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Dense, strongly coupled plasmas in femtosecond laser-matter interactions Forsman, Andrew 1998

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Dense, Strongly Coupled Plasmas Femtosecond Laser-Matter Interactions By Andrew Forsman B. Sc. H. Acadia University M. Sc. University of British Columbia A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Faculty of Graduate Studies .Physics and Astronomy We accept this thesis as conforming to the required standard. THE UNIVERSITY OF BRITISH COLUMBIA March 1998 © Andrew Forsman, 1998 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract The development of laser systems capable of delivering laser pulses lasting merely 15 2 hundreds of femtoseconds, yet having peak intensities exceeding 10 W/cm , has opened new avenues for laboratory studies of the equation of state and transport properties of hot, dense, plasmas in the strongly coupled regime. As the new experiments began to yield data, there arose a need for new methods to interpret their results and to design new experiments. This thesis addresses such needs through numerical modelling of femtosecond laser-matter interactions. Moreover, this thesis shows how numerical modelling is not only indis-pensable for modelling most femtosecond laser-matter experiments but also a guide to bring greater depth and breadth to our understanding of femtosecond laser-matter interactions. Fi-nally, as the continued development of laser technology may reduce the durations of intense laser pulses to tens of femtoseconds, a new class of experiments offering the possibility of measuring the transport and equation of state properties of extremely well-defined, solid-density, strongly coupled plasmas can be realized. ii Table of Contents Abstract ii List of Figures vii Acknowledgements xi 1 Strongly Coupled Plasmas and High-Intensity Short—Pulse Lasers 1 1.1 Strongly Coupled Plasmas 1 1.2 Techniques Used to Create and Probe Strongly Coupled Plasmas 5 1.2.1 Techniques Prior to the Development of High-Intensity Short-Pulse Lasers '. 5 1.2.2 Advent of High-Intensity Short-Pulse Lasers 9 1.3 Present Work 14 2 Modelling the Plasma as a Fluid 15 2.1 Fluid Equations 16 2.1.1 Hydrodynamic Motion 17 2.1.2 Energy Transport and Laser Deposition 19 2.2 Material Models 20 2.2.1 Equation of State . . . 21 2.2.2 Average Ionization Models 25 iii 2.2.3 Electrical and Thermal Conductivity 29 2.2.4 Treatment of Flux-Limited Thermal Conduction 38 3 Applying the Hydrocode to Short-Pulse Laser Experiments 40 3.1 Two-Temperature (Te ^ Ti) Effects 41 3.1.1 Electron-Ion Thermal Equilibration 42 3.1.2 Two-Temperature (T e ^ Ti) Material Models 44 3.1.3 Two-Temperature (T e ^ Ti) Hydrodynamic Model 45 3.2 Electron-Electron and Ion-Ion Thermal Equilibration 47 3.3 Non-Local-Thermodynamic-Equilibrium (Non-LTE) Ionization Models . . 49 3.3.1 Collisional Radiative Equilibrium (CRE) 50 3.3.2 Collisional-Radiative Model (CR) 59 3.4 Electromagnetic Wave Solver 65 3.5 Ponderomotive Force and Wave Breaking 73 3.5.1 Ponderomotive Force 74 3.5.2 Wave Breaking 75 3.6 Testing of Algorithms 76 4 Reflectivity Measurements 78 4.1 Milchberg, Freeman, Davey, and More: A Pivotal Measurement and its Controversy 78 4.1.1 Criticism of the Original Analysis 81 4.1.2 Resolution of the Controversy 87 iv 4.2 Non-Equilibrium Effects 92 4.2.1 Non-Equilibrium Ionization 92 4.2.2 Two Temperature (Te ^ TA Effects 93 4.3 Ponderomotive Density Profile Modification 96 4.3.1 P-polarized Obliquely Incident Laser Light 100 4.3.2 S-Polarized and Normally Incident Laser Light 103 4.4 Towards Shorter Time scales: 120 fs and 20 fs 104 5 Heat Front Propagation 107 5.1 Equilibrium Simulations 109 5.2 Two-Temperature (Te ^ Ti) Effects 119 6 U l t r a - T h i n Targets and 20 fs Laser Pulses 124 6.1 Importance of Pulse Duration 125 6.2 Importance of Laser Intensity 125 6.3 Importance of Laser Wavelength 128 6.4 Ultrathin Foils Irradiated by Ultrashort Pulse Lasers 128 6.5 Proposed Experiment 131 6.6 Idealized Slab Plasma 133 6.7 Method for Interpretation of Experimental Data 140 6.8 Further Use of the Idealized Plasma Slab 144 6.9 Non-Equilibrium Effects 146 6.9.1 Ionization Balance 146 v 6.9.2 Thermal Non-Equilibrium Te±Tx 149 6.10 Possibilities for Execution 149 7 Conclusions 160 7.1 Contributions of Present Work 160 7.1.1 Reflectivity Measurements 160 7.1.2 Heat Front Propagation Measurements 160 7.1.3 A New Experiment for the Coming Generation of Lasers 161 7.2 Other Directions for Further Research 162 Bibliography 163 vi Lis t of Figures 1.1 Regions of the Density-Temperature Plane 4 1.2 Some Previous Experimental Techniques 6 1.3 Transition Region 8 1.4 Pulse Definition 10 1.5 Laser-Target Interaction 13 2.1 Equation of State Comparison 22 2.2 Equation of State Comparison 23 2.3 Equation of State Comparison 24 2.4 Average Ionization Comparison 26 2.5 Average Ionization Comparison 27 2.6 Average Ionization Comparison 28 2.7 Conductivity Comparison 30 2.8 Conductivity Comparison 31 2.9 Conductivity Comparison 32 2.10 Conductivity Comparison 33 2.11 Conductivity Comparison 34 2.12 Conductivity Comparison 35 3.1 Atomic Transitions 53 vii 3.2 Comparison of CRE to Thomas-Fermi average ionization 58 3.3 McWhirter's Criterion 60 3.4 Convergence to CRE average ionization 66 3.5 Electromagnetic Wave Solver 67 4.1 Milchberg et al.'s Conductivity 80 4.2 Evolution of Density Gradients 82 4.3 Reflectivity Histories 84 4.4 Plasma Conditions 85 4.5 Plasma Conditions 86 4.6 Milchberg et al.'s Experiment 89 4.7 Fedosejevs et al.'s Experiment 90 4.8 Sample Plasma Conditions 91 4.9 Milchberg et al.'s Experiment 94 4.10 Fedosejevs et al.'s Experiment 95 4.11 Equilibrium Snapshot 97 4.12 Non-Equilibrium (Te ^ Tt) Snapshot 98 4.13 Ponderomotive Force 99 4.14 Ponderomotive Force 105 5.1 Pump-Probe Technique of Vu et al 108 5.2 Temperature Profiles I l l 5.3 Temperature Profiles 112 5.4 Temperature Profiles 113 viii 5.5 Laser Energy Deposition Profiles 115 5.6 Density Profiles 117 5.7 Heat Front Velocity Histories 118 5.8 Heat Front Velocity Histories 120 5.9 Heat Front Velocity Histories 122 5.10 Heat Front Velocity Histories 123 6.1 Effects of Pulse Duration 126 6.2 Effects of Pulse Duration 127 6.3 Effects of Short Wavelength 129 6.4 Effects of Long Wavelength 130 6.5 Idealized Laser Pulses 132 6.6 Timing Diagram 134 6.7 End of Pump Pulse, 1013 W/cm2 135 6.8 End of Pump Pulse, 1014 W/cm2 136 6.9 End of Pump Pulse, 1015 W/cm2 137 6.10 End of Pump Pulse, 1016 W/cm2 138 6.11 Variation of R and T 139 6.12 Method to Quantify ISP 141 6.13 Comparisons of Reflectivities and Transmissions 142 6.14 Method of Data Reduction 150 6.15 Comparison of ISP with Hydrodynamic Simulation 151 6.16 Comparison of ISP with Hydrodynamic Simulation 152 ix 6.17 Electron Densities 153 6.18 Initial Value for R and T of Expanding Plasma 154 6.19 R and T for Hot, Expanded States 155 6.20 Ionization Histories 156 6.21 Ionization Histories 157 6.22 Electron Densities 158 6.23 Electron Densities 159 x Acknowledgements I would like to thank my research supervisor, Dr. Andrew Ng, for providing the re-sources necessary for the completion of this work and for the countless discussions relating to plasmas and lasers. As well, I thank my supervisory committee, Dr. B. Ahlborn, Dr. B. Birg-ersen, and Dr. G. Jones for their comments on this thesis. I received assistance from Dr. Gordon Chiu in some of the numerical simulations. I also appreciate the theoretical support from Dr. Richard More, Dr. Francois Perrot and Dr. Chandre Dharma-wardana. The former members of our group, Al Cheuck, Kristin Smith, and Steve Dyke provided company and a diverse array of questions which increased the enjoyment of studying. I thank my friends and family for their moral support and company. Finally, I thank my wife Hooban for her encouragement, help, and understanding during the last three years of my program. xi \ Chapter 1 Strongly Coupled Plasmas and High-Intensity Short-Pulse Lasers 1.1 Strongly Coupled Plasmas The study of the physics of matter at high density and temperature has been motivated by research in inertial confinement fusion, geophysics, astrophysics, nuclear explosions* -5 and x-ray sources6. These studies probe the behaviour of matter across a wide range of conditions. Accordingly, a wide range of material models are necessary to describe this behaviour. Figure 1.1 shows a phase diagram of aluminum. At temperatures below the line marked "Z=l" the average ionization will be less than one and this region of the density-temperature plane is pertinent to a weakly ionized plasma. The curve marked "LV coexistence" is the upper bound of the liquid-vapor coexistence region8. The curve marked "T-Fermi" is the locus where the kinetic temperature of the electrons equals the Fermi temperature which is given by where ne is the electron density and m the electron mass. At temperatures below the Fermi temperature the free electron subsystem of the plasma is at least partially degenerate. The most prominent region of Figure 1.1 is the shaded area marked as "strongly coupled plasma". A strongly coupled plasma 7 ' 9 is one in which the potential energy arising 1 Chapter 1. Strongly Coupled Plasmas and High-Intensity Short-Pulse Lasers 2 from the Coulomb interaction between the particles is greater than the kinetic energy of the particles. Traditionally, the ion-ion coupling and the electron-electron coupling are considered separately. For the plasmas of interest the electron subsystem is in general not strongly coupled but the ion subsystem frequently is. As can be seen from Figure 1.1, a considerable range of the density-temperature plane has a strongly coupled ion subsystem. The method used to determine whether or not the ion subsystem is strongly coupled is as follows. The Coulomb energy may be estimated as E< = (1.2) r with r=(— where nj is the ion density, e is the electronic charge, and Z is the average ionization. The ion-ion coupling parameter is therefore defined as and Tn > 1 indicate a strongly coupled plasma. In the plasmas of interest to the current work the ions are treated classically since the deBroglie wavelengths for the ions are smaller than the interparticle spacings9, \JmikBT where rrii is the ion mass. Typically, the more strongly coupled plasmas are found at lower temperatures and higher densities. This trend is illustrated in Figure 1.1 by the Chapter 1. Strongly Coupled Plasmas and High-Intensity Short-Pulse Lasers 3 line marked = 10, which is the locus of points where the ion-ion coupling parameter is equal to ten. In a strongly coupled plasma short-range order starts to develop among the ions and the thermodynamic properties of the plasma deviate from those of an ideal gas. Transport models based on binary electron-ion collisions 1 0 are not appropriate since the electrons interact with the entire spectrum of ions. Even the concept of Debye screening is not applicable since the Debye length becomes comparable to the interionic spacing. In gen-eral, the thermodynamic and transport properties of strongly coupled plasmas are not well understood and hence are the subject of considerable experimental and theoretical s t u d i e s 1 , 2 , 4 , 7 - 9 ' 1 1 - 2 9 . Of particular interest are the transport properties of normal solid density aluminum (p — 2.7g/cm3) as a function of temperature since the transport prop-erties depend heavily on the electron-ion interactions about which little experimental data is available in the strongly coupled regime. Measurements of the transport prop-erties of normal solid density aluminum would yield data pertinent to the electron-ion interactions from the known solid state behaviour of cold aluminum through the strongly coupled region and to the known behaviour of high temperature plasmas. Such a complete measurement of solid density aluminum over the temperature range from 103 to 10 8K is not available. Chapter 1. Strongly Coupled Plasmas and High-Intensity Short-Pulse Lasers 4 Figure 1.1: Regions of the density-temperature plane for aluminum, calculated according to QEOS 8. The normal density of aluminum is 2.7g/cm3. Chapter 1. Strongly Coupled Plasmas and High-Intensity Short-Pulse Lasers 5 1.2 Techniques Used to Create and Probe Strongly Coupled Plasmas 1.2.1 Techniques Prior to the Development of High-Intensity Short-Pulse Lasers Figure 1.2 shows the regimes which are accessible by various traditional techniques. Iso-baric expansion measurements31 have been used to measure the electrical conductivity and sound speed. This technique appears to be limited to maximum pressures of a few kBar and expansions of about 30 % below normal density. Imploding wire array32 and other plasma pinch techniques reach only a fraction of solid density. Also, the denser core region of these plasmas is surrounded by a lower density plasma, which screens the core from direct optical observation. Exploding copper wires confined in glass capillary tubes have been used to prepare strongly coupled plasmas. The use of glass capillaries12,13 main-tains higher densities than in exploding wires and plasma pinches. However, the density reached is at most half solid density. Furthermore, the temperature obtained in these capillary experiments was limited to several eV. Shock wave techniques are used to obtain equation of state and conductivity data along the principal Hugoniot. The principal Hugoniot is the locus of states that can be reached using single, steady shocks 3 3 , 5 5 - 5 8. Shock wave experiments offer the possibility of probing well-defined high density plasmas. A shock wave is typically produced in planar foils by irradiating one side of the foil with a high-power laser. The shock wave propagates through the foil and when it reaches the far side of the foil it releases into the surrounding vacuum. As the shock wave releases the resulting, expanding, plasma may be studied through optical techniques18. Chapter 1. Strongly Coupled Plasmas and High-Intensity Short-Pulse Lasers 6 Figure 1.2: Areas of the density-temperature plane accessible to long pulse laser experi-ments and other non-laser techniques. The checkerboard is the isobaric expansion tech-nique, the grey is plasma pinch, and the cannonballs denote the conditions achieved in a copper wire capillary experiment. The line with the pressures is the principal Hugoniot. Chapter 1. Strongly Coupled Plasmas and High-Intensity Short-Pulse Lasers 7 For studying shock release, optical probing might be able to access dense plasma regimes when applied immediately after the onset of release, before the cool outermost plasma obscures the higher density and higher temperature regions inside. The analysis of shock release techniques must take into account the difference in electron and ion temperatures which arise in a shock wave due to ion viscous heating and the finite time required for thermal equilibration between the electrons and the ions. This complicates the interpretation of measurements from what is otherwise a well-defined initial state. In laser and x-ray driven shock wave experiments care must also be taken to avoid target preheat by x-rays and suprathermal electrons. X-rays and suprathermal electrons cat affect energy transport to the unshocked material ahead of the shock wave 3 0' 3 5 , 3 6 , 4 0 , 7 3 modifying the initial state of the material before it is compressed by the shock wave. The reflectivity and transmissivity of a plasma to optical radiation is strongly de-pendent on the electrical conductivity of the plasma, which in turn is dependent on the electron-ion collision frequency37. This makes optical probing techniques highly desirable16. Figure 1.3 shows a theoretical prediction29 of the electrical conductivity of a normal solid density aluminum plasma as a function of temperature in a particularly important region of the density-temperature plane. This is a transition region where the behaviour of the plasma changes from that of a solid state system, in which the conduc-tivity decreases as a function of temperature38, to the behaviour typical of a high tem-perature plasma in which the conductivity increases as a function of temperature34. The region of minimum conductivity shown in Figure 1.3, which is not expected to be found in lower density plasmas, is governed by the effective electron-ion collision frequency of Chapter 1. Strongly Coupled Plasmas and High-Intensity Short-Pulse Lasers 8 u Q 1 0 I I i I I i i i ' I I I I i i i i i I I i i i i i i i I I i i i i i i i I IO 3 10 4 IO 5 10 6 IO 7 Temperature (K) Figure 1.3: The electrical conductivity of aluminum at solid density as a function of tem-perature, according to Lee and More. This illustrates the region of minimum conductivity. Chapter 1. Strongly Coupled Plasmas and High-Intensity Short-Pulse Lasers 9 a solid-density strongly coupled plasma. The advent of high-intensity short-pulse lasers has enabled new techniques to study this region. 1.2.2 Advent of High-Intensity Short-Pulse Lasers Figure 1.4 illustrates the parameters used to describe a high-intensity short-pulse laser. At the time of peak power, the laser intensity on target would be greater than 1012 W/cm2, and the pulse would last hundreds of femtoseconds or less. These lasers have become readily available since the mid 1980's. Prior to this time, laser-matter interaction studies at high intensities were made using lasers whose pulse width would be measured in tens or hundreds of picoseconds to nanoseconds. High-intensity short-pulse lasers enable the production of hot, solid density, strongly coupled plasmas in a simple experimental arrangement by irradiating a solid target and then measuring the reflected laser light. This yields a time-integrated measurement of reflectivity. The pioneering experiment of Milchberg et al. 1 4 was among the first of such high-intensity experiments. It was an attempt to measure the electron-ion col-lision frequency in solid-density strongly coupled aluminum plasma. The results and their interpretation led to a long-standing controversy28'127 over the effective electron-ion collision frequency in strongly coupled plasmas. Other workers made additional and different measurements39'47-49 which provided further information but did not resolve the controversy. As well, just what constituted a laser pulse short enough to avoid the time-dependent treatment of hydrodynamic expansion in the interpretation of data was also the subject of considerable debate, with the range claimed varying from 250 fs39 Chapter 1. Strongly Coupled Plasmas and High-Intensity Short-Pulse Lasers 10 Time Figure 1.4: Definition of parameters used to describe a laser pulse. The horizontal arrows indicate the pulse width, measured at the half-maximum of peak intensity and the vertical arrows indicate the peak intensity of the laser pulse. Typical pulse lengths for short-pulse lasers are 100—1000 fs and typical intensities are 10 1 2—10 1 8 W/cm2. The highest laser intensity considered in the present work is 10 1 6 W/cm2. Chapter 1. Strongly Coupled Plasmas and High-Intensity Short-Pulse Lasers 11 to 12 ps 4 4-more than an order of magnitude. Throughout the entire period where the techniques of high-intensity, short-pulse laser-matter interactions underwent their initial developments, there was disagreement over the models that should be used to design and interpret the experiments. Generally speaking the design and interpretation of experiments involved assumptions that a r e 4 1 - 5 1 intended to simplify but are difficult to justify. A number of approaches have been employed to deal with hydrodynamic expansion. One is to assume that there is no hydrodynamic expansion. Another is to consider an arbitrarily chosen exponential density profile. A third practice is to choose density profiles and gradient scale lengths to match model results of simulations to experimental reflectivity data. The validity of these approaches is largely untested, either theoretically or experimentally. The nature of the laser plasma interaction is strongly dependent on the spatial profile of the plasma. In order to better understand the limitations of these simplified approaches a brief review of the formation of the plasma in laser-solid experiments is in order. Figure 1.5 shows schematic diagrams of laser-solid interactions for both short-pulse and long-pulse experiments. Laser light is focussed onto the target surface. Initially, the laser light penetrates by skin depth deposition. The energy is predominantly absorbed by the free electrons which rapidly thermalize with the ions, resulting in vaporization of the surface layer of the target. This plasma continues to absorb energy from the laser and expand away from the original target surface. Obviously, a density gradient develops during this process, resulting in plasma densities ranging from vacuum to solid density. Within this range of plasma densities there is a region where the plasma frequency is equal Chapter 1. Strongly Coupled Plasmas and High-Intensity Short-Pulse Lasers 12 to the laser frequency. This is the critical density ncr layer. The plasma frequency125 is where n e is the electron density, me is the electron mass and e is the electronic charge. Generally speaking much of the laser light which penetrates through the plasma and reaches the critical density layer will be reflected at the critical density layer. What is not reflected will propagate into the denser plasma beyond the critical density layer as an evanescent wave 1 2 6. In short-pulse experiments this process still plays a significant role. Although hydrodynamic expansion gives rise to a plasma gradient, femtosecond time scales are too short for the formation of a layer of rarified plasma which may totally dominate the absorption of the laser light. In long-pulse experiments the hydrodynamic expansion of the plasma leads to a large and rarified plasma in front of the solid target, which typically absorbs most of the laser light. Furthermore, as shown schematically in the lower part of figure 1.4 the scale length over which the laser penetrates beyond the critical density layer is small compared to the length of the plasma. In these experiments, the laser-target interaction will be domi-nated by the expanded, low-density plasma. Hence, a reflectivity measurement will not reveal dense plasma properties. Between the critical density layer and the solid target lies the ablation zone where thermal conduction transports energy and heats the target material which then expands outward. The momentum of the expanding plasma drives a compressional wave into the target. In light of the foregoing discussion, the distinction between short-pulse and long-pulse given by (1.5) Chapter 1. Strongly Coupled Plasmas and High-Intensity Short-Pulse Lasers 13 Figure 1.5: Schematic representation of two classes of laser-solid interaction. The top drawing applies to short-pulse laser experiments and the lower drawing applies to long-pulse laser experiments. The arrows denote the direction of the laser, the dotted line is the critical density layer, the snowflakes are the coronal plasma, the gray area is the plasma being heated by the evanescent wave, the wavy lines denote the ablation zone, and the black marble is compressed target material. Chapter 1. Strongly Coupled Plasmas and High-Intensity Short-Pulse Lasers 14 is best dictated by the plasma that is produced. A long-pulse experiment has very little laser-solid interaction, and in a short pulse experiment the laser-solid interaction plays an important role. What length of pulse constitutes a short-pulse also depends on the intensity of the laser and its wavelength. 