Understanding Femtosecond-Pulse Laser Damage Through Fundamental Physics Simulations DISSERTATION

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Understanding Femtosecond-Pulse Laser Damage through Fundamental Physics Simulations

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Robert A. Mitchell III, B.S.

Graduate Program in Physics

The Ohio State University

2015

Dissertation Committee:

Professor Douglass Schumacher, Advisor

Professor Eric Braaten

Professor Gregory Lafyatis

Professor Robert Perry

Robert A. Mitchell III

2015

Abstract

It did not take long after the invention of the laser for the field of laser damage to appear. For several decades researchers have been studying how lasers damage materials, both for the basic scientific understanding of highly nonequilibrium processes as well as for industrial applications. Femtosecond pulse lasers create little collateral damage and a readily reproducible damage pattern. They are easily tailored to desired specifications and are particularly powerful and versatile tools, contributing even more industrial interest in the field.

As with most long-standing fields of research, many theoretical tools have been developed to model the laser damage process, covering a wide range of complexities and regimes of applicability. However, most of the modeling methods developed are either too limited in spatial extent to model the full morphology of the damage crater, or incorporate only a small subset of the important physics and require numerous fitting parameters and assumptions in order to match values interpolated from experimental data.

Demonstrated in this work is the first simulation method capable of fundamentally modeling the full laser damage process, from the laser interaction all the way through to the resolidification of the target, on a large enough scale that can capture the full morphology of the laser damage crater so as to be compared directly to experimental measurements instead of extrapolated values, and all without any fitting parameters. ii

The design, implementation, and testing of this simulation technique, based on a modified version of the particle-in-cell (PIC) method, is presented. For a 60 fs, 1 μm wavelength laser pulse with fluences of 0.5 J/cm2, 1.0 J/cm2, and 2.0 J/cm2 the resulting laser damage craters in copper are shown and, using the same technique applied to experimental crater morphologies, a laser damage fluence threshold is calculated of 0.15

J/cm2, consistent with current experiments performed under conditions similar to those in the simulation.

Lastly, this method is applied to the phenomenon known as LIPSS, or Laser-Induced

Periodic Surface Structures; a problem of fundamental importance that is also of great interest for industrial applications. While LIPSS have been observed for decades in laser damage experiments, the exact physical mechanisms leading to the periodic corrugation on the surface of a target have been highly debated, with no general consensus. Applying this technique to a situation known to create LIPSS in a single shot, the generation of this periodicity is observed, the wavelength of the damage is consistent with experimental measures and, due to the fundamental nature of the simulation method, the physical mechanisms behind LIPSS are examined. The mechanism behind LIPSS formation in the studied regime is shown to be the formation of and interference with an evanescent surface electromagnetic wave known as a surface plasmon-polariton. This shows that not only can this simulation technique model a basic laser damage situation, but it is also flexible and powerful enough to be applied to complex areas of research, allowing for new physical insight in regimes that are difficult to probe experimentally.

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To my family, who stood by me and supported me through more than a decade of higher

education.

Acknowledgments

I am extremely grateful to my advisor Douglass Schumacher, for his guidance, encouragement, and support. I have much to thank him for, starting with picking up a

“high-risk” graduate student such as myself after I was put in the position of looking for a new research advisor in my third year of graduate school. Working with Doug was an honor and provided a great opportunity to learn, as discussions with him were always enlightening. His recommendation to pursue this risky but rewarding topic provided me with immeasurable opportunities, and without his unique combination of supervision and freedom I would not have succeeded as I did.

I would like to thank Eric Braaten, Gregory Lafyatis, and Robert Perry for serving on my thesis committee. I would also like to thank Enam Chowdhury, Kyle Kafka, Vladimir

Ovchinnikov, Chris Orban, Frank King, Matt McMahon, Ginevra Cochran, and all my colleagues for many helpful discussions.

Lastly, I would like to thank the entire High Energy Density Physics group at The

Ohio State University, for sitting through numerous talks and practice talks on a completely foreign topic well outside their regime, and yet always providing constructive feedback and advice.

Vita

May 2004 ...... Newark High School

May 2009 ...... B.S. Physics, University of Delaware

September 2009 to June 2012 ...... Graduate Teaching Associate, Department

of Physics, The Ohio State University

June 2012 to present ...... Graduate Research Associate, Department

of Physics, The Ohio State University

Publications

“Fundamental simulation of laser-induced periodic surface structure,” R. Mitchell, D. W. Schumacher, and E. A. Chowdhury, to be submitted to Phys. Rev. Lett. (2015).

“Modeling crater formation in femtosecond-pulse laser damage from basic principles,” R. Mitchell, D. W. Schumacher, and E. A. Chowdhury, Optics Letters 40, 2189-2192 (2015).

“Using Particle-In-Cell simulations to model femtosecond pulse laser damage,” R. Mitchell, D. W. Schumacher, and E. A. Chowdhury, Proc. SPIE 9237, 92370X (2014).

“Modeling femtosecond pulse laser damage using Particle-In-Cell simulations,” R. Mitchell, D. W. Schumacher, and E. A. Chowdhury, Optical Engineering 53, 122507 (2014).

“Modeling femtosecond pulse laser damage on conductors using Particle-In-Cell simulations,” R. Mitchell, D. W. Schumacher, and E. A. Chowdhury, Proc. SPIE 8885, 88851U (2013).

Presentations

“First principles simulations of laser-induced periodic surface structure using the particle- in-cell method,” R. Mitchell, D. W. Schumacher, and E. A. Chowdhury, SPIE Laser Damage 2015.

“Single-shot femtosecond laser ablation of copper: experiment versus simulation,” E. A. Chowdhury, K. Kafka, R. Mitchell, H. Li, A. Yi, and D. W. Schumacher, SPIE Laser Damage 2015.

“Adapting particle-in-cell simulations to the study of short pulse laser damage,” R. Mitchell, D. W. Schumacher, and E. A. Chowdhury, APS Division of Plasma Physics 2014.

“Using particle-in-cell simulations to model femtosecond pulse laser damage of metals and dielectrics,” R. Mitchell, D. W. Schumacher, and E. A. Chowdhury, SPIE Laser Damage 2014.

“Adapting particle-in-cell simulations to the study of short pulse laser damage,” R. Mitchell, D. W. Schumacher, and E. A. Chowdhury, Ohio Supercomputing Center Statewide Users Group Meeting June 2014.

“A new approach to laser damage using the particle-in-cell method,” D. W. Schumacher, R. Mitchell, E. A. Chowdhury, AFOSR Program Review May 2014.

“Modeling femtosecond pulse laser damage on conductors using particle-in-cell simulations,” R. Mitchell, D. W. Schumacher, and E. A. Chowdhury, SPIE Laser Damage 2013.

“A new technique for fast characterization of intense laser-plasma simulations,” R. Mitchell, C. Orban, V. Ovchinnikov, D. W. Schumacher, and R. Freeman, Ohio Region APS October 2011.

Fields of Study

Major Field: Physics

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Table of Contents

Abstract ...... ii

Acknowledgments...... v

Vita ...... vi

List of Figures ...... xi

Chapter 1: Introduction ...... 1

1.1 Motivation ...... 1

1.2 Brief History of Laser Damage ...... 6

1.3 Laser Induced Periodic Surface Structure ...... 10

Chapter 2: Theory ...... 16

2.1 Laser-Matter Interaction ...... 16

2.1.1 Electron in a Plane Wave ...... 18

2.1.2 Ponderomotive Force ...... 20

2.1.3 Energy Transfer Mechanisms ...... 23

2.1.4 Ionization Processes ...... 27

2.2 Energy Transport ...... 30

2.2.1 Thermal diffusion...... 30 viii

2.2.2 Suprathermal electrons...... 31

2.3 Surface Plasmon Polaritons ...... 32

2.3.1 Dispersion relation ...... 32

2.3.2 Fano and Brewster Modes ...... 35

2.3.3 Grating Coupling ...... 39

Chapter 3: Modeling Laser Damage ...... 41

3.1 Empirical and Rate-Equation Models ...... 41

3.1.1 Two-Temperature Model ...... 42

3.1.2 Thermal Explosion Model ...... 46

3.1.3 Electron Ionization Rate-Equation ...... 48

3.2 Molecular Dynamics Simulations ...... 50

3.2.1 Interatomic Forces ...... 51

3.2.2 Laser Coupling ...... 54

Chapter 4: Particle-In-Cell Simulations ...... 56

4.1 The PIC Method ...... 56

4.2 Stability Requirements ...... 59

4.3 LSP ...... 64

Chapter 5: Modeling Solid Targets with PIC ...... 68

5.1 Early Approaches to Interatomic Forces ...... 69

5.2 Lattice-Based Interatomic Forces Algorithm ...... 74

5.3 Correcting for Discretization Effects ...... 78

5.4 Simple Test Case ...... 82

Chapter 6: A Fundamental Model of Laser Damage ...... 87

6.1 Laser-Target Interaction ...... 88

6.2 Electron-Lattice Thermalization ...... 94

6.3 Atomic Transport ...... 99

6.4 Modeling LIPSS ...... 104

Chapter 7: Conclusion...... 114

References ...... 117

List of Figures

Figure 1: Microstructuring of a stainless steel sheet with 248 nm laser radiation. A) A hole drilled using 25 ns pulses. B) A rectangular hole cut using 120 fs pulses. C) A magnified picture of the exit edge in (B). Image modified from [7]...... 3

Figure 2: Pulsewidth dependence of fused silica damage threshold fluence at 1053 nm

(circles) and 825 nm (diamond). Note the deviation from the τ1/2 scaling for pulse durations shorter than 10 ps. [20] ...... 8

Figure 3: Pulsewidth dependence of a gold-coated grating and mirror at 1053 nm. [20] .. 9

Figure 4: LIPSS generated on a polished stainless steel target, illustrated with a) SEM micrograph, b) AFM micrograph, and c) Fourier transform (both axes are normalized to the laser wavenumber represented by the dotted circle). [31] ...... 11

Figure 5: Illustration of the concept that LIPSS formation is due to the interference between the incoming laser field and a surface scattered wave. [32] ...... 12

Figure 6: Main scenarios for the formation of LIPSS. a) The static model, where a periodic energy input is directly transferred into a corresponding surface corrugation, and b) the dynamic model, where the energy input induces thermodynamic instabilities on the target surface which relaxes via self-organized structure formation. [35] ...... 13

Figure 7: Illustration of the basic processes behind the laser heating of a solid target.

Image modified from [48]...... 17

Figure 8: The coordinate system used to describe the details of SPPs and their generation, shown here to involve p-polarized incident radiation. [57] ...... 33

Figure 9: Illustration of the dispersion curve for an ideal vacuum-metal interface. The

Fano and Brewster modes are plotted in relation to a freespace photon. The shaded region is the region which prohibits coupling to the freespace photons due to momentum conservation...... 38

Figure 10: Pulse length (FWHM) dependence of melting threshold fluences measured experimentally by Stuart et al. [4] on a 200-nm thick Au grating with 1053 nm pulses.

The shaded band results from the TTM, including uncertainties in the constants and the transient change of reflectivity with electron temperature. The dashed line results from

TTM calculations based on constant absorption. [70] ...... 45

Figure 11: The comparison between theoretical results for various ionization models and published experimental data for damage fluence of SiO2 as a function of a) pulse duration and b) laser wavelength. [66] ...... 50

J/cm2, right at the damage threshold for the interacting laser pulse; (g) 500 fs pulse with a fluence of 0.5 J/cm2, double the damage threshold fluence; and (h) 100 ps pulse with a fluence of 0.45 J/cm2, roughly 10% higher than the damage threshold. [81] ...... 53

xii

Figure 13: A) Diagram of the PIC cycle, illustrating the standard order of operations for evaluating a PIC timestep. B) Illustration of the partial discretization of space used in

PIC, with red circles representing the continuous-position macroparticles, and the blue circles representing the nodes on the discretized grid where fields, currents, and densities are evaluated...... 58

Figure 14: Illustration of the Nyquist theory for sampling, shown using (A) a basic sine wave of frequency f, (B) a sampling frequency of f, (C) a sampling frequency of 4f/3, and

(D) a sampling frequency of 2f. The original function is shown as a blue solid line in (A) and a blue dashed line in (B-D), the sampled data points are represented by red circles, and the red solid line in (B-D) represents the function that would be interpolated from the sampled data points...... 61

Figure 15: Organizational diagram of the pressure-based pair-potential algorithm...... 72

Figure 16: Illustration of A) the pair-potential calculation between an atom at a macroparticle’s position (central, blue circle) and its hypothetical nearest neighbors

(outer, dashed circles), and B) the Lennard-Jones pair potential as a function of atomic separation distance...... 77

Figure 17: Color plots of the number density of copper atoms in units of the solid density of copper A) at the start of the simulations shown in (B-C), B) for the simulation without the pair-potential implementation at 200 ps, C) for the simulation with the pair-potential implementation at 200 ps, and D) for a simulation with the pair-potential implementation evaluated for a donut-shaped copper block at 200 ps (initial condition not shown)...... 84

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Figure 18: Illustration of the method proposed by Colombier to handle nonequilibrium electron-electron collisions in metals over a wide range of temperatures as would be experienced in the laser damage process. Shown here is the result of the method for Au.

[115] ...... 92

Figure 19: Illustration of the laser-target interaction for a laser pulse fluence of 2.0 J/cm2, shown via A) the y-component of the incident laser’s magnetic field as it interacts with a flat copper target, the location of which is denoted by a black dashed line, and B) the electron temperature profile in eV immediately after the laser interaction has ceased, approximately 200 fs after the start of the simulation...... 93

The resulting lattice temperature after the electrons and lattice reach thermal equilibrium approximately 80 ps into the thermalization process...... 98

Figure 21: Time resolved dynamics of the ablation plume for a copper target hit with a laser pulse at 2.0 J/cm2 fluence, represented by a density profile in units of the solid density of copper here taken to be 8.5e22 #/cm3. The densities shown correspond to A) the start of the simulation, B) 40 ps, C) 80 ps, D) 120 ps, E) 1.5 ns, F) 2 ns, G) 2.5 ns, and

H) 3 ns...... 102

Figure 22: Density profiles of the resultant damage crater after surface evolution has ceased, in units of the solid density of copper, for A) 0.5 J/cm2, B) 1.0 J/cm2, and C) 2.0

J/cm2...... 103

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Figure 23: Electron temperature after laser excitation of a thin copper foil with a semicircular scratch offset from the laser focus. Horizontal and vertical scales differ to increase clarity of important features...... 108

Figure 24: A) Out-of-plane component of magnetic field after laser excitation showing the signature of surface plasmon-polaritons. B) Electron temperature after laser excitation compared to the time-averaged field magnitude (proportional to the square root of the intensity) along the surface. The laser pulse propagated in the positive-x direction with the incident target surface aligned along x=0...... 109

Figure 25: A) Zoomed in view of the electron temperature at the end of the laser- interaction stage of the simulation. B) Lattice temperature at the end of the TTM stage of the simulation process...... 111

Figure 26: Comparison of A) the initial atomic density profile of the flat copper foil with a semi-circular scratch, and B) The final atomic density profile after the atomic transport stage of the simulation process. All densities are in units of the solid density of copper,

8.5e22 #/cm3...... 112

Chapter 1: Introduction

1.1 Motivation

The development of the chirped-pulse amplification (CPA) technique [1,2] in the mid-1980s allowed for the production of powerful femtosecond laser systems, and brought with it unique challenges and opportunities. One challenge facing researchers today is that increasing the peak power available from these femtosecond laser systems is now largely limited by the damage to key optical components, such as mirrors, polarizers, and diffraction gratings [3]. The tendency of these short-pulse laser systems towards optical damage is not, however, an entirely undesirable situation. Femtosecond laser pulses cause very predictable damage patterns with negligible damage from stress-wave generation, melting, or thermal conduction due to the fact that the characteristic time for energy to be transferred from the hot electrons to the lattice is significantly longer than the interaction time with the laser pulse [4-7], thus minimizing collateral damage and giving considerable control and reproducibility in material damage and ablation. This provides a unique and versatile tool for many applications.

One major application is the micromachining of materials, providing much sharper contours than could be produced via mechanical machining or longer laser pulses [7].

The contrast between drilling through a stainless steel sheet with nanosecond pulses versus femtosecond pulses is illustrated in Figure 1. Figure 1C expands the image on an

1 edge to show the very fine resolution of such a machining method. These sharp contours can be put to many uses. The constant demand for smaller critical dimensions in integrated circuits has required an equivalent reduction in feature size of photomasks, achieved with femtosecond laser pulse removal of defects [8]. A sharp laser focus makes this micromachining a three-dimensional process with a point-like blade, in contrast to the inherent two-dimensional planar nature of a mechanical blade, and has been utilized in writing waveguides within bulk glass for the purpose of three-dimensional optical circuitry [9]. Femtosecond pulses can also be used to create a nano-texturing on the surface of a material, leading to a wide variety of effects such as more efficient photovoltaics [10], the creation of superhydrophobic surfaces [11], and the permanent coloring of metals [12].

There are also numerous medical applications of femtosecond pulse laser damage.

For wavelengths in the visible spectrum, a femtosecond pulse laser is the ideal scalpel for the physically fragile but optically transparent cornea, since its transparent nature allows the energy to be focused wherever it is needed. This allows, for example, corneal transplants without the use of corneal sutures which alter the corneal shape inducing astigmatism [13]. Femtosecond pulse lasers can also be used to improve the laser-assisted in situ keratomileusis (LASIK) procedure, by removing the need for a mechanical blade to slice open a corneal flap to give access to deeper corneal layers, reducing pain, recovery time, and the likelihood of complications [14]. Femtosecond pulse lasers have also seen use in dental work as a replacement for a mechanical drill, decreasing pain from

2 heat and vibration as well as greatly reducing the resulting micro-fractures of the enamel which can act as sources for future dental issues [15].

The broad importance of this field has led to the development of many theoretical tools to aid in the understanding of the underlying physical mechanisms, ranging from fully ab-initio Molecular Dynamics (MD) simulations to simple empirical models. Some

3 of these models have met with considerable success for some aspects of the problem, but prior to this work a fundamental and microscopic model of crater formation had not been demonstrated. Despite the fact that there are already numerous academic and industrial applications of femtosecond pulse damage as well as a wide range of theoretical tools used to study it, there are still many unanswered questions in the field, including: which physical mechanisms are important; under which regime does each mechanism dominate; and what exactly happens at an interface and within the material on the molecular level during laser damage?

The work in this thesis was performed in order to provide a completely new theoretical tool for the study of femtosecond pulse laser damage that serves as a complimentary technique, bridging the gap between the two extremes mentioned above in a way that makes it a unique and versatile tool. The particle-in-cell (PIC) method provides a powerful platform for the development of this new method that directly integrates equations of motion, allowing for zero tunable fitting parameters, but also makes enough simplifying approximations that a full treatment of a realistically sized target and laser is computationally feasible, allowing for direct experimental benchmarking. The result is the first theoretical tool that can handle all phases of the laser damage process, from laser interaction to crater formation, for realistic laser and target parameters that allow for direct experimental verification, and that is flexible enough to model most currently used experimental measures.

The two primary observables of a laser damage experiment are the damage crater, specifically either its appearance or its width and depth, and the damage threshold, the

4 minimum laser fluence required for damage to occur. The damage threshold is clearly important for the case of optical components of high-power laser systems since, by definition, if a laser pulse exceeds this fluence for a particular component then it will be damaged and need to be replaced costing the researcher both money and time.

Understanding this damage threshold then is important for the design of optical components with higher damage thresholds, thus allowing for more powerful laser systems to be built. Further, for the case of precise machining of materials, it is important that the laser intensity be very close to the damage threshold, because the closer the pulse is to the threshold fluence, while still exceeding it, the more precisely material can be ablated from a target due to the width and depth of the damage crater approaching zero

[16]. In special cases, the damage crater can take on interesting structure of its own, such as in laser-induced periodic surface structure (LIPSS) formation [17]. This work will discuss the damage threshold as it relates to various laser and material parameters, as well as the resulting damage crater width, depth, and ablation rate per pulse.

The remainder of this chapter will provide a short overview and history of the field of femtosecond pulse laser damage. Chapter 2 will detail some of the more important physical processes involved in the short-pulse laser interaction. Chapter 3 will then discuss some of the more common theoretical tools currently being utilized to study the laser damage process. Chapter 4 will serve as an introduction to the PIC method, followed by Chapters 5 and 6 which will go into detail about the modifications to the PIC method necessary to tailor it to the laser damage field. After that, Chapter 7 will discuss

5 the most prominent results of this method to date. Finally, Chapter 8 will provide a summary of this work.

1.2 Brief History of Laser Damage

The study of optical ablation and damage of materials began more than two decades before the production of femtosecond laser pulses. Interest began with a study of the internal damage to ruby and Nd:glass crystals which comprised the laser systems of the time [18]. Threshold fluences for which laser damage would appear were measured as a function of many different laser and material parameters in order to pin down the precise mechanism that was leading to the damage [19].

The pulse length, τ, was identified as a key parameter. Numerous experiments [4,20-

25] were done over a laser pulse length range of 10 ps to more than 100 ns long to determine the τ dependence of the fluence threshold for damage of various materials. For pulse lengths greater than approximately 10 ps, experiments showed a reasonable τα scaling with α nominally in the range 0.3 < α < 0.6. The currently accepted explanation for laser damage in this range of pulse lengths involves the incident radiation heating the conduction-band electrons and subsequently transferring their energy to the lattice [20].

For these relatively long pulse durations, the energy in the lattice has time to disperse allowing the material to remain close to being in thermal equilibrium. Damage occurs once the temperature in the lattice becomes high enough to cause melting, boiling, or fracturing in the solid material. Due to the controlling rate being the energy transfer via thermal conduction through the lattice, this model predicts a τ1/2 scaling consistent with

6 most experiments [19,26]. This scaling law can be derived quite simply in one dimension.

