@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix skos: . vivo:departmentOrSchool "Science, Faculty of"@en, "Physics and Astronomy, Department of"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Legault, Lawrence E."@en ; dcterms:issued "2010-07-14T22:45:27Z"@en, "1987"@en ; vivo:relatedDegree "Master of Science - MSc"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description """The generation of hot electrons characterizing the two-plasmon decay (TPD) instability is investigated experimentally both in and out of the plane of polarization of a CO₂ laser incident on an underdense gas target. The results presented here show that, for high intensities (I > ~ 3.5 x 10¹³ W/cm² for a helium target, 1 > ~ 5.5 x 10¹³ W/cm² for a nitrogen target), the electron plasma waves (EPW's) generated by the TPD instability are modified by the electron decay instability (EDI). The relatively short scale lengths at the onset of TPD for these high intensities (<~0.125mm) cause the EPW's propagating towards the higher density regions of the plasma to undergo the EDI resulting in EPW's which contain a vector component perpendicular to the plane of polarization and accelerate electrons by nonlinear Landau damping up to 55° outside the plane of polarization."""@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/26438?expand=metadata"@en ; skos:note "T H E PRODUCTION OF HOT ELECTRONS B Y T H E TWO-PLASMON DECAY INSTABILITY IN A C 0 2 LASER PLASMA INTERACTION By Lawrence E . Legault B.Sc, York University, 1984 A THESIS S U B M I T T E D IN PARTIAL F U L F I L L M E N T O F T H E R E Q U I R E M E N T S FOR T H E D E G R E E O F M A S T E R OF SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES D E P A R T M E N T OF PHYSICS We accept this thesis as conforming to the required standard T H E UNIVERSITY O F BRITISH C O L U M B I A April 1987 © L a w r e n c e E . Legault, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6(3/81) i i A B S T R A C T The generation of hot electrons characterizing the two-plasmon decay ( T P D ) instabil i ty is investigated experimentally both in and out of the plane of polariza-tion of a CO2 laser incident on an underdense gas target. The results presented here show that, for high intensities (I > ~ 3.5 x 1 0 1 3 W / c m 2 for a helium target, 1 > ~ 5.5 x 1 0 1 3 W / c m 2 for a nitrogen target), the electron plasma waves ( E P W ' s ) generated by the T P D instability are modified by the electron decay instability ( E D I ) . The relatively short scale lengths at the onset of T P D for these high in-tensities (<~0.125mm) cause the E P W ' s propagating towards the higher density regions of the plasma to undergo the E D I resulting in E P W ' s which contain a vector component perpendicular to the plane of polarization and accelerate electrons by nonlinear Landau damping up to 55° outside the plane of polarization. iii T A B L E O F C O N T E N T S Abstract ii Table of Contents iii List of Tables v List of Figures vi Acknowledgements ix I Introduction 1 II T h e Generation of Hot Electrons 4 1. Parametric Instabilities 5 2. The T P D Instability 7 3. Nonlinear Landau Damping - Electron Trapping 13 III Experimental Apparatus 17 1. The C 0 2 Laser 17 a. The Hybrid Oscillator 19 b. Short Pulse Generation 19 c. The Amplifier Chain 21 d. Backscattered, Leakthrough, and Self-Lasing Energies 22 e. Laser Output 23 2. The Gas Jet Target , 23 3. The Multi-Frame Interferometer 26 a. Interferometric Theory v . 26 b. The Multi-Frame Interferometer 29 4. The Electron Detectors 35 a. The Electron Spectrometer 35 b. Photographic Detection of Electrons 38 iv I V The Spatial and Temporal Evolution of the Plasma 43 1. Density Contours and General Features 43 2. Plasma Density Distribution 50 3. Scale Lengths and Radial Expansion Rates 50 4. The Laser Plasma Interaction Region 57 V T h e Electrons Produced by the T P D Instability 61 1. The Electron Images 61 2. The Spatial Distribution of the Electrons 73 3. The Energy Spectrum of the Electrons 77 V I Discussion of Results and Conclusions 83 1. The Plasma Temperature 83 2. The Plasma Parameters 87 3. T P D Theory and EPW's in the Plane of Polarization 87 4. E P W Vector Components Perpendicular to the Plane of Polarization . . 89 5. Conclusions and Future Recommendations 94 Bibl iography 97 L I S T O F T A B L E S III-l Calibration factors used to normalize the spectrometer channel signals. vi LIST OF FIGURES 2.1 The two-dimensional wave vector diagram for T P D 11 2.2 The magnitude of (ky/k0)max as a function of (a) peak laser intensity and (b) the threshold intensity 12 2.3 The phase space trajectories of electrons in the potential of an E P W 16 3.1 The C 0 2 laser 18 3.2 The hybrid oscillator and short pulse generator 20 3.3 The Laval gas jet nozzle 24 3.4 The target chamber 27 3.5 The multi-frame interferometer 30 3.6 Top view of the frame delay mechanism 31 3.7 Edge-on view of the frame delay wedge 33 3.8 The electron spectrometer 36 3.9 Signal obtained from uniform illumination of the five scintillator discs by ultra-violet radiation 37 3.10 The film holder 39 3.11 Film density for various foil thicknesses 41 3.12 The continuous energy spectrum 41 3.13 The dependence of film density on electron energy. 42 4.1 The series of interferograms obtained from a single shot where E C 0 2 = 7.25 J , h - 660psec 44 4.2 Plasma density contours for E c o 2 — 6.0 J 45 4.2 continued 46 4.2 continued 47 vi i 4.3 Plasma density contours showing the late emergence of a second plasma island for E C o 2 = 4.75 J 49 4.4 Plasma density contours for EQO2 — 7.75 J 51 4.4 continued 52 4.4 continued 53 4.5 (a) The number of electrons and (b) the plasma volume of a 6.0 J plasma. 54 4.6 (a) The number of electrons and (b) the plasma volume of a 7.75 J plasma. 