Optical Properties of Ultrafast Laser Heated Solid by Tommy Ao B.Sc. (Eng.), University of Alberta, 1998 M . S c , University of British Columbia, 2000 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F THE REQUIREMENTS FOR THE D E G R E E OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Department of Physics and Astronomy) We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F BRITISH C O L U M B I A June 19, 2004 © Tommy Ao, 2004 Abstract ii Abstract The regime of Warm Dense Matter (WDM) has emerged as an interdisciplinary field which has drawn broad interest from researchers in plasma physics, condensed matter physics, high pressure science, astrophysics, inertial confinement fusion, as well as material science under extreme conditions. Warm dense matter represents complex states at the convergence of condensed matter physics and plasma physics where neither conventional theoretical descriptions are valid. However, single-state experimental data for the direct testing of new theoretical models within this regime has been difficult to come by. To examine the W D M state, the optical properties of ultrafast laser heated solids were studied. Experiments were performed utilizing a femtosecond laser pump-probe technique to create and examine single-states of W D M . The isochoric heating of freestanding, ultrathin (30 nm), gold foils by a femtosecond pump laser produced uniform, solid-density states of energy densities from 0.25 to 20 M J / k g . The A C conductivity of such states was determined from reflectivity and transmission measurements of a femtosecond probe as a direct benchmark for transport theory. In addition, observation of the time history of the probe reflectivity and transmission led to the discovery of a quasi-steady-state behavior of the heated sample that suggests the existence of a metastable, disordered phase prior to the disassembly of the solid. To further examine the dynamics of ultrafast laser heated solids, Frequency Domain Interferometry was used to provide an independent observation. Contents iii Contents Abstract ii Contents iii List of Tables vi List of Figures vii Acknowledgements xii 1 2 Introduction 1 1.1 Warm Dense Matter 2 1.2 Electrical Conductivity 4 1.3 The Need for Single-state Measurements 6 1.4 Present Work 12 Electrical Conductivity Theoretical Models 13 2.1 Free electron model . . 13 2.1.1 Free electron gas 14 2.1.2 Electrical conductivity and Ohm's law 15 2.1.3 Frequency response of electrical conductivity 18 2.1.4 Electrical conductivity and optical properties 19 2.2 Statistical conductivity models 20 2.2.1 Collision operator method 20 2.2.2 Boltzmann transport equation 23 Contents 2.3 3 4 26 2.3.1 Ziman theory 26 2.3.2 Kubo-Greenwood formula 33 35 3.1 The concept of Idealized Slab Plasma 35 3.2 Isochoric heating of a freestanding, ultrathin foil by a femtosecond laser 37 3.2.1 State characterization 37 3.2.2 Obtaining A C conductivity 38 Description of the Experiment 46 4.1 Experimental arrangement 46 4.1.1 The pump laser 49 4.1.2 The probe laser 50 The freestanding, ultrathin target 55 Pump-probe Reflectivity and Transmission Measurements . 63 5.1 5.2 6 Quantum mechanical conductivity models Idealized Slab Plasma 4.2 5 iv S-polarized light results 67 5.1.1 Probe reflectivity & transmission 67 5.1.2 A C conductivity results 77 P-polarized light results 83 5.2.1 Probe reflectivity & transmission 83 5.2.2 A C conductivity results 83 5.3 S- & P-polarized light data comparison 91 5.4 Quasi-steady-state 94 Frequency Domain Interferometry Measurements 100 6.1 Theory of operation 100 6.2 Fourier Transform analysis method 105 6.3 F D I Experiment 109 6.3.1 109 Phase shift results Contents 6.3.2 7 Disassembly of the liltrafast laser heated solid v 125 Significance and Impact 138 7.1 Summary 138 7.2 Future work 140 Bibliography 142 A Glossary of abbreviations 149 List of Tables vi List of Tables 5.1 Electron collision time, D C conductivity, and electron density values for gold at normal condition 6.1 80 Summary of electron-phonon coupling constants measured for Au 132 List of Figures vn List of Figures 1.1 Sesame EOS for copper 3 1.2 Warm Dense Matter 5 1.3 Divergence in electrical conductivity models 7 1.4 Milchberg experiment 9 1.5 Milchberg experiment resistivity comparison with models. . . . 10 1.6 Comparsion of simulations to Milchberg experiment 11 2.1 Fermi sphere of an electron gas 17 3.1 1/e deposition length in gold 36 3.2 Coordinate of electromagnetic wavesolver 41 4.1 Illustration of the femtosecond pump-probe experiment 47 4.2 Detailed schematic diagram of femtosecond pump-probe experiment 4.3 Frequency resolved optical gating (FROG) image of 800 nm (luo) laser pulse 4.4 53 Spatial profile of the pump pulse in the focal plane of the off-axis parabola 4.7 52 Measured temporal intensity profile of the lco and calculated 2co pulses 4.6 51 Temporal profile of the pump pulse extracted from the F R O G image 4.5 48 54 Front view of a freestanding ultrathin A u foil mounted on the target holder 57 List of Figures 4.8 viii Side view schematic of the target holder of the freestanding ultrathin A u foil 4.9 58 Calibration of the thickness of the freestanding ultranthin A u foil 60 4.10 Interferogram of A/20 mirror and freestanding A u foil 61 4.11 Extracted variation in height for A/20 mirror and freestanding A u foil 62 5.1 In-situ calibration of probe reflectivity and transmission. ... 5.2 Example of reflected probe light image and slice 65 5.3 Example of reflected probe light image and slice 66 5.4 S-polarized light reflectivity & transmission at the energy density of (4.0 ± 1.8) x 10 J/kg 69 5 5.5 S-polarized light reflectivity & transmission at the density of (1.5 ± 0 . 3 ) x 10 J/kg 70 6 5.6 S-polarized light reflectivity & transmission at the energy density of (1.8 ± 0.4) x 10 J/kg 71 6 5.7 S-polarized light reflectivity & transmission at the energy density of (3.5 ± 1.0) x 10 J / k g 72 6 5.8 S-polarized light reflectivity & transmission at the energy density of (4.2 ± 1.0) x 10 J/kg 73 6 5.9 64 S-polarized light reflectivity & transmission at the energy density of (1.0 ± 0.2) x 10 J/kg 74 7 5.10 S-polarized light reflectivity & transmission at the energy density of (1.2 ± 0.2) x 10 J / k g 75 7 5.11 S-polarized light reflectivity & transmission at the energy density of (2.5 ± 0.5) x 10 J/kg 76 7 5.12 S-polarized A C conductivity results at different energy densities. 78 5.13 A C conductivity results within the quasi-steady-state phase. . 79 5.14 S-polarized results for values of collision time r, D C conductivity cr , and electron density Tt CIS £1 function of energy density. 81 0 e List of Figures ix 5.15 S-polarized results compared with calculations from Rinker conductivity for values of collision time r , D C conductivity OQ , and electron density n as a function of energy density. . . 82 e 5.16 P-polarized light reflectivity & transmission at the energy density of (3.5 ± 1.8) x 10 J/kg 84 5 5.17 P-polarized light reflectivity & transmission at the energy density of (1.4 ± 0.3) x 10 J/kg 85 6 5.18 P-polarized light reflectivity & transmission at the energy density of (1.6 ± 0.4) x 10 J / k g 86 6 5.19 P-polarized light reflectivity & transmission at the energy density of (3.5 ± 1.0) x 10 J/kg 87 6 5.20 P-polarized light reflectivity & transmission at the energy density of (2.5 ± 0.5) x 10 J/kg 88 7 5.21 P-polarized A C conductivity results at different energy densities. 92 5.22 P-polarized results for values of collision time r, D C conductivity t7 , a n 0 d electron density function of energy density. 93 5.23 S- & P-polarized A C conductivity results at different energy densities 95 5.24 S- & P-polarized results for values of collision time r, D C conductivity ao, and electron density clS ci function of energy density 96 5.25 Time duration of quasi-steady-state 97 6.1 Schematic diagram of frequency domain interferometry 102 6.2 Frequency domain interferometry modes of operation 104 6.3 Example of F D I interferogram 108 6.4 Frequency domain interferometry schematic 110 6.5 Example of measured F D I phase shift Ill 6.6 S-polarized light F D I phase shift at the energy density of (4.5± 1.2) x 10 J / k g 5 113 List of Figures 6.7 S-polarized light F D I phase shift at the energy density of (1.0± 0.2) x 10 J / k g 114 6 6.8 S-polarized light F D I phase shift at the energy density of (1.6± 0.3) x 10 J / k g 115 6 6.9 x S-polarized light F D I phase shift at the energy density of (3.8± 0.8) x 10 J / k g 116 6 6.10 S-polarized light F D I phase shift at the energy density of (7.2± 1.6) x 10 J / k g 117 6 6.11 S-polarized light F D I phase shift at the energy density of (1.0± 0.2) x 10 J / k g 118 7 6.12 S-polarized light F D I phase shift at the energy density of (1.3± 0.4) x 10 J / k g 119 7 6.13 P-polarized light F D I phase shift at the energy density of (5.3 ± 0 . 8 ) x 10 J / k g 5 120 6.14 P-polarized light F D I phase shift at the energy density of (1.2 ± 0 . 2 ) x 10 J / k g 6 121 6.15 P-polarized light F D I phase shift at the energy density of (4.0 ± 0 . 7 ) x 10 J / k g 6 122 6.16 P-polarized light F D I phase shift at the energy density of (1.0 ± 0 . 2 ) x 10 J / k g 7 6.17 Phase shift due to motion and density change 123 124 6.18 Comparision of measured and calcualted S-polarized phase shift. 126 6.19 Comparision of measured and calcualted P-polarized phase shift. 127 6.20 Quasi-steady-state behavior observed in 6 diagnostics 128 6.21 Electronic heat capacity 131 6.22 Duration of quasi-steady-state TQSS from S-polarized light phase shift measurements 134 6.23 Temporal history of electron temperature and ion energy density. 136 List of Figures xi 6.24 Comparison of the quasi-steady-state duration TQSS between measured values from S-polarized light phase shift results and 2T model calculations for g = (2.2 ± 0.3) x 10 16 W/m -K. 3 . . 137 Acknowledgements xii Acknowledgements I would like to thank my research supervisor, Dr. Andrew Ng, for his invaluable guidance and support throughout my graduate studies. This work was completed mainly due to his vast contributions and tireless efforts. I also would like thank my supervisory committee, Dr. Paul Hickson, Dr. William Hsieh, and Dr. Janis McKenna for their comments on this thesis. As well, I am greatly indebted to Dr. Klaus Widmann for all of his efforts in implementing and performing the experiments, and interpreting the results. The experiments described here were conducted at facilities which were expertly maintained by Dwight Price, and with targets skillfully produced by A l Ellis. Also, the many useful contributions of Dr. Yuan Ping, Edward Lee, and Heywood Tarn in both performing the experiments and analyzing the data are much appreciated. To my family and friends, I would like to thank them all for their support and encouragement. Finally, I wish to thank Soo Jin for all of her patience and love which made this work possible. The financial assistance of the Natural Sciences and Engineering Research Council is appreciated. Chapter 1. Introduction 1 Chapter 1 Introduction High Energy Density (HED) physics refers to the study of states with energy densities exceeding 10 J/m , which exists in the cores of planets and stars. Such concentrated amounts of energy correspond to states under extreme pressures. For example, a milligram of hydrogen at a temperature of 1 keV confined within a volume of 1 cm is at 1 Mbar of pressure (1 bar — 10 Pa). Within the HED field, the inherently complex states of Warm Dense Matter (WDM) are of particular interest. Warm dense matter consists of states with strong ion-ion coupling, partial ionization, and electron degenerancy, thus making it a difficult regime to describe. In general, WDM states are characterized by comparable thermal and Fermi energies (kT ~ Ep). States with densities ranging from 0.1 to 10-fold solid density and temperatures from 1 to 100 eV have been the main interest of WDM research. 11 3 3 5 While most studies have been focused on equilibrium behavior, thefieldof Non-Equilibrium High Energy Density (NEHED) physics is becoming a new frontier. Non-equilibrium physics is key to the understanding of dynamic behavior under extreme conditions. Examples of such phenomena include ultrafast heating of solids [1, 2], intense shock loading of materials [3], implosion of Inertial Confinement Fusion capsules [4], and plasma formation in the wire array in Z machines [5]. The recent studies on electron thermalization [6], electron-phonon coupling [7, 8, 9], solid-liquid phase transition [10], and electrical conductivity [11, 12] all attempt to examine the excitation and relaxation processes of the non-equilibrium system. Understanding the approach to equilibrium also holds the key to control and optimization of the dynamic behaviors. Chapter 1. Introduction 2 A powerful means to examine the N E H E D physics in the W D M regime is ultrafast laser heating of a solid. Intense femtosecond lasers are readily available to produce the N E H E D states. In addition, the time scales for excitation and relaxation are distinctly different, the latter being considerably longer than the former. Therefore, the associated processes may be resolved from the ultrafast optical measurements of a femtosecond laser pump-probe scheme. 1.1 W a r m Dense M a t t e r Warm dense matter represents the convergence between plasma and condensed matter physics. The theoretical work to describe W D M has been quite challenging. When viewed as a strongly coupled plasma dominated by ion-ion correlation, it cannot be treated with the conventional Debye screening [13] and perturbative approaches. Meanwhile, when viewed as a high-temperature condensed matter, it is a disordered system whose description requires detailed knowledge of excited states, structure factors and the dynamics of strongly interacting electrons and ions. This can best be appreciated from the widely used Sesame tabulated equation of state, a "tour de force" of theoretical effort to provide global physical data over a range of densities and temperatures [14]. In spite of the elaborate combination of models ranging from semi-empirical to first-principle, the W D M regime is described only by interpolations between models, as seen in Fig. 1.1. In plasma physics, the W D M states are described as strongly coupled plasmas. A strongly coupled plasma is one in which the potential energy arising from the Coulomb interaction between the particles is greater than the kinetic energy of the particles. Normally, the ion-ion coupling and the electron-electron coupling are considered separately. In the W D M regime, the ions are generally strongly coupled while the electrons are not. To determine whether or not the ions are strongly coupled, an ion-ion coupling parameter Chapter 1. Introduction 3 Warm Dense Matter I ti" 5 Activity Expansion. 10 2 m CO oT0° SahaBoiteniaJn DL m MC Fits 10- 10 3 • i 1G 1 i i i 10 1 Density (g/cm ) 10 3 3 Figure 1.1: Example of the gaps within the Sesame tabulated equation of state occuring within the Warm Dense Matter regime (box), shown is the Sesame equation of state for copper. 4 Chapter 1. Introduction Tu is calculated, where the kinetic energy is taken to be the thermal energy of the ions. The Coulomb energy may be estimated as Ze 2 Ec = 2 — , (1.2) TO with ro, the effective ion separation given by r„ = (-f) /47mA , . (1.3) - 1 / 3 where Z is the average ionization, e is the electronic charge, and n; is the ion density. Strongly coupled plasmas are defined as having > 1. Experimental studies in W D M have been challenging as well because of the associated high energy densities and extreme pressures, as shown in Fig. 1.2. Numerous pressure loading (static or dynamic) techniques have been developed to create W D M states. The most advanced static technique involves the use of diamond anvil cells [15] to slowly compress a material. Dynamic techniques involve the generation of a rapid compression (shock) wave in the material to be studied. Examples of such techniques include explosive shock tubes [16, 17, 18], flyer-plates [19, 20], two-stage light-gas guns [3], laser-driven shocks [21, 22], and nuclear explosions [23]. Alternatively, if energy can be rapidly deposited into a material and at the same time measurements be made on the created W D M state, the issue of constraining the system is avoided. This of course is the goal of ultrafast laser heating of a solid. The central focus of this thesis is to use this approach to study the properties of the W D M state. 1.2 Electrical Conductivity The optical properties of a material describe its electronic response to electromagnetic radiation, thus they provide an important tool for understanding Chapter 1. Introduction 5 Figure 1.2: Example of the Warm Dense Matter regime (shaded area) for aluminum (QEOS) plotted in the density-temperature plane (po = 2.7 g/cm ). Also, shown are the corresponding isobars 3 (dashed lines), iso-I^ (dotted lines), and the states where the thermal energy equals the Fermi energy (solid line). Chapter 1. Introduction 6 its transport physics. In particular, electrical conductivity is of longstanding interest. For instance, A C conductivity is needed for accurate simulation of laser-matter interaction [24, 25] and high energy density pulse power phenomena [5, 26, 27]. While D C conductivity serves as an important surrogate measure of electron thermal conductivity through the Weidemann-Franz relation [28]. Furthermore, there is the basic need to understand charged particle transport in the complex regime where excited states, partial ionization, electron degeneracy and ion-ion coupling are all important. Several theoretical efforts, which will be discussed in a later chapter, have been developed to describe electrical conductivity within the W D M regime. A common aspect of these conductivity models is the appearance of a minimum in the D C response as the temperature is increased. Generally, this is viewed as the transition from solid-state (electron-phonon scattering) to ideal plasma behavior (electron-ion collision). At lower temperatures, the interaction of electrons with ion acoustic waves dominant and is shown to increase proportionally with temperature [29]. At higher temperatures, the electron-ion collisions are more prominent and are shown to decrease with temperature [30]. There remains significant divergence among these models, as seen in the comparison between two widely used models, a semianalytical model by Lee & More [31] and a partial-wave analysis by Rinker [32], as shown in Fig. 1.3. In order to benchmark theoretical models, precise experimental data are needed. 