any of the predicted shifts. The predicted peak at a shift of 30 A is missing while there is considerable disagreement between the predicted (52 A) and observed (60-80 A) shifts of the remaining peak. The poor agreement in the spectral behaviour of the backscattered and sidescattered light as well as the fact that the sidescatter spectra demonstrated no dependence on the region of the jet sampled, indicate that the original conjecture of three scattering regions is probably incorrect. However, there may have been a common source for the multiple peaks in the backscatter and., sidescatter spectra. This is discussed further in Chapter 6. It remains to suggest a possible source for the spectral peaks with shifts > 100 A in the 90° and 135° sidescatter spectra. The 90° data suggest that this scatter was generated at the far back of the interaction region (no definite source was found for the 135° scatter). If this region was the region around the front of CHAPTER 5: RESULTS 116 the jet, then the density gradients would have to be such as to prevent backscatter but still allow sidescatter. In addition, the blowoff velocities in the region of the jet were likely lower than in the main interaction region located out in front of the jet. If we assume zero blowoff velocity in the region of the jet, then the shifts of the sidescattered spectra are (from Eq. 5-2) ~ 150 A (135°) and ~ 115 A (90°) . The observed shifts are 100-110 A (135°) and > 110 A (90°) . The agreement is reasonable. Refracted Beam Figure 5-6 Refraction of the incident laser beam in an expanding plasma. Finally, we will estimate the plasma blowoff velocity in the interaction region located in front of the jet. A measurement of the radius of the ring of high density plasma (0.3ncr) as a function of time from the interferograms shown in Figs. 4-3, gives a radial expansion velocity of approximately 3 x 105 m/s. However, the radial expansion velocity in the interaction region is expected to be greater than the axial CHAPTER 5: RESULTS 117 expansion velocity since the laser energy was deposited over a large region in the axial direction. Another estimate of the blowoff velocity can be obtained from the blue shift of 80-120 A which was observed in the 90° sidescatter spectrum. This shift may have been the result of refraction of part of the incident beam in the plasma which was expanding away from the jet region (see Fig. 5-6). The required blowoff velocity can then be estimated by treating the refraction as a reflection off a mirror moving at an angle of 135° with respect to the incident beam. The resulting velocity is 1.7-2.5xl05 m/s which agrees well with the blowoff velocities calculated from the observed shifts of the peaks in the backscatter spectrum (see Table V-I) . Therefore, it is likely that the small redshifts in the SBS backscattered spectrum were the result of considerable plasma motion. 5.5 T h o m s o n Scattering The results of the ruby laser Thomson scattering are described in the following sections. Details of the spatial and temporal behaviour of the ion acoustic waves as well as their fluctuation levels and spectral behaviour are presented. 5.5a Spatial and Temporal Behaviour Whereas the SBS backscatter was the result of the combined scatter from the entire interaction region and therefore permitted no spatial resolution, the Thomson scattering was performed almost side-on and hence allowed details of the spatial structure of the ion acoustic waves to be studied. Therefore, the Thomson scattering results were most important in the interpretation of the backscatter observations. Without these results, many of the conclusions concerning the backscatter could not have been made or would have been considerably less certain. The Thomson scattering setup shown in Fig. 4-7 was used to study the tem-poral and spatial behaviour of the ion acoustic waves. The scattering angles and wavelength discrimination were chosen so only scatter from ion acoustic waves with CHAPTER 5: RESULTS 118 k < 4ka could be detected. For the SBS driven wave, k ~ 2ka, resulting in a value for kXrj of 0.15 or a — 6.9. Therefore, the experimental situation was that of col-lective scattering in a regime where the ion feature in the scattered spectrum was well separated from any electron wave features. The predicted spatial resolution in the measurements was limited by the demagnified image of the streak camera pixels. For a magnification factor of 4.4, this coresponds to a resolution of ~ 6 fim in the vertical direction and ~ 20 fim along the CO2 laser axis. However, the full 5 inch aperture of the two focussing lenses was not used in the actual imaging since the interaction region was imaged with the scattered ruby laser light. 8 9 This scat-tered light was observed to have a scattering half angle of ~ 9 mrad (see Sec. 5.5e). Therefore, the spatial resolution was likely no better than ~ 100 fim. Typical streak records which show all the main features of the spatially re-solved Thomson scatter are presented in Figs. 5-7 and 5-8. (Note the difference in the streak speeds for the two figures.) These computer plots were produced by plotting at every point in the 256 x 256 grid of the streak camera system, a square box, the length of which was directly proportional to the measured intensity. The blackened area at each grid point therefore corresponds to the square of the scat-tered intensity. This plotting criteria was compared to others and found to result in the best greytone scale. Also shown in Figs. 5-7 and 5-8 are the position and time scales which correspond to those of the interferograms shown in Figs. 4-3. In addition, Fig. 5-8 shows a plot of the axial plasma density at various times during the period of the laser pulse. These profiles were determined from the interfero-grams, by plotting the density along a light ray which started out on a path 10 fim from the CO2 laser axis (this ray stayed close to the axis until it reached the higher density plasma at the edge of the jet). It is clear from Figs. 5-7 and 5-8 that the enhanced ion fluctuations, believed to be the ion acoustic waves driven by by SBS backscatter, occurred in two regions of plasma. These waves first appeared in the blowoff plasma near the front edge CHAPTER 5: RESULTS 119 t Fiducial Jet | Background Gas Position (mm) <^=co. Laser 5 4 3 2 2 — i • 1 1 • (-r I*'u ink. If' -ft . M R * Enhanced fluctuations 3.7 Joules M.0 Peak of *~Laser Pulse H 5 * • ; :{;-Vf^ -' h2o 3 in. Figure 5-7 Streak record of the spatially resolved Thomson scattered light. of the jet and rapidly spread further into the blowoff (or background gas) plasma. After reaching a length of ~ 1 mm, the interaction region (region of enhanced fluctuations) ceased growing and the fluctuations gradually faded away. About the same time as the fluctuations in this region began to decrease, strong fluctuations were produced in a second region located far in front of the jet in the background gas plasma. Fluctuations in this region soon became dominant and, at low laser energies, persisted nearly until the end of the laser pulse. However, at higher powers, the enhanced ion acoustic wave appeared to die away before the end of the laser pulse. CHAPTER 5: RESULTS Figure 5-8 Streak record of the spatially resolved Thomson scattered light and behaviour of the plasma density in the scattering regions, a.) The axial plasma density as a function of position and time, b.) Streak record. The plasma density along the axis of the CO2 laser beam is shown in Figs. 5-9 for average incident laser energies of 2.8 J , 4.1 J, and 6.7 J . (Fig. 5-8a is repeated as Fig. 5-9c.) The average scattering regions are indicated. Although the rate of expansion of the ionization into the background gas increased with increasing laser energy, the plasma density profile varied very little. At early times in the scattering, the plasma density in the rear scattering region ranged from 0 at the front to ~0.3 ncr at the rear edge. As time progressed, the density at the back edge CHAPTER 5: RESULTS 121 decreased, probably because of plasma expansion and blowoff. At the same time, the density at the front edge of the rear scattering region grew to 0.10-0.15 n c r . Typical density scale lengths (Ln = ne(dne/dl)~x) in this scattering region ranged from Ln = 0.25 mm at the start of the scatter to Ln = 2.5 mm at about the time the scatter switched to the forward scattering region. The density in the forward scattering region typically ranged from 0.05 n c r to 0.15 n c r . Here, the plasma density profile was almost flat. (Most of the ripples in the density are most likely artifacts of the interferometry.) Since the CO2 laser beam was focussed at a point near the centre of the gas jet while the SBS interaction regions were located well in front of the jet, the laser intensities which drove the SBS instability were considerably lower than predicted from the measured focal spot size. Assuming the beam had an approximately Gaussian intensity crossection, with a 1/e radius of 48 fim (see Appendix 1), the intensity in the interaction regions were reduced from the value at the point of best focus by factors of 0.16 (rear scattering region) and 0.06 (forward scattering region). These correction factors to the focussed intensity were applied in the calculation of the SBS thresholds and growth rates. 5.5b Spatial and Temporal Modulat ions So far, we have considered only the large scale spatial and temporal behaviour of the Thomson scattered light. Many of the streak photographs, however, showed large fluctuations in the scattered intensity in both space and time. This behaviour can be seen in Figs. 5-7 and 5-8. Some of the modulation in in the scattered light can be attributed to fluctuations in the probe beam intensity due to beating of the transverse and longitudinal modes. However, in most cases, the observed modulations were uncorrected to the modulations in the fiducial and were of con-siderably larger amplitude and more regular than those observed when the probe beam was scattered off mylar. Therefore the spatial and temporal modulations in CHAPTER 5: RESULTS 122 Position (mm) Scattering Regions — * ' I Time (ns) Position (mm) Figure 5-9 The axial plasma density as function of position and time. (With re-spect to the start of the laser pulse.) The scattering regions are indicated, a.) 2.8 J, b.) 4.1 J, and c.) 6.7 J incident laser energy. CHAPTER 5: RESULTS 123 the scattered intensity can be attributed to actual modulations in the SBS driven ion acoustic wave amplitude. The spatial modulations in the ion acoustic wave amplitude appeared to be quite irregular. On most shots there was little or no periodic behaviour. This irregularity in the wave amplitude indicated that the SBS backscatter likely did not occur off a smooth ion acoustic wave train, but rather off of local concentrations of waves. These regions of large fluctuation amplitude were found to vary in length from < 100 fim (instrument limited) to 200-300 fim. In previous experiments which studied spatially resolved Thomson scatter from SBS driven ion acoustic waves , 8 0 ' 9 0 ' 9 1 the ion acoustic wave amplitude was observed to vary smoothly in space in the manner predicted for convective growth. In the present experiment, it appears that if there was convective growth, then it must have occurred somewhat independently in several parts of the interaction region. While the spatial modulations in the Thomson scattered light appeared to be irregular, the temporal modulations often exhibited a periodic behaviour. (See Figs. 5-7 and 5-8). Two modulation periods were apparent from the streak records. For slow streak speeds, a temporal modulation with a period ranging from 100-200 ps was visible, while at higher streak speeds a second moduation period of 40-70 ps became apparent. These values should be compared to the ion acoustic wave period of ~ 50 ps. Since both modulation periods were occasionly seen in a single streak record, it is believed that they resulted from different mechanisms. The measured periods are plotted in Fig. 5-10 as a function of the incident laser energy and peak intensity (at the centre of the rear scattering region). Three to six modulation cycles were used to determine each of the plotted points. The contributions to the total modulation period from each of the two suspected periods as well as from the intensity fluctuations in the probe beam made it difficult to CHAPTER 5: RESULTS 124 0 2 4 6 8 10 Laser Energy (J) Figure 5-10 The temporal modulation period as function of the incident laser energy. The moduation period predicted from the trapped ion model (see Sec. 6.5b) is shown by the shaded region. determine accurately the modulation period. Therefore, the uncertainty in each point plotted in Fig. 5-10 was estimated to be 20-30%. There was considerable shot to shot variation in the qualitative features of the temporal modulation. For laser energies below ~ 3 J, very little modulation occurred. At higher energies, however, the depth of the long period modulation was usually less than 50-60% while the depth of the shorter period modulation approached 100% on many shots. Both periods were independent of the laser energy CHAPTER 5: RESULTS 125 over the range of energies studied. The temporal modulations were often seen to extend over much of the interaction region. This result was somewhat unexpected since the spatial behaviour suggested that there were several separate regions of ion acoustic waves within the interaction region. However, the growth of waves in one region may have influenced the growth of waves in other regions through the backscattered electromagnetic waves. (This mechanism requires some coherence between the waves in the separate scattering regions.) Temporal modulations in the ion acoustic wave amplitude have been seen in previous experiments. Walsh and Baldis 9 ' observed close to 100% modulation for laser energies above a certain threshold. The modulation period was observed to vary from 100-150 ps at lower incident intensities to ~50 ps at higher intensities. Gellert and Kronast 9 1 have also seen some high frequency (Period ~200 ps) mod-ulations. Possible mechanisms for producing high frequency modulations in the ion acoustic wave amplitude are discussed in Chapter 6. In the following section, the actual amplitudes of the ion acoustic waves are calculated from the intensity of the scattered ruby light. 5.5c The Fluctuation Ampl i tude The SBS ion acoustic fluctuations were shown in the previous section to vary considerably in amplitude across the interaction region. This bunching of the waves implies that the SBS scatter likely did not arise from scatter off a single coherent wave but rather from scatter off of several waves. In addition, since the CO2 laser beam was not comprised of a single plane wave but was rather a focussed beam which contained a distribution of wavevectors, the SBS driven ion acoustic wavevectors likely had a comparable spread in propagation angles. For these reasons, it is reasonable to assume that,the driven ion acoustic waves are better approximated by incoherent rather than coherent waves. Therefore, Eqs. 4-25 and 4-28 were used CHAPTER 5: RESULTS 126 to calculate the ion acoustic wave amplitude. These equations are repeated here for convenience: 2irrfP.fR) _ 2*dP,(R) and n ^ f f i y m r f fa. rieo I \2ir/ neo J J J ( 5 - 4 ) Exp. range of k's In Eq. 5-3, we have chosen to write = PjLz/V where P, is the incident power falling on the scattering region. Therefore, the height and width of the scattering volume, as determined from the demagnified image of the streak camera pixel, dropped out of the calculation leaving only the depth of the scattering volume in the direction of the scattered beam, Lz. This latter quantity was approximately equal to the the diameter of the focussed C O 2 laser beam (~ 100 fim). Before Eqs. 5-3 and 5-4 could be solved, values for the scattering solid angle, dfi, and the integral over the experimental range of k-vectors were required. dQ was estimated to be 2.6 x 10 - 4 sr from the k-spectrum measurements (see Sec. 5.5e). The integral in Eq. 5-4 was solved by assuming that the k-vector directions uniformly filled the cone angle of the C O 2 focussing optics (half angle ~ 4 ° ) and that their magnitudes ranged from 1.86&0 to 2.14A;0 (see the results of the k-spectrum measurements in Sec. 5.5e). Finally, for ne = 0.1n c r, Eqs. 5-3 and 5-4 give the following expression for the time and space averaged fluctuation amplitude: 6n n 2 = ( 6 0 ±10)^51 ( 5 - 5 ) Both peak and spatially averaged values of (5n/n) 2 = |ne(r, t)/neo\2 were calculated from the digitized streak records. The results are shown in Figs. 5-11 and 5-12. By peak (6n/n)rmt, we refer to the absolute maximum fluctuation level observed in a given shot. The peak spatially averaged value of (5n/n) r m , CHAPTER 5: RESULTS 127 2 4 6 8 Laser Energy(J) 10 Figure 5-11 Peak ion acoustic fluctuation levels as function of laser energy. was calculated from the average scattered power from all parts of the interaction region. There is some uncertainty in the spatially averaged fluctuation level due to the uncertainty in the actual interaction length. The error bars in Figs. 5-11 and 5-12 represent the uncertainty in the fluctuation level due to uncertainties in the calibration factors and scattering geometry. Both the peak and spatially averaged fluctuation amplitudes increased with incident laser energy. Up to 4-5 J, the increase was very rapid, while for energies above 5 J, the increase slowed and there was evidence (at least for the spatially averaged data), of saturation. Peak and average fluctuation levels approached 18% CHAPTER 5: RESULTS 128 -i r 2 4 6 8 Laser Energy (J) 10 Figure 5-12 Peak spatially averaged ion acoustic fluctuation levels as function of laser energy. and 8% respectively for laser energies of ~8 J. This energy