54 4.7 The electron density distribution for a 7.75 J plasma 55 4.8 Scale lengths at the quarter critical boundary 56 4.9 Expansion of the quarter critical boundary 58 4.10 Density profiles along the central axis of the plasma for EcOn = 6.0 J 59 4.10 continued 60 5.1 The experimental setup for detection of electrons with photographic film. . 62 5.2 Electron images obtained with a nitrogen target ( E c o 2 = 4.5 J) 63 5.3 Electron images obtained with a nitrogen target (Eco-> = 6.0 J) 64 5.4 Electron images obtained with a nitrogen target ( E c o 2 = 7.0 J) 65 5.5 Electron images obtained with a nitrogen target (Eco-> = 9.0 J) 66 5.6 Electron images obtained with a helium target ( E c o 2 — 5-0 J) 68 5.7 Electron images obtained with a helium target (EQO2 = 6.0 J) 69 5.8 Electron images obtained with a helium target ( E c o 2 — 7.0 J) 70 5.9 Electron images obtained with a helium target ( E c o 2 = 8.5 J) 71 5.10 Electron images obtained with the plane of polarization rotated 60° , a helium target and E c o 2 — 6.0 J 72 5.11 Density contours of the negative shown in Figure 5.6 ( E c o 2 = 5.0 J) 74 5.12 Density contours of the negative shown in Figure 5.7 ( E c o 2 = 6.0 J) 75 5.13 Density contours of the negative shown in Figure 5.8 ( E C o 2 — 7.0 J) 76 5.14 The energy spectrum of hot electrons from a 4.5 J nitrogen plasma 77 v i i i 5.15 The linear relation between In (E~l!2dN/dE) and laser energy verifying a 3-D Maxwel l ian distr ibution 78 5.16 The hot electron temperature as a function of laser energy for a nitrogen target (6 = 4 5 ° ,

is the wave frequency, uip is the plasma frequency, k is the amplitude of the wave vector k, c is the speed of light, fcf, is Boltzman's constant, Z is the atomic number of the plasma material, Te and Tt are the electron and ion temperatures, and m , is the ion mass. In addition to the dispersion relations, the incident and daughter waves must satisfy the frequency and wave vector matching conditions given by u>o ~ u>x ± uj2 (2-4) and k0c?k1±k2 (2-5) where the OJ0 and k0 refer to the incident E M wave while u ; 1 ) 2 and kit2 refer to the daughter waves. These matching conditions arise by assuming that the inter-action between the waves XQ, X\\, and X 2 is given by a set of nonlinearly coupled Chapter II - The Generation of Hot Electrons 7 equat ions: 1 5 f d2 d d2 1 I ~W + dt + ~ C 2 ^ 2 f X 2 K 0 = A 2 X ! ( I , t) X 0 (X, *) (2-6b) where 7X 2 are damping rates, c l j 2 are phase velocities, A i ] 2 are coupling constants, and u>i j 2 are natural frequencies of the unperturbed waves. It can be s h o w n 1 0 that the dr iving forces of the right hand side of (2-6) can excite waves wi th modes Xi(u!,k) and X2( 0. 2. The Two-Plasmon Decay Instability. The T P D instability is an anomalous absorption mechanism which involves an incident E M wave decaying into two plasma waves in the quarter-critical density Chapter II - The Generation of Hot Electrons 8 region (where the local plasma frequency is half that of the E M wave). Nonlinear Landau damping provides the mechanism whereby electrons are accelerated by the E P W ' s and it is the detection of these electrons that indicates the occurence of T P D . T P D as a parametric instability was first discussed by S i l i n , 1 6 G o l d m a n , 1 7 ' 1 8 and J a c k s o n 1 9 wi th subsequent revisions by Lee and K a w , 2 0 L i u and Rosenblu th , 2 1 Lasinski and L a n g d o n , 2 2 and Simon et al.23 Current t heo ry 2 2 holds that T P D along wi th SRS are two branches of the same parametric process where Wave 2 is longitudinal for both processes while Wave 1 may be either longitudinal ( T P D ) , transverse (SRS), or a combination of the two. Wi thout making any assumption as to the polarization of Wave 1, linear analysis yields the vector equation for Ei: (wj - w 2 - 3k%v2) (u2 - w 2 ) Ei - 3u2fc(fci • Ex) + c2kx x ( £ i x Ei) 1 4u>iu;2 (T - \\ , W1 -T2 (kl • v o ) + — V 0 K>2 w 2 k2{v0 -Ex) + —{ki-v0){k1-E1) (2-9) For a non-tr ivial solution for E\\ the secular equation 4 (w 3k2v2) k2 (ki x v0): k2 [u\\ c2k2) (k2-k2)2 (k.-Vo)2 k2 k2 (u, 2 3k2v2) (SRS) ( T P D ) (2-10) must be satisfied. If Wave 1 is longitudinal only the last term in (2-10) is retained and the dispersion relation for the T P D instability is recovered: Chapter II - The Generation of Hot Electrons 9 The E P W ' s produced by T P D can be characterized by growth factors given where A and A0 are the amplitudes of the E P W and the init ial (thermal) wave. J a c k s o n 1 9 shows that two sets of E P W ' s are generated wi th a growth factor pro-portional to [sin(0) cos(0)] where 6 is the angle between the E P W and ka. Each set contains two E P W ' s in the plane of polarization, travelling in opposite directions to each other, perpendicular to the other set, wi th maximum growth at 45° to k0. More recently Lasinski and L a n g d o n 2 2 derive a threshold intensity given by by A = A0t* (2-12) (2-13) and, for k0 = z and E0 — y, a growth factor of (2-14) for which growth is maximized at (2-15) where ky is the component of the E P W along E0. Simon et al.23 characterize the decay process by two parameters 4fc 0 |v 0 (2-16) and . 