Advanced Simulations and Optimization of Intense Laser Interactions

DISSERTATION

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

By Joseph Richard Harrison Smith, B.A., M.S. Graduate Program in Physics

The Ohio State University 2020

Dissertation Committee: Professor Chris Orban, Advisor

Professor Enam A. Chowdhury

Professor Douglass W. Schumacher

Professor Richard J. Furnstahl c Copyright by

Joseph Richard Harrison Smith

2020 Abstract

1This work uses computer simulations to investigate intense laser-plasma interactions. First, we use two-dimensional particle-in-cell (PIC) simulations and simple analytic models to investigate the laser-plasma interaction known as ponderomotive steepening. When normally incident laser light reflects at the critical surface of a plasma, the resulting standing electromagnetic wave modifies the electron density profile via the ponderomotive force, which creates peaks in the electron density separated by approximately half of the laser wavelength. What is less well studied is how this charge imbalance accelerates ions towards the electron density peaks, modifying the ion density profile of the plasma. Idealized PIC simulations with an extended underdense plasma shelf are used to isolate the dynamics of ion density peak growth for a 42 fs pulse from an 800 nm laser with an intensity of 1018 W cm−2. These simulations exhibit sustained longitudinal electric fields of 200 GV m−1, which produce counter-steaming populations of ions reaching a few keV in energy. We compare these simulations to theoretical models, and we explore how ion energy depends on factors such as the plasma density and the laser wavelength, pulse duration, and intensity. We also provide relations for the strength of longitudinal electric fields and an approximate timescale for the density peaks to develop. These conclusions may be useful for investigating the phenomenon of ponderomotive steepening as advances in laser technology allow shorter and more intense pulses to be produced at various wavelengths. We also discuss the parallels with other work studying the interference from two counter-propagating laser pulses. Next we investigate the development of ultra-intense laser-based sources of high energy ions, which is an important goal, with a variety of potential applications. One of the barriers to achieving this goal is the need to maximize the conversion efficiency from laser energy to ion energy. We apply a new approach to this problem, in which we use an evolutionary algorithm to optimize conversion efficiency by exploring variations of the target density profile with thousands of one-dimensional PIC simulations. We then compare this

1Some of this abstract is reprinted from Joseph R Smith, Chris Orban, Gregory K Ngirmang, John T Morrison, Kevin M George, Enam A Chowdhury, and WM Roquemore. Particle-in-cell simulations of density peak formation and ion heating from short pulse laser-driven ponderomotive steepening. Physics of Plasmas, 2019 [1] under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

ii “optimal” target identified by the one-dimensional PIC simulations to more conventional choices, such as with an exponential scale length pre-plasma, with fully three-dimensional PIC simulations. The optimal target outperforms the conventional targets in terms of maximum ion energy by 20% and shows a significant enhancement of conversion efficiency to high energy ions. This target geometry enhances laser coupling to the electrons, while still allowing the laser to strongly reflect from an effectively thin target. These results underscore the potential of this statistics-driven approach for optimizing laser-plasma simulations and experiments. Finally, we present computational fluid dynamic simulations that model the formation of thin liquid targets. These simulations allow us to explore new types of targets that may be beneficial for high repetition rate laser plasma interactions.

iii In memory of my grandmother Theresa ‘Terry’ Harrison and my grandfather Forrest E. ‘Smitty’ Smith

iv Acknowledgments

I would like to thank my advisor Chris Orban for his guidance throughout the marathon that is a PhD. Chris has always been supportive and provided me with the academic freedom to explore a wide range of research projects. His advice is always welcome and I am very grateful that I have had the chance to work with him. Throughout my graduate studies at Ohio State, I have been fortunate to have had an abundance of positive role models and mentors to help me grow as a scientist. I’d like to thank Scott Feister for introducing me to computational plasma physics and it’s always great to work with him on a project-even when the outcome is not what we hoped for. I’d like to thank Gregory Ngirmang for challenging the simulation paradigm in our group, and the field, and for inspiring me to explore paths less taken with my research. Enam Chowdhury is a formidable force in science, and I’d like to thank him for sharing his knowledge and intuition, and for being a strong advocate for those he works with. The past and present scientists working with the Extreme Light Group at AFRL represent a special supportive community that I am honored to have collaborated with. John Morrison is an excellent scientist and I would like to thank him for pushing me to find my own way in research (and of course to graduate in a timely manner). Kevin George exemplifies the qualities of an excellent collaborator, bringing a fresh perspective to every project and pushing us to make excellent work. Mel Roquemore’s unrivaled passion for science and warm attitude towards others encouraged me to be a better scientist. I would also like to thank Kyle Frische for keeping the lab running and warmly welcoming the invasion of theorists from OSU. Joseph Snyder is a talented scientist and provided me with welcome advice for succeeding in graduate school and afterwards. One cannot escape the positive influence of Doug Schumacher and the HEDP group at OSU. Doug encourages people to succeed and his high expectations inspire excellent work. The entire HEDP group is extraordinarily welcoming and helped to broaden my knowledge. Alexander Klepinger, Anthony Zingale, Derek Nasir, Preston Pozderac, and Nick Czapla are always up for a lively discussion on physics or a variety of other topics. It’s a pleasure to work with them and the others in the HEDP group. Of course, my work is only possible due to those like Becky Daskalova, Anthony, Derek, Nick, German, and others in the lab who

v keep those of us playing with computers in check. I would also like to thank Ginny Cochran for all of her effort organizing journal clubs while she was at OSU. The previous members of the group are still making great impressions at their current jobs, and are always welcoming to current students at conferences. I would like to thank my undergraduate research mentors John Ramsay, Jennifer Bowen, and Susan Lehman for introducing me to research and helping me become a competent researcher. I would also like to thank John Grove, Jim Hill, Karlene Maskaly, and the other scientists and students who made my summer in New Mexico enjoyable (along with chances to meet with Alexander Klepinger while he was in Albuquerque, and with other HEDP alumni in the state). I would like to thank Dick Furnstahl for his computational physics and Bayesian methods courses, which were excellent preparations for actual research. I’d also like to thank Andrew Heckler, Tom Gramila, and Ralf Bundschuh for helping me grow as an educator. I would like to thank Chris Porter, Bart Snapp, Jim Fowler, Jon Brown, and others for the surprising quality of work we were able to accomplish with the BuckeyeVR project, despite limited resources. The arduous endeavor of a PhD wouldn’t have been possible without the continual support of my friends and family. My friends have made the last five years a much more enjoyable experience. I’m not sure how I would have made it through the first years of graduate school without the revolving set of officemates Ethel, Jose, Andrés,Daniella, Franz, Emilio, Bruce, and Philip. I’d also like to thank Mike, Estefany, Humberto, Paulo, and Catherine, and the others for welcoming me into the extended Bridge family. I cherish experiences with my friends like semi-spontaneous trips to Canada, to see the total eclipse, and moving a friend across the country. I’d like to thank Jose, Ethel, Bruce, and others for joining me in the culinary exploration of Columbus. I’d like to thank Alexander, Matt, Andrés,and Emily for expanding my repertoire of board games and being wonderful friends! I’d also like to thank my officemates Zach and Derek for interesting conversations. Andrés, thank you for being a great housemate, especially in these times. Alexander, thank you for bringing people together and always having something interesting to talk about. Ethel and Bruce thanks for extending your friendship and being awesome. Jose, thanks for making work interesting and being a great friend. I could keep going, but I shouldn’t, so I’d like to thank all of the others at OSU and elsewhere who have had a strong impact on my life. Lastly I would like to thank my family for their support and encouragement. My mother and father have also been so supportive of me and my goals, I can’t thank you enough for everything they do! I’d also like to thank my brother Jacob, the political scientist, for blazing the path before me, and for my incredibly intelligent sister Heidi who will do great things.

vi Vita

May 2015 ...... B.A., Physics and Mathematics, The College of Wooster, Wooster, Ohio August 2015 - August 2016 ...... Graduate Fellow, The Ohio State Univer- sity (OSU), Columbus, Ohio August 2016 - May 2017 ...... Graduate Associate, OSU

May 2017 - August 2017 ...... DOD HPC Internship Program (HIP) with the Extreme Light Group at the Air Force Research Laboratory (AFRL), facil- itated through the University of Dayton Research Institute (UDRI) August 2017 - December 2017 ...... Graduate Researcher, Innovative Scien- tiﬁc Solutions, Inc. for the Extreme Light Group at AFRL December 2017 ...... M.S. Physics, OSU

January 2018 - May 2018 ...... Graduate Associate, OSU

June 2018 - August 2018 ...... Computational Physics Student Summer Workshop at Los Alamos National Labo- ratory August 2018 - May 2019 ...... Graduate Associate, OSU

June 2019 - August 2019 ...... DOD HIP; Extreme Light Group/UDRI

August 2019 - Present ...... Graduate Associate, OSU

Publications

J. R. Smith, C. Orban, G. K. Ngirmang, J. T. Morrison, K. M. George, E. A. Chowdhury, and W. M. Roquemore, “Particle-in-Cell Simulations of Density Peak Formation and Ion vii Heating from Short Pulse Laser-Driven Ponderomotive Steepening.” Physics of Plasmas 2019.

C. D. Porter, J. Brown, J. R. Smith, A. Simmons, M. Nieberding, A. E. Ayers, and C. Orban. “A Controlled Study of Virtual Reality in First-Year Magnetostatics.” In Proceedings of the Physics Education Research Conference 2019.

K. M. George, J. T. Morrison, S. Feister, G. Ngirmang, J. R. Smith, A. J. Klim, J. Snyder, D. Austin, W. Erbsen, K. D. Frische, J. Nees, C. Orban, E. A. Chowdhury, and W. M. Roquemore, “High Repetition Rate (≥ kHz) Targets and Optics from Liquid Microjets for High Intensity Laser-Plasma Interactions.” High Power Laser Science and Engineering, 2019.

C. Orban, R. M. Teeling-Smith, J. R. H. Smith, and C. D. Porter, “A Hybrid Approach for Using Programming Exercises in Introductory Physics.” American Journal of Physics, 2018.

S. Feister, D. Austin, J. Morrison, K. Frische, C. Orban, G. Ngirmang, A. Handler, J. R. H. Smith, M. Schillaci, J. A. LaVerne, E. A. Chowdhury, R. R. Freeman, and W. M. Roquemore, “Relativistic Electron Acceleration by mJ-class kHz Lasers Normally Incident on Liquid Targets.” Optics Express, 2017.

J. R. Smith, A. Byrum, T. M. McCormick, Nicholas T. Young, C. Orban, and C. D. Porter, “A Controlled Study of Stereoscopic Virtual Reality in Freshman Electrostatics.” In Proceedings of the Physics Education Research Conference 2017, Cincinnati, OH, 2017.

M. Bush, J. Smith, S. Smith-Polderman, J. Bowen, and J. Ramsay. “Braid Computations for the Crossing Number of Klein Links.” Involve – A Journal of Mathematics, 2015.

M. A. Bush, K. R. French, and J. R. H. Smith, “Total Linking Numbers of Torus Links and Klein Links.” Rose-Hulman Undergraduate Mathematics Journal, 2014.

D. Shepherd, J. Smith, S. Smith-Polderman, J. Bowen, and J. Ramsay, ‘The Classiﬁcation of a Subset of Klein Links.” In Proceedings of the Midstates Conference for Undergraduate Research in Computer Science and Mathematics 2012.

Fields of Study

Major Field: Physics

viii Table of Contents

Page Abstract...... ii Dedication...... iv Acknowledgments...... v Vita...... vii List of Figures ...... xii List of Tables ...... xvii

Chapters

1 Introduction...... 1 1.1 Laser-Based Particle Sources ...... 1 1.1.1 Ion Beam Applications...... 2 1.1.2 Neutron Radiography ...... 3 1.1.3 From Proof of Concept to Application...... 5 1.2 This Work...... 6

2 Basics of Laser-Plasma Interactions...... 8 2.1 Particle Motion in an Electromagnetic Field...... 8 2.1.1 Ponderomotive Force...... 9 2.2 Plasma Fundamentals ...... 11 2.2.1 Debye Length...... 11 2.2.2 Oscillations in a Plasma...... 12 2.3 Ion Acceleration ...... 14

3 Computational Methods...... 17 3.1 The Particle-In-Cell Method...... 17 3.1.1 PIC Cycle...... 17 3.1.2 Particle Weighting...... 18 3.1.3 Field and Particle Evolution...... 19 3.1.4 Dimensionality Considerations ...... 20 3.2 Evolutionary Algorithms...... 22 3.3 Computational Fluid Dynamics Simulations...... 24 3.3.1 OpenFOAM and the interFoam Solver...... 24

ix 4 Particle-in-Cell Simulations of Density Peak Formation and Ion Heating from Short Pulse Laser-Driven Ponderomotive Steepening . . 26 4.1 Introduction...... 26 4.2 Ponderomotive Steepening and Ion Acceleration...... 29 4.2.1 A Simple Model for Ponderomotive and Electrostatic Forces in Ponderomotive Steepening...... 29 4.2.2 Timescale of the ion acceleration...... 33 4.3 Particle-In-Cell Simulations...... 36 4.3.1 Model Regime for Parameter Choices ...... 37 4.4 Results...... 38 4.4.1 Peak Formation and Density Proﬁle Modiﬁcation...... 38 4.4.2 Ion Motion ...... 39 4.4.3 Peak Electric Fields ...... 41 4.4.4 Ion Energies...... 42 4.4.5 Transverse Structure and Motion...... 43 4.5 Discussion...... 44 4.5.1 Observational Considerations...... 45 4.5.2 Extensions and Applications ...... 45 4.6 Conclusions...... 47

5 Optimizing Laser-Plasma Interactions for Ion Acceleration using Particle-in-Cell Simulations and Evolutionary Algorithms ...... 48 5.1 Introduction...... 48 5.2 1D PIC Optimization Driven by Evolutionary Algorithms...... 49 5.2.1 Simulation Parameters...... 50 5.2.2 1D Results ...... 51 5.3 Three-Dimensional Simulations...... 52 5.3.1 Simulation Parameters...... 53 5.3.2 Results ...... 53 5.4 Discussion...... 56 5.4.1 The Optimal Target...... 56 5.4.2 Optimization of Laser-Plasma Interactions...... 57 5.5 Conclusions...... 58

6 Computational Fluid Dynamic Simulations of Liquid Target Formation 59 6.1 Dimensionless Numbers in Fluid Mechanics...... 60 6.2 Liquid Sheet Formation ...... 60 6.2.1 Simulation ...... 61 6.2.2 Sheet Simulation Results ...... 63 6.3 Droplet Collisions...... 65 6.3.1 Simulation Parameters...... 65 6.3.2 Drop Collision Results...... 67 6.4 Discussion and Conclusions ...... 68

7 Conclusions ...... 69

Bibliography...... 71

x Appendices

A Additional Considerations for Short Pulse Laser-Driven Ponderomo- tive Steepening ...... 89 A.1 Timescale from Ion Oscillation Frequency...... 89 A.2 Maximum Ion Energies...... 90 A.3 Ion Mean Free Path ...... 90 A.4 Potential Neutron Yield...... 91

B Additional Information for Optimizing Laser Plasma Interactions with PIC Simulations and Evolutionary Algorithms...... 92 B.1 1D Evolution...... 92

xi List of Figures

Figure Page

1.1 Historical and anticipated peak laser intensity and electric ﬁeld. The development of laser techniques such as CPA are noted on the plot. Adapted from Ref. [2]...... 2 1.2 A 150 cm3 model plane engine (left) and corresponding proton (center) and x-ray (right) radiographs. Adapted from Ref. [3]...... 3 1.3 Dose deposition in tissue for a photon beam compared to a monoenergetic proton beam (e.g. proton beam #1) and a ‘Spread Out Bragg Peak’ (SOBP) created by proton beams of multiple energies (12 in this case). This image is reprinted from Ref. [4] under a Creative Commons Attribution-Share Alike 3.0 Unported license (https://creativecommons.org/licenses/by-sa/3.0/deed.en)...... 4 1.4 Neutron radiograph of an analog camera (left) and x-ray radiograph of the same camera (right). Adapted from Ref. [5]...... 4 1.5 Neutron radiograph of a jet turbine blade. The arrow points to defects from the manufacturing process. Adapted from MacGillivray [6]...... 5

2.1 A uniform plasma (left), where the electrons are displaced by a distance x (right). This results in a positively charged region of the plasma, followed by a neutral region in the middle, and a negatively charged region to the right, resulting in an electric ﬁeld that pulls electrons back to the equilibrium...... 13 2.2 Sketch of the TNSA ion acceleration mechanism from Ref. [7]. Electrons are accelerated through the target by the laser and an electrostatic sheath is developed which accelerated ions in a direction normal to the target. . . 15

3.1 Sketch of a 2D PIC simulation grid with positive and negative particles and a simple node structure (in practice the node structure is typically more complex as discussed in Sec. 3.1.3)...... 18 3.2 The general steps completed by a PIC code...... 19 3.3 The intensity as a Gaussian laser beam comes to focus for 1D, 2D, and 3D PIC simulations for an 800 nm laser with a beam waist of 1.5 µm. . . . 21

xii 3.4 The general procedure for an evolutionary algorithm. A population is created and evolved with crossover and mutation, based on a ﬁtness function until a stopping condition is met. For our work, the ‘Evaluate Fitness’ step depends on the output of 1D PIC simulations...... 23

