Electromagnetic Particle-In-Cell Algorithms on Unstructured Meshes for Kinetic Plasma Simulations

Electromagnetic Particle-in-Cell Algorithms on Unstructured Meshes for Kinetic Plasma Simulations

Dissertation

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

Dong-Yeop Na, M.S.

Graduate Program in Electrical and Computer Engineering

The Ohio State University

2018

Dissertation Committee:

Prof. Fernando L. Teixeira, Advisor Prof. Kubilay Sertel Prof. Robert Lee c Copyright by

Dong-Yeop Na

2018 Abstract

Plasma is a signiﬁcantly ionized gas composed of a large number of charged particles such as electrons and ions. A distinct feature of plasmas is the collective interaction among charged particles. In general, the optimal approach used for modeling a plasma system depends on its characteristic (temporal and spatial) scales. Among various kinds of plasmas, collisionsless plasmas correspond to those where the collisional frequency is much smaller than the frequency of interests (e.g. plasma frequency) and the mean free path is much longer than the characteristic length scales (e.g. Debye length).

Collisionless plasmas consisting of kinetic space charge particles interacting with electromagnetic ﬁelds are well-described by Maxwell-Vlasov equations. Electromag- netic particle-in-cell (EM-PIC) algorithms solve Maxwell-Vlasov systems on a computational mesh by employing coarse-grained superparticle. The concept of superparticle, which may represent millions of physical charged particles (coarse-graining of the phase space), facilitates the realization of computer simulations for under- scaled kinetic plasma systems mimicking the physics of real kinetic plasma systems.

In this dissertation, we present an EM-PIC algorithm on general (irregular) meshes based on discrete exterior calculus (DEC) and Whitney forms. DEC and Whitney forms are utilized for consistent discretization of Maxwells equation on general irregular meshes. The proposed EM-PIC algorithm employs a mixed ﬁnite-element

ii time-domain (FETD) ﬁeld solver which yields a symplectic integrator satisfying en-

ergy conservation. Importantly, we employ Whitney-forms-based gather and scatter

schemes to obtain exact charge conservation from ﬁrst principles, which had been a

long-standing challenge for PIC algorithms on irregular meshes.

Several further contributions are made in this dissertation: (i) We develop a local and explicit EM-PIC on unstructured grids using sparse approximate inverse (SPAI) strategy and study macro- and microscopic residual errors in motions of charged particles affected by the approximate inverse errors. (ii) We extend the present EM-PIC algorithm to the relativistic regime with several relativistic particle-pushers and compare their performance. (iii) We implement a secondary electron emission (SEE) processor based on probabilistic Furman-Pivi model and numerically investigate multipactor effects that are resonant electron discharges from conducting surfaces by external RF fields. (iv) We diagnose numerical Cherenkov radiation, which is a detrimental effect frequently found in EM-PIC simulations involving relativistic plasma beams, for the present EM-PIC algorithm on general meshes. (v) We extend the

FETD field solver for the solution of Maxwell’s equations in circularly symmetric or body-of-revolution (BOR) geometries. (vi) Lastly, we combine the EM-PIC algorithm with the BOR-FETD field solver for the efficient analysis of vacuum electronic devices (VED).

iii Dedicated to my beloved wife Da-Young and my family

iv Acknowledgments

First and foremost, I would like to express my sincere gratitude to my advisor,

Prof. Fernando L. Teixeira, for the support, encouragement, and guidance during the years of my graduate study. It has been a great honor and privilege to work with him.

His passion and immense knowledge in electromagnetics, mathematics, and physics, and kindness and commitment to his students will always inspire me.

Besides, I would like to thank Dr. Yuri A. Omelchenko and Prof. Ben-Hur V.

Borges for their helpful discussions and suggestions.

My special appreciation also goes to the members of my doctoral committee, Prof.

Kubilay Sertel and Prof. Robert Lee, for insightful comments.

I would like to thank to many of ESL colleagues, past and present, Haksu Moon,

WoonGi Yeo, Jungwhan Park, Carlos A. Viteri, Cagdas Gunes, Daniel O. Acero, and

Julio L. Nicolini, and my friends, Yun-Shik Hahn, Chunghyun Lee, Jongchan Choi,

Kyoung-Ho Jeong, and Huyngjun Kim.

I wish to thank my family for their constant support and unconditional love.

Last but not least, I would like to share this accomplishment with my beloved wife, Da-Young, and sincerely appreciate her her encouragement, support, and love.

v Financial support from National Science Foundation grant ECCS-1305838, De- fense Threat Reduction Agency grant HDTRA1-18-1-0050, Ohio Supercomputer Cen- ter grants PAS-0061 and PAS-0110, and The Ohio State University Presidential Fel- lowship Program are gratefully acknowledged.

vi Vita

March 30, 1987 ...... Born - Seoul, Korea

Feburary, 2012 ...... B.S. in Electrical and Computer Eng., Ajou University, Suwon, Korea July, 2014 ...... M.S. in Electrical and Computer Eng., Ajou University, Suwon, Korea August, 2014-May, 2017 ...... Graduate Research Associate, ElectroScience Laboratory, The Ohio State University, USA May, 2017-May, 2018 ...... Presidential Fellowship Program, The Ohio State University, USA May, 2018-August, 2018 ...... Graduate Research Associate, ElectroScience Laboratory, The Ohio State University, USA August, 2018-present ...... Graduate Teaching Associate, Electrical and Computer Eng., The Ohio State University, USA

Publications

Jounral Publications

Dong-Yeop Na, Haksu Moon, Yuri A. Omelchenko, Fernando L. Teixeira, “Local, explicit, and charge-conserving electromagnetic particle-in-cell algorithm on unstructured grids,” IEEE Trans. Plasma Sci., 44 (2016) 1353–1362.

Dong-Yeop Na, Yuri A. Omelchenko, Haksu Moon, Ben-Hur V. Borges, Fernando L. Teixeira, “Axisymmetric charge-conservative electromagnetic particle simulation algorithm on unstructured grids: Application to microwave vacuum electronic De- vices,” J. Comput. Phys., 346 (2017) 295–317.

vii Dong-Yeop Na, Haksu Moon, Yuri A. Omelchenko, Fernando L. Teixeira, “Rel- ativistic extension of a charge-conservative ﬁnite element solver for time-dependent Maxwell-Vlasov equations,” Phys. Plasmas, 25 (2018) 013109.

Dong-Yeop Na, Ben-Hur V. Borges, Fernando L. Teixeira, “Finite element time- domain body-of-revolution Maxwell solver based on discrete exterior calculus,” J. Comput. Phys., 376 (2017) 249–275.

Conference publications

Dong-Yeop Na, Fernando L. Teixeira, Yuri A. Omelchenko, “Charge-conserving relativistic PIC algorithm on unstructured grids,” 2016 USNC-URSI National Radio Science Meeting, Boulder, CO, Jan. 6-9, 2016.

Dong-Yeop Na, Fernando L. Teixeira, H. Moon, Yuri A. Omelchenko, “Full-wave FETD-based PIC algorithm with local explicit update,” 2016 IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting, Fajardo, PR, June 26-July 1, 2016.

Dong-Yeop Na, Fernando L. Teixeira, Yuri A. Omelchenko, “Unstructured-grid and conservative electromagnetic particle-in-cell: application to micromachined slow- wave structures,” 2016 IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting, Fajardo, PR, June 26-July 1, 2016.

Dong-Yeop Na, Yuri A. Omelchenko, Fernando L. Teixeira, “An eﬃcient algorithm for simulation of plasma beam high-power microwave sources,” 2017 IEEE MTT-S International Microwave Symposium, Honolulu, HI, June 4-9, 2017.

Dong-Yeop Na, Fernando L. Teixeira, Ben-Hur V. Borges, “Finite-element time- domain solver for axisymmetric devices based on discrete exterior calculus and transformation optics,” 2017 SBMO/IEEE MTT-S International Microwave and Opto- electronics Conference, Aguas de Lindoia, Brazil, Aug. 27-30, 2017.

Dong-Yeop Na, Yuri A. Omelchenko, Fernando L. Teixeira, “Irregular-grid-based particle-in-cell simulations of resonant electron discharges with probabilistic secondary electron emission model,” 2017 XXXIInd General Assembly and Scientiﬁc Sympo- sium of the International Union of Radio Science, Montreal, QC, Canada, August 19-26, 2017.

viii Dong-Yeop Na, Yuri A. Omelchenko, Fernando L. Teixeira, “Discretization of Maxwell-vlasov equations based on discrete exterior calculus,” 2017 XXXIInd General Assembly and Scientiﬁc Symposium of the International Union of Radio Science, Montreal, QC, Canada, August 19-26, 2017.

Dong-Yeop Na, Julio L. Nicolini, Robert Lee, Ben-Hur V. Borges, Yuri A. Omelchenko, Fernando L. Teixeira, “Diagnosis of Numerical Cherenkov Instability in Plasma Simu- lations on General Mesh,” Computational Aspects of Time Dependent Electromagnetic Wave Problems in Complex Materials, The Institute of Computational and Experi- mental Research in Mathematics (ICERM), Providence, RI, June 24-29, 2018.

Dong-Yeop Na, Fernando L. Teixeira, Yuri A. Omelchenko, “Dispersion Analy- sis of Electron Bernstein Waves in Magnetized Warm Plasmas by Finite Element Particle-in-Cell Modeling,” 2018 IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting, Boston, MA, July 8-13, 2018.