1.3 The Need for Single-state Measurements Experiments in W D M have usually been made on systems which are inherently non-uniform and consist of multiple states. Unfortunately, such integrated measurements can lead to misinterpretation of data. The first experiment to test recent electrical conductivity models in the strongly coupled plasma regime was the measurement of the optical reflectivity of shock Chapter 1. Introduction 7 Figure 1.3: Electrical conductivity versus temperature plot showing the divergence in the Lee & More (LM) [31] and Rinker [32] models for aluminum at solid density. Chapter 1. Introduction 8 released states of aluminum by Ng et al. [33]. These results corroborated predictions derived from simulations using the conductivity model developed by Lee & More [31] and a Sesame equation of state [14]. Milchberg et al. [11] later attempted to relate self-reflectivity measurements of femtosecond laser heated aluminum to the electrical conductivity of strongly coupled plasmas, see Fig. 1.4. They claimed to deduced values of D C resistivity (or inverse conductivity) of solid density aluminum as a function of electron temperature. The results were derived from the solution of the Helmholtz equation by assuming an arbitrary constant electron density gradient and uniform electron temperature. The latter was estimated from adiabatic sound speed based on plasma expansion measurements. As shown in Fig. 1.5, the data appear to deviate substantially from the models developed by Lee & More [31], and Rinker [32], as well as those based on the treatment of collision operator by Cauble & Rozmus [34] and density functional theory by Perrot & Dharma-wardana [35]. However, when time-dependent gradient effects in the heated plasma are taken into account [12], the data of Milchberg et al. were found to be in reasonable agreement with simulation results when either the Lee &; More or Perrot &: Dharma-wardana conductivity model was used in conjunction with a Sesame equation of state, as shown in Fig. 1.6. This clearly underlines the intrinsic difficulty of testing theory with integrated measurements on non-uniform systems consisting of multiple states. Therefore, a change in paradigm is needed in benchmarking theory. The direct comparison of theory should be with "single-state" data. In addition, the created state should be characterized by direct measurements. A new approach has been formulated by Forsman et al. [36] to realize single-state measurements. The approach is based on the creation of an Idealized Slab Plasma (ISP). A n ISP is planar plasma that can be considered as a single, uniform state in which all residual non-uniformities have insignificant impact on the measurement of its uniform properties. The proposed Chapter 1. Introduction 9 100 80 ^ 60 > 1 40 * 20 0 UlL 10 I I I 11 id 10 i 12 mil • • • • • ••! 10 13 10 14 ll 10 15 Intensity (W/cm ) Figure 1.4: Results from Milchberg et al. experiment. Using a 308 nm, 400 fs pulse, the self-reflectivity of S- & P-polarized light from an aluminum-vacuum interface was measured as a function of laser intensity. Chapter 1. Introduction 10 Figure 1.5: Comparison of D C resistivity of aluminum found from the Milchberg et al. data with theoretical calculations, as a function of electron temperature. The lines represent the different conductivity models due to Lee & More (LM), Perrot & Dharmawardana (PDW), Rinker, and Cauble & Rozmus (CR). The shaded area represents the derived values for the Milchberg et al. data using a constant electron density gradient and uniform electron temperature. Chapter 1. Introduction 11 Figure 1.6: Comparison of simulation results (lines) to the experiment of Milchberg et al. (shaded areas). Calculations using Lee & More's conductivity (solid lines), Perrot & Dharma-wardana's conductivity (dotted lines), and Rinker's conductivity (dashed lines) have time dependent gradient effects taken into account. Upper and lower sets of curves correspond to S- & P-polarizations, respectively. Chapter 1. Introduction 12 technique is based on the ultrafast, isochoric heating of a solid sample. The created W D M state is characterized directly by its initial mass density and the increase in energy density of the heated solid. 1.4 Present Work In this work, we present the first realization of a single-state measurement of W D M using the novel ISP approach. Single-states of warm dense gold were produced via the isochoric heating of freestanding, ultrathin gold foils with a femtosecond pump laser. The measurement of the reflectivity and transmission of a femtosecond probe laser allowed for the determination of the A C conductivity of single-state W D M . In addition, the time scale associated with the disassembly of the ultrafast laser heated solid was revealed. This work will provide the validation of the ISP approach, and will allow for further development of the technique for investigating other properties of W D M . In Chapter 2, a background on the various theoretical electrical conductivity models that have been developed to describe the transport properties of W D M is given. A detailed description of the ISP approach is presented Chapter 3. In Chapter 4, the experimental scheme used to implement the ISP technique is described. The A C conductivity obtained from the pump-probe reflectivity and transmission experiments is examined in Chapter 5. Corroboration of the ISP reflectivity and transmission results using Frequency Domain Interferometry (FDI) is presented in Chapter 6. A summary of the thesis and future developments is provided in Chapter 7. A glossary of the abbreviations used in this thesis is given in Appendix A . Chapter 2. Electrical Conductivity Theoretical Models 13 Chapter 2 Electrical Conductivity Theoretical Models Currently, a truly complete, consistent theory of the charged particle transport properties in the W D M regime has yet to be established. This chapter will provide historical background of electron transport theory and the current theoretical models being developed. 2.1 Free electron model The physical properties of metals, particularly simple metals, are usually described in terms of the free electron model [29]. In this model, the most weakly bound electrons of the constituent atoms move about freely throughout the volume of the metal. Thus, the valence electrons of the atoms become the conduction electrons within the sample. In the free electron approximation, forces between the conduction electrons and the ion cores are neglected so all calculations proceed as if the conduction electrons were free to move everywhere within the specimen. Prior to the development of quantum mechanics, the interpretation of metallic properties in terms of the motion of free electrons had already been established in classical theory. Some notable successes include the derivation of the form of Ohm's law and the relation between the electrical and thermal conductivity (Weidemann-Franz law) [28]. However, the classical theory completely fails to explain the heat capacity and the paramagnetic susceptibility of the conduction electrons, as well as the occurrence of long Chapter 2. Electrical Conductivity Theoretical Models 14 electronic mean free path within solid state metals. Therefore, more complete description of a free electron gas must include quantum mechanical effects. 2.1.1 Free electron gas In quantum theory, a free electron gas is described in terms of a free particle Schrodinger equation in three dimensions, which is written as h 2 ( d d 2 d \ 2 2 -^(^ v 5Fr + + k ( r ) = £ k * (2 - 1) where m is the electron mass, h is Planck's constant divided by 2TT, ip^ is the electron's wavefunction and is the energy of the eigenstate. If electrons are confined to a cube of edge L, the wavefunction is given as / 7tn x\ i 7vn y\ (Tin z\ / x v z where n , n , n are positive integers, and ib^ is a standing wave. x y z Wavefunctions satistfying the free particle Schrodinger equation and periodic boundary conditions are of the form of traveling plane waves, •0k = exp(zk • r) (2.3) provided that the components of the wavevector k satisfy the contraints k = 0, x ±j-; ±j-; ... and similarly for k and k . The components of k are of the form y z (2.4) 2nir/L, where n is a positive or negative integer, and are the quantum numbers of the state. The energy eigenvalue of the state with wavevector k is given as e k = ^ =^ + * J + ^ < 2 - 5 ) In the ground state of a system of N free electons, the occupied states may be represented as points inside a sphere in k space. The energy at the Chapter 2. Electrical Conductivity Theoretical Models surface of the sphere is the Fermi energy. 15 The wavevectors at the Fermi surface have a magnitude kp such that s F = |-*». (2.6) The total number of allowed states within the sphere of volume 4irkp/S is 47T/4/3 _ V 3 _ , N 2 7 ) The number of states equal to iV is set to the number of electrons, so the Fermi energy is related to the electron concentration by e = ~^N/Vyi\ F (2.8) Im The electron velocity at the Fermi surface is given as m m The number of orbitals per unit energy range or density of states for free electrons is . . D{E) 2.1.2 cLN V /2m\ / 3 2 = n = w? Kin 1 / 2 (210) 6 E l e c t r i c a l c o n d u c t i v i t y a n d Ohm's l a w The basic kinetics of electron transport may be understood as follows. The momentum of a free electron is related to the wavevector by p = mv = hk. In an electric field E and magnetic field B, the force F on an electron of charge —e is given by the Lorentz force, so that Newton's second law of motion becomes „ civ ^ dk /,_ 1 ^\ F = m— = h— = - e I E + - v x B ) . dt dt \ c J l n v _ „. 2.11 ' The Fermi sphere encloses the occupied electron orbital k in the ground state of the electron gas. The net momentum is zero since for every orbital k there Chapter 2. Electrical Conductivity Theoretical Models 16 is an orbital at —k. In the absence of collisions and with no magnetic field (B = 0), the Fermi sphere in k-space is displaced uniformly by an applied electric field, as shown in Fig. 2.1. If the field is applied at time t = 0 to an electron gas which fills the Fermi sphere centered at the origin of k-space, then at later time t, the sphere will be displaced to a new center at eE <5k = k(t) - k ( 0 ) = - — t. (2.12) Due to collisions of the electrons with impurities, lattice imperfections, and phonons within the metal, the displaced sphere may be maintained in a steady state in an electric field. When the applied field is turned off, the collision processes tend to return the system to the ground state. If the collision time is r, the displacement of the Fermi sphere would be given as eE c5k = - ^ r , (2.13) and the incremental velocity or drift velocity is v = -—r. m (2.14) In a constant electric field E , in which there are n electrons of charge q = —e per unit volume, the electric current density is given as ne Er 2 i = nqv = m • . . 2.15 This is in the form of Ohm's law which shows the current density j to be proportional to the electric field E . The electrical conductivity o is defined by j = crE, and from Eq. 2.15 is given as cr = . (2.16) m The electrical resistivity p is defined as the reciprocal of the conductivity, so it is given as P=^T- (2-17) Chapter 2. Electrical Conductivity Theoretical Models 17 Figure 2.1: Displacement of whole Fermi sphere of an electron gas due to the electric field E . Chapter 2. Electrical Conductivity Theoretical Models 2.1.3 18 Frequency response of electrical conductivity The displacement of the Fermi sphere by an applied force F may be written in a general form as ftg i)*k=F, + (2.18) where r is the collision time or relaxation time. In the steady state k is constant, so Eq. 2.18 reduces to k = £ . (2.19) In the absence of collisons (r —>• oo) we recover Eq. 2.12. In terms of the incremental drift velocity or for the drift velocity of the gas v, the equation of motion is given as m f y + i ) v = F. \dt TJ (2.20) Now, suppose the force acting on an electron is an alternating electric field, F(t) = -eEex.p(-iut). (2.21) Then, a solution of Eq.(2.20) is of the form v(t) = v e x p ( - i w t ) , (2.22) so we now have m [ —iijj + (-iv or rewriting, + ^ v = v = -eE, (2.23) E. (2.24) 1er/m — iuir The current density is then written as j = nqv = ^—E. 2.25) 1 — IU1T Therefore, the frequency dependent or A C electrical conductivity is given as . . nov ne r/m a u; = 4 " = 1 — = 2 E 1 — 10JT • 1 + iuir , , 2 1 + (U)T) Z 2.26 Chapter 2. Electrical Conductivity Theoretical Models 19 where cr is the D C conductivity, shown in Eq. 2.16. Equation 2.26 is known 0 as the "Drude formula" in respect to the pioneering work done in electron theory of the transport properties of metals by Drude [37]. In the simple Drude formulation, a constant relaxation time r is assumed which is taken to be the static collision time r(0). 2.1.4 Electrical conductivity and optical properties It is sometimes convenient to express the above result in terms of a complex dielectric rather than as a complex conductivity. Examining the spatial displacement r, then the polarization or dipole moment/volume is -net. (2.27) Now Eq. 2.20 may be rewritten as m U ;s =- ' + r eE - (2 28) or equivalently = -eE. (2.29) , E = ^ E . UJ + iuj/r to Now, the dielectric function is defined as (2.30) m [ -LO -i-)r 2 Thus, the polarization is P = - f , m e = l + 47r^, 7 (2.31) E' so we have the following form of e(uj) = l + 4 7 r i ^ . (2.32) UJ The complex dielectric function is related to a material's optical constants (for a non-magnetic materal, \x — 1) as ep, = 2 n ~\2 = (n + ik) (2.33) Chapter 2. Electrical Conductivity Theoretical Models 20 where h is the index of refraction and k is the extinction coefficient. From the above relations, it's clear that the electron transport behavior manifests itself in the material's optical properties. Now, a brief overview of the theoretical conductivity models developed to describe the transport physics within the W D M regime is given. 2.2 Statistical conductivity models In conventional plasma physics, the Spitzer-Braginskii [30, 38] formulas are widely used to predict the conductivity. However, these formulas are rigorously valid only for fully ionized, nondegenerate plasmas, i.e. for low density and high temperature states. Application of these formulas to warm dense plasmas give erroneous results. The so-called "Spitzer conductivity" is shown to increase proportionally to the temperature as T / . Therefore, when 3 2 e Spitzer's formula is used to calculate electrical conductivity for a plasma at solid density and low temperature (T < 10 eV), it gives results which are shown to be incorrect by a large factor (~ 10 ). Over the years, various 2 theoretical efforts have been attempted to overcome these inconsistencies. 2.2.1 Collision operator method Using kinetic theory based on a memory function formulation and a projection operator method [39], Cauble and Rozmus derived an analytical expression for the electron-ion collision frequency u, or equivalently the collision time (r = 1/v), for strongly coupled plasmas [34, 40]. In addition, the frequency dependence and different electron and ion temperature effects are included in this model. The calculated transport coefficients are expressed in terms of static correlation and interparticle potentials. From Eqs. 2.16 and 2.17, the general relationship between the resistivity Chapter 2. Electrical Conductivity Theoretical Models 21 p, and conductivity o to the collision frequency u may be expressed as 1 P ~ = ATTU (2.34) ~T = where LO is the plasma frequency given as P '47rn e \ 2 1 / 2 P (2.35) The procedure employed by Cauble and Rozmus involves the solution of a kinetic equation by the moment expansion method. The starting point for the calculation is the equilibrium averaged phase space density particleparticle correlation function 'N AI C(l,2;t) = 5 ( n - r f (t)) 6 ( £ - p f (*))) - P l n f l l /£(Pi) 3=1 8 ( r - r f (t)) <5 (p - £ i 2 = 2 f (t))) - n / S ( ) ) 2 P a 2 P 2 (2.36) 1 where 1 = ( r i , p i , a.\), ct\ — e, i (electrons, ions) and (Pi) 1 S t n e Maxwellian distribution function, /£( = (l/2nm T f exp(-pl/2m T ). (2.37) 2 Pl ai ai ai ai The Fourier transform of C ( l , 2; t) is C a i Q 2 (k, P l , P 2 ; t) = J d\r x - r ) exp [-zk • ( r - r ) ] C ( l , 2; t), : 2 2 (2.38) which satisfies an exact kinetic equation of the following form ^C a i a a (k, P l j p ;i) 2 -Ea /o"^i^3* 3 The variable $ a i Q 2 Q l a (k, i, 3;ti)C 2 P P a 3 Q 2 (k, P l , P 3 ; t - i i ) = 0 . (2.39) ( k , i , 3 ; ti) is known as the memory function. It consists P P of three parts: a free streaming term, a Vlasov-like mean field term [13], and a collision term <3>, i.e. c \MF ^ai«2 ^aia2 0:10:2 ai«2' (2.40) Chapter 2. Electrical Conductivity Theoretical Models 22 The free streaming operator is < 5 Q 2 (2-41) - i(k • / m ) < J ( p i - p ) < W ( i ) P l a l 2 and the mean field operator is <a = - * ( • Vi/rn )Q{^)CT {k)5(t) k a (2-42) 2 al where C ^ " is the direct correlation function. The collision operator $ 1 2 c which contains all of the physics of third- or higher-order particle correlations is used in the calculation of the collision frequency. However, to allow the resulting equation to be solvable, approximations are needed but in which much of the physics is still described. The well-studied form of $ c called the disconnected approximation [41], which rigorously "disconnects" two-particle functions from those of higher order, is used in the calculations. Equation 2.39 can be solved by expanding C a i a 2 ( k , p i , p ; t) in moments 2 of momentum, which then reduces into a system of transport equations for hydrodynamic correlation functions. These equations have the form of the Navier-Stokes equations [13], so comparison between the two sets of equations reveal the transport coefficients in terms of momentum moments of the collision operator. By defining an operator P which projects onto the hydrodynamic momentum states, p= £ E W W I . (- ) 2 43 where the |ifj)'s are the zeroth through second moments, and the complementary operator Q = 1 — P, the expression for the collision frequency is given as v = (H \i$ \H;) e c p - Re [(H;\i<f> Q{z - Qi$ Q)- Qi$ \H )] c c l c e p (2.44) From the above equation, the momentum changes for electrons are described explicitly by v. Chapter 2. Electrical Conductivity Theoretical Models 2.2.2 23 Boltzmann transport equation A n electron conductivity model for dense plasmas based on a semi-classical approach was developed by Lee &; More [31]. This model gives a consistent and complete set of transport coefficients including electrical and thermal conductivity. In this conductivity model the transport coefficients are obtained from the solutions of the Boltzmann equation in the relaxation time approximation [42]. The collision operator includes contribution from the scattering of electrons by ions and neutrals. The electron degeneracy effects on the transport coefficents are taken into account by using a Fermi-Dirac distribution for the electrons. In the Boltzmann theory, the electrons are represented by a distribution / which is a function of seven variables (the position r, velocity v, and time t). In general, the distribution function satisfies the Boltzmann equation of the form df df e / v x B\ df df. f n A . where E and B are the electric and magnetic fields, and e is the absolute value of the electric charge, m