4.1 Sketch of the classical ponderomotive steepening process (a-c), and the simplified process considered in this work (d-f). As illustrated in (a), a normally incident laser pulse reflects at the critical density of a plasma and forms a standing electromagnetic wave. This causes the electrons to form peaks near the extrema (separated by ≈ λ/2) of the standing wave via the ponderomotive force (b). The modification of the electron density creates a charge imbalance (sustained by the standing wave), which accelerates ions towards the electron peaks. In time, this modifies the density of the plasma as illustrated in (c). Note that (b) and (c) only include the standing wave region from (a). This paper focuses on the simpler process of a laser traveling through an underdense pre-plasma shelf and reflecting off of an overdense target that behaves like a mirror (d-f)...... 27 4.2 The division between the short pulse regime and long pulse regime as a function of laser intensity and for a variety of wavelengths (for n0 = ncrit/2) for our simple model. The timescale on the vertical axis is the timescale of ion motion from electrostatic forces in ponderomotive steepening. The dashed line represents the timescale for a shelf density of ncrit/20, pertaining to the simulations in this paper. The individual points on the graph represent the full pulse duration for our simulations (scaled by ion mass). The shaded region represents the short pulse regime for a 0.4 µm laser. The exaggerated line widths are used to emphasize that this is a simple model to provide rough estimates for the two regimes we identify in this work. Dotted lines are drawn for a0 & 1...... 35 4.3 Initial conditions for the 2D(3v) PIC simulations. The laser propagates in the +x direction with a rectangular target composed of an extended constant underdense pre-plasma shelf region preceding an overdense region. 2 17 −2 Contours are drawn for at I0/e ≈ 1.35 × 10 W cm for the simulated laser near the shelf, reflecting off of the target at 85 fs, and for a laser at the geometric focus (in the absence of a target)...... 37 4.4 Electron density (z/λ < 0) and ion density (z/λ > 0) near the reflection point for the deuterium simulation. The laser finishes reflecting around 130 fs from the beginning of the simulation (a), although the peaks continue to grow as shown at 260 fs (b) and then begin to dissipate as illustrated at 390 fs (c). The width of the box (in z) represents the region considered in Fig. 4.5 and the entire box represents the region considered for ion trajectories in Fig. 4.6. This density peak growth process is highlighted in a supplemental video (see https://doi.org/10.1063/1.5108811.1). . 38

xiii 4.5 Change in density of the first peak in the ion density. The lines represent the maximum density of the first ion peak averaged over the width of the laser pulse. This is calculated by averaging the maximum density for each value of z in the region -2 µm < z < 2 µm, where error bars represent the standard deviation. We note that the exact density at the peak depends on the cell size (and sharpness of the peak), thus this graph comments more on densities in the region near the peak rather than the peak itself. . 39 4.6 The average trajectories in x for a sample of particles starting in the boxed regions in Fig. 4.4, representing the first three peaks in the ion density. The white vertical lines represent approximately when the laser begins reflecting, reaches its half maxima, and stops reflecting. Shaded vertical lines correspond to the times represented in Fig. 4.4. The ions continue to travel after the standing wave has dissipated, and the observed peaks are created by the crossing ions...... 40 4.7 Longitudinal ion energies for particles starting in the boxed regions in Fig. 4.4. The average kinetic energies for each simulation are plotted in time and the distribution of ion energies in the background corresponds to the deuterium (2H+) simulation (logarithmic grayscale). The maximum energy from this distribution agrees well with the simple model for tSW = 76 fs represented by the dashed line. The energies increase while the charge separation caused by standing EM wave is present...... 41 4.8 The observed longitudinal component of the electric field at 70 fs after the beginning of the deuterium simulation near the center of the laser pulse (PIC) averaged over several cells, as compared to the simple sinusoidal density variation model (Sine), maximum depletion (Max), and the expected ponderomotive force (Eq. 4.4) divided by e for reference (Ep). The electric fields found in the simulation lie between the sinusoidal model and maximum depletion model as expected for this intensity and density. . . . 42 4.9 Ion velocity distribution at 160 fs for the three simulations for a sample of particles with initial positions from -23 µm ≤ x ≤ 15 µm. The maximum transverse velocities from Eq. 4.17 for tsw = 76 fs are given by the dashed lines...... 44

5.1 Template for each of the 1D PIC simulations run by the evolutionary algorithm. The intense 1.2×1019 W cm−2 laser enters the simulation from the left side of the simulation box (x = 0 µm) and interacts with a 5 µm thick ionized hydrogen target composed of ten 0.5 µm thick density bins chosen by the evolutionary algorithm. Ions and electrons are measured as they leave the right side of the simulation box (x = 40 µm). The total energy of these ions is maximized with the evolutionary algorithm. . . . . 51

xiv 5.2 1D PIC simulation optimization results. In (a), all members of the population are plotted, where darker shades represent higher fitness. The best performing density profile is drawn in red. After 50 generations, most members of the population have a similar pattern with a roughly critical density foot at the front of the target (the first two density bins that exceed ncrit), underdense center, and overdense density spike in one of the last two density bins as shown. In (b), the conversion efficiency of measured ions initially increases quickly and then begins to level off with later generations. In (c), the distribution of measured forward going ions for the best performing profile is plotted. This ‘optimal’ density profile is tested with 3D simulations in Sec. 5.3...... 52 5.3 Snapshots of the ion (z < 0) and electron (z > 0) densities for the three 3D simulations (xz plane). The optimal ‘evolutionary algorithm (EA)’ target from the 1D simulations is on the left, the exponential target is in the center, and the thin sheet is on the right. Ions travel farthest for the new EA target, which shows enhanced coupling between the laser and electrons like the exponential target. A contour is drawn at an intensity 2 of Imax/e ; variations come from differences in the focal spot location. . . 54 5.4 The maximum ion energy versus time for all three simulations (a), where the EA target reached the highest energy followed by the Exp and Sheet targets. Spectra of ions with forward going momenta in a 20◦ half angle cone sketched in Fig. 5.5, for the three different 3D simulations at 500 fs (b), where the charge represents the total charge of electrons in a 0.1 MeV energy bin. The total ion conversion efficiency in this cone is included in (c) with the hatched bars showing the difference between the targets if only ions with energies greater than 2 MeV are considered. The overall conversion efficiency is similar, but there is a noticeable enhancement to the population above ∼ 2 MeV for the EA target compared to the other two targets...... 55 5.5 Polar histograms showing the distribution of ion energies for the three 3D simulations. Energy bins have a radial size 0.5 MeV and angular size of 5◦ taken in the xz plane and a 20◦ half-angle cone is sketched for reference. For the evolutionary algorithm target (a), we see a strong forward (laser propagation direction) going component of the ion distribution and enhanced conversion to & 2 MeV ions compared to the other two targets. . 56 6.1 Simulation results demonstrating possible types of liquid sheets for increasing jet velocity. The stable thin section in the middle of the target is advantageous for ion acceleration. From Ref. [8] under a Creative Commons (CC BY-NC-ND 4.0) license (https://creativecommons.org/licenses/by-nc-nd/4.0/)...... 61 6.2 On the left is a schematic of two colliding jets that form a closed rim shape including the formation of a secondary sheet (from Ref. [9]). The two jets collide at an angle of 2φ and the coordinates r and θ are used to describe the location on a thin sheet. Then on the right is an experimental image of a sheet from Ref. [10]...... 62

xv 6.3 Isovolume plots showing the formation of a thin sheet for the OpenFOAM simulation...... 64 6.4 Thickness map for the thin sheet simulation at 88 µs (left). For reference a thickness map from Koralek et al. [11] (no absolute color scale for thickness provided) is included in the center, and Morrison et al. [12] on the right. In all three cases we see decreasing thickness moving farther away from the top of the jet...... 65 6.5 Initial simulation grid for the droplet collision simulations. A 160 µm thick cylindrical shape is used with a quasi-cylindrical geometry as annotated. . 66 6.6 Evolution of the droplet collision simulation compared to experimental images of similar conditions from Ref. [13] ...... 67 6.7 Isovolume plot from the droplet collision at 30 µs (left) and a cross section illustrating the thickness of the drop (right). The small vertical line on the right image represents a thickness of 5 µm...... 68

B.1 Evolution of the population (a-f). The entire population is plotted, with darker lines indicating higher conversion eﬃciencies. The highest performing member of each generation is plotted in red and the classical critical density is plotted for reference...... 93

xvi List of Tables

Table Page

4.1 Maximum ion energies reported in keV from the simulation shortly after the standing wave has dissipated. This is compared to the energies predicted with Eq. 4.17. Because the laser pulse has a temporal profile that is sine squared (rather than square), the time-dependent maximum amplitude makes comparison to the model more ambiguous. We compare the simulation result to the model with three different assumptions for the duration longitudinal electric field caused by the charge separation from the standing wave (tsw)...... 43

xvii Chapter 1 Introduction

When Theodore Maiman made the first laser (“light amplification by stimulated emission of radiation”) in 1960 [14], it was described by some as “a solution looking for a problem” [15, 16]. The laser has since become a ubiquitous ‘solution’ to numerous problems in our daily lives, and a powerful tool for pushing scientific boundaries. Over the last 60 years, the maximum intensity (power per unit area) of laser systems has consistently increased as shown in Fig. 1.1; especially since the development of chirped pulse amplification (CPA) by Strickland and Mourou [17] in 1985 as acknowledged with the 2018 Nobel Prize in Physics2. This technological breakthrough has the potential to deliver a variety of new technologies, but there is still significant fundamental work to do both theoretically and in proof-of-concept experiments to deliver these applications (described in a moment). This dissertation contributes to this wider effort in a few specific ways. Short duration (∼ tens of femtoseconds) high-intensity pulses have the potential to accelerate particles to high energies over very small distances and to create extreme High Energy Density (HED) conditions 11 −3 (& 10 J m ) like those at the center of stars [18]. This dissertation uses computer simulations to model new targets and laser-plasma interactions related to the generation of high energy particles.

1.1 Laser-Based Particle Sources

The high intensity lasers enabled by CPA have oscillating electric fields that exceed Teravolts per meter as shown in Fig. 1.1. Laser-plasma interactions can produce large transverse accelerating fields exceeding 100 GV m−1, compared to traditional radio frequency cavities that can have accelerating fields of ∼ 100 MV m−1 [19]. This makes lasers a promising technology to develop more compact particle accelerators. The potential of laser-based sources has already been demonstrated. For example, electrons can be accelerated to high energies with laser wakefield acceleration as proposed

2One-half of the prize was awarded to Arthur Ashkin for his development of optical tweezers and the other half was shared between Donna Strickland and G´erardMourou for the development of CPA. 1 Figure 1.1: Historical and anticipated peak laser intensity and electric ﬁeld. The development of laser techniques such as CPA are noted on the plot. Adapted from Ref. [2].

by Tajima and Dawson [20]. Recently at the BELLA petawatt laser, a beam of electrons was accelerated to 7.8 GeV [21] over an acceleration length of only 20 cm. Lasers can also be used to accelerate ions, which is the focus of this work. For ion acceleration, lasers interact with thin (micron and sub-micron scale) solid-density targets to accelerate ions to tens of MeV in energy. For example, at the Vulcan laser, protons were accelerated to energies exceeding 94 MeV [22].

1.1.1 Ion Beam Applications

Ions interact differently with matter than electrons or photons, enabling a wide variety of applications for energetic ion beams ranging from cutting edge HED science to medical treatments. Currently, one application of laser-generated ion beams is to probe the electric and magnetic fields in a plasma such as in inertial confinement fusion (ICF), laboratory astrophysics, and other HED experiments (e.g. [23]). There are many other applications for ions beams generated from conventional accelerators; we will discuss a few of those applications here. For example, ion beams from conventional linear particle accelerators (LINAC), such as the 800 MeV LINAC at Los Alamos National Laboratory [24], are used to create radiographic images. In Fig. 1.2, we compare a proton radiograph taken with an 800 MeV proton source to an x-ray radiograph from a 100 keV source. Using a model plane engine as a radiographic test, the 800 MeV proton beam has a penetration depth into the metal that is effective for high-contrast imaging of the components of the engine [3]. Multiple radiographs can be used to capture the evolution of experiments or to provide 3D

2 Figure 1.2: A 150 cm3 model plane engine (left) and corresponding proton (center) and x-ray (right) radiographs. Adapted from Ref. [3].

images through tomography [3]. This facility also uses a 100 MeV proton beam to create radioactive isotopes for medical applications and fundamental science [25]. Another important use of traditional accelerators is hadron therapy for cancer treatment. As illustrated in Fig. 1.3, we can compare the energy deposition of a photon beam to a proton beam in tissue, for proton therapy applications. As a monoenergetic proton beam travels through matter, the protons lose energy ionizing atoms and depositing radiation. As the ions slow down, their cross section increases, and they deposit much of their energy in what is known as a ‘Bragg Peak’ shown in Fig. 1.3. On the other hand, photon beams continue to deposit energy well after the desired target, as illustrated in Fig. 1.3. This technique is useful for cancer treatment because it can reduce the radiation dose to healthy tissue. One challenge of this method is that there are currently fewer that one hundred of these facilities in the world [26], limiting the availability of treatments. Laser-driven proton sources could help compactify and reduce the cost of such facilities, although current laser- based techniques do not have the proper beam characteristics as discussed in Sec. 1.1.3.

1.1.2 Neutron Radiography

Another useful technique for imaging involves beams of thermal neutrons, which have no electric charge and are of a similar mass to a proton. The x-ray scattering cross section increases with the number of electrons, while the neutron cross section depends on the nuclear structure and does not follow a monotonic pattern, providing complementary information [27]. For example in Fig. 1.4, we compare a neutron radiography taken of an analog photography camera to an x-ray radiograph of the same camera. The x-rays are more strongly aﬀected by the higher Z materials like metal and the neutrons are able to reveal details about other materials such as the plastic parts [5]. Neutrons can be used for

3 Figure 1.3: Dose deposition in tissue for a photon beam compared to a monoenergetic proton beam (e.g. proton beam #1) and a ‘Spread Out Bragg Peak’ (SOBP) created by proton beams of multiple energies (12 in this case). This image is reprinted from Ref. [4] under a Creative Commons Attribution-Share Alike 3.0 Unported license (https://creativecommons.org/licenses/by-sa/3.0/deed.en).

Figure 1.4: Neutron radiograph of an analog camera (left) and x-ray radiograph of the same camera (right). Adapted from Ref. [5].

4 Figure 1.5: Neutron radiograph of a jet turbine blade. The arrow points to defects from the manufacturing process. Adapted from MacGillivray [6].

the Non Destructive Evaluation (NDE) of various objects. For example, in Fig. 1.5, we see leftover material from the casting process of a turbine blade. This unwanted material reduces the cooling of the blade, which can cause melting and lead to engine failure. To detect the material, it can be doped3 with an element such as Gadolinium, which has a large neutron cross section [6]. We mention neutrons here, as they can also be created with laser-plasma interactions. In the “pitcher-catcher” scheme, ions are generated with an initial laser-plasma interaction with a thin target, the ‘pitcher’, and then these ions are directed towards a secondary target, the ‘catcher’ to produce neutrons (e.g. Davis and Petrov [28]).

1.1.3 From Proof of Concept to Application

The ion beam properties created with lasers are not yet ideal for many of the applications discussed in the previous subsections. For example, the energies needed for proton therapy treatments range from about 70 to 250 MeV [29], while lasers have only produced proton energies up to about 100 MeV [22]. There are paths to reaching higher energies with more intense lasers [30], but there are many challenges that would need to be overcome before clinical use as elaborated on in Refs. [31, 32]; we will highlight some of them here. The ion beams generated by cyclotrons for medical applications have a very small energy divergence on the order of 0.1% ∆E/E, while laser-driven ion experiments reaching the highest energies have a large energy spread (∼ 100%) with a majority of the ions at lower energies [31]. There are successful eﬀorts to create quasi-monoenergetic (∆E/E . 10% [33, 34]) ion beams, which seems promising as a “spread out Bragg peak” (SOBP) in proton

3or added as part of the inspection process in this case 5 therapy is used to more uniformly irradiate the treatment area, like illustrated in Fig. 1.3. The SOBP is created by applying multiple proton beams of different energies created by sending the ion beam through a device such as a ridge filter or range modulator wheel [35]. In addition to needing higher ion energies, the challenge is obtaining the required flux of ions with laser-driven sources, which is about 1010 protons per second, for 1-3 minutes [32]. For comparison, an experiment at the Titan laser facility produced ∼ 10 MeV protons with an energy spread of ∼ 10% with ∼ 109 particles within a peak, but the laser can only fire twice per hour [36]. Another challenge will be reducing the shot-to-shot variability of laser experiments. Clearly there are a number of technical challenges that remain to be solved. Part of the solution will be developing lasers with both higher intensities and repetition rates, such as with fiber lasers [37]. With our current laser systems there is still important work testing new acceleration mechanisms that can enhance the conversion from laser energy to ion energy, and developing optics and targets that can sustain high-repetition-rate operation, as is the focus of this thesis. Despite these challenges, it is still worthwhile to develop alternate sources of ion beams that can complement traditional sources for a variety of applications as discussed in Sec. 1.1.1. For example, laser-accelerated ions have a very small source size, based on the spot size of the laser. Laser-based sources have the potential to provide a more compact, less expensive4 and more flexible source, where either electrons, ions, neutrons, or x-rays can be produced depending on the target (or combination of targets). The full impact of the pioneering work by Maiman and later development of CPA by Strickland and Mourou is still being realized, and fundamental work will help us better understand laser-plasma interactions and may even reveal new applications.

1.2 This Work

This work is an experimentally-motivated computational and theoretical investigation of intense laser interactions. First, we provide a fundamental background of some of the physics underpinning laser-plasma interactions in chapter2 and then we introduce the computational methods employed in this thesis in chapter3. In chapter4, we use simulations and simple analytic models to investigate ‘ponderomotive steepening’, where the interference pattern of a reflecting laser periodically modifies the density profile of a plasma. We investigate the potential ion energies generated from this interaction and identify relevant timescales of the interaction, which would be useful for developing an experiment to directly observe this phenomenon. The periodic density modulation of the plasma may also be useful for the field of plasma optics and high-repetition-rate

4We do note that as proton therapy becomes more prevalent, the cost and size have greatly improved [32].

6 experiments. In chapter5 we use an optimization technique known as evolutionary algorithms to automatically run thousands of one-dimensional simulations, varying the initial conditions to optimize the conversion from laser energy to acceleration ion energy. The new type of target identified in this optimization process was then compared to conventional targets with more accurate three-dimensional simulations. In chapter6, we use computational fluid dynamic simulations to model the formation of liquid targets for high-repetition-rate laser-plasma interactions. These targets are important for efforts to increase the flux of particles. We use these simulations to explore the formation of liquid sheets by colliding two jets, an increasingly common target, which is appealing for intense laser interactions for a variety of reasons. Then we extend this method to new target geometries that can be created with droplet collisions. Chapter7 provides a summary and broader implications of this work.

7 Chapter 2 Basics of Laser-Plasma Interactions

We begin with an introduction to several fundamental topics in plasma physics and show how lasers interact with plasmas. More extensive discussions can be found in references such as Chen [38] for plasma physics, and Kruer [39], Gibbon [40] or Macchi [41] for laser-plasma interactions.

2.1 Particle Motion in an Electromagnetic Field

A particle in an electromagnetic ﬁeld will experience the Lorentz force dp = q(E + v × B), (2.1) dt where p is the momentum, q is its charge, v is its velocity and E and B are the electric and magnetic ﬁelds. Let us begin by describing an electromagnetic (EM) plane wave, of wavelength λ, in a vacuum propagating in the z direction, which follows the relation

Ex = E0 cos(kz − ωt) (2.2) 1 B = (kˆ × E) → B = B cos(kz − ωt), (2.3) c y 0 where E0 is the electric ﬁeld amplitude, k = 2π/λ is the wave vector, ω is the angular frequency of the wave, t is time, and B0 is the magnetic ﬁeld amplitude with B0 = E0/c.