The local temperature T(x,t) will obey the heat transport equation:

휕푇 휕2푇 (휌퐶) = 푘 + 푆(푥, 푡) (1.1) 휕푡 휕푥2 where 휌퐶 is the heat capacity per unit volume, 푘 is the thermal conductivity, and 푆(푥, 푡) is a source term used to describe the heating due to the absorbed laser energy. If we want to approximate a distance 퐿 over which a temperature increase of ∆푇 will diffuse over a time 휏, we can ignore the source term and replace the derivatives with ratios, leading to:

∆푇 ∆푇 (휌퐶) = 푘 (1.2) 휏 퐿2 which can easily be rearranged to be read as 퐿 = (퐷휏)1/2, where 퐷 = 푘/휌퐶. This shows that the energy deposited by the laser pulse is spread out over a distance of approximately

L, which is proportional to 휏1/2. Increase the energy in the laser pulse, and it will spread out over the same distance 퐿 if the pulse duration is not changed, and if the energy is high enough it can cause melting. If the pulse duration is increased, the distance over which this energy is spread increases thus decreasing the temperature of the material for a given energy, showing that our threshold fluence for damage will have this τ1/2 dependence.

Figure 2 plots several experimental data points to illustrate this dependence for greater than 10 ps pulses. Also exemplified by Figure 2 is the clear distinction between pulses that are greater than a few picoseconds and sub-picosecond in duration.

Figure 2: Pulsewidth dependence of fused silica damage threshold fluence at 1053 nm. Note the deviation from the τ1/2 scaling for pulse durations shorter than 10 ps. Experimental data from [20].

Figure 3: Pulsewidth dependence of a gold-coated grating and mirror at 1053 nm. Experimental data from [20].

The deviation from 휏1/2 scaling is even more distinct in Figure 3. The experiments plotted in Figure 2 and Figure 3 were performed with a 1053-nm Ti:Sapphire CPA laser system, with variable pulse widths from 140 fs to 1 ns focused to a spot size of 0.3-1.0 mm diameter (e-2 intensity) [20]. This deviation from the 휏1/2 dependence, first seen by

Soileau [23] for pulse lengths of 4-10 ps incident on a dielectric, was one of the first clues for researchers to the fact that a different, non-thermal mechanism is responsible for

9 damage in this regime, and sparked the now intense interest in damage mechanisms from femtosecond pulses.

1.3 Laser Induced Periodic Surface Structure

Not long after the invention of the laser an interesting discovery was made in the field of laser damage. In 1965, Birnbaum [27] was the first to observe a phenomenon now known as laser-induced periodic surface structure (LIPSS), also frequently referred to as ripples, by irradiating semiconductors with a pulsed ruby laser. LIPSS is a series of parallel and periodic grating-like surface structures in the damage pattern of a material produced by a single near-threshold laser beam, and is ubiquitous in that it has been shown to form on nearly all materials, confirmed independently for metals [28], semiconductors [29], and insulators [30]. Figure 4 shows an experimental image of

LIPSS formation on a polished stainless steel surface under irradiation with several 300 fs laser pulses at a central wavelength of 1025 nm [31].

However, despite half a century of study, the mechanism by which LIPSS forms is still under debate, and a thorough understanding of the underlying physics is missing.

Historically, the phenomenon of LIPSS formation was attributed to the interference of the incoming laser field with a surface scattering wave [28,32], as illustrated in Figure 5. In

11 this picture, the LIPSS period, Λ, would be equal to the laser wavelength, 휆, at normal incidence, and for arbitrary laser angle of incidence (AOI), 휃, can be represented as:

휆 Λ = (1.3) (1 ± sin θ)

Figure 5: Illustration of the concept that LIPSS formation is due to the interference between the incoming laser field and the scattered wave from a surface defect.

While this picture of LIPSS formation is conceptually and mathematically simple, and thus a rather appealing description of the phenomenon, careful inspection shows it to be inaccurate. Experimentally, LIPSS periods have been measured in certain circumstances to be much smaller than Equation (1.3) would predict, especially near normal incidence [33]. Indeed, it turns out this picture of LIPSS formation is not physically consistent, as at near-normal incidence the scattered wave would have to be

12 longitudinally polarized to interfere with the incoming laser beam, and such waves do not satisfy the Maxwell equations [34].

Currently, there are many competing theories for the generation of LIPSS, often discriminating between two regimes of LIPSS generation, described as low spatial frequency LIPSS (LSFL) and high spatial frequency LIPSS (HSFL), for LIPSS periods

Λ > 0.5λ, and Λ < 0.5λ, respectively. Most models for LIPSS generation can be broken down into one of two categories, the “Static” category, where the LIPSS structure mimics a passively produced modulated energy input, and the “Dynamic” category, where the

LIPSS structure forms via self-organization processes driven by surface instabilities [35].

These two categories are illustrated in Figure 6.

Figure 6: The two main classifications of mechanisms for the formation of LIPSS. A) The static model, where a periodic energy input is directly transferred into a corresponding surface corrugation. B) The dynamic model, where the energy input induces thermodynamic instabilities on the target surface which relaxes via self- organized structure formation.

Discussing first the static models, there is an important distinction between LSFL and

HSFL as no one theory in its most basic form can account for both. The generation of

LSFL, particularly in the case of metals, has long been attributed to the interference of the incoming laser beam with a surface electromagnetic wave (SEW) generated at a rough surface [31,34,36], and recently most often considered to be a surface plasmon polariton (SPP) [33,37-40]. The generation of HSFL is less well understood and many static mechanisms have been proposed to explain it, such as transient changes in the optical properties of a material during laser excitation [41], second-harmonic generation

(SHG) [42-44], and specific types of plasmon modes [45,46]. There is also the dynamic class of models to describe LIPSS generation [35,47]. The most attractive feature of these models is the ability to explain both LSFL and HSFL within one unified model [35].

While it is early in the development of these models, they have been able to show a qualitative agreement with experiment.

The above is not a complete list, but it is clear that there is no real consensus on

LIPSS generation mechanisms despite decades of development. There is a good reason for this though: the laser damage process is quite complex. Short pulse laser damage involves multiple physical mechanisms working together on femtosecond to nanosecond timescales in a highly nonlinear, nonequilibrium regime, with the laser interacting with a target undergoing thermal, electronic, and structural changes. Experimentally, this is a very difficult regime to probe, especially during the laser interaction itself. From a theoretical standpoint, until this work there was no theory that could model the full laser damage process in a fundamental way, and on a large enough scale to capture this

14 phenomenon. Here the term fundamental is being used to mean that standard equations of motion are integrated directly, eliminating any need for tunable or fitting parameters. For this reason, LIPSS generation became one of the first major topics of study using the new simulation method described later in this work.

Chapter 2: Theory

This chapter covers several of the more important basic theoretical elements associated with the laser damage process. Section (2.1) covers the more important elements of the laser interaction, starting with how an electron behaves in an electromagnetic wave, continuing on with a discussion of the methods by which the laser energy couples to the conduction band electrons in a solid, and lastly a discussion of the relevant ionization mechanisms that allow for efficient energy coupling to insulators and semiconductors. Section (2.2) handles how the energy is transported within the solid, discussing thermal diffusion and equilibration, first of the electrons themselves and then of the target as a whole, followed by a discussion of the suprathermal electrons created by the laser interaction and their effect on the target. Finally, Section (2.3) goes into the basic theory of the surface plasmon polariton (SPP) and how it relates to laser damage as one of the leading mechanisms of LIPSS generation. Throughout this thesis, SI units will be utilized unless otherwise specified.

2.1 Laser-Matter Interaction

The interaction of a laser field and a solid target involves many complex processes working together. For metals, the process begins with the laser interacting with the conduction band electrons and rapidly heating them. The electrons then simultaneously

16 begin to spread their energy through the target via thermal diffusion, as well as coupling their energy to the lattice. In the case of insulators or semiconductors, which do not naturally have a population of electrons in the conduction band of the material, for efficient energy transfer the electrons must first be excited to the conduction band, after which the process may continue similarly to that for a metal. This process is illustrated in

Figure 7.

Figure 7: Illustration of the basic processes behind the laser heating of a solid target.

To facilitate an understanding of the physical mechanisms involved in such a process, the following sections each focus on a small aspect of the problem, starting with the most basic of interactions: an electron in a plane wave.

2.1.1 Electron in a Plane Wave

For the discussion in this section, utilizing c.g.s. (Gaussian) units, consider the simple electromagnetic plane wave described by:

퐸⃗ (푟 , 푡) = 퐸0푠푖푛(휔푡 − 푘푧)푥̂ (2.1a)

퐵⃗ (푟 , 푡) = 퐵0푠푖푛(휔푡 − 푘푧)푦̂ (2.1b) where 퐸0 and 퐵0 are the respective magnitudes of the electric and magnetic fields, 퐸⃗ and

퐵⃗ , 휔 is the frequency, and 푘 is the wavenumber. In an electromagnetic field, an electron will experience a force 퐹 described by the Lorentz equation:

푣 퐹 = −푒 (퐸⃗ + × 퐵⃗ ) (2.2) 푐 where 푒 is the electric charge, 푐 is the speed of light, and 푣 is the electron velocity. To evaluate Equations (2.1a), (2.1b), and (2.2) it is helpful to first know if the electron motion is relativistic.

Within the field of plasma physics, it is common to define a normalized vector potential 푎0 to denote the strength of the laser field and determine whether or not the field and subsequent particle interactions will be relativistic, with 푎0 ≪ 1 representing a non- relativistic regime and 푎0 ≥ 1 representing a relativistic regime. The value 푎0 is defined as:

푒퐸 푎 = 0 (2.3) 0 푚휔푐 where 푚 is the rest mass of the electron. By using the definition of the laser intensity 퐼 in terms of the Poynting vector 푆 , 퐼 = 〈|푆 |〉, it is possible using Equations (2.1a), (2.1b), and

(2.3) to derive a relation between 푎0 and intensity:

1 (2.4a) 퐼 = 〈|푆 |〉 = 〈|퐸⃗ × 퐵⃗ |〉 휇0

1 2 (2.4b) 퐼 = |퐸0| 2휇0푐

푎2 (2.4c) 퐼 = 1.37 × 1018 0 [푊 ∙ 휇푚2/푐푚2] 휆2 where 휆 is the wavelength in µm. Rearranging Equation (2.4c), we get:

퐼휆2 푎 = √ (2.5) 0 1.37 × 1018 푊 ∙ 휇푚2/푐푚2

Equation (2.5) provides a measure to determine if the laser interaction is relativistic or non-relativistic, based only on laser wavelength and intensity. While there is no

“standard” value for laser intensity and wavelength in the field of laser damage, a common range of laser parameters used in experiment is 퐼 ≈ 109 푊/푐푚2 to 퐼 ≈

1013 푊/푐푚2, depending on pulse length, and 휆 ≈ 250 푛푚 to 휆 ≈ 10.6 휇푚. The

−6 corresponding value of the normalized vector potential ranges from 푎0 ≈ 5 × 10 to

푎0 ≈ 0.03. Even on the most extreme end of the ranges given here, it is clear that the near-threshold laser damage process of primary interest to academic and industrial uses is well into the non-relativistic regime.

In the highly non-relativistic regime the quantity |푣 |/푐 ≈ 0, and Equation (2.2) can be simplified and with the use of Equation (2.1a) be rewritten as:

퐹 = −푒퐸0푠푖푛(휔푡 − 푘푧)푥̂ (2.6) allowing for an exact analytic solution for the equation of motion of an electron in an infinite plane wave:

푣 = 푣표푠푐푐표푠(휔푡 − 푘푧)푥̂ + ⃗푣⃗⃗ 푖 (2.7) where 푣표푠푐 = 푒퐸0/푚휔, and 푣⃗⃗⃗ 푖 is the initial velocity of the electron. It can be seen from

Equation (2.7) that in the non-relativistic regime an electron initially at rest interacting with a plane wave will oscillate sinusoidally about an equilibrium position, neither gaining nor losing energy on cycle-average. Even in the case where the electron is not stationary initially, it will have a constant drift velocity based on its initial momentum in addition to its oscillatory motion without cycle-averaged energy gain. If the rise and fall time of the laser pulse is slow compared to an optical cycle, this means that no energy can be coupled from the laser field to the electron. In order to examine how a laser heats a target, other effects must be accounted for.

2.1.2 Ponderomotive Force

The above section shows that a monochromatic planewave cannot couple energy to a free electron, but a real laser pulse is not a monochromatic plane wave. For purposes of this discussion, a laser pulse can reasonably be modeled as Gaussian in both spatial and temporal profiles. To examine the effect of a spatial variance in the polarization direction of the field, rewrite Equation (2.6) with the assumption that the electron is initially at rest at the origin so that it now reads:

퐹 = −푒퐸(푥)푠푖푛(휔푡)푥̂ (2.8) where 퐸(푥) is the magnitude of the electric field as a function of our coordinate 푥. If the gradient of 퐸(푥) isn’t too large, then we can separate out the particle motion into a slow component and a fast component, such that 푥 = 푥0 + 푥1, where 푥0 is the slow drift motion and 푥1 represents the rapid oscillations in the field, and 푥0 ≫ 푥1. With this assumption Equation (2.8) can be rewritten and Taylor expanded to read:

푒 푑 푥0̈ + 푥1̈ = − [퐸(푥0) + 푥1 퐸(푥0)] 푠푖푛(휔푡) (2.9) 푚 푑푥0

Using 푥0 ≫ 푥1, and 푥0̈ ≪ 푥1̈ : 푒 푥̈ = − 퐸(푥 )푠푖푛(휔푡) (2.10) 1 푚 0

The above approximations specify that on the timescale that 푥1 oscillates, 푥0 should be nearly constant. Therefore, Equation (2.10) can be integrated directly to arrive at:

푒 푥 = 퐸(푥 )푠푖푛(휔푡) (2.11) 1 푚휔2 0

Equation (2.11) can then be substituted into Equation (2.9), and after time averaging over a laser cycle can be written as:

1 푒 2 푑 푥0̈ = − ( ) 퐸(푥0) 퐸(푥0) (2.12) 2 푚휔 푑푥0 which can be simplified to read:

푒2 푑 2 (2.13) 푥0̈ = − 2 2 [퐸(푥0)] 4푚 휔 푑푥0

Upon being evaluated for 푥 = 푥0, this can ultimately be rewritten as:

2 푒 2 퐹⃗⃗⃗ (푥 ) = − ∇⃗⃗ |퐸⃗ | (푥 ) (2.14) 푝 4푚휔2

Equation (2.14) is called the ponderomotive force, and represents a time-averaged drift force in the direction of lower intensity. It is worth noting that unlike most forces based on electromagnetic interactions the ponderomotive force is not dependent on the sign of the electric charge, but instead will always be directed towards lower field intensities. This phenomenon has a very simple physical explanation. If you imagine an electron near the peak intensity, it will experience a large force outwards towards lower intensity. As that electron is being turned around and sent back towards higher intensities, the field is slightly smaller in magnitude and thus the force is insufficient to restore it to its original position. This results in a slow drift towards lower intensities.

This effect now allows for some energy to be permanently transferred from the laser to the electrons via gradients in the field. For nonrelativistic intensities this effect, as described, can only accelerate a particle in the polarization direction of the field.

However, even though non-relativistic, inclusion of the effect of the magnetic field in a

3D analysis leads to a ponderomotive force with the same magnitude as given, but now acting along any transverse intensity gradient normal to the local propagation direction.

The ponderomotive force can thus establish a characteristic energy scale, but is insufficient to explain electron heating in laser damage. For a pulse incident normal to a planar target, the force direction is along the surface of the target. In the relativistic case the ponderomotive force can cause significant acceleration into the target as well, but in this regime that effect is negligible. At near-grazing incidence, the ponderomotive force acts inward to the target, but the ponderomotive effect often cannot account for the full 22 energy transfer between the laser and the conduction band electrons. To fully understand how a laser transfers its energy to a target, it is necessary to include several more processes.

2.1.3 Energy Transfer Mechanisms

The above sections illustrate the behavior of a single free electron interacting with a laser field, but in a laser damage experiment there will not be only a single electron.

Instead, there will be a target comprised of many electrons and ions all interacting with each other. The laser field will be incident on a step-like gradient between vacuum and the solid density target. For this section, assume that there is a pre-existing population of electrons in the conduction band of the material, as would be the case for a metal target.

When the laser reaches the step-like density gradient between the vacuum and the solid density target, it will be unable to penetrate the material and its intensity will drop exponentially at a rate given by the skin depth of the material, 푙푠, approximated by

푙푠 ≈ 푐/휔푝 [49]. Here, 휔푝 is the plasma frequency of the metal, defined to be:

2 2 푛푒푒 휔푝 = (2.15) 휀0푚푒 where 푛푒 is the electron number density, 푒 is the electric charge, and 푚푒 is the electron mass. If the laser is normally incident on the metal, then the electrons near the surface, roughly within the skin depth, are driven to oscillate together along the surface with the quiver velocity 푣표푠푐 first introduced in Equation (2.7). However, collisional effects between electrons and ions prevent the electrons from oscillating freely, causing the electrons to dephase with respect to the laser field and allowing energy to permanently

23 couple to the electrons. This collisional absorption can be easily approximated. For the case of a laser normally incident on a material in vacuum, the Fresnel equations for reflection simplify to:

1 − 푛 2 푅 = | | (2.16) 1 + 푛

Since any energy not reflected must be absorbed, we can calculate an absorption fraction 휂푐 of the laser light from collision absorption at normal incidence to be:

1 − 푛 2 휂 = 1 − 푅 = 1 − | | (2.17) 푐 1 + 푛

A relevant example of a system similar to that which will be studied in Chapter 6 is a copper plate with reflectance 푅 ≈ 0.95 at a wavelength of 1 µm, leading to an absorption fraction of 휂푐 ≈ 5%.

However, if the laser is incident at some angle 휃 measured with respect to the normal, then there will be a non-zero component of the electric field normal to the surface of the target. In this case, electrons near the surface can be pulled out rapidly into the vacuum and then, as the electric field switches in direction, thrown back into the target where the laser can no longer penetrate. This simple mechanism for transferring energy to the electrons is called Brunel Vacuum Heating, and was first described by Brunel in 1987

[50].

Brunel heating for the low intensities important to laser damage can be described quite effectively with a very simple model. Imagine there is a laser field 퐸⃗ incident on a metal target at an angle 휃 relative to target normal. Assuming a well-polished, perfectly

24 reflecting surface, there will develop a standing wave in the direction normal to the surface with a maximum field strength given by:

퐸푛 = 2|퐸⃗ |sin휃 (2.18)

Now suppose that the field given in Equation (2.18) pulls a sheet of electrons off the front surface of the target some distance ∆푥. This creates a situation similar to that of a capacitor: the surface number density will be given by 휎 = 푛푒∆푥, which will lead to an electric field between this sheet of electrons and the target surface given by:

푒휎 퐸푐 = (2.19) 2휀0

Using the capacitor field in Equation (2.19), and relating it to the standing wave field normal to the surface given by Equation (2.18), we can solve for the surface number density 휎:

4휀 |퐸⃗ | 휎 = 0 sin휃 (2.20) 푒

After these electrons have been pulled off the surface, they will be accelerated back towards the target as the electric field reverses in direction. Upon again reaching the surface, we can approximate their velocity in a way similar to the calculation of 푣표푠푐 as shown in Equation (2.7), giving us an approximate return velocity of:

2푒|퐸⃗ |sin휃 푣 = (2.21) 푟 푚휔

If one further assumes that upon reaching the target all of the electrons penetrate past the skin depth, escaping the laser field, and propagate unhindered into the target, then we can approximate the average energy density absorbed per laser period 휏 as:

휎 1 푃 = 푚 푣2 (2.22) 푐 휏 2 푒 푟

Inserting Equations (2.20) and (2.21) into the average energy density given in

Equation (2.22), we have:

3 4푒휀0 3 푃푐 = |퐸⃗ | sin 휃 (2.23) 휋푚푒휔

This analysis can be taken further to define an absorption fraction of the total laser energy. Given that the energy density in the electric field incident on the target is given by:

2 푃퐿 = 휀0푐|퐸⃗ | cos휃 (2.24) then we can define an absorption fraction 휂퐵 for the Brunel mechanism as:

3 푃푐 4푒 sin 휃 휂퐵 = = |퐸⃗ | (2.25) 푃퐿 휋푚푒푐휔 cos휃 which when rewritten in terms of the laser parameter 푎0 defined in Equation (2.3), becomes:

4 sin3휃 휂 = 푎 (2.26) 퐵 휋 0 cos휃

Equation (2.26) provides an approximation for the fraction of the laser energy absorbed via this Brunel vacuum heating mechanism. A close look at the above equation shows that absorption can surpass 100% for either large 푎0 or large 휃. In the regime of laser damage, a large 푎0 is not a concern, and this equation is reasonably accurate up to angles of approximately 60˚ [49]. A more accurate estimate can be achieved by taking into account both imperfect reflectivity and relativistic effects, but that is beyond the scope of this work. For an example relevant to this work, a situation similar to that which 26 will be shown in Chapter 6 involves a laser strength of 푎0 ≈ 0.002, at an angle of 15˚,

−5 leading to an absorption fraction of 휂퐵 = 4.57 ∙ 10 , or ~0.46%. This is significantly smaller than the example absorption fraction from collisional effects, but from Equation

(2.26), clearly increases rapidly with angle of incidence, becoming dominant at near- grazing incidence.

2.1.4 Ionization Processes

For the case of a laser incident on a metal target with a pre-existing population of electrons in the conduction band, the laser damage process begins with the energy transfer to the electrons described in Section (2.1.3). However, for the laser damage of semiconductors and insulators there is no initial population of electrons in the conduction band of the material. In order for nontrivial energy transfer to occur, the bound electrons must first be excited to the conduction band.

