Novel Avenues of Wakefield Acceleration: Fusion Plasmas and Cancer Therapy

UC Irvine UC Irvine Electronic Theses and Dissertations

Title Novel Avenues of Wakefield Acceleration: Fusion Plasmas and Cancer Therapy

Permalink https://escholarship.org/uc/item/0v88226r

Author Nicks, Bradley Scott

Publication Date 2019

Peer reviewed|Thesis/dissertation

eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA, IRVINE

Novel Avenues of Wakeﬁeld Acceleration: Fusion Plasmas and Cancer Therapy

DISSERTATION

submitted in partial satisfaction of the requirements for the degree of

DOCTOR OF PHILOSOPHY

in Physics

Bradley Scott Nicks Jr.

Dissertation Committee: Professor Zhihong Lin, Chair Professor Toshiki Tajima Professor Roger McWilliams

To Amanda Johnson, my constant anchor on sanity. This achievement is as much your doing as mine.

ii TABLE OF CONTENTS

Page LIST OF FIGURES v

ACKNOWLEDGMENTS ix

CURRICULUM VITAE x

ABSTRACT OF THE DISSERTATION xii

1 Introduction 1 1.1 Particle Trapping and Wakefield Acceleration ...... 1 1.2 Ion-Cyclotron Waves in a Field-Reversed Configuration ...... 5 1.3 Outline for Subsequent Sections ...... 8 2 Modeling Kinetic IC Modes in the SOL Environment 9 2.1 Analytical Modeling of IC Modes ...... 9 2.2 PIC Simulation of the SOL ...... 17 3 Beam-Driven IC Modes in the SOL of an FRC 21 3.1 IC Waves Propagating Parallel to the Magnetic Field ...... 22 3.2 IC Waves Propagating Obliquely to the Magnetic Field ...... 25 3.3 IC Waves Propagating Perpendicular to the Magnetic Field ...... 27 3.4 Analogy to a Proton-Boron-11 Plasma ...... 31 3.5 Summary and Conclusions ...... 33 4 Wakefield Acceleration with IC Resonance 36 4.1 Magnetized Wakefield Acceleration ...... 36 4.2 Cyclotron Resonance Condition ...... 44 4.3 Enhancement from Beam Bunching ...... 45 4.4 Conclusions ...... 48 5 High-Density Laser Wakefield Application to Oncology 50 5.1 Introduction ...... 50 5.2 Acceleration in the High-Density Regime ...... 53 5.3 Laser Intensity Scaling ...... 61 5.4 High-Density LWFA in Fiber Lasers ...... 64

iii 5.5 Electron Tissue Penetration ...... 70 5.6 Conclusions ...... 71 6 Conclusions 75 6.1 Wakeﬁeld Acceleration with Ion-Cyclotron Waves ...... 75 6.2 High-Density Laser-Wakeﬁeld Application to Oncology ...... 77 Bibliography 79

A Perpendicular Integrals of the Beam Velocity Distribution 86 A.1 Summation Forms ...... 86 A.2 Selected Analytical Properties ...... 89 A.2.1 Beam Perpendicular Integral Functions ...... 89 A.2.2 Maxwellian Perpendicular Integral Functions ...... 90 B Plasma Reactivity Calculation 92 B.1 Primary Method ...... 94 B.1.1 Correlation ...... 94 B.1.2 Variable Transform 1 ...... 94 B.1.3 Convolution ...... 95 B.1.4 Variable Transform 2 ...... 95 B.1.5 Final Result ...... 95 B.2 Alternative Method ...... 96

iv LIST OF FIGURES

Page

1.1 A schematic representation of the C-2U plasma, with red representing the core, where plasma density is highest. The magnetic ﬁeld is indicated by the black lines with arrows denoting ﬁeld direction. The red arrow indicated the FRC rotation. . . 7

2.1 A sample dispersion relation from the supplemental approach of numerically solving Eq. 2.1 for the given plasma parameters for a deuterium plasma and hydrogen beam. In this particular case, k = b = 0 and nb∕ni = 0.01. The red points indicate growth (instability), and the blue points indicate damped modes. Black indicates marginal stability. The solid lines indicate the Alfvén speed, and the dashed line indicates the beam resonance lines...... 17 2.2 A schematic representation of the 1D PIC simulation geometry. The one degree ⃗ of spatial freedom is taken as the ̂x direction, with the external magnetic ﬁeld B0 oriented in the x-z plane at an angle k with respect to ̂x to deﬁne the angle of wave propagation. The beam population is given perpendicular and parallel components ⃗ with respect to B0 according to the angle b...... 19

3.1 Dispersion relations for right-handed (3.1a) and left-handed (3.1b) components of ◦ the electric field for purely parallel propagation (k = 0 ) and no beam population. Frequency is normalized with respect to the background ion (deuterium) cyclotron frequency Ωi, and the wavevector is normalized with respect to vA∕Ωi. The space where ! < 0 indicates backwards propagation. The intensity at at particular mode is indicated by the heat-map and is normalized with respect to the maximum value. The Alfvén velocity ! = kvA is indicated as a dashed line. The dotted line indicates the approximate region of strong ion cyclotron damping, ! = 2kvti ± Ωi...... 23 3.2 The dispersion relation for right-handed (3.2a) and left-handed (3.2b) components ◦ of the electric field for purely parallel propagation (k = 0 ) and a purely parallel- ◦ streaming beam population (b = 0 ). The right- and left-handed beam resonance lines (! = k∥vb∥ ± Ωb) are indicated with dotted lines...... 24 3.3 The dispersion relation for right-handed (3.3a) and left-handed (3.3b) components ◦ of the electric field for purely parallel propagation (k = 0 ) and nearly perpendic- ◦ ularly injected beam population (b = 75 )...... 25 3.4 The dispersion relation for the compressional (3.4a) and shear (3.4b) components ◦ of the magnetic field for oblique propagation (k = 60 ) and no beam population. . 27

v 3.5 The dispersion relation for the compressional (3.4a) and shear (3.4b) components ◦ of the magnetic ﬁeld for oblique propagation (k = 60 ) and a beam population ◦ injected at b = 15 . The beam resonance lines (now with all harmonics available) are indicated by the dotted lines...... 28 3.6 The dispersion relation for the longitudinal (electrostatic) component for near-perpendicular ◦ propagation (k = 85 ) without a beam population. The linearly polarized shear Alfvén mode (! = k∥vA), which has phase velocity vA cos k, is indicated by the dash-dot line...... 29 3.7 The dispersion relations for the electrostatic component for near-perpendicular prop- ◦ ◦ agation (k = 85 ) and a beam population injected at b = 0 ...... 30 3.8 The dispersion relations for the electrostatic component for near-perpendicular prop- ◦ ◦ agation (k = 85 ) and a beam population injected at b = 75 ...... 31 3.9 The dispersion relation for the longitudinal (electrostatic) component for a boron-11 ◦ plasma for oblique propagation (k = 75 ) with a proton beam population injected ◦ at b = 60 . In this particular case, k < 0 is used to indicate backwards propagation. The beam velocity is indicated by the dotted line with spacing to show comparison with the other characteristic velocities...... 33

4.1 The maximum fusion enhancement P ∕Ptℎ for a scan over angular parameters k (vertical axis) and b (horizontal axis). Each angular parameter is sampled in mostly ◦ ◦ ◦ 15 increments but also includes 5 and 85 . The bright peak in the upper right ◦ ◦ corner corresponds to k = 85 and b = 75 , which was used in generating figure 3.8...... 37 4.2 A snapshot of the the phase space (heatmap) of ions (D) near peak mode activity overlaid with the electrostatic field (teal) normalized to the Tajima-Dawson saturation Es(n) (Eq. 4.1) for n = 2. The wavelength = 2vA∕2Ωi likewise corresponds to the n = 2 resonance of Ωi with vA. Brighter heatmap colors indicate higher phase space density. The positive x direction is to the right...... 40 4.3 The phase evolution of a single tracer deuteron from t = 0 to t = 12i with the wave phase velocity (vpℎ = vA) indicated. The kick to a higher velocity occurs around t = 8i...... 41 4.4 The vx velocity distribution for the combined background ion and beam ion populations for the initial and final simulation timesteps for the ion-Bernstein mode shown in Fig. 4.2 with the mode phase velocity indicated...... 42 4.5 The normalized ion (D) energy spectrum for various time snapshots, where i is the ion cyclotron period, and Eb is the beam (H) injection energy. At t = 0 (blue curve), all ions are thermal...... 43 4.6 A sample schematic velocity distribution at initial (blue) and later (red) stages of quasilinear saturation...... 44 4.7 Enhanced fusion reactivity comparison. (a) The simulated evolution of the D-D fusion power (neutron branch) normalized to the initial thermonuclear value Ptℎ. (b) Experimental fusion enhancement in C-2U normalized to the thermonuclear value Ptℎ...... 45

vi 4.8 The maximum D-D fusion rate normalized to the initial thermonuclear rate for ◦ ◦ various beam velocities for the case k = 85 and b = 75 . Black points have no beam bunching; red points, bunching at b = vA∕Ωi, corresponding to the resonance at ! = 2Ωi; and blue points, bunching at b = 2vA∕Ωi, corresponding to ! = Ωi. The purple point is the approximate position of the observed fusion enhancement in the C-2U experiment...... 47 ◦ 4.9 A snapshot of the the phase space of ions (B) near peak mode activity for k = 75 ◦ and b = 60 overlaid with the electrostatic ﬁeld with the Tajima-Dawson saturation Es(n), Eq. 4.1, indicated for n = 1 and n = 2. The wavelength = 2vA∕2Ωi likewise corresponds to ! = 2Ωi, approximately the center of the continuum distribution...... 48

5.1 Scaling of the normalized maximum electron energy with density nc∕ne for the laser intensity a0 = 1, compared with the theoretical expression for the energy gain of electrons Δ inLWFA...... 57 5.2 A snapshot of the electron phase space px vs. x (heat-map, with warmer colors representing higher density) and longitudinal Ex (green) and laser Ey (translucent blue) fields for the somewhat typical wakefield case of nc∕ne = 10 (“blue”) and a0 = 1 after the electron acceleration has saturated. The plasma wavelength is given by p = 2c∕!p. The forward edge of the laser pulse is at x∕p = 16...... 58 5.3 Electron phase space and field structure of the high-density (“black”) case nc∕ne = 1 with laser intensity a0 = 1 at early (5.3a) and later (5.3b) stages. The development of the electron streams is observed. Figure 5.3a is zoomed to 0 ≤ x ≤ p from (5.3b). 59 5.4 The specific entropy (“darkness”) index D normalized to its initial value D0 for a scan of the density ratio values nc∕ne ∈ [0.5, 14] for laser intensity a0 = 1 and a resonant pulse length. The plotted index is the mean of the calculated index for the last ten time steps in each run. The black error bars represent one standard deviation of this set of averaged values. Rough indications of the wave type for particular regions are given in text. Because the plasma is initialized with the same temperature in each case, the initial index D0 is the same for every data point. The critical density is marked by a dashed line...... 62 5.5 The electron phase space and field structure of the intermediate (“grey”) case of density nc∕ne = 3 at the laser intensity a0 = 1...... 63 5.6 The electron phase space and field structure of the critical density case nc∕ne = 1 for a laser pulse of length 8p at the laser intensity a0 = 1, showing the “black tsunami” regime. At this snapshot most of the electron acceleration is concluded and most of the laser has exited the domain...... 64 5.7 The maximum electron energy as a function of laser intensity a0 for two density cases: nc∕ne = 10 (5.7a) and nc∕ne = 3 (5.7b). The maximum energies (red dots) are compared with the function g(a0) (blue solid line). The blue dashed lines represent the asymptotic behavior of g(a0) for a0 ≪ 1 and a0 ≫ 1...... 65 5.8 A demonstration of the self-modulation of a laser pulse into resonant pieces in the “blue” regime of nc∕ne = 10. Here l∕p = 5...... 67

vii 5.9 The electron phase space and field structure (left) for the case of a 100 fs pulse at nc∕ne = 10, as well as the laser (Ey) frequency spectrum (right). Fifteen lasers each contributing intensity a0 = 0.01 are coherently added, demonstrating practical parameters for fiber laser applications. In 5.9a, the laser pulse undergoes self- modulation, while in 5.9b, each of the 15 laser contributions is further divided into two components that beat at !p, thus resonantly seeding the wakefield (beat-wave acceleration). Note that here the laser field Ey is normalized with respect to the initial combined amplitude E0 of all the fiber contributions. For the frequency spectra, solid vertical lines indicate the nominal laser frequency (!0∕!p) while dashed lines indicate harmonics, which differ from the nominal frequency by some integer multiple n of the plasma frequency !p...... 69 5.10 Electron penetration in the high-density LWFA regime. (5.10a) shows the normalized electron energy distribution for setup in figure 5.9a, which models a bundle of 15 fiber lasers each with a0 = 0.01 coherently added with plasma density nc∕ne = 10 and a pulse length of 100 fs. (5.10b) shows the resulting normalized distribution of electron penetration depth in the continuous slowing-down approximation (CSDA)...... 71

viii ACKNOWLEDGMENTS

Many people deserve recognition for this work. Firstly, I would like to thank my advisor Professor Toshi Tajima for managing the arduous process of transforming a student into a scientist. I would also like to thank the team at TAE: Aleš Nečas, Richard Magee, Sean Dettrick, and many others, for their guidance through these past six years of summer internships.

Among my UCI brethren, I would like to thank the numerous fellow graduate students who provided helpful advice and injections of sanity into dark hours, in particular Andrew Brandel, Milad Pourrahmani, Michael Seggebruch, Derek Wilson, Gabe Player, Nick Beier, and Sahel Hakimi.

Closer to home, I would like to thank Amanda Johnson, who has always been there to support me when things seemed insurmountable. I would also like to thank my parents, whose support in my academic pursuits reaches back to the early days of Magic School Bus computer games. From weather stations to musical instrument maintenance, I have wanted for nothing.

More prosaically, this work was done at the University of California, Irvine, with the support of the TAE Technologies subcontract. Simulations were conducted chieﬂy on the TAE Technologies high-performance computing cluster.

Portions of this work reproduce material published or intended to be published. The corresponding copyright statements are given below.

Reproduced from “B.S. Nicks, S. Hakimi, E. Barraza-Valdez, K.D. Chesnut, G.H. DeGrandchamp, K.R.Gage, D. B. Housley, G. Huxtable, G. Lawler, D.J. Lin, P. Manwani, E.C. Nelson, G.M.Player, M.W.L. Seggebruch, J. Sweeney, J.E. Tanner, K. A. Thompson, and T. Tajima, Electron Dynamics in the High-Density Laser-Wakeﬁeld Acceleration Regime, Phys. Rev. Accel. Beams, (2019, Submitted)”, © 2019 American Physical Society.

Reproduced from “B.S. Nicks, A. Necas, and T. Tajima, Beam-Driven Ion-Cyclotron Modes in the Scrape-oﬀ Layer of a Field-Reversed Conﬁguration, Phys. Plasmas, (2019, Submitted)”, with the permission of AIP Publishing.

Electronic version of an article published as B. S. Nicks, T. Tajima, D. Roa, A. Necas, and G. Mourou, Laser-Wakefield Application to Oncology, accepted to the International Journal of Modern Physics A, © 2019 World Scientific Publishing Company, https://www.worldscientific.com/loi/ijmpa.

ix CURRICULUM VITAE

Bradley Scott Nicks Jr.

EDUCATION Doctor of Philosophy in Physics 2020 University of California, Irvine Irvine, California Master of Science in Physics 2019 University of California, Irvine Irvine, California Bachelor of Science in Physics 2014 University of Florida Gainesville, Florida

RESEARCH EXPERIENCE Graduate Research Assistant 2014–2020 University of California, Irvine Irvine, California Undergraduate Research Assistant 2012–2014 University of Florida Gainesville, Florida

TEACHING EXPERIENCE Teaching Assistant 2014–2016 University of California, Irvine Irvine, California

x REFEREED JOURNAL PUBLICATIONS

Direct observation of ion acceleration from a beam-driven wave 2019 in a magnetic fusion experiment Nat. Phys. R. M. Magee, A. Necas, R. Clary, S. Korepanov, S. Nicks, T. Roche, M. C. Thompson, M. W. Binderbauer and T. Tajima

REFEREED CONFERENCE PUBLICATIONS Laser-Wakeﬁeld Application to Oncology 2019 Workshop on Beam Acceleration in Crystals and Nanostructures, Fermilab B. S. Nicks, T. Tajima, D. Roa, A. Necas, and G. Mourou

MANUSCRIPTS IN PREPARATION Electron Dynamics in the High-Density Laser-Wakefield Accel- 2019 eration Regime Phys. Rev. Accel. Beams, Submitted B.S. Nicks, S. Hakimi, E. Barraza-Valdez, K.D. Chesnut, G.H. DeGrandchamp, K.R.Gage, D. B. Housley, G. Huxtable, G. Lawler, D.J. Lin, P. Manwani, E.C. Nelson, G.M.Player, M.W.L. Segge- bruch, J. Sweeney, J.E. Tanner, K. A. Thompson, and T. Tajima Wakefield Acceleration with Ion Cyclotron Resonance 2019 Phys. Rev. Lett., Submitted B.S. Nicks, A. Necas, R. Magee, T. Roche, and T. Tajima Beam-Driven Ion-Cyclotron Modes in the Scrape-off Layer of a 2019 Field-Reversed Configuration Phys. Plasmas, in Progress B.S. Nicks, A. Necas, and T. Tajima

xi ABSTRACT OF THE DISSERTATION

Novel Avenues of Wakeﬁeld Acceleration: Fusion Plasmas and Cancer Therapy

Bradley Scott Nicks Jr.

Doctor of Philosophy in Physics

University of California, Irvine, 2020

Professor Zhihong Lin, Chair

Plasma wakefields, whether driven by laser or particle bunch, possess a stable, coherent, and robust structure that can accelerate particles to high energy. Wakefields derive these remarkably properties from a single physical principle: waves with a phase velocity much higher than the bulk speed of a population particles will not strongly couple to the particles. Instead, a small subset of particles are captured by the wake and coherently accelerated to high energy. In typical applications of wakefield acceleration, this property is harnessed to accelerate electrons, but extends widely into other applications. Notably, ion-cyclotron waves in fusion plasmas, when possessed of a fast phase velocity, can generate a fast-ion tail that can dramatically enhance fusion yield without increasing the bulk ion temperature, as has been observed in the C-2U experiment [62]. The first part of this work explores this phenomenon, examining the physics of beam-driven ion-cyclotron waves in the context of the scrape-off-layer (SOL) of a field-reversed-configuration (FRC) plasma. In particular, the nonlinear mechanics of ion acceleration by these waves is examined. Interestingly, this principle of phase velocity can be just as useful when violated. In the field of laser-wakefield acceleration (LWFA), in the regime of near-critical density plasmas, the physics of wakefield acceleration transitions into a sheath acceleration, which is potentially capable of generating a large flux of low-energy electrons. The application of this principle to a novel method of internal cancer therapy is treated in the second part of this work.

xii Chapter 1

Introduction

We begin by laying the foundation for wakeﬁeld acceleration, and then move to the application of these principles to ion-cyclotron waves. Finally, an outline for the remainder of the work is provided.