dependent behaviour is similar to that of the SBS backscatter reflectivity (see Fig. 5-1). It is interesting to compare these incoherent fluctuation levels to those calcu-lated from the coherent formula, Eq. 4-12. This formula can be written as, ( 6ny _ AdP, ( 5 - 6 ) CHAPTER 5: RESULTS 129 As pointed out in Sec. 5.5a, it is very difficult to estimate the actual scattering volume. Instead, we will follow Slusher and Surko 7 0 and assume that the scattering solid angle, dtl, was entirely the result of diffraction from the finite interaction region; dft = \2/ira2. Substituting this value into Eq. 5-6, we obtain, ( 6nV_ 4dP. n) ~ PinlrlL\\r [ b 7 ) For our conditions, this gives the result, 2 dP„ 105 ' Pi ' which is almost twice the result obtained for \6n/n\2. This implies that over regions with an average radius given by a2 = X2/ndQ, the plasma fluctuations are coherent. From the measured scattering solid angle of dCl = 2.6 x 1 0 - 4 sr, we conclude that the average diameter of the coherent fluctuation regions is ~ 50 fim which is considerably smaller than the observed total interaction length of 1-3 mm. 5.5d Late Scatter We conclude this section on the results of the spatially resolved Thomson scatter, by describing a feature present in many of the streak photographs which is not likely associated with SBS. This feature, which will be referred to as "late scatter", is the occurrence of strong Thomson scattering from localized regions in the plasma observed late in the CO2 pulse. An example is shown in Fig. 5-13. Also shown is the plasma density profile at the time of the late scatter. Since the Thomson scattering optics were arranged so as to accept only light scattered from ion fluctuations, the late scattering is an indication of enhanced ion fluctuations. On most shots, this scatter appeared after the end of the SBS induced Thomson scattering and persisted for several nanoseconds. Since the laser intensity was very CHAPTER 5: RESULTS ISO low or zero at this time, it is unlikely that the late scatter was a direct consequence of the interaction of the CO2 laser beam with the plasma. Position frnm) Laser — • — • — ^ — — — — ~ Intensity Fiducial Late Scattering Figure 5-13 A streak record showing late scatter, a.) The plasma density profile late in the laser pulse, b.) Streak record. The late scatter always occurred from the plasma regions of highest density, either near the front or back edge of the jet. (Since the Thomson scattering op-tics viewed the plasma side-on, the radial position of the scattering region was unknown.) The k-spectrum of the late fluctuations was roughly measured with CHAPTER 5: RESULTS 131 various masks which were placed in the Thomson scattered beam. These mea-surements showed that the enhanced fluctuation k-vectors pointed in a variety of directions both parallel and oblique to the C O 2 laser beam. The k-spectrum in the direction of the C O 2 beam extended over a range of at least 2kQ to 4k0. Although there was considerable lack of shot to shot reproducibility in the late scatter, at lower laser energies predominantly larger values of k occured. The fluctuation level, (6n/n)rmt, was roughly estimated from the observed scattered power. Assuming that the scattered solid angle was equal to the solid angle subtended by the col-lection lens (~0.1 sr), and that the scattering k-vectors had a uniform 4JT angular distribution with magnitudes ranging from 2k„ to 4kc, the rms fluctuation level was approximately 10-20%. A mechanism for producing these late fluctuations is discussed in Chapter 6. In the following section the results of the measurements of the k-spectrum of the SBS driven ion acoustic fluctuations are reported. 5.5e T h e k -spectrum of the S B S D r i v e n Fluctuations Thomson scattering measurements of the k-spectrum of the enhanced ion fluctuations were performed to verify that the measured fluctuations were indeed driven by SBS (i.e. k ~ 2ka) and to search for harmonics of the primary SBS driven ion acoustic waves. These harmonics are predicted to occur in certain saturation processes and have been seen in previous work.9* Streak photographs of the Thomson scattered light from ion fluctuations with k-vectors ranging from < 2k0 to ~4A: 0 were obtained with the optical arrangement shown in Fig. 4-8. A typical streak record, showing all the observed k-spectrum features, is displayed in Fig. 5-14. As expected for SBS driven ion acoustic waves, the majority of the scattered light was observed from fluctuations with k ~ 2ka. (In Fig. 5-14, the k-spectrum scale was determined from the measured scattering angles and is for ne = 0.1ncr.) The feature at 2ka was observed to have an energy dependent width. This is illustrated in Fig. 5-15, where the full width, Ak/ka, at CHAPTER 5: RESULTS 132 l o n * p , , , Ml* , , 2 P . acoustic decay of 2u)p decay, il T i 1 ns Harmonic generation Laser Intensity t Fiducial additional 200x atenuation Figure 5-14 A streak record of the k-spectrum of the ion acoustic fluctuations. (Note that the attenuation of the scattered light was different for the two parts of the photograph.) 2ka is plotted versus the incident laser energy. The instrument width, due to both the finite size of the scattering region and the spread in angles in the incident ruby beam, has been subtracted from the observed width of the 2k0 feature for each of the plotted points. Both the scattering solid angle, dd, and the experimental range of k-vectors were determined from the k-spectrum measurements. The k-vector magnitudes, on the average, ranged from 1.86fc0 to 2.14fc0 (see Fig. 5-15). This spectral width CHAPTER 5: RESULTS 133 2 k 6 I Laser Energy (J) 10 Figure 5-15 Spectral width of the feature at 2k0 as function of laser energy. corresponded to a ~ 1° (~ 17 mrad) spread in scattering angles. If it is assumed that the ruby light was scattered into a cone with a 1.0° full cone angle, then the scattering solid angle was 2.6 x 10 - 4 sr. It can be seen by comparing the observed spectral width of the the SBS excited waves to the width of the theoretical growth region as shown in Fig. 2-2a, that the observed spectrum was considerably broader than that predicted for exponentially growing waves. This result is likely an indication that some damping or saturation mechanism was broadening the SBS resonance. The observed k-spectrum can also be compared to that predicted from the observed backscatter spectral width. This CHAPTER 5: RESULTS 184 latter value is difficult to estimate since there were three peaks in the backscatter spectrum. Taking the width as approximately equal to the separation between the two outer peaks (54A), the corresponding frequency spread in the ion acoustic waves was ~ 14 GHz. If we assume that for all the driven ion acoustic waves, the dispersion relation was given approximately by, w,« = K L c t k i a , (5 - 8) where KL ^ 2, is the calculated ratio of the SBS driven ion acoustic wave frequency to 2k0c„ (see Fig. 2-3), then the k-vector spectrum had a width of A k / k a ~ 0.11. This value is about a factor of two smaller than the observed width of A k / k 0 ~ 0.28. However, considering the uncertainty in Eq. 5-8 for the driven waves, the agreement is reasonable. In addition to the strong scatter for k = 2ka, there was also significant scatter corresponding to fluctuations with k-vectors between 3.2A;0 and 4k0. The majority of this scatter appeared after the onset of scatter at 2k0 and is likely evidence of first harmonic generation. However, since the scattered intensity at 4ka was only ~ 5M °f *^at a * 2^ 0) the fluctuation level of the first harmonic was only ~ of the fluctuation level in the fundamental. Such'a low level of harmonic generation would not be expected to result in strong wave steepening. It was pointed out in Chapter 4 that while the measured 2kQ fluctuation laid in the direction of the incident CO2 beam, the measured 4kQ fluctuations propagated at an angle of 3.8°with respect to the CO2 beam. Therefore, it is likely that the first harmonic waves which resulted from the 2k0 waves that propagated in the direction of the CO2 beam, were larger than those measured. Harmonic generation was observed on all shots within the sampled energy range of 3-10 J. Although most of the scatter usually occurred for k ~ 4k0 (the scatter from fluctuations with k > 4k0 was cut off by the fiducial optical fiber), CHAPTER 5: RESULTS 185 significant scatter also occurred for k <— 3.2 — 4.0ko. The source of these fluctuations is not known. Temporal modulations with periods similar to those observed for the 2k0 fluctuations were present in the 4k0 scatter. The results from the present experiment should be compared to results ob-tained in similar Thomson scattering studies of C Q 2 laser driven SBS. Strong har-monic generation with {Sn/n)^ < 0.7(<5n/n)2ito, for 2k0 fluctuation levels ap-proaching 20%, has been seen in a previous experiment by Walsh and Baldis. 9 2 Clayton 8 9 has also seen strong harmonics in the k-vector spectrum. In this experi-ment, the amplitude of the first harmonic was ~25% of the fundamental amplitude for 6n/n levels approaching 10%. However, in another similar experiment, Giles and Offenberger90 detected no harmonic generation for 6n/n levels of greater than 10%. The fluctuation levels measured in the above experiments are spatially av-eraged values. In the present experiment, the spatially averaged fluctuation levels approached 8 % . Therefore, the measured ratio of the fluctuation level at 4k0 to that at 2k0 of >7% is in reasonable agreement with the results of past studies. An interesting feature revealed in the k-resolved streak photographs was the occurance of strong fluctuations at k ~ Z.2k0 — 4kQ (at 0.1 n c r or k ~ 3.5ka — 4.4k0 at 0.25ncr) prior to the start of strong 2ka fluctuations (see Fig. 5-14). This feature will be referred to as early scatter from early fluctuations. These fluctuations are believed to be unrelated to SBS but instead resulted from the ion acoustic decay of 2-plasmon (2up) decay. They persisted for only 100-300 ps and then disappeared before the appearance of the harmonic waves. This short period existence is dis-cussed in the following chapter. CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 186 C H A P T E R 6 D I S C U S S I O N O F T H E E X P E R I M E N T A L R E S U L T S 6.1 Introduction The results presented in the previous chapters will now be discussed. An attempt is made to develop a self-consistent picture of the SBS process as it occurred in the parameter range of the present experiment. Since the results are used to test theoretical predictions, it is important to establish the range of the relevant plasma parameters. Therefore, the ion temperature, which up to now has not been estimated and is the only important plasma parameter still unknown, is first discussed. Knowledge of the ion temperature is then used to calculate the temporal and spatial growth rates of the SBS driven ion acoustic waves which are compared to the measured growth rates. The observed saturation behavior of the SBS instability is discussed and a saturation mechanism (ion trapping) is proposed and shown to be in agreement with observations. Finally, the processes which are thought to be responsible for the early and late ion acoustic fluctuations are discussed. 6.2 The Ion Temperature The calculation of the SBS growth rates requires a knowledge of both the electron and ion temperatures. The electron temperature was experimentally mea-sured and found to be approximately 300 eV throughout the SBS interaction period. CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 187 The ion temperature was not directly measured, but instead, is estimated from the experimental parameters. Since primarily only the electrons are heated directly by the laser and the electron-ion equilibration times are long, the ion temperature at early times in the SBS process is expected to be only a few eV (see Sec. 4.3). However, once the SBS instability is established, the ions can be heated through damping of the driven ion acoustic waves. The primary damping mechanism is either linear or nonlinear Landau damping, which should only change the ion distribution function in the region centered around the wave phase velocity. However, the wave energy that is transferred to the resonant ions in the damping process is eventually distributed to the entire ion population through ion-ion collisions and results in increased ion temperatures. In Sec. 2.8b, two techniques were outlined for estimating the ion temperature as a result the damping of the SBS ion acoustic waves. The first of these techniques (Eq. 2-58) requires a knowledge of the SBS reflectivity while the second technique (Eq. 2-59) requires a value for (8n/n). Both of these parameters have been mea-sured as a function of the incident laser energy (see Figs. 5-1 and 5-12) and are listed in Table VI-I for incident energies of 3 J, 6 J, and 9 J. Eq. 2-58 can be written as, RILZml/2uia netx>0 2/3 ( 6 - 1 ) Substituting the experimental parameters, n e = 0.1n c r = 102 4 m - 3 , w,a = 1.35 x 1011 s - 1 (any corrections due to the presence of the laser are ignored), w0 — 1.78 x 101 4 s - 1 , Zeff = 4, and »7i t e / / = 8mnydrogeii along with the peak laser intensities in the interaction region (chosen to be the back interaction region, since this is where the peak 6n/n occurs), the ion temperatures listed in Table VI-I are obtained. CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 138 TABLE VI-I Laser Energy (J) Peak Laser Intensity (W/cm«) SBS Reflectivity (%) Average (in/n)rm. {%) kBT< „. (Eq. 6-1) (eV) tfa e f t t i n g (ns) (Eq. 2-59) . (eV) *B ' " • ( r a i t i * ) ) = 10 eV (Eq. 2-59) (eV) tB 7i(initi»]) = 50 eV 3 3.1 x 10" 3 3 290 2.7 13 57 6 6.1 x 10" 6 6 740 0.7 26 98 9 9.2 x 10" 7 7 1070 0.4 31 145 Eq. 6-1 was derived with the assumption that the energy is carried off by the heated ions at the same rate as it is deposited by the laser, i.e. a steady state is assumed. Since the SBS interaction period was short, it is unlikely that a steady state ion temperature was achieved. Eq. 2-61 can be used to calculate the time required for strong ion heating (T,- = Te/3) via the production of a heated tail. The heating times, calculated from Eq. 2-61 for an interaction length, L, of 1 mm are listed in Table VI-I. The results clearly indicate that, at least at the lower laser energies, the ion temperature is likely to be considerably lower than that calculated from Eq. 6-1. Eq. 2-59 is applicable to situations where the ion temperature is not steady but changes as a function of time. Since the linear Landau damping rate of the ion acoustic waves, 7,a, depends on T{/ZTe. Eq. 2-59 must be solved numerically. The total integration time, t, was chosen to be 2 ns, which roughly corresponds to the maximum period of the enhanced ion waves at any location. 7,„ was calculated numerically from the plasma dispersion relation (see Chapter 2). A graph of 7,-a/w,-a vs. Ti/ZTe for a He:N plasma having the experimental parameters, is presented in Fig. 6-1. The results of the numerical solution of Eq. 2-59, for initial ion tempera-tures of 10 eV and 50 eV are listed in Table VI-I. Clearly, the final ion temperature depends strongly on the intial temperature. There is considerable disagreement be-tween these results and those obtained from the steady state calculation (Eq. 6-1). It is shown in Sec. 6.5a, however, that for the ion acoustic fluctuation levels present CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 139 in this experiment (6n/n < 18%), nonlinear Landau damping or ion trapping can increase the damping rate significantly and lead to considerably higher predicted ion temperatures. o.oor 1 1 1—1—1—1 I 1 « J\ * V Electron Landau Damping v ; Ion Landau Damping 0.01 0.1 T; /ZTe 1.0 Figure 6-1 The Landau damping rate for ion acoustic waves as a function of Ti/ZTe. The plasma conditions were chosen to be similar to those in the present experiment. For large values of Ti/ZTey most of the damping is due to the ions, while for small Ti/ZTe, the ion Landau damping is negligible and the electrons do most of the damping. In a previous experiment, Gellert and Kronast 9 8 measured the electron and ion temperatures resulting from the interaction of a short pulse (2 ns) CO2 laser beam with a preformed hydrogen plasma. Using large angle ruby laser Thomson scattering, they measured peak electron and ion temperatures of Te = 40—50 eV and 7, = 18 eV compared to initial values of Te ?± T t ~ 10 eV. Since the ion temperature could be predicted from a model involving only classical absorption and conduction, CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 140 little additional ion heating occurred as a result of the damping of the ion acoustic waves in spite of the moderate wave amplitudes (6n/n ~ 1 — 2%). It is therefore likely that the ion temperature remained well below the electron temperature of Te = 300 eV in the present experiment. Therefore, the ion temperature predicted by Eq. 2-59 will be used in the following sections to estimate both the spatial and temporal SBS growth rates. Since the ion temperature during the early stages of the instability was low in any of the models (likely less than 50 eV), any errors in the SBS growth rates due to incorrect values of 7, are likely small. 6.3 Threshold Behavior and Temporal Growth Rates In the following two sections the behavior of the SBS instability at early times in the interaction is discussed. The plasma conditions and laser intensity are shown to be sufficient to lead to absolute growth of the SBS instability very soon after the plasma is formed. 6.3a The S B S Threshold The threshold laser intensity for the SBS process strongly depends on the plasma conditions, especially the velocity and density scale lengths. If we assume for the moment that our plasma was uniform, Eq. 2-47 can be used to calculate the infinite homogeneous threshold. For this purpose, we require the collisional damping rate of the backscattered electromagnetic wave, 7_. This can be written in terms of the electron ion collision frequency, !