9v4k2 Chapter II - The Generation of Hot Electrons 10 and, in the limit of small 0 applicable for low temperature, high intensity, long wavelength interactions, give a threshold value of k0L > 3.094 (2-18) which, in practical units is given as > 61.25 (2-19) where L M is the scale length given by - l (2-20) in microns, A M is the wavelength in microns, Tjtev is the electron temperature in keV, and I14 is the intensity in units of 10 1 4 W / c m 2 . A value for (ky/k0) is derived as Figure 2.1 shows the two-dimensional wave vector diagram for T P D with the loci of maximum u (for A;2 — kx[kx — ka)). For the threshold intensity, the angle between kt and ka is ~ ± 3 4 ° while that between k2 and k0 is ~ 1 8 0 ± 5 6 ° . For higher intensities the angles approach ± 4 5 ° and 1 8 0 ± 4 5 ° , respectively. In practice, strong Landau damping inhibits the growth rates of EPW's for ky/ka > ~ 3.5. Hence, for (2-21) For typical CO2 laser-plasma interactions, = ~300, TkeV = ~0.3, and = 10.591, (2-19) gives a threshold of ~ 0.6 x 10 1 2 W / c m 2 for which (2-22) Chapter II - The Generation of Hot Electrons 11 \\ / -iV. 5 6 ° \\ / *2J/ --Ik. \\ Figure 2.1 — The two-dimensional wave vector diagram for T P D . detectable EPW's, the angles vary from ~ 34° to ~ 40° and from ~ 180 ± 56° to ~ 180 ± 50° . Figure 2.2 shows the range of the magnitudes of kyjk0 as a function of peak laser intensity. Curve (a) represents ( f c y / r c 0 ) m a z derived from (2-21) while curve (b) is the lower bound of the threshold intensity given by (2-22). The shaded area signifies the region where strong Landau damping inhibits the growth of the EPW's. The nature of the incident laser pulse (a 1.2nsec rising front to the peak intensity followed by a 2.8 nsec fall) and the relatively short time scales of T P D , as discussed below, imply that EPW's could be produced at times with the laser intensity equal to the threshold up to the peak intensity. Chapter II - The Generation of Hot Electrons 12 v- I I I I I I I I I f 1 1 1 I I I I 11 • • I .001 .01 .1 1 PEAK LASER INTENSITY (x10UW/cm2) Figure 2.2 — The magnitude of {ky/k0)max as a function of (a) peak laser intensity and (b) the threshold intensity. The shaded region represents values of (ky/k0)max which are heavily damped due to Landau damping. Computational and experimental evidence show that the EPW's generated by T P D do not continue to grow at the rates given by (2 -14) . 2 4 - 2 8 Various mechanisms have been proposed to account for the saturation, quenching and re-occurrence of the EPW's. Of the proposed mechanisms, pump depletion is unimportant for the high intensities available and nonlinear Landau damping cannot account for the low saturation levels observed.28 Hence, the main saturation mechanisms applicable to Chapter II - The Generation of Hot Electrons 13 T P D are those which arise due to pondermotive effects: profile steepening and coupling of electrostatic waves to shorter wavelength ion fluctuations. Pondermotive effects arise due to the interaction of E P W ' s given b y 2 5 fP = - • V {E\\ + El + E\\E2 exp [t (ky - k2) x - i {u} - w2) t] + cc.} . V m w p / (2-23) The first two terms on the right hand side of (2-23) force particles out of the resonant density region resulting in a steepening of the density contours (profile steepening). The last two terms are rapidly varying compared to the time scales of the first two terms, however, wi th W] ~ u>2 these terms become low frequency wi th spatial periodicity of fci - k2 ~ 2k (2-24) since kx ~ — k2. This low frequency allows the pondermotive force to drive ion perturbations wi th fcton, ~ 2k which couple wi th E P W ' s of longer k which are heavily damped. Ion fluctuation coupling occurs wi thin ~50psec after the onset of T P D and is thought to be the main mechanism by which the E P W ' s are saturated. Profile steepening, occuring on longer time scales, eventually quenches T P D for periods of ~120-240psec unti l the density profile relaxes and T P D reoccurs. 3. Nonlinear Landau Damping - Electron Trapping. The presence of electrons characterizing E P W s can be attributed to non-linear Landau d a m p i n g . 2 9 ' 3 0 Particles travelling along wi th the wave that have velocities nearly equal to the phase velocity of the wave wi l l interact wi th the wave. Particles travelling slightly slower than the wave wi l l be accelerated to the phase velocity resulting in a transfer of energy from the wave to the particles. Similarly, particles travelling slightly faster than the wave wi l l be decelerated to the phase velocity resulting in a transfer of energy from the particles to the wave. However, Chapter II - The Generation of Hot Electrons 14 for the Maxwel l ian d i s t r i b u t i o n 3 1 - 3 3 that the electrons have been shown to exhibit , there are more particles traveling at slower velocities than faster. Hence, the net energy transfer is from the wave to the particles and the wave is damped. For low amplitude E P W s linear Landau damping suffices to damp the wave and the electron velocity distr ibution remains essentially unchanged. Conversely, for large amplitude E P W s nonlinear effects become important as electrons become trapped in the wave potential and the electron velocity distribution is significantly altered. Consider the potential of an E P W •[z,t) = 0 cos (ut - kz) (2-25) and an electron wi th velocity v'. In phase space kx = kz-ut, (2-26) the potential is given by 4>{x) = -0cos (kx), (2-27) and the velocity of the electron is given by v — v' — v^. Trapping of the electron wi l l occur when its energy in the wave frame is less than the wave potential. Conservation of energy gives -mv2 - e cos (kx) =W (2-29) and it is clear that an electron wi th energy in the range (2-28) -e