In the standard regime of laser damage, there are multiple pathways to ionize the target and free the electrons to the conduction band. To determine which field ionization method will dominate the process, there exists the Keldysh parameter [51], given by:

휀 1/2 훾 = ( 푖 ) (2.27) 2푈푝 where 휀푖 is the energy of the electronic state, and 푈푝 is the ponderomotive potential defined as 푈푝 = − ∫ 퐹푝푑푥, where 퐹푝 is the ponderomotive force defined in Equation

(2.14). For 훾 ≪ 1, tunneling ionization is dominant, where the electromagnetic field from the laser distorts the Coulomb barrier trapping the electron significantly enough for the electron to tunnel out. On the other extreme, for 훾 ≫ 1 multiphoton ionization dominates

27 for which several photons are absorbed near the same time to bridge the energy gap between the current state of the electron and the conduction band. For 훾 ≈ 1, a combination of these two mechanisms occur together.

Once a population of electrons has been established in the conduction band from photoionization processes, further ionization may occur via avalanche ionization. In this process, free electrons in the conduction band get accelerated directly by the laser field and subsequently collide with bound electrons, transferring some of their energy and freeing the bound electrons. Using a quasi-static approach, it is possible to roughly approximate the energy gained by the initial electrons in the laser field and subsequently the number of electrons excited via avalanche ionization. For an electron initially at rest, the energy ∆푄 acquired via interaction with an electric field is, by the work-energy theorem, equal to the work done by the field on the electron:

∆푄 = 푊 = ∫ 퐹 ∙ 푑푥 ≈ 푒퐸푑 (2.28) where 푑 is the distance over which the electron is accelerated, which can be approximated using basic kinematics to be:

1 1 푒퐸 푑 = 푎푡2 = ( ) 푡2 (2.29) 2 2 푚

Assuming the amount of energy lost in a collision is negligible, the time 푡 over which the electron is accelerated can be rewritten as the average time between collisions 휏푐.

Approximating the number of times a collision has occurred as 푁 ≈ 휏/휏푐, where 휏 is the laser period, and combining Equations (2.28) and (2.29) one can approximate the energy

28 gained by an electron between collision events. Multiplying this energy gain by the number of collisions 푁, the approximate energy gained during a full laser period is:

(푒퐸)2 ∆푄 = 휏푐휏 (2.30) 2푚푒

When the energy gain approximated by Equation (2.30) exceeds the binding energy of the electrons in the material, a collision may trigger ionization. Approximating the power gain over a laser cycle by 푃 = ∆푄/휏, it is possible to define a differential equation to approximate the number of electrons 푁 ionized within the material as:

푑푁(푡) 푓푃 푓 (푒퐸)2 = 푁 = 휏푐푁(푡) (2.31) 푑푡 푊 푊 2푚푒 where 푓 is the fraction of the electron energy that goes into the ionization process and 푊 is the ionization potential of the material. The solution of Equation (2.31) is simply:

푓푃푡/푊 푁(푡) = 푁0푒 (2.32)

Equation (2.32) shows that once an initial population of electrons is present, in the case of an insulator or semiconductor created first from photoionization processes, avalanche ionization will cause the number of electrons present in the conduction band to increase exponentially either until the material is fully ionized or the ionization potential becomes too large for the collisions to further ionize the material.

For a laser fluence above the damage threshold, shortly after the laser pulse begins interacting with the target, the above listed ionization processes will quickly populate the conduction band of the material. This will fulfill the requirement of Section (2.1.3) that there exists a high density electron population in the conduction band, after which the

29 laser may couple energy to the electrons in the same way as it would for a metal as described previously.

2.2 Energy Transport

After the laser has heated the conduction band electrons in the target, which were either already present as in the case of a metal or were first excited to that band as discussed in Section (2.1.4) in the case of a semiconductor or insulator, the energy must be transported within the material and transferred to the lattice. For a long-pulse laser, of several tens of picoseconds or longer in pulse duration, this would happen simultaneously with the electron heating. However, for the short-pulse lasers of subpicosecond duration discussed here, these processes occur on very different timescales such that the electron heating happens much faster than the electron-phonon relaxation time. To discuss the energy transport in the target after a short pulse laser interaction, we must separate out the two distinct processes involved, the thermal processes and the suprathermal processes.

2.2.1 Thermal diffusion

As discussed in Section (1.2), the key feature to short pulse damage is its distinct non- thermal behavior resulting from the fact that the electron and lattice temperatures are far from equilibrium. However, that doesn’t mean that there are no thermal processes involved at all, only that they aren’t as simple as those discussed in Section (1.2). One distinction between long pulse and short pulse damage is that in long pulse damage, the electrons and lattice remain in equilibrium during the pulse, whereas for short pulse damage the electrons and lattice are never in equilibrium during the laser pulse [7].

However, the lattice will always be in a thermal distribution, and not long after the laser interaction the electrons will also form a thermal distribution within themselves via

Coulomb collisions, but at a higher temperature than the ions. So while it is not possible to treat the entire system as if it were in thermal equilibrium, the two components individually are in a thermal configuration and thus thermal effects are prevalent.

As will be detailed in Section (3.1.1), it is possible to make reasonably accurate predictions for damage threshold fluences of materials using only thermal processes

[7,52]. As long as the electrons and lattice are handled separately, the expression for heat flow can still be written as:

푄⃗ 푒,푙 = −휅푒,푙∇⃗⃗ 푇푒,푙 (2.33) where 푄⃗ is the heat flux density, 푇 is the local temperature, and 휅 is the material conductivity, and where the subscripts 푒 and 푙 denote the electron population and the lattice population, respectively.

2.2.2 Suprathermal electrons

Prior to the electrons reaching a thermal equilibrium within themselves, there exists a population of suprathermal electrons accelerated to high energies via the processes described in Section (2.1.3). These so-called “hot” electrons are able to penetrate into the material far deeper than the laser light itself, and often increase the depth of heating by an order of magnitude beyond the expected heating from purely thermal processes.

These hot, ballistic electrons propagate through the material, indirectly heating the nearby electrons and lattice via the formation of a cold return current. In 1955, Bell et al.

[53] argued from a conservation of charge and energy standpoint that this hot electron

31 current traveling through the material would necessarily create a cold, counter- propagating return current, such that the total current would be approximately balanced:

퐽푡표푡푎푙 = 퐽ℎ표푡 + 퐽푟푒푡푢푟푛 ≈ 0 (2.34)

This return current is comprised of a much larger number of electrons moving more slowly than the hot electrons traveling into the target. The hot electrons couple their energy to the return current, but ultimately it is this slow return current that heats the other electrons and the lattice via Coulomb collisions since the Coulomb cross section decreases rapidly as the electron velocity increases [54]. The slow return current will experience far more collisions and thus be able to transfer its energy more efficiently to other electrons and the lattice.

2.3 Surface Plasmon Polaritons

As discussed in Section (1.3), one of the leading theories for LIPSS generation, especially LSFL, is that of Surface Plasmon Polariton (SPP) generation. An SPP is an electromagnetic wave that propagates along the surface of a metal [55]. It is termed a polariton because it is an electromagnetic wave, but it is also termed a plasmon because it couples to a plasma wave within the electrons on the surface. It requires a metallic material, or more precisely a large population of electrons in the conduction band, thus semiconductors and insulators are capable of supporting SPP generation as well during short pulse laser interaction due to the excitation mechanisms discussed in Section

(2.1.4). The following sections discuss the theory in more detail.

2.3.1 Dispersion relation

Following the treatment in [56,57], the discussion will be simplified by considering the planar interface between two semi-infinite, loss-free, non-magnetic, isotropic, and homogeneous media. By definition, an SPP is an electromagnetic wave that is confined to a surface, thus evanescent in both media normal to the surface, but propagates in a wave- like manner along the surface. The coordinate system used in the following discussion is shown in Figure 8.

Figure 8: The coordinate system used to describe the details of SPPs and their generation, shown here to involve p-polarized incident radiation. [57]

Whatever form the surface wave takes, it must satisfy the electromagnetic wave equation in both media, as well as proper boundary conditions. The electric field component of the SPP as defined above must take the form: 33

0 푖(푘1푥푥−휔1푡) (푖푘1푦푦) 퐸⃗ 1(푟 , 푡) = 퐸⃗ 1 푒 푒 (2.35a)

0 푖(푘2푥푥−휔2푡) 푖푘2푦푦 퐸⃗ 2(푟 , 푡) = 퐸⃗ 2 푒 푒 (2.35b) where 퐸⃗ 0 is the electric field at the medium interface, 푘 is the wave vector, and 휔 is the frequency, where the subscripts 1,2 represent medium 1 and medium 2, respectively, and the subscripts 푥, 푦, 푧 denote coordinate direction. Using the two Maxwell curl equations, one can derive the wave equation for the electric field:

휕2퐸⃗ ∇2퐸⃗ = 휇 휀 (2.36) 0 휕푡2 which when Equation (2.36) is applied to Equations (2.35a) and (2.35b) gives the following expressions for the wave vectors:

휔 2 푘2 = 1 휀 (휔 ) − 푘2 (2.37a) 1푦 푐2 1 1 1푥

휔 2 푘2 = 2 휀 (휔 ) − 푘2 (2.37b) 2푦 푐2 2 2 2푥

If we next apply the Maxwell equation ∇⃗⃗ ∙ 퐸⃗ = 0 to Equations (2.35a) and (2.35b), valid for the charge-neutral media considered here, we can obtain the following relations between the various components of the electric field:

푘1푥 퐸1푦 = − 퐸1푥 (2.38a) 푘1푦

푘2푥 퐸2푦 = − 퐸2푥 (2.38b) 푘2푦

Since the SPP is a single wave propagating along the surface but existing in both media, we can determine that 휔1 = 휔2 = 휔, and 푘1푥 = 푘2푥 = 푘푥. Making use of this and applying appropriate boundary conditions [58], specifically that the normal 34 components of 퐷⃗⃗ are continuous and the tangential components of 퐸⃗ are continuous, you

0 0 0 0 get that 퐸1푥 = 퐸2푥 and 휀1퐸1푦 = 휀2퐸2푦. Applying all of this to Equations (2.38a) and

(2.38b), we get the following simple relation between the permittivities and the normal components of the wave vectors in the two media:

휀1 휀2 = (2.39) 푘1푦 푘2푦

Finally, we can combine Equations (2.37a), (2.37b), and (2.39) to arrive at the following dispersion relation for the SPP interface mode:

휔2 휀 휀 2 2 1 2 (2.40) 푘푆푃푃 = 푘푥 = 2 푐 휀1 + 휀2

Equation (2.40) gives the dispersion relation of a surface wave between two media satisfying Maxwell’s equations as well as the boundary conditions. However, there was one additional requirement for an SPP given above: it should be an evanescent wave, thus decaying exponentially into both media.

2.3.2 Fano and Brewster Modes

Taking a closer look at the equations in the above section, it is possible to distinguish between two distinct modes for the surface electromagnetic wave (SEW) allowed by

Equation (2.40). By combining Equations (2.37a), (2.37b), and (2.40), the following relations may be obtained between the permittivities and the wave vectors:

푘2 휀 1푦 1 (2.41a) 2 = 푘푥 휀2

푘2 휀 2푦 2 (2.41b) 2 = 푘푥 휀1

First consider the case where 휀1 and 휀2 are both purely real and positive. Looking at

Equations (2.41a) and (2.41b), it is clear that 푘1푦 and 푘2푦 will also both be real. In that case, it can be seen from Equations (2.35a) and (2.35b) that the generated SEW is not bound to the surface and will instead rapidly lose energy through fields propagating away from the surface. This is called the Brewster mode.

In contrast, consider the case where 휀1 and 휀2 are both purely real and have opposite sign. In this case, Equations (2.41a) and (2.41b) show that 푘1푦 and 푘2푦 will both be imaginary. With 푘1푦 and 푘2푦 both imaginary, Equations (2.35a) and (2.35b) show that the generated SEW will indeed be bound to the surface since the fields will decay exponentially into both media, and thus the SEW propagates without loss along the interface. These modes are known as Fano modes [59], and it is these modes that are most commonly referred to as SPPs.

To gain further insight into the distinction between these two modes, it is convenient to consider an ideal metal-vacuum interface, where the metal is assumed to be perfectly conducting. In this case, the electrons in the metal will behave like a free electron gas, which according to the Lorentz-Drude model will result in the metal’s dielectric constant being:

2 휔푝 휀 = 1 − (2.42) 2 휔2 where 휔푝 is the plasma frequency given in Equation (2.15). Substituting Equation (2.42) into the dispersion relation given by Equation (2.40) results in:

2 휔푝 2 휔 1 − 2 푘2 = ( 휔 ) (2.43) 푥 푐2 휔2 2 − 푝 휔2

Equation (2.43) can then be solved for 휔 as a function of 푘푥. This results in four solutions, two of which can be ignored due to an unphysical negative frequency. The two physical solutions correspond to the Fano and Brewster modes, respectively:

1 2 2 2 4 4 4 휔퐹 = √2푐 푘푥 + 휔푝 − √4푐 푘푥 + 휔푝 (2.44a) √2

1 2 2 2 4 4 4 휔퐵 = √2푐 푘푥 + 휔푝 + √4푐 푘푥 + 휔푝 (2.44b) √2

Figure 9: Illustration of the dispersion curve for an ideal vacuum-metal interface. The Fano and Brewster modes are plotted in relation to a freespace photon. The shaded region is the region which prohibits coupling to the freespace photons due to momentum conservation.

Figure 9 plots Equations (2.44a) and (2.44b) alongside the light line representing a photon in freespace, for which 휔 = 푐푘. For illustration purposes, numerical values for the plasma frequency were obtained for copper, but the results were subsequently normalized using the same plasma frequency to maintain some generality. The light line that divides

Figure 9 represents a maximum momentum in the x-direction for the incoming electromagnetic wave. From a momentum conservation standpoint, since 푘푥 is preserved across the interface, it is easy to observe from Figure 9 that at the appropriate angle of incidence it is possible to couple the incoming light to the Brewster mode, at any frequency. However, the Fano mode is always of greater momentum than the incident wave, thus momentum matching cannot occur for any angle meaning the incident wave cannot directly couple to the Fano mode regardless of frequency.

At first glance this would seem to pose a problem for LIPSS theories based on SPP generation if the SPP must be a Fano type SEW. However, methods have been developed for momentum enhancement that can overcome this problem, most of which fall into one of two categories. One method for momentum enhancement is called prism coupling which makes use of the phenomenon of total internal reflection in a prism placed near a metal surface. Most prism coupling methods are based on either an Otto [60] or

Kretschmann-Raether [61] geometry, though hybrid geometries also exist [56]. The second method for momentum enhancement, of more relevance to this work and thus described in more detail in the next section, is called grating coupling and makes use of a rough surface to couple the incoming light to the SEW.

2.3.3 Grating Coupling

The first observation of grating-assisted coupling to SPPs was reported by Wood in

1902 [62] via the description of anomalous behavior in the diffraction of light, though not realized to be caused by SPPs until much later. While some works distinguish between grating-assisted SPP coupling and coupling to a rough surface or topological defect [55],

39 as others have noted a rough surface can be treated as the superposition of multiple gratings [57], so it is only necessary to treat grating coupling here.

Consider again medium 1 to be a vacuum. In that case, the incident electromagnetic wave, with an angle of incidence 휃 with respect to target normal, would have a wave

⃗ vector in the plane of the surface equal to 푘푥 = |푘|푠푖푛휃. For diffraction order 푛 and a grating spacing of 휆푔, the periodic grating structure allows for momentum enhancement of [63]:

2휋푛ℏ 푝푥 = 푝푥0 ± (2.45) 휆푔

What this means for the generation of Fano type SPP modes is that light incident on a diffraction grating, or equivalently a rough surface which can be interpreted as a superposition of many diffraction gratings, can couple to a Fano mode SEW as long as the following momentum coupling condition derived from Equation (2.45) is met:

2휋푛 푘푆푃푃 = 푘푥 ± (2.46) 휆푔

Equation (2.46) provides the necessary momentum matching for incident radiation on a vacuum-metal interface to couple to a Fano type SPP. This phenomenon of grating- coupling allows for efficient SPP coupling as long as the interface between the two media is rough, ideally with the topological defects sub-wavelength in size [37,55,63].

Chapter 3: Modeling Laser Damage

The work described in this thesis is the development of a completely new theoretical tool to be used in modeling laser damage. Therefore, it is useful to overview many of the more popular methods currently in use. Understanding the current state of the field allows for a comparison between this new model and the existing approaches.

Many theoretical tools have been developed over the last several decades for the study of laser damage [64-68]. Many of these techniques fall into one of two categories: small spatial scale, ab-initio simulation techniques such as the molecular dynamics (MD) method discussed in Section (3.2); or large spatial scale, empirical or rate-equation models such as the Two-Temperature Model (TTM) discussed in Section (3.1.1). Some of these methods have met with considerable success for aspects of this problem, but to the best of the author’s knowledge, until this work a microscopic model of crater formation had not yet been demonstrated. While there are models that do not strictly fall into either of these categories, some notable exceptions being theories based on the

Quantum Kinetic Equation (QKE) [67] and MD-TTM hybrid models [68], most of these are still subject to limitations similar to those of one of the two categories, as will be described in the following sections.

3.1 Empirical and Rate-Equation Models

This section deals with the category of models that are based on either an empirical model, or a rate-equation approach. The major benefit of models that fall into this category is that they are generally simple, both conceptually and computationally, allowing for rapid modeling and analysis. The computational simplicity also allows for modeling very large laser systems and targets. On the other hand, there are two major downsides to these approaches. Since they do not directly integrate equations of motion but instead rely on simple models they generally include many fitting parameters for approximating the effect of various physical phenomena. Also, since they do not treat the particles directly, they necessarily must make an assumption on what constitutes

“damage”. For the selection of models described below, each of these will be described in more detail.

3.1.1 Two-Temperature Model

Of the theoretical tools that fall into this first category, one of the most widely used is the Two-Temperature Model (TTM) [7,65]. The basic TTM consists of separate diffusion equations for electrons and the lattice, with an electron-phonon coupling term proportional to the temperature difference and multiplied by some constant that is representative of the electron-phonon coupling strength. The basic TTM is:

휕푇 퐶 푒 = ∇ ∙ (퐾 ∇푇 ) − 푔(푇 − 푇 ) + 푆(푧, 푡) (3.1a) 푒 휕푡 푒 푒 푒 푙

휕푇 퐶 푙 = ∇ ∙ (퐾 ∇푇 ) + 푔(푇 − 푇 ) (3.1b) 푙 휕푡 푙 푙 푒 푙 where 퐶 and 퐾 are the heat capacities and thermal conductivities, respectively, of the electrons and lattice as designated by the subscript 푒 or 푙, 푔 is the material-dependent

42 electron-phonon coupling constant, and 푆(푧, 푡) is a source term to describe the incident radiation. This equation is set up such that the electrons and lattice do not need to be in thermal equilibrium with each other, as would be the case for long-pulse damage but not for femtosecond-pulse damage. However, the electrons and lattice need to individually be close enough to equilibrium such that a temperature can be defined for each species separately as defined by a Fermi distribution [69].

Classically, the source term in Equation (3.1a) would be defined as:

푆(푟, 휃, 푡) = 퐼(푟, 휃, 푡)(1 − 푅 − 푇) (3.2) where 퐼(푟, 휃, 푡) would be the time-varying spatial profile of the laser incident on the surface, and 푅 and 푇 would be the percentage of the intensity reflected and transmitted, respectively. Equation (3.2) makes use of the fact that any light not reflected from or transmitted through the target must be absorbed, but also makes the rather unreasonable assumption that all of the laser light is absorbed at the very front surface of the target which leads to an underestimation of the damage threshold.

To evaluate Equations (3.1a) and (3.1b) in order to determine a damage threshold or ablation depth it is necessary to first make an assumption on what constitutes damage.

Since this model deals only with temperature, it is natural to define a temperature threshold above which damage is assumed to occur. In most studies, this is often taken to be the melting temperature of the material, such that damage will occur if the front-most surface of the target reaches the melting temperature and the crater depth will be the depth of the target that is above this melting temperature.

There are two important effects that the above definition for the source term does not take into account. The first is the skin depth of the material defined in Section (2.1.3), which is the distance over which the light intensity decreases by e-1. This means that the laser energy will be absorbed over a distance of ~푙푠, not just at the immediate surface.

The second and more important effect [70] is that upon the laser reaching the target, some electrons will be accelerated to suprathermal speeds, as discussed in Section (2.2.2).

These electrons will propagate ballistically through the target, depositing their energy as they go over a much longer distance than the skin depth. Defining the distance over which these electrons travel ballistically as 푙푏, then these effects can be incorporated into the source term by redefining it as [70]:

푒−푧/(푙푠+푙푏) ( ) (3.3) 푆 푟, 휃, 푡 = 퐼(푟, 휃, 푡)(1 − 푅 − 푇) −푑(푙 +푙 ) (푙푠 + 푙푏)(1 − 푒 푠 푏 )

In Equation (3.3) there is one additional term, 푑, representing the thickness of the target. For thick targets, this distance does not change the result. However, for thin targets, where the ballistic electrons may reflect off the back surface of the target and propagate back towards the front surface before losing all their energy, this thickness term plays a critical role as these electrons can heat the same area multiple times as they reflux back and forth.

Figure 10 illustrates the effectiveness of this model, comparing the TTM in Equations

(3.1a) and (3.1b), using the improved source term given in Equation (3.3), to experimental data on damage threshold fluence measurements. The results in Figure 10 also include temperature-dependent coefficients of reflection and transmission. The fit is

44 very good, despite ignoring most of the physics as to how damage is occurring and what physical mechanisms are involved, taking only heating into account and even that only in an empirical manner. The TTM is very computationally simple, capable of being evaluated numerically very rapidly even on a standard desktop computer.