/„-, as 1 0 7- = ( 6 - 2 ) where • ~ 2 x 10 ,oZn, InA |— " ( 6 - 3 ) ,3/2 " e CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 141 Here, In A is the Coulomb logarithm (see Sec. 4.3), Te is measured in eV, and ne is in m - 3 . Substituting the appropriate experimental parameters into Eqs. 6-2 and 6-3, we find, ~ 1.7 x 101 0 s _ 1 and 7_ = 8.7 x 108 s - 1 . Therefore, from Eq. 2-47, the infinite homogeneous threshold intensity is only 1.4 x 107 W / c m 2 . Plasma inho-mogeneities, however, can increase the threshold considerably by limiting the size of the region over which the wavevector and frequency matching conditions apply. Mismatches can result from spatial variations in the plasma density and tempera-ture as well as from nonuniform plasma motion. The magnitudes of the inhomo-geneities are usually given by the temperature, density and velocity scalelengths; LT = Te(dTe/dx)~l, LN = n e ( d n e / d z ) - 1 , and L„ = v(dv/dx)~l respectively. The temperature scale length can be presumed to be long because of the high thermal conductivity (see Sec. 4.3). Ln has been estimated from the interferograms (see Sec. 5.5a) to be 0.25 mm to 2.5 mm in the rear scattering region and > 3 mm in the forward scattering region. A lower limit for Lv of ~ 2 mm is obtained by assuming that the velocity at the high density region of the plasma (near the front edge of the jet) was zero while the velocity in the interaction region was equal to the blow-off velocity calculated from the observed red shift of the central spectral peak (see Table V-I). For convective growth, the threshold intensity is given approximately b y 9 4 €0cmekBTeu* J > tek0u\Lv • (6~4) For a velocity scale length of Lv > 2 mm, this leads to thresholds of approximately 6 x 101 0 W / c m 2 . Absolute growth requires even higher laser intensities:10 4e0cme(jjlkBTe c / kBTeckz0u>0\ _ OJ0L„ \ mew* J c2ka 2u0c u2LT ulLv_ ( 6 - 5 ) CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 142 The term involving Lv dominates this expression and leads to a value for the threshold intensity of 9.7 x 10 1 1 W / c m 2 . The experimental peak intensities (/ ~ 1.1 x 10l2E (W/cm 2 ) and / ~ 4 x 10 1 1 E (W/cm 2) in the rear and forward scattering regions respectively; E is the laser energy in Joules) were typically well above all these thresholds. Examination of Fig. 5-2 reveals that even at the time of the first appearence of any plasma (time of 0.8 J accumulated energy), the threshold inten-sities calculated above were exceeded. However, since the initial plasma was only 100-200 fim in length and had very large density gradients, the SBS thresholds were likely higher than those calculated for the plasma which appeared at later times. The delay of several hundred picoseconds between the formation of the plasma and the beginning of measurable backscatter therefore is reasonable. 6.3b T h e Temporal G r o w t h Rate of the S B S Instability The high temporal resolution of the streak camera permitted the study of the temporal growth of the SBS instability. In this section it is shown that during the initial period of growth, the SBS instability (i.e. the ion acoustic and backscattered waves) is expected to grow absolutely. The absolute growth rate is then calculated and compared to the growth rates of the ion acoustic wave amplitude as measured from the temporal behavior of the Thomson scattered light intensity. As explained in Sec. 2,6, absolute growth of SBS can be prevented either by a limitation of the plasma size due to plasma boundaries and gradients or by strong damping of the backscattered and ion acoustic waves. The plasma size and laser intensity required for absolute instability may be calculated from the relations, Eqs. 2-46, 2-48, and 2-51 which require values for the infinite homogeneous growth rate, K0-For most laser energies, SBS backscattered light appeared prior to the peak of the laser pulse. The laser intensity at the beginning of the SBS process can be roughly estimated by assuming that for all laser energies the plasma first appeared CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 143 after 0.8 J had been delivered by the laser and the SBS process began about 0.2 ns later. For this purpose the laser pulse shape is assumed to be triangular in shape with a 1.2 ns rise and 2.8 ns fall. The predicted time for the start of the SBS and the incident intensity at that time (for the rear scattering region) are listed in Table VI-II. 7 0 was then calculated from the laser intensity by numerically solving Eq. 2-42. These results are also listed in Table VI-II. TABLE VI-H Laser Time of Intensity at Energy Growth Time of Growth (s- 1) (J) (ns) (W/cm 2) 3 1.2 3.4 x 1012 3.4 x 1011 5 1.1 5.1 x 101 2 4.0 x 1011 7 0.9 6.2 x 101 2 4.3 x 1011 9 0.7 7.2 x 101 2 4.5 x 1011 The length of the SBS interaction region required for absolute growth can be calculated from Eq. 2-46. Using Via = c, ~ 1.2 x 10s m/s, V_ = c, and 7 0 = 3 x 1011 8 - 1 we find L > 31 fim which is considerably smaller than either the interaction length which is calculated to result from plasma inhomogeneities (> 50 fim, see Sec. 6.4c) or the length of the modulations in the ion acoustic fluctuation amplitude. If plasma inhomogeneities are present, the length of the interaction region can be limited and the condition for absolute growth is given by Eq. 2-51. It has already been shown from Eq. 6-5 (which is derived from Eq. 2-511 0) that the threshold intensity required for absolute growth as determined from plasma inhomogeneities is I > 9.7 x 101 1 W / c m 2 . The laser intensities at the start of the SBS process were larger than this value. Finally, strong damping of the backscattered and ion acoustic waves can prevent absolute growth. At early times in the interaction, the respective damping rates were ~ 9 x 108 s - 1 and 7,0 ~ 1.4 x 109 s _ 1 . CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 144 Therefore 7C ~ 3.4 x 101 0 s - 1 and 77 1.1 x 109 s - 1 and hence the condition for absolute instability, 7o >^ 7C 2> 7r ('Eq* 2-48) is satisfied. We have therefore shown that the laser intensity at the start of the SBS process is sufficient to exceed the absolute instability thresholds imposed by a finite interaction region, plasma inhomogeneities, and damping of the excited waves. In a finite plasma, the absolute growth rate of the SBS instability changes with time (see Sec. 2.6). This is a consequence of the fact that the backscattered and ion acoustic waves move and carry information concerning the conditions at the plasma boundaries into the interaction region. There are two characteristic time scales involved in the absolute growth of the SBS instability. Up to a time of the order of L/V-, where L is the length of the interaction region, the instability grows at the same rate as it would in an infinite homogeneous plasma (70). The growth rate then decreases. Finally, after a time of the order of L/Via, the growth rate reduces to a value given by Eq. 2-45 and remains there for the rest of the interaction period. In practice, saturation mechanisms will decrease the growth rate before this final stage. In the present experiment, L ~ 1 mm, so L/V- ~ 3 ps and 2»/V,0 ~ 8 ns. Therefore the absolute growth rate should be somewhere between 7 0 and 7 = 27<>^ -v~\ ( ^ ' 2-45). Forslund et. a l . 1 0 state that for an interaction length L » L\ = \ / | V _ V i a | / 7 0 (~ 15 fim in the present experiment) but L < V_/7„ (~ 750 fim) only the growth at the lower rate of 7 is observed. Numerical simulations (see Sec. 2.6) of the growth of the backscattered (BS) and ion acoustic (IA) waves were performed to observe the transition between the growth rates 7© and 7. For an interaction length satisfying the above two conditions, it was observed that the BS wave grew absolutely at the rate 7 almost immediately after a time LR/V-, where LR was the distance between the right hand edge of the interaction volume and the point of observation. The growth rate of the IA wave changed a-bit more gradually but reached a rate of 7 long before a time Li/Via. (Li is the distance between the left hand edge of the interaction volume and the point CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 145 of observation.) In fact, at the time Li/Via, no further change in the growth rates was observed. The reason for this behavior is that for moderately strong coupling between the two waves (large 7C) , the growth of the faster wave almost immediately affects the growth of the slower wave. Therefore, when the growth rate of the faster BS wave was reduced because of the edge effects, the growth rate of the slower IA wave soon also decreased. However, since the BS wave travelled much faster than the IA wave (V-/Via ~ 2500) a given BS wave e-folded far fewer times in crossing the interaction region than a given IA wave. Therefore the edge conditions on the right hand edge of the interaction region (which were communicated to the point of observation at the velocity of the BS wave) had a much stronger effect of the growth rates at the point of observation than the conditions on the left hand edge (which were communicated to the point of observation at the velocity of the IA wave). By the time the IA waves, which started at the left hand boundary, reached the point of observation they had grown through their coupling to the BS to such an extent that any information concerning the left hand boundary conditions was lost. Therefore, in the present experiment, it can be expected that soon after a time L/V- ~ 3 ps, both the ion acoustic and backscattered waves should grow throughout the interaction region at the rate 7. The growth rates of the ion acoustic waves during the early stages of the SBS process were determined by measuring the temporal behavior of the Thomson scattered light. Both the spatially and k-spectrum resolved streak records were used in these measurements. The growth rate was determined for either the position in the interaction region or the k-vector which showed the first signs of growth. (The sampled width in both cases was three streak camera pixels wide.) In the k-spectrum measurements, growth always occurred first at k = (2.0 ± 0.05)fco. The growth rate was calculated for each streak record from the slope of the graph of ln J, vs. time (/, is the scattered intensity). Since 6n/n oc J ^ 2 , the growth rate of the ion acoustic fluctuations was equal to one half this slope. Two typical plots CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 146 of the scattered intensity vs. time are shown in Fig. 6-2. Exponential growth is clearly indicated. The growth rate of the ion acoustic fluctuations as determined from such graphs is plotted in Fig. 6-3 as a function of the incident laser energy. The error bars represent the uncertainties in the slopes of the growth curves. Also shown are the growth rates calculated from Eqs. 2-45 and 2-52. 0 0.1 0.2 0.3 OA 0.5 0.6 0.7 0.8 Time (ns) Figure 6-2 Plot of the measured Thomson scattered intensity as a function of time. Results from both a spatially resolved and a k-spectrum resolved measurement are shown. CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 147 20T I t I I I I 1 r Oi i i i • i i i 1 i 0 1 2 3 4 5 6 7 8 9 10 Laser Energy (J) Figure 6-3 The absolute growth rate of the SBS instability as a function of incident laser energy. The theoretical growth rates were determined for the values of 7<, listed in Table VI-II. During the growth period, the plasma conditions and the laser intensity were not constant. It is therefore possible that the growth rate changed with time. We can estimate the variation in the growth rate due to these effects by noting that 7o a Iljj2nJ2. Although the plasma density varied considerably across the rear interaction region, the density at the centre of this region changed by only ~ 50% during the period of growth. In addition, since the period of growth occurred near the peak of the laser pulse, the laser intensity varied by less that 50%. Therefore, CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 148 the growth rate likely changed by less than 35% during the absolute growth period. This uncertainty is represented reasonably well by the error bars shown in Fig. 6-3. It is clear that there is some discrepancy between the calculated and measured absolute growth rates. Even after allowing for the damping of the backscattered and ion acoustic waves (Eq. 2-52), the calculated growth rate exceeded the measured growth rate by > 50%. It is possible that at the beginning of the interaction when the Thomson scatter from the ion acoustic fluctuations was too small to be detected, the growth rate was larger than that measured. The growth rate at the time of measurement may have been reduced by some form of enhanced damping in addition to the Landau and collisional damping which have already been considered in Eq. 2-52. Since on most shots, the growth rate was measured for times up to the appearence of noticeable saturation, it would not be surprising if some saturation mechanism was lowering the growth rate at times prior to the strong saturation. 6.4 T h e Spatial Behavior of the S B S Instability Temporal growth of the SBS instability is expected to occur only during the early stages of the interaction. As the wave amplitude grows, damping becomes stronger and the instability is expected either to stop growing temporally and only show spatial or convective growth or to saturate. Purely convective growth can be distinguished from saturated behavior by examining the spatial profile of the driven ion acoustic waves. Therefore, particular attention is given in the following sections to comparing the predicted and observed ion acoustic wave profiles. We begin by predicting the spatial growth rate from the measured backscattered reflectivity growth curve. 6.4a T h e S B S Reflectivity For low laser intensities, the SBS backscatter reflectivity was observed to increase approximately exponentially with the incident laser energy (see Fig. 5-1). This behavior is usually interpreted as a sign of convective growth of the SBS CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 149 instability. In such cases, the backscattered power is given by Eq. 2-54, which can be written in terms of the backscattered reflectivity, R, and the noise reflectivity, Rn, as ( 6 - 6 ) The laser energy or intensity enters Eq. 6-6 through the infinite homogeneous growth rate, 7<>; for low laser intensities 7<> a Ell2.10 In order to observe expo-nential growth in reflectivity with incident energy, it is necessary that the plasma conditions be independent of the laser intensity. This condition is satisfied to some extent in the present experiment. The electron and ion temperatures and the SBS interaction length were all essentually independent of the incident laser energy and had respective values of Te = 300 eV, T*,- = 10 — 20 eV (assuming that prior to the SBS interaction T, ~ 10 eV), and L ~ 1 mm. There was some change in the plasma density with energy as a result of the energy dependent expansion of the plasma into the background gas, but as shown in Fig. 5-9, the densities at the centres of the interaction regions was almost energy independent. Therefore, provided that there was convective growth, Eq. 6-6 can be used to calculate the exponential growth rate of the reflectivity. From the numerical solution of Eq. 2-42, the peak infinite homogeneous growth rate in the back interaction region is given in terms of the laser energy by, 7 o ~ 1.8 x l O 1 1 ^ ( 6 - 7 ) Substituting this expression, along with the appropriate linear Landau damping rate, 7,a (from Fig. 6-1), into Eq. 6-6 results in a reflectivity growth rate of 160 J - 1 which is much larger than the experimentally measured value of (4.4 ± 1.3) J - 1 (see Sec. 5.2). If there was indeed convective growth, the ion acoustic wave amplitude should have grown exponentially in space (see Fig. 2-5). The spatial growth rate can be R = Rn exp CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 150 calculated from Eq. 2-50. For an average laser energy of 3 J , the predicted growth rate is 2.4 x 105 m - 1 ; i.e. an e-folding length of ~ 4 fim. On the other hand, the measured reflectivity growth rate of (4.4 ± 1.3) J - 1 results in a spatial growth rate of only 6.6 x 103 m - 1 (i.e. an e-folding length of 150 fim). The streak records for the spatially resolved Thomson scattering do not show any evidence of convective growth at this lower rate. The spatial growth rates appeared to be much higher (in the regions where there was no evidence of saturation). In fact, the growth rates calculated from the spatially resolved streak records were instrument limited (the spatial resolution of the Thomson scattering optics was ~ 100 fim). It is only possible to say that the typical growth rate was greater than 4 x 104 m _ 1 (i.e. an e-folding length of less than 25 fim). As discussed in Sec. 5.5b, the ion wave amplitude was strongly modulated. The SBS backscattered light therefore likely was not scattered from a single wave train as assumed in Eq. 6-6, but rather from several bunches of waves which may or may not have been coherent (with respect to each other). Therefore, the exponential behavior of the backscattered reflectivity was probably fortuitous and not the result of convective growth for the SBS instability. In the following section, the backscattered reflectivity is calculated from the measured ion acoustic fluctuation amplitude. By comparing the calculated and measured reflectivities, we can estimate the size of the ion acoustic wave bunches. 