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Study of half harmonic plasma waves in CO laser-plasma interactions Zhu, Yueqiang 1994

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STUDY OF HALF HARMONIC PLASMA WAVES INCO2 LASER-PLASMA INTERACTIONSByYueqiang ZhuB. A. Sc., Xian E.S.T. University. Xian, PRC 1982M. Sc., University of British Columbia 1987A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPT. OF PHYSiCSWe accept this thesis as conformingto the required standard‘FIlE UNIVERSITY OF BRITISH COLUMI3IAiii i (- I 991( ‘Vueqiang. Zhu, 1994in presenting this thesis in partial fulfilment of tne requirements for an advanceddegree at the University of British Columbia, I agree that theLibrary shall make itfreely available for reference and study. I further agree thatpermission for extensivecopying of this thesis for scholarly purposes may be granted bythe head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall riot be allowed without my writtenpermission.(Signature)_________Department of__________The University of British ColumbiaVancouver, CanadaDate tV4DE-6 (2188)AbstractHalf harmonic plasma waves, which can be generated by the two plasmon decay (TPD)and stimulated Raman scattering (SRS) instabilities in the regions with plasma densitynear n/4, are studied experimentally in the interactions of an intense (I l0’4W/cm2)CO2 laser beam with an underdense plasma. The plasma is produced by focusing theCO2 laser pulse (2 us FWHM) onto a stabilized laminar nitrogen gas jet flowing from aLaval nozzle into low pressure helium. The plasma waves were investigated with rubylaser Thomson scattering. The absolute TPD and SRS instabilities were observed atpump intensities greater than 1.3 x l0’3W/cm2for a plasma with a density scale lengthof 2400 m and an electron temperature of 300 eV. The difference predicted by the theorybetween plasma waves generated by the two plasma decay instability and those due to thestimulated Raman scattering deduced from theory are experimentally confirmed. Twoplasmon decay is found to dominate the generation of half harmonic plasma waves. TheTPD plasma waves appear in series of up to 8 bursts. The intensity distribution of theunsaturated TPD plasma waves is determined to be well described by linear theory. Theintensity distribution of the saturated TPD plasma waves is shown to be governed bythe saturation and quenching mechanisms. The angular distribution of (3/2) emissiondeduced from the intensity distribution of the saturated TPD plasmons agrees well withthe experimental results. The experimental evidence shows that the coupling of plasmawaves to ion acoustic waves and the steepening of the density profile are the dominantsaturation and quenching mechanism for the TPD instability.HTable of Contents1 INTRODUCTION2 Theory of Two Plasmon Decay and Stimulated Raman Scattering2.1 Introduction2.2 Parametric Instabilities in Laser-Plasma Interaction2.3 Parametric Instabilities and Laser Fusion2.4 Ponderomotive Force2.5 Derivation of the Coupling Equations2.5.1 The Electron Fluid Maxwell Equations2.5.2 The Resonant Coupling of Two High Frequency Daughter Waveswith the Pump waveDispersion Relation2.5.3DampingTPD and SRS in an inhomogeneous plasmaSaturation2.8.1 Electron TrappingAbstract iiList of Tables viList of Figures viiAcknowledgements x2.62.72.855591317171821242833331112.8.2 Mode Coupling and Profile Modification . . 343 EXPERIMENTAL APPARATUS AND SET-UP 373.1 Introduction 373.2 CO2 Laser System 373.3 First CO2 laser system 393.4 The Upgraded CO2 Laser System 413.4.1 The Hybrid oscillator 433.4.2 Short pulse generator 453.5 The amplifier chain . 483.5.1 The Preamplifiers . . 483.5.2 Amplifiers 493.5.3 The Optical Set Up for the Laser System . 503.6 Operating procedure 523.7 The Gas Jet Target 533.8 Streak Camera 563.9 Thomson Scattering 593.10 Experimental Set Up for Thomson Scattering Measurements 653.11 Synchronization 744 RESULTS 774.1 Introduction 774.2 Maximization of the scattered light intensity 774.3 Results from the experimental set-up with the original CO2 laser system 804.3.1 Spatially resolved Thomson scattering measurements 824.3.2 Wave-number spectra of EPW’s 874.4 Results from experimental set-up with the upgraded CO2 laser system . 91iv4.4.1 Spatially Resolved Thomson Scattering 914.4.2 Wave-vector Resolved Thomson Scattering Measurements 964.4.3 The distribution of the intensity of EPW’s in wave-vector space 1054.4.4 Measurements with S-polarized pump radiation 1205 DISCUSSION 1245.1 Introduction 1245.2 Experimental conditions 1255.3 The nature of TPD and SRS instabilities 1255.4 TPD and SRS instabilities 1285.5 Intensity distribution of the TPD plasmons in the wave-number space 1355.5.1 Temporal behaviour 1355.5.2 Intensity distribution of unsaturated plasmons 1395.5.3 Intensity distribution of saturated plasmons 1425.5.4 TPD and (3/2)c emission 1445.6 Saturation 1475.6.1 Mode-coupling 1505.6.2 Plasma density profile modification 1536 CONCLUSIONS 1576.1 Summary and conclusions 1576.2 Original Contributions 1586.3 Suggestions for further work 159Bibliography 160vList of Tables4.1 Optimization of the jet pressnre 78viList of Figures2.1 Locations of instabilities in a corona plasma 102.2 Feedback leading to the parametric instabilities. . 132.3 Wave-vector matching condition for TPD and SRS 232.4 A mechanical analogue for Landau damping 273.1 Overview of the experimental apparatus 383.2 The layout of the old CO2 laser system 403.3 The layout of the new CO2 laser system 423.4 Hybrid oscillator and short pulse generator 443.5 The shape of the CO2 laser pulse . . 483.6 Laval nozzle 543.7 Target chamber 573.8 The principle of operation of a streak camera. . . 583.9 The scattering coordinate system 613.10 Simultaneous wave-vector matching conditions for TPD and Thomsonscattering 673.11 The location of possible detected EPWs in wave-vector space 683.12 Set up for spatially resolved Thomson scattering 703.13 Projection effects on spatially resolved Thomson scattering 723.14 Synchronization of lasers and associated instruments 754.1 Photographs of typical streak records of wave-number resolved Thomsonscattering 81vii4.2 Streak records of spatially resolved Thomson scattering measurements . 834.3 More streak record of spatially resolved Thomson scattering measurement 844.4 Plasma density profile 864.5 A photo of a streak record of spatially resolved Thomson scattering measurement 874.6 Contour plots of EPW’s as a function of time and wave-number 894.7 Time-integrated wave-number spectra of EPW’s 904.8 Streak image of spatially resolved EPW’s 924.9 More streak image of spatially resolved EPW’s 934.10 Time-integrated spatial distributions of EPWs’ intensity 954.11 Spatially-integrated temporal evolution of EPW’s intensity 974.12 Photographs of typical streak images of EPW’s from different geometries 984.13 More photos of streak images of EPW’s 994.14 First set of streak images of wave-number spectra of EPW’s 1004.15 Second set of streak images of wave-number spectra of EPW’s 1014.16 Third set of streak images of wave-number spectra of EPW’s 1024.17 Fourth set of streak images of wave-number spectra of EPW’s 1034.18 Fifth set of streak images of wave-number spectra of EPW’s 1044.19 Temporal evolution of EPW’s displayed in Figure 4.14 1064.20 Temporal evolution of EPW’s displayed in Figure 4.15 1074.21 Temporal evolution of EPW’s displayed in Figure 4.16 1084.22 Temporal evolution of EPW’s displayed in Figure 4.17 1094.23 Temporal evolution of EPW’s displayed in Figure 4.18 1104.24 First set of the averaged wave-number spectra of EPW’s 1124.25 Second set of the averaged wave-number spectra of EPW’s 1134.26 Third set of the averaged wave-number spectra of EPW’s 114viii4.27 . . .. 1154.28 . . .. 1164.29 . . .. 1174.30 . . .. 1184.31 . . .. 1194.32 . . .. 1214.33 1224.34 123The wave-number ranges for the TPD and SRS plasmons 130Growth rate contours of the TPD plasmons 133Exponential growth of TPD plasmons observed at = 19° 136Growth rate of TPD plasmons at = 19° 137Intensity distribution of unsaturated TPD plasmons 141Intensity distribution of saturated TPD plasmons and the threshold contourl43Expectation of (3/2) harmonic emission 145Unsaturated plasmons and (3/2)w0 emission 146Intensity distribution of saturated TPD plasmons 148The angular distribution of (3/2) emission intensity 149Peak ion acoustic fluctuation levels as a function of pump energy 154Fourth set of the averaged wave-number spectra of EPW’sFifth set of the averaged wave-number spectra of EPW’sThe number-plot of the intensity distributionDistribution of the intensity of EPW’s integrated over 100 ps.Distribution of the intensity of EPW’s integrated over 2Ops. .Streak records of EPW’s for 5- and P-ploarized pump at 3 = 19°Streak records of EPW’s for S- and P-ploarized pump at 3 = 55°Streak records of EPW’s for 5- and P-ploarized pump at = 127°5.15.25.35.45.55.65.75.85.95.105.11ixAcknowledgementsI would like to express my sincere thanks to my supervisor, Dr. Jochen Meyer, for hisguidance and encouragement in the course of this work. Special thanks are extended toDr. Luiz DaSilva for sharing his streak image data-collecting program, and to Al Cheuckand Hubert Houtman for providing technical support and for teaching me technical skills.I would also like to thank my fellow students Abdul Elezzabi, Michael Hughes, Dr. MichelLaberge, Steven Leffler, Ross McKenna, and Mostafa Sadeghi for hours of interestingdiscussions. A special thank you note is extended to Ross McKenna for the monumentaltask of correcting my English. Finally, I would like to thank my wife, Yuling Li, for herconstant support and encouragement.xChapter 1INTRODUCTIONMotivated by the prospect of laser fusion, laser-plasma interaction studies have becomea very active research field since the 1960’s. After three decades of study, widespreadapplications have been developed using its principles[Kruer 91]. Besides its applicationto plasma heating both in laser fusion and in magnetic fusion, it also has many otherapplications including X-ray lasers, plasma spectroscopy, particle accelerators (beatwaveand wakefield accelerators), X-ray sources, and the general study of high-energy densityphysics. There are many different processes, such as parametric instabilities, in the interactions which compete to determine the coupling of an intense laser beam to a plasma.The significance of parametric instabilities in laser fusion and laser plasma interactionexperiments has long been established[Nuckolls 72, Kruer 81, Max 82, IKruer 88]. Parametric instabilities, such as stimulated Raman scattering (SRS) and two plasmon decayinstabilities (TPD), can generate c/2 plasmons in the quarter critical density region.,i.e., the region where the laser frequency, , is equal to twice the plasma frequency.These plasmons are electrostatic waves and can be driven to large amplitudes to createsuperhot electrons which have detrimental effects ill laser fusion. This dissertation willstudy these plasmons and determine their characteristics.In the present study, /2 plasma waves are studied in the interaction of an intenseCO2 laser beam with an underdense gas jet plasma. It is unlikely that laser fusion willuse CO2 lasers. However, the long wavelength of a CO2 laser implies that the plasmadensities at which TPD aild SRS instabilities occur are low, and therefore the plasmaChapter 1. INTRODUCTION 2waves can be studied with a visible laser beam Thomson scattering. On the other hand,the plasma waves due to TPD and SRS in plasmas generated by visible and UV laserradiation of relevancy to laser fusion are not readily detectable themselves because ofthe present lack of suitable far-UV lasers for Thomson scattering. Thus, knowledge ofTPD instability in short-wavelength laser experiments has been principally limited tothat which can be derived from the study of half-harmonic (primarily (3/2)w0 ) emission[Baldis 91]. Hence, the experimental study of TPD instability in CO2 laser radiationcan provide data as a test bed for theoretical and numerical studies of TPD instability,especially the nonlinear saturated TPD regime in short-wavelength laser experiments.The motivation for this study is to gain a better understanding of the features of(3/2)w0 harmonic emission. The w0/2 plasmons can scatter photons of an incident laserbeam and generate (3/2)c harmonic radiation. Since the (3/2) radiation is mucheasier to detect than the plasmons, the observations of (3/2)w0 emission provide an important diagnostic for the presence of w0/2 plasmons in laser fusion experiments. Thespectrum of (3/2)w0 emission has been suggested to provide a potential thermometer forcoronal plasmas. However, the predictions derived from linear theory are not consistent with experimental observations of the spectra[Seka 85, Turner 84] and the angulardistribution of the emission[Meyer, Zhu, and Curzon 89]. Many difficulties remain in understanding the relationship between (3/2)wo and TPD, especially, saturated TPD. Infact the mere link between them has to be shown experimentally. This motivated theauthor to study /2 plasmons with ruby laser Thomson scattering and to determinetheir properties.The plasmons were studied by time resolved Thomson scattering measurements oftheir spatial distributions and wave-number spectra. The important questions to beanswered in this study are:Chapter 1. INTRODUCTION 31. Both stimulated Raman scatteriug aud two plasmon decay instabilities can generatew0/2 plasmons. Which instability dominates the generation?2. What is the intensity distribution of w0/2 plasmons in the wave-number space?3. What is the dominant saturation mechanism of w0/2 plasmons?4. Do saturated or unsaturated w0/2 plasmons dominate the generation of (3/2)w0harmonic emission?This dissertation is organized as follows. In the next chapter, the theoretical background for two plasmon decay and stimulated Raman scattering instabilities is given.First, the general features of parametric instabilities in laser plasma interactions are reviewed. Then the relation between laser fusion and parametric instabilities is discussed todemonstrate the importance of the study of parametric instabilities in the development oflaser fusion. The driving force for the parametric instabilities, the ponderomotive force,is derived. With this force the physical picture of the parametric instabilities is easilyconstructed. The rest of the chapter is devoted to TPD and SRS instabilities. The coupling equations for these instabilities are derived using linear theory for a homogeneousplasma. From these equations the dispersion relations and the growth rates are obtained.Since in practice, an experimental plasma is inhomogeneons, the models for TPD andSRS in an inhomogeneous plasma are reviewed. Chapter 2 concludes with the discussionof the saturation mechanisms for the instabilities.The experimental apparatus and set up are presented in chapter 3. Two different highintensity pulsed CO2 laser systems used for the study are described and their differencesare discussed. The target chamber and the target consisting of a gas jet are examinedand it will be shown that the plasma conditions are mainly controlled by the pressureof the jet. As all experimental results are obtained from ruby laser Thomson scatteringChapter 1. INTRODUCTION 4measurements, the principles of Thomson scattering are reviewed and the relation between the scattered ruby laser beam intensity and the plasma wave intensity is derived.The experimental set ups for spatially and temporally resolved Thomson scattering aredescribed at the end of the chapter.In chapter 4, the experimental results are presented. As the experiments were conducted using two CO2 laser systems, the results from the different systems are presented separately. Experiments done with the first system were designed to answerthe question: “Among TPD and SRS instabilities, which one dominates the generationof /2 plasmons?”, and the results are presented at the beginning of the chapter. Theresults from the other CO2 laser system are presented in the remainder of the chapter toanswer the rest of the defined questions.The experimental results are discussed and compared in Chapter 5. Here the natureof the instabilities, absolute or convective, is determined. The instability dominating thegeneration of w0/2 plasmons is determined. The intensity distribution of /2 plasmonsin the wave-number plane is discussed. It is shown that TPD is well described by lineartheory before it is saturated. The relation between TPD and (3/2)w0 harmonic emission is then demonstrated and discussed. Finally, after discussing the saturation andquenching processes for TPD, the dominant saturation mechanism is proposed.In the last chapter, the results are summarized and the important conclusions aildthe original contribution of this study are pointed out. Some suggestions for future studyend the chapter.Chapter 2Theory of Two Plasmon Decay and Stimulated Raman Scattering2.1 IntroductionThe theoretical background for two plasmon decay and stimulated Raman scatteringinstabilities is presented in this chapter. First, the general features of parametric instabilities in laser plasma interactions are reviewed. Then, the relationship between theparametric instabilities and laser fusion is discussed. The driving force for the instabilities, the ponderomotive force, is derived. The rest of the chapter is devoted to the twoplasmon decay and stimulated Raman scattering instabilities.2.2 Parametric Instabilities in Laser-Plasma InteractionWhen a pulsed high power laser beam is focused on a solid or a gas target, a plasma israpidly formed on the surface or in the gas as the extremely high electrical field strengthin the focal spot of the laser beam triggers avalanche ionization after initial free electronshave been generated via multi-photon ionization. When this laser beam or anotherpulsed high power laser beam then propagates through and interacts with the plasma,the laser beam may resonantly excite the natural modes of the plasma and couple energyto them. This type of resonant coupling is called a parametric instability . Here resonant‘The parametric instabilities are so called because of an analogy with parametric amplifiers orfrequency converters based on the properties of a nonlinear reactance, usually capacitive [Chen 88,Daglish 68]. In a nonlinear capacitance the charge is not linearly related to the applied voltage. Whenseveral alternatiug voltages are applied simultaneously to the capacitance, frequency mixing takes placeand the capacitance is capable of transferring energy from one frequency to another. In the parametricamplifier, this property is used to take energy from a ‘pump’ source at high frequency to amplify a signal5Chapter 2. Theory of Two Plasmon Decay and Stimulated Raman Scattering 6means that the waves satisfy the frequency and wave-vector matching conditions. In thecase of three-wave parametric instabilities, the pump wave resonantly interacts with onedaughter wave to drive another daughter wave. Assigning the subscript 0 to the pumpwave frequency () and wave number (k), and 1 and 2 to the daughter waves, the resonantcoupling conditions are:= (2.1)k0 = k1 + k2. (2.2)These conditions maximize the strength of the coupling that allows the two daughterwaves to efficiently gain energy from the pump.Although the pump is a high frequency electromagnetic wave (EMW) whose polarization is strictly transverse, i.e. E0 L k0, the daughter waves can be EMW’s or either oftwo kinds of waves which can exist in a nonmagnetized plasma: high frequency electrostatic or electron plasma waves (EPW) and low frequency longitudinal ion acoustic wave(lAW), both with a strictly longitudinal polarization, i.e. E k. The latter two representthe high frequency and low frequency responses of the plasma to density fluctuations.Only electrons are involved in high frequency waves. Because of their large mass, ionscannot take part in high frequency oscillations. The dispersion relations for these threekinds of waves in a homogeneous plasma respectively are:EMW: w2 = w + c2k (2.3)EPW: = w + 3vk (2.4)JAW: = ck2 for T, << ZTe, (2.5)where w is the frequency, k the wave-vector magnitude, c the speed of light, Ti(Te) ision(electron) plasma temperature, and Z is the charge states of ions. The electron plasmaat a lower frequency. In a plasma, the pump at high frequency and the signal at a lower frequency maybe different types of waves.Chapter 2. Theory of Two Plasmon Decay and Stimulated Raman Scattering 7frequency (wv) is given (in cgs units) by:= 47rn5e2 (2.6)meHere —e is the charge of an electron, me its mass, and fle the electron plasma density.The electron plasma thermal speed, vth is defined as:Vth = \/KBTe/me= 1.32 x 1O(cm/sec).Here TkeV is the electron plasma temperature being measured in keV and B Boltzmann’sconstant. The ion-acoustic or sound speed, c3 is given by:Cs ZKBTe/Mi1 >< lO/ZTev t(cm/sec), (2.7)where Z is the charge state of the ions, T measured in eV, and = ni/M is the ratioof proton to ion mass.It can be seen from eq. 2.3 that w, is the minimum frequency for propagation of anEMW in a plasma, as k becomes imaginary for ‘ <,. Since the characteristic responsetime for a plasma is w1, the plasma shields out the field of an EMW for .‘ <,. Sincecx c, = defines the maximum plasma density in which an EM wave offrequency can propagate. This is termed the critical plasma density (ne), and is givenby:wmflc = 4ire= 1.1 x 1021 /)(cm3, (2.8)where . is the free-space wavelength of the EMW in units of sum.Chapter 2. Theory of Two Plasmon Decay and Stimulated Raman Scattering 8If we restrict ourselves to three-wave interactions, and consider that each wave hasto satisfy it’s own dispersion relation and the resonant coupling conditions, then we findthat four kinds of parametric instabilities are possible: 1) an EMW decays into an EPWplus an JAW, referred to as the parametric decay instability; 2) both daughter wavesare EPWs, called the Two Plasmon Decay (TPD) instability; 3) one daughter waveis an EMW and another is an EPW, the resulting parametric instability is defined asStimulated Raman Scattering (SRS); and 4)in the Stimulated Brillouin Scattering (SBS)instability, one daughter wave is an EMW and another is an lAW. These processes canbe summarized as follows:EMWO —* EPW + JAW (Parametric Decay)EMWO —* EPW1 + EPW2 (TPD)EMWO —* EMWS + EPW (SRS) (2.9)EMWO —* EMWS + JAW (SBS).The parametric instabilities with two EM- or two IA-daughter waves are not possiblebecause the dispersion relations and the resonant coupling conditions can not be satisfiedsimultaneously.Considering the dispersion relations and the resonant coupling conditions , we findthat each of these four parametric instabilities can only occur at certarn plasma densityregions [Goldman 86]. We can write the plasma wave dispersion relation as= w(l + 3k2)’12 (2.10)where = (2)112 is the Debye length. Since plasma waves with kAD 0.3 areheavily Landau damped (we will discuss Landau damping in detail in a later section),we can make the approximation that w wi,. As cx e, it is clear that plasma wavesChapter 2. Theory of Two Pla.smon Decay and Stimulated Raman Scattering 9of a given frequency are very localized in density. Hence those instabilities involving anelectron plasma wave can only occur in very specific density regions.Fig. 2.1 shows the locations of parametric instabilities in a corona plasma in a solidtarget. The density profile shown is one of generic character in front of a solid target.The origin of X is in the surface of the target. Since the ion acoustic wave frequencyis always very small compared to w0, then for parametric decay .