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Laser-matter interactions at sub-micron laser wavelengths Pasini, Daniel 1984

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L A S E R - M A T T E R INTERACT IONS A T SUB-MICRON LASER W A V E L E N G T H S by DANIEL PASINI B.Sc, McGill University, 1978 M.Sc. Institut National de la Recherche Scientifique, Universite du Quebec, 1980 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept this thesis as confirming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1984 © Daniel Pasini, 1984 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of vW S \ c S  The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date OcTofte^. I S - V^flH E - 6 ( 3 / 8 1 ) Abstract In recent years there has been considerable interest in the use of sub-micron wave-length lasers for target irradiation in laser fusion experiments. We present here an experi-mental investigation of laser-matter interaction in this short wavelength regime. We have irradiated planar targets with 0.532, 0.355 and 0.266 /im laser light with pulse lengths of 2 ns and at intensities ranging from IO 1 2 to 5 X 10 1 3 W/cm2. The results show that for these conditions inverse bremsstrahlung is very effective and leads to very high absorption ( ~ 95%). The X-ray spectrum also show that the absorbed energy is well thermalized and only a small number of energetic, non-thermal electrons are present. Also a very low level of scattered light was measured indicating that laser-plasma collective processes are not important. Energy transport was investigated using multilayer targets. Anomalous laser penetration depths in the target were obtained in these measurements. This has been successfully explained in a simple hydrodynamic model where Uvo-dimensional effects were taken into account. In the study of the ablation process wavelength and intensity scaling laws for the mass ablation rate and ablation pressure have been obtained. These results show the inadequacy of the present analytical theories of the laser-driven ablation process. ii Table of contents Chapter 1 Introduction 1 1 — 1 Introduction to controlled fusion research 1 1 — 2 Basic concept of laser fusion 2 1 — 3 Recent developments in laser-matter interaction studies 4 1 — 4 Present investigation 5 1 — 5 Thesis outline 6 Chapter 2 Absorption of laser light in a laser produced plasma 7 2 — 1 Inverse bremsstrahlung absorption 8 2 — 2 Resonance absorption 14 2 — 3 Absorptive and scattering instabilities 18 Chapter 3 Electron thermal transport in laser produced plasmas 22 3—1 Basic concepts 23 3 — 2 Spitzer-Harm theory of thermal conductivity: small temperature gradients 25 3 — 3 Thermal transport in laser produced plasmas: large temperature gradients 27 3 — 4 Thermal transport inhibition mechanisms 28 3 — 4 — A Self-generated magnetic field 29 3 — 4 — B Ion acoustic turbulence 29 Chapter 4 Scaling laws for the mass ablation rate and ablation pressure: one dimensional steady-state ablation theory 32 iii 4— 1 Structure of the laser induced plasma flow 32 4 — 2 One dimensional steady-state ablation theory 35 4 — 2 — A Conservation equations and model assumptions 35 4 — 2 — B Flow solutions in the plasma corona: isothermal expansion 38 4 — 2 — C Flow solutions in the conduction zone: steady state flow 41 4 — 2 — D Scaling laws 42 4 — 2 — E Discussion of model assumptions 45 Chapter 5 Experimental facility and diagnostics 48 5— 1 Neodymium-glass laser system and the experimental arrangement 48 5 — 2 Irradiation conditions 50 5 — 3 Diagnostics 56 5 — 3 — A Integrating sphere 56 5 — 3 — B Faraday cup 58 5 — 3 — C Differential calorimeter 60 5 — 3 — D X-ray crystal spectrometer 60 5 — 3 — E An in-situ X-ray film calibration technique 62 5 — 3 — F X-ray pinhole camera 74 5 — 3 — G Multichannel X-ray detector 83 5 — 4 Multilayer target fabrication 90 5 — 4 — A Fabrication of CH-glass multilayer targets 90 5 — 4 — B Fabrication of Mg-, Sn-, Ag- and Au-glass multilayer targets 91 Chapter 6 Experimental measurements of laser energy coupling to target 93 6— 1 Absorption measurements 93 iv 6 — 2 Plasma electron temperature 95 6 — 3 Laser induced plasma instabilities 102 Chapter 7 Investigation of electron energy transport using multilayer targets 106 7— 1 Experimental conditions 107 7 — 2 Experimental results 110 7 — 3 A hydrodynamic model for laser penetration depth 118 Chapter 8 Experimental scaling laws for the mass ablation rate and the ablation pressure 121 8— 1 Measurement principle and experimental set-up 121 8 — 2 — A Angular distribution of the plasma energy and velocity 124 8 — 2 — B Experimental scaling laws for the mass ablation rate and the ablation pressure 124 8 — 3 Discussion of the results 126 Chapter 9 Summary and conclusions 133 9— 1 Experimental effort 133 9 — 2 Experimental results and conclusion 133 9 — 3 New contributions 134 9 — 4 Suggestions for future work 136 Bibliography 136 Appendix 143 V List of Figures 1 — 1 Schematic of laser-driven implosion of a pellet containing deuterium-tritium fuel. 3 2 — 1 Schematic picture of electron density profile in a laser produced plasma, showing locations of major coupling processes 9 2 — 2 Percentage absorption of laser energy in an exponential plasma density profile for different laser wavelengths 13 2 — 3 Schematic representation of resonance absorption for 'p' polarized light obliquely incident on a plasma density gradient 16 2 — 4 Feedback mechanism for growth of stimulated Brillouin scattering 19 3— 1 Electron and ion distribution in the presence of an electron heat flux Q 30 4— 1 Structure of the steady-state plasma flow: density and temperature profiles 34 4 — 2 Space-time diagram for the densities associated to the shock front, the ablation surface and the absorption surface 36 4 — 3 Density profile for isothermal plasma expansion 40 5— 1 Schematic of the laser system and oscilloscope trace of the oscillator laser pulse. 49 5 — 2 Experimental set-up 51 5 — 3 Backscatter signal as function of the lens position 53 5 — 4 Intensity distribution acsross the diameter of focal spot 54 5 — 5 Time integrated two-dimensional image of the focal spot intensity distribution... 55 5 — 6 Experimental arrangement for the laser energy absorption measurement 578 5 — 7 (A) Faraday cup electrical connections. (B) Oscilloscope display of a Faraday cup signal 59 vi 5 — 8 (A) Schematic of a differential calorimeter. (B) Oscilloscope display of a differential calorimeter signal 61 5 — 9 Experimental set-up used in the X-ray film calibration technique 65 5 — 10 Silicon X-ray spectrum obtained with a three-channel step-wedge filter 66 5 — 11 Silicon spectrum transmitted through different filters 68 5 — 12 Film transmission {%) for the four silicon lines used 70 5 — 13 Relative intensity calibration curve obtained with silicon spectrum 71 5 — 14 Intensity of a group of silicon lines as a function of CH thickness 73 5 — 15 Aluminum X-ray spectrum obtained with a four-channel step-wedge filter 76 5 — 16 Film transmission (%) for each channel for the six aluminum lines used 77 5 — 17 Relative intensity calibration curve obtained with an aluminum spectrum 78 5 — 18 X-ray pinhole photograph of plasma 80 5 — 19 X-ray pinhole photograph of plasma 81 5 — 20 Series of X-ray pinhole photographs obtained by irradiating thin strip of aluminum foils of different thicknesses 82 5 — 21 (A) Schematic diagram of the electron temperature monitor. (B) Theoretical X-ray response of the Reticon array 85 5 — 22 (A) Block diagram of synchronization circuit. (B) Timing sequence of synchronization circuit 86 5 — 23 (A) Relative response of diode elements of the Reticon array. (B) Output waveform for a nine-channel electron temperature monitor 88 5 — 24 Linearity of response of the Reticon detector 89 5 — 25 Experimental arrangement used in the fabrication of CH-multilayer targets 92 6—1 Absorption fraction as a function of the laser intensity for different laser wavelengths. 94 vii 6 — 2 Relative intensity of X-ray continuum transmitted through different filters 97 6 — 3 Scaling of electron temperature as a function of laser intensity for 0.532 fim radiation. 98 6 — 4 X-ray continuum spectrum for 0.532 radiation 100 6 — 5 X-ray continuum for 0.266 /xm radiation 101 6 — 6 Brillouin reflectivity as a function of incident laser intensity for 0.532 /im radiation. 104 6 — 7 backscatter reflectivity as a function of laser intensity for 0.532 /im radiation 105 7—1 Experimental arrangement for energy transport measurement 108 7 — 2 Silicon line spectrum 109 7 — 3 Intensity of the silicon line group between 6.65 and 6.74 A as function of CH layer thickness 112 7 — 4 Intensity of the silicon line group between 6.65 and 6.74 A as function of Sn layer thickness 113 7 — 5 Intensity of the silicon line group between 6.65 and 6.74 A as function of Ag layer thickness 114 7 — 6 Intensity of the Si XIII 1 « 2 ( 1 5 0 ) — ls2p(1P1) line as a function of Mg layer thickness. 115 7 — 7 Intensity of the Si XIII ls2(150) — l «2p ( 1 F 1 ) line as a function of Au layer thickness. 116 7 — 8 Laser penetration depth as a function of target density 117 7 — 9 Schematic showing the hydrodynamic shearing of the target layer 119 8—1 Experimental arrangement to measure energy and velocity of the expanding plasma. 119 8 — 2 Energy and velocity distribution of the expanding plasma 125 viii 8 — 3 Mass ablation rate as a function of absorbed flux for 0.532, 0.355 and 0.266 fim laser light 127 8 — 4 Ablation pressure as a function of absorbed flux for 0.532, 0.355 and 0.266 / i m laser light 128 8 — 5 Wavelength dependence of mass ablation rate at absorbed irradiances of 2.6 X 1012,6.6 X 1012 and 101S W/cm 2 120 8 — 6 Wavelength dependence of ablation pressure at absorbed irradiances of 2.6 X 1012,6 X 1012 and 1013 W/cm 2 130 ix List of Tables 2—1 Partial list of the instabilities which can be driven by laser light propagating in a plasma 20 5 — 1 List of the foils used for three channel step-wedge filter 64 5 — 2 List of the silicon lines used for constructing the calibration curve and the transmis-sion factors through different channels 69 5 — 3 List of foils used for four-channel step-wedge filter 74 5 — 4 List of aluminum lines used for constructing the calibration curve and their trans-mission factors through the different channels 75 5 — 5 X-ray filters for various channels of the electron temperature monitor 87 X Acnowledgements I w i s h t o t h a n k m y s u p e r v i s o r D r . A n d r e w N g f o r p r o p o s i n g t h e e x p e r i m e n t a n d f o r h is he l p a n d g u i d a n c e t h r o u g h o u t t h i s w o r k . H i s deep i n v o l v e m e n t , d e d i c a t i o n a n d u n d e t e r r e d e n t h o u s i a s m were m o s t a p p r e c i a t e d . T h i s was a l so an o p p o r t u n i t y f o r me t o w o r k i n f r u i t f u l c o l l a b o r a t i o n w i t h o t h e r peop le . I w i s h t o expres s m y deep a p p r e c i a t i o n t o P e t e r C e l l i e r s , L u i z D a S i l v a , J oe K w a n a n d D e a n P a r f e n i u k f o r t h e i r a s s i s tance a n d f r i e n s h i p . T h e i r p resence m a d e t h e w o r k m u c h m o r e s a t i s f y i n g a n d p l ea san t . D r . S i d n e y K a s t n e r r e a d t h e m a n u s c r i p t o f t h i s thes i s as i t wa s w r i t t e n . I w a n t t o t h a n k h i m f o r h i s c o r r e c t i o n s a n d suggest ions . T h i s l ed t o severa l i n t e r e s t i n g a n d v a l u a b l e d i s cu s s i on s . T h a n k s are due t o t he m e m b e r s o f m y s u p e r v i s o r y c o m m i t t e e : D r . A . J . B a r n a r d , D r . F.L. C u r z o n , D r . D.S. B e d e r a n d D r . H .P . G u s h . It has been a p l e a su r e t o be a s soc i a ted w i t h t h e f r i e n d l y a n d s t i m u l a t i n g peop le c o m -p r i s i n g t he p l a s m a phy s i c s g r o u p a t U B C . In p a r t i c u l a r I have h a d m a n y u s e f u l l d i s cu s s i on s w i t h J o h n B e r n a r d , H u b e r t H o u t m a n a n d R o m a n P o p i l . A l s o , t he t e c h n i c a l a s s i s tance of A l l a n C h e u c k was g r e a t l y a p p r e c i a t e d . X i Chapter 1 Introduction 1-1 Introduction to controlled thermonuclear fusion For the past thirty years there has been intense research towards controlled ther-monuclear fusion. The easiest fusion reaction to initiate is: D + T — • <He(3.5 MeV) + n(14.1 MeV) where a deuterium nucleus fuses with a tritium nucleus to produce an a particle and a neutron. The energy release when such a reaction takes place is in the form of the kinetic energy of the reaction products. This energy can then be recovered in a heat cycle. The difficulty in attaining the fusion reaction is to have the deuterium and tritium nuclei collide with sufficient energy to overcome their repulsive Coulomb potential barrier. One way to achieve this is to heat up the reacting nuclei in a plasma to a temperature of approximately 108 K. As only a small fraction of the collisions between the nuclei leads to fusion events, the high temperature fusion fuel , must be held together long enough for sufficient fusion reactions to occur. Therefore to achieve thermonuclear fusion two problems must be solved: (1) produc-tion and heating of a plasma to thermonuclear temperatures, and (2) confinement of the reacting plasma to produce a positive energy yield, that is, more energy generated than expended in the heating and confinement of the fuel. These two requirements impose a condition on the plasma density n , the confinement time r and the plasma temperature T. This condition is usually expressed through the Lawson1 criterion which reflects the balance between the input energy to the plasma, the thermonuclear energy production, the plasma radiation losses and recovery of the thermonuclear and radiation energy through 1 a t h e r m a l c y c l e w i t h a n e f f i c i ency of 33 %. T h e L a w s o n c r i t e r i o n f o r t h e D - T r e a c t i o n a t T « * 108 K r equ i r e s t h a t nr > 1014 c m - 5 s f o r " b r e a k - e v e n " . T w o d i f f e ren t a p p r o a c h e s have been i n v e s t i g a t e d t o con f i ne t h e p l a s m a . In t h e m a g -ne t i c c o n f i n e m e n t s cheme, t h e a t t e m p t is t o c on f i ne t h e p l a s m a (at n ~ 1014 — 1016 c m " 8 ) f o r r e l a t i v e l y l o n g t i m e ( r ~ 0.1 — 1 s) i n a s u i t a b l y s h a p e d m a g n e t i c field. T h i s m e t h o d is o n t h e t h r e s h o l d o f a c h i e v i n g " b r e a k - e v e n " . In t h e i n e r t i a l c o n f i n e m e n t a p p r o a c h , t h e o b j e c t i v e is t o c o m p r e s s t h e f u e l t o v e r y h i g h den s i t i e s 102B c m - 5 ) a n d heat i t t o t h e r -m o n u c l e a r t e m p e r a t u r e s e x t r e m e l y r a p i d l y 100 ps) so t h a t a p p r e c i a b l e t h e r m o n u c l e a r r e a c t i o n e n e r g y w i l l be p r o d u c e d be fo re t h e f u e l d i s i n t eg r a te s b y r a p i d e x p a n s i o n . T h i s c o u l d be a c h i e v e d , i n p r i n c i p l e , b y f o cu s s i n g an i n tense la ser b e a m (or b e a m o f c h a r g e d p a r t i c l e s ) o n t o s m a l l d e u t e r i u m - t r i t i u m s p h e r i c a l pe l l e t s . T h i s a p p r o a c h w i l l now be d i s -cu sed f u r t h e r i n o r d e r t o i n t r o d u c e l a s e r - m a t t e r i n t e r a c t i o n s tud ie s d e s c r i b e d i n t h i s thes i s . 1-2 Basic concept of laser fusion T o s a t i s f y t h e L a w s o n c r i t e r i o n w i t h p r a c t i c a l l a ser ene r g y ( < 1 M J ) t h e d e u t e r i u m -t r i t i u m f ue l m u s t be c o m p r e s s e d t o 10,000 t i m e s i t s l i q u i d d e n s i t y 2 . F u e l c o m p r e s s i o n t o s u ch e x t r e m e l y h i gh den s i t i e s m a y be a c h i e v e d b y l a s e r - d r i v e n i m p l o s i o n o f s p h e r i c a l f u e l pe l l e t s as s c h e m a t i c a l l y r ep r e s en ted i n F i g u r e 1 — 1. T h e pe l l e t c o n t a i n i n g d e u t e r i u m -t r i t i u m f ue l is u n i f o r m l y i r r a d i a t e d w i t h i n tense laser l i g h t ( ~ 1018 — 1016 W / c m 2 ) . T h e laser e n e r g y is a b s o r b e d i n t he o u t e r a t m o s p h e r e o f low d e n s i t y p l a s m a , p r o d u c e d b y t h e v a p o r i z a t i o n of m a t e r i a l a t t h e pe l l e t su r face . T h e ene r g y is t h e n c a r r i e d i n w a r d b y e l e c t r o n t h e r m a l c o n d u c t i o n . T h i s hea t flux causes a b l a t i o n of t he t a r g e t m a t e r i a l a n d the r e s u l t i n g h o t , dense p l a s m a r a p i d l y e x p a n d s o u t w a r d . T h e r e c o i l m o m e n t u m genera te s a s t r o n g p re s su re wave w h i c h p r opaga te s i n w a r d s , a c c e l e r a t i n g a n d c o m p r e s s i n g t h e m a t e r i a l b e h i n d i t . C o n s e q u e n t l y , t h e s h o c k - c o m p r e s s e d o u t e r s he l l beg ins t o i m p l o d e . W h e n the i m p l o d i n g she l l s t agna te s at t h e cen te r o f t h e pe l l e t , i t s k i n e t i c ene r g y is t r a n s f o r m e d i n t o hea t , i n i t i a t i n g a t h e r m o n u c l e a r b u r n . T h e t h e r m o n u c l e a r b u r n f r o n t t h e n p r o p a g a t e s o u t w a r d , b u r n i n g t h e c o m p r e s s e d f ue l t o d e p l e t i o n before i t d i sa s sembles . U s i n g t h i s s i m p l i f i e d p i c t u r e , t h e i n t e r a c t i o n phy s i c s l e a d i n g t o t h e r m o n u c l e a r b u r n c a n be c a t e g o r i z e d a c c o r d i n g t o t h e v a r i o u s processes i n v o l v e d : 2 Ablation Ablation surface Low density plasma Laser light Compressed shell Shock wave Uncompressed shell D-T fuel Figure 1-1 Schematic of laser-driven implosion of a pellet containing deuterium-tritium fuel. 3 (i) absorption of laser energy in the low density plasma, (ii) transport of energy by electron thermal conduction from the absorption region to the target surface, (iii) compression of the target material by the ablation process, and (iv) ignition and nuclear burn. 1-3 Recent developments in laser-matter interaction studies Present lasers lack the energy required to both compress and heat the fuel mass sufficiently to produce large thermonuclear energy yields. Experiments to date3 have only produced yields approximately 3 X 10 - 6 the input laser energy. To optimize the design of fusion targets and determine the characteristics of future fusion drivers a detailed under-standing of laser-matter interaction is most vital. Until a few years ago experiments in laser fusion were done using almost exclusively 1.06 /im radiation from a Nd-glass laser or the 10.6 /im radiation from a C 0 2 laser. Two particularly challenging problems for laser fusion were identified in these experiments3. Firstly at high laser intensities (especially at 10.6 fim) a large fraction of the absorbed energy was coupled to very energetic electrons. Because of their long range for energy deposition in the target, these "hot"electrons would heat the core of the fuel pellet before it was compressed, that is, preheating the fuel. Much larger laser energy would then be needed to compress the fuel to the required density. Secondly energy transport by electron thermal conduction was found to be much less than theoretically calculated. As a result much higher laser energy than previously assumed would be necessary to achieve the required compression. Short-wavelength lasers offer several possible advantages over long-wavelength lasers2,4 At short wavelengths classical collisional absorption is more effective, and this should lead to higher absorption efficiency. Deposition of the laser energy also occurs at higher plasma densities, leading to higher mass ablation rate and ablation pressure. Furthermore most of the collective laser-plasma interaction processes which lead to scattering of the incoming laser light and which are responsible for the production of hot electrons, will be mitigated because of the higher laser intensity thresholds required at shorter wavelengths. 4 Experiments performed at Ecole Polytechnique84 (1980) were the first to demonstrate increased laser absorption, higher mass ablation rate, and reduced hot electron energies when using sub-micron wavelength laser radiation. These preliminary results sparked an increasing interest in the use of short-wavelengths for target irradiation. The process of laser-driven ablation in particular received considerable attention and analytical models predicting the scaling of the mass ablation rate and ablation pressure with laser irradiance and wavelength were proposed8 6 - 6 7. At the same time many laboratories6-16,68 including our own intiated intense investigations specifically to study the effect of the laser wavelength in laser-matter interactions. 1-4 Present investigation The main objective of this work was to study the effect of the laser wavelength in laser-matter interactions for sub-micron laser light. Specifically the physics issues that we adressed in our work were: 1. coupling of laser energy to the target plasma, 2. energy transport from the absorption region to the high density target interior by-electron thermal conduction and 3. dependence of the mass ablation rate and ablation pressure on laser intensity and wavelength. For this work we have used sub-micron laser wavelengths (0.532, 0.355 and 0.266 fim) at moderate irradiances (1012 — 5 X 1018 W/cm2) and with long pulse length (2ns FWHM). To allow a clear comparative study of the effect of the laser wavelength, identical irradiation conditions were used for each laser wavelength. For this short wavelength and relatively low intensity regime, it is expected that classical collisional absorption is the dominant absorp-tion mechanism and laser-plasma collective processes should not be important. Therefore the production of hot electrons should be kept to a minimum and energy transport into dense matter should be mainly through classical electron thermal conduction. Also using long laser pulses the plasma flow from the target suface should reach a steady-state. Thus the results should provide in principle a valid comparison with predictions from classical, steady-state theories. 5 R e l a t e d w o r k s s t u d y i n g t h e d e p e n d e n c e of t he mass a b l a t i o n r a t e a n d a b l a t i o n p res -su re o n laser w a v e l e n g t h a n d i n t e n s i t y has been c a r r i e d o u t c o n c u r r e n t l y a t o t h e r l a b o r a t o -r ies. In t he w o r k of F a b r o et a l . 