Dong-Yeop Na, Fernando L. Teixeira, Yuri A. Omelchenko, “Numerical Cherenkov Radiation Eﬀects from Grid Dispersion in Finite Element Particle-in-Cell Simulations of Relativistic Electron Beams,” 2018 IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting, Boston, MA, July 8-13, 2018.

Fields of Study

Major Field: Electrical and Computer Engineering

Studies in: Electromagnetic theory Computational electromagnetics Antennas Mathematics

ix Table of Contents

Page

Abstract ...... ii

Dedication ...... iv

Acknowledgments ...... v

Vita ...... vii

List of Tables ...... xiv

List of Figures ...... xvi

1. Introduction ...... 1

1.1 Background and motivation ...... 1 1.2 Contribution of this dissertation ...... 6 1.3 Organization of this dissertation ...... 7

2. Local, Explicit, and Charge-conserving EM-PIC on Unstructured Mesh . 11

2.1 Explicit FETD-PIC Algorithm ...... 13 2.1.1 Mixed E − B FETD scheme ...... 14 2.1.2 Gather-scatter and particle pusher steps ...... 16 2.1.3 Discrete continuity equation ...... 17 2.2 Sparse Approximate Inverse (SPAI) strategy ...... 19 2.2.1 Discrete Gauss’ law ...... 20 2.3 Numerical Results ...... 21 2.3.1 Single-particle trajectories ...... 21 2.3.2 Plasma ball expansion ...... 27 2.3.3 Electron beam in a vacuum diode ...... 30 2.3.4 Electron Bernstein waves ...... 33

x 2.4 Conclusion ...... 35

3. Relativitic Extension of Particle-Pusher ...... 36

3.1 Particle-pushers in the relativistic regime ...... 37 3.1.1 Relativistic Boris pusher ...... 38 3.1.2 Vay pusher ...... 40 3.1.3 Higuera-Cary pusher ...... 41 3.2 Numerical results ...... 41 3.2.1 Synchrocyclotron ...... 41 3.2.2 Harmonic oscillations in Lorentz-boosted frame ...... 44 3.2.3 Relativistic Bernstein Modes in Magnetized Pair-Plasma . . 47 3.3 Conclusion ...... 55

4. Multipactor ...... 57

4.1 Irregular-Grid EM-PIC Algorithm integrated with Furman-Pivi model 60 4.2 Charge-conserving scatter near conducting surface ...... 61 4.3 Furman-Pivi SEE model implementation ...... 62 4.4 Numerical Results and Discussion ...... 63 4.4.1 Verification of SEE model in EM-PIC simulations ...... 63 4.4.2 Multipactor on copper versus stainless steel surfaces . . . . 66 4.4.3 Surface treatment effects ...... 68 4.4.4 Multipactor susceptibility to RF voltage amplitude . . . . . 70 4.4.5 Multipactor saturation effects ...... 72 4.5 Conclusion ...... 75

5. Numerical Cherenkov Radiation and Grid Dispersion Eﬀects ...... 79

5.1 Numerical Cherenkov Radiation in the FDTD-based EM-PIC Algo- rithm ...... 81 5.2 Numerical Cherenkov Radiation in ﬁnite-element-based EM-PIC Al- gorithms ...... 85 5.2.1 SQ Mesh ...... 88 5.2.2 Triangular-element-based FE meshes ...... 92 5.3 Numerical Experiments ...... 100 5.4 Conclusion ...... 109

6. Finite-Element Time-Domain Body-of-Revolution Maxwell-Solver . . . . 111

6.1 Formulation ...... 113 6.1.1 Exploration of transformation optics (TO) concepts . . . . . 113 6.1.2 Field decomposition ...... 115

xi 6.1.3 Mixed FE time-domain BOR solver ...... 116 6.1.4 Symmetry axis singularity treatment ...... 123 6.2 Numerical Examples ...... 125 6.2.1 Cylindrical cavity ...... 126 6.2.2 Logging-while-drilling sensor simulation ...... 129 6.3 Conclusion ...... 137

7. Axisymmetric Electromagnetic Particle-in-Cell Algorithm: Application to Microwave Vacuum Electronic Devices ...... 140

7.1 Spatial dimensionality reduction ...... 145 7.1.1 Exterior calculus representation of Maxwell’s equations . . . 145 7.1.2 Cylindrical axisymmetry constraints ...... 146 7.1.3 Modiﬁed Hodge star operator ...... 147 7.2 Validation ...... 150 7.2.1 Metallic hollow cylindrical cavity ...... 150 7.2.2 Space-charge-limited (SCL) cylindrical diode ...... 152 7.3 Numerical examples ...... 156 7.3.1 Relativistic backward-wave oscillator (BWO) ...... 158 7.4 Conclusion ...... 166

Appendices 168

A. Basics of Plasmas ...... 168

A.1 Fundamental parameters ...... 168 A.2 Quasi-neutrality in plasma ...... 170 A.3 Plasma oscillation ...... 171 A.4 Collisions in plasmas ...... 172

B. Kinetic Plasma Description ...... 174

B.1 Plasma kinetic equation ...... 174 B.2 Vlasov equation for collisionless plasmas ...... 178 B.3 Superparticle: Coarse-grained f (x, v, t) ...... 179 B.4 Maxwell-Vlasov or Poisson-Vlasov systems ...... 181

C. Discrete Exterior Caclulus (DEC) ...... 183

C.1 Whitney forms ...... 183 C.2 Pairing operation ...... 184 C.3 Generalized Stokes’ theorem ...... 184

xii C.4 Discretization of Maxwell’s equation ...... 184 C.4.1 Cartesian coordinates case ...... 184 C.4.2 Body-of-revolution case ...... 185 C.5 Incidence Matrices ...... 187 C.6 Discrete Hodge matrix ...... 189 C.7 Barycentric dual lattice relations ...... 193

D. Cartesian-like PML implementation ...... 195

E. Stability Conditions ...... 201

Bibliography ...... 203

xiii List of Tables

Table Page

2.1 Number of elements in Meshes 1, 2, and 3 ...... 22

2.2 Convention used for particle trajectory visualization...... 24

3.1 Veriﬁcation of discrete Gauss’ law for the non-relativistic case (Fig. 3.2a)...... 45

3.2 Veriﬁcation of discrete Gauss’ law for the relativistic case without syn- chronization (Fig. 3.2b)...... 46

3.3 Veriﬁcation of discrete Gauss’ law for the relativistic case with syn- chronization (Fig. 3.2c)...... 46

4.1 Multipactor simulation parameters for the parallel waveguide in Fig. 4.7a. 69

4.2 Triangularly-grooved surface parameters...... 70

4.3 Mesh parameters...... 70

4.4 Spectral amplitude of output voltage signals for high-order harmonics. 73

5.1 Basic meshes properties...... 102

6.1 Maximum time-step intervals for various cases in the simulation of cylindrical metallic cavity...... 127

6.2 Eigenfrequencies for the cylindrical cavity and normalized errors between numerical and analytic results...... 132

7.1 Estimation of the run time of EM-PIC simulations based on FETD and FDTD at each time-update...... 149

xiv 7.2 Resonant frequencies for axisymmetric cavity modes and normalized errors between numerical and analytic works...... 153

7.3 Mesh information for diﬀerent SCSWS cases ...... 163

xv List of Figures

Figure Page

2.1 Basic steps in a EM-PIC algorithm. On unstructured meshes, conventional ﬁeld solvers are implicit, requiring the solution of a (large) linear system at each time step...... 14

2.2 Charge-conserving gather and scatter steps [1]. (a) Interpolation of E and B at the position of the particle by edge-based (left) and face- based degrees of freedom contributions (right) (weighted by the Whit- ney functions) in the gather step. (b) Exact charge-conserving scatter scheme. The sum of the two colored areas in the left, representing the magnitude of the edge currents, is equal to the blue area in the left, representing the charge variation at node 1 during one time step. . . 16

2.3 Relative position diﬀerence (RPD) of the various test particles w.r.t. the standard particle placed at the origin, in a polar diagram where the radial distance is represented in logarithmic scale...... 23

2.4 Results for a circular particle trajectory on 3 diﬀerent meshes. (a) (b) (c) Particle trajectory histories. (d) (e) (f) RPDs versus time for the four test particles. (g) (h) (i) Normalized RPD bands for the four test particles...... 25

2.5 Results for a trajectory with drift. (a) (b) (c) Particle trajectory history. (d) (e) (f) RPDs versus time for the four test particles. (g) (h) (i) Normalized RPD bands for the four test particles...... 27

2.6 Radial current versus radius coordinate for the expanding plasma at time step n = 9 × 104 using the LU-based implicit ﬁelds solver and the SPAI-based explicit ﬁeld solver with k = 2, 4, and 6...... 28

xvi 2.7 (a) Normalized residuals of the discrete continuity equation for the plasma ball expansion example using diﬀerent ﬁeld solvers, at t = 2×104∆t. (b) Similar results for the discrete Gauss’ law. (c) Averaged normalized residuals for the discrete Gauss’ law versus time step index. 29

2.8 Results for the accelerated electron beam at t = 6×104∆t. (a) (b) Par- ticle distribution snapshot from charge-conserving EM-PIC algorithms using an LU-based implicit solver and a SPAI-based (k = 2) explicit solver, respectively . (c) Particle distribution snapshot from a conventional (non-charge conserving on the unstructured grid) EM-PIC algorithm with an LU-based implicit solver. (d) (e) (f) Corresponding electric-ﬁeld proﬁle distributions...... 31