1.1 Particle Trapping and Wakeﬁeld Acceleration

Laser Wakefield Acceleration (LWFA) [98] is a compact method to accelerate charged particles to high energies that was first purposed by Tajima and Dawson [95] in 1979. While the accelerating electric field in a conventional linear accelerator is limited by the breakdown threshold of the de- vice walls, plasma-based accelerators can access far higher electric fields, owing to the inherently broken-down nature of plasma. Consequently, plasma-based accelerators can access far higher accelerating gradients than those available to conventional accelerators, reaching potentially GeV per 18 2 cm or higher. The original proposal for LWFA called for a laser intensity of 10 W∕cm , but this regime of intensity only became accessible after the invention of Chirped Pulse Amplification (CPA) [19], and LWFA was experimentally verified shortly thereafter [71, 72]. Since then, pro-

1 lific experiments demonstrated this technique in different regimes, and the field has grown steadily. Wakefield acceleration has achieved electron acceleration up to 8 GeV [42] and has expanded to encompass such diverse applications as generating electron beams for cancer therapy [39] and ex- plaining the highest-energy cosmic rays [29].

Wakeﬁelds can be driven by a laser, as was the case in the original conception, but can also be driven by bunches of particles, such as electrons [85] or ions [14]. In displacing the surrounding plasma, the driver excites a coherent plasma wave structure that forms a long train of waves, similar to the wake generated by a boat moving in water. The speed of the driver is typically near the speed t 2 of light c. For a laser driver, the precise group velocity is given by vg = c 1 − (!p∕!0) , where !p

and !0 are respectively the plasma frequency and laser frequency. For the typical case of !0 ≫ !p, vg ≈ c. This group velocity then defines the phase velocity of the wakefield train following the bunch, while the group velocity of the wake train is near zero. As a consequence of the latter property, the energy deposited in the plasma by the driver largely remains fixed in space. More importantly, however, because the phase velocity of the wakefield is less than the speed of light, vpℎ < c, it is possible for particles in the plasma to be trapped and accelerated by this wave. In contrast, the phase velocity of the accelerating field in conventional cavity accelerators is typically superluminal (vpℎ > c), leading to relatively inefficient particle trapping.

Particle trapping occurs when the velocity of a particle in the frame of an electrostatic (longitudinal) wave is less than or equal to the trapping velocity vtr, which is given by

u qE v = , tr mk (1.1)

where q and m are respectively the charge and mass of a particle (ion), E is the wave electric ﬁeld, and k the wavenumber of the wave [79]. The condition for trapping particles from a thermal √ distribution with thermal velocity vt = T ∕m, where T is temperature, is then given by

vt ∼ vpℎ − vtr. (1.2)

2 A trapped particle is either accelerated or decelerated depending on whether the initial particle velocity is less than or greater than the wave speed. Slower particles extract energy from the wave, and faster particles donate energy to the wave. In the wave frame, a trapped particle is then locked in closed orbits in phase space, which in real space manifests as an oscillation in the phase of the particle with respect to the wave.

Exchange of energy between the particles and the wave inevitably alters both. If the phase velocity of the wave is comparable to the thermal velocity of the particles, where the derivative of the particle distribution function is negative, )f(v)∕)v < 0, there are more particles with velocities slower than the wave than those with velocities faster than the wave. Consequently, the net energy flow is out of the wave, and the wave is damped. Meanwhile, the particle distribution function is flattened in the vicinity of the wave phase velocity. More interestingly, if the thermal distribution possess a “bump-on-tail”, and the wave phase velocity lies within the region where )f(v)∕)v > 0, then the wave can grow exponentially. Yet, the positive slope is eventually flattened and the wave loses its source of energy. The wave also typically is broadened in its spectrum during this interaction. These mechanics are formulated rigorously in quasilinear theory [27, 3], which describes the saturation amplitude of the wave and its eventual decay.

Suppose, however, that the phase velocity is much faster than the thermal velocity (vpℎ ≫ vt), as is the case in typical wakeﬁeld acceleration. In this regime, there are essentially no particles available to exchange energy with the wave. Instead, the wave grows in amplitude until its trapping velocity begins to reach within the thermal distribution. With vpℎ ≫ vt, the thermal velocity can be neglected in the trapping condition to give

vtr ∼ vpℎ. (1.3)

Retaining the ﬁnite ratio vt∕vpℎ and combining equations 1.1 and 1.2 in the limit vt∕vpℎ → 0 with

3 the deﬁnition of phase velocity, vpℎ = !∕k, leads to the saturation ﬁeld

m!vpℎ E = . s q (1.4)

¨ For ﬁnite vt∕vpℎ, a correction factor must be included. The adjusted saturation level Es is then

m!v 0 12 ¨ pℎ vt Es = 1 − . (1.5) q vpℎ

Thus, the wave excitation and saturation depart from those of the quasilinear case. In particular, the saturation amplitude is now determined primarily by the phase velocity of the wave rather than the distribution function of the particles. This expression for the saturation field Es is a generalization of the wakefield amplitude of the Tajima-Dawson field [95], hereafter abbreviated at the TD field where relevant. At this field two additional phenomena are known to follow: (a) the perturbed plasma density associated with this wakefield approaches the total density, and (b) the phase space orbit approaches wave-break in the relativistic case [22]. Only once this saturation level is reached can the distribution flattening and wave broadening of quasilinear mechanics begin to occur.

Yet, the interaction with the thermal distribution is so minimal that the wave maintains a robust amplitude for a long lifetime. Even more importantly, the wave does not disrupt the bulk plasma structure. The phenomena of turbulence, anomalous transport, and general disordering of the plasma does not occur. In contrast, waves with phase velocity comparable to the thermal velocity, such as drift waves [48, 60] in fusion plasmas, lead to loss of conﬁnement, allowing particles to freely ⃗ ⃗ dislocate their positions via E × B drifts and magnetic islands.

Such properties may be considered as isomorphic to the stable, self-sustained, elevated energy level of the Landau-Ginzburg state in a double humped potential structure [59]. (The Higgs’ state [47, 4] was also proposed as such an energetically elevated state.) Another physical metaphor is offered by water waves. The wake trailing a moving boat and an offshore tsunami wave both exhibit a robust, nearly stationary state afforded by their relatively fast phase velocities. In the case of a tsunami,

4 √ the phase velocity is given by vpℎ = gℎ, where g is the constant of gravitational acceleration and ℎ is the water depth [58]. Offshore, where the wave phase velocity is high, the tsunami leaves objects such as boats essentially unaffected. Near the shore however, where the amplitude of the wave rises and its phase velocity approaches the “thermal” velocity of bulk objects, these stationary objects become “trapped” catastrophically. Furthermore, the tsunami wave near the shore becomes turbulent and dredges significant sediment from the sea floor, forming a visibly black wave. This momentum transport of sediment (and objects) indicates the presence of an effective viscosity. In contrast, the deep-sea wave does not cause substantial dredging of sediment and remains visibly blue. Similarly for a plasma, to prevent turbulence, transport, and destruction, the wave must be in the “blue” regime; the phase velocity must lie far from the plasma bulk motion.

As will be shown, this physics extends far beyond wakefield acceleration of electrons, particularly with regard to ions. In typically wakefield acceleration, only the electrons are effectively mobile; the ions, slow and massive, are essentially stationary. For ion waves, however, such as the ion- cyclotron waves that can exist in a magnetized plasma, the relative importance of the species is reversed. Ions instead resonate with the wave and become accelerated, while electrons respond adiabatically. It is to this topic that we now turn.

1.2 Ion-Cyclotron Waves in a Field-Reversed Conﬁguration

In a field-reversed magnetic configuration (FRC), a compact toroid of (reversed) closed field lines is embedded in a magnetic mirror [88]. In addition to providing a relative technological simplicity, this arrangement produces several distinct beneficial plasma properties, such high plasma beta ( ) and large particle orbits. One strategy of augmenting this base conception of an FRC is the injection of neutral beams nearly perpendicularly to the axial magnetic field, as in the C-2U machine [8]. In this case, the beam ion pressure is equal to or greater than the thermal plasma pressure, and the fast beam ions possess machine-sized orbits. This intense fast-ion pressure forms an internal spine that

5 stabilizes the FRC against macro-instabilities. Such a scheme was the vision of Norman Rostoker [9]. A schematic representation of the C-2U FRC is shown in Fig. 1.1.

While the beam stabilizes harmful macro-scale modes, it can also drive microscopic modes, warranting an examination of beam-plasma interaction with regard to micro-instabilities. An interesting class of these modes which may be preferred are those with frequencies in the vicinity of the ion-cyclotron (IC) frequency or its harmonics. In the high-beta FRC core, the red regions of the torus in Fig. 1.1, the preferred of such modes is the Alfvén-ion cyclotron (AIC) mode [96], but in the lower-beta scrape-oﬀ-layer (SOL), the region radially just outside the torus, electrostatic modes such as ion-Bernstein modes [28, 20] or other IC modes [63] may be excited.

Among the various regions of an FRC plasma, multitudinous types of waves are possible, necessi- tating a means of cutting this body of modes down to a more manageable level. The requirement of the generation of a fast-ion tail without bulk heating provides such a means and discriminates electrostatic IC modes in the SOL as the chief object of interest. As will be shown in more detail later, the most signiﬁcant beam-driven IC waves in this context have the phase velocity

√ vpℎ ∼ vA ≈ vti∕ , (1.6)

where vA is the Alfvén speed and vti is the ion thermal speed. Using equations 4.3 and 1.6, the saturation amplitude in Eq. 1.5 can be approximately cast in terms of , giving

m!v √ 2 E¨ = A 1 − . s q (1.7)

Consequently, in the SOL, where ∼ 0.2, vpℎ ≫ vti, allowing the robust excitation of electrostatic IC wave and the acceleration of a fast-ion tail without bulk ion heating, which has important bearing for fusion enhancement [62]. In contrast, in the core, ∼ 1, and thus vpℎ ∼ vti, resulting in bulk ion heating, rather than non-thermal tail formation, and a return to quasilinear saturation physics. Such bulk heating is typically anisotropic and can lead to instabilities, but because the ﬁeld lines in the

6 Figure 1.1: A schematic representation of the C-2U plasma, with red representing the core, where plasma density is highest. The magnetic field is indicated by the black lines with arrows denoting field direction. The red arrow indicated the FRC rotation. core are closed, these modes are essentially benign. Furthermore, the low value in the SOL gives electrostatic modes prominence relative to the fully electromagnetically dominated core region. We thus confine our attention to the SOL for this work and assert the importance of electrostatic modes.

Despite this robust excitation of electrostatic IC waves, the plasma confinement does not show discernable degradation, indicating that the large-amplitude waves do not induce anomalous diffu- sivity. The observation of a robust ion-cyclotron instability without loss of confinement demands a deeper theoretical understanding. In this work, we first examine the process by which near- perpendicular injection of a strong proton beam into deuteron plasma in the SOL causes a robust beam-driven ion-cyclotron instability. We then analyze the saturation mechanism of the wave and its nonlinear coupling to the plasma ions, relating these issues to fusion enhancement in kinetic simulation and theory.

Ion-cyclotron waves can be broadly divided into two categories: electrostatic and electromagnetic. While the former tend to propagate perpendicularly to the magnetic field, the latter propagate parallel to the magnetic field. A large quantity of work on electrostatic IC modes has focused on instabilities from a fast beam population, both in the case of particles streaming along the magnetic field [7, 56, 28, 101, 20, 104] and particles with a substantial perpendicular component [43, 10, 17]. In particular, IC emission driven by fast fusion ions via the magnetoacoustic instability has been

7 studied extensively [23, 25, 15, 24]. In this same context, IC heating has been pursued as a means of heating beam ions [45]. However, substantial acceleration of thermal ions through beam-driven ion-Bernstein waves (contrasted with collisional bulk heating) has, to our knowledge, not been considered previously. Observations of accelerated ion populations in the magnetosphere and iono- sphere [55, 16, 78] has also motivated study of electrostatic modes. Electromagnetic IC modes have similarly been studied in the contexts of space [41, 80, 38, 37, 36] and fusion [21, 97, 96] physics. In this study, we marshal this large body of knowledge to build an understanding of the beam-driven IC physics possible in the SOL of the C-2U FRC. To our knowledge, this has not been thoroughly treated previously.

To reach a clear theoretical understanding of the mechanisms by which IC waves in the SOL of the C-2U experiment generate a benign tail of fast ions, we employ both the tools of simulation and semi-analytical methods. Both consider a simpliﬁed system so that the underlying physics is revealed unambiguously.

1.3 Outline for Subsequent Sections

From this basis, the first subject matter addressed is the acceleration of ions by IC waves in the SOL of the C-2U FRC experiment, which is the main component of this work. Within this topic, first the modeling of these waves is described in chapter 2. Next, a broad survey of the linear physics of different modes that can be excited by a beam population is presented in chapter 3. One category of these modes, ion-Bernstein waves, distinguishes itself in this survey, and the nonlinear mechanics by which this mode generates a tail of fast ions are examined in chapter 4. This chapter also treats the topic of seeding this mode for increased ion acceleration. Shifting focus, chapter 5 then presents preliminary theoretical work on the use of LWFA for applications in endoscopic cancer treatment. Finally, chapter 6 closes this work with concluding remarks.

8 Chapter 2

Modeling Kinetic IC Modes in the SOL Environment

This chapter details the methods by which the IC modes under consideration are modeled in analytical and simulation methods.

2.1 Analytical Modeling of IC Modes

The foundation of this work is the modeling of kinetic ion-cyclotron waves in an experimentally relevant geometry and parameter regime. To this end, we make use of the essentially 1D geometry of the SOL of the C-2U FRC and consider a homogeneous plasma in a uniform magnetic ﬁeld. This situation can be described formally in analytical theory, from which the major properties can be extracted. The kinetic beam eﬀects are of particular concern, and so in general it will be necessary to treat an arbitrary beam injection and wave propagation angle. Additionally, to obtain a relatively realistic picture of the robustness of excitations in various regimes of these angular parameters, the beam is given a thermal spread in addition to its drift component. This concern requires a

9 substantial analytical eﬀort and is a somewhat novel feature of this study. Such an eﬀort is not simply an abstract exercise, but in fact will be shown to have experimentally relevant consequences. We thus set now to tackling the analytical formalism underlying this work.

⃗ ⃗ ⃗ The analytical theory of linear plasma waves begins with the relation D = ⃡ ⋅ E [90, 50], where D is the displacement electric ﬁeld in a medium (plasma), and ⃡ is the dielectric tensor. (It is assumed ⃗ that the medium has a linear response to the electric ﬁeld E in the limit of low-amplitude waves.) By expressing Maxwell’s equations in terms of complex Fourier amplitudes, it can be shown that ⃗ ⃡ determines the allowable waves in a plasma. For a wave with wavevector k and frequency !, the general plasma dispersion relation is given by

⃡ ⃗ det ðΔ(k, !)ð = 0, (2.1) where

0 12 0 1 !∕k k⃗k⃗ Δ(⃡ k,⃗ !) = ⃡(k,⃗ !) − I⃡ − . c k2 (2.2)

⃡ ⃗ ⃗ The electric ﬁeld polarization is given by solving for the eigenvectors of Δ ⋅ E = 0 once k and ! ⃗ ⃗ ⃗ are determined by Eq. 2.1. The magnetic ﬁeld polarization is then given by B = (c∕!)(k × E).

⃗ The dielectric tensor ⃡ is thus chief object of interest for determining the allowed k and !. While a simpler, scalar function containing only information about electrostatic (longitudinal) waves can be ⃗ ⃗ ⃗ ⃗ 2 found by (k, !) = [k ⋅ ⃡(k, !) ⋅ k]∕k , the relatively high plasma beta considered here ( = 0.2) necessitates the inclusion of the full tensor, which contains the full wave information, including both electrostatic and electromagnetic waves or hybridized combinations thereof. The full tensor can be conveniently divided among the contributions from each plasma species according to

⃗ ⃡ É ⃡(k, !) = I − ⃡, (2.3)

10 where

L ∞ H I ⃡ M ! 2 É nΩ )f )f Π(v⟂, v∥; n) ⃡ (k,⃗ !) = I⃡ + + k d3v ! v )v ∥ )v n k v ! i (2.4) n=−∞ Ê ⟂ ⟂ ∥ Ω + ∥ ∥ − −

⃗ ⃗ for a homogeneous plasma in a uniform magnetic ﬁeld B0 ∥ ̂z with wavevector k = k⟂ ̂x + k∥ ̂z. ◦ ◦ ⃗ ⃗ The angle 0 ≤ k ≤ 90 denotes the direction of k with respect to B0, with which k⟂ = k sin k

and k∥ = k cos k. Note that this coordinate frame is diﬀerent from that used in the simulation scheme, as will be shown. The characteristic frequencies for each species are the species plasma t 2 frequency ! = 4nq∕m and cyclotron frequency Ω = qB0∕mc, where q, m, and n are respectively the charge, mass, and number density of a species. The total plasma frequency is given 2 ∑ 2 2 ⃡ by !p = ! ≈ !e. The tensor Π is given by

2 ⎡ nΩ 2 nΩ ¨ nΩ 2 ⎤ Jn iv⟂ JnJn v∥ Jn ⎢ k⟂ k⟂ k⟂ ⎥ ⃡ ⎢ nΩ ¨ 2 ¨ 2 ¨⎥ Π = −iv J J v (J ) −iv v J J , ⎢ ⟂ k n n ⟂ n ∥ ⟂ n n⎥ (2.5) ⎢ ⟂ ⎥ nΩ 2 ⟂ 2 2 ⎢ v∥ Jn iv∥v⟂JnJn v∥Jn ⎥ ⎣ k⟂ ⎦

¨ where Jn ≡ Jn(z), z ≡ k⟂v⟂∕Ω, and Jn ≡ dJn(z)∕dz. The term represents a small, adiabatic turning-on of a ﬁeld perturbation. The integral is done over all velocity space, with in general 3 d v = 2v⟂ dv⟂ dv∥.