6 8 , t h e mass a b l a t i o n r a te s have been d e t e r m i n e d f o r 1.06, 0.53 a n d 0.26 /im l a ser i r r a d i a t i o n o f t h i n fo i l s . T h e re su l t s were ba sed o n a l i m i t e d n u m b e r of d a t a po i n t s o b t a i n e d ove r a w i d e r ange o f l a se r i n t en s i t i e s a n d pu l se l eng th s . C o l l e c -t i v e p l a s m a processes w h i c h b e c o m e i m p o r t a n t at h i g h i r r a d i a n c e s ( 1018 W / c m 2 ) a n d l o n g w a v e l e n g t h s (1.06 /im) m a y affect t h e o v e r a l l w a v e l e n g t h dependence o f t h e a b l a t i o n r a t e . A l s o , t he a b l a t i o n process w i l l r e a c h s t e a d y - s t a t e c o n d i t i o n s o n l y f o r t h e l o n g pu l se ( > I n s ) e x p e r i m e n t s . K e y et a l . 1 4 o b t a i n e d b o t h w a v e l e n g t h a n d i n t e n s i t y s c a l i n g l aws f o r t h e mas s a b l a t i o n r a t e a n d a b l a t i o n p re s su re u s i n g 1.06, 0.53 a n d 0.35 /im laser r a d i a t i o n . T h e la ser i n t e n s i t y o n t a r g e t was v a r i e d b y c h a n g i n g t h e f o c a l spo t s izes. T h e r e s u l t s a p p e a r e d t o be d o m i n a t e d b y s i g n i f i c an t l a t e r a l ene rgy t r a n s p o r t o u t o f t h e f o ca l s po t a rea , p a r t i c u l a r l y f o r t he l onge r w a v e l e n g t h r a d i a t i o n . M o s t r e c e n t l y , M e y e r a n d T h i e l l 1 5 have m e a s u r e d t h e t o t a l a b l a t e d mass i n p l a n a r t a r ge t s i r r a d i a t e d w i t h 1.05 a n d 0.35 fim l a ser l i g h t . A g a i n , l a t e r a l ene r g y t r a n s p o r t i n t h e l onge r w a v e l e n g t h m e a s u r e m e n t m a d e w a v e l e n g t h s c a l i n g c o m p a r i s o n s d i f f i c u l t . In o u r w o r k , u s i n g i d e n t i c a l laser f o c a l spo t s izes a n d pu l se l eng th s , m o d e r a t e i r r a d i a n c e s a n d s u b - m i c r o n w a v e l e n g t h la ser r a d i a t i o n , t h e effect o f c o l l e c t i v e p l a s m a processes a n d l a t e r a l ene rgy t r a n s p o r t o n t h e s c a l i n g s tud ie s we re m i n i m i z e d . 1-5 Thesis outline C h a p t e r s 2, 3 a n d 4 p re sen t a t h e o r e t i c a l b a c k g r o u n d o n l a s e r - p l a s m a c o u p l i n g , e lec -t r o n t h e r m a l t r a n s p o r t a n d s t e a d y - s t a t e a b l a t i o n . E s t i m a t e s o f t h e i m p o r t a n c e of t h e d i f fe rent p h y s i c a l m e c h a n i s m s w i l l be m a d e f o r t y p i c a l c o n d i t i o n s i n o u r w o r k . In C h a p -t e r 5 we de s c r i be o u r e x p e r i m e n t a l s e t - up , t h e e x p e r i m e n t a l c o n d i t i o n s a n d the d i f f e ren t d i a g n o s t i c s w h i c h have been used. T h e e x p e r i m e n t a l r e su l t s a n d t h e i r i n t e r p r e t a t i o n s are p re sen ted i n C h a p t e r 6, 7 a n d 8. F i n a l l y i n C h a p t e r 9 we s u m m a r i z e t h e re su l t s a n d c o n -c l u s i o n s of o u r w o r k , h i g h l i g h t t h e o r i g i n a l c o n t r i b u t i o n s a n d g ive sugges t ions f o r f u t u r e w o r k . 6 Chapter 2 Absorption of laser light in a laser produced plasma In this chapter we describe the principal mechanisms through which laser light can interact with a plasma and which lead to absorption and/or scattering of the radiation. This will constitute a general background for the interpretation of our measurements on the coupling of the laser energy to the target, that we present in Chapter 6. First let us introduce some basic properties which characterize laser light propagation in a plasma. The laser radiation is considered as a plane wave described by a transverse electric where U'L and kL are the frequency and wave number of the laser light in the plasma. The propagation of laser light in a plasma will be governed by its dispersion relation1 6 : where t is the plasma dielectric constant, ve{ is the electron-ion collision frequency which leads to a damping for the incident laser light, c is the velocity of light in free space and wp is the "plasma frequency" which is the natural frequency of oscillation of the electrons in a plasma, given by : where ne represents the plasma electron density, e and m e the electric charge and mass of an electron respectively and e0 the permitivity of free space. The plasma frequency depends on the electron density. Thus as laser light propagates from lower to higher density in a plasma its wave number as given by (2 — 1) becomes smaller. Finally at the "critical density" the plasma frequency wp becomes equal to the laser light frequency wL and the laser light field E = E 0 expi{wLt - k L r ) (2-1) 7 can no longer propagate ; for n e > ner the wave number becomes purely imaginary and the light wave is evanescent. Thus the critical density is the reflection point for laser light of wavelength \i in a plasma. Figure 2 — 1 shows a schematic of the electron density profile in the expanding or "blow-off" plasma of a laser irradiated solid. It also identifies different processes through which laser light interacts with the plasma. Since the laser radiation cannot penetrate to densities above ncr all laser light absorption and reflection processes occur at electron densities ne < ner , that is, in the so called underdense plasma. In section 2 — 1 we will discuss inverse bremsstrahlung absorption where the electrons oscillate in the electric field of the laser light and by colliding with ions convert their coherent energy of oscillation into random thermal energy. The laser light can also excite plasma waves which in turn can heat the plasma or scatter the incident laser radiation. (There are two plasma waves of interest in an unmagnetized plasma ; the first is a high frequency electron plasma wave which occurs at the natural frequency with which electron oscillates, the second is a low frequency ion acoustic wave which is the analog of a sound wave in a ordinary gas). In particular in section 2 — 2, we will describe a process known as "resonance absorption" where the electric field of the laser light resonantly excites electron plasma waves (Langmuir waves) at the critical density surface. In section 2—3 we will discuss laser driven plasma instabilities . These processes which includes the parametric instability, the two-plasmon decay instability, the Raman and Brillouin scattering instabilities lead to the excitation of plasma waves and can occur throughout the underdense plasma as shown in Figure 2—1. 2-1 Inverse bremsstrahlung absorption The main absorption mechanism for laser light in a plasma is inverse bremsstrahlung or collisional absorption. We recall that bremsstrahlung radiation is emitted when a free electron is accelerated in collision with an ion. The reverse process is also possible where radiation is absorbed by accelerating an electron in the field of an ion (the presence of an 1.1 X 10 2 1 Xfi/jm) (2-3) 8 i—M.B. Absorption and Brillouin scattering Figure 2—1 Schematic picture of electron density profile in a laser-produced plasma, showing locations of major coupling processes. 9 ion is necessary to conserve momentum). Let us consider the motion of an electron in the oscillating electric field of an incident electromagnetic wave (the incident laser light). As the electron oscillates in this field, it collides with the ions, thereby converting its coherent energy of oscillation into random thermal energy. In other words, the incident laser light drives electron currents and the resistivity represented by electron-ion collisions leads to a heating of the plasma. This implies a spatial attenuation of the incident wave as it propagates in the plasma, a process that we will now analyse in more detail. In describing the spatial attenuation of an electromagnetic wave in a plasma, the parameter normally used is the "linear absorption coefficient". It is defined in terms of the laser light intensity IWL. For a wave travelling in the z direction, IU,L — Io,wL exp— (aWLz) • Since the intensity IWL is proportional to the square of the electric field E = E 0 e\ipi(wit — kLz), we have : aWL = -2Im(kL) . (2-4) The imaginary part of is obtained by solving equation (2 — 1) from which we find : B,L = (^ )(4) } . (2-5) By using the appropriate expression for the electron-ion collision frequency ve< the following expression for the absorption coefficient is finally obtained1 7 - 1 8 for a Maxwellian velocity distribution • • i - s 4 « < » ™ . H - . ) « M „ - . . ; / „ * ) " < ( S ' ° n , t s ) • ( 2 " 6 ) where < Z > is the average ion charge, Te the electron temperature, k the Boltzmann constant, In A the Coulomb logarithm. A is the ratio of the maximum and minimum distance through which an electron interacts with an ion. It is given by : A = Xi)/dmin y w 10 Thus, rs/2 T In A = Min[ln(2.58 X 1 0 " , ln(2.83 X 1 0 1 0 ^ ) ] (2 - 7) Zne ne'" where the units of Te and n e are in eV and c m - 3 respectively. Equation (2 — 6) shows that inverse bremsstrahlung is strongest for low-temperature, high-density and high-Z plasmas. For the absorption of laser light in a density gradient it is convenient to rewrite equation (2 - 6) as : ZlnA ne 2 / 1 " e v - i / 2 0 , a"L=tl^j2Ti( — ) ( 1 - — ) > (2-8) (2K)^2 m\l2c2t2 Let us consider a plasma with an exponential density profile x. ne = ncrexp-{-) , (2-9) where L is the density gradient scale length, and calculate the absorption for a laser beam propagating at normal incidence to this gradient. The absorption fraction up to the density ne = 0ncr (where /? is a number less or equal to 1) is given by : A(0ner) = 1 - exp -( / aw{x)dx) (2-10) Substituting equations (2 — 8) and (2 — 9) in this equation yields : Atfn„) = 1 - exP(-p^ Z ^ 1 " ' « P "f(1 - CXP T)_1/2 dX) (2 " U ) The integral is easily evaluated to give : A(0ner) = l-exP-[i-r^J^F(0)\ (2-12) J Te' Xf, where F(0) = 1 - (1 - fi)1^ + 1) 11 or in practical units A{Pner) - 1 - exp-[15.33 f l D ^ y - F ( f > ) ] (2-13) This expression shows that inverse bremsstrahlung absorption in an exponential den-sity profile is stronger for low temperature (low laser intensity), large scale length, high Z material, short laser wavelength and high electron densities (the function F(fi) increases rapidly with fi = ne/ncr). Figure 2 — 2 shows the evolution of the percentage absorption of laser radiation of different wavelengths as it penetrates an exponential density profile. This calculation was done for typical conditions encountered in our experiments namely : Z ==: 11 (using aluminum targets), L = 80/tm (determined by our laser focal spot according to equation (4 — 35) of chapter 4) and plasma temperatures of 500 eV at 0.532 /im, 390 eV at 0.355 /im, 330 eV /tm at 0.265 fi-m. (according to formula (4 — 36) of chapter 4 for an incident irradiance of 101S W/cm 2 , which yields good agreement with our experimental measurements of plasma temperature). The Coulomb logarithm was calculated using equa-tion (2 — 7) for the critical density value at each laser wavelength. Our simple calculation shows that the absorption coefficient is so high for these experimental conditions that in fact most of the laser energy will be deposited before reaching the critical density. This last point will have an important consequence on the scaling of the plasma parameters with laser intensity and wavelength as will shall see in chapter 4. For high laser fluxes the non-linear processes can reduce the effectiveness of collisional absorption. In one effect, discussed by Silin19, the electron collision frequency is modified by the coherent oscillation of the electrons in the laser electric field. Thus rather than having only a thermal velocity Vte = {kTe/me)ll2 the electrons have an effective velocity ve/f given by : V;„ = V* + V0\e , (2-14) where V0,e = (2-15) mewL is the oscillatory velocity of an electron in the electric field EL of the laser light. Because the absorption coefficient aWL scales as T~s^2 a V~^, (2 — 8) the absorption coefficient becomes 12 Figure 2-2 Percentage absorption of laser energy in an exponential plasma density profile for different laser wavelengths. 13 aw,(non linear) —» aWL(linear) (2-16) 1 + UeEL/(mewLVu)f at high laser intensities. Another non-linear effect, discussed more recently by Langdon20 is due to the fact that for high laser intensity the electron velocity distribution can be far from Maxwellian. This will occur when the rate of laser energy absorption by the electrons becomes larger than the relaxation rate of the electron distribution towards a Maxwellian, a condition which can be expressed as : where vei is the electron-ion collision frequency and f e e ( ~ ve{\Z) the electron-electron colli-sion frequency. It is those electrons with electron-ion collision frequency vei (V) close to the laser frequency Wi that contribute most to the absorption process. Because vei{Y) <* V~z and since generally i/ei (Vte) < < wL, the electrons responsible for absorption have relatively low velocity compare to Vte. However when equation (2 — 17) is satisfied (for high laser intensities), the electron distribution becomes distorted and has relatively less low velocity electrons than in a Maxwellian distribution. For this non-Maxwellian velocity distribution the amount of absorption is reduced by a up to a factor of two compared to what is pre-dicted with the Maxwellian distribution usually assumed (eq. 2 — 6). The relevant parameter in these two non-linear processes is V ^ / V , 2 . This ratio may be expressed as V 0 2 c / V , 2 = 4 X 10 - 1 6/i,X 2/T , e where IL is the intensity in W/cm 2 , \ L is the laser wavelength in /im and Te is the electron temperature in keV. Typical conditions in our experiments are IL = 101S W/cm 2 , \ L = 0.532 nm, Z = 11 and Te = 500 eV. For these values V2te/Vfe « 2 X 10~s < < 1. Therefore non-linear effects will not be important in the inverse bremsstrahlung absorption process in our experiment. 2-2 Resonance absorption Resonance absorption21-28 is the simplest absorption process via electron plasma wave. Consider 'p' polarized (Et vector parallel to the plane of incidence) laser light VeiVL > UeeVt2e <=• Z{ eEL (2-17) me wL Vte 14 obliquely incident on a plasma with a density gradient VTJ€. The component of the laser electric field parallel to the density gradient produces a charge separation and drives a charge density fluctuation 6ne at the laser frequency wL. At the critical density surface, this driven fluctuation is at the plasma frequency (wL = wp) and an electron plasma wave is resonantly excited. The laser energy is thus transferred to the electron plasma waves. The damping of these waves then completes the energy transfer to the thermal energy of the plasma. Figure 2 — 3 illustrates this process in more detail. For light incident on a plasma density gradient at an angle $ to the normal, Snell's law and the dispersion relation (2 — 1) can be combined to show28 that the the classical turning point for the light wave occurs at a density ne($) = nercos0 . Beyond this point, the wave is evanescent. However, if the electric field vector of the wave is in the direction of the density gradient, part of this field can penetrate through to the critical-surface region where it can resonantly drive electron plasma waves. We can show24 this rather easily from Maxwell's equations which can be written as V-[e(z)E] = 0 , (2-19) w (x) where t(x) = 1 ^ [see eq. (2 - 1)] . (2-20) Expanding this we get t(x)VE + (EV) f(a;) = 0 . (2-21) Now, from Poisson's equation we have V-E = - ^ , (2-22) where Sne is the electron density perturbation. Since Ve(a:) a V n e , comparison with (2-21) gives ^ « (2-23) fo f(z) This simple result illustrates the two main characteristics of resonant absorption: the density perturbations are only excited by a light wave with a component of its electric field 15 LARGE ELECTRON PLASMA OSCILLATIONS INCIDENT LASER LIGHT AT ANGLE 6 TO NORMAL REFLECTED LASER LIGHT Figure 2—3 Schematic representation of resonance absorption for 'p' polarized laser light obliquely incident on a plasma density gradient. 16 in the direction of the density gradient, and these perturbations are resonantly excited only for f « 0 , that is, where wi = wp. The actual amount of absorption for 'p' polarized light will depend on the angle of incidence 6 (Fig. 2 — 3). For small 6 the electric field is almost parallel to surfaces of constant electron density and the resulting density fluctuation is small. For large 0, the cut-off and the resonance regions are widely separated and the driving electric field is greatly attenuated in the evanescent region. The absorption fraction maximizes at about 60 % for an angle $ which satisfies s i n 0 ^ O . 8 ( - ^ - ) 1 / s ( 2 - 2 4 ) where c is the speed of light in vacuum and L the density gradient scale length. Computer simulations of resonance absorption25-28 show collisionless heating of elec-trons traveling through the resonant electron plasma waves from high densities towards low densities. Electrons with a velocity close to the phase velocity of a wave see nearly a con-stant electric field. A large energy transfer then occurs as the electrons are efficiently accelerated by the wave. Hence, faster electrons are preferentially heated, which leads to a heated velocity distribution characterized by an energetic tail. This velocity distribution can be described using a two-temperature Maxwellian distribution. The lower energy ther-mal electrons are characterized by a temperature Tcou and the very energetic resonantly heated electrons by a temperature Thot. The average temperature Thot has been estimated to be27 Thot = Teold + 4.3 X l0-*T?o%(ILl\L/l.0Q}2)042 (2 - 25) where the units for T, IL and \ L are keV, W/cm 2 and /im respectively. Computer simulations have also shown the importance of plasma density profile-steepening in resonance absorption. This steepening can be due to the ponderomotive force (light pressure in a plasma29), but the dominant force is frequently due to the resonant plasma waves which accelerate hot electrons down the density gradient during resonance absorption28. To balance the momentum they carry away, the density profile adjusts near critical and steepens there. Such profile steepening then makes resonance absorption more 17 effective as it is less sensitive to the angle of incidence because there is now less distance to penetrate to reach the critical density surface. In addition, laser-driven plasma instabilities near the critical surface (discussed in the next section) are strongly limited since there is only a very small region of plasma in which these instabilities can operate. Resonance absorption will be the dominant absorption mechanism for high plasma temperature (high laser intensity), long laser wavelength and short plasma scale-length, that is, under conditions for which inverse bremsstrahlung absorption is inefficient and for which most of the laser radiation reaches the critical density surface. Evidently with very strong inverse bremsstrahlung absorption, resonance absorption is not expected to be important in our experiment. 2-3 Absorptive and scattering instabilities At high intensities the laser light can excite natural modes of oscillation of the plasma (electron plasma wave and ion acoustic wave) by laser-driven plasma instabilities80. These waves in turn can lead to absorption or scattering of the incident laser light. As an example the stimulated Brillouin scattering instability will be described. The feedback loop leading to instability is illustrated in Figure 2 — 4. Consider a small density fluctuation 6n associated with an ion acoustic wave. The electric field EL couple with the density fluctuation 6n and produces a transverse current a 6nE[,, due to the oscillatory mo-tion of electrons in the laser electric field. This transverse current produces a reflected light wave with field ER. The ponderomotive force V(ELERI?>TT) of the incident and reflected light waves in turn enhances the density fluctuation. If such enhancement in 6n exceeds its damping losses, the process becomes unstable. In this case, it will lead to exponential growth of both the ion acoustic wave and the scattered light wave. 18 Ion wave 6n Force V(EREL/8n) Current a 6nEL Reflected wave ER Figure 2-4 Feedback mechanism for growth of stimulated Brillouin scattering Most of the laser-driven plasma instabilities can be described as the resonant decay of an electromagnetic wave into two other waves. The parametric-decay instability represents the decay of the incident light wave {wi, k^) into an ion-acoustic wave (wia , kia) plus an electron plasma wave (wpe, kpe). This takes place near the critical density ncr as shown in Figure 2 — 1. If the two waves product are two electron-plasma waves, we have the two-plasmon-decay (2wpe) instability. This occurs near (l/4)n c r. The Raman scattering instability represents the decay of the light wave into a scattered light wave (w,c) plus an electron plasma wave, a process which occurs for ne < (l/4)nc r. The analogous scattering process in which the electron plasma wave is replaced by an ion-acoustic wave is called the Brillouin scattering instability and occurs for n e < ncr. In the ion-acoustic and two-plasmon decay instabilities part of the laser energy will be transferred to plasma waves and this represents energy absorption by the plasma. On the other hand in Raman and Brillouin scattering, an electromagnetic wave is also produced which can escape from the plasma. This represents reflection of the incident laser radiation by the plasma. For strong coupling of the laser light wave to the excited waves, the following match-ing or "resonance" conditions must be satisfied (2 - 26) ki + k2 (2 - 27) 19 where wL and kL are the frequency and wave number of the laser light and Wj , w2, kj , k2 the frequencies and wave numbers of the excited waves. The frequency matching conditions will determine the region of plasma densities for which the various instabilities occur. Also in order for the excited waves to grow, the laser field must feed energy into the waves u/j and w2 faster than the natural rate of energy dissipation or damping of these two modes. The latter requirement determines a "threshold intensity" for the laser "pump" field below which Wi and w2 are not excited. For any given process the threshold will depend on the strength of the coupling and on the damping of the natural modes of oscillation. These thresholds can be calculated theoretically as a function of the plasma parameters30 and are given in Table 2 — 1. Tab le 2-1 Partial list of the instabilities which can be driven by laser light propagating in a plasma name process density threshold intensity31 3 3 W/cm 2> parametric-decay wL —» wpe + wia at ne ~ nCT 6.72 X 1019T^2\2B^ instability X L _ 2 / 8 n J / 2 o two-plasmon- wL — wpe + wpe at ne ~ (l/4)nc r 3 X 1013Z T~1/2\ls decay instability Raman scattering wL —* w,c + wpe at ne < (l/4)nc r 5 X 1017L7/^3X7^2/3 Brillouin scattering wL -* w.e + wia at ne < ner 7.5 X 10127;X7;1£r1(^) where Te, ne, \L and L are in eV, cm , fim and /im respectively. Evaluation of the threshold intensities for typical conditions in our experiment, namely , Te = 500 eV, ne = 4 X 1021 cm - 3 , Z = 11, L = 80/im and \ L = 0.532/im 20 yields 9.8 X IO10 W/cm 2 for the parametric-decay instability, 9.8 X 101S W/cm 2 for the two-plasmon-decay instability, 2.2 X 1015 W/cm 2 for the stimulated Raman scattering in-stability and 8.8 X 101S W/cm 2 for the stimulated Brillouin scattering instability. Since typical laser intensities of 101S W/cm 2 are used in our experiments, laser-driven plasma instabilities are not expected to play a significant role. The parametric-decay instability could be excited but because of the strong inverse bremsstrahlung absorption only very little laser energy will reach the critical density where the instability is excited. 