2.9 Number density and average velocity of particles across a transversal section of the electron beam at t = 3 × 103∆t, after steady-state has been reached...... 32

2.10 Simulated ω × k dispersion diagram for the X mode propagation and

for electron Bernstein waves in a magnetized warm plasma. Here ωpe is the plasma frequency and ∆x is the grid spacing, chosen uniform. The analytical results are indicated by the red dots in the diagram. Note that the use of a charge-conserving scatter step in PIC algorithm as described in [1] reduces the numerical noise and yields cleaner spectral bands in the numerically generated band diagrams. In addition, a charge-conserving scatter step mitigates the spurious DC ﬁeld cause by spurious charge accumulation on the grid nodes, as observed at the bottom of the zoomed plots. Overall, a very good agreement is observed between the numerical and the analytical results...... 34

3.1 (a) Cyclotron configuration. (b) Computational domain, where the blue vertical strip indicates the region where an external longitudinal RF electric field is applied. The DC magnetic field is applied in the whole computational region except for the RF acceleration gap (red). 42

3.2 Electron trajectories on a cyclotron: (a) Non-relativistic, (b) Relativis- tic, unsynchronized, and (c) Relativistic, synchronized...... 43

3.3 Orbital frequency and relativistic factor for the case shown in Fig. 3.2c. 44

3.4 Comparison of electron velocity magnitudes of the three cases shown in Fig. 3.2...... 45

xvii 3.5 Motion of harmonic oscillator of a single positron inverse-Lorentz- transformed into Laboratory frame...... 47

3.6 Dispersion relations for classical (non-relativistic) electron Bernstein modes of PIC results (Parula colormap) and analytic predictions [2] (dashed red line)...... 48

3.7 An isotropic 2D Maxwell-Boltzmann-J¨uttnervelocity distribution, f0 (p) for η = 1/20: (a) Speed distribution and (b) relativistic velocity distribution...... 51

3.8 Dispersion relations for plasma waves propagating in magnetized relativistic pair-plasma for η = 1/20: Comparison of PIC results and analytic prediction...... 52

3.9 Normalized residuals versus nodal index for (a) discrete continuity equation (DCE) and (b) discrete Gauss law (DGL)...... 53

4.1 Schematic illustration of a typical SEE process in an irregular-grid-based EM-PIC simulation. Note that electric current densities by the primary or secondaries are deposited on red- or blue-highlighted edges, respectively. . 58

4.2 Comparison of simulation and experimental results for SEE on copper [(a) and (b)] and stainless steel [(c) and (d))] surfaces. Figures (a) and (c) illustrate SEY δ versus the primary incident energy. Figures (b) and (d) show the emitted-energy spectrum dδ/dE...... 59

4.3 Geometrical illustration of exact charge conservation on irregular grids for a primary impact (also applicable for secondary electrons emitted on the opposite way) at PEC surfaces during ∆t. Plot (a) depicts the charge variation rate at jth node. Plot (b) depicts the divergence of current on jth node, which is equal to the sum of ith and kth currents...... 61

4.4 Angular dependence of δ on a copper surface...... 65

xviii 4.5 PIC results for probabilistic SEE model. (a) Superparticle population versus time (RF voltage periods). (b) and (c) Snapshots of particle’s trajectories for copper and stainless steel cases, respectively. These trajectory snapshots

are taken during four successive half-periods of the RF signal, i.e.: t/TRF ∈ (0, 0.5), t/TRF ∈ (0.5, 1), t/TRF ∈ (1, 1.5), and t/TRF ∈ (1.5, 2), where TRF = 1/fRF = 0.96 [ns]...... 65

4.6 Particle trajectory snapshots on the phase space. The coordinate axes represent x/10 [m], y [m], and the normalized speed of the particles (|vp| /20c). Each plot corresponds to a half-period of the RF signal, as in Fig. 4.5. . . 66

4.7 Multipactor in parallel plate waveguides. (a) Schematics of the problem geometry. (b) Flat surface waveguide meshing. (c) Triangular-grooved waveguide meshing...... 68

4.8 RF voltage amplitude susceptibility at fRFDpp = 4 [GHz·mm] for ﬂat and grooved copper surfaces...... 71

4.9 RF voltage cycle versus population ampliﬁcation, An for both surfaces at a VRF = 1, 143.16 V...... 72

4.10 Output signals for both surfaces (a) in time-domain and (b) frequency domain. 74

4.11 Particle position snapshots taken over a half RF period during the saturation regime. The RF voltage period is 0.5 ns. Plots (a)-(f) are for the ﬂat surface and plots (g)-(l) are for the grooved surfaces...... 76

4.12 External-field and self-field snapshots taken over a half period during the saturation regime. The RF voltage period is 0.5 ns. Plots (a)-(f) are for the flat surface and plots (g)-(l) are for the grooved surfaces...... 77

4.13 Snapshots of vy [m/s] versus y [m] taken over a half RF period during the saturation regime. The RF voltage period is 0.5 ns. Plots (a)-(f) are for the ﬂat surface and plots (g)-(l) are for the grooved surfaces...... 77

4.14 Snapshots of vx [m/s] versus y [m] taken over a half RF period during the saturation regime. The RF voltage period is 0.5 ns. Plots (a)-(f) are for the ﬂat surface and plots (g)-(l) are for the grooved surfaces...... 78

xix 5.1 Numerical grid dispersion of the 2-D Yee’s FDTD scheme on a structured mesh. (a) The red color surface represents the dispersion diagram of the normalized frequency ω∆t/π versus the normalized numerical wavenumber κ˜h in radians. The olive color surface represents the light cone. The contour levels at the bottom represent the normalized phase errors (with respect to the color bar). (b) Wavenumber magnitude versus frequency for diﬀerent wave propagation angles with respect to the o o x axis, φp ∈ [0 , 45 ]...... 82

5.2 Analytic NCR predictions on a structured FDTD grid for a bulk beam

velocity vb = 0.9c. (a) 3-D numerical dispersion diagrams (in red) and beam planes (fundamental plane in green and aliased beams in transparent yellow). (b) Trajectories of NCR solutions projected onto the 2-D κ˜-space...... 84

5.3 Schematic illustration of the four types of mesh considered in this study. (a) Square regular (SQ) elements in both FDTD and FETD, (b) right-angle triangular (RAT) elements in FETD, (c) isosceles triangular (ISOT) elements in FETD, and (d) highly-irregular triangular (HIGT) elements in FETD...... 86

5.4 Schematic of SQ mesh. There are two characteristic edges (A and B) directed along the y and x and colored in red and blue, respectively. . 88

5.5 Numerical grid dispersion for the FETD scheme on the SQ mesh. (a) The red color surface represents the dispersion diagram of the normalized frequency ω∆t/π versus the normalized numerical wavenumber κ˜h in radians. The olive color surface represents the light cone. The contour levels at the bottom represent the normalized phase errors (with respect to the color bar). Note that the normalized phase error is always negative in this case because of a slightly faster-than-light numerical phase velocity. (b) Projected dispersion curves for diﬀerent o o wave propagation angles with respect to the x axis φp ∈ [0 , 45 ]. . . . 91

5.6 Analytic prediction of NCR for the FETD algorithm on the SQ mesh

when vb = 0.9c. (a) 3-D dispersion diagram. (b) NCR solution contours projected onto the ﬁrst Brillouin zone in the κ˜-space...... 92

xx 5.7 A periodically-arranged triangular grid. It has three characteristic edges denoted by A, B, and C. Labels inside circles denote global facet indexes and labels inside rectangles and pentagons denote local edge and node indexes, respectively...... 93

5.8 Numerical grid dispersion for the FETD scheme on the RAT mesh with the CFL number equal to one. Unlike the FDTD or FETD-SQ cases, this diagram exhibits an additional (upper) dispersion band. (a) The red (lower band) and blue (upper band) color surfaces represent the dispersion diagram of the normalized frequency ω∆t/π versus the normalized numerical wavenumber κ˜h in radians. The olive color surface represents the light cone. The contour levels at the bottom represent the normalized phase errors (with respect to the color bar). (b) Projected dispersion curves for diﬀerent wave propagation angles with o o respect to the x axis φp ∈ [−45 , 45 ]...... 96

5.9 (a) The vector proxy of a Whitney 1-form associated with the edge −→ AB on a triangular mesh. (b) Tangential component along edge. (c) Normal component to the edge direction...... 97

5.10 Analytic prediction of NCR for the FETD-based EM-PIC scheme on

the RAT mesh assuming a plasma beam with bulk velocity vb = 0.9c. (a) Dispersion diagram. (b) NCR solution contours projected onto the ﬁrst Brillouin zone in the κ˜-space...... 98

5.11 Numerical grid dispersion for the FETD scheme on the ISOT mesh with CFL number equal to one. Unlike the FDTD or FETD-SQ cases, this diagram exhibits an additional (upper) dispersion band. (a) The red (lower band) and blue (upper band) color surfaces represent the dispersion diagram of the normalized frequency ω∆t/π versus the normalized numerical wavenumber κ˜h in radians. The olive color surface represents the light cone. The contour levels at the bottom and top represent the normalized phase errors (with respect to the color bar). (b) Projected dispersion curves for diﬀerent wave propagation angles o o with respect to the x axis φp ∈ [26.57 , 90 ]...... 98