is the electron mass, and c is the speed of light. The term on the right df/dt\ u co describes the rate of change in the distribution function due to scattering processes. Without the contribution from electronelectron scattering, the collision operator can be simply represented as %U OI = - ^ T (2-46) where / is taken to be a local-equilibrium Fermi-Dirac distribution function 0 which depends on the electron temperature T and density n . The electron e relaxation time r is calculated with contributions from both electron-ion and electron-neutral scattering, - = — + — (2.47) where T = ; ei niVo i e r en = (2.48) n va Q en Chapter 2. Electrical Conductivity Theoretical Models are the electron-ion and electron-neutral collision rates, and 24 is the ion density, no is the neutral density, o t is the electron-ion momentum transfer e cross section, and o en is the neutral momentum transfer cross section. For efficiency in the calculations, a Coulomb cross section with appropriate cut-off parameters was used, and gave good approximation to the transport coefficients calculated from numerically derived cross sections. The cutoff parameters were obtained by comparing the Coulomb cross section to the results from a partial wave calculation. In the partial wave calculations, the electron-ion cross section is obtained by numerical solution of the Schrodinger equation for Thomas-Fermi potentials [43]. The Coulomb cross section employed in the conductivity model is 4n(Z*) e In A 0~tr = mv 2 A 2 i (2.49) where Z* is the ionization state, and In A is the Coulomb logarithm given by (2.50) where b max and b i are upper and lower cutoffs on the impact parameter for m n Coulomb scattering. The minimum impact parameter is the value allowed by the uncertainty principle, while the maximum impact parameter is the Debye-Huckel screening length [44], corrected for electron degeneracy. The Coulomb logarithm is set to have a minimum value of 2.0 to overcome an inherent difficulty in the model that the calculated electric field screening length can become less than the interionic spacing. This minimum value of the Coulomb logarithm was justified by comparing with the partial-wave calculations by Green [45] and Lee [46]. To calculate the transport cefficients, one solves the Boltzmann equation for the electron distribution function and then uses the solution to calculate the flux of electrical current and energy. Since only the steady state transport processes are of interest here, all the time derivatives are equated to zero, Chapter 2. Electrical Conductivity Theoretical Models 25 thus the Boltzmann equation (Eq. 2.45) becomes ^M-L^M ar m av = JlzM, r (2 .51) Note, the magnetic field effects are ignored here to simplify the discussion. Rewriting Eq. 2.51, we have For a plasma with weak electric field and small temperature and density gradients, the change in the distribution function is small and the secondorder terms in / may be neglected. Then, to the first-order approximations, we have ,,53) . - (rM-UM). f h T Eq. 2.53 can be rewritten as / = /o - rf , • (-eE + ^ e VT) (2.54) where the energy is e = ^mv , (2.55) 2 and [i is the chemical potential. The electrical current and energy fluxes are then given as j = -ef fv/(v), r2d pmv* 2 ^ Q „. , ' = I^— (2.56) 3 . (2 57) - - v/(v) Following Spitzer [30] and Braginskii [38], the heat current is of the form Q = Q. i ( H - Q , + (2,8) Using Eq. 2.54, both j and Q are found to be linear functions of the temperature gradient and electric field: j = er(E-SVT), (2.59) Chapter 2. Electrical Conductivity Theoretical Models Q = TSj - KVT, 26 (2.60) where o is the electrical conductivity, K is the thermal conductivity, and S is the thermoelectric power. These transport coefficients are then given as o = eK, (2.61) 2 0 K \T ' (2 62) -W - 8 w here 2.3 = (2 63) Quantum mechanical conductivity models 2.3.1 Ziman theory For solid and liquid states, strong ion-ion correlation can have dramatic effects on the electron relaxation time. A theory for electrical conductivity which takes ion-ion correlation into account using the structure factor was developed by Ziman [47]. In this formulation, the collision rate is evaluated using the Fermi Golden rule. This rule was formulated by Dirac, and dubbed the "Golden rule" by Fermi, enables one to write down the rate (e.g. the scattering rate) of a quantum mechanical process in terms of the interaction and the product of the initial and final densities of states available for the process. Chapter 2. Electrical Conductivity Theoretical Models 27 Once again in the relaxation-time approximation, one assumes that the perturbed Fermi distribution / ( k ) for electrons with momentum k relaxes towards the equilibrium distribution /o(k) according to the equation df T t _ \ c o /(k) - / ( k ) ^ ) u- 0 • (2.65) If we consider an electron scattered elastically from state k to state k', with k = |k| = |k'| and e = £ ', the k n e k t scattering rate is the difference of the two processes (k —¥ k') — (k' —¥ k). The initial and final densities of states for the processes k —> k' process is / ( k ) and (1 — /(k')). Using the Fermi Golden rule, the rate of this process is R(k -> k') = [2-K/K) £ \T <\ 6(£ 2 kk k - e ,)f(k)(l k - /(k'))- (2.66) The interaction term T < is called the "T-matrix" and describes the scatkk tering of an electron by the whole ion-distribution. If the electron-ion interaction is weak, the effect of the full ion distribution from individual scattering effects could be built up from individual scattering effects. If we have a set of point ions with charge Z at instantaneous positions R / , then the interaction of an electron at r is of the form / |*v - r| The matrix element between the initial state k and the final state k', with q = k — k' and q = 2k (l — cos#), is the Fourier transform of Eq. 2.67, and 2 2 is given as ^(q) = E ^ - ^ e x p ^ q - R , ) , (2.68) where v is the Fourier transform of the Coulomb potential, q v = ^ (2.69) q and e = — 1 for simplicity. If the electron-ion interaction is really of the form Z/\Ri — r| then the interaction can be very strong, so a weak pseudopo- tential is used instead. This is achieved by replacing the v (—Z) term by q Chapter 2. Electrical Conductivity Theoretical Models a M(q)v (—Z) q 28 term where M(q) is a form factor defining the effect of the pseudopotential. The idea of the pseudopotential is to hopefully avoid the issue of higher power (than second order) terms in the interaction potential. Now, combining Eqs. 2.65 and 2.66 gives / ( k ) r ~ (° = f ( k ) k / /(kO]|T - ^)[/(k) - fcfc f. (2-70) The result of the integration gives the inverse of the relaxation time to be = 2^5{e -e )\T \\l k kl - cosO). kk (2.71) To obtain the resistivity, the inverse relaxation time l / r ( k ) for an electron of momentum k, or equivalently l / r ( e ) for the energy, e = k /2m, is averaged 2 over all electron energies, P=- {l/r{e)). (2.72) 2 The most familiar form of the Ziman formula applies to plasmas and liquid metals where the scatters are either weak or strong but isolated. The resistivity is given as h r°° P=V-W^T 6iiZe ri Jo r^ / dqq S(q)a (q), Jo 3 sc l (2.73) where n is the mean electron density, S(q) is the ion-ion structure factor, f'(e) is the derivative of the Fermi-Dirac distribution for the electrons, and osc is the differential scattering cross section which depends on the incident electron momentum k and the transferred momentum q. In the regime of strong ion-ion interactions where the system cannot be treated as a sum of independent scattering centers, be they weak or strong scatters, the resistivity corresponds to a definite configuration c of ions and is written as P c = 3 ^ J o d £ f ' { £ ) Jo d q q " T ^ - ( 2 - 7 4 ) The actual measured resistivity is obtained by averaging over all ion configuations, p = (p ). c Chapter 2. Electrical Conductivity Theoretical Models 29 Ziman theory and density functional theory Using density functional theory and Ziman theory, first-principle calculations of electrical conductivity were derived by Perrot and Dharma-wardana [35, 48]. Density functional theory allows the ground-state properties of a manyelectron system to be determined exactly through the electron density. The calculation of resistivity (or conductivity) involved the following steps: 1. solve the Kohn-Sham equations [49, 50] for the electrons and ions; 2. use the Kohn-Sham results to form pseudopotentials and scattering cross-sections; 3. use the pseudopotentials to form pair potentials and pair distribution functions 4. calculate the equation of state, ionization balance etc., to obtain effective ionic charges Z etc., and iterating to self-consistency 5. calculate the resistivity using the Ziman formula Kohn-Sham equations A first-principle theory is obtained by beginning with the bare nuclei and then constructing the electronic and ionic structure of the plasma. To solve the electronic structure around a nucleus placed in an electron fluid, the density and temperature of the plasma is required. The interaction of the nucleus with the electron fluid is a highly nonlinear process and attempts using perturbation theory, such as Green's function techniques [51], are quite difficult. Alternatively, one may use the Kohn-Sham technique [49, 50], and construct the nonlinear charge density around the nucleus. The Kohn-Sham equation is similar to the Hartee equation, except that an exchange-correlation potential is included because of the many-body effects, and has the form of Chapter 2. Electrical Conductivity Theoretical Models 30 a one-particle Schrodinger equation V - where VKS( ) T 1 2 T + V {T) KS </>„(r) = e „ ^ ( r ) (2.75) is the Kohn-Sham one-body potential and ip {*) e the Kohna r u Sham eigenstates. For an electron, the Kohn-Sham one-body potential has the form V& (r) = V ^ W + / s where V£ XT < T ]~_ ^ Z V , ) dv' + V£(r) + V?{T) (2.76) is an external potential, the 2nd term is the electrostatic potential, V£ is the electron exchange potential, and V* is the correlation potential % C which describes the many-body effects on the electron arising from the ion distribution p(r). Similarily, for an ion we have V j s t o = Ct(r) -2J " ^ y W + V?(r) + V*(r) (2.77) where V™ is the correlation effect of the ion-ion many-body interactions on an ion, and V™ is the correlation contribution to the potential felt by an ion due to the electron distribution n(r). In a plasma, the bound states as well as the continuum states are required to determine the charge density around the nucleus. The nucleus, surrounded by its charge cloud of bound electron states and continuum (free) electron states is decribed as a "neutral pseudo-atom" (NPA). A key result of the Kohn-Sham technique is the charge-density "pileup", n(r), around the nucleus that essentially screens the nucleus. Using the N P A , phase shifts suffered by the continuum electrons when they scatter from the nuclei provide the information needed for constructing the scattering cross sections (or pseudopotentials) which describe the electron-ion interaction. The psedopotential has an effective ionic charge Z and it behaves as — Z/r for large r. For most situations, it is possible to construct a soft pseudopotential Vie(q) which is weak in the sense that it is possible to recreate the nonlinear Chapter 2. Electrical Conductivity Theoretical Models 31 electron-density "pileup" n/(r), obtained via the Kohn-Sham equation, by simply using linear response theory. In other words, Vi (q) is choosen such e that n (r) f = -V (q)x(q), (2-78) ie where x{o) is the electron linear-response function, which depends on the electron density and temperature. Thus, the ion-ion pair potential can be written as _ VM = ^4- 2 + \V (q)\ x(q)2 ie (2.79) Q Given Vu(q), the ion-ion distribution function gu(r) can be calculated using the Kohn-Sham equation for classical particles, otherwise known as the hyper-netted-chain (HNC) equation, with certain choices of the ion-ion correlation potential, namely no exchange potential since the ions are classical particles. Once the gu(r) is known, the ion structure factor, Su can be obtained by Fourier transforming (gu — 1). In summary, by using the Kohn-Sham procedures, the ion-electron pseudopotential Vi (q), the structure factor Su(q), and the charge density e n (r), ie and an effective ionic charge Z which enters into the pseudopotential are all obtained. The phase shifts are used to construct a scattering cross section. These are incorporated into the Ziman formula which allow for the calculation of resistivity (or conductivity). Ziman theory and Atom-in-jellium model Also following the Ziman theory, Rinker developed a theoretical model for electrical conductivity using a more heuristic approach [32]. Rinker's calculations were motivated by the need to calculate the conductivity with moderate accuracy for a wide regime of plasma conditions, thus some simplications were employed to achieve a relatively rapid computational procedure. The "average-atom" approximation is used in which the material of interest consists of a homogeneous medium into which is imbedded a static Chapter 2. Electrical Conductivity Theoretical Models 32 and statistically distributed collection of identical, nonoverlapping, spherical scattering centers. For the isolated atom, the most successful theory is based upon the self-consistent, relativistic Hartree-Fock-Slater mean field approximation (MFA). In this approximation, every electron moves in the potential created by the nucleus plus the average potential of all the other electrons. This assumption leads to the independent-particle model, which essentially reduces the many-electron problem to the problem of solving a number of coupled single-electron equations. The desired ionic potential is required to approach this limit in the low-density, low-temperature limit. For high temperatures and densities, the corresponding limit is the temperature dependent Thomas-Fermi-Dirac (TFD) approximation [52]. The T F D or statistical theory marks a change in approach from Hartree-Fock theory in that the electronic charge density is the fundamental variable instead of the wavefunction. For intermediate regions of temperature and density, neither of these approximations is appropriate. This model tries to provide a physical in- terpolation between M F A and T F D limits and is usually referred to as an atom-in-jellium (AIJ) model. For calculation simplicity, in the A I J model a procedure is adopted which uses T F D potentials to provide the dependence upon temperature and density, combined with a M F A potential to establish the low temperature, low density limit. The numerical mixing formula used is V(r) Vo(r)+£V (r) (2.80) T where Vo(r) is the appropriate low temperature, low density potential and Vr(r) is the T F D potential for the temperature and density of interest. The mixing parameter £ is given by (2.81) £ = (T/To)* + (p/ ) Po where T and p are parameters of order 100 eV and 100 g/cm , and the 3 0 0 exponents £i and £ are normally taken to be unity. These choices are made 2 33 Chapter 2. Electrical Conductivity Theoretical Models so that the mixing is done smoothly as a function of temperature and density and the correct limits are obtained. For temperatures near the melting point, the Percus-Yevick structure factor is shown to accurately fit neutron-diffraction data [53]. While at high temperatures, the ion-ion correlations are described by the Debye-Huckel structure factor of an ideal gas. In the model, a correlation function is used which gives a structure factor that is close to the Percus-Yerick solution on the average, and approaches the limit of the Debye-Huckel structure factor at high temperatures. The electrical conductivity is calculated using the Ziman formula with the above structure factor and the previously described ionic potential model. 2.3.2 Kubo-Greenwood formula Density functional theory based molecular dynamics simulations were used by Desjarlais et al. [54] to improve conductivity model calculations. The molecular dynamics (MD) simulations were performed in conjunction with the plane wave density functional code V A S P (Vienna A b initio Simulation Program). Ion configurations for the conductivity calculations were obtained by performing the "ab initio" M D simulations within the framework of the finite temperature density functional theory. For the M D runs, the ions and their respective core wavefunctions are modeled using the Vanderbilt ultra-soft pseudopotentials [55]. The D F T exchange and correlation functionals are calculated at the level of the generalized gradient approximation. Schrodinger's equation is solved for an effective potential that includes the contribution from the ions V , the classical contribution of the electrons V i , and the ext c ass quantum-mechanical exchange and correlation contribution V . xc The electron density n is then constructed from the wavefunctions e n = £ | ^ | e 2 (2.82) Chapter 2. Electrical Conductivity Theoretical Models and the equations are iterated until convergence is obtained. 