21 Chapter 3 Electron thermal transport in laser produced plasmas As we have seen ihthe irradiation of a solid by a high power laser, the laser energy can only be absorbed in the "blow-off" plasma. It is mainly the heated electrons which transport the energy from the laser-absorption region towards the solid interior thereby controlling the mass ablation rate and the ablation pressure generated at the solid surface (in the case of high Z target material, the X-rays can also contribute significantly to the transport of energy). Electron thermal conduction thus play a very important role. For high laser intensities very strong density and temperature gradients exist in the plasma. The classical theory of heat transport in plasmas can no longer be applied and an exact description of energy transport becomes very complex. Furthermore coherent fluctuations in the ion charge density (ion turbulence) and large self-generated d c.magnetic fields in the plasma may also affect the heat flow. Some of these considerations are described here as background for laser experiments in general. Some basic concepts of electron thermal conduction are introduced in Section 3 — 1. In Section 3 —2 we review the classical theory of Spitzer-Harm on the thermal conductivity of a plasma and show that it is valid only in plasmas with small temperature gradients. In Section 3 — 3 we discuss the problem of heat transport in laser-produced plasmas which are characterized by large temperature gradients. Finally in Section 3 — 4 we will mention two possible mechanisms which may contribute to a reduction of the heat flux in laser produced plasma. In our work attempts were made to ascertain the role of electron thermal conduction in the series of multilayer targets experiments presented in Chapter 7. However as will be seen in Chapter 7 our results revealed anomalies which precluded such interpretation in our experiment. 22 3—1 Basic concepts A temperature gradient in a plasma gives rise to a heat flux which tends to equalize the temperature difference. This heat flux represents an energy transfer by particles which have a thermal velocity given by Vt = (kT/M)1^2, where T is the temperature of the particles contributing to the heat flux and M their mass. Because of their lower mass, the electrons have a much higher thermal velocity than the ions and therefore play the main role in heat transport. Electrons with higher thermal energy will tend to move down a temperature gradient. This produces an electric field which then gives rise to a return electron current. As a result, there is a net flow of thermal energy towards the colder layers of plasma with no net current flowing. This heat flux is proportional to the relative temperature drop, and the heat flux crossing a unit plane can be described by : where K is the plasma thermal conductivity. Since the electrons contributing to the heat flux at a given plane come from a distance of the order of their mean free path, equation (3 — 1) is strictly valid only if the temperature gradient is constant over a distance of few mean free paths. Otherwise, the heat flux at a given plane will not depend only on the local thermal conductivity and temperature gradient, as implied by equation (3— 1), but will also depend on energy transport by electrons many free paths away. This condition is most commonly expressed as X e /Lr < 1 where Xe is the mean free path of energy carrying electrons and LT is the temperature gradient scale length. An expression for the plasma thermal conductivity can be obtained using a simple electron thermal transport theory. Let us imagine a plane perpendicular to a temperature gradient dTjdx and calculate the heat fluxes passing through it in both directions. Each of the fluxes is : where ne is electron density and Vle their thermal velocity. These heat fluxes are due to the motion of the electrons which travel, prior to collision, a distance which is about their (3-1) Q ~ neVu X meV2 neV,e Te (3-2) 23 mean free path Xe. Since the heat fluxes in both directions are due to the particles moving on different sides of the plane where the temperatures are different, the net flux is : Q ~ neVu AT (3 - 3) Here A T is the temperature difference over a distance of the order of the mean free path (the characteristic size of the region from which the electrons get to the plane without collisions with other particles, that is, without energy exchange). Hence, we obtain AT ~ \edT/dx and : dT Q ~ neVte Xe — (3-4) A comparison with equation (3 — 2) yields the following dependence for the thermal con-ductivity : K ~ neVte\e = ne^ (3-5) where i>et- is the electron-ion collision frequency. Since t/et- ~ ne /T e 3 /2 (see eq. 2 - 5 and 2 — 6 of chapter 2) and Vle ~ (Te/me)1^2 equation (3 — 5) can be rewritten as : K ~ - e — . (3-6) me The thermal conductivity of a plasma is therefore an extremely sensitive function of tem-perature (for comparison, in an ordinary gas34 K ~ T\l2 jM since the mean free path X —' l/N where N is the density of the gas particles). Also we see that the thermal con-ductivity is independent of the density of the electrons. Indeed, an increase in electron density gives rise to a proportional increase in the number of electrons transporting heat and to a proportional decrease in the mean free path of the electrons. These two effects cancel each other. Using this simple theory we can only obtain a rough estimate for the thermal con-ductivity. A more correct treatment of heat transport in a plasma requires the velocity distribution of the electrons to be taken into consideration as discussed in the next section. 24 3—2 Spitzer-Harm theory of thermal conductivity: small temperature gradients In the kinetic theory of plasma the electrons are represented through their velocity distribution function /(r, v, t). By definition the electron density n(r, t) at point r is given by : n(r,t) = / f(r,v,t)d3v where d*v represents a three dimensional volume element in velocity space. For equilibrium conditions f(r,v,t) is a Maxwellian distribution function. A temperature gradient in the plasma distorts the velocity distribution and a heat flux Q appears which is given by8 5 : Q = J f(r,v,t)?^-vd\ . (3-8) Equation (3 — 8) is similar to equation (3 — 2) but here an average over the velocity dis-tribution f(r, v,t) is taken. The evolution of the distribution function is described by the Boltzmann equation: df dt + v / + ~ = ~ ~ , 3 - 9 or m. ov \ot/con where (dfJdt)con is a Fokker-Planck88 collision term which represents the electron-ion and electron-electron interactions in the plasma. E is a self-consistent electric field , related to the temperature gradient by the expression for the current density in a plasma : J = crE + a V T = 0 , (3-10) where a and a are appropriate coefficients86. The temperature gradient produces a current in the plasma and as a result a secondary electric field E builds up which draws a return current such as to maintain a zero net current. The net current has to be equal to zero as otherwise an electrostatic field would rise without limit. The thermal conductivity of a plasma using the kinetic theory was first calculated by Spitzer and Harm8 6. In their treatment they assumed a small temperature gradient which produces a small perturbation on the velocity distribution function which they write in the form : /(v) ==/>) + /> (v) with r«f (3-11) 25 where /°(v) is the unperturbed Maxwellian velocity distribution and /^v) is a small per-turbation due to the temperature gradient. After substitution in the Boltzmann equation (3 — 9) and neglecting second order terms, a second order differential equation for the per-turbation function / !(v) was obtained. The equation was solved numerically and values were obtained for P{v) in terms of the parameter \e/Lr where Xe is the mean free path of the electrons and LT the temperature gradient scale length. Using these values of P(v) the heat flux carried by the distribution function was then evaluated from equation (3 — 8) (only the perturbation contributes to the heat flux ). Comparison with equation (3—1) then yielded the following expression87 for the thermal conductivity K. R / 2 \ s / 2 c*k(kTef/2 /0.095(Z + 0.24)\ , r t l . , * = 2X10W r^LA (1-roW) (S,UmtS) ' P",2) where Z is the average charge of the ions, c the speed of light and In A is the Coulomb logarithm (see eq. (2 — 7) of chapter 2). Substituting the physical constants, this can also be expressed as : K -186 x l°-°lkx{f^) K H <3-13> The Spitzer-Harm theory was derived under the assumption that / :(v) < < /°(v). However, Gray and Kilkenny88 have pointed out that the perturbation function / !(v) in-creases with \e/LT and at some velocity, depending on \e/LT, /'(v) can become greater than /°(v). Then, the assumption that ^ ( v ) is a small perturbation is no longer valid. Also, since the function P{v) can take negative values the distribution function /(v) may then become negative. This is clearly unphysical. It comes about because higher order terms are neglected. However, they also pointed out that a negative value for /(v) would not matter in the velocity range whrere the heat flux is very small. Their analysis then showed that the heat flux was carried mainly by electrons whose velocity is between 2.3 to 3 times the thermal velocity and that the distribution function /(v) became negative in this velocity range for X e / L T » .02 . Plasmas produced by high-power lasers frequently exhibit very sharp temperature gradients over distance comparable to the mean free path of electrons. In the next section, 26 we discuss how the heat flux is usually treated in a ser-heated plasma with steep temper-ature gradients and also discuss the theoretical work which are aimed at improving the understanding of electrons thermal transport in such situation. 3-3 Thermal transport in laser produced plasmas : large temperature gradients The Spitzer-Harm heat conduction law is not a correct description of heat trans-port when too many of the heat-carrying electrons have mean free paths exceeding the local temperature gradient scale length. In a laser heated plasma this condition predomi-nates in the high density region, immediatly beyond the laser energy deposition zone. The Spitzer-Harm conductivity greatly overestimates the heat flux under such circumstances. In computer hydrodyuamic simulations of laser-plasma experiments, it has been customary to artificially limit the heat flux to a maximum value which corresponds to free particle flow. The heat flux is then described by a law of the type: Q = min(QsH.. QFS ) , (3 — 14 — A ) or Q = (<2SH + Q F S ) " 1 , ( 3 - 1 4 - / 3 ) where QSH is the Spitzer-Harm heat flow and Qrs is the "free-streaming flux" : QFS = FnekTeVte , ( 3 - 1 5 ) representing the energy carried by a flow of free streaming electrons. F is called the flux limiter and is a parameter less than unity, which is adjustable in the numerical simulation. Heuristic arguments suggest39 that F ~ 0.2 — 0.6 . However40-41 in numerical simulations, a better fit to many experimental results is obtained if F is reduced to about 0.03 . This has been interpreted as an indication that electron thermal transport is strongly inhibited in laser-produced plasmas. Some mechanisms have been proposed to explain such inhibition (see next section). Since the "free-streaming limit" is essentially a crude attempt to estimate an upper bound to the heat flux when the classical theory of Spitzer-Harm breaks down, it was also suggested42 that a correct treatment of electron thermal transport for large 27 temperature gradients might reveal a much lower conductivity than as previously been supposed. A more accurate treatment of heat transport in steep temperature gradients was given by Bell, Evans and Nicholas43. In their work, the Boltzmann equation (eq. (3 — 9)) was solved numerically, by expressing the distribution function /(v) as an expansion of Legendre polynomials. They then calculate the heat flux , using equation (3 — 8), for a one-dimensional static plasma (no ion motion) with prescribed temperatures TY and T2 across a large temperature gradient. Their results show, as expected, that in the presence of a steep temperature gradient the heat flux at any point is not simply a function of the local plasma state (as assumed in the classical conduction theory of Spitzer-Harm ) but is also determined by the state of the surrounding plasma over a region which is several mean free path thick. Also, on the main body of the heat front, the heat flux is an order of magnitude less than that given by the Spitzer-Harm theory. This is equivalent to a free streaming heat flux with a flux limiter F = 0.1 . Similar work 4 4 - 4 6 of other groups have led to the same general conclusion. Thus a more correct treatment indeed shows a reduction in the heat flux compared to the Spitzer-Harm theory or the classical free-streaming flux limiter F ~ 0.2 — 0.6 . However , this is still much larger than the phenomenological free-streaming flux (see above) with F ~ .03 which reproduces experimental results better. 3-4 Thermal transport inhibition mechanisms Two mechanisms have been proposed41 to account for a reduction of the heat flux in laser-produced plasmas. These are self-generated magnetic fields and ion acoustic tur-bulence. It should be noted that in our experiment using moderate laser intensities and long pulse irradiation the temperature gradient is not expected to be large compare to the mean free paths of the electrons. Therefore in Chapter 4 which describes a simple analytical model of the ablation process, the classical expression for the plasma thermal conductivity (eq. 3 — 12) will be used. Self-generated magnetic fields and ion acoustic turbulence are not included in this simple model. They are described here for completeness. 28 3-4-A Self-generated magnetic field Megagauss self-generated magnetic fields B have been observed in laser produced plasmas4 7 - 4 8. The heat flux is not modified when the temperature gradient is parallel to the magnetic field36. However electron thermal transport across the field lines will be reduced when the electron Larmor radius PL = emeVte /(eB) is less than the collisional mean free path Xe. Under these conditions the step-size for random walk of the electrons across a field line becomes pi instead of Xe and the cross-field heat flux is reduced by the factor R = [1 + {\E/PL)2}-1 • For example, in plasmas where Z = 4, B = 1 MG, ne = 1021 c m - 3 and kTe = 1 keV, R <~ 0.05 which shows that this effect becomes significant. 3-4-B Ion acoustic turbulence In the presence of large coherent fluctuations in the ion charge density (ion turbu-lence), electrons will be scattered from the coherent ion charge bunches. This effectively increases the electron-ion collision frequency38 vti . Since the plasma thermal conductivity is inversely proportional to uei (see eq. (3 — 5)), these fluctuations reduce the heat flux. It has been proposed3 8'4 9 - 6 0 that the electron heat flux itself could give rise to ion turbulence via the ion-acoustic instability. The process can best be understood with refer-ence to Figure (3— 1). In this figure we have represented, in the frame of reference in which the mean particle velocity is zero, the reduced distribution F(v||) = / f(v)dvj_ (v\\ is the velocity in the direction of the heat flux) for the ions and the electrons. The ions are taken to be considerably colder than the electrons, so their distribution F,-(v||) is narrow. The electron distribution Fe(v\\) is distorted for vx < 0 because heat is being carried in the —x direction by electrons with speeds \vx\ > vte . To avoid a net current flow, this current of high velocity electrons is balanced by a return current of low velocity electrons which causes a shift in the peak of the distribution function. The shift in the peak is given by3 8 : vD = i 0.43—-vte (3 - 16) The ion acoustic wave velocity is approximately (3-17) 29 and if ZTe >> T{ this can be well above the thermal ion velocity vti which is - = ( - ) ' V m,- / Thus there are insufficient ions moving at the wave velocity to cause significant ion Landau damping (Landau damping or Landau growth of waves arises from an exchange of energy between the wave and the particles travelling with velocities close to the phase velocity of the wave). Conversely if vD >> via, the slope of the electron velocity distribution is positive near the ion acoustic velocity v,-a and this causes unstable amplification of the ion waves by Landau growth. From equation (3 — 16) and (3 — 17) the criterion for instability is : (3 - 19) or ^ > 2 .3 ( ^ Y / 2 ^0.05 (for Z = 1) (3-20) In most laser produced plasmas this threshold is exceeded and ion acoustic turbulence is certainly excited. However from the point of view of transport inhibition one must also know the effect of a given amount of ion turbulence on the plasma transport coefficients. Many authors have suggested that the turbulence has little effect on electron thermal transport61. Other authors find that significant inhibition can occurs 6 2 - 6 3. This is clearly an area where more theoretical work is needed. (3-18) 30 F(v)t h - vD —»i heat flow Q Figure 3-1 Electron and ion distributions in the presence of an electron heat flux Q. When VD >> Via and ZTe > > T{, the ion -acoustic instability grows rapidly (see text). 31 Chapter 4 Scaling laws for the mass ablation rate and ablation pressure: one dimensional steady-state ablation theory In this chapter, using a simple analytical model, we study the plasma flow which develops in the irradiation of a solid with a high power laser. We will be particularly interested in finding how the plasma temperature, the mass ablation rate and the ablation pressure generated by the ablation process at the solid surface vary with the laser intensity and wavelength. Such dependences on the laser parameters are generally referred to by the inclusive term "scaling laws". In Chapter 6 and 8, these theoretically determined scaling laws will be compared with experimental results. Several analytical studies of laser-driven ablation of solid exist and references 15, 07 and 68 review the subject. The principal goal of these analytical models is to shed light on the dominant physical processes, which are sometimes obscured in computer hydrody-namic simulations. The most crucial assumption which affects the scaling laws, made in these models15, concerns the absorption regime (strong or weak inverse bremsstrahlung ab-sorption) which determines the electron density where most of the laser energy absorption occurs. In this chapter we use the analytical model of Manheimer and Colombant66, which assumes laser energy deposition at the critical density n c r, to establish scaling laws in the weak inverse bremsstrahlung absorption regime and the model of Mora6 7, which assumes laser energy deposition at a density n0 which is less than the critical density ner, to estab-lish scaling laws in the strong inverse bremsstrahlung absorption regime. In Section 4 — 1 we give a physical description of the plasma flow stucture to be analysed. Section 4 — 2 contains a description of the analytical model, together with a derivation of the scaling laws and a discussion of the model assumptions. 4-1 Structure of the laser induced plasma flow When intense laser light is focussed on a solid in vacuum a thin layer at the surface of 32 the solid is vaporized and ionized, producing a high density plasma. The very rapid temper-ature increase at the solid surface causes the laser-produced plasma to expand or "blow-off" towards the laser beam. The dispersion relation of an electromagnetic wave in a plasma (see eq. (2 — 1) of Chapter 2) does not allow wave propagation in a plasma region where the electron density is higher than the critical density. As the laser radiation propagates through the "blow-off" plasma towards the solid, it encounters regions of increasing plasma density . It will penetrate up to a maximum electron density n 0 ( "with corresponding mass density p0> see eq. 4 — 4) which for strong inverse bremsstrahlung absorption conditions can be much less than the critical density (see for example Figure 2 — 2 of Chapter 2). We call the layer where the electron density ne = n0 the "absorption surface". The thermal gradient which exists between the hot plasma at the absorption surface and the cold solid, will drive a flow of energy towards the solid by electron thermal conduction. (At high laser intensity a significant portion of the energy can also be transported by fast, non-thermal electrons or in the case of high Z plasma, by radiation. However we will limit our description to cases where these two mechanisms are not important). This will cause more material to be heated up and ablated from the solid and there will be a continuous flow of new material from the ablation surface as long as the laser irradiation process continues. This flow of material carries away momentum with it and by Newton's third law of motion there is a corresponding force exerted on the remaining solid. This force,designated as ablation pressure, drives a shock wave into the solid and compresses and accelerates sucessive layers of the solid material as new layers of the solid are ablated8 7 - 6 8. A schematic of this global structure is shown in Figure 4 — 1 . It represents a snapshot of the density and temperature profiles, in the solid and the "blow-off" plasma. We can distinguish four different regions: i) the plasma corona which extends from the absorption surface (p = p0) into the vacuum, ii) the conduction zone bounded by the ablation surface and the absorption surface, iii) the shock-compressed and -heated solid, and iv) the unperturbed solid material ahead of the shock front. 