ü, =—w0. Thismeans that the parametric decay instability can only occur very close to n.The TPD instability is also quite localized. The two plasma waves generated in TPDinstability are produced at the same location, and therefore the frequency matchingcondition can be written as:= ‘l + 2 2w.Thus, the TPD instability can only occur in the density region around n/4. In contrast,the SBS and SRS instabilities are not localized. Because the ion acoustic waves and EMwaves can exist anywhere if the plasma density is below n, the SBS instability can occuranywhere at e < n. Thus, it is usually very difficult to determine the location of theSBS instability. Considering the frequency matching condition for the SRS instability, wecan see that SRS can only occur in the density region ne n/4. Scattering at densitiesabove n/4 leads to a scattered EM wave with a frequency less than the local plasmafrequency, and thus it cannot propagate.2.3 Parametric Instabilities and Laser FusionWe will later demonstrate, that the longer the plasma density scale-length(to be given aprecise definition later on), the higher are the growth rates for parametric instabilities.Thus, the long scale-length plasma encountered in the corona of laser fusion target provides ideal conditions for the occurrence of parametric instabilities. To understand theChapter 2. Theory of Two Plasmon Decay and Stimulated Raman Scattering 10Parametric Decayn(x)ssSRS4xFigure 2.1: Locations of instabilities in a corona plasma.Chapter 2. Theory of Two Plasmon Decay arid Stimulated Raman Scattering 11relevance of parametric instabilities to laser fusion, we must first examine what some ofthe requirements are for achieving laser fusion and what consequences the parametricinstabilities produce.Controlled thermonuclear fusion offers the promise of large scale nuclear energy production without the problems of radioactive waste that are inherent in fission reactors.Nuclei will fuse together at reasonable rates to generate energy only if their kinetic energyis high enough to overcome their mutual Coulomb barriers. The plasma temperature required for deuterium and tritium (D-T) fusion, the easiest fusion to achieve, is about108K (10 keV), and a plasma at such a high temperature cannot be contained in anymaterial vessel. The two most promising approaches to controlled fusion are either tocontain the plasma by a magnetic field (commonly called Magnetically Confined Fusionor MCF), or to make the reaction occur so fast that no confinement is necessary otherthan the inertia of the hot dense plasma (termed Inertially Confined Fusion or ICF).Laser fusion is one of the most studied approaches to ICF.The concept of laser fusion is as follows: a laser beam strikes the surface of a laserfusion target (spherical geometry), and deposits energy into the corona plasma whichis rapidly formed. Even though the laser light penetrates only to the critical densitysurface, the energy is transported inwards by thermal conduction into the “ablationsurface”, where the pressure is a maximum and the electron velocity changes sign fromthe inward acceleration to the outward ablation. The rapid outward expansion of theheated plasma produces a compression wave propagating towards the center of the targetcapsule. The compression of the capsule is designed so that the final fuel density achievedis ‘—‘ iO times the original liquid density of the D-T fuel. The fuel ignites at 10 keV andyields many times the driver input energy. Two of the fundamental requirements of highgain laser fusion are: (1) the absorption of most of driving laser energy in the target orhigh absorption efficiency; (2) the efficient or adiabatic compression of the capsules. OneChapter 2. Theory of Two Plasmon Decay and Stimulated Raman Scattering 12would therefore like to avoid any loss of driving energy via scattering and the capsuleremains as cold as possible before and during the compression. The term used to describethe adverse premature heating of the capsule prior to sizable compression is “preheat”.One of the processes that can preheat the capsule is the generation of energetic (alsocalled hot or superthermal) electrons.Both the SRS and SBS instabilities reduce absorption efficiency. As much as 30% ofthe laser light energy has been detected as SR scattered light [Kruer 91]. SBS has beenobserved to approach 100%, although not under laser fusion conditions [Handke 81]. Inaddition, high amplitude electrostatic EPWs generated by parametric decay, SRS andTPD instabilities produce superthermal electrons with energies from tens to hundredof keV [Meyer and Zhu 90, Villeneuve 84, Drake 84, McIntosh 86]. The time-scale overwhich these parametric instabilities develop is so short (on the order of a picosecond,while hydrodynamic time-scales are in the hundreds of picosecorids) that they will precede hydrodynamic compression and make it much less efficient. Parametric instabilitiestherefore can have detrimental effects on laser fusion [Baldis 92]. It is essential to understand under what conditions parametric instabilities are excited in order to find criteriato avoid or suppress them.On the other hand, parametric instabilities can be put to very good use as plasmadiagnostic tools if they are weak, so that they do not significantly affect the gain. Becauseof the extreme temperature, density, energy, and pressure values in a laser-fusion target,and the use of short wavelength lasers, there are very few simple methods to diagnose theperformance of the target. Spectroscopic studies of the electromagnetic radiation emittedfrom a laser-produced plasma can be used to find some of the plasma parameters, suchas temperature, density and scale-length. Post experimental studies have been verysuccessful at extracting the temperature inside the corona plasma, the density scalelengths at various locations, by measuring the signature of various parametric instabilitiesChapter 2. Theory of Two Plasmon Decay and Stimulated Raman Scattering 13V(öEiEo) onE0OEliFigure 2.2: Feedback leading to the parametric instabilities.within the framework of a specific theoretical model.2.4 Ponderomotive ForceThe basic mechanism for the parametric instabilities is the oscillation of electrons inthe electric field of the pump laser light propagating through a plasma with a densityfluctuation [Kruer 92]. An illustration of the feedback leading to the instabilities isshown in Fig. 2.2. Let there be a density perturbation (thermal level) with amplitudeSn and an intense pump wave with electric field amplitude E0 and frequency w0. Byforcing the electrons to oscillate with speed Vc, = eEo/(mec’o), the pump wave producesa current SJ = —e6nv0 cx SnE0. This current then generates an EM wave or an electricwave with electric field amplitude SE. In return this electric field beats with the pumpwave to generate variations in electric field pressure. This gradient in the field pressuregives rise to a force, the so called ponderomotive force ‘- V (8E), which acts toChapter 2. Theory of Two Plasmon Decay and Stimulated Rarnan Scattering 14enhance the original density perturbation. If the density fluctuation Sn and the electricfield SE are characterized by k1) and (w2,k2) respectively, Sn and SE will growexponentially if the resonant coupling conditions given by eq. 2.1 and 2.2 are satisfied.We see that the poilderomotive force is the driving force for the instabilities. In addition,in terms of the ponderomotive force, many nonlinear phenomena occurring in the laser-plasma interactions have a simple explanation and the ponderomotive force will be usedextensively in later sections. Hence we will briefly discuss ponderomotive force in rest ofthis section.The ponderomotive force [Chen 88, Niu 88] is a nonlinear force that arises due to thegradient in the electric field pressure. To derive this nonlinear force we start with theequation for the motion of an electron in the field of an EM wave with no dc field:= —e[E(r) + x B(r)], (2.11)where r is the instantalleous position of the electron, the thermal motion and the frictiondue to collisions with ions are neglected. To the first order, the electric field is assumedto be uniform in space and the second term in eq. 2.11 is small, so the resulting electronvelocity lies in the direction parallel to E. The nonlinearity of the force arises from twosecond-order effects. The first is caused by spatial non-uniformities in the EMW’s electricfield. These produce a net drift in the electron motion since an electron moves further inthe half cycle when it is moving from a strong field region to a weak field region than viceversa. The result is that the electrons move from regions of high intensity to regions oflow intensity. The second source of the nonlinearity is due to the Lorentz force. Assumean oscillating electric field of the form:E = E8(r) coswt,where E3(r) contains the spatial dependence. We can expand E3(r) about r0, the initialChapter 2. Theory of Two Plasmon Decay and Stimulated Raman Scattering 15position of the electron:E8(r) = E(r0)+ (Sr1 V)Er=r0+= E1+E2•. (2.12)Considering the first order terms in eq. 2.11, we getmdvei/dt = —eE(r)cost (2.13)Vel = —(e/mL)E(r0)sin wt (2.14)Sr1 = (e/mw2)E(rcoswt. (2.15)With Faraday’s equation:laB (2.16)we find B1:B1 = —c (v x E(r0)sinwt). (2.17)Then with all these first order values, we can find the second order force asm2= —e[(5r1 V)E3(r) + x Bifirr0= ————[(E3(r) V)E(r)cos2t+r)x (V x Es(r)) sin2wt]rr0.(2.18)By averaging over many cycles of the EMW, we get:m K2)=[(E3 . V)ES + E8 x (V x E3)]= --—VE4 mw2= fNL. (2.19)This is the so called ponderomotive force. We can see that the ponderomotive force isdue to the variation in the wave pressure. A direct effect of NL is the self-focusing andChapter 2. Theory of Two Plasmon Decay and Stimulated Raman Scattering 16subsequent filamentation of laser light in a plasma. For example, if a laser beam has aGaussian cross-section, then this force moves electrons out of the high intensity region ofthe beam, and causes the electron density inside the beam to be less than that outsidethe beam. As the plasma dielectric constant = 1— ne/nc, the plasma acts as a convexlens, focusing the beam to a smaller diameter. This self-focusing process may also causea laser beam with spatial intensity modulations to break into several beams or filaments.The measurement of emission [Meyer and Zhu 86] indicates that the self-focusing orfilamentation does exist in the present set of experiments. Thus the pump intensity maybe modified. The self-focusing or filamentation process may interact or effect parametricinstabilities. However, the direct measurement of self-focusing and filametation is a bigproject and will not be conducted in the present study.For TPD and SRS instabilities, the total electric field E(x, t) includes the pump field,E0(x, t), and daughter wave fields, Ei,2(x t) (daughter field can be a transverse scatteredelectric field or a longitudinal electric field):E(x,t) = E0(x,t) + Ei(x,t) +E2(x,t).Assume each field of form:E(x, t) = E exp [i(k . x—wit)] + c.c.,where i = 0, 1, 2. With the help of eq. 2.18 we can see that the resulting force consists ofcomponents of different frequencies. Neglecting the components of high frequencies, wecan derive the low frequency force which can enhance the original density fluctuationsby considering the cross terms. The result is:fNL = {[V(E0)2 + V(E1)2+ V(E2)2]d + [V(E. E2)]1k+ [V(E0 E1)]2k+ [vE1 . E2) + V(E) + V(E)]}. (2.20)Chapter 2. Theory of Two Plasmon Decay and Stimulated Raman Scattering 17The first three terms are the d.c. drifts independently caused by each of the threewaves. Both fourth and fifth terms enhance the original density fluctuations in the TPDinstability, or one enhances the original density fluctuation and another enhances thescattered light in the SRS instability. The last three terms, which occur at (w0, k0), arenegligible since E1,2 << E0.2.5 Derivation of the Coupling EquationsHaving described the basic principles of TPD and SRS instabilities, we now are ready toderive the coupling equations describing both the TPD and SRS instabilities around the(1/4)n plasma density region. The derivation is based on a number of references withthe important ones being: [Drake 74, DuBois 74, Forsiund 75, Kruer 88, Langdon 73,Lasinski 77, Liu 73, Liu 76a].2.5.1 The Electron Fluid Maxwell EquationsBecause the ion response to a high frequency EM field of a light wave is less than theelectron response by a factor of Zm/M, where Z is the charge state of the ion, and m(M)is the electron(ion) mass, we treat the massive ions as a fixed, neutralizing backgroundand describe the electrons as a warm fluid. Then we can write Maxwell’s equations andthe standard fluid equations (in cgs-Gaussian units) as:V. E = —47rene (2.21)V x E = (2.22)VB = 0 (2.23)laE 47rVxB = ——+—J (2.24)+ V. (nev) =c(2.25)Chapter 2. Theory of Two Plasmon Decay and Stimulated Haman Scattering 18ôv e vxB 3vç + (v. V)v = ——(E + ) — Vfle, (2.26)at m cwhere e and m are the density and mass of the electrons, respectively, and v is theelectron fluid velocity. The first four of these equations are the standard Maxwell’sequations describing the way in which an electric charge and current distribution giverise to electromagnetic fields. Eq.2.25 is the continuity equation. It is a statement of the(nonrelativistic, and fixed ionization state) assumption that no electrons may be createdor destroyed. The last expresses Newton’s second law or the law of momentum balance.The damping is not included at this stage. The thermal speed of the electron, vth, must besmall compared to the phase velocity of the waves which we wish to consider. This groupof equations given above describes the self-consistent evolution of the electromagneticfield and the electron fluid.2.5.2 The Resonant Coupling of Two High Frequency Daughter Waves withthe Pump waveConsider a polarized EM pump wave with the electric field Ec, propagatillg in a homogeneous plasma. The equilibrium consists of electrons oscillating with high velocity in thepump field with the ions forming a stationary background.We now disturb the equilibrium and study the evolution of the disturbance usinglinearized equations. Imagine a propagating density disturbance Sn associated with anelectrostatic wave perturbing this equilibrium. As the electrons oscillate in the electromagnetic field, which includes the field of the pump wave and the field of the electrostaticwave, a current J is generated. This current produces another electrostatic wave or anEMW with electric field E1. Here we only study three-wave parametric instabilities andput w1 =—w2 and k1 = k0 — k2. By taking the time derivative of eq. 2.24 and theChapter 2. Theory of Two Plasmon Decay and Stimulated Raman Scattering 19curl of eq. 2.22 we can get the wave equation for E1 asV2E1— V(V. E1) —___= (2.27)The Fourier transformed wave equation can be written as[(w—c2k)I + ck1k] = —i4irw1J, (2.28)where I is the unit dyadic. Since we assume the ions as stationary neutralizing background, we neglect the ion current and obtainJ(r,t) = erieV, (2.29)andJ(k1,Wi) = eff n(ki, w!)v(kii)6(k!i — ki — ki)8(wfi — / —W1)d3kFd//dw!dw!!. (2.30)In a finite plasma the integrals are replaced by sums over all ki and k/F which satisfy thematching conditions.We now linearize the above equations. We take v = v0 + ye, ne = n0 + Sfle, andE =E0+E1.Here v0 is the quiver velocity of electrons in the pump field, n3 is the uniformbackground electron plasma density, and Ve, 6ne and E1 are infinitesimal quantities. Wecan then write the curreilt asJ(k1,w1) = —e[nove(ki, w1) + Sn(k2,W2)v0(k,w0)], (2.31)and the linearized forms of eq. 2.25 and eq. 2.26 as8Sne+ (v0 V)ne + n0V v = 0 (2.32)+ V(v0 ye) = (E1 + V0 X B) — 3kBTeV (2.33)m c menoChapter 2. Theory of Two Plasmon Decay and Stimulated Rarnan Scattering 20We use eq. 2.33 to calculate Ve. The Fourier transformed form of eq.2.33 without considering the Lorentz force term may be written asZWiVe(ki, Wi)+iki[v0(k,w0).v(k2,W2)] = —E1(k,c’— i3kB Tk 6n(ki, w1) (2.34)72 men0Here v(k2,w)is due to the density fluctuation Sn(k2,W)and can be written ask2 Sn(k2,W)v(k2,w)= (2.35)“2andik1 E(k1,w)Sn(ki,Wi) =— 4 , (2.36)has been calculated using Poisson’s equation. Substituting v(k2,W)and 6n(ki,Wi) intoeq. 2.34 we obtaine e 3kBTkikiEi k1c Sn(k2,W)Ve(ki,Wi) = —[———-—E1—_________________— 2 (v0 k1) 1. (2.37)w1 me men0 4K k2 noHere we take k2 = k0— k1 and k0 . v0 = 0. As we mainly study TPD and SRS instabilitiesin the quarter-critical plasma density region, we may take w2 and Wi = — w.Then after substituting Ve into the expression for the current and combining with eq. 2.28we obtain— c2k —w)I+ (c2 —3Te)klkj = [kv0 — (v0 ki)ki]Sn(k2,w,m 2(2.38)The Right Hand Side (RHS) of eq. 2.38 is the current generated by the coupling of thepump field and the density fluctuation. If we neglect this coupling current, we recoverthe dispersion relations for a high frequency longitudinal plasma wave( k1 . E1 = k1E)or a transverse E.M. wave( k1 . E1 = 0) propagating in a uniform plasma:2 2 Be2 .W1 = W + k1 (Longitudinal)me= W + c2k (Transverse).Chapter 2. Theory of Two Plasmon Decay and Stimulated Raman Scattering 21Next we use eq. 2.32 and eq. 2.33 to derive an equation for the density fluctuation6n(k2,w) associated with an electron plasma wave. We may use the ponderomotiveforce to simplify the the derivation and write eq. 2.33 asaVe--- + V [v0 . ye]= —E+fNL—3kBTeV6 (2.39)me menowheree2fNL— V[E0E1]2.menowlTaking the divergence of eq. 2.39 and the time derivative of eq. 2.32, we can cancel eand get the Fourier transformed equation for the density fluctuation Sn(k2,w2) as—— 3kBTC kn(k = “° [kv0 — (v0 . k1)] . E. (2.40)This equation describes the generation of a plasma density fluctuation by the couplingof the pump wave and the scattered wave(SRS) or another plasma wave(TPD).2.5.3 Dispersion RelationEquations 2.38 and 2.40 describe the coupling of the electrostatic and electromagneticwaves involved in the SRS and TPD instabilities. To derive the dispersion relations, wesubstitute eq. 2.40 into eq. 2.38 and get the vector equation for E1 as——c2k)I + (c2 — 3v1)kiki] .= 4 2 2 2 [kv0—(v. i]4k2 W2—— 3Vthk2{[kv0- (v k1)] . E1}. (2.41)This expression without the 3v?hk2 term is the same as that given by [Langdon 73].It can be shown that:if (akk + bI) . E = cE’ (2.42)Chapter 2. Theory of Two Plasmon Decay and Stimulated Raman Scattering 221 kk kkthen E = c[(I— -b-) ++ k2)k2]E’, (2.43)where a, b, and c are constants. With this identity, we can get the condition for anontrivial solution E1 to exist, the so called dispersion relation, as:4( —— 3vhk) — k k1 x v2+(k — k)2 (k1 v0)2 (2 44)— k ( — — c2k) k?k (w? — — 3vh2k?)Using the dispersion relation we can determine some of the features of the TPD and SRSinstabilities. When k1 is not too small, the important term in the right hand side ofeq. 2.44 can be determined by examining the respective denominators. When the firstterm dominates, the Raman instability is recovered. By letting w = Wlr + Z7, L’)2— i7, assuming 7/ <<1 where -y is the growth rate, and takingw. = + 3vhkand= +c2k,we can find the growth rate for the SRS as70 x v. (2.45)The cross product on the right hand side of eq. 2.45 implies that there should be no scattered EM wave in the plane parallel to the polarization plane of the linearly polarizedincident E0 and that the scattered EM wave should be strongest in the plane perpendicular to E0. Hence, the EPW associated with the density fluctuation 6n(w2,k2) dueto SRS has the similar feature according to the wave-vector matching condition. Themaximum growth rate is kv0/4 if we assume k1,2 >> k0.When the second term of eq. 2.44 dominates, we find the dispersion relation for theTPD instability. Its growth rate with zero frequency mismatch is1 172 1270 — ; j 1 Vo ii — .1’’1 “‘2Chapter 2. Theory of Two Plasmon Decay and Stimulated Raman Scattering 23k-//k2 !Yk0Figure 2.3: Wave-vector matching condition for TPD and SRS.Here the dot product means that the best plane for viewing the electron plasma wavesgenerated by the TPD instability is the incident plane with p-polarization. If we considerthe wave-vector matching condition as shown in Fig. 2.3, then we can write the growthrate as:= k0vo1+ [kkk0/2]2}. (2.47)We see from this expression that the growth rate is a maximum, i.e.,’y0 = if— k0) = k, (2.48)and is zero when k = 0.When k1 is small with respect to k0, eq. 2.41 reduces to(w-- 3vhk)(w - = wkv0(v . E1). (2.49)This shows that E1 is always parallel to v0 regardless of the direction of k1, and thegrowth rate is (k0v)/4. The resulting wave is a mixed transverse-longitudinal mode.Chapter 2. Theory of Two Plasmon Decay and Stimulated Raman Scattering 24This is sometimes referred as the SRS-TPD hybrid instability and is studied with a newmodel by [Afeyan 93]. Equation 2.45 or 2.46 correctly describes the case of small k1 onlywhen k1 is parallel or perpendicular to k0, respectively.2.6 DampingIn the last section we derived the growth rates and the dispersion relation for TPD andSRS without considering damping for simplicity. Now we study the damping and thethresholds for the instabilities introduced by the damping.Because the electrostatic plasma waves are simply plasma density fluctuations andtheir associated electric fields, they do not intend to escape from a plasma [Kruer 88].Their energy is basically transferred to the particles via either linear or nonlinear dampingprocesses. Here we simply discuss the linear damping processes: collisional damping andlinear Landau damping, and leave the discussion of the nonlinear damping to a latersection when we study the saturation mechanisms for the instabilities.As a plasma consists of both electrons and ions, electron plasma waves are naturallydamped via electron-ion collisions. The coherent motion of oscillating electrons in thefield of the electron plasma wave is converted to random(or thermal) motion at the ratewhich electron-ion collisions occur. To balance the energy dissipated, the energy of thewave then damps at the rate 7E:7EE2 rimv87r = Z/ei 2 (2.50)where Ve = eE/(m) is the quiver velocity of electron in the field, E is the amplitudeand w is the frequency of the electrostatic field associated with the plasma waves, andV6j is the electron-ion collision frequency. If we assume that the electron velocities haveChapter 2. Theory of Two Plasmon Decay and Stimulated Raman Scattering 25a Maxwellian distribution, then11ei 3 lo_6l1Ae5)), (2.51)where A is the ratio of the maximum and minimum impact parameters and given by[Chen 88]A =A = 127rn)3-. (2.52)This factor represents the maximum impact parameter, in units of r0 =e2/(4Tr0mv),and r0 is the minimum impact parameter, averaged over a Maxwellian distribution. Forthe plasma conditions encounted in our experiment, A 10. Substituting ye into eq. 2.50,we obtain= 1Jj. (2.53)wFor an electron plasma wave, w= ,hence the damping rate of the amplitude of theplasma wave is given by = = i.e.,7c = e(31)= 1.5 x 1o_61nA)). (2.54)We can see that the collisional damping rate is larger for a plasma with higher densityand lower temperature.Besides the collisional damping, the electron plasma waves can also be damped bycollisionless damping, or so called Landau damping [Kruer 88], at a rate given byL)7L = ——f(), (2.55)where f is the distribution function for the velocities of electrons. From this expressionwe can see that if the velocity distribution function increases with velocity, that is, thereChapter 2. Theory of Two Plasmon Decay and Stimulated Raman Scattering 26w/kFigure 2.4: A mechanical analogue for Landau damping.are more fast electrons than slow ones, then )‘L is positive, and the wave gains energyfrom particles, and vice versa. For an individual electron, whether it gains or loses energydepends on whether its velocity (v) is larger or smaller than the phase velocity ( ) ofthe wave. If v > , it loses energy. If v < ,it gains energy from the wave. Thiscan be shown by a mechanical analogue given by Kruer[Kruer 88]. Consider a group ofboxes traveling at a velocity equal to u/k. Inside the boxes are uniformly-distributedparticles, some moving slightly slower than /k , some moving slightly faster. As shownin Fig. 2.4, those particles moving slower than /k are overtaken by the wall to their leftand gain energy as they are bounced off. Similarly, those particles moving faster thanc’/lc overtake the right wall and lose energy as they are reflected. For a period less thanthe transit time of a particle through the box, the net energy change simply depends onwhether more particles are initially moving faster or slower than w/k. If we again assumeChapter 2. Theory of Two Plasmon Decay and Stimulated Raman Scattering 27that the velocity distribution of the electron plasma is Maxwellian, then we have:1WpeLL)2_______7L = —/— exp(— 2 2) (2.56)8 1k Vh 2k VthIt is obvious from eq. 2.56 that the Landau damping of an electron plasma wave is a strongfunction of its phase velocity. The damping becomes sizeable whenever w/k 3v i.e.,when ADk > 0.3 whereAD = Vth/LI.)AD = 740 [Tev/n(cm)]112cm (2.57)is the electron Debye length. EPW’s with short wave lengths experience strong Landaudamping