Let us consider the motion of a single electron with mass me in this ﬁeld. For v c, we can neglect the magnetic ﬁeld contribution to the Lorentz force. This gives the velocity and position of the electron to be

eE0 vx(t) = sin(kz − ωt), (2.4) meω eE0 x(t) = 2 cos(kz − ωt). (2.5) meω

8 If we compare the maximum velocity in this trajectory to the speed of light, we get a dimensionless number known as the normalized vector potential

1/2 eE0 λ I a0 = ≈ 0.85 18 −2 , (2.6) meωc 1 µm 10 W cm

2 where I = cε0E0 /2 is the laser intensity and ε is the vacuum permittivity. This dimensionless quantity provides a useful ﬁgure of merit for discussing intense lasers, where a0 & 1 is the threshold for relativistic eﬀects (clearly the velocity in Eq. 2.4 is no longer valid approaching this regime as a0 > 1 would imply superluminal speeds). In chapter4 we deal with a nonrelativistic laser where in the interaction region, a0 ≈ 0.35.

In chapter5, we consider lasers with a0 = 2.3 and a0 = 4.3. There are much more intense lasers with a0 > 50 (e.g. see Fig. 1.1), but there are still interesting effects to investigate at moderate intensities accessible to hundreds of facilities worldwide [42]. The Lorentz factor γ = (1 + p2/m2c2)1/2 of an electron can be estimated from the cycle averaged velocity from Eq. 2.2 as [41, 43] q 2 γ ≈ 1 + a0/2. (2.7) As the electron velocities increase, the v × B term in the Lorentz force begins to have a more significant impact on the particle’s motion and relativistic effects need to be considered for a0 & 1. When considering relativistic particles, we use the relativistic momentum p = γmev d 2 in Eq. 2.1 and energy conservation dt (γmec ) = −e(v · E). As discussed in Gibbon [40], Macchi [41], the v × B provides a longitudinal force that causes a net displacement of the electron in the laser propagation direction that increases with a0, but the particle does not gain any energy from the magnetic field. Even though the laser has strong accelerating fields, no energy is gained by the particle, which is a consequence of the Lawson-Woodward Theorem [44–46]. It may seem that all hope is lost for using the large electric fields of high intensity lasers (Fig. 1.1) to accelerate particles, but luckily the assumptions of this theorem are easily violated by intense laser-plasma interactions. The assumption of the theorem include: the laser is in a vacuum with no walls or boundaries, that the interaction region is infinite, that there are no static electric or magnetic fields, and nonlinear effects are neglected [40, 47].

2.1.1 Ponderomotive Force

When moving from an ideal plane wave to a real focusing laser in an experiment, we begin to break assumptions of the Lawson-Woodward Theorem. If we imagine an electron oscillating in a focusing (e.g. Gaussian) laser beam, the maximum ﬁeld decreases as it moves farther away from the laser axis. This means that the cycle-average force on the electron will be nonzero; this gives rise to the ‘ponderomotive force’. We provide a simple illustration of

9 this principle and then cite the more general result.

Let us consider the EM wave from Eq. 2.2, but with a spatial variation Ex = E0(x) cos φ, where we let φ = (kz − ωt). For simplicity we consider the nonrelativistic case, we assume that a0 1, the spatial variation of electric field is larger than the laser wavelength λ, and we consider an electron near the center of the laser pulse. Taylor expanding the electric field gives us that ∂E (x) E ≈ E (x) cos φ + x 0 cos φ + ... (2.8) x 0 ∂x Then, solving the Lorentz force for the first order term, will recover the result from Eq. 2.4. This gives the first order ‘fast’ oscillating behavior x(1). Then we use this value to solve the second order term of the force giving

∂E (x) F (2) = −ex(1) 0 cos φ (2.9) x ∂x eE0(x) ∂E0(x) = −e 2 cos φ cos φ (2.10) meω ∂x 2 2 −e ∂E0 (x) 2 = 2 cos φ. (2.11) 2meω ∂x Then to ﬁnd the ponderomotive force, we take the cycle average, which gives

2 2 e ∂E0 (x) Fx = − 2 . (2.12) 4meω ∂x This helps provide an intuitive idea of where the ponderomotive force

e2 F = − ∇E2(x), (2.13) p 4mω2 comes from.5 We do note that ions also experience this force, although due to their greater mass and the short timescales of the laser pulses considered here, it is often neglected. In chapter4, we consider how the ponderomotive force of a standing EM wave from a reﬂecting laser modiﬁes the electron density of a plasma, and subsequently the ion density due to the charge separation created by the laser.

It is often useful to describe things in terms of a ponderomotive potential Up, where

Fp = −∇Up (e.g. see Fig. 1.1). For relativistic lasers this is often used to estimate the hot electron temperature r ! a2 U = k T = m c2(γ − 1) ≈ m c2 1 + 0 − 1 (2.14) p B hot e e 2 where we use the averaged Lorentz factor from Eq. 2.7. This scaling is often referred to as Wilks Scaling [43].

5For a formal proof, see Macchi [41]. For relativistic considerations see Quesnel and Mora [48].

10 2.2 Plasma Fundamentals

Plasma is a state of matter that can be thought of as “a quasineutral gas of charged and neutral particles which exhibits collective behavior,” as deﬁned by Chen [38]. The intense lasers considered in this work quickly ionize a target6 into a plasma and the particles interact with each other and the laser ﬁeld electromagnetically. First, we see what quasineutral means by showing how a plasma reacts to a local charge imbalance.

2.2.1 Debye Length

A plasma will shield an isolated charge and this provides a useful length scale for plasmas known as the Debye length, as developed by Debye and H¨uckel [49]. We begin by describing a plasma where the electrostatic forces and the pressure of the electron ﬂuid are balanced, or

−eneE = kBTe∇ne, (2.15) where we have used the equation of state p = nekBTe, with kB being the Boltzmann factor

and Te the temperature of the electrons. Then writing the electric ﬁeld as E = −∇φ, in terms of the potential φ, we can rearrange Eq. 2.15, as

eφ ∇ − ln ne = 0, (2.16) kBTe where we can readily see the solution

eφ/kB Te ne = n0e , (2.17) where n0 is the background density. For an isolated positive charge (at the origin), we can write the one-dimensional Poisson equation as

2 e e ∇ φ = − (ni − ne) − δ(x), (2.18) ε0 ε0 where δ(x) is the Dirac delta. Then if we Taylor expand Eq. 2.17, we get

eφ ne ≈ n0 + n0 + .... (2.19) kBTe

We assume that the plasma is singly ionized and the ion background (ni = n0) is immobile

(mi me). Then, away from the origin, we can rewrite the Poisson equation to ﬁrst order

6Often the inherent pre-pulse of these lasers is enough to partially ionize a target, or the rising edge of the pulse will ionize the target. In this work we will not closely consider the ionization process. The reader can see a discussion in Gibbon [40].

11 as 2 2 d φ e eφ(x) ∇ φ = 2 ≈ − n0 − n0 + n0 (2.20) dx ε0 kBTe n e2 = − 0 φ(x), (2.21) ε0kBTe which has a solution

−|x|/λD φ(x) = φ0e , (2.22) where the Debye length is deﬁned to be r ε k T λ = 0 B , (2.23) D ne2 where n is the background density of the plasma. This provides a good order of magnitude length-scale over which the charged is screened to maintain a ‘quasineutral’ plasma. One can also deﬁne a Debye sphere (of radius λD), where for an ideal plasma we require the 3 number of particles in a Debye sphere ND = 4/3 πλD 1.

2.2.2 Oscillations in a Plasma

Now we look at electron oscillations in a plasma. We start with a uniform plasma as sketched in Fig. 2.1(left), and assume that there is no thermal motion of the particles, the much more massive ions are assumed to be ﬁxed in place, and the system can be described in 1D: motion only occurs in one direction, which we will call x and the plasma slab is inﬁnite in extent. Then we displace the electrons in the plasma by a distance x, as sketched in Fig. 2.1(right). We can treat this like a parallel plate capacitor. There is a neutral center

region, and each charged region has a charge density with magnitude σ = enex, giving an electric ﬁeld in the x direction of

σ enex Ex = = , (2.24) ε0 ε0 which follows with a simple application of Gauss’s Law. The electric force on an electron in this ﬁeld is then 2 d x enex F = me 2 = −e , (2.25) dt ε0 which we see has the form of a harmonic oscillator. The angular frequency of the oscillation is what we call the plasma frequency s 2 nee ωpe = . (2.26) meε0

The plasma frequency is an important quantity to determine how a plasma responds

12 Uniform Plasma E +- +- +- +- +- +- + + +- +- +- +- - - +- +- +- +- +- +- + + +- +- +- +- - - +- +- +- +- +- +- + + +- +- +- +- - - +- +- +- +- +- +- + + +- +- +- +- - - +- +- +- +- +- +- + + +- +- +- +- - - +- +- +- +- +- +- + + +- +- +- +- - - +- +- +- +- +- +- + + +- +- +- +- - - +- +- +- +- +- +- + + +- +- +- +- - - x

Figure 2.1: A uniform plasma (left), where the electrons are displaced by a distance x (right). This results in a positively charged region of the plasma, followed by a neutral region in the middle, and a negatively charged region to the right, resulting in an electric ﬁeld that pulls electrons back to the equilibrium.

to a laser of frequency ω. The dielectric function for a plasma can be written as ε(ω) = 2 2 ε0(1 − ωp/ω )[39, 41], which gives the index of refraction of the plasma to be s s 2 ε(ω) ωp n = = 1 − 2 , (2.27) ε0 ω which we see becomes complex when ωp > ω. To take a closer look we consider the dispersion relation for electromagnetic wave in the plasma. For an EM wave propagating in the z direction of the form E(r) = E0 exp(i(kz − ωt))ˆx, we can apply the wave equation

∂2E ∇2E = ε(ω)µ , (2.28) 0 ∂t2

2 2 2 2 √ where solving this gives us that k = ε0(1 − ωp/ω )µ0ω , or (with c = 1/ µ0ε0)

2 2 2 2 k c = ω − ωp. (2.29)

Here we see that for ωp > ω, the wavevector k becomes complex, which means that the wave can no longer propagate and becomes evanescent. For a given laser wavelength, the

13 density where this occurs, known as the critical density of a plasma is given by

2 2 2 4π εomec 1 µm n = = 1.1 × 1021 cm−3 (2.30) crit λ2e2 λ

We refer to densities greater than the critical density as ‘overdense’ and those less dense as ‘underdense.’ When electrons begin to approach relativistic velocities, the classical critical density is increased by a factor of gamma, or ncrit → γncrit. This means that relativistic lasers can propagate into higher density plasmas. 2 2 −1/2 The electric ﬁeld for the evanescent wave drops as exp (−z/ls), with ls = c(ωp −ω ) ; for ωp ω, we have c ls ≈ , (2.31) ωp which is known as the skin depth [41]. While the laser cannot propagate into the overdense plasma, the electric and magnetic ﬁelds can still be quite large (dropping by 1/e over the skin depth), which can drive electron heating.

2.3 Ion Acceleration

Now that we have introduced some of the basic terminology and concepts underlying laser- plasma interactions we consider the acceleration of ion beams. While ions can be directly accelerated by the laser field, their mass is orders of magnitude (& 1800 ×) greater than that of electrons, and they are not greatly effected during the tens-of-femtosecond duration laser pulses. Instead, the laser is used to heat the electrons and cause a charge separation between the ions and electrons, which is used to accelerated ions. There are a variety of ion acceleration mechanisms (e.g. see Macchi et al. [50]), although our focus will be one of the most studied methods known as Target Normal Sheath Acceleration (TNSA) [51–54], which is most relevant to the intensities and target thicknesses considered in this work. A sketch of the TNSA process in shown in Fig. 2.2. First, an intense laser interacts with a target, generating a pre-plasma at the front of the target (left side of Fig. 2.2). Then electrons are accelerated by the laser and travel through the target. Some of the highest energy electrons escape, which charges the target, but most of the electrons are accelerated back towards this target and reflux (oscillate) through the target. An electrostatic sheath is developed on the surfaces of the target (front and back), which ionizes and accelerates the ions from the surface of the target in a direction normal to the surface (in experiments there is typically a layer of contaminants such as hydrocarbons or water on the surface of the target and protons from this layer are accelerated first). There are a variety of models to help estimate the accelerating fields and ion energies produced by TNSA. For example, Mora [53] models the expansion of a plasma into a vacuum with a quasi-static model assuming the electrons density has the form of Eq. 2.22. This

14 Figure 2.2: Sketch of the TNSA ion acceleration mechanism from Ref. [7]. Electrons are accelerated through the target by the laser and an electrostatic sheath is developed which accelerated ions in a direction normal to the target.

model finds the peak strength of the electrostatic field that accelerates ions to be k T E ∼ B hot , (2.32) λD where we see that the field can be increased by increasing Thot or decreasing the Debye

Length. We can achieve this by increasing the intensity of the laser to increase Thot (e.g. 2.14)7. Alternatively, increasing the density of the sheath (decreasing the Debye length) can be achieved by improving the conversion eﬃciency from laser energy to electron energy as we will see in chapter5. A more sophisticated dynamic model can be found in Fuchs et al. [30], which can better predict the accelerated ion energies. This dynamic model also improves based on the hot electron temperature and number of hot electrons. TNSA experiments routinely accelerate ions to tens of MeV (e.g. [55]) and the leading experimental results can reach up to nearly 100 MeV [22] (in a hybrid acceleration scheme

7 The Debye length in the denominator does have a dependence on Thot, but it is inside of a square root. Also this can allow the electrons to stay hot for a longer period of time.

15 involving TNSA). While it is fairly straightforward to increase the electron temperature, √ there are diminishing returns with increased laser intensity, for example Thot ∼ I (according to Eq. 2.14). With this in mind, in chapter5 we ﬁx the laser intensity and try to optimize the conversion from laser energy to ion energy by altering the target that the laser interacts with. There are other ion acceleration methods that may scale more favor- ably [50], but there is still an abundance of theoretical and experimental work that must be completed before laser-generated ion beams are ready for applications, as discussed in Sec. 1.1.3.

16 Chapter 3 Computational Methods

In this chapter, we describe the computational methods employed throughout this thesis. First in Sec. 3.1 we introduce the particle-in-cell (PIC) method, which is used to model laser plasma interactions in chapters4 and5. Then in Sec. 3.2 we introduce evolutionary algorithms, which are used to optimize laser plasma interactions in chapter5. Lastly in Sec. 3.3 we introduce computational ﬂuid dynamic (CFD) simulations which are used in chapter6 to model the formation of liquid targets for high-repetition-rate laser systems.

3.1 The Particle-In-Cell Method

As the name suggests, particle-in-cell (PIC) simulations model the charged particles in a plasma with discrete particles that each have a specific location in space, and a computational grid is used to solve Maxwell’s equations as sketched in Fig. 3.1. Dawson [56] was one of the first to use this type of technique to simulate plasma in 1962. For the dense targets we model, there are many more particles in the plasma than feasibly could be represented with a computer, so instead the ions and electrons are represented statistically with a much smaller number of ‘macroparticles’. Using a computational grid further simplifies the problem, where rather than calculating the forces between all particles, electric fields are interpolated to the particles from a (typically) much smaller number of cells. More extensive introductions are available in Birdsall and Langdon [57], Hockney and Eastwood [58]. For this work, we use the commercial PIC code LSP [59].

3.1.1 PIC Cycle

We begin a PIC simulation by discretizing space into a simulation grid (i.e. deﬁning the grid dimensions and resolution) and then initializing the particles on the grid, based on the desired density and velocity distributions. We also deﬁne the boundary conditions (e.g. a laser is introduced to our simulations as a time-dependent boundary condition). Then the system is allowed to evolve based on an algorithm, which we will refer to as the PIC

17 nodes - +- - ++ - cell + - particles + - +

Figure 3.1: Sketch of a 2D PIC simulation grid with positive and negative particles and a simple node structure (in practice the node structure is typically more complex as discussed in Sec. 3.1.3).

cycle. The general steps for a single time step of a PIC simulation are included in Fig. 3.2. The forces on each particle (Eq. 2.1) are calculated based on their position and then their position and velocity are updated. Next the particles positions and velocities are mapped to nearby nodes to determine the charge density and current. These quantities are then used to update the electric and magnetic ﬁelds, which in turn lets us calculate the forces on the macroparticles. An additional step of a PIC simulation can include additional physics such as ionization or collisions.

3.1.2 Particle Weighting

We begin by discussing how the particles are mapped to the grid and how the electric and magnetic fields are interpolated back to the particles. Each macroparticle in a PIC simulation has a continuous location in space that must be mapped to the computational grid. In our simple sketch of a PIC simulation (Fig. 4.1) we see a particle that lies on the boundary between two cells, showing the need to define a ‘size’ for our macroparticles. The simplest ‘zeroth-order’ model maps the density of a particle to the closest node. In one-dimensional (1D) the particle has an effective width of ∆x, the node spacing. This ‘nearest grid point’ (NGP) method is computationally inexpensive but it is not often used in practice due to numerical noise arising from sharp changes in density and the electric field as particles move between nodes [57]. The first-order method used in this work is the Cloud-in-Cell (CIC) developed by Birdsall and Fuss [60], where a particle’s density is now spread out over multiple nodes. The particles take a triangular shape of width 2∆x, where the particle is spread out over two nodes (in

18 Calculate force on particles from E & B

Update E & B Fields t Advance Particles

Map Particles to Grid

Figure 3.2: The general steps completed by a PIC code.

each dimension). For a particle of charge qi located at xi, the charge contribution to adjacent nodes located Xj and Xj+1 is given by

X − x x − X q = q j+1 i , q = q i j+1 . (3.1) j i ∆x j+1 i ∆x

The ﬁelds are interpolated back to the particles with the same scheme. Higher order schemes are also available (e.g. [61]), which can reduce noise and the growth of certain instabilities. The per-step computation time is increased with these schemes, although they may allow for coarser grids depending on the problem [57].

