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Shock waves generated by intense femtosecond lasers Ao, Tommy 2000

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SHOCK WAVES G E N E R A T E D B Y INTENSE FEMTOSECOND By Tommy A o B . Sc. (Eng.) University of Alberta,  1998  A THESIS S U B M I T T E D IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF M A S T E R OF SCIENCE  in T H E FACULTY OF G R A D U A T E STUDIES D E P A R T M E N T OF PHYSICS A N D A S T R O N O M Y  We accept this thesis as conforming to the required standard  T H E UNIVERSITY OF BRITISH C O L U M B I A  September 2000 © T o m m y A o , 2000  LASERS  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 of Physics and Astronomy The University of British Columbia 6224 Agricultural Road Vancouver, B.C., Canada V 6 T 1Z1  Date:  Sepf-  7  ZooQ  Abstract  The advent of intense femtosecond lasers has created the exciting possibility of accessing regimes of extreme high pressure using a relatively small laser system.  This  stems from the lack of significant hydrodynamic expansion during the process of laser deposition i n a solid v i a skin-depth absorption, which leads to extremely high energy densities in the irradiated sample. After the short-pulse laser energy has been absorbed, the laser-heated material begins to be released which drives a shock wave into the sample. However, unlike previous long-pulse laser driven shock waves, the shock wave driven by a intense short-pulse laser rapidly decays as it propagates through the sample. Before adopting such a shock wave as a new approach in the study of high density plasmas, its unique characteristics must be understood. A one-dimensional hydrodynamic code which is coupled to an electromagnetic wave solver is used to elucidate the basic properties of shock waves generated by intense femtosecond lasers. Using a unique experimental scheme, the electrical conductivity of silicon in the dense, plasma state can also be studied. Calculations were performed i n which a shock wave was driven. into a silcion sample by a pump laser with a wavelength of 400 nm, pulse length of 120 fs ( F W H M ) and irradiances ranging from 1 0  14  —  10 W/cm , 15  while rear-side optical measurements were made by a 800 nm, 120 fs probe laser.  ii  2  Table of Contents  Abstract  ii  Table of Contents  iii  List of Tables  v  List of Figures  vi  Acknowledgements 1  Introduction  1  1.1  Strongly Coupled Plasmas  1  1.2  Laser-Driven Shock Waves  2  1.2.1  Long-Pulse Laser Experiments  3  1.2.2  Short-Pulse Laser Experiments  6  1.3 2  viii  Present W o r k and Thesis Outline  7  M o d e l l i n g of Femtosecond-Laser D r i v e n Shock Waves  9  2.1  Shock Wave Theory  9  2.2  M a t e r i a l Models  10  2.2.1  Equation of State and Average Ionization  13  2.2.2  T h e r m a l and Electrical Conductivities  2.3  .  15  Fluid Model  16  2.3.1  16  Hydrodynamics  iii  2.3.2  3  4  Energy Transport  18  2.4  Electromagnetic Wave Solver  22  2.5  Other Effects  28  Discussion of Femtosecond-Laser D r i v e n Shock Results  30  3.1  Proposed Scheme  31  3.2  Results  36  3.2.1  Basic Considerations  36  3.2.2  Characteristics of the Decaying Shock Wave  37  3.2.3  Optical Probing  57  Conclusions 4.1  Summary  4.2  Future W o r k  79 :  79 80  Bibliography  82  iv  List of Tables  3.1  Dielectric values for solid and liquid silicon for 800 nm light  v  58  List of Figures  1.1  Schematic diagram of a solid target irradiated by a laser at high intensities.  4  2.1  Schematic diagram of an idealized shock transition  11  2.2  Schematic diagram of Hugoniot curve on the pressure-density plane. . . .  12  2.3  Coordinate system used by the electromagnetic wave solver  23  2.4  A schematic diagram of an inhomogeneous medium modelled as a stack of stratified media  27  3.1  Schematic diagram of the proposed scheme  33  3.2  Temporal profile of 120 fs laser pulse used in the simulations  35  3.3  T h e Si off-Hugoniot states on the pressure-compression plane for a femtosecondlaser driven shock wave  3.4  39  T h e Si off-Hugoniot states on the electron temperature-compression plane for a femtosecond-laser driven shock wave  40  3.5  E a r l y time pressure spatial profiles  42  3.6  E a r l y time electron temperature spatial profiles  43  3.7  E a r l y time ion temperature spatial profiles  44  3.8  E a r l y time density spatial profiles  45  3.9  Late time pressure spatial profiles  48  3.10 Late time electron temperature spatial profiles  49  3.11 Late time ion temperature spatial profiles  50  3.12 Late time density spatial profiles  51  3.13 Decay of the pressure at the shock front  53  vi  3.14 Decay of the electron temperature at the shock front  54  3.15 Decay of the ion temperature at the shock front  55  3.16 Decay of the density at the shock front  56  3.17 Penetration of the electric field strength of the probe laser beyond the shock front  59  3.18 Dielectric functions at the shock front  60  3.19 Fresnel reflectivity at the shock front and at the free surface  64  3.20 Reflectivity at $ = 5 x 10 W/cm 14  : Fresnel and E M wave solver  2  . . . .  65  3.21 Penetration depth of probe light beyond shock front  67  3.22 Dielectric functions at shock front (at 10 ps)  68  3.23 Dielectric functions at shock front (at 20 ps)  69  3.24 Dielectric functions at shock front (at 40 ps)  70  3.25 Phase shift  72  3.26 Shock speed of decaying shock wave  73  3.27 Similar reflectiviies at different irradiances  75  3.28 Similar CT given by the Lee and More model at different irradiances. . . .  76  0  3.29 Sensitivity of electrical conductivity at the shock front on the Fresnel reflectivity  at the free surface  78  vii  Acknowledgments  I would like to thank my research supervisor, Dr. Andrew Ng, for his invaluable guidance and support throughout the course of this work. I would also like to thank the former members of the group, Dr. Andrew Forsman and Dr. Gordon Chiu, for their introductory assistance with the computer code. The editorial comments and suggestions made by Dr. Stuart Watson were greatly appreciated. Finally, I thank my family and friends for all their love and support which made this work possible. The financial assistance of the Natural Sciences and Engineering Research Council is appreciated.  viii  Chapter 1 Introduction  1.1  Strongly Coupled Plasmas  W i t h i n the field of plasma physics the area of strongly coupled plasmas ( S C P ) remains the least understood.  The study of these plasmas has been motivated by research i n  geophysics, astrophysics, x-ray sources, and nuclear explosions, to name a few. Yet, the main driving force behind the study of S C P continues to be the goal of achieving inertial confinement fusion. In this regime of high density and modest temperature, the plasma cannot be described soley by the statistical behaviour of a screened Coulomb system as i n the case of a rarefied plasma; effects due to strong particle correlations must be addressed. Simply stated, when the potential energy due to the Coulomb interaction between the ions is greater than kinetic energy of the ions, the plasma is considered strongly coupled. In order to create a strongly coupled plasma, great pressures must be achieved. T h e methods employed are usually divided under the headings of static pressure-loading and dynamic pressure-loading. T h e most advanced static techniques involve the use of d i amond anvil cells to slowly compress a solid. However, the highest pressure achievable in this manner (< 2 M b a r ; 1 T P a = 10 M b a r ) is ultimately limited by the finite yieldstrength of the diamond anvils [1]. Thus, in order to reach higher pressures dynamic pressure-loading methods must be used. T h e basic principle in dynamic techniques i n volves the generation of a rapid compression (shock) wave in the material to be studied. In the early days of shock physics, explosive systems'such as shock tubes were employed  1  Chapter 1. Introduction  2  to achieve pressures of a few hundred kbars [2, 3, 4]. T h i s method was later refined into the use of explosive-driven flyer-plates allowing precise measurements (< ± 2 % pressure uncertainty ) to be made at pressures near 10 M b a r [5, 6]. T h e current state-of-the-art precision measurements are obtained using two-stage light-gas guns [7]. Even higher pressures can be achieved v i a nuclear explosions (hundreds to thousands of Mbars) [8]. However this method is quite expensive and unavailable to most researchers, not to mention the obvious political and environmental ramifications associated w i t h it.  1.2  Laser-Driven Shock Waves  W i t h the advent of high-power pulsed lasers in the early 1970's, a new avenue for achieving ultra-high pressures became available (10-100 M b a r ) [9, 10]. M u c h of this early research involved heating and compressing small fuel pellets i n inertial confinement fusion schemes [11, 12, 13]. In fact, current advances i n high-power pulsed laser technology remain mainly due to the work on achieving inertial confinement fusion. However, other uses for these lasers soon became evident, and over the past two decades an array of research has been carried out [14, 15, 16]. In addition to the advantage of reaching pressures unattainable w i t h the other methods, the laser-driven shock technique has the added advantage of being able to easily control the pressure by simply varying the laser irradiance on the target. T h i s technique is intrinsically different from those previously described i n several key aspects.  Unlike the other methods where the shock is produced by an impact or  explosion, it is formed through the laser ablation (vaporization) process. T h i s process is discussed below, but essentially the shock arises out of conservation of momentum. Experiments of this nature are inherently spatially microscopic and short-lived (less than a few nanoseconds), thus requiring diagnostics w i t h high spatial and temporal resolutions. Experiments involving high-power pulsed lasers to create a strongly coupled plasma  Chapter 1. Introduction  3  can be separated according to the pulse-width of the laser used. P r i o r to the m i d 1980's, all high-power pulsed laser research employed lasers w i t h pulse-widths ranging from tens or hundreds of picoseconds up to nanoseconds, and are considered "long"-pulse lasers. In these long-pulse laser experiments, as the target is irradiated by the laser pulse, the ablated surface expands outwards.  In the duration of the laser pulse a considerable  amount of material is released into the vacuum. T h e subsequent interaction between the released material and the laser pulse greatly reduces the efficency of laser energy coupling to the target. However a new class of lasers, whose pulse-widths last hundreds of femtoseconds or less, soon became available which avoided this hydrodynamic expansion problem. In these "ultra-short-" (or simply "short-") pulse laser experiments, the laser energy is deposited before any significant expansion of the target occurs, and results i n a more efficient deposition of laser energy to the target. Therefore, using a relatively small short-pulse laser system, it became possible to access regimes of extreme high pressure that are traditionally only attainable using massive long-pulse laser systems. Since the features of short-pulse laser experiments differ considerably from that of long-pulse laser experiments, new approaches for their use i n studying strongly coupled plasmas are required.  1.2.1  Long-Pulse Laser Experiments  T h e process of laser-driven ablation using long-pulse lasers is shown i n Figure 1.1. T h e laser light is focussed onto the target surface, where initially v i a skin depth deposition the laser energy is absorbed predominately by free elections. T h r o u g h electron-ion collisions, the ions are rapidly thermalized, causing the surface layer of the target to be vaporized. A s the vaporized material expands away from the surface, a rarified or coronal plasma is formed. A s a result a large density gradient, ranging from vacuum to solid density, is formed. Associated w i t h a l l plasmas is a physical quantity called the plasma frequency,  Chapter  1.  