Figure 10: Pulse length (FWHM) dependence of melting threshold fluences measured experimentally by Stuart et al. [4] on a 200-nm thick Au grating with 1053 nm pulses. The shaded band results from the TTM, including uncertainties in the constants and the transient change of reflectivity with electron temperature. The dashed line results from TTM calculations based on constant absorption. [70]

While the TTM is capable of fitting experimental data rather well, it does have a few major downsides. The first is that even though it can provide a match to data, it provides little physical insight into the problem. This means that while its results may be useful in tabulating damage thresholds for industrial efforts under simple conditions, it does not help researchers to learn about the problem.

The second major downside is that many of the physical constants in the model are often treated as fitting parameters to help it match experimental data, meaning they do not always correspond to their actual physical values. On the one hand this is necessary, as temperature-dependent values for all the physical constants are not always well- known, and traveling distances for ballistic electrons depend on a wide variety of experimental parameters and is often not known at all. On the other hand, this creates a potential avenue for the model to match experimental results without actually relating to the physics involved in the damage process, by simply involving enough fitting parameters.

3.1.2 Thermal Explosion Model

The Thermal Explosion (TE) model is based in uneven heating near an imperfection in the material near the surface and for this reason is often referred to as the inclusion- initiated thermal explosion model. The TE model is based on a heat transfer equation very similar to Equations (1.1) and (3.1a), written here as:

휕 (퐶푇) = ∇ ∙ (퐾∇푇) + 푄(퐼, 푇) (3.4) 휕푡

In Equation (3.4) the intensity- and temperature-dependent source term 푄(퐼, 푇) represents the laser absorption from the inclusion, and has been shown to take the form

[71]:

푇−푇 (휉 0) 푄(퐼, 푇) = 푄(퐼)푒 푇0 (3.5) where 휉 is a material parameter, and 푇0 is the initial temperature. Solutions of Equation

(3.4) exhibit an “explosive” character near a threshold intensity such that 푇 → ∞ as

퐼 → 퐼푡ℎ.

While this damage model is thermal in nature, it does not have the same √휏 damage threshold dependence as the thermal model mentioned in Section (1.2). In fact, a careful analysis of the TE model shows that the pulse-width dependence of the laser-induced damage threshold (LIDT) depends on the pulse shape [72]. For example, in the cases of a rectangular pulse and a Gaussian pulse, the pulse-width dependence of the LIDT (for

휏 ≥ 휏푟) take the forms [64]:

퐼 퐼푆푞푢푎푟푒 = 푡ℎ (3.6a) 퐿퐼퐷푇 (1 − 푒−휏/2휏푟)

2 퐺푢푎푠푠 (휏푟/휏) 퐼퐿퐼퐷푇 = 퐼푡ℎ푒 (3.6b) where 휏푟 is the heat relaxation time of the medium. One of the major successes of this model is the experimental observation of these pulse-width dependences for certain materials [73].

While the TE model has had this success and shares the computational simplicity of the TTM, it also shares the downsides of the TTM. Similarly to the TTM, temperature- dependent values of the material constants, especially in the local vicinity of the

47 inclusion, are often not known and must become fitting parameters to match experiment.

The model also ignores the specifics of how the damage mechanism occurs, making it a useful tool for industrial pursuits but less helpful in determining the microscopic mechanisms behind the laser damage process.

3.1.3 Electron Ionization Rate Equation

Another notable class of models for laser damage are those based on an ionization rate equation, the analysis of which will continue the discussion started in Section (2.1.4).

In order to make predictions for the laser damage fluence threshold, this class of models must make an assumption on when damage occurs. Generally speaking, damage is assumed to occur if the electron density in the target exceeds the critical density for the laser wavelength in use such that the laser can no longer propagate freely in the material and nontrivial energy absorption can begin as described in Section (2.1.3).

With the assumption that damage occurs if the electron density 푁 reaches some threshold value 푁푡ℎ by the end of the laser pulse, Equation (2.32) can be solved to determine the damage threshold intensity 퐼푡ℎ, which scales as:

푓푃휏/푊 푁(푡 = 휏) = 푁0푒 (3.7a)

푁 (3.7b) 푡ℎ = 푒푓푃휏/푊 푁0

2 푁푡ℎ 푓(푒퐸) (3.7c) 푙푛 ( ) = 푡푐휏 푁0 2푚푒푊

2푛휀0푚푒푐푊 푁푡ℎ (3.7d) 퐼푡ℎ = 2 푙푛 ( ) 푓푡푐휏푒 푁0

In the derivation above, Equation (3.7b) combined with Equation (2.31) leads to

2 Equation (3.7c), which then combined with the definition of intensity 퐼 = 휀0푛푐〈퐸 〉 leads to Equation (3.7d), showing the scaling of the damage threshold intensity if avalanche ionization is the primary physical mechanism responsible for the damage process.

This particularly simple model breaks down when analyzing Equation (3.7d), as there is no regime where the LIDT scales as 휏−1. This model can be expanded upon, however, to include the effects of both photoionization as well as electron-ion recombination. The basic form of this model would be:

푑푁(푡) = 푊 (퐼(푡)) + 푊 (퐼(푡))푁(푡) − 푊 (푁(푡), 푡) (3.8) 푑푡 푃퐼 퐴푉 푅 where 푊푃퐼 is the intensity-dependent photoionization rate often calculated from Keldysh theory [51], 푊퐴푉 is the intensity-dependent avalanche ionization rate for which the best model is currently disputed [66], and 푊푅 is the density dependent electron recombination rate often approximated as 푊푅 = 푁/푡푅, where 푡푅 is an experimentally measurable, material dependent relaxation constant [74-76].

The results of applying this model to SiO2 are illustrated in Figure 11, using a variety of different approaches to evaluating the avalanche ionization rate. It can be seen from

Figure 11 that the more realistic models scale more accurately than the simple model shown in (3.7d). While the scaling is better, it is also clear that there is a big difference between the various avalanche ionization models, as well as a reasonably large difference even from one experiment to another. The limit of this class of models in representing simple targets then seems to be the accuracy of the individual ionization models. In addition, given the restricted nature of this class of models, it is difficult to account for 49 anything but ideal targets, meaning no non-uniform surfaces, no structural or electronic defects, and no non-uniform materials.

There is one notable extension to this class of models, those based on the Quantum

Kinetic Equation (QKE) [67]. This extension solves a fully quantum equation of state to evaluate a rate equation as opposed to the individual empirical models described previously. While this model is more closely connected to the physical mechanisms involved in the laser damage process, it too must make an assumption on what constitutes damage since it does not take the target’s structure and morphology into account.

3.2 Molecular Dynamics Simulations

When greater physical insight is required, molecular dynamics (MD) simulation methods are available tools at the cost of significant computational complexity. The MD

50 method is well adapted to modeling every atom or molecule in a target, along with all of the interatomic forces keeping the target together [77]. The MD method is in principle ab-initio, limited only by the accuracy of the interatomic potentials modeled. However, the evaluation of the time-evolution of even a small target (generally 10-100 nm in each dimension) is computationally very complex, requiring large supercomputing clusters running for days or weeks. That said, MD simulations have provided interesting insight into ablation dynamics and cluster formation that is not accessible by any other existing simulation method [81].

3.2.1 Interatomic Forces

At the most basic level, MD simulations solve a classical system of equations to determine the trajectory in both position and momentum space of an ensemble of simulation particles. Generally speaking, an MD code models every atom or molecule in the system of interest, evaluating the equations of motion for every particle pair, leading to a complexity scaling of the order O(N2) in “Big Oh” notation [78], where N is the number of particles in the system. The standard equations of motion evaluated take the form:

푑2푟 푚 푖 = 퐹⃗⃗ , 푖 = 1,2, … , 푁, (3.9a) 푖 푑푡2 푖

퐹⃗⃗ 푖 = −∇⃗⃗ 푈푖(푟 1, 푟 2, 푟 3, … , 푟 푁) (3.9b) where 푚푖, 푟⃗⃗ 푖, and 퐹⃗⃗ 푖 are the mass, position, and total force on particle 푖, respectively, and

푈푖 is the position-dependent interatomic potential at particle 푖 from all other particles.

Unlike the models discussed in Section (3.1), the MD technique needs no assumption for

51 when damage occurs, has no inherent fitting parameters, and the only real input is the form of the potential applied between the particles. Fewer user inputs for the simulation means there are fewer knobs for fine-tuning a simulation to match expected results, either accidentally or intentionally, which adds validity to an MD simulation if it reproduces experiment. Applying this method to optical damage allows for the analysis of spatial and temporal evolution of certain thermodynamic properties such as temperature and pressure, as well as structural properties such as phase transitions and defect formation, all important for damage mechanisms. Indeed, much of the research involving MD simulations is devoted towards understanding better the phase transitions occurring during the laser damage process [77,79,80]. An illustration of the MD method is shown in Figure 12.

As the potential 푈 is the only adjustable input parameter in the MD technique, it is worth some consideration. While it is possible to solve a fully quantum many-body potential for the entire ensemble of particles, it is extremely complicated computationally in an already computationally expensive simulation. Often, something as simple as the

Morse pair-potential or Lennard-Jones pair potential is used. The Lennard-Jones pair potential takes the form:

12 6 푟0 푟0 푈퐿−퐽(∆푟푖푗) = 퐷 (( ) − 2 ( ) ) (3.10) ∆푟푖푗 ∆푟푖푗 where 퐷 is the dissociation energy of the material, 푟0 is the equilibrium atomic distance, and ∆푟푖푗 is the distance between a pair of atoms. These pair potentials are

52 computationally very simple, and for the hexagonal close-packed lattices of most metals

[82], they have been shown to be reasonably accurate [83,84].

Figure 12: Illustration of the laser damage process on a Si(100) surface by pulses of length 500 fs and 100 ps at a wavelength of 266 nm. Green represents atoms of crystalline silicon and red represents atoms of liquid silicon, phase determined by particle energy. The pulse begins interacting at t=0. (a-f) 500 fs pulse with a fluence of 0.225 J/cm2, right at the damage threshold for the interacting laser pulse; (g) 500 fs pulse with a fluence of 0.5 J/cm2, double the damage threshold fluence; and (h) 100 ps pulse with a fluence of 0.45 J/cm2, roughly 10% higher than the damage threshold. [81]

When further accuracy is required, there is a class of pair-potentials that incorporate many-body effects empirically with an extra term. This class of pair-potentials is called the embedded-atom method (EAM) [85]. With this method, the total energy of a system is given by:

퐸푡표푡 = ∑ 푈(∆푟푖푗) + ∑ 퐹(휌푖) (3.11) 푖,푗; 푖≠푗 푖 where 푈 is a pair potential such as those discussed above, 퐹 is what is called the embedding energy which depends on the so-called electron density 휌푖 at an atom, given by:

휌푖 = ∑ 푔(∆푟푖푗) (3.12) 푗≠푖 where 푔(∆푟푖푗) is the contribution of atom 푗 to the total electron density at atom 푖. The details of 퐹(휌푖) and 푔(∆푟푖푗) can be found in Ref. [86].

3.2.2 Laser Coupling

While the MD method is a powerful computational tool, there is a distinct limitation of the method: there is no simple way of directly modeling electrons, and without electrons no direct way to handle laser interactions. Methods have been developed to include the effect of electrons and laser in an indirect manner, such as through Monte

Carlo (MC) methods [81], and a combined MD-TTM approach [87]. In the MC approach, it is common to randomly increase the energy of an atom within the skin depth of the material by the energy of a single photon during the modeled laser interaction, the frequency of which increases as the intensity increases. In the combined MD-TTM approach, the TTM is used to handle the electrons and the laser interaction, and a new 54 term is added to Equation (3.9a) to model the energy transfer from the electron population to the lattice. The new set of equations of motion is:

푑2푟 푚 푖 = 퐹⃗⃗ + 휉푚 푣 푇 (3.13a) 푖 푑푡2 푖 푒 푖

⃗⃗ ⃗⃗ (3.13b) 퐹푖 = −∇푈푖(푟 1, 푟 2, 푟 3, … , 푟 푁)

푐푒푙푙 ∑푁 푚 (푣 푇)2 푐푒푙푙 푖=1 푖 푖 (3.13c) 푇푙 = 푐푒푙푙 3푘퐵푁 휕푇 퐶 푒 = ∇ ∙ (퐾 ∇푇 ) − 푔(푇 − 푇 ) + 푆(푧, 푡) (3.13d) 푒 휕푡 푒 푒 푒 푙 푇 where the 휉푚푒푣 푖 term takes the electron-phonon interaction into account in an MD approach as opposed to the TTM approach through the coupling constant 푔. Otherwise,

Equations (3.13a) and (3.13b) are the same as (3.9a) and (3.9b), and Equation (3.13d) is the same as the TTM model’s Equation (3.1a). Finally, Equation (3.13c) is designed to take the velocities of all the atoms in a discrete segment, or cell, of the simulation to calculate a thermodynamic temperature for the lattice in that cell to be inserted into the

TTM component of the equations.

Though the TTM and MC approaches to modeling electrons allow the MD technique to be used to model a laser interaction, necessary for fundamental and robust studies of laser damage, they bring their own troubles with them. Prior to their inclusion, the MD method is inherently ab-initio, but with them there is now the issue of multiple competing models, such as is an issue with ionization rate-equation approaches, and the issue of multiple fitting parameters associated with the TTM.

Chapter 4: Particle-In-Cell Simulations

All of the theoretical tools currently in use for the study of femtosecond laser damage have their advantages, but there is a gap in capability between the very small spatial scale

MD simulations discussed in Section (3.2) and the empirical/rate-equation approaches discussed in Section (3.1) that, as this work will show, particle-in-cell (PIC) codes can fill. PIC codes were in use as early as 1955 for plasma simulation [88]. PIC simulations operate at a fundamental level directly integrating the Maxwell equations and the Lorentz equation, while at the same time making enough simplifying assumptions to model large- scale phenomena. PIC simulations include a self-consistent treatment of fields and particles and in principle zero fitting parameters, but still allow for realistic target sizes giving access to the full damage morphology. However, until this work PIC codes had never been used to study laser damage so the basics of the PIC cycle are outlined in the following sections.

4.1 The PIC Method

The two primary simplifying assumptions used by PIC codes are the use of macroparticles, and the partial discretization of space and time. A macroparticle is a single simulation particle that represents a distribution of real particles, and is typically a cluster on the order of 106-1010 real particles, where the number of real particles a

56 macroparticle represents is known as its weight. The macroparticle has a continuous, well-defined velocity vector and position vector and has the same charge/mass ratio as the particles it represents and, thus, the same equation of motion. Using macroparticles instead of modeling every real particle is a technique for sampling the full 6D phase space of real particles, and as with any sampling may converge on the exact solution with sufficient resolution. The benefit of this technique should be clear: a simulation using macroparticles that takes approximately 1 second to complete would take about 1 year to complete if all of the ~108 real particles per macroparticle were modeled instead.

Although the macroparticle position is continuous, real space is broken up into a grid consisting of nodes and cells. The grid is used to discretize fields and currents which are evaluated only at the nodes [88]. The benefit of this is that instead of solving for the forces on a particle from every particle-particle pair which, as described in Section (3.2) when discussing MD forces is an O(N2) process, each particle only contributes to the fields and currents at the nearest-neighbor nodes (in the simplest case). To evaluate forces, these fields are then interpolated back to the particles, making the entire process an O(N) algorithm. For a typical simulation where N is on the order of 108 macroparticles, the improvement in speed is significant. An illustration of this partial discretization of space is shown in Figure 13B.

The standard PIC cycle during one timestep is illustrated with a diagram in Figure

13A. First, the cycle begins by mapping electric currents to the nodes by linearly interpolating the current contribution from each particle to each of the particle’s nearest surrounding nodes. Once currents have been calculated, the electromagnetic fields at each

57 node are advanced using a finite-difference time-domain (FDTD) discretization of

Maxwell’s equations, often employing a spatial grid configuration termed a Yee lattice

[89,90] that incorporates two separate spatial grids, offset by half a spatial cell so the electric field and magnetic field are not evaluated at the same location, giving a higher- order solution for the Maxwell curl equations. The third step is the interpolation of the updated electromagnetic fields back to the simulation particles, followed by the evaluation of the particle equations of motion to determine new momenta and positions.

Lastly, other various physical models are evaluated at the end of the timestep, including particle collisions, ionization, and as will be discussed later in this work, pair-potential physics.

Figure 13: A) Diagram of the PIC cycle, illustrating the standard order of operations for evaluating a PIC timestep. B) Illustration of the partial discretization of space used in PIC, with red circles representing the continuous-position macroparticles, and the blue circles representing the nodes on the discretized grid where fields, currents, and densities are evaluated.

Ultimately, the PIC method provides a powerful tool for the fundamental simulation of laser damage. The approximations PIC makes allows for the simulation of realistic target sizes, reasonably handling targets as large as 100 µm per dimension [90], and in the right situation even larger. This is in stark contrast to MD simulations, which typically model targets over a thousand times smaller in each dimension. Unlike the empirical models and rate-equation approaches of Section (3.1), PIC modeling of laser damage does not require assumptions on when the target damages, since it directly models target atoms allowing for a fully dynamic surface structure and, in principle, it requires no fitting parameters since it directly integrates equations of motion. Lastly, unlike any of the approaches mentioned in Sections (3.1) or (3.2), since the PIC method directly integrates Maxwell’s equations and the Lorentz equation, it allows for a fully self-consistent treatment of charged particles and electromagnetic fields, including the propagation of light in a bounded or partially bounded region of space.

4.2 Stability Requirements

When employing the PIC method, there are several spatial, temporal, and algorithmic considerations that must be taken into account for an accurate and stable simulation. For instance, different PIC codes use different approaches to interpolating particle quantities to nodes and node quantities back to the particles. While the exact method of interpolation can have important consequences, a key consideration is whether the interpolation from particles to node and node to particles is done self-consistently, that is, the same interpolation is used in both directions. If this interpolation is not done self-

59 consistently, self-forces can arise and break momentum conservation. This may or may not be acceptable depending on the simulation. The rest of this section will discuss some details of the considerations that must be made for accurate PIC simulation.

There are two distinct approaches to discretizing the equations of motion for the system, termed explicit solution methods and implicit solution methods. With explicit solution methods, physical quantities at the current timestep can depend only on quantities from the previous timestep. Computationally, this class of solutions is the simplest to evaluate, since at the current timestep the values from the previous timestep are all known. With the implicit solution methods, physical values for the current timestep can be dependent on a combination of known values from the previous timestep as well as unknown values from the current timestep, often requiring complicated matrix inversion techniques or some manner of prediction and correction algorithm to evaluate.

The reason a code would use an implicit method, despite its computational complexity, is to significantly relax spatial and temporal resolution requirements as implicit algorithms are, in general, more stable [91].

At a very minimum, the resolution requirements for numerical accuracy are given by the Nyquist theory [92], stating that a wave must be sampled at least twice in a period to resolve it. The sampling of a sine wave is shown in Figure 14 to illustrate this theory.

Applying the Nyquist theory to PIC simulations gives the following conditions:

휏 ∆푡 < (4.1a) 2

휆 ∆푥 < (4.1b) 2

60 where ∆푡 is the timestep, ∆푥 is the grid size, 휏 is the smallest temporal period in the simulation, and 휆 is the smallest spatial period in the simulation. However, a sampling frequency of 휏/2 or 휆/2 is the limiting case for minimally resolving a wave; for simulation purposes a sampling frequency much higher is required for numerical accuracy.

Figure 14: Illustration of the Nyquist theory for sampling, shown using (A) a basic sine wave of frequency f, (B) a sampling frequency of f, (C) a sampling frequency of 4f/3, and (D) a sampling frequency of 2f. The original function is shown as a blue solid line in (A) and a blue dashed line in (B-D), the sampled data points are represented by red circles, and the red solid line in (B-D) represents the function that would be interpolated from the sampled data points.

For the case of an explicit simulation method, stability requires resolving all important temporal and spatial scales. For instance, the upper limit for the timestep may 61 be given by the Courant-Friedrich-Lewy condition [94], sometimes simply referred to as the Courant limit, which places upper limits on the timestep based on the fastest element in the simulation, such that for the one-dimensional explicit case:

∆푥 ∆푡 < (4.2) 푢 where 푢 is the highest velocity in the simulation. When simulating a laser or charged particles the fastest element is the electromagnetic wave, requiring ∆푡 < ∆푥/푐. The simple interpretation of this condition is that explicit PIC codes become unstable when anything can cross over an entire cell in one timestep. In some cases, the limiting factor on the timestep might be the resolution of either the plasma frequency or the cyclotron frequency, as an analysis of an electron in a plane wave can show that explicit finite- differenced equations of motion for charged particles are unstable unless 휔푝∆푡 < 2 and

휔푐∆푡 < 2 [95].

As for the spatial grid size, there are similar constraints. To resolve an electromagnetic wave and allow it to propagate, at a minimum Equation (4.1b) must be satisfied. However, to sufficiently resolve an electromagnetic wave without large numerical error the spatial grid must be much finer that this limit, often at least 8 times smaller than the wavelength. For many situations however, this is not the limiting factor for the spatial grid. One of the more stringent spatial grid restrictions for explicit PIC is the need to resolve the Debye length, given by:

휀 푘 푇 0 퐵 푒 (4.3) 휆퐷 = √ 2 푛푒푞푒

62 where 휀0 is the vacuum permittivity, 푘퐵 is Boltzmann’s constant, 푇푒 is the electron temperature, 푛푒 is the electron density, and 푞푒 is the electric charge. The Debye length is the distance from a charged particle beyond which the effect of the particle is electrically screened. For the simulation of low-temperature, high-density solids, as would be modeled for laser damage, this Debye length can be on the order of the atomic separation distance. This can put significant constraints on the spatial grid resolution of an explicit

PIC simulation. Without resolving the Debye length, an explicit PIC simulation will tend to artificially heat until the Debye length is on the order of the cell size, ruining energy conservation.