in a hot electron plasma.EMW’s are mainly damped by collisional damping (inverse bremsstrahlung) with adamping rate given by‘Yem = VeiWp2/(2L)em). (2.58)The threshold condition due to these damping rates thus is‘Yo 7L + Yc (TPD) (2.59)7 (7i + 7c)yem (SRS). (2.60)2.7 TPD and SRS in an inhomogeneous plasmaIn the previous section, we assumed that the plasma is homogeneous and infinite, andthat the pump beam is sinusoidal and infinite when we derived the growth rate for puretemporally growing instabilities. However in laser-plasma interaction experiments, witha few notable exceptions, the plasma is inhomogeneous and finite, and the pump beamalso has a profile of finite extent. In this section we will briefly study the effects due to theChapter 2. Theory of Two Plasmon Decay and Stimulated Raman Scattering 28inhomogeneity and the spatial limits on the instabilities. A detailed discussion of theseinstabilities in inhomogeneous plasmas can been found in [Lee 74, Liu 76b, Simon 83].Due to the finite extent of the plasma and the pump beam or the inhomogeneity ofthe plasma, the instabilities are spatially very localized as the instabilities can only occurin certain density regions. The waves may propagate out of the interaction region beforethey grow to large amplitudes if their group velocity, Vg is so large that l/Vg (1 is thesize of interaction region) is much smaller than the instability growth time. Then wedo not have a pure temporally growing instability (i.e. an absolute instability), but aconvective instability instead 2 For an absolute instability, the waves excited in a locallyunstable region grow indefinitely in the local region until some nonlinear processes setsin to saturate them. For a convective instability, the waves can propagate out of theunstable region before they grow to sufficiently large amplitude and the instability canbe saturated linearly. An absolute instability is possible in an inhomogeneous plasma ifthe group velocity of the unstable wave is so small that the time it needs to propagate outof the interaction region is much longer than the interaction time. As the group velocityis a function of plasma density, whether an instability is absolute or convective dependson the plasma conditions. It is also obvious that the threshold for an instability dependson whether it is absolute or convective. Therefore, determining the manner in which theinstabilities evolve is a very important aspect of the study of parametric instabilities inan inhomogeneous plasma.The study of parametric instabilities in a finite inhomogeneous plasma with a finitepump beam is very complicated because there are additional variables associated with2lnstabilities are defined as absolute or convective according to the manner in which they evolve frominitially small-amplitude perturbations of a given equilibrium [Bers 84]. An instability is defined to beconvective if the unstable wave can grow and then propagate away from its origin so that eventually ata fixed point in space the wave decays with time. If the unstable wave can encompass more and moreof space, so that eventually at every point in space the wave grows with time, then the instability isabsolute.Chapter 2. Theory of Two Plasmon Decay and Stimulated Haman Scattering 29the inhomogeneity, and the finiteness of the plasma and the pump beam. To simplifythe problem, assumptions are made, such as that the plasma temperature is high, orthat Landau damping is not significant, et cetera, when one tries to study parametricinstabilities in an inhomogeneous plasma. Hence each theory is valid only for a specificset of plasma conditions. Here I summarize the results from theories which are valid forthe plasma conditions found in our experiment or for very well known models.There are three common approaches to the study of TPD and SRS instabilities ininhomogeneous plasmas. The simplest model that includes two basic stablization mechanisms, namely damping due to daughter waves and plasma gradients, was introducedby Rosenbiuth [Rosenbiuth 72]. The model consists of the following pair of differentialcoupled equations for the slowly varying daughter wave amplitudes a(i = 1, 2):a a(+Vi—+v)ai =-yoa2exp[q(x)] (2.61)anda( + V2— + v2)a = 0a1 exp [—zq(x)] (2.62)where V are the group velocities and v are the damping rates of daughter waves, isthe growth rate derived in the previous section for homogeneous plasma, and q(x) is thephase mismatch arising from the inhomogeneity. The phase mismatching is defined as:(x)= f k!(x!)dxi, (2.63)where the spatial variation of the wave-numbers of the coupled modes is given by:kF(xf) = k0(x!) — ki(xi) — k2(xF). (2.64)It can be shown using WKB analysis that for a linear phase mismatch, i.e., k(x) = kix,only convectively unstable modes exist. The threshold for a CO2 pump600I14LJTkeV > 1, (2.65)Chapter 2. Theory of Two Plasmon Decay and Stimulated Raman Scattering 30and the spatial amplification factorexp[27/V12ki] (2.66)are obtained in the absence of damping. Here ‘14 is the pump intensity in units of1014W/cm2,L, the density scale length in units of sum, and TKeV plasma temperature inunits of KeV,However, Lee and Kaw [Lee 74], avoiding the WKB approximation, found an integraltransform solution for the linearized coupling equations in an inhomogeneous plasma witha linear density profile, and showed that TPD can be absolute even in an inhomogeneousplasma. To simplify the coupling equations, they kept only lowest order terms in k0/k,assuming the latter to be small. They obtained thresholds for TPD asv0> (c/4)(k — yL)1. (2.67)In the n/4 density region, where a w0/2 scattered EMW is near its cutoff density, theSRS instability is absolute and its threshold is relatively high,v0 > 0.52c(kL)213 (2.68)However, once the thresholds are exceeded, both instabilities grow at nearly the samegrowth rate,-y =k0v/4.Later Liu, Rosenbluth and White [Liu 76] used a different method (commonly referredas the LRW model) to study TPD in inhomogeneous plasmas in more detail but withoutassuming k0/k to be small. They converted the coupling equations to a Schrödingerequation in wave-number-space and solved for the ground eigenvalues of the equationby a local expansion method. They found that TPD is absolute under the followingcondition:\/(1cL)(v0> 1. (2.69)Chapter 2. Theory of Two Plasmon Decay and Stimulated Raman Scattering 31The threshold is‘(‘0)2kL> 1. (2.70)3 VhIn practical units the threshold can be expressed as:5.5Ii3ApLp/Te > L (2.71)Here ) is the wavelength of the pump beam in units of m. The growth rate for themost unstable modes is:= kovo— kL (2.72)Subsequently, Lasinski and Langdon [Lasinski 77] revised Liu’s result with the aid of anumerical calculation. The threshold is given by [Lasinski 77] as:(v0/vth)21cL> 24V’, (2.73)and the growth rate as:— k0v11 10.3v7‘1 WpV08For a given plasma condition(vth and L )the growth rate is maximized when=l2P(kL)_1/2 (2.75)VthA final growth rate for an arbitrary plasma wave component (not just the peak angle),including damping, can be written as:k0v k — k)____ ____=4 k12 — kL — Vvk exp (2k2 2)— v, (2.76)where v is the collision damping rate.Most recently Simon et al. refined the model by using the LRW model to rederive theSchödinger equation and solve for the ground state eigenvlaues analytically, semianalytically and numerically for various parameter regimes. They investigated both small andChapter 2. Theory of Two Plasmon Decay and Stimulated Raman Scattering 32large k/k0, short and long laser wavelengths, and hot and cold plasmas. Their resultsshowed that the threshold for absolute TPD instability varies little between these different regimes and is quite close to the threshold given by [La.sinski 77]. In practical units,the threshold conditions are given as:L)\,iIl4/TkeV > 61.25, (2.77)(k/k0)2= 0.l9(I14)/TkeV), (2.78)for L = 300, TkeV = 0.3, and ) = 10.6.Compared with absolute TPD, convective TPD has a much lower threshold. In someregimes, where the threshold for absolute TPD is too high, convective TPD can be thedominant mode of TPD. Powers and Berger used a kinetic model to study TPD in a hightemperature (T > 1 keV) weakly inhomogeneous plasmas, and found that convectiveTPD becomes dominant as Landau damping increases the threshold for absolute TPDto prohibitively high values of intensity. As our experimental conditions are not includedin this regime, this model will not be further examined.2.8 SaturationIn previous sections we discussed TPD and SRS instabilities in a homogeneous and aninhomogeneous plasma, the thresholds due to damping mechanisms and inhomogeneity,and the growth rates for linear theory. Once the pump intensity exceeds the thresholds,the instabilities start to grow. In convective instabilities, the daughter waves propagateout of the growth region before they are amplified to large amplitude, and are saturatedby linear processes, such as damping and their convection, which we have discussed. However, in absolute instabilities the daughter waves can grow to an amplitude sufficientlylarge that they can trigger other instabilities and nonlinear processes, such as electronChapter 2. Theory of Two Plasmon Decay and Stimulated Raman Scattering 33trapping, wave breaking, mode coupling, et cetera and eventually are saturated by theseprocesses. Since we are dealing with pump intensities well above the threshold values,pump depletion is not effective as a means of saturating the instabilities. In this sectionwe will discuss three main saturation mechanisms: electron trapping, profile modification, and mode coupling of absolute TPD, for which experimental evidence is presentediii this study and in previous work conducted at UBC.2.8.1 Electron TrappingElectron trapping is a nonlinear effect resulting from strong particle-wave interactions.In a previous section when we discussed linear Landau damping, we assumed that thewave amplitude is small and is damped only by those electrons with a velocity quiteclose to its phase velocity. However, in a large amplitude electron plasma wave, it ispossible for a significant number of electrons to be trapped in the wave potential troughand be efficiently accelerated by the wave, thus reducing the energy of the wave. Thiskind of damping is known as nonlinear Landau damping or electron trapping. If thewave amplitude is so large that the oscillation velocity of an electron in the field of thewave is equal to the phase velocity, i.e., eE/(m) = , then the wave breaks and thewave energy is suddenly damped as the slow electrons are accelerated by falling into thepotential troughs of the wave.Consider an electron plasma wave with an electrostatic potentialq(x, t) = çS sin (kx — ‘vet).Trapping of an electron with velocity v occurs when its energy in the wave frame issmaller than the wave potential, i.e., wheneq > m(v — v)2 (2.79)Chapter 2. Theory of Two Plasmon Decay arid Stimulated Raman Scattering 34where v, is the phase velocity of the wave. The eq. 2.79 clearly shows that small waves,i.e., e = 0, will trap only those electrons moving at speeds near v. To trap an initiallycold, main body electrons(v 0), would requireeq = rnv= m()2. (2.80)The electrostatic potential can be related to the EPW’s amplitude, Sn/n via Poisson’sequatione= -(-). (2.81)me nAt large amplitudes EPWs can trap an appreciable number of electrons and acceleratethem to sufficiently high energies to be emitted from the plasma. As a result, EPWs aredamped.2.8.2 Mode Coupling and Profile ModificationBoth mode coupling and profile modification are due to the ponderomotive effect andhave been identified in simulation [Langdon 79] and experiments [Baldis 83, Meyer 85]as the dominant saturation mechanisms for the TPD instabilities. The principles ofthe mode coupling and the profile modification can be described as follows. Once theEPW’s grow to a large amplitude, they become unstable to modulations driven by theirown ponderomotive force, JNL and consequently are quenched or saturated. The twoEPW’s produced by TPD can be written as E(x, t) = E exp [i(k . x—wit)], and theponderomotive force due to the superposition of the counter propagating EPW’s becomes:f = —4V{E + E +mc0E12exp {i[(ki — k2)x— (‘‘i —2)t]} + c.c.}. (2.82)Chapter 2. Theory of Two Plasmon Decay and Stimulated Raman Scattering 35The first two terms on the RHS generate a large scale ponderomotive force pushingelectrons out of the resonant region, steepening the density profile, and reducing thedensity scale length. From a previous section we know that the threshold due to theinhomogeneity is inversely-proportional to the scale length. Hence, as the steepeningcauses the scale length to decrease, the threshold increases; when the threshold is higherthan the pump intensity, the instability is quenched. Once the instability has beenquenched the EPW’s are damped and the profile is relaxed due to the Coulomb force,and consequently the instability can reoccur.The last two terms on the RHS in eq. 2.82 are oscillatory terms. For — w2 small,these terms are of low frequency with a wavenumber k1 — 2k1, since k1 = —k2.This force can effectively drive ion acoustic fluctuations with k0 = 2k1, which in turncouple to EPW’s with large k that are heavily damped by Landau damping. Resultantlarge amplitude ion-acoustic fluctuations have been seen in simulations [Langdon 79] andexperiments [Baldis 83, Meyer 85] and have been identified as the dominant saturationmechanism for the TPD instability.The ratio of the component of the ponderomotive force which drives the profile steepening to the component which drives the mode coupling is ()epw/L) (n/Sn)2. The relativetime scales over which the effects of these two components become important is the inverse of this ratio [Baldis 83], i.e.,= (L/epw) (Sn/n)2. (2.83)Here Aepw is the wavelength of the electron plasma wave; tL and t. represent long andshort time scales, respectively.Chapter 3EXPERIMENTAL APPARATUS AND SET-UP3.1 IntroductionThe experimental apparatus and the details of the experimental set up for the Thomsonscattering measurements are described in this chapter. The arrangement of the experimental apparatus is displayed in Figure 3.1. A 2 ns, 10 J, 10.6 m laser pulse generatedby a CO2 laser is focused onto the gas jet in the target chamber, and generates and interacts with an underdense plasma. During the interaction, TPD and SRS instabilities canoccur. Thomson scattering of ruby laser light is used to investigate these instabilities.The scattered ruby laser light is detected by a streak camera, and the streak images aresent to a computer to be stored for analysis. Among the above-mentioned experimentalapparatus, the CO2 laser system, the gas jet, the target chamber, and the streak cameraswill be described. In addition, the principles of Thomson scattering and the details ofthe experimental set ups will be discussed.3.2 CO2 Laser SystemIt has been shown in Chapter 2, that the growth rates of TPD and SRS instabilities areproportional to v0, the quiver velocity of electron in the pumping EM field, and thatvgwhere I is the intensity and A0 the wavelength of pump beam. It is clear that a pumpbeam with high intensity and long wavelength is desirable, in order to produce strong36Chapter 3. EXPERIMENTAL APPARATUS AND SET-UP 37PC Streak CameraC02 Laser Target Chamber(2ns, 1 OJ pulse)Ruby Laser(6ns, O.5J Pulse)Figure 3.1: Overview of the experimental apparatus.Chapter 3. EXPERIMENTAL APPARATUS AND SET-UP 38parametric effects in laser-plasma interactions. The CO2 laser is a well developed, longwavelength = 10.6 pm) laser capable of high intensities (over 105W/cm2)which iswell over the threshold intensities of parametric instabilities intended to be studied in thislaboratory. In addition, the long-wavelength, low-frequency pump beam has a low criticalplasma density, which is transparent to a visible beam. Hence the plasma features, suchas density and temperature, and parametric instabilities can be studied with visible light,which can be measured with a high speed and high sensitivity streak camera. Thereforea CO2 laser was chosen and had been built in this laboratory. In order to maximize thepower which can be delivered to the target without damaging the optical components,and also to simplify the hydrodynamic phenomena of the laser-plasma interaction, a shortsingle laser pulse of 2 ns was designed.During this study, two CO2 laser systems have been used. At the beginning, someexperiments were done with the CO2 laser system (hereafter referred to as “first CO2 lasersystem”) which had been developed over many years in the Hennings building. We thenmoved to the new Chemistry-Physics building and upgraded the CO2 laser system. Mostof the experiments have been done with the upgraded laser system. The differencesbetween the first CO2 laser system and the upgraded CO2 laser system are: 1) differenthybrid oscillators; 2) different preamplifiers; and 3) different optical set ups for the 3-stageamplifier. These systems will be described separately in next two sections.3.3 First CO2 laser systemThis system had been described in detail elsewhere[Popil 84, Bernard 85]. Here it will bebriefly reviewed. The physical layout of the system is shown in Figure 3.2. The hybridoscillator operates on a single longitudinal and a single transverse mode, and produces atemporally-smooth, linerly-polarized 100 ns pulse propagating in the form of a Gaussian1 CD CD IiC) IK103HYBRIDOSCILLATORSECTIONHIGHPRESSURESECTION3-STAGESF6CE112ccChapter 3. EXPERIMENTAL APPARATUS AND SET-UP 40beam. Then a 0.15 mJ, 2 ns pulse is switched out at the peak of the 100 us pulse by thePockels cell. This 0.15 mJ, 2 us pulse passes through the amplifier chain consisting of apreamplifier and two amplifiers and is amplified to about 10 J.3.4 The Upgraded CO2 Laser SystemAfter the move to the new building the CO2 laser was modified. The main reason wasthat we wanted to operate the CO2 laser without a potentially carcinogenic material,tri-n-propylamine (CH3C2)N. Tri-n-propylamine had to be added to the gas mixfor the K103 amplifier and the high pressure section of the hybrid oscillator to preventarcing of the pump discharge. The other reason arose due to the limited space in the newbuilding. We installed a new hybrid oscillator and replaced the K103 amplifier with amore compact home made amplifier, which could operate with a gas mix not containingtri-n-propylamine. We also wanted to improve the reliability of the short pulse generatorwhen we upgraded the laser system. For this reason we replaced the single Pockels cellshort pulse generator with a double Pockels cell short pulse generator [Laberge 90]. Thesingle Pockels cell short pulse generator was very sensitive to the room temperatureand therefore caused the pulse contrast ratio to change frequently. Thus the upgradedCO2 laser system is different from the original CO2 laser system in the following parts:1) the hybrid oscillator; 2) the short pulse generator; 3) the preamplifier; and 4) theoptical set up.The layout of the upgraded CO2 laser system is as shown in Figure 3.3. The hybridoscillator operates on a single longitudinal and a single transverse mode and producesa temporally smooth, horizontally polarized, 200 mJ, 100 ns pulse. Then a 0.25 mJ,2 ns pulse is gated out at the peak of the 100 us pulse by the short pulse generator.This 0.25 mJ, 2 ns pulse travels through the preamplifier four times and is amplifiedChapter 3. EXPERIMENTAL APPARATUS AND SET-UP 41a0/,a/I //Il /7_____-‘ I III C0CC,)—CIC. —Ip — ——— (ID* C)cD L C.IoCC’)II i:* EFigure 3.3: The layout of the new CO2 laser system.Chapter 3. EXPERIMENTAL APPARATUS AND SET- UP 42to 45 mJ. After passing through the 3-stage amplifier twice, the pulse is amplified to 4J. Finally the pulse is amplified up to 10 J after a double-pass through the Lumonicsamplifier. In the following sections, the hybrid oscillator, the short pulse generator, the 4-pass preamplifier, and the optical set up, the new components in the upgraded CO2 lasersystem, will be discussed in detail.3.4,1 The Hybrid oscillatorAs shown in Figure 3.3, the hybrid oscillator is a laser cavity made up of a high pressurepulse section and low pressure continuous wave (CW) section. The 150 cm long cavity isformed by a gold coated concave mirror, M1 (R = 4m) and an uncoated germanium fiat,E, acting as an etalon. The CW section in the oscillator has two functions: 1) It forcesthe oscillator to operate in a single longitudinal and single transverse mode in order toproduce a temporally smooth Gaussian output pulse; 2) It simplifies the alignment ofthe amplifier chain, since the 2 ns pulse laser beam follows the identical path.The uncoated germanium flat, E, also acts as a exit coupler of the oscillator. It is temperature controlled to force the oscillator to lase on the P(20) line at ,\ = 10.6 um at alltimes. Due to abundant vibrational states and rotational substates of the CO2 molecule,the oscillator can lase on many lines. But the P(20) line at 10.6 pm is one of three lineswhich have the strongest gain. In addition, the germanium fiats used in the short pulsegenerator are set up in the Brewster angle. The value of the Brewster angle depends onthe refraction index which is a function of wavelength. The flats are set up for the laserbeam at 10.6 m . The lasing wavelength, or the longitudinal mode, of the oscillator isdetermined by the refraction index of the etalon. Tuning the temperature of the etalonchanges its refractive index, causing the oscillator to oscillate on a single wavelengthand in a single longitudinal mode. The circular aperture, SF1 as shown in Figure 3.4,inside the cavity is used to produce high loses on all but the lowest-order transversect3C)04ZChapter 3. EXPERIMENTAL APPARATUS AND SET-UP 43to Spectrum AnalyzerM 2Short Pulse GeneratorHigh Pressure Section Low Pressure SectionFigure 3.4: The hybrid oscillator and the double Pockels cell short pulse generator.Chapter 3. EXPERIMENTAL APPARATUS AND SET-UP 44mode and thus to force the oscillator to operate on a single transverse mode. In orderto obtain linearly polarized laser radiation the intracavity windows are installed at theBrewster-angle.The CW section operating at low pressure (26 torr) on the P(20) line at ). = 10.6 tmproduces a cw beam of about 3 W power using gas mix of (CO2 : N2 : He) = (15 : 15: 70).The output of the high pressure (3 atms) pulsed section is about 200 mJ in a 100 ns fullwidth of half maximum (FWHM) pulse using a mix of (CO2 : N2 : He) = (8 : 8 : 84).With a gas mix of (CO2 : N2 : He) = (15 : 15 : 70) for the high pressure section,the hybrid oscillator produces larger energy, 300 mJ energy over 100 ns. However atthis energy the intracavity windows consisting of NaC1 fiats and Ge fiat, are subject tofrequent damage. Therefore a gas mix with low percentage of CO2 and N2 has to beused. Both 6% and 8% of CO2 gas mix were tried, and the oscillator functioned wellwithout damaging the windows.A typical output of the hybrid oscillator is a 100 ns FWHM, 200 mJ, single longitudinal, single transverse mode laser radiation of horizontal polarization. The electricalcircuit for the hybrid oscillator can be found in [Laberge 90].3.4.2 Short pulse generatorThe hybrid oscillator generates a 100 ns FWHM pulse. As mentioned before, a short pulsehas to be sent into the amplifier chain in order to maximize the power and to protectthe optical system from severe damage. A double Pockels cell short pulse generator[Laberge 90], as shown in Figure 3.4, was used to gate a 2 ns pulse from the peak of the100 ns pulse. The short pulse generator is made up of Pockels cells P1 and P2, and Gefiats, F1, F2, F3, and F4. Anti-reflection coated GaAs crystals are used in both Pockelscells. With no voltage applied to the Pockels cells, the horizontal polarization of the laserbeam is unchanged except for a slight effect due to residual birefringence in the GaAsChapter 3. EXPERIMENTAL APPARATUS AND SET-UP 45crystal. The beam passes through the flat F1 with very little reflection and is analysedin an Optical Engineering CO2 Spectrum Analyser, which is used to monitor the laserbeam spectrum to ensure that the oscillator output is at P(20) line at A = l0.6itm. Ifthe flat F1 is detuned slightly from the Brewster angle due to an index shift induced bya temperature change, then a small amount of the laser beam of long duration, 100 nscan be reflected by the Ge flat F1 and directed to the amplifier chain without the secondPockels