1  Introduction  Ablation Front  Shock Front  Figure 1.1: Schematic diagram of a solid target irradiated by a laser at high intensities. T h e laser is incident from the right. The light gray area is the coronal plasma, the black line is the critical density n layer, the dotted area is the ablation zone, and the dark gray area is the shock compressed target material. cr  Chapter 1. Introduction  5  which is defined as  ,2\ 1/2 (1.1)  where n is the electron density, m e  e  is the electron mass and e is the electronic charge.  W i t h i n this density gradient lies a particular region with a critical density n  cr  i n which  the plasma frequency equals the laser frequency. T h i s region is called the critical density layer. For long-pulse laser experiments, long after the coronal plasma is created, laser energy continues to be absorbed by the plasma, but only up to the critical density layer. A n y light which reaches the critical density layer w i l l generally be reflected back and is absorbed by the coronal plasma. A n y light which does penetrate beyond the critical density layer w i l l be an evanescent wave. Therefore, i n these experiments the laser-matter interactions are dominated by the rarefied plasma, so any optical measurements made on the target would not provide any clear information about the dense plasma of interest. Beyond the critical density layer lies a transition region between the hot coronal plasma and the cold, solid target material known as the conduction or ablation zone. In the ablation zone a large temperature gradient exists, where thermal conduction transports energy from the hot, coronal plasma to the cold, solid region. A s the solid target material is rapidly ablated, it expands outward, and as a result of conservation of momentum, a large ablation pressure is produced near the surface causing a compression wave to propagate into the solid region. A s the laser continues to irradiate the target and ablate material, the ablation pressure increases and causes compression waves of increasing magnitude to be launched into the target. T h e speed of propagation for these wave is given by the local isentropic sound speed c : s  (1.2) where P is the pressure, and p is the material density. For adiabatic compression the  Chapter 1. Introduction  6  sound speed can be written as  1/2 (1.3)  where 7 is the ratio of specific heat. It can be shown that for most materials, the sound speed increases with density (for an ideal gas, c  s  oc p V  -  with 7 > 1), therefore the  subsequent, stronger waves will catch up w i t h preceding, weaker waves and leads to a steepening of the wave front.  Eventually, a sharp discontinuity of the thermodynamic  variables between the uncompressed and compressed material is formed.  T h i s is the  general picture of a shock wave.  1.2.2  Short-Pulse Laser Experiments  In short-pulse laser experiments, the process of driving a shock into a material is similar to that of the long-pulse experiments except for a few key aspects. Because time scales are too short for the creation of a layer of expanded plasma to dominate the lasermatter interaction, more energy is directly transferred to the solid density material. D u r i n g the extremely brief laser pulse, the energy is absorbed v i a skin depth depostion almost entirely by the electrons, and due to the finite time required for energy exchange between electrons and ions, the latter initially remain cold. W i t h i n a thin solid layer of material at the target surface, the electrons can reach temperatures of over a hundred eV. Consequently, while the electrons begin to thermalize with the ions, a large thermal flux drives a supersonic heat wave into the bulk of the solid target [17, 18]. A s the ions are heated, the surface is ablated and material expands into the vacuum. Once again, because of conservation of momentum a compression wave ahead of the ablation front is generated and a shock front is formed. A s the thermal wave moves away from the surface it slows and the shock wave soon overtakes it.  Chapter 1. Introduction  7  In long-pulse laser experiments a relatively steady shock wave is formed, but i n shortpulse laser experiments the shock wave is quite transient. A t the end of the short laser pulse no more energy is absorbed at the surface. Hence the pressure, density and temperature of the shock wave decrease rapidly resulting in a decaying shock wave. Nevertheless, as long as the shock states does not change considerably while measurements are being made, the non-steady nature of the short-pulse laser driven shock wave would not preclude their use i n the study of strongly coupled plasmas. However i n order to properly make use of the short-pulse laser method, detailed understanding of such decaying shocks and new experimental approaches are required.  1.3  Present W o r k and Thesis Outline  Because of the transient nature of the shock wave produced by short-pulse lasers, most earlier works [19, 20, 21] have avoided studying the actual shock wave. Instead, they have focussed on the interaction between the femtosecond laser pulse w i t h the near-solid density plasma formed just after the target is irradiated [19, 20]. Nevertheless, issues related to hydrodynamic expansion and plasma scale-lengths still remain pertinent. T h e use of ultra-thin foils (~ few hundred A ) along w i t h femtosecond lasers has been suggested as a new approach to produce a well-defined solid density plasma, or slab plasma [23]. Recently, optical probing of femtosecond-laser driven shock waves in aluminum were performed by Evans et al.  [24].  In their work, the shock states were inferred from  the simultaneous measurement of the particle and shock velocities along the principle Hugoniot curve, as is normally done in long-pulse laser experiments.  However, it will  be shown i n this thesis that for shock waves driven by femtosecond lasers the shock states cannot be interpreted in the usual manner. T h e focus of the present work is to delineate the unique charasteristics of the shock wave which is produced i n short-pulse  Chapter 1. Introduction  8  laser experiments. Numerical simulations are used to study the features of these decaying shock waves and to analyze a new approach for obtaining information on strongly coupled plasmas, such as the electrical conductivity. A description of the hydrodynamic model and electromagnetic wave solver used in the calculations is presented in chapter 2. In chapter 3, results from the simulation of a proposed experimental scheme are discussed. A summary of the thesis is provided in chapter 4.  Chapter 2  M o d e l l i n g of Femtosecond-Laser D r i v e n Shock Waves  Due to the complex nature of laser-matter interactions, computer modelling of laser induced phenomena have become standard. A finite element model i n which the plasma is treated as a fluid is used. T h e hydrocode L T C (Laser Target Code) used in the present work was adapted from the laser fusion code M E D U S A [25] i n 1987 by Peter Celliers. A s various investigations have been pursued, L T C has undergone considerable modifications due to the changing requirements. In this chapter, a brief overview of the operation of the code is presented. A more complete description can be found elsewhere [26, 27].  2.1  Shock Wave T h e o r y  A n idealized shock transition between initial uncompressed material and final compressed material is illustrated in figure 2.1. T h e locus of thermodynamic states obtained from the compression of a material by a single shock wave is uniquely determined by the equation of state ( E O S ) . T h i s locus, usually referred to as the shock adiabat or the Rankine-Hugoniot curve (or simply the Hugoniot), can be calculated in a completely general manner by considering the conservation of mass, momentum and energy across the shock front.  T h e respective mass, momentum and energy conservation equations  across the shock front are PoU = pi(U - Up) s  Px-P  s  0  = PoUsU  P  9  (2.1) (2.2)  Chapter 2.  Modelling of Femtosecond-Laser Driven Shock Waves  E -E 1  0  =  l(P ^  x  + P ) ( - - -) \Po Pi/ 0  10  (2.3)  where p is the mass density, P the pressure, E the internal energy, and Us and Up are the shock speed and the particle speed. The subscripts of 0 and 1 denote the unshocked and shocked regions, respectively. For the shock state, this gives three equations w i t h five parameters: Us, Up, Pi, pi, and Ei. If the E O S , P = P(p,E),  is known for the material, then it may be added as a  fourth equation which reduces the system of equations to a single equation and one free parameter.  T h e resulting single equation gives a locus of points attainable by a  single shock for specific initial conditions (p ,P ,E ) 0  0  0  known as the Hugoniot curve, see  figure 2.2. Alternatively, if the E O S of the material is not known it can be measured in shock wave experiments (e.g. Us and Up measurements). F r o m the three conservation equations and simultaneous measurement of two shock parameters the E O S parameters P i , pi and Ei (along the Hugoniot) can be calculated [28].  2.2  Material Models  T h e E O S along w i t h other material models used in the present work is briefly described. In order to simulate plasma phenomena, the hydrocode requires the equation of state (pressure P and temperature T), average ionization Z*, thermal conductivity K and electrical conductivity a for the material of interest.  These are usually given by the  Chapter 2.  Modelling of Femtosecond-Laser Driven Shock Waves  11  Pi, P i , Ex  Ui  PO' 0 ' o E  p  u  P  =o X  Figure 2.1: Schematic diagram of an idealized shock transition. T h e particle speed i n the uncompressed material is assumed to be negligible.  Chapter 2. Modelling of Femtosecond-Laser Driven Shock Waves  Figure 2.2: Schematic diagram of Hugoniot curve on the pressure-density plane.  12  Chapter 2.  Modelling of Femtosecond-Laser Driven Shock Waves  13  various models as functions of p and E,  2.2.1  T  =  T(p,E)  P  =  P(p,E)  Z*  =  Z*(p,E)  K  =  K(P,E)  cr  =  tr(p,E).  (2.4)  E q u a t i o n of State and Average Ionization  For this work, a quotidian equation of state ( Q E O S ) based on the work by More et al. [29] is used. Q E O S is a general purpose equation of state model. It is a self-contained theoretical model that requires no external data base, and the inputs are simply the material composition and the cold solid properties (such as its density, bulk modulus or sound speed). In this model, the electrons and ions are described separately, thus the total internal energy and pressure of the system is, to a sufficient degree of accuracy, assumed to be the sum of the electron and ion contributions. The thermodynamic variables can be expressed i n the following form: T  = T (p,E ),  P =  P (p,E ) e  (2.5)  T = T (p,E ),  P =  P [p,E )  (2.6)  e  e  i  e  i  i  e  i  e  i  i  T h e physical properties of the ions are treated completely independent of the electrons except for the constraint that the charge densities are equal. The ion equation of state is based on a Cowan fluid model [30] which combines the Debye, Griineisen, and liquid scaling-law theories. The electron equation of state is based on the Thomas-Fermi theory formulated by Feynman, Metropolis, Teller [31] for hot dense plasmas. In the Thomas-Fermi model, the electrons are treated as a charged fluid surrounding the nucleus. T h e properties of this  Chapter 2. Modelling of Femtosecond-Laser Driven Shock Waves  14  electron gas are obtained from finite temperature F e r m i - D i r a c statistics. Effects due to the plasma environment are introduced using the ion-sphere model where each nucleus is located at the centre of a spherical cavity of radius (2.7) and the cavity contains enough electrons to be electrically neutral while other ions are assumed to remain outside the sphere. T h e ion density n , is defined as  where A is the average atomic mass number, and m  p  is the proton mass. In addition,  the model is modified by a semi-empirical treatment due to Barnes [32] to account for the chemical bonding energy, so that the pressure at solid density at 300 K is not unrealistically large (several M B a r ) , but only a few k B a r . T h e free electron density n is given e  as n.  (2.9)  In Q E O S , the Thomas-Fermi theory is also used to obtain the average ionization of the plasma which is assumed to be in local-thermodynamic-equilibrium ( L T E ) [33, 34].  