33 ablation surface temperature perturbed compressed zone solid solid corona Figure 4-1 Structure of the steady-state plasma flow: density and temperature profiles. 34 Figure 4 — 2 shows an x — t diagram of the flow stucture identifying different density surfaces associated with the shock front, the ablation surface and the absorption surface. The laser is turned on at t = 0 and its intensity is assumed to remain constant thereafter. After some initial transients, for t > ti, the distance x0 between the absorption surface and the ablation surface remains constant indicating that a steady state regime has been reached in the conduction zone. That this is to be expected can be seen as follows. If the thickness of the conduction zone starts to increase (the laser intensity being constant), there will be less energy reaching the surface of the solid. Therefore less plasma is produced and the thickness of the conduction zone will start to decrease. If instead, the thickness of the conduction zone starts to decrease (again for constant laser intensity), the reverse process will take place. The thickness of the conduction zone is therefore self-regulated and adjusts itself to the laser intensity. In Figure 4 —2 we have also depicted the trajectory of a particle of the solid. Initially at rest, it is first accelerated by the shock wave. At a later time it is overtaken by the ablation surface, heated and accelerated in the conduction zone. Finally it is ejected with large velocities away from the solid surface. In the following section we will show how the characteristics of the laser radiation, its intensity and wavelength, and the properties of the solid material affect the rate at which material is ablated from the solid and the ablation pressure which is generated by this process at the solid surface. 4-2 One dimensional steady-state ablation theory 4 -2-A Conservation equations and model assumptions The basic properties of the plasma flow occurring as a result of the irradiation of a solid with intense laser light can be described by applying the conservation equations for mass, momentum and energy. To obtain simple analytical solutions for these equations we will assume that: i) The plasma flow from the solid can be described using one-dimensional planar ge-ometry. ii) The flow in the conduction zone has reached a steady state and thus the width the zone is constant and equal to x0-35 ablation surface Figure 4-2 Space-time diagram for the densities associated to the shock front, the ablation surface and the absorption surface. The laser pulse is a step function starting at t = 0. For t > ti the conduction zone has reached a steady state. 36 Conditions for the applicability of these two assumptions will be given in section 4 - 2 - E . In one-dimensional form, the conservation equations for mass, momentum and energy can be written64 as : £ + £ w - 0 , (4-1) ^ ) + ^ ( P + pF 2) = 0 , (4-2) where: • V is the plasma flow velocity ; • p represents the mass density, given by : . neAmp p = Tj ,Am p + neme ^  — - — , (4-4) where n,- is the ion density, ne the electron density, A the atomic mass number of the ion and Z its average electric charge, me is the electron mass and m p the proton mass; • C represents the sound speed in a plasma , given by : • P is the plasma pressure which is defined through the equation of state : P = nckTe + mkTi = p{^-^)kT = pC2 , (4-6) 37 where we have assumed that the electron temperature Te is equal to the ion tem-perature Ti . This simplifying assumption, which is certainly reasonable in the high density conduction zone, will not affect the general application of the results to be derived16-60; • Q represents the heat flux ; it is given by equation (3—1) and (3 — 12) of chapter 3 which we rewrite in the form : Q = -K0T&'2~ ; (4-7) • IAHX — x0) is a delta function which represents the absorption of laser energy at the absorption surface located at x — x0 It is assumed that this energy is rapidly distributed in the plasma by two heat flux <2i n and Q0nt, originating at x = x0, and flowing towards and away from the solid. In our description of the plasma flow, the flow structure is divided into the three regions identified on Figure 4 — 1 .These are the plasma corona, the conduction zone and the shock compressed target. Solutions for the conservation equations (4 — 1) to (4 — 3) can be found in each of these regions. Appropriate boundary conditions must then be used to connect the solutions between the different regions. We will limit our treatment only to the flow solutions which are essential to derive scaling laws for the mass ablation rate and the ablation pressure. For this purpose it will be sufficient to solve the conservation equations only in the plasma corona and the conduction zone. 4—2—B Flow solutions in the plasma corona : isothermal expansion The coronal plasma is expanding into a vacuum and therefore it cannot reach a steady state66. This expansion will be described by the time dependent conservation equa-tions 4 — 1 to 4 —3. To obtain simple analytical solutions we will assume that the plasma is isothermal throughout the expansion region. This implies that the plasma thermal conduc-tivity must be sufficient to maintain the plasma temperature constant against cooling by 38 expansion. Since the plasma temperature is high in the plasma corona and since the ther-mal conductivity is an extremely sensitive function of temperature (a T^2) this assumption is not unreasonable. We will use one-dimensional planar geometry to describe the expansion of the plasma. We introduce a spatial coordinate system x' attached to the unperturbed plasma (see Figure 4 — 3). At t = 0 the plasma occupies the half-space —oo < x1 < 0 and starts to expand. We assume that 7V) = T =constant so that = ( f ^ ) W 0 = C2^, and therefore equations (4—1) and (4—2) can be solved for the dynamical variables of plasma density p and velocity V. These two equations do not contain any characteristic length or characteristic time. The time t and the coordinate x1 can appear in the solutions of such a problem only in the combination x'/t. It is therefore convenient to use the variable f = x1 ft. Thus, A — A . d — — ziA d~x1~tdt: a n dl and substitution in equations (4 — 1) and (4 — 2) gives: <"--«3i + ' f - 0 ' « " 8 > and The two partial differential equations (4 — 1) and (4 — 2) have thereby been reduced to two ordinary differential equations which can be solved for V and p to give: V' = C+j , (4-10) and P = pexp-{l + —) , (4-11) p being the density of the unperturbed plasma. The solution (4 — 11) for the density profile is represented in Figure 4 — 3. It shows an exponential density profile which extends to infinity in the vacuum region and includes 39 Figure 4-3 Density profile for isothermal plasma expansion. a rarefaction wave which propagates into the unperturbed plasma at the sound velocity —C. Let us rewrite these equations in a new coordinate system x attached to the ablation surface and in which the absorption surface is located at the position x = x0 (see Figure 4 — 1). The transformation equation from the coordinate system x1 to the coordinate system x is x1 —• (x — x0) — C0t , where the subscript "0" is used to denote values for the variables at the absorption surface (where x = x0). In this new frame of reference the solutions (4 - 10) and (4-11) take the form : V(x) = C0 + °^ , (4-12) and ^ x ) - p o e x p ( ^ ^ ) (4-13) . These solutions imply that for the isothermal rarefaction wave in the coronal plasma to stay in equilibrium with a steady state conduction zone, the plasma flow velocity at the boundary between these two regions must be equal to the local sound speed V(x0) = C0. 40 Let us now calculate the heat flux necessary to maintain isothermal conditions in the expanding coronal plasma. Equation (4 — 3) describe the energy balance in this region. The first term represents the rate of energy flux in the rarefaction wave. The second term an enthalpy flux coming from the conduction zone and the third term the heat flux Q o u t supplied by the absorption of laser energy at the absorption surface x = z 0 (third term). To evaluate Q o u t we integrate the energy equation (4 — 3) from x = x0 to infinity. Making use of the flow solutions (4 — 12) and (4 — 13) we get: This result will be used later in the laser energy balance equation. 4-2-C Flow solutions in the conduction zone: steady state flow To describe the flow in the conduction zone we choose the same frame of reference in which the solutions (4 — 12) (4 — 13) are written. In this frame of reference, for steady state condition, the conservation equations (4 — 1) to (4 — 3) become: Qout — a . (4-14) UpV) - 0 ox (4-15) (4-16) and ) + Qin] + IA6{x-Xo) = 0 (4-17) Integrating equations (4 — 15) and (4 — 16) give : pV = p0C0 = m (4-18) and pC* + PV2 = 2p0C2 = P4 (4 - 19) 41 Here we have made use of the fact that the plasma flow velocity must be equal to the sound speed at the absorption surface to connect smoothly with the solutions in the coronal region of the flow (see eq. (4 — 12)). Equation (4 — 18) shows that the mass flux is constant throughout the conduction zone and equal to to the mass ablation rate m . Equations (4 — 18) and (4 — 19) can be solved for the plasma flow velocity to give: V = C0-(C2-C2)1'2 (4-20) The total outward flow of plasma energy (l/2)pVi + (5/2)pVC2 (see eq. (4 - 17)) through the absorption surface is 3p0C$, (obtained by setting V = C 0 at the absorption surface). This flow is balanced, as indicated by equation (4 — 17), by an inward heat flux Qin which provides the energy for the ablation process. We have already shown (see eq. (4 — 14)) that there is an outward heat flux QOUT equal to PQC^ which maintains isothermal conditions in the coronal region. Therefore energy balance requires that the absorbed laser intensity is equal to: lA = Qin + Qo»t=4poCZ = 4(^-)n0CZ . (4-21) 4-2-D Scaling laws Using equations (4 —5),(4 —18),(4— 19) and (4 — 21), the following expressions for the plasma temperature, the mass ablation rate and the ablation pressure are obtained: To = 4-^(j)l/i^fn-^lT , (4-22) m = 4 - 1 ' ^ ) 8 ' V y « » S ' ' , (4-23) and P, = r 2 / ' ( ^ ) 1 / ! / f n y (4-24) The density n0 will depend on the absorption regime. Two different cases can be considered. 42 i) Weak inverse bremsstrahlung absorption : n 0 — n,.,. For long laser wavelength and high intensity, inverse bremsstrahlung absorption is relatively negligible (see section 2 — 1 of chapter 2) and absorption of the laser energy occurs mainly at the critical density through resonant absorption (section 2 — 3). In this case n 0 is equal to the critical density which is given by: _ _ 47rVme€0 " ° ~ U " = e*X2 substitution in equations (4 — 22), (4 — 23) and (4 — 24) yields: T(eV) = 5.9 X 102(|),/S/1/SX[/S , (4 - 27) m(gcm- 2s- 1) = 7.1 X lO^f'^V , (4-28) P^(Mbar) = 5.1(|) 1 / 8/l / SX- 2 / S (4 - 29) where the units of IA and XL are respectively 101S W/cm 2 and /im . ii) Strong inverse bremsstrahlung absorption : n 0 < n c r For short laser wavelength and low intensity, inverse bremsstrahlung absorption is high and the energy can be totally absorbed before the critical density (see Figure 2 — 2 of chapter 2). It is possible to evaluate the maximum density n 0 beyond which very little laser energy is transmitted (Mora67). The absorption of laser energy through the plasma is (eq. (2— 10) of chapter 2): /+oo au{x)dx] (4-30) The absorption coefficient o u for inverse bremsstrahlung is given by the expression (2 — 8) of chapter 2 which we rewrite in the form : « „ ( * ) = a„e,re-s'2(^  )2(1 - ^ r<2 (4 - 31) "er "cr 43 (4-26) A = 1 - exp{-Qncrr0 '(—) / (1 -exp — — I exp where a is a constant which can be obtained by comparison with eq. (2 — 8). Using the expression for the density profile, equation (4 — 13), for an isothermal expansion in the plasma corona (T = T0), together with equation (4 — 4), equation (4 — 30) can be rewritten as: {x-x0)Y1/2 2(Z-X 0) . , -c~T dx} (4-32) Assuming (n0/ncr)exp — << 1 we can easily evaluate the integral to get: -aT->>*ner(^f^=ln(l-A) . (4-33) To evaluate the density no, we can assume67 that A = 0.86 , noting that a change of its numerical value will affect the absolute value of n0 but not its scaling with other parameters. Thus from equation (4 — 33) The density gradient scale length L = C0t for short laser pulse is roughly equal to C 0rt , where rL is the laser pulse length. On the other hand , if the laser pulse length is so long that C0TL becomes larger than the diameter of the focal spot Dapot, the spherical divergence of the plasma flow will limit L to a length of the order of D,pot 1 6 . Therefore L = i min (C0TL,Dspoi) (4 - 35) Finally substitution of equation (4 — 34) in equations (4 — 22), (4 — 23) and (4 — 24) yields the scaling laws for strong inverse bremsstrahlung absorption : T(eV) = 89(ZlnA) 2 / 9 (^ ) 2 / 9 L 2 / 9 Xl / 9 /y 9 , (4-36) m(gcm- 2s- 1) = 4.5 X 106(Zln A ) - 2 / 9 ( ^ ) 7 / 9 L - 2 / 9 X^ / 9rl / 8 , (4-37) ZJ 44 and PAbl (Mbar) = 6.6(Z In A)-1^)7^^''^917A/9 (4 - 38) where the units of IA are 1013 W/cm 2 and the units of X/, and L are fim . 4-2-E Discussion of model assumptions In this section we examine the model assumptions (one-dimensional flow and steady-state conduction zone) and derive criteria which delineate the regions of parameter space where they can be applied. i) One-dimensional flow In the model it was assumed that the plasma flow can be described using one-dimensional planar geometry. This will be true only if the lengths of interest are small com-pared to the focal spot size, a distance beyond which the flow will definitely be divergent16. In particular the scaling laws were determined from the values of the hydrodynamic vari-ables at the absorption surface. Therefore a reasonable criteria for a one-dimensional description to be valid would be that the distance from the ablation surface to the absorp-tion surface (length of the conduction zone) must be much smaller than the laser focal spot diameter. The length of the conduction zone can be obtained by integrating the energy equation (4 — 17) for x < x0. We find ^ 2 + y ) + Qi, = 0 , (4-39) where the constant of integration is equal to zero since the outward flux of kinetic energy and enthalpy is balanced by the inward heat flux (the amount of energy transmitted to the solid through the shock wave is small compared to the energy in the conduction zone). Using expression (4 — 20) for the plasma flow velocity and expression (4 — 7) for the heat flux, we can rewrite equation (4 — 39) 2K0Thl*dT dx = - 5- . 4 - 40) m 5C 2 ( T ) + (C70 - (C 2 - C2(D i/ 2) 2 45 Numerical integration of this equation from T — 0 at the ablation surface to T = T0 at the absorption surface gives60 : — " ^ i g f c f * « - « » This can be rewritten using equations (4 — 4), (4 — 5), (4 — 21) and (3 — 13) as : Let us evaluate this expression for typical conditions in our experiments, namely: IA = 101S W/cm 2 , A = 27, Z = 11 ; r»0 = 0.27 nCT and In A = 2.4 for X t = 0.266/tm ; n 0 =0.36n c r and lnA = 2.9 for \ L = 0.355 /im ; n 0 =0.55n e r and In A = 3.7 for X/, = 0.532/tm ; where n0 is given in Figure 2 — 2 of Chapter 2. This yields z 0 ~9-6/tm at 0.266/tm ; io ~ 16.3/tm at 0.355/tm ; and i 0 ~ 32/tm at 0.532/tm Thus J 0 is strongly dependent on the laser wavelength. Since our laser focal spot diameter is 80 /tm for 90 % energy content (see section 5 — 2 of chapter 5) the one-dimensional flow assumption is only marginally satisfied for our experimental conditions. 46 ii) Steady state flow in the conduction zone The plasma flow structure and the hydrodynamic variables p,v,T all depend on the laser intensity and therefore a complete steady state cannot exist when using a laser pulse whose intensity varies with time. However if the laser intensity does not change sharply on a time scale shorter than the times it takes the plasma flow to adjust to intensity changes then the flow is at all times in a " quasi-steady-state" i.e. in equilibrium with the changing laser intensity. Time-avaraged values of the measured ablation parameters can then be related to the time-averaged value of the laser intensity. A more precise criteria for this situation to exist65,80 is that the laser intensity should not change significantly in the time it takes a fluid element to flow from the ablation surface to the absorption surface. This flow time is given by: /'° dx . h • (4-42) Using equation (4 — 20) for V(x) and (4 — 40) for dx ,we fin by numerical integration of equation (4 — 42): Using equations (4 — 4) and (3 — 13) we can rewrite this expression r, = 1 . 3 0 X , 0 = . ^ ( ^ ) ^ (S .uoi , , (4-44) Evaluating this expression for the same experimental parameters that we used to evaluate expression 4 — 41 yields r, = 230ps at 0.266/xm ; r/=350pa at 0.355 fim ; and r/ = 600p« at 0.532/xm Since our laser pulse is near-gaussian with a full-width at half-maximum intensity of 2ns (see Section 5 —3 of Chapter 5), a steady state regime will exist for most of the interaction in our experiment. 47 Chapter 5 Experimental facility and diagnostics In this chapter we describe the experimental facility, experimental conditions and diagnostics which were used in our experiments. The laser system and the experimental arrangement are described in Section 5—1. Target irradiation conditions are detailed in Section 5 — 2. In Section 5 —3 we describe various diagnostics used. Finally in Section 5 — 4 we describe the procedure used in the fabrication of multilayer targets used in some of our experiments. 5-1 Neodymium-glass laser system and the experimental arrangement The experimental facility is based on a Quantel neodymium-glass laser system (model NG-34), which includes a Nd-YAG (neodymium-yttrium aluminum garnet) oscillator, a Nd-YAG preamplifier and two Nd-glass amplifiers. The aperture of the final amplifier rod is 25 mm in diameter. The laser oscillator is passively Q-switched with a dye-cell and provides a single laser pulse at 1.064 /im in the TEM 0o mode.The temporal pulse shape is approximately gaussian with a full width at half maximum (FWHM) intensity of 2 ns. The laser system also includes three KDP (potassium dihydrogen phosphate) crystals for harmonic generation at 0.532, 0.355 and 0.266 /zm. The system is capable of delivering 12 joules at 1.064 fim, 6 joules at 0.532 fim, 2 joules at 0.355 fim and 1.5 joule at 0.266 fim. A schematic of the laser system and an oscilloscope trace of the oscillator laser pulse are shown in Figure 5 — 1. A schematic of the experimental set-up is shown in Figure 5 — 2. The collimated laser beam (25 mm diameter) of 0.532, 0.355 or 0.265 fim wavelength is brought to the target chamber through a series of three or more dichroic mirrors. The laser light is then focussed on target using //10 optics. The use of dichroic mirrors, together with the chromatic aberration of the focussing lens, reduces the intensity of unwanted harmonic laser radiation on target to negligible level (less than one part in 106). As sh own in Figure 5 — 2, a beam 48 1.06 urn YAG oscillator i 1 YAG preamplifier con crystal >nvertor(s) 0.266 urn 0.355um 0.532 urn Glass amplifier Glass amplifier Figure 5-1 Schematic of the laser system and oscilloscope trace of the oscillator laser pulse. 49 splitter is inserted in the laser beam path for diagnostic purposes. It reflects part of the incident laser light towards a Gentec laser energy meter (model ED-200) which measures the incident laser energy,and part to a fast photodiode (Hamamatsu model R1193U, 350 ps rise time) which monitors the laser pulse shape. The same beam splitter is used to sample part of the backscattered light, which is monitored with similar energy and power detectors. The wavelength of the back scattered light was identified by placing different interference filters in front of the photodiode monitor. The overall transmission of the relay optics for the incident laser beam was measured for each laser wavelength. Also shown on Figure 5 — 2 is the optical arrangement used to image the laser focal spot onto the photocathode of a streak camera. Details of this measurement will be given in Section 5 — 2. The target chamber is a cylindrical vacuum vessel two feet high and tv/o feet in diameter. It is equipped with various ports to allow laser beam input, access for optical diagnostics and electrical connections. It can be evacuated with a forepump to less than 100 mTorr (1 Torr=132 Pascals) in a few minutes. When necessary a vacuum of the order of 10 - 6 Torr (monitored with an ionization gauge) can be attained within several hours using a 15 cm aperture diffusion pump attached to the chamber. The targets were mounted near the center of the vacuum chamber on a target wheel placed at the focus of the lens. The target wheel was driven by a stepping motor which could be operated from outside the chamber. It was possible in this way to perform a large number of laser shots on different targets without having to break the vacuum seal. The target normal was oriented 10° from the laser beam axis to prevent direct specular reflection feeding back into the laser chain. It also rendered the axis of symmetry of the plasma "blow-off" accessible to measurements. 