There are further considerations that arise due to spatial discretization and the use of macroparticles. For instance, the impact parameter for Coulomb collisions will often be much smaller than a cell size and particles are sampled at approximately a rate of 1 in every 100 million, meaning collisions will be significantly under-resolved. For this reason, Coulomb collisions must be handled separately, often using a Monte Carlo approach. Also, since all particle-particle interactions are handled via grid nodes, the force between two particles is no longer translationally invariant within a cell.

All of the considerations mentioned above can be handled with the appropriate choice of timestep and grid size. For the case where the resolution requirements are too strict for reasonable computation, many of the constraints can be relaxed or even removed completely with the appropriate choice of interpolation scheme and algorithm, particularly with the use of implicit algorithms. For that reason, for any given set of simulations it is necessary to weigh the extra computing time per timestep against the

63 relaxed resolution requirements associated with higher-order or implicit algorithms.

Further, each algorithm must be considered for how quickly it converges to a solution as resolution increases, as well as its ability to conserve energy, momentum, and charge. For this reason, the PIC code used in this work, known as LSP [96], offers a wide variety of particle advance, field advance, collision, and error-correction algorithms.

4.3 LSP

The PIC code LSP has many options for numerical solution of the particle and wave equations, but for this work a variation on the direct-implicit particle and field solutions first derived in [97,98] are used. Specifics of how these algorithms are implemented in

LSP can be found in [99], but will be outlined in this section.

The particle equations of motion are evaluated using a leapfrog approach, where the particle positions are evaluated at half a timestep offset from the velocities, and the velocities are evaluated at half a timestep offset from the accelerations, meaning the positions and accelerations are evaluated at the same point, which allows for a second- order integration requiring no more computation than a first-order integration [100]. In this implicit scheme, the equations of motion of the particles derived from the Lorentz force are:

(푝 푛−1/2 + 푝 푛+1/2) 푞 푝 푛+1/2 = 푝 푛−1/2 + ∆푡 [푎 푛 + × 퐵⃗ 푛(푥 푛)] (4.4a) 2 훾푛푚푐

1 푞 푎 = [푎 + 퐸⃗ (푥 )] (4.4b) 푛 2 푛−1 푚 푛+1 푛+1

64 where 푝 is the particle’s momentum and 푎 is the particle’s acceleration, and the subscripts denote timestep number. Note the explicit distinction in Equations (4.4a) and

(4.4b) between the acceleration and position which are evaluated on the timestep, and the momentum which is evaluated on the half-timestep and time-averaged to get a value for on the timestep. In practice, Equations (4.4a) and (4.4b) are broken up into two steps, first with only the old acceleration “pushing” the particle for half a timestep, and second with only the new acceleration pushing the particle half a timestep. The fields are evaluated in between the two particle pushes, with the following set of equations:

퐸⃗ 푛+1 − 퐸⃗ 푛 = ∇⃗⃗ × 퐵⃗ − 퐽 1 − 〈푆〉 ∙ 퐸⃗ (4.5a) 푛+1 푛+ 푛+1 ∆푡 2

퐵⃗ − 퐵⃗ 푛+1 푛 = −∇⃗⃗ × 퐸⃗ (4.5b) ∆푡 푛+1

휌푞∆푡 〈푆〉 = (〈푇〉 − 푣 푛+1/2⨂푣 푛+1/2) (4.5c) 2훾푛+1/2푚

2 1 + Ω푥 Ω푥Ω푦 + Ω푧 Ω푥Ω푧 − Ω푦 1 2 〈푇〉 = 2 [Ω푥Ω푦 − Ω푧 1 + Ω푦 Ω푦Ω푧 + Ω푥] (4.5d) 1 + |Ω⃗⃗ | 2 Ω푥Ω푧 + Ω푦 Ω푦Ω푧 − Ω푥 1 + Ω푧

푞퐵⃗ Δ푡 Ω⃗⃗ = 푛 (4.5e) 2훾푛푚푐 where 〈푆〉 is the so-called susceptibility tensor, 〈푇〉 is the magnetic field rotation tensor, and 퐽 is the electric current. The susceptibility terms are used to correct for the missing acceleration from the two-step particle push, and were determined by a perturbation analysis shown in [99]. Equations (4.5a) and (4.5b) are the Maxwell curl equations. It is

65 worth noting that LSP only evaluates the two curl equations, not the full set of Maxwell equations. For exact solutions of the two curl equations, assuming that the initial conditions for the simulation are charge-neutral, charge will be conserved throughout the integration and the two curl equations are sufficient for a full treatment of the wave dynamics. However, in the discrete treatment of these equations, the integration is not exact and charge will not be conserved. However, within LSP this discrepancy was approximated and corrected for on each timestep in this work.

While Equations (4.5a)-(4.5e) accurately evaluate electromagnetic fields on the size of a cell or larger, within a cell the fields are not resolved enough to handle some short- range effects. In particular, Coulomb scattering of charged particles within a Debye length of each other is significantly under-resolved, and so a separate algorithm must be developed to handle this process. For this work, the algorithm used in LSP to evaluate particle scattering is the Jones algorithm [101]. Within the Jones algorithm, collisions between particles of the same species (intraspecies) and between particles of different species (interspecies) are handled in two different manners, as described briefly below.

For interspecies collisions, the Jones model calculates Coulomb collisions of particles off of a grid-based “collision field”. The algorithm begins by first calculating an average momentum and temperature for each cell, assuming a Maxwellian distribution for each species such as:

∑푛∈푖 푤푛푝 푛 〈푝 〉푖 = (4.6a) ∑푛∈푖 푤푛

2 1 ∑푛∈푖[푤푛(푝 푛 − 〈푝 〉푖) ] 푘퐵푇푖 = (4.6b) 3푚푖 ∑푛∈푖 푤푛 where 푤 is the weight, 푝 is the momentum, and 훾 is the relativistic factor of the macroparticle, and where the subscripts 푖 and 푛 denote species and particle, respectively.

Each particle in the cell is then scattered by sampling this Maxwellian distribution. As

LSP is packaged, collision rates for this model can be taken from either the Spitzer model

[102] or the Lee-More-Desjarlais (LMD) model [103,104]. There is a flaw in the original

Jones model in that each pair of collisions was modeled independently. LSP however uses a modified Jones algorithm that treats all of the particle collisions in a timestep simultaneously which improves energy conservation. This is done by writing the system of equations describing momentum transfer in matrix form and evaluating them as an eigenvalue/eigenvector problem.

While the above approach works for interspecies collisions, it does not work for intraspecies collisions [101], and thus a separate approach must be used. For technical reasons based on the application of the scattering field, the eigenvalue solution of the matrix would result in a zero force between all scattered particles. Instead, a Monte-Carlo scattering model is used that is based off of the Langevin equation, which has been shown to be a good approximation for modeling same-species collisions in the theory of

Brownian motion [105].

Chapter 5: Modeling Solid Targets with PIC

As was discussed in Chapter 3, prior to this work there was a large gap in laser damage modeling capability between the ab-initio MD simulations and the large scale rate-equation or empirical-model based approaches. While there were some notable exceptions discussed to these two limiting categories, most of these exceptions still fell victim to one or the other’s limitations. There are two primary observables for the experimentalist, the laser damage threshold and the crater morphology, and while all the methods discussed in Chapter 3 were capable of calculating a damage threshold, none of them were capable of directly modeling crater morphology for the average experiment.

MD methods are too small scale to capture the entirety of most damage craters, and the rate-equation or empirical model approaches don’t account for surface deformation.

The purpose of this work is to show that PIC codes could fill this gap, not only being able to calculate a damage threshold but also able to handle the entire crater morphology allowing for complete and direct experimental verification. Being able to model the full structural modification to the surface directly from fundamental principles gives PIC simulations the ability to model phenomenon that have not previously been modeled in a fundamental way, such as LIPSS formation discussed in Section (1.3), the cause of which has been debated for decades. Finally having a theoretical tool capable of modeling this phenomenon directly allows for a direct examination of its cause.

The first problem that appears when attempting to model laser damage with a standard PIC code is its inability to maintain a target’s structure. Being designed for plasma simulation, the macroparticles are inherently free to move around and only long- range electromagnetic effects are considered. If a metal target is initialized in a spatial grid and modeled for a long enough time-scale to allow for nontrivial atomic motion, it will not behave as a solid but instead will slowly disperse as if in a gaseous state. There has never been a need to include interatomic forces in a PIC code, since even the lowest temperature plasmas are hot enough that the binding forces that maintain a solid’s structure are negligible. This problem is even worse than it appears at first; it also means that once the damage process begins, it will never end. There are no forces in place to allow a target to resolidify; no means of a crater forming. The damage would continue until no atoms remained in the simulation, whether damage should have occurred or not.

For this reason, the first hurdle that needed to be overcome was the inclusion of a PIC- consistent interatomic forces algorithm, something that, to the best of our knowledge, had never been attempted before. For the purpose of rapid development, a simple pair- potential was used to model interatomic forces since they are computationally simple and have been shown to be reasonably accurate [106,107]. However each of the below PIC implementations of interatomic forces is fully capable of being modified to use more accurate many-body potentials in the future.

5.1 Early Approaches to Interatomic Forces

Prior to developing a working, energy-conserving, momentum-conserving interatomic forces algorithm for PIC, there were a few approaches that were attempted that did not work, two of which will be discussed briefly in this section. This is done because the resulting failure modes help indicate where the primary challenges were and could be useful for future development.

The first approach attempted was a direct application of the Morse pair-potential to simulation macroparticles. The Morse potential is given by:

푈(푟) = 퐷[푒−2푎(푟−푟0) − 2푒−푎(푟−푟0)] (5.1) where 푟 is the distance between two atoms, 푟0 is the equilibrium distance between those two atoms where the force is zero, 퐷 is the dissociation energy or depth of the potential well, and 푎 is a lattice constant associated with the width of the potential well. For a direct application of Equation (5.1) to macroparticles, several factors must be taken into account. First of all, the sampling of approximately 1 in every 100 million particles means there is a lot of vacuum between each particle. The equilibrium distance and lattice constant must be adjusted accordingly, such that the initial distance between simulation macroparticles is their equilibrium distance and such that the force does not fall off too fast as they separate from each other. The benefits of this approach are that it is simple to apply and it conserves energy and momentum exactly.

There were complications with using this algorithm in a PIC code, however. First and most-importantly, it is not PIC-consistent. Strange numerical artifacts tend to manifest when PIC loses its self-consistency, such as charge non-conservation where the divergences of the electric and magnetic fields no longer properly correspond to the

70 amount of charge on the grid. Second, the physical effects of applying this potential in such a manner are not obvious. The potential becomes much longer range, but the depth of the potential doesn’t change, meaning that the potential changes more gradually with increased or decreased distance between particles. Third, since this potential is being applied to every particle pair directly it is an O(N2) process, making it very slow to evaluate. Lastly, it is incompatible with parallel processing methods that do not share memory, since it will have no information about particles belonging to a separate computing node. This becomes an issue when two particles of separate computing nodes are close enough that this potential is nontrivial between them. While this can be accounted for, it further increases the already significant computation time drastically.

The second attempt at a pair-potential implementation within PIC attempted to model the effect of the potential as a pressure and is shown in detail in [108] and illustrated in

Figure 15. Each cell had a pressure associated with it calculated from the density in the cell, and this pressure could be positive or negative depending on whether the density in the cell was over-dense or under-dense. Using the same Morse potential, a force was calculated as:

|퐹 | = 2푎퐷[푒−2푎(푟̅−푟0) − 푒−푎(푟̅−푟0)] (5.2a)

21/6 푟̅ = (5.2b) 푛1/3 where 푟̅ is the average distance between real particles in that cell, 푛 is the atomic density, and the factor of 21/6 comes from the assumption that the lattice is a hexagonal close- packed lattice, specifically in a face-centered cubic unit cell arrangement.

Figure 15: Organizational diagram of the pressure-based pair-potential algorithm.

The force calculated in (5.2a) was mapped to each of the nodes surrounding the cell as a vector force, pointed directly towards (for a negative pressure) or directly away from

(for a positive pressure) the center of the cell. For a reminder of the discretization geometry, see Figure 13B. Each node then will have 2, 4, or 8 (in 1-D, 2-D, or 3-D, respectively) vector contributions from the surrounding cells that get added together to form a single vector force at each node.

The next step in the process is to correct for self-forces. A simple thought experiment reveals that they clearly exist with this implementation: put a single macroparticle in a cell surrounded by nothing but vacuum; it will still experience forces using the algorithm as described so far, since even a single simulation particle contributes to the density that is used to determine average separation distance. These self-forces will cause artificial heating of the simulation, as energy is not conserved. The solution to this problem

72 involves removing the contribution of a given macroparticle prior to applying the force on that macroparticle. Rewriting the force in Equation (5.2a) as 퐹(푛, 푛0), to make explicit the dependence on the cell density 푛, and on solid density (or equilibrium density) 푛0. The self-forces correction factor 훿퐹푖 is then calculated as:

훿퐹푖 = 퐹(푛, 푛0) − 퐹(푛 − 푛푖, 푛0 − 푛푖) (5.3) where 푛푖 is the density contribution of macroparticle 푖 to the current cell. For simulations with macroparticles representing nonuniform numbers of real particles (i.e. nonuniform macroparticle weights), this must be calculated for each individual macroparticle. For simulations where all macroparticles represent the same number of real particles, this value may be calculated once for each cell to increase computational efficiency. The correction factor then gets applied as a separate pressure term during the interpolation phase.

The last step of the implementation is interpolating the force from the nodes back to the particles. To be consistent with the interpolation performed by LSP for electromagnetic forces and fields, this was done as a bilinear interpolation only from the nearest neighboring nodes to the particle. This vector force was then used to update the particle’s momentum. This entire process was performed at the end of the PIC cycle, along with all other algorithms supplementing the PIC equations of motion given in

Equations (4.4a), (4.4b), and (4.5a)-(4.5e), such as ionization and collision processes.

Since the density is calculated on the timestep, so is the force, and since the momentum is calculated on the half timestep this algorithm is second-order in ∆푡 and consistent with the leapfrog time-integration scheme.

This algorithm also had its strengths and its weaknesses. Its biggest strength is that it was self-consistent with all the other PIC algorithms being evaluated. Given that the force was calculated via the nodes, it was also an O(N) process, making it much faster to evaluate than the direct particle-particle potential. Treating the force as a pressure between cells makes the physical effects of such a force obvious, as the idea of a pressure is fairly basic. However, despite the successes that it had, it also had some weaknesses.

Energy and momentum were no longer conserved exactly, since the particles do not transfer momentum directly. Within the bulk, they were still conserved on the average, however at the target-vacuum interface artificial heating would occur over long periods of time. In addition, since the particles are not treated individually, the hottest atoms near the target surface are allowed to travel ballistically through the material, creating a situation similar to that discussed in Section (2.2.2), where there would be “fast” atoms traveling through the material and a “slow” return current of atoms taking their place.

This is obviously not physical and has the effect of greatly decreasing the damage done on the surface of the target since the energy absorbed by the laser will be spread further into the target than physically reasonable.

5.2 Lattice-Based Interatomic Forces Algorithm

With the lessons learned in the work described in the previous section, the pair- potential algorithm ultimately developed in this work was based on calculating the approximate positions of the real particles in the system. For the simulation results

74 highlighted later, the Lennard-Jones pair potential, illustrated in Figure 16B, served as the starting point for the model, given by:

푟 12 푟 6 푈 (∆푟) = 퐷 (( 0 ) − 2 ( 0 ) ) (5.4) 퐿−퐽 ∆푟 ∆푟 where ∆푟 is the separation distance between real atoms, 푟0 is the material-dependent equilibrium distance between atoms, and 퐷 is the material-dependent dissociation constant representing the depth of the potential well. Similarly to the pressure-based model discussed in Section (5.1) above, for computational simplicity a cubic lattice structure was assumed, though this method can easily be expanded to use a hexagonal close-packed lattice structure or a dynamically varying structure, as well as a more accurate many-body potential.

In contrast to the pressure-based model above, this model is particle-based in that it uses the local environment at a macroparticle’s position to approximate the conditions on a real particle in its position. Using the local environment at a macroparticle, bilinearly interpolated from the surrounding nodes, as well as the lattice structure assumption, it is possible to calculate where the nearest-neighbor real particles would be relative to a real particle residing at the macroparticle’s position. Using the cubic lattice assumption, the average distance between atoms as well as the change in separation distance between atoms when a gradient in the density 푛 is present can be represented by:

푟̅ = 푛−1/3 (5.5a)

1 ∇⃗⃗ 푟̅ = − 푛−4/3∇⃗⃗ 푛 (5.5b) 3

With Equations (5.5a) and (5.5b), and using the approximation that ∇⃗⃗ 푟̅ is constant in the near vicinity of an atom, the distance to each nearest-neighbor atom can be calculated.

For example, for the two nearest-neighbor atoms in the x-direction, we would get separation distances of:

1 휕푟̅ −1 ∆푟 = 푟̅ (1 ± ) (5.6) 1,2 2 휕푥 where ∆푟1 (+) is the distance from the central atom to the nearest-neighbor in the positive x-direction and ∆푟2 (-) is the distance from the central atom to the nearest-neighbor in the negative x-direction. Equation (5.6) provides locations for nearest-neighbor real atoms to a real atom located at the macroparticle’s position, meaning that the Lennard-Jones pair- potential in Equation (5.4) can be applied directly to the hypothetical real particles as shown in Figure 16A. The force that would be applied to this atom located at the macroparticle’s position is then applied directly to the macroparticle to update its position. Using the local environment at the macroparticle’s position to perform the calculation, which was bilinearly interpolated from the surrounding nodes, means that the algorithm is fully self-consistent with other PIC algorithms, thus preventing numerical artifacts.

Figure 16: Illustration of A) the pair-potential calculation between an atom at a macroparticle’s position (central, blue circle) and its hypothetical nearest neighbors (outer, dashed circles), and B) the Lennard-Jones pair potential as a function of atomic separation distance.

Although similar in appearance to the pressure model discussed in Section (5.1), insofar as it calculates a force from the density, this algorithm is far more accurate and stable than the one mentioned previously and it converges on a solution almost an order of magnitude faster, allowing for much larger time steps and cell sizes leading to faster evaluation. Though computationally more complex to implement with this algorithm, the prescription in Equation (5.3) for eliminating self-forces still applies. Also, although neither energy nor momentum is conserved exactly, they are both conserved on the average. As a figure of merit, the total energy in a simulation employing only this algorithm (i.e. no electromagnetic dynamics) such as those discussed in the next chapter will decrease by approximately 1% after a nanosecond. In terms of energy conservation in PIC simulations, where energy non-conservation on the order of 10-20% is common, this is extremely good. 77

With this PIC-based pair-potential algorithm, this simulation method is fully capable of modeling a wide variety of laser damage situations, with various target structures, laser parameters, and materials. All the model needs is to be initialized to a temperature profile indicative of its thermodynamic state after the laser interaction, and this initialization is completely flexible. The initial temperature profile can come from any number of sources, including but not limited to an empirical laser absorption model, the TTM described in Section (3.1.1), an ab-initio calculation, or even an experimental measurement. For this work it was decided that, since PIC was already being used to model the atomic transport associated with laser damage after the laser interaction, the

PIC code LSP would also be used to model the laser interaction. Since PIC codes have long been used to model lasers and their interaction with various materials, this was a straight-forward extension to the method. Section (6.1) details the laser interaction portion of the simulation process, as well as the requirements to make a standard PIC code accurately handle a laser interaction in the laser damage regime. However, despite the much improved stability and accuracy of this algorithm, in its current form there are still a few issues that arise due to the spatial discretization of the system that must be accounted for that are detailed next.

5.3 Correcting for Discretization Effects

One of the discretization issues is that the particle density as calculated on the spatial nodes is effectively smoothed over with the large cell sizes that permit evaluation of a laser damage simulation on a reasonable time scale. This issue actually causes two

78 separate problems. The first is that the spatial smoothing of the density under-estimates the repulsive component of the force that should occur when the target density exceeds solid density, or in terms more suitable to a pair potential, when the separation distance between atoms is less than the equilibrium distance. This issue was indicated briefly in

Section (5.1), as it is what allows there to be a “fast” atomic current flowing freely into the material with a “slow” return current of atoms maintaining equilibrium density in the material in analogy with the discussion of hot electron generation in Section (2.2.2). The solution to this problem used in this work was to use a modified Lennard-Jones potential that is infinite at distances closer than the equilibrium distance. In terms of actual implementation, this was done in the code by performing a check any time a particle is advanced on a timestep and pushed into a different cell: if the particle’s weight would push its new cell above the equilibrium density then that particle, instead of advancing into the new cell, reflects off the cell wall back into its original cell. This makes the equilibrium density in a cell a hard limit, and prevents the “fast” atomic current from propagating into the material in an unphysical manner.

The second issue associated with the smoothing of the density due to large cell sizes is that the range of the attractive component of the pair-potential force is over-estimated.

This can be seen if one imagines a single particle leaving the target and entering the vacuum. The target will apply an attractive force to the particle over the entire distance of a cell, instead of the much smaller actual range of the force typically on the order of the atomic separation distance. The effective range of the force is then a factor of ∆푥/푟0 too large which, for example with the simulations that will be discussed below, means the

79 range is ~100-1000 times too large. Partially balancing this is the fact that the smooth density under-estimates the magnitude of the attractive component of the force in the same way that it under-estimates the repulsive component. The resolution of this discrepancy in this work was done by introducing a constant multiplicative factor to the force calculated by integrating the energy needed for a single atom to escape the target surface, essentially calculating the atomic work function of the material. This integration was done for the exact, continuous analytical system as well as for this PIC implementation of a pair potential, with the multiplicative factor on the PIC force used to set the work functions equal to each other. This equality was found to be a physically reasonable way of ensuring the laser damage threshold of the material was matched. For the case of the copper system outlined in the rest of this chapter, this multiplicative factor was ~1/3.