However, i n order for plasma to be considered i n L T E , the following conditions  must be satisfied: (a) collisional ionization, recombination, excitation, and de-excitation are dominant over a l l radiative transitions; (b) the electron and ion temperatures are equal; and (c) the electron and ion velocity distributions are near Maxwellian. W h e n these conditions are not satisfied, a n o n - L T E ionization model such as a collisionalradiative equilibrium ( C R E ) model [35, 36], or a time-dependent collisional-radiative ( C R ) model [37, 38], usually has to be employed. T h e C R E model is used to calculate the average ionization for the case when radiative atomic transitions compete with collisional transitions, but the plasma conditions are changing slowly enough so that the distribution  Chapter 2. Modelling of Femtosecond-Laser Driven Shock Waves  15  of ionization states can adapt to the changing plasma conditions. O n the other hand, the C R model is used to calculate the case when the plasma conditions are changing faster than the rate at which the ionization state distributions can adapt. However for the present work, the effect of non-equilibrium ionization is not included, thus ionization values used are those given by Q E O S .  2.2.2  T h e r m a l and Electrical Conductivities  T h e r m a l and electrical conductivities are described by the conductivity model developed by Lee and More [40]. In this model, the electron transport coefficients are obtained from the solution of the B o l t z m a n n equation i n the relaxation time approximation [39]. In this approach, ones solves the B o l t z m a n n equation for the electron distribution function and then uses the solution to calculate the flux of electrical current and energy. T h e electron relaxation time is calculated from the electron collision frequency which only includes contributions from electron-ion and electron-neutral collisions. Calculation of the electron transport coefficients requires the knowledge of the momentum transfer cross sections. In the conductivity model, the Coulomb cross section 4^(Z*) e lnA 2  °tr =  ^  2  4  2.10  4  is employed using appropriate cut-off parameters. T h e Coulomb logarithm In A is given by \nl\= -\n{l l  where 6 , „ and m  b  max  Coulomb scattering.  (2.11)  + b Jbl ) 2  m  in  are the upper and lower cut-offs on the impact parameters for T h e m i n i m u m electron-ion impact parameter  b  min  is the value  allowed by the uncertainty principle, while the m a x i m u m impact parameter  b  rnax  is the  Debye-Hiikel screening length [41], corrected for electron degeneracy. In accordance w i t h other numerical results [42] a m i n i m u m value of 2 is set for the Coulomb logarithm i n  Chapter 2.  Modelling of Femtosecond-Laser Driven Shock Waves  16  order to overcome the inherent difficulty in the model for the case when the calculated electric field screening length become less than the interionic spacing.  2.3  Fluid Model  2.3.1  Hydrodynamics  L T C solves the one-dimensional Langrangian fluid equations involving the conservation of mass, momentum, and energy. Initially, the target is partitioned into a mesh of fluid cells. In the Langrangian description, the calculation follows the time evolution of the cells in which the mass of each cell is kept constant. In this scheme, the size of the cells decrease or increase to respond to the respective processes of compression and rarefraction of the target. T h e independent variables are time t, and the Lagrangian coordinate of mass m (kg/m ) 2  which is defined in terms of the density profile p(r,t)  as (2.12)  where i?o is a reference position, taken to be zero, and r is the cell position in the laboratory frame. The target material is treated as a compressible fluid composed of electron and ion components each described by a separate temperature. A more complete treatment of the model would include a radiation component, with a corresponding separate temperature, as well. However, in the present work the radiation component is negligible, thus effects due to radiation transport are omitted. In Lagrangian coordinates the fluid equations are  du dt  dV_  du  ~dt  dm  d{P  e  + P) l  dm  (2.13)  = 0  0  (2.14)  Chapter 2. Modelling of Femtosecond-Laser Driven Shock Waves  17  Equation 2.13 represents the fluid continuity equation, where V = 1/p is the specific volume, and M is the fluid velocity. Conservation of momentum of the fluid is represented by equation 2.14, where P  e  and Pi are the pressures due to the electrons and ions,  respectively. Energy conservation i n the fluid is represented by the two equations 2.15 and 2.16, where E and Ei are the internal electron energy and ion energy, respectively. e  The heat source terms Q and Qi for the corresponding electron and ion components of e  the fluid are discussed i n the next section. For each time step i n the simulation, the conservation equations are solved i n the following manner. First, the mesh coordinates r and velocities u are advanced i n time by solving a simplified version of equations 2.13 - 2.16 which is represented by the following set of equations, dV  du  ,  -at - a s = du  0  dp  de  ( 2  "  1 7 )  ,  d(up)  The total energy of the fluid is defined as e = E + E, + —, e  and the total hydrodynamic pressure is given as p = P + P. e  In addition, the heat source terms, QE and Q  l  i}  are eliminated, and the fluid is considered  to be of a single temperature. Since gradients exist between adjacent cells, the piecewise  Chapter 2. Modelling of Femtosecond-Laser Driven Shock Waves  18  parabolic method [45] is used to interpolate values at cell boundaries from the average values of the cells. A t each interface, a R i e m a n n shock tube problem [46, 47] is solved to yield average values for the pressure and velocity which are used to obtain hydrodynamic momentum and energy fluxes. H y d r o d y n a m i c motion is described by using the solutions from the R i e m a n n problem at each interface to advance i n time the cell coordinates and velocities. However, this yields only an adiabatic solution which must be corrected by solving the full energy equations which include the heat source terms.  2.3.2  E n e r g y Transport  The energy equations 2.15 and 2.16 can be written i n the form, dT ~dt  dp ~di  +  +  dE ~&t \  h y d r o  ~ ^  ( °) 2,2  where the coefficients Cy and Br are defined by the E O S , _dE C  v  - d T  dE W  -T -  Bt  >  W  P  T h e t h i r d term on the left-hand side of equation 2.20 represents the adiabatic work which was obtained i n the above hydrodynamic phase of the calculations, while on the righthand side, Q represents the heat sources arising from thermal conduction, electron-ion equilibration and laser energy deposition. A s i n M E D U S A [25], the Crank-Nicholson iterative time-centred differencing scheme [48] and Gauss elimination[48] were used to solve the energy equations.  B o t h the electrons and ions are treated using the same  numerical method w i t h the only difference being i n their respective heat sources, Q = H + X e  e  ei  + A  (2.21)  Qi = Hi — X  (2.22)  ei  where H and Hj are the respective electron and ion thermal fluxes, X e  energy exchange, and A is the laser energy deposition.  ei  is the electron-ion  Chapter 2.  Modelling of Femtosecond-Laser Driven Shock Waves  19  Flux-Limited Thermal Conductivity Energy transport due to thermal conduction is given as H = -VK-VT  (2.23)  P  where K is the thermal conductivity and T is the temperature. However, comparison w i t h observations [49, 50, 51] show that this classical Fourier heat flow law 2.23 overestimates the thermal flux in areas of the plasma where the mean free path of the electrons are greater than the thermal gradient scale length of the shock front. Therefore, the non-local thermal flux would cause an unsubstantiative preheat ahead of the shock wave. To reduce thermal flux where steep temperature gradients exist, a harmonic mean flux-limited [52] thermal conductivity K, is used: - = where  K dei mo  + 7777  2.24  is the thermal conductivity given by the conductivity model, and / is a free  parameter called the flux limiter. W h e n applied to equation 2.23, W refers to the heat flux that could arise from free streaming electrons, and is given as dx W = —n k Tv e  (2.25)  B  where x is the position, T is the temperature, n  e  is the electron density, k  B  is the  B o l t z m a n n constant, and v is the greater of the mean thermal velocity or the Fermi speed. T h e theoretical m a x i m u m value of / is 0.6 since 0.6n ksTv e  corresponds to the  free streaming l i m i t which is the m a x i m u m heat flux that can be carried by electrons with a Maxwellian velocity distribution.  Electron-Ion Equilibration Because of the short time scales of high intensity short-pulse laser experiments, thermal equilibrium between the electrons and ions within the plasma cannot be assumed. The  Chapter 2. Modelling of Femtosecond-Laser Driven Shock Waves  20  traditional description of thermal equilibration between electrons and ions is dT  (T - Ti)  dt  r  t  e  (2.26)  eq  where T and Tj are the respective electron and ion temperatures, and r e  eq  is the equili-  bration time. In numerical calculations, it is more convenient to consider the energy transferred between the electron and ion subsystems during a single time step, AE =  C  v  ^  T  - -  ^  T  A  (2.27)  t  where cyi is the isochoric ion heat capacity and A t is the time step. For weakly coupled plasmas cyi = 3riik/2 where k is the B o l t z m a n n constant. non-degenerate plasmas, r  eq  For weakly coupled and  is given by an expression by Spitzer [53],  where m,; is the ion mass, iii the ion density, Z* the average ionization of the ions, e the electron charge, and In A the Coulomb logarithm. Brysk [54] extended the Spitzer formula to derive an expression for degenerate plasmas,  e q  ~ 8m2(Z*) e lnA 3  4  l  +  e  -/V^)  [] s  (2.29)  ,  where m is the electron mass, p the chemical potential, and h = h/2n where h is the e  Planck constant. However, both equations 2.28 and 2.29 for r  eq  are valid only for plasmas  where the ion-ion correlation is weak. For strongly correlated systems, these expressions underestimate the equilibration times by two to three orders of magnitude compared to observations [55, 56, 57, 58, 59]. Unfortunately, no simple analytical expressions are available for r  eq  for strongly coupled plasmas.  Chapter 2. Modelling of Femtosecond-Laser Driven Shock Waves  T h e factor cvi/r  eq  21  is more conveniently replaced by a single constant g which is left  as a free parameter. T h e phenomenological approach of the electron-ion energy exchange is modelled as AE=  ^--^-(T -Ti)At Po Po  [J/kg]  e  (2.30)  where g is the electron-ion coupling coeffecient, and the density dependence of the electron-ion equilibration time is incorporated by scaling the density of the plasma p w i t h the solid density p of the material of interest. 0  Laser E n e r g y Deposition For electrons, i n addition to the energy transport due to thermal conductivity and electron-ion equilibration, energy is absorbed from the laser. T h e mechanics of laser energy deposition is modelled using an electromagnetic wave solver. T h e plasma is treated as an inhomogeneous dielectric medium w i t h the complex dielectric function, e(u) = l + i  ^  (2.31)  where O(LU) is the electrical conductivity at the laser frequency u. T h e Drude model [62] is used to construct the electrical conductivity, a(u) =  ^  .  [5" ]  (2.32)  1  Z*n,e  2  (2.33)  ma e  where cu is the plasma frequency, v p  ei  0  is the electron-ion collision rate, o is the tabulated  D C electrical conductivity, Z* is the average ionization, m  0  e  i =  is the electron mass, and  \T l. z  T h e procedure used to solve the wave equations follows treatment given by B o r n and Wolf [61] for stratified media, and is briefly described in the next section.  Laser  Chapter 2. Modelling of Femtosecond-Laser Driven Shock Waves  22  energy deposition v i a O h m i c heating based on the electrical conductivity of the plasma is calculated as A =< E • J >= ^Re(a)\E\  (2.34)  2  where E is the electric field vector, and J = <JE is the current density.  2.4  Electromagnetic Wave Solver  In the electromagnetic wave solver, Maxwell's equations are solved using the Helmholtz formulation i n the following manner [61].  