6.4b Estimates of the S B S Backscattered Reflectivity In this section, the backscatter reflectivity is estimated from the ion acoustic fluctuation amplitudes which were calculated from the intensity of the Thomson scattered light. The reflectivity, R, is given in terms of the interaction length and fluctuation amplitude by Eq. 2-53. Since the fluctuation amplitude was not constant across the interaction region, Eq. 2-53 must be solved for the local Sn/n and the contributions added to give the total reflectivity. The way in which this addition CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 151 should be carried out depends on whether the ion acoustic waves were coherent or incoherent. For incoherent waves, the total bacscattered intensity is just the sum of the intensities backscattered from each part of the interaction region. For this purpose, it was assumed that the ion acoustic waves were coherent over a length, Lp, equal to the total interaction length divided by the number of streak camera pixels, but that each of these regions was randomly incoherent with respect to all similar regions. The resulting reflectivity from Eq. 2-53 can then be written as where C is a constant determined from the calibration of the Thomson scattering optics, (Sig),- is the number of counts registered at the t ' th pixel and N is the total number of pixels. (Note that the approximation, tanhx » z, valid for x < 0.5, has been made.) If the ion acoustic waves were coherent across the entire interaction region and only varied in amplitude, then the total backscattered intensity can be found by adding together the amplitudes of the scattered light from each scattering region and squaring the result. Again, a length of Lp is used for each scattering region. The resulting reflectivity from a single coherent wave can then be written as The predicted reflectivities off incoherent and coherent fluctuations were calculated using Eqs. 6-8 and 6-9 and the experimentally measured Thomson scattered inten-sities and are plotted in Fig. 6-4 vs. the measured peak backscattered reflectivity. (Since the simple formula, Eq. 6-9, neglects depletion of the incident beam as it is reflected, calculated reflectivities can be greater than 100%.) Since the measured ( 6 - 8 ) ( 6 - 9 ) CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 152 reflectivity was partially time averaged (as a result of the ~ 500 ps risetime of the backscatter detection system), the calculated reflectivity was also time averaged. This was accomplished by moving a square 500 ps window through the summed results of Eqs. 6-8 and 6-9. Also shown for comparison in Fig. 6-4 is the line representing the case where the calculated and measured reflectivities are equal. Calculated Time Averaged Reflectivity (%) Figure 6-4 The calculated backscatter reflectivity vs the measured reflectivity. The uncertainty in the measured reflectivity was approximately ± 1 0 % . CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 158 Neither the incoherent nor the coherent reflectivities agree with the measured reflectivity. This is not surprising since it is unlikely that the ion acoustic wave was coherent over either the entire interaction length or over only the length imaged by a single pixel. It is more likely that the waves were coherent over distances close to the length of the observed spatial modulations in the ion wave amplitude. In order to estimate the length of each of the coherent regions, we assume that each region has the same length, x £ p , where x is an integer > 1. Then Eq. 6-8 can be written as ir2x2L2„ n2 fl_nLy1i; R = 4 A 2 2 i rM (6 - 10) where M = N/x and [Sig], is the number of counts at each pixel in the group. A reasonable estimate of [Sig],- is (Sig)1 + ( S i g ) 2 + - - + (Sig)a etc. Therefore Eq. 6-10 becomes: IT2 xL2 n2 /? - " —JP • N «=i (6-11) In order to bring the reflectivities calculated from Eq. 6-8 (incoherent calculation) into agreement with the measured reflectivities, we require x ~ 6. This implies that the length of the coherent scattering regions was ~ 120 fim compared to a measured length for the spatial modulations in the ion waves of 100-300 fim. The estimate of 120 fim is only an average value for the length of the coherent regions. The Thomson scattering spatial resolution of ~ 100 fim did not permit the observation of such small structures and so would tend to show only the large scale modulations. In addition, the observed length of the interaction region of CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 154 100-300 fim does not necessarily correspond to the length of the coherent region but probably represents only an upper bound. There are several sources of error involved in the calculation of the backscatter reflectivity from Eqs. 6-8 and 6-9. These include errors in the calibration of the Thomson scattering optics (the estimated uncertainty has been included in the error bars shown in Fig. 6-4) or an underestimation of the total interaction length. If the interaction length was underestimated, the resulting sums in Eqs. 6-8 and 6-9 would be too low and hence the calculated reflectivities would also be low. The calculated fluctuation amplitudes and the resulting reflectivity would also be low if the Thomson scattering and SBS interaction regions were not perfectly aligned. Athough great care was taken to align the two regions, some misalignment may have occurred as is revealed from a comparison of the measured and calculated periods of the backscattered radiation. The temporal behavior of the backscattered light was calculated from the streak records of the Thomson scattered light using Eqs. 6-8 and 6-9. For the same shots, fast scope traces of the backscattered intensity were obtained. After correcting the calculated reflectivities for the difference in detector risetimes, it was found that the measured backscatter lasted for a period approximately 1.5 times the calculated period. This discrepancy may be due to a misalignment of the Thomson scattering optics, in which case the amplitude of the ion acoustic waves at the edge of the interaction region was measured. (These waves would not last as long as those near the axis of the interaction region where the laser intensity was highest.) If this was the case, the measured fluctuation amplitudes and the calculated reflectivities would be low. Another possible cause for the discrepancy between the measured and calculated backscatter periods is that the backseattering optics were more sen-sitive than the side-on Thomson scattering optics to scatter from long regions of low amplitude fluctuations. Therefore measurable backscatter could have occurred off waves from which there was no measurable Thomson scatter. Since the period CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 155 of the reflectivity calculated from the k-resolved Thomson scattering measurements (for which the alignment of the SBS and Thomson scattering regions was not crit-ical) agreed well with the period calculated from the spatially resolved data, it is concluded that the latter process was likely responsible for the difference between the measured and calculated backscatter periods. Therefore, the calculated values of the fluctuation amplitude probably apply to those fluctuations along the axis of the interaction region. Finally, with regard to this last point, the calculated reflec-tivities may be too high since the laser beam intensity and hence the amplitude of the driven ion acoustic waves varied across the width of the interaction region. If this was the case, then the calculated length of the coherent regions is greater than 120 fim. 6.4c T h e S B S Interaction Length The question remains as to what caused the incoherence in the ion acoustic waves. A possible cause for the strong spatial modulations in the SBS instability amplitude is that the region over which a coherent ion acoustic wave could be driven was limited by mismatches in the k-vectors of the three interacting waves. These mismatches can be a result of strong gradients in the electron temperature, plasma density, or plasma blowoff velocity. The interaction length, / , n t , is typically limited by mismatches to a length such t h a t 8 0 , 9 6 Jo £-where K(X) = ka(x) — kia(x) — k-(x). For a linearly varying mismatch (*c(x) = K'X) the interaction length is given by (6 - 12) (6 - 13) CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 158 «' can be roughly estimated from the measured plasma conditions. For temperature, density, and velocity scale lengths of Lj, Ln, and Lv respectively, K' becomes9 8 Using the experimental scale lengths of Ln = 0.25 — 2.5 mm, Lv ~ 2 mm, and X j , large, K' is found to range from (3.9 — 5.5) x 108 m - 2 , i.e. / I n t =40 — 50 fim. Even if velocity gradients were negligible, the interaction length is still not likely to have exceeded 250 fim (at least in the rear scattering region; / , n t was likely somewhat larger in the forward scattering region because of the long density scale lengths). T A B L E V I - I H Diagnostic Coherent Interaction Length Measured Size of the Thomson Scattering Region (Sec. 5.5b) 100 - 300 fim Calculated Size from Reflectivity Measurements (Sec. 6.4b) > 120 fim Calculated Size Limited by Inhomogeneities (Sec. 6.4c) 40 - 250 fim Calculated Size from Thomson Scattered Intensity (Sec. 5.5c) ~ 50 fim The reasonable agreement between the measured and calculated lengths of the coherent SBS interaction regions, as summarized in Table VI-III, leads us to conclude that the strong spatial modulations in the ion acoustic wave amplitude were due to plasma inhomogeneities which had the effect of limiting the size of the SBS interaction regions. To conclude this section on the spatial behavior of the CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 157 driven ion acoustic waves, we now discuss the origin of the two prominent scattering regions. The long period modulation in the backscatter reflectivity (see Fig. 5-2) and the appearence of two scattering regions (see Figs. 5-7 and 5-8) are likely associated with the changing plasma conditions. The SBS scattering occurred first in the rear scattering region, near the front of the jet, and rapidly extended to a length of ~ 1 mm as the background gas was ionized. Scattering later appeared well in front of the jet in a region ~ 1 mm long after the background plasma was formed. At about the time strong SBS scattering started in the forward scattering region, the ion acoustic waves dropped in amplitude in the rear scattering region. This behavior is likely due to either conditions in the rear region becoming unfavourable to SBS or to a strong reduction in the light intensity. From the interferograms, it can be seen that the plasma conditions in the rear scattering region became more uniform with time. However, since the ion temperature may have increased due to strong ion trapping, it is possible that strong damping in the rear region led to the decay of the SBS. However, it is probably more likely that absorption, reflection, and refraction of the incident laser beam in the forward scattering region reduced the intensity sufficiently to stop SBS in the rear scattering region. This concludes our discussion of the spatial and temporal growth of the SBS instability. In the following sections, the saturation behavior is discussed. 6.5 Saturation Clear evidence of saturation of the SBS instability was observed in both the backscattered light and the behavior of the driven ion acoustic waves. The backscat-tered reflectivity was limited to less than 10% while the peak spatially averaged ion acoustic fluctuation level was observed to apparently saturate at (6n/n)rm$ ^ 7—8% for laser energies of greater than 6 J. The peak value of (6n/n)rmi reached 17-18% at laser energies of ~ 8 J . (It is not clear from Fig. 5-11 whether or not the peak CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 158 fluctuation level saturated. However, since the average ( « 5 n / n ) r m , was observed to saturate and the spatial behavior of the scattering ion wave did not change much with energy, the peak fluctuation amplitude of 17-18% probably represents a satu-rated value.) It is clear from the complicated temporal and spatial behavior of the scattering ion acoustic waves, that the SBS did not saturate simply as a result of the ion acoustic waves reaching a maximum fluctuation level and then remaining at that level with further increases in the incident laser intensity. Rather, the ion wave amplitude fluctuated in both time and space. Any saturation mechanism proposed for the present experiment must explain this behavior as well as the experimentally observed saturation levels. 6.5a Ion T r a p p i n g The saturation mechanism which was likely dominant in the present exper-iment is ion trapping. Ion trapping can limit the amplitude of the excited waves through both strong damping and wavevector mismatches. In addition, the tempo-ral behavior of the ion acoustic waves and the spectral features in the backscattered spectrum can be explained as effects of trapped ions. The ion acoustic fluctuation amplitude required for strong ion trapping can be estimated using a waterbag model for the ion distribution function as outlined in Sec. 2.8c. From Eq. 2-63 we find that for kBTi = 20 eV, Sn/n = 0.32. If the ion temperature is increased, the fluctuation level required for strong trapping is decreased. For example, A^T, = 50, 100, or 300 eV results in Sn/n = 0.25, 0.19, or 0.10. Since ion temperatures greater than 50 eV are unlikely to have occurred early in the interaction, strong ion trapping only occurred for Sn/n > 0.25. In order to estimate the extent of the trapping for 6n/n < 0.25, the true Maxwellian distribution for the ion velocities must be used. The number of trapped ions can then be calculated by assuming that all those ions whose velocity component in the direction of the ion acoustic wave propagation lies within the trapping region, CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 159 as given by Eq. 2-60, will be trapped. The fraction of the total number of ions which are trapped is shown in Fig. 6-5 as a function of the fluctuation amplitude and ion temperature. It is clear from these results that for low ion temperatures the number of trapped ions is small. However, for Sn/n greater than ~ 15%, the number of trapped ions increases very rapidly with the fluctuation amplitude. For the conditions in the present experiment | of the total number of ions can become trapped for 6n/n > 25%. Such high levels of trapping can lead to strong damping of the ion acoustic waves and significant changes in their propagation characteristics. on/n (%) Figure 6-5 The fraction of the total number of ions which are trapped vs the fluctuation amplitude. Curves are shown for = 20, 50, and 100 eV. A value of Zeff = 4 was chosen. The ions which are trapped by the wave potential are accelerated to the wave phase velocity. As a result, the ions, on the average, gain energy while the ion acoustic wave is damped. Damping as a result of ion trapping, however, is not a CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 160 steady process as is linear Landau damping but only occurs while the ions are being accelerated. Since the ions initially have a Maxwellian distribution of velocities, there are more ions which move slower than the wave phase velocity than faster than the phase velocity. Therefore, when the trapping begins, most of the trapped ions lie on the parts of the phase space orbits shown in Fig. 2-€a which are at velocities less than v^. These ions are accelerated by the wave potential until they reach the tops of their orbits at which time they begin to transfer their energy back to the wave. Therefore, after the initial acceleration, the ions bounce back and forth within the troughs of the wave potential and there is a periodic exchange in energy between the ions and the wave. The bounce period of the trapped ions can be roughly calculated by assuming that they execute simple harmonic motion in the troughs of the sinusoidal potential distribution. The result for small oscillations is Since the bounce periods of the trapped ions are different for the various orbits, the trapped ions phase mix and the amplitude variations in the ion acoustic wave eventually disappear. The ion wave amplitude is expected to vary in time as shown in Fig. 6-6. 9 7 Such behavior has been seen in experiments on large amplitude grid driven ion acoustic 9 8 and electron plasma 9 9 waves. In order to calculate the approximate damping due to ion trapping in the present experiment, the ions are assumed initially to have a Maxwellian velocity distribution. The trapping is assumed to accelerate (or decelerate) all those ions within the trapping region given by Eq. 2-60 to the wave phase velocity in a time approximately equal to one bounce period. The resulting loss of wave energy to the ions is plotted in Fig. 6-7 as a function of the fluctuation amplitude. For the purpose of comparing the damping due to ion trapping to the linear Landau damping, the (6 - 15) CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS Figure 6-6 The ion acoustic wave amplitide as function of time. The bounce period is shown. wave energy decrement in Fig. 6-7 has been divided by the corresponding linear damping loss. (The time scale was chosen to be much smaller than The large increase in the nonlinear damping rate with 6n/n for fluctuation amplitudes greater than 15% clearly demonstrates that the increased damping due to ion trapping was likely a major mechanism in the observed saturation of the ion acoustic wave amplitude during the early stages of the interaction. Several simplifications have been made in the preceeding analysis: 1. The ions that were trapped in the wave potential were asssumed to remain trapped. Collisions between ions, however, would eventually have led to their escape from the potential troughs. 