3.1.3 Field and Particle Evolution

Now we discuss how the particles and ﬁelds are evolved. The force on the particles is given by the Lorentz force (Eq. 2.1), and Maxwell’s equations are used to evolve the ﬁelds. There are a variety of methods to numerically integrate these equations to evolve the system, but the general two categories are explicit and implicit techniques. For an explicit method, the quantities for the next time step are calculated based only on current state of the system. Whereas implicit methods use information from the current state of the system and algebraically depend on a later time step of the system. For explicit PIC codes, the leap-frog method is often used to calculate the trajectory of the particles. It is a second order method where the velocity and position are updated alternately every half time step. This provides reasonable accuracy at a small computational cost compared to higher order methods (e.g. fourth order Runge Kutta), which have better accuracy, but at the expense of additional computational and memory requirements. For

19 the electromagnetic ﬁeld evolution, the electric and magnetic ﬁeld grid are often staggered in space to form a Yee [62] Grid (creating a more complex node structure than the sketch in Fig. 3.1). We refer the reader to Arber et al. [61] for an extensive discussion about the explicit PIC code EPOCH. For this work, we use an implicit algorithm in LSP; it uses a version of a direct implicit algorithm [63] discussed in [59, 64]. For a detailed discussion of LSP’s implicit algorithms, see the graduate theses by Ovchinnikov [65], King [66].

Explicit vs. Implicit PIC Codes

Here we consider some of the practical differences between explicit and implicit codes. Explicit algorithms are generally less computationally expensive per time step and easier to implement than implicit ones, although they are more susceptible to instabilities. For example, if the Debye length is not sufficiently resolved by the gird the simulation, the particles can ‘self-heat,’ adding nonphysical energy to the simulation [61]. Thus very small cell sizes can be required for these simulations. In practice, to circumvent this instability the plasma temperature is often increased, to tens of keV [22] (or more), or the plasma density is reduced, to increase the Debye length, although these choices can cause nonphysical effects. Implicit simulations ease this restriction. We do note that there are energy-conserving explicit algorithms (including in the LSP code) that also effectively address this problem. The Courant-Friedrichs-Lewy (CFL) Condition is an important consideration for selection the simulation time step [67, 68]. This restriction can be thought of as limiting the distance traveled by the fastest entity (light for laser simulations, or fluid velocity for CFD simulations as discussed later later) in the simulation to be less than one cell width during a single time step. For example in 1D, we would consider C = v∆t/∆x, where ∆t is the time step and ∆x is the grid spacing. For these purposes, we would require C, the Courant number, to be less than one. There are other important timescales to resolve, depending on the problem, such as the plasma frequency (Eq. 2.26). The stability of implicit codes is very useful for modeling these systems, although we must also keep in mind that stability does not guarantee accuracy.

3.1.4 Dimensionality Considerations

Despite the simplifying assumptions, PIC simulations can still be quite computationally expensive, particularly for three-dimensional (3D) simulations which can require in excess of 100,000 CPU hours each (1 CPU hour = 1 core running for one hour). As such, there are limits on the number of 3D simulations that can be run and often lower dimensional two-dimensional (2D) and 1D simulations are used. These lower dimensional simulations are much less computationally expensive where; for example, some of the high resolution 1D simulations included in this dissertation took less than one-half hour on a single core.

20 1.00

0.75 x a m

I 0.50 / I zR

0.25 1D 2D 3D 0.00 15 10 5 0 5 10 15 z ( m)

Figure 3.3: The intensity as a Gaussian laser beam comes to focus for 1D, 2D, and 3D PIC simulations for an 800 nm laser with a beam waist of 1.5 µm.

In this work we employ 1D(3V), 2D(3V), and 3D simulations with Cartesian grids8 to explore different phenomena. The ‘3V’ indicates that particles are confined to one (or two) spatial dimension, but particle velocities are considered in three dimensions. Lower dimensional simulations do not capture all of the physics of a 3D simulation, but they are necessary to explore larger parameter spaces and can be useful in finding trends that extend to higher dimensions. For example, the electric field of a charge drops off as 1/r2 in 3D (point charge), 1/r1 in 2D (line charge), and does not drop off (1/r0) in 1D (sheet of charge) [57]. Another difference is the focusing of the laser pulse. As a Gaussian beam comes to focus 2 2 2 in 3D, the intensity on axis goes as I3D = I0/(1+z /zr ), where zr = πw0/λ is known as the Rayleigh length with w0 being the beam waist (the radial distance away from the axis at focus where the intensity drops by e2). The intensity of the laser drops by half at a Rayleigh length away from focus as shown in Fig. 3.3. In 2D there is one fewer dimension to focus p 2 2 in, and the intensity goes as I2D = I0/ 1 + z /zr , and in 1D the laser does not focus. Figure 3.3, illustrates this difference, where the laser comes to focus more slowly in 2D. Some of the implications of these differences are illustrated in Ngirmang et al. [69], Stark et al. [70]. For example, in 2D choosing whether the laser is polarized in a physical or virtual dimension has significant effects [70].

8Although not considered here, one may also exploit other symmetries with another coordinate system such as cylindrical or spherical.

21 3.2 Evolutionary Algorithms

In chapter5, we use a type of optimization known as an evolutionary algorithm to select the initial conditions for a series of PIC simulations. Evolutionary algorithms are a broad class of metaheuristics inspired by the biological theory of evolution [71–74]. Within this class are “genetic algorithms” and in this study we use an evolutionary algorithm called “differential evolution” [75, 76], which is specifically designed to deal with continuous vari- ables. Evolutionary algorithms seek to optimize a ‘fitness function’ (or ‘objective function’) by testing many different candidate hypotheses creating a ‘population’ that reproduces and evolves over many generations [77]. For our work, the population is composed of many 1D PIC simulations and the ‘genome’ represents the search space, where each ‘gene’ is a parameter corresponding to one density bin throughout the depth of the ten-dimensional target density profile. The general procedure for an evolutionary algorithm is sketched in Fig. 3.4 and is ex- plained in depth in Refs. [72, 77]. We begin by initializing the population, typically a random sampling of the search space. Then the fitness of each member of the population is evaluated. If the maximum fitness of the population is within some threshold, or if it has reached the maximum number of iterations, the algorithm is complete. Otherwise, we proceed to selection, where the ‘parents’ of the new generation are selected based on their fitness (the initial population may be used in whole as the first parents). Next ‘crossover’ occurs, where two or more parents are mated to form a ‘child’. Then mutation occurs, where the genes of some children are modified. Finally, we evaluate the fitness of the children and the process repeats until the stopping condition has been satisfied. The stopping condition is triggered if the fitness reaches some predetermined value. In practice, if there is no stopping condition selected, the user may manually stop the evolution based on performance. For a full description of the differential evolution algorithm, we refer the reader to Refs. [75, 76]; but we provide a summary of the process here. In differential evolution, four parents are used in the crossover/mutation steps, where to create the mutation vector, the parameter from one of the parents is perturbed based on the difference between the value of two other parents. The algorithm begins by initializing NP (population size) D-

dimensional vectors randomly sampling the parameter space, which we will call xi where i = 1, 2,...NP . In our case, we use 10-dimensional vectors, where each element corresponds to part of the density profile of a target (see figure 5.1 on page 51). Then to find the test vectors for the next generation, we loop through all members of the population. For each i, we generate a mutant vector

mi = xn1 + F · (xn2 − xn3 ), (3.2)

where xn1 , xn2 , xn3 are mutually distinct members of the population (also distinct from xi

22 Figure 3.4: The general procedure for an evolutionary algorithm. A population is created and evolved with crossover and mutation, based on a ﬁtness function until a stopping condition is met. For our work, the ‘Evaluate Fitness’ step depends on the output of 1D PIC simulations.

which is used in (3.3)), and F ∈ [0, 2] is a factor that controls the weighting of the diﬀerential

evolution [75]. Then to form the test vector ti for the next generation, we select a crossover rate CR ∈ [0, 1]. Next for each gene the crossover rate represents the chance that a gene is

selected from ti. To do this, we generate loop over the genes j = 1, 2,...D and generate a random number between 0 and 1 (Rand[j]) to see determine crossover occurs, or  mi[j], if Rand[j] ≤ CR ti[j] = (3.3) xi[j], otherwise.

If no genes have been selected from mi, one is automatically chosen to prevent testing of the same point twice (some implementations of the algorithm automatically switch one gene). The ﬁtness is then calculated (a 1D PIC simulation is run in our case) and if the

fitness of ti is better than xi, it becomes a member of the next generation. We use F = 0.5, and CR = 0.9, which are initial parameter choices recommended by [75]. Often a population size of ten times the dimension size is used [75], we slightly exceeded this with NP = 120, which was a convenient choice as the computer system used had 40 cores per node. While typical values proved to have good performance for our problem, they can require significant tuning in practice, which would be an important consideration for problems with higher computational (or experimental) costs. For our work, by far the largest computational expense in this process is the 1D PIC simulations (which determines the fitness of each member of the population). The mutation of the population and selecting from these mutations to create a new population requires negligible computational time by comparison. We implemented the evolutionary algorithm in Python. It selects the density profiles

23 and then creates data files that are read by the PIC simulation code LSP. The simulation runs are initiated directly from the Python code with a system call to run an compiled LSP executable.9 Following the completion of all PIC simulations, the Python script reads the output files from LSP to calculate the fitness.

3.3 Computational Fluid Dynamics Simulations

In chapter6, we use computational ﬂuid dynamic (CFD) simulations to model the formation of liquid targets for high-repetition-rate laser experiments. We use the open source CFD code OpenFOAM [78] with the interFoam solver [79, 80].

3.3.1 OpenFOAM and the interFoam Solver

The interFoam solver considers two isothermal, non-mixing fluids and uses a volume of fluid (VOF) method to model the interface between the fluids. Each cell in the simulation has a phase fraction α ∈ [0, 1], where α = 1 represents one fluid of density ρ1 and α = 0 represent the other fluid of density ρ2. The total density of the cell ρ is given by

ρ = ρ1α + ρ2(1 − α). (3.4)

Fluid Equations

For a CFD simulation, the boundary conditions and initial conditions are set for the system and the ﬂuid equations, introduced next, are used to evolve the system. A more in depth discussion of the equations and solver with source code is included in Ref. [80], we will provide a short summary here. The continuity equation for incompressible ﬂow is given by

∇ · u = 0, (3.5) where u is the ﬂuid velocity. The momentum equation is given by ∂u + ∇ · (ρu ⊗ u) = −∇p + (∇ · (µ∇u) + ∇u · ∇µ) + ρg + F(σ), (3.6) ∂t where F(σ) represents the force from surface tension that depends on the interface using a continuum surface force model from Brackbill et al. [81], µ is the viscosity (which depends on the viscosity of each ﬂuid and the phase fraction), p is the pressure, and g is the force of gravity [79]. Also to evolve the phase fraction α (Eq. 3.4), we have the equation

∂α + ∇ · (αu) = 0. (3.7) ∂t 9For increased computational demands, or different allocation structures, it may be beneficial to submit the PIC simulations to run as a separate job files.

24 A ﬁnite volume method is used to evolve the system. The time step of the simulations is adaptive based on the Courant number (see Sec. 3.1.3). An evaluation of the performance of interFoam including a summary of veriﬁcation tests is presented in Deshpande et al. [79].

25 Chapter 4 Particle-in-Cell Simulations of Density Peak Formation and Ion Heating from Short Pulse Laser-Driven Ponderomotive Steepening

4.1 Introduction

10Ultra-intense laser interactions with dense targets represent an interesting regime, both from a fundamental and an applied perspective, that has not yet been exhaustively explored. One less explored phenomenon in this regime is the formation of electron and ion density peaks due to a laser pulse that strongly reflects from a dense target. There are papers that discuss this process – sometimes called ponderomotive steepening – going back to Estabrook et al. 1975 [82]. Figure 4.1 provides a qualitative sketch of the physics involved in this laser-plasma interaction. First, a normally incident, linearly polarized laser makes a strong reflection from a dense plasma. The interference between the incident and reflected pulse produces a standing wave pattern (Fig. 4.1a). The ponderomotive force associated with this standing wave has a strong effect on the electron distribution (Fig. 4.1b) and, over time, peaks form in the density of both the electrons and ions (Fig. 4.1c). Readers who are familiar with Kruer’s 1988 textbook [39] will recall the discussion of this phenomenon there. Ponderomotive steepening also draws many parallels to theoretical and computational work that considers the standing electromagnetic (EM) wave formed by crossing two laser pulses to generate plasma optics such as plasma gratings [83, 84] and so-called transient plasma

10Much of this chapter is reprinted from Joseph R Smith, Chris Orban, Gregory K Ngirmang, John T Morrison, Kevin M George, Enam A Chowdhury, and WM Roquemore. Particle-in-cell simulations of density peak formation and ion heating from short pulse laser-driven ponderomotive steepening. Physics of Plasmas, 2019 [1] under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). 26 Plasma Density Plasma Density Ion Density Reflecting Electron Laser Density /2 /2 Incoming Laser /2 |E| 2

(a) (b) (c) Plasma Density Plasma Density Ion Density Reflecting Electron Laser Density 2 |E| /2

/2 /2

(d) (e) (f)

Figure 4.1: Sketch of the classical ponderomotive steepening process (a-c), and the simplified process considered in this work (d-f). As illustrated in (a), a normally incident laser pulse reflects at the critical density of a plasma and forms a standing electromagnetic wave. This causes the electrons to form peaks near the extrema (separated by ≈ λ/2) of the standing wave via the ponderomotive force (b). The modification of the electron density creates a charge imbalance (sustained by the standing wave), which accelerates ions towards the electron peaks. In time, this modifies the density of the plasma as illustrated in (c). Note that (b) and (c) only include the standing wave region from (a). This paper focuses on the simpler process of a laser traveling through an underdense pre-plasma shelf and reflecting off of an overdense target that behaves like a mirror (d-f).

photonic crystals [85–87] which are phenomena that may have useful applications in the future (see discussions in Refs. [84, 86]). Drawing parallels to this work, we consider the simplified case of a laser propagating through a constant density pre-plasma shelf and reflecting off of a highly-reflective overdense target as illustrated in Fig. 4.1d-f. From an experimental point of view, ponderomotive steepening only requires one laser pulse, and the high densities near the critical surface allow for larger transverse electric fields than with counter-propagating lasers in low density media. We are motivated to return to this topic with fresh eyes in part due to the matura- tion of technologies to produce intense laser pulses at mid-infrared (IR) wavelengths (2 µm . λ . 10 µm) [88]. This presents an opportunity to examine the wavelength dependence of intense laser-matter interactions to see if theoretical models developed from studying laser

27 interactions at shorter wavelengths remain valid at longer wavelengths (e.g. Ref. [89], and ongoing research efforts [90]). As discussed later, the density peaks that form with ponderomotive steepening are separated by approximately half the laser wavelength. It is therefore challenging to detect and resolve these density peaks in near-IR or shorter-wavelength laser interactions. There have been many experiments that confirm that the ponderomotive force does steepen the plasma profile near the target as expected (e.g. Fedosejevs et al. [91], Zepf et al. [92], Audebert et al. [93], Ping et al. [94], Gong et al. [95]) and researchers have found evidence in experiments with counter-propagating near-IR laser pulses that the interference shapes the plasma distribution in a low density medium (e.g. Suntsov et al. [96]). How- ever, multiply-peaked ponderomotive steepening [39, 82, 97–100]: where multiple electron and ion density peaks are formed in the pre-plasma, has not yet been directly observed with interferometry or by other means. We aim to provide useful analytic insights for experimentalists working to demonstrate this effect. A challenge for connecting theory to observation is that multiply-peaked ponderomotive steepening is simplest to model and has larger longitudinal electric field strengths when the laser interactions are at normal incidence, whereas at the highest intensities, normal incidence experiments are rare because of the potential damage that the reflected pulse could do to optical elements. There are, however, methods to protect optics from the reflected pulse. Normal incidence experiments were conducted, for example, at ≈ 1018 W cm−2 peak intensities with ≈ 3 mJ pulses at a kHz repetition rate in Refs. [101, 102]. Although the present paper is not tied to modeling interactions from a particular laser system, it is important to note that normal incidence experiments can be performed. As will be discussed, the initial dynamics of the ions in the pre-plasma for multiply- peaked ponderomotive steepening are not accounted for in steady state models [39, 82, 97– 100], but some ions do reach significant energies due to the charge separation caused by the ponderomotive force; and experiments could investigate this regime. According to estimates that agree with our 2D(3v) PIC simulations, under the right plasma conditions and laser parameters these interactions have the potential to accelerate ions to energies exceeding 100 keV. Experiments of this kind would also be interesting as a new type of code validation experiment for high intensity laser-plasma interactions. Both during and after the laser interaction, ions move and the electron and ion density profiles change over time which can be investigated with interferometry [103–105] and measurements of escaping ion energies (e.g. Ref. [12]). The simplicity and symmetry of normal incidence interactions would be helpful for comparing experiment to simulation and theory in a straightforward way. In Sec. 4.2, we provide a brief review of the physics of ponderomotive steepening and identify the relevant timescales for ion motion using simple analytic models. In Sec. 4.3 we describe 2D(3v) PIC simulations that exhibit multiply-peaked ponderomotive steepen-

28 ing. In Sec. 4.4 the simulation results are presented and compared to the analytic models discussed in Sec. 4.2. Finally, we address implications of our results in the concluding sections.

4.2 Ponderomotive Steepening and Ion Acceleration

The traditional analytic approach for ponderomotive steepening considers a steady state solution to the fluid equations, to which a term for the ponderomotive force is added. The electric field is then assumed to take a particular form based on the geometry of the problem and to allow for numerical solutions or approximate solutions [39, 82, 97–100]. These approximations limit the validity of the conclusions, and the steady state solution provides little insight into the dynamics of the phenomenon. We investigate these dynamics by developing a simple model to estimate the longitudinal electric fields experienced by the ions and comparing the predictions to PIC simulations. Our simple model is similar in many ways to a more sophisticated analytic model described in a recent paper by Lehmann and Spatschek [87] that considers the dynamics of the electron motion for the case of lower intensity counter-propagating laser beams in a low density medium with 1D Vlasov simulations. Our paper is complimentary to theirs because we consider standing waves that form from the normal incidence reflection of intense laser pulses from an overdense target, rather than counter-propagating laser pulses. While we probe the same physics, we consider much shorter pulses, higher densities11, and higher intensity lasers, as discussed in Sec. 4.3.1. Also, we focus on the dynamics of the ions, which move much more quickly because of these parameters. Our 2D(3v) PIC simulations include the focusing of the laser and transverse extent of the density perturbations, which are not captured in 1D Vlasov simulations. Where appropriate, we provide comments for those wishing to compare our work to Lehmann and Spatschek [87]. The timescales and intensity thresholds we develop are very similar to their models. We also acknowledge that these electron density perturbation models could be modified to account for different pulse shapes following the work by Gorbunov and Frolov [106, 107].