34 The forces on the ions are computed and the molecular dynamics simulations are performed. Next, a total of ten to twenty configurations are selected from an equilibrated (in an average sense) portion of the M D run. For each of these configurations, the electrical conductivity are calculated using the KuboGreenwood formula. The Kubo-Greenwood formula [51, 56, 57] gives the electrical conductivity directly from the electron wavefunctions, thus it avoids the difficulties of calculating or modeling, independently and consistently, the population of free electrons, the various relaxation times between the electrons and other species, and the proper form of the pseudopotential and screening models. In addition, it includes contributions to the conductivity from electron-atom, electron-ion, and electron-electron interactions. The Kubo-Greenwood formula for the electrical conductivity as a function of the frequency to for a particular k point in the Brillouin zone of the simulation supercell may be written as 27re ft 2 im 2 LOIL N J N = 1 I 3 = 1 A = 1 (2.83) where e and m are the electron charge and mass. The i and j summations are over the ./V discrete bands included in the triply periodic calculation of the cubic supercell volume element Q. The a sum is over the three spatial directions. F(e u) is the Fermi weight corresponding to the energy it for the ith band at k; NT/^k is the corresponding wave function. In general, the integration over the Brillioun zone is performed using the method of special k points [58], ^) = E ^ M k ) , (2-84) k where W(k) is the weighting factor for the point k in the Brillouin zone. The final result is obtained by taking the average of o(cu) over the ensemble of configurations sampled. Chapter 3. Idealized Slab Plasma 35 Chapter 3 Idealized Slab Plasma 3.1 The concept of Idealized Slab Plasma The new approach to realize single-state measurements on W D M is based on the ISP concept. A n ISP is a planar plasma that can be considered as a single, uniform state in which any residual non-uniformity has insignificant impact on the measurement of its uniform properties. In addition, the state can be characterized by direct measurements without the need for model interpretations. In the ISP approach, an ultrafast energy source is used to rapidly heat a solid sample. The basic idea is to limit the heating process to femtosecond time scale, thus mitigating hydrodynamic expansion to produce near isochoric conditions. A t the same time, to produce uniform heating, the thickness of the sample is either matched to the deposition range of the energy source or the sample's characteristic scalelength for thermal conduction. Accordingly, the thickness of such an ISP can be tailored to vary between tens of nanometers, with the use of optical lasers to micrometers using soft X-rays and to millimeters using hard X-rays or energetic electrons and ions, as indicated in Fig. 3.1. Warm dense matter states have intrinsically high energy densities, and are thus at extreme pressures. As a result, the ISP of warm dense matter would only last for a very brief time before expansion of the sample dominates. Moreover, in order to measure the properties of the single W D M state of interest, precise synchronization of diagnostics with the production of the ISP must be achieved. Chapter 3. Idealized Slab Plasma 36 Figure 3.1: 1/e deposition length of various energy sources in solid gold at normal condition [59, 60, 61]. Chapter 3. Idealized Slab Plasma 3.2 37 Isochoric heating of a freestanding, u l t r a t h i n foil by a femtosecond laser Among the various ultrafast energy sources, high-intensity femtosecond lasers are the most developed and accessible. Hence, the first ISP approach that has emerged is the isochoric heating of a freestanding, ultrathin foil by a femtosecond laser [36]. Originally, this approach was introduced using the example of a 100 A aluminum foil heated by a pump laser pulse of 20 fs (FWHM) at a wavelength of 400 nm. However, using simulations it was shown that by choosing the appropriate pump laser intensity, the concept could be made applicable for longer laser pulse lengths (~ 100 fs) and thicker foils (~ 300 A), both of which are readily available in the laboratory. A n examination of the process of ultrafast laser-solid interaction reveals how uniform heating of the solid may be achieved [62]. First, the electrons are optically excited within the skin depth of the material. These non-equilibrium electrons move ballistically with velocities readily exceeding 10 m/s [8, 63, 64]. For gold at normal conditions, the Fermi speed of the 6 conduction electrons is 1.4 x 10 m/s. The mean free path of these electrons 6 would be ~ 390 A, so the transit time through a 300 A sample would be only ~ 23 fs compared to the pump pulse duration of 150 fs. In addition, the use of a freestanding foil avoids the problem of a thermal gradient at the substrate interface, thus axial uniformity would be ensured. Meanwhile the lateral uniformity of the heated sample is achieved by making sure that the intensity variations are small within a large central region of pump laser focal spot. 3.2.1 State characterization In past W D M research, temperature is commonly used to describe the state of the system. However, temperature requires knowledge of the equation of state of a material, which is model dependent. Instead, to characterize 38 Chapter 3. Idealized Slab Plasma directly the state of the ISP, its mass density p (kg/m ), and energy density 3 e (J/kg) are the practical quantities to use. Since hydrodynamic expansion of the ultrafast laser heated solid is negligible during the measurement of the resulting ISP, the mass density of the state is simply the initial solid density p of the sample. In previous studies 0 using femtosecond laser heating of thick foils or films deposited on substrates, the actual energy density of the heated material is not known. Instead, only the incident or the absorbed laser fluence (mJ/cm ) or intensity (W/cm ) 2 is measured. 2 However, for a heated ultrathin foil the energy density can be obtained in a straight-forward manner. First, the energy of the incident (Ei ), n specularly reflected (E f), re diffusely scattered (E ), and transmitted scat laser pulse are determined in spatially resolved measurements. From (E ) tran these measurements, the energy absorbed by the sample (E ) abs Eabs — E{ Ej n re E i sca is given by (3-1) E\'tran • In conjunction with the inital mass density and heated volume V of the sample, the increase in energy density Ae is obtained, (3.2) Ae = Although, the initial energy density of the unheated sample eo is known only from theoretical models, it is generally negligible. For example, eo of gold is 3 x 10 J / k g as given by the Sesame equation of state tabulated data, while 4 Ae pertinent to the studied W D M states exceed 5 x 10 J/kg. Thus, the 5 W D M state can be specified by p and A e . 0 3.2.2 Obtaining A C conductivity In earlier laser heated solid experiments [11, 65] non-uniform states were created, thus a "forwardcasting" methodology must be used to analyze the results. Specifically, hydrodynamic simulations using theoretical conductivity Chapter 3. Idealized Slab Plasma 39 models and equation of states were used to predict experimental observables. This yielded only convolved tests of theory [12]. For this work instead, a novel "backcasting" methodology is employed which uses reflectivity (R*) & transmission (T*) measurements directly to solve the Helmholtz equations for an electromagnetic wave propagating through a dielectric medium of uniform sample of thickness d [66]. This yields the complex dielectric function e of the W D M state. As seen in Eq. 2.32, the dielectric function may be written as € W 1 + 47TZ^. = (3.3) LO From this the A C conductivity is given as = £ ~ - 1) • (3-4) The direct benchmark of conductivity models for a known single, well-defined plasma state can be done using the measured cr (Ae,p ) value. w 0 Additional information about the transport properties of the W D M state may be found by assuming a free electron gas behavior. Accordingly, the A C conductivity can be describe by the Drude model [37] as <7 (1 + 0 1 + W T 2 2 (3.5) The collision time r is then given by the measured ratio of the imaginary to the real part of c r , w cy \ioJ Furthermore, the D C conductivity o is evaluated from 0 a = o ( l + toV) 0 r (3.7) and the electron density n is given by e n = e (3.8) Chapter 3. Idealized Slab Plasma 40 where e and m are the electron charge and mass, respectively. e For completeness, an overview of the electromagnetic wavesolver algorithm used to obtain the dielectric function is presented here. In the algorithm, the Maxwell equations are solved using the Helmholtz formulation in the following manner [66]. Consider a plane electromagnetic wave propagating through a dielectric medium with the plane of incidence taken to be the y^-plane, see Fig. 3.2. A n incident electromagnetic wave may be resolved into two linearly polarized orthogonal waves: a transverse electric (TE) wave and a transverse magnetic (TM) wave. For a T E , or S-polarized, wave incident on the target medium, E = E = y z 0 and Maxwell's equations reduce to the following scalar equations: dH dH z ieuo y ~o a dy ^ 1 dz dH dH x z 0 n — dz = c ^- Jk d o = dx =0 (3.9) ox dy icop H =0 c dE loop T x y ~o dz dE ~H = H c iw/i x 1 H u y z = - u dy c Eliminating H and H yields the following differential equation y z dE dE 2 -W d(lo 2 x x + ^ 2 + n k ° 2 E x = gAt ) dE x "^~dz- ( 3 - 1 0 ) where n is index of refraction of the medium, while k and A are respectively 0 0 the wave number and wavelength of the incident wave in vacuum, given by n = 6^, fco = - = ^ . c A 2 0 (3.11) Chapter 3. Idealized Slab Plasma 41 dielectric medium Figure 3.2: Coordinate system used in the electromagnetic wavesolver for describing the propagation of a plane electromagnetic wave incident on a dielectric medium at an angle 6 to the z-axis, which is perpendicular to the medium's surface. 42 Chapter 3. Idealized Slab Plasma For non-magnetic materials, the magnetic permeability (i is equal to one. Through separation of variables, Eq.(3.10) can be replaced by the following second-order differential equations: dU dz d(\o p) ^ dz 2 g 2 dV k l { #og(e - f)] dV 2 dz dz 2 n _ 2 a 2 ) u = Q + k (n -cx )V 2 dz 2 ( 3 1 2 ) = 0 2 where E — U(z) exp[i(k ay — cot)] H = V(z)exp[i(k ay x y H z 0 -cot)} 0 = W(z)exv[i(k ay (3.13) — cot)]. 0 The functions U, V and W are related by the following equations: U' = V aU + pW ikopV = ik (e-f)U (3.14) 0 = 0. For the case when the wave is a plane wave a = n sin 9 — constant (3.15) where 6 is the angle which the normal to the wave makes with the z-axis. Similarly for a T M , or P-polarized, wave incident on a target medium (H y = H z — 0), a corresponding set of expressions can be found. As a consequence of the symmetry of Maxwell's equations, the substitution rule of exchanging e and —p gives the following equations: dU d (log e)dU 2 dz dz 2 dV dz + k (n 2 d[log(/x- f )}dV 2 dz dz 2 dz - a )U = 0 2 2 + k (n 2 2 (3.16) - a )V = 0 2 where H x — U(z) exp[i(k ay - cot)] E y = -V(z)exp[i(k ay-cot)] = -W{z) exp[i(k ay - cot)]. E z 0 0 0 (3.17) Chapter 3. Idealized Slab Plasma 43 Now, the functions U, V and W are related by the following equations: U' = V aU + eW ik eV 0 = (3.18) ik (p-f)U 0 = 0. The solutions, subject to the appropriate boundary conditions, of the differential equations (3.12) and (3.16) and various theorems relating dielectric medium, can most conveniently be expressed in terms of matrices [66]. In the case of a homogeneous dielectric film, e, p, and n — y/eji are constants. For a S-polarized wave, the solutions to Eq.(3.12) are given as U(z) = A cos(k nz cos 0) + Bsin(k n cos 8) V(z) = 0 0 \.f^cos6{Bcos(k nzcos6) — 0 Asm(k nzcos9)} (3.19) 0 with the boundary conditions (at 2 = 0) £7(0) = U , 0 (3.20) V(0) = V . 0 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 the medium (3.21) M(z) where M(z) = cos(k nz cos 9) 0 _ rri2i m 2 _ 2 ip sin^nnz cos 0) -s'm(k nz cos0) 0 cos(k nz cos0) (3.22) 0 with p — \l cos 0. (3.23) For a P-polarized wave, the same equations hold, with p replaced by COS0. (3.24) Chapter 3. Idealized Slab Plasma 44 A n inhomogeneous target medium can be treated as a stack of thin homogeneous films by multiplying the characteristic matrices of all the dielectric media. In other words, the plasma is modelled as a succession of dielectric media extending from 0 to z^, 0 < z < Z\, zi < z < z , • • •, Zflf-i < z < ZN 2 with the characteristic matrix = M (z )M (z M(z ) N 1 1 2 - z - ). - z i ) • • • MN_(Z 2 N n (3.25) X Thus, consider a plane wave incident upon a dielectric medium that extends from z = 0 to z = z^ and that is bounded on each side by a homogeneous, semi-infinite medium. Applying 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 U 0 = Ei + E , V 0 = po(Ei-E ), r r U ZN = V ZN = E, ^ t p E, N t where Po = 4/ —cos0 , PN = J— o V cos8 . (3.27) ZN V ^0 The amplitudes of the electric field vectors of the incident, reflected and transmitted waves are Ei, E and E , respectively. Furthermore, e , Po r e, N t 0 a n d are the dielectric 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 with the ^-direction (direction of stratification). The reflection and transmission coefficients of the dielectric medium are E (m + m p )p - (m + m p ) r = —— = T r r Ei (mn + mi p )p + (m i + m p ) r n l2 2 N N 0 0 2l 22 N 2 22 N {6.2a) Chapter 3. Idealized Slab Plasma 2p t - — _ Ei ( m n + m p )Po + (wi2i + 22PN)' 0 45 (3 29) m 12 N Thus, the reflectivity and transmissivity (transmission), in terms of r and t are R=\r\ , 2 T = PN UI2 (3.30) t and the phase of the reflected and transmitted waves relative to the incident wave are Im(r) tan - l [Re(r)J 4> t — tan 1 Im(t) Re(t) For the case of a P-polarized wave the quantites p and 0 (3.31) are replaced by 9o cos#o, e 0 cos 6N q N y (3.32) *-ZJV in which case r and t are then the ratios of the amplitudes of the magnetic field vectors. Chapter 4. Description of the Experiment 46 Chapter 4 Description of the Experiment Experiments were performed using the Ultra-Short Pulse (USP) laser facility at Lawrence Livermore National Laboratory (LLNL) to measure the optical properties of a single-state of W D M . In this chapter, the laboratory setup of the experiments is described and the important experimental considerations are discussed. As described in the previous chapter, the experiment is conceptually very simple. However successful implementation of this completely new technique requires careful attention to many details of the experiment. 4.1 Experimental arrangement A schematic of the experimental setup is shown in Fig. 4.1. A more detailed layout of the experimental setup is presented in Fig. 4.2. In the vacuum chamber, the freestanding, ultrathin foil target is heated by an intense 400 nm pump laser pulse at normal incidence. Meanwhile, a low energy 800 nm probe laser pulse illuminating the foil at an angle of 45° is used to measure the optical properties of the heated foil. Using a pump-probe technique, the A C conductivity for a single-state of warm dense matter was obtained directly from probe reflectivity and transmission measurements {R*,T*}. Both the pump and probe laser pulses are derived from a chirped pulse amplification Ti:sapphire laser system. A commercially available mode- locked, cw-pumped Thsapphire oscillator produces low energy (nJ) pulses 1 at a wavelength of 800 nm with a pulse length of about 100 fs ( F W H M ) . 1 Tsunami, Spectra Physics Chapter 4. Description of the Experiment 47 CAM-PUR Probe 150fs 800nm PD2< ^300A ^ u R CAM-PUT Pump 150fs 400nm PD1 PPD3 ^^^^^ CAM-PRR r* CAM-PRT Figure 4.1: Illustration of the femtosecond pump-probe experiment. Shown are the photodiode mounted on integrating spheres for measuring the input(PDl), reflected (PD2) & transmitted (PD3) pump energy, and the C C T V cameras which are used to record the images of the pump reflectivity ( C A M - P U R ) & transmission ( C A M P U T ) , and the probe reflectivity ( C A M - P R R ) & transmission (CAM-PRT). Chapter 4. Description of the Experiment 48 1/2 Figure 4.2: Detailed schematic diagram of femtosecond pump-probe experiment. Shown are the photodiodes mounted on integrating sphere for measuring the input (PD1), reflected (PD2) & transmitted (PD3) pump energy, and the C C T V cameras which are used to record the images of pump reflectivity ( C A M - P U R ) & transmission ( C A M - P U T ) , and the probe reflectivity ( C A M - P R R ) k transmission ( C A M - P R T ) . Also, shown are the A/2 waveplate used to adjust the polarization (S- or P-) of the probe pulse, and the K D P crystal used to produce the 2OJ pump pulse. Chapter 4. Description of the Experiment 49 These pulses are temporally stretched (chirped) to about 400 ps using a single diffraction grating pulse stretcher [67]. One of these chirped pulses is injected into a regenerative amplifier where its energy is raised to the pJ level. Next, a "bow-tie" Tksapphire amplifier stage boosts the stretched pulse up to its final energy (mJ). This amplified pulse is then recompressed to approximately 150 fs F W H M in a parallel grating compressor. The 800 nm probe pulse corresponds to the fundamental wavelength (lu;) of the laser system. To produce a 400 nm pulse, the orginal lui is passed through a birefringent, potassium-dihydrogen phosphate ( K D P ) crystal where part of the pulse is converted into the 2nd harmonic (2co). The 2co beam was used as the heating or pump pulse because of better absorption efficiency of laser light at the shorter wavelength of 400 nm. In addition, the 2co beam has a much better contrast in the pulse shape than the leu beam, as shown in the next section. The energy of the pump pulse was controlled by angle turning of the K D P crystal. 4.1.1 The pump laser The temporal profile of the pump pulse is obtained from the frequency resolved optical gating (FROG) [68, 69] measurements of the loo pulse. The F R O G apparatus uses a non-linear optical medium in the autocorrelation of a lu pulse. Since the K D P crystal used to obtain the 2co pump is a non-linear optical medium as well, a similar effect on the shape of the laser pulse would occur. The temporal profile of the pump pulse is obtained by taking the F R O G image (Fig. 4.3) and spectrally integrating it. The pump pulse has a full width at half maximum (FWHM) of (150 ± 10) fs with a base width of about 600 fs at 1% of peak intensity (Fig. 4.4). From previous autocorrelation measurements, a comparison of the peak intensity to the intensity values at 1 ps before and after the peak gives a contrast ratio of better than 10 for 7 the pump pulse, see Fig. 4.5. The pump pulse is focused using an off-axis parabolic mirror with a focal length of 600 mm that produces a focal spot Chapter 4. Description of the Experiment 50 diameter of (64 ± 2) pm (FWHM) as presented in Fig. 4.6. Furthermore, within the central spot diameter of 20 pm of the pump beam shows only a 3% variation in intensity. Referring back to Fig. 4.2, the energy of the incoming pump pulse, as well as the energy specularly reflected off and transmitted through the foil were measured simultaneously in two ways. First, outside of the chamber a set of photodiodes (PD1, PD2 & PD3), which are calibrated with respect to a energy meter, measured the total temporally integrated energy. Also, three photodiodes within the chamber observed that less than 1% of the pump energy was scattered by the foil. At the same time, the spatial profiles of the reflected & transmitted pump beams are imaged by f/8 optics onto a pair of C C T V solid-state cameras , refered to as C A M - P U R and C A M - P U T , 2 respectively. At 400 nm the f/8 optics yield a spatial resolution of about 8/im. Using a frame grabber card and a P C the images from the C C T V cameras 3 4 are digitized and recorded. By incorporating the measured spatial images of the C C T V cameras with the total energy measurements of the photodiodes, a spatially resolved absorbed energy density of the heated foil was obtained. 