2. Trapping can lead to enhanced ion heating and hence an increase in the linear Landau damping rate as well as an increase in the number of trapped ions. In fact, CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 162 OH 1 1 1 1 0 5 10 15 20 25 Jn/n (%) Figure 6-7 Calculated energy loss for an ion acoustic wave due to ion trapping as a function of Sn/n. To illustrate the enhanced damping, AE due to ion trapping has been divided by the energy loss due to linear Landau damping. (The linear Landau damping rate was taken as 7,„ = 1.4 x 109 s _ 1 .) solving Eq. 2-59 as in Sec. 6.2 with an enhanced damping of 20 x 7,a for the first 0.2 ns of the interaction, results in a final ion temperature of T» a* 400 eV (initial Ti = 20 eV). While this value is likely an overestimate (because of the finite ion equilibration times) it does indicate that ion heating due to ion trapping was likely significant in the present experiment. 3. The nonlinear damping occurs only during the initial acceleration of the trapped ions. Once the ions are accelerated, the damping effectively saturates and the ion wave amplitude varies periodically. In addition, since the ion wave is continually driven by the ponderomotive force, the losses in the wave energy due to damping can be overcome very quickly. Other effects due to the trapped ions besides enhanced CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 168 damping are probably responsible for limiting the SBS instability at later times. These effects will now be discussed. One effect of the trapped ions on the wave is to cause the walls of the potential troughs to steepen. 1 0 0 This leads to the production of harmonics and a decrease in the amount of the SBS backscattered light. This in turn lowers the ponderomotive force which drives the ion fluctuations. The moderate first harmonic production observed in the k-spectrum resolved streak records (see Sec. 5.5e) probably resulted via this mechanism. The trapped ions can also affect the phase velocity of the ion acoustic wave through the nonlinear frequency s h i f t . 1 0 1 ' 1 0 ' The trapped ions weaken the wave potential and hence the restoring force on the ions which make up the ion acoustic wave. Therefore the wave frequency and the resulting phase velocity are reduced from their zero trapping values. The effect on the initial value problem in which the wavenumber, k, is fixed is that the frequency of the wave changes by an amount Su oc — I^n/nl 1 / 2 , while for the more common experimental case in which the frequency is fixed, the wavenumber changes by an amount 6k oc \6n/n\ 1/ 2. Ikezi et. a l . 1 0 ' find that the frequency or wavenumber shift is given by, 6 k -6u> v%c lc» tin kia VgUia vg 6n n 1/2 f>4>) - , (6 - 16) where is the wave phase velocity, vg is the group velocity and f"(v^) is the second derivative of the ion velocity distribution evaluated at t; = tty. For u»ta <£. <Jpt-(the ion plasma frequency) ~ vg ~ ct. Evaluating /"(tty) at kBT{ = 20, 50, and 100 eV, we find that .for 6n/n = 10%, 6k/kia = 2.8 x 10" 1 0,2.0 x 10"3, and 1.4 x 10 - 1 respectively; the shift in the wavenumber is a strong function of the ion temperature. This shift in wavenumber can result in a phase shift of the ion acoustic wave across the SBS interaction region. For instance, if the wavenumber is k in the part of the interaction region where the ion wave is small (no trapping) CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 164 while the wavenumber in the region with large trapping is k + Sk, then in a distance, x, there is a phase shift in the ion acoustic wave of 6k x x. This phase shift has two main effects on the SBS. First of all, backscattered light from one region of plasma may be of the wrong phase to drive (through the ponderomotive force with the incident light) the ion acoustic waves in another region. Secondly, backscattered light originating from two parts of the plasma may not be coherent. Both of these processes can have the effect of limiting the growth of large amplitude SBS driven ion acoustic waves. The length of the SBS interaction region limited by the nonlinear frequency shift can be roughly estimated as that distance over which the accumulated phase shift amounts to 180°. For 6n/n = 20% and fcjjT, = 20 eV, the interaction length is much larger than the plasma length, while for kBTi = 50 — 100 eV, the interac-tion length is 14-1000 fim. Since ion temperatures of greater than 50 eV are not unreasonable (if some heating due to ion trapping occurred) it is possible that the nonlinear frequency shift may have contributed to the saturation of the growth of the SBS ion acoustic waves. Besides the evidence already presented that ion trapping could saturate the ion acoustic waves at the observed level, there are also indications from the temporal behavior of the ion acoustic waves and the spectrum of the backscattered light that ion trapping occurred. 6.5b T h e Observed Effects of Ion Trapping The bouncing back and forth of the ions trapped in the potential troughs of the large amplitude ion acoustic waves has two main effects on the wave. First of all, the interchange of energy between the wave and the ions causes the wave amplitude to be modulated in time at the bounce frequency as illustrated in Fig. 6-6. Secondly, this amplitude modulation leads to the production of sidebands in the CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 165 ion acoustic wave spectrum which are separated from the fundamental ion acoustic frequency by the bounce frequency.84 The period of the slower temporal modulations in the ion acoustic wave am-plitude which were seen in the present experiment are in good agreement with the bounce period of the trapped ions. This is illustrated in Fig. 5-10 where the bounce period, Tp, as predicted from Eq. 6-15, is plotted along with the experimentally measured modulation periods. Values for Sn/n were estimated from the measured peak fluctuation amplitudes shown in Fig. 5-11. The observed appearance of three peaks in the backscattered spectrum (see Sec. 5.3) is in agreement with the proposed generation of sidebands for the ion acoustic wave. These sidebands have previously been seen for large amplitude electron plasma waves 1 0 8 and recently in SBS backscatter spectra. 8 4 We propose that in the present experiment, the central spectral peak with a red shift of 33±5A corresponded to SBS backscatter from the fundamental ion acoustic wave while the peaks at red shifts of 7 ± 5A and 61 ± 5A corresponded to backscatter from the two sidebands. This model requires a plasma blowoff velocity towards the incident laser beam of ~ 1.9 x 105 m/s which is in reasonable agreement with the blowoff velocity calculated from the interferometric measurements of the plasma motion and the observed blue shift of the sidescattered light. The average separation of either sideband from the fundamental was 27 ± 7 A or 7.2 ± 1.9 GHz. This value is in good agreement with the expected shift of 6-10 GHz estimated from Eq. 6-15 (see Fig. 5-10). The large shot to shot changes in the relative strengths of the three spectral peaks is unexplained. Since most of the backscatter spectra were time integrated, changes in the spectrum during the interaction period may have resulted in a com-plicated average spectrum. The fact that the sidebands were occasionally as strong as (or even stronger than) the central spectral feature indicates a strong (~ 100%) modulation in the amplitude of the fundamental. The relative strengths of the two CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 186 sideband peaks depends on the population of trapped ions . 1 0 4 If the trapped ions are located primarily in the bottoms of the potential troughs so that their number at velocities around the wave phase velocity exceeds the number predicted from the linear theory, then the higher frequency sideband will be larger. On the other hand, the lower frequency sideband will be larger if most of the trapped ions are located around the rims of the potential troughs, in which case the number of ions at t; = is less than for the unperturbed Maxwellian. In the present experiment the high frequency sideband was usually larger indicating that the trapped ions were probably located near the bottoms of the potential troughs. We have shown that ion trapping should occur for our plasma conditions and can lead to a saturation of the ion acoustic wave amplitude at approximately the observed level. In addition, both the observed temporal modulation of the ion acoustic waves and the sidebands in the backscattered spectrum can be explained by the effects of ion trapping. We therefore conclude that ion trapping was the major mechanism in the saturation of the SBS in the present experiment. The effects of various other saturation mechanisms are discussed in the following sections. 6.5c P u m p Depletion Since the backscattered reflectivities were limited to less than 10%, it is unlikely that pump depletion had a major effect on either the growth rate or the saturated level of the ion acoustic waves. Absorption and refraction of the laser light were probably more effective than reflective losses in reducing the flux reaching the interaction region. Losses through these mechanisms was probably responsible for the termination of SBS in the rear scattering region soon after the formation of plasma far in front of the jet. 6.5d Ion Heating It was shown in Sec. 6.2 that ion heating through linear Landau damping of the driven ion acoustic waves should have resulted in ion temperatures of much CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 167 less than 100 eV. However, additional heating due to ion trapping may have led to higher temperatures (see Sec. 6.5a). The SBS backscatter reflectivities, limited only by pump depletion and the damping which would be present at these tem-peratures, can be estimated from Eq. 2-57. For typical parameters of L = 1 mm and leaser = 5 J we find for T, = 300 eV that the reflectivity, R, should be greater than 60%. (A value for the noise level of the backscattered light of B = 10 - 6 was chosen.) For smaller ion temperatures the calculated reflectivity would be even larger. Since the observed reflectivities saturated at less than 10%, it is clear that ion heating was not responsible for the saturation. In addition, ion heating limits the backscatter reflectivity primarily by forcing the SBS to grow convectively; a higher ion temperature leads to a lower convective growth rate. Since the observed ion acoustic wave amplitude did not vary exponentially with position over the en-tire interaction region but rather appeared to be discontinuous and showed signs of saturation, it is unlikely that ion heating was a dominant factor in determining the spatial behavior of the ion acoustic waves. We therefore conclude that ion heating was unimportant in the observed saturation of the SBS instability. 6.5e Harmonic Generation The threshold ion acoustic fluctuation amplitudes required for harmonic and subharmonic generation are given by Eqs. 2-65 and 2-67 respectively. For plasma conditions of Xj)e = 1.3 x 10 - 7 m and 7", < 50 eV, the resulting threshold fluctua-tion levels are Sn/n > 0.02 (first harmonic generation) and Sn/n > 0.015 (daughter wave production). Since the measured fluctuation levels were well above these val-ues, both harmonic and subharmonic generation should have occurred. The SBS backscatter reflectivity limited by harmonic generation is given by Eq. 2-66. Solv-ing this expression for L < 1 mm and c < 1 0 - 4 results in R < 10%, in good agreement with the observed reflectivity. The reflectivity limited by subharmonic production as given by Eq. 2-68, however, is typically less than 1%. Therefore, it CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 168 is unlikely that subharmonic generation was occurring in the present experiment at the level predicted by theory. The actual extent of the subharmonic production was not measured. However, previous experiments 9 0 ' 9 3 involving Thomson scat-tering measurements of the SBS driven ion acoustic waves showed no evidence of subharmonics for 8n/n levels comparable to those in the present experiment. The fluctuations at k ~ Ak0 which were observed in the k-spectrum resolved Thomson scattering measurements indicate that some first harmonic production occurred. However, since {8n/n)^0 < 0.1(6n/n)2k0, the energy contained in the first harmonic was less that of that contained in the fundamental. Therefore, unless this first harmonic wave was damped (through Landau damping or further harmonic generation) at a rate much higher than the Landau damping rate of the fundamental, very little energy in the fundamental wave was lost through harmonic production. Therefore, it is unlikely that harmonic generation can account for the SBS saturation observed in the present experiment. Indeed, the fact that the ratio of the harmonic to fundamental fluctuation levels remained the same for laser energies of 3-10 J (the saturation began at 4-6 J) practically rules out harmonic generation as a possible saturation mechanism. The harmonic production was probably a result of the ion trapping (see Sec. 6.5a). Simulations9 4 have shown that ion trapping can lead to wave steep-ening and eventually wavebreaking. Experimental evidence of this process has been seen by Walsh and Baldis 9 2 who observed both strong harmonic production and a periodic quenching of the ion acoustic waves. However, the low harmonic levels in the present experiment cannot produce significant wave steepening, and hence, wave breaking likely did not occur. 6.5f Profile Modif icat ion and Self Focussing The interferograms showed no clear evidence of profile steepening in the SBS interaction regions. If anything, the density profile became less steep as time CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 169 progressed; probably as a result of hydrodynamic motion. There were signs of profile steepening in the dense (^O^Sricr) plasma located near the front edge of the j e t 1 0 6 probably as a result of the stimulated Raman scattering and two-plasmon decay interactions. This steepening, however, occurred before the start of the SBS and in a region of plasma downstream from the SBS interaction region. Therefore, such steepening likely did not affect the SBS. Self-focussing of an intense laser beam in a plasma can be caused by both thermal self-focussing and filamentation. Thermal self-focussing is caused by local heating of the plasma along the axis of the laser beam. This plasma expands, leaving a region of lower density and higher refractive index. The beam therefore refracts towards the axis leading to even greater intensities and further self-focussing. In the case of filamentation,106 a similar self-focussing occurs but the plasma expansion is caused by the ponderomotive force (see Sec. 2.4a). Both thermal self-focussing and filamentation therefore can lead to increased laser intensities and hence increased SBS growth rates (and lower thresholds). 1 0 7 Since the threshold is usually l o w , 1 0 6 filamentation can occur in interactions with moderate laser intensity. Evidence for filamentation has been seen in enhanced x-ray emission from small regions of p l a s m a 1 0 8 ' 1 0 9 as well as from the refraction of a probe beam by the filament walls . 1 1 0 The level of self-focussing or filamentation in the present experiment likely did not significantly affect the SBS. Interferometry and schlieren photographs5 5 showed no evidence of plasma channels while x-ray pinhole photographs showed no local regions of enhanced emission.5 6 (The low x-ray flux did not allow x-ray pinhole photographs to be obtained in a single shot.) Since the focal spot intensity was reasonably uniform (no prominent hot spots), only full beam self-focussing should have occurred. However, in a previous experiment with similar plasma conditions, 1 1 0 no evidence of full beam (diameter ~ 200 /xm) filamentation was observed for laser intensities below 101 3 W / c m 2 . Since the beam CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 170 diameter at the interaction region was greater than 200 fim and Ziage, < 101 3 W / c m 2 even full beam filamentation likely did not occur in the present experiment. This concludes our discussion of the saturation of the SBS process. In the following section, two phenomena which were unrelated to SBS but which led to strong Thomson scattering are discussed. 6.6 T h e E a r l y and Late Scattering Both the early and late ion acoustic fluctuations which were observed in the Thomson scattering measurements (see Sees. 5.5d and 5.5e) appear to be unrelated to the SBS. Rather the temporal, spectral, and spatial behavior of these fluctuations indicate that they are a consequence of two-plasmon decay (2wp-decay). (The two-plasmon decay measurements were primarily the work of J. Meyer and H . Houtman and therefore only the important results will be discussed.) Evidence that 2wp-decay occurred in the present experiment is provided by ob-servations of enhanced Thomson scattering from the excited electron plasma waves ( E P W ' s ) , 1 1 1 backscattered light at | ( j o , 1 0 6 ' l i a and high energy electrons ejected from the interaction r e g i o n . 1 0 6 ' 1 1 2 The EPW's produced in 2u>p-decay predicted to have the largest growth rates are two waves which propagate in opposite direc-tions and lie in the plane of E 0 and k 0 at an angle of ~ 45° with respect to these vectors (see Fig. 6-8). Experimental measurements of the directions in which the fast electrons were ejected from the plasma indicated that the largest amplitude EPW's did indeed lie in these directions. 