The species velocity distribution function is given denoted by f, and we must now provide a notation to diﬀerentiate species. Let the subscripts i, e, and b respectively indicate background ions, electrons, and beam ions. References henceforth to simply “ions” refer to the background ions. The ions and electrons are given Maxwellian distributions with isotropic temperature. For ∈ {i, e},

H 2 2 I v v⟂ + v∥ f (v , v ) = ⟂ exp − , ⟂ ∥ 3∕2 3 2 (2.6) (2) vt 2vt

11 √ where vt = T∕m is the characteristic thermal speed for a species with temperature T. 3 This distribution obeys the normalization ∫ f(v⟂, v∥) d v = 1. Substituting the ion and electron expressions for f into Eq. 2.4 and carrying out the integration over velocity space yields (for ∈ {i, e})

L ∞ M ! 2 É ⃡ (k,⃗ !) = − z2 ̂ẑz + z (z ) ⃡( , z ; n) , ! 0 0 n n (2.7) n=−∞ where

! − nΩ zn ≡ , (2.8) ðk∥ðvt

2 2 and ≡ (k⟂vt∕Ω) = (k⟂) (where for ∈ {i, e}, the Larmor radius is deﬁned as = vt∕Ω). The modiﬁed tensor ⃡ is given by

2 ⎡ n ¨ zn ⎤ Λn( ) inΛn( ) n 1∕2 Λn( ) ⎢ ⎥ ⃡( , z ; n) = ⎢ ¨ n2 ¨ 1∕2 ¨ ⎥ , n −inΛn( ) Λn( ) − 2 Λn( ) −izn Λn( ) (2.9) ⎢ ⎥ ⎢ ⎥ zn 1∕2 ¨ 2 ⎢n 1∕2 Λn( ) izn Λn( ) znΛn( ) ⎥ ⎣ ⎦

−x where Λn(x) = In(x)e and In(x) is the modiﬁed Bessel function of index n. The function (z) is a modiﬁed version of the plasma dispersion function [34] and is given by

∞ − 2∕2 1∕2 1 e z 2 (z) = d = −21∕2F + i e−z ∕2,  1∕2 1∕2 (2.10) (2) Ê−∞ − z − i 2 2

where F (x) is the Dawson function.

The beam distribution requires more intensive treatment. The beam velocity distribution is a √ generalized Maxwellian with temperature Tb and drift speed vb = 2Eb∕mb directed at an angle ◦ ◦ 0 ≤ b ≤ 90 with respect to the external magnetic ﬁeld, where Eb is the beam drift energy. The

parallel and perpendicular drift components are thus given as vb∥ = vb cos b and vb⟂ = vb sin b, re-

12 spectively. The beam Larmor radius is deﬁned as b = vb⟂∕Ωb. The perpendicular drift is uniformly sampled from all directions perpendicular to the external magnetic ﬁeld, giving the perpendicular distribution function

H I L 2 M 1 vb v v⟂ − vb⟂ f (v ) = v Λ ⟂ ⟂ exp − , b⟂ ⟂ 2 ⟂ 0 2 2 (2.11) 2vtb vtb 2vtb

∞ This function obeys the normalization 2 ∫0 fb⟂(v⟂)dv⟂ = 1. Note that in the limit vtb → 0, this function converges to the more familiar ring distribution fb⟂ → (v⟂ − vb⟂). The distribution function for the beam velocity parallel to the external magnetic ﬁeld is simply a drifted Maxwellian:

L 2 M 1 v∥ − vb∥ f (v ) = exp − , b∥ ∥ (2)1∕2v 2 (2.12) tb 2vtb

∞ which obeys the normalization ∫−∞ fb∥(v∥)dv∥ = 1. While this beam distribution is likely similar to that of freshly injected beam particles in the C-2U experiment, the beam overall in experiment has a slowing-down distribution, which would tend to blunt the growth of the beam-driven modes examined here.

Substituting fb into Eq. 2.4 does not yield closed-form solutions as is the case with the ions and

electrons. While the integration over v∥ can still be expressed in terms of the  function with argument

! − nΩb − k∥vb∥ zbn ≡ , (2.13) ðk∥ðvtb

the integration over v⟂ has no closed form. These integrals can be evaluated numerically, but can

also be expressed as sums over Laguerre polynomials. The tensor ⃡b can be expressed compactly

13 as

T ∞ U ! 2 É ⃡ (k,⃗ !) = − b A⃡ + B⃡ ◦ z C⃡ + (z − z ⃡) (z ) , b ! bn b0 bn  bn (2.14) n=−∞

⃡ ⃡ where the ◦ operators denotes an element-wise (Hadamard) product. The matrices A and B are given by

⎡ 2 0 − 2 tan2 ⎤ ⎢ ⟂ ⟂ k ⎥ 1 ⎢ ⎥ A⃡ 0 2 0 ≡ 2 ⎢ ⟂ ⎥ (2.15a) ⎢ 2 2 2 2 2 ⎥ ⎢− tan k 0 2(zb0 + ∥) + tan k⎥ ⎣ ⟂ ⟂ ⎦ and

2 z ⎡ n i n n bn+ ∥ ⎤ 1∕2 1∕2 ⎢ b b b ⎥ B⃡ ⎢ n ⎥ , ≡ −i 1∕2 1 −i(zbn + ∥) (2.15b) ⎢ b ⎥ ⎢ z + ⎥ ⎢n bn ∥ i(z + )(z + )2 ⎥ 1∕2 bn ∥ bn ∥ ⎣ b ⎦

⃡ ⃡ where ⟂,∥ ≡ vb⟂,∥∕vtb. The matrices C and  contain the perpendicular integration, are given by

⎡P R P ⎤ ⎢ n n n ⎥ C⃡ ⎢ ⎥ ≡ ⎢Rn Un Rn⎥ (2.16a) ⎢ ⎥ ⎢Pn Rn Pn ⎥ ⎣ ⎦

and

⎡ ⎤ ⎢n n n ⎥ ⃡ ⎢ ⎥ .  ≡ ⎢n n n⎥ (2.16b) ⎢ ⎥ ⎢n n n ⎥ ⎣ ⎦

14 The functions that make up the elements of these matrices are distinct perpendicular integrals:

∞ 2 1∕2 2 − ⟂∕2 2 − ∕2 Pn ≡ e I0 ⟂ Jn b e d, (2.17a) Ê0

∞ 2 1∕2 2 − ⟂∕2 2 − ∕2 n ≡ e I0 ⟂ − ⟂I1( ⟂) Jn b e d (2.17b) Ê0

∞ 2 1∕2 1∕2 2 − ⟂∕2 2 ¨ − ∕2 Rn ≡ e I0 ⟂ Jn b Jn b e d, (2.17c) Ê0

∞ 2 1∕2 1∕2 2 − ⟂∕2 ¨ − ∕2 n ≡ e I0 ⟂ − ⟂I1( ⟂) Jn b Jn b e d (2.17d) Ê0

∞ 2 2 1∕2 2 − ⟂∕2 3 ¨ − ∕2 Un ≡ e I0 ⟂ Jn b e d, (2.17e) Ê0 and

∞ 2 2 1∕2 2 − ⟂∕2 2 ¨ − ∕2 n ≡ e I0 ⟂ − ⟂I1( ⟂) Jn b e d. (2.17f) Ê0

Properties of these integrals and their evaluations as sums are given in appendix A. In the cases of ◦ ◦ purely perpendicular or parallel propagation (k = 90 or k = 0 ), special care must be taken to re-derive the expressions in Eqs. 2.7 and 2.14.

By approximating ⃡, certain closed analytical forms for modes can be found. For this plasma beta ∑ 2−1∕2 regime ( = 0.2), electromagnetic modes, which involve the Alvén speed vA = c (!∕Ω) ,

15 are highly prominent. In particular, the modes most relevant to this work follow ! = kvA, while one mode follows ! = k∥vA as a variant to this expression. Perhaps the most important simple analytical expression is motivated by Eq. 2.13: the beam resonance condition, given by

! = k∥vb∥ ± pΩb, (2.18)

where p is an integer. Across various regimes of propagation, this relation will reappear. For p = ±1, however, the k = 0 intercept of this relation is more precisely the hybrid ion resonance

[12] at !∕Ωi = Ωb(niΩb + nbΩi)∕(niΩi + nbΩb) ≈ 1.9. Because nb ≪ ni here, this distinction is not of great importance for the mode structure.

More generally, however, Eq. 2.1 must be solved numerically, particularly for cases where the entire dielecic tensor is needed to produce the examined modes, such as will be seen in the case of near-perpendicular propagation (chapter 3). Indeed, while simulations are the primary tool for this study, numerical solutions to Eq. 2.1 help to further clarify and complement the results from simulations. Following this logic, numerical (semi-analytical) methods were used in this work to conﬁrm and better understand the simulation results.

Equation 2.1 can be solved numerically for choice of plasma parameters and the angles k and

b. To do so, a 2D grid of real and imaginary frequencies and a list of k values for the relevant physics are first defined. For each k value, each box in the frequency sample grid is mapped as a contour through the dispersion function, and the winding number of the resultant mapping indicates whether a solution is present. In most cases, the beam density is lowered to 10% of that of the simulation beam density to ensure that the semi-analytical growth rate remains in the linear regime. Generally the mode structure is not strongly affected by this consideration. The linear dispersion relation, polarization and growth or damping rate for a wave mode for arbitrary propagation and beam injection angle can thus be determined.

A sample output of this numerical solving method is shown in Fig. 2.1 for k = b = 0 and nb∕ni =

16 Figure 2.1: A sample dispersion relation from the supplemental approach of numerically solving Eq. 2.1 for the given plasma parameters for a deuterium plasma and hydrogen beam. In this particular case, k = b = 0 and nb∕ni = 0.01. The red points indicate growth (instability), and the blue points indicate damped modes. Black indicates marginal stability. The solid lines indicate the Alfvén speed, and the dashed line indicates the beam resonance lines.

0.01. The other plasma parameters are those deﬁned above. The red points indicate growth (instability), and the blue points indicate damped modes. Black indicates marginal stability. This particular excitation will be elaborated in chapter 3.

2.2 PIC Simulation of the SOL

Particle-in-cell (PIC) simulations can, in a sense, serve as an exact solver for the types of linear waves excited in a plasma and their nonlinear coupling and evolution over time. In the exemplary case of the observed fusion enhancement in the C-2U experiment, we encounter the wakeﬁeld physics introduced in chapter 1 in the atypical context of a magnetized plasma. Understanding this physics thus requires a fully nonlinear treatment. Examining this case, we model IC waves in a deuterium (D) plasma with a hydrogen (H) beam in the C-2U SOL environment, corresponding to

17 a radius of 45 cm in the C-2U FRC. The near-uniform axial magnetic field in the SOL readily allows dissection of the important physics. We demonstrate the main kinetic dynamical processes via a PIC simulation with one spatial dimension and three velocity dimensions using the electromagnetic particle simulation code LSP [102]. The SOL plasma is modeled as a locally homogeneous deuterium (D) and electron plasma with a proton (H) beam in a uniform (axial) magnetic field. Collisional effects are minimal compared with collective processes. The spatial dimension of the simulation is labeled the ̂x direction. The external magnetic field is given a strength of B0 = 750 ◦ ◦ G and is directed at an angle 0 ≤ k ≤ 90 with respect to the x axis to define the direction of the ⃗ wavevector k. A schematic representation of the simulation geometry is shown in Fig. 2.2.

The plasma considered has three components: background thermal ions, background electrons, and beam ions. Each is designated with the subscripts i, e, and b, respectively, and designates an arbitrary species. In the case of the C-2U FRC, which is our primary interest in this work, the thermal ions are deuterium (D) and the beam ions are hydrogen (H). We will also brieﬂy consider the case of a boron-11 (B) thermal plasma with hydrogen beam, however.

The ions and electrons are given a temperature of Ti,e = 200 eV, and the beam is given Tb = 500 eV. All temperatures are isotropic; instabilities deriving from anisotropic temperature are not considered here (except in the sense that a beam population adds velocity anisotropy). The beam drift energy is Eb = 15 keV. The plasma component densities n are deﬁned relative to the ion density 12 −3 ni = 7 × 10 cm . The beam density is 10% of this value: nb = 0.1ni, and the electron density is determined by these two quantities to preserve quasineutrality.

Because the relevant length scale is that of ion cyclotron motion (i), which for the SOL environment is taken here to be much larger than the Debye length D, an implicit algorithm is adequate[35], allowing a grid spacing Δx ≫ D. In particular, the size and resolution of the spatial domain is such that the modes of interest are well-resolved in k space and the total domain size is much larger than the relevant mode wavelength to prevent aﬀects from the periodic boundary condition. The timestep Δt resolves all major frequencies lower than the plasma frequency !p, including the ion

18 Figure 2.2: A schematic representation of the 1D PIC simulation geometry. The one degree of ⃗ spatial freedom is taken as the ̂x direction, with the external magnetic field B0 oriented in the x-z plane at an angle k with respect to ̂x to define the angle of wave propagation. The beam population ⃗ is given perpendicular and parallel components with respect to B0 according to the angle b. plasma frequency !i and the electron cyclotron frequency Ωe), but we only concern ourselves with frequencies approximately ! ≤ 10Ωi, where Ωi is the ion cyclotron frequency. To further assist with the timestep, the electrons are given a mass of 20 times their realistic mass, with care taken to ensure than doing so does not perceivably affect the results. Because the collision timescale is much longer than the ion cyclotron timescale, collisions are neglected.

To analyze the mode structure of the waves in the simulation, a 2D FFT over the spatial and temporal ⃗ ⃗ dimensions of the perturbed fields is taken to give E(k, !) and B(k, !) in Cartesian polarizations. From these the circular polarizations are calculated, with handedness defined such a right-handed mode circulates in the same direction as an electron, and a left-handed mode circulates in the same direction as an ion, regardless of propagation direction. From the 1D nature of the simulation, ⃗ ⃗ Bx = 0 to ensure ∇⋅B0 = 0. Consequently, Bz always represents the magnetically compressional component of waves. To isolate the compressional component of the electric field, however, the ⃗ fields must be rotated. Additionally, as ̂y ⟂ B0, By always represents the magnetically shear component of waves. Meanwhile, Ex thus represents the electrostatic (longitudinal) component

19 ⃗ of waves. For the purposes of the dispersion plots that follow in this paper, ðkð ≥ 0, with ! < 0 ⃗ representing backwards propagation. While an alternative scheme where ! is kept positive and k is allowed to be reversed would also suﬃce, the present convention will aid in the illustration of mode structure in some instances.

20 Chapter 3

Beam-Driven IC Modes in the SOL of an FRC

Having now established the scheme for modeling IC waves, we can now proceed to a survey of beam-driven modes that can potentially exist in the C-2U SOL environment. In this section, the ◦ ◦ angular variables k and b are each scanned from 0 to 90 and the mode activity in each regime is examined. For each regime, the nonlinear acceleration of background ions is also examined. In particular, a distinction is made between modes that increase the bulk ion temperature and those that merely generate a tail of fast ions, the former being generally conducive to anomalous transport, instabilities, and turbulence, and the latter being relatively benign. The nonlinear physics is kept cursory in this chapter, however, and a deeper treatment is made in chapter 4. Most of the signiﬁcant beam-driven modes examined here easily comprehensible from the theoretical foundation of linear physics.

21 3.1 IC Waves Propagating Parallel to the Magnetic Field

First, consider the regime of wave propagation parallel (or nearly parallel) to the external magnetic ◦ ﬁeld, k ≈ 0 . In contrast to the case of perpendicular propagation, here only the cyclotron fun- damentals ! = ±Ω, as well as an unmagnetized contribution, enter into the dispersion relation. Discounting the plasma oscillation, this regime features three chief modes: two shear Alfvén modes √ and the ion acoustic mode, which in this regime obeys ! = kvs, where vs = (Te + 3Ti)∕mi is the ion sound speed. While the former two modes are transverse and have shear polarizations, the latter is electrostatic and longitudinal. However, in the present parameters, the ion acoustic mode is strongly damped and is not of concern. In contrast, the shear Alfvén modes are potentially excitable.

These two transverse modes are circularly polarized, one right-handed and the other left-handed. At low frequency, ! ≪ Ωi, both branches obey ! = vAk, but begin to diverge once ! approaches Ωi.

The right-handed branch thus does not resonantly couple to ions and passes through Ωi unimpeded. The mode in this frequency regime is termed a Whistler [89] before ﬁnally resonating the electrons at ! = ðΩeð. In contrast, the left-handed branch cannot pass through Ωi and instead resonates with ions. This mode is thus termed the Alfvén-ion-cyclotron (AIC) mode [97] and can be excited by a temperature anisotropy in the ions [96]. In mirror machines, such a mode can lead to scattering of ions into the loss cone. In a plasma composed of only thermal ions and electrons with purely parallel ◦ propagation (k = 0 ), the mode structure is shown in Fig. 3.1 where the right- and left-handed components are distinguished. The resonance of the AIC mode with the ion cyclotron resonance cone ! − Ωi = kvti is visible. While only the electric ﬁeld can impart work on ions, for the plasma beta in this case ( = 0.2), it should be noted that the magnetic energy content of these waves dominates the electric energy content.

With the addition of an energetic beam population, either of these shear modes can become excited depending on the beam resonance condition. Fig. 3.2 shows the case of purely parallel propagation ◦ and beam injection (k = b = 0 ) with the beam resonance lines indicated. As consequences of the

22 (a) (b)

Figure 3.1: Dispersion relations for right-handed (3.1a) and left-handed (3.1b) components of the ◦ electric field for purely parallel propagation (k = 0 ) and no beam population. Frequency is normalized with respect to the background ion (deuterium) cyclotron frequency Ωi, and the wavevector is normalized with respect to vA∕Ωi. The space where ! < 0 indicates backwards propagation. The intensity at at particular mode is indicated by the heat-map and is normalized with respect to the maximum value. The Alfvén velocity ! = kvA is indicated as a dashed line. The dotted line indicates the approximate region of strong ion cyclotron damping, ! = 2kvti ± Ωi. purely parallel propagation, only the p = ±1 beam resonance lines enter into the dispersion relation, and these resonance lines have their maximum possible slope. With the present beam energy, these conditions cause the p = −1 line to excite the right-handed mode in the forward direction, and, to a lesser extent, the left-handed mode in the backward direction, both at roughly ð!ð ≈ Ωi. This mode structure is also elucidated in the semi-analytical solution in Fig. 2.1. There is furthermore a mode associated with the beam resonance lines themselves, for which the p = +1 line is left-handed, and the p = −1 line is right-hand. Interestingly, the ! < 0 portion of the right-handed waves are also bent “into line” with the p = −1 resonance, crossing ! = 0. The right-handed excitations bear resemblance to the right-hand resonant and non-resonant modes studied previously [38, 37]. The excitation of the right-handed mode with ! > 0 is extremely robust and may lead to inefficient but finite coupling to ions, as an increase in the ion perpendicular temperature to 275 eV is produced in this case. It is also possible that the much smaller left-handed excitations may be producing this warming, which could do so much more efficiently.