5—2 Irradiation conditions In order to maximize the laser intensity, the laser light was sharply focussed onto the target. To position properly the focussing lens, the backscattered laser energy was moni-tored as the position of the lens was varied. For a constant laser energy the power density on target is highest at the best focus and the backscattered laser energy will consequently be maximum for this lens position. Typical results for the 0.532 /tm experiment are shown in Figure 5 — 3. For this particular measurement, the incident laser energy was 1.2 joule 50 Figure 5-2 Experimental set-up. 51 and 130 fim thick glass slides were used as targets. The same procedure was used to focus laser radiation of 0.355 and 0.266 fim on targets. Characterization of a laser-plasma interaction experiment requires the measurement of the spatial and temporal distributions of the laser energy at the target. This was made by imaging the laser focal spot onto the photocathode of a streak camera ( Hamamatsu Temporal Disperser System , model C1370-01) which yielded either a time-resolved, one-dimensional image or a time-integrated, two-dimensional spatial distribution of the laser intensity at focus. The focal spot was imaged using both a single and a double-lens imaging system (see Figure 5 — 2). It is important that the imaging optic does not introduce aberrations on a scale larger than the details we want to measure. To establish the spatial resolution and magnification of our imaging system, we imaged a reticle placed at the target plane inside the target chamber. The resulting image indicated a spatial resolution better than 10 fim with a depth of field of 100 fim. With lens configurations we obtained overall magnifications ranging from 30 to 45. Figure 5 — 4, shows the intensity distribution across the diameter of the focal spot as a function of time. This corresponds to a 0.355 fim laser pulse of 0.8 joule. Figure 5 — 5 shows a time integrated two dimensional image showing the focal intensity distribution. This corresponds to a 0.532 fim laser pulse of 3.4 joules. Detailed analysis in similar measurements showed that 90% of the laser energy is contained within a spot of 80 fim diameter and 60% of the energy within a spot of 40 fim diameter. Time resolved (30 ps resolution) measurements showed spatial intensity modulations (4 fim resolution) of less than 30 %, and time integrated measurements showed modulations of less than 10%. The laser pulse length was close to 2 ns (FWHM). Variations up to 30% Avere observed in the focal spot size and laser pulse length when the laser energy on target was changed. Changes in the spot size were attributed to a thermal induced lensing effect ocurring in the laser rods since the laser energy on target was varied by changing the flashlamp pumping of the laser amplifier rods. On the other hand, the non-linear property of the frequency conversion crystal introduced distortion in the laser pulse shape for high laser intensity. The incident irradiances <i>L specified throughout this work were defined according to the following formula 52 10 Figure 5-3 Backscatter signal as function of the lens position, \L = 0.532/im EL = 1.2J. 53 LASER SPOT STREAK CAMERA SLIT > x 54 55 where EL is the incident laser energy, TL the laser pulse length taken at FWHM and D the focal spot diameter containing 90 % laser energy. 5-3 Diagnostics To facilitate the experimental investigation a comprehensive system of diagnostics was developed in-house. These include an integrating-sphere photometer (Section 5 — 3 — A), Faraday cups* (5 — 3 — B), differential calorimeters* (5 — 3 — C), an X-ray crystal spectrometer (5 — 4 — D), an X-ray film calibration technique (5 — 3 — E), X-ray pinhole cameras (5 — 3 — F) and a multichannel X-ray detector (5 — 3 — G). In particular the X-ray film calibration technique69 and the multichannel X-ray detector70 constitute new and original developments and are described in detail. 5-3-A Integrating-sphere The simplest way to determine the fractional absorption of laser energy by the target is to measure the laser energy that is reflected or scattered from the target. It is easy to measure the energy reflected back through the focussing lens. However in laser produced plasmas , a large fraction of the unabsorbed laser light may be scattered in a rather diffused manner. Hence, a precise measurement of the energy reflected from the target requires a detector with a 47T collection angle. An integrating-sphere (or Ulbricht photometer) is ideal for this purpose. It consists essentially of a hollow sphere with its inner surface coated with a diffuse reflecting layer. A diagram of the photometer developed for our experiments is shown in Figure 5 — 6. It is made of two aluminum hemispherical cavities of 7.6 cm radius. The surface of the cavity was coarsely sand-blasted and then coated with a diffusing paint (Eastman Kodak white reflectance coating71 CAT 118 1759). The reflection coefficient72 of the paint ranges from 94% to 99% between 2500 and 9000 A. Separate apertures are provided for the incoming beam, the target holder and the radiation detectors. Their overall area is less than 10% of the total surface area of the sphere. Radiation scattered from the target is multiply * A description of the Faraday cup (Section 5 — 3 — B) and differential calorimeter detectors (section 5 — 4 — C) has been given in Reference 59. These two detectors were developed by A.Ng and P.Celliers. 56 57 reflected by the diffuse reflector surface and monitored by two calorimeters (Gentec model ED-200) positioned at angles of 45°and -135° with respect to the laser beam and in the plane of incidence. They are covered with a glass slide which blocks the plasma particles and X-rays and transmits only the laser light reflected by the sphere (most of the radiation energy emitted by the plasma is in the soft X-ray energy range, therefore mainly laser light, with negligible plasma radiation is seen by the detectors). An attractive feature of the Ulbricht photometer is that the measured intensity is independent of the angular distribution of the source73. The calibration of the sphere is therefore a straightforward procedure in which the target is removed and the laser light directly illuminates the rear wall of the sphere. The sphere efficiency, defined as Edet/Ein where Edet is the laser energy detected by the two detectors in the sphere and 2?,-„ is the input laser energy, is typically of the order of 4%. Absorption measurements using the integrating sphere are presented in Chapter 6. 5-3-B Faraday cup A set of Faraday cups were used to determine the velocity of the ions in the expanding plasma. This is done by measuring the time of flight of the ions from the target to the Faraday cup detector which is positioned a known distance from the target. A schematic of a Faraday cup with its electrical connections is given in Figure 5 — 7 — A. As shown in this figure the plasma which enters the detector first passes through a grid connected to ground and then encounters another grid which is biased at -215 V. This biased control grid rejects the electrons which do not have enough kinetic energy to overcome the grid potential. The ion stream then enters the cup through a 7 mm hole and strikes the collecting surface of the detector which is connected, via a 50 ohm coaxial cable, to an oscilloscope. A current pulse is produced which is proportional to the current density of the ion stream. The geometry of the cup serves to restrict the escape of secondary electrons which are generated when the ions strike the rear end of the cup. Any secondary electrons escaping from the cup are reflected by the negatively biased grid at the entrance. Figure 5 —7 —B shows an example of an ion current trace for an aluminum target irradiated with 0.355 fim laser light at 1013 W/cm 2 . The detector was positioned along the target normal, 35 cm away from the target. The ringing at the beginning of the trace is 58 OSCILLOSCOPE SIGNAL 50 Q I •215:1: I BIAS ft OTHER DETECTORS PLASMA (A) ( B ) Figure 5-7 (A) Faraday cup electrical connections. (B) Oscilloscope display of a Faraday cup signal. 59 produced by the X-ray burst at the time the laser light is incident on the target, ejecting secondary electrons from the collector surface. This provides an excellent zero time marker for the time of flight analysis. Measurements obtained with these detectors are described in Chapter 8. 5-3-C Differential calorimeter A set of differential calorimeters was used to measure the energy of the expanding plasma. The differential calorimeter consist of two minicalorimeters placed side by side and connected in a differential configuration (see Figure 5 — 8). One minicalorimeter measures all incident ion, electron, X-ray and light energy. The other minicalorimeter is covered with a glass slide and collects only the visible light energy. The difference in the signal of the two minicalorimeters constitutes the output of the detector. Evaluation of the X-ray contribution to the signal is then necessary in principle to obtain the particle energy. For our experimental conditions the X-ray contribution to the detector signal was assumed to be small and was neglected. The active element in each calorimeter is a Peltier effect thermoelectric juction in which a temperature gradient accross the device produces a voltage difference which can be recorded. The thermoelectric junction was attached on one side to an aluminum block which serves as heat sink, and on the other side to a small aluminum plate which serves as an absorber for incident plasma energy. The device is calibrated by discharging a known amount of electrical energy through a resistor embedded in the aluminum absorber plate. Measurements made with the differential calorimeters are discussed in Chapter 8. 5-3-D X-ray crystal spectrometer An X-ray crystal spectrometer was developed to monitor X-ray line radiation from highly ionized silicon plasma. The instrument was used in an investigation of electron thermal transport (see Chapter 7), and was also used in a new calibration technique for X-ray films (see Section 5 —3 —E). A schematic of the X-ray crystal spectrometer is shown in Figure 5-9. In this figure we show the use of X-ray film (covered with a step-wedge X-ray filter for intensity calibration purpose) as radiation detector; we have also used other detectors such as a PIN diode (see section 5 — 3 — E) and a plastic scintillator coupled to 60 HEAT SINK \ EES THERMOELECTR IC ABSORBER DEVICE GLASS SLIDE CAL IBRAT ION RESISTORS ( A ) ELECTRICAL CONNECTOR MOUNTING POST KSiMittMMHn ESSHBSa ( B) Figure 5-8 (A) Schematic of a differential calorimeter. (B) Oscilloscope display of a differential calorimeter signal. 61 a photomultiplier tube (see Chapter 7). Typical spectra obtained with the instrument can be seen in Figures 5 — 10, 5 — 15 and 7 — 2. The key components of the spectrometer are a beryllium entrance window which protects the crystal from the plasma "blow-off", a flat potassium acid phthalate (KAP) crystal and an X-ray detector. The target-to-crystal distance was typically 12 cm with the crystal-to-detector distance of the order of 10 cm. The crystal length is 5 cm (2.5 cm wide and 3 mm thick). It is inclined 13.2° with respect to the line of sight to the target. Radiation from the target can fall on the crystal at angles of incidence varying from 10.9° to 16.6°. At each point on the crystal the local glancing angle of incidence and the Bragg diffraction condition X = 2dsin0 (5-2) determine a unique value of photon wavelength that will be diffracted towards the X-ray detector with high efficiency. The 001 crytal planes, which have an interplanar distance of 13.31 A, are used to reflect the X-rays. The reflection coefficient98 of the crystal is of the order of 7 X IO - 6 in the wavelength range covered by the instrument (from 5 to 7.5 A). No slit was necessary between the target and the spectrometer, and good spectral resolution could be obtained because of the small size of the source (typically less than 100 fim). 5-3-E A n in-situ X-ray film calibration technique X-ray films have been widely used for the study of laser-produced plasmas74""80, both in X-ray imaging and spectroscopy. They offer the advantages of excellent spatial resolution and good dynamic range. However, for quantitative measurements the response of the film has to be known and a characteristic curve relating film optical density to exposure has to be determined. The exact shape of this characteristic curve depends on many factors. First, it is wavelength dependent81 so that calibration must be performed in the spectral region that the film is to be used for. It may also be affected by the film development procedures. The type of chemicals used, processing times and temperatures have to be well controlled to obtain consistent and reproducible results. Furthermore, it is well known that even for the same exposure the optical density of the film when measured with microdensitometers will depend on the type and the setting of the instrument82. Finally, changes in sensitivity between film batches and possible changes in film composition by manufacturers83 can 62 modify the characteristic curve as well. Because of these considerations, users of X-ray films in general have to perform their own calibrations. Film calibration is usually carried out in an independent measurement. A charac-teristic curve relating optical density to exposure is determined using a standard radiation source such as a fluorescent radiation source 8 1' 8 5 - 8 4. This is then used to analyse film ex-posure obtained in other measurements. Because of the aforementioned factors, extreme care must be exercised to ensure the applicability of the characteristic curve. Thus, an in-eitu calibration technique would be most advantageous, since it would also eliminate the need of additional calibrations. We here describe an in-situ, relative X-ray film calibration technique. The technique is based on film exposure measurements of an X-ray spectrum transmitted through a step-wedge absorption filter. To demonstrate the technique we will discuss the construction of a characteristic curve for Kodak SC5 film using laser-produced silicon and aluminum plasmas as radiation sources. Results of the method showed good agreement with that obtained using an X-ray PIN diode as the recording device. 5-3-E-i Calibration technique and characteristic curve In principle, the technique is a direct extension to the X-ray domain of the neutral-density step-filter technique used for film calibration in the visible spectrum86. The only difference is that transmission through X-ray absorption filters is wavelength dependent and the spectral distribution of the source needs to be known. Accordingly, the radiation to be studied is first spectrally resolved. The radiation is then passed through a step-wedge absorption filter and recorded on film. The wavelength of each spectral line can be identified and the attenuation in intensity through each channel of the step-wedge filter calculated from tabulated data in the literature. Hence, for an N-channel filter, there will be N different exposures on film for each line, from which a characteristic curve can be composed. The process can be repeated for each spectral line and a set of characteristic curves corresponding to different wavelength can be obtained. In general, these lines all lie within a very narrow spectral range so that the curves can be combined to form one single characteristic curve which is applicable throughout the spectral region of interest. Such a combination process is not absolutely necessary. It will, however, reduce the number of 63 channels in the step-wedge filter required to provide film exposures covering its full dynamic range since different spectral lines will have different intensities. A schematic of the experimental set-up is shown in Figure 5 — 9. Laser radiation of wavelength 0.53 /im and pulse duration 2 ns (FWHM) was focussed on planar, solid targets with //10 optics. The angle of incidence was 10° off target normal and the laser intensity on target was about 101S W/cm 2 . Targets used include aluminum foils, glass slides and layered targets of polystyrene-on-glass. The X-ray emission from the laser-produced plasma was analysed with a slitless, flat KAP crystal spectrometer (see section 5 — 3 — D) at 15° off target normal. X-rays diffracted by the crystal were recorded on film through a step-wedge absorption filter as shown on Figure 5 — 9. A three-channel step-wedge filter was used for the study of X-rays from silicon plasmas in glass or layered targets. The absorption filter used in each channel is listed in Table 5-1. In this experiment, Kodak SC5 X-ray film86 was used. Table 5-1 List of foils used for three-channel step-wedge filter. channel filters) 1 10 /im Be 2 10 /im Be + 2/xm Al 3 10 fim Be + 5/tm Al A typical X-ray spectrum obtained from a single laser shot on a planar glass target is presented in Figure 5 — 10. This shows the silicon spectrum from 5.1 to 7.1 A. The three intensity bands normal to the dispersion axis correspond to the different exposures due to the three-channel filter. The transmission of the exposed film was measured with a microdensitometer87. Linearity of the instrument was verified using a set of calibrated neutral density filters. A microdensitometer scan, made over each exposure band on the film, is shown in Figure 5—11. We have arbitrarily defined the signal obtained when no film was placed between the light source and the photomultiplier in the microdensitometer as 100% transmission. The zero level was established by placing an opaque cover in the 64 Figure 5-9 Experimental arrangement used in the X-ray film calibration technique. 65 Channels (2) (1 ) ( 3 ) 9 SATELLITES — INTERCOMBINATION — Si XLH 1s2-1s2p-^ Si XE" L a 1s-2p Si XDI 1s-1s3p SiXHI 1s-1s4p — Figure 5 -10 Silicon X-ray spectrum obtained with a three-channel step-wedge filter 10 /im Be, (2) 10 /tm Be + 2 /tm A l , (3) 10 /tm Be + 5 /tm A l . 66 microdensitometer beam path. Transmission through the unexposed part of the film was about 55%. Four silicon lines have been used for compiling the film characteristic curve. These are (i) Si XIII Is^1 S0) - l84p(lP1), (ii) Si XIII U2{1S0)-U3p{1P1), (iii) Si XIV La and (iv) Si XIII ls2(150) — ls2p(1Pl). For each of these, three different values of film transmission corresponding to three different levels of exposure can be obtained. The relative exposure for a given line depends on the attenuation of intensity of the filters in the various channels. The list of silicon lines used, their wavelength X and the transmission factor exp /*t(X)p,r,] in each channel are given in Table 5-2. Here /i,-(X) is the mass absorption coefficient88, p{ the density and ry the thickness of the filter used in each channel. Film transmission as a function of relative exposure can be determined for each spectral line. The results are shown in Figure 5 — 12, unit exposure corresponds to that in Channel 1 (10 tun beryllium filter). As mentioned earlier, because of their difference in intensity, each spectral line give a set of points covering different sections of the characteristic curve. Given that the lines all lie within a narrow spectral range (5.40 to 6.65 A) it is reasonable to assume that the characteristic curve remains unchanged in this this interval. So, irrespective of their wavelength, the set of data points are shifted along the relative exposure axis to form a single continuous line with a characteristic elongated S-shape. The final curve thus obtained is shown on Figure 5 — 13. Accuracy of the composite curve depends on the number of channels used in the step-wedge X-ray filter and on the number of spectral lines available for analysis. It also depends on the accuracy of the values of the thicknesses of the X-ray absorption filters and their corresponding mass absorption coefficients and on the accuracy of the microdensitometer used. 5-3-E-ii Evaluation of technique In general, once the characteristic curve is obtained it can be used to determine the relative intensity distribution of the X-ray spectrum recorded by the film. The technique therefore serves as an in-situ calibration which is obtained simultaneously with the data. The calibration technique was used in a preliminary investigation of electron ther-mal energy transport in laser produced plasmas. The principle of the latter measurement 67 CO CO CO < (A ) ( B ) (C) Figure 5—11 Silicon spectrum transmitted through different filters : (a) 10 /im Be, (B) 10 /im Be + 2 /im Al, (C) 10 /tm Be + 5 /im Al. (S) satellites, (1) Si XIII la2^ S0)-lsSp^P,). (2) Si XIII U^So) - l «2p ( 1 F 1 ) . (3) Si XIV La. (4) Si XIII ls2{'S0) - l «3p ( 1 P 1 ) and (5) Si XIII l « 2 ( 1 S 0 ) - l « 4 p ( 1 / , 1 ) . 68 Table 5-2 List of silicon lines used for constructing the calibration curve and their transmission factors through the different channels Spectral X(A) Transmission: exp J2i[~t1i(^)pir>] line channel 1 channel 2 channel 3 10 /xm Be 10 / t m Be 10 fim Be -f 2 fim Al +5 fim Al Si XIII Is2 - ls4p 5.405 0.915 0.382 0.103 Si XIII Is2 - ls3p 5.681 0.903 0.331 0.074 Si XIV La 6.182 0.878 0.249 0.038 Si XIII 1« 2 - ls2p 6.649 0.848 0.185 0.019 69 RELATIVE EXPOSURE Figure 5-12 Film transmission % for the four silicon lines used. Exposure is normalized to the exposure in channel (1) 10 /im Be; (•) Si XIII l 2 - ls2p; (•) Si XIV L0;(o) Si XIII l 2 - ls3p; (•) Si XIII Is2 - ls4p. 70 £ 30 RELATIVE EXPOSURE Figure 5-13 Relative intensity calibration obtained with a silicon spectrum. 71 together with more complete results are given in Chapter 7. Here we briefly describe this measurement as an independent evaluation of the film calibration technique. Glass slide targets coated with different thicknesses of polystyrene were irradiated with 0.53 / t m laser light. In this experiment we monitored the silicon spectrum (5.1 to 7.1 A) emitted. As the CH layer thickness was increased, intensities of the silicon lines decreased and eventually disappeared. The experimental set-up was similar to that in Figure 5 — 9. In addition to using the X-ray film to record the silicon spectrum, an X-ray PIN diode89 was also used to provide an independent measurement. However, the PIN diode only monitered the group of silicon lines centered around 6.7 A . These included the Si XIII ls2(150) — l « 2 p ( 1 F i ) resonance line at 6.65A, the intercombination line at 6.69 A a n d the satellite lines from 6.6 to 6.74 A (see Figure 5 — 10). The integrated intensity of this group of silicon lines mea-sured by the X-ray film and the diode as a function of CH layer thickness is presented in Figure 5 — 14. Intensities recorded by the film were analysed using the calibration technique described above. The results show good agreement with that obtained using the diode. As pointed out above, accuracy of the film characteristic curve obtained using this technique depends on the number of channels used in the step-wedge X-ray filter and the number of spectral lines available for analysis. As a further demonstration of the technique, we have also obtained a relative intensity calibration using a four-channel filter and six spectral lines from aluminum plasmas. For this measurement, the absorption foils used in each channel of the step-filter are shown in Table 5-3 and the spectral lines used, their wavelengths and the transmission factor i n each channel are listed in Table 5-4. A typical aluminum spectrum recorded on film is presented in Figure 5 — 15. Figure 5—16 shows the film transmission as a function of relative exposure for each spectral line, with the relative exposures normalized to unity. A composite curve can then be obtained as shown in Figure 5 — 17. Evidently, this provides a more detailed measurement of the film characteristic curve. It is in good agreement with that obtained trom the silicon line spectrum (Figure 5-13). 