A third major issue with this pair-potential algorithm that must be addressed is the unphysical situation it naturally creates at the target-vacuum interface: the outermost node of the target contributes a density gradient that is equal to the solid density of the material divided by the cell size. This extremely large gradient is then linearly interpolated across the first cell into the target causing even small deviations away from the equilibrium density, which naturally occur due to the statistical nature of a PIC code, to create large unphysical forces on the particles in that first cell ruining energy conservation. For the first cell into the vacuum region of the simulation this situation is fine: a strong restorative force is physically reasonable for a solid metal target and the strength of this restorative force is corrected as prescribed in the paragraph above. Within

80 the target however, when considering a simple metal block in vacuum, there are no physical gradients. The fact that the particles in that first cell feel a strong gradient is entirely a numerical artifact of the PIC approach. Therefore, the solution implemented in this work was to selectively ignore unphysically large gradients within the first cell of the target without affecting any physical gradients. Within the code, this is implemented when calculating the force on a particle by ignoring the gradient at any node that is above

75% of the solid density of the material per cell if the particle’s cell is above 75% of the solid density of the material. These two checks, requiring a very large density as well as a very large density gradient, will only ever occur at the interface between a solid target and the vacuum. As soon as the damage process begins on a target’s surface, the irradiated area quickly leaves this regime and the forces are handled normally. This check is done separately for each spatial dimension, so small gradients in the transverse direction will still affect the particles in the first cell, but the large gradient normal to the target’s surface will not be taken into account.

Lastly, it is clear from Equation (5.6) that there is a singularity at |휕푟 /휕푥| = 2, a situation that can only occur at this location of unphysically large gradients. Unlike the situation above, however, this situation can only occur when the gradient is large and the density is small, such as would occur in the first cell of vacuum outside the target. The correction to this used in this work is simply to set a hard cap such that |휕푟 /휕푥| ≤ 1/2.

Similar to the conditions in the previous paragraph, this situation is only created by a sharp target-vacuum interface; almost immediately after the damage process begins, this correction is no longer necessary near the damage region. For a rough baseline of when

81 these last two corrections apply, it is approximately when the density discrepancy between two adjacent cells is more than 80% of the solid density of the material (e.g. one cell is at >90% of solid density and its neighbor cell is at <10% of solid density).

With this we have a complete description of atomic forces within a target represented by a pair potential and corrections to the numerical artifacts created by the discretization of space and under-resolving the atomic separation distance. The next step is to test the model to ensure that it indeed will allow a solid target to maintain its structure, and that additionally it will allow a target to resolidify after the damage process, or “heal”, to create a sharp, well-defined damage crater. After ensuring the model recreates the expected basic behavior, the next step is to benchmark the algorithm against published experimental results. The rest of this chapter will focus on testing this method and the next chapter will describe the application of the method and its benchmarking against experiment.

5.4 Simple Test Case

The first test of this model was simply designed to determine if the pair-potential model will indeed maintain a target’s structure for an extended period of time when there are no outside forces interacting with it. In reality, a block of solid copper in free space experiencing no external interactions should remain static. If the simulation grid is taken to reside in the rest frame of the copper block, then the copper block should remain motionless, its structure should remain intact, and no interesting changes should occur.

However, for a simulation employing only the standard PIC algorithms, this would not be

82 the case. As discussed at the beginning of this chapter, since PIC codes are designed to handle plasmas which are inherently electromagnetic and do not retain a structure the way a solid does, a block of electrically neutral copper in vacuum would not remain static. Instead, each atom would propagate ballistically out from the target in whichever direction its velocity was initialized to. This would result in the target dispersing as if in a gaseous state. Checking that this situation is corrected thus makes for an appropriate first test of the pair-potential model.

Physically, the simulations performed for this test modeled a two-dimensional, 1 µm

× 1 µm copper block in a perfect vacuum initialized to a temperature of 0.05 eV (580 K), or roughly twice room temperature. The simulations included no external forces or fields, and no laser interaction. Computationally, this was achieved with a 1 µm × 1 µm block

84 consisting of neutral copper atoms at the solid density of copper, ~8.5e22 #/cm3, placed at the center of an otherwise-empty 2 µm × 2 µm spatial grid with absorbing boundaries.

The cell size was 0.01 µm × 0.01 µm, and there were 3600 neutral Cu macroparticles per cell corresponding to a macroparticle weight of ~2.4e7 atoms. Note that no charged particles (electrons or ions) were used for this simulation. The timestep was 11.79 fs, and the simulation was allowed to evolve for 200 ps. Three separate simulations were performed: the first was a pure PIC simulation without a pair-potential model; the second included the pair-potential model; and the third simulation cut a square hole in the center of the copper block, creating a donut-shaped target, to show that even non-trivial structures are maintained. The results of these simulations are shown in Figure 17.

Figure 17A shows the initial target geometry for the simulations shown in Figure 17B and Figure 17C in the form of a color map of the number density of copper atoms, in units of the solid density of copper. Figure 17B shows the density of copper atoms after the simulation without the pair-potential algorithm is allowed to evolve for 200 ps. A quick comparison to Figure 17A shows a distinct difference in the target structure from the start of the simulation to 200 ps later as the atoms begin to disperse as if in gaseous form, confirming the previous statement about the nature of the PIC simulation process.

Figure 17C shows the density of copper atoms after the simulation with the pair-potential algorithm is allowed to evolve for 200 ps. In contrast to the previous simulation, this simulation shows negligible difference between the initial conditions and the state of the target after 200 ps, validating this pair-potential implementation at least for this limited set of conditions. Lastly, Figure 17D goes one step further and shows that, even for non-

85 trivial surfaces, there is negligible surface modification after 200 ps with this pair- potential algorithm for this set of conditions.

This test may seem minor at first glance, after all it is a simple chunk of metal in vacuum where the desired result is that nothing happens, but it actually represents an important step in developing a PIC-based approach to modeling laser damage and represented a significant challenge during development. This test was the very first use of a PIC code in modeling a material in its solid phase, and with the inclusion of the pair- potential algorithm developed here it seems to do so rather well.

The next step in testing this algorithm was to take this same copper target, and hit it with a femtosecond-pulse Gaussian laser beam. This next step is even more important, since adding this one feature to the simulation now means that all at once we are testing the damage fluence threshold, testing the target’s solidity under highly non-equilibrium conditions, as well as testing the target’s ability to resolidify and heal after the damage process is done. Chapter 6 will go through the details of evaluating a full laser damage simulation from start to finish using the PIC method developed in this chapter.

Chapter 6: A Fundamental Model of Laser Damage

This chapter will show the results of the first fundamental simulations of the full laser damage process, from the incoming laser interacting with the conduction band electrons, to the electron-lattice thermalization, and then to the atomistic dynamics leading to damage crater formation. Until this work, PIC had never been used in this regime, and there is good reason for that: there were many challenges to developing this technique.

PIC codes were developed ideally to handle high-temperature (compared to the scale of condensed-matter physics), low-density plasmas. The simulation of laser damage requires the modeling of relatively low-temperature, high-density solids undergoing phase changes. At high densities relative to the critical density, PIC is also best at short timescales, femtoseconds to picoseconds, and notoriously unstable at long time-scales. The laser damage process, however, requires the target to be at solid density, which is many times the critical density, and simulated over time-scales on the order of nanoseconds or longer.

The series of simulations described in this chapter represents a complete simulation of the physics of laser damage. In order to overcome all the complications mentioned previously throughout this work, a strategy was developed for simulating the laser damage process by dividing it into three steps, each step optimized for a different time scale and the processes that dominate at each scale. This optimization required

87 consideration of multiple factors, the two most important being meeting the requirements for numerical stability and accuracy while at the same time maintaining reasonable runtime, and separating the short pulse laser damage process into its three natural timescales: the femtosecond laser interaction; the picosecond material thermalization; and the nanosecond atomic transport and resolidification. Completing the simulation in this manner allows for high spatial and temporal resolution when needed, specifically during the laser interaction, but also allows for rapid evaluation of the simulation when this high resolution is not necessary, specifically when modeling the atomic motion. The following sections will each focus on one of these three regimes. The last section will apply this method to the study of LIPSS formation as discussed in Section (1.3) in an attempt to solve the question of the physical mechanism(s) behind their generation.

6.1 Laser-Target Interaction

The first regime is the laser-target interaction itself. While this amounts to less than

0.0025% of the total time simulated, it takes more than half of the total computation time due to the high spatial and temporal resolution required to accurately resolve the laser parameters, such as the laser period and the laser interaction region at the sharp interface between the solid density target and the vacuum region. While the PIC method is fully capable of running in 3D, for the purpose of rapid development all of the following simulations were performed in a 2D geometry known as 2D3V. In 2D3V, all vector quantities maintain all 3 spatial components, but the particles are only allowed to propagate in two spatial dimensions. The third spatial dimension is known as a virtual

88 dimension, and exists to allow self-consistent electromagnetic wave propagation which cannot happen in a purely 2D environment. Thus light can freely propagate in the non- virtual dimensions or reflect at an interface, and electrons can acquire momentum in the virtual dimension, establishing a current which generates a magnetic field. However, the electrons will not have a displacement in the virtual dimension.

For the first tests of this method’s validity for modeling a laser damaging a material, the technique was applied to a flat, thin copper film. This is because the geometry is basic enough for the results to be easily understood and easily compared to existing experiments, as copper is a very common material both in laser damage studies [109,110] and previous applications of PIC [111,112]. Additionally, PIC codes have long been used to model lasers interacting with metals, since they are well approximated on short timescales by a plasma as the electrons in the conduction band are naturally free to move around within the metal and the atomic motion is negligible on femtosecond timescales

[113].

The laser modeled for this first study was a 60 fs FWHM pulse with a Gaussian envelope in the transverse direction and a sine-squared envelope in the direction of propagation. The laser pulse was centered at a wavelength of 800 nm and was focused to a 1 µm waist at normal incidence to the target for a peak fluence that varied between 0.1

J/cm2 and 2.0 J/cm2 for these tests. The laser was propagated onto a spatial grid with a resolution of 1/128 µm in the propagation direction and 1/32 µm in the transverse direction, and with a timestep of 5.5 attoseconds, which for the prescribed grid is 0.3 times the electromagnetic Courant-limited timestep. For stability considerations, the

89 computation grid was bounded on three sides with a conductor with an adjacent Perfectly

Matched Layer (PML) of 3 cells in thickness. A PML is an absorbing layer designed such that it does not reflect waves at its interface, creating a strongly absorbing surface for electromagnetic waves originating inside the computational grid [114]. The boundary with the incoming laser was an open, absorbing boundary, but not employing a PML. The laser was incident on rectangular targets which were between 4.5 µm and 6.0 µm wide and between 0.75 µm and 1.0 µm thick, depending on incident laser intensity. For each tested laser intensity, the target size was picked to be as small as possible for computational efficiency but still large enough to capture the full damage crater so as to not lose generality. The targets were modeled with 900 macroelectrons and 25 macro Cu+ ions per cell. For these early studies, ionization effects were not included, however for a dielectric target where ionization dynamics are far more important and cannot be neglected, LSP is fully capable of including both field and avalanche ionization processes.

Collisions between particles were handled with the Jones model described in Chapter

4. As mentioned previously, as LSP is packaged there is an option between Spitzer or

LMD scattering rates. Given that the Spitzer model for collision rates is designed for hot plasmas and scales as 푇−3/2, it is inappropriate for use with room-temperature solid targets as it significantly over-estimates the collision frequency between particles.

However, the LMD rates are designed to calculate collision frequencies for electron-ion collisions only, not for ion-ion or electron-electron collisions. This is problematic as accurately resolving electron-electron collisions is crucial to accurately modeling the

90 laser interaction, as was discussed in Chapter 2. A lot of the important physics for the laser damage process depends on the electron collision frequency, including the electromagnetic skin depth, absorption and reflection coefficients, and electron thermalization, all of which will have a large effect on the resulting damage spot.

On the other hand, the electrons in the system do not remain at room temperature for long after the laser has been turned on. The laser will rapidly heat the electrons and they will exist in a highly non-equilibrium state for most of the laser interaction, rapidly switching from a cold thermal distribution of electrons, to a hot non-thermal distribution of electrons, and then slowly settling to a warm thermal distribution of electrons. As it turns out, there is no existing single physics-based model that can accurately describe electron collision frequencies throughout all of these regimes that occur during the laser damage process of a material starting at room temperature, nor is there a reliable method to measure these rates directly [115]. A solution to this problem was proposed by

Colombier et al. (2008), where the efficacy of a simple polynomial interpolation between the two well-understood scattering regimes of ultracold metals and hot plasmas was tested. This interpolation is shown in Figure 18.

The interpolation uses a cubic polynomial in temperature in order to simultaneously match both the magnitude and slope of the cold solid model and the plasma model at each endpoint. The interpolation regime is defined by Colombier to start at the Fermi energy of the material, and go to twice the Fermi energy of the material. Indirect testing of numerical results to experimental measurements indicates that this method is indeed a reasonable approximation to the electron-electron scattering rates. As such, as part of this

91 work, this model was incorporated into LSP to calculate electron-electron collision rates during the laser interaction. For electron-ion collisions, the LMD model was used as it has been shown to accurately handle collisions in this regime. Lastly, ion-ion collisions on the timescale of the laser interaction are negligible, and as such the discussion of them will be postponed until Section (6.3).

The results of one of the simulations of a laser interacting with a copper target are shown in Figure 19. Figure 19A shows the laser coming in at normal incidence, hitting the front surface of the target, and subsequently decaying rapidly over the skin depth as it propagates further into the copper target. Figure 19B shows the resulting temperature profile of the electrons in the simulation after the laser of fluence 2.0 J/cm2 has left the simulation grid, either through reflecting off the surface or being absorbed into the material. The large temperature gradient between the hottest electrons at nearly 4 eV and the room-temperature electrons at 0.025 eV illustrates well the nonequilibrium nature of this interaction. It is worth noting that at the time this electron temperature profile is taken, the ion temperature in the target has not yet begun to increase by an appreciable amount.

6.2 Electron-Lattice Thermalization

Once the electrons have relaxed into a thermal distribution within themselves, the second stage of the simulation process can begin. For the timescale of electron-lattice thermalization, generally on the order of 100 ps, the lattice is still relatively static. Using

Figure 12 as an example one can see that indeed target evolution has begun, however, within the resolution of a PIC cell, generally around 10 nm or larger for this work, the atomic motion is still minimal but large enough to be a potential source of error, the effect of which will need to be studied in future work. However, this work makes the assumption that the target evolution can be neglected during this process.

With that assumption, the most computationally efficient way to handle the electron- lattiace thermalization is through the use of the two-temperature model (TTM). Recalling

Chapter 3, many issues were brought up with using the TTM to model the entire laser damage process, including many fitting parameters, an assumption on what constitutes damage, and little to no physical insight into the problem. Using the TTM in this manner, just to evaluate the thermalization of the target in between PIC simulations of the laser interaction and the atomic transport, however, eliminates all of these problems. Modeling the laser interaction with PIC eliminates the need for a source term, where most of the fitting parameters enter the equation. The atomic transport simulation after the electrons and lattice have thermalized eliminates the need to make an assumption on what constitutes damage. And both of the other stages of the simulation process add physical insight back into the modeling. Without a source term, and making the standard

94 assumption that the lattice thermal conductivity is negligible compared to the electron thermal conductivity [52,116-118], Equations (3.1a) and (3.1b) reduce to

휕푇 퐶 푒 = ∇ ∙ (퐾 ∇푇 ) − 푔(푇 − 푇 ) (6.1a) 푒 휕푡 푒 푒 푒 푙

휕푇 퐶 푙 = 푔(푇 − 푇 ) (6.1b) 푙 휕푡 푒 푙 where just as before 퐶 is the heat capacity, 푇 is the temperature, 퐾 is the thermal conductivity, and 푔 is a coupling constant, and the subscripts 푒 and 푙 denote the electrons and the lattice, respectively.

In order to complete this stage as accurately as possible, temperature-dependent values for all of the constants were taken from [118], where a detailed density-of-states

(DOS) calculation was performed to calculate these values for various metals. Equations

(6.1a) and (6.1b) were then evaluated on the same spatial grid described in Section (6.1) that was used to evaluate the laser interaction. Due to the simplicity of the TTM calculation, specifically both its lack of computational complexity as well as the monotonicity of the solution, it was sufficient to use a first-order explicit finite-difference time-domain approach to numerically solve these equations on the grid. Instead of using the same timestep as was used in the laser interaction simulation, the timestep for this stage was adjusted to ensure numerical convergence of the solution but also allow for rapid evaluation. In stark contrast to the laser-target interaction stage, the thermalization stage accounts for ~2.5% of the simulated time and only ~0.5% of the computation time.

For the simulations shown here, the timestep used was 2 fs. For this finite differencing, shown in 1D for simplicity, the resulting set of equations takes the form:

푛+1 푛 (퐾 ∇푇 )푛 − (퐾 ∇푇 )푛 푛 푛 (푇푒)푖 (푇푒)푖 푒 푒 푖+1/2 푒 푒 푖−1/2 푛 ((푇푒)푖 − (푇푙)푖 ) = + 푛 − 푔푖 푛 (6.2a) ∆푡 ∆푡 (퐶푒)푖 ∆푥 (퐶푒)푖

푛+1 푛 푛 푛 (푇푙)푖 (푇푙)푖 푛 ((푇푒)푖 − (푇푙)푖 ) = + 푔푖 푛 (6.2b) ∆푡 ∆푡 (퐶푙)푖

(퐾 )푛 + (퐾 )푛 (푇 )푛 − (푇 )푛 (퐾 ∇푇 )푛 = 푒 푖 푒 푖+1 푒 푖+1 푒 푖 (6.2c) 푒 푒 푖+1/2 2 ∆푥 where in addition to the parameters listed before, ∆푡 is the timestep size, ∆푥 is the spatial grid size, and 푖 and 푛 are the indices denoting spatial node and timestep number, respectively.

Implicit in Equations (6.2a), (6.2b), and (6.2c) is that the system is considered loss- less, such that no energy is lost to the environment outside of the target. In the case of a laser damage experiment performed in vacuum, this is nearly exact. In the case of the experiment being done in air, this assumption is still approximately true as thermal losses to air on picosecond timescales are still negligible. During the numerical evaluation of these equations, each boundary of the target is mirrored such that all spatial derivatives are zero across the boundary. This results in zero thermal energy transfer across boundaries, ensuring that all thermal energy remains within the target. This simulation process is continued until the average electron temperature across all nodes and the average lattice temperature across all nodes are within 0.1% of each other, at which time the target is considered to be sufficiently thermalized. This condition is necessary, as it has been noted [52] that the electrons and lattice don’t equilibrate exactly on these timescales, but instead the electrons will continue to spread the thermal energy

96 throughout the target over microsecond timescales until the entire target is one uniform temperature.

The results of applying this TTM model to the simulation discussed in Section (6.1) is shown in Figure 20. Figure 20A shows the electron temperature distribution at the end of the laser-target interaction stage of the simulation process, as was also shown in Figure

19B. This temperature distribution is used to initialize Equations (6.2a), (6.2b), and

(6.2c), and the initial lattice temperature is assumed to be room temperature, 0.025 eV.

This assumption is approximately true for the actual lattice temperature at the end of the laser-interaction stage; however, the temperature distribution at the end of the first stage contains significant numerical noise inherent to PIC codes when using only a few macroparticles. The Cu+ ions needed to be simulated in the first stage for the target to be electrically neutral, but they do not play a major role in the process otherwise due to the short time scale, so for computational simplicity only a few macroparticles were used.

Therefore, it was decided that this approximation was more physical than using the numerical results from the first stage directly.

Figure 20: Results of the second stage of the simulation process: the TTM calculation of the electron-lattice thermalization for a 2.0 J/cm2 fluence laser pulse incident on a flat copper target. A) The initial electron temperature used as input to initialize the TTM. B) The resulting lattice temperature after the electrons and lattice reach thermal equilibrium approximately 80 ps into the thermalization process.

Figure 20B shows the resulting lattice temperature when the electrons and lattice reach thermal equilibrium as defined above. It can be seen that a small amount of thermal spreading occurs during this process, and the lattice temperature at the surface is considerably lower than the initial peak electron temperature at the surface. This temperature distribution is subsequently used to initialize the third simulation stage, the atomic transport stage using a modified PIC method that includes the pair-potential algorithm described in Chapter 5.

Finally, we note that this process can be modeled using PIC if collisions are included using the collision models discussed previously. The advantage of this is that the small amount of ion motion that occurs on the picosecond time scale required for equilibration

98 is included. However, otherwise, PIC is an unnecessarily complicated tool for the relatively simple processes occurring and the TTM is significantly more efficient.

6.3 Atomic Transport

The third and last stage of the simulation process is the atomic transport stage, performed with the modified PIC method initialized with the results of the TTM calculation above and allowed to evolve using the pair-potential algorithm described in

Chapter 5. For this stage of the simulation process, all of the simulation parameters are once again adjusted to balance performance and accuracy. In comparison to the other three stages, the atomic transport stage accounts for ~97.5% of the simulated time and just under half of the total computation time, making it still far more computationally efficient than the laser-interaction stage, but not as efficient as the TTM stage.

The ions and electrons from the TTM simulation were assumed to have recombined and were replaced with neutral copper atoms. This is essentially the approach taken by molecular dynamics simulations. Our PIC model can accommodate the simultaneous presence and interaction of electrons, ions, and atoms, including recombination, for scenarios where that is required and this will be explored in the future.