5.12 Analytic prediction of NCR for the FETD-based EM-PIC scheme on

the ISOT mesh assuming a plasma beam with bulk velocity vb = 0.9c. (a) Dispersion diagram. (b) NCR solution contours projected onto the ﬁrst Brillouin zone in the κ˜-space...... 99

xxi 5.13 Initial velocity distributions for a relativistic pair plasma beam with

bulk velocity vb = 0.9c (γb ≈ 2.3). (a) Phase space in the beam rest frame. (b) Velocity distribution in the beam rest frame. (c) Phase space in the laboratory frame. (d) Velocity distribution in the laboratory frame...... 101

5.14 (a) HIGT mesh. (b) Histogram of the edge lengths. (c) Histogram of the triangular element angles...... 102

5.15 B ﬁeld amplitude distribution (log scale) over the ﬁrst Brillouin zone in the κ˜-space as measured from EM-PIC simulation snapshots at 47 µs. (a) and (c) plots correspond to FDTD- and FETD-based EM-PIC simulations on the SQ mesh, respectively. In (b) and (d), the analytical predictions are superimposed to the numerical results...... 104

5.16 B ﬁeld amplitude distribution (log scale) over the ﬁrst Brillouin zone in the κ˜-space as measured from EM-PIC simulation snapshots at 47 µs. (a) and (c) plots correspond to FETD-based EM-PIC simulations on the RAT and ISOT meshes, respectively. In (b) and (d), the analytical predictions are superimposed to the numerical results...... 105

5.17 B ﬁeld amplitude distribution (log scale) over the ﬁrst Brillouin zone in the κ˜-space as measured from FETD-based EM-PIC simulation snapshots at 47 µs on the HIGT mesh...... 107

5.18 The qualitative comparison of the B ﬁeld amplitude distribution (log scale) on the κ˜-space between FDTD and FETD-HIGT cases. (a)

shows the spectral amplitude of B versusκ ˜yh at some ﬁxed values of κ˜yh and vice-versa in (b)...... 107

5.19 Evolution of the magnetic energy Wm due to NCR on various meshes. 108

5.20 Snaphots of the magnetic ﬁeld distribution resulting from EM-PIC simulations of a single electron-positron pair moving relativistically. The snapshots are taken at 75.2 ns, 112.8 ns, and 150.4 ns, as indicated. The results correspond to: (a-c) FDTD-based EM-PIC simulation on SQ mesh , (d-f) FETD-based EM-PIC simulation on SQ mesh, (g- i) FETD-based EM-PIC simulation on the RAT mesh, (j-l) FETD- based EM-PIC simulation on ISOT mesh, (m-o) FETD-based EM-PIC simulation on HIGT mesh...... 110

xxii 6.1 Depiction of an axisymmetric structure...... 113

6.2 (2+1) setup for ﬁelds on (a) primal and (b) dual meshes at the meridian plane. The vertical axis is ρ and the horizontal axis is z...... 116

6.3 Vector proxies of various degrees of Whitney forms on the mesh: (a) (1) (2) (0) (RWG) Wj , (b) Wk , (c) Wi , and (d) Wj . Note that tj is a unit vector tangential to j−th edge and parallel to its direction and nk is a unit vector normal to k−th face...... 119

6.4 Field boundary conditions on the primal mesh for the TEφ field with (a) perfect magnetic conductor (m = 0) and (b) perfect electric conductor (m 6= 0) and for the TMφ field with (c) perfect magnetic conductor (m 6= 0) and (d) perfect electric conductor (m = 0). Dashed lines indicate Dirichlet boundary condition, for example edges on the z axis representing a perfect electric conductor boundary for TEφ field in (b), or nodes on the z axis representing a perfect electric conductor boundary for the TMφ field in (d)...... 124

6.5 Schematic view of the simulated cylindrical cavity with perfect electric conductor (PEC) walls. The cavity dimensions are a = 0.5 m and h =1m...... 127

6.6 Normalized spectral amplitude for E, showing the eigenfrequencies of the cavity. Black solid lines correspond to the present FETD-BOR result. Red solid and blue dashed lines are analytic predictions for the

TEmnp and TMmnp eigenfrequencies, respectively...... 129

6.7 Transient snapshots for Ez inside the cylindrical cavity at (a) 1.0024 [µs], (b) 1.0028 [µs], (c) 1.0032 [µs], and (d) 1.0036 [µs]...... 130

6.8 Transient snapshots for Bz inside the cylindrical cavity at (a) 1.0024 [µs], (b) 1.0028 [µs], (c) 1.0032 [µs], and (d) 1.0036 [µs]...... 131

6.9 Logging-while-drilling sensor problem geometry (from inner to outer features): metallic mandrel, transmit (Tx) and receive (Rx) coil antennas, mud-ﬁlled borehole, and adjacent geological formation. . . . . 133

xxiii 6.10 Logging-while-drilling sensor responses. (a) First scenario: the conductivity of the adjacent geological formation is varied. (b) Second scenario: the sensor moves downward through a borehole surrounded by a geological formation with three horizontal layers...... 135

6.11 Computed (a) AR and (b) PD (in deg.) by a logging-while-drilling sensor surrounded by homogeneous geological formations with diﬀerent conductivities. This corresponds to the ﬁrst scenario in Fig. 6.10. The results from the present algorithm are compared against FDTD and NMM results [3] (see more details in the main text)...... 136

6.12 Computed PD (deg.) between the two receivers of the logging-while- drilling sensor versus the z position of the transmitter coil antenna. This corresponds to the second scenario in Fig. 6.10. The results from the present algorithm are compared against FDTD and NMM results [3] (see more details in the main text)...... 137

6.13 Electric ﬁeld distribution during the half period for zTx = (a) −50 inch, (b) −25 inch, (c) 5 inch, (d) 25 inch, (e) 50, and (f) 70 inch. Note that

zTx = 0 at the interface between ﬁrst (5 S/m) and second (0.0005 S/m) formations...... 138

7.1 Schematics of two examples of axisymmetric vacuum electronic devices. (a) Backward-wave oscillator producing bunching eﬀects on an electron beam. Wall ripples are designed to support slow-wave modes in the device. (b) Space-charge-limited cylindrical vacuum diode...... 142

7.2 A charged ring travels inside an axisymmetric object bounded by PEC: (a) a 3D view, (b) the meridian plane...... 146

7.3 The original problem shown in Fig. 7.2 is replaced by an equivalent 2D problem in the meridian plane as depicted above, which considers TEφ-polarized EM fields on Cartesian space with an artificial inhomogeneous medium. The variable coloring serves to stress the dependency of the artificial medium parameters on ρ...... 149

7.4 Snapshots for electric ﬁeld distribution at 2 µs. Note that RGB colors and white arrows indicate magnitudes and vectors of the electric ﬁelds, respectively...... 151

7.5 Spectrum for resonant cavity modes from 1 MHz to 1 GHz...... 152

xxiv 7.6 Schematics for divergent and convergent ﬂows in the cylindrical diode. 154

7.7 Space-charge-limited current density for various Lz/ρo and comparison between present EM-PIC simulations and KARAT by [4]...... 155

7.8 Electric ﬁeld intensity of self- and external ﬁelds at the instant of virtual cathode formation...... 156

7.9 Shematics of backward-wave oscillator with an instant particle distribution snapshots at t = 21.50 ns...... 157

7.10 Electric potential distribution (contour plots) and corresponding electric ﬁelds (vector plots) between the cathode and the anode...... 157

7.11 A zoomed-in region of four rightmost corrugations of Fig. 7.9 with RGB color scales reﬂecting particle velocities...... 159

7.12 Phase-space plot at 24.00 ns...... 160

7.13 A snapshot of steady-state self-ﬁelds (76.00 ns)...... 160

7.14 Output signal analysis in (a) time and (b) frequency domains. . . . . 161

7.15 Veriﬁcation of charge conservation at nodes along time (at time-steps of 7.5 × 104, 9 × 104, 12 × 104) by testing NR levels of (a) DCE and (b) DGL...... 162

7.16 3D velocity plots for an electron beam with the BFS magnetic ﬁeld of 0.5T...... 162

7.17 SCSWS boundary proﬁles for all cases...... 163

7.18 Field signal at the output port in (a) SCSWS and (b) staircased SC- SWS in the time domain...... 164

7.19 Normalized spectral amplitude at the output port in SCSWS and staircased SCSWS...... 165

7.20 Dispersion relations from “cold tests”...... 166

xxv C.1 Example (primal) unstructured mesh...... 187

C.2 Incidence matrices for (a) curl [Dcurl] and (b) gradient [Dgrad] operators for the mesh in Fig. C.1...... 188

C.3 Sparsity patterns for discrete Hodge matrices corresponding to the toy 0→0 1→1 −11→1 mesh depicted in Fig. C.1: (a) [?] , (b) [?] , (c) ?µ , and 2→2 (d) [?µ−1 ] ...... 192

xxvi Chapter 1: Introduction

1.1 Background and motivation

Plasma is a signiﬁcantly ionized gas, known as the fourth state of matter, composed of a large number of charged particles such as electrons and ions [5]. A distinct feature in characterizing most plasmas originates from collective interactions among all charged particles through the long-range behavior of Coulomb forces [6] rather than binary interactions or hard collisions (between every two particles) which dominate molecular dynamics of neutral gases [7]. At low densities, plasmas behave classically and its underlying dynamics includes particle kinematics and electromagnetism.