cos(fcx) + W}1/2 (2-31) yields the phase space trajectories shown in Figure 2.3. Whi le untrapped electrons decelerate and accelerate when passing over the potential (without reversing direc-t ion) , electrons in the shaded region (trapped electrons) oscillate in the troughs of the wave in phase space and are carried along wi th the wave at the phase velocity in the lab frame. Chapter II - The Generation of Hot Electrons 16 Figure 2.3 — a) The phase space trajectories of electrons in the b) potential of an E P W . 17 C H A P T E R III EXPERIMENTAL APPARATUS The measurements reported in this thesis utilize five basic apparatus to in-vestigate the hot electrons generated by a CO2 laser produced plasma: l) the CO2 laser which produces and then interacts with the plasma; 2) the gas jet target where the plasma is formed; 3) a multi-frame Mach Zehnder interferometer which charac-terizes the plasma; and 4) the electron spectrometer and 5) the film holder which are used to analyze the electrons. 1. The C 0 2 Laser. A CO2 laser developed over the years 1 3 was chosen to investigate the paramet-ric effects evident when an intense laser beam interacts with a plasma. As discussed in §2.1, the ratio of the electron quiver velocity to the thermal velocity v0/ve, which is proportional to IX2,, indicates the level of parametric instabilities that result in a laser plasma interaction. The high intensities and long wavelengths available with CO2 lasers provide the ideal parameters to drive the instabilities. Figure 3.1 shows the laser used. The hybrid oscillator consists of a folded continuous wave section and a high pressure pulsed section. The lOOnsec gain switched pulse produced by the hybrid oscillator is then linearly polarized before Figure 3.1 — The C 0 2 laser. Chapter III - Experimental Apparatus 19 passing through a short pulse generator which switches out a 2nsec pulse at the peak of the 100 nsec pulse (the short duration of the pulse minimizes damage to the optics in the remainder of the system). The 2 nsec pulse is then passed through an amplifier chain before delivering up to 10 Joules to the target. a. The H y b r i d Oscillator. Temporally smooth Gaussian pulse shapes are desirable for laser produced plasma experiments. Mode beating, which causes severe temporal oscillations, can be eliminated with a single longitudinal, single transverse mode pulse. Hence, the hybrid oscillator (Figure 3.2), which produces pulses of this type, is used. In the continuous wave section, the single transverse mode is attained simply due to the large length to diameter ratio. The continuous wave section runs on a single longitudinal mode since the medium gain profile is homogeneously broadened. A temperature controlled germanium flat etalon at the exit of the continuous wave section controls the rotational line. The single -longitudinal, single transverse mode continuous wave beam then mode locks the pulse from the high pressure pulse section and the output of the oscillator is then effectively a single longitudinal, single transverse pulse. At room temperature, CO2 lasers achieve greatest gain operating at the P(20) transition of the 10.6/xm band (A = 10.591 firn, E = 0.117 eV). The continuous wave section, operating at low pressure (26 Torr), at P(20) produces ~ 3 W with a mix of (He:C0 2 :N 2 ) = (15:15:70)%. The output of the high pressure (latm) pulsed section is 1/4 - 1/2 J in a 100 nsec (FWHM) pulse with a mix of (He:C0 2 :N 2 ) = (76:13:11) % flowing at a rate of 2.61/min. b. Short Pulse Generation. The 100 nsec pulse produced by the hybrid oscillator would, if passed through the amplifier chain, heat the plasma and cause severe damage to the optics in the remainder of the system. To prevent this a Pockels cell is used to switch out a 2 nsec pulse which is then fed to the amplifier Chapter III - Experimental Apparatus 20 to s p e c t r u m a n a l y z e r spatial_ filter F1 F2 F 3 P o c k e l s cel l § JL c o •c to 2: (A Of L_ o 1[ r4m focal length mirror c o • MS u I/) a* V) (A Q_ J C c n to a m p l i f i e r --, c h a i n S h o r f P u l s e G e n e r a t o r r - 8 - 8 o u E u H y b r i d Oscillator Figure 3.2 — The hybrid oscillator and short pulse generator. Chapter III - Experimental Apparatus 21. chain. The cell is made from a GaAs crystal which, when no voltage is applied to the crystal, transmits a linearly polarized signal with little significant change to the signal. However, when voltage is applied to the crystal, a birefringence is produced which alters the polarization of the pulse from linear to elliptical. Figure 3.2 shows the short pulse generater. The pulse from the hybrid oscil-lator is already partially polarized due to two intercavity KC1 Brewster windows in the continuous wave section. Three germanium flats (F l , F2, and F3) positioned at the Brewster angle ensure maximum polarization prior to the beam entering the Pockels cell. With no voltage applied, the beam is transmitted by a fourth germanium flat (F4) at the Brewster angle to an Optical Engineering CO2 Spec-trum Analyzer. With applied voltage, the elliptically polarized beam is partially reflected by F4 and passed to the amplifier chain. A 2 nsec high voltage pulse (28 kV) applied to the Pockels cell near the peak of the 100 nsec pulse results in a symmetric, 2 nsec (FWHM), vertically polarized output pulse with power ~70kW and energy ~0.15mJ reflected off F 4 . The polarization of the pulse is then flipped to the horizontal plane by mirrors M4 and M5 before it is passed to the amplifier chain. c. The Amplif ier C h a i n . The 2 nsec pulse output by the short pulse generator is passed through three amplifiers prior to impacting on the target. Each amplifier operates in the saturated regime and, since only 60% of the available energy is absorbed in a single pass, each amplifier is double passed. The first amplifier is a Lumonics model K103 preamplifier (operating with the same gas mix as the high pressure section of the hybrid oscillator) which has a measured gain of 280 over an active length of 282 cm (~2 %/cm). The second amplifier is a home built three-stage amplifier (operating at 1 atm with a gas mix of (He:C02:N2) = (63:25:12) % flowing at a rate of 8.51/min) which has a gain of ~200 over an active length of 180 cm (~3%/cm). The final amplifier is a Lumonics model TEA600A Chapter III - Experimental Apparatus 22 amplifier (operating with the same gas mix as the three-stage) which has a gain of ~20 over an active length of 100cm (~3%/cm). Typical output energies for an input pulse of 0.15 mJ into the chain are 0.042 J from the K103, 4.0 J from the three-stage, and 10.4 J in a 7.0 cm diameter beam from the TEA600A amplifier. d. Backscattered, Leakthrough, and