23 (a) (b)

Figure 3.2: The dispersion relation for right-handed (3.2a) and left-handed (3.2b) components of ◦ the electric ﬁeld for purely parallel propagation (k = 0 ) and a purely parallel-streaming beam pop- ◦ ulation (b = 0 ). The right- and left-handed beam resonance lines (! = k∥vb∥ ± Ωb) are indicated with dotted lines.

For near-perpendicular beam injection, the excitation shifts predominantly to the backward-propagating ◦ right-handed mode, as is shown in Fig. 3.3 for b = 75 , but is much less robust than in the case of parallel beam injection. This mode generates similar bulk heating as in the previous case, how- ◦ ever. If the beam injection angle is increased further to b = 90 , mode activity diminishes greatly, the beam resonance lines coupling poorly with both shear Alfvén modes. Instead, the dominant mode activity falls along the beam resonance lines themselves at the hybrid ion resonance [12] at

!∕Ωi ≈ 1.9. This mode is left-hand polarized for ! > 0 and right-hand polarized for ! < 0.

In both of these cases, a signiﬁcant portion of the wave activity driving ion acceleration occurs for phase velocities within reach of the ion thermal distribution, particularly for perpendicular beam injection. The resulting increase in perpendicular temperature opens the possibility of the AIC instability caused by temperature anisotropy, and in a mirror (or FRC SOL) situation could lead to scattering of ions into the loss cone. In the C-2U experiment, the beam injection angle is close ◦ to b = 75 , and so the near-perpendicular injection case examined here (Fig. 3.3) likely presents a closer picture to the experimental reality. However, evidence of such modes, such as loss of

24 (a) (b)

Figure 3.3: The dispersion relation for right-handed (3.3a) and left-handed (3.3b) components of ◦ the electric ﬁeld for purely parallel propagation (k = 0 ) and nearly perpendicularly injected beam ◦ population (b = 75 ).

confinement of ions, has not been seen in this experiment, suggesting that either the dominance of other modes or that the resultant perpendicular heating is too subtle to observe. Modes propagating parallel to the external field may also be fundamentally difficult to observe given the configuration of this experiment.

3.2 IC Waves Propagating Obliquely to the Magnetic Field

If the angle of wave propagation is increased, the circularly polarized shear modes and ion acoustic mode explored in the previous section gradually transition into linearly polarized shear and com-

pressional Alfvén modes. The former mode follows ! = k∥vA and is restricted to ! < Ωi. This ◦ ¨ mode can therefore not propagate if k = 90 . The latter mode follows ! = kvA in the limit of long t ¨ 2 2 wavelengths (as is relevant to this work), where vA = vA + vs ≈ vA in this parameter regime, and converges to the lower-hybrid frequency !LH for short wavelengths. This mode is called variously the magnetoacoustic, magnetosonic, or extraordinary mode.

25 The details of this evolution are partially determined by the ion sound speed vs. If vs > vA, the ¨ ion acoustic mode migrates to the new phase velocity vA and adopts a magnetically compressional polarization while the right-handed shear mode “wilts”, shifting to follow ! = k∥vA and becoming strongly damped. Conversely, if vs < vA, the right-handed shear mode shifts to the phase velocity ¨ vA and becomes a compressional mode while the ion-acoustic mode “wilts”, following ! = k∥vs and inevitably becoming increasingly strongly damped. In the present parameter regime, the latter ¨ case prevails, and as vs ≪ vA, we hereafter take vA = vA.

In addition to its magnetically compressional character, the compressional mode possesses a hybrid electric character, having both a longitudinal and transverse component. This longitudinal component will prove highly eﬀective at generating a fast ion tail, as is explored in section 3.3 and more thoroughly in chapter 4. However, the shear mode also possesses a longitudinal electric component, ◦ which for k < 90 may generate ion heating.

◦ For k ≈ 90 , the compressional mode features resonances with ion-Bernstein harmonics, particularly those of the beam population. In this section, however, our interest is the mode structure ◦ for k ≈ 60 , where the Alfvén modes become dominantly linearly polarized, but without ion- Bernstein resonances. The mode structure without a beam population is shown in Fig. 3.4. Some of the remnants of the parallel-propagation mode structure can be seen, such as in the slight up- ward or downward curvature of the compressional mode for ! > 0 and ! < 0, respectively, and the presence of a shear component of the nominally compressional mode.

The addition of the beam population excites these base modes. In this parameter regime, the excita- ◦ tion is comparable between the compressional and shear modes, as is shown in Fig. 3.5 for k = 60 and b = 15. Because we have moved away from the parallel-propagation regime, all harmonics of the beam resonance condition are now available. For near-perpendicular beam injection, the mode activity generally diminishes and shifts to a compressional mode that may be due to the n = 0 beam resonance line. In the present case, substantial perpendicular bulk heating of the ions occurs, with the perpendicular ion temperature approximately doubling. In the case of near-perpendicular beam

26 (a) (b)

Figure 3.4: The dispersion relation for the compressional (3.4a) and shear (3.4b) components of ◦ the magnetic ﬁeld for oblique propagation (k = 60 ) and no beam population. injection, there is essentially no heating of background thermal ions.

3.3 IC Waves Propagating Perpendicular to the Magnetic Field

◦ In the regime k ≈ 90 , cyclotron harmonics, typically in the form of ion-Bernstein modes, assert themselves over the continuum modes of the previous section. As these modes are typically electrostatic (longitudinal) in nature, electrostatic physics with regard to wave-particle coupling becomes particularly important. The case of purely perpendicular propagation is not examined because both mode excitation and wave-particle coupling tend to be poor. A small but finite k∥ typically makes a dramatic difference, possibly because particles can stream along with the wave to a small degree. ◦ The mode structure for the near-perpendicular regime k = 85 is shown in Fig. 3.6. The compressional mode forms the spine of the wave activity, but resonances with the even cyclotron harmonics can now be seen in the form of ion-Bernstein modes, in contrast to the case of the previous section. These resonances are greatly amplified by the presence of a beam population and create ion acceleration of sufficient interest that the nonlinear interaction of these modes with the background

27 (a) (b)

Figure 3.5: The dispersion relation for the compressional (3.4a) and shear (3.4b) components of ◦ ◦ the magnetic ﬁeld for oblique propagation (k = 60 ) and a beam population injected at b = 15 . The beam resonance lines (now with all harmonics available) are indicated by the dotted lines. ions is given a separate chapter. Also visible is the linearly polarized shear Alfvén mode, clinging to life in the near-perpendicular propagation regime.

When a beam population is added to the plasma, the mode structure in Fig. 3.6 is modified in distinct ways depending on the beam injection angle. These differences broadly manifest according to the relevance of the cyclotron motion of the beam ions. For purely parallel injection, there is essentially no beam cyclotron motion. Thus, the background ions solely dictate the IC mode structure. The case of perpendicular waves driven by a streaming population has been investigated extensively [7, 56, 28, 101, 20, 104], though mostly in a low-beta regime. In such a regime, many cyclotron harmonics are excited with the mode growth rate broadly monotonically decreasing for higher harmonics. The excitations, while sharp in frequency space, spread over a large range in k at essentially constant frequency. In the present case, however, where = 0.2, the electromagnetic ◦ ◦ contribution is substantial, as is shown in Fig. 3.7 for k = 85 and b = 0 . Notably, the first few harmonics are modified by an excitation of the shear Alfvén mode. The excitation of these modes is remarkably weak compared with the other regimes examined, and as a partial consequence produce weak ion heating. Of course, the principle of heating plasma by injection of a streaming species is

28 Figure 3.6: The dispersion relation for the longitudinal (electrostatic) component for near- ◦ perpendicular propagation (k = 85 ) without a beam population. The linearly polarized shear Alfvén mode (! = k∥vA), which has phase velocity vA cos k, is indicated by the dash-dot line. well-established, but for higher beta values, this strategy may merely become less eﬀective.

In the opposite regime, that of near-perpendicular beam injection, the magnetoacoustic instability may be excited, as has been investigated previously [23, 25, 15, 24]. The exemplary case of ◦ ◦ k = 85 and b = 75 is shown in Fig. 3.8. The intersections of the beam resonance lines and the compressional mode determine the dominant excitations. Because both k∥ and vb∥ are ﬁnite, Doppler shifting causes the resonant points to deviate from the pure beam cyclotron harmonics. For the angular parameters use in Fig. 3.8, the result is a coherent, additive summation of harmonics in the positive propagation direction, creating a Dirac-comb-like wave, and a dominant beam cyclotron fundamental in the backwards propagation direction. The fundamental frequency for both is approximately the beam cyclotron frequency, ! = 2Ωi. The wavelength of this mode is roughly 40 cm. Additionally, because the phase velocity of this mode is much faster than the ion thermal

29 Figure 3.7: The dispersion relations for the electrostatic component for near-perpendicular propa- ◦ ◦ gation (k = 85 ) and a beam population injected at b = 0 . speed (vA∕vti ≈ 4), there is little coupling to the bulk motion of ions [76]. Consequently, in real space, the electrostatic component of this mode structure manifests as coherent, periodic sharp peaks propagating in the forward direction that ride atop a backwards-propagating sinusoidal wave. This field configuration is highly amenable to allowing the nonlinear generation of a fast-ion tail. This electrostatic component of the mode also produces density fluctuations in the plasma with an essentially identical mode structure. The mechanics and consequences of the nonlinear coupling of this mode to the ions is treated in chapter 4. Also visible are secondary resonances between the beam resonance lines and the ion-Bernstein modes, as well as perhaps a small resonance between the p = 0 beam mode and the linearly polarized shear Alfvén mode.

30 Figure 3.8: The dispersion relations for the electrostatic component for near-perpendicular propa- ◦ ◦ gation (k = 85 ) and a beam population injected at b = 75 . 3.4 Analogy to a Proton-Boron-11 Plasma

One of the ultimate goals of the successor experiments of C-2U is to achieve a burning proton- 11 boron-11 plasma, a conﬁguration hereafter abbreviated pB . It is thus worthwhile to brieﬂy consider how the physics examined thus far for a deuterium plasma and proton beam applies to this case. The boron-11 (the a fully ionized charge of +5) is treated as the background plasma, and the protons are treated as the beam. Compared to the previous cases then, essentially the only change is that the background ion species is swapped for boron-11, with the electron density increased to preserve quasineutrality.

With the aim of reproducing the favorable fast-ion tail generation seen for a deuterium plasma for ◦ ◦ k = 85 and b = 75 , an intuitive comparison of boron-11 and deuterium may suggest that both

31 should yield similar physics, as the cyclotron frequencies are nearly identical, as well as the ratio

of Alfvén speed to ion thermal velocity. While qi∕mi is nearly the same in both cases, however, the species plasma frequencies !i are distinct, owing to the +5 charge of boron-11. With the subscripts 2 B and D respectively denoting boron-11 and deuterium, (!B∕!D) = 5. In Eq. 2.4, this squared ratio boosts the contribution of the background ions in the plasma dielectric tensor and is eﬀectively equivalent to lowering the density of the beam population. Consequently, the resonances of the compressional mode and beam resonance lines seen in Fig. 3.8 are muted, and the magnetoacoustic instability is overall much less robust.

Instead, electrostatic activity is found more robustly in the slightly more oblique propagation regime considered in section 3.2. An example case is shown in Fig. 3.9. The excitation broadly falls along the compressional Alfvén mode. In real-space, this mode structure manifests as a wave two length scales: a smaller wavelength corresponding to roughly = vA∕Ωi and a longer length (about 5) over which the excitation is small. This behavior can be understood as a consequence of the low-frequency component of the excitation in Fig. 3.9; in the limit of a continuous sum over all frequencies, the wave tends towards a Dirac-comb function with an inﬁnite length between peaks. The incomplete summation in this case may lead to the long, but ﬁnite, stretches between wave peaks.

Regarding enhancement of the fusion rate, some ion acceleration is seen, similar to that seen in section 3.2. In this case, however, the fusion is of the beam-target nature, so any enhancement is more modest compare to the baseline level. It should also be noted that this simulation has been made 11 with parameters mimicking C-2U, while those of future pB experiments would have significantly different parameters, particularly in external magnetic field, plasma temperature, and beam energy. This work seeks only draw attention to a fundamental difference between boron-11 and deuterium and a background plasma species. Additionally, these results suggest that maintaining a relatively low beta of = 0.2 in the SOL of these experiments may be beneficial for fusion enhancement, despite the otherwise different parameters. Excitation of the magnetoacoustic mode in a boron-11

32 Figure 3.9: The dispersion relation for the longitudinal (electrostatic) component for a boron-11 ◦ ◦ plasma for oblique propagation (k = 75 ) with a proton beam population injected at b = 60 . In this particular case, k < 0 is used to indicate backwards propagation. The beam velocity is indicated by the dotted line with spacing to show comparison with the other characteristic velocities. plasma may also require a stronger beam presence.

3.5 Summary and Conclusions

This section has surveyed in simulation many of the beam-driven modes that can potentially exist in the SOL of the C-2U FRC experiment for a deuterium plasma and hydrogen (proton) beam, ◦ ◦ scanning the wave propagation and beam injection angles between 0 and 90 . The most significant beam-driven modes examined in this scan are understandable in the context of linear wave theory. The observation of an enhanced fusion rate in this experiment provides the chief impetus for this study. In particular, the mode responsible should preferably not cause turbulence or bulk temperature increase, as such eﬀects would be observable through various means such as loss of conﬁnement. The capacity for modes in each angular regime for generating fusion enhancement though ion accelerating is thus evaluated. As the fundamental linear physics is treated, these results are also potentially applicable to devices of similar parameters. Within this experimentally

33 motivated set of plasma parameters, the linear physics of each regime can be brieﬂy summarized.

◦ The regime of parallel propagation (k ≈ 0 ) predominantly features two circularly polarized shear Alfvén modes, either of which may be excited depending on the beam resonance condition, which in turn depends on the beam injection angle. In both cases, left-handed modes strongly resonate with ions and can lead to a bulk increase in the perpendicular temperature of the ion population. The mode excitation is generally less robust for perpendicular beam injection, but an increased preference for left-handed modes in this case partially compensates for this effect with regard to ion acceleration. In the case of the C-2U experiment, where beam injection is nearly perpendicular to the external magnetic field, it is expected that the latter situation would be more likely. These modes have also been used extensively for ion-cyclotron resonance heating (ICRH), both increasing the plasma temperature and plugging a mirror geometry. Such heating normally incurs the AIC instability (which goes by the same name as the beam-driven mode examined here but is caused by temperature anisotropy), however, which scatters particles into the loss cone. Significant perpendicular heating and loss of confinement in this manner has not been observed on the C-2U experiment. If this mode is present in the SOL in this case, its effects are (thankfully) sufficiently subtle to neglect.

◦ In the regime of oblique propagation, k ≈ 60 , the circularly polarized shear Alfvén modes are replaced by linearly polarized compressional and shear Alfvén modes. While the former mode can be prominent even in near-perpendicular propagation, the latter cannot propagate in the limit ◦ k → 90 . Ion acceleration in the oblique propagation regime is generally modest, but is strongest for parallel beam injection. For the C-2U experiment then, and similar devices, it may be possible to neglect the eﬀects of these modes.

◦ The most interesting regime occurs for near-perpendicular propagation, k ≈ 90 . Here, electrostatic ion-Bernstein modes rise to prominence at the ion cyclotron harmonics and alter the dispersion of the compressional Alfvén mode. With perpendicular beam injection, the beam resonance lines form Doppler-shifted resonances with the compressional Alfvén mode, which we identify

34 as the magnetoacoustic instability. The exclusively fast phase velocity and discreteness in mode structure for this mode aﬀord eﬃcient acceleration of a fast-ion tail without increasing the ion temperature.

11 One of the future goals of the successor experiments to C-2U is the achievement of a burning pB plasma, warranting a brief examination of the analogy of this magnetoacoustic mode in such a plasma. While boron-11 and deuterium have nearly identical cyclotron frequencies, a difference in the species plasma frequencies owing to the +5 charge of boron-11 effectively diminishes the role of the beam, muting the beneficial magnetoacoustic mode seen in the deuterium plasma case. Instead, continuum modes, such as the compressional Alfvén mode, are favored for electrostatic activity. Some ion acceleration is seen, but the contribution of such to the fusion rate is likely small relative to the beam-target contribution. However, this examination was done for C-2U parameters, 11 and the physics is likely somewhat different for a pB -burning plasma. Future work should more closely at realistic parameters for these plasma species.

While linear plasma wave physics is a well-trod path [90, 50], the microscopic beam-driven phenomena of the SOL in an FRC geometry warrant a clear theoretical understanding. In addressing this goal, the accelerator physics at the heart of the C-2U and related experiments perhaps inevitably comes to the fore. These results suggest that the ﬁeld of accelerator physics may be further harnessed to in the support of fusion eﬀorts.

35 Chapter 4

Wakeﬁeld Acceleration with IC Resonance

The survey of beam-driven IC modes in the previous chapter revealed the ion-Bernstein modes of the near-perpendicular propagation regime to be of particular interest for benign ion acceleration. Here, we ﬁrst examine more closely the nonlinear physics of how this mode is able to accelerate ions eﬃciently, including a cyclotron resonance condition [76]. Next, a scheme for seeding this mode with beam density bunches is explored over a wide range of beam energies.

4.1 Magnetized Wakeﬁeld Acceleration

Our scan of IC wave activities by the beam over the angular parameters k and b shows approx- ◦ ◦ imately k = 85 and b = 75 as the domain of most active fast-ion generation. Deﬁning P and fusion power and Ptℎ as the initial thermonuclear fusion power, Fig. 4.1 indicates the prominence of this parameter regime. The ion acceleration is the result of beam-driven IB modes, which take the form of resonances of the beam ion-cyclotron harmonics and the Alfvén speed according to the beam resonance condition ! = k∥vH∥ + pΩb, where p is an integer (related to the harmonic number n in Eq. 4.1 via p = (Ωi∕Ωb)n) and vb∥ = vb cos b is the parallel component of the beam velocity.

36 Figure 4.1: The maximum fusion enhancement P ∕Ptℎ for a scan over angular parameters k (verti- ◦ cal axis) and b (horizontal axis). Each angular parameter is sampled in mostly 15 increments but ◦ ◦ ◦ also includes 5 and 85 . The bright peak in the upper right corner corresponds to k = 85 and ◦ b = 75 , which was used in generating ﬁgure 3.8.

We may then state simply vpℎ = vA. After growing from noise, the mode reaches peak activity between roughly t = 8i and t = 10i, where i = 2∕Ωi, where i = 2∕Ωi.