1.3 Present Work The present work uses numerical simulations to show that the laser-solid coupling, hydro-dynamic expansion, and the subsequent laser interaction with the expanding below-solid density plasma are important processes for understanding the behavior of femtosecond-laser heating of a solid. This is the regime where much of the experimental work in the field of strongly coupled plasmas has been done. A hydrodynamic model implemented in computer code is used as a tool to simulate high-intensity laser-solid interactions. A description of the hydrodynamic model is pre-sented in chapters 2 and 3. Chapters 4 and 5 use numerical simulations to provide deeper and broader understanding of existing high-intensity short-pulse laser-solid experiments. Chapter 4 also resolves a controversy surrounding the interpretation of Milchberg et al.'s experiment. Chapter 6 is a new idea for a different class of experiments to be performed with the new generation of high-intensity short-pulse lasers — ones that produce intense pulses that are tens of femtoseconds long, instead of hundreds of femtoseconds long. Chapter 2 Modelling the Plasma as a Fluid The interaction of the laser with the target and the response of the target to the laser are modelled using a finite element model represented in a computer code, hereafter referred to as the hydrocode. Hydrocodes are commonly used in the studies of plasmas, including shock waves in solids, inertial confinement fusion experiments, and laser-solid exper iments 5 2 ' 5 9 , 7 4 ' 7 7 , 7 9 , 9 3 , 1 3 7 ' 1 3 8 , 1 4 6 . The hydrocode used in the present work was adopted from M E D U S A 5 3 in 1987 by Peter Celliers 5 4. Since then, it has evolved to meet the changing demands of various investigations involving shock waves77 and high-intensity short-pulse laser-matter interactions 5 2 , 1 3 7. This has resulted in a hydrocode of substantial complexity, and having interacting physical models required to describe non-local and nonequilibrium processes in a manner consistent with the hydrodynamic motion of the plasma. The clearest way to describe the hydrocode is to split the description into two parts. The first part discusses a basic hydrodynamic model 5 9 in which the plasma is assumed to meet the conditions of local thermodynamic equilibrium 9 0 and the evolution of the plasma is governed by the principles of conservation of mass, momentum, and energy. In this basic model the plasma is thermally conducting and the laser energy deposition is specified as a source term. This basic model is the framework to which models of 15 Chapter 2. Modelling the Plasma as a Fluid 16 physical phenomena not included in the basic model are added to extend the capability of the hydrocode. These additional models form the second part of the discussion of the hydrocode and include but are not limited to the modelling of the deposition of the laser energy and electron-ion thermal nonequilibrium (Te ^ T j ) 4 6 ' 4 7 . Chapter 2 describes the basic hydrodynamic model and chapter 3 describes the roles that the additional models play in simulating the interaction of high-intensity short-pulse lasers with matter. 2.1 F l u i d Equations The hydrocode solves the one-dimensional fluid equations which include the conservation of mass, momentum, and energy in a Lagrangian formulation 3 3. In the model the plasma is divided into cells. The time evolution of the cells is followed and the mass of each cell is kept constant. The physical size of the cell changes to represent compression or rarefaction. The Lagrangian coordinates are mass and time and are related to the Eulerian coordinates by where r is position, t is time and p is the mass density. RQ is a reference position, and in the current work it is taken to be zero. In Lagrangian coordinates the fluid equations are that of continuity, dV du . conservation of momentum, (2.1) Chapter 2. Modelling the Plasma as a Fluid 17 and conservation of energy, dE dV where V = 1/p is the specific volume, u is the fluid velocity, P is the pressure, E is the internal energy and Q is a source term, representing thermal conduction H and laser deposition A. Q = H + A (2.5) Energy transport due to thermal conduction in or out of each cell is calculated as H=-VK-VT (2.6) P where K is the thermal conductivity and T is the temperature. The conservation equations are solved for each time step in the simulation. In each time step, the solution proceeds in three operations. First, hydrodynamic motion and its associated quantities p, u, and E are calculated. Then the transport and deposition processes are calculated. Finally, the internal energy E is adjusted taking transport and deposition into account, and the temperatures and pressures are calculated from the equation of state, using the mass density and internal energy as independent variables. This entire procedure is repeated for every time step of the simulation. The time steps are sufficiently small that the error introduced into the calculation by not simultaneously solving for all variables is negligible. 2.1.1 Hydrodynamic M o t i o n The cell coordinates r and velocities u are advanced in time by solving a simplified version of the conservation equations 2.2-2.4. The simplification is to ignore thermal conduction Chapter 2. Modelling the Plasma as a Fluid 18 and laser deposition. From conservation of mass, dV_ ~dt du dm = 0, (2.7) from conservation of momentum du dP dt dm (2.8) and conservation of energy, de d(up) dt dm = 0 (2.9) where is the total energy of the fluid. The plasma may contain gradients in quantities such as pressure, density, temperature, and fluid velocity. In the finite-element analysis used to represent the plasma in the model, each gradient is described by the differences in the physical quantity between adjacent cells. When there is a gradient in pressure the boundaries of the cells of the mesh will move and this represents the fluid motion. The task is to evaluate the motion of the cell boundaries in a manner consistent with the physical conditions of each pair of adjacent cells. The value of a physical quantity that is assigned to a cell is an average over the domain of the cell. In order to improve the accuracy of evaluation of the motion, the piecewise parabolic method60 is used to interpolate the values at the cell boundaries from the average values of the cells. Following this procedure, the the solution of this Riemann problem61'62 is to calculate the average pressure and velocity of each cell interface during the time step that yields identical conditions within a prescribed distance of the cell Chapter 2. Modelling the Plasma as a Fluid 19 interface61. This distance is equal to the product of the sound speed in the cell and the duration of the time step. The average pressure P and particle velocity u during the time step at each interface are then used to calculate the momentum and energy changes in each cell. The cell coordinates and velocities are then advanced in time using the solutions to the Riemann problem at each interface, r a"+ 1 = rna + Atua (2.10) £n+l = £n + ^L(paua _ pbub) (2.13) Am where the subscripts a and b denote respectively the edge positions of the cell under consideration and the superscripts n and n + 1 denote respectively the old and new time steps. The total adiabatic change in the internal energy E of each cell is the difference between the total hydrodynamic work e done on the cell and the change in the kinetic energy for the time step being performed, AE\hydro = en+1 -en- \[{un+lf - {unf)\ . (2.14) 2.1.2 Energy Transport and Laser Deposition The energy equation is Cv— + BT-£ + AE\hydro = Q (2.15) where Cv and BT are defined Chapter 2. Modelling the Plasma as a Fluid 20 and Q (equation 2.5) is energy flux due to thermal conduction and laser deposition. The procedure used to model the laser energy deposition in each cell is discussed in chapter 3. The Crank-Nicholson67 iterative time-centered differencing scheme and Gauss elimination67 are used to solve the energy equation 2.15 for each cell simultaneously. Each iteration yields a temperature for each cell and these temperatures are used to calculate the thermal energy flux from cell to cell. The new internal energy of each cell is then evaluated according to Convergence of the solution is tested on the internal energy of each cell. An iterative technique is necessary since several quantities, such as ionization, conductivity, and the specific heat capacities depend on the temperature. 2.2 Ma t e r i a l Models Models of the equation of state, thermal and electrical conductivities, and of the ionization of aluminum are required. The equation of state, thermal conductivity, electrical conduc-tivity, and average ionization Z are derived from various sources and used in tabular form as function of p and E, En+1 = En + AE\hydro + QAt. (2.16) T T(P,E) P K K(P,E) (2.17) o a(p,E) z Z(p,E). Chapter 2. Modelling the Plasma as a Fluid 21 The emphasis of this thesis is on the methodology of the design and interpretation of experiments. Therefore only very brief descriptions of the material models will be given here. 2.2.1 Equation of State In the simulations presented in the current work two equations of state were used, namely QEOS 8 and SESAME 6 9 ' 7 0 . This provides a test of the sensitivity of the predictions to the equation of state model. Figure 2.1-2.3 show comparisons for the pressure and heat capacities for the two equations of state. A third equation of state, the ideal gas law, is also shown for purposes of comparison. The average ionizations used in calculating the ideal gas equation of state are derived from a Thomas-Fermi model8'68. SESAME refers to the SESAME 6 9 ' 7 0 data table package. It is widely used in laser-matter interaction studies. The SESAME data table is composed of seven different theo-retical and semi-empirical equation of state models, applied to seven different regions of the density-temperature plane. The seven models are connected to each other by an in-terpolation scheme satisfying the condition that the state functions are continuous except at phase transitions, and that the isothermal bulk modulus and heat capacity be positive. QEOS 8 stands for Quotidian Equation Of State. In this model the electrons and ions are described separately and the total internal energy and pressure are treated as sums of the electron and ion contributions. The electron equation of state is based on a Thomas-Fermi model which has been modified near solid density by a semi-empirical treatment due to Barnes71 to account for the chemical bonding energy so that the pressure at solid Chapter 2. Modelling the Plasma as a Fluid 22 10 u e 10 PQ S 10 3 U CM -2 v3 10 10" -4 • • I I I I I I m m ! h • • - m H I J ^ m a^m M M • • m mm^t ; w ^ A Y1 J^TTZ..^ M. » I • • • • | 70 10 4 Pressure QEOS S E S A M E - A — Ideal Gas i i o 2 » & U -4—> d o> X i .a o o o 10 s Temperature (K) 10 e 10 7 Heat Capacity QEOS S E S A M E — •-•-Ideal Gas Figure 2.1: Pressures and heat capacities for three equations of state at l /100 t / l solid density. Chapter 2. Modelling the Plasma as a Fluid 23 IO1 10° 10" £ i o -S3 10" 10" 10" • I llllll 1 m m • " • / .'l \ ~f • s r , A • • J J\j— • • m>£x \ • j £ \ •// - • v * / (PL //• i _# i- f. - •. • • • a • _ • • : / • • • / / 1 B " / / I „ •_ • m m I /• • • • • • • • q 10 2 10 J Pressure -o— QEOS - • — SESAME -A— Ideal Gas J 10 ^ bo o rt cu as U rt m X o 'C o j= o o 10' Temperature (K) 0.1 7 10" 10 Heat Capacity QEOS SESAME Ideal Gas Figure 2.2: Pressures and heat capacities for three equations of state at l/10th solid density. Chapter 2. Modelling the Plasma as a Fluid 24 Figure 2.3: Pressures and heat capacities for three equations of state at solid density. Chapter 2. Modelling the Plasma as a Fluid 25 density and 300 K is not several MBar, but only a few KBar. For the short time scales and MBar pressures typical of the simulations considered in the present work, an additional few kBar static pressure occurring at solid density is inconsequential. The ion equation of state is derived from a modified Cowan72 model. For the current work, QEOS has two practical advantages over SESAME. The use of separate electron and ion models facilitates the application of QEOS to simulations where electron and ion temperatures are different. Furthermore, the thermodynamic functions in QEOS are numerically smoother than those derived from the SESAME tables since there is no need to interpolate between different equation of state models. This is apparent from the plots of the isochoric heat capacity, particularly in the low temperature range of the plot of solid-density data (figure 2.3). However, SESAME is believed8 to be more accurate than QEOS although in the present calculations both SESAME and QEOS often yield similar results. In the plots of the isochoric heat capacity for 10 % normal density (figure 2.1) and 1 % normal density (figure 2.2), a large local peak can be seen in in the low temperature range for both the QEOS and SESAME models. This peak corresponds to the liquid-vapor coexistence region which is shown in figure 1.1. 2.2.2 Average Ionization Models Knowledge of the free electron density is essential in the calculation of many plasma properties, including electrical and thermal conductivity. Figures 2.4-2.6 show results of the three models used in the present work. The ionization model used most heavily here is the Thomas-Fermi68 model in QEOS 8. Chapter 2. Modelling the Plasma as a Fluid 26 — Lee and More ••— Rinker • A — Perrot and Dharma-wardana Figure 2.4: Average ionizations for three models at 1/100"1 solid density. Chapter 2. Modelling the Plasma as a Fluid 12 10 D 6 (JO rt > < 4 2 • • • i o 3 10 4 10 5 10 6 IO 7 Temperature (K) — • — Lee and More — • — Rinker — • — Perrot and Dharma-wardana Figure 2.5: Average ionizations for three models at l /10 t / l solid density. Chapter 2. Modelling the Plasma as a Fluid 12 10 • • / * • • • • A A . A A A A i k A A _ _ . • • — 10 3 104 10 5 Temperature (K) 10° 10 7 — Lee and More -•— Rinker - A - — Perrot and Dharma-wardana Figure 2.6: Average ionizations for three models at solid density. Chapter 2. Modelling the Plasma as a Fluid 29 This ionization model is the same model which is in Lee and More's thermal and electrical conductivity models 2 9. Another promising candidate for the calculation of average ioniza-tion is that due to Perrot and Dharma-wardana 2 7. This application of density functional theory 7 5 ' 7 6 to strongly coupled plasmas is significant since the model contains no freely adjustable parameters. The third model is that due to Rinker 2 5 which is included in the S E S A M E data tables 7 0 , 1 3 9 . Each of these three models assumes that the plasma is in local thermodynamic equilibrium, 7 8 a condition that is satisfied when (i) collisional ionization, recombination, excitation, and de-excitation are dominant over all radiative transitions, (ii) the electron and ion temperatures are equal, and (iii) the electron and ion velocity distributions are Fermi-Dirac. 2.2.3 Electrical and Thermal Conductivity Figures 2.7-2.12 show results of the equilibrium conductivity models due to Lee and More 2 9 , Perrot and Dharma-wardana2 7, and Rinker 2 5 . Although the conductivity model of Spitzer 1 0 is not used in the present work, it is included in the figures for comparison. The average ionization used in Spitzer's model is calculated using a Thomas-Fermi model 8. The conductivity model of Lee and More is based on the Boltzmann relaxation-time approximation 8 0. The collision frequency, governed by both electron-ion and electron-neutral collisions, enters the model through the calculation of the relaxation time. In order to make the model applicable over a wide range of density-temperature conditions, Chapter 2. Modelling the Plasma as a Fluid 30 10 3 10 4 10 5 10 6 10 7 Temperature (K) - • — Lee and More • • — Rinker - A — Perrot and Dharma-wardana J * — Spitzer Figure 2.7: DC Electrical Conductivities for four different models, at l/lOO*71 solid density. Chapter 2. Modelling the Plasma as a Fluid 31 IO3 10 4 IO5 IO6 IO7 Temperature (K) • • — Lee and More M— Rinker - i t — Perrot and Dharma-wardana — Spitzer Figure 2.8: DC Electrical Conductivities for four different models, at 1/10"1 solid density. Chapter 2. Modelling the Plasma as a Fluid • • — Lee and More • • — Rinker - i k — Perrot and Dharma-wardana — Spitzer Figure 2.9: D C Electrical Conductivities for four different models, at solid density. Chapter 2. Modelling the Plasma as a Fluid 33 Temperature (K) - • — Lee and More • — Rinker - A — Perrot and Darma-wardana — Spitzer Figure 2.10: Thermal Conductivities for four different models, at l /100 4 / l solid density. Chapter 2. Modelling the Plasma as a Fluid 34 — Lee and More M— Rinker • A — Perrot and Dharma-wardana — Spitzer Figure 2.11: Thermal Conductivities for four different models, at l / 10 i / l solid density. Chapter 2. Modelling the Plasma as a Fluid IO3 Temperature (K) - • — Lee and More - • — Rinker - A — Perrot and Dharma-wardana — Spitzer Figure 2.12: Thermal Conductivities for four different models, at solid density. Chapter 2. Modelling the Plasma as a Fluid 36 certain limits are imposed on some physical quantities. The minimum electron-ion im-pact parameter bmin is the value allowed by the uncertainty principle. The maximum impact parameter bmax is the Debye-Hiickel screening distance, corrected for electron degeneracy29. The Coulomb logarithm, \nA = Un(l + b2max/b2min) (2.18) used in the calculation of the electron-ion collision cross section is set to have a minimum value of two. This minimum is chosen to overcome an inherent difficulty in the model that the calculated electric field screening length can become less than the interionic spacing. As the application of the Coulomb logarithm to solids and liquids is questionable the model is extended into those regimes by taking the relaxation time as the ratio of the interionic spacing to the mean electron velocity. This approximation is used where the predicted electron mean free path is less than the interionic spacing. In other regions of liquid and solid aluminum an approximation due to Ziman 8 1 is used to calculate the electron mean free path. The ion-ion structure factor9, which can be written as a function of the ion-ion coupling constant Tu , does not explicitly enter Lee and More's conductivity model. Lee and More justify the minimum value of the Coulomb logarithm by comparing the results of their conductivity model to a partial-wave calculation 8 4, where the ion-ion structure factor would enter the calculation explicitly. Perrot and Dharma-wardana have constructed a model for electrical conductivity based on density functional theory. 7 5 ' 7 6 It is a first-principle calculation with no free parameters. The model solves self-consistently the Kohn-Sham equation 8 0, an effective one-particle Schrodinger equation for the electron density distribution in the presence of Chapter 2. Modelling the Plasma as a Fluid 37 the potential of the ions, coupled with the screened hypernetted chain equation 8 3 which governs the ions. The Kohn-Sham equation provides the electron density distribution, the electron-ion scattering phase shifts used in calculating the transport coefficients, and the average ionization. The screened hypernetted chain equation yields the ion-ion structure factor. Perrot and Dharma-wardana did not provide a thermal conductivity model with the electrical conductivity. In the present work the thermal conductivity K is derived from the electrical conductivity o via the Wiedemann-Franz law 8 0 , Rinker's model 2 5 is used in the S E S A M E data base. It considers the plasma as a homo-geneous medium in which the free electrons are treated as plane waves propagating among a fixed group of scattering centers which represent the ions. The scattering centers are statistically distributed and have non-overlapping spherical forms. The relation between any two scattering centers is determined by the ion-ion structure factor. The potential in the interstitial region is constant. The scattering of the electrons is assumed to occur from one site at a time and treated through partial-wave analysis 8 4. The electron-ion scatter-ing cross sections are calculated using mean field theory for a single scattering center. As the plasma approaches high temperature and low density the ionic potentials used in the scattering cross sections are required to approach the Hartree-Fock-Slater potentials for an isolated atom. In the limit of high temperature and high density the potentials are required to approach the Thomas-Fermi-Dirac potentials. For the intermediate regime the potentials are calculated using a model developed by Liberman 8 5 , which incorporates (2.19) Chapter 2. Modelling the Plasma as a Fluid 38 exchange effects and shell corrections. In strongly coupled plasmas this model may break down since multiple site scatterings become important. 