cell, P2. Even though this leakage beam is very weak, about 2% of the incidentbeam power, because its duration is quite long compared to the main pulse of 2 ns, itdepletes a lot of the energy of tile amplifier chain prior to the main pulse of 2 ns. Worse,the backscattered long pulse of the same frequency as the incident pulse will be amplifiedto very high energy because of its long duration and will damage the optics when ittravels back to the oscillator and its size get smaller. Hence the long pulse should be cutto minimum, i.e. the 2 us pulse contrast ratio should be higher. In the first CO2 lasersystem a single Pockels cell short-pulse generator was used. The contrast ratio of the 2ns pulse was 200 : 1. With the double Pockels cell short-pulse generator, most of thehorizontal polarized component of the leaked beam from the first Pockels cell is reflectedby the Ge Brewster flats, F2 and F3; the vertical polarized component transmits throughall the three Ge flats, F2,F3 and F4. The leakage beam from the double Pockels cell shortpulse generator is about 0.1% of the incident beam and goes into amplifier chain.When a high voltage pulse is applied across the GaAs crystal, a birefringence isproduced which changes the polarization of the incident beam from linear to elliptical.If the voltage is equal to the half wave voltage, V112 the horizontally polarized beam ischanged to a vertical polarized beam. The half wave voltage is given as [Yariv 89]:(3.1)where,Chapter 3. EXPERIMENTAL APPARATUS AND SET-UP 46is the vacuum wavelength of the incident beam (10.6 am),n is the refraction index of GaAs crystal (3.3),r41 is the electro-optic coefficient of the crystal (1.6 x l02m/V),d is the crystal thickness between the electrodes,and 1 is the crystal lengthThe dimensions of the crystals used in the two Pockels cells are 6 x 6 x 39 mm3 and7 x 7 x 80 mm3. V112 for the former crystal is 14.2 kV and for latter one is 12.5 kV.The 2 ns high voltage pulse at c’-’ 14.5 kV used for gating the Pockels cell is producedby discharging a 20 cm length of RG8/U 50 cable charged to 29 kV. A high pressuren.y. triggered spark gap is used to switch the cable discharge. The 14.5 kV voltage pulsepasses first through the first Pockels cell with the 6 x 6 x 39 mm3 dimension crystal. Itis then reduced to 12.5 kV by a 50 attenuator, and passes through the second Pockelscell after it travels through a 50 1 cable, which is used to compensate the time for thelaser beam to travel from the first Pockel cell to the second Pockels cell. The 12.5 kVvoltage pulse is finally sent through a 501 cable to trigger the streak camera.The laser pulse that emerges from the first Pockels cell, P1 has a 2 ns FWHM and ispolarized in vertical direction. If the pulse has any horizontal component, it is eliminatedby the Ge flats, F2 and F3 before the beam passes though the second Pockels cell. Thebeam is changed back to horizontal polarization after it passes through the second Pockelscell and is directed to the amplifier chain. The part of the beam transmitted throughF4 is directed to the RF screen room to be used to monitor the output of the hybridoscillator.The resulting laser pulse emerging from the short pulse generator is of a 2 us FWHMas shown in Figure 3.5 and is horizontally polarized. The contrast ratio is 1000 : 1. Atypical output power and energy for the laser pulse are 125 kW and 0.25 mJ respectively.Chapter 3. EXPERIMENTAL APPARATUS AND SET-UP 47Figure 3.5: The shape of the CO2 laser pulse.3.5 The amplifier chainThe amplifier chain as shown in Figure 3.3 consists of a four-pass preamplifier and two-double passed amplifiers in succession along with an optical system and absorber cells.In the following sections these components are discussed.3.5.1 The PreamplifiersThe preamplifier is a home-made amplifier. The detailed construction and operatinginformation of this amplifier can be found in Lab Report No 1O1[Liese 84]. It is operatedin the small signal gain regime. It is originally designed to be used as five-pass amplifier.But with 5-passes, its output energy is over 100 mJ which is too high for the nextamplifier, the 3-stage amplifier. The designed input for the 3-stage amplifier is only42 mJ. If the input beam energy is over 42 mJ, the beam creates an air-spark in theChapter 3. EXPERIMENTAL APPARATUS AND SET-UP 48focal position of 2 m focal length lens which was part of a optical shutter used to stopbackscattering. Due to the air-spark, the beam loses energy and changes shape. Hencethe air-spark should be avoided. At the beginning, in order to avoid the airspark, wechanged the optical set up for the 3-stage amplifier: sending a collimated beam insteadof an expanding beam to 3-stage amplifier. Hence the optical shutter was removed andthe curvature mirror with f = 1.5 m in the rear end of the 3-stage amplifier is replaced bya flat mirror. With this change, the output of the laser system increases from 10 J to 15J. This is desired. But this also produces a serious problem: the back scattered light isvery strong and damages some windows. After we tried many ways and failed to stop orweaken the back scattered light to avoid damage to the optics, we were forced to reducethe output of the preamplifier from 100 mJ to 40 mJ by reducing the number of passesto 4. The preamplifier is operated with a gas mix of(C02 : N2 : He) = (15 : 15 : 70). Theaverage single pass gain factor is about 4.3.5.2 AmplifiersThe first amplifier is a home made three-stage amplifier operating near the saturatedgain regime at one atm pressure with a gas mix of (He : CO2 : N2) = (63 : 25 : 12) at atotal flow rate of 8.5 liters/m. The same gas mix subsequently flowed through the finalamplifier. Both amplifiers are taken from the first laser system. Hence, tri-n-propylaminemust be still added to the gas mix in order to eliminate arcing and reduce streamers.Part of the gas mix flows over liquid tri-n-propylamine(vapor pressure=6 Torr at 20°contained in a closed container). The active double passed length of the 3-stage is 360cm. The 2 ns pulse of 40 mJ emerging from the preamplifier as input is amplified to 4.0J.The last amplifier is a Lumonics model TEA 600A amplifier. It operates in thesaturated gain regime. Its active double passed length is 200cm. The output is a 10 JChapter 3. EXPERIMENTAL APPARATUS AND SET- UP 49beam with 3 inch diameter with a input of 4.0 J.3.5.3 The Optical Set Up for the Laser SystemThe optical system is designed in such a way that the maximum amount of energy isextracted from the amplifiers in a uniform beam and the optical damages are eliminatedfrom the system. For this reason, the preamplifier is four passed, 3-stage and TEA-600Aamplifiers are doubled passed, and the beam is kept as wide as possible through theamplifiers. In addition two optical shutters and two absorption cells are inserted in thepath to avoid the self lasing in the amplifier chain and prevent the optics from severedamage due to the backscattering. Figure 3.3 shows the optical set up and the beampath.The 2 ns pulsed laser beam emerging from the double Pockels cell short pulse generatorfirst passes through the first optical shutter consisting of lens L1 and L2, and is directedinto the preamplifier by the mirror, M7. Then the beam is reflected back out of thepreamplifier by the mirror, M8. Subsequently the beam is reflected by the convex mirror,M9 and enters the preamplifier through the side of M7 as it expands. Then the beamcollimated by M8 is sent out and directed to 3-stage amplifier by M10. After the beampasses though the second optical shutter consisting of lens L3 and L4 and the spatialfilter, SF2, it is directed into the 3-stage amplifier by M12 and reflected back out by M13.The beam passes through above M12 and is forwarded to the last amplifier. Here thebeam is further expanded by M17 as it enters the TEA-600A amplifier. The beam isfinally collimated by M18 and is sent to the target chamber.Up to 10% of the laser beam energy can be reflected from the plasma in the backwarddirection through the stimulated Brillouin scattering [Bernard 85]. This back scatteredlight can be amplified to very high energy when it travels back through the amplifier chainsince its wavelength is often shifted less than 1OA from the incident beam wavelength andChapter 3. EXPERIMENTAL APPARATUS AND SET-UP 50therefore falls within the 3.5GHz of the P(20) transition. The back scattered light cancause severe damages to the various optical components. The damages are eliminated by2 passive optical shutters and spatial filter, SF2 as follows.The back scattered beam is amplified to several Joules after it double passes throughthe TEA-600A and 3-stage amplifiers. When it is focussed by L4 onto the second spatialfilter, SF2, it creates a long air spark which starts from the point 30 -‘ 40cm away fromSF2 and is stopped by SF2. This big air spark exhausts a lot of the amplified backscatteredbeam and increases the size of the remaining beam. Then most of the remaining beamis blocked by SF2 due to its tiny pinhole, 1mm diameter. A very small amount ofthe beam transmitted through SF2 is amplified by the 4-pass preamplifier and producesan airspark at the sharp focus of L2 in the first optical shutter. Thus, even thoughthe backscattered beam reaches the second Pockels cell, but it is not strong enough todamage the GaAs crystal. The size of the pinhole should be as small (-‘- 1 mm) aspossible, otherwise the beam transmitting through the pinhole is still too strong. Afterit only double passes through the preamplifier, it is amplified to be strong enough todamage M9 when it is focussed by M8 onto M9.Because of the high gain and many optical components in the amplifier chain, it ispossible for a random noise signal inside the amplifier chain to cause amplifier self-lasing.The seif-lasing depletes the pump gain of the amplifier chain prior to the 2 ns pulse andcan lead to the damage of some optical components, therefore it must be eliminated. Twoabsorption cells, which are not shown in Figure 3.3, SF6 cell#1 located in between the3-stage amplifier and the preamplifier, and SF6 cell#2 located at the back of the 3-stageamplifier are used to stop self-lasing in the amplifier chain. Both cells contain 2-3 TorrSF6, 20 Torr ethanol, 100 Torr freon-502 and 640 Torr helium. The SF6 strongly absorbssmall noise signal in the vicinity of 10.6 1um but is bleached by the much stronger mainpulse. Freon-502 and ethanol are strong absorbers in the 9-10.3 region but are only weakChapter 3. EXPERIMENTAL APPARATUS AND SET-UP 51absorbers of 10.6 tm radiation. Helium is added to aid in the recovery of the bleachedSF6.In addition, all intracavity NaC1 windows inside the amplifier chain are mountedunder a wedge angle of approximately 5 degrees to prevent seif-lasing building up fromreflections off the surfaces. For the same reason, the mounts of the lenses are paintedblack and beam-size masks cover all mirrors. A particular caution has to be taken forthe second spatial filter, SF2 because of its tiny size, 1mm pinhole. It should be heldseveral degrees off the 90 degree position with respect to the beam and painted black toavoid reflection.A typical output sequence of the new laser system: The hybrid oscillator produces a100 ns pulse with 200 mJ energy. A 2ns pulse with 0.25 mJ energy and contrast ratio,1000:1 is gated out from the peak of the 100 ns pulse by the double Pockels cell shortpulse generator. The diameter of the beam is -.-4/4 cm. The beam is polarized in thehorizontal plane. After amplification by the 4-pass preamplifier, the beam is expandedto 3 cm and its energy increases to 45 mJ. After passing through the 3-stage amplifier,its energy grows to 4.0 J. Then the beam is amplified by the TEA 600A amplifier to 10J and it is further expanded to 7 cm in diameter.The polarization of the CO2 laser pulse can be switched to vertical polarization byinserting a half wavelength plate in the path after the short pulse generator.3.6 Operating procedureThe CO2 laser system is a quite complicated system. The laser pulse travels a long distance, about 70m, from the oscillator to the target chamber. The alignment of optics hasto be perfect. In addition, many spark gaps are used in the pulse oscillator, preamplifier,and amplifiers to generate a pulsed gain medium for the laser pulse. The trigger pulsesChapter 3. EXPERIMENTAL APPARATUS AND SET-UP 52for the spark gaps, in principle, determine when the spark gaps break down. But thecondition of electrodes of a spark gap can cause big time jitter in breaking down. Hence,to operate the CO2 laser consistently for maximum energy, it is essential to flush allspark gaps at the beginning of the operation day by largely opening the exhaust valvesand increasing the original pressure by 10 to 20 psi to blow out the dust which is insidethe spark gap cells. This dust is created on the electrodes due to the high voltage breakdown. By cleaning out the dust it is found that the reliability of the CO2 laser is muchimproved.3.7 The Gas Jet TargetThe laminar gas jet target used is similar to that in [Popil 84]. It has been designedin such a way that a underdense plasma with long scale length and low bulk plasmamotion is created to optimize the effect of parametric instabilities. The target is a pulsednitrogen gas jet flowing out of a convergent-divergent(Laval) nozzle. The flow is stabilizedto a laminar flow by ambient helium gas at a pressure of a few Torr. The 1.2 mm thick1cm wide laminar jet of variable density simulates plane solid targets. Considering therequired plasma density which is underdense to CO2 laser light and low reservoir pressure,nitrogen has been chosen as the target gas. Helium is used as background gas to stabilizethe flow to ensure a low density plasma compared to that formed from nitrogen in theevent of upstream breakdown. The main features of the Laval nozzle are shown inFigure 3.6. The details of the design of the nozzle can be found in reference [Popil 84].The critical design feature of the Laval nozzle is the radius of curvature in the divergingpart, which is related to the desired Mach number as shown by Shapiro[Shapiro 53]. TheLaval nozzle consists of a pair of stainless steel jaws set in a Lucite cone. The nozzlethroat is 70 ,um across and the mouth opens to a final width of 1.2 mm. This sits atop aChapter 3. EXPERIMENTAL APPARATUS AND SET-UP 531.2Gas Jet(a)1cm//[=.5inFigure 3.6: Laval nozzle.Chapter 3. EXPERIMENTAL APPARATUS AND SET-UP 54short brass pipe connected to a solenoid valve which is connected to the 0.5 liter volumehigh pressure nitrogen reservoir. When firing, the solenoid valve is opened electronicallyand the nitrogen gas flows through the Laval nozzle. A piezo detector located near thenozzle senses the pressure change. After the jet is stabilized(’-’-40 us), the output pulsefrom the piezo detector triggers a delay pulse generator. The pulse from this delay unitis sent to trigger the delay units controlling the CO2 laser system, ruby laser, and streakcamera systems.The plasma density is proportional to the density of the neutral nitrogen gas jet.Thus the plasma density can be controlled by varying the jet pressure. The ratio of thenitrogen pressure in the reservoir to the ambient helium pressure is determined by theLaval nozzle design as shown in the following. Assuming adiabatic and isentropic flowin a perfect gas, it can be derived that the ratio of the pressure at any point inside thenozzle, P to the pressure in the reservoir, Po is related to the local Mach number byL= (1 + 1 —7M2)_/(_1), (3.2)P0 2where y is the ratio of specific heats(7 = 1.4 for N2). The pressure and the Mach numberis also related to the cross section, a througha— m (‘—(7+1)/27 (3 3— M’ypoPo “P01Here m is the mass flow rate. For a Laval nozzle, the Mach number is fixed at a value ofone at the throat. Substituting M = 1 at the throat and the ratio of cross section at thethroat, at = 1.1 x 0.07mm2 to the cross section at the mouth, am = 1.1 x 1.2mm withthe assumption that the mass flow rate is constailt in the ilozzle, the Mach ilumber atthe mouth, Mm can be found with/ 2 1 ‘) / — 1 \ (‘y+1)/(’y—1)i+ M2 34at) M 7+1k 2 m .Chapter 3. EXPERIMENTAL APPARATUS AND SET-UP 55Then the design pressure ratio for the Laval nozzle can be determined with the eq. 3.2.Thus the pressure ratio must be kept the same when changing the nitrogen reservoirpressure to varying the plasma density in order to get a laminar jet with a particularLaval nozzle.At the beginning of this program the gas jet target was tested at different nitrogenreservoir pressures and corresponding helium pressure, and it was found that the effect ofthe TPD and SRS instabilities was the strongest at 24.8 PSI nitrogen reservoir pressureand vanishes at 18.6 PSI or lower pressure. As the interferometric results show, themaximum plasma density is lower than 0.25n when the nitrogen pressure is equal to orlower than 18.6PSI[Mclntosh 83, Bernard 85]. Therefore in this program, a gas jet targetfrom 24.8 PSI nitrogen reservoir pressure has been used.The target chamber used is the same as that set up by G. McIntosh [McIntosh 87].The Laval nozzle is located in the center of the chamber as shown in Figure 3.7. Thechamber is a 60 cm diameter, 38 cm high aluminum cylinder. Twelve ports are evenlydistributed around the circumference, at 18°interval. The four 10 cm diameter portspositioned every 90° and the sixteen 5cm diameter ports in between allow access for laserbeam, optical diagnostics, electric and mechanic feedthrough. A f/5 50 cm KC1 lensdesigned for minimum spherical aberration held in 20 cm extension through one of the10 cm ports focuses the 7.0 cm beam down to 100 um onto the gas jet attaining intensitiesup to 104/cm2. The target chamber and the nitrogen reservoir are automatically filledto the desired pressure with an automatic gas handling system.3.8 Streak CameraA streak camera is an instrument used to detect low intensity visible light at high timeresolution and high sensitivity. The principle of the operation of a streak camera canChapter 3. EXPERIMENTAL APPARATUS AND SET-UP 56C0250cm f/5 lens5cm port30cmGas Jet 10cm portIFigure 3.7: Target chamber.Chapter 3. EXPERIMENTAL APPARATUS AND SET-UP 57Incident ,lightFigure 3.8: The principle of operation of a streak camera.be described with the aid of Fig 3.8. The incident light is collected on the horizontalslit which is imaged onto the photocathode of the streak tube. At the photocathode theoptical slit image is converted into an electron image. Then the electrons are acceleratedby the accelerating electrodes, and pass through the deflection plates. A high sweepvoltage ramp is applied to the plates at an appropriate time to cause a sweep of theelectron image from top to bottom. The horizontal axis image information remainsunchanged. When the amplified electron image impinges upon the phosphor screen, it isreconverted into a light image, with an associated gain of iO due to the acceleration.The phosphor image is then recorded by a digital video camera. As a result, the imagerecorded is the streak image swept from top to bottom, representing a time axis, and theslit axis representing one coordinate axis.The streak cameras used were Hammamatsu C979 and C1370 Temporal Dispersersattached to a Cl000 video camera and a C1440 Frame Memory Image Analysis System.Phosphor screenPhotocathode Channel plateChapter 3. EXPERIMENTAL APPARATUS AND SET-UP 58The best time resolution of the C979 model is 10 ps while that of the C 1370 model is 3ps.Both have five MCP(multi channel plate) voltage(gain) settings. Gain 3 was generallyused. The sensitivity of the C1370 model after a new photocathode was installed, ismuch higher than that of the C979 model. The length of the entrance slit of both streaktubes is 2.4cm. But the real length corresponding to 256 pixel depends on the opticsused and is less than 2.4cm. The calibrated length of the slit corresponding to 256 pixelswas 2cm for C1370 streak tube and 1cm for C979 streak tube.The shutter behind the slit for the streak tube can be opened manually in randommode or automatically by a trigger signal in gate mode. As the scattered light is a veryshort pulse, about 500ps, the shutter was opened before firing the lasers and then closedmanually in this study.The sensitivity of the streak tube is not uniform from center to side. Hence onlythe center part, about half of total length is used and the sensitivity can be consideredconstant.3.9 Thomson ScatteringThe most commonly used diagnostic for the study of the parametric instabilities inlaser-plasma interaction has been the scattering of an EM wave from one of the longitudinal waves resulting from the instabilities. The EM wave can be the pump beamitself or an independent beam. Typical examples of pump beam scattering are the scattered light associated with SRS and SBS instabilities. Another example is the w/2 and(3/2)t0 emissions, resulting from the Thomson scattering of the pump beam from theelectron plasma waves generated by the TPD and SRS instabilities in n/4 density region. But because the pump frequency is generally very close to the plasma frequency,Chapter 3. EXPERIMENTAL APPARATUS AND SET-UP 59the pump intensity is not constant in the interaction region due to the parametric instabilities. Hence it is not possible to use the scattering of the pump beam to directlystudy the longitudinal waves. In addition, if the pump beam is of 10 1um wavelength,the wavelength of the scattered beam is in the range of [5 tm to 20 tm ], for which thehigh sensitivity detector, high quality filter, and high resolution streak camera are notavailable; therefore it is not possible to study the microscopic features in ps temporalscale and of ,um spatial scale.The scattering of an independent beam as a probe beam has been proven a morepowerful technique. The frequency of the probe beam can be chosen much higher thanthat of the pump beam. The plasma then can be considered transparent to the probebeam and therefore the intensity of the probe beam can considered constant in the proberegion. In this study ruby laser Thomson scattering was used as a diagnostic to investigatethe electron plasma waves generated by the stimulated Raman scattering and the twoplasma decay instabilities around the quarter critical plasma density region. Becauseall experimental results to be presented in this dissertation were obtained from thisdiagnostic, a brief discussion of the application of Thomson scattering as a diagnostictechnique is given. Reviews of this technique can be found in (Sheffield 75, Evans 69,DeSilva 70, Kunze 68].The use of Thomson scattering as a diagnostic method is based on a relationshipexisting between the scattering characteristics and the plasma parameters. Under theaction of an incident EM radiation, the plasma electrons begin to oscillate and radiatein all directions. It is this radiation that produces the scattering effects and includes theinformation about the plasma density, temperature, and the plasma waves within theplasma. To derive the relationship between the scattered beam and the plasma waves westart with a single charge. For a single charge with velocity v << c, the scattered powerChapter 3. EXPERIMENTAL APPARATUS AND SET-UP 60ObserverIncident Wave S0Scattering volume VFigure 3.9: The scattering coordinate system.in the solid angle d is given in cgs unit as:2dP = q x ( x , (3.5)47rc c retwhere q is the charge, is the unit vector in the direction from the oscillating charge tothe observer as shown in Figure 3.9, and r is the acceleration of the charge due to theincident EM wave, calculated at the retarded time at which the radiation left the charge.Consider the case of a plane monochromatic incident EM wave,E1(r,t) =E0cos(k1 r — wtF),Chapter 3. EXPERIMENTAL APPARATUS AND SET-UP 61andB1(r, ti) = - xE0cos(k r —where ti is retarded time, 1(k) is incident wave frequency(wave vector). As v << c wemay neglect the effect of the field B1(r, t!), and assume no other forces to act on thecharge. Then the equation of motion of the charge is:(dv’\m = qE10cos[k . r(t/) — wit!].The scattered power in the solid angle (do) is given as:dP= 432 x ( xE10cos[k . r(t!) — Wjt!j)Iet (3.6)It is clear from the mass dependence that we may neglect the scattering from the positiveions in comparison to that from the electrons. Then eq. 3.6 can be written asdP= rodQ[x ( x E10)]2, (3.7)where r0 =q2/(mec)= 2.82 x 10—13 