This mode manifests microscopically as periodic coherent peaks in the longitudinal ﬁeld Ex with fundamental cyclotron harmonic n = 2 (i.e., ! ≈ 2Ωi). Resonance between the n = 2 harmonic and the Alfvén velocity yields the approximate wavelength = 2vA∕2Ωi. The presence of higher harmonics likely contributes to the Dirac-comb-like structure of the peaks. These waves accelerate ions as they pass through the plasma, as is shown in Fig. 4.2, which overlays particle phase space

(vx vs. x in the lab frame) with the longitudinal electric ﬁeld Ex for a few wavelengths at a snapshot in time when mode activity is near saturation. The mode travels in the forward direction (+ ̂x, to the right) with vpℎ = vA. Also present is a backwards-propagating (− ̂x, to the left) single harmonic mode at ! ≈ 2Ωi with smaller amplitude than the forward-propagating mode. The interaction of the forward- and backward-propagating components manifests as periodic amplitude modulation of the forward-propagating peaks at approximately twice the mode frequency. In Fig. 4.2, the forward-propagating peaks have nearly reached the point of full constructive interference with the backwards-propagating component, but nonetheless have achieved essentially their maximum sat-

37 uration amplitude. This time step has been chosen to more clearly show the ion acceleration, which

is modulated at the same frequency as Ex but with some phase delay.

The fast phase velocity vpℎ = vA ≫ vti of the magnetoacoustic mode shown in Fig. 3.8 allows the mode saturation level to depart from that predicted by quasilinear theory. Instead, the waves grow until reaching a robust saturation without strong damping on the bulk plasma. The ionic √ wave trapping velocity [79] vtr = qiE∕mik, where E is the electric ﬁeld of the wave and k is the mode wavevector, grows with the wave amplitude until reaching nearly the wave phase velocity.

For vti ≫ vpℎ, the saturation amplitude converges to the Tajima-Dawson ﬁeld (Eq. 1.4) [95], which

is commonly encountered in wakeﬁeld acceleration and can be given for ions as Es = mi!vpℎ∕qi. In the present case, where the mode frequency approximately corresponds to an integer multiple n

of the ion cyclotron frequency and vpℎ = vA, this expression becomes

v E (n) = nB A . s 0 c (4.1)

Including the correction from a ﬁnite value of vti∕vpℎ, the adjusted saturation level (from Eq. 1.5) is

0 12 ¨ vA vti Es(n) = nB0 1 − . (4.2) c vA

In terms of , using Eq. 1.6,

v √ 2 E¨(n) = nB A 1 − . s 0 c (4.3)

Once reaching this saturation level, the wave can begin damping on the fastest thermal particles, accelerating them to much higher energies. With the ratio vpℎ∕vti ≈ 4.3, the wave is largely isolated from the bulk plasma for small amplitudes and saturates near the Es(n) ﬁeld in Eq. 4.1 with n = 2, when its trapping width stretches into the ion thermal distribution. The maximum saturation level indicated in Fig. 4.2 and lies between that given by Eq. 4.1 for n = 1 and n = 2, suggesting

38 that the saturation mechanism is indeed that of wakeﬁeld physics. However, the ﬁniteness of the

ratio vpℎ∕vti likely acts to reduce the saturation level from the maximal predicted value. Indeed, 2 the correction factor (1 − vtℎ∕vpℎ) ≈ 0.6 brings the simulation saturation level approximately into ¨ line with what expected from Es(2) in Eq. 4.2. In supplemental runs where the ions are initialized with a colder temperature, raising this ratio, the saturation level indeed converges to that predicted

by Es(2) (Eq. 4.1).

Modulation of the ion acceleration results in discrete packets of accelerated ions rather than a continuous stream. Near each of the wave peaks exists such a packet of ions accelerated in the forward

direction, the fastest of which reaching roughly vx = 2vpℎ. These ions with vx > vpℎ consequently

appear slightly in front of the peaks as well. On the vx < 0 side, similar packets of fast ions are

visible with approximately vx = −2vpℎ. A typical accelerated ion begins by executing several un- perturbed cyclotron revolutions, and then receives a strong kick from the mode, expanding to a larger Larmor radius with only modest shifting of the gyro-center. The phase-space evolution of

a single sample deuteron is shown in Fig. 4.3. After saturating at v ≳ vpℎ, its new gyro-center remains essentially ﬁxed. A similar eﬀect can occur in the y position of the ion. Because the kick

to the ion gyro-center does not occur after saturation (v ∼ vpℎ), the wave does not cause a systematic drift, a novel feature as compared with the case of non-magnetized wakeﬁeld [95]. Meanwhile, the ﬁeld Ex produces only transient periodic shifts in the bulk distribution of vx. The wave can therefore exist with the robust TD amplitude while maintaining macroscopic plasma stability.

The efficient ion acceleration leads to the accumulation of a large fast-ion tail, as is shown in Fig. 4.5. The long energy tail derives almost entirely from the ion perpendicular velocity distribution; the parallel distribution is largely unaffected. Because the accelerated tail does not substantially change the bulk perpendicular temperature of the ions, however, the AIC temperature anisotropy instability [96] is not seen. The fastest ions saturate around the beam velocity, which is in general agreement with the experimental results [62]. The acceleration of a large tail while leaving the bulk thermal distribution unchanged is characteristic of wakefield acceleration. Meanwhile, a schematic

39 Figure 4.2: A snapshot of the the phase space (heatmap) of ions (D) near peak mode activity overlaid with the electrostatic ﬁeld (teal) normalized to the Tajima-Dawson saturation Es(n) (Eq. 4.1) for n = 2. The wavelength = 2vA∕2Ωi likewise corresponds to the n = 2 resonance of Ωi with vA. Brighter heatmap colors indicate higher phase space density. The positive x direction is to the right. qualitative sample of the evolution of a velocity distribution in a quasilinear saturation case is shown in Feg. 4.6 for a wave with phase velocity in the region of positive slope. In this case, the thermal part of the distribution is modiﬁed while leaving the tail essentially unchanged.

This fast-ion tail translates into D-D fusion enhancement. Figure 4.7a shows the evolution of the normalized fusion power P ∕Ptℎ during the simulation for the neutron-producing D-D branch. As the mode builds the fast-ion tail, the normalized fusion power grows from the thermonuclear base- 4 line (unity) to more than 10 times this initial value. (The method for calculating fusion rate from the simulation is given in appendix B.) The growth in fusion rate lags somewhat behind the growth of ﬁeld energy in the mode, as is expected. Figure 4.7b oﬀers a qualitative comparison of this result with the fusion enhancement observed for a typical shot on the C-2U (D plasma, H beam) experi- 2 ment. The experimental data indicates a fusion rate of order 10 over that expected from the bulk ion temperature and shows qualitative and parametric agreement between experiment and theory. These results suggest that the IC mode under consideration is capable of generating the anomalous neutron signal seen in experiment. Various realities of the experiment would tend to produce lower

40 Figure 4.3: The phase evolution of a single tracer deuteron from t = 0 to t = 12i with the wave phase velocity (vpℎ = vA) indicated. The kick to a higher velocity occurs around t = 8i.

fusion enhancement than those in this simulation study, such as a slowing-down distribution for the beam, the slow accumulation of beam density, and the presence of other modes.

While the ions strongly interact with the wave in this fashion, the electrons see an essentially ⃗ ⃗ static ﬁeld and respond adiabatically. The electrons brieﬂy experience an E × B drift given by

⃗vEB ≈ −2vA ̂y as each wave passes. Such a mean shift in the electrons (by 2vA∕vte ≈ 0.6) is indeed seen in the simulation following the wave peaks. Furthermore, because the electrons respond to-

gether in this drift, the mean of the electron vy distribution merely shifts by vEB; turbulence is not created.

The growth of this mode also deviates from the usual picture of quasilinear physics. In such a model, a positive slope in the velocity distribution function ()f∕)v > 0) provides free energy for the wave to grow. More particles are decelerated by the wave than accelerated, and the region of positive slope ﬂattens. In the present case, however, for the total distribution of beam and thermal ions, the phase velocity vA lies within a region where )f(vx)∕)vx < 0, and indeed the mode can

41 Figure 4.4: The vx velocity distribution for the combined background ion and beam ion populations for the initial and ﬁnal simulation timesteps for the ion-Bernstein mode shown in Fig. 4.2 with the mode phase velocity indicated. still grow exponentially even with warmer ions, reinforcing the negative slope at the phase velocity.

The initial and final vx distributions for the combined population of background and beam ions is shown in Fig. 4.4. The resolution of this situation is that the free energy is coming not from a driver at the phase velocity, but rather at the beam velocity, which has vb ≫ vA. The coupling of the beam resonance with the compressional mode allows the free energy of the beam to be channeled into a wave with phase velocity vpℎ = vA. The beam population contains sufficient such free energy to drive an exponentially growing mode even in a region of velocity space that is quasilinearly stable. The mode then grows until saturating via Tajima-Dawson wakefield physics. The population of fast beam particles then remains as a reservoir of free energy for the mode. Over the course of a simulation, the beam population gradually loses energy to the mode and thermal plasma, until the free energy is depleted, causing the mode to dissipate.

Another distinction from typical quasilinear saturation is in the realm of saturation through coupling to other modes. Some considerations suggest that such a mechanism does not play a substantial role in the present case. First, the excitation of higher beam cyclotron harmonics is a fundamental feature of this particular mode and, in this case, the harmonics coherently add together. A composite

42 Figure 4.5: The normalized ion (D) energy spectrum for various time snapshots, where i is the ion cyclotron period, and Eb is the beam (H) injection energy. At t = 0 (blue curve), all ions are thermal. coupling between the wave and particles is formed; the higher harmonics are not given the chance to damp individually. Rather than dissipating energy, the higher harmonics merely increase the sharpness of the waveform until the wakeﬁeld saturation level is reached. This conclusion is further strengthened by the supplemental simulations with a colder ion population, suggesting that the relationship between the wave phase velocity and ion thermal speed are the crucial factors determine the saturation amplitude. If dissipation through higher harmonics occurs, it does not seem to appreciably aﬀect the saturation level in this case.

The principle of high phase velocity is critical for the nondestructive ion acceleration shown here for deuterium, but also important, and implicitly satisfied in the present case, is a cyclotron resonance condition. This condition, involving the harmonic relationship between the wave frequency and ion cyclotron frequency, plays a significant role in the efficiency of acceleration, which to our knowledge has not been considered previously.

43 Figure 4.6: A sample schematic velocity distribution at initial (blue) and later (red) stages of quasilinear saturation. 4.2 Cyclotron Resonance Condition

In an unmagnetized plasma, a trapped particle can effectively ride along with the wave indefinitely. In contrast, in a magnetized plasma for waves propagating perpendicularly to the external magnetic field, a particle can only experience periodic trapping. Particles are effectively fixed in space by cyclotron motion but can be accelerated periodically by passing wave peaks. This periodic acceleration is most efficient if the wave frequency is an integer n ≥ 1 multiple of the particle cyclotron frequency: !∕Ωi = n.

If this condition is satisfied, an ion accelerated to near vpℎ then encounters a wave peak every cyclotron period, allowing multiple sessions of acceleration until the ion reaches v ≈ 2vpℎ, at which wave-particle coupling becomes poor. The fundamental harmonic n = 1 is the most efficient case, but modes with n > 1 (as in the case of n = 2 examined here) can also produce efficient acceleration because the wave-particle coupling is negligible while the ion is traveling in the opposite direction

of the wave (the “back-swing”), during which time it passes n − 1 peaks. Indeed, if !∕Ωi = n, the

44 (a) (b)

Figure 4.7: Enhanced fusion reactivity comparison. (a) The simulated evolution of the D-D fusion power (neutron branch) normalized to the initial thermonuclear value Ptℎ. (b) Experimental fusion enhancement in C-2U normalized to the thermonuclear value Ptℎ. efficient recapturing by the wave likely compensates for any losses incurred losses during the “back- swing”. For n ≫ 1, however, efficiency increasingly suffers. If n is not an integer, an ion does not encounter a wave peak every cyclotron revolution, impeding acceleration. (One can also imagine the perverse case where the ratio !∕Ωi is irrational, creating the least efficient cast of acceleration.) In practice, however, the finite width of wave peaks allows some tolerance for slightly non-integer n. As a consequence of this mechanism, a deuterium plasma and proton beam is well-optimized for ion acceleration, and thus the wakefield physics in Fig. 4.2 can operate nearly unimpeded (and perhaps even assisted) by cyclotron resonance concerns.

4.3 Enhancement from Beam Bunching

The fast-ion tail generated by this mode is potentially beneficial for efforts at fusion energy. The benefit and efficiency of this process may be even further increased if this mode is excited directly, obviating the need for the mode to grow spontaneously from noise from the free energy provided by the beam. A potential means of doing so is indicated by the beam and ion density fluctuations also excited by the electrostatic component of this mode, similar to the self-modulation instability in ion-driven wakefield acceleration [81, 1, 65], where an injected ion beam becomes bunched at

45 the plasma wavelength, improving wakeﬁeld development. Seeding the beam density with bunches [14] with the same wavelength as that of the mode could thus immediately and robustly excite the mode, akin to playing a musical instrument at a particular note.

If the beam density is seeded with square wave bunches with period b = vA∕Ωi, corresponding to the fundamental resonance of the mode at ! = 2Ωi, such an eﬀect is indeed seen. The fundamental resonance at ! = 2Ωi, as well as its higher beam harmonics, become sharply dominant in the dispersion relation, and the real-space structure of the mode becomes more robust, consistent, and coherent. Additionally, a signiﬁcant boost to the fusion rate growth is also seen, likely a result of this cleaner mode structure.

◦ To further examine the beneﬁt to fusion rate from beam bunching, the run with k = 85 and ◦ b = 75 is repeated for a scan of beam velocity values in the range vb∕vti ∈ [4, 128]. Concerns of numerical stability deﬁne the upper limit. The cases of no bunching, bunching corresponding to the resonance at ! = 2Ωi, and bunching corresponding to ! = Ωi at each beam velocity are treated. The maximum D-D fusion rate normalized to the initial thermonuclear rate in each run is then tabulated in Fig. 4.8 and compared with the approximate result from the C-2U experiment.

The singular case considered previously roughly corresponds to vb∕vti = 16.

The points with no bunching indicate a power-law relationship in the fusion enhancement of approx- 5.6 imately P ∕Ptℎ ∝ (vb∕vti) at this ion temperature. With bunching corresponding to the mode at

! = 2Ωi, there is a further enhancement of fusion power over the un-bunched case for vb∕vti ≲ 16.

In the same range, the fusion enhancement from bunching corresponding to ! = Ωi for comparison is indistinguishable from the un-bunched case. This result makes sense. Not lying along a beam resonance line, ! = Ωi is not spontaneously excited in the un-bunched case, and so attempting to do so with bunching effectively produces inaccessible free energy. This example demonstrates the importance of choosing the correct mode at which to bunch the beam; with Doppler-shifting of frequencies caused by finite k∥ and vb∥, this effort may be nontrivial in some cases.

46 Figure 4.8: The maximum D-D fusion rate normalized to the initial thermonuclear rate for various ◦ ◦ beam velocities for the case k = 85 and b = 75 . Black points have no beam bunching; red points, bunching at b = vA∕Ωi, corresponding to the resonance at ! = 2Ωi; and blue points, bunching at b = 2vA∕Ωi, corresponding to ! = Ωi. The purple point is the approximate position of the observed fusion enhancement in the C-2U experiment.

For the higher range of beam velocities, vb∕vti ≳ 16, bunching at both ! = Ωi and ! = 2Ωi converges in the resulting fusion enhancement. This departure from the lower-energy beam regime is likely a result of a change in the excited mode structure for beams of such energy. Rather than the ion-Bernstein resonances as in the case of Fig. 3.8, the n = 0 beam resonance mode become dominant. As this mode has ! < Ωi and lies along a continuum, bunching at all harmonics ! = nΩi likely becomes equivalently ineffective. Nonetheless, for the regime considered in the C-2U experiment, bunching of the beam population, perhaps by pre-exciting the magnetoacoustic mode with a uniform beam or with RF techniques, may allow a systematic increase in fusion efficiency. Here, knowledge from the field of plasma accelerators may be of significant assistance.

11 It is also useful here to consider the case of a pB plasma, whose linear physics was brieﬂy considered in the previous chapter. For comparison, the real-space electric ﬁeld and ion phase phase space corresponding to Fig. 3.9 is shown in Fig. 4.9. Also visible is the dual length scale described in the previous chapter. Because of the continuum mode structure seen in Fig. 3.9, there is no particularly

47 ◦ Figure 4.9: A snapshot of the the phase space of ions (B) near peak mode activity for k = 75 and ◦ b = 60 overlaid with the electrostatic ﬁeld with the Tajima-Dawson saturation Es(n), Eq. 4.1, indicated for n = 1 and n = 2. The wavelength = 2vA∕2Ωi likewise corresponds to ! = 2Ωi, approximately the center of the continuum distribution. preferred resonance at which to tailor the beam bunching. Consequently, beam bunching does not eﬃciently enhance the fusion power with the angular and plasma parameters considered.

4.4 Conclusions

Neutral beam injection, in addition to providing macro-stability to an FRC, can potentially provide a means for increasing the fusion rate through microscopic collective eﬀects. We have seen experimentally and theoretically that a wave with phase velocity much greater than the thermal velocity of the relevant plasma particles does not induce bulk disruptions in the plasma. The relatively low value of in the SOL allows electrostatic IB modes to accelerate a non-thermal fast-ion tail. As is shown in Eq. 4.3, as → 1, the saturation amplitude retreats from that of wakeﬁeld physics and returns to that of quasilinear physics. Thus, the core of the FRC and its associated modes can be excluded in favor of the SOL.

These beam-driven IB modes also are essentially the same process as the self-modulation instabil-

48 ity in ion-driven wakefield acceleration [1], where an injected ion beam becomes bunched at the plasma wavelength, improving wakefield development. This phenomenon suggests the possibility of seeding this mode with bunched beams. A further enhancement of fusion power is revealed when such bunching is performed. This technique can be considered analogous to playing a musical instrument at a particular note; rather than letting the desired mode grow slowly from noise, the plasma is struck and rings strongly at this desired mode. Despite such strong, targeted violence, the fast phase velocity of the mode prevents deleterious effects to the plasma.

Experimentally, such seeding might be accomplished with a method analogous to the self-modulation instability seen in proton-driven wakefield acceleration. In such an optimized setup, ion-cyclotron radio-frequency (ICRF) waves [96] may even fulfill part of the role of the beam. These approaches may allow a method of directly energizing ions of a given type by self-modulation and its associated wakefield physics. This topic will be explored in future work and suggests further cooperation between the fields of accelerator and fusion physics.