In general, the approach used for modeling a plasma system depends on its characteristic (temporal and spatial) scales [8]. The simplest one is magnetohydrodynam- ics (MHD), which is computationally eﬃcient, based on the assumption of plasmas behaving like ﬂuids [5], but only captures large-scale phenomena, and some of the physics such waves and instabilities are not described. The most accurate model is of course to microscopically account for the dynamics of all charged particles. This is impractical though since, as noted, usual plasmas consist of large numbers of charged particles.

1 Among various kinds of plasmas, collisionsless plasmas correspond to those where the strong binary Coulomb collisions are almost negligible for their description [6,

9, 10]. This occurs if the collisional frequency is much smaller than the frequency of interest (e.g. plasma frequency) and the mean free path is much longer than the characteristic length scale (e.g. Debye length). The main focus of this dissertation will be on the study of collisionless plasmas.

The behavior of collisionless plasmas is governed by Vlasov equation which describes nonlinear evolution of the phase space distribution function, viz. the number density over the 6-dimensional phase space (position and momentum) [9]. The

Maxwell-Vlasov system is the combination of the Vlasov equation with Maxwell’s equations in a multi-physical system involving (1) Maxwell’s equation, (2) Newton’s law of motions, and (3) Lorentz force acting on each particle [10]. In this system each particle will be tracked in 3-dimensional Euclidean space in response to Lorentz forces. In modeling collisionless plasmas, a coarse-graining of the phase distribution function (relatively macroscopic treatment) is employed to make the number of simulated particles not too large. In this case, superparticles are employed, each representing typically several millions of actual charged particles [11–13].

Electromagnetic particle-in-cell (EM-PIC) algorithm is a numerical approach to solve the Maxwell-Vlasov system by temporally tracking all superparticles over the

Euclidean space [11–13]. From their kinetic movement, the algorithm calculates their equivalent currents with the (direct or Galerkin) projection onto a grid (cell complex) which reconstructs the original problem domain. Subsequently, EM ﬁelds driven by the currents are to be solved by applying conventional computational electromagnetic

(CEM) techniques, speciﬁcally, discrete counterparts of EM ﬁelds are updated on the

2 grid. Then, the updated discrete ﬁelds are interpolated at the superparticles’ positions so as to evaluate Lorentz forces acting on superparticles. Finally superparticles are accelerated and pushed to the new positions by solving Lorentz force equation and Newton’s law of motion. The above describes a fundamental cycle that EM-PIC algorithm conducts at each time step and this is repeated through the desired simulation time window. The four steps in each cycle are called scatter, ﬁeld-solver, gather, and particle-pushers, respectively [11–13].

Most previous EM-PIC algorithms have employed a structured grid with the use of the finite-difference time-domain (FDTD) algorithm [11–14] or the pseudo-spectral time-domain (PSTD) algorithm [15] and here demonstrated successful performances on various practical applications. Apart from the historical origin, the main reasons to use the structured grids are that (1) its formulation and implementation is rather simple but robust enough, (2) it is relatively straightforward to introduce finite size superparticles (shape factors) onto the grid that can alleviate some numerical artifacts [12], (3) discrete charge conservation can be achieved for arbitrary orders of shape factors [16–18], and (4) superparticles can be easily tracked along the grid.

Nevertheless, structured grids present two fundamental drawbacks: (1) staircasing errors and (2) poor numerical dispersion properties [19, 20]. The former severely degrades the geometric fidelity while modeling realistic devices that may include curved and slanted boundaries. In addition, local mesh refinement (to capture locally find features) is hampered. Furthermore, it becomes difficult to accurately model secondary electron emission process from curved surfaces. As a result, structured grids necessitate the use of special treatments such as ad-hoc cut cell methods or conformal

ﬁnite-diﬀerence approaches [21] that may violate energy and charge conservation.

3 A natural alternative is to use unstructured grids based on the ﬁnite-element method (FEM). Such grids are devoid from staircasing errors and provide better performance w.r.t. numerical grid dispersions [22]. Moreover, unstructured grids enable a greater degree of space adaptivity using mesh reﬁnement techniques. Conventional

FEM to solve for electromagnetic fields [23] are mostly based on vector wave equation in the frequency domain and implemented using either a weighted residual method or variational principle. In this dissertation, we shall utilize FEM applied for transient plasma problems on the time-domain. In this case, the time-varying Maxwell’s curl equations are discretized based on compatible discretization principles, yielding the so called mixed E −B finite-element time-domain (FETD) scheme [24–26]. In order to do that, the discrete exterior calculus of differential forms shall be utilized, shedding light on clearer geometrical meaning for all Maxwell dynamic variables hidden behind vector calculus [27–36].

A long-standing challenge for EM-PIC simulations on unstructured grids has been violation of charge conservation which requires a posterior corrections based on costly

Poisson’s solvers. Based on compatible discretization principles, a novel EM-PIC method on unstructured grids has been proposed in [1] which makes use of Whitney forms for the scatter and gather algorithms, guaranteeing exact charge conservation from ﬁrst principles.

Nevertheless, there are still important challenges when using unstructured grids such as: (1) the resulting Maxwell ﬁeld solver is implicit on the time domain, requiring a sequential linear solver at each time step and (2) A full analysis of the grid numerical dispersion remains necessary to evaluate grid-heating-eﬀects and numerical Cherenkov instabilities in unstructured grids.

4 In this dissertation, we ﬁrst develop a local and explicit EM-PIC on unstructured grids using a sparse approximate inverse (SPAI) strategy. We study the perturba- tions in the motion of charged particles induced by the approximate inverse error. In addition, we extend the EM-PIC algorithm on unstructured grids to the relativistic regime using several types of relativistic particle-pushers (Boris, Vay, and Higuera-

Cary pushers [37–39]). Their performance is compared analytically and numerically.

We implement realistic particle boundary conditions for secondary electron emission

(SEE) based on the probabilistic Furman-Pivi model [40] and study multipactor effects associated to avalanched electrons resonant with external RF voltages frequently observed in high power microwave applications. In addition, we investigate numerical

Cherenkov radiation (NCR) or instability, which is a detrimental eﬀect frequently found in EM-PIC simulations involving relativistic plasma beams.

Another motivation of this dissertations is in the development of the EM-PIC solvers in circularly symmetric or body-of-revolution (BOR) geometries, which is important a plethora of applications involving analysis and design of high power microwave devices, directed energy devices and other applications. In the cylindrical coordinate system, azimuthal field variations can be described by eigenmodal ex- pansions, where the modal field solutions is reduced a 2-dimensional problem in the meridian ρz-plane. In this dissertation, we shall explore transformation optics (TO) principles [26, 41–45] to map the original 3-D BOR problem to a 2-D equivalent one in the meridian ρz-plane based on a Cartesian coordinate system where cylindrical metric is fully embedded into the constitutive properties of an effective inhomogeneous and anisotropic medium that fills the domain. On the meridian plane, the

ﬁelds are decomposed into TEφ and TMφ polarizations. In this way, a Cartesian

5 2-D FETD code can be easily retrofitted to this problem with no modifications necessary except to accommodate the presence of the artificial medium. We validate the algorithm against analytic solutions for resonant fields in cylindrical cavities and pseudo-analytical solutions for the radiated fields by cylindrically symmetric antennas in layered media.

We combine the EM-PIC algorithm with the BOR-FETD scheme into an axisymmetric EM-PIC algorithm optimized for the analysis of vacuum electronic devices

(VED) [46–50]. These typically employ corrugated cylindrical or coaxial waveguides, called slow-wave structure (SWS), that interact with an energetic electron beam to produce high power microwaves. We use the algorithm to investigate the physical performance of VEDs designed to harness particle bunching eﬀects arising from the coherent (resonance) Cherenkov electron beam interactions within micro-machined

SWSs.

1.2 Contribution of this dissertation

Main contributions of this dissertation are:

• Integration of a local and explicit FETD scheme with the sparse approximate in-

verse (SPAI) strategy with the previous charge conservative EM-PIC algorithm

on unstructured grids and investigation of the approximate inversion error in-

ﬂuencing on the motion of charged particles.

• Extension of the algorithm to the relativistic regime with Boris, Vay, and

Higuera-Cary particle-pushers and comparison of their relative performance.

6 • Numerical analysis of parallel-plate multipactor eﬀects based on probabilistic

Furman-Pivi model for the estimation of secondary electron emission process.

• Evaluation of numerical Cherenkov radiations (or instabilities) present in rela-

tivistic EM-PIC simulations with a generalized grid dispersion analysis account-

ing for diﬀerent mesh element shapes.

• Development of a new FETD Maxwell solver for the general analysis of body-

of-revolution (BOR) geometries based on transformation optics concepts.

• Development of the EM-PIC algorithm optimized for the analysis of axisymmet-

ric vacuum electronic devices such as cylindrical vacuum diodes and backward-

wave oscillators.

More details for each contribution are presented in the next subsection.

1.3 Organization of this dissertation

This dissertation is organized as follows.

In Chapter 2, we present a charge-conserving EM-PIC algorithm on unstructured grids based on a FETD methodology with explicit ﬁeld update, i.e., requiring no linear solver [51]. The proposed explicit EM-PIC algorithm attains charge conservation from

first principles by representing fields, currents, and charges by differential forms of various degrees, following the methodology put forth in [1]. The need for a linear solver is obviated by constructing a SPAI for the FE system matrix, which also preserves the locality (sparsity) of the algorithm. We analyze in detail the residual error caused by SPAI on the motions of charged particles and beam trajectories and

7 show that this error is several order of magnitude smaller than the inherent error caused by the spatial and temporal discretizations.