4.1.2 The probe laser The temporal profile of the probe pulse is not directly measured but inferred to be similar to that of the pump pulse and simply taken to have a F W H M of about 150 fs as well. The probe is a collimated beam with a diameter greater than 625 pm. Since the probe light is incident at 45° to the target normal, either S-polarization or P-polarization measurements can be obtained. Spolarized light is considered better suited to the ISP approach since it is free from the complexity of resonant absorption. In this study, experiments were performed first with S-polarized probe light for measuring the A C conduc2 3 4 Hitachi Denshi C C T V cameras (KP-101A, KP-140U) Coreco Viper-Quad framegrabber Dell Optiplex computer Chapter 4. Description of the Experiment 51 Time Wavelength Figure 4.3: Frequency resolved optical gating (FROG) image of the 800 nm (lu;) laser pulse. Chapter 4. Description of the Experiment 0.001 I -600 -400 -200 0 200 52 I 400 600 Time (fs) Figure 4.4: Temporal profile of pump pulse extracted from the F R O G image (see Fig. 4.3). Chapter 4. Description of the Experiment 53 Figure 4.5: Measured temporal intensity profile of the lu and calculated 2u pulses [2]. The non-linearity of frequency doubling improves the pulse contrast (measured at 1 ps prior to the intensity peak) by about 3 orders of magnitude. Chapter 4. Description of the Experiment 54 * • • ' j »' i < ; • ' i »i ' i i »i i i i i | i i i i i i i > i | i i i i | i i 0 Ii ' i' *• • 0 10 1 • 1 i 1 20 ' •l ' 1 30 i • ( ' 1 40 < i 1 • • ' i 50 1 ' • i ' 60 1 • • • • 70 1 11 j i i i i • • • ' < • • ' • I 80 90 100 Radial distance (nm) Figure 4.6: Spatial profile (6 shots are overlaid) of the pump pulse in the focal plane of the off-axis parabola (f=600 mm). Chapter 4. Description of the Experiment 55 tivity of ISP states. P-polarized probe light was then used to confirm the validity of the ISP approach. After the ISP of W D M is created, it will evolve over time. The time history of the W D M state is tracked by varying the path length of the probe pulse relative to the pump pulse. The temporal uncertainty of the measurements consists of the reproducibility of this time delay (~ 20 fs), the width of the pulses (~ 150 fs), and fluctuation in the optical path lengths of the probe and pump beam (< 50 fs) due to air disturbances. Thus, the overall uncertainity in the timing of the pump and probe pulses is estimated to be ~ 250 fs. The probe light reflected off and transmitted through the foil were imaged by f/3 optics onto another pair of C C T V cameras (3 [im resolution), C A M P R R and C A M - P R T , respectively. The signals from the heated region are compared in-situ with those from an unheated region of the foil. The latter values are absolutely calibrated and the procedure is described in the next section of this chapter. 4.2 The freestanding, ultrathin target The targets for these ISP experiments were freestanding, 300 A thick gold foils. Gold was chosen as the material of study for several reasons. First, most metals when exposed to air oxidize rapidly. Even a thin layer of oxide on the surface of a ultrathin foil would compromise the single-state measurements. Since gold is an inert metal, the oxidation complications would be avoided. In addition, because gold at normal conditions has no interband transitions at 800 nm, the probe light interaction would be with the conduction electrons and so a Drude model may be applied to the conductivity measurements. Finally, ultrathin samples of normal solid density (19.3 g/cm ) with high 3 purity may be obtained commercially. 5 Schafer Corp., Livermore, CA 94550, U.S.A. 5 Chapter 4. Description of the Experiment 56 More importantly, to achieve an ISP state the ultrathin foil targets were required not to be in contact with any substrate, i.e. they had to be freestanding. If the foil were laid upon a substrate, thermal transport across the foil-substrate interface would result in a non-uniformly heated state. To obtain the freestanding target, a ultrathin layer of gold was deposited onto a substrate which had a "release" layer on top of it. By immersing the gold deposited substrate in water, the release layer dissolves and the ultrathin gold foil is freed from the substrate. Then, the foil is floated onto the frontside of stainless steel support plate with circular openings of about 625 Lim in diameter. The steel plate is about 2.5 cm by 5 cm in size and 1 mm in thickness, and with 36 openings drilled through it. On the frontside of the steel plate, burrs around the openings allow the foil to adhere to the edges and stretch flat across the openings. A n example of such a typical freestanding foil is shown in Fig. 4.7. On the rearside, each of the openings are beveled to allow for an unobstructed view of the 45° probe transmission beam, as illustrated in Fig. 4.8. Absolutely calibrated reflectivity & transmission measurements of the unheated foil were obtained, for both S-polarized and P-polarized light 800 nm probe light at 45°. One C C T V camera records the 45° probe reflectivity ( C A M - P R R ) , while the other records the 45° probe transmission ( C A M PRT). The ratio of intensities of the captured C A M - P R R images for light reflected off the foil to that off a calibrated mirror gave the absolute reflectivity of the unheated foil RQ. Similarly, the ratio of intensities of the captured C A M - P R T images of light with and without the foil in place gave the absolute transmission of the unheated foil T *. 0 Values for the density and thickness, 99.9% solid density and (300 ± 20 A) respectively, of the foils are supplied by the manufacturer. However, these are values given for the foils before they are floated off the manufacturer substrate onto our target mount. The absolutely calibrated reflectivity & transmission (RQ,TQ) at 800 nm at 45° were compared to values calculated Figure 4.7: Microscope photo of front view of a freestanding ultrathin Au foil mounted on the target holder. Chapter 4. Description of the Experiment 58 Figure 4.8: Side view schematic of the target holder of the freestanding ultrathin A u foil and the beam path for the probe light. Chapter 4. Description of the Experiment 59 for various thicknesses using tabulated dielectric values for gold at normal density and room conditions. As seen in Fig. 4.9, the measured reflectivity and transmission values correspond to thicknesses in agreement with the manufacturer's value. Numerous ultrathin gold foil targets were used in the experiments. Each target was calibrated to obtain the initial cold reflectivity & transmission values, and foil thickness. The flatness of the foil over the opening is determined by interferometric means using a Michelson interferometer and a 632.8 nm HeNe laser. Shown in Figs. 4.10 & 4.11 are the interferograms and the extracted variations in height across a A/20 mirror and a A u foil. The flatness of the foil is observed to be better than A/10 which is of mirror quality in terms of surface finish. Chapter 4. Description of the Experiment 25 1 • 1 i 1 15 I H 10 P 1 1 1 i 1 i 1 0 ^-250 A 20 l £ 1 1 • • i 1 1 i • 1 P-pol -£—i-a 250 A S-pol 350 A 5 I 0 70 60 X 350 A • • • • 1 75 • • • • 1 • • • 80 • • 1 85 R* 90 sd _i i i 95 i_ 100 (%) o v ' Figure 4.9: Calibration of the freestanding ultrathin A u foil. Comparison of the absolutely calibrated reflectivity and transmission measurements of a cold gold (Au008) foil (data points with error bars represent the average of the measured values) for 800 nm light at 45° with values calculated using tabulated dielectric values for solid gold at normal conditions (lines). From the comparison, this foil has a thickness of (280 ± 20) A. Chapter 4. Description of the Experiment 61 A/20 mirror 300 A Au Figure 4.10: Interferograms of A/20 mirror (top) and freestanding A u foil (bottom). Chapter 4. Description of the Experiment A/20 62 & o mirror « -0.2 eg 200 500 300 x(um) 300 A Au s o as > 0 100 200 300 400 500 XQLUH) Figure 4.11: Extracted variation in height for A/20 mirror (top) and freestanding A u foil (bottom) taken from the highlighted boxes in Fig. 4.10. Chapter 5. Pump-probe Reflectivity and Transmission Measurements 63 Chapter 5 Pump-probe Reflectivity and Transmission Measurements In this chapter, the results of pump-probe measurements performed to determine the A C conductivity of single-state warm dense matter are presented. The probe pulse detects changes in the optical properties of the gold foil as it is heated by the pump laser. Changes in the reflectivity & transmission signals are recorded by the corresponding C C T V cameras. Shot-to-shot variations in the probe pulse were mitigated by the insitu calibration of the probe reflectivity and transmission, as illustrated in Fig. 5.1. To obtain the reflectivity R* for the heated gold, the ratio of the signal from this heated central region R* divided by the signal from the unH heated cold gold foil region R* is multiplied by the the absolutely calibrated c reflectivity of the cold gold foil J?Q, i.e. nr = (5.1) Similarly, the transmission T* for the heated gold is given by the ratio of the signal through the heated region T^ divided by the signal through the unheated cold foil region T£ multiplied by the the absolutely calibrated transmission of the cold gold foil T *, i.e. 0 T* = ^T*. (5.2) Figure 5.2 shows an example of the camera image of the reflected probe light off the heated foil. The spatial uniformity of the heated foil corresponds Chapter 5. Pump-probe Reflectivity and Transmission Measurements 64 800 nm Heated region Figure 5.1: In-situ calibration of probe reflectivity and transmission. The probe pulse (large hatched area) illuminates the gold foil. The small hatched area shows the region heated with the pump pulse. The locations used to determine the probe reflectivity & transmission of the heated and cold foil are shown as the small circles. Chapter 5. Pump-probe Reflectivity and Transmission Measurements 65 110 : 100 t 0 I I I I 40 I I I I 80 I I x (nm) I I 120 1 1—> 1 160 L Figure 5.2: Example of reflected probe light image recorded by C A M - P R R , and a slice of the central heated region, shown with data (circles) and best fit (line). Chapter 5. Pump-probe Reflectivity and Transmission Measurements 66 I 0 i I i I 40 i I i I i I i I 80 120 x (um) i I i I i 160 Figure 5.3: Example of transmitted probe light image recorded by C A M P R T , and a slice of the central heated region, shown with data (circles) and best fit (line). Chapter 5. Pump-probe Reflectivity and Transmission Measurements 67 to the spatial profile of the pump pulse. Since the probe spot size is much larger than the pump spot size, the probe reflectivity & transmission measure spatially resolved range of energy density states. A slice of the image of the heated region show that within a central 20 pm diameter region, the measured change in reflectivity is uniform within 5%. A similar example of the transmitted probe light through the heated foil and the associated slice is shown in Fig. 5.3. 5.1 5.1.1 S-polarized light results Probe reflectivity & transmission First, to minimize the sensitivity to any gradients in the laser heated sample, a set of S-polarized probe measurements were obtained. A key objective in these experiments was to record the temporal history of the creation and evolution of the ISP state. During a set of measurements, the pump laser is maintained at a certain energy level. Meanwhile, probe reflectivity and transmission measurements were made in which the probe pulse was systematically delayed with respect to the pump pulse. Figures 5.4- 5.11 show S-polarized light reflectivity and transmission data at different energy densities. Note that for each figure the average energy density of the data set is given. In all of these figures, "zero" time delay is taken arbitrarily to be the onset of observable changes in the reflectivity and transmission signals. The dashed line at the beginning represents the inital absolutely calibrated values of the unheated gold foil. The minor variation in the initial reflectivity and transmission values for different data sets corresponds to the slight variation in the thickness of each ultrathin foil used. In addition, in each of these figures a point located at about the end of the pump pulse heating is shown with the error bars representing the uncertainity in the measured signals. Both reflectivity and transmission responses seem to follow a general three Chapter 5. Pump-probe Reflectivity and Transmission Measurements 68 phase delevopment which could be catergorized as: (1) laser heating, (2) "quasi-steady-state", and (3) hydrodynamic expansion. First, the heating phase is characterized by the initial rapid drop in the reflectivity and transmission signals. This is attributed to the energy deposition by pump pulse to the gold foil. This heating phase lasts for about 1 ps which corresponds approximately to the pulse length of the pump pulse (1/e peak value of the pulse « 600 fs). The absorption of the pump pulse 2 laser energy is mediated by the electrons which are free to move within the solid gold target. During this heating phase, the initial conduction electrons become highly energized, which in turn allows more electrons to be ionized. Meanwhile, the ions within the gold foil remain in their initial state until the electrons begin to transfer energy to them. The next phase begins when the initial rapid changes in reflectivity and transmission level off and a "plateau" in the signals is observed. Depending on the energy density of the heated foil, this quasi-steady-state lasts between about 2 to 10 ps. It is during this stage that the energetic electrons transfer energy to the cold ions of the lattice. The quasi-steady-state phase is discussed further at the end of this section. It is only when the ions gain enough energy to break the lattice bonds, i.e. melting, that ion movement occurs and the target expands. This expansion phase results in a continuous change in both the reflectivity & transmission signals. As the plasma expands into the vacuum, the probe pulse now interacts with a range of states and thus the ISP approximation is no longer valid. The reflectivity drops until eventually no specular reflection off the expanded states would be detected. The transmission on the other hand steadily increases as the plasma expands and allows more and more light through. Chapter 5. Pump-probe Reflectivity and Transmission Measurements 69 100 I • 1 I • 1 1 ' I 1 1 1 I 1 1 I 1 ' I 1 1 90 80 70 cu 60 50 40 h i . , , • • • • • • • • • • • • • • • • • • = (ZJ c U H 2 4 6 8 10 12 14 Time Delay (ps) Figure 5.4: S-polarized light reflectivity & transmission at the energy density of (4.0 ± 1.8) x 10 5 J/kg. Chapter 5. Pump-probe Reflectivity and Transmission Measurements 70 100 I ' 1 I 1 1 1 1 I ' 1 1 I 1 ' 1 I ' 1 I 1 1 1 1 I ' 1 • 1 I 1 90 ..(••• 80 . •I. ••.«".. 70 60 50 : 40 r. • • 6 I 1 • • i 1 1 I i • • • i i i i i i 1 • • i • i • i • 5 4 3 mf 2 • 1 I 0 r 0 j - 2 i 4 6 8 10 12 14 Time Delay (ps) Figure 5.5: S-polarized light reflectivity & transmission at the density of (1.5 ± 0 . 3 ) x 10 J/kg. 6 Chapter 5. Pump-probe Reflectivity and Transmission Measurements 71 100 i i i i i < i i i i • • • i 1 • • i • • •i 90 80 70 60 50 40 L i ... i • i i • • • • • • • • • • • j—i i i—i i i—i \ i i i i 6f^r 5 i 4 3 2 0 r0 • 2 • • • • 4 6 8 10 Time Delay (ps) 12 i 14 Figure 5.6: S-polarized light reflectivity & transmission at the energy density of (1.8 ±0.4) x 10 J / k g . 6 Chapter 5. Pump-probe Reflectivity and Transmission Measurements 72 100 i 1 1 i 1 1 1 1 i 1 1 i 1 90 80 70 C 60 X 50 : 40 *" • • i • • i i i • • i • • i i i • • i • • • i i i • i • • i i i • * i • • i i i • 1 • • • • • • i • i 5 c o "35 CO C 4 3 2 u H 1 0 •• f • • •» • m • • Jr I • " • • • • 9 V llllll o 4 6 8 10 Time Delay (ps) 12 14 Figure 5.7: S-polarized light reflectivity & transmission at the energy density of (3.5 ± 1.0) x 10 J/kg. 6 Chapter 5. Pump-probe Reflectivity and Transmission Measurements 73 100 1 I • • 1 1 1 I 90 80 v CJ CU CJ P6l 70 60 50 ' 6 i • • • i • • 1 I I I 1 i • • I I I i 1 I I 1 I 1 1 I 5 fc-* s c H 3 : 2 i 1 i 0 _1 0 2 I I I I I ' 1_ 4 6 8 10 Time Delay (ps) 12 I 14 Figure 5.8: S-polarized light reflectivity & transmission at the energy density of (4.2 ± 1.0) x 106 J/kg. Chapter 5. Pump-probe Reflectivity and Transmission Measurements 7 4 100 1 90 i 1 1 1 i i 1 1 1 i 1 1 1 i 1 1 1 i •» • 80 70 60 : 50 4Q r 6 . i ... i ... i ... i ... i ... i ... i . i ii• • i • • • i • 1 • i • • • i 1 • • i • • • i • • • i • • • i 3 SB c 2 1 0 •' i o 4 6 8 10 Time Delay (ps) 12 14 Figure 5.9: S-polarized light reflectivity & transmission at the energy density of (1.0 ± 0 . 2 ) x 10 J/kg. 7 Chapter 5. Pump-probe Reflectivity and Transmission Measurements 75 100 90 T-i—i-r-p-i—i—i—|—i-T-r-i—i—i—i—pi—i—i—|- ri 80 70 cu cu 5= cu 60 50 I 4Q r . i . . . i . . . i . . . i • i i i i ' ••• • 5Mc tf2 s H 4t 2 1 0 -I 0 I I I 2 I I I • • • • 4 6 8 10 Time Delay (ps) 12 14 Figure 5.10: S-polarized light reflectivity & transmission at the energy density of (1.2 ± 0 . 2 ) x 1 0 J/kg. 7 Chapter 5. Pump-probe Reflectivity and Transmission Measurements 76 100 +^ • PP > • PN cu cu i i • • •i • • • i i i i i i iI i i • i 1 1 5t s p "55 •pp Cft 3 C 2 scn ea H 1 1 o 0 2 • • • • 4 6 8 10 Time Delay (ps) i . . . 12 i 14 Figure 5.11: S-polarized light reflectivity & transmission at the energy density of (2.5 ± 0 . 5 ) x 10 J/kg. 7 Chapter 5. Pump-probe Reflectivity and Transmission Measurements 77 5.1.2 A C conductivity results At early times, after the end of the pump pulse, the ISP approximation is expected to be valid. The backcasting procedure is applied for the {R*,T*} measurements made at the end of the heating phase, when the ISP approximation should apply, to obtain the A C conductivity {o ,Oi} r of a single, well-defined state (p, Ae) of warm dense matter. Note that in the inversion procedure, the measured thickness of the ultrathin foil for each set of reflectivity and transmission measurements was used. The real and imaginary parts of the A C conductivity at different energy densities are shown in Fig. 5.12. These results will allow for the key benchmarking of the assorted theoretical electrical conductivity models mentioned earlier. Also, note that the A C conductivity results taken from the reflectivity and transmission measurements during the quasi-steady-state phase are consistent with those taken at the end of the heating pulse, see Fig. 5.13. Using the Drude approximation for a (nearly) free electron gas, additional information about the electron transport may be inferred. For gold this approximation may be justified given: (1) the absence of interband transitions at 800 nm at normal condition, (2) the likely excitation of perhaps only one 5d electron at the highest energy density considered, and (3) the electrical conductivity is effected by conduction electrons near the Fermi surface. The evaluated results for the electron collision time r, D C conductivity cr , and 0 electron density n are shown in Fig. 5.14. These results appear to follow e the general expectation that r and <J are less than their values at normal 0 conditions, while the average ionization increases from the normal conditions value of Z = 1. The normal condition values for gold are listed in Table 5.1. Several processes contribute to the collision time of the conduction electrons. For a metal at room temperature, the mean free paths for electronelectron collisions are longer than 10 A. Two factors are responsible for these 4 long mean free paths. The first is the screening of the Coloumb interaction between electrons. The second, more dominant factor is the Pauli exclusion Chapter 5. Pump-probe Reflectivity and Transmission Measurements 78 8 l i i i i i i i | l l l i l l i i | T 1 A—•— G K 2 <>_ 1 l—(ji—l 0 • 8 -i— • 1 • 1 | i—m—i 6 o — i-tfir 4 1 b~ 2 OL 10 £ I 10 1(T I I I I I 8 10 Ae (J/kg) Figure 5.12: S-polarized results for the real o and imaginary a* part of the r A C conductivity as a function of energy density. The values are obtained from data taken at the end of the heating phase. The arrows point to the conductivity values for gold at normal conditions. Chapter 5. Pump-probe Reflectivity and Transmission Measurements 79 8 4r 0 0 2 3 4 Time Delay (ps) Figure 5.13: S-polarized light results for the real o (circles) and imaginary r Oi (square) parts of A C conductivity within the quasi-steadystate phase for the energy density of (3.5 ± 1.0) x l O J/kg. 6 The dashed lines correspond to the results obtained at end of the heating pulse. Chapter 5. Pump-probe Reflectivity and Transmission Measurements 80 principle. Consider a two-body collision ei + e 2 —> e + e> v 2 (5.3) between an electron in the excited orbital 1 and an electron in the filled orbital 2 in the Fermi sea. Because of the Pauli exclusion principle the orbitals 1' and 2' of the electrons after the collision must lie outside the Fermi sphere since all orbitals within the sphere are already occupied. Thus, only a small fraction of the final orbitals which are compatiable with conservation of energy and momentum are allowed. When a metal is heated by a laser, electrons are rapidly excited to states above the Fermi level. The distribution of the electron Fermi gas is altered allowing more electron-electron collisions to occur which results in a rapid drop in the electron collision time. Meanwhile, the ion structure factor of the metal is modified as phonon excitations begin to increase. Both the electron-electron and electron-ion interactions contribute to the total electron collision time which was measured in the experiment. However, the separation of the individual contributions in the measurements was not possible. 