2.2.4 Treatment of Flux-Limited Thermal Conduction The flux-limited model of thermal conduction is a means of reducing the thermal heat flux in areas of the plasma model where the thermal gradient is extremely strong. The flux-limited approach is used since the classical Fourier heat flow law fails, predicting a thermal heat flux that is larger than consistent with observations 8 8 , 8 9 ' 1 3 6 , 1 4 0 , when the temperature gradient is extremely strong. The value of K in equation 2.6 is the flux-limited86 thermal conductivity, defined as 1 1 +Jn (2-20) K Kmodel Wf W = v%—nekhT dT where /cm0dez is the thermal conductivity predicted by the conductivity model, v is the greater of the mean thermal velocity or the Fermi speed, and T is the temperature. When used in equation 2.6, W is proportional to the heat flux that could arise from free streaming electrons. It represents the maximum possible (saturated) heat flux. This is equal to the mean thermal velocity of the electrons multiplied by the energy of each electron. / is the flux-limiter. Its theoretical maximum value is 0.6 since 0.6nevkbT is the maximum heat flux that can arise due to a free-streaming flow of electrons88, provided that the electrons have a Maxwellian velocity distribution. Some workers 8 7 ' 8 8 have used values of / as low as 0.03. Chapter 2. Modelling the Plasma as a Fluid 39 It should be realized that the flux limiter model is a crude attempt to apply a local thermal conductivity model to a non-local regime of thermal transport. In the present work it would appear that non-local thermal conduction is usually not important, and as such it is frequently the case that Kmodei ~ K. These effects are discussed further in chapters 4 and 5 along with the relevant simulations. Chapter 3 Applying the Hydrocode to Short-Pulse Laser Experiments The fluid equations and the models of equation-of-state, conductivity, and ionization are described in chapter 2 are basic components of the hydrocode. However, additional components are required in order to model the non-local and non-equilibrium phenomena which generally occur in and may even dominate the behavior of high-intensity short-pulse laser-solid experiments 1 7 ' 5 2. The plasma may not be accurately modelled using hydrodynamic and material models based on the existence of local thermodynamic equilibrium 9 0. This is because the electron temperature may be different from the ion temperature9 6 and electron-ion collisions 9 0 ' 9 1 may not be sufficiently rapid to dominate the distribution of ionic states, or to even achieve a steady-state distribution of ionic states. Furthermore, the interaction of the laser with the target is complex and must be mod-elled using techniques more sophisticated than the elementary inverse Bremsstrahlung models 9 3 commonly applied to long-pulse laser-matter interactions. Ponderomotive den-sity profile modification 6 5 and limiting of the electric fields in the plasma due to wavebreaking deserve consideration, although they are rarely significant for the experimental conditions considered in this thesis. Some phenomena are not considered in the present work. Parametric instabilities 9 5 40 Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 41 and exotic absorption mechanisms such as vacuum heating 9 6 and anomalous absorption 9 7 are not considered relevant to this thesis. This is due to three factors, namely the short optical laser wavelengths, the modest laser irradiance range (10 1 3 — 10 1 6 W/cm2), and the short plasma scale lengths (50-500 A) resulting from the short laser pulse lengths. Previous work 9 4 suggests that these absorption mechanisms and instabilities are expected to be seen in femtosecond experiments with optical lasers only when the intensity exceeds 10 1 6 W/cm2. In the design of long pulse laser-solid experiments a commonly used criterion for the suppression of parametric instabilities 9 8 is * A 2 < 10 1 4 (3.1) cm1 where $ is the laser intensity and A is the laser wavelength. Generally speaking, this criterion is rarely exceeded in the present work and even then, it is only exceeded for the highest irradiances. Radiative energy transport is also not considered in this thesis since the plasma temperatures corresponding to the irradiation conditions considered are generally too low to yield a significant radiative energy flux99. We now turn to the discussion of the modifications to a basic hydrocode which are necessary to the present work. 3.1 Two-Temperature (Te ^ Ti) Effects In high-intensity short-pulse laser-solid experiments, the electron and ion components of the plasma are not necessarily in thermal equilibrium with each other. The laser en-ergy is initially absorbed by the electrons, which then thermalize with the ions through Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 42 collisions. In the process of absorbing energy from the laser, the electron velocity distribu-tion may also be driven away from its equilibrium velocity distribution 1 4 0 ' 1 4 1 . As a shock wave forms in solid targets, the ions are directly heated through viscous ion heating 3 3 and the ion temperature may exceed the electron temperature. Moreover, the electron and ion subsystems of the plasma interact and do work on each other and therefore the hydrodynamic treatment must be modified. 3.1.1 Electron-Ion Thermal Equilibration A more important issue is the process of thermal equilibration between the electron and the ion subsystems. It should be noted that, although electron-ion collisions are the de-termining factor in both thermal equilibration and conductivity, thermal equilibration is treated separately from transport in the hydrocode to facilitate the use of validated con-ductivity models such as those described in chapter 2. There are no validated models for electron-ion thermal equilibration for the range of temperature and density encountered in high-intensity short-pulse laser-solid experiments18. This range is typically below solid density and below 10 7K, and is contained in the portion of the density-temperature plane in figure 1.1 marked as strongly coupled plasma. The thermal equilibration process may be described by 1 0 dTx Te-Ti , . « = - I T < 3 ' 2 ) where rei is the time scale for electron-ion thermal equilibration. In numerical schemes, it is more convenient to consider the energy transferred between the electron and ion Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 43 subsystems during a single time step, AE = C V l ( - T e T i ) A t (3.3) where At is the time step and cy, is the isochoric ion heat capacity. Since both Te and Ti will be changing in time, and since cvi depends on Tj while rei depends on both Te and Ti, the time step must be kept sufficiently small so that the electron and ion temperatures do not undergo large changes in a single time step. The greatest difficulty lies in the calculation of rei. Spitzer gave an expression valid for weakly coupled plasmas 6.5 x 10 3 T e 3 / 2 T « = n ^ l n A ( S ) ( 3 - 4 ) where rii is the ion density in cm~3 and Z* is the average ionization. Brysk has also derived an expression1 0 1 for the electron-ion thermal equilibration time, where p is the chemical potential. Brysk's expression attempts to take into account the effects of electron degeneracy. Whereas equations 3.4 and 3.5 may satisfactorily describe electron-ion thermal equilibration for weakly coupled plasmas, they are inappropriate when applied to strongly coupled plasmas. Both equations 3.4 and 3.5 predict ion-electron thermal equilibration times a couple of orders of magnitude too short 1 0 3. Hence, using either equation 3.4 or 3.5 to describe electron-ion thermal equilibration in solid density plasma may erroneously lead one to believe that the electron and ion temperatures are equal. Previous work 1 8 has indicated that the factor cvi/rei of equation 3.3 may more conveniently be replaced by a single Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 44 constant "g". This constant "g", is substituted into equation 3.3 as AE = ±(Te - Ti)At, (3.6) Po where po is the normal solid density of aluminum. The choice of <7 = 5 x l 0 1 6 (3.7) has been corroborated by experimental data 1 8. To mimic the density dependence of the electron-ion equilibration time prescribed by Spitzer's model 1 0 the energy exchange equation is further modified to become AE = ^(Te-Ti)At (J/kg) (3.8) Po Po where p0 is normal solid density. This is only a phenomenological and simplistic approach. Fortunately, as will be discussed later in chapters 4, 5 and 6, the impact of electron-ion thermal non-equilibrium (T e ^ Ti) on the numerical simulations turns out to be relatively minor for the experimental conditions considered in this thesis. 3.1.2 Two-Tempera tu re (T e ^ Tj) M a t e r i a l Mode l s Owing to the great difference in mass between the electrons and the ions, the time scales for electron-ion thermal 1 0 ' 1 0 0 equilibration can be thousands of times longer than the electron self-collision time (equation 3.11). They can also be greater than the time scales of hydrodynamic motion. Since conductivity and equation of state properties depend on both the electron and ion temperatures, one would ideally have material data and models describing the behavior of the aluminum plasma as a function density, electron temperature, and ion temperature. Unfortunately, no general and rigorous treatment of Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 45 material models of strongly coupled plasmas across the entire range of plasma conditions encountered in high-intensity short-pulse laser-solid experiments is available. A two-temperature version of QEOS 8 is used in two-temperature simulations. As was discussed in Chapter 2.2.1, QEOS is treated as the sum of electron and ion fluids. For the equilibrium equation of state the electron and ion temperatures are equal. In the two-temperature application of QEOS, the contributions to the Helmholtz free energy92 from the electrons and ions are calculated separately according to their respective temperatures and then the contributions are summed together. The same conductivity models25'27,29 used in the equilibrium calculations are used in the two-temperature calculations. The conductivity is assumed to be a function of the electron temperature and mass density. The justification of this assumption is that it is the electrons which are primarily responsible for conduction, owing to their comparatively small mass and consequently greater mobility. 3.1.3 Two-Temperature (Te ^ Tt) Hydrodynamic Model In the single-temperature hydrodynamic model the electrons and ions are treated as a single fluid. In the two-temperature hydrodynamic model they are treated as a single fluid for purposes of calculating hydrodynamic motion but the electrons and ions are treated as two separate fluids for purposes of calculating their internal energies. It is assumed that all energy deposited by the laser is absorbed by the electrons and that shock heating33 affects only the ions. In addition to collisional electron-ion energy exchange the electron and ion fluids may do mechanical work on each other. For electrons the conservation Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 46 equation for the internal energy of each cell, equation 2.15, becomes dt dt dt po po kg and for ions the equation becomes Cvi^ + Bj£ + Pe^- + A E \ h y d r o = H- ^ ( T e - Ti) (3.10) dt dt dt Po Po kg The C's and B's are defined in a fashion similar to that for the single-temperature model, equation 2.17, using the electron or ion portions of the equation of state. AE\hydro is the hydrodynamic work term defined in equation 2.14. The hydrodynamic work is assigned to the ion fluid since it incorporates the portion of the energy that will be given up to or come from the bulk motion of the plasma, and ions make up nearly all of the mass of the plasma. This implicitly assigns the energy of shock heating to the ion fluid thereby rendering artificial viscosity 5 3 treatments unnecessary. The term A refers to the laser energy deposition, and the H1 s refer to the electron and ion thermal conduction. In this model we assert that the electron and ion fluids move together. Since the electron and ion pressures, temperatures, and momenta are generally different the two fluids therefore do mechanical work on each other. The PdV term in equations 3.9 and 3.10 represents this work. This mechanical work can be thought of as arising from the electric field between the electron and ion fluids. For example, in a laser-heated plasma expanding from a solid into a vacuum, the electron fluid pressure is higher and maintains the expansion process. Hence, the electrons "pull" the ions and this is modelled through treating it as mechanical work. In this case, PdV is positive and equations 3.9 and 3.10 show that the electrons do work on the ions. This model assumes that charge separation Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 47 and hence electric fields occur over length scales smaller than the Debye screening length and that the Debye length is small compared to the hydrodynamic scale lengths in the plasma. In the dense plasmas considered in this thesis the Debye length is typically measured in Angstroms. In a shock wave, the situation is reversed since the shock wave acts first on the ions, and then the ions "drag" the electrons along behind them. Now, PdV is negative and equations 3.9 and 3.10 show that the ions do work on the electrons. Throughout the entire discussion of two-temperature (T e ^ 7$) the assumption is made that the electrons and the ions may be described as having equilibrium, be that Maxwellian or Fermi-Dirac, velocity distributions. A brief note on this assumption is in order. 3.2 Electron-Electron and Ion-Ion Thermal Equilibration Electron-electron collisions are predominantly responsible for maintaining an equilibrium 1 0 velocity distribution among the electrons. In a laser-heated plasma, when the equili-bration time is short compared to the laser duration, the electrons will have a well de-fined temperature. Determining the electron-electron thermal equilibration time in dense, strongly coupled plasmas is non-trivial. The shortest duration high-intensity laser-solid experiment reported is that of Price et a l . 1 3 5 , an intense 120 fs laser-solid reflectivity measurement. The successfull interpretation of the data involved the use of a hydrocode, using the Q E O S 8 equation-of- state, and Lee and More's conductivity models 2 9. The interpretation of the experiment depends on accurate modelling of the laser-solid interac-tion and is therefore dependent on the conductivity model. Although Price et al.'s result depends on modelling physical processes other than equation of state and conductivity, Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 48 such as the electron-ion thermal equilibration time, it suggests that the heating time of 120 fs is longer than the electron-electron relaxation time. As discussed in chapter 1, the ion subsystem of the plasma is strongly coupled (Tn > 1) and the electron system is weakly coupled (T e e < 1) for the density-temperature regime of concern to the present work. In a weakly coupled plasma a Maxwellian electron velocity distribution will be restored by electron-electron collisions, following a perturbation, on a time scale given approximately by 1 0 3 '. e e e ne In A >] (3 - n ) where the electron temperature Te is in Kelvin, the free electron density ne is in c m - 3 , and A is the Coulomb logarithm. This is the electron self-collision time 1 0 . Equation 3.11 shows that electron-electron equilibration rates are extremely rapid and can be expressed in femtosecond time scales. For example, in an aluminum plasma at 106 K for l /100 i / l solid density r e e « 100 fs and for solid density r e e « 1 fs. In view of the lack of theoretical and experimental information on strongly coupled plasmas, definitive conclusions cannot be made pertaining to electron-electron thermal equilibration rates in strongly coupled plasmas. However, it is reasonable to assume that the electrons may be considered to have equilibrium velocity distributions in all aspects of the model pertaining to equation of state, and collisional ionization, excitation, de-excitation, and recombination rates. Following the same procedure10 leading to equation 3.11, we arrive at an expression for the self-collison time for weakly coupled aluminum ions, 59.2Tf / 2 (Z*Yn% In A Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 49 where Z* is the average ionization state. Given that the average ionization of the alu-minum ions is typically greater than 2, the ion-ion equilibration time Tu is comparable to the electron-electron equilibration time. However, TH is shorter for strongly coupled plasmas than for weakly coupled plasmas. If Tu > 1/3 the ratio of the equilibration times is decreased by a factor equal to the square root of the average ionization of the ions 1 0 0 . The assumption made with regard to the ions in the present work is that they may be described by equilibrium velocity distributions. 3.3 Non-Local-Thermodynamic-Equilibrium (Non-LTE) Ionization Models There are two kinds of non-LTE ionization models discussed in this thesis. First, there is the model used to calculate the average ionization for the situation which exists when radiative atomic transitions compete with collisional transitions but the plasma conditions are changing slowly enough so that the distribution of ionization states can adapt to the changing plasma conditions. This model describes steady-state behaviours and is known as a collisional-radiative equil ibrium 1 0 4 ' 1 0 5 (CRE) model. Second, there is the model used to calculate the average ionization for the situation which exists when the plasma conditions are changing faster than the rate at which the ionization state distributions can adapt. In this case the average ionization state of the plasma is dictated not only by the instantaneous plasma conditions but also by the plasma conditions that existed within a finite interval before the time under consideration. This model takes into account time-dependent behaviour and is referred to here as a collisional-radiative (CR) model 1 0 6 ' 1 0 7 . The primary use of non-LTE ionization models is in the simulation of the interaction Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 50 of extremely short laser pulses with matter where the average ionization states of the plasma change too slowly to be in concert with the plasma conditions. Consequently, the CR model will be necessary to assess the effects of non-steady-state ionization in these cases. The steady-state non-LTE (CRE) model is used to provide a good reference for the comparison of the non-steady-state (CR) ionization calculations. The application of the atomic rate coefficients used in this thesis to the calculation of plasma ionization states may not be valid when the electron temperature is below the Fermi temperature84 = - ^ r {K) (3-12) since none of the atomic rate coefficients used in this work incorporated Fermi electron velocity distributions. Hence, for low temperatures an equilibrium Thomas-Fermi ioniza-tion model8 was used. However, this does not present a limitation to the present work since the plasma is heated very rapidly to temperatures above the Fermi temperature. 3.3.1 Collisional Radiative Equilibrium (CRE) The portion of the computer code used to perform the CRE calculation has already been extensively documented elsewhere105 and will only be reviewed here. Application of a CRE model involves determining the transition rates due to all significant collisional or radiative processes and then using them to formulate a set of rate equations, ohi j = i ~ l m ~ = RJinJ ~ J2 R V n i + RjiKj, (3-13) where is the population fraction of the state i, Rij is the transition rate from state i to state j, and the available atomic states are numbered 1, • • •, m. For the CRE solution Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 51 the population fractions are assumed to be constant, so that drii/dt = 0 and the final rate equation is replaced by an equation representing the conservation of matter to provide closure to the set of rate equations. The system of rate equations follows that due to Duston and Davis 1 0 4 . Their set of 104 rate equations incorporated data from 181 atomic states. The reason for the apparent discrepancy between the number of atomic states and the number of rate equations is that the very close-lying states were averaged together. The multiplicity of each state used in forming an averaged state is used in weighting the average of the transition rates to and from the averaged state. The number of atomic states is augmented1 0 8 to 207 and the calculation is performed with 125 rate equations. However, at high densities continuum lowering 1 0 9 raises the energy levels of the upper states and ultimately removes them. Continuum lowering is a consequence of local electrostatic fields arising from high electron and ion densities in a plasma. Continuum lowering is accounted for by reducing the ionization potential of each state xZ °f ionization Z by an amount Axz as calculated by Stewart and Pyat t 1 0 9 to be where Axz and Te are in eV and the ion density is in c m - 3 . Figure 3.1 summarizes the atomic transitions considered in the calculation of the rate coefficients i?