72 10 100 CH COATING THlCKNESS(/4,m) Figure 5-14 Intensity of a group of silicon lines as a function of CH thickness. 73 Table 5-3 List of foils used for four-channel step-wedge filter. channel filter(s) 1 15 fim Be 3 2 15 fim Be + 2 fim Al 15 fim Be + 4/im Al 4 15 fim Be + Qfim Al Finally, this technique can also give an absolute film calibration since the film expo-sures are correlated with signals from PIN diodes whose absolute response can be calibrated independently. 5-3-F X-ray pinhole camera X-ray images of the laser-target interaction in our experiment were obtained using pinhole cameras. In an X-ray pinhole camera a small hole in an otherwise opaque substrate is used to provide two-dimensional imaging. The geometric resolution of a pinhole camera is given by9 0 where d is the spatial resolution of the pinhole camera, D is the diameter of the pinhole, and M is the magnification (the camera magnification is given by the ratio of the image distance from the pinhole to the object distance to the pinhole). Figure 5 — 18 shows an X-ray pinhole photograph of the plasma produced in our experiments. This particular image was obtained by irradiating 50 fim thick aluminum foil target with 1.45 joule of 0.355 fim laser light. The pinhole camera was positioned at 5° from target normal. Imaging was obtained using a 15 fim diameter pinhole covered with a 10 fim beryllium filter. Kodak SC5 X-ray film, calibrated with the technique described in section 5 — 3 — E was used for the measurement. The image magnification on the film was d = D{l + 1/M) (5-3) 74 T a b l e 5-4 List of aluminum lines used for constructing the calibration curve and their transmission factors through the different channels. Spectral X(A) Transmission: exp 2Z.[—/*<(M/'<7'<] line channel 1 channel 2 channel 3 channel 4 15 nm Be 15 /im Be 15 /im Be 15 /im Be + 2 /im Al +4 /tm Al +6 //m Al Al XIII L 7 5.739 0.854 0.314 0.116 0.043 Al XIII Lp 6.052 0.833 0.261 0.082 0.026 Al XII la2 - Isbp 6.175 0.821 0.237 0.068 0.020 Al XII la2 - laip 6.314 0.8312 0.223 0.061 0.017 Al XII Is2 - ls3p 6.635 0.786 0.178 0.040 0.009 Al XIII Lc 7.172 0.727 0.122 0.020 0.003 75 Channels (4) (3K2M1) f — A l XHI La 1s-2p j « — A l XE 1s-1s3p — A l XE 1s-1sAp — A l XE 1s2-1s5p \ — A l XUI lyj1s-3p — AIXJJI ly 1s-Ap Figure 5-15 Aluminum X-ray spectrum obtained with a four-channel step-wedge filter. 76 o I—I CO CO CO < RELATIVE EXPOSURE Figure 5 - 1 6 Film transmission (%) for each channel for the six aluminum lines used. Exposure is normalized to the exposure in channel (1) 15 / t m Be; ( ) Al XIII La; ( ) Al XII l 2 - ls3p; ( ) Al XII Is2 - \sAp; ( ) Al XIII Lfi; ( ) Al XII Is2 - l«5p; ( ) Al XIII L 7 . 77 o CO CO CO < RELATIVE EXPOSURE Figure 5-17 Relative intensity calibration curve obtained with an aluminum spectrum. 78 5.7 . This together with the pinhole size yielded a spatial resolution of 18 /im. Only X-ray with energies greater than 1 Kev (12.4 A) were transmitted by the X-ray filter. The X-ray image of Figure 5 — 18 shows a circular uniform region of X-ray emission. Also shown on this Figure is a microdensitometer scan of the image across its diameter. Analysis shows that the diameter of the X-ray emission region at the 0.1 intensity points is approximately 32 /im. This indicates that the X-ray emission is mainly coming from that part of the target irradiated by the high intensity central region of the focal spot. For the quantitative analysis of the X-ray image it was necessary to use the X-ray film in its linear response region. When the X-ray film is operated in its saturation region even the low intensity X-ray radiation emitted by the peripheral plasma appears on the X-ray image. This can be seen in Figure 5 — 19. This pinhole photograph was obtained for experimental conditions very similar to that of Figure 5 — 18, except in this case, the camera magnification was only 2.1, increasing the X-ray intensity on the film. From the microdensitometer scan of the X-ray image the maximum size of the X-ray emission region is estimated to be 155 /im in diameter. In figure 5 — 20 a series of X-ray images of the plasma using two pinhole cameras is shown. For this series of pictures, thin strips (1 mm wide) of aluminum foils of different thicknesses were irradiated with 3 joules of 0.532 /im laser light. One pinhole camera was facing the plasma at target normal, while the other camera was viewing the target edge-on. The front side camera had a 15 /im pinhole and a 5 /tm aluminum filter and the other one had a 10 /im pinhole and a 10 /im beryllium filter. In both cases the camera magnification was approximately two. To obtain a high contrast image the Kodak SC5 X-ray film employed was used in its saturation region. In this series of pictures we clearly see a plasma developing at the back of the target even for aluminum foil as thick as 9 /tm. This was attributed to the rapid two dimensional expansion of the foil (because of the small focal spot used in our experiments) after it has been traversed by a shock wave. The laser light can then propagate into the rarefied target material which is transformed into a hot X-ray emitting plasma. For most of our experiments 50 /im thick aluminum foil were used to eliminate the effect of target disassembly on the measurements. 79 AL= 0.355 um EL=1.45J. TL= 2 ns 50 um A l target magnified image (X175) of X-ray pinhole photograh. pinhole size = 15 um 10um Be filter . camera magnification : 5.67 Figure 5-18 X-ray pinhole photograph of the i 1 1 1 r 100 200 300 um microdensitometer scan asm a. 80 0~ 200 400 600 um microdensitometer scan Figure 5-19 X-ray pinhole photograph of plasma. The film is saturated showing the full size of the X-ray emission region. 2 um A l 5urn A l 9 u m Al (A) (B) Figure 5-21 Series of X-ray pinhole photograph obtained by irradiating thin strip (1mm wide) of aluminum foils of different thicknesses. (A) camera facing tha plasma at target normal, (B) camera viewing the target edge-on. The laser is coming from the right. \ L — 0.532nm, EL = 3J. 82 5-3-G Multichannel X-ray detector In experimental studies of laser plasma interactions, it is of great interest to mon-itor the plasma electron temperature in every laser shot on a target. Such a diagnostic may provide information on laser absorption and plasma heating mechanisms. For these high temperature plasmas the X-ray continuum has been commonly used to determine the electron temperature91. Typically, the X-rays emitted are n. insured using either a sin-gle detector (such a PIN diode or a scintillator-photomultiplier combination) scanning the X-ray spectrum for different shots, or an array of detectors, each of which monitors simul-taneously X-rays of a particular energy. The single detector method is relatively simple and inexpensive, but data must be taken over many laser shots in order to obtain a spectrum. The multi-detector array method, on the other hand, can yield an electron temperature measurement in a single shot and can be used for an on-line diagnostic system. Yet, such an array also has several disadvantages, namely, (i) the large number of individual detectors and recording devices implies complexity and high system cost, and (ii) the size of the array makes the measurement particularly sensitive to angular anisotropy of X-ray emission from the plasma. We have developed a simple electron temperature monitor70 which consists of a step-wedge X-ray filter and a charge-coupled device (CCD), modified to be sensitive to X-rays. This monitor offers the advantages of (i) compactness, (ii) simplicity of operation, (iii) versatility in the number and configuration of different X-ray channels and (iv) relatively low cost. 5 — 3 — G — i System design The electron temperature monitor is basically a multichannel X-ray analyser. The different energy channels are defined by the steps of a step-wedge X-ray filter. Such a filter can be fabricated by stacking foils (beryllium or aluminum, for example) of varying length and thicknesses (Figure 5 — 21 — A). This filter is then matched to the X-ray sensor, an EG&G Reticon RL512S92. To achieve the desired X-ray sensitivity in the energy range of 1-10 KeV, the thickness of the Si0 2 layer over the silicon diode elements is limited to 1 /im. The Reticon is a self-scanning array of 512 discrete photodiode elements. Center-to-center spacing of the diode elements is 25 /im, and the aperture height of each element is 2.5 mm. 83 The large aperture height is chosen to provide high output. The X-ray response82 of such an array is shown in Figure 5 — 21 — B. The internal functions of the Reticon array can be divided into two essential parts: the photodiode array and the shift registers. The charge produced on each photodiode element is proportional to the integrated radiation intensity impinging on the photodiode sensitized surface. Under normal operation (video mode) the accumulated charge on each photodiode element is transferred simultaneously into its associated shift register element by a transfer pulse ("start pulse" in the EG&G literature92 from an external clock). The charges in the shift register elements are then scanned out serially after the trailing edge of the transfer pulse. An integration circuit92 (EG&G RC 1024) further provides a sample-and-hold output waveform with improved dynamic range. Since the CCD is self-integrating, its dark current will cause saturation of the light sensitive elements in approximately 0.5 second. The device has to be continuously cleared to eliminate such saturation. In order to process single-shot events (eg. laser shots) an additional circuit is required to provide synchronization between the laser pulse, the video output of the array and the recording instrument. The synchronization circuit* is shown in Figure 5 — 22 — A and the associated timing diagram in Figure 5 — 22 — B. It is allowed to run in a continuously scanning mode with a scan frequency of 250 kHz. The timing of the laser pulse is asynchronous with the scanning of the Reticon array. A photodiode picks up part of the laser light and sets a latch "A" whose output raises the data input of the D-type flip-flop "B". Simultaneously, the charges produced by impinging X-rays from the laser-induced plasma are stored in the diode array elements for time interval Atx. These charges are then transferred into the associated registers by the transfer pulse (time interval At2). The trailing edge of the transfer pulse is used to clock the flip-flop UB" and to reset the latch "A" . At the same time the internal circuitry of the Reticon board activates the scanning clock and the video information is scanned out (time interval At3). Flip-flop "B" is reset by the end-of-line (EOL) pulse generated by the Reticon board. Hence, a pulse which is time coincident with the video output appears as the output of "B" in the scan period immediately after the laser pulse. This synchronization pulse is at present used to trigger a storage oscilloscope which displays the video waveform from the array. * Designed by K. Fong 84 Figure 5-21 (A) Schematic diagram of the electron temperature monitor. (B) Theoretical X-ray response of the Reticon array. loser CCD E6 & 6 R e t i c o n board photodiode sensor (7) video signal EOL © transfer © pulse •f> pulse shaper to scope or A/D CP CP syn pulse (D B (b) (j) laser pulse (2L © latch A output j i transfer © pulse n © syn 5V,200ns © EOL - Integration-time Figure 5-22 (A) Block diagram of synchronization circuit. (B) Timing sequence of syn-chronozation circuit. 86 5 — 3 — G — II System calibrations Two calibrations have to be made before the system can be used for electron tem-perature measurements: (i) relative response of the Reticon photodiode elements and (ii) linearity of the response. These calibrations were performed using X-rays produced from the irradiation of aluminum foil targets (25 >im thick) with 0.53 /im laser radiation of 2 ns (FWHM) pulse duration. Laser intensities on target were varied between 1012 and 101S W/cm 2 . For the calibration of the relative X-ray response of the Reticon elements, the step-wedge was replaced with a single beryllium filter 10 /im thick, which corresponds to a cut-off energy t of 1 keV (c is defined by K{i)d = 1, where K{t) is the absorption coefficient as a function of t and d the absorber thickness). Results of the measurement are shown in Figure 5 —23 —A in which the output signal from each element is displayed. The rising and falling edges correspond to the first and last exposed elements, respectively. The radiation shield (see Figure 5 — 21 — A) blanks off approximately 30 elements at each end of the linear array. Evidently, the response is flat to within ± 5 % accross the entire length of the array used. To measure the linearity of response, a 9-channel step-wedge filter was used in the electron temperature monitor. The filter was obtained by stacking on top of a beryllium support foil (10 /im thick) one 5 /im and eight 2 /im thick aluminum foils. The resulting X-ray filters in the different channels are listed on Table 5-5. Table 5-5 X-ray filters for various channels of the electron temperature monitor Channel No. 1 2 3 4 5 6 7 8 9 Al filter (/im) 5 7 9 11 13 15 17 19 21 Be filter (/im) 10 10 10 10 10 10 10 10 10 87 > Q > (A) 1ms /DIV. (B) 1 ms/ DIV. Figure 5-23 (A) Relative response of diode elements of the Reticon array. (B) Output waveform for a nine-channel electron temperature monitor. 88 2.5 0 0.1 0.2 0.3 0.4 0.5 P-I-N D i o d e s i g n a l ( v o l t s ) Figure 5-24 Linearity of response of the Reticon detector. 89 Each channel extends over approximately 50 photodiode elements in the Reticon array. A typical output waveform, representing the X-ray signals in the nine channels of the monitor, is shown in Figure 5—23—B. The X-ray emission was also measured simultaneously by three PIN diodes (Quantrad 100-PIN-250) with aluminum filters of 18.5, 37.5 and 50 /im respectively. These PIN diodes are linear for output signals up to 30 volts. The linearity of the 9-channel system is demonstrated in Figure 5 — 24 in which the output signals from channel 1 are plotted against that from the PIN diode with a 50 /im aluminum filter. The response of the array is therefore linear for output signals up to 2.3 volts. With an observed noise level of 10 mv due to transient switching noise this corresponds to a dynamic range of 230 to 1. In our experiment the detector was used to measure the dependence of the electron plasma temperature on laser intensity. The results of these measurements are given in Chapter 6. 5-4 Multilayer target fabrication In Chapter 7 we discuss an investigation of electron energy transport using multilayer targets. The target fabrication procedure is described here. 5 — 4 — A Fabrication of CH-glass multilayer targets For the fabrication of plastic film (CH) targets, we used a technique described in reference 94. It is schematically represented on Figure 5 — 25. The plastic solution was prepared by dissolving polystyrene98 into toluene. Glass slides (130 thick) were held vertically in the solution by spring clips. The solution was then allowed to drain away through the tap (Figure 5 — 25). The initial level of the film solution in the funnel was arranged to be well above the top of the glass slides so that steady flow conditions were achieved before the meniscus reached the specimen. The rate of fall of liquid level down the length of the glass slide is approximately constant and this results in a fairly uniform film. Various film thicknesses ranging from less than a 1 /im to more than 100 /im were obtained by changing the concentration of the film solution and repeating the coating procedure. Rapid drying of the film coating with a heat gun was found to be a good technique to obtain good quality 90 coating. The film thicknesses were then measured using a Talysurf stylus instrument, or for the thicker film a digital micrometer. 5 — 4 — B Fabrication o f Mg-, Sn-, Ag- and Au-glass multilayer targets The metallic coating targets were made by an evaporation technique. The coating thickness was monitored during the evaporation process using a crystal thickness monitor. It was difficult to obtain film thicker than 3 um during one evaporation run and repeated procedure was necessary to produce the thicker film necessary for some of our targets. 91 I (X) Tap V polystyrene solution glass slide Figure 5-25 Experimental arrangement used in the fabrication of CH-niultilayer targets. 92 Chapter 6 Experimental measurements of laser energy coupling to targets 6-1 Absorption measurements In our experiment the absorption of laser light by the target was determined by measuring the incident and scattered light, the difference being the absorbed laser energy. The experimental arrangement has been shown in Figure 5 — 6 of Chapter 5. The incident laser energy was measured using a Gentec laser energy meter (model ED-200). The scat-tered light measurement consisted of measuring both the light scattered back through the focussing lens and the light scattered diffusely in a 4n geometry. The backscatter energy was monitered by a Gentec energy meter and the diffuse scattered light by a calibrated integrating sphere which enclosed the target. A description of the integrating sphere has been given in section 5 — 3 of Chapter 5. Targets consisting of 50 /im thick aluminum foils, 25 /tm thick gold foils and 130 /im thick polystyrene (CH) foils were irradiated with laser pulses of 2 ns (FWHM) at 0.532, 0.355 and 0.266 /tm wavelength. The targets were sufficiently thick to prevent target disassembly while the absorption is still in progress (see Section 5 — 3 — F). The laser light was focused on target using f/10 optics, at an angle of incidence of 10° from target normal. At focus, 90 % of the laser energy was contained in a spot of 80 /im diameter. The laser beam was linearly polarized with its electric field inclined at 45° with respect to the plane of incidence. In Figure 6 — 1 we have plotted the percentage of energy absorption as a function of the incident laser intensity. The error in the absorption measurement is estimated to be less than 4 %. In this figure data points with error bars represent averages of multiple shots at nearly identical irradiance. (All error bars given in this thesis correspond to standard deviation in a single set of measurements). Data points with no error bars represent single laser shot measurements. The data show very high level of absorption for all laser wavelengths, laser intensities and targets of different Z. For 0.266 and 0.355 93 100, • D O • °a C. 90 50um A l 80 25Mm AUG 130 urn CH A oAL= a266 pm • AL= 0.355 Mm • XL= 0.532 Mm JAp; 0.355 um J L 1 J L 6 8 10 ,13 6 8 10' INCIDENT LASER INTENSITY (W/cm) Figure 8-1 Absorption fraction as function of laser intensity for different laser wave-lengths. 94 for all laser wavelengths, laser intensities and targets of different Z. For 0.266 and 0.355 /im laser radiation and aluminum targets we observed ~ 95 % absorption for intensities between 1012 - 2 X 101S W/cm 2 . The same absorption level was also measured at 0.355 /im when instead polystyrene (CH) foils or gold foils were used as target. At 0.532 /im and for aluminum targets, ~ 95% absorption was also observed for incident laser intensities less than 1013 W/cm 2 . However, for higher intensities the absorption started to decrease and at 5 X IO13 W/cm 2 it was around 85 %. All of these observations are consistent with inverse bremsstrahlung being the dom-inant absorption mechanism. As discussed in Chapter 2 no other absorption mechanism can produce such high level of absorption. In Section 2 — 1 of Chapter 2, using equation (2 — 13), the absorption was estimated to be 100 % for our experimental conditions. In this calculation we assumed a plasma with a constant temperature and constant density gradi-ent scale length. No scattering of the laser light was taken into account. These assumptions are not completely realistic and may explain the reason for the lower absorption fraction (~ 95%) observed in our experiment. We recall that inverse bremsstrahlung absorption in an exponential density profile (see equation 2 — 13 of chapter 2) is stronger for short laser wavelengths, low plasma temperature (low laser intensity) and high Z plasmas. For laser intensities higher than 1013 W/cm 2 we have indeed observed stronger absorption for 0.266 and 0.355 /im than for 0.532 /im. At 0.532 /im, we have also observed a decrease in the absorption for increasing laser intensity (increasing plasma temperature). No dependence on the target Z value was observed at 0.355 /im because the absorption was nearly complete even for low Z plasmas. In the next section we discuss measurements of the plasma electron temperature which provided further evidence that inverse bremsstrahlung is the dominant absorption mechanism in our experiment. 