Self-Lasing Energies. Since up to 10 % of the laser energy can be backscattered and amplified on the return trip, precautions must be taken to prevent optical damage. Two spatial filters and a sharp focus between lenses L i and L 2 effectively absorb the reflected beam. The high intensities of the beam at the sharp focus break down the surrounding atmosphere dissipating the energy in an air spark. Two absorption cells are used in the system to counter the effects of self-lasing and to enhance the contrast ratio of the Pockels cell. The gases used are freon-502 and ethanol which are strong absorbers in the 9-10.3/mi region, and SF6 which absorbs weak signals at 10.6 nm but bleaches for strong signals. Hence the strong main pulse is only weakly attenuated by the bleached SF6 while other signals are strongly absorbed. Helium is added to each cell to aid in the recovery of the bleached S F 6 . A small portion of the linaerly polarized lOOnsec pulse leaks through the short pulse generator when no voltage is applied to the Pockels cell. Despite the low power of this'leakthrough (~0.5 % of the 2nsec main pulse), the time scale is considerably longer and, hence, the energy is comparable to that of the main pulse (~25 %). Cell C j , containing ~12torr SFQ and ~748torr helium, dramatically reduces the level of leakthrough to acceptable levels through absorption of the low power leakthrough by the S F 6 . Parasitic oscillations arising due to random noise in the amplifier chain lead to amplifier self-lasing. Cell C 2 , located at the rear mirror of the three-stage, filled with ~1.2 torr S F 6 , ~28torr ethanol, ~80torr freon-502, and ~651torr helium, Chapter III - Experimental Apparatus 23 effectively eliminates this problem. The oscillations in the 9-10.3 /zm band are absorbed by the freon-502 and the ethanol while those at 10.6 /xm are absorbed by the SFe before they grow to appreciable strength. e. Laser Output. Typically, output pulses of 0-10 J in a 7.0 cm diameter beam at 10.591 with a roughly triangular pulse shape are observed. The rapid rise time of 1.2 nsec compared to a fall time of 2.8 nsec is attributed to the effects of the SF6 in the absorption cells and to the saturation effects in the amplifiers. These pulses focused to intensities of up to 10 1 4 W / c m 2 easily provide sufficient intensities to initialize non-linear parametric processes. 2. The Gas Jet Target. Investigations of interactions occuring in an underdense plasma are easily accomplished with gas jet targets. The long scale lengths, the precise control over densities and the low bulk plasma movement makes the choice of a laminar jet flowing out of a planar Laval nozzle advantageous. Both helium and nitrogen were chosen as target gases flowing into a low pressure background helium gas. The Laval nozzle, shown in Figure 3.3, has a fixed 70 firn throat and a mouth (controlled with stainless steal jaws) of 1.2 mm. The target gas is stored in a high pressure reservoir until a solenoid valve is triggered. The pressure change in the nozzle is detected by a piezo detector which sends a signal to fire the laser after a suitable delay to allow the jet to stabilize (> 10 msec). The maximum density reached in the gas target is controlled by the pressure in the reservoir. Given the basic assumptions of 1) constant mass flow rate in the nozzle, 2) isentropic and adiabatic flow in a perfect gas, and 3) the first law of thermodynamics, it can be shown 3 4 ' 3 5 that the ratio of pressure at any point in the QPter /// H 1cm H Jet Chapter III - Experimental Apparatus 25 nozzle p to the pressure in the reservoir p0 is related to the local Mach number M by where 7 is the ratio of specific heats, and that the cross-sectional area a is given by m / p \\ - ( T + i ) / 2 7 My/lP0p0 \\p0/ where m is the mass flow rate. The Laval nozzle is designed such that the Mach number at the throat equals one. Hence, the Mach number at the exit M e can be determined from the ratio of cross sectional areas at the throat and exit ' O g N 2 1 7 + ( 7 + l ) / ( 7 - l ) (3-3) Once Me is determined, the design pressure ratio p0/Pe c a n be found from equation (3-1). The nozzle used has a throat to exit area ratio of 17 and a design pressure ratio of 648 for helium (7 = 5/3) and of 285 for nitrogen (7 = 7/5). Assuming complete ionization, a density of 1.17 nc would be reached with a nitrogen jet in a 5 torr background while a density of 0.537 nc would be reached with a helium jet in an 8 torr background. A maximum of 100 psi imposed by the limits of the piezo detector did not allow higher densities to be reached for the helium jet. Interferometric results, reported by others 1 3 , 1 4 and in Chapter IV, show that densities in the range 0.4 nc < n < 0.5 nc are generally achieved with both gases. The fact that the observed densities are lower than those predicted is attributed to plasma expansion. This expansion has less effect on the helium jet due to the fact that, despite the higher ionization potential of helium compared to nitrogen, helium ionizes at a Chapter HI Experimental Apparatus 26 faster rate due to a cascade process resulting from the collisions between the helium and energetic electrons. 3 0 The nozzle is enclosed in the target chamber shown in Figure 3.4. The chamber is a 60 cm diameter, 38 cm long cylinder standing on end with a 20cm extension holding a f /5, 50cm KC1 lens which focuses the 7.0cm beam down to 50/ im at the gas jet thus attaining intensities up to l ' O 1 4 W / c m 2 . Three large (10cm) and 16 smaller (5 cm) ports arranged symmetrically about the cylinder allow access for optical diagnostics. Automatic cont ro l s 3 7 regulate the evacuation and filling of the chamber and the high pressure reservoir. 