^. We consider only the coupling between each excited level and the ground [3(Z' + 1)K + l p - 1 2(Z' + 1) Z' = <Z2> <Z> (3.14) Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 52 state of the next ionization stage. For the lower electron densities (ne < 10 2 2 c m - 3 ) this is a good approximation 8 2 since then the coupling of the excited states to the ground state of the next ionization stage is negligible. At high densities (ne > 10 2 3 cm"3) this approximation may break down. However, even if the average ionization calculated with this model disagrees with the true average ionization by an entire ionization stage, the fractional difference in the average ionization state of the plasma will not exceed 20—30%, which is acceptable for the purposes for which the C R and the C R E models are used in the present work. Collisional Ionization Rate Coefficient Collisional ionization is the process whereby an ion of charge Z and in state j loses an electron through collision with another electron Ion-ion collisions can also lead to ionization but they are not expected to be important at temperatures below 20 K e V 1 1 0 . Ionization from both the ground state and the excited states of each ionization stage of the ions is included in the model. The results of M . A . Lennon et a l . 1 1 1 are used to calculate the collisional ionization rates. Three-Body Recombination Three-body recombination, Nz{j) + e -r Nz+1{0) + e + e. (3.15) Nz (0) + e + e ^ Nz~l(j) + e, (3.16) Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 53 Figure 3.1: Schematic representation of the atomic transitions considered. The thick hor-izontal lines represent the ground states of the ionic stages, labeled "Z" . The arow labeled "S" represents collisional ionization. "A" is spontaneous emission, OJ3B is three-body re-combination, OLR is radiative recombination, a^i is dielectronic recombination, and " X " is collisional excitation. Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 54 is the inverse process of collisional ionization. The three-body recombination rate is calculated from the collisional ionization rate using the principle of detailed balance. The non-recombining electron is necessary for the conservation of momentum and the resulting recombination rate scales as the square of the electron density. Three-body recombination is included for electron capture to both the ground and excited states of each ionization stage, a™={^M) 29(z,o) ^ e x p - f c r T [ ~ ] ( 3 - 1 7 ) where Z is the ionization stage, S is the collisional ionization rate, j is the level of the ionization stage and j = 0 refers to the ground state of the ionization stage, and g is the multiplicity. Radiative Recombination Radiative recombination occurs when an ion captures a free electron and a photon is emitted. i V z + 1 ( 0 ) + e-> Nz(j) + hu (3.18) The rate due to Seaton 1 1 2 is used, 3 (aR)zz-Qltj = 5.2 x 1(T 1 4 Z<^ (0.43 + 0.5 In 0 + 0.470^ [^-] (3.19) where </> = xZ/^Te and x is the ionization potential. Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 55 Dielectronic Recombination An electron incident on an ion is resonantly captured in a level with a large principal quantum number and the excess energy is spent exciting a bound electron, so that Nz(j) + e-^Nz-1{j',k'). (3.20) If the electron energy is equal to the excitation energy of state j' minus the binding energy of state k', the capture process proceeds without radiation. The two-electron excited state is unstable since it is above the ionization limit of the recombined ion. It can either undergo the inverse process of auto-ionization or the excited electrons can radiatively decay to lower levels. Burgess and Mer t ' s 1 1 3 , 1 1 4 rate coefficient is used and the ion is assumed to be in ground state before the process begins. ( ^ z ) f - 1 = 2A*Z9B(Z)j:f0jA(Z,])e-^ [C^,eV] (3.21) J-e 7 where B^ = WT^S z~20 (3'22) E = W ( l + ™ ~ l y (3.23) and A(Z,j) a; 1 / 2 /^ + 0.105a; + 0.015a;2) A n = 0 0.5x x/ 2/(l + 0.21a; + 0.30a;2) A n ^ 0 (3.24) x = Mn+Z). (3.25) XH Here xoj and f0j are the excitation energy and oscillator strength of the ground state to level j transition in the recombining ion of charge Z. Although the sum is over all the Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 56 excited states j it is usually sufficient115 to limit the calculation to states for which the principal quantum number n is below the collision limit nc. The collision limit is the principal quantum number above which collisional transitions dominate over radiative transitions. Collisional Excitation and De-excitation and Spontaneous Emission Collisional excitation and its reverse process de-excitation are transitions in the state of the bound electrons of an ion, due to electron-ion collisions. Nz{j) + e-+ Nz(k) + e. (3.26) Spontaneous emission, Nz(j) -> Nz(0) + hv (3.27) is the Einstein coefficient and the values come from a variety of sources 1 1 6 - 1 2 0. The rate coefficients for collisional excitation are calculated using the semi-classical method of impact parameters134 and are related to the oscillator strength by 3 X(i,j) = 1.578 x l O ' 5 ^ < % > e - x ^ T ' [—,eV] (3.28) where < > is the thermally averaged Gaunt factor and the oscillator strength is the value of the allowed transition to the level with the same princpal quantum num-ber. A Gaunt factor is proportional to the probability of an incident electron inducing a transition142 and a thermally averaged Gaunt factor is the average of the Gaunt factors over a Maxwellian electron velocity distribution. The value of the thermally averaged Gaunt factor is calculated according to a four-parameter interpolation formula due to Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 57 Mewe 1 2 2 , < % >= A + [B<f> - Ctf + D]e't>El(</>) + C<f> (3.29) where <j> = Xij/(kTe), E\ is the exponential integral, and the parameters for the various transitions are: • for allowed transitions A n = 0, A = 0.60, B = C = 0, D = 0.28 • for allowed transitions A n ^ 0, A = 0.15, B = C = 0, D = 0.28 • for forbidden monopole or quadrupole, A = 0.15, B — C = D = 0 • for spin flip transitions, A — 0, B = 0, C = 0.1, D — 0. For H—,He—,Li—, and A^e—like isoelectronic sequences, these parameters give rates within a factor of two of the available experimental and theoretical data. For other isoelectronic sequences the accuracy is reduced to within a factor of three. The de-excitation rate coefficient is calculated using the principle of detailed balance, X-\j,t) = ^ e ^ X ( t , j ) . (3.30) General C R E Results Having calculated the rate coefficients and set up the system of rate equations, the popula-tion fractions of each excited state are calculated by applying standard matrix techniques to the system of rate equations. Figure 3.2 shows a comparison of the C R E calculation of average ionization to that provided by the Thomas-Fermi model in QEOS 8 . It is useful to note that the C R E model is not necessary for all regimes of density and temperature encountered in high-intensity short-pulse laser-matter experiments. Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 5 8 Temperature (K) * — CRE .27, g/cc V — Q E O S , 2.7 g/cc * — C R E , 2.7g/cc Figure 3.2: Comparison of C R E to Thomas-Fermi average ionization. -O—QEOS, .027 g/cc — CRE, .027 g/cc QEOS, .27 g/cc Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 59 McWhirter 7 8 gives a necessary but not sufficient condition that the free electron den-sity must satisfy in order to justify the application of a local thermodynamic equilibrium model: Ne > 1.8 x l O ^ / V U , k) [cm-% (3.31) where x(j, k) in eV is the largest transition energy available in the system being considered and Te in eV is the electron temperature. Figure 3.3 shows a phase diagram detailing where, according to the average ionization predicted by QEOS, McWhirter's condition is or is not satisfied. In this application of McWhirter's criterion, the transition energy X is taken to be the highest transition energy available for the ionization stage which is immediately above the average ionization. 3.3.2 Collisional-Radiative Model (CR) The rate coefficients and the system of atomic levels considered in the C R calculation is identical to that used in the C R E solution. There are two differences in the system of rate equations between the C R and the C R E models. First, the left hand sides of equation 3.13 are not set to zero since obviously the population fractions of each ionic state are not steady-state (dNi/db ^ 0). Second, an equation representing the conservation of matter does not replace the final rate equation and the solution for the state populations is calculated in the iterative Crank-Nicholsen loop 5 3 ' 6 7 (chapter 2.1.2), along with the cell temperatures, internal energies, laser deposition, and thermal conduction. The approach to finding the solution strategy follows that used by Gauthier 1 0 6 et al. Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments Figure 3.3: McWhirter's Criterion may not be satisfied above the line. Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 61 If F is the vector of the population fractions of each state then dF — = S{Te,ne)-F (3.32) where S is the matrix of rate coefficients. Since ionization, recombination, excitation, and de-excitation absorb and release energy and thereby affect the energy balance the C R ionization calculation is included in the Crank-Nicholsen loop. The energy liberated from or lost to atomic processes is treated as a source or sink term and is included in the calculation of transport by modifying equation 2.15, CV—£~ + BT-^ + AE\hydro — AE\IONIZATION = Q (3.33) where AE\ionization — F~*~ • X^~ F X (3.34) is the source or sink term and x is the vector of energy levels. The superscripts + and — denote whether the superscripted quantity is from the new or the old time step, respec-tively. In order to update F from the old to the new time step an implicit differencing scheme67 is used, F+ = F~ + S(T+) • F+ (3.35) where the time step A t has been multiplied into the rate coefficient matrix S_. Thus, we have F+ = [1 - S(T+)]~1 • F- (3.36) upon which we can perform first-order Taylor expansion to get F+ = F* + (Te+ - T - ) [ l - S ( T - ) ] - 1 • • F* (3.37) Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 62 where F* = [1 - S ( T - ) ] - 1 • F-. (3.38) Here, Te is the electron temperature in Kelvin, and equation 3.37 is derived in the same way as shown by Gauthier et a l . 1 0 6 . The plasmas they considered in their study were at most l/10th solid density, and therefore they are able to deal simply with energy conservation using an ideal gas equation of state. However, ideal gas equation of state is not valid throughout much of the density and temperature range of concern in this thesis, and therefore cannot be used. Instead, we have used QEOS, which incorporates a Cowan model for the description of ions and a modified Thomas-Fermi model for the treatment of electrons8. We can simplify the task of accounting for energy conservation by assuming that the electrons contain the bulk of the internal energy. There is, however, a basic problem in applying Thomas-Fermi theory to the C R nonequilibrium system. In the Thomas-Fermi model 6 8 matter is divided into identical spherical Wigner-Seitz cells, each cell having an atomic nucleus at the center. The nucleus carries the positive charges, considered to occupy no volume while the electrons are modelled as negative charges which are distributed continuously and with spherical symmetry throughout the rest of the Wigner-Seitz cell. The total negative charge in each Wigner-Seitz cell is equal to the nuclear charge. Changes in the density or temperature of the matter being modelled manifest themselves through changes in the distribution of the negative charge within the cel l 6 8 . The Thomas-Fermi model is valid only when each density-temperature condition is accompanied by a unique distribution of negative charges. Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 63 In high intensity laser-solid interaction experiments where the laser pulse is very short, say 20 fs, heating and hydrodynamic expansion of the plasma may occur before the atomic processes adjust the atomic state populations to the new conditions of the heated matter. During this process, equivalent conditions of density and temperature may have different population distributions among the atomic states. Consequently, if we attempt to model this situation using the Thomas-Fermi picture of matter, then there will be more than one distribution of negative charge in the cell for a given temperature and density. Without re-deriving Thomas-Fermi theory so that density, temperature, and atomic state vector are independent variables it is impossible to deal with equation of state properties such as pressure, entropy, internal energy, and heat capacity in a consistent manner. A n approximation is used here. For each new ionic state vector F+ (equation 3.37) calculated by the C R portion of the simulation, we perform a C R E calculation and obtain the vector H+ describing the distribution of population fractions among the ionic states that would exist if the atomic system had time to come to steady-state. The quantity (H+ - F+) • x+ - (H~ - F~) • x- (3-39) is the energy that did not go into ionization or come from recombination but eventually would if the system is left unperturbed long enough for the ionic population distribution to reach steady-state, assuming that the entropy also remains constant. The energy given by equation 3.39 is added to the internal energy prior to the calculations of the temperatures and pressures from the equation of state, but this "shift" in internal energy is not propagated through the rest of the hydrodynamic calculation. This approximation is Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 64 made in order to include the effects of non-steady-state ionization balance on temperature and pressure. A simple-minded picture of the situation is as follows. When the true (CR) ionization is lower than the Thomas-Fermi average ionization, there are fewer electrons to absorb the energy that has been deposited by, for instance, the laser. This yields a higher temper-ature and pressure. When the C R ionization is greater than the Thomas-Fermi average ionization the situation is reversed. Shifting the internal energy by the amount given by equation 3.39 when calculating temperatures and pressures raises or lowers the tempera-ture and pressure according to the difference in internal energy between the steady-state and non-steady-state predictions for the population state distribution among the ions. With this approximation, the C R solution proceeds as follows: 1. The matrices S(T~), d[S(T-)]/dt, [1 - S(T-)]~l, and the vector F* are calculated before the Crank-Nicholsen loop. This is done to reduce computational time. 2. Using F* as an initial approximation to F+, the average ionization, laser deposition, and thermal transport are calculated. 3. Using equation 3.37, F+ is calculated. Then equations 3.33 and 3.34 are used to calculate the energy balance, and equation 3.39 is used to calculate the internal energy difference between the steady-state and non-steady-state predictions. 4. The final step in the Crank-Nicholsen scheme is to test the convergence of the solution using the internal energy, the fractional change in the entries of F+, and the magnitude of F+. Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 65 5. If convergence is achieved the calculation proceeds to the next time step. If not, the next Crank-Nicholsen iteration begins with the calculation of a new average —* ionization using the vector F+ so that the laser deposition, and thermal transport may be re-calculated (step 2). Generally 2-10 iterations were required. As a test case, the convergence of a plasma starting at lOeV and being instantaneously heated to 200eV was studied. Figure 3.4 shows the convergence of the average ionization to the C R E prediction. 3.4 Electromagnetic Wave Solver Here we describe the procedure used to model laser energy deposition in plasmas and the interaction of laser probes with plasmas. The treatment is similar to that found in Born and Wolf 3 7 for stratified media. Figure 3.5 is a schematic representation of the interaction of the laser with the target. The interaction is modelled by solving the Maxwell equations. A l l material information is contained in the plasma dielectric function, e(u>) = 1 + (3.40) cu and o(co) is obtained using the Drude model and the tabulated D.C. conductivity, o(co) = [s-1] (3.41) Z*nie2 ,„ = — (3.42) mea0 where up is the plasma frequency, OQ is the D.C. conductivity, Z* is the average ionization, i = and me is the electron mass. At the resonance layer, where the plasma frequency Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 66 .027 g/cc — — - .27 g/cc 2.7 g/cc Figure 3.4: Demonstration of the C R solution converging to the steady-state C R E pre-diction of average ionization in an aluminum plasma instantaneously heated to 200 eV at time 0. Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 67 Target Figure 3.5: Coordinate system used for the electromagnetic wave solver. The path of the laser is shown schematically as a dashed line. In this case, the path corresponds to the use of a laser pulse long enough and intense enough so that an extended coronal plasma has developed. The different densities of shading in the target represent the density gradient of the target. Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 68 Up is equal to the laser frequency u, the real part of the dielectric function can be seen to approach zero, u>2 ivtiJi e = 1 - 2 p . + . " p 3.43 being kept from disappearing by the electron-ion collision frequency 2 5 ' 2 7 , 2 9 , 3 8 vei. At the resonance layer the electron-ion collision frequency is typically small compared to the laser frequency. In the case of P-polarized laser light the small value of the real part of the dielectric function leads to a divergence in the calculated electric field at the resonance layer, and in this way the phenomenon of resonance absorption 1 3 6 is included in the model. The simulations are based on a finite-element model, and therefore the plasma is represented as discrete cells. The divergence of the electric field strength at the resonant layer will be represented only in the cell containing the resonant layer unless the model is modified to prevent this. This would be the case no matter how thin the cell is. Since the laser energy deposition is proportional to the electric field strength squared there would be a corresponding divergence in the power absorbed in the cell containing the critical density layer (equation 1.5). Since this cell can be made arbitrarily small, the solution of Maxwell's equations for obliquely incident P-polarized light in the region of the critical density layer is not physical. Clearly, one must set a minimum physical scale length to the absorption process. This is achieved by modifying the local dielectric function of equation 3.40 through spatial averaging 1 /--W2 enl = — / e(z + z')dz' (3.44) where \ s is the screening length of the electric field in the plasma, taken to be the larger of Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 69 the degeneracy corrected Debye length 2 9 or the mean interatomic distance. This nonlocal dielectric function is a crude, phenomenological model of a minimum physical scale length in a plasma, but it is sufficient for our purposes. Inverse Bremsstrahlung laser absorption can cause the electron velocity distribution to deviate from Maxwellian, which may lower the absorption of laser light in the plasma. Langdon 1 2 3 has calculated a factor 1 - 0.553/[l + (°-§^r5} (3.45) to modify vei in order to take into account the effects of the non-Maxwellian distribu-tion on laser deposition. Ve is the average thermal electron velocity, VQ = eE0/corne is the maximum quiver velocity of the electrons in the laser field, and as before Z* is the average ionization of the plasma. This effect, however, is not large since in high-intensity femtosecond laser-matter interaction the dominant portion of laser deposition occurs at densities greater than critical density (equation 1.5) where vei is comparable to the laser frequency. Typically the inclusion of the Langdon factor in the simulations of a high-intensity short-pulse laser incident on an aluminum target changes the predicted reflectivity by only a few percent. The Maxwell equations are solved in the Helmholtz formulation. Following Born and Wolf, and using the coordinate system of figure 3.5, for S-polarized light we have dHz dHy ieuj ~^ a 1 ^ = U oy oz c dHx dHz n — - = 0 dz dx dHv dHx „ . 4 . Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 70 ^ E x = 0 c dEx iup ~x Hv = u dz c dEx iujjj, ~x H z = u oy c and after the appropriate substitutions we have a differential equation describing the electric field in the plasma, d^.d^ _ d (log y)dEx dy2 + dz2 + U k ° E x ~ dz dz ( 3 ' 4 7 ) where \i is the magnetic permeability, equal to one in all cases in the present work, k0 = 2TT/X0 is the wave number corresponding to the incident wave outside the target and n = y/eJL is the index of refraction. Through separation of variables we get d2U _ d(\og/i) dU dz2 dz dz + kzQ(n2 - a2)U = 0 (3.48) d2V d[\og(e - f)] dV *.'J"' + k2(n2-a2)V = 0 dz2 dz dz where Ex = U(z)e^kooiy-^) Hz = W(z)ei(koay~UJt) and aU + pW = 0. (3.50) The separation of variables procedure leads to the constant a which can be shown to be a = n 0 sin#0 (3.51) Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 71 where n 0 and 90 are the index of refraction and angle of incidence of the laser light before it strikes the target. In all cases in the present work, n 0 = 1. In order to obtain a similar set of expressions for P-polarized light we exchange e and —p, and obtain d2U _ rf(loge) dU ~ozY + k2(n2 - az)U = 0 dz dz (PV d [ log(n- ^)}dV (3.52) dz2 dz dz + kz0{n2-a2)V = 0 where Hx = U(z)ei^oay-u'^ Ey = -V{z)ei{koay-wt) Ez = -W(z)eiVloay-ut) (3.53) aU + eW = 0. For a stratified medium the solution to the differential equations 3.48 or 3.52 can be expressed as matrices where each slice of the stratified medium is represented by a characteristic matrix. If UQ and V 0 a r e the quantities of equations 3.48 and 3.52 at the vacuum-target interface and we define the characteristic matrix M{z) cos(k0nz cos 9) sin^onz cos 6) where ip sin(konz cos 9) cos(k0nz cos 9) p = J — cos 0 for S-polarized light V M p = y^cos# for P-polarized light (3.54) (3.55) then we can write \ V ° J < u l \v>) (3.56) Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 72 Thus, we can relate the electric and magnetic fields of the laser at the boundary of a uniform plasma slab having thickness z to the electric and magnetic fields at a depth z in the slab. In order to model a nonuniform target we treat it as a stack of thin uniform slabs and we can obtain a description of the entire plasma by multiplying the characteristic matrices M for all the slabs together into one matrix. The thickness z in this case is replaced by the thickness Az for each slab. To calculate the laser fields at a specific point in the plasma, we can multiply together all the characteristic matrices for the slabs between the target-vacuum boundary and the desired point in the plasma and then apply the boundary conditions. The boundary conditions are, for S-polarized light, UQ = A + R V 0 = P L ( A - R ) UZMAX = T (3.57) where pl= /£°-cos# 0 (3.58) ' f cos 0-max and R and T are the reflectivity and transmission of the entire target, respectively, A is the amplitude of the incident wave, and zmax is the position of the opposite target surface. Equations 3.57 and 3.58 allow us to calculate the total reflectivity and transmission of the Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 73 target in terms of the matrix elements 3.54 and the factors p\ and p2. UQ and V0 are calcu-lated using equation 3.57. Equation 3.56 allows us to calculate UZ and VZ for any position inside the target. This in turn allows us to calculate the laser field strength throughout the target (equation 3.49). Subsequently the laser energy deposited is evaluated via the electrical conductivity of the corresponding plasma as, X l a s e r = R e [ a ) 2 l E l . (3.59) As a check on the accuracy of the calculation, the deposited energy Xiaser is summed over the cells used to model the target, and conservation of laser energy, ^meshXlaser + R + T = 1, (3.60) is shown to be satisfied to within 2 % for every step. For the case of P-polarized laser light the factors p\ and p2 become Pi = \— cos#0, (3.61) V eo and P2 = J cos6>0. y ^Zmax In all cases we are treating the plasma as a non-magnetic medium so that p = 1 and we are considering the target to be placed in a vacuum so that eo = 1 and eZmax = 1. 3.5 Ponderomotive Force and Wave Breaking These two phenomena may be important in extremely high-intensity (<pL > 10 1 6 W/cm2) short-pulse laser-solid experiments or in experiments using obliquely incident P-polarized Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 74 laser light. Loosely speaking, ponderomotive force arises due to gradients in the laser electric field. Wave breaking is the tendency for the electric field to limit itself due to large amplitude electron motion in the electric field. These two effects are related to each other since the strength of the ponderomotive force depends on the strength of the electric field. 3.5.1 Ponderomotive Force The ponderomotive force arises due to a gradient in the laser electric field. The force can be shown 1 2 5 to be col 4 m 3 where tup is the plasma frequency defined in equation 1.5, a;^  is the angular laser frequency, E is the laser electric field. In the calculation of the field gradient care must be taken so that the derivative is not calculated over a region having an arbitrarily short length. As described in the preceding section a length equal to the greater of the Debye length or the interatomic spacing is used as a minimum distance over which the derivative of \E\2 may be calculated. To incorporate the ponderomotive force into the procedure used to calculate hydro-dynamic motion, the approach of Colella and Woodward 6 0 is followed in which the cell interface pressures are modified prior to their use as input data to the Riemann problem. PL = PL + AtC^F^ (3.63) PR = P R - MCri9htF^ht PL is the pressure immediately to the left of the cell interface under consideration, PR is the pressure immediately to the right, Cle^ is the sound speed in the cell left of the Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 75 interface, and F^1 is the ponderomotive force acting on the cell left of the interface. A t = tn+1 — tn is the duration of the time step in the hydrodynamic simulation. Solving the Riemann problem in the fashion described in chapter 2.1.1, the average cell interface pressures P and velocities u are calculated and used in a modified version of equations 2.10—2.13 to evaluate the hydrodynamic velocities and energy fluxes A t , _ ^ AtF„ u un + —(Pa-Pb) + p- (3.64) A m p £ n + l = £ n + ^L(PaWa- P f t W ) + ( 3 . 6 5 ) A m 2 p where as before the subscripts a and b refer to the interfaces on each side of the cell in question. 3.5.2 Wave Breaking When the free electron density is equal to the critical density, the component of the electric field of the laser light which is parallel to the density gradient can resonantly excite electron plasma waves and the electric field at the critical density layer can become extremely large 1 2 6. The mechanism which ultimately limits the strength of the electric field for P-polarized light at the critical density layer in the plasma is wave breaking 9 4 of the electron plasma waves. If g(z) is the function describing the displacement g of the electrons due to the electric field along the z axis, then the condition 1 3 6 for wave breaking is \dg/dz\ > 1. The breaking of the electron plasma waves damps the waves, and transfers their energy predominantly to the electrons in the high-energy "tail" of the Maxwellian velocity distribution. This adds a suprathermal component to the electron velocity distribution 1 3 6. Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 76 A prescription to ensure that the electric field does not exceed the wave breaking limit is to set a minimum value to the electron-ion collision frequency vei (equation 3.42), i w = ( - ^ r ) 1 / 2 , (3-66) meL whereL is the density gradient scale length defined as i = l / i ^ (3.67) pdx and p is the mass density. This prevents the dielectric function from becoming extremely small and consequently prevents the electric field strength from diverging. Generally speaking, neither ponderomotive force nor wavebreaking are significant for the simulations in this thesis. This is due to the fact that most of the scenarios considered involve moderate intensity (4>L < 10 1 6 W/cm2) or normally incident lasers. 3.6 Testing of Algorithms Where possible the algorithms used to implement the physical models in the computer code were tested by comparison with analytical models. For example, the thermal con-duction and electron-ion thermal equilibration routines were tested by comparing the predictions of the algorithms in the hydrocode with the predictions of analytical solu-tions of the heat equation or to electron-ion thermal equilibration 1 4 3. Accordingly, the solution of the Riemann problem of the hydrodynamic motion was tested by comparing the predictions of the hydrodynamic model in the hydrocode to the exact solution of the Riemann problem 5 4. In addition to testing the algorithms, it is necessary to test the equation-of-state, conductivity, ionization, and hydrodynamic models of the laser-plasma interaction. This Chapter 3. Applying the Hydrocode to Short-Pulse Laser Experiments 77 is done by simulating experimental measurements and forms the subject of the next chapter. Chapter 4 Reflectivity Measurements One of the more straightforward techniques used to measure the conductivity of strongly coupled plasmas is to irradiate a thick planar target with a high-intensity short-pulse laser and measure the laser energy that is reflected back from the target. In addition to measuring the reflected laser energy Doppler shift measurements have been used in an attempt to deduce the expansion velocity of the plasma 1 4 , 6 5 . The proper modelling of the plasma expansion and the non-local interaction of the laser with the plasma is crucial. This is one of the contributions in the present work to the field 5 2 . To understand its significance a critique of a previous method used to interpret reflectivity measurements is in order. 4.1 Milchberg, Freeman, Davey, and More: A Pivotal Measurement and its Controversy The first reflectivity measurements of high-intensity short-pulse laser-solid interactions and the first attempt to relate the reflectivity to the electrical conductivity of strongly coupled plasmas is due to Milchberg, Freeman, Davey and More 1 4 . Figure 4.1 shows the results for plasma conductivity which they extracted from their data. Their claim is that they measured the conductivity of solid density aluminum as a function of temperature 78 Chapter 4. Reflectivity Measurements 79 throughout the important region of the conductivity minimum. Also shown in figure 4.1 is the conductivity model of Lee and More 2 9 . Apparently Lee and More's conductivity model, and the other conductivity models shown in figure 2.9, is in disagreement with the data. The data presented in Milchberg et al.'s paper are shown as resistivity, which is the inverse of conductivity. During the following years, new theoretical models 2 8 , 1 2 7 appeared and fur ther 3 9 , 4 6 - 5 1 experiments were done, but the discrepancy was not resolved. However, the discrepancy appears to hinge heavily on the interpretation of the experimental data. Milchberg et al. interpreted the reflectivity data by assuming that the plasma had a steady-state density gradient characterized by a constant scale length, p(x) = p s ( l - f ) « (4.1) L — avEXPT where p is the mass density, the suffix s denotes solid density (for aluminum 2.7 g/cm3), T is the duration of the laser pulse, vexp is the plasma expansion velocity, x is the position, and a and q are free parameters. The expansion velocity was calculated from Doppler shift measurements of the reflected laser light. Milchberg et al. further assumed that the electron-ion collision frequency vei in the plasma depended linearly on the density and could be normalized to the electron-ion collision frequency at solid density, vs. = vs-1J- (4.2) Ps The collision frequency was used in a Drude dielectric function, as in equation 3.41, and the dielectric function in the Helmholtz wave equations37 to calculate a reflectivity. The solid density collision frequency vs would be adjusted so that the calculated reflectivity would Chapter 4. Reflectivity Measurements 80 104 105 106 Temperature (K) Figure 4.1: Conductivity of solid density aluminum. The shaded region is the conductivity values calculated from the resistivities measured by Milchberg et al.. The solid line is Lee and More's model. Chapter 4. Reflectivity Measurements 81 match measurements. This yielded a "measured" value for vs. The plasma temperature was calculated from the expansion velocity assuming that the expansion was adiabatic 1 2 8, 2 ZkT Vexp= ~\ 4.3 7 - 1 V TUi where 7 is the Gruneisen gamma 1 2 4 , k is Boltzmann's constant, T is the temperature, and mi is the ion mass. Once the temperature was known, the D.C. conductivity was calculated from the collision frequency using the Drude model 3 8 , rpy, Z(P: T)nje2 < W , T ) = . (4.4) meuei 4.1.1 Criticism of the Original Analysis The interaction of the short-pulse laser is strongly dependent47 on the density profile of the plasma. The original analysis of the data was based on the assumption that the density profile had a constant exponential form. Furthermore, the use of two freely adjustable parameters to determine this form (equation 4.1) leaves the reliability of this interpretation open to question. Assuming a constant density gradient is hardly justifiable. Figure 4.2 shows the evo-lution of the density gradient scale length at critical density derived from simulations of Milchberg et al.'s experiment. The scale length is defined as Clearly, the density gradient undergoes considerable evolution. The reflectivity is strongly dependent on the scale length 4 7. Obviously, the plasma is not uniform spatially and is not constant temporally. The effects of the non-steady nature of the plasma are also Chapter 4. Reflectivity Measurements 82 Figure 4.2: Evolution of density gradient scale lengths for both S-polarized and P-polarized driver pulses. Calculated for laser irradiances of 10 1 4 W/cm2 and a 400 fs pulse duration. The laser is incident on a solid target at 45 degrees. Chapter 4. Reflectivity Measurements 83 apparent in the reflectivity history of the plasma during the interaction of the laser with the plasma. Figure 4.3 shows the evolution of the reflectivity of the plasma throughout the duration of the laser pulse. Initially the reflectivity is high, consistent with polished aluminum 1 3 1 . Then, as the plasma temperature increases the reflectivity drops due to the decreasing conductivity. As the plasma continues to expand and its temperature continues to increase, a marked difference appears between the behavior of the self-reflectivity S-polarized driver laser light and the behavior of the self-reflectivity of P-polarized driver laser light. In the case of the S-polarized light the reflectivity again increases, corresponding to the increase in conductivity as the plasma temperature increases (figure 2.9) above 105 K. In the case of the P-polarized light resonance absorption plays an important role thus lowering reflectivity. Ultimately, the reflectivity for both S-polarized and P-polarized light will decrease as the plasma continues to expand resulting in absorption of the laser radiation in the low-density plasma corona, a phenomenon commonly observed in long-pulse experiments1 3 2. Clearly, in high-intensity short-pulse laser-solid experiments a time-integrated reflectivity measurement does not pertain to a single plasma gradient. Figures 4.4 and 4.5 show snapshots of the calculated density, temperature, and laser energy deposition profiles at the peak laser intensity for conditions typical of Milchberg et al.'s experiment. Maximum laser deposition occurs at 190 A below the target surface (figure 4.4). This is due to resonance absorption of the P-polarized laser light. Figures 4.4 and 4.5 also show that one would expect to see different plasmas being formed if one uses an obliquely incident P-polarized laser, as opposed to an S-polarized laser. Chapter 4. Reflectivity Measurements 84 Figure 4.3: Instantaneous reflectivity of the plasma for plasmas created with P-polarized laser light and plasmas created with S-polarized laser light. The peak laser intensity is 10 1 4 W/cm2. The laser is incident on the target at 45 degrees. Chapter 4. Reflectivity Measurements 85 104 C Q IO3 10" 10' 10 u 1 • * k •1 •1 m !i-• • / \ • * \ • • \ i \ i * 1 1 1 "X-" / / * 1 — . 1 1 1 1 1 1 \ • • ' •1000 -800 -600 -400 -200 0 Position (A) 200 400 10s 107 106 105 104 103 IO2 10 600 tso Cu o CH -a u X3 < g 2 3 a 4) Cu B H Mass density — — - Temperature (K) Absorbed power (pW/g) Figure 4.4: Plasma conditions and laser energy deposition profile at the peak a laser pulse typical of Milchberg et al.'s experiment. The irradiance is 10 1 4 W/cm2 and the laser light is P-polarized. 0 is the position of the original target surface, and the laser is incident on a solid target at 45 degrees. Chapter 4. Reflectivity Measurements 86 104 10 J 10z 10'L 10 u • •j • • / ' ____ — / / 0 I I I I I I & § t 1 * > 108 io 6 r o a. •a 10 ' 10 4 10 10 < 3 g 3 2 rt <D a, £ 1 -1000 -800 -600 -400 -200 0 200 400 600 Position (A) Mass density — — - Temperature (K) Absorbed power (pW/g) Figure 4.5: Plasma conditions and laser energy deposition profile at the peak a laser pulse typical of Milchberg et al.'s experiment. The irradiance is 10 1 4 W/cm2 and the laser light is S-polarized. 0 is the position of the original target surface, and the laser is incident on a solid target at 45 degrees. Chapter 4. Reflectivity Measurements 87 What is also evident from spatial profiles of the absorbed power in figures 4.4 and 4.5 is that the laser interacts with nearly the entire plasma gradient. Reflectivity is a non-local phenomenon. Therefore, due to the density and temperature gradients in the plasmas produced in high-intensity short-pulse laser-solid experiments, even an instantaneous reflectivity measurement does not yield information pertaining to only one density and temperature condition of the plasma. Due to hydrodynamic expansion, there is a velocity gradient in addition to the density and temperature gradients. The outer regions of the expanding plasma move faster than the inner, high-density, regions of the plasma. Therefore, since the reflectivity measure-ments of Milchberg et al. do not represent localized density-temperature conditions, the Doppler shift measurement of the frequency spectrum of the reflected laser light is not a meaningful determination of the expansion velocity. The temperature measurements are hence questionable. Elements of this critique apply to the analysis techniques used by other workers 4 6 - 5 1 as well. The present work demonstrates the need for the calculation of laser deposition in a manner consistent with hydrodynamic expansion in numerical simulations of laser matter interactions in order to properly understand the physical processes governing the outcomes of these experiments. 4.1.2 Resolution of the Controversy Figure 4.6 shows the simulation results obtained in the present work for the experimental conditions described in the paper by Milchberg et a l . 1 4 . Figure 4.9 shows the results Chapter 4. Reflectivity Measurements 88 of three sets of calculations. The first is performed using QEOS 8 and the conductivity and ionization models due to Lee and More29. This offers the advantage that the same calculation of average ionization is used in both the equation of state and in the thermal and electrical conductivity. The second set is obtained using QEOS and the electrical conductivity due to Perrot and Dharma-wardana27 while the thermal conductivity is obtained from the electrical conductivity using the Lorentz number38. The difference between the average ionizations used in QEOS and that included in Perrot and Dharma-wardana's conductivity model is small, as can be seen from figures 2.4-2.6. The advantage of this calculation is that the electrical conductivity is calculated using density functional theory which is a first principle calculation with no freely adjustable parameters, as discussed in 2.2.3. Rinker's models25 of conductivity and ionization and QEOS are used in the third set of simulations. This generally gives a poorer agreement. Rinker's conductivity model generally yields higher values of conductivity, as shown in figures 2.7-2.9. Clearly the use of numerical simulations to model the experiment demonstrates that the reflectivity data from Milchberg et al.'s experiment can be explained with existing conductivity models. The numerical calculation is further validated by simulating the experiment due to Fedosejevs et al. 3 9. In this experiment, 250 fs laser pulses with a wavelength of 248 nm and a peak intensity of 1014 W/cm2 were focussed onto thick planar aluminum targets. The two variables in the experiment were the angle of incidence and the polarization. Figure 4.7 shows measured reflectivity and results of simulations. The equation of state is again QEOS and the three models for conductivity and ionization Chapter 4. Reflectivity Measurements 89 Incident Laser Intensity ( W / c m ) - • — L M S-polarized — P D W S-polarized - • — Rinker S-polarized L M P-polarized P D W P-polarized Rinker P-polarized Figure 4.6: Comparison of measured reflectivities with three sets of simulated reflectivities. The experimental data is shown in grey. The upper grey band corresponds to experimental data obtained with S-polarized light and the lower grey band corresponds to data obtained with P-polarized light. Chapter 4. Reflectivity Measurements 90 1 1 1 1 1 1 1 1 1 1 1 I I I I 1 1 1 1 1 1 1 1 1 1 1 ^. : • 1 S * K • = ^ 1 1 1 1 1 1 . • ' • ' 0 10 20 30 40 50 60 70 80 Incident Angle (degrees) • Expt. S-polarized O Expt. P -polarized — • — - L M S-polarized — 0 — L M P-polarized —*— - P D W S-polarized —ti—PDW P-polarized — • — - R i n k e r S-polarized —V—Rinker P-polarized Figure 4.7: Comparison of measured reflectivities with three sets of simulated reflectivities. Chapter 4. Reflectivity Measurements 91 Figure 4.8: Range of temperatures and densities typically encountered in simulating Milchberg et al.'s experiment. The line represents McWhirter's criterion. Chapter 4. Reflectivity Measurements 92 are again due to Lee and More, Rinker, and Perrot and Dharma-wardana. These models are used in the same manner as they are for simulating Milchberg et al.'s experiment. Again, the use of the existing dense-plasma conductivity models apparently corroborates the experimental results. That the simulation accurately reproduces the experimental results, including the minimum reflectivity for P-polarized light at the 60-degree angle of incidence, lends further credence to the treatment of resonance absorption and the scale length calculated in the simulation. 