49 Chapter 5

High-Density Laser Wakeﬁeld Application to Oncology

We now turn to another application of wakeﬁeld acceleration where new insights may be gained: oncology. In particular, by pressing the high phase velocity principle of typical wakeﬁeld acceleration to its limits, a qualitatively distinct acceleration mechanism is found, which has a potentially novel application. This chapter presents a preliminary study on this topic with an attempt to address experimental realities.

5.1 Introduction

The treatment of cancer remains one of the most pressing concerns of medical research. One promising avenue of cancer treatment is brachytherapy, in which a source of radiation is brought inside the body close to the tissues requiring treatment [54]. This technique localizes the radiation dose to its source, limiting collateral damage to surrounding healthy tissue. In contrast, more conventional external sources of radiation can cause signiﬁcant damage to intervening tissues. Typ-

50 ically, small quantities of radioisotopes provide the dose source for brachytherapy, but such sources suﬀer from decay, which lengthens treatment times and increases shielding costs. The use instead of an electron beam as the radiation source would eliminate the ﬁrst challenge and greatly mitigate the second.

Accelerators have many applications in our current society, one of the most important being their use in radiation therapy. Particle beams in the form of X-rays, electrons, and protons can be used to treat cancer. These energetic beams can ionize molecules which in turn damage the DNA. Cells with damaged DNA cannot reproduce and are eliminated through natural processes in the body. The type of beam used depends highly on the size and location of cancer being treated. For example, X-rays and electrons deposit most of their energy in the surface layers. On the other hand, protons can be controlled to deposit their energy at a speciﬁc depth due to Bragg peak phenomena [33, 100]; therefore, the damage to healthy cells is reduced dramatically. Proton therapy contends with other limitations, however [13].

A potential means for generating the more desirable option of an electron beam is Laser Wakefield Acceleration (LWFA) [98], a compact method of accelerating electrons to high energies which was first proposed by Tajima and Dawson [95] in 1979. Research in the use of LWFA to generate electron beams for medical applications has proceeded for at least two decades now. Initially, these efforts focused on generation of high-quality electron beams with energies roughly in the range 6- 25 MeV, as would be applicable for conventional, external sources of radiation for cancer therapy [93, 18, 53, 40, 39, 74, 70]. Recent innovations in the field of fiber lasers has offered a new leap forward in this effort: the Coherent Amplification Network (CAN) [68], in which many individual micron-scale fiber lasers are coherently combined and amplified to provide both high-rep rate and high power. This innovation allows medical LWFA to proceed into a new regime of applications, though a CAN laser certainly is applicable as well to the traditional medical effort at producing an external electron beam.

As we see below, we ﬁnd that even at very modest intensity, LWFA can produce electrons that

51 are relevant for tissue penetration and delivery of beams of ionizing radiation. With this insight, in combination with the compactness afforded by the recent developments in fiber lasers, we are led to consider the new situation of in situ radiation sources (of electrons). In this vision, we see three chief schemes in which the wakefield electron source could be brought directly to the cancer. First, the laser-wakefield accelerator could be inserted in an intraoperative fashion [40, 39], which involves surgically opening the intervening tissues and can presently be used for LINAC sources in some instances. A surgeon may also use such an operation to remove any residual cancer or clean affected tissues by hand. Less invasive is brachytherapy, where the laser is injected discreetly into the body, such as through a blood vessel or directly through tissue. Finally, it may be possible to carry the laser into the body in a endoscope, whereby the surgeon could potentially both diagnose and treat the cancer simultaneously. It is our goal here to devise a way in which any or all of these methods might be possible with LWFA.

In each of these cases, the electrons in the beam need only have shallow penetrative power, as they need not traverse the body before reaching the tissues to be treated. The desirable energy 2 for an electron beam is then reduced to the order of 10 keV. We thus seek a means of producing 2 low-energy electrons. The electron energy gain from LWFA is given by Δ = 2g(a0)mec (nc∕ne), where a0 is the normalized laser intensity, g(a0) represents the function dependence of energy gain on a0, nc is the laser critical density, and ne is the plasma density [95]. This relation suggests a path to low-energy electrons through a high plasma density and modest laser intensity, parameters that are also favorable to ﬁber lasers, as will be discussed. To provide a target material near the critical 21 −3 density of an optical laser (nc = 1.11 × 10 cm for a laser wavelength of 1 micron), it may be best to use a porous nanomaterial [94, 69, 105], such as porous alumina or carbon nanotubes. Such a target material for irradiation by the laser would also avoid the presence of ionized gas inside the body.

While the low-density, high-energy regime of LWFA has been studied extensively, the high-density, low-energy regime has been less explored in detail [91]. Indeed, the expression for electron energy

52 gain given above was established by studying the low-density regime. In the following, we study

the scaling laws of electron energy gain Δ over the parameters of plasma density, intensity a0, and laser pulse duration, as well as investigate the mechanics of electron acceleration in the high-density regime [75]. We also study ways to create a laser setup more amenable to ﬁber lasers [77].

5.2 Acceleration in the High-Density Regime

If the plasma density is near the laser critical density, the typical physics of LWFA transitions into a qualitatively distinct regime where analytic extensions of conventional wakeﬁeld physics √ [99] may become insuﬃcient. In particular, the laser group velocity vg = c 1 − ne∕nc, which is approximately equal to the wake phase velocity, approaches zero, and the pump depletion and L L a2 n n dephasing lengths, which are on the order of d ∼ p ∼ p 0( c∕ e), become shorter than the

plasma wavelength p = 2c∕!p for a0 = 1. The length scale of laser interaction with the plasma

becomes better described by the plasma skin depth, given by c∕!p. Consequently the laser-plasma

interaction primarily occurs within one p, and the laser couples significantly to the bulk motion of √ the plasma, which is characterized by the plasma thermal velocity vT = T ∕m, where T and m here are most relevantly those of the electrons but can in general correspond to ions or electrons. This situation contrasts starkly with that of low-density wakefield physics, in which the laser (having vg ≈ c) penetrates deeply into the plasma without coupling to the bulk motion. In this case the phase velocity of the wakefield becomes vpℎ = vg ≈ c, allowing the laser to build a long wake train

(Ld ∼ Lp ≫ p) robustly and stably. When electron injection occurs, the wakeﬁeld can then skim a small population of electrons from the bulk and accelerate them to high energies. This sharp divide in fundamental physics requires examination of the high-density regime on its own terms and a qualitative understanding apart from that of conventional wakeﬁeld acceleration.

The diﬀerences between the qualitative physics of the high- and low-density cases, which respectively represent waves of low and high phase velocity relative to the plasma thermal motion, extends

53 to many general features of plasma physics. For waves with vpℎ ∼ vT , the wave couples to the bulk thermal motion of the plasma, typically producing macro-instabilities and turbulence. Plasma structures are thus fragile to such waves, and may disintegrate from wave-induced transport. In contrast, waves with vpℎ ≫ vT do not couple to the bulk thermal motion of the plasma, and the plasma and wave are insulated from each other. With regard to the wave fields, the wave can then reach a robust saturation amplitude before particle trapping begins to occur. In this limit, the wave trapping velocity [79] becomes approximately equal to the wave phase velocity, leading to the characteristic Tajima-Dawson saturation amplitude ETD = m!vpℎ∕q, where ! is the wave frequency and m and q are respectively the mass and charge of the relevant species. At this saturation amplitude, wave-particle interaction manifests as the acceleration of a tail of extremely fast particles, and the thermal distribution remains intact. Indeed, under the influence of the free energy of such a wave, plasma structures can built rather than destroyed. The plasma is durable against the wave, rather than fragile. Because the original LWFA concept was built on this high phase-velocity paradigm and departs from the sheath-forming, high-density regime [64], most or all works on LWFA have to date avoided the high-density (low phase-velocity) regime. Thus it is the purpose of the present work to first qualitatively distinguish these two regimes and then quantitatively characterize the differences in their most important dynamics. To do so, we isolate the longitudinal spatial dynamics and show their overwhelming influence on the departure of the physical characteristics of the above two regimes. In the analysis below, we casually apply these labels of “black” and “blue” for the high- and low-density wakefield regimes, respectively, as well as “grey” for the transitional regime.

To study the distinct physics of the high-density regime and its transition from the low-density regime, we employ the particle-in-cell (PIC) code EPOCH. Establishing a firm conceptual foundation in this physics first requires understanding of the case where one spatial dimension (and three velocity dimensions) are employed. This work is also primarily interested in the phase-space structure of accelerated electrons, rather than the real-space evolution of the laser pulse and wakefield. Thus, 2D phenomena such as strong self-focusing [82] and hole-boring of ions [103] are not examined. Such 2D effects beget other effects, such as collapse of the laser pulse [91], which can lead to

54 an expanding cloud of electrons. These eﬀects and others, such as magnetic vortex physics [92] and prominent ion motion, are also more typical of the regime of ultra-intense pulses, which is not an emphasis of this work. Peak ion energy achieved in this study is typically 0.1 MeV. Comparisons to ion-acceleration schemes, such as TNSA and RPA [87, 31] are not addressed here.

For this simulation setup, the laser is injected from vacuum through an impedance-matching boundary into a uniform plasma of electrons and protons with temperature T = 100 eV. Such a configuration represents a simple and idealized case that can be analyzed easily, yet contains the crucial physics. This scheme contrasts with some past efforts in which the laser was injected into a density ramp that peaked near the critical density [91]. In this case, the laser pulse collapsed before reaching the region of critical density owing to important 2D effects. Here a uniform density is used to examine the physics of laser-plasma interaction at the full critical density. Additionally, as opposed to initializing the laser inside the plasma, the vacuum injection scheme may better reflect the experimental reality for the high-density case near the critical density if we take as a target a porous nanomaterial [94, 69, 105], which for the laser can provide vg ∼ 0. Care is taken that the essential laser-plasma physics is unchanged between the cases of vacuum injection and laser ini- tialization inside the plasma. The laser is taken to have a vacuum wavelength of one micron and a resonant pulse length with functional dependence Ey = E0 sin (kx − !t − )ℎ(x, t), where ℎ(x, t) is a resonant flat-top profile, and is an optional phase. By default is set to zero. (The oscillatory component of Ey thus grows from zero as the laser enters the simulation domain.) A resonant pulse is half the length of the plasma wavelength p. The remaining laser and plasma quantities are then controlled through two parameters: the laser intensity (a0) and the critical density ratio (nc∕ne). Position in the simulation domain is indicated by the value x. The laser is injected at x = 0, and “forward” and “backward” correspond to the directions + ̂x and − ̂x, respectively.

A resonant laser pulse in a plasma near the critical density (nc∕ne ≤ 2) must necessarily be sub- cycle, or at most single cycle. While sub-cycle lasers have been demonstrated experimentally [83, 44], such a setup would be generally diﬃcult to implement, particular in a ﬁber laser [68]. Instead,

55 a long, self-modulating pulse would be much more amenable to a fiber laser application, as is discussed below. For simplicity and consistency with the low-density regime, however, here a resonant pulse is used. It will be shown that with a longer pulse the essential physics remains unchanged. Another concern arising from a sub-cycle laser pulse is the influence of the initial laser phase . Care has been taken to ensure that choice of does not affect the essential physics, even

for nc∕ne = 1. One can show analytically that the initial ponderomotive kick felt by the electrons lies in the same direction as the laser Poynting vector, creating a tendency toward uniformity across various values of .

With the simulation scheme established, the ﬁrst step in examining the high-density regime is then to approach it from the better-understood low-density limit. Such an approach is done by scanning

over the density values nc∕ne ∈ [1, 10], taking a modest intensity of a0 = 1. The maximum en- 2 ergy gain in LWFA, where 1D theory applies, is given by Δ∕mec ≈ 2(nc∕ne) for a0 = 1. The maximum acceleration found over the scan of density values, shown in Fig. 5.1, agrees well with this proportionality. For at least the highest-energy electrons this scaling is evidently obeyed up to the critical density. However, a crucial assumption of this expression for Δ is that the wake-

field amplitude remains constant. For low densities (nc∕ne ≳ 10), this assumption begins to break down significantly, resulting in longer distance required for acceleration and somewhat lower final electron energy than that given by Δ above. 2D effects may also become important in this limit.

We may now take a closer look at each regime. For the “blue” wave case of nc∕ne = 10, Fig. 5.2 shows a snapshot of the electron phase space overlaid with the longitudinal and laser electric ﬁelds. The train of accelerated electrons are clearly visible, and the highest-energy electrons reach

max √ 2 the theoretically expected momentum of px ≈ mec (2g(a0)nc∕ne) − 1 [95]. The coherent wake structure is also seen, with saturation in the longitudinal ﬁeld reaching the expected value [30] of Emax∕ETD ≈ 0.4. The wake is gradually diminished as it imparts energy to the electrons, but recovers after the dephasing length Lp ≈ 8p.

In contrast, the “black” wave case of ne = nc, shown in an initial stage in Fig. 5.3a and ﬁnal stage

56 Figure 5.1: Scaling of the normalized maximum electron energy with density nc∕ne for the laser intensity a0 = 1, compared with the theoretical expression for the energy gain of electrons Δ in LWFA. in Fig. 5.3b, exhibits quite different behavior. Here, vg = 0, and Lp,d ≲ p, restricting the laser- plasma interaction to within one plasma wavelength. The long train of trapped electrons becomes replaced by streams of low-energy (Δ ∼ 100 keV) electrons ejected from the site of oscillation roughly every plasma period. The generation of a net bulk momentum indicates the presence of an effective viscosity. The mechanics of acceleration in the case of nc∕ne ≈ 1 represent a distinct qualitative regime than that in the typical wakefield case of nc∕ne ≫ 1. Having vg ≪ c, the laser couples strongly to the bulk motion of the electrons upon entering the plasma, pushing out a much larger spike in electron density (ne∕ne ≈ 3.5) than in the low-density case while the ions are not substantially affected. This density spike creates a longitudinal electric field of approximately twice the strength as in the low-density case and causes the reflection of a substantial portion of the laser. This powerful initial kick to the plasma establishes a strong longitudinal oscillation of electrons in the range 0 ≤ x ≤ p.

57 Figure 5.2: A snapshot of the electron phase space px vs. x (heat-map, with warmer colors representing higher density) and longitudinal Ex (green) and laser Ey (translucent blue) ﬁelds for the somewhat typical wakeﬁeld case of nc∕ne = 10 (“blue”) and a0 = 1 after the electron acceleration has saturated. The plasma wavelength is given by p = 2c∕!p. The forward edge of the laser pulse is at x∕p = 16.

The strong restoring electric field then causes this density spike to rebound, and simultaneously those electrons having vx ≪ c are accelerated to near the expected energy Δ from 1D wakefield theory. The restoring motion of the electron density spike expels many electrons through the left edge of the domain, establishing a sheath [64] at the boundary of strength comparable to the initial laser amplitude. This sheath, representing the remnants of the original electron density spike, sub- sequently oscillates longitudinally, with each oscillation accelerating a stream of electrons to low energy (< 100 keV). These electrons are then ejected from the site of oscillation in the forward direction. The presence of the sheath insures that these electron streams travel nearly exclusively in the forward direction. (If the sheath is not present, a significant number travel backwards.) Fig- ure 5.3a, a snapshot of the electron phase space and fields zoomed to the range 0 ≤ x ≤ p, shows the beginning of this process, where the first streams of accelerated electrons, as well as secondary, fin-like streams, are visible. As the oscillation continues, more streams accumulate, building up the

58 (a) (b)

Figure 5.3: Electron phase space and ﬁeld structure of the high-density (“black”) case nc∕ne = 1 with laser intensity a0 = 1 at early (5.3a) and later (5.3b) stages. The development of the electron streams is observed. Figure 5.3a is zoomed to 0 ≤ x ≤ p from (5.3b).

phase-space distribution in Fig. 5.3b, which shows the full simulation domain. In the later stages, the electron acceleration becomes increasingly turbulent until the oscillation is ﬁnally exhausted after about 30 plasma periods. While the individual energy of the accelerated electrons is low compared to that of the low-density case, the total energy imparted to the accelerated electrons can potentially be higher. This total imparted energy represents about 12% of the total injected laser energy. This understanding of the electron dynamics associated with sheath formation should also be useful for the understanding of related ion acceleration dynamics [98, 33]. The sheath acceleration mechanism observed here is also reminiscent of that in cases examined previously [103].

The transition to sheath acceleration from typical wakefield acceleration represents a sharp divi- sion of qualitative regimes of wakefield physics. Pursuing this point further, we can attempt to find a quantitative means of discriminating these regimes. Visually comparing figures 5.3 and 5.2 suggests a quantitative index related to the entropy of the phase-space structure; the electron phase space in the low-density case is highly ordered compared with that of the low-density case. In general, however, comparing the entropy of two distributions also requires an account- ing for the mean kinetic energy of the distributions. Thus, as an index for discriminating the

59 “blue” and “black” regimes, we propose a “darkness” metric D. This quantity D is defined as ∞ the specific momentum entropy D = S∕⟨K⟩, where S = − ∫−∞ ln [f(Px)]f(Px)dPx is the differ- ential Boltzmann entropy of the longitudinal momentum distribution f(Px), where Px = px∕pT √ is the longitudinal momentum normalized to the thermal momentum pT = meTe, and ⟨K⟩ = t ∞ 2 2 ∫−∞[ 1 + Px ∕(mec∕pT ) − 1]f(Px)dPx is the average electron kinetic energy per particle nor- 2 malized to mec considering only the contribution from px. The distribution f(Px) is normalized ∞ according to ∫−∞ f(Px)dPx = 1. This index D does not take into account the dependence of the

total laser energy content on the plasma wavelength p; as p increases for lower densities, the laser pulse length increases to maintain a resonant length. Instead, by keeping the laser always to a resonant pulse length, the plasma excitation mechanism is held constant.

For a scan over the density values nc∕ne ∈ [0.5, 14] at a0 = 1, Fig. 5.4 shows the approximately ﬁnal

values for the index D normalized to the initial value D0, which is the same for each data point and

has an analytic form for a Maxwellian distribution. One point in the overdense regime (nc∕ne = 0.5)

has been added as well for cautious comparison with the “black” regime. As nc∕ne → ∞, visually

s∕s0 → 0, while as nc∕ne → 0, D∕D0 climbs to a large value. These limits reflect the substantive difference in each regime. For “blue” waves, the average electron energy grows much faster than the momentum entropy, owing to the development of a typical wakefield phase-space structure. For “black” waves, in contrast, disorder in phase space dominates growth in average kinetic energy and

becomes ever more severe for increasing plasma density. The “darkness” index D∕D0 for “blue” waves thus asymototes to zero while that for “black” waves is characterized by ﬁnite size tending to a large value. Between these two extremes, a “grey” wave state can exist, as is shown for example in

Fig. 5.5 for the case of nc∕ne = 3, which shows both aspects of bulk ﬂow and traditional wakeﬁeld acceleration.