4.2.1 A Simple Model for Ponderomotive and Electrostatic Forces in Pon- deromotive Steepening

As sketched in Fig. 4.1, the laser creates a charge imbalance due to the ponderomotive force on the electrons, which in turn creates a longitudinal electric ﬁeld to accelerate the ions towards the electron peaks. We develop a simple model that balances the ponderomotive

11The ﬁrst simulation presented in Lehmann and Spatschek [87], does use higher densities, but the second simulation in the nonlinear regime we investigate uses a much lower density as compared in Sec. 4.3.1

29 force with the Coulomb force associated with the charge separation. A charged particle in an inhomogeneous EM ﬁeld experiences the ponderomotive force, which is a cycle-averaged force that models the motion of these particles on timescales larger than the laser period. For a particle of mass m with charge e and an electric ﬁeld with frequency ω and amplitude E, the ponderomotive force is given by

e2 F = − ∇E2(x), (4.1) p 4mω2 where the electric field is cycle-averaged (see Sec. 2.1.1 for a more detailed introduction). While this effect is experienced by both electrons and ions, for the laser intensities we are concerned with here, the much more massive ions are hardly affected by the ponderomotive force. We consider a linearly polarized plane electromagnetic wave propagating in the +x direction and reflecting off of a semi-infinite overdense plasma at x > 0. Similar to Refs. [108, 109], we assume that the plasma is a perfect conductor and reflects 100% of the light, resulting in a standing EM wave (x<0) with electric and magnetic fields described by

2πx Ez = 2E0 sin λ sin(ωt) (4.2) 2πx By = 2B0 cos λ cos(ωt). (4.3) where E0 is the electric ﬁeld strength of the laser if there had been no reﬂection and

B0 = E0/c. In real experiments we do not expect 100% reflectivity. Yet, the reflectivity can be high for laser interactions near, but not significantly above, the threshold for relativistic effects since the high temperatures produce a nearly collisionless plasma, but relativistic absorption is not yet pronounced [110, 111]. Inserting Eq. 4.2 into Eq. 4.1 yields the longitudinal ponderomotive force associated with the standing wave,

2 2 λe E0 4πx Fp = − 2 sin , (4.4) 2πmec λ which will be compared to the Coulomb force associated with the charge separation.

Sinusoidal Density Variation Model

We assume that before reaching the overdense plasma at x > 0 the laser travels through a constant, sub-critical density shelf. We estimate the strength and spatial dependence of the electrostatic force in this shelf by choosing a distribution to perfectly balance the ponderomotive force when integrated with the one-dimensional Poisson equation. This produces a sinusoidal electron density modulation of the form

4πx n = n + n cos , (4.5) ele 0 e λ

30 where n0 is the average electron density in the plasma (i.e. the electron density at that location in the plasma before the laser pulse arrives) and ne describes the amplitude of the density modulation. Equation 4.5 is useful for gaining qualitative insight into the ponderomotive steepening process. We remind the reader that the ponderomotive force is time averaged, so this simple model does not fully capture the physics involved. For a more accurate description of the electron density distribution, we refer the reader to the methods discussed in Gorbunov and Frolov [106, 107], which can account for the pulse shape and temporal profile. Note that because the local electron density must always be greater than or equal to zero, ne in Eq. 4.5 must not exceed n0 as you cannot remove more electrons than are available in the plasma. Since the laser only travels in the sub-critical-density region of 2 2 2 2 the plasma, n0 must also be less than the critical density, ncrit = 4π εomec /λ e , and the maximum electron density is limited12. Integrating Eq. 4.5 with the one-dimensional Poisson equation (for experimental and simulated beam profiles, we assume that the laser spot size is much larger than λ/2) results in a quasi-static electric field in the longitudinal (x) direction of the form

en λ 4πx E = − e sin . (4.6) 4πε0 λ According to Eq. 4.6 the peak longitudinal electric ﬁeld is expressed by

2 ne πmec Emax = (4.7) ncrit eλ which is equivalent to 12 −1 ne 1 µm Emax = (1.6 × 10 V m ) × . (4.8) ncrit λ

Note that ne depends on the intensity of the laser. If we equate the peak ponderomotive force (Eq. 4.4) to the electrostatic force from (Eq. 4.6), one ﬁnds that for this model the laser is limited to displacing electron densities up to 4I ne,max = 3 mec I = (1.6 × 1021 cm−3) × (4.9) 1018 W cm−2

2 where the laser intensity I = cε0E0 /2 is used to simplify the expression and ne,max is less than the initial electron density in the plasma. For this model, the density modulation is

12Relativistic intensity lasers can travel through classically overdense plasmas [112], but we ignore this eﬀect because these arguments do not extend to arbitrarily high intensities (a0 1). Also, the correction to the model for a0 . 1 is small.

31 saturated with a critical intensity of

m c3n I = e 0 crit 4 2 n0 1µm = (6.8 × 1017 W cm−2) × . (4.10) ncrit λ We note that the critical intensity does not directly depend on λ, instead it comes by

comparing the density n0 to the critical density. It will appear here (and in some later equations) to more easily examine the behavior of diﬀerent wavelength lasers with plasmas at the same fraction of critical density (e.g. near the critical surface of an expanding plasma).

For this critical intensity, the normalized vector potential a0 for the laser (a0 = eE0/meωc), is r n0 a0,crit = , (4.11) 2ncrit

p 2 or in terms of the electron plasma frequency, ωpe = nee /meε0 (using ne = n0), a0,crit = p 1/2 ωpe/ω. Since a0,crit . 0.7, it is clear that the applicability of this model does not extend to the strongly relativistic regime. Intensities somewhat above this limit are considered in Sec. 4.2.1. We note the similarity of this estimate with the wave-breaking limit in laser wakeﬁeld acceleration [20], and in Lehmann and Spatschek [87] this intensity threshold relates to the transition between what they call the “collective electron” regime to the “single electron bouncing” regime. For high electron temperatures, this type of model could be extended by considering the Bohm-Gross frequency [113] like in Ref. [87]. Laser driven instabilities would also play a role in certain regimes [18, 39] and other eﬀects will be relevant for higher intensities (e.g. [114]).

Maximum Depletion Limiting Case

At laser intensities significantly above the critical estimate derived in the previous subsection, the electrons are more strongly peaked than predicted by Eq. 4.5 and our sinusoidal model breaks down, as demonstrated by simulation results that will be presented later. Al- though the sinusoidal model breaks down, there is a simple way to determine the maximum longitudinal electric fields in this limiting case. If all of the available electrons are evacuated to the peaks, the maximum electric field in the maximum depletion regime is a factor of π greater than the sinusoidal model. This comes from integrating the charge density in the depletion region (en0 × λ/4), giving the maximum longitudinal electric field to be

2 2 en0 λ n0 π mec Emax = = ε0 4 ncrit eλ ne 1 µm = (5 × 1012 V m−1) × . (4.12) ncrit λ

32 This result is notable simply in that it implies that the longitudinal electric ﬁeld is enhanced

(relative to Eq. 4.8) at intensities slightly exceeding Icrit from Eq. 4.10, rather than being suppressed.

4.2.2 Timescale of the ion acceleration

This subsection determines a timescale for ion motion (for ions to reach an electron peak), which will be useful for comparison to the duration of the laser pulse. If the laser pulse duration is shorter than the timescale for ion motion, then we characterize this as the ‘short pulse’ regime. If instead, the laser pulse duration is signiﬁcantly longer than this timescale, we label this as the ‘long pulse’ regime. We assume the plasma to be an initially neutral mixture of electrons, and ions with charge +Ze where Z is the average ionization. Now we consider the electrostatic force on

an ion of mass mi between two of the electron peaks. Following the sinusoidal model, we focus on the ions at a distance of λ/8 or less away from an electron peak as they will reach the electron peak more quickly (the farthest away ions are considered in Appendix A.2), and we approximate the electric ﬁeld as linear in this region (matching the slope of Eq. 4.6 near its root), or 4Ze2I F = − 3 x. (4.13) mec ε0 This results in simple harmonic motion with an angular frequency of s 4Ze2I ωion = 3 . (4.14) mimec ε0

We use this equation to compute the oscillation period of the ion motion. Since we are primarily interested in the dynamics of the ion density peak growth, we are concerned with the timescale for an ion to move from its initial location to the electron density peak. This timescale is equivalent one-quarter of the ion oscillation period (Appendix A.1), which is r π m m ε c3 τ = i e 0 , (4.15) ion 4 Ze2I

where we note this formula is only valid for I ≤ Icrit where Icrit is given by Eq 4.10. For higher intensities, as discussed in Sec. 4.2.1, the maximum electron density that the laser displaces is limited by the number of available electrons and critical density. This results in a minimum timescale of r π miε0 τion,min = 2 2 Ze n0 n 1/2 λ m 1/2 = (51 fs) crit i (4.16) n0 1 µm 2Zmp

33 where in the approximation we have for convenience assumed Z = 1 and mi ≈ 2mp where mp is the mass of a proton. Equations 4.15 and 4.16 are represented in Fig. 4.2 which illustrates the division between the short pulse regime and long pulse regime as a function of laser intensity and wavelength. As examined earlier, we do not expect the timescale to be signiﬁcantly de- creased for I Icrit due to the depletion of electrons. This limit is represented with the horizontal lines. We also note that relativistic eﬀects should be considered in this region for a0 & 1. The laser wavelength does not appear in Eq. 4.15, which is why at low intensities in Fig. 4.2 the timescale does not depend on wavelength. At higher intensities our minimum timescale (Eq. 4.16) depends on the initial plasma density, where ncrit does depend on wavelength. Above this critical intensity, according to the sinusoidal model, the maximum electron density that the laser could displace exceeds the available number of electrons in the plasma near the electron peak. It should be noted that Eq. 4.15 has the same scaling with parameters as the time estimate in Lehmann and Spatschek [87] for an ion “grating” to develop in the standing wave.

Maximum ion velocities and energies

We assumed simple harmonic motion to obtain τion in Eq. 4.15. This approach also provides a characteristic ion energy which can be compared to our simulations. Assuming simple harmonic motion with an amplitude of λ/8 and an available electron density of ne we have an ion velocity that increases with time as r r π Zme ne π t vion ≈ c sin , (4.17) 4 mi ncrit 2 τion where t ≤ tSW is the time elapsed since the standing wave fields began. Ions are assumed to move at a constant velocity after the standing wave dissipates (t > tSW). Although Eq. 4.17 does not explicitly depend on the laser wavelength, as mentioned earlier our expression for τion is only valid at laser intensities below the critical intensity (Eq. 4.10) which does depend on wavelength. Since shorter wavelength lasers have a higher critical intensity, one can reach much smaller values of τion as illustrated in Fig. 4.2, which would allow the ion velocity (Eq. 4.17) to grow more quickly. But this growth is limited by the duration of the laser pulse if we are considering the short pulse regime (tSW τion). For a sufficiently long duration laser pulse the standing wave fields will last long enough

that tSW approaches τion. From Eq. 4.17, it is straightforward to show that this implies a

34 10 4 10.0 µm Long Pulse 3.0 µm 1.1 µm Regime 0.8 µm 0.4 µm 10 3

2 10 Simulations Short Pulse Regime

10 1 10 14 10 16 10 18

Figure 4.2: The division between the short pulse regime and long pulse regime as a function of laser intensity and for a variety of wavelengths (for n0 = ncrit/2) for our simple model. The timescale on the vertical axis is the timescale of ion motion from electrostatic forces in ponderomotive steepening. The dashed line represents the timescale for a shelf density of ncrit/20, pertaining to the simulations in this paper. The individual points on the graph represent the full pulse duration for our simulations (scaled by ion mass). The shaded region represents the short pulse regime for a 0.4 µm laser. The exaggerated line widths are used to emphasize that this is a simple model to provide rough estimates for the two regimes we identify in this work. Dotted lines are drawn for a0 & 1.

35 maximum kinetic energy exceeding 100 keV,

2 π ne 2 2 π tSW KEmax ≈ Zme c sin 32 ncrit 2 τion n π t ≈ 157.6 keV × Z e sin2 SW . (4.18) ncrit 2 τion Interestingly, this expression is independent of wavelength except for the wavelength dependence of ncrit.

4.3 Particle-In-Cell Simulations

Multiply-peaked ponderomotive steepening is examined numerically with implicit 2D(3v) PIC simulations performed with the LSP PIC code [59]. The initial conditions are such that we are in the short pulse regime of our model and we have exceeded the critical intensity for our model (Fig. 4.2). For these simulations, an x − z Cartesian geometry is used, where the laser propagates in the +x direction and the polarization is in the z direction. The simulations have a spatial resolution of 25 nm × 25 nm (λ/32 × λ/32) and were run for 400 fs with a 0.1 fs time step. To isolate the dynamics of the ion peak formation process we consider an idealized geometry of a rectangular target with an extended pre-plasma shelf. The plasma is assumed to be singly ionized with fixed ionization. This choice is made to prevent the critical surface from moving significantly due to ionization caused by the laser pulse. Ponderomotive steepening still occurs in simulations when the critical surface moves forward from ionization (e.g. Refs. [89, 111]) but we ignore this effect in order to focus on the electron and ion dynamics. In the laser propagation direction, the target consists of a 7 µm long constant 19 −3 sub-critical density plasma shelf (n = 8.594 × 10 cm ≈ ncrit/20) with a sharp interface 23 −3 between an 15 µm overdense target (n = 10 cm ≈ 60ncrit) as illustrated in Fig. 4.3. In the polarization direction, the target is 30 µm wide. Ion-ion collisions are not considered for reasons discussed in Appendix A.3. We describe three different simulations with targets composed of fully ionized hydrogen, deuterium, or tritium ions in order to investigate different charge-to-mass ratios. These simulations all keep the laser intensity and initial target electron and ion number densities constant. The overdense region is given a number density similar to our group’s previous work [111]. The simulations were initialized with 81 particles per cell for the electrons and 49 particles per cell for the ions with initial thermal energies of 1 eV. We consider an 800 nm wavelength, normally incident laser pulse propagating in the +x direction that would reach a peak intensity of 1018 W cm−2 if no target were present. The pulse duration is 42 fs full width at half maximum (FWHM) with a sine-squared envelope

and a Gaussian spot size of 1.5 µm (FWHM). The laser has Gaussian beam radius of w0

36 0 8.594 10 19 10 23

-30 incoming laser

-20 reflecting laser

-10 x (µm) x

0 vacuum focus

-20 0 20 z (µm)

Figure 4.3: Initial conditions for the 2D(3v) PIC simulations. The laser propagates in the +x direction with a rectangular target composed of an extended constant underdense 2 pre-plasma shelf region preceding an overdense region. Contours are drawn for at I0/e ≈ 1.35 × 1017 W cm−2 for the simulated laser near the shelf, reﬂecting oﬀ of the target at 85 fs, and for a laser at the geometric focus (in the absence of a target).

2 = 1.27 µm and Rayleigh length (zr = πw0/λ) of 6.3 µm. These parameters are similar to those of the Ti:Sapphire kHz repetition rate laser system described in Refs. [101, 102, 111].

We use these parameters to explore the short pulse regime of our model with tSW < τion. The laser focus is set at the back of the target, as shown in Fig. 4.3, in order to create a larger transverse region over which ponderomotive steepening can occur. For these 2D(3v) p 2 2 simulations, the laser comes to focus as I2D = I0/ 1 + x /zr , which is less sharply than 2 2 3D Gaussian beams (I3D = I0/(1 + x /zr )), as illustrated in Ref. [89]. This results in an intensity of ≈ 2.7 × 1017 W cm−2 at the beginning of the underdense shelf (x = −23 µm) and ≈ 3.8 × 1017 W cm−2 where the laser reﬂects (x = −15.5 µm).

4.3.1 Model Regime for Parameter Choices

In our simulations the intensity near the reﬂection point is measured to be roughly ≈ 17 −2 2.6×10 W cm . According to Eq. 4.10, for our wavelength and plasma density Icrit ≈ 5×1016 W cm−2. Our simulations therefore explore the regime where the intensity is about ﬁve times larger than this threshold. Regarding the timescale of ion motion, for these p simulations τion = 129 fs × mion/mp. This timescale in all three simulations is longer than the 42 fs FWHM laser pulse (and even the full simulated 84 fs pulse with a sine-squared

37 Figure 4.4: Electron density (z/λ < 0) and ion density (z/λ > 0) near the reflection point for the deuterium simulation. The laser finishes reflecting around 130 fs from the beginning of the simulation (a), although the peaks continue to grow as shown at 260 fs (b) and then begin to dissipate as illustrated at 390 fs (c). The width of the box (in z) represents the region considered in Fig. 4.5 and the entire box represents the region considered for ion trajectories in Fig. 4.6. This density peak growth process is highlighted in a supplemental video (see https://doi.org/10.1063/1.5108811.1).

envelope), making these interactions well within the short pulse regime as illustrated in Fig. 4.1. We are able to explore the interesting nonlinear regime identiﬁed in Lehmann and Spatschek [87], but with a much denser plasma (≈1/20th of critical density compared to −4 10 ncrit), much higher intensity laser pulse (a0 ≈ 0.35 compared to their a0 = 0.03), and a much shorter laser pulse duration (42 fs versus 3200 fs).

4.4 Results

4.4.1 Peak Formation and Density Proﬁle Modiﬁcation

Figure 4.4 shows snapshots of the electron density and ion density from the deuterium simulation at three different times. The standing EM wave causes the electrons to form peaks which, over time, produce peaks in the ion density separated by approximately λ/2 throughout the underdense region. The hydrogen and tritium simulations show similar behavior, with the growth of the ion peaks happening sooner for lighter ions and later for more massive ions. In all three simulations we observe more than 10 ion density peaks in the 7 µm long underdense region. As discussed in the next subsection, an examination of the ion trajectories confirm that ions accelerated from both sides of the peak are streaming past each other. Figure 4.5 shows this happening in all three simulations, albeit on different timescales. For each simulation, 38 3.0

2.5

2.0

1.5

1.0

0.5

0.0 0 100 200 300 400

Figure 4.5: Change in density of the ﬁrst peak in the ion density. The lines represent the maximum density of the ﬁrst ion peak averaged over the width of the laser pulse. This is calculated by averaging the maximum density for each value of z in the region -2 µm < z < 2 µm, where error bars represent the standard deviation. We note that the exact density at the peak depends on the cell size (and sharpness of the peak), thus this graph comments more on densities in the region near the peak rather than the peak itself.

the peak ion density increases to ≈ 2.5 × 1020 cm−3 (approximately three times the initial density), which lasts for tens to hundreds of femtoseconds, and then begins to decrease. Multiply-peaked ponderomotive steepening in this short pulse regime is therefore a highly transient eﬀect.