8-2 Plasma electron temperature The absorbed laser energy is thermalized in the plasma. Therefore more information can be gained about the absorption mechanisms by measuring the plasma temperature and its dependence with the laser parameters. The plasma electron temperature can be obtained by measuring the X-ray line or continuum emission from the plasma119. In our experiment 95 suggested by Jahoda et al 9 1 . In this technique the X-ray continuum transmitted through different X-ray filters is measured and the result is compared to calculated curves for different plasma temperatures. Details of these calculations are described in Appendix A. The dependence of the plasma electron temperature on laser intensity was measured in a series of experiments in which aluminum targets were irradiated with laser pulses of 2 ns at 0.532 /im at irradiances of 3 X 1012 - 1.1 X 101S W/cm 2 . The measurements were made using the multichannel X-ray detector described in Section 5 — 3 — G of Chapter 5. The detector was used in a 9 channels configuration where the X-ray filters for each channel consisted of a 10 um beryllium support foil plus aluminum foils ranging in thickness from 5 um to 21 um in steps of 2 um. Using this detector it was possible to monitor the plasma electron temperature for every laser shot on target. In the analysis of the results it was assumed that the observed signals were solely due to continuum emisssion with no significant X-ray line contribution. This assumption is not unreasonable because all the X-rays lines emitted by the aluminum plasma ( Figure 5 — 15 of Chapter 5 shows the most energetic lines of the spectrum) have energies below the lowest cut-off energy ( energy for e - 1 transmission) of the filters used and are therefore strongly attenuated. Because of the high intensities of certain lines it may still be possible that some of the X-ray line radiation contributed significantly to the signals in the 5 um and 7 fim aluminum filter channels. However, since the transmitted spectrum is determined from a large number of channels this would have little effect on the final result. Figure 6 — 2 shows a typical distribution of the relative intensity of X-ray contin-uum transmitted through different filters in the nine-channel detector. Laser intensity on target for this shot was 1.3 X 101S W/cm 2 . The plasma electron temperature was deter-mined by comparing the distribution with theoretical curves for different temperatures. As shown in this figure the best-fit theoretical curve yields an electron temperature of 550 eV. Uncertainties in the curve fitting limit the accuracy of the temperature measurement to ± 5 0 eV. The measured temperature represents a spatially and temporally integrated value. Since the X-ray continuum emission increases with the plasma temperature and density, this measurement is weighted towards the peak of the laser pulse and towards the higher plasma density region. In Figure 6 — 3 we have plotted the measured plasma electron 96 Figure 8—2 Relative intensity of X-ray continuum transmited through different filters. 97 Figure 6-3 Scaling of electron temperature as laser intensity for 0.532 /tm radiation. 98 plasma density region. In Figure 6 — 3 we have plotted the measured plasma electron plasma temperature as a function of the absorbed laser intensity. Each data point in this figure represents an average of many laser shots. The corresponding standard deviations are indicated. In this figure we have also plotted (solid line) the dependence of the electron temperature with absorbed laser intensity as predicted by equation (4 — 36) of Chapter 4 which was derived for a strong inverse bremsstrahlung absorption regime. This line is in good agreement with our data and further verifies that inverse bremsstrahlung is indeed the dominant absorption mechanism. As discussed in Chapter 2, inverse bremsstrahlung absorption produces a thermal distribution of electrons while other absorption mechanisms, such as resonance absorption and various laser-driven plasma instabilities, produce a small population of very energetic electrons. These energetic electrons produce "harder" X-rays than thermal electrons and therefore will show their signatures in the X-ray spectrum. To asses the existence of other absorption mechanisms in our experiment the measurement of the X-ray continuum was extended to include the more energetic part of the X-ray spectrum. For this measurement the X-ray emission was monitored using a set of three X-ray PIN diodes89. (The multi-channel X-ray detector could not be used for this measurement as its sensitivity rapidly drops for X-rays with energies higher than 5 keV). Results of the measurement are shown in Figure 6 — 4 which includes X-ray emissions obtained at 5° and 30° from target normal. This corresponds to aluminum targets irradiated with 0.532 fim laser light for an average absorbed irradiance of 3 X 101S W/cm 2 . The X-ray spectrum in Figure 6 — 4 was mea-sured from many laser shots at nearly identical irradiance and the data points represent the average values with the corresponding standard deviations. The X-ray spectrum obtained indicates the presence of two electron temperature distributions. The dominant component represents the thermal plasma with a temperature of 550 ± 50 eV. However X-ray emissions near target normal (data points for -5 ° with electron-sweeping magnets in front of the diode) show a "hot" component of about 1.5 keV. The presence of energetic electrons is also evident from the data points for —5° without the sweeping magnet. Equation (2 — 25) of Chapter 2 gives the predicted temperature of the electrons produced by resonance absorption. For our experimental conditions (Te=500 eV, IA = 3 X 1013 W/cm 2 and \ L = 0.532 fim) it gives Thot = 1.6 keV which is in good 99 CUTOFF ENERGY(KeV) Figure 6-4 X-ray continuum spectrum for 0.532 /im radiation, IA = 3 X 1013 W/ cm2. 100 Figure 6-5 X-ray continuum spectrum for 0.266 /tm radiation, IA = S X 101S W/cm 2 . 101 X-ray components and from the fact that the X-ray intensity is proportional to the square of the electron density, the number of the hot electrons is roughly estimated to be < 1% of that of the thermal electrons. The electron plasma temperature was also measured at 0.266 fim for an absorbed irradiance of 3 X 101S W/cm 2 . The results of the measurement is shown in Figure 6 — 5. These indicate a plasma temperature of 200 eV. It is qualitatively in agreement with equation 4 — 36 of Chapter 4, which predicts a lower plasma temperature for shorter laser wavelength. The X-ray spectrum did not show the presence of hot electrons. This was expected because we were operating below the threshold of laser-driven plasma instabilities. Also, with very strong inverse bremsstrahlung absorption, virtually no laser energy reached the critical density surface where resonance absorption could occur. 6-3 Laser induced plasma instabilities As we have seen in Chapter 2 parametric instabilities are not expected to be im-portant for our experimental conditions. To see if any instabilities were excited, the light backscattered through the focussing lens was monitored using a laser energy meter and a fast photodiode with different interference filters. The experimental arrangement has been shown in Figure 5 — 2 of Chapter 5. A low level (< 4%) of Brillouin backscatter was observed in the 0.532 fim experi-ments. The Brillouin reflectivity as a function of incident laser intensity is shown in Figure 6 — 6. The instability was observed to grow from noise at an intensity of about 8 X 1012 W/cm 2 . Using the formula given in Table 2 — 1 of Chapter 2 the threshold intensity for this instability is calculated to be 8 X 1018 W/cm 2 . This apparent discrepancy may be due to the fact that we observe a spatially- and temporally-averaged intensity whereas the instantaneous intensity at the peak of the laser pulse and in the central part of the focal spot is much higher. (3/2) wL light was also observed in the 0.532 fim experiment. The source of (3/2) wL light is not well understood97. It is generally assumed to arise from the presence of electron plasma waves near ner/4. The two possible sources of these waves are the two-plasmon decay instability and the stimulated Raman scattering instability. The (3/2) wL light is a second order process involving either the coalescence of three electron plasma waves or 102 a second order process involving either the coalescence of three electron plasma waves or the coalescence of an electron plasma wave and an electromagnetic wave. In our experi-ment the (3/2) wL light is probably due to the two-plasmon decay instability, as it has a lower threshold intensity. Figure 6 — 7 shows the backscattered intensity of (3/2) wL light as a function of the laser intensity. The maximum intensity measured corresponds to a reflectivity of less than 5 X 10 - 9. Our measured threshold intensity for the excitation of the instability is 1.1 X 101S W/cm 2 which is again smaller than the theoretically predicted value of 9.9 X 101S W/cm 2 . This may be due to the same reason discussed above. In the 0.355 and 0.266 //m experiments, no (3/2) wL emission was observed. Only marginal signals from Brillouin backscatter were detected. This was expected because the intensity thresholds for all the laser driven plasma instabilities are higher for shorter laser wavelengths. 103 o UJ - J Lu LxJ CH ZD O cr CD 0 1 2 3 4 5 6 INC IDENT L A S E R INTENS ITY ( 10 1 3 W/CM 2 ) Figure 6-6 Brillouin reflectivity as a function of incident laser intensity for 0.532 fim radiation. 104 10 INCIDENT L A S E R I N T E N S I T Y (10 1 3 W/CM 2 ) Figure 6-7 fit;/, backscatter reflectivity as a function of incident laser intensity for 0.532 fim radiation. 105 Chapter 7 Investigation of electron energy transport using multilayer targets As discussed in Chapter 3, the extreme temperature and density gradients inherent in laser-produced plasmas render the exact theoretical description of heat transport difficult. There are many unresolved problems in correlating theory and experimental data. Thermal transport measurements are therefore important to gain a better understanding of this important aspect of the laser-target interaction. An observable quantity closely related to thermal transport is the mass ablation rate, m , which is the rate at which material is ablated from unit target area. An average value for rn can be defined by M m = - — , (7-1 ATAbl where M represents the total mass ablated during the interaction, A the ablation surface area and rAbl the average ablation time. In Chapter 8 we will present measurements of m based on a direct evaluation of the parameters of equation (7 — 1). The total mass ablated M can also be expressed as M = pAd (7 - 2) where p represents the initial target density and d the "ablation depth", that is, the thick-ness of material ablated during the interaction. In principle, the ablation depth can be obtained by measuring X-ray line radiation from the substrate layer of multilayer targets. This diagnostic technique was first reported by Seka et al.98 and have been used in many experimental investigations of thermal transport 1 1 ' 1 2 ' 7 7 , 7 8 , 9 9 - 1 0 6 . Targets are made of a tracer substrate material coated with varying thicknesses of target material. During laser irradiation a thermal front penetrates into the solid, ablating the target material and even-tually, for sufficiently thin target layer, reaches the substrate causing characteristics X-ray 106 lines to be emitted. By measuring the X-ray lines emission from the substrate as a function of the target layer thickness it is possible to study the propagation of the thermal front into the target material. The thickness of target layer at which the X-ray line emission disappear is usually taken as the laser ablation depth. In this chapter we report on an investigation of thermal transport using such multi-layer targets. In section 7—1 the experimental conditions are described. This is followed in section 7 — 2 by a summary of the results of this study, which revealed anomalies that indicate a limitation of the multilayer techniques for energy transport measurements. A simple hydrodynamic model which can account for such anomalies is presented in Section 7 - 3 . 7-1 Experimental conditions In our experiment glass slides of thickness 130 microns were used as the substrate material. To study the effect of the target density on thermal energy transport, five different coating materials were selected. These included polystyrene (CH), magnesium (Mg), tin (Sn), silver (Ag) and gold (Au). Details about the procedures for target fabrication are given in Chapter 5, section 5 — 4. Silicon lines emitted from the glass (Si02) substrate were spectrally resolved using a flat KAP crystal spectrometer. The X-ray spectrometer is described in Chapter 5, Section 5 — 3 — D. The instrument was oriented at 15° off target normal with a crystal-to-target distance of 12 cm (see Figure 7 — 1). A typical silicon spectrum from the glass substrate is shown in Figure 7 — 2. The spectrum was recorded using Kodak SC-5 X-ray film. It includes hydrogen-like and helium-like lines which are present only for plasma temperatures of several hundred electron volts106. The X-ray detector used was a plastic scintillator (NE102) coupled through fiber optics to a photomultiplier tube (RCA 8575). The scintillator surface exposed to the X-ray radiation was covered with a 25 fim thick beryllium foil to block off visible light. The combined response time of the plastic scintillator, photomultiplier tube and oscilloscope was of the order of 10 ns. Since the X-ray line emission is known to last for the duration of the laser pulse104 (2 ns) only, the measured intensities were effectively time integrated. In order to discriminate properly between the X-ray emission from the glass substrate and the X-ray emission from the upper layer, an adjustable slit was positioned 107 Af0.355 um 1=1.2X10 W/c spectrometer slit beryllium foil plastic scintillator fiber optic photomulti-plier tube To oscilloscope Figure 7-1 Experimental arrangement for energy transport measurement. 108 M A ) 6.74 6.65 6.18 5.68 5.41 fc • . > • • / I S A T E L L I T E S INTERCOMBINATION sixnr i s 2 - i s 2 p - S i X E Let 1s-2.p - S i X K 1s 2-1s3p - SiXDI 1s 2-1s4p Figure 7-2 Silicon line spectrum. 109 at the rear of the spectrometer to accept only a selected part of the spectrum. For the CH-, Sn- and Ag-glass multilayer targets, the slit was adjusted to accept only that part of the spectrum between 6.65 and 6.74 A. As shown on Figure 7 — 2 this includes the Si XIII ls 2( 1S0)—ls2p(1Pi)resonance line, the intercombination line 1« 2 ( 1 5 0 )— l «2p( s P! ) and some weaker satellite lines (the satellite lines are due to transitions in ions with outer shielding electrons). For the Mg- and Au-glass targets, emissions from the Mg or the Au layer were so strong that in order properly to isolate the substrate emission, the slit was narrowed to accept only the Si XIII IS^SQ)- ls2p(1P1) line. This series of experiments was done using 0.35 /im laser light. The irradiance con-ditions were kept constant with 90% of the energy contained within a focal spot of 80 /im diameter, giving an intensity of 1.2 X 101S W/cm 2 at the target surface. 7.2 Experimental results Figure 7 — 3, 7 — 4 and 7 — 5 show the observed intensity of the silicon line group between 6.65 and 6.74 A for CH-, Sn- and Ag-glass multilayer targets as a function of the target layer thickness. Figure 7 — 6 and 7 — 7 show the intensity of the silicon resonance line Si XIII la2( !5o) — l «2p( 1 P 1 ) as a function of the target layer thickness for the Mg- and Au-glass multilayer targets. The intensity is given in arbitrary units, normalized to the intensity observed for zero coating thickness. Each data point on these figures represents a group average from many laser shots, with error bars indicating the corresponding standard deviations. The given target layer thicknesses are estimated to be precise within 10%. In Figures 7 —3 to 7 —7 we observe an expected drop in the intensity of the X-ray line emission from the glass substrate, for increasing target layer thickness. The intensity then reaches a plateau which corresponds to the basic noise level of each type of target, which was established by irradiating pure targets of the coating material. We here define as "laser penetration depth" the thickness of target layer for which X-ray line emissions from the substrate layer disappear. Figure 7 — 8 shows this laser penetration depth as a function of target material density. The best fitted line yields a density dependence of p~lb for the penetration depth. As discussed in the introduction of this chapter the usual interpretation of the multilayer target diagnostic101 is that the X-ray emission from the substrate layer is detectable only if the overcoat target layer has been completely ablated during the laser 110 pulse. However we will show that our results are inconsistent with this interpretation, and that the measured penetration depth differs significantly from the conventionally defined "ablation depth" which is a true measure of the thickness of material vaporized from a pure target. The ablation depth itself was measured using another experimental diagnostic. Far away from the target all the thermal energy of the plasma has been transformed into kinetic energy of expansion. By measuring both the energy and asymptotic velocity of the expanding plasma we can obtain the total mass ablated, which is given by IE M = ^w> • (7~3) where E is the total plasma energy and < V2 > its mean square velocity. This was done for aluminum targets (details of the measurement are discussed in chapter 8; the experimental results are presented in Figure 8 — 3 for different irradiance conditions). Now the ablation depth is simply obtained from equation (7 — 2) as d=^A , (7-4) pA where p is the target density and A the ablation surface area. Using this formula, the value for the total mass ablated at 1.2 X 101S W/cm 2 from Figure 8 — 3, the density of aluminum and assuming the ablation crater to be of cylindrical geometry with a diameter the size of the focal spot, yields a value of 2.4 ± 0.8 /im for the ablation depth in aluminum. This ablation depth for Al is indicated on Figure 7 — 8. In this figure, the dashed line indicates the actual penetration depth obtained. The ablation depth measured for Al clearly lies well below the penetration depth curve (X-ray intensity measurements on Al-glass targets could not be made because it was not possible to isolate silicon diagnostic lines from the intense nearby aluminum lines). Also, the density dependence given by equation (7 — 4) is only p~l, in marked disagreement with the observed density dependence of p - 1 ' 6 . Two dimensional effects such as spatial inhomogeneities in the laser beam or thermal self-focusing18 and filamentation18 in the plasma could conceivably cause the laser energy to be channelled into a small area, drastically increasing the local irradiance. Thus the mass ablation rate would be greatly increased in a local region smaller than the laser focal spot, leading to large ablation depth. However no evidence for this was seen in the image 111 3 < r— i—i CO z: L U C L O Q: C D U J co 1000 CH LAYER THICKNESS ( um ) Figure 7-3 Intensity of the silicon line group between 6.65 and 6.74 A as function of CH layer thickness. 112 ZD < CO •z. U J DL ZD o c r CD UJ z: n CO 0.1 — 0.01 V \ S \ \ \ \ \ -1 \ \ ', _ T... 1 1 [ r Sn strip 1 1 0.1 1 10 100 Sn LAYER THICKNESS ( um ) Figure 7-4 Intensity of the silicon line group between 6.65 and 6.74 A as function of the Sn layer thickness. 113 CO 3 < co 2 : U J CL 3 O Ql CD U J 1 10 100 Ag LAYER THICKNESS ( um ) Figure 7-5 Intensity of the silicon line group between 6.65 and 6.74 A as fuction of Ag layer thickness. 114 10 100 Mg LAYER THICKNESS (um) 1000 Figure 7-6 Intensity of the Si XIII ls^So) - ls2p{lPi) line as a function of Mg layer thickness. 115 0.1 1 10 Au LAYER THICKNESS ( um ) 100 Figure 7-7 Intensity of the Si XIII 1 « 2 ( 1 5 0 ) - ls2p(1P1) line as a function of Au layer thickness. 116 100 CL g 10 < r — LU Z UJ CL cr LU < - £ C H - 1 N \ \ \ - V r Mg A \ \ \ \ \ \ — \ \ \ \ \ \ — T Sn \ r A 9 i 1 Al 1 \ \ \ \ — I i i 1 1 1 2 4 6 8 10 20 40 TARGET DENSITY (gm/cm 3 ) Figure 7-8 Laser penetration depth as a function of target density : ((^  multilayer target measurements , (•) ions measurements. 117 of the focal spot, or in X-ray emission pictures of the plasma (see Chapter 5, Sections 5 — 2 and 5 — 3 — F). Moreover even if such variations in the local irradiance could explain the large penetration depth, it could not explain the measured density dependence. To explain these anomalies we propose in the next section a simple hydrodynamic model in which the ablation process is modified by an edge effect due to the finite, small focal spot. 7-3 A hydrodynamic model for laser penetration depth In the irradiation of a solid by a high intensity laser, as indicated in Figures 4 — 1 and 4 — 2 of Chapter 4, a shock wave propagates into the solid with a characteristic velocity Us. It is followed by an ablation front (boundary layer between the shock compressed solid and the resultant plasma) moving with a characteristic velocity V 0 . This is shown schematically in Figure 7 — 9. To allow formulation of a tractable physical model, we have assumed, as depicted in Figure 7 — 9, that: (i) the lateral distribution of the laser intensity in the focal spot can be approximated as trapezoidal, (ii) the resulting ablation front A also has a trapezoidal profile and (iii) transient effects due to the density discontinuity at the target-substrate interface are not present and the ablation front A, contact surface B and the shock front C all propagate with steady velocities. The density discontinuity can cause the shock wave to be reflected at the interface if the substrate material is more dense than the target material, or produce a returning rarefaction wave if the substrate material is less dense than the target material107. Such detailed interactions, however will be neglected in this simple model. Given that the measured density dependence (Figure 7 — 8) appears to be insensitive to the density ratios of the target to substrate it is reasonable to assume that the effects of density discontinuity at the target-substrate interface may be neglected. As illustrated in Figure 7 — 9, the hydrodynamic motions of the ablation surface and the contact surface produce a thin, stretched layer of target material along the edge of the irradiated spot. From the simplified geometry of trapezoidal profiles, shown in Figure 7 — 9 it follows that the thickness of the stretched target layer is simply where L is the projection of the rising edge of the laser focal spot spatial profile (Figure 7 — 9), T is a time of the order of the laser pulse length, Us and Va are the shock and ablation (7-5) 118 R A-ABLATION FRONT B-CONTACT SURFACE C-SHOCK FRONT Figure 7—9 Schematic showing the hydrodynamic shearing of the target layer. 119 front speeds and De is the original thickness of the target layer. Shock compression of this layer is further assumed to be negligible. If such an edge distortion effect occurs before the end of the laser pulse, ablation of the stretched target layer by lateral thermal conduction can produce X-ray emissions from the substrate material. Thus, for an axial penetration depth D0, there is a corresponding lateral ablation depth of A , indicated in Figure 7 — 9. To see that this model predicts the observed behavior, Equation 7 — 5 can be rewritten as in which L , r and ( V a / U s ) are independent of the target density110 p. From conservation equations107 Us varies as p~0B, while the ablation depth A varies as p~x. Therefore, the penetration depth will scales as p~xi , in agreement with the measured scaling. Further evidence for the model is obtained by considering the penetration depth in aluminum as inferred from the measured density dependence. Interpolation of the data in Figure 7 — 8 yields a penetration depth in aluminum of 16±3 um. For an irradiance of 1.2 X 101S W/cm 2 , a shock pressure of 4.3 Mbar was measured (see Figure 8 — 4). For such a shock pressure, calculations108-109 showed that Us 2 X 10° cm/s and ( V a / U s ) «=> 0.6. T is taken as 2 ns which is the full-width-half-maximum duration of the laser pulse. Measurements of the laser intensity distribution in the focal spot indicated that L ~ 15 um. Substituting these values into equation (7 — 5), one obtains A = 4.0 ± 0.8/im. This is reasonably consistent with the ablation depth of 2.4 ± 0 . 