Accurate modeling of relativistic particle motions is essential for physical predictions in many problems involving vacuum electronic devices, particle accelerators, and relativistic plasmas. In Chapter 3, we extend the local, explicit, and charge-conserving

FETD-PIC algorithm to the relativistic regime by implementing and comparing three relativistic particle-pushers: (relativistic) Boris, Vay, and Higuera-Cary [52]. We illustrate the application of the proposed relativistic FETD-PIC algorithm for the analysis of particle cyclotron motion at relativistic speeds, harmonic particle oscillation in the

Lorentz-boosted frame, and relativistic Bernstein modes in magnetized charge-neutral

(pair) plasmas.

In Chapter 4, we combine a novel FE-based EM-PIC algorithm for the solution of Maxwell-Vlasov equations on unstructured grids together with the Furman-Pivi probabilistic model governing the secondary electron emission (SEE) process [53].

The Furman-Pivi probabilistic model [40] is based on a broad phenomenological fit to experiment data to obtain accurate simulations of SEE process (rather than a conventional monoenergetic one). The algorithm is suited for the analysis of resonant electron discharging phenomena (multipactor effects) in high-power RF devices since the use of unstructured grids enables local mesh refinement and simulation of complex geometries with minimal geometrical defeaturing. We apply the algorithm to model multipactor effects on waveguides with flat or corrugated walls and contrast the evolution of the electron population in various cases and investigate the respective saturation process arising from self-field counterbalance effects.

8 In Chapter 5, we investigate numerical Cherenkov radiation (NCR) or instability which is a detrimental effect frequently found in EM-PIC simulations involving relativistic plasma beams [54]. NCR is caused by spurious coupling between electromagnetic-field modes and multiple beam resonances. This coupling may result from the slowdown of poorly-resolved waves due to numerical (grid) dispersion and from aliasing mechanisms. NCR has been studied in the past for finite-difference- based EM-PIC algorithms on regular (structured) meshes with rectangular elements.

In this chapter, we extend the analysis of NCR to finite-element-based EM-PIC algorithms implemented on unstructured meshes. The influence of different mesh element shapes and mesh layouts on NCR is studied. Analytic predictions are compared against results from FE-based EM-PIC simulations of relativistic plasma beams on various mesh types.

In Chapter 6, we present a FETD Maxwell solver for the analysis of BOR geometries based on discrete exterior calculus (DEC) of diﬀerential forms and TO concepts [55]. We explore TO principles to map the original 3-D BOR problem to a

2-D one in the meridian ρz-plane based on a Cartesian coordinate system where the cylindrical metric is fully embedded into the constitutive properties of an effective inhomogeneous and anisotropic medium that fills the domain. The proposed solver uses a (TEφ, TMφ) field decomposition and an appropriate set of DEC-based basis functions on an irregular grid discretizing the meridian plane. A symplectic time discretization based on a leap-frog scheme is applied to obtain the full-discrete marching-on-time algorithm. We validate the algorithm by comparing the numerical results against analytical solutions for resonant fields in cylindrical cavities and

9 against pseudo-analytical solutions for ﬁelds radiated by cylindrically symmetric antennas in layered media.

In Chapter 7, we present a charge-conservative EM-PIC algorithm optimized for the analysis of cylindrically-shaped VEDs, which typically employ corrugated cylindrical or coaxial waveguides, called slow-wave structure (SWS), with an energetic electron plasma beam to produce high power microwaves [56]. Present Maxwell field solver is a specific version of the BOR-FETD scheme, viz. only accounting for only the zeroth azimuthal eigenmode, combined with the Cartesian EM-PIC algorithm. The previous advances including charge conservation, local and explicit field update, relativistic extension of particle-pusher, and the BOR-FETD scheme, have made possible this work, which is motivated by the demand to accurately capture realistic physics of beam-SWS interactions in complex geometry devices. The algorithm is validated considering cylindrical cavity and space-charge-limited cylindrical diode problems.

We use the algorithm to investigate the physical performance of VEDs designed to harness particle bunching eﬀects arising from the coherent (resonance) Cherenkov electron beam interactions within micro-machined slow wave structures.

10 Chapter 2: Local, Explicit, and Charge-conserving EM-PIC on Unstructured Mesh

In the past few decades, electromagnetic particle-in-cell (EM-PIC) algorithms coupled to time-dependent Maxwell’s equations [11, 13, 57] have been applied to a variety of problems involving charged particles and beam-wave interaction, including plasma-based accelerators [58–61], inertial conﬁnement fusion [62], and vacuum electronic devices [46,63]. Historically, EM-PIC codes have been using regular grids and

finite-difference approaches [14], such as the celebrated Yee’s finite-difference time- domain (FDTD) algorithm [64]. However, complex geometries involving curved (such as conformal cathodes and curved waveguide sections) or very fine geometrical features cannot be accurately modeled by regular grids because of ensuing ‘staircase’

(step-cell) effects [65]. Although many studies have been done to ameliorate staircase errors in finite-differences, including the use of conformal finite-differences [21, 66], heterogeneous grids [67], and subgridding [68, 69], the most general solution to this problem is to employ irregular, unstructured grids (meshes). The finite-element (FE) method is a better option in this case because it is naturally suited for such type of grids. In addition, FE also enables a greater degree of space-adaptivity (using mesh refinement techniques) in a systematic fashion and can also be applied for transient problems using FE time-domain (FETD) algorithms [27,70].

11 However, existing FE-based EM-PIC codes based on unstructured grids have three important drawbacks. First, FE-based EM-PIC algorithms tend to numerically violate charge conservation due to the fact that the continuity equation leaves residuals at the discrete level on unstructured grids. Past efforts to enforce charge conservation have included adding a posterior correction steps by Poisson’s solvers [14] or pseudo-currents [71]. However, the former approach requires a time-consuming linear solver at each time step and the latter introduces a diffusion parameter that may alter the physics. A recent charge-conserving PIC algorithm based on second-order vector wave equation for the electric field that does not require introduction of correction terms is described in [72,73]. However, the solution space of the second-order vector wave equation in the time-domain includes spurious solutions with secular growth of the form t∇φ, which are not physical admissible solutions to Maxwell’s equations and can pollute the numerical results [1,74,75]. More recently, a novel gather-scatter algorithm with exact charge conservation on unstructured grids was described in [1], based on concepts borrowed from differential geometry [30,35] and discrete differential forms [28, 76]. Charge-conserving PIC algorithms were also developed under similar tenets in [77,78]. A second challenge for unstructured-grid EM-PIC algorithms is that the field solver is implicit, i.e., it requires the repeated solution of a linear system of equations sequentially at each time step [27,79]. Finally, a third challenge (shared by

FDTD-based algorithms as well) is that their performance is hindered by the global

Courant stability bounds on time steps used to advance ﬁelds and particles.

In order to overcome the second challenge noted above, a sparse inverse approximation (SPAI) strategy for unstructured meshes [26, 33] is incorporated here into an explicit FETD-based EM-PIC algorithm with exact charge-conserving properties

12 developed in [1]. For a given mesh, the resulting SPAI explicit solver obtains an approximation for the inverse of the FE system matrix based on (powers of) the sparsity pattern of the original FE system matrix. This is done once-and-for-all for any given mesh i.e., independently from any ﬁeld excitation and particle distribution, and decoupled from the ﬁeld update. The SPAI explicit solver is easily parallelizable and produces exponential convergence of the approximate inverse matrix to the exact inverse matrix as the density (sparsity) of the former is increased (reduced) [33].

Importantly, since sparsity is retained, the algorithm remains local [26]. The explicit and sparse nature of the resulting EM-PIC algorithm enable integration with asynchronous time stepping techniques [80–82] designed to overcome the third challenge indicated above. We investigate in detail here the eﬀect of the approximate inverse on the particle dynamics by comparing particle trajectories computed with the new proposed algorithm against analytical solutions (when available) and a conventional implicit EM-PIC algorithm employing a direct LU-solver. We show that the error caused by the SPAI approximation is several order of magnitude smaller than inherent space and time discretization errors.

2.1 Explicit FETD-PIC Algorithm

A typical EM-PIC algorithm consists of four basic steps [1]: (1) field solver (consisting of electric and/or magnetic field updates from Maxwell’s equations), (2) gather step (fields interpolation at each particle position), (3) scatter (assigning currents to grid edges and charges to grid nodes from the particle positions and velocities), and

(4) particle acceleration and push (governed by Lorentz force and Newton’s law of

13 B update ( Faraday’s law )

Gather

Particle Acceleration & Push (Lorentz force / Newton’s law of motion)

Scatter

E update ( Ampere’s law )  Implicit

Figure 2.1: Basic steps in a EM-PIC algorithm. On unstructured meshes, conventional ﬁeld solvers are implicit, requiring the solution of a (large) linear system at each time step.

motion). These four steps are sequentially performed at each time step, as illustrated in Fig. 2.1.