As with the laser-target interaction stage, accurate modeling of scattering rates is important to the over-all accuracy of the simulation. For a completely self-consistent treatment of the particles in this stage, the atomic scattering rates on spatial scales shorter than the cell size were determined by analogy with the way PIC codes handle electromagnetic scattering. An exact solution of Maxwell’s equations will, in principle,

99 provide a complete description of the electromagnetic fields and forces of interest to this problem. However, the approximation PIC uses to partially discretize space means that evaluating Maxwell’s equations only resolves interactions at the scale length of a cell size or larger. Thus, Coulomb collisions must be incorporated separately in order to handle interactions below the scale length of a cell size, in this case in the form of the Jones model. Atomic collisions for this stage of the simulation process were handled in the same fashion. While the Lennard-Jones pair-potential, in principle, represents the combination of all the relevant interactions between neutral atoms, because of the implementation here the model will only accurately resolve interactions at the scale length of a cell size or larger. Since a collision rate for these neutral atoms would be designed to add in the effect of this potential at length scales smaller than a cell size, it is only natural that the collision rate should come from the Lennard-Jones pair-potential.

To that end, a small MD code was developed in order to model the dynamics of a few copper atoms experiencing only the effects of the Lennard-Jones pair-potential, for a given temperature, over a short period of time. For simplicity, the lattice was initialized first to a cubic structure and the velocities were sampled from a Maxwellian distribution for the given temperature. Since the cubic lattice is an unstable equilibrium for a pair potential, the atoms quickly rearranged themselves into a hexagonal close-packed lattice, with the extra energy going into heating the atoms. A snapshot of this new distribution was taken, and used to re-initialize a new MD simulation that maintained the atomic positions, but replaced the velocities again with the desired temperature distribution. This process was iterated on until the full 6D phase space was consistent with a hexagonal

100 close-packed lattice at a given temperature. Using this method, the “scattering rate” for the Lennard-Jones pair-potential model for neutral copper atoms was determined, using the definition of scattering rate as being the amount of time necessary such that the total magnitude of the many small angular deflections would add up to 휋/2. In this way, a temperature-dependent scattering rate can be tabulated for a given material. This scattering rate was used in the Jones model to provide self-consistent atomic scattering in this stage of the simulation process.

The PIC simulation itself is performed with a spatial resolution of 10 nm × 10 nm cells, 12 fs timesteps, and 3600 neutral Cu atoms per cell. For this stage, the boundaries in the transverse direction are periodic, and the boundaries parallel to the target surface are absorbing. The timestep used here corresponds to 500 times the electromagnetic

Courant-limited timestep for this grid size, but since this stage of the process includes only neutral atoms and no electromagnetic fields this timestep is fully converged and stable. Because of this electrically neutral situation, the reference speed for the relevant

Courant limit is the atomic velocity, not the speed of light, putting this timestep at approximately 0.002 times the atomic motion Courant-limited timestep, well into the stable timestep regime. The target sizes remain constant from the previous two stages, ranging from 4.5-6 µm wide and 0.75-1.0 µm thick, depending on laser fluence. These simulations were allowed to evolve for 2-4 ns, allowing the simulation to capture all relevant ablation and resolidification. At the end of this stage, there will still be a hot, low-density gas of atoms filling the vacuum outside the target which would take several microseconds to fully evacuate, but this gas has no effect on the resulting target structure

101 and thus will not be discussed further. For all density plots in the remainder of this work, in order to preserve the clarity of the plots there will be a lower cutoff at 1% of solid density below which it will not be plotted so as to exclude this hot gas from the relevant results.

Figure 21: Time resolved dynamics of the ablation plume for a copper target hit with a laser pulse at 2.0 J/cm2 fluence, represented by a density profile in units of the solid density of copper here taken to be 8.5e22 number/cm3. The densities shown correspond to A) the start of the simulation, B) 40 ps, C) 80 ps, D) 120 ps, E) 1.5 ns, F) 2 ns, G) 2.5 ns, and H) 3 ns.

Figure 21 shows the temporal evolution of one of the copper targets upon being hit with a 2.0 J/cm2 fluence laser pulse. Figure 21A through Figure 21D show the ablation plume as it begins to form at early times, and Figure 21E through Figure 21H shows the ablation plume dissipating and eventually leaving a clean, well-defined damage crater which has ceased evolving and has resolidified. This well-defined damage crater can be compared directly to experimental data, and is the first comprehensive laser damage simulation capable of being compared directly to all relevant experimental

102 measurements. Figure 22 shows the resulting damage crater for a flat copper target for

0.5 J/cm2, 1.0 J/cm2, and 2.0 J/cm2 laser fluences, explicitly showing the size difference in the targets.

Figure 22: Density profiles of the resultant damage crater after surface evolution has ceased, in units of the solid density of copper, for A) 0.5 J/cm2, B) 1.0 J/cm2, and C) 2.0 J/cm2.

With the damage patterns from several fluences, it is possible to now compare to experimental data. Since these simulations used a tight focus geometry and focal spot smaller than that in a typical experiment to allow for rapid development of the method, the most direct comparison to experiment will involve using a procedure designed to extract the damage threshold fluence from experimental crater morphologies and applying it to the simulated crater morphologies shown in Figure 22 [109]. The approach is based on interpolation of the crater width, using the relation:

Γ 2 퐹 = 퐹푒푥푝 (− ( ) ) (6.3) 푡ℎ 푎 where Γ is the crater diameter, 푎 is the laser beam diameter, 퐹 is the incident laser fluence, and 퐹푡ℎ is the damage threshold fluence. Using the FWHM of the modeled crater

103 and beam widths illustrated above with Equation (6.3) results in a predicted damage threshold fluence of 0.15 J/cm2, consistent with published experimental measurements of copper thresholds for the modeled laser system [109,119]. Though many experiments have been performed with copper it is well known that damage thresholds depend strongly on laser parameters, thus the experimental measurements referenced here were chosen for their close match to the simulated laser parameters.

This is a big success for the model: not only is the simulated behavior consistent with the expected behavior of a target during the laser damage process allowing for qualitative predictions, but the resulting craters predict a damage threshold that is within experimental error bars, meaning the method may be useful for quantitative predictions as well.

6.4 Modeling LIPSS

The results of the previous section show promise for this method, but the real test of the power and versatility of the method will come from more complicated situations, such as those involved in the generation of LIPSS. As discussed in Section (1.3), LIPSS is a periodic corrugation in the damage pattern on the surface of a target, and though it has been studied experimentally for decades, there is still debate on the mechanism by which it forms. As discussed in Section (2.3), one of the leading theories for the generation of

LIPSS is through the excitation of an evanescent surface electromagnetic wave called a surface plasmon-polariton (SPP). Given that an SPP is inherently an electromagnetic phenomenon, none of the theoretical tools discussed in Chapter 3 would be capable of

104 modeling it directly. However, PIC codes have long been used to model electromagnetic phenomenon, making this a prime candidate for a test of this technique.

Motivated by the experimental results of [120], it was decided that the target for this test would be similar to that of Sections (6.1-6.3), with the addition of a pre-existing, subwavelength scratch along the surface, offset somewhat from the focal point. As was discussed in Section (2.3.3), a laser beam cannot couple directly to a given SPP mode, but adding a scratch to the surface allows for the grating coupling mechanism to give rise to SPP generation. Given the nature of LIPSS generation as well as experimental measurements for its periodicity, the target size was doubled in the transverse direction and the beam waist was doubled to match. This provides a larger area for LIPSS generation, and the ability to observe multiple periods in the damage spot for more accurate measurement.

A review of Section (2.3) will reveal that one of the most important factors in SPP generation is the dielectric function of the material. The SPP is a natural consequence of

Maxwell’s equations at an interface, if the permittivities support SPP modes. While accurately modeling the collisions within the target will accurately represent the imaginary component of the dielectric function, it is necessary to manually insert the real component of the dielectric function since PIC codes represent point particles and do not take dipole interactions into account.

There is an issue with inserting the real component of the dielectric function onto the grid, however. As it turns out, for reasons beyond the scope of this discussion, most

FDTD field solvers are unstable at an interface between positive and negative values of

105 the real component of the dielectric function, as would be the case for a metal-vacuum interface. Regarding the Alternating Direction Implicit (ADI) FDTD technique used in

LSP, it too is unstable for negative values of the dielectric function, as can be seen by substituting a negative value into the stability analysis done in [121]. For this work, it was decided that the absolute value of the dielectric function would be used instead.

There are two primary consequences of this: the SPP mode generated is no longer the standard Fano-type, but instead it is of the Brewster-type as differentiated in Section

(2.3.2); and the wavelength of the expected LIPSS pattern changes, though this is easily accounted for with the SPP theory. The effect of the difference in mode is that the

Brewster mode is not bound to the surface as the Fano mode is, therefore any excited SPP will decay more rapidly, but since the two modes have the same dispersion relation, it is believed that their behavior will otherwise be similar such that the primary consequence of this would be to underestimate the electron heating and subsequent damage. Other consequences of this will need to be examined in future works, or a stable field solver under these conditions must be developed and implemented. For the laser-interaction stage of the simulation, values for the magnitude of the real component of the dielectric function of copper were taken from [122], and numerically inserted onto the grid in every cell that contained Cu+ ions, linearly scaling with density up to its room temperature, solid density value. A relative permittivity of 24 was used for this work for copper.

For this set of simulations, the laser modeled was a 60 fs FWHM laser pulse,

Gaussian in the transverse direction and sine-squared in the propagation direction, with a central wavelength of 800 nm focused to a 2 μm waist at a 15 degree angle of incidence

106

(AOI), for various intensities. The target was a 12 μm wide, 1 μm thick Cu foil with a semi-circular scratch of radius 0.5 μm offset 1 μm from the laser focal point. For the laser interaction stage, the number of macroelectrons per cell was increased to 2500, and for the atomic transport stage of the simulation the number of macroatoms per cell was increased to 4225. Otherwise, simulation parameters, spatial and temporal resolutions, and physics models for each stage are identical to those described in Sections (6.1-6.3).

107

The electron temperature after completion of the first stage of the simulation process,

200 fs after the start of the simulation, is shown in Figure 23 for a laser pulse with a fluence of 6.0 J/cm2. At this point in the simulation, it is clear that there is no discernable alteration to the target surface, as there has been negligible energy transfer to the lattice

108 and insufficient time for the ablation process to occur. However, there is already a clear periodicity in the system in the form of the electron heating, consistent with Low Spatial

Frequency LIPSS (LSFL).

To take this analysis further, Figure 24A shows the out-of-plane component of the magnetic field (By) after the laser has completely vacated the simulation grid. The fields in Figure 24 show an SEW consistent with the SPP model, with an average wavelength of

810 nm which, within the spatial resolution of the simulation, matches the theoretical expectation of 816 nm for an SPP in this regime [17]:

1/2 휀푚 + 휀푑 휆푠 = ( ) (6.4) 휀푚휀푑

109 where 휀푚 is the real component of the dielectric function of the metal, 휀푑 is the real component of the dielectric function for the dielectric material (here, vacuum), and 휆푠 is the wavelength of the SPP.

For our particular setup it is clear that the laser couples to an SPP through the surface scratch. To further explain the electron heating, we compare it to the time-averaged field intensity. For a laser interaction in this regime, it is well known that the electron temperature will scale ponderomotively with intensity as √퐼 [123]. Figure 24B compares the final electron temperature along the surface of the target with the square root of the time-averaged field intensity during the laser pulse along the surface of the target. It is clear there is a direct correlation, indicating that the interference between the SEW and the incoming laser is indeed responsible for the periodic heating in the electrons.

Although a periodic heating profile in the electrons is compelling evidence on its own, the benefit of our simulation approach is the ability to obtain the primary experimental observable: the damage morphology. Accordingly, we next allow the electrons and lattice to approach equilibrium through the second stage of the simulation process as described in Section (6.2), the TTM calculation. Figure 25B shows the results of the TTM calculation. Notice that, even though the temperature profile spread throughout the target to make the periodic heating less distinct, the periodic nature of the heating was still preserved.

110

Figure 25: A) Zoomed-in view of the electron temperature at the end of the laser- interaction stage of the simulation. B) Lattice temperature at the end of the TTM stage of the simulation process.

Once the TTM stage was finished, after approximately 80 ps, the atomic transport simulation was initialized with the temperature profile shown in Figure 25 as prescribed in Section (6.3). The target was allowed to evolve until the surface resolidified and ceased evolving, for the case shown here after 3 ns. The resulting damage profile is shown in Figure 26B, with Figure 26A showing the initial target structure for comparison. From this final copper density profile, the periodic damage pattern can be seen and measured explicitly, and thus directly compared to theoretical predictions and experimental measurements.

111

For the 6.0 J/cm2 laser pulse damage pattern shown in Figure 26, and with the measured SEW wavelength of 810 nm, the expected periods of the damage pattern from

SPP theory is 1098 nm and 642 nm in the laser propagation and anti-propagation directions, respectively. Since a very tight laser focus was used for rapid development of this method, only a few ripples formed along the surface of the target in each direction from the scratch, so for a more accurate measurement of the period of the damage, the electron temperature profile in Figure 24B was used along with the final density profile in

Figure 26B, combining to give a measured damage period of 1057 nm and 643 nm in the laser propagation and anti-propagation directions, respectively, once again providing a close match within a few cell sizes of the theoretical expectations for the SPP theory.

These results are also consistent with recent experimental measurements under similar

112 laser and target conditions [124,125]. However, since these experiments did not start with a pre-determined surface deformation, the measured LIPSS period was only for the anti- propagation direction SPP and measured to be an average of approximately 640 nm for these conditions, again providing a good match.

Using all three stages of the laser damage simulation technique described in this work, we have demonstrated the first fundamental model of the formation of LIPSS. The model was implemented without fitting parameters or employing assumptions on when damage occurs. The results provide evidence confirming the SPP generation mechanism for LSFL structure in metal targets irradiated with femtosecond laser pulses. Further work will be needed to address LSFL in nonmetallic targets, but this simulation method provides the basic tools necessary to pursue this as well. Treatment of HSFL may also be possible, though additional models, perhaps treating the incorporation and evolution of defects, may be needed.

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Chapter 7: Conclusion

Presented here is the first fundamental simulation method capable of modeling the full laser damage process, from laser interaction and electron heating, to the electron- lattice thermalization, to material ablation and crater formation, all the way through resolidification. The method is based on a three-stage simulation process, employing a standard PIC approach with regime-appropriate scattering rates to simulate femtosecond- timescale physics, followed by a TTM calculation using the first stage PIC calculation of the electron temperature as input to model the picosecond timescale physics, and lastly with a modified PIC method using the stage two TTM calculation for initialization to handle the atomic transport on nanosecond timescales employing the PIC-consistent pair- potential algorithm developed here. Being fundamental in nature, this technique has no fitting parameters and no tuned parameters, meaning no knobs were used adjust the results of the final answer. A given input results in a fixed output.

The PIC-consistent pair-potential algorithm used to evaluate the third stage of the simulation process uses the Lennard-Jones pair potential for the results here, but the general algorithm is fully consistent with the use of any pair potential, or for greater accuracy at the cost of computational complexity even a quantum many-body potential.

Making use of the local environment at each macroparticle in the system, nearest- neighbor real atom locations relative to a virtual real atom located at that macroparticle’s

114 position are extrapolated from the density and the density gradient with an assumption on lattice structure. Once nearest-neighbor particle locations are determined, the Lennard-

Jones pair potential can be calculated directly for each particle pair. The force on the macroparticle is then determined to be equivalent to the force on the virtual real atom located at the same position as the macroparticle.

In Chapter 6 the results of several test simulations are presented. Shown in this work were the results of a 60 fs FWHM, 1 μm wavelength laser pulse incident on a flat copper target for fluences of 0.5 J/cm2, 1.0 J/cm2, and 2.0 J/cm2. The resulting damage craters, using an empirical scaling law that is normally applied to experimental crater morphologies, extrapolate down to a damage threshold fluence of 0.15 J/cm2, consistent with recent experimental results under similar conditions.

The last test case shown recreates an experimental case where a periodic damage structure, known as LIPSS, is formed on a copper target that is smooth other than for a single pre-existing sub-wavelength surface scratch. Despite decades of experimental study, the LIPSS formation mechanism is still debated. With this new simulation technique, this work presented the first fundamental simulation of LIPSS formation, allowing for close scrutiny of the physical mechanisms behind its creation. For the particular case analyzed here, which created Low Spatial Frequency LIPSS (LSFL), it was determined that an evanescent surface electromagnetic wave known as a surface plasmon-polariton (SPP) was responsible for the formation of the LIPSS pattern through interference with the laser pulse, confirming the SPP-generation theory for LSFL in metals. Theoretically, an SPP for the conditions simulated here should have a wavelength

115 of 815 nm and in the simulation was measured to have a wavelength of 810 nm, falling well within the spatial resolution of the simulation. Using this value for the wavelength of the SPP, theoretical predictions for the damage pattern periodicity in the laser propagation direction and counter-propagation direction are 1098 nm and 642 nm, respectively. Measured values for damage pattern periodicity were 1057 nm and 643 nm, also reasonably close to spatial-resolution accuracy.

The PIC method has been in use for decades and has proven itself fully capable of modeling a wide variety of materials from metals, to dielectrics, to plasmas and, with self-consistent propagation of electromagnetic fields, provides a flexible treatment for how electromagnetic radiation is propagated onto the grid and how it interacts with the target. The pair-potential algorithm developed here is fully consistent with neutral atoms or ions, indicating that an extension of this method to dielectric materials should be possible upon implementing regime-appropriate ionization mechanisms and molecular dissociations. In its current form this model should be capable of handling any metal, insulator, or semiconductor that can reasonably be considered monatomic, where monatomic here refers to the viability of modeling the material with a single macroparticle representing the lattice, though that macroparticle could very well represent a complex molecule consisting of several elements. An example of this would be modeling fused silica with a single SiO2 macroparticle representing the atomic structure. It is reasonable to imagine extensions of this method to mixed materials or to layered targets, which could broaden its applicability even further.

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References

1 D. Strickland, and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Comun. 56, 219-221 (1985). 2 P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398-403 (1988). 3 M. D. Perry, and G. Mourou, “Terrawatt to petawatt subpicosecond lasers,” Science 264, 917-924 (1994). 4 B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Short, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53, 1749-1761 (1996). 5 X. Liu, D. Du, and G. Mourou, “Laser ablation and micromachining with ultrashort laser pulses,” IEEE J. Quantum Electron. 33, 1706-1716 (1997). 6 Y. M. D. Perry, B. C. Stuart, P. S. Banks, M. D. Feit, V. Yanovsky, and A. M. Rubenchik, “Ultrashort-pulse laser machining of dielectric materials,” J. Appl. Phys. 85, 6803-6810 (1999). 7 S.-S. Wellershoff, J. Hohlfeld, J. Gudde, and E. Matthias, “The role of electron- phonon coupling in femtosecond laser damage of metals,” Appl. Phys. A 69, S99- S107 (1999). 8 R. Haight, D. Hayden, P. Longo, T. Neary, and A. Wagner, “Implementation and Performance of a Femtosecond Laser Mask Repair System in Manufacturing,” Proc. SPIE 3546, 477-484 (1998). 9 K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a femtosecond laser,” Opt. Lett. 21, 1729-1731 (1996). 10 M. Halbwax, T. Sarnet, Ph. Delaporte, M. Sentis, H. Etienne, F. Torregrosa, V. Vervisch, I. Perichaud, and S. Martinuzzi, “Micro and nano-structuration of silicon by femtosecond laser: Application to silicon photovoltaic cells fabrication,” Thin Solid Films 516, 6791-6795 (2008).

117

11 B. Wu, M. Zhou, J. Li, X. Ye, G. Li, and L. Cai, “Superhydrophobic surfaces fabricated by microstructuring of stainless steel using a femtosecond laser,” Appl. Surf. Sci. 256, 61-66 (2009). 12 A. Y. Yorobyev, and C. Guo, “Colorizing metals with femtosecond laser pulses,” Appl. Phys. Lett. 92, 041914 (2008). 13 Y. Cheng, E. Pels, and R. Nuijts, “Femtosecond-laser-assisted Descemet’s stripping endothelial keratoplasty,” J. Cateract Refract. Surg. 33, 152-155 (2007). 14 T. Juhasz, F. H. Loesel, R. M. Kurtz, C. Horvath, J. F. Bille, and G. Mourou, “Corneal Refractive Surgery with Femtosecond Lasers,” IEEE J. Quantum Electron. 5, 902-910 (1999). 15 J. Kruger, W. Kautek, and H. Newesely, “Femtosecond-pulse ablation of dental hydroxyapatite and single-crystalline fluoroapatite,” Appl. Phys. A 69, S403-S407 (1999). 16 Y. White, X. Li, Z. Sikorski, L. M. Davis, and W. Hofmeister, “Single-pulse ultrafast-laser machining of high aspect nano-holes at the surface of SiO2,” Opt. Exp. 16, 14411-14420 (2008). 17 M. Huang, F. Zhao, Y. Cheng, N. Xu, and Z. Xu, “Origin of Laser-Induced Near- Subwavelength Ripples: Interference between Surface Plasmons and Incident Laser,” ACS Nano 3, 4062-4070 (2009). 18 P. Avizonis, and T. Farrington, “Internal Self-Damage of Ruby and Nd-Glass Lasers,” Appl. Phys. Lett. 7, 205 (1965). 19 E. S. Bliss, “Pulse duration dependence of laser damage mechanisms,” Opto- Electronics 3, 99-108 (1971). 20 B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Optical ablation by high-power short-pulse lasers,” J. Opt. Soc. Am. B 13, 459-468 (1996). 21 W. H. Lowdermilk, and D. Milam, “Laser-induced surface and coating damage,” IEEE J. Quantum Electron. 17, 1888-1903 (1981). 22 T. W. Walker, A. H. Guenther, and P. E. Nielsen, “Pulsed laser-induced damage to thin-film optical coatings – Part I: Experimental,” IEEE J. Quantum Electron. 17, 2041-2052 (1981). 23 M. J. Soileau, W. E. Williams, E. W. Van Stryland, T. F. Boggess, and A. L. Smirl, “Temporal dependence of laser-induced breakdown in NaCl and SiO2,” Laser-Induced Damage of Optical Materials: 1982, H. E. Bennet, A. H.