1 1 3 Thomson scattering from these elec-tron plasma waves revealed enhanced fluctuations near the front of the jet in the region of 0.25n c r. The EPW's appeared only at early times in the laser pulse (before the start of the SBS) and occurred in a short burst lasting only 50-100 ps. Often the enhanced fluctuations were observed to quench and then reappear again after ~ 100 ps. CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 171 Figure 6-8 The k-vector matching for the 2u/p-decay process. The electron plasma waves lie in the plane containing both E 0 and k0. a) and b) show the two strongest pairs of electron plasma waves. 6.6a Ear ly Scattering Two-plasmon decay can drive EPW's to amplitudes (6n/n ~ 10 — 15% in the present experiment 1 1 1 ' 1 1 2 ) at which several processes can occur which have the effect of removing energy from the waves and saturating the instability. It is possible for a large fraction of the electrons to become trapped in the potential troughs of the EPW's and accelerated to approximately the wave phase velocity. As a consequence, the wave is damped. Such trapping is likely the source of the 50-100 keV electrons (average kBTe) observed in the present experiment. The two-plasmon decay and S R S 6 0 instabilities together accelerated more than 1% of the electrons in the focal volume to energies, kBTe > 50 keV. The ponderomotive force resulting from the large amplitude EPW's can force electrons out of the interaction region and lead CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 172 to saturation through profile steepening. Profile steepening resulting from 2up-decay has been seen in both simulations*8 and experiments. 1 0 5 , 1 1 8 In addition, through the ponderomotive force, two high frequency EPW's can couple together parametrically and drive low frequency ion acoustic waves. 8 8 ' 1 1 4 These ion acoustic waves can in turn be coupled to one of the EPW's and drive heavily damped high frequency (larger k) E P W ' s . 8 8 Hence the original EPW's are damped. Langdon et. a l . 8 8 have shown through simulations that the EPW's couple most efficiently to ion acoustic waves with fcta = 2fcEP\v-The parametric coupling which drives the ion acoustic fluctuations can be understood from the wavevector and frequency matching conditions: (6 - 17) k i = k2 +k,-a, and Fig. 6-9. Here, (u;i,ki) and (w2,k2) are the frequencies and wavevectors of the two electron plasma waves. For the situation shown in Fig. 6-9a (where we have chosen the pair of electron plasma waves shown in Fig. 6-8a), the resulting ion acoustic wave has a k-vector of magnitude ~ 2ki and propagates at an angle of ~ 45° with respect to the incident laser beam. However, it is also possible for EPW's from two different 2wp-decay pairs to couple and drive ion acoustic waves of magnitude \ /2fcEPW which propagate in the direction of the incident laser beam. This is shown in Fig. 6-9b. This latter coupling is likely the source of the early ion acoustic fluctuations which were observed in the k-resolved Thomson scattering spectrum. 1 1 6 The k-spectrum resolved Thomson scattering measurements of the 2wp-decay driven EPW's showed that these waves were driven to fluctuation levels of 6n/n ~ 10% over the entire k-vector range from less than 2\/2k0 to greater than Z\/2k0. Such large fluctuations can then couple to produce ion acoustic waves with k-vectors ranging from less than 4k0 to greater than 6k0 propagating in the direction of the CHAPTER 8: DISCUSSION OF THE EXPERIMENTAL RESULTS 173 Figure 6-9 The k-vector matching for generating ion acoustic waves from the decay of the 2o;p-decay EPW's. a) The coupling which produces ion acoustic waves which propagate at 45° with respect to k O J and b) in the direction of k. incident laser beam. The observed k-vector spectrum of the early ion acoustic fluctuations extended from 3.5fc0 to greater than AAk0 (at 0.25ncr). Therefore, the k-vector spectrum of the early ion acoustic fluctuations as well as their time of occurrence and short temporal behavior are in agreement with the behavior expected for ion acoustic waves which result from the decay of 2up-decay. Therefore, ion acoustic decay of 2wp-decay was probably the source of the early fluctuations. It is interesting to calculate the growth rate and fluctuation level of the early fluctuations. The growth rates were calculated from the streak records in a manner similar to the calculation of the SBS growth rates (see Sec. 6.3b). The result, 7 ~ 2.5 x 101 0 s - 1 , indicates that the ion acoustic waves grew by a factor of only 1.3 during the short (~ 10 ps) period in which the 2wp-decay EPW's saturated. 1 1 1 CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 174 Therefore, the 2u;p-decay process was saturated by some process other than ion acoustic wave generation. Finally, we will calculate the fluctuation amplitude of the early fluctuations. We can roughly estimate 6n/n for the early ion acoustic fluctuations from the observed scattered intensity and k-spectrum. The total scattered power detected from the early fluctuations was only 350 — IM °* *^at m e a s u r e d from the SBS driven waves (6n/n ~ 17%). However, the scattering volume was only of the SBS scattering volume while the spread in k-vectors (if the early ion acoustic waves ranged from 4k0 to 6&0) was more than 10 times greater. Therefore, using Eqs. 4-25 and 4-28 (with neo = 0.25ncr), we find that the level of the early fluctuations was (6n/n)rmt > 7.5 — 10%. In a previous similar experiment, Baldis and W a l s h 1 1 4 observed ion acoustic waves resulting from 2wp-decay of 6n/n ~ 25%. (The elec-tron plasma waves reached 6n/n ~ 17%.) Considering this previous work and the observed E P W fluctuation level of 6n/n ~ 10—15%, the observed ion acoustic level is not unreasonable. 6.6b Late Scattering At the time of the occurrence of the late Thomson scatter, both the 2a>p-decay and the SBS instabilities were completed and the laser intensity had dropped to a very low (or zero) level. Therefore, the late enhanced ion acoustic fluctuations were probably caused by some plasma process which was not directly driven by the laser beam. A likely candidate is the current driven ion acoustic instability.8 The current required to drive this instability can be provided by the return current which flowed into the interaction region to balance the charge lost via fast electrons (i.e. those electrons accelerated to high energy by the 2wp-decay and SRS instabilities). Before discussing the experimental results, we will first describe the general features of the current driven ion acoustic instability. CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 175 The mechanism which drives the ion acoustic instability can be understood from Fig. 6-10. Here, the electrons and ions both have Maxwellian velocity distribu-tions corresponding to their temperatures, but there is a relative shift between the two distributions due to a net current. (Note that for Te = 300 eV and 7, = 50 eV, the ion distribution function would be much narrower than shown in Fig. 6-10.) This shift means that the phase velocities (±t/ ,„) of both ion acoustic waves can lie on the positive slope portion of the electron distribution function. Just as Landau damping of a wave is a consequence of the larger number of electrons (or ions) at velocities less than the wave phase velocity than greater than this velocity, the ion acoustic instability results from the larger number of electrons at velocities greater than Via than ^ e 8 a than  via- Therefore, provided that the Landau growth of the ion acoustic wave due to the electrons exceeds the Landau damping due to the ions, one of the ion acoustic waves will take energy from the electron distribution and grow. The other ion acoustic wave will remain damped. For ZTe » T,-, instability requires that the drift velocity of the electrons with respect to the ions, vj, exceeds the ion acoustic phase velocity, t/,0. For vj » t/,a, waves can be driven which propagate at large angles with respect to the drift direction. 1 1 7 The cone of unstable waves has a half angle of ~ v < f / v , a . In 1-D, the ion acoustic instability can saturate by essentually trapping all the electrons in the vicinity of v, a . This leads to the formation of a plateau in the electron distribution function at v, a . However, when wave propagation at a variety of angles is taken into account, no significant plateau is formed and the main saturation mechanism becomes ion trapping. 1 1 8 The waves driven by the instability can decay through mode coupling to higher and lower frequency waves . 1 1 8 ' 1 ' 0 This process leads to a turbulent spectrum of ion acoustic fluctuations often referred to as the Kadomtsev spectrum. 1 1 9 The amplitudes of the fluctuations are expected to decrease as Jfc-3 at high frequencies due to the increased Landau damping at short wavelengths. At low amplitudes, the peak of the turbulence spectrum is found to CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 176 vd * Figure 6-10 Electron and ion distribution functions which lead to the ion acoustic instability. The relative width of the ion distribution and the shifts of the two ion acoustic velocities have been expanded for clarity. be in the vicinity of kiaXr)e ~ 0.5 — 0.7, but as the fluctuation level increases and electron and ion trapping become significant, the peak in the fluctuation spectrum moves to kiaXDe ~ 0.04 - 0 . 2 . m In the present experiment, Thomson scattering from the late ion acoustic fluctuations was observed over a broad range of angles corresponding to k{aXj)e ~ 0.08 — 0.17 (limited by the finite size of the collection optics). This result is in-dicative of a turbulent ion spectrum as predicted to result from the ion acoustic instability. The large fluctuation amplitude of [6n/n)rmt ~ 10 — 20% (see Sec. 5.5d) is likely limited by strong ion trapping (similar to the case of the SBS driven ion acoustic waves). Significant fluctuation levels (6n/n > 10% ) resulting from the current driven ion acoustic instability have also been seen in previous laser plasma CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 177 experiments. 1 " ' 1 1 8 In these experiments, return currents were generated to bal-ance the charge loss from the interaction region which resulted from the electron transported thermal flux. In the present experiment, the return current is required to balance the charge lost via fast electrons. Both the magnitude of the return current and the drift velocity of the electrons can be expected to vary in time as the charge imbalance is neutralized. This may explain why the enhanced ion acoustic fluctuations did not appear until some time after the fast electrons escaped from the interaction region. The growth rate of the ion acoustic instability is predicted to be largest when vd ~ ve = (fcpTe/me)1/2 (the electron thermal velocity). 1* 4 This is due to the fact that the growth rate depends on both the magnitude (number of resonant electrons) and the slope of the electron distribution function at v, 0 ; for large drift velocities feo(via) is small, while for small drift velocities f'eo(via) is small. Therefore the drift velocities in the original flux of fast electrons (energies> 50 keV) and the initial return current (electron energies > 50 keV) were probably too large to drive effectively the ion acoustic instability. However, at later times, the potential driving the return current was reduced and the drift velocity fell to a value for which the instability was strongly driven. Since the late fluctuations occurred only in the plasma regions of high density, it appears that the growth of the ion acoustic instability had a density dependence. It is possible that the fluctuations which occurred near the front edge of the jet were a consequence of the high return current density in the region from which the fast electrons had escaped. However, since no 2wp-decay electron plasma waves were detected in the plasma region near the back edge of the jet, some other mechanism must account for the large ion acoustic waves in this region. No definite process has been found, but it is possible that since the plasma Debye length was smaller in the regions of higher density, the damping of the ion acoustic fluctuations was reduced in these regions. CHAPTER 6: DISCUSSION OF THE EXPERIMENTAL RESULTS 178 Finally, we can compare the measured and predicted growth rates of the late fluctuations. From Ichimaru,8 the approximate growth rate of the ion acoustic instability in the limit of small k;a and for (fc^/m,)1/2 < t/,a < (kBTelme)ll 2 -\-Vd is, For kBTe = 300 eV, kBTi = 50 eV, k = 3^ , and vd = ve we find 7,a = 1.2 x 1011 s - 1 which is about an order of magnitude larger than the measured growth rate of Ha = (1.0 ± 0.4) x 101 0 s - 1 . Agreement between the theoretical and measured growth rates requires kBTi ~ 230 eV. These results therefore indicate that the ion temperature may have risen considerably during the laser pulse. The source of this heating was likely the ion trapping which saturated both the SBS process and the ion acoustic instability itself. CHAPTER 7: CONCLUSIONS AND SUGGESTIONS FOR FURTHER WORK 1 7 9 C H A P T E R 7 C O N C L U S I O N S A N D S U G G E S T I O N S F O R F U R T H E R W O R K Stimulated Brillouin scattering (SBS) in the interaction of an intense CO2 laser beam with an underdense plasma has been investigated in this work. The SBS interaction was studied through measurements of the intensity and the tempo-ral and spectral behavior of the backscattered light as well as through time resolved ruby laser Thomson scattering measurements of the spatial and spectral behavior of the SBS generated ion acoustic waves. In addition, measurements of the SBS sidescattered light at 90° and 135° and time resolved ruby laser interferometry of the plasma were performed. The information obtained from these measurements was used to determine the spatial and temporal growth characteristics and the sat-uration behavior of the SBS instability. In addition to the laser plasma interaction experiments, the author also made a major contribution to the development of the short pulse CO2 laser system used in these studies. The SBS interaction occurred in the low density, long scale length plasma which was formed in the background gas in front of the gas jet target. In this region, the plasma was composed of both helium and nitrogen ions and had an electron density of 0.05 — 0.25ncr and an electron temperature of 300 eV. Two main SBS interaction regions were detected. SBS in a region near the front of the jet occurred first and grew in extent as the ionization spread into the background gas. At a later time in the laser pulse, SBS in this region ceased, probably as a result of CHAPTER 7: CONCLUSIONS AND SUGGESTIONS FOR FURTHER WORK 180 lowered laser intensities, and the majority of the stimulated scattering took place in a region of plasma ~ 2 mm in front of the jet. The SBS backscatter reflectivity was limited to less than 10% in spite of the moderate fluctuation levels of the SBS driven ion acoustic waves and the greater than 1 mm total interaction length. Agreement between the measured and cal-culated reflectivity was obtained if the ion acoustic wave in the SBS interaction region was not coherent along its entire length but rather was made up of mutually incoherent waves, each with a coherence length of ~ 120 fim. Calculations of the fluctuation amplitude using the Thomson scattering data indicated that the lengths of the coherent regions were approximately 50 fim. The Thomson scattering mea-surements indeed showed a strong spatial modulation in the amplitude of the ion acoustic waves, typically with lengths of between 100 and 300 fim. The probable cause of the observed breakup of the ion acoustic wave was wavevector mismatches in the SBS process as a result of plasma inhomogeneities. Density gradients as well as possible gradients in the plasma velocity were shown to limit the interaction length through mismatches to 50 — 250 fim. An exponential temporal growth for the SBS driven ion acoustic waves of ap-proximately (5—10) x 109 s - 1 was measured from both the spatially and k-spectrum resolved Thomson scattering measurements. The growth rates were compared to theoretical growth rates which were calculated from a numerical solution of the plasma dispersion relation. Even after allowing for damping of the excited waves and plasma inhomogeneities, the theoretical growth rates were almost a factor of two larger than those measured. At laser energies below 4 J, the peak SBS backscatter reflectivity increased approximately exponentially with the incident laser energy. The usual explanation that this behavior is the result of a purely convective growth for the SBS instability is not valid in this experiment. The convective growth rate calculated from the measured slope of the reflectivity vs. incident energy curve is only 7 x 103 m - 1 , CHAPTER 7: CONCLUSIONS AND SUGGESTIONS FOR FURTHER WORK 181 compared to a predicted growth rate of 2 x 105 m - 1 and a measured growth rate of at least 4 x 104 m _ 1 . In addition, the spatial profile of the ion acoustic wave amplitude did not show the expected exponential shape over the entire lengths of the postulated coherent wave trains but rather showed a spiky behavior with signs of saturation. Saturation was also seen in the behavior of the peak power and energy backscat-tered reflectivity as well as in the peak and spatially averaged ion acoustic fluctua-tion amplitudes. Both backscatter reflectivities saturated at between 7 and 10% for laser energies above 6 J . The spatially averaged fluctuation amplitude, {8n/n)rmi) saturated at approximately 7%, also for laser energies above 6 J, while the peak fluctuations saturated at ~ 17 — 18%. The most likely cause for the observed sat-uration was shown to be trapping of the ions in the potential troughs of the ion acoustic waves. Both the number of trapped ions and the resulting enhanced damp-ing were shown to increase significantly for fluctuation levels approximately equal to the observed saturated levels. It was also shown that ion trapping could have sat-urated the SBS instability through