This index may then provide a guide to the most appropriate regime in which to operate for a partic-

ular application. For the generation of a mono-energetic, high-energy electron beam, D∕D0 ≪ 1 is desirable, as a “blue” wave will cleanly accelerate a small population of electrons to extremely high

60 energy. Being mindful of the dephasing length, one can then ensure that the accelerated electrons are captured at peak energy. In contrast, for some medical applications, such as in a cancer-treatment scheme where a source of radiation is brought directly to the cite of a tumor, a signiﬁcant dose of low-energy (shallow-penetrating) electrons is desirable, with less constraint on the beam quality and electron energy distribution. One is attracted in this case to the “black” regime, which has

D∕D0 ∼ 1. For long, low-intensity laser pulses, as would be amenable to a ﬁber laser, this regime

can be accessed even at moderate plasma densities (nc∕ne ≈ 10) due to Raman forward scattering [51, 32, 67].

The maximum electron energy in the phase-space distribution shown in Fig. 5.3b is roughly 1 MeV, which is encouraging for the medical aims in this study. A naïve next step to increase the electron dose and alleviate the difficulty with a sub-cycle pulse might then be to increase the length of the laser pulse to produce a larger dose of electrons. Such a case is shown in Fig. 5.6 for nc∕ne = 1 and a laser pulse length of lp∕p = 8 for a snapshot at time t∕p = 88, where p = 2∕!p is the plasma period. At this time the laser interaction has finished, as well as most of the electron acceleration, and thus this snapshot represents the near-final behavior. The overall interaction with the plasma

is much more violent. For x∕p ≳ 32 a large body of overlapping electron streams is seen, which originate in the initial laser interaction. For x∕p ≈ 2 a longitudinal sheath oscillation similar to that in the resonant pulse case is formed which generates subsequent electron streams. This scheme is also more energetically eﬃcient than that of Fig. 5.3b, indicating that both the quantity and quality of low-energy electron production is improved. However, this case relies on the relatively strong

intensity of a0 = 1, which is likely untenable for ﬁber lasers.

5.3 Laser Intensity Scaling

Apart from plasma density, the second chief parameter determining the nature of the wakeﬁeld

response is the laser intensity a0. To understand wakeﬁeld physics at high density, it is thus also

61 Figure 5.4: The speciﬁc entropy (“darkness”) index D normalized to its initial value D0 for a scan of the density ratio values nc∕ne ∈ [0.5, 14] for laser intensity a0 = 1 and a resonant pulse length. The plotted index is the mean of the calculated index for the last ten time steps in each run. The black error bars represent one standard deviation of this set of averaged values. Rough indications of the wave type for particular regions are given in text. Because the plasma is initialized with the same temperature in each case, the initial index D0 is the same for every data point. The critical density is marked by a dashed line.

necessary to understand the scaling of accelerated electron energy Δ with respect to a0. For medical applications, it is also necessary to understand the electron energies available for various laser intensities, particularly a0 < 1. Two cases are considered: the low-density case of nc∕ne = 10 and the moderate-density case of nc∕ne = 3. For each case a0 is scanned logarithmically over the range a0 ∈ [0.1, 10] and the maximum electron energy is recorded.

The expected functional dependence of Δ(a0) has thus far been represented simply by g(a0). Here t g a a2 a we compare the case ( 0) = 1 + 0 − 1 to the results of the scan over 0. Of particular interest for comparison with the simulation results, this function has two slope regimes: ln g∕ ln a0 = 2 for a0 ≪ 1 and ln g∕ ln a0 = 1 for a0 ≫ 1.

The results of the scan are shown in Fig. 5.7. First for the low-density (“blue”) case (Fig. 5.7a), the transition in slope is indeed seen, and the simulation results are in general agreement with g(a0).

The moderate-density (“grey”) case (Fig. 5.7b) also shows rough agreement with g(a0). Notably,

62 Figure 5.5: The electron phase space and ﬁeld structure of the intermediate (“grey”) case of density nc∕ne = 3 at the laser intensity a0 = 1.

for the low-density case, a sharp transition in maximum electron energy is seen around a0 = 1. This transition may be indicative of the “switching on” of electron trapping that occurs once the laser amplitude enters the relativistic regime at a0 = 1; for a0 ≪ 1, substantial electron trapping does not occur for typical LWFA. In contrast, at higher density (5.7b), the transition is both less prominent

and electron energy for a0 ≪ 1 is higher than that given by g(a0). Because the electron acceleration mechanism in this regime has shifted more to the sheath acceleration typified in Fig. 5.3b, which does not have an intensity-based “switching on” transition, but can instead accelerate electrons even at very low intensities owing to the slow laser group velocity vg ≲ vT , a less abrupt transition around a0 = 1 in this case is expected. As an additional consequence, the sheath acceleration is able to accelerate electrons with comparative efficiency in the regime a0 ≪ 1, leading to the apparent acceleration enhancement above g(a0) in Fig. 5.7b. In the regime of very large a0, it should also a2 be noted that other work [61] has found that the scaling of electron energy should follow 0 from ulta-relativistic effects.

63 Figure 5.6: The electron phase space and ﬁeld structure of the critical density case nc∕ne = 1 for a laser pulse of length 8p at the laser intensity a0 = 1, showing the “black tsunami” regime. At this snapshot most of the electron acceleration is concluded and most of the laser has exited the domain.

These results suggest that an a0 between 0.1 and 0.8 gives the best results for achieving electrons with Δ < 1 MeV while still allowing the density ratio to be varied. In general an a0 value greater than 1 results in greater electron energies than what we have in mind for medical applications.

5.4 High-Density LWFA in Fiber Lasers

Fiber lasers, particularly in coherent networks, offer the potential for high-power and high-rep rate. Yet, individual fibers are also subject to a number of constraints not shared by large, conventional lasers. Perhaps most importantly, the material properties of fibers, which are typically made from silica, place a much more stringent limit on the intensity of the transmitted laser pulse [2]. Certainly for any fiber application inside the body, it is paramount that material damage to the fibers be avoided, as this damage would then harm the surrounding tissue as well, with potentially severe

64 (a) Emax vs a0 for ﬁxed density ratio of 10 (b) Emax vs a0 for ﬁxed density ratio of 3

Figure 5.7: The maximum electron energy as a function of laser intensity a0 for two density cases: nc∕ne = 10 (5.7a) and nc∕ne = 3 (5.7b). The maximum energies (red dots) are compared with the function g(a0) (blue solid line). The blue dashed lines represent the asymptotic behavior of g(a0) for a0 ≪ 1 and a0 ≫ 1. consequences. In addition to the requirement of avoiding ionization of the fiber, an even more restrictive condition may be the level of tolerable accumulation of nonlinear phase in the laser pulse, the severity of which grows with laser intensity. Given this background from the field of fiber optics, a acceptable order of magnitude for laser intensity that safely avoids these issues may 14 −2 be 10 W cm . We take this value as a starting point, but subsequent technical and material efforts may require a more conservative value. This intensity corresponds to a0 ≈ 0.01, which is far below the relatively powerful value of a0 = 1 considered thus far. The expression for electron g a m c2 n n g a a2 a ≪ g a energy gain Δ = 2 ( 0) e ( c∕ e), where ( 0) = 0∕2 for 0 1 if ( 0) takes the form of the ponderomotive potential, suggests that one means of compensating for this low individual fiber laser 2 intensity to attain 10 keV electrons is simply to use a coherent network of ∼ 100 lasers, though in a endoscopic application this number may be difficult to achieve. Phase-matching each fiber would also be technically difficult, but may be possible making use of the partial reflection of the laser pulse at the boundary between the fiber and high-density material to be irradiated. However, by lowering the target material density, one can use fewer coherent lasers, such as with ∼ 10 lasers at 2 nc∕ne = 10. Another alternative is the use of a hollow fiber, which can have a diameter of order 10 times larger than that of a typical fiber, thereby accommodating much more pulse energy.

65 Another chief concern with ﬁber lasers is the pulse length. While thus far the simulations in this

work at nc∕ne ≈ 1 have used resonant laser pulse lengths of ∼ 2 fs (a diﬃcult, sub-cycle pulse in its own right), the shortest pulse practically achievable for a ﬁber laser is likely around 100 fs.

Letting l be the laser pulse length, a resonant pulse has l∕p = 0.5, where in contrast at the critical density, a 100 fs pulse has l∕p ≈ 30. An attractive alternative is provided by the phenomenon of self-modulation [5, 57, 72, 73, 66], where a large laser pulse (l∕p ≫ 1) becomes spontaneously

broken into units of length p and reproduces the desired wakefield behavior. This regime is called self-modulated LWFA or SM-LWFA, and is likely much more amenable to fibers than the resonant pulse case. Stated simply, fibers prefer a longer, low-amplitude pulse to a short, intense pulse.

To demonstrate the onset of SM-LWFA, a low-density plasma of nc∕ne = 10 and a0 = 1 is injected with laser pulse with length (l∕p = 5), as is shown in Fig. 5.8. As progressive peaks pass a point in the plasma, the wakefield is strengthened, and ultimately the strength of the wakefield is enhanced in the self-modulated cases compared to that of the resonant case. Electron acceleration is also seen following the wakefield.

Applying this technique to a ﬁber laser case, Fig. 5.9a shows the case of 15 coherently added ﬁber

lasers, each at a0 = 0.01 with nc∕ne = 10. The pulse is now Gaussian, also for practical considerations, with FWHM a pulse length of 100 fs. For self-modulation, it is desirable that the laser power be greater than the critical power [30], while in this case, without considering laser guiding 2 conditions, this condition may not be satisﬁed. (A rough calculation using a laser spot size of p gives a laser power of 7.5 GW, while the critical power is given by Pc = 17(nc∕ne) GW = 170 GW.) Nonetheless, substantial acceleration in Fig. 5.9a is seen up to nearly 1 MeV.

Many low-energy electrons are accelerated as well. This coupling to bulk electrons may be caused by the generation of lower-frequency laser components through Raman forward scattering, which is expected in the presence of a long laser pulse [51, 32, 67]. A spectral analysis of the laser

ﬁeld in Fig. 5.9a reveals the presence of down-shifted frequency components at ! = !0 − !p and

! = !0 − 2!p, as well as a small up-shifted component at ! = !0 + !p, where !0 is the nomi-

66 Figure 5.8: A demonstration of the self-modulation of a laser pulse into resonant pieces in the “blue” regime of nc∕ne = 10. Here l∕p = 5.

nal laser frequency. These components arise almost immediately after the laser enters, and begins √ interacting with, the plasma. With nc∕ne = 10 in this case, !0∕!p = 10 ≈ 3.2. A frequency

down-shifted from this value in multiples of !p thus becomes very close to resonance with !p; the

two down-shifted components ! = !0 − !p and ! = !0 − 2!p are equivalent to a laser with nom-

inal density ratios of nc∕ne = 5 and nc∕ne = 1.25, respectively. The latter of these values is ﬁrmly within the “black tsunami” regime of ﬁgures 5.3b and 5.6. Further down-shifting is suppressed

because frequencies lower than !p would not be able to resonantly excite the Raman forward scattering instability and would be immediately absorbed by the plasma.

Another approach using practical ﬁber laser parameters is the laser beat-wave accelerator [84, 95], which was used historically in the early years of LWFA. In this scheme, two lasers with frequencies

diﬀering by !p create a modulation at the plasma frequency and resonantly excite the wakeﬁeld.

Exploration of this possibility at nc∕ne = 10 with a 100 fs Gaussian pulse and 15 a0 = 0.01 lasers, each divided into two equal components separated in frequency by !p, shown in Fig. 5.9b, yielded

67 slightly more efficient acceleration, with electrons reaching energies slightly in excess of 1 MeV. This relatively clean acceleration is expected given the seeded plasma oscillation and provides a confirmation of the physics of the pre-modulated laser field. As in the self-modulating case, here a lower laser harmonic equivalent to nc∕ne = 4 is seen, as well as harmonics higher than the nominal frequency, the former perhaps aiding bulk acceleration of electrons and the latter pulling the highest-energy electrons past the energies reached in self-modulated case (Fig. 5.9a).

In these examples, an initially “blue” wave is converted into a “black” wave that can efficiently accelerate low-energy electrons even at very low laser intensity. Together with a variable number of coherently added fibers, this effect may provide substantial practical flexibility for a medical fiber laser application. For instance, if an optimized setup were to require an even lower individual laser 14 −2 intensity than 10 W cm , the target density could be modestly lowered, preserving the bulk flow of electrons in the desired energy range. Furthermore, if a beat-wave laser is possible, a potentially cleaner electron beam can also be produced if desired. It is remarkable that these benefits derive from the requirement of a long laser pulse, one of the “limitations” of fiber lasers.

68 (a) Self-Modulation

(b) Beat-Wave

Figure 5.9: The electron phase space and field structure (left) for the case of a 100 fs pulse at nc∕ne = 10, as well as the laser (Ey) frequency spectrum (right). Fifteen lasers each contributing intensity a0 = 0.01 are coherently added, demonstrating practical parameters for fiber laser applications. In 5.9a, the laser pulse undergoes self-modulation, while in 5.9b, each of the 15 laser contributions is further divided into two components that beat at !p, thus resonantly seeding the wakefield (beat-wave acceleration). Note that here the laser field Ey is normalized with respect to the initial combined amplitude E0 of all the fiber contributions. For the frequency spectra, solid vertical lines indicate the nominal laser frequency (!0∕!p) while dashed lines indicate harmonics, which differ from the nominal frequency by some integer multiple n of the plasma frequency !p.

69 5.5 Electron Tissue Penetration

We may now consider more closely the interaction of an electron population like those in figures 5.6 and 5.9 with human tissue for radiation therapy. Conventional radiation therapy typically relies on exposing the body to an external source of radiation, whether X-ray, gamma-ray, protons, or electrons. In this process, the radiation passes through a significant depth of healthy tissue, causing collateral cellular damage. Three techniques to avoid this collateral damage include intraoperative radiation therapy (IORT) [40], where the source of radiation is surgically brought to the tumor, brachytherapy, in which the laser is injected into the body, or endoscopic radiation therapy (ESRT), where a small endoscope is internally brought to the tumor site. Consequently, in all of these cases, the radiation produced need only penetrate a short distance, perhaps millimeters or less. In such a scheme, the distribution in figures 5.6 and 5.9, possessing a large spread of low-energy electrons, may be particularly fitting. The recent development of coherent networks of fiber lasers (CAN) [68] has allowed LWFA to branch into this new and distinct application.

The penetration depth in human tissue can be approximated by integrating the stopping power of electrons in water, giving the stopping distance in the continuous slowing-down approximation (CSDA) [6, 11]. At the critical density, the distribution of low-energy electrons in Fig. 5.9a has the energy distribution shown in Fig. 5.10a. This distribution f() corresponds to a maximum

penetration depth xCSDA in water of ≲ 1 mm, as is shown in Fig. 5.10b as a function of xCSDA. Tuning the plasma density allows control of the penetration depth. Similarly, in an experimental implementation, by changing the density of the irradiated material, the penetration depth can be tuned to the desired value; lower material density will give deeper penetration. The laser intensity a0 or number of fibers can also be tuned for the desired electron energies produced. As an additional benefit, near the critical density, a significant acceleration of the bulk population of electrons occurs, potentially creating a far larger overall dose of radiation than would occur for more typical wakefield acceleration. This combination of a large dose and shallow, yet tunable, penetration may be ideal for intraoperative, brachytheraputic, and endoscopic medical applications.

70 (a) (b)

Figure 5.10: Electron penetration in the high-density LWFA regime. (5.10a) shows the normalized electron energy distribution for setup in ﬁgure 5.9a, which models a bundle of 15 ﬁber lasers each with a0 = 0.01 coherently added with plasma density nc∕ne = 10 and a pulse length of 100 fs. (5.10b) shows the resulting normalized distribution of electron penetration depth in the continuous slowing-down approximation (CSDA). 5.6 Conclusions

The majority of eﬀorts involving plasma wakeﬁeld acceleration have focused on producing ever higher-energy electron beams, particularly for applications in particle accelerators. Notable experiments in this area include BELLA, FACET, and AWAKE (mentioned in such as [30, 1]), which aim to reach TeV energies. To reach such high acceleration gradients, experiments typically use

low-density plasmas (nc∕ne ≫ 1) and a0 > 1.

The opposite regime, that of LWFA near the critical density, remains comparatively unexplored, but holds potential for innovative approaches to cancer therapy, such as LWFA-powered endoscopic electron therapy (or intraoperative radiation therapy). For these applications the source of radiation is brought close to the tissues to be treated, removing collateral damage to other tissues. Conse- quently the radiation need only have limited penetrative power. With electrons used as the source of radiation, the desired energies are then ≲ 1 MeV, which yields a penetrating depth on the order of hundreds of micrometers. The precise electron energy can be tuned through modiﬁcation of the

71 plasma density and laser intensity to achieve a speciﬁc penetration depth. Here we have shown though a preliminary study that LWFA can indeed produce such electrons.

To do so, the plasma density must be close to the critical density (nc ∼ ne). In this regime the group velocity of a laser pulse becomes increasingly slow, and the laser-plasma interaction range reduces nearly to a single wakefield oscillation. As a consequence of these properties, the wakefield, rather than skimming a small number of electrons from the bulk distribution and accelerating them to high energy, instead dredges somewhat more deeply from the bulk, creating an effective viscosity and momentum transport. This situation manifests as a “black tsunami” in analogy to beach wave physics and represents a qualitative departure from typical wakefield physics. Upon slowing down near the shore, the wave begins to break, creates turbulence, and dredges the sea floor, creating a

visibly black wave. Wakeﬁeld physics in the limit of nc∕ne = 1 manifests in a similar way, resulting in a churning wave of relatively low-energy electrons which can then be harnessed as a beam.

The gradual transition between the “black” and “blue” wakefield regimes, corresponding respectively to high and low plasma density, has been shown, along with the linear scaling of peak electron energy. The specific entropy metric D has been proposed as a quantitative index for the regime of wakefield physics under consideration. In the low-density limit (“blue” waves), D → 0, while near the critical density (“black” waves), D becomes finite, possibly tending to a large value. Fur- thermore, we have found that the self-modulation or Raman forward-scattering process allows a conversion of the “blue” or “grey” regimes into the “black” regime, providing efficient generation of a bulk flow of low-energy electrons despite the presence of a density ratio nc∕ne greater than unity. Along with the invention of the Coherent Amplification Network (CAN) [68] fiber laser technology, these dynamical characteristics in the appropriately chosen regime of operation open a pathway to creating far more compact electron radiation sources through LWFA and thereby a radically new radiotherapy using compact electron sources. With regard to laser intensity, a scan of maximum electron energy Δ over a range of intensities a0 reveals a general agreement with t g a a2 the function ( 0) = 1 + 0 − 1, derived from the ponderomotive potential, despite additional

72 complexities and the limitations of a 1D simulation geometry. The diﬀerence in the mechanism of acceleration between the high- and low-density regimes is also manifested in the maximum electron energies attained in the two cases in Fig. 5.7.