2.1.1 Mixed E − B FETD scheme

In the language of diﬀerential forms for the electromagnetic ﬁeld [83], the electric

ﬁeld E and the (Hodge dual of the) current density ?J are represented as 1-forms, and the magnetic ﬂux density B is represented as a 2-form [24]. On a mesh, 1-forms and

2-forms are associated to mesh edges and facets, respectively [30, 35]. Accordingly,

14 in order to discretize Maxwell’s equations, the FETD algorithm expands E and ?J in terms of Whitney 1-forms associated with edges of the mesh, and B in terms of

Whitney 2 forms associated with faces of the mesh [1,24].

Next, using the generalized Stoke’s theorem to obtain semi-discrete equations fol- lowed by a leap-frog discretization in time (second-order symplectic time integration), the following full-discrete FETD scheme is obtained [1,33]:

n+ 1 n− 1 n [B] 2 = [B] 2 − ∆t [Dcurl] · [E] (2.1)

1 1 n+1 n T n+ 2 n+ 2 [?] · [E] = [?] · [E] + ∆t [Dcurl] · [?µ−1 ] · [B] − [J] . (2.2)

where ∆t is the time step increment, the superscript n denotes the time step index,

and [B], [E], and [J] are column vectors representing B on each face, and E and ?J

on each edge, respectively. In addition, [Dcurl] is the incidence matrix representing

the discrete exterior derivative (or, equivalently, the discrete curl operator distilled

from the metric, that is, with elements in the set {−1, 0, 1}) on the mesh [30,33], and

[?] and [?µ−1 ] are discrete Hodge (mass) matrices whose elements are given by the

volume integrals [33,76]

Z (1) (1) [?]J,j = WJ · Wj dΩ (2.3) Ω Z −1 (2) (2) [?µ−1 ]K,k = µ WK · Wk dΩ (2.4) Ω

(1) (2) where Wj , j = 1,...,N1 and Wk , k = 1,...,N2 are the vector proxies of Whitney

1- and 2-forms [30] that span the set of N1 edges and N2 faces of the mesh, respectively. T h ˜ i It can be shown that [Dcurl] = Dcurl , the incidence matrix on the dual mesh [1,30,

35,84]. Eqs. (1) and (2) constitute an implicit ﬁeld solver because [?] is nondiagonal:

in order to update the electric ﬁeld from eq. (2) it is necessary to solve a large linear

15 ( ) + 𝑛𝑛 1 𝑛𝑛 1 1 ( ) 1 𝑝𝑝 𝑛𝑛+ 𝑛𝑛− edge 𝔼𝔼 𝑊𝑊1 ⃗𝑟𝑟 ( ) 2 2 2 𝔹𝔹 𝑖𝑖 𝔹𝔹 𝑖𝑖 2 𝑛𝑛 𝑛𝑛 1 𝑛𝑛 𝑊𝑊𝑖𝑖 ⃗𝑟𝑟𝑝𝑝 1 3 3 𝑝𝑝 𝑛𝑛 𝔼𝔼 𝑊𝑊 ⃗𝑟𝑟 edge 𝑝𝑝 edge 𝑛𝑛 face ⃗𝑟𝑟 ⃗𝑟𝑟𝑝𝑝 ( ) 2 3 𝑖𝑖 𝑛𝑛 1 𝑛𝑛 𝔼𝔼 2 𝑊𝑊2 ⃗𝑟𝑟𝑝𝑝 (a)

node

1 𝑛𝑛+ 2 1 edge 𝕁𝕁 𝑛𝑛+1 𝑛𝑛 𝑛𝑛+1 ℚ 1 − ℚ 1 𝑛𝑛+1 𝑝𝑝 𝑝𝑝 edge 1 ⃗𝑟𝑟 edge ⃗𝑟𝑟

2 𝑛𝑛 3 node 𝑛𝑛 ⃗𝑟𝑟𝑝𝑝 ⃗𝑟𝑟𝑝𝑝 1 1 𝑛𝑛+ 2 node 𝕁𝕁 2 (b) 3

Figure 2.2: Charge-conserving gather and scatter steps [1]. (a) Interpolation of E and B at the position of the particle by edge-based (left) and face-based degrees of freedom contributions (right) (weighted by the Whitney functions) in the gather step. (b) Exact charge-conserving scatter scheme. The sum of the two colored areas in the left, representing the magnitude of the edge currents, is equal to the blue area in the left, representing the charge variation at node 1 during one time step.

system of equations at every time step. The explicit scheme proposed here is detailed in Chapter 2.2 below.

2.1.2 Gather-scatter and particle pusher steps

In the gather step, Whitney forms are used to determine the electric and magnetic

ﬁeld values at the position of each particle, as depicted schematically in Fig. 2.2a.

n n+ 1 n− 1 Speciﬁcally, from the values of [E] on edges and [B] 2 and [B] 2 on faces, vector

16 proxies of Whitney forms are used to interpolate En(x) and Bn(x) at any ambient

point x, and in particular at the charged particles’ locations, by

N1 n X n (1) E (x, n∆t) ≡ E (x) = Ej Wj (x) (2.5) j=1

N2 X 1 n+ 1 n− 1 (2) B (x, n∆t) ≡ Bn(x) = 2 + 2 W (x) (2.6) 2 Bk Bk k k=1

1 n n n+ 2 where Ej denotes the j-th element of the column vector [E] and likewise for Bk 1 n− 2 and Bk . This is illustrated schematically in Fig. 2.2a. In the scatter step, we compute the particle current densities mapped to the edges of the mesh, i.e. to the

n+ 1 mesh-based quantity [J] 2 , for incorporation back into the ﬁeld solver. We adopt here the charge-conserving scatter for unstructured grids recently proposed in [1]. By

n n+1 referring to Fig. 4.3, given the initial xp and ﬁnal xp locations of a particle p with charge qp during a time step ∆t, the associated current ﬂowing along edge 1 is written as

n+1 Z xp n qp (1) qp n n+1 n+1 n J1 = W1 (x) · dl = λ1(xp )λ2(xp ) − λ1(xp )λ2(xp ) (2.7) ∆t n ∆t xp

where λ1(x) and λ2(x) are the barycentric coordinates of point x w.r.t vertices 1 and 2

respectively (the boundary points of edge 1 in consideration). Analogous assignments

follow for the other edges of the mesh.

2.1.3 Discrete continuity equation

As demonstrated in [1], the above scatter algorithm yields exact charge conserva-

tion at the discrete level because the variation of the charge at any node of the mesh

exactly matches the total current ﬂowing in or out of that particular node. In other

17 words, the discrete continuity equation (DCE) below holds

n+1 n h i n+ 1 [Q] − [Q] D˜ · [ ] 2 + = 0 (2.8) div J ∆t

h ˜ i where Ddiv is the incidence matrix associated to the discrete divergence operator

T n in the dual mesh, which is also equal to [Dgrad] [1, 30, 35, 84], and [Q] denotes the column vector with the charge associated to each node of the mesh1. Note that the

nodal charge at any node i us obtained from the sum of the nearby particle charges

weighted by their corresponding barycentric coordinates w.r.t. at that particular

node, that is

n X n Qi = qpλi(xp ). (2.9) p

Barycentric coordinates can be identiﬁed as Whitney 0-forms associated to a par-

(0) n n ticular node i, i.e. Wi (xp ) = λi(xp ) [30, 35]. We provide a geometrical illustration of (2.8) in Fig. 4.3. From eq. (2.9), the charge variation at node 1 due

n+1 n to a charged particle movement during ∆t is proportional to λ1(xp ) − λ1(xp ). This quantity is represented by the blue-colored area in Fig. 4.3. At the same

time, from eq. (2.7), the current ﬂowing along edge 1 is associated with the factor

n n+1 n+1 n λ1(xp )λ2(xp ) − λ1(xp )λ2(xp ), which is equal to the red-colored area in Fig. 4.3. A similar factor is present for edge 2 which is indicated by the green-colored area.

From the area equivalences, it is clear that the sum of the current ﬂow out of node 1

along edges 1 and 2 is equal to the charge variation on node 1.

The particle push step computes the Lorentz force acting on each charged parti-

cle given the (interpolated) electric and magnetic ﬁelds at the particle location and

1 h i T h i T The equivalence between D˜div and [Dgrad] , and similarly between D˜curl and [Dcurl] is up to a sign, depending on the relative orientation chosen for the primal and dual meshes [30].

18 its velocity, and applies Newton’s force law to accelerate the particle. This step is

implemented here by extending the particle push described in [1] to the relativistic

regime based on the methodology put forth in [38].

2.2 Sparse Approximate Inverse (SPAI) strategy

As noted above, a linear solve (implicit time-update) is required in (2.2) due to

n+1 the presence of [?] multiplying the unknown [E] on the l.h.s. Naively, this linear

−1 solve could be avoided by pre-multiplying both sides of (2.2) by [?] , leading to

n+1 n −1 h i n+ 1 n+ 1 ˜ 2 2 [E] = [E] + ∆t [?] · Dcurl · [?µ−1 ] · [B] − [J] . (2.10)

−1 This multiplication is, of course, wholly impractical for large problems because [?] is dense and such a direct inversion is computationally very costly and scales poorly

−1 with size. Even for relatively small problems, the fact that [?] is dense makes the algorithm non-local and unsuited for asynchronous time-update algorithms [80].