the associated non-linear frequency shift which has the effect of limiting the plasma length over which the SBS wavevector matching conditions are satisfied. Additional evidence for ion trapping was the observed temporal modulations of the ion acoustic wave amplitude and the appearance of sidebands in the spectrum of the backscattered light. Both features can be caused by the bouncing back and forth of ions within the potential troughs of the ion acoustic wave resulting in a periodic temporal modulation of the wave amplitude at the bounce frequency of the trapped ions. The observed modulation period and the frequency shift of the sidebands are both in quantitative agreement with the predictions of this model. An alternative explanation, that these features were the result of multiple ion acoustic modes due to the presence of several ion species, was shown to be incorrect. Solu-tions of the plasma dispersion relation in the presence of a strong electromagnetic CHAPTER 7: CONCLUSIONS AND SUGGESTIONS FOR FURTHER WORK 182 wave for a plasma consisting of two ion species showed that only one mode, deter-mined primarily by the e.m. wave and the effective ion Z/m,- ratio, was excited and grew. Finally, two mechanisms, unrelated to SBS, which produced enhanced ion acoustic fluctuations were investigated. The temporal behavior as well as the fluc-tuation level and k-vector spectrum of the ion acoustic waves which appeared early in the laser pulse indicated that these fluctuations resulted from the ion acoustic decay of the two-plasmon decay instability. Fluctuations which appeared either after or late in the laser pulse were also attributed to 2up-decay. These late fluc-tuations can be explained to be the result of the ion acoustic instability driven by the return current which was required to balance the charge lost from the inter-action region via fast electrons generated by the 2up-decay and stimulated Raman scattering instabilities. In conclusion, the growth rates and saturation behavior of the stimulated Brillouin scattering instability have been studied. The instability initially grew absolutely at a rate close to that predicted for a finite inhomogeneous plasma. Saturation occurred at low laser intensities and can be explained as being due primarily to ion trapping. The observed backscatter reflectivity was shown to be limited by the break-up of the interaction region due to plasma inhomogeneities. Finally, the early and late Thomson scattering are shown to be consequences of the 2wp-decay instability. The important original contributions of this work are now summarized. In previous experiments it has often been necessary to assume that the SBS interaction length is very small in order to explain the observed reflectivities. In this experiment, it has been shown that the low backscatter reflectivity is a conse-quence of the incoherence in the driven ion acoustic wave which results from plasma inhomogeneities. CHAPTER 7: CONCLUSIONS AND SUGGESTIONS FOR FURTHER WORK 183 The temporal growth rate of the SBS instability in a laser-plasma interaction was measured for the first time in this experiment. The initial growth was shown to be exponential and the measured growth rate was found to agree within a factor of two with the theoretically predicted growth rate for a finite interaction region. It has been clearly shown that ion trapping can be a predominant saturation mechanism for SBS. It was shown that ion trapping should occur for the plasma conditions present in this experiment and that the trapping can account for the observed level of saturation. Additional supporting evidence for ion trapping was provided by: 1. the observation of harmonics in the ion acoustic wave spectrum, 2. temporal modulations in the ion acoustic waves, and 3. sidebands in the backscat-tered spectrum. The dispersion relation for the SBS process in a plasma with multiple ion species was solved for the first time (published in Ref. 19). It was shown that only one ion acoustic mode grows and that the characteristics of this mode are determined primarily by the laser coupling. Finally, enhanced ion acoustic waves, not associated with the SBS instability were observed to occur. These fluctuations were shown to be due to: 1. the ion acoustic decay of two-plasmon decay and 2. the ion acoustic drift instability driven by return currents, (published in Ref. 116) Suggestions for Further Work In this work, the ion temperature was estimated indirectly from theoretical heating models and from its effect on the measured growth rates and thresholds. Since the growth and saturation characteristics of the SBS driven ion acoustic waves depend quite strongly on the ion temperature, the understanding of these processes would be aided considerably if the ion temperature could be measured directly. This probably could best be accomplished by spectrally resolving the ion component in a Thomson scattering experiment. For the case of moderate ion temperature, it should be possible to measure the width of the ion feature if the scattering optics CHAPTER 7: CONCLUSIONS AND SUGGESTIONS FOR FURTHER WORK 184 are arranged so the scattering parameter, or is > 1. In such cases, the width is determined approximately by the ion thermal velocity. However, for larger ZTe/Ti, it may be difficult to use this technique since the ion feature would then show mainly the ion acoustic resonances, the positions of which are determined predominantly by the electron rather than the ion temperature. The image dissector used in the measurement of the backscattered spectrum permitted only gross temporal changes in the spectrum to be observed. There-fore, most of the spectra were time integrated, and hence, changes in the plasma and scattering conditions could not be observed. Recently, an optical Kerr cell technique 1 2 6 has been developed which allows the measurement of time resolved backscatter spectra. This thechnique could be applied to the present experiment without great difficulty. Finally, the determination of the plasma parameters in the interaction region would be made easier if the extent of the plasma was limited to the region of the jet itself. By lowering the density of the background gas, it may be possible to prevent the formation of a dense plasma in front of the jet. B I B L I O G R A P H Y 1. Hughes, T.P. , Plasmas and Laser Light, (Adam Hilger, 1975). 2. McCall, G . H . , Plasma Phys. 25, 237 (1983). 3. Salpeter, E .E . , Phys. Rev. 120, 1528 (1960). 4. Fejer, J .A. , Can. J . Phys. 38, 1114 (1960). 5. Dougherty, J.P. and Farley, D .T . , Proc. Roy. Soc. A259, 79 (1960). 6. Nishikawa, K. , J. Phys. Soc. Japan 24, 916 (1968). 7. Fried, B.D. , White, R.B., and Samec, T . K . , Phys. Fluids 14, 2388 (1971). 8. Nakamura, Y. , Nakamura, M . , and Itoh, T . , Phys. Rev. Lett. 37, 209 (1976). 9. Hack, R.J. and Johnson, E .A. , Phys. Rev. Lett. 44, 142 (1980). 10. Forslund, D.W. , Kindel, J . M . , and Lindman, E.L. , Phys. Fluids 18, 1002 and 1017 (1975). 11. Drake, J.F., Kaw, P.K., Lee, Y . C . , Schmidt, G . , Lui, C.S., and Rosenbluth, M . N . , Phys. Fluids 17, 778 (1974). 12. Ichimaru, S., Basic Principles of Plasma Physics — A Statistical Approach, (Benjamin, 1973). 13. Chen, F.F. , Introduction to Plasma Physics and Controlled Fusion, Vol. 1: Plasma Physics, 2nd ed., (Plenum Press, New York, 1984). 14. Ruckdeschel, F.R., Basic Scientific Subroutines, vol. 2 (Byte Books, McGraw Hill, 1981). 15. Offenberger, A . A . , Cervenan, M.R. , Yam, A . M . , and Pasternak, A .W. , J . Appl. Phys. 47, 1451 (1976). 16. Mayer, F.J., Busch, G.E . , Kinzer, C M . , and Estabrook, K . G . , Phys. Rev. Lett. 44, 1498 (1980). 17. Decoste, R., Lavigne, P., Pepin, H. , Mitchel, G.R., Kieffer, J .C. , J. Appl. Phys. 53, 3505 (1982). 18. Walsh, C.J. , Meyer, J., Hilko, B., Bernard, J .E. , and Popil, R., J . Appl. Phys. 53, 1409 (1982). 19. Barnard, A.J . and Bernard, J .E. , Can. J . Phys. 63, 354 (1985). 20. Pesme, D. , Laval, G . , and Pellat, R., Phys. Rev. Lett. 31, 203 (1973). 21. DuBois, D.F. , Forslund, D.W., and Williams, E . A . , Phys. Rev. Lett. 33, 1013 (1974). 22. Rosenbluth, M . N . , Phys. Rev. Lett. 29, 565 (1972). 23. Forslund, D.W., Kindel, J . M . , and Lindman, E.L. , Phys. Rev. Lett. 30, 739 (1973). 24. Estabrook, K. , Harte, J., Cambell, E . M . , Ze, F., Phillion, D.W., Rosen, M . D . , and Larsen, J .T. , Phys. Rev. Lett. 46, 724 (1981). 25. Kruer, W.L. , Estabrook, K . G . , and Sinz, K . H . , Nucl. Fusion 13, 952 (1973). 26. Vinogradov, A . V . , Zel'dovich, Ya. B., and Sobel'man, J E T P Lett. 17, 195 (1973). 27. Kruer, W.L. , Phys. Fluids 23, 1273 (1980). 28. Phillion, D.W., Kruer, W.L. , and Rupert, V . C . , Phys. Rev. Lett. 39, 1529 (1977). 29. Dawson, J . M . , Kruer, W.L. , and Rosen, B. in Dynamics of Ionized Gases, ed. M . J . Lighthill, I. Imai, and H . Sato, (Univ. Tokyo, 1973), p.47. 30. Kruer, W.L. and Estabrook, K . G . in Laser Interaction and Related Plasma Phenomena, ed. H.J . Schwarz, H . Hora, M.J . Lubin, and B. Yaakobi (Plenum Press, New York, 1981), Vol. 5, p. 783. 31. Karttunen, S.J. and Salomaa, R.R.E. , Phys Lett. 88A, 350 (1982). 32. Heikkinen, J .A. , Karttunen, S.J., and Salomaa, R.R.E. , Phys. Fluids 27, 707 (1984). 33. Karttunen, S.J., McMuIlin, J .N. , and Offenberger, A . A . , Phys. Fluids 24, 447 (1981). 34. Karttunen, S.J. and Salomaa, R.R.E. , Phys. Lett. 72A, 336 (1979). 35. Karttunen, S. J., Plasma Phys. 22, 151, (1980). 36. Kruer, W.L. , Valeo, E.J., and Estabrook, K . G . , Phys. Rev. Lett. 35, 1076 (1975). 37. Estabrook, K. and Kruer, W.L. , Phys. Fluids 26, 1892 (1983). 38. Langdon, A .B . , Lasinski, B.F., and Kruer, W.L. , Phys. Rev. Lett. 43, 133 (1979). 39. Estabrook, K . G . and Kruer, W.L. , Phys. Rev. Lett. 40, 42 (1978). 40. Bernard, J .E. , Plasma Physics Laboratory Report #104, University of British Columbia (1985). 41. Heckenberg, N.R. and Meyer, J., Opt. Comm. 16, 54 (1976). 42. Yariv, A. , Introduction to Optical Electronics, 2nd. ed. (Holt, Rinehart, and Winston, New York, 1976). 43. Alcock, A.J . , Fedosejevs, R., and Walker, A . C . QE-11, 767 (1975). 44. Levine, A . K . , Lasers, Vol. 2 (Marcel Dekker, New York,1968). 45. Abrams, R.L. , Appl. Phys. Lett. 25, 609 (1974). 46. Schappert, G.F . , Appl. Phys. Lett. 23, 319 (1973). 47. Feldman, B.J., QE-9 , 1070 (1973). 48. Stark, E .E . , Reichelt, W.H. , Schappert, G . T . , and Stratton, T . F . , Appl. Phys. Lett. 23, 322 (1973). 49. Lietti, A.J . , J . Appl. Phys. 49, 4674 (1978). 50. Morckawa, E. , J . Appl. Phys. 48, 1229 (1977). 51. Houtman, H . and Walsh, C.J . , Plasma Physics Laboratory Report #79, Uni-versity of British Columbia (1981). 52. Burak, I., Steinfeld, J .L, and Sutton, D . G . , JQSRT 9, 959 (1969). 53. Rheault, F., Lachambre, J.L., Lavigne, P., Pepin, H. , and Baldis, H.A. , Rev. Sci. Instrum. 46, 1244 (1975). 54. Offenberger, A . A . and Ng, A. , Phys. Rev. Lett. 45, 1189 (1980). 55. Popil, R., Ph.D. Thesis, University of British Columbia (1984). 56. Francis, J .R.D. , Fluid Mechanics for Engineering Students, 4th ed., (William Clowes and Sons Ltd., London, 1975). 57. Jahoda, F . C . and Sawyer, G . A . in Methods of Experimental Physics, Vol. 9, Part B, (Academic Press, 1971). 58. Krall, N.A. and Trivelpiece, A.W. , Principles of Plasma Physics, (McGraw-Hill, 1973). 59. Houtman, H . , Meyer, J., and Hilko, B., Rev. Sci. Instrum. 53, 1369 (1982). 60. Mcintosh, G . , M.Sc. Thesis, University of British Columbia, (1983). 61. Burnett, N.H. , Baldis, H.A. , Corkum, P.B., and Samson, J .C. , Phys. Can. 34, 26 (1978). 62. Smith, D . C . and Meyerand, R . G . in Principles of Laser Plasmas, ed. by G . Bekefi (Wiley, 1976), pp. 475-507. 63. Spitzer, L. Jr., Physics of Fully Ionized Gases, 2nd ed. (Wiley, New York 1962). 64. Sheffield, J., Plasma Scattering of Electromagnetic Radiation, (Academic Press, 1975) 65. DeSilva, A . W . and Goldenbaum, G . C . in Methods of Experimental Physics, Vol. 9 Part A, (Academic Press, 1970). 66. Evans, D . E . and Katzenstein, J., Rep. Prog, in Phys. 32, 207 (1969). 67. Kunze, H.J . in Plasma Diagnostics, ed. by Lochte-Holtgreven, W., (North Holland, Amsterdam, 1968). 68. Landau, L . D . and Lifshitz, E . M . , Electrodynamics of Continuous Media, (Perg-amon Press, 1960), Chap. 15. 69. Jackson, J .D. , Classical Electrodynamics, (Wiley, New York, 1975), Chap. 14. 70. Slusher, R.E . and Surko, C M . , Phys. Fluids 23, 472 (1980). 71. Turechek, J.J. and Chen, F.F. , Phys. Rev. Lett. 36, 720 (1976). 72. Massey, R., Berggren, K. , and Pietrzyk, Z.A. , Phys. Rev. Lett. 36, 963 (1976). 73. Herbst, M.J . , Clayton, C . E . , and Chen, F.F. , Phys. Rev. Lett. 43, 1591 (1979). 74. Turechek, J.J. and Chen, F.F. , Phys. Fluids 24, 1126 (1981). 75. Burnett, N.H. , Baldis, H.A. , Enright, G . D . , Richardson, M . C . , and Corkum, P.B., J . Appl. Phys. 48, 3727 (1977). 76. Offenberger, A . A . , Ng, A. , and Cervenan, M.R. , Can. J . Phys. 66, 381 (1978). 77. Ng, A. , Pitt, L. , Salzmann, D., and Offenberger, A . A . , Phys. Rev. Lett. 42, 307 (1979). 78. Handke, J., Rizvi, S .A.H. , and Kronanst, B., Appl. Phys. 26, 109 (1981). 79. Baldis, H . A . and Walsh, C.J. , Phys. Fluids 26, 3426 (1983). 80. Ripin, B .H. , McMahon, J . M . , McLean, E .A. , Manheimer, W . M . , and Stam-per, J .A. , Phys. Rev. Lett. 33, 634 (1974). 81. Grek, B., Pepin, K, and Rheault, F., Phys. Rev. Lett. 38, 898 (1977). 82. Grek, B., Pepin, H . , Johnston, T . W . , Leboeuf, J .N. , and Baldis, H.A. , Nucl. Fusion 17, 1165 (1977). 83. Turner, R.E . and Goldman, L . M . , Phys. Rev. Lett. 44, 400 (1980). 84. Handke, J., Rizvi, S .A.H. , and Kronast, B., Phys. Rev. Lett. 61, 1660 (1983). 85. Chen, F.F. , in Laser Interaction and Related Plasma Phenomena, ed. by H.J . Schwarz and H . Hora (Plenum, New York, 1974), Vol. 3A, p. 291. 86. Eidmann, K. and Sigel, R. in Laser Interaction and related Plasma Phenom-ena, ed. by H.J. Schwarz and H . Hora (Plenum, New York, 1974), Vol. 3B, p. 667. 87. Stamper, J .A. , Barr, O.C. , Davis, J. et. al. in Laser Interaction and Related Plasma Phenomena, ed. by H.J . Schwarz and H . Hora (Plenum, New York, 1974), Vol. 3B, p. 713. 88. Lehmberg, R . H . , Phys. Rev. Lett. 41, 863 (1978). 89. Clayton, C . E . , Ph.D. thesis, University of California, Los Angeles (1984). 90. Giles, R. and Offenberger, A . A . , Phys. Rev. Lett. 60, 421 (1983). 91. Gellert, B. and Kronast, B., Appl. Phys. B33, 29 (1984). 92. Walsh, C.J . and Baldis, H.A. , Phys. Rev. Lett. 48, 1483 (1982). 93. Gellert, B. and Kronast, B., Appl. Phys. B32, 175 (1983). 94. Lui, C.S., Rosenbluth, M . N . , and White, R.B., Phys. Fluids 17, 1211 (1974). 95. Rosenbluth, M . N . , White, R.B., and Liu, C.S., Phys. Rev. Lett. 31, 1190 (1973). 96. Liu, C.S., Rosenbluth, M . N . , and White, R.B., Phys. Rev. Lett. 31, 697 (1973). 97. Nicholson, D.R., Introduction to Plasma Theory, (Wiley, New York, 1983). 98. Sato, N. , Ikezi, H. , Yamashita, Y. , and Takahashi, N . , Phys. Rev. Lett. 20, 837 (1968). 99. Malmberg, J . H . and Wharton, C.B. , Phys. Rev. Lett. 19, 775 (1967). 100. Kono, M . and Mulser, P., Phys. Fluids 26, 3004 (1983). 101. Morales, G.J . and O'Neil, T . M . , Phys. Rev. Lett. 28, 417 (1972). 102. Ikezi, H . , Schwarzenegger, K. , Simons, A . L . , Ohsawa, Y. , and Kamimura, T . , Phys. Fluids 21, 239 (1978). 103. Wharton, C.B. , Malmberg, J .H. , and O'Neil, T . M . , Phys. Fluids 11, 1761 (1968). 104. Mima, K. and Nishikawa, K. , J . Phys. Soc. Japan 33, 1669 (1972). 105. Meyer, J., Bernard, J .E. , Hilko, B., Houtman, H . , Mcintosh, G . , and Popil, R., Phys. Fluids 26, 3162 (1983). 106. Chen, F.F. , in Laser Plasmas and Nuclear Energy, ed. by H . Hora (Plenum Press, New York, 1975). 107. Sodha, M.S. , Ramamarthy, K. , and Sharma, R.P., Plasma Phys. 24, 223 (1982). 108. Ng, A. , Salzmann, D. , and Offenberger, A . A . , Phys. Rev. Lett. 43, 1502 (1979). 109. Haas, R.A. , Boyle, M.J . , Manis, K.R. , and Swain, J .E. , J . Appl. Phys. 47, 1318 (1976). 110. Baldis, H .A. and Corkum, P.B., Phys. Rev. Lett. 45, 1260 (1980). 111. Meyer, J . and Houtman, H . , Phys. Rev. Lett. 63, 1344 (1984). 112. Meyer, J., Bernard, J .E. , Hilko, B., Houtman, H . , Mcintosh, G . , and Popil, R., Phys. Rev. A 2 9 , 1375 (1984). 113. Baldis, H.A. , Samson, J .C. , and Corkum, P.B., Phys. Rev. Lett. 41, 1719 (1978). 114. Baldis, H . A . and Walsh, C.J. , Phys. Fluids 26, 1364 (1983). 115. Karttunen, S.J., Phys. Rev. A23, 2006 (1981). 116. Meyer, J . and Bernard, J .E. , Phys. Fluids, in press. 117. Sagdeev, R.Z., and Galeev, A . A . , Nonlinear Plasma Theory, (Benjamin, New York, 1969), p. 94. 118. Biskamp, D. and Chodura, R., Phys. Rev. Lett. 27, 1553 (1971). 119. Kadomtsev, B.B., Plasma Turbulence, (Academic Press, London, 1965), p. 68. 120. Watanabe, S., J . Phys. Soc. Japan 35, 600 (1973). 121. Estabrook, K. , Phys. Rev. Lett. 47, 1396 (1981). 122. Gray, D.R., Kilkenny, J .D. , White, M.S., Blyth, P., and Hull, D. , Phys. Rev. Lett. 39, 1270 (1977). 123. Martin, F., Baldis, H.A. , and Walsh, C.J . , Phys. Rev. Lett. 52, 196 (1984). 124. Jackson, E .A. , Phys. Fluids 3, 786 (1960). 125. Baldis, H.A. , Walsh, C.J . , and Benesch, R., Appl. Opt. 22, 2217 (1983). 126. Goodman, J.W., Introduction to Fourier Optics, (McGraw-Hill, San Francisco, 1968). 127. Fowles, G.R., Introduction to Modern Optics, (Holt, Rinehart, and Winston, New York, 1975). 128. Baldis, H.A. , Burnett, N.H. , and Richardson, M .C. , Rev. Sci. Instrum. 48,173 (1977). 129. Wolfe, W.L. and Zissis, G.J. , The Infrared Handbook, (Dept. of the Navy, Washington, 1978). 