118

Guenther, D. Milam, and B. E. Newman, eds., Natl. Bur. Stand. Spec. Publ. 669, 387-405 (1984). 24 S. R. Foltyn, and L. J. Jolin, “Long-range pulselength scaling of 351 nm laser damage tresholds,” Laser-Induced Damage of Optical Materials: 1986, H. E. Bennet, A. H. Guenther, D. Milam, and B. E. Newman, eds., Natl. Bur. Stand. Spec. Publ. 752, 336-343 (1988). 25 D. Du, X. Liu, G. Korn, J. Squier, and G. Mourou, “Laser-induced breakdown by impact ionization in SiO2 with pulse widths from 7 ns to 150 fs,” Appl. Phys. Lett. 64, 3071-3073 (1994). 26 J. R. Bettis, R. A. House II, and A. H. Guenther, “Spot size and pulse duration dependence of laser-induced damage,” in Laser-Induced Damage in Optical Materials: 1976, A. J. Glass and A. H. Guenther, eds., Natl. Bur. Stand. Spec. Publ. 462, 338-345 (1976). 27 M. Birnbaum, “Semiconductor surface damage produced by ruby lasers,” J. Appl. Phys. 36, 3688-3689 (1965). 28 A. K. Jain, V. N. Kulkarni, D. K. Sood, and J. S. Uppal, “Periodic surface ripples in laser-treated aluminum and their use to determine absorbed power,” J. Appl. Phys. 52, 4882-4884 (1981). 29 A. Borowiec, and H. K. Haugen, “Subwavelength ripple formation on the surfaces of compound semiconductors irradiated with femtosecond laser pulses,” Appl. Phys. Lett. 82, 4462-4464 (2003). 30 F. Costache, M. Henyk, and J. Reif, “Surface patterning on insulators upon femtosecond laser ablation,” Appl. Surf. Sci. 208, 486-491 (2003). 31 S. Graf, and F. A. Muller, “Polarisation-dependent generation of fs-laser induced periodic surface structures,” Appl. Surf. Sci. 331, 150-155 (2015). 32 D. C. Emmory, R. P. Howson, and L. J. Willis, “Laser mirror damage in germanium at 10.6 µm,” Appl. Phys. Lett. 23, 598-600 (1973). 33 J. Bonse, J. Kruger, S. Hohm, and A. Rosenfeld, “Femtosecond laser-induced periodic surface structures,” J. Laser Appl. 24, 042006 (2012). 34 J. E. Sipe, J. F. Young, J. S. Preston, and H. M. van Driel, “Laser-induced periodic surface structure. I. Theory,” Phys. Rev. B 27, 1141-1154 (1983). 35 J. Reif, O. Varlamova, and S. Uhlig, “On the physics of self-organized nanostructure formation upon femtosecond laser ablation,” Appl. Phys. A 117, 179-184 (2014).

119

36 K. Okamuro, M. Hashida, Y. Miyasaka, Y. Ikuta, S. Tokita, and S. Sakabe, “Laser fluence dependence of periodic grating structures formed on metal surfaces under femtosecond laser pulse irradiation,” Phys. Rev. B 82, 165417 (2010). 37 M. Huang, F. Zhao, Y. Cheng, N. Xu, and Z. Xu, “Origin of Laser-Induced Near- Subwavelength Ripples: Interference between Surface Plasmons and Incident Laser,” ACS Nano 3, 4062-4070 (2009). 38 R. A. Ganeev, “Optical Modification of Semiconductor Surfaces Through the Nanoripples Formation Using Ultrashort Laser Pulses: Experimental Aspects,” Opt. Spectrosc. 117, 320-340 (2014). 39 T. J. Y. Derrien, J. Kruger, T. E. Itina, S. Hohm, A. Rosenfeld, and J. Bonse, “Rippled area formed by surface plasmon polaritons upon femtosecond laser double-pulse irradiation of silicon: the role of carrier generation and relaxation processes,” Appl. Phys. A 117, 77-81 (2014). 40 F. Garralie, J. P. Colombier, F. Pigeon, S. Tonchev, N. Faure, M. Bounhalli, S. Reynaud, O. Parriaux, “Evidence of surface plasmon resonance in ultrafast laser- induced ripples,” Opt. Express 19, 9035-9043 (2011). 41 Q. Wu, Y. Ma, R. Fang, Y. Liao, Q. Yu, X. Chen, and K. Wang, “Femtosecond laser-induced periodic surface structure on diamond film,” Appl. Phys. Lett. 82, 1703-1705 (2003). 42 T. Q. Jia, H. X. Chen, M. Huang, F. L. Zhao, J. R. Qiu, R. X. Li, Z. Z. Xu, X. K. He, J. Zhang, and H. Kuroda, “Formation of nanogratings on the surface of a ZnSe crystal irradiated by femtosecond laser pulses” Phys. Rev. B 72, 125429 (2005). 43 J. Bonse, M. Munz, and H. Sturm, “Structure formation on the surface on indium phosphide irradiated by femtosecond laser pulses,” J. Appl. Phys. 97, 013538 (2005). 44 D. Dufft, A. Rosenfeld, S. K. Das, R. Grunwald, and J. Bonse, “Femtosecond laser-induced periodic surface structures revisited: A comparative study on ZnO,” J. Appl. Phys. 105, 034908 (2009). 45 G. A. Martsinovskii, G. D. Shandybina, D. S. Smirnov, S. V. Zabotnov, L. A. Golovan, V. Y. Timoshenko, and P. K. Kashkarov, “Ultrashort excitations of surface polaritons and waveguide modes in semiconductors,” Opt. Spectrosc. 105, 67-72 (2008). 46 G. Miyaji, and K. Miyazaki, “Origin of periodicity in nanostructuring on thin film surfaces ablated with femtosecond laser pulses,” Opt. Express 16, 16265 (2008).

120

47 E. L. Gurevich, “Self-organized nanopatterns in thin layers of superheated liquid metals,” Phys. Rev. E 83, 031604 (2011). 48 C. Momma, S. Nolte, B. N. Chichkov, F. V. Alvensleben, A. Tunnermann, “Precise laser ablation with ultrashort pulses,” Appl. Surf. Sci. 109, 15-19 (1997). 49 P. Gibbon, “Short Pulse Laser Interactions with Matter: An Introduction,” 1st edition (2005). 50 F. Brunel, “Not-so-resonant, resonant absorption,” Phys. Rev. Lett. 59, 52 (1987). 51 L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” JETP 20, 1307-1314 (1965). 52 J. K. Chen, D. Y. Tzou, and J. E. Beraun, “A semiclassical two-temperature model for ultrafast laser heating,” Int. J. Heat Mass Transfer 49, 307-316 (2006). 53 A. R. Bell, J. R. Davies, S. Guerin, and H. Ruhl, “Fast-electron transport in high- intensity short-pulse laser-solid experiments,” Plasma Physics and Controlled Fusion 39, 653-659 (1997). 54 E. Everhart, G. Stone, and R. J. Carbone, “Classical Calculation of Differential Cross Section for Scattering from a Coulomb Potential with Exponential Screening,” Physical Review 99, 1287-1290 (1955). 55 W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824-830 (2003). 56 J. R. Sambles, G. W. Bradbery, and F. Yang, “Optical excitation of surface plasmons: an introduction,” Contemp. Phys. 32, 173-183 (1991). 57 A. P. Hibbins, “Grating coupling of surface plasmon polaritons at visible and microwave frequencies,” Ph.D. Thesis, University of Exeter (1999). 58 J. D. Jackson, “Classical Electrodynamics”, 3rd edition (1998). 59 U. Fano, “The Theory of Anomalous Diffraction Gratings and of Quasi-Stationary Waves on Metallic Surfaces (Sommerfeld’s Waves),” J. Opt. Soc. Am. 31, 213- 222 (1941). 60 A. Otto, “Excitation of Nonradiative Surface Plasma Waves in Silver by the Method of Frustrated Total Reflection,” Z. Phys. 216, 398-410 (1968). 61 E. Kretschmann, and H. Raether, “Radiative Decay of Non Radiative Surface Plasmons Excited by Light,” Z. Natur. 23, 2135-2136 (1968). 62 R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Phil. Mag. 4, 396-402 (1902). 121

63 R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, “Surface-plasmon resonance effect in grating diffraction,” Phys. Rev. Lett. 21, 1530-1533 (1968). 64 A. A. Manenkov, “Fundamental mechanisms of laser-induced damage in optical materials: today’s state of understanding and problems,” Optical Engineering 53, 010901 (2014). 65 S. I. Anisimov, B. L. Kapeliovich, and T. L. Perelman, “Electron emission from metal surfaces exposed to ultrashort laser pulses,” JETP 39, 375-377 (1974). 66 X. Jing, Y. Tian, J. Zhang, S. Chen, Y. Jin, J. Shao, and Z. Fan, “Modeling validity of femtosecond laser breakdown in wide bandgap dielectrics,” App. Surf. Sci. 258 (10), 4741-4749 (2012). 67 A. S. Epifanov, A. A. Manenkov, and A. M. Prokhorov, “Theory of avalanche ionization induced in transparent dielectrics produced by an electromagnetic field,” JETP 70, 728-737 (1976). 68 D. S. Ivanov, and L. V. Zhigilei, “Combined atomistic-continuum modeling of short-pulse laser melting and disintegration of metal films,” Phys. Rev. B 68, 064114 (2003). 69 J. K. Chen, D. Y. Tzou, and J. E. Beraun, “A semiclassical two-temperature model for ultrafast laser heating,” Int. J. Heat Mass Transf. 49, 307-316 (2006). 70 J. Hohlfeld, S. Wellershoff, J. Gudde, U. Conrad, V. Jahnke, and E. Matthias, “Electron and lattice dynamics following optical excitation of metals,” Chem. Phys. 251, 237-258 (2000). 71 Y. K. Danileiko, A. A. Manenkov, V. S. Nechitailo, A. M. Prokhorov, and V. Y. Khaimov-Mal’kov, “The role of absorbing impurities in laser-induced damage of transparent dielectrics,” JETP 36, 541-543 (1972). 72 M. F. Koldunov, A. A. Manenkov, and I. L. Pokotilo, “Pulse-width and pulse- shape dependencies of laser-induced damage threshold to transparent optical materials,” Proc. SPIE 2714, 27th Annual Boulder Damage Symposium: Laser- Induced Damage in Optical Materials: 1995, 718 (1996). 73 M. J. Soileau, W. E. Williams, E. W. Van Stryland, T. F. Boggess, and A. L. Smirl, “Picosecond damage studies at 0.5 and 1 µm,” Opt. Eng. 22, 424-430 (1983). 74 M. Li, S. Menon, J. P. Nibarger, and G. N. Gibson, “Ultrafast electron dynamics in femtosecond optical breakdown of dielectrics,” Phys. Rev. Lett. 82, 2394-2397 (1998).

122

75 P. N. Saeta, and B. I. Greene, “Primary relaxtion processes at the band edge of SiO2,” Phys. Rev. Lett. 70, 3588-3591 (1993). 76 P. Audebert, Ph. Daguzan, A. Dos Santos, J. C. Gauthier, J. P. Geindre, S. Guizard, G. Hamoniaux, K. Krastev, P. Marton, G. Petite, and A. Antonetti, “Space-time observation of an electron gas in SiO2,” Phys. Rev. Lett. 73, 1990- 1993 (1994). 77 L. V. Zhigilei, Z. Lin, D. S. Ivanov, E. Leveugle, W. H. Duff, D. Thomas, C. Sevilla, and S. J. Guy, “Atomic/Molecular-Level Simulations of Laser-Materials Interactions,” Chapter 3 of Laser-Surface Interactions for New Materials Production: Tailoring Structure and Properties, Springer (2010). 78 S. Arora, and B. Barak, “Computational Complexity: A Modern Approach,” (2009). 79 L. V. Zhigilei, Z. Lin, and D. S. Ivanov, “Atomistic Modeling of Short Pulse Laser Ablation of Metals: Connections between Melting, Spallation, and Phase Explosion,” J. Phys. Chem. C 113, 11892-11906 (2009). 80 D. S. Ivanov, and L. V. Zhigilei, “Combined atomistic-continuum modeling of short-pulse laser melting and disintegration of metal films,” Phys. Rev. B 68, 064114 (2003). 81 P. Lorazo, L. Lewis, and M. Meunier, “Thermodynamic pathways to melting, ablation, and solidification in absorbing solids under pulsed laser irradiation,” Phys. Rev. B 73, 134108 (2006). 82 T. H. K. Barron, and C. Domb, “On the cubic and hexagonal close-packed lattices,” Proc. R. Soc. Lond. A 227, 447-465 (1955). 83 G. Ziegenhain, A. Hartmaier, and H. M. Urbassek, “Pair vs many-body potentials: Influence on elastic and plastic behavior in nanoindentation of fcc metals,” J. Mech. Phys. Solids 57, 1514-1526 (2009). 84 V. V. Zhakhovskii, N. A. Inogamov, Y. V. Petrov, S. I. Ashitkov, and K. Nishihara, “Molecular dynamics simulation of femtosecond ablation and spallation with different interatomic potentials,” Appl. Surf. Sci. 255, 9592-9596 (2009). 85 M. S. Daw, M. I. Baskes, “Embedded-atom method: derivation and application to impurities, surfaces, and other defects in metals,” Phys. Rev. B 29, 6449 (1984). 86 Y. Mishin, M. J. Mehl, D. A. Papaconstantopoulos, A. F. Voter, J. D. Kress, “Structural stability and lattice defects in copper: ab initio, tight-binding, and embedded-atom calculations,” Phys. Rev. B 63, 224106 (2001).

123

87 X. Liu, W. Zhou, C. Chen, L. Zhao, and Y. Zhang, “Study of ultrashort laser ablation of metals by molecular dynamics simulation and experimental method,” J. of Mat. Proc. Tech. 203, 202-207 (2008). 88 F. H. Harlow, and M. W. Evans. "A machine calculation method for hydrodynamic problems." LAMS-1956 (1955). 89 J. M. Dawson, "Particle simulation of plasmas,” Rev. Mod. Phys. 55, 403 (1983). 90 K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. of Ant. And Prop. 14, 302- 307 (1966). 91 H. Cai, K. Mima, A. Sunahara, T. Johzaki, H. Nagatomo, S. Zhu, and X. T. He, “Prepulse effects on the generation of high energy electrons in fast ignition scheme,” Physics of Plasmas 17, 023106 (2010). 92 B. I. Cohen, A. B. Langdon, and A. Friedman, “Implicit Time Integration for Plasma Simulation,” J. of Comp. Phys. 46, 15-38 (1982). 93 H. Nyquist, “Certain Topics in Telegraph Transmission Theory,” Trans. of the Amer. Inst. of Elec. Eng. 47, 617-644 (1928). 94 R. Courant, K. Friedrich, and H. Lewy, “On the Partial Difference Equations of Mathematical Physics,” IBM J. of Res. and Dev. 11, 215-234 (1967). 95 A. B. Langdon, “Analysis of the Time Integration in Plasma Simulation,” J. Comp. Phys. 30, 202-221 (1979). 96 D. R. Welch, D. V. Rose, B. V. Oliver, and R. E. Clark, “Simulation techniques for heavy ion fusion chamber transport,” Nucl. Instrum. Methods Phys. Res. A 464, 134-139 (2001). 97 A. Friedman, “A Second-Order Implicit Particle Mover with Adjustable Damping,” J. Comp. Phys. 90, 292-312 (1990). 98 D. W. Hewett, and A. B. Langdon, “Electromagnetic Direct Implicit Plasma Simulation,” J. Comp. Phys. 72, 121-155 (1987). 99 D. R. Welch, D. V. Rose, R. E. Clark, T. C. Genoni, and T. P. Hughes, “Implementation of an non-iterative implicit electromagnetic field solver for dense plasma simulation,” Comp. Phys. Comm. 164, 183-188 (2004). 100 C. K. Birdsall, and A. B. Langdon, “Plasma Physics via Computer Simulations,” (1985).

124

101 M. E. Jones, D. S. Lemons, R. J. Mason, V. A. Thomas, and D. Winske, “A Grid- Based Coulomb Collision Model for PIC Codes,” J. Comp. Phys. 123, 169-181 (1996). 102 L. Spitzer, and R. Harm, “Transport Phenomena in a Completely Ionized Gas,” Phys. Rev. 89, 977 (1953). 103 Y. T. Lee, and R. M. More, “An electron conductivity model for dense plasmas,” Phys. Fluids 27, 1273-1286 (1984). 104 M. P. Desjarlais, “Practical Improvements to the Lee-More Conductivity Near the Metal-Insulator Transition,” Contrib. Plasma Phys. 41, 267-270 (2001). 105 S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15, 1 (1943). 106 T. H. K. Barron, and C. Domb, “On the cubic and hexagonal close-packed lattices,” Proc. R. Soc. Lond. A 227, 447-465 (1955). 107 G. Ziegenhain, A. Hartmaier, and H. M. Urbassek, “Pair vs many-body potentials: influence on elastic and plastic behavior in nanoindentation of fcc metals,” J. Mech. Phys. Solids 57, 1514-1526 (2009). 108 R. A. Mitchell, D. W. Schumacher, and E. A. Chowdhury, “Modeling femtosecond pulse laser damage using particle-in-cell simulations,” Opt. Eng. 53, 122507 (2014). 109 M. Hashida, A. F. Semerok, O. Gobert, G. Petite, Y. Izawa, and J. F-. Wagner, “Ablation threshold dependence on pulse duration for copper,” Appl. Surf. Sci. 197, 862-867 (2002). 110 S. Y. Wang, Y. Ren, C. W. Cheng, J. K. Chen, and D. Y. Tzou, “Micromachining of copper by femtosecond laser pulses,” Appl. Surf. Sci. 265, 302-308 (2013). 111 J. A. King, K. U. Akli, R. R. Freeman, J. Green, S. P. Hatchett, D. Hey, P. Jamangi, M. H. Key, J. Koch, K. L. Lancaster, T. Ma, A. J. MacKinnon, A. MacPhee, P. A. Norreys, P. K. Patel, T. Phillips, R. B. Stephens, W. Theobald, R. P. J. Town, L. Van Woerkom, B. Zhang, and F. N. Beg, “Studies on the transport of high intensity laser-generated hot electrons in cone coupled wire targets”, Phys. Plasmas 16, 020701 (2009). 112 G. Pretzler, Th. Schlegel, E. Fill, and D. Eder, “Hot-electron generation in copper and photopumping of cobalt,” Phys. Rev. E 62, 5618 (2000). 113 W. L. Kruer, “The Physics of Laser Plasma Interactions,” (2003). 114 J-. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Phys. 114, 185-200 (1994). 125

115 J. P. Colombier, P. Combis, E. Audouard, and R. Stoian, “Transient optical response of ultrafast nonequilibrium excited metals: Effects of electron-electron contribution to collisional absorption,” Phys. Rev. E 77, 036409 (2008). 116 A. M. Summers, A. S. Ramm, G. Paneru, M. F. Kling, B. N. Flanders, and C. A. Trallero-Herrero, “Optical damage threshold of Au nanowires in strong femtosecond laser fields,” Optics Express 22, 004235 (2013). 117 B. H. Christensen, K. Vestentoft, and P. Balling, “Short-pulse ablation rates and the two-temperature model,” Appl. Surf. Sci. 253, 6347-6352 (2007). 118 Z. Lin, L. V. Zhigilei, and V. Celli, “Electron-phonon coupling and electron heat capacity of metals under conditions of strong electron-phonon nonequilibrium,” Phys. Rev. B 77, 075133 (2008). 119 S. Nolte, C. Momma, H. Jacobs, A. Tunnermann, B. N. Chichkov, B. Wellegehausen, and H. Welling, “Ablation of metals by ultrashort laser pulses,” J. Opt. Soc. Am. B 14, 2716-2722 (1997). 120 K. R. P. Kafka, D. R. Austin, H. Li, A. Y. Yi, J. Cheng, and E. A. Chowdhury, “Time-resolved measurement of single pulse femtosecond laser-induced periodic surface structure formation induced by a pre-fabricated surface groove,” Opt. Exp. 23, 19432-19441 (2015). 121 W. Fu, and E. L. Tan, “Stability and dispersion analysis for ADI-FDTD method in lossy media,” IEEE Trans. Ant. Prop. 55, 1095-1102 (2007). 122 A. D. Rakic, A. B. Djurisic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37, 5271- 5283 (1998). 123 S. C. Wilks, W. L. Kruer, M. Tabak, and A. B. Langdon, “Absorption of ultra- intense laser pulses,” Phys. Rev. Lett. 69, 1383-1386 (1992). 124 K. Okamuro, M. Hashida, Y. Miyasaka, Y. Ikuta, S. Tokita, and S. Sakabe, “Laser fluence dependence of periodic grating structures formed on metal surfaces under femtosecond laser pulse irradiation,” Phys. Rev. B 82, 165417 (2010). 125 S. Sakabe, M. Hashida, S. Tokita, S. Namba, and K. Okamuro, “Mechanism for self-formation of periodic grating structures on a metal surface by a femtosecond laser pulse,” Phys. Rev. B 79, 033409 (2009).

126