cm is the classical electron radius.The pattern of EM wave scattering from a plasma is determined by the parameter:1= 4rDsin(q/2)’where k = — k0 is the scattering induced change in wave vector, ) is the wavelengthof the incident beam, is the angle between k and k3, and=(tT/(41re2n)h/2= 740[Tev/n(cm3)J”cm, (3.8)is electron plasma Debye length. Here Te is the plasma temperature measured in eV,and n(cm3) is the plasma density in units of cm3. For a << 1, that is the incidentChapter 3. EXPERIMENTAL APPARATUS AND SET-UP 62beam wavelength is much smaller than Debye length, the wave “sees” the charge on ascale length in which they appear to be free; the plasma collective effects are thus notimportant. The total scattering would be obtained from N (the total number of electronsin the scattering volume) randomly distributed electrons. This is called incoherent scattering. The important feature of this incoherent scattering is that the scattered spectrumreflects the shape of the electron velocity distribution at the frequency shift 6w = k v.For c >> 1, on the other hand, the incident wave interacts with the charges which arenot free from other charges, and the plasma effects are important. This is called coherentscattering. The plasma wave(fluctuation) generated by instabilities in the plasma in thecoherent scattering regime play a predominant role. Since in our experiments, >> 1 issatisfied, next only the coherent scattering will be discussed.If we define the Fourier transform of the scattered field by+00ES(R, w)= f dt/E(R, t)e_stI,and takeR •rtft——+——c cwhere R is defined in Figure 3.9, then the scattering power from an electron plasma wavewith amplitude 6ne(r, t) into a solid angle d1 isE2 2d1.P(R,w)= [ ‘° Jdr x ( x E0)J2f+oo. s•r RJ dt6ne(r, t)exp[—zw3(t— — + —)]cos(k. r — wit) . (3.9)—00 C CWe substitute the electron density fluctuation in terms of the Fourier space and timetransform:6ne(r, t)= f ()3 f 6fle(k, w)exp(—i(k. r — wt).Chapter 3. EXPERIMENTAL APPARATUS AND SET-UP 63into eq.3.9. Then in the first term of the exponent in eq. 3.9 we haveW3—z[k.r—wt-Fwt —s.r+—R+k.r—wtJC C= i[(w— ( — — (k — (k8 — kg)) r — k3R], (3.10)where k is scattered EM wave vector. In the second term we have(-‘ — + w))t — (k — (k + k)) r — kR.After integrating over r and t, we get= x ( xHere P =c0Era is incident beam power, a0 is the radius and L is the length of thescattering volume in the direction of the incident beam. This scattered power is mainlydue to electron plasma wave with frequency,W L’s + L’,and wave-vector,k = k5 + k1.When the incident beam is linearly polarized and the experiment is set up in such a waythat E10J, that is the polarization of the incident beam is in the direction perpendicularto the scattering plane sox ( x E) = 1,then= (3.11)This clearly shows that the scattered power is proportional to the square of the plasmawave amplitude. However in our experiments, it is the scattered beam intensity, notChapter 3. EXPERIMENTAL APPARATUS AND SET-UP 64power, that was detected. As the scattered beam intensity, I = we can easilyget:=22(3.12)where 10 = is the incident ruby beam intensity. Therefore with eq. 3.12 we shouldderive the amplitude of the electron plasma waves from a measurement of the scatteredbeam intensity after calibrating the streak camera response.3.10 Experimental Set Up for Thomson Scattering MeasurementsTwo kinds of, spatially resolved and wave-number resolved, Thomson scattering experiments were performed. One spatially and temporally resolved Thomson scattering experiment was set up with one geometry and used to determine the location, the size, andtime evolution of the electron plasma waves driven by TPD and SRS instabilities. Inorder to determine the intensity distribution of the plasma waves in wave vector plane,another wave-vector and temporally resolved Thomson scattering experiment was preformed with nine different geometries. In both cases, the frequency matching conditionwas chosen such that only electron plasma waves generated by TPD and SRS instabilities around the n/4 were detected. Two 670 nm-interference filters with 11 nm FWHMpassband were used to achieve the frequency selection. The anti-Stoke scattered rubylight from w0/2 plasma waves would have wavelength of 672.3 urn which is within thepassband of the filters. The most significant noise to the signal are stray ruby light andscattered ruby light by ion acoustic waves. Both are much stronger than the signal buthave wavelength near 694.3 urn which is well outside the passband of the filter. Themeasured cutting ratio of ruby light by the two filter combination is more than iO. Theratio was measured in the following way. Without firing the CO2 laser, the noise wasrecorded when the ruby was fired. The plasma was simulated by a piece of mylar putChapter 3. EXPERIMENTAL APPARATUS AND SET-UP 65on the top of the nozzle. Then the interference filters were replaced by neutral densityfilters. The same level of noise was recorded when equivalent number 7 neutral densityfilter was used.The simultaneous wave-vector matching condition for Thomson scattering and TPDor SRS instability is shown in Figure 3.10. Here /3 is the angle between the incidentruby wave-vector, kr and the CO2 laser wave-vector, k0, kr+,r_ are wave-vectors of thescattered ruby beam by the plasma waves with k > 0 and the scattered ruby beamby the plasma waves with k < 0. The direction of x-axis is the same as k0, Thesewave-vectors are related by the matching conditions:kr+ kr+k+ikr_ kr+k_2k0 = k+, + k_,where i = 1, 2. The radius of the dashed arc is equal to the wavenumber of the scatteredruby beam. Hence the detected scattered ruby beam is due to the /2 plasma waveswhose wave-vector lie on the arc. It is indicated in the figure that for a given angle ofincidence against k0, /3, the plasma waves with k > 0 can be diagnosed by putting thecollecting mirror on the right side of the incident ruby beam and the plasma waves withk <0 can be diagnosed by the mirror positiolled on the left side.In order to find the intensity distribution of EPWs in wave-vector space, the wavevector resolved Thomson scattering experiments were conducted at nine geometries. Inprinciple, Thomson scattering experiments can be set up with 22 possible geometries inthe target chamber in our experimental system, as there are 11 ports on one side in thechamber, but because of the position of the incident CO2 laser beam, and the positionof the optics guiding the scattered beam out the target chamber, it is hard to set upThomson scattering experiments in 13 of the 22 geometries. The range of EPWs, whichChapter 3. EXPERIMENTAL APPARATUS AND SET-UP 66Figure 3.10: Simultaneous wave-vector matching conditions for TPD and Thomson scattering. kr is the wave-vector for the incident ruby beam; kr_,r+ represents the scattered ruby beams due to EPWs with k < 0 or /e > 0; k0 is the wave-vector of theCO2 beam; k_,+ correspond to EPW’s. They are related by wave-vector matching conditions: kr+ = kr + k+i, kr_ = kr + k_2, and k0 = k+ + k_i. Here ?i = 1 or 2.+krk1k±2Chapter 3. EXPERIMENTAL APPARATUS AND SET-UP 67ttCFigure 3.11: The location of possible detected EPWs in wave-vector space. Each curverepresents one position specified by/3. Here R stands for right, and L for left.Chapter 3. EXPERIMENTAL APPARATUS AND SET-UP 68can be detected, in wave-vector space is shown in Figure 3.11. Each curve correspondsto one geometry specified by 3, the sides of the incident ruby beam, R or L. Here Rstands for right, and L stands for left. All these curves are calculated by assuming thatthe interference filters transmit only one wavelength radiation which is the probe beamscattered by the w0 EPW. But in fact the filters have hum pass band and also transmitprobe beam scattered by EPWs whose frequency is close to w0. Thus this will introducesome error, which will be estimated at the end of this section, to the wave-vector of theelectron plasma waves.The experimental set up for spatially resolved Thomson scattering measurementsis shown in Figure 3.12. A 6 ns, 694.3 nm pulse produced by a Q-switched, cavitydumped ruby laser system[Houtman 85] was used as a probe beam. The probe beam isastigmatically focused by a f/100 glass lens to a 5 mm horizontal focus which covers theCO2 laser focal volume in the gas jet and to a vertical focus 25cm beyond the jet, endingin a beam dump consisting of razor blade stack. A large f/2.5 mirror with f=25 cmpositioned in the vertical focal plane collimates the scattered probe radiation collectedover the scattering angle 1° a 18° which is determined by the 2.5cm x 8cm maskon the mirror. The mask has a uniform vertical width over the collecting area and asuitable horizontal size such that the collected scattered beam passes through the exitwindow without any loss. A black paper mask is also positioned in the entrance windowof the ruby laser beam to block the stray ruby light outside the target chamber to reachthe collecting mirror. The collimated scattered radiation is guided out of the targetchamber towards the slit of streak camera. Inserting an appropriate imaging lens orlens combination in the collimated scattered radiation path produces either a magnifiedimage of the CO2 laser focal volume or a demagnified image of the collimating mirror onthe slit of the streak camera. The former set up is used for spatially resolved Thomsonscattering measurements, while the latter is used for wave-number resolved ThomsonChapter 3. EXPERIMENTAL APPARATUS AND SET-UP 69Plane Mirror 2 Ruby,,!l f/100// I IScattered // ‘I,Light - 1 56°IfI, I/j// I/Plane Imaged Jet f/7C021’conto Slit //Col1imati/12 11!18° IMirror f/2.5 Beam IDump’ L’—Imaging LensI I1 , II Ij+/-’---- Interference FiltersIII’Ij‘— Streak Camera SlitFigure 3.12: Set up for spatially resolved Thomson scattering.Chapter 3. EXPERIMENTAL APPARATUS AND SET- UP 70scattering measurements. In both cases, two interference filters are included to select thefrequency of the scattered light. A cylindrical lens with f = 5 cm is also positioned inthe front of the slit in the wave-number resolved Thomson scattering set up to reducethe image size vertically without affecting the horizontal(along the slit) size to increasethe intensity of the scattered radiation into the slit.To monitor the intensity of the ruby laser beam, about 5% of the main beam was sentto the streak camera slit through a single-mode optical fiber to produce a fiducial on thestreak record. The fiducial was synchronized with the main ruby beam incident upon thegas jet with the streak camera. The fiducial was also utilized to set the suitable delaytime between the ruby laser and the CO2 laser system. It was shown from experimentsthat the scattering process starts about 0.5 ns earlier than the time when the visibleplasma light is detected. The output from the streak camera is imaged onto a digitalvideo camera and the resulting picture is displayed on a monitor as well as digitized in a256 x 256 array. The picture on the monitor could be photographed for quick referenceand the digitized array is transferred to a PC to be stored in disk files for later analysis.Because the scattered radiation is very weak,I/I0 iO, and pump and probe lasersare involved in the experiment, accurate alignment is required. To align the incidentCO2 laser focal position with the ruby pulse horizontal focal line on the gas jet, first apinhole in an aluminum foil, held on a copper washer located at the jet, was made by firingCO2 laser system without the last amplifier. To avoid strong backscattering which mightdamage the optics in the CO2 laser system when making the pinhole, the aluminum foilshould be positioned about 15° off the position perpendicular to the CO2 laser beam. Wethen center the HeNe laser beam, which simulates the ruby beam, onto the pinhole anduse the diffracted light from the pinhole to align the optics to the streak camera with thestreak camera in the focal mode. The copper washer was originally manually removedfrom the gas nozzle after opening the target chamber. However, it was found that openingChapter 3. EXPERIMENTAL APPARATUS AND SET-UP 71Figure 3.13: Projection effects on spatially resolved Thomson scattering.the chamber to remove the washer changes the alignment. Without a perfect alignment,the observed intensity of the scattered light is very weak, which is one of reasons thatin a previous experiment[Mclntosh 87], weak scattered light was observed. To keep thealignment the target chamber should not be opened. The washer may instead be ejectedsimply by firing the gas jet.In the spatially resolved Thomson scattering experiments, since we are observinga two dimensional region from an angle, there are projection effects which cause theobserved length to differ from the real interaction length. According to Figure 3.13, thereal interaction length, 1, relates to the observed length as:L1= . —w/tan,6sin (3.13)In the streak records to be presented, the interaction length is the observed interactionlength, L. Then according to eq. 3.13, the real interaction length can be calculated.k0—’ --onkrChapter 3. EXPERIMENTAL APPARATUS AND SET-UP 72Most experiments were performed in the horizontal plane, with P-polarized pumpradiation, i.e., the incident plane of pump radiation determined by k0 and E0 is alsoa horizontal plane. For some experiments a A/2 plate was inserted into the CO2 laserbeam path to switch the polarization of the pump radiation from horizontal to verticalpolarization. The Thomson scattering measurement then was conducted with S-polarizedpump radiation.The interaction length of the plasma was derived from the spatially resolved Thomsonscattering measurement in the following way. The magnification factor of the set up forthe measurement was first determined. The length of the slit of the streak camera whichcorresponds 256 pixels was calibrated, and the total number of pixels in the horizontalaxis recording the scattered light was coverted to the length of the image of the interactionregion, and subsequently the length of the interaction region was found. This observedinteraction length is substituted into eq. 3.13 and the real interaction length then can befound.To determine the value of the wave-number of the EPWs from the wave-numberresolved Thomson scattering records, the scattering angle, a, as shown in Figure 3.10must be determined first. As the range of the scattering angle determined by the maskon the collimating mirror is [1° — 18°], and the size of the image of the mask can bemeasured, then the total number of pixels corresponding to the size of the image can befound after the length of the slit corresponding to 256 pixels was found. Hence if theposition of a = 1° is found, then the absolute value of a represented by each pixel canbe found. This position was found by putting a piece of mylar on the top of the nozzleto simulate the gas jet and firing the ruby, the position was clearly shown in the monitorand was recorded. Then according to Figure 3.10 the wave-number of EPWs are foundChapter 3. EXPERIMENTAL APPARATUS AND SET-UP 73as:k1 = k+k—2krkscosa (3.14)0 = (3.15)where icr and k3 is calculated with eq. 2.3 by substituting e = n/4, W9 Wr + L)epw,and LL = . 0 is angle between k0 and k+1 and q5 between k and k+1.The wave-vector resolution of the experimental set up for the wave-number resolvedThomson scattering experiment is determined by the resolution of the streak camera,and is 0.06k The experimental error to the wave-number of EPWs is mainly causedby the error in calculating the magnification factor of the optics, /k = 0.16k, theerror in determining the boundary positions of the scattered signal in the streak image,Lk& = 0.11k, and the error caused by the band pass filter, /..kf = 0.20k The totalerror is /ic + /k + zk = 0.27k3.11 SynchronizationIn Thomson scattering experiments, appropriate synchronization of CO2 laser, ruby laser,target chamber and streak camera are required since the durations of laser pulses and thesweeping time of streak camera are of nanosecond scale. Figure 3.14 demonstrates thefiring sequence of the devices. When the CO2 laser, ruby laser, streak camera, and targetchamber are ready, the 6 ns ruby laser is charged. The charging takes about one and onehalf minutes. Then, the Lumonics amplifier of CO2 laser is charged. Half a minute later,the gas nozzle is fired. A 40 volt pulse is generated in the target chamber control unitabout 10 ms after it is fired. The 40 volt pulse triggers the ruby laser control unit whichfires the flash lamps for the ruby oscillator and amplifier, and sends out a 40 volt pulseto trigger the 16 channel delay unit for CO2 laser. Then the CO2 laser pulse oscillator,preamplifier, amplifiers, and the 2 ns, 14.5 kV pulse generator are fired at certain delayChapter 3. EXPERIMENTAL APPARATUS AND SET-UP 74Gas Jet I RubyControl Delay Unit Oscillator• 6ns U2nsSpark GapSpark GapEifie1_____16 ChannelPockels I Delay UnitCelliPockels I Streak III. ICell 2 CameraFigure 3.14: Synchronization of lasers and associated instruments.Chapter 3. EXPERIMENTAL APPARATUS AND SET- UP 75times. One of two 2 ns, 14.5 kV pulses from the generator travels along a RG8/U 50 lcable through the Pockels cells to the streak camera, another pulse through another 50Q cable goes to trigger the spark gap for ruby laser, to switch out a 6ns ruby laser pulse.The synchronization is achieved by adjusting the length of cables to change the delaytimes between lasers and streak camera. The time jitter of this arrangement is about0. 3ns.Chapter 4RESULTS4.1 IntroductionThe experimental results of my research are presented in this chapter. The techniqueused to maximize the scattered signal by optimizing the plasma condition is describedfirst. Then the results of spatially and wave-number resolved Thomson scattering experiments using the original CO2 laser system and the streak camera with C979 TemporalDispersers are presented. The results of Thomson scattering measurements made withthe upgraded CO2 laser system and the streak camera with better time resolution arepresented and compared with the previous results. The intensity distribution of w0EPW’s is then inferred from wave-number resolved Thomson scattering measurements.Where suitable, the results of these experimental studies are compared to those fromprevious works conducted iii this laboratory and other laboratories. Some commentsabout the results are also given in this chapter. A more detailed discussion is left to thenext chapter.4.2 Maximization of the scattered light intensityIt was previously shown that the scattered light intensity from Thomson scattering experiments, conducted to measure the w0 plasma waves generated by TPD and SRS atthe n/4 density region, was quite weak and very hard to detect[Mclntosh 87]. Therefore,it was logical to maximize the scattered signal by optimizing the plasma conditions and76Chapter 4. RESULTS 77[iressure (P/P0) 0.4 0.6 0.8 1 1.6 2.0Signal no weak strong weak weaker noTable 4.1: Optimization of the jet pressure.improving the experimental procedure at the outset of this work.As shown in eq. 3.12 in the previous chapter, the scattered intensity in Thomsonscattering depends on: 1) the square of the plasma wave amplitude, and 2) the size ofthe scattering volume or interaction volume. From chapter 2 we know that the largerthe plasma density scale length, the larger the growth rate and the lower the threshold.The scale length is proportional to the size of the interaction volume. Hence the plasmawave amplitude also depends on the size of the interaction volume. Therefore, the sizeof the interaction volume is a crucial factor determining the intensity of the scatteredbeam. In our experiment, the size of the interaction volume, is controlled by the pressureof the gas jet, or the pressure of the reservoir at a given pump energy. Therefore byconducting Thomson scattering experiments at the different gas jet pressures enables usto find the conditions providing maximum scattered light intensities. The results areshown in Tab. 4.1. As pointed out in the last chapter the ratio of the- reservoir pressureto the background pressure is determined by the design of the nozzle, and must be keptconstant in order to get a laminar jet. Here P is the pressure of the reservoir for thegas jet. P0 = 31.5 psi was the pressure originally used [McIntosh 87, Bernard 85] withthe background pressure Pb = 5Torr inside the target chamber. It was found that thescattered light intensity is a maximum at P/PO = 0.8. For P/PO < 0.6 or P/P0 > 2,there is no scattered light detected, which implies that there are no plasma waves asthe scattered beam intensity is proportional to the intensity of the plasma waves. Theseresults can be explained in the following way.Chapter 4. RESULTS 78As we know from Chapter 2, w0 plasma waves are generated by TPD or SRS instability at plasma densities around the n/4. When P/PO < 0.6, there are no w0 plasmawaves. This means the plasma density is significantly lower than n/4. This agrees withthe result obtained from interferometry measurements [McIntosh 83]: when P/PO < 0.6the plasma density is always below n/4. This is also consistent with the findings of(3/2) harmonic measurements[Zhu 87]: when P/PO < 0.6 no 3/2 c’ signal was detected. It therefore can be inferred that for P/PG < 0.6 no n/4 density region exists,and TPD cannot occur.The growth rate, and therefore the amplitude, of electron plasma waves increasewith the plasma density scale length. For P/PO = 0.8, the scattered light signal is thestrongest. This means that the scale length is the longest, which is expected for the TPDinstability when the peak plasma density is 0.25n. Hence, we can conclude that thepeak plasma density is-‘ 0.25n at P/PO = 0.8.When the jet pressure increases, the scale length will decrease as the peak plasmadensity gets higher and the size of the plasma keeps constant. This causes the growthrate to decrease and the threshold to increase. When P/PO > 2, even though a n/4density region exists, -c plasma waves are not generated because the thresholds for theTPD and SRS instabilities are higher than the pump intensity. This also agrees with theresult of 3/2 w0 harmonic measurements [Zhu 87]: no (3/2)w0 harmonic emission wasdetected when F/PQ > 2.In addition to optimizing the plasma conditions to maximize the effect of the TPD andSRS instabilities, improvement of the experimental procedure, especially the procedurefor the alignment of the optics for the Thomson scattering experiments, contributed alot to improving the scattered light signal. As pointed out in the last chapter, the copperwasher, used to align the ruby laser and pump CO2 laser, was blown away from the gasnozzle without opening the target chamber. In previous experiments, the washer wasChapter 4. RESULTS 79removed from the gas nozzle manually after opening the target chamber. Opening thetarget chamber was shown to cause a change of the alignment, and therefore the observedscattered light was quite weak.A photograph of a typical streak record from wave-number resolved Thomson scattering experiments conducted in the present study, and a photograph from a previous study[McIntosh 87] with a similar experimental set-up, are shown in Figure 4.1(a) and (b)respectively. Comparing these two photographs indicates clearly that as a result of theoptimized plasma condition and with improved experimental procedure, much strongerplasma waves were observed.Scattered light was observed for all laser energies over 2 J. This means that theinstabilities started at the time during the pump beam pulse when the intensity on thetarget surpassed 1 x 1013 W/(em)2,which is well above the threshold for absolute TPDand SRS instabilities. While in previous experiments [McIntosh 87], the TPD and SRSinstabilities were observed when the pump energy was over 4 J.4.3 Results from the experimental set-up with the original CO2 laser systemWith the original CO2 laser system and a Hamamatsu model C979 streak camera with a10 ps best time resolution, we conducted the spatially resolved and wave-number resolvedThomson scattering measurements in one scattering geometry (3 = 55°). With thespatially resolved Thomson scattering measurements we try to determine the interactionlength and to estimate the average plasma density scale length. With the wave-numberresolved Thomson scattering measurements we try to find the wave-number spectrum,determine the plasma density