4.4.2 Ion Motion

To better understand the dynamics of the peak formation process, we consider the motion of the ion macroparticles in the simulation. In particular, if we consider the ion trajectories (Fig. 4.6) we see that the ions are accelerated towards the electron peaks while the standing wave is present. Later, as the standing wave dissipates, the inertia of the ions allows them to continue to travel with a roughly constant velocity. We see from Fig. 4.6 that many of the ions travel through the peak before the end of the simulation, which produces the broadening observed in Fig. 4.4. We note that the transverse movement of the ions is negligible compared to the spot size of the laser. The energy distribution of the ions is represented in Fig. 4.7 which highlights results from the deuterium simulation and overlays the average ion energies from the hydrogen and tritium simulations. In all three simulations, the ions are accelerated while electron

39 -21.5

-21.0

-20.5

100 200 300 400 Time (fs)

Figure 4.6: The average trajectories in x for a sample of particles starting in the boxed regions in Fig. 4.4, representing the first three peaks in the ion density. The white vertical lines represent approximately when the laser begins reflecting, reaches its half maxima, and stops reflecting. Shaded vertical lines correspond to the times represented in Fig. 4.4. The ions continue to travel after the standing wave has dissipated, and the observed peaks are created by the crossing ions.

40 Figure 4.7: Longitudinal ion energies for particles starting in the boxed regions in Fig. 4.4. The average kinetic energies for each simulation are plotted in time and the distribution of ion energies in the background corresponds to the deuterium (2H+) simulation (logarithmic grayscale). The maximum energy from this distribution agrees well with the simple model for tSW = 76 fs represented by the dashed line. The energies increase while the charge separation caused by standing EM wave is present.

density peaks from the standing wave are present, reaching keV energies. The average ion energy decreases slightly as the standing wave dissipates. The transverse motion of the ions is considered in Sec 4.4.5. The conversion eﬃciencies from laser energy to (> 100 eV) ion energy were approximately 0.10%, 0.06%, and 0.04% for the three simulations at 200 fs respectively. We did not include ion-ion collisions in these simulations, which could potentially change the behavior of the ions and potentially lengthen the duration of the peak. In Appendix A.3 we determine that the mean free path of ion-ion collisions for our conditions is larger than the scale of the peak for the higher energy ions in the shelf region.

4.4.3 Peak Electric Fields

Figure 4.8 shows a line out of the longitudinal electric ﬁeld along the laser axis from the deuterium simulation compared to various models for context. As mentioned, the intensity

41 5 4 3 2 1 0 -1 -2 PIC -3 Sine Max -4 E P -5 -21 -20.5 -20

Figure 4.8: The observed longitudinal component of the electric ﬁeld at 70 fs after the beginning of the deuterium simulation near the center of the laser pulse (PIC) averaged over several cells, as compared to the simple sinusoidal density variation model (Sine), maximum depletion (Max), and the expected ponderomotive force (Eq. 4.4) divided by e for reference (Ep). The electric ﬁelds found in the simulation lie between the sinusoidal model and maximum depletion model as expected for this intensity and density.

of this standing wave exceeds Icrit by about a factor of 5, which means that we do not expect the sinusoidal model to be accurate in this case. As seen in Fig. 4.8, the peak sustained longitudinal electric fields along the laser axis in the simulation are close to 2 × 1011 V m−1 which is larger than one would expect in this case from the sinusoidal model (1011 V m−1) by about a factor of 2. This is still somewhat below the peak electric field of the “maximum depletion” model shown in Fig. 4.2 which is near 3.1 × 1011 V m−1. This model is described in Sec. 4.2.1 and it concludes that the peak electric fields are up to a factor of π larger than the sinusoidal model as a limiting case. The results from the simulation lie between these two bounds. At the critical surface, where there are more available electrons, larger fields are present as shown in Fig. 4.8, although there are oscillations in the field. When moving further away from the laser axis, there are oscillations in the longitudinal electric field.

4.4.4 Ion Energies

Equation 4.18 estimates the maximum ion energies from the interaction that we compare to the PIC simulations, however this estimate requires some assumption for how long the

42 Table 4.1: Maximum ion energies reported in keV from the simulation shortly after the standing wave has dissipated. This is compared to the energies predicted with Eq. 4.17. Because the laser pulse has a temporal profile that is sine squared (rather than square), the time-dependent maximum amplitude makes comparison to the model more ambiguous. We compare the simulation result to the model with three different assumptions for the duration longitudinal electric field caused by the charge separation from the standing wave (tsw).

Simulation Model

tsw = 42 fs tsw = 76 fs tsw = 84 fs 1H+ (keV) 4.9 1.9 5.0 5.7 2H+ (keV) 3.0 1.0 2.9 3.4 3H+ (keV) 2.0 0.7 2.0 2.4

standing wave is in place (tSW). This is a difficult number to uniquely establish because the intensity envelope of the laser pulse is sine-squared and there is no abrupt turn on and turn off of the standing wave. From considering the results of Fig. 4.7, using the 42 fs FWHM of the laser pulse as the duration of the standing wave is too short because the ion energies continue to grow even 42 fs after it begins to rise. Using the 84 fs full pulse duration of the laser pulse as the duration of the standing wave is too long, both empirically from Fig. 4.7 and from the reality that the standing waves are created by the overlap of the forward and reflected laser pulse. One could also do a more careful calculation following Refs. [106, 107]. In Tab. 4.1 we therefore consider both of these timescales for the standing wave in our model in order to bracket the possible ion energies. We empirically find that choosing tSW to be 76 fs yields particularly accurate estimates for the max ion energies in all three simulations.

4.4.5 Transverse Structure and Motion

One-dimensional models of ponderomotive steepening and transient photonic crystal growth provide useful insights to explain simulation results and plan experiments, but there are higher dimensional eﬀects indicated by our simulations that must also be considered. The focusing of the laser limits the longitudinal extent and uniformity of density perturbations as shown in Fig. 4.4, which is especially important for high intensity lasers. We do observe density peaks throughout the underdense region with fairly signiﬁcant transverse widths as shown in Fig. 4.413. One-dimensional models do not account for the transverse acceleration of ions, which

13As discussed in Sec. 4.3, lasers come to focus diﬀerently in 2D PIC simulations compared to 3D. Con- sequently, the transverse extent of density peaks from our Fig. 4.4 can be interpreted as upper bounds for the width of the peaks. 43 1

0.5

-0.5

-1 -0.2 0 ±0.2 0 ±0.2 0 0.2

Figure 4.9: Ion velocity distribution at 160 fs for the three simulations for a sample of particles with initial positions from -23 µm ≤ x ≤ 15 µm. The maximum transverse velocities from Eq. 4.17 for tsw = 76 fs are given by the dashed lines.

is non-zero as shown in Fig. 4.9, where we see a similarly shaped distribution for all three simulations. We observe that the maximum transverse velocity is roughly one-fifth of the maximum longitudinal velocity, although comparing these results to models, and determining how this generalizes to different systems is the subject of future work. For the timescale of our simulations, there is minimal transverse movement of the ions compared to the transverse width of the density peaks as mentioned previously. These effects may be important for plasma grating research considering much longer pulses.

4.5 Discussion

A multiply-peaked density modulation is observed in our simulations throughout the underdense shelf region for these initial conditions. The short pulse regime for ponderomotive steepening identified in this theoretical work shows large longitudinal electric fields (potentially up to ≈ 1012 V m−1 for 800 nm light near the critical density) that accelerate ions to tens to hundreds of keV in energy when the above conditions are satisfied. The consequences of these conditions seem to be overlooked in the literature. From a peak ion energy standpoint, this mechanism is not as appealing as conventional laser-based acceler-

44 ation schemes such as Target Normal Sheath Acceleration (TNSA) [51, 115] and Radiation Pressure Acceleration (RPA) [22, 116], but because the energies are still suﬃcient to produce fusion, experiments of this kind may be useful, for example, for producing neutrons with a very small source size.

4.5.1 Observational Considerations

Largely because the spacing between density peaks is close to λ/2, features like these have not yet been observed in optical interferometry. By using intense mid-IR laser systems to produce these modulations this may be possible, so long as one is careful to consider that the peaks are a highly transient effect and that in a freely expanding plasma the density will drop off significantly if the interaction region is extended to tens to hundreds of microns. In our simulations with 800 nm wavelengths, 42 fs FWHM pulse durations, and peak intensities near 1018 W cm−2 the features persist for less than a picosecond. While there are interferometric systems that can operate at this short timescale (e.g. Ref. [103]), experiments at longer wavelengths, lower intensities, and involving ions with lower charge- to-mass ratios can be designed to make the ion acceleration happen over a longer timescale in order to study the evolution of these peaks. The interferometric data would be useful as a novel validation test of kinetic plasma codes, especially if the experiment can be performed at normal incidence. There are papers in the literature that study the presence of periodicity in the density distribution from the overlap of two crossed laser pulses (e.g. Sheng et al. [84], Suntsov et al. [96]) because this produces a kind of transient “plasma grating” that can be detected with probe light. The growth of this plasma grating is similar in many ways to the peaks that form via ponderomotive steepening and we outline a number of parallels in the present paper to recent work by Lehmann and Spatschek [87] who consider overlapping laser pulses through a low density medium. This phenomenon has interesting potential applications as discussed in Refs. [84, 86].

4.5.2 Extensions and Applications

Compared to approaches with counter-propagating laser pulses, there are some advantages to producing these density modulations through the reflection of laser light from an overdense target. Specifically, less total laser energy is required because the reflected laser pulse interferes with itself and there is no need to carefully time the overlap of the pulses since the laser naturally reflects from an overdense surface. The other advantage of overdense targets, as we have explored in this paper, is simply that the density of the shelf or medium the laser travels through can be significantly larger than counter-propagating laser experiments would allow. Larger densities allow for significantly larger longitudinal electric fields for

45 accelerating ions. The density of the medium in experiments with overlapping laser pulses is typically a few orders of magnitude below critical density because of the need to avoid intensity dependent index of refraction effects. Experiments with overdense targets are not as constrained by this because irradiating an overdense target with an appropriate “pre- pulse” produces a few-to-many-micron sub-critical density plasma in front of it. Besides increasing the peak ion energies, the other advantage of producing density modulations in a higher density medium is that the difference between the peak and minimum density will be larger, which should produce more easily detectable fringe shifts in efforts to perform interferometric imaging. We have emphasized the novelty of performing experiments of this kind in the mid-IR (2 µm . λ . 10 µm). Our results also imply that it would be interesting to investigate ponderomotive steepening with shorter wavelengths as well. Shorter wavelength lasers are able to propagate into denser regions and, as previously discussed, denser plasmas produce larger peak electric fields which are advantageous for accelerating ions. This detail is important for the possibility of using experiments of this kind to create a neutron source with a very small source size because, as is well known, neutron yields increase significantly with ion energy [28]. In a suitably designed experiment, one could try to produce neutrons from the collision of counter-streaming ions in the density peaks. However, as considered in Appendix A.3, the mean free path for these collisions is large compared to λ/2. Neutron-producing fusion reactions are more likely to come from ions that stream towards the first peak near the target and continue into the overdense region. This would be a “pitcher-catcher” type configuration where the pitcher and catcher are separated by only ≈ λ/2. We note that the expected yield is likely less than conventional schemes as shown in Appendix A.4 but could be detected and is still interesting from a fundamental perspective.

Model Limitations

We note that our model is limited as it neglects motion of ions that are initially further than λ/8 from an electron peak, which require a longer pulse for maximum energy. We also approximate the field as linear. Alternatively, one could find the maximum energy from the work done by the electric field, this is included in Appendix A.2 which predicts slightly larger velocities. When considering more realistic laser pulses, there are additional considerations. For example, our model ignores the transverse pulse shape (e.g. see Ref. [107]) and does not account for the temporal pulse shape of the incoming beam. In practice our model can be extended by replacing the timescale with an effective timescale (as in this work), or similarly the ponderomotive force could be averaged over the pulse to get an effective intensity to be used in our model. A more accurate model could expand of the work by Gorbunov and Frolov [106, 107] and Lehmann and Spatschek [87] to better capture the dynamics of the

46 electron density distribution. Our 1D model assumes that the spot size is much larger than λ/2 and also is the most applicable near the reflection point of the laser and along the laser axis where we have the full effect of the standing wave. Hence, our model should only serve as a rough estimate (or upper bound) for potential ion velocities, particularly for experiments, which have additional considerations. A crucial assumption of this work is that the plasma remains highly reflective. This is certainly true of our simulations, but it is well known that the intensity and wavelength of the laser are important factors for the reflectivity. To make more reliable extrapolations to shorter and longer wavelengths and smaller and larger intensities than we consider in the simulations we present here, one would need to carefully consider the scaling of the reflectivity with various parameters (e.g. Levy et al. [110]). While it is outside the scope of the present work, this remains an important priority for future investigations.

4.6 Conclusions

The formation of multiply-peaked density modulations associated with ponderomotive steepening is of fundamental interest as a basic plasma process and of practical interest as a means to modify the density profile of a target and to accelerate ions. Our PIC simulations indicate that these peaks are especially transient, lasting less than a picosecond after the end of a short-pulse laser interaction. This is important to factor into the design of future experiments to detect this phenomenon. We also find that the large longitudinal electric fields that are produced in these laser interactions accelerate ions to few keV energies in short pulse laser interactions, and potentially up to hundreds of keV energies in longer duration interactions. In our simulations these fields reach 2 × 1011 V m−1. We outline a simple model to estimate the timescale of ion motion and peak energies of ions in these interactions. This model matches the peak ion energies in our simulations reasonably well. We also comment on extensions to this model that provide some insight even when the laser intensity exceeds a critical value. The model indicates that higher field strengths are achieved with shorter wavelength interactions due to the increased critical density. Ion acceleration should be much less pronounced in longer wavelength interactions, but this may still be an interesting regime to perform interferometric imaging as a novel validation test of plasma codes if the experiments are performed at normal incidence. Multiply-peaked ponderomotive steepening has many parallels to studies of counter- propagating laser pulses which is a phenomenon with interesting potential applications for the field [84, 96]. A key difference is that interference from reflection occurs at a com- paratively higher density. As a result, the longitudinal electric field strengths are much larger, as just mentioned, and there are important subtleties to analytically modeling this phenomenon and challenges in experimentally probing it that we have outlined.

47 Chapter 5 Optimizing Laser-Plasma Interactions for Ion Acceleration using Particle-in-Cell Simulations and Evolutionary Algorithms

5.1 Introduction

Ultra-intense laser-based sources of energetic ions hold great potential to compactify and make more widely available the technology needed to accelerate ions to many MeV energies and higher [22, 51, 52, 117]. Recently, up to 2 MeV proton acceleration was demonstrated with a kHz repetition rate laser system at the Air Force Research Lab (Morrison et al. [12]). In a subsequent perspectives article “Paving the way for a revolution in high-repetition- rate laser-driven ion acceleration,” Palmer [118] comments on Morrison et al. and prior studies [119–127], arguing that experiments have now reached the point where these high- repetition-rate laser systems can be explored “to provide compact accelerators for research and industry.” With the rapid advancement of laser technology, the capability to accelerate signiﬁcant numbers of many MeV ions from a compact source is becoming feasible for a variety of applications, including proton imaging [24], hadron therapy for cancer treatment [128, 129] and materials science. An important next step in the translation from proof-of-concept experiments to these applications is to achieve more control over the properties of the laser-accelerated ion beam. Due to the complexity of ultra-intense laser interactions, rather than explore the large simulation parameter space essentially by hand or some other means, instead we use an evolutionary algorithm with a series of thousands of one-dimensional (1D) particle-in-cell (PIC) simulations to optimize the laser-plasma interaction. The wider ﬁeld of plasma physics

48 is beginning to embrace statistical methods for various problems such as inertial confinement fusion [130–133], magnetic fusion [134, 135], x-ray production [136], laser-wakefield acceleration [137, 138], and to optimize the laser focus for electron or ion acceleration experiments [102, 139]. To our knowledge, the present study is the first to directly optimize laser-based ion acceleration with such an approach. Increasing the peak energy of the ions, while important, is only one of the properties of the ion beam that needs to be improved. In this paper we consider what can be done to increase the conversion efficiency between short-pulse laser energy to energetic ions (& 3 MeV). In particular we explore ion acceleration using different target density profiles, while keeping the laser parameters fixed, in thousands of 1D PIC simulations as described in Sec. 5.2. Then, in Sec. 5.3, we describe results from a few three-dimensional (3D) PIC simulations. These simulations allow us to more realistically examine whether the optimum target density profile found from 1D simulations in Sec. 5.2 will indeed enhance ion acceleration compared to more conventional targets. Our results point to a novel type of target for enhancing ion acceleration, showing the potential of this method as discussed in Sec. 5.4.

5.2 1D PIC Optimization Driven by Evolutionary Algorithms

To perform this optimization, we use an evolutionary algorithm known as differential evolution as discussed in Sec. 3.2. This approach allows us to explore a large parameter space in a highly parallelizable way, but there are several limitations that must be kept in mind. First, evolutionary algorithms are not guaranteed to find the global maximum and simulation choices further restrict the search space. Second, in order to quickly perform simulations we use 1D(3V) PIC simulations (one spatial dimension and three particle velocity dimensions) that are known to not be as realistic as 2D or 3D PIC simulations (e.g. see [69, 70] for differences between 2D and 3D simulations). Notably, 1D(3V) PIC simulations do not capture the focusing of the laser or the drop off of the electric field. To address this, we later present results from a 3D simulation that uses the optimal target from the 1D simulations. Despite these limitations, the prior success of evolutionary algorithms in related fields [102, 131, 134, 136–139] demonstrates the power of this approach, and in this work, we find it to be advantageous for optimizing ion acceleration. For this work, we use a population size of 120 and let the algorithm evolve for 50 generations14, resulting in a total of 6,000 simulations. Each simulation ran on a single core of a 2.4 GHz (Intel Xeon 6148) 20 core processor for approximately 30 minutes, resulting in a total execution time of about 2 days for all 50 generations. By starting with 1D simulations,

14The population size was chosen for computational and methodological considerations as discussed in 3.2, and the number of generations was based on convergence results (Sec. 5.2.1).

49 we explore orders of magnitude more target conﬁgurations than would be possible with two- dimensions with the same spatial and temporal resolution.