130. Jenkins, F . A . and White, H.E . , Fundamentals of Optics, 4th ed., (McGraw-Hill, New York, 1976). APPENDIX 1: MEASUREMENT OF THE FOCAL SPOT SIZE 193 A P P E N D I X 1 M E A S U R E M E N T O F T H E F O C A L S P O T S I Z E In order to determine the focussed laser intensity at the location of the plasma it is necessary to measure the size and shape of the focus. In the infrared, where ordinary photographic techniques are not available, focal spot size measurements are usually performed by: 1. Measuring the transmission through a pinhole of known size, 2. Measuring the x-ray emitting region from x-ray pinhole photographs, or 3. Measuring the size of burnmarks. The first technique has several disadvantages. First of all, it is difficult to position a small (~ 100/im) pinhole accurately at the laser focus. Secondly, in order to prevent the formation of a plasma at the edge of the pinhole, the laser beam must be strongly attenuated either by placing attenuating filters in the beam or by not firing the later stages in the laser amplifier chain. The only attenuators available were polyethylene sheets which were irregular in shape and therefore likely to introduce changes in the wavefront coherence and so affect the properties of the focus. The focus would also be affected if the later amplifier stages were not fired since the saturating properties of the power amplifiers can significantly change the beam profile. Finally, transmission measurements give only the average intensity but no information on the shape of the focus. The second technique could not be used in this experiment because the low x-ray flux coming from the plasma made it impossible to obtain accurate pinhole photographs. A variation of the third technique was used in the present experiment. APPENDIX 1: MEASUREMENT OF THE FOCAL SPOT SIZE 194 A self-calibrating method was developed which relied on the Fourier trans-forming properties of lenses and allowed the determination of the focal spot size and shape in a single shot. The technique is illustrated in Fig. A l - 1 . The incident laser beam passes through a 2-D multiple slit mask located directly in front of the main focussing lens and is then focussed onto a sheet of aluminized mylar produc-ing a diffraction pattern burnmark. The mask serves two purposes; first of all, it attenuates the incident beam and secondly, it produces a diffraction pattern which allows the calibration of the response of the aluminized mylar target. The theory for this technique will be briefly discussed in order to make clear the analysis which permitted contour maps of the focal spot intensity to be obtained. Incident C02 Laser Beam Figure A l - 1 The experimental arrangement for measuring the intensity distribu-tion at the focal spot. APPENDIX 1: MEASUREMENT OF THE FOCAL SPOT SIZE 105 We start with the mathematical expression of the Huygens-Fresnel principle: 1 2 8 U(fl>) = -k f f U ( P 1 ) ^ P J * r ° 1 c o s ( n ) r 0 i ) ds, ( A l - 1) 3 A J J roi E where U(Po) is the complex amplitude of the wave at point PQ due to the com-plex amplitude U(Pi) located at the diffracting aperture roi is the vector from Po to P i , n is the normal to the surface of the aperture, and k is the magnitude of the wavevector of the incident monochromatic radiation. The spherical Huy-gens wavelets are replaced by quadratic surfaces (Fresnel approximation) and the complex amplitude at point PQ can be written as + 0 O + O O I I ( U ( x l i y i ) e x p [ ^ ( x ? - r r f ) ] ) — oo —oo e x p l . — A T ^ z ° Z i + y m ^ \ d x i d y u ( A 1 ~ 2 ^ where U(xi ,y i ) is the complex amplitude at the position of the aperture. It has been assumed that z is much greater than the linear dimensions of the aperture. In the present experiment, where the mask was located directly in front of the lens, U ( x i , j/i) is made up of three factors: 1. j4ob(xi, j / i ) . . .the amplitude of the incident laser beam, 2. m(xi, yi).. .the transmission characteristics of the mask, and 3. exp — 2j(x\ + J/i)j • • -the effect of the lens on the phase (a constant phase factor has been neglected). APPENDIX 1: MEASUREMENT OF THE FOCAL SPOT SIZE 196 Substituting U(zi , j / i ) = i4 o b(xi J j / i)m(xi,yi)exp|^-0(ii+yi)j into A l - 2 , setting z = / , and neglecting a constant phase factor we have U(x / ,y / ) = exp[0(x} + yj)] A 0 + 00 +0O J J b ( x , y ) m ( x , y ) e x p [ - ^ ( x x / + yy/)] dxdy. ( A l - 3) The integral in A l - 3 is just the 2-D Fourier transform of the product b(x, y)m(x, y) with the spatial frequencies given by vx = x//Xf and uy = y / / A / . Now 7 b(x,y)m(x, y) = B(vx,vy) *M(vx,uy) where B(vx,i/y) = j [b(x ,y) and M(ux,vy) = T m(x,y) . Therefore, with the mask in place, the amplitude of the focal spot intensity distribution, B(i / X , p y), is convoluted with the Fourier transform of the mask transmission, M(vx,vy). For the 2-D multiple slit mask used m(xu yx) = ($(zi) * /(x x)) (g[yi) * / (yi ) ) , where and 9xW ~ \ o if |x| > ax/2; oo where ax and hx are the slit widths and separations respectively. Similar forms apply for gy(y) and fy(y). Now, since the x and y components can be separated f[m(x,y)] = T[gx(x) * fx{x)]?[gy(y) * /„ (y ) ] . = f[fo(x)]f[/,(x)]/[^(y)]7[/,(y)]; APPENDIX 1: MEASUREMENT OF THE FOCAL SPOT SIZE 197 where r 1 1 sin Tvxax . 7 0*(x) = = sinc(i/*a*), ( A l - 4) L J ir t^a* and from Goodman, 1 ' 8 /.(*)] = £ - f )• (A1 -5) n=—oo x etc. Therefore, the convolution in Eq. A l - 3 becomes (considering the x-component only), sinc(rax) | f ; 6 (<r - }B(I/X - 0 * - f ; - c ( ^ ) B ( . . - i ) . ( A l - 6 ) n=—oo The intensity in the focal plane is proportional to the square of this amplitude. If the slit separation is small enough, then there is negligable overlap between the various orders, and hence, the cross terms in the square of Eq. A l - 6 are negligable. Therefore, the intensity is given by (2-D): m=—oo n=—oo " " Hence, the size and shape of each order in the multiple slit diffraction pattern are determined by the focus spot intensity distribution. The overall intensity of each order is determined by the single slit diffraction pattern. A photographic enlargement of the burn pattern on the aluminized mylar tar-get is shown in Fig. A l - 2 along with the single slit diffraction pattern for yj = 0 and Xf = 0 which formed the intensity envelope of the observed pattern. The entire laser chain was fired. The mask used to obtain this pattern had separations of APPENDIX 1: MEASUREMENT OF THE FOCAL SPOT SIZE 198 Figure A l - 2 An enlargement of a typical burnmark. The single slit diffraction patterns used in the calibration of the response of the mylar are also shown. hx = hy = 10.0 mm, and widths of ax = 0.8 mm and ay = 1.0 mm. A certain laser intensity is required to burn the aluminum coating off the mylar, lower intensities leave the coating intact while higher intensities cause little additional change to the mylar after burning off the aluminum coating. Therefore, the intensity at the edge of each burnmark is always the same. From the product of the single slit diffrac-tion patterns it was possible to determine the relative intensities of the radiation producing all the burnmarks and so calibrate the response of the aluminized mylar. A contour map of the focal spot intensity distribution obtained from the bunmark of Fig. A l - 2 is shown in Fig. A l - 3 . Fig. A l - 4 is a plot of ln(Calculated relative intensity) vs. (Measured radius)2 from which a waist of o>o = 49 fim (radius) is determined for the roughly Gaussian intensity profile. This value is about twice the diffraction limited result of 24 fim for a Gaussian beam with a 1/c diameter of APPENDIX 1: MEASUREMENT OF THE FOCAL SPOT SIZE 199 7.0 cm focussed by a 50 cm f.l. lens. However, on some shots, part of a separate ring was detected around the main burn mark. This indicates that the burnmark was in reality an Airy pattern. Indeed, the central Airy disk which would be pro-duced by a 7 cm diameter constant amplitude wave incident on the focussing lens has an intensity profile, out to the first minimum, very similar to a Gaussian with a 1/e radius of 49 /im. Since the last two laser amplifiers were saturated, it is not surprising that the beam reaching the focussing lens had an intensity profile more like a constant intensity wave than a Gaussian. H 100 jmm i Figure A l - 3 Contour map of the focal spot intensity distribution. It was observed that the shape of the burnmark changed somewhat from shot to shot. This was likely due to small variations in the incident beam profile possibly caused by changes in the gain uniformity of the amplifiers. It should also be noted that the mask had the undesirable property of filtering out the high spatial APPENDIX 1: MEASUREMENT OF TEE FOCAL SPOT SIZE 200 2500 5000 7500 Radius 2 (jam2) Figure A l - 4 Graph of ln(Calculated relative intensity) vs (Measured radius)2. frequency variations (spatial frequency > 1 c m - 1 ) in the incident beam profile. If this occurred, the measured intensity profile would differ significantly from the actual profile. A burnmark, produced by focussing the beam at the location of the main focussing lens is shown in Chapter 3. The structure in the beam is accentuated by the nonlinear response of the bum paper. Since the slits in the mask had a spacing of lO.O.mm, some of the high frequency structure was likely filtered out by the mask. APPENDIX i: RAY TRACING 201 A P P E N D I X 2 R A Y T R A C I N G Geometrical ray tracing was performed on the density maps in order to deter-mine the effects of refraction on the laser intensity in the plasma. The ray tracing technique relied on the fact that in a medium of refractive index fi the phase velocity is given by c/ft where c is the speed of light in a vacuum. The refractive index for a plasma is given by: /x = ( l - n e / n c r ) 1 / 2 where n e is the plasma electron density and ncr is the critical density (101 9 c m - 3 for A = 10.6 fim). The procedure used for the ray tracing is illustrated in Fig. A2-1. At point Pi the ray is travelling in the direction shown. The ray is given an imaginary width, /, and the refractive indicies, fi^ and fig are calculated at the two endpoints, A and B by linear interpolation of the density grid determined from interferometry. In a time step i , the edges of the ray then move the distances ct/fi^ and ct/fiff to the points C and D. The points C and D are then connected by a straight line whose perpendicular bisector defines the new position and direction of the ray. The procedure is then repeated at point, P2t and so on. The technique requires a careful selection of the ray width, /, and the time step, t. Both / and t were decreased in magnitude until further changes did not affect the resulting ray paths. The values of / and t were then chosen to satisfy this APPENDIX I: RAY TRACING 202 B • n(p,q) • n(p*1,q) Figure A2-1 The geometry assumed in the ray tracing. condition as well as to minimize the resulting computer time. In practice, the rays were given a width of 0.5 fim and the time step was 5 x 10~ 1 4 s. The operation of the ray tracing program was checked on a problem that could be solved analytically. A density distribution was chosen that gave a refractive index profile that varied linearly from 0.5 on axis to 1.0 at a radius of 1.5 mm. Snell's law gives fi cos 8 = K where i f is a constant and 0 is the angle between the x axis (perpendicular to the density gradient) and the ray direction. The ray path can then be found from, The results are shown in Fig. A2-2. The dashed curve is the analytical result and the solid curve is from the ray tracing program. The small shift between the two curves is likely due to slightly different starting points. APPENDIX t: RAY TRACING 3 0 3 0 I l 1 1 1 1 1 1 ~ I * 0 OA 0.8 12 1.6 2.0 x-position (mm) Figure A2-2 A comparison of the ray paths calculated theoretically (dashed curve)and with the ray tracing program (solid curve). APPENDIX 3: THE IMAGE DISSECTOR 304 A P P E N D I X S T H E I M A G E D I S S E C T O R The image dissector used in this experiment was similar in construction to that descibed by Baldis et. a l . 1 ' 8 Combined with an infrared spectrometer, the image dissector permitted the measuring of spectra in the infrared region in a single shot while still using only one detector. The apparatus used in this experiment is shown in Fig. A3-1. A magnified (4.4 times) image of the spectrum at the spectrometer output plane is produced at the top edge of the square mirror, Af 1, by lens #1 (2 in diameter, 15 cm f.l.). The mirrors, M l and M 2 , are separated from Af3 by their common radius of curvature (1 m) as shown such that Af2 produces a series of images of the spectrum progressing down and towards the left edge of Af 1 while M 3 produces images of the spectrum progressing down and to the right on Af 1. The left edge of Af 1 is sharp and a piece of the spectrum is sliced off each time the pulse reaches this edge. By properly aligning the mirrors, 10 or more 'channels* can be obtained. Since each channel involves 4 reflections, a total of more than 40 reflections in the dissector are possible. For this, reason, the optical quality of Af 1, Af 2, and Af3 was high (A/10) and the mirrors were gold coated giving a single surface reflectivity of ~ 99%. The spectrometer-image dissector combination was aligned with the aid of a HeNe laser. The 16th order reflection from the grating was sufficiently bright for this purpose. Final alignment and focussing of the lenses, however, had to be performed using 10.6 fim radiation because of the difference in the refractive indices APPENDIX S: THE IMAGE DISSECTOR 205 Figure A3-1 The image dissector. of KC1 at 10.6 fim (1.454) and at 6328 A(1 .488) . m For this purpose, the C W C 0 2 laser beam was focussed onto the spectrometer slit. The series of slices of the spectrum were displaced vertically from one another along the left edge of M l but their rays appeared to come from a horizontal strip across M 2 . The "object" was therefore highly astigmatic and the second lens (2 in diameter, 15 cm f.l.) had to be tilted in order to bring all the channels to a common focus at the detector. The required position and tilt angle of lens #2 was roughly calculated 1 8 0 and then final adjustments were made with the aid of the C W CO2 laser which was sent into the spectrometer and manually scanned across the output plane to simulate a broad spectrum. The alignment of the image dissector was adjusted before each experimental run. The amplified 2 ns pulse emerging from the K103 amplifier was sent to the spectrometer and was used for calibrating the individual channel responses. To APPENDIX S: THE IMAGE DISSECTOR 206 simulate the broad spectrum, the spectrometer grating was scanned so as to move the monochromatic output across the exit plane. A typical response curve is shown in Fig. A3-2. The expected effect of reflection losses on the higher channels is partially masked by misalignment effects. The instrument width was determined from the number of channels in the response to the monochromatic input. It was typically 2-3 channels and was uniform across the 16-20 channels used. 1 0 i — i — | — i — i — i — i — i — i — • — i — i — i — i — i — " — i — i — r Channel Number Figure A3-2 The measured intensity response of the image dissector as a function of channel number. Either a Cu:Ge detector or a (Cd,Hg)Te detector was used in the experimen-tal runs. The Cu:Ge detector had a large detector crystal (4 mm) and the highest sensitivity but had to be cooled with liquid helium. The (Cd,Hg)Te detector, how-ever, could be operated at room temperature but was not as sensitive as the Cu.Ge APPENDIX 3: THE IMAGE DISSECTOR 307 Figure A3-3 A scope trace of the output of the image dissector showing two temporally separated backscatter spectra. detector and had only a 1 mm square detector crystal which made alignment more difficult. A typical scope (Tektronix 7104) trace, obtained for backscatter is shown in Fig. A3-3. The pulses are separated by 13.3 ns (4.0 m). Many of the pulses show double peaks (not to be confused with a shoulder produced by the detector) indicating at least two different temporally separated spectra. Therefore, in certain cases, the image dissector can provide temporal information as well as producing single shot spectra. L i s t of Publications John Edward Bernard 1. J.B. Hutchings and J.E. Bernard "A Spectroscopic Reinvestigation of the Massive Binary HD698", Publ. Astron. Soc. Pac. 90, 179 (1978). 2. J.B. Hutchings, J.E. Bernard, D. Crampton, and A.P. Cowley "Spectroscopy and Possible Orbital Periods for HDE245570) (=A0535+26)", Astrophys. J . 223, 530 (1978). 3. J.B. Hutchings, J.E. Bernard, and L. Margetish "Spectroscopic Studies of Nova V1500 Cygni I I . Models for the 3 Hour Profile Modulations", Astrophys. J . 224, 899 (1978). 4. J.B. Hutchings, J.E. Bernard, L. Margetish, and M. McCall "Spectroscopic Data from (Nova) V1500 Cygni 1975", Publ. Dom. Astrophys. Obs. (Canada) 15, 73 (1978). 5. J.E. Bernard, F.L. Curzon, and A.J. Barnard "Experimental Stark Profiles of. He II 3203 and He II 4686", accepted for publication in J . Quant. Spec, and Rad. Trans. (1980) but published instead in Spectral Line Shapes; Proc. of the Fifth International Conf.  Berlin 1980. B. Wende (Walter de Gruyter, 1981). 6. C.J. Walsh, J . Meyer, B. Hilko, J.E. Bernard, and R. Popil "Observations of Stimulated Brillouin Scattering and Enhanced Ion Wave Damping in a Z-Pinch Plasma", J . Appl. Phys. 5_3, 1409 (1982). 7. J . Meyer, J.E. Bernard, B. Hilko, H. Houtman, G. Mcintosh, and R. Popil "Quenching of Two-Plasmon Decay and Stimulated Raman Scattering Instabilities by Profile Modification", Phys. Fluids 26_, 3162 (1983) . 8. J . Meyer, J.E. Bernard, B. Hilko, H. Houtman, G. Mcintosh, and R. Popil "Observation of Energetic Electrons Produced in Laser-Irradiated Plasmas at Quarter C r i t i c a l Density", Phys. Rev. A29, 1375 (1984). 9, A.J. Barnard and J.E. Bernard "Calculated Frequency Shifts in Stimulated Brillouin Backscattering from Homogeneous Plasma with Multiple Ion Species", Can. J . Phys. 63, 354 (1985). |QJ. Meyer and J.E. Bernard "Large Amplitude Ion Acoustic Fluctuations Generated by the Two Plasmon Decay Instability", Phys. Fluids in press. 

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