A medium whose properties are constant  throughout each plane perpendicular to a fixed direction is called a stratified  medium.  Consider a plane electromagnetic wave propagating through this statified medium with the plane of incidence taken to be the yz-plane, as shown i n figure 2.3. A n y incident electromagnetic wave may be resolved into two linearly polarized orthogonal waves: a transverse electric ( T E ) wave and a transverse magnetic ( T M ) wave. For a T E , or S-polarized, wave incident on the target medium, E  y  = E  z  = 0 and  Maxwell's equations reduce to the following scalar equations: dH IF oy z  1  dH ~ I oz T  ieuj,_, + — = c*  dH — dz  dH  v  x  E  z  n  = 0  dx  0H dH —^-—^ = 0 ox ay V  x  2.35  c dE  x  ~5 oz dE  x  dy  iufj. — v H  c tup c  0  • =  u  Chapter 2. Modelling of Femtosecond-Laser Driven Shock Waves  •  23  x  Stratified Medium  Figure 2.3: Coordinate system used by the electromagnetic wave solver. The propagation of the plane electromagnetic wave of the laser incident to the stratified medium at an angle 9 to the z-axis is shown.  Chapter 2. Modelling of Femtosecond-Laser Driven Shock Waves  24  E l i m i n a t i n g H and H yields the following differential equation y  z  dE  dE  2  ^  „  2  x  x  2 l  - ^  +  +  >  B  t  d (log fi)dE  x  e  = - L l « _ ^  (2.36)  where the n is index of refraction of the medium, while k and A are the respective wave 0  0  number and wavelength of the incident wave outside the medium, arid are given as U> 27T k = -=.— . c Ao  o  n = eu.  :  (2.37)  0  For a l l cases i n the present work, the magnetic permeability n is equal to one since only non-magnetic materials are studied. T h r o u g h separation of variables, equation 2.36 can be written as the following second-order differential equations: ^  -  ^  + * „ V - « V = 0  + k (n 2  dz  dz  2  dz  2  (2.38)  - a )V = 0 2  where Ult)  Ex  V(z)e ( ° ~-ut)  Hy = H  z  =  i k  ay  W(z)e  (2.39)  --wi)  i{koay  The functions U, V and W are related by the following equations:  U' =  ikonV  V  =  ik (e-f)U  alJ + pW  =  0.  0  (2.40)  For the case when the wave is a homogeneous plane wave a = n sin 9 = constant where 9 is the angle which the normal to the wave makes with the z-axis.  (2-41)  Chapter 2. Modelling of Femtosecond-Laser Driven Shock Waves  25  Similarly, for a T M , or P-polarized, wave incident on a target medium (H = H = 0), y  z  a corresponding set of expressions can be found. A s a consequence of the symmetry of Maxwell's equations, the substitution rule of exchanging e and —p gives the following equations: dU d(\oge)dU , , w - * - * ' S < - ' *  ,  2  9  +  dV 2  v2 >  d[\og{p-^)]dV  dz  , + k (n t  dz  2  w  n  dz  | 2  2 2  2  -  t  =  v  2  0  ( 2  -  4 2 )  . a )V 2  where H  x  =  E  y  =  E  z  u(z)e - - ^ l(  koay  UJt  -V(z)e^ -^  (2.43)  koay  =  -W{z)e - - . l(  koay  UJt)  Now, the functions U, V and W are related by the following equations: U'  =  ik eV  V  =  ik (p-^)U  aU + eW  =  0.  0  (2.44)  0  The solutions, subject to the appropriate boundary conditions, of the differential equations ( 2.38 and  2.42) and various theorems relating stratified medium, can most  conveniently be expressed in terms of matrices. In the case of a homogeneous dielectric film, e, p and n = yfep are constants. For a S-polarized wave, the solutions to 2.38 are given as U(z)  =  A cos(k nz cos 6) + B sin(A,- n cos 0)  V(z)  =  ^J^cosO{Bcos(k nzcos9)  0  0  0  —  (2.45) Asm(k nzcos9)} 0  w i t h the boundary conditions (at z = 0) U(0) = U , Q  F(0) = V . 0  (2.46)  Thus, the x and y components of the electric (or magnetic) vectors in the plane z = 0 are related to components in an arbitrary plane z by the characteristic  matrix M ( z ) of  Chapter 2. Modelling of Femtosecond-Laser Driven Shock Waves  the medium,  ( )  26  UZ  f u A  M(z)  \v ) 0  (2.47)  where M(z) =  mn  mi2  mi  cos(k nz cos 9)  0  ip sin(k nz cos 9)  m  2  — ^ sin(k nz cos 9)  0  —  0  22  (2.48)  cos(k nz cos 9) 0  with p — J — cos 9.  (2.49)  For a P-polarized wave, the same equations hold, w i t h p replaced by q=J^cos9. Ve  (2.50)  T h e inhomogeneous target medium can be treated as a stack of thin homogeneous films by m u l t i p l y i n g the characteristic matrices of all the stratified media. In other words, the plasma is modelled as a succession of stratified media extending from 0 < Z < Zi, Zi < z < z , • • •, Z j v - l < z < ZN 2  w i t h the characteristic matrix M{z ) N  = M (z )M {z 1  1  1  - zi) • • • MN_{Z  2  - z _i).  n  N  (2.51)  Thus, consider a plane wave incident upon a stratified medium that extends from 2 = 0 to z = Z[ and that is bounded on each side by a homogeneous, semi-infinite medium, see figure 2.4 A p p l y i n g the boundary conditions allow for the calculation of the electric (or magnetic) field strength within the target medium.  For a S-polarized wave the boundary  conditions are Uo =  Ei + E ,  U  V  po(Ei-E ),  V  0  =  r  r  Z{  Zl  — =  E,  (2.52)  t  P l  E  u  Chapter 2. Modelling of Femtosecond-Laser Driven Shock Waves  27  Figure 2.4: A schematic diagram of an inhomogeneous medium modelled as a stack of stratified media.  Chapter 2. Modelling of Femtosecond-Laser Driven Shock Waves  28  where p = J — cos9 , Po 0  Pi = . —cos 03,. V t**i  0  (2.53)  T h e amplitudes of the electric field vectors of the incident, reflected and transmitted waves are Ei, E and E , respectively. Furthermore, CQ, po and ti, pi are the dielectric r  t  constants and magnetic permeabilities of the first and last media (semi-infinite), while 9Q and 9{ are the angles which the normals to the incident and transmitted waves make w i t h the ^-direction (direction of stratification). T h e reflection and transmission coefficients of the stratified medium are E  r  (mn + m Pi)po - {m \ + m Pi) 12  2  (mn + m pi)p 12  0  E,  + (m  22  + m pi)  2i  2po (mn + rn pi)p 12  0  + (m  2i  (2.54)  22  + m pi)'  (2.55)  22  Thus, the reflectivity and transmissivity, i n terms of r and t are T  R  (2.56)  Po  and the phase of the reflected and transmitted waves are arctan  Re(r)  arctan  Im(r)  Re(t) Im(t)  (2.57)  For the case of a P-polarized wave the quantites p and pi are replaced by 0  [Po 9o = A / — c o s 0 , O  Qi =  Z[  Pz,  cos 9  t  (2.58)  in which case r and t are then the ratios of the amplitudes of the magnetic field vectors.  2.5  Other Effects  In high-intensity, short-pulse laser experiments, effects due to radiation (x-rays) transport and suprathermal ("hot") electrons [65] must be addressed. The m a i n concern is the issue  Chapter 2. Modelling of Femtosecond-Laser Driven Shock Waves  29  of preheat of the cold material ahead of the shock front by x-rays a n d / o r hot electrons, thus modifying the i n i t i a l state of the material before it is compressed by the shock wave [66, 67, 68, 69, 70]. W i t h i n the t h i n layer of material heated by the intense short-pulse laser, the electrons may reach extremely high temperatures (100-1000 e V ) , and emit high-energy photons which penetrate deep into the target. However, for laser irradiances less than  10 W/cm , 15  2  the electron temperatures of the plasma would be about 100 e V or less, and the power density of the frequency-integrated emission is only about 1 % of the laser radiation even if the plasma radiates as a blackbody. Therefore, the effect of radiation transport would be m i n i m a l . Hot electrons generated in the laser heated layer have large diffusion lengths allowing them to penetrate into the solid and heat the material. H o t electron sources include the processes of spatial dispersion and wave breaking [71] along w i t h the absorption mechanisms of laser energy deposition such as resonant absorption [49], vacuum heating [72], and forward-scattering instabilities [73]. In this thesis, the aforementioned effects are considered negligible since the situations studied involved relatively moderate pump laser intensities ($£, < 10 W'/cm ) at normal incidence to the target. 15  2  Chapter 3  Discussion of Femtosecond-Laser D r i v e n Shock Results  Studying plasma transport properties such as electrical conductivity is of fundamental importance to the understanding of collision processes w i t h i n a dense plasma. In shock produced plasma experiments, optical probing is a widely used and crucial diagnostic. Studies usually involve the examination of rear surface luminous emission and reflectivity as the shock wave is released at the vacuum-target interface [74]. However, as the release wave expands into the vacuum a critical density layer, which is cooler and of lower density than the shock state, is formed and acts as an absorbing layer to radiation passing through it in either direction. Thus, subpicosecond time resolution is required to allow accurate data collection before the critical density layer expands a few optical depths away from the hot, shock-compressed material. W i t h the advent of intense femtosecond lasers, a slightly different approach for producing and studying dense plasmas has been developed. The target is irradiated by an intense short-pulse beam to produce near-solid-density plasma with a very short gradient scale-length near the target surface [19]. Because of the limited plasma expansion during the heating pulse, it was thought that the reflectivity of the laser could be calculated by considering an electromagnetic wave propagating through a plasma with a steady-state gradient which is assumed to be linear [20], exponential [20, 21], or given by a power law [22]. From the reflectivity measurements the electrical conductivity of the dense plasma could then be inferred. T h e initial results of experiments done on aluminum by Milchberg et al. [19] using this method showed severe deviations [75] from an existing dense plasma  30  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  31  conductivity model [40]. A s a result, several experimental investigations [20, 21] were performed and theoretical models [75, 76] were developed to examine the discrepancy. However, the apparent discrepancy seemed to hinge heavily on the interpretation of the experimental data. Given the extremely transient nature of the heating pulse, the assumption of a constant gradient scale-length generated by a steady-state ablative flow is questionable. Using a hydrodynamic code to self-consistently treat the plasma dynamics and the laser-plasma interactions, the controversy was resolved by N g et al. [77]. It was shown that the assumption of a single density scale-length throughout the duration of the laser pulse is clearly invalid; it was also revealed that the electromagnetic wave of the laser interacts w i t h the target material over a wide range of plasma conditions. Thus, by incorporating the aforementioned effects, the experimental reflectivity measurements were shown to agree quite closely w i t h the model predictions.  3.1  Proposed Scheme  In order to limit effects due to gradient scale-length issues associated with the aforementioned approach, it is ideally better to look at a shock produced plasma state which is still situated within a material. Traditionally, rear-side optical probing is used to examine the dense plasma state in a shock wave as the shock front approaches the rear surface of a target. Since most studies of dense plasmas deal with materials which are metallic in their cold, solid state, the electromagnetic wave of the probe laser penetrates only a very short skin depth w i t h i n the material. Hence, observation of the shock-state is quite brief before the shock is released into the vacuum and thereby destroying the desired dense plasma state. To deal with this problem, the rear surface of the target is usually placed on a window substrate such as quartz or glass to prevent the shock plasma from releasing, but then effects arising from the interface between the window substrate and the target  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  32  material must be addressed. It seems that it would be simpler to study a material w i t h a relatively low absorption coefficient for optical light i n the cold, solid state. A t normal conditions, silicon has an indirect band gap of 1.2 e V and its absorption coefficient is < 1 0 c m 4  - 1  for wavelengths > 0.5pm [78]. Another important characteristic  of silicon is that its small band gap closes at a relatively low pressure (~ 12 G P a ) , thus at higher pressures silicon is expected to be metallic [79] and interband transitions need not be addressed. Silicon also has the practical advantage of being readily available as high-purity, ultrathin wafers w i t h optical-quality surfaces. A proposed experimental scheme is shown i n figure 3.1. O n the right, a 400 n m pump pulse with a 120 fs full-width half-maximum ( F W H M ) pulse-width is incident normally onto the target's front surface. O n the left, a 120 fs F W H M , 800 n m probe laser is incident normally onto the target's rear surface. Generally speaking, shorter wavelengths for the pump pulse offer greater absorption and reduce hot-electron production compared to longer wavelengths [69]. A practical target is presented which consists of a 0.1//m thick aluminum layer, and a l . O ^ m thick silicon layer situated on a quartz window substrate.  