The recent development of coherent networks of fiber lasers (CAN) [68] has allowed LWFA research to branch into a new field of medical applications. However, two chief limitations must be addressed for a medical fiber laser system: laser intensity and pulse length. Fiber lasers have stringent intensity limitations, with a maximum allowed individual fiber intensity likely less than 14 −2 10 W cm . This limitation can be mitigated through the use of many coherently added fibers and

by retreating from the critical density to a more modest plasma density (such as nc∕ne = 10). Even with these factors, the ultimate intensity would likely remain in the regime of a0 < 1. Limitations on pulse length are also stringent; the shortest pulse length likely achievable in a ﬁber laser system is around 100 fs, which is several times longer than was used in Fig. 5.6. Fortunately, Raman scattering eﬀects and self-modulation may allow the “black”, low-energy electron regime to be ac-

cessible for a long pulse, even at very low intensity (a0 ≪ 1) and low density (nc∕ne = 10). Here we have shown a higher-intensity example of self-modulation for nc∕ne = 10, and similar results have been shown elsewhere [86].

Other challenges remain. This work has addressed only the most fundamental aspects of LWFA near the critical density. Treatment of higher-dimensional effects such as focusing and hole-boring will be necessary for any ultimate medical application, and thus must therefore await future work. However, we emphasize that to show this remarkable specific normalized entropy jump approach- ing the critical density may not have been noticed with the inclusion of higher-dimensional effects in a mixed effort. In this sense, our study has extended the original vision of Boltzmann by his formulation of entropy [52]. Similarly, finding a fitting material to be irradiated, as well as estimating radiation dose, will require further study, particularly of 2D and 3D effects. In absolute terms, the 21 −3 critical density for a 1-micron laser is approximately 10 cm . To achieve such a density, and to avoid the use of gas ionization inside the body, one possibility is the use of nanomaterials with a

73 significantly open structure, such as carbon nanotubes [94]. Such a medium would also provide the benefit of guiding the laser and wakefield. It might also be possible to tailor the design of the nanomaterial to suit the desired plasma density. Another possible issue is that the population of accelerated electrons generated by LWFA at high density is non-monoenergic and probably of high emittance. We may also strive to further increase the efficiency. Toward such a purpose we may wish to employ a graded density of plasma to control the phase gradation of the wakefield [49, 26]. Nonetheless, interesting physics has already emerged from these efforts, and the richness of a new regime is evident. Within the last decade, the technology needed to realize endoscopic electron therapy through LWFA has come of age, and serious endeavors for implementation should now proceed.

74 Chapter 6

Conclusions

This work has examined two novel avenues of wakeﬁeld acceleration, one pertaining to ionic waves, and the other to acceleration of low-energy electrons. Both of these avenues suggests new applications and innovations in the future for the ﬁelds of fusion energy and cancer therapy. Here, some of the most important points covered in this work are reiterated, and some topics for future work are mentioned.

6.1 Wakeﬁeld Acceleration with Ion-Cyclotron Waves

The observation in the C-2U experiment of an enhanced fusion rate forms the foundational motiva- tion of this work. The subsequent theoretical effort to explain this observation led to the formulation of a mechanism based on the principles of wakefield acceleration. Namely, the low value in the SOL ( ∼ 0.1) allows beam-driven electrostatic IC modes to propagate with the fast phase velocity vpℎ ≈ vA ≫ vti. Specifically, these modes manifest as beam-driven ion-Bernstein modes in an excitation of the magnetoacoustic instability. These properties allow these modes to accelerate a small population of ions to extremely high energies, up to the beam injection energy. In doing so,

75 the mode does not increase the ion bulk temperature, nor does it induce turbulence or anomalous transport. We thus take this mode to be responsible for the experimental observations. Moreover, this result potentially vindicates a class of “instabilities”; rather than destroying the plasma, these modes improve its performance.

This favorable mechanism can additionally be seeded by density bunches in the beam corresponding to the mode wavelength. When such is done, the mode reaches saturation instantaneously, rather than growing from noise. Despite this violent excitation, the bulk plasma is still left intact. This result suggests that this mode could be seeded in experiment with injection of the mode directly through ICRF, which would induce the desired density bunches in the beam and jump-start the mode. Rather than some traditional schemes of ICRF injection, which use shear Alfvén modes, here the desired polarization would be the compressional Alfvén mode at the appropriate wavelength to couple to the beam resonance condition. Future work should pursue this topic, as it oﬀers the potential for substantially increasing the energy yield of fusion devices.

As the conception of this mode is thought to propagate azimuthally in the FRC SOL, future work should also extend these results to a 2D geometry, where the full dynamics of beam orbits and their eﬀect on the mode could be seen. Because the wavelength of this mode (roughly 40 cm) is comparable to the radius in the FRC at which this mode occurs, an eigenmode structure may develop. It would be extremely interesting to observe these dynamics. Additionally, in the ultimately desir- 11 able case of a pB plasma, likely in the form of a hydrogen beam and boron background plasma, more realistic plasma parameters should be considered for the impact of this mode on the relevant beam-target fusion to be properly accounted.

76 6.2 High-Density Laser-Wakeﬁeld Application to Oncology

The principles that motivated the work above also led to a somewhat different line of thinking with regard to wakefield acceleration: can benefit be gained from violating the high-phase-velocity condition? This work treated such a case as a potential application for internal cancer therapy. In contrast to external sources of radiation, which cause collateral damage to intervening tissues, radiation brought directly to the site of the tumor can limit the radiation dose to the target tissue. Using a near-critical density target to be irradiated by a laser, a burst of low-energy, shallow-penetrating electrons can be produced. These electrons desirably only damage tissues to a depth of ≲ 1 mm. This depth can also be tuned as desired by adjusting the laser intensity or target density.

These low-energy electrons are produced by sheath acceleration at the point of laser entry into the plasma. Indeed, there is a gradual transition between the highly ordered wake train structure of the typical low-density regime of wakeﬁeld acceleration and the highly disordered sheath structure of the high-density regime. The entropy in longitudinal momentum of the accelerated electrons provides a quantitative means of the distinguishing these two qualitatively distinct regimes.

The invention of the CAN [68] laser provides the crucial technological foundation for this idea, as fiber-optic transmission of the laser provides a means of bringing the laser inside the body in a compact, relatively non-intrusive manner. While fiber lasers posses stringent restrictions on laser intensity and pulse length, nature, as well as the ability to coherently add individual fiber lasers, provides a ready compensation. Namely, Raman scattering of the laser pulse can generate frequency components near the plasma frequency, and thus even a target with 10% of the critical density can still dredge a large population of low-energy electrons. The number of fibers, target density, pulse length, and laser intensity can all be adjusted for the desired electron production.

While experimental eﬀorts at producing a coherent network of lasers are underway, the next step in theoretical work should be the extension of these results to a 2D spatial geometry. With the fundamental qualitative physics elucidated in 1D, a 2D simulation can then build upon this under-

77 standing and capture the detailed wake dynamics. Doing so is particularly important to model the channeling eﬀects of using a nanomaterial such as carbon nanotubes as the target material, as is the current goal. Also crucial, a 2D or 3D simulation would allow for more accurate estimation of the delivered dose of radiation. Despite these concerns, it is clear again that a class of “instabilities” is vindicated, this time that of chaotic sheath acceleration. Harnessed properly, instabilities can strengthen the performance of a plasma rather than degrade it.

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85 Appendix A

Perpendicular Integrals of the Beam Velocity Distribution

The perpendicular integrals involved in the dielectric tensor for a beam population with finite temperature (Eq. 2.17) cannot be expressed in a closed form for easy evaluation. Instead, they must be either integrated numerically or cast in the form of an infinite summation. To aid their evaluation, other properties of these integrals are worthwhile to examine. Section A.1 presents the infinite summation forms, and section A.2 lists analytical properties of both the beam and Maxwellian perpendicular integral function properties.

A.1 Summation Forms

The perpendicular integrals in Eq. 2.17 can be cast into the form of an inﬁnite summation with the aid of the Bessel-function identity

∞ É (−1)m+ðnð(2m)! x2m J 2(x) = . n 2 (A.1) m=n (m − ðnð)!(m!) (m + ðnð)! 2

86 This identity can be proven by manipulating the product of the series expansions of two factors

of Jn(x) and converting these into an inﬁnite and ﬁnite sum, the latter of which can be evaluated. The absolute value on n enforces that these functions are even with respect to n. If the absolute value was not explicitly included, the denominator diverges, pulling the function to zero, but while satisfying analytically, this fact is numerically intractable.

The goal is then to express the Jn(x)-type factors in the integrals as summations, leaving only powers of in the integrand. First consider Pn and n. From these, the other perpendicular integrals can be found via parametric diﬀerentiation. With the identity Eq. A.1, Pn becomes

∞ (−1)m+n(2m)! m ∞ − 2 ∕2 É b 2m+1 −2∕2 P = e ⟂ I e d. n 2m 2 0 ⟂ m=n 2 (m − n)!(m!) (m + n)! Ê0

The remaining integration can then be evaluated using

∞ H 2 I 2 2 ⟂ 2(m+l)+1 − ∕2 m+l ⟂∕2 0 I0 ⟂ e d = 2 (m + l)!e Lm+l − , (A.2) Ê0 2

a where l is an integer and Lb(x) is a generalized Laguerre polynomial of ordinary index b and asso-

ciated index a. Similarly, n becomes

∞ (−1)m+n(2m)! m ∞ − 2 ∕2 É b −2∕2 = e ⟂ I − I ( ) e d. n 2m 2 0 ⟂ ⟂ 1 ⟂ m=n 2 (m − n)!(m!) (m + n)! Ê0

The ﬁrst term in the remaining integration can be evaluated using Eq. A.2, and the second term is evaluated as

∞ H 2 I 2 2 ⟂ 2(m+l) − ∕2 m+l−1 ⟂∕2 1 I1 ⟂ e d = 2 ⟂(m + l − 1)!e Lm+l−1 − . (A.3) Ê0 2

With some simplifying, the functions Pn and n can then be expressed as summations:

∞ 0 1m H 2 I É (−1)m(2m)! P = (−1)ðnð b L0 − ⟂ n (m − n )!m!(m + n )! 2 m 2 (A.4a) m=ðnð ð ð ð ð

87 and

∞ 0 1m H 2 I É (−1)m(2m)! = (−1)ðnð b L0 − ⟂ . n (m − n )!m!(m + n )! 2 m−1 2 (A.4b) m=ðnð ð ð ð ð

The other perpendicular integrals can be found by parametric diﬀerentiation of Pn and the Laguerre jL0 x jL0 x xL1 x R three-point identity j−1( ) = j ( ) + j−1( ). The n and n functions are

∞ 0 1m H 2 I (−1)ðnð É (−1)mm(2m)! R = b L0 − ⟂ n 1∕2 (m − n )!m!(m + n )! 2 m 2 (A.5a) b m=ðnð ð ð ð ð and

∞ 0 1m H 2 I (−1)ðnð É (−1)mm(2m)! = b L0 − ⟂ . n 1∕2 (m − n )!m!(m + n )! 2 m−1 2 (A.5b) b m=ðnð ð ð ð ð

Finally, the Un and n functions are

∞ m 2 2 0 1m H 2 I (−1)ðnð É (−1) [2m (m − 1) + n ](m − 3∕2)! U = b L0 − ⟂ n 1∕2 m (A.6a) b (m − n )!m!(m + n )! 2 2 m=ðnð ð ð ð ð and

∞ m 2 2 0 1m H 2 I (−1)ðnð É (−1) [2m (m − 1) + n ](m − 3∕2)! = b L0 − ⟂ . n 1∕2 m−1 (A.6b) b (m − n )!m!(m + n )! 2 2 m=ðnð ð ð ð ð

88 A.2 Selected Analytical Properties

A.2.1 Beam Perpendicular Integral Functions

In deriving Eq. 2.14, summations over the perpendicular integral functions are frequently used. Notable of these are

∞ É Pn = 1, (A.7a) n=−∞

H I ∞ 2 É 2 ⟂ n Pn = b 1 + , (A.7b) n=−∞ 2

∞ É − 2 n = e ⟂ , (A.7c) n=−∞

∞ É 2 n n = b, (A.7d) n=−∞

∞ ∞ É É Rn = n = 0, (A.7e) n=−∞ n=−∞

∞ ∞ É É nRn = nn = 0, (A.7f) n=−∞ n=−∞

89 and

∞ 2 É ⟂ Un = 1 + . (A.7g) n=−∞ 2

The limiting behavior as vb → 0 is also useful for its connection to the Maxwellian species terms (Eq. 2.7) in the dielectric tensor, which for each function is

lim Pn = lim n = Λn( b), (A.8a) vb→0 vb→0

1∕2 ¨ lim Rn = lim n = b Λn( b), (A.8b) vb→0 vb→0 and

2 n ¨ lim Un = lim n = Λn( b) − 2 bΛ ( b). v v  n (A.8c) b→0 b→0 b

A.2.2 Maxwellian Perpendicular Integral Functions

−x The Maxwellian perpendicular integrals chieﬂy involve the function Λn(x) ≡ In(x)e . Some useful properties of this function are

1 Λ¨ (x) = Λ (x) − 2Λ (x) + Λ (x) , n 2 n−1 n n+1 (A.9a)

¨ ¨ Λn = Λ−n, (A.9b)

90 ∞ É Λn(x) = 1, (A.9c) n=−∞

∞ É ¨ Λn(x) = 0, (A.9d) n=−∞ and

∞ É 2 n Λn(x) = x. (A.9e) n=−∞

This identities derive from more fundamental Bessel-function identities.

91 Appendix B

Plasma Reactivity Calculation

For a uniform plasma, where we assume that reactivity has no spatial dependence, the reactivity of the entire simulation domain can be calculated using the overall velocity distribution functions of the relevant species. The general expression for fusion emissivity s [46] (reactions per second per unit volume) for two populations a and b is

nanb sab = ⟨v(v)⟩, (B.1) 1 + ab where ab = 0 if a ≠ b and ab = 1 if a = b and

t 2 2 2 v = ð⃗vð = ð⃗vb − ⃗vað = (vbx − vax) + (vby − vay) + (vbz − vaz) (B.2)

is the magnitude of the diﬀerence between the two general velocity vectors ⃗vb and ⃗va. Thus, if a population is reacting with itself, a factor of one-half is needed to prevent double-counting. The expectation value ⟨v(v)⟩, the heart of this calculation, is given by

3 3 ⟨v(v)⟩ = v(v)fa(⃗va)fb(⃗vb) d vb d va, (B.3) Êa Êb

92 f (⃗v ) where  is the full volume for velocity space and X X is the velocity distribution function for a species X. It is assumed here that distribution functions can be broken into Cartesian components:

fX(⃗vX) = fXx(vXx)fXy(vXy)fXz(vXz). (B.4)

−∞ +∞ For Cartesian distributions,  ranges from to for all integrals. The distributions are assumed to have normalization

3 fX(⃗vX) d vX = 1. (B.5) ÊX

The goal of this eﬀort is to reduce the number of variables in the expression for v from six to one, thereby making numerical integration of s far faster. To do so, a series of transforms of the distribution functions is taken in four steps to create a single distribution function f(v). At each step, the transformed distribution function must be renormalized (a requirement for transforming on discrete data), which is omitted from the description of each step. Additionally, each fsi must be discretized on the same array of bins, giving dataset each the same values and length N. In particular, let the minimum and maximum bin values be v∨ and v∧, respectively. The algorithm then proceeds in the following steps. Two schemes are shown, a primary method and an alternative, likely more numerically robust method.

93 B.1 Primary Method

B.1.1 Correlation

First, for each direction i, the pairs of raw distribution functions fsi for each species are combined

via Fourier correlation to give three functions fi(vi ≡ v2i − v1i)):

∞ fi(vi) ≡ f2i ⋆ f1i = f2i(t)f1i(t + vi) dt. (B.6) Ê−∞

The resulting discrete domain of vi has length 2N − 1 and spans the range v∨ − v∧ ≤ vi ≤ v∧ − v∨. Taking this step alone, ⟨v(v)⟩ can be expressed in terms of three integrals.

B.1.2 Variable Transform 1

2 Next, each function fi is re-binned according to the change of variable ui ≡ vi to create a new set of three functions gi(ui). This new set of bins has arbitrarily deﬁned length and maximum. Let these quantities be M and u∧, respectively. The minimum bin is ﬁxed at u∨ = 0 (left edge). For continuous functions, gi is related to fi by

1 √ √ gi(ui) = √ fi + ui + fi − ui , (B.7) 2 ui

and the re-binning for discrete data approximates this functional form.

94 B.1.3 Convolution

Now, the remaining three functions gi can be combined into one function g(u ≡ ux + uy + uz) through a two-step Fourier convolution:

g(u) = (gx ∗ gy) ∗ gz. (B.8)

The resulting discrete domain of u has length 2(2M − 1) − 1 = 4M − 3 and spans the range 0 ≤ u ≤ 3u∧. With this step, ⟨v(v)⟩ is reduced to a single integral. If one’s fusion cross section is a function of energy, rather than relative velocity, one can stop at this step.

B.1.4 Variable Transform 2

√ Finally, g is re-binned according to the change of variable v ≡ u, makign a new function f(v). As in the ﬁrst variable transform, the new set of bins for v has arbitrary length and maximum, but a ﬁxed minimum at v∨ = 0. For continuous functions, f is related to g by

2 f(v) = 2vg v . (B.9)

As before, the re-binning for discrete data approximates this functional form.

B.1.5 Final Result

With f(v) in hand, ⟨v(v)⟩ can be expressed simply as

∞ ⟨v(v)⟩ = v(v)f(v) dv, (B.10) Ê0

95 which evaluates numerically much more quickly than the six integrals in the original expression for ⟨v(v)⟩. The emissivity is then

∞ nanb sab = v(v)f(v) dv. (B.11) 1 + ab Ê0

B.2 Alternative Method

The above method may be prone to numerical errors because of the divergence of equation B.7. An alternative approach is to retain three integrals for ⟨v(v)⟩ and carry out the convolutions explicitly. These convolutions can be expressed as two integrals over variables and , giving

∞ ∕2 ∕2 nanb 3 sab = v (v) cos Fx(v cos sin )Fy(v cos cos )Fz(v sin ) d d dv, 1 + ab Ê0 Ê0 Ê0 (B.12)

where Fi(x) ≡ fi(x) + fi(−x). This form has the beneﬁt of numerical robustness relative to the previous method or simply integrating after the ﬁrst correlation therein.