2.8 x l O " * 1 4 T 4.1 x 1 0 s 17 n _1 5.9 x 10 cm,22 e 1 3 Table 5.1: Electron collision time, D C conductivity, and electron density values for gold at normal condition [29]. Using the only available conductivity model developed by Rinker [32], a first comparison of our results with theory was made (Fig. 5.15). There were obvious discrepancies between model calculation and the observation, although both appear to follow a similar behavior as the energy density is increased. A t this time, we are awaiting comparisons with the first-principle Chapter 5. Pump-probe Reflectivity and Transmission Measurements 81 I I I I I I I 11 Vi IT) 1 l _i_uJL 0 ' T I !- J—<y-i -A—i Vi •• 1 £ I— i—«•—I 0 12 10 "a T r r- IT| t- 8 CJ o 21 0 10 10= 10 e 7 10° AE (J/kg) Figure 5.14: S-polarized results for values of collision time r, D C conductivity o" , and electron density 0 function of energy density. Values are derived from the A C conductivity measurements using the Drude approximation. Chapter 5. Pump-probe Reflectivity and Transmission Measurements 82 i © i 11111 i i i i 11111 i i 11111 it 0 1 1 • 1 1 1 •'''' i 0 • * 1 12 1 i 1 1 1 1 I I I II I I 11 J I I I II I lOt "a *t o I 0 10 s ( H 10° Ae(J/kg) 10' 10 B Figure 5.15: S-polarized results compared with calculations from Rinker conductivity (lines) for values of collision time r, D C conductivity cr , and electron density n as a function of energy density. Val0 e ues are derived from the A C conductivity measurements using the Drude approximation. Chapter 5. Pump-probe Reflectivity and Transmission Measurements 83 models based on density functional theory [35] and quantum molecular dynamics [54]. 5.2 5.2.1 P-polarized light results Probe reflectivity & transmission Additional experiments were performed using P-polarized light to probe the gradient effects on the ISP state. Figures 5.16-5.20 show P-polarized light reflectivity and transmission data at similar energy densities to the S-polarized light data. Once again, in these figures "zero" time delay is taken arbitrarily to be the onset of observable changes in the reflectivity & transmission signals, and the dashed line at the beginning represents the initial absolutely calibrated values of the unheated gold foil. The three stages of laser heating, quasi-steady-state, and hydrodynamic expansion are observed in the P-polarized reflectivity & transmission data as well. 5.2.2 A C conductivity results Applying the backcasting inversion technique again, A C conductivity values were obtained for the P-polarized data, see Fig. 5.21. The corresponding collision time, D C conductivity, and electron density for the P-polarized data are shown in Fig. 5.22. {<7,<7;} r However, at the highest energy density a set of values for the measured set of {R*,T*} data was not found by the E M wavesolver. This suggests that at the higher energy density there are noticeable gradient effects, which the P-polarized light would strongly interact with, and the validity of the ISP approximation becomes less certain. Normally gradient effects are due to the expansion of the heated material. However, the A C conductivity results are obtained from measurements at the end of the heating phase and at start of the quasi-steady phase, therefore the foil should not have disassembled and expanded yet. Chapter 5. Pump-probe Reflectivity and Transmission Measurements 84 100 i • • 1 i • • i • • • i • • • i • • • i • * • i * • 1 1 i 80 60 40 CU 20 I 0 20 • • • • • • • • • • • • • • • • • • • • r- 15 = • o scz; PN 10 C 83 S. H 0 4 6 8 10 Time Delay (ps) 12 14 Figure 5.16: P-polarized light reflectivity & transmission at the energy density of (3.5 ± 1.8) x 10 J/kg. 5 Chapter 5. Pump-probe Reflectivity and Transmission Measurements 85 100 i • • • i • 1 1 i 1 1 • i • • • i • • • i • • • i • • • i • 80 60 CU 40 5= cu 20 o r 111111 20 • 11111111111111 i • • • i • • • i • • • i • • • i • • • i • • • i • • • i 15 C O •— 10 cn c ca S- H 0 L-L 0 2 4 6 8 10 12 14 Time Delay (ps) Figure 5.17: P-polarized light reflectivity & transmission at the energy density of (1.4 ± 0 . 3 ) x 10 J/kg. 6 Chapter 5. Pump-probe Reflectivity and Transmission Measurements 86 100 I 1 1 1 I 1 ' 1 I 1 1 1 I I 1 1 1 I 80 • 60 cu 40 cu 20 a 0 20 • • • • • • •i • i • • • i • • • i • • • i' • • • • 1 • • • • • i • • • i • • • i • • • i' 15 c *tz5 «3 lOt c u H 5L 0 • •.. i 0 2 • • • • • 4 6 8 10 Time Delay (ps) • • 12 • • • 14 Figure 5.18: P-polarized light reflectivity & transmission at the energy density of (1.6 ± 0 . 4 ) x 10 J/kg. 6 Chapter 5. Pump-probe Reflectivity and Transmission Measurements 87 100 I • 1 1 I ' 1 I 1 1 1 1 i I 1 • • i 80 60 # 40 » EM cu 20 L 0 L-L 20 i i I • • • • i • • • i • i i 1 1 i 15 a *5! 10 CC S cs H 5T 0 > | •••••• # • • • • • • i i • • • • • • 0 2 • 4 6 8 10 Time Delay (ps) • • • • 12 14 Figure 5.19: P-polarized light reflectivity & transmission at the energy density of (3.5 ± 1.0) x 10 J/kg. 6 Chapter 5. Pump-probe Reflectivity and Transmission Measurements 88 100 I 80 • • • I • • I 1 1 ' 1 I • 1 I • • • I 1 I I I I •# 60 cu 40 cu 20 c *#•* • #g • 0 20 • r r - I . . . i . . . i • • 1 1 I 1 1 1 I 1 1 1 I 1 15 10 6B C ca H 5 0 l i 0 • • • 1 2 1 1 1 1 • 1 1 1 1 • 1 1 1 1 4 6 8 10 Time Delay (ps) 12 14 Figure 5.20: P-polarized light reflectivity & transmission at the energy density of (2.5 ± 0 . 5 ) x 10 J/kg. 7 Chapter 5. Pump-probe Reflectivity and Transmission Measurements 8 9 The more likely cause of such gradient effects is the creation of an "electron sheath" around the foil. When the foil is heated, the excited electrons will have large thermal velocities and will naturally try to escape from the foil. As electrons begin to move away from the foil, the un-equilibrated ions of the lattice will remain in place thus producing a surplus of positive charge in the foil. A "space charge" field would be created which prevents the electrons from moving too far away from the foil [70, 71]. The result is a "steady-state" electron sheath surrounding the foil and altered dielectric values at the edges. For S-polarized light this gradient effect would be small but for P-polarized light it would be large. To understand the effect of gradients on the propagation of an optical wave incident on the plasma, we examine Maxwell's equations with perturbation up to first order solutions [72]. The electron density and velocity are written respectively as n — n + n + n + ... 0 x (5.4) 2 v = v +V!+v 0 + ... 2 (5.5) where n and v are the respective unperturbed values. Maxwell's equations 0 0 are rewritten as V • E = -Airerii V (5.6) V •B = 0 (5.7) x E = — B c (5.8) V x B = — (-en vi) - — E c c (5.9) 0 where the Fourier transform of the electron density, electric and magnetic fields are denoted by the overhead tilde, e.g. E = E(CJ). In addition, the equation of motion is given as (—icu + u)virio = —u Vfii 2 n E 0 (5.10) Chapter 5. Pump-probe Reflectivity and Transmission Measurements 90 where v is the electron-ion collision time, u is the mean square thermal 2 velocity of the electrons, e and m are the electron charge and mass, respectively. Using Eq. 5.6 to substitute for h\ in Eq. 5.10, the first order current density is given as Ji = , , % . , (co E - « V ( V • E ) ) 2 4TT(W + iv) V 2 p 't V v (5.11) ' where co is the plasma frequency. Now combining Eqs. 5.8, 5.9, and 5.11, p the electric field is described by 2 2 -——T-T-VV(V • E ) V x V x E = ^ - e E 4c l (LO + (5.12) c l IV) where the Drude dielectric function e= l - ° Jp l ., LO{LO + (5.13) IV) is used. Both "transverse" electromagnetic and "longitudinal" quasi- electrostatic waves may be supported in plasmas. Thus the electric field may be decomposed into two components, E = E + E J (5.14) ( where E ' is the transverse, electromagnetic part ( V • E ' = 0, V x E* ^ 0), and E is the longitudinal "electrostatic" part ( V • E* ^ 0, V x E* = 0). l Two separate equations for the transverse and longitudinal electric fields are obtained by applying V - and V x on Eq. 5.12, (A + ^ ( A ( V x E*) = - ^ V e x ( E ' + E ' ) + ^)(V-E ) = -^V 1 E .(E' + E«). (5.15) (5.16) Chapter 5. Pump-probe Reflectivity and Transmission Measurements 91 Since V x E* is essentially the magnetic field, and V • E ' is the electric charge density, Eq. 5.15 and 5.16 are recognized as the wave equation of the electromagnetic waves and the electric charge density waves, respectively. The latter are the well-known electron plasma waves or Langmuir waves [13]. For a homogeneous dielectric medium, the two waves are independent of each other since Ve = 0. However, for an inhomogeneous dielectric medium where Ve ^ 0 the electromagnetic and electron plasma waves are coupled. Therefore, an electromagnetic wave incident on a plasma can excite electron plasma waves. The process of transferring energy from the electromagnetic wave to the electron plasma waves is called "resonant absorption". In the situation of the ultrafast laser heated foil target, an electron sheath would provide such a gradient. Referring back to Fig. 3.2, if the probe light is incident in the y — z-plane at an angle 6 = 45° with respect to the surface normal, z-direction, then the y-axis is parallel to the surface, and the x-axis is perpendicular to the plane of incidence. At the center of the heated area, we may assume that the plasma density varies only in the z-direction. Recall that the S-polarized light is perpendicular to the plane of incidence while the P-polarized light is parallel to the plane of incidence. For S-polarized light no interaction with electron plasma waves takes place since it's electric field E is orthogonal to the dielectric gradient Ve, so E • Ve = 0. However, for P-polarized light E • Ve ^ 0, so resonant absorption occurs thus transferring the elecromagnetic wave's energy to an electron plasma wave. 5.3 S- &; P-polarized light data comparison A comparison of the S- & P-polarized light A C conductivity results are shown in Fig. 5.23. At lower energy densities, the A C conductivity values obtained from S- & P-polarization give quite good agreement with each other. As the energy density is increased the two measurements begin to give diverging Chapter 5. Pump-probe Reflectivity and Transmission Measurements 92 8 in © i i i i 11M i T i 41 2 0 8 1 1 1 , 1 i i i i i 05 | < r - -5-i i — i b~ 2 0 10 1 10 1 10 6 i 7 i i i 11 10 J A E (J/kg) Figure 5.21: P-polarized results for the real o and imaginary <7; part of the r A C conductivity as a function of energy density. The values are obtained from data taken at the end of the heating phase. The arrows point to the conductivity values for gold at normal conditions. Chapter 5. Pump-probe Reflectivity and Transmission Measurements 93 i vi 1/1 pH I © 1 i i i i 11111 i i i 11111 I 0 I I I 11 I 2 1 t r o a r 1 _1_LJJ_ 12 © J I 10 : 8• 6• 4 2£ 0 10 I I I I I | I l l l l l l i - OO H 10 f 10 6 7 10° Ae(J/kg) Figure 5.22: P-polarized results for values of collision time r, D C conductivity cr , and electron density Tig clS ct function of energy density. 0 Values are derived from the A C conductivity measurements using the Drude approximation. Chapter 5. Pump-probe Reflectivity and Transmission Measurements 94 results because of the effect of the electron sheath. 5.4 Quasi-steady-state Conventional thinking would assume that once the ultrathin foil has been heated by the pump pulse, it will immediately begin to expand. However, this of course neglects the issue of equilibration between the initially hot electrons with the initially cold ions as stated earlier. The discovery of a quasi-steady-state in both the reflectivity and transmission measurements provides an unexpected and intriguing contradiction to the intuitive result. As seen in both cases, after the inital heating of the sample, there was a definite time duration in which no considerable changes occur. This duration TQSS was found to decrease as the energy density of the ISP was increased, see Fig. 5.25. In earlier work, Schoenlein et al. [7] used femtosecond pump and continuum probe techniques to examine the non-equilibration electron-heating dynamics in gold. A low energy 65 fs pump pulse at a wavelength of 630 nm generated non-equilibrium electron temperatures which relaxed through equilibration with the lattice. Transient-reflectivity of the continuum probe pulse (wavelength range of 580-450 nm) showed that the electron temperatures cooled to the lattice on a time scale of 2-3 ps. More recent work by Hohlfeld et al [8, 73] and Hostetler et al. [9] examined the electron-phonon coupling of femtosecond laser heated gold, and found comparable equilibration times. In addition, using electron diffraction, Siwick et al [10] studied the solid-to-liquid transition of femtosecond laser heated aluminum. Interestingly, they found that following the phase transition, which occurred around 3.5 ps after heating, a long lasting metastable state was observed in which the electron diffraction pattern persisted (up to 50 ps). Recently, Ivanov and Zhigilei examined the non-equilbrium melting of femtosecond heated ultrathin A u foils through numerical simulations [74]. Chapter 5. Pump-probe Reflectivity and Transmission Measurements 95 8 • 1111 1 ' • « " "| i i -i—i—i i 6L vi 4L i—®- T ^21 •fez 0 8 I I III I i 6t b~ ®—I i—e=f VI 4 111ii W— i — i - i ^ - i 21 0 10' 1 • • 10 10° 10 J Ae (J/kg) Figure 5.23: The real and imaginary part of the A C conductivity as a function of energy density. Results for both S-pol (solid circles) and P-pol (open circles) light are given. The arrows point to the conductivity values for gold at normal conditions. Chapter 5. Pump-probe Reflectivity and Transmission Measurements 96 C/5 m pH i—£7l I o 1 1—I 0 rr| T r t/! so o r 0 _ujL 10" 10' 10° Ae (J/kg) Figure 5.24: S- & P-polarized results for values of collision time r, D C conductivity ao, and electron density Tig clS cl function of energy density. Results for both S-polarized (solid circles) and P-polarized (open circles) light are given. Values are derived from the A C conductivity measurements using the Drude approximation. Chapter 5. Pump-probe Reflectivity and Transmission Measurements 97 Figure 5.25: Time duration of quasi-steady-state as a function of increase in energy density as seen in the reflectivity and transmission signals. Both polarizations are shown, S-polarized reflectivity (open circles) & transmission (solid circles), and P-polarized reflectivity (open squares) & transmission (solid squares). Note, due to the limited data points at later times for the lower energy densities (See Fig. 5.4, & 5.16), only a lower bound on the duration of the quasi-steady-state could be inferred. Chapter 5. Pump-probe Reflectivity and Transmission Measurements 98 Using a hybrid atomistic-continuum computational model, the kinetics and microscopic mechanisms of laser melting and disintegration were investigated. This hybrid approach combines the advantages of the two-temperature model and molecular dynamics (MD) method. The two-temperature model provides an adequate description of the laser energy absorption in the electron system, energy exchange between the electrons and phonons, and fast electron heat conduction in the metals, while the M D method is appropriate for simulation of the non-equilibrium processes of lattice superheating, melting, and ablation. The laser excitation and subsequent relaxation of the conduction band electrons are described within a finite difference model. In this formulation the diffusion equation for the electron temperature T is given as e C (T )^ = e e (tf (T )^T ) - g(T e e e e - T<) + S(z,t) (5.17) where C and K are the electron heat capacity and thermal conductivity, e e g is the electron-phonon coupling constant, and S(z,t) is the source term which describes the laser energy deposition during the laser pulse duration. Meanwhile, the lattice ions are described within the M D formulation through the integration of the equation of motion of the ions, given by dr 2 m -~ i where = F +tm vJ i i (5.18) and r-j are the mass and position of an ion i, and F$ is the force act- ing on ion i due to the interatomic action. The formulation also distinguishes between the thermal velocities of the ions vf and the velocities of their collective motion. The coefficient £ is responsible for the energy exchange between the electrons and the lattice, and is related to the electron-phonon coupling constant g by i iZUgVMJl - Ti) E » mi(yj) where Vjv is the volume of each M D cell, and the summation is performed over all the atoms in the cell. In one of the simulations, a 500 A A u foil is Chapter 5. Pump-probe Reflectivity and Transmission Measurements 99 heated with a 200 fs laser pulse resulting an increase in its energy density of Ae ~ 10 J/kg. It was observed that the electron heating of the lattice took 6 ~ 10 ps, after which the entire foil melts within ~ 2 ps. These values seem to correspond reasonably well to time scale of the quasi-steady-state in our reflectivity and transmission measurements. Finally, it has been suggested that perhaps this quasi-steady-state is not actually a non-changing state. Instead, after the sample is heated, it expands in such way that the resulting changes in density and volume within the plasma mimic that of a quasi-steady-state. In other words, the cumulative effect of the expanded multiple states and their associated dielectric values on the propagation of an electromagnetic wave is identical to the effect due to the heated single, uniform state. Such a scenario although not impossible seems to be quite unlikely. In order to further examine this quasi-steady-state phenomenom, supplementary experiments were performed using the diagnostic of frequency domain interferometry (FDI) [75, 76]. Because of its sensitivity to detect very small changes in phase shifts (~ 10~ rads) F D I has been used to determine 3 the gradient scale-length of laser produced plasmas [77, 78], thus making it ideal to investigate the onset of any net expansion in our ISP experiments. Chapter 6. Frequency Domain Interferometry Measurements 100 Chapter 6 Frequency Domain Interferometry Measurements Corroboration of the pump-probe reflectivity & transmission results was performed using Frequency Domain Interferometry (FDI). Applying F D I provides the means to resolve the dynamics of the ultrafast laser heated solid and the subsequent hydrodynamic expansion. Simultaneous measurement of both the amplitude and the phase difference between a pair of femtosecond probe pulses induced by changes in the index of refraction of the heated solid and changes in the path length due to expansion the plasma may be obtained with high spatial and temporal resolution [75]. For our purpose, the additional measurement of phase shift represented the key to elucidating the quasi-steady-state. 