In addition, anti-  reflective coatings are placed at the window-silicon interface and at the window-vacuum interface. T h e anti-reflective coatings are needed to prevent interference effects due to probe laser light reflected off the shock front and these surfaces. In the proposed scheme, effects arising from the interface between the window and the silicon are avoided by examining only the results i n which the shock wave is still well within the silicon layer. For the present work, the window-silicon interface is considered to be the rear (free) surface of the target. Because silicon is relatively transparent to optical laser light, the thin aluminum layer is needed to allow for efficient absorption of the pump laser energy in order to initiate and drive a shock wave into the silicon layer. In addition, because the 1/e skin depth of  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  33  Anti-Reflective Coating  Quartz Window Substrate  Si 1.0 |im  Al 0.1 \im  Figure 3.1: Schematic diagram of the proposed scheme. T h e thicknesses of the anti-reflective coatings and quartz window substrate are not drawn to scale.  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  34  aluminum is quite short (~ 7nm at A = 400nm) transmission of the intense femtosecond laser pulse into the silicon target is precluded thereby preventing preheat of the cold silicon. After the aluminum is heated by the pump pulse, a thermal wave propagates into the silicon layer which is then overtaken by a compression wave generated by the ablation of the front surface. As the shock wave moves through the silicon layer, the probe pulse incident on the rear surface propagates through the cold silicon material and interacts with the shock front. Since the probe pulse is extremely short in duration, the measured reflectivity of the shock front is considered almost instantaneous. For the silicon layer, a thickness of 1.0pm was chosen for several reasons. First, in contrast to previous studies [80, 81] where thinner targets were used in order to study the thermal wave produced by intense femtosecond lasers, the target used in the present work needed to be thick enough to allow the formation of the shock wave. On the other hand, because there is still a small amount of absorption of the probe laser light by the cold solid silicon, the target cannot be too thick, otherwise the measured reflected probe light will be too attenuated. Therefore, the silicon samples used in this approach would be somewhat restricted to thicknesses ~ lpm. The 120 fs laser pulses employed in the experiment above can be readily obtained from a Titanium:Sapphire (Ti:Al 03) laser system [82]. The 800 nm probe pulse would 2  correspond to the fundamental frequency (la;) beam of the T i : A l 0 2  400 nm pump pulse, whose irradiance ranges from 10 W/cm u  2  laser, while the  3  to 10 W/cm , 15  2  would  correspond to the frequency doubled (2a;) beam. In the present work, the temporal profiles of pump and probe pulses are modelled after the laser pulse reported in a recent experiment [83] as shown in figure 3.2. In reality, the wing structure of the pulse extends for a considerable amount of time away from the peak of the pulse, but at such low intensities the target is effected negligibly. For the sake of efficiency, in the simulation the laser pulse is truncated at intensities below 10~ of the peak of the pulse laser intensity. 6  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  -600  -400  -200  0  200  35  400  600  Time ( f s )  Figure 3.2: Temporal profile of 120 fs laser pulse used in the simulations.  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  3.2  36  Results  3.2.1  Basic Considerations  Recently, work has been done to further shorten the pulse duration down to about 35 fs, and over the last decade many experimental studies have used slightly longer pulse duration (up to ~ 500 fs). For the present work the choice of pulse duration is not too critical. A s long as the pulse is short enough to avoid hydrodynamic expansion during the heating of the target by the pump pulse, efficient laser energy deposition would be achieved. In addition, the probe pulse duration should be short enough such that the shock state of the changes negligibly during the interaction with the probe pulse, so it may be considered to be an instantaneous measurement of a shock state. Slight changes in the pump pulse duration have the effect of simply decreasing or increasing the amount of energy that the target absorbs, but adjustment of the pump pulse duration represents a rather inefficient means of laser energy control. A more suitable control over laser energy deposition would be the varying of the irradiance of the pump pulse while keeping the pulse duration constant. The pump irradiances used i n the present work were chosen for several reasons. They were limited to a m a x i m u m value of 1 0 W / c r a l D  2  in order to avoid preheat of the un-  compressed material in front of the shock by radiation transport a n d / o r hot electrons. Meanwhile, a m i n i m u m value of 10 W/cm u  2  was employed in order to avoid phase tran-  sition issues. In addition to the fact that silicon become metallic at 12 G P a [79], the sequence of silicon phase transitions include cubic diamond to body-centre tetragonal at 11 G P a [84, 85, 86], to simple hexagonal at 13-16 G P a [87], to an intermediate phase of undetermined structure at 34 G P a , to hexagonal-close packed at above 40 G P a [88], and to face-centre cubic above 78 G P a [89]. Plasmas produced by lower laser irradiances  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  37  would have shock pressures comparable to that associated with the these phase transitions, thus complicating matters. For laser irradiances above \0 W/cm , u  2  the shock  pressures are substantially above these phase transition pressures. Also, for irradiances less than 1 0 W/cm , the initial shock wave is relatively weak (~ a few hundred kbars) 14  2  and does not propagate deep into the material before the thermal wave actually catches back up to the shock wave and extinguishes the shock structure. A l t h o u g h a harmonic-mean flux limiter model [52] is incorporated in the calculations, the heat flux i n the absorption region or the shock front never reaches saturation even for a flux limiter of 0.03 because of the large collision frequency and the limited temperature difference across the heat or shock fronts.  In all calculations, a flux limiter of 0.6 was  used, thereby avoiding legislation of heat flow. Since the electrons and ions are treated separately in the calculations and there exists no simple analytical expression for electron-ion energy exchange, the thermal equilibration between the electrons and ions was treated using a phenomenological coupling constant. Based on previous studies by N g et al. on silicon [43], an electron-ion coupling coefficient of g = 10 W/m K 16  3  was used for all calculations in this thesis. Changes in g up  to an order of magnitude showed only slight changes in the electron and ion temperatures at the shock front.  3.2.2  Characteristics of the Decaying Shock Wave  Off-Hugoniot States In shock physics, thermodynamic information about the shock state is usually described by the Rankine-Hugoniot curve (or Hugoniot). For given initial conditions, the final state reached due to compression of the initial state by a single shock will lie on the Hugoniot. For a spatially and temporally constant shock propagating through a sample  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  38  all compressed material behind the shock front w i l l be i n the same shocked state. In order for a constant shock wave to continue to propagate, a steady or quasi-steady energy source must drive the shock, as i n the case of long-pulse laser experiments.  Obviously,  in short-pulse laser experiments the shock wave is not driven continuously, so it decays in magnitude as it propagates through the sample. Because of the influence of the initial thermal wave in the short-pulse laser experiments, the shock states of the decaying shock wave will lie off the Hugoniot.  Figures 3.3 and  3.4 show the pressure and electron  temperature at the shock front of a decaying shock wave driven by an intense femtosecond laser in silicon, along w i t h the corresponding silicon Hugoniot states. The end of the pump pulse is represented by the point A where 99.99% of the laser energy has been deposited in the target. A t this point in time, the ions at the front surface of the target are still relatively cool while the electrons are heated to their m a x i m u m temperature thus producing a very large thermal pressure at the surface of the target. From point A to point B , a thermal wave propagates into the silicon followed by a compression wave, where at point B the compression wave has overtaken the thermal wave and steepened to form a shock front. After point B , the magnitude of the shock rapidly decays as the front surface releases into vacuum. Initially, the state at the shock front is well off the Hugoniot due to the effect of the thermal wave on the material. A s the decaying shock propagates further into the target, the state at the shock front can been seen to asymptotically approach the Hugoniot states. To obtain more information about the femtosecond-laser driven shock, careful examinations of the thermodynamic spatial profiles of the decaying shock at different times are needed.  Spatial Profiles of the Decaying Shock Wave Figures 3.5 - 3.12 represent the spatial snapshots of the thermodynamic variables of pressure, density, electron temperature and ion temperature at different times. E x a m i n a t i o n  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  39  Figure 3.3: The Si off-Hugoniot states on the pressure-compression plane for a femtosecond-laser driven shock wave. The solid density of Si is p = 2.33 g/cm . Time proceeds from point A to B to C . 3  0  40  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  Figure 3.4: The Si off-Hugoniot states on the electron temperature-compression plane for a femtosecond-laser driven shock wave. The solid density of Si is p = 2.33 g/cm . Time proceeds from point A to B to C . 3  0  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  41  of these profiles w i l l illustrate the unique evolution and characteristics of shock waves generated by intense femtosecond lasers. Results for a pump irradiance of 5 x  10 W/cm u  2  is used as the standard case for discussion. For other pump irradiances, the shape of the profiles are quite similar to that of the standard case while the magnitudes of the profiles scale w i t h the irradiance used. T h e features of the decaying shock wave are revealed by studying the profiles at early times shortly after the end of the p u m p pulse, figures 3.5 - 3.8, and at later times when a definite shock wave has been formed, figures 3.9 - 3.12. T i m e zero corresponds to the time of the peak of the pump pulse laser intensity that is incident on the target. Referring to the early time profiles, figures 3.5 - 3.8, the highest pressures occur at the front of the target while it is being heated by the pump pulse.  