5.2.1 Simulation Parameters

There are now more than one-hundred ultra-intense laser facilities in the world [42]. Rather than simulate some futuristic laser system, we chose to model a laser similar to the kHz repetition-rate laser described in Ref. [12] with an 800 nm wavelength, 1.2 × 1019 W cm−2 peak intensity, and 42 fs full width at half maximum (FWHM) pulse duration. Electrons oscillating in these laser fields will experience significant relativistic effects and this intensity is sufficient to accelerate ions using the Target Normal Sheath Acceleration (TNSA) mechanism [51–54] and potentially other acceleration processes. Figure 5.1 on the next page illustrates the blueprint of the 1D(3V) implicit PIC simulations run with the LSP PIC code [59]. The laser enters the 40 µm wide simulation box at x = 0 and propagates towards a 5 µm thick target density profile that is generated by the evolutionary algorithm for each simulation. The target is composed of fully ionized hydrogen (protons and electrons) for simplicity and computational speed. As shown in Fig. 5.1, the density profile has ten independent 0.5 µm thick density bins. Although there is important work being done to create new kinds of targets for high-repetition-rate laser systems [11, 12, 140, 141], we did not limit the search based on the current practicalities of what kinds of density profiles can be made in the lab15. These bins are initialized randomly, by sampling from a uniform distribution, with a density up to 5 × 1021 cm−3. For comparison, 2 2 2 2 21 the classical critical density for laser propagation is ncrit = 4π ε0mec /λ e , or 1.7 × 10 cm−3 for an 800 nm laser. For high intensity lasers, relativistic effects increase this density

to γncrit where γ is the Lorentz factor for the electrons, but for the intensity we consider here the relativistic critical density is still generally below 5 × 1021 cm−3. To optimize the conversion efficiency from laser energy to ion energy, the fitness function was the total energy of ions that leave the right edge of the simulation boundary (Fig. 5.1). This choice ignores the backwards (i.e. leftward) going ions. Also, due to the finite simulation time, ions with less than ∼ 3 MeV may not reach the right edge of the simulation box. As the algorithm evolves the targets, we do not limit the maximum density to 5 × 1021 cm−3 and instead allow the densities to grow above this value with no upper bound. If the evolutionary algorithm selects a negative density value for one of the density bins, that density is set to zero. While it is generally advisable to allow the initial random parameter selection to span the entire search space, we found that this skewed the initial population to significantly overdense targets that were not as conducive to ion acceleration. The simulations have a spatial resolution of 10 nm (λ/80) with 64 particles per cell for each species. The macroparticles are initialized with thermal temperatures of 1 eV and the

15Various plasma shaping techniques will also facilitate new high-repetition-rate targets (e.g. [1, 87, 96]). 50 6 8 0 μ μ 0

7 5

0 0 9 z 3 2 4 0 0 0 0 0 0 0 0 0 μ

Figure 5.1: Template for each of the 1D PIC simulations run by the evolutionary algorithm. The intense 1.2×1019 W cm−2 laser enters the simulation from the left side of the simulation box (x = 0 µm) and interacts with a 5 µm thick ionized hydrogen target composed of ten 0.5 µm thick density bins chosen by the evolutionary algorithm. Ions and electrons are measured as they leave the right side of the simulation box (x = 40 µm). The total energy of these ions is maximized with the evolutionary algorithm.

simulation time is 1,000 fs with a time step of 0.05 fs. The spatial scale does not resolve the Debye length for all possible target configurations, however the implicit field solver and energy conserving algorithms of LSP limit artificial grid heating. Collisions are allowed in the code, although turning off collisions does not make a significant difference when tested with the optimal conditions in 2D simulations.

5.2.2 1D Results

Figure 5.2 on the following page(a) shows the population after 50 generations, which con- verges to a general density profile. Density profiles from earlier generations are presented in B.1. For the optimal density profile shown in Fig. 5.2(a) (drawn in red), there is a classically overdense foot at the front of the target for the first two density bins. This is followed by classically underdense bins in the center of the target and an overdense spike in the last density bin. The dark lines in Fig. 5.2 show that all of the other members of this final generation follow a similar trend with reduced density in the center of the target and an overdense spike one of the last two density bins. Figure 5.2 on the next page(b) shows how the conversion efficiencies of the simulations improve with each generation. In the initial generation, the best performing target has close to 10% conversion efficiency. The following generations improve upon this result, eventually reaching nearly 25% in the 50th generation, with most of this improvement coming from the first ten or so generations. The last ten generations only improve the conversion efficiency by

51 (b)

%) (c) 0.20 1

0.15 1 1 ncrit 1 0.10 Best Average 0.05 1 Worst (normalized) Charge

1 0.00 1 0 10 20 Energy (MeV)

Figure 5.2: 1D PIC simulation optimization results. In (a), all members of the population are plotted, where darker shades represent higher fitness. The best performing density profile is drawn in red. After 50 generations, most members of the population have a similar pattern with a roughly critical density foot at the front of the target (the first two density bins that exceed ncrit), underdense center, and overdense density spike in one of the last two density bins as shown. In (b), the conversion efficiency of measured ions initially increases quickly and then begins to level off with later generations. In (c), the distribution of measured forward going ions for the best performing profile is plotted. This ‘optimal’ density profile is tested with 3D simulations in Sec. 5.3.

a small amount, which is part of our rationale for ending the generations at 50. Figure 5.2(c) shows for the optimal target (generation 50) the distribution of energies for ions ejected from the target. The highest energy ions exceed 20 MeV, which is quite high for these laser parameters, but the approximations made by 1D simulations typically overpredict the maximum energy so we will hold our comparison to typical targets until the higher dimensional simulation results are presented.

5.3 Three-Dimensional Simulations

To better understand this new type of target identiﬁed with the 1D PIC simulations, we performed a series of 2D(3V) and 3D PIC simulations. For brevity, we focus on the results of the 3D simulations for a 4 × 1019 W cm−2 laser with the longitudinal density proﬁle of the best target from the evolutionary algorithm, represented with a 20 µm wide target. We compared these simulation results to the more conventional targets of a thin 0.5 µm sheet (with the same density as the last density bin of the evolutionary algorithm target) and a target with a 1.5 µm exponential scale length in front of the sheet. A 2D slice of all three targets is shown in Fig. 5.3.

52 5.3.1 Simulation Parameters

The simulation parameters for the 3D simulations closely match those of the 1D simulation in Sec. 5.2.1 to test whether the same behavior persists in higher dimensional simulations. The 3D simulations were conducted with 4 × 1019 W cm−2 lasers so more interesting ion energies could be explored, while staying on the frontier of capabilities of current kHz laser systems. Earlier in Sec. 5.2.1 we did not specify a spot size or the position of peak focus for the laser pulse because focusing is not accounted for in 1D PIC simulations. For the 3D simulations, we assume a Gaussian spot size of 1.5 µm (FWHM) and we set the peak focus at the front of the target (x = 17.5 µm) which allows most of the laser pulse to propagate through the classically overdense section of the target there via relativistic transparency. For the exponential scale-length target, the focus was set near the critical density at x = 19.3 µm and for the sheet it was set at x = 22.5 µm. The spatial resolution of these simulations is 50 nm (λ/16) in the laser propagation (x) direction and 100 nm (λ/8) in the transverse directions and 125 particles per cell were used for each species. This is a much lower resolution than the 1D simulations, although 2D tests indicated that these conditions are suﬃcient to model the process. Despite this lower resolution, one 3D simulation required over 30 times more computational resources than all 6,000 1D simulations.

5.3.2 Results

Figure 5.3 on the next page shows snapshots of the ion and electron densities at several points in time throughout the three simulations. We observe that the ions are able to travel noticeably farther away from the back of the target in the simulation with the evolutionary algorithm target, suggesting higher maximum energies than the conventional targets, as explored shortly. For the evolutionary algorithm and sheet target, the laser does not reach the critical density until it reﬂects at the last density bin near the back of the target allowing the laser to interact with an eﬀectively thinner target. The sheet target has many fewer electrons expanding from the target than the other two simulations as shown Figure 5.3 on the following page. The total electron energy during the simulation rises to a maximum of 44% of the total incident laser energy for the evolutionary algorithm target, 56% for the exponential target and only 12% for the sheet target. Figure 5.4 on page 55(a) shows the maximum ion energy versus time for each simulation. Around 150 fs, the evolutionary algorithm target begins to outperform the other targets in terms of maximum ion energy and for later times exhibits sustained growth similar to the exponential target for the rest of the simulation. The targets have maximum ion energies of about 4.8 MeV for the sheet, 5.3 MeV for the exponential target, and 6.4 MeV for the evolutionary algorithm target. We compare the spectra of forward going ions at

53 Ion Density (cm 3) Electron Density (cm 3)

1016 1017 1018 1019 1020 1021 1022 1016 1017 1018 1019 1020 1021 1022 0 EA Exp Sheet 10

m) 20 x ( 30 060 fs 060 fs 060 fs 0

m) 20 x ( 30 150 fs 150 fs 150 fs 0

m) 20 x ( 30 210 fs 210 fs 210 fs 0

m) 20 x ( 30 500 fs 500 fs 500 fs 40 20 10 0 10 20 10 0 10 20 10 0 10 20 z ( m) z ( m) z ( m) Figure 5.3: Snapshots of the ion (z < 0) and electron (z > 0) densities for the three 3D simulations (xz plane). The optimal ‘evolutionary algorithm (EA)’ target from the 1D simulations is on the left, the exponential target is in the center, and the thin sheet is on the right. Ions travel farthest for the new EA target, which shows enhanced coupling between the laser and electrons like the exponential target. A contour is drawn at an intensity of 2 Imax/e ; variations come from diﬀerences in the focal spot location.

54 10 9 5 (a) (b) (c) 6 EA 10 10 Exp 4 5 Sheet 10 11 4 3 12 >2 3 10 MeV 2 13 2 EA Charge (C) 10 Exp 1 10 14 1 Sheet Maximum Ion Energy (MeV) 0 10 15 Ion Conversion Efficiency (%) 0 200 400 0 2 4 6 EA Exp Sheet Time (fs) Energy (MeV)

Figure 5.4: The maximum ion energy versus time for all three simulations (a), where the EA target reached the highest energy followed by the Exp and Sheet targets. Spectra of ions with forward going momenta in a 20◦ half angle cone sketched in Fig. 5.5, for the three different 3D simulations at 500 fs (b), where the charge represents the total charge of electrons in a 0.1 MeV energy bin. The total ion conversion efficiency in this cone is included in (c) with the hatched bars showing the difference between the targets if only ions with energies greater than 2 MeV are considered. The overall conversion efficiency is similar, but there is a noticeable enhancement to the population above ∼ 2 MeV for the EA target compared to the other two targets.

55 Backwards Backwards Backwards 10 8 6 MeV 6 MeV 6 MeV 9 4 MeV 4 MeV 4 MeV 10

2 MeV 2 MeV 2 MeV Target Left Target Left Target Left 10 10

10 11

10 12 Charge (C) Target Right Target Right Target Right

10 13

14 Forwards Forwards Forwards 10 (a) Evolutionary Algorithm (b) Exponential (c) Sheet

Figure 5.5: Polar histograms showing the distribution of ion energies for the three 3D simulations. Energy bins have a radial size 0.5 MeV and angular size of 5◦ taken in the xz plane and a 20◦ half-angle cone is sketched for reference. For the evolutionary algorithm target (a), we see a strong forward (laser propagation direction) going component of the ion distribution and enhanced conversion to & 2 MeV ions compared to the other two targets.

the end of each simulation in Fig. 5.4 for the 20◦ half-angle cone sketched in Fig. 5.5, which shows that higher ion energies are obtained with the new target geometry. From Fig. 5.4(b), we see that the exponential target has a slightly larger population of ions at lower energies, but drops off more quickly for higher ion energies. The higher energy ions are of interest for many applications [142–145]. As illustrated in Fig. 5.4(c), the exponential and evolutionary algorithm targets had similar overall conversion efficiency in this cone of about 4.8%, but there were significant differences when considering higher energy ions. For example, conversion efficiency to > 2 MeV ions is 0.47% for the evolutionary algorithm target, 0.33% for the exponential target and 0.12% for the sheet target. As shown in Fig. 5.5, the highest energy ions are traveling in the forward (laser propagation) direction and are primarily contained within a 20◦ half-angle cone. While not the focus of this work, Fig. 5.5 also shows more significant back directed ions from the evolutionary algorithm target than the other cases. One can also see that the exponential target has more significant semi-isotropic ion acceleration (i.e. at large angles) than the other targets, which is not surprising due to the higher laser-electron coupling.

5.4 Discussion

5.4.1 The Optimal Target

In this subsection we comment on the distinct features of the optimal 1D target (Fig. 5.2(a) shown in red) that make it an interesting new candidate for ion acceleration. There are

56 three basic elements of the optimal target: classically overdense foot, near-critical density cavity, and an overdense spike. The classically overdense foot at the start of the target becomes relativistically transparent, allowing a majority of the laser pulse to pass. This first phase of the interaction is reminiscent of studies where ion acceleration is enhanced by relativistic transparency (e.g. [22, 54, 55]). Next, the laser propagates through the near-critical density cavity, transferring significant energy to electrons. However, unlike the targets in the aforementioned studies, the laser reaches an overdense spike where it makes a strong reflection because the density of the spike significantly exceeds the relativistic critical density. The outgoing pulse continues to transfer significant energy to electrons as it passes through the near-critical density cavity a second time. Then the pulse reaches the foot of the target and escapes. In the 1D simulations, some of the laser pulse appears to become trapped in the cavity, although this effect is not significant in the 3D simulations, which is likely due to 3D considerations such as the focusing of the laser pulse. Through the use of thousands of 1D simulations and an optimization method, we identified a new type of target, not yet explored with experiments, that employs commonly studied laser-plasma effects. Because of the strong reflection, the “optimal” target is com- parable to efforts that use reflection to better confine the laser energy and enhance coupling. So-called plasma half cavity targets use a hemispheric reflecting surface to direct laser light back to the interaction region [146]. So-called “escargot” targets use reflections to direct the laser light into a kind of spiral [147]. Both of these approaches involve much larger scales than the few-micron thick targets we consider here. On smaller scales, two studies that consider electron heating in high-reflectivity laser interactions are [148] and [111]. Although there is both constructive and destructive interference where the laser pulse overlaps, the constructive interference can enhance the population of hot electrons, which is well known to play an important role in TNSA. Recent simulation work with nanostructured double- layer targets, such as a random forest of nanowires on a thin target, shows enhanced ion acceleration due to increased laser-to electron coupling like our work [149].

5.4.2 Optimization of Laser-Plasma Interactions

We searched a 10-dimensional parameter space, considering 5 µm thick targets with rather course 0.5 µm thick density bins and using a single laser intensity and pulse duration. Even with this limited search space, we identified a new type of target that seems to match or outperform conventional targets in terms of maximum ion energy and conversion efficiency to higher energy ions. There is still a vast parameter space of laser-plasma interactions to be explored with this technique and others. We have also generated a data set of 6,000 simulations that is being examined to find additional trends16. This proof of concept illus-

16The parameter search is inherently biased by the evolutionary algorithm, but still reveals some interesting features.

57 trates the potential beneﬁt of using many 1D simulations to discover new target geometries for laser-plasma interactions. This method is not limited to ion acceleration may be useful for tasks such as optimizing pulse shapes for inertial conﬁnement fusion [150].

5.5 Conclusions

With the small computational cost of one-dimensional simulations and an optimization routine utilizing evolutionary algorithms to run thousands of 1D simulations, we identified a new type of target for enhancing ion acceleration. This new target was then examined with 3D simulations and showed enhancement compared to conventional targets. One limitation of this approach is that 1D PIC simulations are much less realistic than 2D or 3D simulations. Future efforts using large numbers of 1D PIC simulations to guide efforts to optimize laser-plasma interactions may not be as generally successful as was demonstrated in our study. We noticed, for example, that 1D simulations saw the “trapping” of the laser pulse due to a second reflection inside the cavity inside the target whereas this phenomenon was not seen in 3D simulations. There are also targets with complicated geometries that are not amenable to running 1D simulations (e.g. [151]). This work highlights the potential for using evolutionary algorithms and other statistical methods to study laser-plasma interactions in both experiment and simulation. There are many outstanding challenges in this field that may benefit from this approach such as increasing the maximum ion energy with current laser systems and efforts to produce monoenergetic ion beams.

58 Chapter 6 Computational Fluid Dynamic Simulations of Liquid Target Formation

Ultra-intense high-repetition-rate laser experiments are an important step in the path towards making laser-based ion sources viable for many applications. Ion acceleration experiments typically use a target such as a thin sheet of foil, that is destroyed with a single experiment. Some efforts allow continuous operation with solid density targets such as a tape drive [127, 139] or rotating target [152, 153], but factors such as debris accumulation and limited surface area restrict the operation time [141]. Very thin (tens of nm) liquid crystal film targets can also be formed at a moderate repetition rate (∼ Hz), solving many of these limitations [140, 154]. For high-repetition-rate (& kHz) experiments, free-flowing liquid targets provide a continually refreshed target.17 A relatively simple liquid target is a tens-of-micron diameter liquid jet flowing at tens of meters per second. The targets may be liquid metal [120, 156], water [69, 101, 111], or another liquid such as ethylene glycol [141]. For ion acceleration, thinner sub-micron scale targets are more advantageous. An emerging high-repetition-rate target type is sub-micron thick liquid sheets created by colliding liquid jets [12, 141], or using a specialized gas- dynamic nozzle [11, 157, 158]. We investigate the formation of these sheets computationally using methods described in chapter3. Reduced-mass micron-scale liquid drops are also used for experiments [121, 141, 159, 160] and can be created with a piezoelectric nozzle synchronized with a laser. In Sec. 6.3 we investigate new candidates for targets, such as thin pancake-like targets created by colliding droplets [141]. The target types investigated in this chapter with CFD simulations are not new, but they do provide new insights, and help introduce new computational tools for studying laser-plasma interactions. This type of simulation provides the freedom to test different liquids and flow parameters, that may require custom ordered micron-scale components to

17Liquid and liquid-crystal targets can also serve as disposable optics such as plasma mirrors [141, 154, 155]. 59 evaluate in the lab. There are also measurements that are much easier to extract from the simulation than to take experimentally18, and the simulations can help guide whether experimental investigation of a particular configuration is worth pursuing. We can also use these simulations to explore new classes of targets that would require extensive experimental reconfiguration. Our work represents an initial effort into using these tools for high-repetition-rate targets and we discuss how the simulations can be refined in the future with additional comparisons to theory and experiment.

6.1 Dimensionless Numbers in Fluid Mechanics

In fluid dynamics, there are a number of dimensionless numbers that provide useful information about the system, similar to how a0 (Sec. 2.1) informs the regime of the laser-plasma interaction. One useful parameter for flowing jets is the Reynolds number (Re), which compares the strength of inertial forces to viscous forces, or ρvD vD inertial forces Re = = ∼ (6.1) µ ν viscous forces where ρ is the density of the fluid, v is the velocity, D is the diameter of the jets (or more generally the length scale of the problem), µ is the dynamic viscosity, and ν = µ/ρ is the kinematic viscosity. This number helps determine whether the fluid exhibits laminar or turbulent flow with lower numbers corresponding to laminar flow and higher numbers turbulent flow. For example, flow in a pipe exhibits laminar flow for Re . 2,000 [161]. Similarly, the relation between inertia and surface tension gives rise to another quantity known as the Weber number We, or

ρv2D inertia We = ∼ , (6.2) σ surface tension where σ is the surface tension, D is the characteristic length (e.g. the diameter of colliding droplets), and v is the relative velocity of the colliding ﬂows. This is useful in determining how colliding ﬂuids interact and, in our case, how they spread out.