range, and study the dependence of w0/2 plasmons on thepolarization of the pump radiation. The polarization dependence is used to determinewhether TPD or SRS instability dominates the generation of /2 plasmons. The resultsChapter 4. RESULTS 80(a)(b)Figure 4.1: Photographs of typical streak records of wave-number resolved Thomsonscattering. (a) from the present study and (b) reproduced from McIntosh’s thesis.Chapter 4. RESULTS 81are presented in the following subsections.4.3.1 Spatially resolved Thomson scattering measurementsFor a finite inhomogeneous plasma, the interaction length, which relates to the plasmadensity scale length, is an important parameter in determining whether the instabilitiesare absolute or convective. The interaction length can be determined from the spatiallyresolved Thomson scattering measurements. In principle, the plasma density scale lengthcan be estimated if the plasma density profile is known. But we do not know the density profile, hence we only can estimate the average density scale length by assumingthe plasma has a linear density profile for comparison with previous results. The bestway to measure the scale length is to measure the plasma density using interferometryBernard 85, McIntosh 87]. However, because in the present experiments the peak plasmadensity is quite low (O.25n), the fringe shift is small. Thus, it is difficult to measure theplasma density accurately using interferometry.Two consecutive streak records from spatially resolved Thomson scattering measurements are shown in Figure 4.2. The spatial resolution of the experimental set-up forthe spatially resolved Thomson scattering measurements is determined by the spatialresolution of the streak camera and the magnification factor of the image of the plasma.The resolution calculated was 16 m , which corresponds to a streak camera pixel. Thisagrees with the observation that 16 um features, or pixel features, can be seen in theenlarged streak image shown in Figure 4.3. X axis is in the direction of the pump beam,k0. The origin of X is arbitrary. The total distance along the X axis is the observedlength of the interaction region, represented by L in eq. 3.13. The real length of theinteraction region can be calculated with eq. 3.13 by taking w = 2r0, where r0 is the radius of focal spot of pump CO2 laser beam which is equal to 50 m. It was determinedusing a diffraction technique that the pump beam has a Gaussian intensity profile in theChapter 4. RESULTS 82(c)4o0EZ20000400H200Figure 4.2: Streak records of spatially resolved Thomson scattering measurements. Contour plots of intensity of plasma waves for (a) S-polarized and (b) P-polarized pumpradiation.C00400 800 1200X(m)Chapier 4. RESULTSX(um)830 590 1q00 15p0k0Figure 4.3: Streak record of spatially resolved Thomson scattering measurement. Thepump radiation is P-polarized with a pulse energy of 5.3J.0250500:.7501 0001250V.1 500Chapter 4. RESULTS 84focal spot with a 1/e radius of 49 m [Bernard 85]. Substituting the values for w, L,and 3 into eq. 3.13, we get an interaction length of 1.9 mm. This long interaction lengthresulted from optimizing the plasma conditions to maximize scattered light intensity. Inprevious experiments[Meyer 85, McIntosh 87] the interaction length was about 400 imFigure 4.2(a) was obtained with S-polarized pump radiation of 3.1J. Here “S-polarizedpump radiation” means that E0 is perpendicular to the plane of the observation governedby the incident probe beam wave-vector, kR and the scattered probe beam wave-vector,k. Figure 4.2(b) and Figure 4.3 were obtained with P-polarized pump radiation of4 J, and 5.3 J respectively. When the energy of the pump radiation was over 5 J,multi-bursts as shown in Figure 4.3 were observed in all shots with P-polarized pumpradiation, but not observed with S-polarized pump radiation. The burst feature was alsoobserved in previous experiments[Baldis 83, Meyer 85]. It is also shown in Figure 4.3 thatplasma waves appeared in two regions of the plasma. This two island feature was alsoobserved previously in interferometric measurements [Bernard 85]. The plasma wavesfirst appeared in the region around X = 1500 tm and were quenched after 100 ps.Another 100 ps later strong plasma waves were produced in multi-bursts in a secondregion around X = 500 m.According to eq.2.46 and eq. 2.45, the EPW’s due to TPD are maximum in the planedetermined by k0 and E0, which is detected with the P-polarized pump radiation, whilethe EPW’s due to SRS are maximum in the plane perpendicular to the plane of (k0,E0),which is detected with the S-polarized pump radiation. Figure 4.2 indicates that bothabsolute SRS and TPD EPW’s occupy the same spatial region. But the SRS plasmonsand the TPD plasmons maximize at different locations.Once the interaction length is known, the average plasma density scale length canbe estimated by assuming that the plasma has a triangular density profile as shown inFigure 4.4 to simplify the calculation. However, the real density profile is more likelyChapter 4. RESULTS 85Figure 4.4: Plasma density profile.parabolic and could be described as:= n(1- )‘ (4.1)where L is the density scale length for a parabolic density profile. The location of thehighest plasma density region was found by the following experiment. By covering thepart of the collecting mirror near the probe beam, we blocked the scattered light dueto EPW’s with wave-number around k0, that is EPW’s in n/4 region and found a gap.Thus we identified the position of the n/4 density region on the streak image from thespatially resolved Thomson scattering measurement. A photo from this measurementis shown in Figure 4.5. We see that the gap is located at the center of the interactionregion. Since 0.20 — 0.25n is the plasma density range bounded by the experimentalset-up, m can be set to 0.25n and n0 to 0.20n. Using the definition of the density scalennoXm xChapter 4. RESULTS 86Figure 4.5: A photo of a streak record of spatially resolved Thomson scattering measurement. The plasmons from n/4 are blocked.length:L = (-)-‘dx nfloXm— no’we estimated the average density scale length to be about 2.4 mm. In previous experiments [McIntosh 87] the scale length was inferred to be 400 m from spatially resolvedThomson scattering measurements and was 300 m as obtained from interferometrymeasurements which were done for a gas jet with a nitrogen reservoir pressure 31 psi.4.3.2 Wave-number spectra of EPW’sFrom the dispersion relations for EPW’s and EM waves, we know that the EPW’s due toSRS will have very narrow wave-number spectrum because one of the daughter waves istime.k0Chapter 4. RESULTS 87a EM wave which is a strongly dispersive wave. On the other hand EPW’s due to TPDhave a much wider wave-number spectrum since the two daughter waves are plasma waveswhich are not strongly dispersive. This can be tested by conducting the wave-numberresolved Thomson scattering experiments both with P-polarized and S-polarized pumpradiation. Figure 4.6 shows the wave-number distribution of EPW intensity observed intwo consecutive experiments, (a) for a S-polarized pump of 7.3 J and (b) for a P-polarizedpump of 6.3 J. Figure 4.7 shows the temporally integrated wave-number spectra ofFigure 4.6. For S-polarized pump radiation the fluctuations concentrate near k =as expected for the absolute SRS instability. The sharp drop in scattered intensity atk k0 corresponds to the edge of the collection optics. At longer wave numbers thescattered intensity drops off much more slowly than expected for SRS plasma waveswithin the detected frequency range. Tis is consistent with the wave-number spectrumof SRS plasmons observed in a previous experiment [Villeneuve 85]. It was suggestedthat the SRS plasmons with larger wave numbers may correspond to the coupling of SRSplasmons to ion acoustic waves. In present experiment, besides the coupling, this maybe caused by the insertiion of the /2 plate in the CO2 laser system. If conversion fromP to S polarization is not 100%, curve S as well displays a small amount of radiationscattered off TPD plasmons, which are of larger wave numbers.For P-polarized pump radiation we observed a continuous EPW wave-number spectrum from k = k0 to ‘- 5.7k0(this indicates that the plasma density range is indeed in0.20 — 0.25n), the detectable wave-number range determined by the collecting optics,confirming the speculation indicated above: EPW’s generated by TPD have a wavenumber spectrum which covers a very wide range. There is no gap between the SRS andTPD regions, rather we observed a distinct sharp maximum near k = 2k0. For both 5-and P-polarized pump radiation the EPW intensity first grows at small wave numbersand then spreads rapidly to longer ones. The quenching of the instability after 100’— 300Chapter 4. RESULTS 88200CJDS1000 1 2 3 4 5 6(b)200..‘:. -‘SS-.••5S.S10000 5 6kdk0Figure 4.6: Contour plots of EPW’s as a function of time and wave-number. (a)S-polarized and (b) P-polarized pump radiation.(C)JJ 0Chapter 4. RESULTS 896-54ct(ID1 I’S—>I—s0123456ke/koFigure 4.7: Time-integrated wave-number spectra of EPW’s. Curves S and P correspondto Figure 4.6 (a) and (b), respectively.Chapter 4. RESULTS 90ps has previously been shown to be the result of profile steepening at n/4[Meyer 85]and the mode coupling of EPW’s to IAWs[Baldis 83]. Previous results seem somewhatcontradictory as some experiments produced very wide wave-number spectra for EPW’sMeyer 85], while others measured very narrow spectra [McIntosh 87].4.4 Results from experimental set-up with the upgraded CO2 laser systemAll the following results were obtained with the faster streak camera (Hamamatsu modelC1370 capable of a 3 ps best time resolution) and the upgraded CO2 laser system.4.4.1 Spatially Resolved Thomson ScatteringThe spatially and wave-number resolved Thomson scattering measurements were repeated at 9 = 55° with the faster streak camera and the upgraded CO2 laser system.The time resolution of the model C979 streak camera did not give us the ps scale temporalinformation about EPW’s that was required. As well, it was necessary to conduct a groupof experiments with the same streak camera and laser system to simplify the analysis ofthe intensity distribution of EPW’s in wave-vector space. The spatial resolution of theexperimental set-up for the spatially resolved Thomson scattering measurement shownin Figure 3.12 with the faster streak camera is 18 um . Figures 4.8 and 4.9 show thestreak record of spatially resolved EPW’s for P-polarized pump radiation of 7.5 J and 6.7J, respectively. Again, a long interaction length ( mm) was observed. This agreeswith the results obtained from the experiments conducted with the original CO2 lasersystem. Comparing the results of spatially resolved Thomson scattering measurementsusing the original and the upgraded laser system, an important difference can be seen:the results from the first laser system show that the plasma appears in two regions of theplasma as shown in Figure 4.3, but with the upgraded system, the plasma is confinedX(um)Figure 4.8: Streak image of spatially resolved EPW’s intensity as a function of time forpump radiation of 7.5 J.IChapter 4. RESULTS 910 590 1q00(.1200atCta••— 1Sf)I 5(.)(.).4: •‘ ••••‘k0Chapier 4. RESULTS 92X(um)I 400k0Figure 4.9: Streak image of spatially resolved EPW’s for pump radiation of 6.1 J.Chapter 4. RESULTS 93around the 1.2mm N2 jet. The two island feature of the plasma in the initial experiments agrees with the previous interferometry results[Bernard 85], but does not agreewith observations of a confined plasma in an oxygen jet [Offenberger 80, Burnett 78]. Itis now clear that previous arguments that the two-island feature resulted from the use ofnitrogen gas jet were not correct since in the experiments with the upgraded CO2 lasersystem, a nitrogen jet was also used. An improved explanation of the two-island featurein the plasma is thus required.After studying the differences between the original and the upgraded CO2 laser system, it is believed that the two-island feature was the result of laser prepulse. In theoriginal CO2 laser system, a single Pockels cell switch short pulse generator was used,while a double Pockels switch short pulse generator has been implemented in the upgraded CO2 laser system. Hence the contrast ratio of the laser pulse from the upgradedsystem is much higher than that of the pulse from the original CO2 laser system. If along pulse leaks though the short pulse generator , since it is very weak, it will oniy bestrong enough to breakdown the gas jet and produce a plasma at the focal point. Thisprepulse plasma at the focal point appears earlier and is smaller than that produced bythe main pulse which is much stronger and generates a plasma before it reaches the focalpoint.The streak images shown in Figures 4.8 and 4.9 also indicate that the plasma waveamplitude ripples spatially. This is clearly shown in Figure 4.10, the time-integrated plotsof the above streak images. As the amplitude ripple is different from burst to burst,we time-integrate only the signal of one burst. The scale of the the amplitude rippleestimated from the Figure 4.10 is about 200 jtm , which is much longer than the longestdetectable EPW’s wavelength, -‘ 20 ,um. These ripples are possibly caused by JAW’sdriven by the SBS instability. JAW’s have a 200 tm spatial modulation[Bernard 85].The temporal features of the new results are very different from the previous ones,Chapter 4. RESULTS 94(a)I I I I I I=(b)I I I I I I0 250 500 750 1000 1250X(um)Figure 4.10: Time-integrated spatial distributions of EPWs’ intensity. (a) correspondsto Figure 4.8 and (b) to Figure 4.9.Chapter 4. RESULTS 95as many more bursts, up to 8, were observed. In the previous results, at most 4 burstswere observed. This results from the use of the higher time resolution streak camera. Inprevious experiments [Baldis 83, Meyer 85] only two bursts were observed. The temporalresolution determined by the setting of the speed of the streak camera is 5 ps. Thespatially-integrated temporal evolution of the EPW’s intensity is shown in Figure 4.11.We can see that the interval between the bursts is not a constant in a single shot. It alsovarys from shot to shot.4.4.2 Wave-vector Resolved Thomson Scattering MeasurementsThe wave-number resolved Thomson scattering measurements were conducted with ninegeometries to determine the intensity distribution of EPW’s in wave-vector space. Thephotos of the typical streak images of EPW’s from each geometry are shown in Figure 4.12and Figure 4.13. The boxes in the right-bottom corner of Figure 4.12 identify theposition of each photo taken and the arrows indicate the direction of pump radiation andthe time axis. From these photos we can clearly see the difference between the plasmawaves with k > 0 (indicated by R as diagnosed by putting the collecting mirror inthe RIGHT side of the incident probe ruby beam) and the plasma waves with k, < 0:plasma waves with kT < 0 show multi-bursts in all positions except 19L position whilethe plasma waves with k > 0 show monoburst in all positions except one position. Themore detailed and accurate comparison can be made from the the enlarged calibratedimages of the typical streak images of EPW’s from each geometry shown in Figures 4.14and 4.18. The zero of the time axis is again arbitrary: the time of the start of thesignal is taken to be the origin. The wave-number of the EPW’s is labeled at the top ofthe image. For EPW’s with k > 0 the wave-axis starts from the right, while for EPW’swith k < 0 it starts from the left side. The corresponding x-y components of the wavenumber are marked at the bottom of the image. The shades of gray of the images are notChapter 4. RESULTS 96==(a)I I I I==(b)I I I I0 250 500Time(ps)Figure 4.11: Spatially-integrated temporal evolution of EPW’s intensity. (a) correspondsto Figure 4.8 and (b) to Figure 4.9.Chapter 4. RESULTS 97—I ( U — 24• i;,’SSIIJFT•SC*I_E —II OFFSET •T G*IU —64• LEFTHU( NOD€ plcwI —z555RFigure 4.12: Photographs of typical streak images of EPW’s from different geometries.The boxes in the right-bottom corner indicate the geometry of each photo taken. Thearrows show the time axis and the direction of k0.37R 37LChapter 4. RESULTS 98127R 127LFigure 4.13: More photos of streak images of EPW’s.1DHr\crU)U)0ci)ci)rcc0OTime(pS)r:,,.CTime(pS)SIke/ko 0.77 1.770 I. I • I2.48 3.19-1.24 -2.01Chapter 4. RESULTS 100ke/ko 4.77 3.77 2.77 1.77 0.770(37R)100ID200ky/kokx/koI • I • I • I3.76 2.81 1.87 0.94 -0.172.93 2.51 2.04 1.50 0.752.77 3.77 4.77I . I(37L)100200‘-‘ 300-400-kylko 0.67kx/ko 0.371.70-0.493.85-2.81Figure 4.15: second set of streak images of wave-number spectra of EPW’s.kelko 0.770—(1O9J200ky/ko -0.56kxlko -0.53-0.06-1.770.41-2.74RESULTS4.77 3.77 2.77I — I— I1.771010.77Chapter 4.ke/ko0(55R)r:j100•ky/kokxfkopI I I I2.68 1.90 1.15 0.433.95 3.26 2.52 1.721.77-0.400.662.77 3.77I — I4.77300-400-500-600-0’I I0.92 1.48-3.66 -4.53Figure 4.16: Third set of streak images of wave-number spectra of EPW’s.Cc;c?NN::.;i:;vrNc-’—J4:;::::——4..C\CN....c1:.:•...-I-I-I-I-I.CCCCCC’’CCCCCCCCNoc-c1)20000c)Cl)CIf)VSV-4--,Cl)C4-,VCl)4-,VCtI—ICCCC-Time(ps)Time(ps)‘IChapter 4. RESULTS 103ke/ko 0.770—(145L)1002003005001.77 2.770.99-1.474.773.77400ky/ko -0.15kx/ko -0.76I I • I1.94 2.90 3.86-1.97 -2.41 -2.80Figure 4.18: Fifth set of streak images of wave-number spectra of EPW’s.Chapter 4. RESULTS 104calibrated with respect to each other, i.e., the same shade of gray does not represent thesame intensities in different images. Therefore the differences between the intensities ofEPW’s are not clearly evident in these images. All of these streak images were obtainedwith P-polarized pump radiation.From these images we can see that: 1)the wave-number spectra and the temporalbehaviors of EPW’s observed from different geometries are different; 2)the EPW’s withk <0 are different from the EPW’s with k > 0. The EPW’s with k <0 exhibit morebursts and last longer than the EPW’s with kr > 0, as was expected according to thelinear theory: the plasma waves with k > 0 generated by TPD instability have largerwave-number and suffer larger Landau damping and hence appear in shorter periodscompared with the EPW’s with k < 0. But we also observe that most EPW’s withk <0 have longer wave-numbers in the later bursts than the EPW’s with k > 0, whichis not what is expected from the linear theory. This must arise from a new mechanismwhich cannot be inferred from the linear theory, and this will be discussed in detail innext chapter.The time evolution of EPW’s is more clearly displayed in the wave-number integratedplots shown in Figures 4.19- 4.23. From these plots we can see that the forward EPW’slast about 200 ps, while the backward EPW’s last about 600 Ps. In previous experiments[Baldis 83, Meyer 85] in which only forward EPW’s were investigated, they were foundto last about 200ps.4.4.3 The distribution of the intensity of EPW’s in wave-vector spaceIn this subsection we try to find the intensity distribution from the wave-number resolvedThomson scattering measurements conducted with those nine geometries. Because of thedifference in the duration of EPW’s, the signals are time integrated over the intervals.First we integrate the signal over the first 100 ps since during 100 ps at least one burstChapter 4. RESULTS 1050 100Time(ps)0 100Figure 4.19: Temporal evolution of EPW’s observed at 19R and 19L.Chapter 4. RESULTS 106(37R)=II I Io ioo 200(37L)CI I I I I0 100 200 300 400Time(ps)Figure 4.20: Temporal evolution of EPW’s observed at 37R and 37L.Chapter 4. RESULTS 1070 1000 100 200 300 400 500 600Time(ps)Figure 4.21: Temporal evolution of EPW’s observed at 5511. and lO9L.Chapter 4. RESULTS 108- (127R)=1:OE(127L)II I I I I I I0 100 200 300 400 500 600Time(ps)Figure 4.22: Temporal evolution of EPW’s observed at 127R and 127L.Chapter 4. RESULTS 109- (145L)I I I I I Io 100 200 300 400 500Time(ps)Figure 4.23: Temporal evolution of EPW’s observed at 145L.Chapter 4. RESULTS 110was observed for all shots. We chose the starting point of the signal as the origin ofthe 100 ps interval. Because the wave-number and temporal behavior of EPW’s are notexactly the same from shot to shot, we average the time-integrated intensity for all shotsfrom the same geometry. The wave-number spectra of EPW’s integrated in the firstloops averaged over all shots from each geometry are shown in Figures 4.24- 4.28.Because the data from the measurements conducted with those nine geometries arenot enough to use any available computer graphic software to draw a 3-D intensitydistribution plot, we draw a special plot as shown in Figure 4.29 to display the measuredintensity distribution. In this plot the numbers forming the line correspond to intensityranges, i.e. 0 represents intensities from 0 to 99, 1 corresponds to intensity of 100 to 199and so on. Since the intensity should be the same where the curves overlap, we calibratecurves 55R, 37R, l9R, 19L, 37L and 145L with respect to curve 127R. Curve 109L andl27L are not calibrated as they do not interact with the other calibrated curves. Usingthe calibrated intensities, we draw the distribution of the intensity of the EPW’s as shownin Figure 4.30.With the spectra integrated over the first 100 ps, we face some difficulties becausethe EPW’s have saturated already in the last part of the 100 ps. Thus we cannotuse linear theory to explain these nonlinear processes, and there are no sophisticatednonlinear theories which explain all these nonlinear processes. To avoid these problems,we integrate the signal over a short interval, which we chose to be 20 ps. The time beforethe plasmons are saturated is chosen as the end of 20 ps interval. The distribution of theintensity of EPW’s integrated over 20 ps is shown in Figure 4.31. The intensity of thisplot is not calibrated as Figure 4.30.Chapter 4. RESULTS 111ke/ko077 1.77 2.77 3.77 4.77I • I • I •(19R)600500400‘‘300ro2001000 I I I0.07 1.36 2.41 3.45 4.49 kylko0.77 1.14 1.36 1.52 1.62 kx/kokelko077 1.77 2.77 3.77 4.77I • I I I(19L)100 -0 I • I I I0.53 1.77 2.74 3.66 4.53 kylko0.56 0.06 -0.41 -0.92 -1.48 kx/koFigure 4.24: First set of the averaged wave-number spectra of EPW’s.Chapter 4. RESULTS 112ke/ko0 77 1.77 2.77 3.77 4.77I • • I I(37R)600 -500 -400 -200 -100 -0-0.17 0.94 1.87 2.81 3.76 ky/ko0.75 1.50 2.04 2.51 2.93 kx/kokeJko0.77 1.77 2.77 3.77 4.77I • I I I(37L) fi300 -200 -100-.67 1.70 2.48 3.19 3.’85 ky/ko0.37 -0.49 —L24 -2.01 -2.81 kx/koFigure 4.25: Second set of the averaged wave-number spectra of EPW’s.Chapter 4. RESULTS 113kelko077 1.77 2.77 3.77 4.77I • I • I • I.(55R)500400c3002001000 I I I I-0.40 0.43 1.15 1.90 2.68 kylko0.66 1.72 2.52 3.26 3.95 kx/koke/ko0 77 1.77 2.77 3.77 4.77I I I • I(109L)1000 I I I I-0.56 -0.06 0.41 0.92 1.48 ky/ko-0.53 -1.77 -2.74 -3.66 -4.53 kx/koFigure 4.26: Third set of the averaged wave-number spectra of EPW’s.Chapter 4. RESULTS 114kelko0.77 1.77 2.77 3.77 4.77I • • I(1 27R)700600‘500!400200100 -0 I • I • I0.75 1.50 2.04 2.51 2.93 ky/ko-0.17 0.94 1.87 2.81 3.76 kx/kokelko0 77 1.77 2.77 3.77 4.77(127L) A200-100 -0 I I I I-0.37 0.49 1.24 2.01 2.81 ky/ko-0.67 -1.70 -2.48 -3.19 -3.85 kx/koFigure 4.27: Fourth set of the averaged wave-number spectra of EPW’s.Chapter 4. RESULTS 115Figure 4.28: Fifth set of the averaged wave-number spectra of EPW’s.Chapter 4. RESULTS 1164t\o)J/A)J I C-—c- /-‘:1--./NFigure 4.29: The number-plot of the intensity distribution.CD C.3C (‘7 0 C C CD CD (‘7 C 07 CD CD C CD C C (aCD C C CD CD 0 CD CD 0 CD J)5. 4. 3. 