A t this time,  no compression of the target has taken place, and this high pressure is attributed to the large thermal pressure of the electrons heated by the pump laser. Since only the electrons absorb energy from the laser, the electron temperature at the surface reaches up to about 70 e V while the ions remain relatively cold. A s the electrons start to thermalize the ions, the surface of the target begins to be ablated causing material to expand into the vacuum. Meanwhile, an electron thermal wave [81, 90] begins to propagate through the target. T h e pressure at the target front surface quickly decays due to both the release of the hot material into the vacuum and the propagation of the thermal wave into the target. Because no additional energy is provided for the thermal wave as it moves further into the target, it quickly begins to weaken and slow down. A t the same time, because of conservation of momentum, the rapidly expanding ablated material leads to compression of the material near the surface. T h e compression wave follows the thermal wave into the target and as it moves from the aluminum layer to the silicon layer, its density profile starts to steepen into a discontinuity and a shock wave is formed. T h e shock wave overtakes the initial thermal  42  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  1O \ i i ' 1  i  I  T—i—r  -r-i—i—i  | i i i i | i i i—i | i i i i d  10 2L_ 1  10  1  10  1  u 3  Vi Vi  u  10  y  10  s  • • • i I i i i—i—I—i—u  0.5  -0.4  -0.3  '  -0.2  1  •  •  I  -0.1  I  I  0  I  I  I  L  0.1  x (|_im )  0  2 ps  4ps  6ps  8 ps  Figure 3.5: Early time pressure spatial profiles. The position x = 0 represents the initial location of the target front surface, and the pump pulse is incident from the right. For early times, the profiles remain near the front surface and the material not shown (x < — 0.5pn) is still cold, solid Si.  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  43  Figure 3.6: Early time electron temperature spatial profiles. The position x = 0 represents the initial location of the target front surface, and the pump pulse is incident from the right. For early times, the profiles remain near the front surface and the rest of target material which is not shown (x < —0.5/j.m) is still cold, solid Si.  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  F~i—i—i—i—|—i—'—i—i—|—i——i——|—————|—r T—I—I—I—I—I—I—r  1 0  1  1  1  1  ;t-..'s  1  1  J.  fc4 u  44  I ' l l  10 \r 4  3 *-< OS s-  a>  a  CD  H  s  !  10 fcJ  /  o  10 2 I • 0.5 J  • • • I '  -0.4  I  I  I  1  I  -0.3  I  I  I  I  I  I  -0.2  I  l__L J _ J  I  -0.1  I  I 0  I  I  I  L  0.1  x (fim )  0  - 2 ps  4 ps  6 ps  8 Ps |  Figure 3.7: Early time ion temperature spatial profiles. The position x = 0 represents the initial location of the target front surface, and the pump pulse is incident from the right. For early times, the profiles remain near the front surface and the rest of target material which is not shown (x < —0.5pm) is still cold, solid Si.  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  6  —|—i—i—i—i—|—i i i—r—  \i  1  • 5  i  x  3  -  —  'I 4  45  • i •  \  -  «.i \  1  It -  N\ \ \  —  —  2 1 0 0.5  1  1  1 _. I  1  -0.4  1  1  1  1  1_  -0.3  .1  1  1  1  1  L  -0.2  _I_J  _  1  1  1  1  -0.1  1  0  0.1  x (fim)  — 0  - 2 ps  4 ps  6 ps  8 ps  Figure 3.8: Early time density spatial profiles. The position x — 0 represents the initial location of the target front surface, and the pump pulse is incident from the right. For early times, the profiles remain near the front surface and the rest of target material which is not shown (x < —0.5pm) is still cold, solid Si.  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  46  wave w i t h i n several picoseconds, but due to the finite equilibration time between the electrons and ions an definite two-temperature  shock structure is produced.  A s the  electrons are thermalizing the cold ions, electron thermal flux across the shock front heats the electrons ahead of the shock wave. A s a result an inherent electron thermal foot [44] of heated electrons i n front of the shock wave is produced. T h e less coupled the electrons and ions are to each other, the greater the electron thermal flux thus the larger the electron thermal foot. Due to the release of the front surface, a rapid tailing off of the pressure and density of the material behind the shock front is seen. Meanwhile, the decrease i n the temperature of the ions behind the shock front is retarded by the electrons as thermal equilibration between the two species continues. Referring to the late time profiles, figures 3.9 - 3.12, the shock wave can be seen propagating through the silicon layer of the target, and just like the thermal wave, it is seen to weaken and slow down.  In addition to the decrease i n the magnitude of  the shock wave, a relaxation i n the gradients at the shock front can be observed. T h e electron temperature profile shows the most noticeable smoothing of the gradient while the other profiles show less pronounced relaxation effects. A s the electron temperature profile relaxes, its gradient scale length increases which causes the electron temperature foot ahead of the shock to become larger; i n other words more electrons ahead of the shock are heated. Consequently, ahead of the shock more energy is transferred from the electrons to the ions. A s a result, a noticeable ion temperature foot begins to appear ahead of the shock front, too. A t earlier times, when a strong shock wave with a steep shock front is observed, almost a l l of the material i n front of the shock wave is described as cold silicon. However the appearance of the ion temperature foot modifies that view, since material ahead of the shock front which has ion temperatures above the melting temperature of silicon would have to be considered as liquid silicon. Therefore as time goes on, the notion of the idealized discontinuity between a cold solid and a hot, dense  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  47  plasma becomes less realistic. Also, whereas in early times the material behind the shock font is seen to drop off very steeply, for later times it appears that the tail behind the shock front has relaxed as well. Finally, at even later times (not shown in the figures) the shock wave reaches the rear surface of the target. However, i n order to neglect effects due to interaction of the shock wave w i t h the rear edge of the target, only times when the shock wave is still relatively far away from the rear surface are pertinent to this work. Accordingly, based on the observed features of the spatial profiles of the thermodynamic variables at various times, the following remarks can be made. First, some time is required before a definite shock wave is formed. Hence, probing of the target at too early a time would reveal information mainly about the electron thermal wave structure instead of about the desired shocked states. O n the other hand, probing of the target too late a time would be on a relaxed shock wave where results are effected by the melted material ahead of the shock front.  Consequently, in order to avoid the early and late  time effects, study of the shock states produced by an intense femtosecond laser within a certain time frame in between would be optimal. This is not too much of a concern since the probe pulse is also of femtosecond time duration, hence a fair amount of data could be attained during this optimal time window.  T e m p o r a l History of T h e r m o d y n a m i c State at the Shock Front A s previously mentioned, for an idealized shock wave, the shock front represents the discontinuity between the thermodynamic states of the material ahead of and behind the shock wave. A l l material ahead of the shock front is described by a single thermodynamic unshocked state, while all material behind the shock front is described by a single thermodynamic shocked state. In reality, no matter what method is used to drive a shock wave, gradients in the thermodynamic variables, with associated scale-lengths, always  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  48  Figure 3.9: Late time pressure spatial profiles. The position x — —l.lfim represents the location of the rear free surface of the target (window-silicon interface), and the probe pulse is incident from the left.  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  49  Figure 3.10: Late time electron temperature spatial profiles. The position x = —1.1pm represents the location of the rear free surface of the target (window-silicon interface), and the probe pulse is incident from the left.  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  50  Figure 3.11: Late time ion temperature spatial profiles. The position x = — \.lu.m represents the location of the rear free surface of the target (window-silicon interface), and the probe pulse is incident from the left.  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  51  Figure 3.12: Late time density spatial profiles. The position x = — 1.1pm represents the location of the rear free surface of the target (window-silicon interface), and the probe pulse is incident from the left.  Chapter 3.  Discussion of Femtosecond-Laser Driven Shock Results  52  exists. In addition, the material behind the shock front is never i n a single, shock state since the compressed material asymptotically settles towards equilibrium.  To further  complicate matters, in the case of femtosecond-laser driven shock experiments, because the shock wave is not being continuously driven, it is followed by the release wave which results i n the rapid relaxation of the shocked state behind the shock front. Thus, as soon as the target material is compressed to a strongly coupled plasma state of interest, it quickly relaxes into an undesireable hot expanded plasma state. However, one of the key features about the proposed scheme relies upon interaction of the probe pulse with the shock front. In fact, as it w i l l be shown i n the next section, the probe laser light propagates a very short distance beyond the shock front, thus only the desired dense plasma states near the shock front are actually measured. Referring to the previous figures ( 3.5 - 3.12 ), at each instant i n time the shock wave is described by the various thermodynamic profiles. A shock wave is generally interpreted as a supersonic compression wave, thus identification of the shock front of the decaying shock wave relies mainly on the examination of the density profiles. To simplify matters, in this work the shock front is represented by the point w i t h i n each density snapshot at which m a x i m u m compression of the material occurs.  T h e corresponding pressure,  electron temperature, ion temperature, and density represent the state of the shock front at that instant in time. Figures 3.13 - 3.16 track these thermodynamic variables in time as the shock wave propagates through the target material. Because of the aforementioned criterion of m a x i m u m compression utilized for identifying the shock front, for early times the values given correspond to the thermal wave and the steepening of the compression wave. The shock wave is considered to be fully formed near the time at which the density of the "shock front" reaches the m a x i m u m compression. Once the shock wave is formed the common feature about the temporal histories is that all the thermodynamic variables decay in time. In addition, there is a  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  0  10  20  30  Time (ps)  Figure 3.13: Decay of the pressure at the shock front.  53  40  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  Figure 3.14: Decay of the electron temperature at the shock front.  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  10  I  \ Q3 I  0  I  I  I I  I  I  I  I  I |  I  I  I  I  I  I  I  I  I  I  10  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  I  I  I  I  I  I  I  20  I  55  I  I  I  I  I  I  30  Time (ps)  Figure 3.15: Decay of the ion temperature at the shock front.  I I I  I  I  I  I  40  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  Figure 3.16: Decay of the density at the shock front.  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  57  striking similarity between the decay of the shock front pressure and electron temperature. T h i s reveals that for femtosecond-laser driven shocks, the pressure of the shock front has a strong electron temperature dependence.  