6.2 Liquid Sheet Formation

Thin liquid sheets have shown to be reliable targets for high-repetition-rate experiments [12]. Liquid sheets formed by colliding two laminar-flowing jets of water have a long history dating back to Sir Geoffrey Taylor [162] in 1961 and before (e.g. Dombrowski and Fraser [163]). The type of sheet created by two impinging jets depends on their fluid parameters and collision properties such as the incident flow velocity and collision angle. Different sheet types are

18Also, we can gain conﬁdence in simulation results by matching easier to measure observables such as the curvature of a sheet.

60 Figure 6.1: Simulation results demonstrating possible types of liquid sheets for increasing jet velocity. The stable thin section in the middle of the target is advantageous for ion acceleration. From Ref. [8] under a Creative Commons (CC BY-NC-ND 4.0) license (https: //creativecommons.org/licenses/by-nc-nd/4.0/).

shown in Fig. 6.1, ranging from low to high impact velocities. As illustrated in Fig. 6.2, two jets collide at an angle of 2φ between them and form a thin sheet perpendicular to the plane of the collision. For the ‘liquid chain’ and ‘closed rim’ shapes Fig. 6.1 we observe subsequent smaller sheets can form at the ‘tip point’ where the rim closes. When the velocity is increased, the Weber number also increases and the inertia begins to dominate, creating a larger sheet before the streams recombine. If the velocity is increased again, the area of the sheet does increase, but then an open rim forms (Fig. 6.1(c)) and other less stable configurations are possible (Fig. 6.1(d-e)). A model developed by Hasson and Peck [164] predicts that the thickness of a sheet formed by colliding jets should drop as 1/r (for a fixed value of θ), where r is the radial distance from the collision of the jets. This model shows that the sheet thickness is reduced towards the bottom of the sheet and towards the sheet edges. There are more complex models that predict the shape of the sheet and account for the flux of fluid between the sheet and rim [10]. We are interested in thinner targets because in some circumstances they can be used to produce higher energy ions (e.g. [55]).

6.2.1 Simulation

The simulation box extent is 300 µm in the sheet width (x) direction, 600 µm in the ﬂow direction (y), and 60 µm in the thickness direction (z) (see Fig. 6.3). The initial cells were cubes with a side length of 1.0 µm, and a maximum reﬁnement level of 2 for a

61 Figure 6.2: On the left is a schematic of two colliding jets that form a closed rim shape including the formation of a secondary sheet (from Ref. [9]). The two jets collide at an angle of 2φ and the coordinates r and θ are used to describe the location on a thin sheet. Then on the right is an experimental image of a sheet from Ref. [10].

minimum cell size of 0.25 µm. Refinement was performed every time 20 time steps with refinement based on the phase fraction α with refinement occurring between 0.001 and

0.999. The two fluids in the simulation are water and air, with fluid properties νwater = −6 2 −1 −3 −5 2 −1 −3 10 m s , ρwater =1,000 kg m , νair = 1.48 × 10 m s , ρair = 1 kg m and the surface tension between the fluids is σ = 0.07 N m−1. Gravity was included in the simulation (in the +y) direction. The water jets have a diameter of 25 µm and flow velocity of 10.4 m s−1. They are introduced to the simulation as a boundary condition on the xy plane with an elliptical cross section in the xy plane19 (see Fig. 6.3) using the swak4Foam library. The angle of impact φ (see Fig. 4.1) is 45◦. A uniform velocity profile was utilized for simplicity, a parabolic profile could also be employed (e.g. Ref. [9]). These jets have a Reynolds number of 260, well within the laminar flow region. The Weber number of the interaction is 77

19The cross section is circular in the plane perpendicular to its ﬂow.

62 (using the relative jet velocity and circular diameter for the length scale), where inertia is more dominant than surface tension, but not too drastically.

Simulation Considerations

One of the challenges in developing these simulations is the large width and height of the sheets compared to the sheet thickness, which must be sufficiently resolved. Adaptive resolution reduces the number of cells in the simulation, but the time step depends on the worst-case Courant-number, greatly increasing the computation time required with increased refinement. One could utilize symmetry in the problem coupled with appropriate boundary conditions, although we did not take advantage of these as we are focused on systems with less convenient symmetries moving forward (e.g. [12, 141]). For a sense of scale, this simulation ran on ∼ 2,000 processors for more than twelve hours with ∼ 2.5×107 total cells at the end of the simulation. Simulations may be useful for initially optimizing and measuring sheet qualities such as thickness, which can be experimentally difficult to measure due to high spatial resolution required and the sub-micron thicknesses that must be resolved.

6.2.2 Sheet Simulation Results

The evolution of the simulation over 88 µs is presented in Fig. 6.3, where we see the formation of a ‘closed rim’ shape and we see that a second sheet begins to form, although the fluid reaches the side and bottom boundaries of the simulation box before the end of the simulation. A sheet thickness map is extracted from the simulation as shown in Fig. 6.4 (created by summing the phase fraction α through the target). This is then compared to thickness maps extracted from laser experiments, which although they have different generation procedures (liquid gas nozzle and obliquely incident jets, and different initial conditions), show similar behavior. In particular, the sheet is thinner near the bottom, and we observe thinner regions near the rims. Although we expect the thickness of the sheet to be smaller the further away the fluid is from the collision point [164], the sheet near the rim seems to be particularly thin. This interesting result deserves a closer examination in future work. There may be numerical issues that affect the accuracy of the solution. We noted previously that the sheet interacted with the boundary of the simulation box; the boundary conditions allow the fluid to flow out of the simulation, although there may be unintended behavior. Also with the reduced thickness near the edges of the sheet, additional convergence testing could be completed to better understand this behavior. We also see some fluctuations in the sheet in later times that may not be physical. However to a great extent the simulated sheet qualitatively matches experimental expectations (Figures 6.2 and 6.4),

63 Figure 6.3: Isovolume plots showing the formation of a thin sheet for the OpenFOAM simulation.

so we regard the simulations as a successful proof of concept for using this type of method to model laser targets.

64 Figure 6.4: Thickness map for the thin sheet simulation at 88 µs (left). For reference a thickness map from Koralek et al. [11] (no absolute color scale for thickness provided) is included in the center, and Morrison et al. [12] on the right. In all three cases we see decreasing thickness moving farther away from the top of the jet.

6.3 Droplet Collisions

Another goal of this simulation work is to develop the capability to test new types of targets not yet explored in the lab. As mentioned previously, liquid drops can be synchronized with the laser [121, 141, 159, 160]. These types of targets are interesting because they have a reduced mass, which may be useful for study warm dense matter conditions and may enhance certain laser-plasma interactions. In this section we look at the collision of tens-of-micron-diameter drops that can form thin sheets and other useful geometries.

6.3.1 Simulation Parameters

The general grid setup (before reﬁnement) for the simulation is shown in Fig. 6.5, where a cylinder mesh is utilized based on Ref. [165]. Starting 7 µm away from the center, a cylindrical geometry is used where cells are 0.5 µm thick in the radial direction and 9◦ in the angular coordinate. The resolution in the axial direction is 0.4 µm. An alternate mesh

65 Figure 6.5: Initial simulation grid for the droplet collision simulations. A 160 µm thick cylindrical shape is used with a quasi-cylindrical geometry as annotated.

is used in the center to avoid prohibitively small cells and to accommodate the hexahedral- shaped cells of the simulation. In the central region there is a circular shape with a radius of 7 µm with 10 × 10 cells that is constructed to align with the cylindrical outer grid. The same refinement parameters from the sheet simulation in Sec. 6.2.1 were used, although refinement was performed on every time step. This results in a minimum thickness of 0.1 µm in the axial direction. Gravity was not included in the simulations to retain symmetry, and because the drops are isolated and their velocity would not change significantly during the timescale of the simulation. The two fluids in the simulation are air and an ethylene glycol-like fluid20 (ν = 1.78 × 10−6 m2 s−1 (see footnote) ρ = 1, 097 kg m−3) with σ = 0.0473 N m−1. The 50 µm diameter spherical drops are initialized completely inside of the simulation grids (one micron apart) with an initial velocity of 3.536 m s−1. This gives a Weber number of 58, which is the same as for the collision shown in the second figure of Pan et al. [13], which we will use for comparison.

20The density and surface tension are similar, although due to a transcription error, the kinematic viscosity is about an order of magnitude too small. Despite this error, information can be learned from the droplet collision process as our main comparison will only involve the Weber number, which is not altered.

66 Figure 6.6: Evolution of the droplet collision simulation compared to experimental images of similar conditions from Ref. [13]

6.3.2 Drop Collision Results

As illustrated in Fig. 6.6, the global behavior of the simulation closely matches an experiment with the same Weber number (with a larger spatial scale). As the drops collide, they form a ring that expands perpendicular their initial velocity. This ring begins to expand as the drops continue to collide. As the ring expands outwards, a toroidal-type shape is formed with a thin sheet in the center. In later times, surface tension draws the ring inward and the shape begins to elongate in the axial direction. From these experimental images, it is difficult to determine the behavior near the center, but from mass conservation, we can infer that there is limited material in the center ring. In our simulation, a thin sub-micron sheet is formed in the center for most of the collision as shown in Fig. 6.7, where the outer ring is being drawn back towards the center. This type of thin sheet would be useful for ion acceleration experiments, and with increased Weber numbers, larger, potentially thinner, sheets could be obtained [13]. Alternately, this is a candidate for creating high-repetition-rate ‘structured targets’ (e.g. [151]) for enhancing ion acceleration. We do note that later in the simulation, this thin sheet does break up shortly before recombining, although that may be in part due to resolution limitations and potentially the presence of some air bubbles trapped between the droplets that may alter the sheet behavior. These factors can be removed or evaluated with different initial conditions or potentially a different fluid solver. Nonetheless, this is a promising proof of concept simulation for this type of target that can be refined with different fluid and collision parameters.

67 Figure 6.7: Isovolume plot from the droplet collision at 30 µs (left) and a cross section illustrating the thickness of the drop (right). The small vertical line on the right image represents a thickness of 5 µm.

6.4 Discussion and Conclusions

Liquid targets are an emerging technique for supporting & kHz repetition-rate laser-plasma interactions. There are many opportunities to refine current targets to obtain thinner sheets to access different ion acceleration mechanisms, and to explore new target geometries that may be beneficial for other physical effects. The sheets between the rims exhibited reduced thickness further away from the interaction point and near the rims, which could be investigated with further simulations and experiments. Droplet collisions at other Weber numbers can provide additional candidates for thin reduced-mass targets [13], which can be used for ion acceleration. Also, there are existing geometries such as water bells [166] and more exotic shapes [141] that can be explored in the future. Simulations allow us to more easily extract a variety of measurements from these targets and can be used synergistically with experimental work to explore new target regimes. These simulations can be fairly computationally expensive, although once developed can require relatively little human input compared to testing in the lab. OpenFOAM is a powerful open source tool for modeling these targets that scales very well to thousands of processors in our testing. Additional work in the future could compare these simulation results to experiment or other simulation codes or methods (e.g. a smoothed-particle hydrodynamics (SPH) [167, 168] CFD code). Adding CFD codes to the computational tool-set of laser-plasma physicists will provide new opportunities as more intense high-repetition-rate facilities come online.

68 Chapter 7 Conclusions

Intense laser-plasma interactions represent a rich field of physics with a variety of potential applications including the generation of energetic particle beams. This work contributes to the areas of ion acceleration and high-repetition-rate laser-plasma interactions with efforts to address many of the limitations discussed in Sec. 1.1.3. While this work is theoretical and computational in nature, the explored parameter spaces are inspired by current experimental capabilities. The underlying thread of this work is building upon established methods and techniques, often from other fields, and using them to advance the study of laser-plasma interactions. In chapter4 we revisited the classical laser-plasma interaction of ponderomotive steep- ing by considering the impact of shorter, more intense pulses at various wavelengths. This work also helped connect the related, but surprisingly disjointed, research efforts of ponderomotive steepening and the formation of plasma gratings with overlapping laser pulses. We investigated the dynamics of this process with simple analytic models and PIC simulations, which helped reveal the transient nature of this phenomenon and higher dimensional considerations. We also show that sub-MeV ions are accelerated during the peak formation process. The models in this work help constrain the parameter space for experimental confirmation of this behavior and for potential applications in plasma optics. Then in chapter5 we used an evolutionary algorithm to run thousands of 1D PIC simulations to optimize ion acceleration. While evolutionary algorithms are a well-established optimization technique, they are relatively new to our field, and novel for directly optimizing ion acceleration. Through the demonstration of this technique, we also identified a new type of target for ion acceleration. This target showed enhanced laser to ion conversion efficiency and maximum ion energy compared to conventional targets in more realistic 3D simulations. We addressed advances in high-repetition-rate laser systems in chapter6 by performing CFD simulations of current and potential future thin liquid targets that are advantageous to ion acceleration. Developing this new capability can work hand in hand with future

69 experimental efforts for evaluating new high-repetition-rate targets. As we have illustrated, there is an existing body of fluid dynamics work that can be drawn upon for inspiration in future work. There have been great advances in laser technology since its inception [14], including the development of CPA [17], and in this simulation work we have explored a variety of parameters not yet tested with experiments. We have developed models and generated simulation results that will help guide future experimental efforts to more effectively utilize limited resources. The potential applications of intense lasers have not been fully realized, but the future is bright.

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88 Appendix A Additional Considerations for Short Pulse Laser-Driven Ponderomotive Steepening

This appendix (A) is reprinted from Joseph R Smith, Chris Orban, Gregory K Ngirmang, John T Morrison, Kevin M George, Enam A Chowdhury, and WM Roquemore. Particle- in-cell simulations of density peak formation and ion heating from short pulse laser-driven ponderomotive steepening. Physics of Plasmas, 2019 [1] under a Creative Commons Attri- bution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

A.1 Timescale from Ion Oscillation Frequency

The timescale for ion motion found in Sec. 4.2.2 makes a linear approximation for the force on ions a distance λ/8 or closer to the peak. Alternatively, to represent the timescale of the ion motion, we could consider the ion plasma oscillation frequency s 2 2 niZ e ωpi = . (A.1) miε0

We replace Zni with ne and then use Eq. 4.9 to write this as a function of laser intensity, s 4Ze2I ωpi = 3 , (A.2) mimec ε0 and providing a timescale of (one-quarter of the ion oscillation period) r π m m ε c3 τ = i e 0 , (A.3) ion 4 Ze2I which agrees with the timescale found in Sec. 4.2.2.

89 A.2 Maximum Ion Energies

To calculate the maximum ion energy, one may also calculate the work done by the electric ﬁeld on an ion traveling from a valley to an electron peak. For the sinusoidal model, this results in a maximum energy of 1 2 ne KEmax = mec Z × 2 ncrit n = 255.5 Z × e keV, (A.4) ncrit which, as expected, is slightly higher than predicted with the energy predicted from the

linear approximation in Eq. 4.13. If ne ≈ n0, then the ﬁeld would to shrink at later times, due to ion movement, reducing this maximum energy. Similarly, if we calculated the energy from the maximum depletion model, we ﬁnd

2 π 2 ne KEmax = mec Z 8 ncrit n = 630.4 Z e keV, (A.5) ncrit where we see that the ion acceleration associated with ponderomotive steepening appears to be primarily a sub-MeV acceleration mechanism.

A.3 Ion Mean Free Path

Following the ion-ion mean free path estimate from Park et al. [169] and Trubnikov [170], the ion-ion mean free path for colliding ﬂows with mass number A and velocity v before the collision is approximated as

s4 A2 v4 λ [cm] = 5 × 10−13 , (A.6) mfp cm6 Z4 n assuming the Coulomb logarithm is ten21 and that the temperature of the counter streaming flows is much smaller than the energy of the ions due to the bulk flow velocity. For example, if we look at the deuterium simulation, (Az = 2,Z = 1), and consider the maximum peak density to be ≈ 2.5 × 1020 cm−3 and average velocity to be ≈ 3 × 105 m/s, then we find

λmfp ≈ 65 µm, which is orders of magnitude larger than the width of the peaks. This mean free path is shorter for the low energy ions, although these are of less interest for this work. For higher densities, such as in the bulk of the target, or with a higher density pre-plasma, collisions would be more signiﬁcant.

21A more precise treatment could be used in the calculation (e.g. Ref. [171]), but 10 should be suﬃcient for this approximation.

90 A.4 Potential Neutron Yield

The accelerated ions from this phenomenon could be used to create neutrons with a “pitcher- catcher” type configuration. One could make an experiment where there is a pre-plasma ‘pitcher’ in front of the solid density target which would act as the catcher. For this case the pitcher and catcher are only separated by ≈ λ/2 (for the first peak). For an order of magnitude estimate of potential neutron yield, we consider a deuterated polyethylene (CD2) target as discussed in Davis and Petrov [28]. From Eq. 4.18, we expect the maximum energy deuterons from this phenomenon to be ≈ 100 keV, which would have a yield of ≈ 4 × 10−8 neutrons per deuteron [28]. From these simulations, we make a very generous estimate of a conversion efficiency of 10−4 for laser energy to 100 keV ions. This results in O(102) neutrons per Joule of laser energy. This is a few orders of magnitude lower than typical schemes producing ∼ 105 − 107 neutrons per Joule of laser energy [28]. This suggests that this phenomenon would not outperform conventional schemes in terms of neutron yield, but it may be useful for certain applications.

91 Appendix B Additional Information for Optimizing Laser Plasma Interactions with PIC Simulations and Evolutionary Algorithms

B.1 1D Evolution

Figure B.1 on the following page includes additional snapshots of the population throughout the evolutionary algorithm. For the initial generation, we see a relatively uniform sampling of the parameter space, but with further generations patterns begin to emerge and the population becomes more homogeneous. After twenty or so generations, the front of the target is typically classically overdense, then the center of the targets in the population are primarily underdense and there is typically an overdense spike in one of the last two density bins.

92 0 0

00 ncrit 0

00 0 0

000 μ

00 0 0

000 0 0 0

Figure B.1: Evolution of the population (a-f). The entire population is plotted, with darker lines indicating higher conversion eﬃciencies. The highest performing member of each generation is plotted in red and the classical critical density is plotted for reference.