4.2 Non-Equilibrium Effects 4.2.1 Non—Equilibrium Ionization Figure 4.8 shows the region of the density-temperature plane occupied by plasmas typical of the current work and a line determined by McWhirter's criterion (equation 3.31). The formulation of McWhirter's criterion is identical to that described in section 3.2.1. The average ionization of the portions of the plasma above the line in figure 4.8 are adequately modelled by the L T E models. As can be seen from figure 3.2 the C R E average ionization is comparable to the L T E average ionization in this region of density and temperature. Consequently, C R E models are not necessary to model the high-intensity short-pulse laser-solid reflectivity experiments considered in this thesis. In order to better illustrate this point, let us consider the effect of a 10% change in the free electron density, for the region above the line in figure 4.8. Based on the average ionizations shown in figure 3.2, 10% is a reasonable estimate for the greatest difference between the average ionization predicted by the C R E and the L T E models, for the region Chapter 4. Reflectivity Measurements 93 of plasma above the line in figure 4.8. The equation of state (QEOS) 8 is not significantly sensitive to a 10% change in electron density. Comparisons of figures 2.1 and 2.2 show that a change in density by a factor of 10 with correspondingly a factor of approximately 10 change in the electron density will lead to factor of 10 change in heat capacity. Hence, a 10% change in free electron density will cause the heat capacity to change by no more than 10%, which is not important here. The same factor of 10 change in density will lead to a factor of 10 change in pressure. Hence, a 10% change in electron density will lead to approximately a 10% change in pressure. This is not large enough to significantly affect the outcome of the simulations. 4.2.2 T w o Temperature (T e ^ T{) Effects Figures 4.9 and 4.10 show the effects of non-equilibrium between electrons and ions in simulations of Milchberg et al. 's 1 4 and Fedosejevs et al. 's 3 9 experiments. For clarity, only QEOS and Lee and More's conductivity and ionization models are used and only one g-factor, equal to 5 x 10 1 6 W/Km3, was used to describe the electron-ion energy exchange. If g is increased to 10 1 9 the equilibrium result is recovered. The calculation of electron-ion thermal equilibration follows the prescription set out in chapter 3.1. However, whereas the constant ^-factor approach of equation 3.8 may be justifiable for aluminum at solid density, these simulations involve scenarios where electron-ion thermal equilibration must be considered across a wide range of densities. Hence the two-temperature simulation results for Milchberg et al.'s and Fedosejevs et al.'s experiments may not be reliable, due to inaccurate modelling of the electron-ion energy exchange rates. Chapter 4. Reflectivity Measurements 94 Figure 4.9: Electron-ion thermal nonequilibrium (Te ^ Ti) effects on simulation of Milch-berg et al.'s experiment. QEOS and Lee and More's conductivity and ionization models are used in the simulation and the electron-ion exchange parameter g is 5 x 10 1 6 W/m?K. Chapter 4. Reflectivity Measurements 95 100 30 40 50 Incident Angle (degrees) • Equilibrium P-polarized •Equilibrium S-polarized • T e * T i P-polarized • T e * Ti S-polarized 80 Figure 4.10: Electron-ion thermal nonequilibrium (Te ^ Ti) effects on simulation of Fedosejevs et al.'s experiment. QEOS and Lee and More's conductivity and ioniza-tion models are used in the simulation and the electron-ion exchange parameter g is 5 x 10 1 6 W/m3K. Chapter 4. Reflectivity Measurements 96 The differences between the equilibrium (Te = Ti) and non-equilibrium (T e 7^ Ti) results are most pronounced for the lower intensities of Milchberg et al.'s experiment, figure 4.9. This arises from the difference in the electron and ion temperatures result-ing from assigning the laser deposition to the electrons and the choice of a p-value of 5 x 10 1 6 W/m3K. This g-value dictates that the electrons and ions do not come to equilibrium on the time scale of the laser pulse. Since the electron and ion heat capacities are separated in the two-temperature (Te ^ Ti) model, the electrons may rise to higher temperatures at earlier times, enhancing electron thermal conduction deeper into the tar-get. Since the electrical conductivity is calculated using the electron temperature the conductivity of the target will be different, leading to a difference in reflectivity. Figures 4.11 and 4.12 illustrate this process. Figure 4.11 shows a snapshot from an equilibrium simulation. The position at which the temperature is 1000 K occurs at -600 A in the tar-get. Figure 4.12 shows a non-equilibrium (T e ^ TJ-) simulation and the electron heat front has penetrated twice as far into the target. The difference in reflectivities are strongest for the lower intensities because the front-side plasma expands more slowly and hence more of the laser energy is coupled to the deeper parts of the target, enhancing the effectiveness of this process in changing the reflectivity of the plasma. 4.3 Ponderomotive Density Profile Modification The ponderomotive force can exert pressure on the expanding plasma and restrict the hydrodynamic expansion of the plasma. This may lead to steepening of the density profile of the plasma. The ponderomotive force is important in short-pulse laser-solid Chapter 4. Reflectivity Measurements 97 Figure 4.11: Snapshot of plasma conditions predicted by equilibrium simulation at the peak of a laser pulse. The peak intensity is 10 1 5 W/cm2, the pulse width is 400 fs, the laser is P-polarized, and incident on a solid target at 45 degrees. 0 is the position of original target surface. Chapter 4. Reflectivity Measurements 98 Figure 4.12: Snapshot of plasma conditions predicted by non-equilibrium (Te ^ Ti) sim-ulation at the peak of a laser pulse. The peak intensity is 10 1 5 W/cm2, the pulse width is 400 fs, the laser is P-polarized, and incident on a solid target at 45 degrees. 0 is the original target surface. Chapter 4. Reflectivity Measurements 99 Figure 4.13: Snapshot of plasma conditions predicted by simulation at the peak of a laser pulse. The peak intensity is 10 1 5 W/cm2, the pulse width is 400 fs, the laser is P-polarized, and incident on a solid target at 45 degrees. 0 is the position of the original target surface. This simulation shows the effect of ponderomotive force. Chapter 4. Reflectivity Measurements 100 experiments where the laser intensity is extremely high ( > 10 1 7 W/cm2 ) or when the laser is P-polarized and incident on the target at an oblique angle. Ponderomotive effects in laser produced plasmas have been the subject of numerous experimental 6 5 ' 6 6 and theoretical 6 3 ' 6 4 studies. One of the first short pulse laser-solid experiments where the effects of the ponderomo-tive force were evident was carried out by Liu et a l 6 5 . The Doppler frequency shifts of laser light reflected from a laser produced plasma were measured for two types of obliquely inci-dent laser irradiation: S-polarized and P-polarized. In the S-polarized case, the Doppler shift measurements showed that the reflected light was subject to a monotonically increas-ing frequency shift throughout the duration of the laser pulse, which is consistent with an expanding plasma being heated by the laser. However, when the plasma was produced using the P-polarized laser the frequency shift of the reflected light was depressed during the peak of the laser pulse. Liu et al. attributed this to ponderomotive force at the critical density layer, enhanced by resonance absorption. Other workers66 have used the ponderomotive force to attempt to restrict the expansion of the plasma. 4.3.1 P-polarized Obliquely Incident Laser Light Generally speaking for normally incident or S-polarized laser light and for the range of intensities considered in this thesis, the ponderomotive force is never strong enough to become significant. However, for Milchberg et al. 's 1 4 experiments corresponding to an angle of incidence of 45°, P-polarization and laser intensities greater than 10 1 4 W/cm2 the ponderomotive force may become important. Figure 4.11 shows a snapshot of target Chapter 4. Reflectivity Measurements 101 conditions at the peak of the laser pulse for an intensity of 10 1 5 W/cm2. This calculation ignores the ponderomotive force. Figure 4.13 shows the results of a simulation for the same conditions, except that the ponderomotive force is included. The difference in the plasma density gradient and temperature in the region of the critical density layer is small, but apparent. One of the salient features of this comparison is that the ponderomotive force affects the density profile only near the critical density layer. Where the laser absorption is very high and where the laser light is reflected most strongly the gradient of the electric field will be correspondingly larger, leading to a strong ponderomotive force. Since the laser absorption is extremely strong at the critical density layer due to resonance absorption and laser light is reflected at the critical density layer, it is at the critical density layer that the ponderomotive force will be greatest. The ponderomotive force rapidly becomes insignificant at densities greater than critical density since the electric field is an evanescent wave, and the ponderomotive pressure becomes less than the hydrodynamic pressure in the low density plasma. The major controlling factor of the density gradient length near the critical density layer is the length over which the laser energy is deposited, which in this case is con-trolled by the the non-local approximation to the plasma dielectric function (equation 3.44). The minimum of this scale length is set to the longer of either the Debye length or the interionic spacing. Since resonance absorption is strengthened by strong density gradients, the steepening of the density gradient due to the ponderomotive force leads to a further increase in resonance absorption at the critical density layer, which in turn Chapter 4. Reflectivity Measurements 102 leads to an increase in the ponderomotive force. This leads to further steepening of the density gradient. The minimum scale length over which the laser energy is deposited in the plasma usually limits the steepening of the density gradient at the critical density layer of the plasma. In simulations corresponding to the conditions of Milchberg et al.'s experiment using P-polarized light and a peak laser intensity of 1015W/cm2 the electric fields are eventually limited by wavebreaking (equation 3.66). If the minimum scale length over which the non-local approximation to the plasma dielectric function is changed, the plasma profiles change accordingly, leading to changes in the outcome of the simulation. For example, the predicted reflectivity of the laser pulse can change by several percent if the minimum scale length used in the calculation is quadrupled. The flux limiter used in the calculation of thermal conductivity also affects the gradient scale length at the critical density layer. Small flux limiters, say 0.06, reduce the thermal flux across the critical density layer. This steepens of the density gradient because the heat flux to the higher density plasma beyond the critical density layer is lower, thereby reducing the hydrodynamic pressure. The second salient feature is the high temperature in the region of the critical density layer, due to the extremely strong laser deposition at the critical density layer. Also, the temperature gradient is extremely large across the critical density layer. This result is probably unphysical, for two reasons. First, in the high temperature regions in the figure 4.13 radiation transport would become important", leading to smoother temperature gradients. Second, resonance absorption is typically accompanied by the production of suprathermal electrons, which can transport significant amounts of energy throughout the Chapter 4. Reflectivity Measurements 103 plasma 1 3 4 . The calculated reflectivities for Milchberg et al.'s experiment for P-polarized lasers at intensities above 10 1 4 W/cm2 are sensitive to the screening length, ponderomotive force, and the flux limiter. The agreement between the measured and calculated reflectivities for P-polarized laser light at intensities above 10 1 4 W/cm2 may be fortuitous. 4.3.2 S-Polarized and Normally Incident Laser Light For normally incident or purely S-polarized laser light, resonance absorption does not come into play and thus many of the foregoing problems associated with obliquely inci-dent P-polarized laser drivers are avoided. Some workers attempt to use the pondero-motive force to modify the density profiles while avoiding the problems associated with resonance absorption. This is done by using a S-polarized or normally incident lasers having intensities 6 5 , 6 6 typically in excess of 10 1 7 W/cm2. Although this technique has had success in reducing the hydrodynamic expansion of the above-critical density region expected to dominate x-ray spectroscopic measurements, it is not obvious that it can be successfully applied to optical measurements. This is because optical measurements are more sensitive to the expanding low-density plasma than x-ray measurements are. First, even if the laser irradiation conditions yield an idealized one-dimensional plasma and even if the ponderomotive force were to prevent hydrodynamic motion, there will still be a temperature gradient in the solid-density plasma and with it gradients in average ionization. This would produce a non-uniform electron density which is as detrimental to localized conductivity measurements as a mass density gradient. Chapter 4. Reflectivity Measurements 104 Second, high-intensity short-pulse lasers do not instantaneously rise to peak inten-sity. The 120 fs laser pulse described in a recent measurement135 has a roughly Gaussian temporal pulse shape. As an example, figure 4.14 shows a snapshot calculated for a 120 fs S-polarized laser pulse incident at 45 degrees on a solid aluminum target. The peak laser intensity will be 5 x 10 1 7 W/cm2 and the snapshot is taken at 100 fs before the peak of the laser pulse. Since the simulations used in this thesis are not considered reliable above intensities of 10 1 6 W/cm2 the snapshot is taken at the time when the laser intensity is 10 1 6 W/cm2. The important point is that an expanding plasma is produced before the laser reaches peak intensity, and consequently before the ponderomotive force becomes great enough to slow the expansion of the plasma. Although measurements66 of x-ray emission spectra from such experiments have been done and used to infer the existence of a hot solid density plasma, there is no direct measurements to indicate that no hydro-dynamic expansion has occurred. A spatially resolved x-ray spectroscopy measurement may reveal the full extent of the hydrodynamic expansion in the presense of significant ponderomotive force, but this has not been observed. 4.4 Towards Shorter Time scales: 120 fs and 20 fs Since 400 fs and 250 fs laser pulses are not short enough to alleviate the effects of hydro-dynamic expansion and render the interpretation of reflectivity measurements straight-forward, one may ask whether or not shorter laser pulses might succeed. Other workers have performed experiments with a 120 fs laser 1 3 5. By following the technique52 used to re-analyze the data from Milchberg et al.'s and Fedosejevs et al.'s experiments, they Chapter 4. Reflectivity Measurements 105 Figure 4.14: Snapshot of plasma conditions predicted by simulation 100 fs before the peak of a laser pulse. The laser intensity at the time of this snapshot is 10 1 6 W/cm2. The laser will reach a peak intensity of 5 x 10 1 7 W/cm2, the pulse width is 120 fs, the laser is S-polarized, and incident on a solid target at 45 degrees. 0 is the original target surface. This figure shows the plasma conditions before the ponderomotive force becomes significant. Chapter 4. Reflectivity Measurements 106 successfully simulated their 120 fs experiment. This shows that hydrodynamic modelling is still necessary at 120 fs time scales. On the other hand simulations do suggest that 20 fs laser pulses may be short enough to simplify the need for hydrodynamic modelling of high-intensity short-pulse laser-solid experiments. This is discussed in chapter 6. Chapter 5 Heat Front Propagation Heat front propagation measurements were performed by Vu et a l . 5 0 ' 5 1 to determine the thermal conductivity of glass at solid density. Figure 5.1 shows a schematic diagram of the experiment. Initially, a high-intensity short-pulse laser (616 nm, 100 fs, 5 x 10 1 4 W/cm2) is incident on a carbon coating on top of a glass microscope slide. As well, a low-intensity short-pulse probe laser (616 nm, 100 fs) is directed into the opposite surface of the glass slide. Thermal conduction from the carbon layer launches a heat wave into the glass. As it heats the glass, the average ionization state and hence the free electron density of the glass increases. When the free electron density exceeds the critical electron density the glass becomes reflective to the probe laser. As the heat wave propagates through the target, a moving critical density layer is produced by the associated ionization wave. By measuring the Doppler shift in the reflected probe light, the speed of the heat wave can be determined. In the interpretation of this data, Vu et al. assumed that the energy of the heating laser (the pump laser) is deposited entirely in the carbon layer. The carbon layer then acts as a "hot plate" transferring energy to the glass via thermal conduction. A nonlinear heat wave model 5 1 is used to relate the observed heat front propagation velocity to the thermal conductivity of the medium. In this model thermal conduction is treated in a manner 107 Chapter 5. Heat Front Propagation 108 •*.••'."••.•• •••••••••• • • • • • • • • • • •".^•.•••.^ •••••••••• • • • • • • • • • • •;••*.••;•• a w Figure 5.1: Schematic representation of the experimental arrangement used by Vu et al. The high-intensity pump laser is represented by the solid black arrow. The carbon layer is represented by the charcoal layer, the glass substrate by the clear block, and the heated glass by the wavy lines. The grey arrows represent the incident and reflected probe beams. Chapter 5. Heat Front Propagation 109 similar to equation 2.6. Vu et al. found that the use of Spitzer 1 0 thermal conductivity yielded results in agreement with observation. This is surprising, since strictly speaking, Spitzer's calculation is expected to be valid only for low density, high temperature or weakly coupled plasmas. Wide-ranging reliable models of the equation of state and conductivity of glass are not available. Simulations of the experiment by Vu et al. can not be performed. To examine and understand the basic process of femtosecond laser-driven heat waves, however, one can utilize simulations of intense femtosecond lasers with aluminum targets. 5.1 Equilibrium Simulations Using numerical simulations to examine the formation and propagation of a heat front produced by an intense femtosecond laser incident normally on a thick aluminum target we will illustrate the complex interplay between laser absorption, electron thermal con-duction, and hydrodynamics. Our results will show that the initial propagation of the heat front is dominated by the deposition of the laser radiation in the solid as an evanescent wave. This process is most sensitive to the electrical conductivity of the dense plasma. As the temperature of the heated solid increases, electron thermal conduction becomes important and leads to steepening of the heat front. At later times, hydrodynamics begins to take effect driving a compressional wave which coalesces with the thermal wave and propagates as a decaying shock. The calculations will be limited to peak laser pulse inten-sities of 10 1 5 W/cm2 and maximum laser pulse lengths of 500 fs. The laser wavelength for the simulations is 400 nm, corresponding to the second harmonic of Ti-Sapphire lasers. Chapter 5. Heat Front Propagation 110 These parameters are chosen because a reasonable agreement between numerical simula-tions and the observed reflectivity of intense femtosecond lasers from aluminum targets have been obtained in this regime, as discussed in chapter 4. As observed from results of simulation of the reflectivity experiments thermal flux inhibition does not play a significant role. For 100 fs laser pulses having irradiances of 10 1 5 W/cm2, heat flux does not reach saturation even for a flux limiter as low as 0.03. To illustrate the dominant mechanisms affecting the heat front at various times, figures 5.2, 5.3, and 5.4 show snap-shots of calculated temperature profiles in a planar aluminum target irradiated with a 500 fs laser pulse having a peak intensity of 10 1 5 W/cm2. The conductivity and ionization models of Lee and More 2 9 , and Q E O S 8 are used in this simula-tion. Time zero corresponds to the time of peak laser intensity. The temporal behavior of the heat fronts can be categorized into three regimes identified by different characteristic features. On the rising edge of the laser pulse between -700 to -300 fs (figure 5.2) heating becomes increasingly stronger but the temperature gradients remain similar. Near the peak of the laser pulse (figure 5.3) steepening of the heat front can be seen. During the falling edge of the laser pulse (figure 5.4), the heat fronts steepen slightly more and the shape of the heat front approaches steady-state. The development and shape of the heat front during the rising edge of the laser pulse may be seen, from figures 5.2 and 5.5, to result primarily from the dependence of the electrical conductivity on temperature and the increasing laser intensity as the laser power rises towards peak intensity. For example, one may compare the snap-shots at -300 fs. Within a depth of -100 A from the target surface, the plasma temperature Chapter 5. Heat Front Propagation Figure 5.2: Temperature profiles. 