3. The Mul t i -Frame Interferometer A thorough understanding of the plasma is necessary in order to explain the processes occurring in laser-plasma interactions. Interferometers are simple tools which can measure scale lengths, densities, expansion rates and temperatures of a plasma. Knowledge of these parameters are essential in the application of the theory presented in §2.2. The use of multi-frame interferometry provides the advantages that the temporal separation between the frames is accurately known to wi th in 1% and that the four frames follow the history of a single laser-plasma interaction. a. I n t e r f e r o m e t r i c Theory. Interferometric techniques are based on the phenomenon of interference fringes which result when two monochromatic, coherent beams are superimposed after travelling paths of different optical lengths . 3 8 The relative phase shift between the two beams gives rise to an interference term in the intensity of the superimposed beams which, in turn, accounts for the interference fringes observed. The presence of any localized variation of the optical index of refraction wil l modify the phase shift of the section of the scene beam passing through the variation relative to the rest of the beam. This phase shift is evident as Figure 3.4 — The target chamber. Chapter HI - Experimental Apparatus 28 a shift in the fringes of the interference pattern. This shift / is easily related to the difference between the actual length of the variation L and the optical path length: L/2 (3-4) -L/2 where A is the wavelength of the ruby beam in vacuum and p is the index of refraction given by 1 0 where ne is the electron density and nc is the critical density. Hence, the plasma density can be determined from measurements of the fringe shifts obtained. Since we are dealing with densities below critical density, p will be less than one and, hence, the fringe shifts will be negative — the fringes will shift towards the incoming CO2 laser. The fringe shift given by (3-4) is the result of the integrated sum of infinitesi-mal density variations along the path lengths perpendicular to the CO2 laser beam. Since the plasma density in the path is not constant through the plasma, the fringe shifts must be unfolded. If cylindrical symmetry is assumed, (3-4) can be trans-formed to cylindrical coordinates and integrated over the half diameter 1/2 (3-5) or (3-6) (3-7) Chapter III - Experimental Apparatus Abel inversion3 9 of (3-7) gives 29 M - l = - (3-8) or, by (3-6) 2 ne = nc ~ nc 1 — (3-9) b. The Mult i -Frame Interferometer. A Mach-Zehnder interferometer40 (Figure 3.5) was chosen due to the ability to attain large separations between the scene and reference beams. This allowed for large plasma expansions to take place with out interaction of the plasma with the reference beam. The strength of any E M wave plasma interaction is dependent on the ratio between the plasma frequency and the wave frequency. Hence, a 50psec, Q-switched, mode-locked, cavity dumped ruby laser 4 1 (wavelength 694.3 nm) was chosen as the probe beam. The interference fringes were recorded on polaroid film which allowed instant data analysis. The frame delay mechanism, which provides the multi-frame attributes of the interferometer, consists of an uncoated BK7A crown glass flat (/x = 1.50) which has a wedge angle e = 30 ± 5 min oriented with its narrowest part up. Figure 3.6 shows the top view of the wedge where the wedge angle is directed down into the page. Four spatially and temporally separated beams arising from reflections at the surfaces of the wedge are utilized while other reflections are spatially filtered out of the apparatus by the neutral density filter holder in Figure 3.5. Beams 1 and 2 are temporally separated by a difference in optical path length of 2/xa -c = 2t {u2 - sin 2 B) 1/2 = 2.85 cm (3-10) Chapter III - Experimental Apparatus 30 50 ps ft Ruby v Laser Pulse ,Frame#1 Monitor Frame Delay Mechanism L_ Final Image co 2 Laser \" 1 » I 3 »* 2 A *• 6.4 X Overall Magnification Figure 3.5 — The multi-frame interferometer. Chapter III - Experimental Apparatus Ruby Probe ure 3.6 — Top view of the frame delay mechanism. Chapter III - Experimental Apparatus 32 or a temporal delay of 95psec for t = 12.45 mm, 6 — 74.7°. Similarly beams 3 and 4 are also separated by 95psec. These calculations assume the thickness of the wedge is constant over the paths travelled. In fact, the paths in Figure 3.5 are not in the plane of the page. However, with a wedge angle of 30 min, the difference in path lengths is less than 0.002% and is neglected. The difference in path lengths from the wedge to mirrors M l and M2 allow temporal adjustments to be made for the two sets of beams such that the time delay between all four beams is 95psec. Spatial separation of the beams is easily achieved with the wedge. Figure 3.7 shows the edge on view of the wedge where all four beams are directed out of the page. For small t and a, ct = fi8 7 = fi (2c + 8) (3-H) 6 = n{e + 3) and the angular separation between the two sets of beams on each side of the wedge is given by -)-a = — 6.0 J where Figures 4.2a and 4.2b are frames 3 and 4 of one shot while Figures 4.2c through 4.2f and Figures 4.2g through 4.2j are two sets of four frames from two other shots. The Chapter IV - The Spatial and Temporal Evolution of the plasma 44 Figure 4.1 — The series of interferograms obtained from a single shot where Eco2 — 7.25 J , t\\ = 660psec. CO2 laser is directed from the origin along the z-axis and the solid lines represent density steps of 0.1 nc while the dashed line is the quarter-critical density boundary. The earliest observed fringe shifts result in the formation of a plasma island at the front of the jet at t = 50psec as shown in Figure 4.2a. The lack of any fringe shifts in the previous interferogram frame and the correlation between the cross sectional areas of the island and the incident laser beam indicate this is the initial formation of the plasma