Meanwhile, the aforementioned  process of continual thermal equilibration behind the electrons and ions is indicated by the slightly less rampant drop off i n ion temperature at the shock front. However, whereas the pressure, electron and ion temperatures experience rapid rates of decline, the density of the shock front decreases rather slowly. W i t h i n a certain window in time, from 10 ps to 30 ps after the pump pulse, a comparison between the relative change i n the electron temperature and the density provide useful information about the decaying shock wave. Whereas the electron temperature drops by more than 50% from its initial value, the density decreases only by about 10%. Consequently, proper measurements of the decaying shock wave could be used to scan the temperature dependence of dense plasma properties along an approximately constant density region. More specifically, the following section demonstrates how optical probing of the decaying shock wave could be used to assess the electrical conductivity of a strongly, coupled plasma.  3.2.3  Optical P r o b i n g  In chapter 2, laser energy deposition to the target material by the pump pulse is modelled using an electromagnetic ( E M ) wave solver. Similarly, the interaction of the probe pulse with the decaying shock wave is described using a post-processor version of the E M wave solver.  In the latter case, because the energy of the probe pulse is considerably less  than the pump pulse, the target material is not modified by the probe pulse. Therefore, the hydrodynamic profiles generated by the hydrocode for each instant in time can be directly converted into the corresponding complex dielectric function profiles and used to calculate the reflectivity, transmission and phase change of the probe pulse.  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  58  A s previously mention i n chapter 2, the dielectric function is calculated v i a the Drude prescription using electrical conductivities based on the model given by Lee and More [40] which is valid for a wide range of plasma conditions. However for the unshocked material ahead of the shock wave, and for the material at the foot of the shock front, the model gives values for the dielectric function which differ considerably from that of reported values.  T h i s is to be expected since in the former case the material is still  just cold, solid silicon, while i n the latter case the material is the uncompressed silicon which has been melted by the thermal foot just ahead of the shock front.  In both  cases, the dense plasma conductivity would not be valid and the associated dielectric function would be incorrect, hence the dielectric values for solid silicon [91] and liquid silicon [92, 93] (See Table 3.1) are substituted i n their place. It should be noted that no relevant temperature dependent dielectric values for solid and liquid silicon are given in the literature. Therefore, regardless of the temperature of the solid silicon, the dielectric value at room temperature (298 K ) is used, and regardless of the temperature of the liquid silicon, the dielectric value at the melting temperature (1687 K ) is used.  Solid S i (Ti = 298K)  L i q u i d Si (T, = 1 6 8 7 ^ )  e = (13.601, 0.0443)  e = (-21.48, 45.01)  Table 3.1: Dielectric values for solid and liquid silicon for 800 n m light.  The interaction of the probe pulse w i t h the shock wave is best presented by overlaying the electric field strength of the probe laser on the corresponding dielectric functions of the shock wave, see figures 3.17 and 3.18. T h e probe laser is incident from the left while the shock wave moves within the target from right to left. W i t h i n the cold, solid silicon, the electromagnetic wave of the probe light propagates with little attenuation. However, once probe light reaches the shock front, penetration of the electromagnetic wave is very  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  x  59  (fim)  Figure 3.17: Penetration of the electric field strength of the probe laser beyond the shock front. The normalized E for the 800 nm probe light at 20 ps after the peak of the pump pulse is shown. 2  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  100 [_ 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  60  i i. i i i i I i i  80 60 t/3  s o  40  u  c 3  20 0 -20  5  -40  60 i i ii 1 -0.9  •  1  •  -0.8  '  i  -0.7  i  l  i  -0.6  i  i  i  l  i  -0.5  i  i  i  i  i  -0.4  i  i  i  i  i  i  -0.3  x ((im )  Figure 3.18: Dielectric functions at the shock front. T h e real part of e (solid line), and the imaginary part of e (dashed line) of the silicon target for 800 n m light at 20 ps after the peak of the pump pulse is shown.  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  61  short, hence any interaction w i t h the shocked material behind the shock front would be rather weak.  If this is the case, then the interface between the cold, sold silicon and  the shock front of the decaying shock wave may be approximated as a classical Fresnel surface.  Fresnel Reflectivity A p p r o x i m a t i o n In most experimental endeavors, the measurement of a definite or single state is usually desired.  Since the probe light does not "see" very far beyond the shock front, an ap-  proximation may be made where the effect of the shock states in the tail of the decaying shock wave is neglected. To compare the discrepancy between the results given by the E M wave solver on the decaying shock wave and the results for the case of the idealized single shock state the following approximation is used. For each instant i n time the shock profile is simply divided into two regions: (1) a non-conducting, linear medium and (2) a conductor. T h e boundary between these two regions is given by the shock front of the shock wave. Recall that the location of the shock front is defined as the point in which m a x i m u m compression of the material occurs. T h e density and electron temperature of that point are used to obtain the corresponding dense plasma electrical conductivity which, v i a the Drude model, gives a dielectric value. Since the probe laser does not penetrate much beyond the shock front, all material behind is assumed to have the same dielectric value. Meanwhile, all material ahead of the shock front is assumed to have the dielectric value corresponding to that of cold, solid silicon, thereby neglecting the effects due to gradient of the shock front and the melted material ahead it. Under these approximations, for each instant in time the interaction of the probe light w i t h the shock wave is reduced to a Fresnel surface problem that is easily solved using the Fresnel equations. For normal incident light, the Fresnel reflectivity and transmittivity  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  62  [94] are given as  Rp —  1 + /?  (3-1)  1+ P  where P i n*2  n.  Pi = p  2  (3.2)  = 1.  P2«T  T h e respective media complex refractive indices n*, n* and complex dielectric values t\, 2  e are related by 2  n* = n ( l + in) = y/e  (3.3)  where n is the real refractive index and K is the attenuation index or extinction coefficient. Recall that a wavelength of 800 n m was chosen for the probe laser because the absorption at that wavelength i n cold, solid silicon is m i n i m a l . Nevertheless, some attenuation of the probe light w i l l occur as the incident electromagnetic wave travels through the cold material to the shock front, and then as the reflected electromagnetic wave travels back through the cold material to the free surface. For a plane, time harmonic wave, the energy density w of the wave is proportional to the time average of E , so it follows that 2  'to decreases in accordance w i t h the relation [61] -x(r-s)  (3.4)  where x is the absorption coefficient given by  X =  4TT  2u  c  UK  —  A  —UK  (3.5)  0  and A is the probe laser wavelength in the vacuum. A p p l y i n g the above attenuation 0  effect to the calculated Fresnel reflectivity of the shock front, the anticipated reflectivity observed at the free surface would be  R =  R e~ F  2Axx  (3.6)  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  63  where A x is the distance from the free surface to the shock front at any given instant in time as seen in figure 3.19. It should be noted that the calculation of transmittivity using the Fresnel approximation would be invalid since behind the shock front the expanded plasma would absorb almost a l l light propagating beyond it, so only reflectivity results are analyzed. For early times when the shock wave is not fully formed yet, the Fresnel approximation would not be valid, hence those results are not included. After the shock wave is formed, it can be considered a moving mirror whose refractive index changes w i t h time. A s time goes on, the reflectivity of the moving mirror steadily seen at the free surface increases due to two effects. First, as time passes the shock wave moves further into the silicon target so there is less material between the shock front and the free surface which results in less attenuation of the rear probe light. However the increase in the reflectivity is mainly due to the change in the plasma state at the shock front as the shock wave decays. For the dense plasma model used, the electric conductivity at the shock front steadly increases as both the density and electron temperature at the shock front decreases, thus the observed increase in reflectivity. T h e above approximation is used for comparison w i t h the result given by the E M wave solver on the hydrodynamic profiles, see figure 3.20. In the latter case, for each plasma state within the decaying shock wave the corresponding dielectric value is used and the effect due to melted silicon ahead of the shock front is included. The following remarks can be stated about the reflectivity of the decaying shock wave calculated by the E M wave solver. First, at the peak of the pump pulse the boundary between the aluminum pusher layer and the silicon target layer is still intact, thus we see the corresponding reflectivity due to an aluminum-silicon interface. A s the thermal wave and then the compression wave moves from the aluminum into the silicon, the reflectivity drops until the shock wave  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  i i i i i i i i i | i  64  i i i i i i i i |I i i i i i i i i i  0.8  •£  0.6  Pi  0.4  0.2  0  •  0  •  •  •  •  •  1  •  •  1  •  *  1  >  •  •  •  10  •  •  1  •  •  •  •  i  20  i  i  i  i  i  i  i  i  3 0  Time (ps) R (@ free surface)  R  F  (@ shock front)  Figure 3.19: Fresnel reflectivity at the shock front and at the free surface.  40  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  1 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  65  i  i  i  i  i  i  T T  0.8  0.6  0.4  0.2  0 »• • • • • • • • • • • • • •• • • • • • • • • • • • • 0 10 20 30 1  1  1  1  •• • 40  Time (ps)  Figure 3.20: Reflectivity at $ = 5 x 10 W/cm : Fresnel and E M wave solver. The dotted line is the reflectivity calculated using the Fresnel approximation. The solid line is the reflectivity caluated using the E M wave solver where effects due to the thermal foot and interaction of the probe light beyond the shock front are included. u  2  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  66  begins to form. Once the shock wave begins to form and propagates into the silicon, the reflectivity is seen to increase w i t h time as anticipated. It is observed that while the reflectivity given by the E M wave solver is always lower that the reflectivity calculated using the Fresnel approximation, the amount of discrepancy varies w i t h time. Initially, the wave solver reflectivity is lower than the Fresnel reflectivity by a fair amount, as time goes on the wave solver reflectivity begins to approach the Fresnel reflectivity. However, at later times the wave solver reflectivity diverges from the Fresnel reflectivity as expected based on analysis of the spatial profiles. T h e above behaviour of the reflectivity may be better understood by directly examining the probe laser interaction with the decaying shock wave. In other words, the electric field strength of probe laser and the complete dielectric functions of the shock structure is examined. Using the wave solver, the probe light penetration of the probe light beyond the shock is seen to remain fairly constant (~ 80 A ) as a function of time, see figure 3.21. Therefore, the deviation of the wave solver reflectivity from the Fresnel reflectivity is traced to changes i n the dielectric functions of the melted silicon foot ahead of the shock front and the silicon plasma material behind it as time progresses, see figures 3.22 - 3.24. A t early times, right after the shock front is formed, the dielectric function of the thermal foot is rather small i n length so its effect on the reflectivity would be minor. However, behind the shock front the dielectric values of the plasma material differ significantly from that of the shock front. A s the shock wave moves further into the target, the dielectric function behind the shock front is seen to level off somewhat, resulting i n the wave solver reflectivity to move closer to the Fresnel reflectivity. A s time goes on, the thermal foot grows larger and its effect becomes prominent. Also, the relaxation of the shock front is seen i n the late time dielectric functions. These two effects are responsible for the divergence of the Fresnel reflectivity and the E M wave solver reflectivity at late times.  