6.1 Theory of operation In contrast to interference in the spatial domain, interference in the frequency domain can be observed even when the two pulses are displaced in time by more than the pulse duration. Twin light pulses separated by a time A t are reflected off a laser heated target and imaged upon the entrance slit of a spectrometer, see Fig. 6.1. The two pulses interfere because of the dispersion of the spectrometer grating. The dispersive element act as a pulse stretcher and thus broadens the pulses to make them overlap in time. The first or Chapter 6. Frequency Domain Interferometry Measurements 101 'reference" pulse's electric field is given by E ( t ) = E (t) 0 0 exp(ito t). 0 (6.1) During the time A t the target undergoes a change in refractive index. Therefore, the second or "probe" pulse reflected off the target undergoes a phase change A $ and its amplitude is reduced by a factor R (an effective reflection coefficient). Its electric field is then given by Ei(t) = E (t - At)y/Rexp{i[u) (t 0 0 - At) + A $ ] } . (6.2) The combined intensity of the two pulses is the square of the Fourier transform of the sum of the two fields, I(OJ) = I (u)[l 0 + R(CJ) + 2y/R(u) cos(wAt + A$)] (6.3) which is shown to be the spectrum IQ(UJ) of a single probe pulse modulated by a cosine function. Thus, the reflection coefficient and the phase difference may be simultaneously taken from the amplitude changes and the peak (or valley) shifts of the fringes, respectively. Note that the number of fringes observed is related to the time separation of the two pulses, namely, the further apart the pulses are, the more fringes are seen. However, the time separation cannot be made arbitarily large because as less and less of the pulses overlap in time, the contrast in the fringes decreases until eventually no interference occurs. Chapter 6. Frequency Domain Interferometry Measurements 102 pump pulse Figure 6.1: Schematic diagram of frequency domain interferometry. The reference pulse reflects off the gold foil before it is heated by the pump pulse. The probe pulse is reflected off the heated gold foil. The probe and reference pulses interfere after passing through a dispersive element, e.g. a spectrometer. Chapter 6. Frequency Domain Interferometry Measurements 103 There are two modes of operation possible in FDI, see Fig. 6.2. In the first mode, known as the "absolute mode", the time delay t D between the pump and probe pulse is shorter than the time separation A t of the reference and probe pulses. Thus, the measured phase shift stems from the comparison of the unperturbed to the heated target, and is referred to as an absolute phase shift. In the second mode, known as the "relative mode", to is longer than At, and the measured phase shift is the result of comparing two different heated states, and is referred to as a relative phase shift. To obtain the absolute phase shift of the second heated state compared to the unperturbed state, the relative phase shift must be added to the phase shift associated with the first heated state which the reference pulse interacts with. Chapter 6. Frequency Domain Interferometry Measurements 104 ~*»Pump D Probe Absolute Mode Reference A Direction of Propagation At Pump Reference Probe A A At Relative Mode J Direction of Propagation Figure 6.2: Frequency domain interferometry modes of operation. Chapter 6. Frequency Domain Interferometry Measurements 6.2 105 Fourier Transform analysis method Various techniques have been developed for fringe analysis over the years. The Fourier Transform (FT) method has been shown to be less susceptable to noise due to its inherent separation of phase information from background variation [80]. A fringe-analysis program based on the F T method was created which converted the intensity information from the F D I interferogram (see Fig. 6.3) into a phase map, which allows for the extraction of phase shifts. The F T method of fringe analysis assumes an interferogram with highfrequency fringes. Rewriting Eq. 6.3 and adding the spatial dependence, the intensity of the F D I interferogram is given as / ( C J , y) = B(UJ, y) + A{u, y) cos[u;At + A$(w, y)\ (6.4) or /(*,,) = B( ,v) + ^ W x {exp[zA$(u;, y)} exp[zwAt] + exp[-zA$(w, y)} exp[-zu;At]} (6.5) where A$(w, y) is the desired phase shift value. The terms B(co, y) and A(uj,y) are related to the target reflectivity R(u,y) and laser illumination io(u,y), B{u,y) = I {u,y)[l A(oj,y) = 2I (uj,y)^R(uj,y). 0 + R{co,y)}, 0 (6.6) (6.7) The fringes are perpendicular to the frequency domain, and the separation between fringes is given by the time delay A t between the two pulses. By letting = i^exp[iA$( ,y)], (6.8) C*(co,y) = ^^expHA^y)] (6.9) C{u,y) W Chapter 6. Frequency Domain Interferometry Measurements 106 where C* is a complex conjugate, the intensity simplifies to I(u, y) = B{u, y) + C(u, y) expfiwAt] 4- C*(w, y) exp[-iwAt]. (6.10) Therefore, the application of a one-dimensional (ID) Fourier Transform along the w-direction to Eq. 6.5 yields I(t, y) = FT[/(w, y)] = B(t, y) + C(t - At, y) + C*(t + At, y) (6.11) where F T is the Fourier Transform operator. Next, the background variation B(t, y) is taken out by using a bandpass filter to isolate C(t — A t , y), and the "carrier frequency" A t is removed to give I'(t,y) = BP[I(t,y)] where B P is the bandpass operator. = C(t,y) (6.12) Finally, taking the inverse Fourier Transform we get r{u,y) = TF[I'(t,y)] = C(u,y) (6.13) Now, the phase shift can be obtained from I'(w,y), A$(w, y) — tan" lm{I'(uo,y)} _Re{I'(u,y)} (6.14) However, since the range of values for the arctan function is (—IT, IT), the issue of discontinuities in the calculated phase values has to be addressed [81]. To overcome this problem, a phase unwrapping procedure is applied to obtain a continuous function along the spatial y-direction. To begin, as a reference point the first row is initially given a phase value of zero. Then, the phase difference 5 (assumed to be less than 2iv) between neighbouring points in the y-direction is calculated, 5(A${oj,y)) = A$(u,y)-A${u,y-1) = tan - 1 Im{I'(u,y)/I'{cj,y-l)} [Re{I'(u,y)/I'{u},y-l)} (6.15) Chapter 6. Frequency Domain Interferometry Measurements 107 These phase differences are accumulated along the y-direction to obtain the phase shift profile A$(ui, y) while avoiding the problem of discontinuities. In addition, the phase shifts at each location y is averaged across the o;-direction to obtain a smoother profile, A<%) = (A$(u;,y)) (6.16) Finally, due to distortions from imaging aberrations, the fringes have a slight curvature which is imprinted on the phase shift profile. This curvature contribution is removed by simply subtracting the phase shift profile taken from an interferogram of the initial cold foil target. Hence, the final phase shift profile as a function of the spatial y-direction is given as A $ ( y ) = A$ {y) s A$i(y) (6.17) where AQ (y) is the phase shift of the heated "shot" target and A $ j ( y ) is s the phase shift profile of the "initial" cold target. Chapter 6. Frequency Domain Interferometry Measurements 108 Figure 6.3: Example of F D I data. The interferogram was recorded at the output plane of the spectrometer (shown in Fig. 6.4). Chapter 6. Frequency Domain Interferometry Measurements 6.3 109 FDI Experiment The experimental setup for F D I is quite similar to the original reflectivity and transmission measurements. The modifications for the F D I measurements are shown in Fig. 6.4. A Michelson interferometer is used to split the 800 nm probe pulse into a double pulse of approximately equal intensity with a fixed time separation of A i . As in the original setup, both probe pulses travel along the same path and illuminate the target at a 45° angle. The first pulse is reflected from the unperturbed gold target. The second pulse arrives at the target shortly after it has been heated by the pump pulse. Both specularly reflected probe pulses pass through the entrance slit of the spectrometer which is oriented perpendicular to a the plane of incidence and 1 images a 3 pm slice of the heated target. At the exit of the spectrometer, a C C D camera records the interference pattern, see Fig. 6.3. 2 6.3.1 Phase shift results The phase shift is extracted along the spatial dimension of the interferograms using the F T algorithm. Figure 6.5 shows an example of the typical phase shift profile. On the edges are the unheated regions of gold and so the phase shift between the two pulses is zero as expected. The middle of the phase shift profile corresponds to the central region of the gold foil heated by the pump pulse. As in the reflectivity & transmission measurements the center region shows only a small variation in the phase shift. Once again, by varying the path lengths between the probe beam to the pump beam a temporal history of the phase shift is obtained. Figures 6.66.12 show the temporal history of the S-polarized phase shift for various energy densities. Corroboration of the reflectivity & transmission results is Acton Research Corporation SpectaPro-500: 0.5 Meter Triple Grating Monochroma- 1 tor/Spectagraph (1200 grooves/mm) Photometrics CC 200 system: CH230 CCD camera head (384x576 pixels), CE200 2 camera electronics unit, Nu200 CCD cameral controller Chapter 6. Frequency Domain Interferometry Measurements 110 Figure 6.4: Schematic diagram of femtosecond pump-probe frequency domain interferometry experiments. The twin probe pulses from the Michelson interferometer arrive at the A u foil separated by time A t , and are reflected into the spectrometer where the inteferogram is recorded by a C C D camera. Also, shown are the integrating sphere photodiodes for input (PD1), reflected (PD2) & transmitted (PD3) pump energy, and the C C T V cameras for pump reflectivity ( C A M - P U R ) & transmission ( C A M - P U T ) . Chapter 6. Frequency Domain Interferometry Measurements 111 Figure 6.5: Example of measured F D I phase shift, shown with data (circles) and best fit (line). Chapter 6. Frequency Domain Interferometry Measurements observed in the phase shift measurements. 112 First, the temporal behaviour of the phase shift corresponds well with the previous two measurements. Namely, a quasi-steady-state can clearly be seen to exist between the laser heating and hydrodynamic expansion stages in the phase shift as well. For completeness, P-polarized phase shifts for a few energy densities were obtained as well, see Fig. 6.13- 6.16. The time-resolved phase shift measurements reveal additional information that the previous reflectivity &; transmission results could not observe, especially at later times. First, at the end of the quasi-steady-state phase, the material expands away from the surface, a rarified plasma is created. As a result a density gradient, ranging from vacuum to solid density, is formed. Within this density gradient lies a particular region with a "critical density" n cr in which the plasma frequency uo equals the laser frequency UJ. p Most of the probe light will be reflected from this critical density layer. As the plasma continues to expand, the critical layer moves toward the laser source resulting in the phase shift to increase. Normally, in experiments on laser heated bulk targets the phase shift will continue to increase as more material is ejected from the target. However, the heated ultrathin foil expands outwards at both sides. Two rarefraction waves propagate inward, and as the density of the foil decreases its dielectric value also changes. The individual effects on phase shift due to the motion of the critical density layer and the change in density were modeled using the electromagnetic wavesolver. First, a block of gold is simulated to "move" toward a light source and the phase shift is calculated. Second, the density of a "stationary" block of gold is decreased as a function of time. Using the associated conductivity given by the Rinker model the phase shift is also calculated. The simulated phase shift results are shown in Fig. 6.17. While the motion of the block causes an increase in phase shift, the drop in the block's density will result in a decrease in phase shift. Comparing with our measured phase shift results at later times, we see that during the inital expansion the motion Chapter 6. Frequency Domain Interferometry Measurements 150 • • i 1 1 1 • i 1 1 1 • • i 1 • • 1 • • 1 1 1 1 • 1 • • i 1 113 • •• 100 CQ | 50 t . T • 0 < • • • « « ••I* • • •• • • «• ••>• • • • • • • • •« • • • •• •• • • • • • 0 -50 • 0 • . . . i . 5 . . • • • * • 111 • 10 * 15 20 25 30 Time Delay ( p s ) 35 40 Figure 6.6: S-polarized light F D I phase shift at the energy density of (4.5 ± 1.2) x 10 J/kg. 5 Chapter 6. Frequency Domain Interferometry Measurements 150 11111111111111111111111111111111111 100 I• C/3 u 114 ••• • •• • • • w • X » • • m mm •m* • • • • • mm* * 50 £ < 0L •50 11 • i 0 • 5 10 • 15 20 25 30 Time Delay (ps) • • • • • 35 • • • 40 Figure 6.7: S-polarized light F D I phase shift at the energy density of (1.0 ± 0.2) x 10 J/kg. 6 Chapter 6. Frequency Domain Interferometry Measurements 150 100 | 115 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I I •• • • > • • T •I r * • •» • • « w * *** m • • 50 0 < OL •50 0 5 10 15 20 25 30 Time Delay (ps) 35 40 Figure 6.8: S-polarized light F D I phase shift at the energy density of (1.6 ± 0.3) x 10 J/kg. 6 Chapter 6. Frequency Domain Interferometry Measurements 300 T—'—'— — —I— 1 1 116 r 250 ^200 I 150 100 50 0 •50 J L. j 0 L 5 J L 10 Time Delay (ps) 15 20 Figure 6.9: S-polarized light F D I phase shift at the energy density of (3.8 ± 0.8) x 10 J/kg. 6 Chapter 6. Frequency Domain Interferometry Measurements 400 1— ->——r r 1 117 i— —•— 1 r 300 tc 1 200 • T f • • • £ % 100 0 0 5 10 15 Time Delay (ps) 20 Figure 6.10: S-polarized light FDI phase shift at the energy density of (7.2 ± 1.6) x 10 J/kg. 6 Chapter 6. Frequency Domain Interferometry Measurements 400 1—r T — r 118 -T—i—i—r 300 1 200 | 100 0 L. J 0 J L. 5 l _ 10 Time Delay ( p s ) 15 20 Figure 6.11: S-polarized light F D I phase shift at the energy density of (1.0 ± 0.2) x 10 7 J/kg. Chapter 6. Frequency Domain Interferometry Measurements 700 T — 119 r 600 500 | 400 £300 % 200 100 \ of.. J 5 L. 10 15 Time Delay (ps) 20 Figure 6.12: S-polarized light F D I phase shift at the energy density of (1.3 ± 0.4) x 10 J/kg. 7 Chapter 6. Frequency Domain Interferometry Measurements 150 i T — ' — 100 • • & m i i 120 i % «3 ta 50 e < 0 •50 J i i i i 0 L 5 j 10 Time Delay (ps) L 15 20 Figure 6.13: P-polarized light F D I phase shift at the energy density of (5.3 ± 0.8) x 10 J/kg. s Chapter 6. Frequency Domain Interferometry Measurements 121 250 5 10 15 Time Delay (ps) 20 Figure 6.14: P-polarized light F D I phase shift at the energy density of (1.2 ± 0.2) x 10 J/kg. 6 Chapter 6. Frequency Domain Interferometry Measurements 122 400 300 I I 200 < 100 OL 5 10 Time Delay (ps) 15 20 Figure 6.15: P-polarized light F D I phase shift at the energy density of (4.0 ± 0.7) x 10 J/kg. 6 Chapter 6. Frequency Domain Interferometry Measurements 123 700 5 10 15 Time Delay (ps) 20 Figure 6.16: P-polarized light F D I phase shift at the energy density of (1.0 ± 0.2) x 10 J/kg. 7 Chapter 6. Frequency Domain Interferometry Measurements i i i i I i i i i I i i i i I i i i i I i i 124 i time(ps) Figure 6.17: Phase shift due to motion towards the observer and density change. Calculations are for S-polarized light at 45° using the Rinker conductivity model. Chapter 6. Frequency Domain Interferometry Measurements 125 of the critical density layer seems to be the dominant effect, but as the foil continues to expand the effect due to the change in the density takes over. For the lower energy densities, the increase in phase due to motion seems to be small suggesting a rather slow expansion of the foil before the effect due to density drop takes hold. Meanwhile, a comparsion of the measured phase shift values may be done using the A C conductivity results derived from the reflectivity & transmission measurements. Now, inputting those A C conductivity or dielectric values into the electromagnetic wavesolver, we obtain the phase shifts expected at the various energy densities. Figure 6.18 shows the comparison between the measured and expected phase shifts using the A C conductivity value obtained from the S-polarized light results. At lower energy densities the measured and calculated phase shifts agree well, while at higher energy densities they show noticeable divergence. Once again, the validity of ISP approximation is shown to be be questionable at the higher energy densities. Interesting, the energy density where the measured and calculated phase shifts diverge corresponds to about the same energy density where the S-polarized and P-polarized A C conductivity results diverge as well ( ~ 2 x 10 J/kg). The 6 measured P-polarized phase shifts gave a similar comparison to expected phase shifts, see Fig. 6.19. The quasi-steady-state behavior has been confirmed in 6 different diagnostics, see Fig. 6.20. Using the measured time duration of the quasi-steady-state TQSS, 6.3.2 the disassembly of the ultrafast laser heated solid was investigated. Disassembly of the ultrafast laser heated solid Two-temperature model After the pump laser has excited the electrons, the target consists of a hot electron bath surrounding a cold lattice of ions, which