0 is the position of the original target surface. Chapter 5. Heat Front Propagation Figure 5.3: Temperature profiles. 0 is the position of the original target surface Chapter 5. Heat Front Propagation 113 Figure 5.4: Temperature profiles. 0 is the position of the original target surface. Chapter 5. Heat Front Propagation 114 drops to about 5 x 105 K while laser absorption shows a rapid but non-exponential decrease. This can be traced to the temperature dependence of the electrical conductivity (figure 2.9) for Lee and More's model 2 9, in the temperature range between 5 x 105 and 106 K . At these temperatures the conductivity increases with temperature so the hotter plasma has a higher energy deposition rate than the plasma below 5 x 105 K . Deeper in the target, between -100 and -400 A, the plasma temperature varies between 5 x 10 5 -4 x 104 K . The corresponding absorption profile shows a constant exponential gradient since the electrical conductivity remains approximately constant (figure 2.9). Still deeper in the target the plasma temperature drops from 4 x 104 K , decreasing to 10 3 K at -600 A. The rapid and non-exponential decrease in laser absorption is due to the increase in electrical conductivity at low temperatures (figure 2.9). Below 103 K , the constant electrical conductivity assumed in the model yields another constant exponential decrease in laser absorption. Thus, temporal variations in the heat front are the result of local heating by an evanescent wave whose intensity is increasing with time. This leads to an apparent propagation of the heat front. In figure 5.5 the laser deposition profiles for times between -200 to 0 fs are also plotted. These appear to be receding towards the target front surface. This indicates that the deposition process starts to decouple from the heat front as absorption by the expanding plasma in the vacuum becomes dominant. At the same time, the temperature of the target front surface reaches a maximum which enhances thermal conduction into the solid. The increasing thermal conductivity above 104 K , shown in figure 2.12, leads to steepening of the heat fronts of figure 5.3. On the other hand, hydrodynamic effects also become Chapter 5. Heat Front Propagation 115 6 0 o PH -a JO < 1 0 3 E 10* t 1 0 J I 102fc 10' 1 0 u f 10 1 i 1 1 ! . • ^ • • \ E y A ' yr m m / y'i 1 / i I / l J ' ...J • • fi / A y * / / > / / y • / . , . i . . . 1 -0.1 -0.08 -0.06 -0.04 Position (um) •0.02 - -700 fs - -600 fs - -500 fs - -400 fs -300 fs -200 fs -100 fs - 0 fs 0.02 Figure 5.5: Laser energy deposition profiles. 0 is the position of the original target surface. Chapter 5. Heat Front Propagation 116 noticeable. A compressional wave begins to form behind the heat front as shown in figure 5.6. At later times, the compressional wave overtakes the heat front giving rise to a shock wave. However, since the energy deposition process has already ceased this shock wave will decay as it propagates into the solid. To provide a more direct comparison with experiments, figure 5.7 shows the instan-taneous velocity of the 104 K point on the heat front as a function of time for three different plasma conductivity and ionization models. The 104 K point on the heat front is chosen since in most materials sufficient ionization will occur at this temperature to yield a critical density plasma for optical frequencies. The first prominent feature of the velocity history is a broad peak between -700 to -300 fs. As discussed above, the change in temperature gradients during this period results from local heating by a laser field which is increasing in time. The apparent propagation of the heat front is dictated by the penetration of the laser field which is determined primarily by the electrical conductivity in the dense plasma. For a broad range of temperatures between ~ 104 — 105 K , Lee and More's model (shown in figure 2.9) yields the lowest conductivity values. This leads to the least screening of the laser field and hence the highest apparent propagation velocity of the heat front. On the other hand, Rinker's model 2 5 (as shown in figure 2.9) gives the highest conductivity values and thus the lowest propagation velocity. At the end of this broad velocity peak, the laser deposition process begins to decouple from the heat front. This together with the significant temperature rise on the target front surface now renders electron thermal conduction the dominant mechanism driving Chapter 5. Heat Front Propagation 117 -200 fs lOOfs 800 fs 100 fs 200 fs 1000 fs 0 600 fs 1200 fs Figure 5.6: Density profiles. 0 is the position of the original target surface. Chapter 5. Heat Front Propagation 118 Lee and More — — - Perrot and Dharma-wardana Rinker Figure 5.7: Velocity histories of the 104 K heat front, created by a 500 fs pump pulse. Chapter 5. Heat Front Propagation 119 the heat front. This accounts for the increase in velocity between -200 and +200 fs. For temperatures of ~ 104 — 105 K , Lee and More's model yields the lowest thermal conductivity which in turn leads to the slowest increase in the heat front velocity. On the other hand, Perrot and Dharma-wardana's models 2 7 give the highest thermal conductivity values, producing the strongest competing effect against the laser deposition process. At later times, the compressional wave induced by reaction to the momentum of the plasma expansion overtakes the heat wave. The propagation is then dominated by hydrodynamic effects and becomes insensitive to plasma conductivities. Figure 5.8 is similar to figure 5.7 except that pulse length of the driving laser is 100 fs. As seen from figures 5.7 and 5.8, there is a velocity spike at the very beginning of the formation of the heat front. This is attributed to the combined effect of laser deposition and thermal conduction. However, such a short-lived transient is beyond the temporal resolution of current diagnostics. Clearly, heat front propagation measurements offer the possibility of determining both the electrical and thermal conductivity of a solid density plasma. 5.2 Two-Tempera tu re (Te ^ Tj) Effects As discussed in chapter 4, thermal non-equilibrium between the electrons and ions may lead to a deeper penetration of the electron heat front than would be expected in the equilibrium case. This electron heat front results from both laser deposition and thermal conduction from the outer regions of the plasma and develops over the same time scale as the laser pulse. Chapter 5. Heat Front Propagation Figure 5.8: Velocity histories of the 104 K heat front, created by a 100 fs pump pul Chapter 5. Heat Front Propagation 121 In the two-temperature plasma model the electron heat capacity is decoupled from the ion heat capacity. When the thermal equilibration time is longer than the laser pulse duration a two-temperature simulation yields a higher heat front propagation speed. Figure 5.9 shows a comparison of the heat front propagation speeds derived from one-temperature and two-temperature simulations for a 100 fs driving pulse. Figure 5.10 is similar to figure 5.9 except that a 500 fs driving pulse was used. Until the electron-ion coupling parameter g gets large, such as 10 1 9 W/m3K, the initial propagation velocity is substantially higher for the two-temperature case. Since the heat front is being tracked by the position at which the electron temperature reaches 10 4K, the apparent propagation of a heat wave appears at an earlier time in the two-temperature (T e ^ 7*) calculations because of the decoupling of the electron and ion heat capacities. At later times, approx-imately 600 fs in figure 5.9 and 400 fs in figure 5.10 the heat front propagation becomes dominated by the compressional wave. Now, the two-temperature simulation predicts a slower rise in propagation velocity. This can be attributed to the deeper penetration of the corresponding electron heat front. More time is required for the compressional wave to overtake the heat wave. The differences between the heat front propagation velocity histories for the single-and two-temperature simulations are not negligible as is evident from figures 5.6 and 5.7, and from figures 5.8 and 5.9. Although this experiment could be performed in a partially transparent material such as silicon, the results of the two-temperature simulations per-formed for aluminum targets suggest that electron-ion thermal equilibration effects may complicate the interpretation of the data. Chapter 5. Heat Front Propagation 122 8 IO7 -200 0 200 400 600 800 1000 Time (fs) Equilibrium T e * T i g=5xl016 Te* Ti g=5xl017 Figure 5.9: Velocity histories of the 104 K heat front, created by a 100 fs pump pulse, following a non-equilibrium (Te / Tj) simulation. This figure illustrates the sensitivity of the velocity history to two different values of the electron-ion coupling parameter g (W/Km3). Chapter 5. Heat Front Propagation 123 3.5 IO7 3 IO7 1 107 L l i .. • t - < \ • l> - :> • Lj •» I 1 ! \ \ ;| . . . . . . -800 -400 0 400 800 Time(fs) Equilibrium T e * T i g=5x l0 1 6 T e * T i g=5xl( r 7 Figure 5.10: Velocity histories of the 104 K heat front, created by a 500 fs pump pulse, following a non-equilibrium (Te ^ Tj) simulation. This figure illustrates the sensitivity of the velocity history to two different values of the electron-ion coupling parameter g (W/Km3). Chapter 6 U l t r a - T h i n Targets and 20 fs Laser Pulses Chapter 4 uses published experimental data 1 4 ' 3 9 to illustrate the value of numerical simu-lation of hydrodynamic and non-local phenomena relating to laser-matter interaction in the design and interpretation of high-intensity short-pulse laser-solid experiments and to benchmark conductivity models 2 7 , 2 9 . Chapter 5 examines an experimental configuration involving the use of pump and probe lasers to measure the propagation speed of heat waves driven by femtosecond lasers 5 0 , 5 1 to further demonstrate the application of numer-ical simulations to experiments. Both chapters 4 and 5 contain detailed discussions of the key physical processes involved in the interaction of high-intensity short-pulse lasers with matter. In attempting to design the next generation of experiments in which measurements can be related to a well-defined solid density plasma, the use of laser pulses which are short enough to render hydrodynamic expansion unimportant appears to be necessary. However, as is apparent from the discussions in chapters 4 and 5, using a thick target will always produce temperature gradients and hence electron density gradients regardless of the pulse duration. Therefore, the use of an ultra-thin target is critical. Such a target allows the evanescent wave to heat the entire target simultaneously. However, prior to exploring these concepts a discussion of the importance of laser wavelength, pulse 124 Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 125 duration, and laser intensity to the laser-target coupling process is in order. 6.1 Importance of Pulse Duration For the purposes of studying solid-density plasmas, the duration of the laser pulse is probably the most important parameter. Figure 6.1 shows the fractions of the pump laser light absorbed by aluminum target material which is less than solid density, and by the target material which is solid density or greater as a function of the pump laser pulse length. The peak laser intensity is 10 1 5 W/cm2 and the target is thick in all cases. If the width of the pulse is 20 fs, then nearly 90 % of the energy is absorbed by solid-density matter. On the other hand, if the width of the laser pulse is 20 ps, then virtually none of the laser deposition occurs at solid density. 6.2 Importance of Laser Intensity As the laser intensity increases the coronal plasma tends to develop more quickly and the coupling of the laser to a solid-density plasma deteriorates more rapidly. Conversely, at lower intensities the coupling of the laser to a solid-density plasma persists over longer period. Figure 6.2 shows the fraction of laser energy deposited in regimes with above-solid density and that in regimes with below-solid density matter as a function of the peak laser intensity. The pulse width is 100 fs, and the laser is incident normally on a thick aluminum target. Although pulses at the higher intensities couple less energy to the solid density regions of the target, laser intensity is clearly not as important as the pulse length. Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 126 Figure 6.T. Fraction of laser energy absorbed at solid density and above, or below solid density, as a function of the pulse width. The pulse has a peak intensity of 10 1 5 W/cm2 and a wavelength of 400 nm and is normally incident on a thick aluminum target. Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 127 80 -e 40 < 20 0 • • — — — — ^ ^ ^ ^ - * p 1012 1013 1014 1015 10 Laser Intensity (W/cm 2) — — Solid and above — — - Below solid Figure 6.2: Fraction of laser energy absorbed at solid density and above, or below solid density, as a function of the peak laser intensity. The pulse has a width of 100 fs and a wavelength of 400 nm and is normally incident on a thick aluminum target. Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 128 6.3 Importance of Laser Wavelength Generally speaking, shorter wavelengths are more desirable due to the correspondingly greater critical densities (Equation 1.5). This slows the process of decoupling the laser interaction from the solid-density regions of the target as hydrodynamic expansion starts to occur. Figure 6.3 shows a snapshot of the profiles of density, temperature, and laser deposition at the peak of a 100 fs, 200 nm laser pulse incident normally on an aluminum target at an irradiance of 10 1 5 W/cm2. The broad peak in the laser deposition profile shows that considerable laser energy deposition occurs at solid density. Figure 6.4 shows the equivalent information for a simulation using a laser wavelength of 1 pm. In this case, the sharp peak in the laser deposition profile occurs at a position corresponding to much lower-density plasma and hence less laser energy is deposited in solid density matter. In the optical wavelength range the attenuation depth of the light in solid density aluminum is slightly sensitive to the wavelength of the light, ranging between -100 A and -800 A as shown in figures 6.3 and 6.4. However, in all the simulations done in the present work this variation is of little consequence. Generally speaking, shorter optical wavelengths are preferable to longer ones. 6.4 Ultrathin Foils Irradiated by Ultrashort Pulse Lasers Simply using short-pulse lasers cannot yield a uniform, hot plasma at solid densities. However, if one uses a 20 fs laser pulse to irradiate a 100 A foil, the evanescent wave will heat the entire target and mitigate the detrimental effects of gradients. Thus, an idealized slab plasma of uniform, solid density can be produced. A new method of interpreting the Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 129 10" 10 J u a, H 102 c a 10' 10" -1000 q 10° • • / y F~ ' / • « f • !• % * 1 • 1 • 1 • i 1 * i i • • / ' / » / / • • • I"1" 11 1—i • / / / . v. / t • • • • 1 i i i 1 . . . . . . -800 -600 -400 -200 Position (A) • Density (kg/m 3) •Temperature (1000 K) =£ o 1 Q 4 PH o X! X) < J 10' 10 z 200 Absorbed power (pW/g) Figure 6.3: Snapshot of target conditions for irradiation of thick aluminum by a 200 nm laser. Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 130 3 IS 1) C M 10 J t H io 2 c Q 10' 10" 1—«—*—»—1—•—•—•—1 • t • _ • • _ ^ . __ , • • • • [ • . 1 • ! • 1 • 1 a 1 I 1 I / 1 1 If /,/, , • -1000 -800 -600 -400 -200 Position (A) • Density (kg/m 3) •Temperature (1000K) 10" 103 J10 o 4 OM -a < 10 J 10 2 200 Absorbed power (pW/g) Figure 6.4: Snapshot of target conditions for irradiation of thick aluminum by a 1 pm laser. Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 131 optical measurements of an idealized slab plasma is proposed. The two key differences between irradiating thick slabs versus ultrathin foils with ultrashort pulse lasers are that in the latter the laser energy is deposited in a well defined quantity of matter and both transmission and reflectivity measurements may be made. The concurrent use of the reflectivity and transmission diagnostics on such a slab plasma allows data interpretation without having to resort to any hydrodynamic models. This is a significant advantage. 6 . 5 Proposed Experiment A 20 fs, 400 nm pump laser pulse is incident normally onto a 100 A thick aluminum foil. The laser pulse shape is assumed to be similar to that of a 400 nm, frequency doubled, 120 fs laser pulse reported in a recent experiment 1 3 5. The principle reason for using aluminum is the broad range of available experimental and theoretical data on its physical properties. In the simulation the laser pulse is truncated at intensities below 1 0 - 6 of the peak laser intensity, shown by the solid line in figure 6.5. A pulse length of 20 fs is chosen to minimize hydrodynamic motion. As discussed in chapter 4, normal incidence enhances coupling of the laser light to the dense region of the plasma. This foil thickness corresponds to the scale lengths of the evanescent wave and electron thermal conduction in the dense plasma. Once the plasma is produced by the pump pulse, it is then probed with a 260 nm, 20 fs probe laser with a pulse shape identical to that of the pump laser. The time delay between the peaks of the pump and probe pulses is chosen to correspond to the time when 99.999% of the energy of the pump pulse has reached the target. A schematic of the timing sequence is illustrated in figure 6.6. The probe laser is incident normally on Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 132 Time (fs) Figure 6.5: Idealized laser pulses. The solid line is the 20 fs laser pulse scaled from a 120 fs frequency doubled laser pulse 1 3 5. The dashed line is scaled from the fundamental pulse of the same laser 1 3 5, and is used in the present work to study the effects of broader wing structure. Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 133 either the front or the back of the target. 6.6 Idealized Slab Plasma In order to interpret this experiment without using a hydrocode we need to be able to model the plasma as an ideal slab 100 A thick and having solid density. In this set of calculations the equation of state is QEOS 8 , the conductivity models of Lee and More 2 9 are used, and the Drude model 3 8 is again used to obtain the A C conductivity from the tabulated DC conductivity. The rationale for considering the laser-heated ultrathin foil as a slab plasma can be found first by examining the electron density and temperature profiles at the end of the pump laser pulse, shown in figures 6.7-6.10. At the lowest intensity (figure 6.7) the electron density is most slab-like and the temperature is reasonably uniform. With increasing irradiances plasma expansion becomes more noticeable but thermal conduction renders the temperature of the plasma increasingly more uniform. A further indication of the slab behavior of the heated foil is seen in figure 6.11 which shows the reflectivity and transmissions of the probe pulse as a function of the time delay between the pump and probe pulses. That the reflectivity and transmissivity of the probe pulse remains nearly constant between 50 and 100 fs is an indication that the plasma is changing very slowly on these time scales. Figure 6.12 illustrates schematically the method used to quantify the idealized slab plasma (ISP) behavior of the femtosecond-laser heated thin foil. On one hand we perform a complete hydrodynamic simulation of the interaction of the pump laser pulse with the Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 134 Figure 6.6: Timing diagram of the basic proposed experiment. The time delay of 62 fs between the pump and probe pulses corresponds to the time when 99.999 % of the pump laser energy is incident on the target. Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 135 2 10 1.5 i o " L p c <u • o c o 1 10" L 5 10" L -200 -150 •100 -50 0 Position (A) 50 100 Figure 6.7: Snapshot of target conditions at the end of a 10 1 3 W/cm2 pump pulse. Circles represent temperature, triangles represent electron density. Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 136 Figure 6.8: Snapshot of target conditions at the end of a 10 1 4 W/cm2 pump pulse. Circles represent temperature, triangles represent electron density. Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 137 Figure 6.9: Snapshot of target conditions at the end of a 10 1 5 W/cm2 pump pulse. Circles represent temperature, triangles represent electron density. Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 138 Figure 6.10: Snapshot of target conditions at the end of a 10 1 6 W/cm2 pump pulse. Circles represent temperature, triangles represent electron density. Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 139 100 I I I I I I I I I I I I I I I I I I I I I I I 80 s£ 60 40 20 • • ' i i i i i i I i 50 100 Probe Pulse Delay Time (fs) 150 • Reflection • Transmission Figure 6.11: Reflectivity and Transmissivity of the target as a function of the delay time of the pump laser. 62 fs is considered the optimum delay time. Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 140 target to yield the reflectivity R and transmission T of the pump pulse. A comparison of the reflectivities (RF, RR) and transmissions (Tp, TR) of front (F) and rear (R) side probes provides an assessment of the axial symmetry of the plasma, as illustrated in figure 6.13. Alternatively, the calculated R and T can be used to evaluate the energy deposited by the pump laser pulse in the foil. Assuming that the heated foil behaves as an uniform slab at solid density, the plasma temperature can be determined using an equation of state. The reflectivity Rs and the transmissivity T 5 of the probe pulse for such a slab plasma can then be calculated by solving equation 3.48 for the electromagnetic wave. For a more stringent test, we have treated the slab plasma as being static throughout the probe pulse. The agreement with Rp, RR, Tp, and TR derived from the full hydrodynamic calculation yields a measure of the validity of the idealized slab plasma model. As shown in figure 6.13, the laser-heated thin foil behaves as an idealized slab plasma over the entire range of irradiances considered. 6.7 Method for Interpretation of Experimental Data Next, we will describe how a hypothetical experiment using similar pump and probe laser pulses and ultrathin foils may be performed and interpreted. First, the reflectivity and transmission of a probe pulse are measured for time delays extending over several probe pulse-lengths. The relatively constant values will be an indication of the suitability of the idealized slab plasma approximation, as expected from results of full hydrodynamic calculations. Additional support for the slab model can be obtained from symmetry considerations be comparing the reflectivity and transmission of the front and rear side Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 141 Full Hydrodynamic Calculation Hydrocode PUMP • (R, T) QEOS; L M ; Drude Hydrocode FRONT/REAR 1 • ( R , T ) / (R, T ) PROBE QEOS; L M ; Drude F R Idealized Slab Plasma Approximation ^ ISP EMS (R, T) • Absorption t» n ,T @ p • (R, T) QEOS e e 0 L M ; Drude S Figure 6.12: Schematic illustration of the method used to quantify the idealized slab plasma (ISP) behavior of the target. "R" is reflectivity and "T" is transmissivity. " L M " is Lee and More's conductivity model, "Drude" is the Drude model, "EMS" is electro-magnetic wave solution, "QEOS" is the equation of state, "ISP" is idealized slab plasma. Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses Figure 6.13: Comparison of front-side (F), rear-side (R), and idealized slab plasma reflectivities and transmissions. Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 143 probes at the end of the pump pulse, as previously discussed. The method for further data reduction is illustrated in figure 6.14. We begin by using the measured reflectivity RF (or RR) and transmission TF (or TR) of a front (or rear) side probe laser pulse in the electromagnetic wave solver routine to solve for the dielectric constant tsu of the slab plasma. Knowing eSa, we can then calculate the A C conductivity, osw To determine the condition of the idealized slab plasma, we will turn to the measured values of reflectivity R* and transmission T* of the pump laser pulse. These yield the energy deposited in the foil. Using an equation of state we can derive the electron density and temperature of the slab plasma which is assumed to remain solid. This procedure thus leads to a direct measurement of the A C conductivity at a specific electron density and temperature. If the conductivity model of the plasma of interest is known, it can be used to calculate an A C conductivity, a W ) for comparison with o~su-To simulate this procedure, we will use the results of our full hydrodynamic calcula-tions. That is, R* and T* are replaced with R and T while R*F and TF (or R*R and TR) are replaced with Rp and TF (or RR and TR). A comparison of the corresponding ow and o$u is presented in figures 6.15 and 6.16. Figure 6.17 shows the corresponding electron densi-ties. The close agreement is not a test of the validity of the plasma conductivity model 2 9 since the hypothetical experimental data used are obtained from simulations based on the same conductivity model. Rather, such an agreement demonstrates the validity of the ide-alized plasma slab approximation and illustrates the feasibility of obtaining experimental A C conductivity data for very dense plasmas. In an experiment, the accuracy in determining ne and T e of the idealized slab plasma Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 144 will depend on the accuracy of the accuracy of the measured R* and T* values for the pump pulse. Uncertainties in the measured values of R*F, Tp, R*R, and Tp will lead to uncertainties in the derived values of o~su- To test such sensitivities, we evaluate the maximum errors assuming that all reflectivity and transmission measurements can be made with either ±10% or ± 5 % relative accuracies. The corresponding error bars are shown in figures 6.15 and 6.16. As a test of the sensitivity of the results to variations in the pulse shape of the laser, calculations have also been performed for 20 fs pump and probe pulses with a broader wing structure similar to that of an experimental 120 fs pulse at the fundamental frequency, as shown by the dashed line in figure 6.5. The results show that the idealized slab plasma approximation remains valid and the A C conductivity can be measured to better than 35%. If the durations of the frequency doubled pump and probe laser pulses were increased to 30 fs, the accuracy of the A C conductivity measurement would deteriorate to about 60% at the highest pump irradiance of 10 1 6 W/cm2. On the other hand, if an aluminum foil 200 A thick is used with a 20 fs frequency doubled pump laser, the A C conductivity can be determined to 25% accuracy. 6.8 Further Use of the Idealized Plasma Slab In addition to allowing the measurement of A C conductivity in the strongly coupled regime, the idealized slab plasma can also serve as an initial value problem for studying hot, expanded states. Once the slab plasma is formed at the end of the pump laser pulse, it undergoes free expansion from a well defined initial state. The significance of an initial Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 145 value problem approach is that the initial plasma conditions can be determined from the absorbed pump laser energy and an equation of state model, thus avoiding the need to calculate accurately the interaction of the pump pulse with the target. By extending the time delay of the probe pulse into the picosecond regime, the reflectivity and transmission of the probe can be used to test models describing hot, expanded states. As an example, we present results of a simulation for a slab plasma produced by irradiating a 100 A aluminum foil with a 400 nm, 20 fs pump laser pulse incident normally at an irradiance of 10 1 5 W/cm2. As the slab expands, the reflectivity and transmission of a 260 nm, 20 fs normally incident probe laser pulse can be calculated with the hydrodynamic code using as an initial condition either (a) the plasma density and temperature profiles derived through a full hydrodynamic simulation of the interaction of the pump laser with the foil or (b) the idealized slab plasma density and temperature derived from the absorbed pump laser energy and an equation of state. As shown in figure 6.18, such calculations incorporating Lee and More's conductivity 2 9 models and Q E O S 8 are essentially identical, supporting the treatment of the expanding plasma as an initial value problem. Results of full hydrodynamic simulation using the S E S A M E data 6 9 for the equation of state and Rinker's conductivities2 5 are shown in figure 6.19. The difference between these two results is striking and readily measurable in experiments. This offers a new opportunity to test our theoretical understanding of the properties of hot, expanded states. Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 146 6.9 Non-Equilibrium Effects 6.9.1 Ionization Balance Generally speaking, non-equilibrium ionization has little effect in the cases studied here. As shown in figure 3.2, the average ionization derived from the C R E model is fairly close to that obtained from the Thomas-Fermi model in Q E O S 8 , or the Perrot and Dharma-wardana's models 2 7. However, given the short time scales prescribed in this proposal, whether or not the plasma comes to a steady-state ionization balance should be consid-ered. As shown in figure 3.4, the average ionization in solid-density plasmas converges to the steady state solution rapidly. At lower densities, the convergence becomes slower as the collision frequency is reduced. A situation may exist where the C R and the C R E predictions of average ionization are drastically different and the plasma state may be closer to that described by the C R model. In the present work, detailed atomic configurations are not considered in the equation-of-state or conductivity models. They are only considered in the calculation of the average ionization and the internal energy of the plasma. If the average ionization predicted by the C R model is drastically different from the C R E values the equation of state models and transport models used may be inappropriate. Therefore, it is useful to compare the average ionizations predicted by the C R and the C R E models. Figure 6.20 shows the evolution of the C R and C R E average ionization values of four of the 200 cells which represent the target in the simulation. The four cells are chosen according to where they are located in the initial cold foil. If the cold foil has thickness L, then the cells come Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 147 from the positions L/2, L/4, L/8, and 0. Thus, if we follow our convention of making the right-hand surface of the foil the front surface and assigning it the position 0, the four chosen cells are initially at -50 A, -25 A, -12.5 A, and 0 A. As foil expands, the cells move with the plasma and hence the curves in figure 6.20 are the histories of four specific pieces of the target. The pump laser intensity of the simulation of figure 6.20 is 10 1 6 W/cm2. Figure 6.21 is similar to figure 6.20 except that the pump laser intensity is 10 1 5 W/cm2. It may be seen from both figures 6.20 and 6.21 that throughout most of the target the average ionization predicted by the C R model converges to that predicted by the C R E model before the probe pulse reaches its peak intensity at 62 fs. It is also apparent that attempts to study the plasma properties exclusively through the reflectivity and transmission of the pump pulse will necessarily involve accounting for the effects of non-steady-state ionizations on the equation of state and conductivities — a monumental task, especially in a high-density strongly coupled plasma. This suggests that the pump-probe idealized slab plasma scheme has another strength: non-steady-state average ionization during the pump(heating) pulse doesn't matter because the average ionization converges to steady-state by the time the probe pulse measures the heated, idealized slab plasma. The only measurement performed during the heating is the reflectivity and transmission of the pump pulse, to determine how much energy was deposited in the foil. Figures 6.20 and 6.21 show that the outermost portions of the target do not reach the C R E average ionization. The reason that the outermost portions of the target lag behind in ionization is that the collisional ionization rate decreases as density decreases, and it is the outermost portions of the target which experience the fastest hydrodynamic Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 148 expansion. This means that the electron density will be lower in the expanded portions of the target than the steady-state or the equilibrium models would suggest. This is illustrated in figures 6.22 and 6.23. Figure 6.22 shows the electron density profile of a 100 A aluminum foil irradiated by a 10 1 6 W/cm2 laser pulse, at the time of the peak of the probe pulse. This time corresponds to 62 fs on figure 6.20. That at this irradiance the idealized slab plasma approximation starts to break down is apparent from the peaked electron density profiles of both the C R and the C R E cases. Figure 6.23 shows the results of a similar calculation except that the pump laser intensity is 10 1 5 W/cm2. Where the electron density is low, the interaction of the plasma with the probe laser will be reduced. Although, without a complete re-working of the conductivity models it is impossible to estimate the magnitude of the effect that this will have on the reflectivity and transmission of the probe pulse, the trend will be to reduce the importance of the expanded portions of the target in the measurements of the probe pulse. This is an advantage, not a weakness since the goal here is to measure solid-density plasma. Fortunately, aluminum has a relatively narrow range in which the ionization can vary, from 3 times ionized to about 10 times ionized for the conditions considered here, and this sets an upper limit on the departures which the equation-of-state properties and conductivities may take from their equilibrium values. If one is going to use other target materials, such as gold or uranium, the average ionization can change by as much as 50 and in these cases the effects of non-steady-state ionizations may be much stronger. Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 149 6.9.2 Thermal Non-Equilibrium Te ^ Tt On a 20 fs time scale, simulations predict that the ions essentially remain unheated. Thus, the highly ionized and cold aluminum ions may form an extremely strongly coupled plasma (equation 1.3), possibly even a crystalline state9. Material models covering the range of conditions required to properly treat such strongly coupled systems are not available and this limits the degree to which thermal non-equilibrium (Te ^ Ti) can be considered here. At the laser intensities used and the resulting plasma temperatures most of the internal energy is carried by the electrons. Two-temperature simulations yield results similar to equilibrium simulations. 6.10 Possibilities for Execution Recent development in high-intensity short-pulse lasers has yielded a laser 1 4 4 at the Uni-versity of Michigan which may be capable of performing this experiment. Workers 1 4 5 at Lawrence Livermore National Laboratory have succeeded in making 100 A thick alu-minum foils. Thus, the two greatest obstacles to performing this experiment appear to have been overcome. Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 150 PROBE PUMP Figure 6.14: Schematic illustration of method used to analyze experimental data. This method does not require the use of a hydrodynamic model. Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 151 i i i i I i i i i i i i 11 i i i i i 11 i 104 10s lO 6 Electron Temperature (K) Figure 6.15: Comparison of the real part of the A C conductivity derived from the idealized slab plasma approximation (solid circles) to the corresponding values derived from Lee and More's conductivity model (open squares). The small error bars correspond to ± 5 % relative uncertainties in the reflection and transmission measurements, and the larger error bars correspond to ±10%. Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 152 I - f ^ ^ j ^ p B j p i j f j m m m W M H ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ _ l I I L_ I I I 10 4 IO5 IO6 Electron Temperature (K) Figure 6.16: Comparison of the imaginary part of the A C conductivity derived from the idealized slab plasma approximation (solid circles) to the corresponding values derived from Lee and More's conductivity model (open squares). The small error bars correspond to ± 5 % relative uncertainties in the reflection and transmission measurements, and the larger error bars correspond to ±10%. Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 153 Figure 6.17: Electron densities in the idealized plasma slab. Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 154 0 _ l 1 1 1 1 1 0.5 i 1 i _ E 1 Time (ps) 1.5 — O — T (hydro) - - • - - T (slab) — • — R (hydro) - - • - - R (slab) Figure 6.18: Comparison of the reflectivity and transmission of a long-time delayed probe pulse in an expanding plasma for results obtained from full hydrodynamic simulations (solid and open squares) to the results obtained using the idealized slab plasma as an initial value problem (solid and open circles). Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 155 Figure 6.19: Comparison of the reflectivity and transmission of a long-time delayed probe pulse in an expanding plasma for results obtained from full hydrodynamic simulations based on the material models of Lee and More (solid and open circles) to the results obtained using the Sesame data base (solid and open squares). Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 156 Figure 6.20: Comparison of the histories of the average ionization predicted by the C R and by the C R E models for four different pieces of the target. Time 0 is the peak of the pump pulse, and the pump laser intensity is 10 1 6 W/cm2. Chapter 6. Ultra-Thin Targets and 20 fs Laser Pulses 157 Figure 6.21: Comparison of the evolution of the average ionization predicted by the C R and by the C R E models for four different pieces of the target. Time 0 is the peak of the pump pulse, and the pump laser intensity is 10 1 5 W/cm2. Figure 6.22: Comparison of the histories of the electron densities predicted by the CR and by the CRE models at the peak of the probe laser pulse. The pump laser intensity is 1016 W/cm2. Figure 6.23: Comparison of the histories of the electron densities predicted by the C R and by the C R E models at the peak of the probe laser pulse. The pump laser intensity is 10 1 5 W/cm2. Chapter 7 Conclusions 7.1 Contributions of Present Work 7.1.1 Reflectivity Measurements Numerical simulations have been used to contribute to the understanding of high-intensity short-pulse laser-solid experiments. The first contribution 5 2 the present work made to the field was to resolve the controversy surrounding the measurement due to Milchberg et a l 1 4 . The relations among the physical processes which govern the reflectivity of intense femtosecond laser pulses incident on massive aluminum targets were explored. The new interpretations of the experimental results provided additional support for the dense-plasma conductivity models of Lee and More 2 9 , and Perrot and Dharma-wardana 2 7. Furthermore, it was shown that it is unlikely for any reflectivity measurement of this kind to sample a single, well-defined, plasma condition. This is a serious drawback to the study of strongly coupled plasmas by femtosecond lasers. 7.1.2 Heat Front Propagation Measurements The next contribution 1 3 7 the present work made to the field was to explain the nature of femtosecond laser-driven heat front propagation in aluminum using numerical sim-ulations. It was shown that it may be possible to achieve extremely rapid heat front 160 Chapter 7. Conclusions 161 propagation speeds governed by the electrical conductivity of the plasma and by the tem-poral dependence of the laser intensity. Although direct comparison to the measurement of Vu et a l . 5 0 , 5 1 is not possible since they used glass targets, it is interesting to note that they measured propagation speeds similar to our simulations. If this pump-probe mea-surement of heat front propagation could be done using a 100 fs probe laser to provide the necessary temporal resolution during the initial heat front velocity, then the effects of electron-ion thermal non-equilibrium could be also studied. However, the interpretation of the results still requires the use of a hydrodynamic model and the plasma conditions sampled still contain temperature and electron density gradients. 7.1.3 A New Experiment for the Coming Generation of Lasers Two ideas are resoundingly demonstrated in the studies of self-reflectivity and heat front propagation. First, calculation of the laser-plasma interaction must be performed in a manner consistent with hydrodynamic expansion. This is necessary for the interpretation of optical measurements from the high-intensity short-pulse laser-solid experiments done so far. Second, these measurements do not sample uniform, well defined states of matter. The final contribution the present work makes to the field is to propose an experi-ment to overcome the difficulties of the previous experiments, a new experiment which measures well-defined, hot, solid density aluminum plasma and whose interpretation can be performed without the use of a hydrocode. In addition to proposing the experiment, theoretical predictions of the outcome are made. This experiment would substantially advance our understanding of strongly coupled plasmas. Chapter 7. Conclusions 162 7.2 Other Directions for Further Research The treatment of electron-ion thermal equilibration needs to be comprehensively ad-dressed. Data on electron-ion equilibration rates is needed over a wide range of densities and temperatures. Wide-ranging two-temperature (Te ^ Ti) conductivity models are required. There is the question of what minimum physical scale lengths characterize the phe-nomena in strongly coupled plasmas. 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