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  67  Figure 3.21: Penetration depth of probe light beyond shock front. The penetration depth di/ is defined as the distance beyond the shock front in which the normalized E of the probe light has fallen to 1/e of the value at the shock front. Note: at 5 ps the shock front has not steepened entirely thus the larger penetration depth. 2  e  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  100  1/5  i  i  i  I  L  i  i  i  I  L  i  i  68  T  1  I  T  1  I  J  I  L  J  I  L  50  a O  c s  fa  0  S-i CJ  0> "a> Q  50  100  1  J  3000  J  -2800  -2600  -2400 x  2200  2000  (A)  Figure 3.22: Dielectric functions at shock front (at 10 ps). The solid line is the real part of t and the dashed line is the imaginary part of e  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  Figure 3.23: Dielectric functions at shock front (at 20 ps). The solid line is the real of e and the dashed line is the imaginary part of e  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  70  Figure 3.24: Dielectric functions at shock front (at 40 ps). The solid line is the real part of e and the dashed line is the imaginary part of e  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  71  Phase Shift W h i l e the above discussion dealt only w i t h the measurement of the magnitude of the reflected probe laser light, the following deals w i t h the phase shift $ of the reflected probe laser light. T h e phase shift is defined as  $ = $• - $ where  and $  r  (3.7)  r  are the incident and reflected phase of the probe light. T h e change  in the phase of the probe light can be attributed to the following effects: (1) changes in the optical properties of the reflecting surface, (2) changes i n the refractive index of the material between the reflecting surface and the observer, and (3) the motion of the reflective surface [95]. In order to analyze the phase shift results, the following approximation is made. The decaying shock wave is treated once again as a moving mirror where the associated motion of the shock front results i n Doppler phase shifting which dominates the other two effects.  If this is the case, then the phase shift of the reflected probe light would  provide a measurement of the shock speed Us of the decaying shock wave. For the simple case of a mirror moving, its speed is calculated as  Us =  A  A$ —-— Ann At  ,  (3.8)  0  v  N  ;  where n is the refractive index of the medium (cold silicon) which the mirror is moving through, Ao is the probe light wavelength i n the vacuum (800 nm), and A $ is the change in the phase shift i n a given At time interval. T h e time history of the phase shift of the probe laser light is presented i n figure 3.25. Initially, the phase shift remains constant as the probe laser reflects off only the aluminum-silicon interface. T h e phase is observed to change drastically once the thermal wave reaches the silicon layer and continues to increase as the shock wave propagates through the silicon. T h e phase shift is seen to  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  0  10  20  Time (ps)  Figure 3.25: Phase shift.  30  72  40  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  25  I  i or 5  i  i  I  i  N  i  i  | i  i  10  i  i  i  i i | i  i  i  i  15  i  i i | i  i  i  i  i  i  20  i  i  i  i  i  i  i  | i  i  i  25  i  i  i  i  i  i  73  | i i i  i  30  i  i  i  i  i  i  35  i  i  11  40  Time (ps)  Figure 3.26: Shock speed of decaying shock wave. The solid line is the calculated shock speed using the hydrodynamic profiles, and the dashed line is the calculated shock speed using the phase shift of probe laser.  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  74  slowly curve over indicating the decrease i n shock speed as expected for the case of a decaying shock wave. The shock speed is calculated using equation 3.8 and compared w i t h the shock speed calculated directly using the hydrodynamic profiles, see figure 3.26. It is seen that the phase shift calculated shock speeds are i n relatively close agreement w i t h that of actual shock speeds. In shock wave experiments, measurement of the shock speed along w i t h measurement of the particle speed are used i n conjunction to obtain an E O S measurement.  However, as stated earlier the decaying shock wave initially lies well off the  Hugoniot and asymtotically approaches it. Hence for femtosecond-laser driven shocks, measuring the shock speed would not allow any direct quantitative access to the actual state of the shock front, rather it may be used as a qualitative check on the steepness of the shock front, thus allowing the use of the moving mirror or Fresnel surface approximation. T h e state of the shock front would still have to be attained through computer simulations based on the experiment parameters and plasma models.  Electrical C o n d u c t i v i t y The proposed scheme is supposed to be applicable w i t h i n the pump pulse laser irradiances of lO W/cm? u  to 10 W/cm . lo  2  T h e wave solver results for the two irradiance limits are  shown along w i t h the standard case, see figure 3.27. T h e different irradiances produce shock waves of corresponding magnitudes and shock speeds as seen in the different times in initial drop in reflectivities signifying the their emergence into the silicon. However, once the shock waves are formed, not only do the observed reflectivities follow the same rate of increase, but the magnitudes of the reflectivities have very similar values. This does not necessarily mean that the proposed scheme lacks the capability to distinguish between the obvious different plasma states which would be produced by the various irradiances. Rather it has to do w i t h the fact that in this regime the dense  75  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  0  10  20  30  40  Time (ps) *  10  14  W/cm  2  —^5xl0  1  4  W/cm  2  ~ *~ " 1 0  Figure 3.27: Similar reflectivity at different irradiances.  15  W/cm  2  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  Figure 3.28: Similar cr given by the Lee and More model at different irradiances. 0  76  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  77  plasma model used give electrical conductivities which are very similar, see figure 3.28, thus producing the similar reflectivities seen. Given that only one electrical conductivity model for silicon was available, further discussion about the optical probing results is somewhat l i m i t e d . However, as a final remark, the sensitivity of the electrical conductivity on the magnitude of the reflected probe light is discussed. To simplify matters, the Fresnel reflectivity is used to examine the change i n reflectivity based on the change i n electrical conductivity at the shock front only. Suppose that, at the shock front, the plasma state remains the same but the actual associated electrical conductivity is a factor of 2 higher or lower than that predicted by the model. T h e effect on the reflectivity would be rather significant (~ 10%) and would be quite measurable, as seen i n figure 3.29. If another electrical conductivity model is provided then the proposed scheme could be used to provide a method for distinguishing between models.  Chapter 3. Discussion of Femtosecond-Laser Driven Shock Results  78  Figure 3.29: Sensitivity of electrical conductivity at the shock front on the Fresnel reflectivity at the free surface. ($ = 5 x 10 W/cm ) The solid line represents the case when electrical conductivity oo given by the Lee and More model is used, the dotted line represents the case when 2 x uo is used, the dashed line represents the case when ao ^ 2 is used. 14  2  Chapter 4  Conclusions  4.1  Summary  Using numerical simulations, the characteristics of a shock wave generated by an intense femtosecond laser were investigated. A new approach was developed to study the electrical conductivity of silicon in the dense plasma state. The unique experimental scheme involved the use of a 400 nm, 120 fs pump laser pulse as the shock driver and a 800 nm, 120 fs probe laser pulse as the probe. T h e new understanding of the crucial characteristics of femtosecond-laser driven shocks points to an existing controversy in the literature [24] in which the effects of the thermal wave were not addressed and the resulting off-Hugoniot states were not recognized. It was shown that the shock states of shock wave generated by an intense femto-second laser cannot be described by the use of the principle Hugoniot curve as in previous long-pulse laser experiments. Spatial profiles of the thermodynamic variables at various times showed the resulting shock wave to be quite transient.  In addition, the temporal histories of states at the  shock front revealed that although the pressure, electron temperature and ion temperature decayed quite rapidly, the mass density at the shock was shown to decrease at a considerably slower rate. W i t h this information, we suggest that the transient nature of the femtosecond-laser driven shock can be used advantageously. Namely, a temperature scan of the electrical conductivity of dense plasma at a relatively steady density can be  79  Chapter 4. Conclusions  80  performed by optically probing the decaying shock wave as it propagated inside silicon. T h e concept of measuring definite shock states as a function of time was addressed using a Fresnel surface approximation. W i t h i n a relatively long time window the approximation appears to be reasonably valid and state of the shock front may be discriminated from the other shock states of the decaying shock wave. T h e results of this thesis has recently been presented at the International Workshop on W a r m Dense Matter [96] and the I E E E International Conference on P l a s m a Science [97].  4.2  Future W o r k  Another electrical conductivity model based on density functional theory ( D F T ) [98, 99] has been constructed by Perrot and Dharma-wardana [76]. In this model all the ingredients needed for the calculation are computed from first principle in a self-consistent manner, using a density functional description of the electrons and ions in the plasma. For aluminum [27], in some regimes the electrical conductivities given by the D F T model have been shown to differ noticably from the results given by the Lee and More model. Acquisition of electrical conductivities calculated using the D F T model for silicon is being pursued. Using the new electrical conductivities and employing the proposed scheme, reflectivity calculations will be performed. These results may then be compared w i t h the results given in this thesis, thereby allowing for a comparison between the two conductivity models. For this work, optical probing of the decaying shock wave was limited to a normally incident probe pulse on the rear surface of the target.  A s a result, only forecasting  calculations were possible, that is, the predicted reflectivity of the probe is compared with the measured value. T h i s allows the inference of electrical conductivity through  Chapter 4. Conclusions  81  the numerical simulation. 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