after ~ 0.6 J of temporally integrated energy has been delivered to the target. The island expands and increases in density until t — 1070 psec in Figure 4.2g whence the rate of expansion is greater than the Chapter IV - The Spatial and Temporal Evolution of the plasma 45 O o. CD o o CM • . O o o. — = 2.5X10 1 8 cm' 3 a) t = 50psec i i i i - » — i — i i i i « i i i i t i I i i GAS JET b) t = 145 psec i i i i i i i i \" i | 1 \"i i i i i i i i i | i > CD O c) t = 360 psec CO d-X10 1 8cm\" 3 d) t = 455 psec 1.0 Z (mm) Figure 4.2 — Plasma density contours for Eco2 — 6.0 J . Chapter IV - The Spatial and Temporal Evolution of the plasma 46 — =2.5X1018cm3 e) t = 550 psec Z (mm) Figure 4.2 — Continued. Chapter IV - The Spatial and Temporal Evolution of the plasma 47 — =2.5X1018cm\"3 cs d-co d ' GAS JET i) t = 1260 psec E E ^ rr o CM CO j) t = 1355psec X10 1 8 cm' 3 Z (mm) Figure 4.2 — Continued. Chapter IV - The Spatial and Temporal Evolution of the plasma 48 density increase. As a result, the density profile assumes a doughnut like shape where the greatest densities lie off the central axis of the laser beam. This high density profile typifies shock wave formation described by blast wave theory with heat transfer. 3 7' 5 1 Also evident in Figure 4.2g is the emergence of a second island near the rear of the jet. The late emergence of this island is attributed to refraction, reflection, and absorption of the early part of the pulse by the front island. 1 3 This prevents the build up of sufficient energy to break down the plasma at the center of the jet until this later time. In the preceding paragraph it was stated that a second island emerged at a later time relative to the first island. Since Figures 4.2f and 4.2g are frames from two different shots it may be argued that this second island did not emerge late but rather it is present at all times for some shots and is absent in other shots even at later times. Figure 4.3 refutes this argument showing four frames of a single shot where the second island is absent in the first frame then emerges and grows in the last three frames. Figure 4.4 outlines the evolution of the plasma through the later stages of the 2nsec C 0 2 pulse for energies of ~ 7.75 J. Figures 4.4a through 4.4d and 4.4e through 4.4h are two sets of four frames from two separate shots while Figures 4.4i and 4.4j are frames 1 and 3 of a single shot and Figures 4.4k and 4.41 are frames 2 and 4 of still another shot. Figure 4.4a shows that the second island near the rear of the jet forms much earlier compared to its emergence in Figure 4.2g simply because there is more energy available at t — 660 psec from a 7.75 J shot than a 6.0 J shot. Both the front and rear islands then expand and increase in density until they assume the same doughnut like shape exhibited in Figure 4.2. At later times (> ~ 1400psec) the majority of the energy from the laser pulse has been delivered to the target and the main features of the denser regions of the plasma remain relatively unchanged. However, diffusive effects greatly affect the lower density regions as the 0.1 nc boundary of the Chapter IV - The Spatial and Temporal Evolution of the plasma 49 Z (mm) Figure 4-3 — Plasma density contours showing the late emergence of a second plasma island for Eco2 = 4.75 J . Chapter IV - The Spatial and Temporal Evolution of the plasma 50 . two islands merges and expands well beyond the region of the jet. This expansion is especially noticable at the front of the jet in Figures 4.4i through 4.41. 2. Plasma Density Distribution. The study of T P D requires the presence of densities above 0.25 nc. Consider-ations in §3.2 showed that, without allowing for expansion of the plasma, densities up to 0.537 nc could be achieved wi th the hel ium jet in an 8 torr background. The dashed line representing the 0.25 nc boundary is clearly evident for all the den-sity contours throughout Figures 4.2, 4.3 and 4.4, thus ensuring the proper ini t ial condit ion for T P D is met. Figures 4.5a and 4.6a show the total number of free electrons and the number of electrons enclosed by the quarter critical boundary for the two series of density contours shown in Figures 4.2 and 4.4. Clearly evident is that, while the total number of electrons increases linearly over the time intervals shown, the number of electrons bounded by the quarter crit ical boundary reaches a maximum value at t > ~ 0.6nsec in Figure 4.5a (Eco2 = 6.0 J) and at i > ~ 1.0nsec in Figure 4.6a ( £ c o 2 = 7.75 J ) . Figures 4.5b and 4.6b show the volume enclosed by the 0.1 nc and 0.25 nc boundaries for the two series. The two volumes in Figure 4.5b are expanding at rates of 1.1 m 3 / s e c and 0.16m 3 / sec for the 0.1 nc and 0.25 n c boundaries, respectively. For the higher energies the boundaries expand at rates of 1.7m 3 /sec and 0.52m 3 / sec . The fact that the number of electrons enclosed by the quarter cri t ical boundary is constant at later times indicates a shift in the electron density distribution. Indeed this is evident in Figure 4.7 which shows a series of historgrams for the 7.75 J series and a shift in the distribution from higher to lower densities. 3. Scale Lengths and Radial Expansion Rates. The scale length L is easily derived from (2-20) and the density contours of §4 .1 . Figure 4.8, which shows the scale length determined along the quarter Chapter IV - The Spatial and Temporal Evolution of the plasma 51 -—=2.5 X10 1 8cm' 3 a) t = 660 psec b) t = 755 psec | ^ GAS JET • Z (mm) Figure 4.4 — Plasma density contours for Eco2 — 7.75 J. Chapter IV - The Spatial and Temporal Evolution of the plasma 52 e) t = 1070psec =2.5 X 1018 CfTf 3 f) t = 1165 psec W GAS JET • ) Z (mm) Figure 4.4 — Continued. Chapter IV - The Spatial and Temporal Evolution of the plasma 53 : i)t = i 8 o o P s e c —=2.5X1018crrf3. k) t = 2195 psec Z (mm) Figure 4.4 — Continued. Chapter IV - The Spatial and Temporal Evolution of the plasma 54 X CO X: O: 0.10 nc boundary 0.25 ne boundary — i 1 1— 0.6 1.2 T I M E (nsec) Q ro E E k: 0.10nc boundary o: 0.25 nc boundary 06 , 1.2 T I M E (nsec) Figure 4 .5 — The number of electrons and the volume of a 6.0 J plasma. x: o: OO to s ° \" I— o UJ 1 0.10ne boundary 0.25 ne boundary LU rsi O O O 1 1 1 1 — 0.0 1.0 2J0 T I M E (nsec) 3 o ro ro E E - J • Q J O O x: 0.10ne boundary o: 0.25 n c boundary x x X 1 x / */ V x o 0.0 1.0 2.0 T I M E (nsec) Figure 4 . 