2 1•IIIII-5-4-3-2-112345kfk0IIIIIChapter 4. RESULTS 1194.4.4 Measurements with S-polarized pump radiationThe preceding subsections reported the results from measurements conducted using theupgraded CO2 laser system with P-polarized pump radiation. We also carried out somemeasurement with S-polarized pump radiation. For comparison Figures 4.32-4.34 displayone streak image with P-polarized pump radiation and one with S-polarized pump radiation from each geometry, where measurements were carried out with both P- andS-polarized pump radiation. The scattered light was not detected for k < 0 at=370 andl27° using S-polarized pump radiation, even though with these geometriesthe strongest scattered light was detected using P-polarized pump radiation. Consequently we did not conduct measurements using S-polarized pump radiation at 19L,109L, and 145L, since the scattered light signal is very weak even for the P-polarizedpump radiation.rl).HC,)EChapter 4. RESULTS 120:‘ “;(a)(I •‘ II ‘ II II IVsit,I I :“. I I I50-0100Figure 4.32: Streak images of EPW’sradiation at 19R.2.77 3.77ke/kofor (a) S-polarized and (b) P-polarization pumpChapter 4. RESULTS 121100(a)_SCI) \ I50-0 I I I I I150(b)00.77 1.77 2.77 1 4,77JSe/ISoFigure 4.33: Streak images of EPW’s for (a) S-polarized and (b) P-polarization pumpradiation at 55R.Chapter 4. RESULTS 122100c’DH0HI III I Ig: .::s,(_:II It SI, ‘sIII’I.., (a)I I‘I‘Ske/177Figure 4.34: Streak images of EPW’s with S-polarized and Ppolarization pump radiationat 127R.Chapter 5DISCUSSION5.1 IntroductionIn this chapter the results presented in the last chapter will be discussed and comparedwith theoretical predictions and other relevant experimental results. In the discussion Iwill attempt to develop a self-consistent picture of the intensity distribution of the TPDplasmons in the wave-number plane. To allow a comparison of our results with theoryand with other experimental results, the plasma conditions must be specified. Hence inthe next section the experimental parameters, are summarized to avoid repetition in thesections which follow. To determine what kinds of instabilities we observed, the thresholds for TPD and SRS instabilities will be estimated in Section 5.3. As both TPD andSRS can generate w0/2 plasmons, in Section 5.4 we will discuss the difference betweenTPD and SRS, and distinguish TPD plasmons from SRS plasmons. In Section 5.5 wewill discuss the intensity distribution of the TPD plasmons. As the intensity distributionevolves with time, the discussion will begin with an analysis of the temporal behaviourof TPD plasmons. The features of the intensity distributions in the unsaturated andsaturated phases of the TPD instability are then discussed. The relation between theintensity distribution of the saturated TPD plasmons and the angular distribution of(3/2) harmonic emission will be examined. As the intensity distribution of the saturated TPD plasmons is determined by the dominant saturation mechanism, the potentialsaturation mechanisms will be discussed in the last section.123Chapter 5. DISCUSSION 1245.2 Experimental conditionsIn this section, the experimental parameters, which will be referred to quite oftpn laterin this chapter, are listed to avoid repetition. These experimental parameters have beenderived from the present and some related previous experiments conducted in this laboratory.I%IBT = 300 eV [Popil 84, Bernard 85],L = 2400 tm (inferred from the spatially resolved Thomson scatteringmeasurements in the present study),v0/c = 0.05 (for a 5 J incident pump radiation pulse),Z = 4 [Popil 84, Bernard 85].Here PBT is the electron plasma temperature in units of eV, L is the average plasma density scale length, v0/c is the ratio of the quiver speed, v0 of electrons in the pump fieldto the speed of light in a vacuum, c, and Z is the average charge state of the ions.5.3 The nature of TPD and SRS instabilitiesIn this section we attempt to determine whether the instabilities observed were absoluteor convective by comparing the thresholds necessary for absolute instabilities derivedfrom theories and the range of pump intensities used in our experiments.As pointed out in Chapter 2, a certain threshold intensity must be exceeded before theinstabilities start to grow. Eqs. 2.68 and 2.71 show that the thresholds are determined bythe nature of the instabilities and the plasma conditions, especially the plasma temperature and the density scale length. We will estimate the thresholds with the parametersgiven in the last section and the theories discussed in Chapter 2.We will use several theories to estimate the thresholds. The simplest theory is the oneChapter 5. DISCUSSION 125for an infinite, homogeneous plasma in which the thresholds are determined mainly bythe dissipation mechanisms of collisional damping and Landau damping. The collisionaldamping rate is given by eq. 2.54 and Landau damping rate by eq. 2.56. Substitutingour experimental parameters into eq. 2.54 , the collisional damping rate, yc is:= 1.5 x i0, (5.1)and substituting into eq. 2.56, the Landau damping rate, yj. is:7L/Wo = 4 x io. (5.2)Here we calculate the Landau damping rate by taking ke = 5.66 k0, which is the maximumwave-number that the detectable EPWs can have. However, the Landau damping ratedecreases exponentially as the wave-number decreases; for example,7L/Wo = 4.5 x 10’°, (5.3)for ke = 2k0. Hence in our experiments the collisional damping dominates the Landaudamping for the observed EPWs with wave-numbers in the range of 0.77— 5.66 k0. Thuswe will neglect the Landau damping and consider only the collisional damping. Thethresholds for TPD and SRS at n/4 are thus70 > (5.4)or in practical units, the infinite homogeneous threshold is onlyI > 107W/cm2. (5.5)However, TPD and SRS instabilities have never been observed at such low pump intensities due to the inhomogeneity of experimental plasmas. The inhomogeneity increasesthe threshold to a much higher level by limiting the size of region over which the wavevector and frequency matching conditions, or resonant conditions, are satisfied. TheChapter 5. DISCUSSION 126interferometrical results [McIntosh 83, Bernard 85] indicate that indeed the laser produced plasma encountered in this study has density gradients, which are measured by adensity scale length. Next we estimate the thresholds for a finite, inhomogeneous plasma.A more recent, complete model for the TPD instability in an inhomogeneous plasmais the one given by Simon et al. [Simon 83]. The result obtained with this model for theplasma conditions encountered in our experiments is similar to that derived using theLRW model [Liu 76, Lasinski 77]. Hence in the following we will use eq. 2.73 to estimatethe threshold. In an inhomogeneous plasma the TPD instability is absolute when thecondition given by eq. 2.69 is satisfied. We rewrite eq. 2.69 as:(5.6)Substituting our experimental parameters into eq. 5.6, we get:lc > 0.01k (5.7)This condition is satisfied in most of the experimental region as k is in the range [0 —5.6k0]. The threshold obtained from eq. 2.73 for the absolute TPD instability is:I> 3 x 1O”W/cm2. (5.8)In comparison, the threshold for absolute SRS calculated with eq. 2.68,I> 102W/cm2, (5.9)is higher by a factor 3.We mentioned in the last chapter that we observed TPD and SRS instabilities withpump energies over 2 J, which represents a peak intensity about 1.3 x 10’3W/cm2.Thepump pulse is assumed to have a triangular shape with a rise time of 1.6 ns and a fall timeof 2.4 ns to simplify the calculations. This intensity does not determine an experimentalChapter 5. DISCUSSION 127threshold because our experiments are conducted with a single beam: the pump createsand interacts with the plasma. It was found [Bernard 85] that about 1 J energy wasrequired to form the plasma. When the pump energy is low, it takes a longer time forthe pump pulse to create the plasma. The plasma density drops due to the expansion ofthe plasma. The density of a plasma created by a low energy pump pulse likely dropstoo low for the SRS and TPD instabilities to be observed. This may be why there are noplasmons observed when the pump energy is below 2 J, even though the pump intensityis well above the thresholds.The intensity of the pump used is of order of 1O’3W/cm2which is well above calculated thresholds for the TPD instability in an inhomogeneous plasma. In addition, thecondition for absolute TPD, eq. 2.69 is satisfied. Therefore, we can conclude that theabsolute TPD and SRS instabilities were observed in this study.Summary: The absolute TPD condition is satisfied; the pump intensity is muchhigher than the thresholds for absolute TPD and absolute SRS instabilities, and thus theabsolute TPD and SRS instabilities were observed.5.4 TPD and SRS instabilitiesIn Chapter 4, experimental results showed that: 1) the SRS plasmons have a narrow wavenumber spectrum concentrated at k = k0, while the TPD plasmons have a wide wavenumber spectrum covering the whole detectable wave-number range [0.77k — 5.66k0] (seeFigures 4.6, 4.7, 4.32, 4.33, and 4.34); 2) the SRS plasmons observed have wave-vectorswith k > 0, while TPD plasmons with both k > 0 and k < 0 are detected; 3) the peakintensity of SRS plasmons occurs at a different location than that for the TPD plasmons(see Figure 4.3); 4) the SRS plasmons appear in a single burst while the TPD plasmonsappear in multibursts; and 5) the SRS plasmons have a smaller intensity than the TPDChapter 5. DISCUSSION 128plasmons (see Figures 4.6, 4.7, 4.32, 4.33, and 4.34). Based upon points 2 and 3, it wasconcluded that the TPD instability dominates the generation of w0/2 plasmons. Here theTPD and SRS plasmons are distinguished as the plasmons observed driven by P- andS-polarized pump radiations, respectively. In this section, an attempt is made to explainthe differences between w0/2 SRS plasmons and TPD plasmons with the theories givenin Chapter 2.First, let us find the possible wave-number ranges for the observed TPD and SRSplasmons set by the experimental parameters given in Section 5.2. Considering thedispersion relations for EM waves and EPW’s, and the matching conditions for theTPD and SRS instabilities, we find the wave-number range for the SRS plasmons to be[0—2k] and the range for the TPD plasmons to be [0—25k] for the plasma temperatureTe = 400eV. The wave-number range for the TPD plasmons is much larger than thatfor the SRS plasmons because Vth << c. Considering further the band-pass interferencefilters and the collecting mirror used in the Thomson scattering measurements, we get thepossible wave-number range for the observed SRS plasmons to be [0.77k — 1.42k0] andfor the observed TPD plasmons to be [0.77k — 5.66k0] as shown in Figure 5.1. The TPDplasmons may have wave-numbers in the region between circle a and circle b. Accordingto the wave-vector matching condition for the SRS instability, the SRS plasmons canonly have wave-vectors with k > 0. Hence the SRS plasmons may have wave-numbersin the region included in circle c but at the same time excluded from circle a. The radiusof circle a is the minimum wave-number which the observed plasmons may have due tothe geometry of the collecting mirror. The maximum wave-number for the SRS plasmonsis mainly determined by the pass bands of the interference filters, while the maximumwave-number for the TPD plasmons is mainly limited by the size of the collecting mirror.In our experiments we see that the SRS plasmons have a narrow wave-number spectrum centered at k0 as shown in Figures 4.7, 4.32-4.34. The long wave-number end ofChapter 5. DISCUSSION 129CDCDFigure 5.1: The wave-number ranges for the TPD and SRS plasmons. The radius forcircles a, b, and c are 0.77k, 5.66k0 and 0.42k, respectively. The TPD plasmons mayhave wave-numbers which are outside circle a and inside circle b, and the SRS plasmonsmay have wave-number outside circle a and inside circle c.4CN0ENChapter 5. DISCUSSION 130the spectrum drops slowly and extends to wave-numbers out of the range for the SRSplasmons. This perhaps results from the fact that the conversion of the pump radiationfrom P-polarization to S-polarization is not 100%. Thus the TPD plasmons are alsoobserved.From the above discussion, we can conclude that since the plasma temperature is low,the TPD plasmons have a wider wave-number spectrum than the SRS plasmons have;according to the the dispersion relations and the matching conditions, the SRS plasmonscan only have wave-vectors with k > 0, while the TPD plasmons can have wave-vectorswith either k > 0 and k < 0. Next we will discuss why the peak intensity of the TPDplasmons occurs at a different location than that for SRS plasmons.Based upon Figure 4.3, the observed distance between the peak intensity of the TPDplasmon and the peak intensity of SRS plasmons is about 300 gum. The real distance,z, calculated with eq. 3.13 is equal to 310 tm. If we assume that SRS plasmons reach amaximum intensity at the focal spot, then TPD plasmons peak at a point 310 um fromthe focal spot. Since the pump beam has a Gaussian intensity profile centered at thefocal spot with a 1/e radius r0 of 49tm [Bernard 85], the pump beam intensity at theposition 310 m from the focal spot on the beam axis changes only by 4% from that atthe focal point. Thus we can conclude that the fact that the TPD and the SRS plasmonsreach a maximum intensity at different locations is not due to the different thresholds,as the pump intensity changes very little in the interaction region, but rather to someother effects.Due to the interference filters, the plasmons may have frequencies in the range[0.45w—0.5w]. A single filter has a 11 nm pass-band at FWHM transmission centeredat 670 nm. The combination of two filters has a 9 nm pass-band at FWHM transmission.The corresponding density range for the SRS and the TPD plasmoris is [0.20n — 0.25n].However, the SRS instability is absolute and has maximum growth rate in n/4 densityChapter 5. DISCUSSION 131regions. On the other hand, if we plot the growth rate of the TPD instability, calculatedaccording to eq. 2.76 without collisional damping, we can see that the TPD plasinonswhich experience a maximum growth rate have wave-numbers about 3k0, which impliesthat they are from the region of plasma density around 0.225n, as shown in Figure 5.2.As discussed in Chapter 4, the 0.25n density region is located in the center of theinteraction region as shown in Figure 4.5. Figure 4.3 indicates that the SRS plasmonsreach a maximum intensity at 0.25n and the TPD plasmons peak at about 0.25n, justas expected based upon the above discussion. Thus we can conclude that the peak intensities of TPD plasmons and SRS plasmons occur at different locations in the interactionregion because the instabilities peak at different plasma densities.As plasma expands, the plasma density decreases. When the maximum density islower than 0.25 n, the SRS instability is convective. Then the threshold for SRS increasesby a factor of (k0L)’/3 and is much higher than that for TPD. Hence, both the TPDand the SRS instabilities occur at the beginning of the interaction when the maximumplasma density is about 0.25n. After about 50 ps, the SRS and TPD are quenched bythe ion-acoustic waves. The ion-acoustic waves are then damped and the density scalelength increases. At the same time, the plasma expands and the density drops belown/4. When the density scale length becomes so large that the threshold for the TPDinstability is lower than the pump intensity, the TPD reoccurs and the amplitude of theion-acoustic waves grows again. Thus, before the density scale length becomes so largethat the threshold for the convective SRS is lower than the pump intensity, the scalelength start to decreases. Hence, the SRS cannot reoccur. Therefore, the SRS plasmons,which are observed with S-polarized pump radiation, appear in a monoburst. This mayexplain why no scattered radiation corresponding to the SRS instability at 0.20 — 0.24ndensity regions were observed with S-Polarized pump radiation [McIntosh 86].1324CN0Chapter 5. DISCUSSIONIl/i\\ N\/t N /Z2zI I I I—CN 0Figure 5.2: Growth rate contours of the TPD plasmons. The growth rate contours forTPD plasmons are calculated with eq. 2.75 without including the collisional dampingrate. The interval of the contours equals 0.26 x 10_2.Chapter 5. DISCUSSION 133From the above discussion, we can see that the plasmons observed with P- and Spolarized pump radiations do indeed bear the features deduced from the theory for TPDand SRS plasmons. Now we would like to comment on the plasmons observed with Ppolarized pump radiation. In the streak images of the plasmons having wave-vectors withk < 0 as shown in Figures 4.14-4.18, it can be seen quite clearly that there is a progressionin time from small wave-number to large wave-number. This is expected when the plasmaexpands with time. According to the dispersion relation for the electron plasma waves,the cL0/2 plasmons have larger wave-numbers in lower density regions. However, thestreak images of plasmons having wave-vectors with k > 0, as shown in Figures 4.14-4.18, do not show this progression. The plasmons with wave-numbers ke = k0 last about200 Ps without apparent shift. These plasmons must be due to the SRS instability.As we indicated previously, the plasma density drops below 0.25n after about 100 Ps.This means that the plasmons with ke = k0 after 100 ps are generated in density regionbelow 0.25n, where the SRS plasmons are not expected to reoccur. In addition, theseplasmons are not observed with S-polarized pump radiation, where the SRS plasmonsare expected to have a maximum intensity. Therefore, these plasmons must be due tosome process which has not been identified. It is clearly shown in Figure 4.15 that boththe TPD plasmons and the SRS plasmons disappear at the same time. Hence, maybethe SRS instability is seeded by the TPD instability as the latter has a low thresholdand a maximum growth rate in the density region around 0.225n. The TPD instabilityhas a maximum growth rate in the incident plane of the pump radiation. If seeded bythe TPD instability, the SRS instability would then also have a maximum growth ratein the incident plane, instead of in the plane perpendicular to the incident plane.Summary: SRS plasmons reach a maximum intensity in the n/4 density region,while TPD plasmons peak in the 0.225n density region; SRS plasmons have a narrowwave-number spectrum, while TPD plasmons have a wide wave-number spectrum; SRSChapter 5. DISCUSSION 134plasmons are weaker in intensity than TPD plasmon. The TPD instability dominatesthe generation of the w0/2 plasmons.5.5 Intensity distribution of the TPD plasmons in the wave-number spaceThe experimental results (see Figures 4.14, 4.15, 4.16, 4.17, and 4.18) indicate thatthe wave-number spectrum of the TPD plasmons evolves with time. In this section wewill first discuss the temporal behaviour of the TPD plasmons. We will then discussthe intensity distribution of the unsaturated and the saturated TPD plasmons. Thissection concludes with a discussion of the relation between the intensity distribution ofthe saturated TPD plasmons and the angular distribution of the (3/2)w0 emission.5.5.1 Temporal behaviourAs pointed out in Section 5.3, the plasmons observed were due to the absolute TPD instability since the pump intensity is about 2 order of magnitude higher than the thresholdand the condition is as well satisfied for the absolute TPD instability. Absolute instabilities are characterized by exponential growth. The plot of e-fold intensity of TPDplasmons versus time, shown in Figure 5.3, indeed indicates exponential growth in theearly stages of TPD growth. From such plots, we can determine the growth rate wellabove threshold of the TPD instability in its early stages. The growth rate for TPDplasmons measured at 19R is plotted in Figure 5.4. Since most of our experiments wereconducted on a lower speed-setting of the streak camera and the time-resolution is nothigh enough for a precise measurement, we have not presented the growth rate for TPDobserved in other geometries. On the high speed setting, the time jitter of the streakcamera is too large to conduct any experiments. The large time jitter is caused by electronic noise due to electrical discharges within the pump and probe lasers. Because ofChapter 5. DISCUSSION 1358o ke=3.1 6ko+ ke=3.7Oko +x ke=5.1 2ko +060 0+x+x xC— 0+x02 Xxx0I I I i I I i I • I •0 2 4 6 8 10 12 14 16t(ps)Figure 5.3: Exponential growth of TPD plasmons observed at 19R.Chapter 5. DISCUSSION 136+N+ N++++ N+ N+++ 0+ -+ ci)++ N+++ (N++++++NI NLU (N OC)(sd/ [)JFigure 5.4: Growth rate of TPD plasmons at 19R.Chapter 5. DISCUSSION 137the electronic noise, the streak camera also has a significant dark current. Here the darkcurrent refers to the noise that the streak camera records when the laser system and otherelectronic devices are fired, but the shutter at the entrance slit of the streak camera isleft closed.Considering the noise, we calculate the growth rate in the following manner. Weassume a common time for initial growth of plasmons regardless of wave-number. Wethen determine the time required to reach the maximum signal level, and thus calculatethe growth rate. This assumption introduces a large systematic error, since plasmonswith different wave-numbers require different thresholds, and thus must start at differenttimes. The measured growth rate for TPD is about the same as that measured in aprevious experiment [Meyer 85].After about 10 Ps of exponential growth, the plasmons become saturated, and aresubsequently quenched by coupling to ion acoustic waves. We will try to demonstratein the next section that coupling of TPD plasmons to ion acoustic waves acts as thedominant saturation mechanism for the instability. The plasmons then reappear andthe process can repeat up to 8 times. The double-burst feature has been observed insimulations [Langdon 79, Langdon 84] and in experiments [Baldis 83, Meyer 84]. In thepresent study, the fact that the TPD plasmons appear in so many bursts is as a resultof the optimization of the plasma conditions for growth of the TPD instability. Previously only plasmons with k > 0 have been investigated. In the present study, plasmonswith both k > 0 and k < 0 are investigated. The temporal behaviour of plasmonswith different wave-vectors is measurably different. Plasmons with k < 0 last about500 + 100 ps, while plasmons with k > 0 last about 200 + 50 ps. This is due to thedifference in the magnitude of the wave-vector of the plasmons when they are generated.In addition, it can clearly be seen in Figure 4.15 that there are two different periodsmodulations: one is about 5Ops in duration, and the other one lasts about 100 ps. ThisChapter 5. DISCUSSION 138likely results from the different frequency components of the ponderomotive force due tothe TPD plasmons. The short period modulation is due to the coupling of TPD plasmons to ion acoustic waves, and the long period modulation is due to the density profilesteepening. Similar temporal modulations were also observed in the temporal behaviourof ion acoustic waves [Bernard 85]. The occurrence of two distinct periods of modulation has been demonstrated in simulations [Langdon 79] and observed in experiments[Baldis 83, Meyer 83].Summary: Initially, the TPD instability does grow exponentially as predicted by thelinear theory. The TPD instability is then saturated and quenched. The wave-numberspectra of the plasmons evolve with time.5.5.2 Intensity distribution of unsaturated plasmonsFrom the last section we know that the TPD instability initially grows exponentially aspredicted by the linear theory. In this section the intensity distribution of the unsaturatedTPD plasmons will be discussed in the context of the linear theory.From the wave vector matching condition for TPD shown in Figure 2.3, we can seethat one of the pair of EPWs generated by TPD is in the direction of k > 0 with a longerwave vector, while the other is in the direction of k < 0 with a shorter wave vector.Thus we should observe a matched pair in the wave vector space; i.e., if we observe amaximum in the distribution of the intensity of EPWs in wave vector space at (ks, ku),then we should also observe the matched maximum at (k — k0, ku). Next we will tryto determine the location of the matched maximum pair in wave-vector space for theplasma conditions encountered in our experiments.Because the plasma temperature is low, the Bohm-Gross dispersion relations, eq. 2.4,Chapter 5. DISCUSSION 139for TPD plasmons can be written as:= w(1 +k,2)h/2‘p(5.10)Then the frequency matching condition for TPD can be written as:CL)0 =3 v2 (5.11)“-‘pBy rewriting the above equation as:2 2wk1 + k (w — 2w)—-, (5.12)we can see that for given experimental conditions(T(v), w(n), we), the RHS is a constant; i.e., the TPD plasmons are located on a circle in the wave-vector plane. Examiningthe wave-vector matching condition for TPD shown in Figure 2.3, we can see that theTPD plasmons in the k — k plane have to be localized on a circle centered at k = 0.5 k0and k = 0 of radius R given by(R2 1 (c (n1/2 (1- 2(n/nc)h/2 - (513k0) 3 “V) ‘ni \\ 1—n/ne ) 4Here n and n,, are the electron and critical electron densities, respectively. For thegiven conditions, the circles for n/n, = 0.235 and 0.225 are indicated in Figure 5.5. Inaddition, the net growth rate for the TPD instability maximizes on