may be described by two separate temperatures, namely T and Tj, respectively. A simple e Chapter 6. Frequency Domain Interferometry Measurements i • ° 300 1 i i i i i i i i "T T-—r* Calculated Measured 200 I—w-H < 126 100 0 10' II it I T J - II • • •' 1 10* 10 7 10 ? Ae (J/kg) Figure 6.18: Comparison of measured and calculated S-polarized phase shift for a (320 ± 20) A foil. Calculated phase shifts are based on the conductivity values which were obtained using S-polarized reflectivity & transmission results. Chapter 6. Frequency Domain Interferometry Measurements • 300 • ° i i 127 1111 Calculated Measured I —* 1 « 200 < 100 0 10' I _l_Li_ 10 10 6 7 10* Ae (J/kg) Figure 6.19: Comparison of measured and calculated P-polarized phase shift for a (320 ± 20) A foil. Calculated phase shifts are based on the conductivity values which were obtained using S-polarized reflectivity & transmission results. Chapter 6. Frequency Domain Interferometry Measurements 100 I ' ' I 1 1 1 1 100 ' 1 1 •XI I I I 1 I I I 1 I I I i I I I I 60 DC -It j •J" §0 "5 40 ca. 20 3.5xl0 J/kg iiiiii 6 0 12 11 11111 11111 3.5xl0 J/kg 6 — 1 5 10 Time Delay (ps) 20 0 * . . < .i 5 10 15 Time Delay (ps) 20 25 11111 3.5xl0 J/kg 10 3.5xl0 J/kg 6 6 20 H 15 6 • I SO <S0 40 I I 128 2 4 10 2 0 1 0 - • • • < • • • • t ' i 11 i 5 10 15 Time Delay (ps) I II I II I i i I l i l t l l n l 20 0 400 300 250 200 3 150 ft < 100 501- ok. -50 0 3.8xl0 J/kg 6 200 Figure 6.20: Quasi-steady-state 20 T ' i ' • 'j i i "i 1 | i i 'i i - 100 • • * 4.0xl0 J/kg 6 (it- i i i i i i 5 10 15 Time Delay (ps) i i i I r 20 300 J- e 5 10 15 Time Delay (ps) 0 5 10 15 Time Delay (ps) 20 behavior observed in 6 diagnostics: S- polarized light reflectivity, transmission, & phase shift, and Ppolarized light reflectivtiy, transmission & phase shift. Also, shown are the corresponding energy densities and quasi-steadystate duration (lines). Chapter 6. Frequency Domain Interferometry Measurements 129 two-temperature (2T) model may be used to calculate the energy exchange between the electrons and the lattice ions. Cooling of the electrons proceeds by electron-phonon coupling and by electron diffusion. The latter effect may be neglected since the foil is uniformly heated by ballastic transport of the electrons. The heated gold foil would retain its solid structure until melting occurs. However, the liquid gold would not necessarily begin to flow freely outwards due to the constraints of short-range ordering and surface tension. When vaporization occurs the short-range ordering and surface tension are lost and the gold would be able to expand freely. Assuming the electron-phonon coupling may be decribed by one linear coupling term, usually called g [82], the temperature relaxation may be modeled by the two coupled equations: C * ^ t i ~ = ~ g [ T e { t ) " T i ( t ) ] + 5 W ' C ^=g[T {t)-T {t% e 6 - 1 8 ) (6.19) d l ( l where C and Ci are the heat capacities of the electrons and ions, repectively. e The term S(t) describes the laser energy deposition and is given as & (t) = Aep -t 2 Au z=- exp L T (6.20) 2 'p where Ae is the excitation energy density, PA is the mass density of gold U (19.3 g/cm ), and T = £ i / / \ / r n 2 in which ty 3 P 2 2 is the full width ait half maximum of the laser pulse (150 fs). The heat capacity of a system at constant volume is defined as where U is the internal energy and T is the temperature. The heat capacity of the lattice ions is governed by the phonon excitation. At ion temperatures above the Debye temperature 9 , the phonon heat capacity is given by the D Chapter 6. Frequency Domain Interferometry Measurements 130 classical Dulong and Petite value. Since for gold OD = 165 K , in our model calculations the heat capacity of the ions is simply Cj = 2.5 x 10 J / m - K 6 3 [83]. On the other hand, the heat capacity of the conduction electrons is more complex. Classical statistical mechanics predict that a free particle should have a heat capacity of 3/2fc, where k is the Boltzmann constant. For a system of n = N/V atoms per unit volume where each atom contributes one valence electron to the electron gas, the electronic heat capacity should be 3/2nk. However, when a solid is heated from absolute zero not every electron gains an energy ~ kT as expected classically. Only those electrons in orbitals e within an energy range of kT of the Fermi level are excited thermally and e gain an energy of the order kT . As a result, at low electron temperatures e (kT <C EF), the electron heat capacity is given by C = A T , where A = 68 e E J/m -K 3 2 e e e [84] for gold. As the electron temperature increases the electron heat capacity should approach the classical value of 3/2nk, which for gold is C = 1.2 x 10 J / m - K . The temperature dependence of the electron heat 6 3 E capacity was obtained from the electron internal energy (6.22) where the chemical potential p, or the Fermi level, varies with electron temperature [85]. As shown in Fig. 6.21, the calculated electron heated capacity follows both the expected low and high electron temperature limits. The electron-phonon coupling constant of metals has been extensively studied. Various techniques such as transient reflectivity and transmission [86], time-resolved photo-emission spectroscopy [87], time-resolved surface plasmon polariton resonance [88], and time-resolved probe beam deflection of surface expansion [89] have been applied. For gold, g has been reported to be as low as 1.1 x 10 16 W / m - K and as high as 4 x 10 3 16 W/m -K. A 3 detailed study the electron-phonon coupling of gold has been compiled by Hohlfeld et al. [8, 62, 73] using various techniques, input energies, and sample Chapter 6. Frequency Domain Interferometry Measurements 131 Figure 6.21: Electronic heat capacity as a function of electron temperature. The dotted lines show the low and high temperature limits of the heat capacity of the electrons. Chapter 6. Frequency Domain Interferometry Measurements 132 thicknesses. They obtained an average electron-phonon coupling constant for gold of g = (2.2 ± 0.3) x 10 16 W / m - K . Their results are summarized in 3 Table 6.1. Thickness (nm) (10 16 g W/m -K) 3 Expermental h(-Opump 1 pump technique (eV) (mJ/cm ) 2 10-500 2.1 ± 0.3 R (2.48 eV) 3.10 2.9 10-500 2.3 ± 0.3 R & T (2.26 eV) 3.10 0.22 100 2.0 ± 0.2 R (2.48 eV) 3.10 0.49-4.20 20-2000 2.0 ± 0.4 damage threshold 3.10 8-120 2500 2± 1 surface expansion 1.55 5.6 Table 6.1: Summary of electron-phonon coupling constants measured measured for A u by Hohlfeld et al. pump, F pump hiO pump — photon energy of = pump fluence, R = transient reflectivity, T = transient transmission, surface expansion = transient surface expansion measured by time-resolved probe beam deflection. However, the use of the 2T model in the form of Eqs. 6.18 & 6.19 would not be completely valid once a phase change occurs. During the process of melting there would not be a smooth increase in the ion temperature as the above coupled equations would predict. Instead, energy transfered from the electrons is needed to break the lattice bonds so the ion temperature would be stationary until melting is complete. Then, the ion temperature would increase until the vaporation occurs. Once again, energy is needed to vaporize the liquid so a pause in the ascension of the ion temperature would occur. Thus, the latent heats of fusion and vaporization should be included in the analysis. Unfortunately, the literature values for the latent heats of fusion and vaporization are for a system in which the electrons and ions are Chapter 6. Frequency Domain Interferometry Measurements 133 already in equilibrium. These values should not be applied to a strongly overdriven non-equilibrium system like the case of the ultrafast laser heated solid. Alternatively, the change in the energy density of the ions, which would steadily increase, is examined. The above ion temperature is treated as an "effective" ion temperature corresponding to the ion energy density, ei = (6.23) PAu Using Eq. 6.23, the coupled equations of the 2T model are modified as: T (t) e d£i(t) (t) + S(t), PA £i (6.24) PAu (6.25) Ci J ' and used to examine the disassembly time of an ultrafast laser heated solid. T (t)-Ei{t) e S-polarized light phase shift and 2T model comparison Because of the sensitivity of F D I and completeness of the data set, the Spolarized light phase shift measurements were used for comparison with the 2T model results. Figure 6.22 shows the quasi-steady-state duration TQSS as a function of the change in energy density Ae observed in the S-polarized light phase shift measurements. In the 2T model, the electron temperature and ion energy density are tracked as a function of time, as shown in Fig. 6.23. A t the end of heating, time t\, the electron temperature will reach its maximum value. A t the later time £2, the ions will reach a critical energy density SD m start to disassemble. The time difference between t d t\ is inferred to be QSS- T AE a n 2 which the foil will For a chosen disassembly energy density sp, a curve of TQSS versus is calculated. This curve is then compared to the measured TQSS values. By varying ED, a best fit to the S-polarized phase shift data is obtained, as Chapter 6. Frequency Domain Interferometry Measurements 134 tc CC CC O 10 10* Ae (J/kg) 10 7 10* Figure 6.22: The duration of quasi-steady-state TQ S from S-polarized light S phase shift measurements as a function of change in energy density Ae. Chapter 6. Frequency Domain Interferometry Measurements 135 shown in Fig. 6.24. Thus, the disassembly energy density of the ions for the ultrafast laser heated gold was found to be er, = (3.1 ± 0.3) x 10 J/kg. 5 Chapter 6. Frequency Domain Interferometry Measurements 136 Figure 6.23: Temporal history of electron temperature and ion energy density for an excitation energy density of Ae = 3.8 x 10 J/kg. 6 Chapter 6. Frequency Domain Interferometry Measurements 137 Figure 6.24: Comparison of the quasi-steady-state duration TQ S between S measured values from S-polarized light phase shift results and 2T model calculations for g = (2.2 ± 0.3) x 10 16 W/m -K. 3 Chapter 7. Significance and Impact 138 Chapter 7 Significance and Impact 7.1 Summary The physics of Warm Dense Matter (WDM) is a frontier science. Warm Dense Matter describes states with expanded to compressed densities of a normal solid and with comparable thermal and Fermi energies. On the one hand, W D M represents condensed matter states at finite temperatures. On the other hand, they are high density, strongly coupled plasmas. Therefore, these complex states are at the crossroads between conventional condensed matter physics and plasma physics where neither theoretical treatments are valid. Understanding the W D M regime requires the exploration of new theoretical models which need to be benchmarked by measurements utilizing novel experimental techniques. One of the greatest challenges in the study of W D M is to obtain measurements of the physical properties of uniform, well-defined states for unambiguous test of theory when no physical means is available to confine such states. To avoid misinterperting measurements made on non-uniform W D M states, a change in paradigm is needed. The benchmarking of theory will be based on direct comparison with single-state data. Physical quantities characterizing the state must also be obtained from direct measurements. To meet this challenge, the Idealized Slab Plasma (ISP) concept was devised. This refers to a high density plasma in planar geometry which can be considered an uniform slab of W D M in which all residual non-uniformities have insignificant impact on the measurement of its properties. In this work, we have successfully demonstrated the ISP approach as a vi- Chapter 7. Significance and Impact 139 able technique to produce single, uniform, well-defined states of W D M based on isochoric heating of ultrathin freestanding, gold foils by a femtosecond pump laser. The states are characterized by their mass and energy densities obtained from direct measurements. This approach has enabled us to study gold at normal solid density with energy densities ranging from 0.25 to 20 M J / k g , that is, from the solid to the plasma regime. Data collected over such a wide range provide extremely stringent tests of theory. The obtained A C conductivity data further allows us to derive values of electron-ion collision time, D C conductivity, and electron density. As a first test of theory, the results are compared with Rinker's conductivity model. While there are noticeable discrepancies between the experimental values and the model results, the similarity in their functional dependence on energy density is interesting given that the Rinker conductivities are based on equilibrium calculations whereas the observation is for a non-equilibrium system of hot electrons and cold ions. More importantly, our data provides an impetus for the development of new first-principle models calculations based on density functional theory and quantum molecular dynamics in the W D M regime. A n exciting development of this project was the observation of the quasisteady-state behavior in the optical properties of femtosecond laser heated gold. The duration of the quasi-steady-state, TQSS, appears to last for 2-20 ps, depending on the excitation energy density. This is completely unexpected since conventional thinking assumes that after the solid is heated hydrodynamic expansion occurs in a time scale of about 1 ps. This intriguing discovery leads to the consideration of the non-thermal melting process which is induced by the excitation of electrons from femtosecond laser heating and the subsequent disassembly of the foil when a liquid-vapor transition occurs. Our results show an initial rapid change in the optical properties, occuring ~ 1 ps, followed by the quasi-steady-state phase where the optical properties remained constant, and then the onset of hydrodynamic ex- Chapter 7. Significance and Impact 140 pansion. This temporal behavior is reminiscent of the non-thermal melting shown in femtosecond laser heated aluminum measurements made by Guo et al. [6]. Whereas thermal melting takes place within several picoseconds, nonthermal melting occurs within the timescale of the electron excitation. The former depends on the electron-phonon equilbriation rate, while the latter is due to the softening of the lattice bonds by the excited electron population. According, the measured single-state within the quasi-steady-state phase is identified to be a non-equilibrium liquid. To provide a detailed examination of the time scale of the quasi-steadystate and disassembly of the ultrafast laser heated solid, we used Frequency Domain Interferometry measurements of phase shift as the diagnostic with the heightened sensitivity to density gradient of hydrodynamic expansion. It is quite remarkable that the observed duration of the quasi-steady-state, QSS, T can be explained with a simple two-temperature model. Our data points to a liquid-plasma transition at a critical disassembly energy density of (3.1 ± 0.3) x 10 J/kg. This is the first observation of the liquid-plasma 5 transition in the ultrafast laser heated solid. In addition to determining the transport properties of finite-temperature condensed matter, this work has also provided insight into its equation-ofstate in terms of ionization and phase transitions. 7.2 Future work Because of the success of this work, extension of the ISP approach may now be considered. First, the frequency dependence of conductivity that has been examined in the theoretical work should now be measured over a range of frequencies to further validate the conductivity models. By focusing a femtosecond probe laser onto a dispersive medium such as water, fused silca [90], or optical fiber [91], a supercontinuum (broadband) [92] femtosecond laser pulse can be produced. Alternatively, high harmonics can be obtained from Chapter 7. Significance and Impact 141 focusing ultrashort laser pulses onto monomer gases, molecules, and solid surfaces [93]. These are readily available probes for broadband measurements. 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F W H M - Full width at half maximum 18. H E D - High Energy Density 19. ISP - Idealized Slab Plasma 20. K D P - Potassium-dihydrogen phosphate 21. K S - Kohn-Sham 22. L L N L - Lawrence Livermore National Laboratories 23. L M - Lee k More 24. M D - Molecular dynamics 25. M F A - Mean field approximation 26. N E H E D - Non-equilibrium High Energy Density 27. N P A - Neutral pseudo-atom 28. QEOS - Quotidian equation-of-state 29. QSS - Quasi-steady-state 30. T E - Transverse electric 31. T F D - Thomas-Fermi-Dirac 32. T M - Transverse magnetic 33. V A S P - Vienna A b initio Simulation Program 34. W D M - Warm Dense Matter 35. USP - Ultra-short pulse 150
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Optical properties of ultrafast laser heater solid Ao, Tommy 2004
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Title | Optical properties of ultrafast laser heater solid |
Creator |
Ao, Tommy |
Date Issued | 2004 |
Description | The regime of Warm Dense Matter (WDM) has emerged as an interdisciplinary field which has drawn broad interest from researchers in plasma physics, condensed matter physics, high pressure science, astrophysics, inertial confinement fusion, as well as material science under extreme conditions. Warm dense matter represents complex states at the convergence of condensed matter physics and plasma physics where neither conventional theoretical descriptions are valid. However, single-state experimental data for the direct testing of new theoretical models within this regime has been difficult to come by. To examine the WDM state, the optical properties of ultrafast laser heated solids were studied. Experiments were performed utilizing a femtosecond laser pump-probe technique to create and examine single-states of WDM. The isochoric heating of freestanding, ultrathin (30 nm), gold foils by a femtosecond pump laser produced uniform, solid-density states of energy densities from 0.25 to 20 MJ/kg. The AC conductivity of such states was determined from reflectivity and transmission measurements of a femtosecond probe as a direct benchmark for transport theory. In addition, observation of the time history of the probe reflectivity and transmission led to the discovery of a quasi-steady-state behavior of the heated sample that suggests the existence of a metastable, disordered phase prior to the disassembly of the solid. To further examine the dynamics of ultrafast laser heated solids, Frequency Domain Interferometry was used to provide an independent observation. |
Extent | 13788500 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-12-02 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085729 |
URI | http://hdl.handle.net/2429/16096 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2004-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
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