8 u m obtained from the ion energy and velocity measurements. A direct test of the model would be the observation of the ring of emission in a temporally resolved, spatial image of the Si line. The insufficient X-ray flux, however precluded this observation in our work. (7-6) 120 Chapter 8 Experimental scaling laws for the mass ablation rate and the ablation pressure In this chapter we present experimental measurements of the ablation pressure and mass ablation rate using planar aluminum targets irradiated with 0.265 and 0.532 fim laser pulses of 2 ns (FWHM) and at irradiances of 1012 - 5 X 101S W/cm 2 . To complete the analysis on wavelength dependence, we have also included the measured intensity scalings of the mass ablation rate and ablation pressure for 0.355 fim radiation from the work of P. Celliers69. In Section 8 — 1 we discuss the measurement principle and the experimental set-up. The experimental results are then presented in Section 8 — 2. Further discussions and comparison with theoretical predictions are given in Section 8 — 3. 8—1 Measurement principle and experimental set-up By measuring the energy and velocity of the expanding plasma, time averaged values for the mass ablation rate and the ablation pressure can be obtained . This is possible because far away from the target all the plasma internal energy has been converted to kinetic energy of expansion and we can write: where E is the total plasma energy, M the total mass ablated and < V2 > the mean square velocity of the expanding plasma. The total mass ablated Af and the momentum > (8-1) Pj_ carried off perpendicular to the target can be evaluated from: (8-2) and (8-3) 121 Here 6 represents the angular position from the target normal and cylindrical symmetry about the target normal is assumed. Average values for the mass ablation rate in and the ablation pressure PAbt are then evaluated using the expressions: Af (8-4) m = and P± (8-5) where AAbi is the ablation area and rAbi the ablation time. In this analysis the ablation area was assumed to be the same size as the laser focal spot of 80 /tm diameter for 90% energy content (Chapter 5, Section 5 — 2). The ablation time is of the same order as the laser pulse length but slightly longer because both rh and PAbl scale as the laser intensity to a power less than unity. To estimate rAbl we have used the width at the 1/e intensity points of our laser pulse taken to the 7/9 power (theoretical intensity scaling for ablation pressure). This was 2.7 ns. To evaluate expressions 8—4 and 8 — 5 the energy E and average velocity < V > of the expanding plasma are required. The plasma energy was measured using differential calorimeters. A description of the differential calorimeters has been given in Chapter 5, Section 5 —3 —C. The plasma calorimeters measure both the ion energy and the energy of the X-ray radiation emitted by the plasma. For an aluminum plasma the total amount of energy radiated in the X-ray range is expected to be of the order of 15% for our experimental conditions16'111. These X-rays are emitted isotropically in a 47r angle. In our analysis the X-ray contribution to the detector signal was assumed negligible. The possible effect of this assumption on our derived scaling laws is discussed below. The velocity of the expanding plasma was measured using Faraday cups which have been described in Chapter 5, Section 5 — 3 — B. A typical output signal is shown in Figure 5 — 7 — B. It shows a strongly peaked ion current signal indicative of a single narrow velocity distribution112. In our analysis the velocity given by the peak of the current trace was taken to approximate the average velocity of the expanding plasma. The experimental set-up for the ion measurements is shown Figure 8 — 1. A total of seven differential calorimeters and seven Faraday cup collectors were used, arranged in a horizontal plane at angles between 5° and 80° from the target normal. The Faraday cups 122 Figure 8-1 Experimental arrangement to measure energy and velocity of the expanding plasma. 123 were mounted directly above the calorimeters and therefore at the same angular position. The detector-pairs were placed at distance varying from 20 to 35 cm from the target. As indicated in Figure 8—1, 50 pm thick aluminum foils were used as targets and the pressure in the target chamber during these measurements was of the order of 10~8 Torr. Such vacuum is required to eliminate any significant charge exchange processes114 which would modify the charge state of the ion flow arriving at the detector. 8-2-A Angular distributions of the plasma energy and velocity Typical angular distributions of the plasma energy and velocity measured by the detectors are shown in Figure 8 — 2. The plasma energy is given in units of joules per steradian and the velocity in units of 107 cm/s. The angles in degrees are measured from the target normal. This particular energy and velocity distribution was observed for an incident irradiance of 6 X 101S W/cm 2 and 0.532 /im laser light. Continuous functions representing the angular energy and velocity distributions were obtained by fitting cubic spline functions through the individual experimental data points. These functions were then used in expressions 8 — 4 and 8 —5 to evaluate the mass ablation rate and the ablation pressure for each laser shot on target. 8-2-B Experimental scaling laws for the mass ablation rate and the ablation pressure Figures 8 — 3 and 8 — 4 show the observed mass ablation rate in units of g/cm2-s and the ablation pressure in units of Mbar (10e bars) as a function of the absorbed laser flux for different wavelengths. Each individual point represents one laser shot. Best-fitted lines to the data (solid lines on the figures) yielded the empirical scaling laws: pAbl = 4.4(/ A / l0 l s f 9 2 Mbar for XT = 0.266/im , ( 8 - 6 - a ) pAbl = 3.7(JV101s)0-84 Mbar for XL = 0.355 um , (8 - 6 - b) pAbl = 3.0(/A/10ls)0-78 Mbar for X T = 0.532/im , ( 8 - 6 - c ) 124 ANGLE (DEGREES) Figure 8-2 Energy and velocity distribution of the expanding plasma. IA = 6 X 101S W/cm 2 . 125 and m = 3.3 X 108(/A/1018)0-72 g/cm2-s for = 0.266 fim , (8 - 7 - a) m = 2.3 X 10 5(/A/10 1 8) 0" g/cm2-s for \ L = 0.355 fim , (8 - 7 - b) m = 1.3 X 10 6(7A/10 1 8) 0" g/cm2-s for \ L = 0.532fim . ( 8 - 7 - c ) The results show the ablation pressure and the mass ablation rate to increase for shorter wavelength radiation. This is expected since PA = 2n0(^f£)C'o a n ^  m == " 0 ( ^ - 5 ^ )Co (eq. 4 — 18 and 4 — 19 of Chapter 4). For shorter laser wavelength the laser light pene-trates to higher density n 0 (see eq. 4 — 34 and 4 — 26) with corresponding increases in PA and tii. In general, wavelength scaling laws can then be derived from formula (8 — 6) and (8 — 7). However, at different laser wavelengths, both m and PAbi exhibit slightly different dependences on the absorbed flux. This phenomenon was also observed in other scaling studies14'104. Consequently, the wavelength dependences of th and PAbi may also be differ-ent at different irradiances, i.e. the intensity and wavelength scalings are not completely independent, in contradictions to steady-state ablation theories (see Chapter 4). To illus-trate the wavelength dependences, experimental values of m and PAbi are plotted against \ L at absorbed fluxes of 2.6 X 1012, 6.6 X 1012 and 1018 W/cm 2 (Figures 8 - 5 and 8 - 6) In these figures the data points represent average values for nearly identical laser intensities with the corresponding standard deviations. The dashed line represent the best-fitted line through the data points. For these irradiances, the wavelength scaling of tin varied from X]T118 to XT;1"87 whereas that of PAbl varied from XT;0 8 4 to X ^ 0 6 7 . 8-3 Discussion of the results Absorption measurements and the measured intensity scaling of the plasma elec-tron temperature have indicated that inverse bremsstralung absorption is the dominant absorption mechanism in our experiments (Chapter 6). As described in Chapter 4, in this 126 10 10 L 0.53 Mm 10 G 10 10 I 0.35 um 107 106 10 10 L 0.27um J l l l l l l l l I I I 10 J1 10 12 10 10 ABSORBED FLUX (W/cm) Figure 8-3 Mass ablation rate as a function of absorbed flux for 0.532, 0.355 and 0.266 /tin laser light. 127 ABSORBED FLUX (W/cm) Figure 8-4 Ablation pressure as a function of absorbed flux for 0.532, 0.355 and 0.266 /im laser light. 128 5 4 2k 1 .8 .6 D 2.6x10W/cm2, A ' L ' 3 7 • 6.6 x 1(fw/cm2, Af5 ° 1 x ifw/cm2. A"L115 .15 .4 .6 WAVELENGTH (um) .8 1.0 Figure 8—5 Wavelength dependence of mass ablation rate at absorbed irradiances of 2.6 X 1012, 6.6 X 1012 and 1013 W/cm 2 . 129 8 .8 — °2.6x10 W/cm 2, X L ° , 3 A — 12 o .OA6 • 6.6x10 W/cm2, V o 1 x103W/cm2, A; 0- 5 2 r ] K 3 ^  i 1 S J 1 1 1 1 1 1 1 .15 A .6 WAVELENGTH (Mm) .8 1.0 Figure 8-8 Wavelength dependence of ablation pressure at absorbed irradiances of 2.6 X 1012, 6.6 X 1012 and 101S W/cm 2 . 130 absorption regime the following scaling laws for the mass ablation rate and the ablation pressure are expected: (8-9) PA ~ H ' V • Evidently, the intensity scalings of both rh and PA are in reasonable agreement with the measured scalings (eq. (8 — 6) and (8 — 7)). On the other hand the predicted wavelength dependence of m is much weaker than the measured scalings of X7J1'16 to \~ [ l i 7 whereas the theoretical scaling for PA is also weaker than the observed X^ 0 ' 8 4 to X^ 0" 5 7 dependences for irradiances of 2.6 X 1012,6.6 X 1012 and 10 1 8W/cm 2. Most recently, Meyer and Thiell1 8 observed that the total ablated mass measured in planar targets irradiated with 0.35 and 1.05 fim laser light also scaled as XTJ1'8. They further pointed out that in Mora's model for strong inverse bremsstrahlung absorption (see Chapter 4 Section 4 — 2 — D—ii), the density gradient length L in the coronal plasma was assumed constant whereas numerical simulations from a hydrodynamic code describing the laser-target interaction showed a steepening of the density gradient for decreasing wavelength. This resulted in a different scaling for no. the maximum density for laser absorption, namely: "o ~ n„l)[* . (8-10) using this new expression for n0 the scaling laws for strong inverse bremsstrahlung absorp-tion becomes: P ~ r7/9v-2/S (8-11) These modified scalings relations are in almost complete agreement with our measurements except for the wavelength dependence of PA, particularly at irradiances near 1012 W/cm 2 . Similar variation of n 0 with ner was also observed in our one-dimensional hydrody-namic simulations118. However when the computer simulation was run to reproduce the 131 experimental conditions it indicated the following dependence for m and PAbt: • r0.58\ — 0.62 rn ~ 1A \ L 8-10) r> r 0 . 7 8 \ - 0 . 2 8 iTAbl ~ *A  XL for IA = 2.6 X 101S W/cm 2 . These predictions do not agree with our measurements. (Curiously, they reproduce the results of Mora's model which should have been wrong because, as it has been pointed out, the density n 0 is not properly estimated.) One possible cause for these discrepancies in the wavelength scaling of the m a s 3 ablation rate and ablation pressure is the X-rays emitted by the plasma. The conversion efficiency of laser radiation to X-rays is known7 to be higher for shorter laser wavelengths as the laser energy deposition occurs at higher plasma densities. However, energy transport by X-rays was not included in the hydrocode simulations used to obtain the scaling laws given by equation 8— 10. Also the X-ray contribution to the calorimeter signals was not taken into account in our analysis of the measurements. Neglect of the X-ray emission could lead to an underestimate of the theoretical wavelength scaling law exponents and to an overestimate of the experimental wavelength scaling law exponents. Let us first consider the effect of the X-ray radiation on the ablation process. The X-rays transport energy to the ablation surface more efficiently than the electrons and if more laser energy is converted into X-rays this will lead to a higher mass ablation rate and ablation pressure. Therefore, if a greater amount of laser energy is converted into X-rays for the shorter laser wavelength, this would lead to a stronger wavelength dependence for the mass ablation rate and the ablation pressure. Preliminary results11* obtained by including the effect of X-ray radiation in our hydrocode showed, as expected, higher values for the mass ablation rate and the ablation pressure. They also revealed a stronger wavelength scaling. As for the effect of X-ray radiation on the experimental measurements, if the X-ray contribution to the calorimeter signal is important, the ion energy will be overestimated. Consequently, the mass ablation rate and the ablation pressure (see eq., 8 — 2 and 8 — 3) would also be overestimated. This would lead to an overestimate in the wavelength scaling for the mass ablation rate and the ablation pressure, since the X-rays emission is higher for shorter laser wavelength. 132 Chapter 0 Summary and conclusions 9-1 Experimental effort In the course of this work, many diagnostics have been developed. These include a spherical photometer, a multichannel X-ray detector, pinhole cameras and an X-ray crystal spectrometer. An in-situ calibration technique for X-ray films as well as a procedure for the fabrication of multilayer thin film target were also developed. 9-2 Experimental results and conclusions As theoretically predicted, inverse bremsstrahlung absorption is very effective for short wavelength laser radiation. This was established from the high level of absorption measured (> 90 %) as well as the scaling of electron temperature with laser intensity. From the observed X-ray spectrum, the absorbed energy was found to be well thermalized, with a cooler plasma temperature for the shorter laser wavelength and insignificant amount of hot electrons produced. Parametric processes did not play any significant role as evidenced by the low level of scattered light measured. All these measurements showed good agreement with theory and confirmed the results of many other workers. The advantageous effects of short wavelength radiation on energy coupling are evident. An experimental investigation of electron thermal transport using multilayer targets was performed at 0.355 /im. This experiment represented the first measurement of the effect of target density on energy transport. However, because of two-dimensional hydrodynamics, the measurement was dominated by an edge effect yielding an apparently much larger laser ablation depth in the target. This has been successfully explained using a simple model. Since layered target measurements are widely used by other workers, the identification and the explanation of a problem associated with this diagnostic are important. 133 Scaling laws for the mass ablation rate and the ablation pressure were obtained at 0.532, 0.355 and 0.266 /im for intensities ranging from 1012 to 5 X 101S W/cm 2 . This consti-tuted the first detailed measurements in a strictly classical regime where direct comparison with an analytical model is possible. The measurement showed that much more efficient energy transport is obtained for shorter laser wavelength as indicated by the increase in the mass ablation rate and ablation pressure. A very interesting observation is that the ob-served wavelength dependence is much stronger than predicted. The X-ray energy radiated by the plasma is identified as a plausible cause for this stronger wavelength dependence. 9-4 New contributions The multichannel X-ray detector70 (Rev. Sci. Inst. 54, 1091 (1983)) used for plasma temperature measurements and the insitu X-ray film calibration technique68 (Appl. Opt. 23, 762 (1983)) constitute new and original diagnostic developements in this work. In terms of the physics understanding in laser-matter interactions, we have identified and successfully explained the anomalous laser penetration depths in layered targets116 (Appl. Phys. Lett. 44, 713 (1984)) where edge-effects are important. Furthermore we have obtained the first detailed measurements of the scaling laws for both the mass ablation rate and ablation pressure for laser-driven ablations in a classical and steady-state regime118 (Appl. Phj's. Lett. (1984)). The results have indicated the inadequacies of the existing analytical theories. 9-3 Suggestions for future work Future fusion drivers will be operating at intensities ranging from 1014 to 1016 W/cm 2 . In this intensity regime the plasma temperature will be higher and inverse bremsstrahlung may become less effective. Also the threshold for the parametric processes will be exceeded, leading to increased scattering and to the production of hot electrons. It will be important to pursue the study in this regime. Such studies can be made possible by upgrading the existing laser facility. For a better understanding of the laser-target interaction the knowledge of the actual amount of energy radiated as soft X-rays is essential. In particular the X-ray emission has been identified as the possible cause for the differences between our measured wavelength 134 scaling laws and that predicted from analytical and numerical models. As a large amount of the X-ray energy emitted by the plasma is in the soft X-ray range, spectral information on the X-ray emission can be obtained using a grazing incidence vacuum spectrometer and temporal information using an X-ray streak camera. 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A - l Transmitted X-ray spectrum for different plasma temperatures The X-ray intensity behind an absorber foil of thickness D is obtained by integrating over wavelength the product of the absorber's transmission function and the intensity of the incident continuum, Itran. = j hnc (X, Te) eX P [-/i(X)D] d\ (A-l) where 7( r a „ , is the transmitted intensity, I{ne the incident intensity, fi(\) the absorption coefficient of the filter used and D the filter thickness. The following expressions were used for the absorption coefficient of aluminum foils: = 4.776 - 7.32X + 3.84X2 + 13.84X3 - 0.755X4 cm2/g for X < 7.9 A (.4-2) M(\) = 42 - 35X + 6X2 + 0.3838X4 cm2/g for X > 7.9 A where X is the wavelength in A. These polynomials were obtained88 from least square fit to values of the absorption coefficient between 100 eV and 12 keV tabulated by Henke117. For an optically thin homogeneous plasma of length /, the intensity I is related to the plasma emissivity t by I = tl (A — 3) 143 The plasma emissivity is due both to recombination radiation and bremsstrahlung contin-uum emission it can be expressed as118 : t(X,r e) = 4.08 X 10" (EH\Z2N?ne \kTt) X 2 e x p ( - \kTe he ) 2 [a// + tjb 2Z2Eh n*kTt H exp( )] (erg s 1 cm s sr 1 A 1 ) (A - 4 ) EZ-l,n <he/\ where EH is the ionization potential of hydrogen, k the Boltzmann's constant, ne the electron number density (cm - 5), Te the electron temperature (eV) , Nf the number density (cm - 8) of ion of charge Z calculated assuming a coronal equilibrium model, X is the photon wavelength, EZ,N the ionization potential in eV of an electron in the n-th shell of an ion of charge state Z, n the lowest level that has a vacancy for radiative recombination and 9/f >9/b the Gaunt factors for bremsstrahlung and recombination radiation respectively. Using equations (A — 2), (A — 3) and (A — 4), repeated numerical integrations* of expression (A — 1) for the various foil thicknesses D yielded curves of the relative trans-mitted X-ray intensity versus the absorber thickness for different electron temperatures. These were then compared to our experimentally measured spectrum. In the evaluation of expression (A — 4) for calculating the ground state ion densities, a steady state coronal equilibrium was assumed. We wish now to discuss the validity of this assumption. A -2 Applicability of the Coronal model The most complete model to calculate the ion populations in a plasma is the collisional radiative model where all the possible excited states of the ions are taken into consideration. However because of the complexity of a full collisional radiative model computation, simpler approximate models are more commonly used119. In particular the Coronal model is used if the electron density is such that * The computer program to perform these calculations was written by K. Fong1 1 7 n e < 1.5 X 1 0 1 0 r 4 X - 1 / 2 (A -5) 144 where ne is the electron density (cm - 3), Tt is the electron temperature in eV and x  13 the ionization potential of an ion of charge Z. At the other extreme, the local thermodynamic equilibrium modele (LTE) is used if nt > 1.7 X 1014Ty*x*ip,q) (cm"3) (A — 6) where \(p, q) is the largest energy gap in the term scheme of the ion considered. Typical values for the plasma in our experiments were : n e ~ 4 X 1021 c m - 3 , Te ~ 500 eV and x(p,q) ~ X ~ 2000 eV . The conditions (A — 5) and (A — 6) for these plasma conditions becomes: n e < 2.1 X 10ig (cm - 3) for Coronal model ne > 3.1 X 1026 (cm - 3) for LTE model Thus we see that our plasma with ne ~ 4 X 1021 c m - 3 lies between the coronal and LTE regions and an accurate calculation of the ionization distribution should in principle employ the collisional radiative model. In the following table we compare the calculation of the relative ion density popu-lations for an aluminum plasma at Te = 500 eV using computed values from our coronal model and published20 collisional radiative values. T a b l e A - l Comparison between Coronal and Collisional radiative model calculations Relative ion Coronal model Collisional density population radiative model N +10/NTotai -006 .008 N +ll/NTolal .853 .74 N +12/NTotat .136 .24 N +1*/NTotai .005 .005 145 A Coronal model approximation is probably rather crude for calculating the line emission where the population of the bound states have to be known. However for calculating the recombination radiation, only the total relative population density of a given type of ion need to be known and the results of the above table show that it is well justified to use a Coronal model calculation. 146 

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