the hyperbola d andd’, as shown in Figure 5.5. For a homogeneous plasma with a given n/nc, we wouldtherefore expect TPD waves to grow at the intersections of the appropriate circle withthe hyperbola in the k — k plane denoting the maximum growth rate.The contour of TPD plasmon intensity during the first 2Ops of TPD growth shown inFigure 5.5, clearly demonstrates the growth of the TPD plasmon pair with k0 = k1 + k2Chapter 5. DISCUSSION 140oFigure 5.5: Intensity distribution of TPD plasmons during the first 20 ps of TPD growth.The threshold contour a for TPD is calculated with eq. 2.75 without the collisionaldamping rate. At densities n/ne = 0.225 and 0.235, TPD plasmons are localized on thecircles b and c respectively. TPD growth maximizes on the hyperbola d and d’. Theshaded contours are those of plasmon intensity integrated over the first 20 Ps.4CN0Chapter 5. DISCUSSION 141at the positions expected from linear theory when the density is adjusted to producemaximum growth of the TPD instability, as attempted in our experiments. To ourknowledge, this is the first experimental confirmation of this fundamental basis of linearTPD theory. It shows that initially the development of TPD is well described by lineartheory and is not altered by mode-coupling and other nonlinear processes. However, thesituation changes dramatically when TPD saturates. Next we will discuss the intensitydistribution of the saturated plasmons.5.5.3 Intensity distribution of saturated plasmonsThe difference between the the measured intensity distribution of the saturated TPDplasmons and the theoretical intensity distribution of the TPD plasmons deduced fromthe growth rate can be seen in Figure 5.6. For comparison, we have overlapped thecontours, for the saturated TPD plasmon intensity in k-space, obtained from the streakrecords after integrating over the first 100 ps, and the growth rate contours obtained fromeq 2.76 without the collisional damping. The interval is arbitrary for the former contoursand is 0.0026w for the latter contours. As shown in Figure 5.6, the TPD plasmons aredetected over a wide region in wave-number space extending out to and in places beyondthe threshold. Three regions of maximum TPD plasmon intensity are found in bothforward (+k) and backward (—ks) directions and their positions cannot be associatedwith a matching condition corresponding to k + k_ = k0. Obviously the saturationprocess, presumably arising from a coupling to ion-acoustic waves [DuBois 93, Meyer 92],generates a wide spectrum of unstable plasmons and affects their maximum amplitude.Large amplitude plasmons are detected at positions where the TPD growth rate is zero,at regions in the vicinity of (kr, lc) = (0.5k,k0) and around (ks, k) = (k0,0). The latterhas to be associated with the SRS instability as we discussed before.Chapter 5. DISCUSSION 142oFigure 5.6: Intensity distribution of saturated TPD plasmons and the threshold contour.The thick solid-line contours are those of the saturated TPD plasmori intensity obtainedfrom the streak records after integrating over the first 100 ps. The thin line contours arethose of the growth rate for TPD calculated with eq. 2.73. The interval is 0.0026.4CN0ENChapter 5. DISCUSSION 1435.5.4 TPD and (3/2)w0 emission(3/2) harmonic emission is commonly observed in laser-plasma interactions. It is generally thought that the (3/2) harmonic is generated by Thomson scattering of thepump photons from /2 plasmons. As we discussed in the previous section, TPD isconsidered to dominate the generation of w0/2 plasmons. Hence, the (3/2)w0 harmonicemission is commonly adopted as a diagnostic of the TPD instability. In this sectionwe will discuss the relation between the intensity distribution of w0/2 plasmons and theangular distribution of (3/2)w0 harmonic emission.Considering the frequency and wave-vector matching conditions, and the dispersionrelation for EM waves, (3/2) harmonic emission arises due to w3/2 plasmons thathave wave-vectors localized on a circle in the k — k plane of radius (8/3)’/2k0around(kr, k) = (—k0,0) as shown in Figure 5.7 by the dashed half circle. The radius is theamplitude of the wave-number for (3/2)w0 harmonic emission in the n/4 density region.In Figure 5.7, the solid-line contours represent the threshold contours for the TPD instability, calculated for conditions encountered in our experiments using eq. 2.76 withoutcollisional damping take into account. On the hyperbola b and b’, the growth rate forTPD reaches a maximum. As we discussed in the preceding section, the intensity of theunsaturated u.0/2 plasmons is indeed a maximum on the hyperbola. Hence, if the unsaturated w0/2 plasmons dominate the generation of (3/2)w0 emission, we would expectthat (3/2) harmonic emission be strongest in the direction indicated by arrow c ato = 98°, where 0 is the angle between c and +k-axis. But the observed pair of maximafor the unsaturated /2 plasmons, as shown by the shaded contours in Figure 5.8, arenot along the matching circle for (3/2) harmonic emission. Thus, we can conclude thatthe unsaturated TPD plasmons do not contribute to the generation of (3/2)c40 emission.Chapter 5. DISCUSSION 144(0CNII IFigure 5.7: Expectation of (3/2)w0 harmonic emission. Contour a is the threshold contour for TPD. TPD growth maximizes on hyperbola b and b’. Plasmons on the dashsemi-circle generate (3/2)wo emission. (3/2)w0 emission is expected to maximize in thedirection indicated by vector c.Chapter 5. DISCUSSIONO)JJi()T n/_I_c / —/_0In145IIFigure 5.8: Unsaturated plasmons and (3/2)c emission. The shaded contours are thoseof the plasmon intensity integrated over the first 20 Ps.Chapter 5. DISCUSSION 146On the other hand, if we draw the matching circle for (3/2) emission and the contours of saturated TPD plasmons plotted in Figure 4.30 together, as shown in Figure 5.9,we can then see that the measured saturated TPD plasmon intensity reaches a maximumon the matching circle for (3/2) emission into two ranges of angles centered aroundO = 350 and 0 115°, as indicated by arrows a and b respectively in Figure 5.9. This canbe compared with the measurements of the angular distribution of the (3/2)w0 emissionshown in Figure 5.10 [Zhu 87, Meyer, Zhu, and Curzon 89]. If the generation mechanism does indeed consist of the processes in which the pump photons are scattered offTPD plasmons, then the (3/2)w0 intensity is proportional to the plasma wave intensityand the Thomson scattering cross-section th Dividing the values of Figure 5.10(a) bycos2 O(cx °ti) will therefore generate the TPD plasmon intensity on the matching circle as shown in Figure 5.10(b). At 0 = 90°, (3/2)w0 intensity would be zero in theabsence of a finite detected angular range and the TPD plasmon intensity becomes indeterminable. Therefore, no point is shown at 90° in Figure 5.10. The good agreement withthe Thomson scattering results of Figure 5.9 represents, to our knowledge, the first directexperimental evidence of the generation of (3/2)w0 harmonic emission by TPD plasmonsvia scattering of pump and back scattered pump photons.5.6 SaturationThe TPD instability was shown to cycle through the exponential growth, saturation, andquenching processes up to eight times in some of our experiments. In this section we willdiscuss the saturation and quenching mechanisms.As pointed out in Chapter 2, ponderomotive effects have been shown numerically[Langdon 79, Langdon 84, Kolber 93], experimentally [Baldis 83, Meyer 84], and analytically [Karttunen 80] to be the dominant saturation mechanism for the TPD instability.Chapter 5. DISCUSSION 1474Figure 5.9: Contours of the saturated TPD plasmon intensity in k-space obtained fromstreak records after integrating over the first 100 ps. Plasmons on the circle around= —k0, k = 0 generate (3/2)c emission and the arrows indicate the wave vectors ofthis emission due to plasmons with maximum amplitudes.Chapter 5. DISCUSSION 148(a)+I t I ‘P1 • I(b)I--I++ ++— I t I I I0° 36° 72° 1080 1440 0Figure 5.10: The angular distribution of (3/2)w0 emission intensity (a) and generatingw0/2 plasmon intensity (b). Points in (b) are obtained after dividing the measurementsin (a) by the Thomson cross section. Horizontal error bars indicate the accepted angularrange and vertical error bars the standard deviation of five or more experiments.Chapter 5. DISCUSSION 149In eq. 2.82, the ponderomotive force due to TPD plasmons includes two componentscharacterized by different frequencies. The first one changes rapidly with time and candrive ion acoustic waves. The second one changes much more slowly and can cause profile modification. We will discuss mode-coupling and profile modification in the followingsubsections.5.6.1 Mode-couplingMode-coupling of the TPD plasmons to the ion acoustic waves has demonstrated to beone of the dominant saturation and quenching mechanisms for the TPD instabilities.Computer simulations [Langdon 79, Langdon 84] show that the temporal behaviour ofthe energy of the TPD plasmons is related to the ion inertia, that is, the damping rateof the ion acoustic waves. The wave-number of the ion acoustic waves, according tothe dispersion relation, is proportional to the square root of the ion inertia. The rateof Landau damping, which is the main damping mechanism for the ion acoustic waves,depends on the wave-number. Experiments [Baldis 81] demonstrated that there is a correlation between the amplitude modulation of the TPD plasmons and the ion acousticwaves, and that initial saturation and quenching of the TPD instability take place overtimes which are less than 50 ps, which corresponds well with our experimental results(see Figures 4.14-4.18). Saturation of the TPD plasmons occurs at the same time thathigh levels of ion fluctuations are observed. In our experiments, the temporal modulation of the intensity of the TPD plasmons correlated well with previous observations ofthe variation in the intensity of ion acoustic waves [Bernard 85]. Therefore, we expectthat mode-coupling is the dominant saturation and quenching mechanism for the TPDinstability.Recently, by solving the coupling equations for SRS and lAWs numerically, Kolberet al. [Kolber 93, Rozmus 87], found that the coupling of the SRS plasmons with lAW’sChapter 5. DISCUSSION 150via the parametric decay instability is the primary saturation mechanism for the SRSinstability, and that the temporal behavior of SRS is closely associated with the ratioof the growth rate to the damping rate of the ion acoustic waves. If the ratio is large,SRS appears in bursts; if the ratio is small it appears only once. This implies thatwheii the ratio is large, a large amplitude ion acoustic wave is generated in a short timeand the SRS instability is quenched; SRS then reoccurs after the ion acoustic waveshave been damped to sufficiently small amplitudes, and the process can repeat manytimes giving rise to a series of bursts of SRS activity. When the ratio is small, theamplitude of the ion acoustic waves can not reach the level necessary to quench theSRS instability before the interaction is over and SRS appears only once unless otherquenching mechanisms take effect. In the n/4 density region, TPD dominates SRSin the generation of /2 plasmons. Hence it is expected that TPD instability can besaturated and quenched in a similar way. In contrast to the analyses of the couplingof the SRS plasmoris to ion acoustic waves [Kolber 93, Rozmus 87], the necessarily two-dimensional nature of TPD instability complicates the situation. This is why no similarnumerical calculation describing the coupling between ion-acoustic waves and TPD isreported.In previous experiments [Baldis 83, Meyer 84], the TPD plasmons have been observedto appear most often in double bursts, while in the present set of experiments, theTPD plasmons are observed to appear in multi-bursts of up to 8 bursts. Based on theexperimental results, especially on our particular results, and the simulation results, wepropose a physical picture of the saturation and the quenching process for the TPDinstability as follows.Large amplitude TPD plasmons acting as pump waves couple to ion acoustic wavesvia the parametric decay instability. The ion acoustic waves thus generated can saturateTPD in two ways. Firstly they directly saturate TPD by taking energy from TPDChapter 5, DISCUSSION 151plasmons. Secondly they saturate TPD by decreasing the growth rate. The ion acousticwaves oscillate much slower than the TPD plasmons. Thus the oscillation of the ionacoustic wave iii space acts as a static density ripple from the point of view of theplasmons. Due to the ripple, the density scale length decreases and the inhomogeneitydamping rate increases. Therefore, the growth rate of TPD decreases. In addition thethreshold increases when the density scale length decreases. When the threshold becomeshigher than the pump, TPD is quenched. Without a driving force, the lAW’s are rapidlydamped. Consequently, the density ripple is reduced and the scale length increases. TPDcan then reoccur and the process can repeat many times. In a simulation [Langdon 84j, itwas shown that the TPD instability is arrested by the density fluctuation due to the ionacoustic waves before the density profile steepening can have any effect on the instability.We can also test the above proposed physical picture by examining the relation between the maximum amplitude of the ion waves and the maximum pump intensity. It isclear that the larger the amplitude of the ion waves, the smaller the density scale-length,and the higher the threshold for the plasmons. To find the maximum amplitude of theion acoustic waves, we start with a calculation of the density scale length of the densityripples due to ion acoustic waves of amplitude dn/n0. To simplify the calculation, weassume that the ion waves have a triangular wave shape. Then the density scale lengthdue to the ion waves isL=dn/n04’ (5.14)Here A, is the wave-length of the ion wave. When the threshold exceeds the pumpintensity, TPD is quenched and the TPD plasmons are rapidly suppressed by damping.Without a driving force, ion waves also diminish rapidly. Hence, the ion waves reach amaximum amplitude when the threshold for the TPD instability is equal to the pumpChapter 5. DISCUSSION 152intensity. According to eq. 2.73, we find:dn O.O63xEx)2= 4 x 61.25’(5.15)where E is the energy in units of J and ) is the wave-length in units of ,um for the pumpbeam. We have taken ) = )/2, which corresponds to the most unstable ion mode[Bernard 85]. A comparison of eq. 5.15 with the peak ion acoustic fluctuation levels asa function of the pump energy obtained in a previous experiment [Bernard 85] is shownin Figure 5.11. The agreement is quite good considering the large uncertainty in themeasurement and the fact that only one ion acoustic mode is examined. In addition, thepump intensity can be raised locally by self-focusing or filamentation processes. Experimental evidence [Meyer and Zhu 86] suggests that the self-focusing or filamentation doesoccur in the present set of experiments.5.6.2 Plasma density profile modificationAs pointed out in Chapter 2, the ponderomotive force due to the TPD plasmons, inaddition to the high frequency component, has a low frequency component driving density profile modification. Density profile steepening has been modelled in simulations[Langdon 79] and observed in experiments [Baldis 81, Meyer 83]. Thus we expect thatdensity profile steepening occurs in our experiments. However, there is a significantdifference in plasma conditions between the previous experiments and the present experiment. In the previous experiments, the maximum plasma density was about O.4n.Hence, the density gradient was large and, more importantly, the O.25n density regions,where TPD may occur, are separated. Then the physical picture of the density profile steepening is straight forward: electrons are pushed out of the O.25n regions whereTPD occurs, and thus the density profile of the O.25n regions becomes steepened. Inthe present experiments, the maximum density is O.25n and the regions where TPDChapter 5. DISCUSSION 15318 • —a-xo0.6’ T a41 12’0 2 4 6 8 10Laser Energy (J)Figure 5.11: Peak ion acoustic fluctuation levels as a function of pump energy. The datapoints are from Bernard’s thesis. The solid line is obtained with eq. 5.15.Chapter 5. DISCUSSION 154occurs are not separated (see Figures 4.2 and 4.8). If we then claim that density profilesteepening occurs, it remains to be explained where the electrons in the center region go.Examining the eq. 2.82, we see that the component of the ponderomotive force whichcauses density profile steepening is proportional to the gradient of the square of theamplitude of the electrostatic field, or equivalently the gradient of the intensity of theplasmons. If the intensity of the plasmons is constant throughout the interaction region,then the ponderomotive force is zero and density steepening can not occur. However ourexperimental results (see Figures 4.2 and 4.9) show the intensity of the plasmons is notconstant; but instead display regions of high illtensity. Hence, the ponderomotive force isnot constant in the interaction region. The electrons are thus pushed out of the regionswhere the ponderomotive force is large to the areas where the force is small. Consequently,we expect density profile steepening does occur in our experiments. Further more, wecan demonstrate from some experimental evidence that density profile steepening occursin our experiments.In Figure 4.15 it was clearly shown that after about 200 ps, backward (—ks) propagating TPD plasmons reappear with longer wave-numbers, and forward (+k) plasmons areno longer observed. This can be interpreted in the following way if we assume that thedensity profile steepening occurs. Steepening of the density profile reduces the densityscale length until the threshold for TPD exceeds the pump intensity, and the instabilityis quenched. The ponderomotive force then diminishes, and the density profile steepening relaxes. Consequently, the scale length increases and TPD reoccurs. Howeverdue to expansion, the plasma density is now lower than it was during the first occurrence of the instability. Therefore, even though TPD can still occur, the daughter wavesnow have larger wave-numbers and suffer severe Landau damping. Since according tothe wave-vector matching condition the forward propagating TPD plasmons have largerwave-numbers than the backward plasmons do, they suffer stronger Landau dampingChapter 5. DISCUSSION 155than their matching partners. Hence, their growth rate is lower and their amplitude issmaller than that of backward plasmons. In addition, the noise is now higher as randomplasma light becomes stronger due to recombination processes in the expanding plasma.The forward plasmons are therefore not detected. Because the TPD plasmons are nowweakened due to strong Landau damping, they generate weak JAW’s via parametricdecay instability. This is why we see the temporal modulation of the TPD plasmonsincrease. When the plasma expands further, the plasma density becomes too low andthe TPD instability is finally quenched. It was shown in other experiments [Berilard 85]that JAW’s also have a temporal modulation lasting 100 200 ps.The ratio of the short and long wavelellgth components of the ponderomotive forcein eq. 2.82 is given as (L/)epw)(dfl/)2.The relative time scales over which the effects ofthese two components become important is the inverse of the ratio given by eq. 2.83:= (L/) (Sn/n)2. (5.16)For t = 50P8, tL = 200P3, Aepw = )/3, and L =24OOtm, we can infer that Sn/n = 0.14.The maximum amplitude of the TPD plasmons was measured to be ‘-- 15% in a previousset of experiments [Meyer 84].Summary: Experimental evidence suggests that ponderomotive effects dominate thesaturation and quenching of the TPD instability.Chapter 6CONCLUSIONSThis chapter summarizes the contents of this dissertation, and the important conclusionsthat were made. The original contributions of this study are outlined. Finally, thesuggestions for further study are presented based on questions arising in the course ofthis study.6.1 Summary and conclusionsThe /2 plasma waves, which can be generated by the TPD and the SRS instabilities inregions where the plasma is near n/4, have been investigated in this study. Through temporally resolved ruby laser Thomson scattering measurements of their spatial behaviourand wave-number spectra, the properties of the c/2 plasmons have been studied.When intense CO2 laser radiation interacts with an underdense plasma with a largedensity scale length, the absolute TPD and SRS instabilities grow in regions where theplasma density is near ri/4 and can generate large amplitude w0/2 plasma waves. TheTPD plasmons can be distinguished from those generated by SRS. Because the plasmatemperature is low, TPD plasmons have a much wider wave-number spectrum than SRSplasmons. The peak intensity of the TPD plasmons is in the incident plane of thepump beam. The SRS plasmons produced in the O.24n — O.25n density regions have amaximum intensity in the plane perpendicular to the incident plane. In regions where theplasma density is below O.24n, SRS is likely seeded by TPD, which has a lower thresholdand a maximum growth rate near O.225n, and thus peaks in the incident plane. The156Chapter 6. CONCLUSIONS 157TPD instability dominates the SRS instability in the generation of w0/2 plasmons.In its early stages, the TPD instability grows exponentially. After the TPD plasmonsgrow to large amplitudes, they couple to ion-acoustic waves via the parametric decayinstability and also drive a density profile modification. Consequently, the TPD plasmonsare saturated and then quenched by the large amplitude ion acoustic waves and by thedensity profile steepening. Once the ion acoustic waves diminish or the density profilerelaxes, the TPD instability can reoccur. This process has been observed to repeat up toeight times. The intensity distribution of the unsaturated TPD plasmons is well describedby linear theory. The matching pair related by the wave-vector matching condition canbe clearly observed in the initial stage of growth. The intensity distribution of thesaturated TPD plasmons is governed by the dominant saturation mechanisms. When aTPD plasmon decays into an ion acoustic wave and another plasmon, the wave-number ofthe TPD plasmon is modified. As a result, c/2 plasmons are observed with wave-numberoutside the TPD threshold contour.(3/2)w0 harmonic emission is generated by the Thomson scattering of the pump radiation by /2 plasmons. The angular distribution of (3/2)w0 emission deduced fromthe intensity distribution of the saturated TPD plasmons agrees well with the measureddistribution of (3/2) emission.6.2 Original ContributionsThe experimental confirmation of the differences between the characteristics of TPD plasmons and SRS plasmons was first achieved by the author [Meyer and Zhu 90] (PhysicalReview Letters 64(22), 2651 (1990)).The experimental observation of the matched TPD plasmon pair and the demonstration that the saturated TPD plasmons dominate the generation of (3/2)w0 emission wereChapter 6. CONCLUSIONS 158first made in this study [Meyer and Zhu 93] (Physical Review Letters 71(18), 2915(1993)).In addition, the observation of experimental evidence that the saturation of TPDis due to both long wavelength and short wavelength ponderomotive forces was initiallymade by the author. The repeated occurrence of growth and decay of the TPD instabilityin series of up to 8 bursts was observed for the first time in this study.6.3 Suggestions for further workExperimental evidence from this study suggests that SRS is seeded by TPD in the regions where the plasma density is in the range O.2On — O.24n, which would explain theobservation of the peak intensity of the SRS plasmons occurring in the incident plane.This speculation could be confirmed by measuring the scattered pump radiation dueto the SRS instability in the regions where the plasma density is below O.24n. If thisspeculation proves to be correct, it will provide an important clue towards the completeunderstanding the frequency spectra of SRS scattered radiation.Bibliography[Afeyan 93] Afeyan, B.B., A variational approach to parametric instabilities in in-homogeneous plsamas, PhD Thesis , Unversity of Rochester(1993).[Baldis 81] Baldis, H.A. and C.J. 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