6 — The number of electrons and the volume of a 7.75 J plasma. Chapter IV - The Spatial and Temporal Evolution of the plasma 55 o in 3 § t=660psec £ ° 0 . 0 if) CO o o LU LU vp o o CM O O O t=1990psec 0 0 0.2 m o eg O t = 1070 psec 0.4 0.6 0D 0.2 DENSITY (X1019cm\"3) 0.4 0.6 Figure 4.7 — The electron density distribution for a 7.75 J plasma. Chapter IV - The Spatial and Temporal Evolution of the plasma E E - l LU —I < O to • X 6.0J • 7.75 J o.o } -I— 0.5 1 1.0 1.5 TIME (nsec) 1 3> i 2.0 2.5 Figure 4.8 — Scale lengths at the quarter critical boundary. Chapter IV - The Spatial and Temporal Evolution of the plasma 57 density boundaries from both series of contours in Figures 4.2 and 4.4 with the error bars representing one standard deviation of the values derived along the boundary. Indicated here is the variance in scale length over the lifetime of the laser pulse, from < ~50 um up to ~750 /xm. For a cylindrically symmetric plasma it is more appropriate to refer to radial expansions rather than the volume expansion of the previous section. Figure 4.9 shows the expansion rate of the quarter critical boundary in the front island of the two series of contours. This boundary approaches a constant expansion rate of 1.9 x 106m/sec for times greater than ~525 and ~900psec for the 6.0 and 7.75 J plasmas, respectively. 4. The Laser Plasma Interaction Region. Sections 4.2 and 4.3 discuss the properties of the plasma as a whole. However, despite the scattering effects of the cross-sectional area of the laser beam by the plasma, the effect of plasma expansion carries the bulk of the plasma outside the laser plasma interaction region. Hence, the volume of the plasma along the central axis requires a closer inspection. Figure 4.10 shows the density profile of the plasma along the central axis for a single laser plama interaction with t = 1070 psec to t = 1355 psec and Eco2 = 6-0 J. Of special interest here is the effect of profile steepening evident in Figures 4.10b through 4.10d. The relatively gentle density gradient across the front island of Figure 4.10b changes dramatically as the density increases past the 0.25 nc level in Figure 4.10c until Figure 4.10d where the density gradient has attained a much steeper profile. Also evident in Figure 4.10a are the presence of peaks and valleys showing that the plasma density does not increase uniformly along the axis. Chapter IV - The Spatial and Temporal Evolution of the plasma 58 Figure 4.9 — Expansion of the quarter critical boundary. Chapter IV - The Spatial and Temporal Evolution of the plasma 59 Figure 4.10 10 2.0 Z (mm) Density profiles along the central axis of the plasma for Eco2 = 6.0 J . 3.0 Chapter IV - The Spatial and Temporal Evolution of the plasma 60 Z(mm) Figure 4.10 — Continued. 61 C H A P T E R V T H E E L E C T R O N S P R O D U C E D B Y T H E T P D I N S T A B I L I T Y As discussed in Chapter 1, the main objective of this thesis is to study the elec-trons generated by the T P D instability; to examine the three dimensional spatial distribution of these electrons and to determine their energy distribution. Photo-graphic film, as discussed in §3.4b, was used to study the spatial distribution while the electron spectrometer of §3.4a was used to determine the energy distribution. 1. The Electron Images. The film holder of §3.4b was oriented over the gas jet as shown in Figure 5.1 where the wave vector of the incident laser beam is directed along the x-axis (k0 — x) and the beam is polarized in the xy-plane (Eco2 — $)• As discussed in §2.3, the TPD-produced EPW's will accelerate electrons along the direction of the wave vectors of the EPW's. Hence, the areas of the film exposed by the electrons can be related to the wave vectors of the EPW's which, as discussed in Chapter VI, can be compared to the theory of §2.2. The direction of the wave vectors of a particular wave vector k3 can be conve-niently expressed in terms of two angles 6 and u2, &i > k2 and geometric considerations show that

kj^. With ki < k j A the geometric considerations are different and 2 — UIA (6-14) and, since UJ\\ < u2, kt < k2. Geometric considerations show that tp will be maxi-mized when the angle between ki and k i A equals 90° . Hence, k2 — k\\ kjA, (6-15) Chapter VI - Discussion of Results and Conclusions 92 and tan £>max = ( 6 \" 1 7 ) The magnitude of the ion acoustic wave vector kjA contributes little to the range of values of frnax since, with v2 = kiTe/me, 5.9 x l O - 9 ^ < kIA < 2.6 x l O - 9 ^ (6-18) for 100 < Te < 500 eV and T{ — 2 eV. However, there is a wide range of values for 'Pmax because of the dependence of k\\ on the electron density ne. The magnitude of the wave vector of an E P W varies as the wave travels through an inhomogeneous plasma. With wj = — , (6-19) t 0 m e where e0 is the permittivity of free space and e and me are the electron charge and mass, for a given wave frequency and electron thermal velocity the wave vector magnitude from (6-5) is k2 = J _ ( u } 2 _ ^ \\ ( 6 _ 2 0 ) Zvj \\ e0meJ Since u;2 ~ u2 for an E P W , any small increase in ne results in a large decrease in k and the maximum density n m a x into which the E P W can propagate (the density for which u)\\ = u2) is quickly reached. Of the two frequency matching conditions (6-9) yields values of m a z < 11 .0° . Experimentally, values of