Instead, to obtain an explicit and local ﬁeld update algorithm, we explore the fact

−1 that, in the continuum, not only ? but also ? is a strictly local operator [26,

−1 76, 85]. This indicates that, although dense, [?] should be well approximated

−1 by a sparse approximate inverse (SPAI), which we denote [?]a . Each column of

−1 [?]a can be obtained independently (and in parallel fashion) once a suitable sparsity

−1 pattern for [?]a is chosen. Since the sparsity pattern of [?] encodes nearest-neighbor

−1 k edge adjacency, good candidates for the sparsity pattern of [?]a are [?] for k = 1, 2,..., which would encode k-nearest neighbor adjacency among edges (with larger

k providing better accuracy but denser matrices). A parallel algorithm for computing

−1 [?]a along these lines is detailed in [33], where it is also shown that the Frobenius

19 −1 −1 norm of the diﬀerence matrix k [?]a − [?] kF has exponential convergence to zero for increasing k.

−1 Once [?]a is precomputed, the explicit and local SPAI-based ﬁeld update simply writes

n+1 n −1 h i n+ 1 n+ 1 ˜ 2 2 [E] = [E] + ∆t [?]a · Dcurl · [?µ−1 ] · [B] − [J] . (2.11)

2.2.1 Discrete Gauss’ law

h ˜ i h ˜ i By premultiplying both sides of (2.11) by Ddiv ·[?]a, where Ddiv is the incidence matrix representing the discrete divergence operator on the dual grid, and using the h i ˜ ∗ identity Ddiv · [Dcurl] = 0 [30,35,84], we obtain

h i n+1 h i n h i n+ 1 ˜ ˜ ˜ 2 Ddiv · [?]a · [E] = Ddiv · [?]a · [E] + ∆t Ddiv · [J] . (2.12)

This last equation can be rearranged as

n+1 n ! h i [E] − [E] h i n+ 1 D˜ · [? ] · = − D˜ · [ ] 2 , (2.13) div a ∆t div J which, using (2.8), can be rewritten as ! h i [ ]n+1 − [ ]n [ ]n+1 − [ ]n D˜ · [? ] · E E = Q Q . (2.14) div a ∆t ∆t

Eq. (2.14) implies that residuals of the discrete Gauss’ law (DGL) at any two successive time steps remain the same, in other words

h ˜ i n+1 n+1 h ˜ i n n Ddiv · [?]a · [E] − [Q] = Ddiv · [?]a · [E] − [Q] , (2.15) and by induction,

h ˜ i n n h ˜ i 0 0 Ddiv · [?]a · [E] − [Q] = Ddiv · [?]a · [E] − [Q] (2.16) | {z } | {z } resn res0

20 h ˜ i 0 0 for all n, so that if initial conditions have Ddiv · [?]a · [E] = [Q] , then the DGL is veriﬁed for all time steps.

In the next Section, we analyze the error incurred by the above SPAI approx-

imation to obtain an explicit ﬁeld solver for EM-PIC simulations on unstructured

grids.

2.3 Numerical Results

In order to investigate the error caused by the SPAI-based explicit solver in EM-

PIC simulations, we consider in this Section examples involving single charged particle

trajectories, a plasma ball expansion, and an accelerated electron beam.

2.3.1 Single-particle trajectories

Typical PIC simulations comprise an ensemble of superparticles eﬀecting a coarse-

graining of the phase-space. As such, instantaneous errors in individual particle trajec-

tories may not always be relevant when computing grid-averaged physical quantities.

Nevertheless, it is of interest to examine the secular trends on the particle trajectory

discrepancies.

We investigate the motion of a single charged particle initially positioned at the

ext ext origin in the presence of an external magnetic ﬁeld Bz and electric ﬁeld Ey . In this case, the exact solution can be written as [86]

ext ext vy,0 vx,0 qpEy qpEy vy,0 x (t) = cos ωct + + 2 sin ωct − 2 t + (2.17) ωc ωc mpωc mpωc ωc ext ext vy,0 vx,0 qpEy qpEy vy,0 y (t) = sin ωct − + 2 cos ωct + 2 + (2.18) ωc ωc mpωc mpωc ωc

where vx,0 and vy,0 are the initial velocity components.

21 Table 2.1: Number of elements in Meshes 1, 2, and 3

Mesh 1 Mesh 2 Mesh 3 Edge # 951 2168 6036 Face # 610 1408 3960 Node # 342 761 2077 ∆lav [m] 0.1160 0.0590 0.0300

We examine two types of single-particle trajectories. The ﬁrst corresponds a pure cyclotron motion (Bz 6= 0 and Ey = 0) and the second includes a drift motion as well

−15 (Bz 6= 0 and Ey 6= 0). We assume a superparticle with qp = −1.6×10 [C] and mass

−27 8 mp = 9.1×10 [kg]. In both cases, the initial velocity is set equal to 2×10 [m/s]. We consider three unstructured meshes labeled, from coarsest to ﬁnest, as 1, 2, and 3, all of which discretize the domain Ω = {(x, y) ∈ [0, 1]2}. Table 2.1 provides information about the number of elements and other properties of the meshes considered. The parameter ∆lav indicates the average edge length, which roughly halves for each mesh index increment.

The boundaries of the solution domain are truncated using a perfectly matched layer (PML) [74, 75]. The time increment is chosen as ∆t as 10, 5, and 2.5 [ps] for meshes 1, 2, and 3, respectively, and the simulation is terminated at t = 150 [ns].

An implicit solver based on LU decomposition is used as reference. Charged particle trajectories calculated by using such LU solver are referred to standard trajectories. On the other hand, particle trajectories obtained by the SPAI-based explicit

field solver are designated as test trajectories. The effect of the inverse approximation error can be quantified by examining the discrepancy between standard and test trajectories. This discrepancy can be further compared to the discrepancy in particles’

22 𝒏𝒏 𝐑𝐑𝐑𝐑𝐑𝐑𝐭𝐭𝐭𝐭𝐭𝐭𝐭𝐭𝟏𝟏

𝒏𝒏 𝐭𝐭𝐭𝐭𝐭𝐭𝐭𝐭𝟐𝟐 𝒏𝒏 𝐑𝐑𝐑𝐑𝐑𝐑 𝐭𝐭𝐭𝐭𝐭𝐭𝐭𝐭𝟒𝟒 𝐑𝐑𝐑𝐑𝐑𝐑 𝒏𝒏 𝐑𝐑𝐑𝐑𝐑𝐑𝐭𝐭𝐭𝐭𝐭𝐭𝐭𝐭𝟑𝟑

(a)

Figure 2.3: Relative position diﬀerence (RPD) of the various test particles w.r.t. the standard particle placed at the origin, in a polar diagram where the radial distance is represented in logarithmic scale.

trajectories between that result from the LU-based solver and the analytic solution, which measures the inherent numerical (space and time) discretization error.

To quantify the error, we define the relative position difference (RPD), which is the ratio of the magnitude of the difference between the standard and test position vectors at certain time step n to the total travel length of the standard particle up to time step n, i.e.,

n n n xp,test − xp,std |d | j testj RPDn = = (2.19) testj Pn i i−1 i=1 xp,std − xp,std Lstd where RPDn is the RPD for the j-th test particle at time instant n, and xi and testj p,std xi are the standard and test particle position, respectively, at time step i. p,testj

23 Table 2.2: Convention used for particle trajectory visualization.

solver test particle number symbol used analytical sol. 1 SPAI k = 2 2 + SPAI k = 4 3 × SPAI k = 6 4

For visualization purposes, we plot the RPD in a polar graph as shown in Fig. 2.3,

with the radial coordinate represented in a logarithmic scale. The standard trajectory

points computed by the implicit LU-based solver are indicated by 4 and placed at the origin of the RPD for all times. The symbols , +, ×, and represent, in turn, the relative position of test particles’ 1, 2, 3, and 4 w.r.t. to standard trajectory points, as given by the vector dn /L . As summarized in Table 2.2, these four testj std sets of points correspond, respectively, to the exact trajectory points obtained via an analytic solution and to the trajectory points obtained using the SPAI-based explicit

ﬁeld solver with k = 2, 4, and 6.

24 (a) (b) (c)

-4 -4 -4 10 10 10

space and time discretization error -8 -8 -8 10 10 10

-12 -12 -12 10 10 10 RPD [a.u.] RPD [a.u.] RPD RPD [a.u.] RPD

-16 Analytic -16 Analytic -16 Analytic 10 10 10 inverse SPAI w/ k=2 SPAI w/ k=2 SPAI w/ k=2

approx. SPAI w/ k=4 SPAI w/ k=4 SPAI w/ k=4

-20 error SPAI w/ k=6 -20 SPAI w/ k=6 -20 SPAI w/ k=6 10 10 10 2 4 6 8 10 12 14 2 4 6 8 10 12 14 2 4 6 8 10 12 14 -8 time [sec] -8 time [sec] -8 time [sec] x 10 x 10 x 10 (d) (e) (f)

Analytic Analytic Analytic SPAI w/ k=2 SPAI w/ k=2 SPAI w/ k=2 SPAI w/ k=4 SPAI w/ k=4 SPAI w/ k=4 SPAI w/ k=6 SPAI w/ k=6 SPAI w/ k=6

(g) (h) (i)

Figure 2.4: Results for a circular particle trajectory on 3 diﬀerent meshes. (a) (b) (c) Particle trajectory histories. (d) (e) (f) RPDs versus time for the four test particles. (g) (h) (i) Normalized RPD bands for the four test particles.

25 Oscillatory motion

ext −3 2 ext In this case Bz = 5.085 × 10 [Wb/m ] and Ey = 0 [V/m] so that a pure

2 cyclotron motion with angular frequency ωc = 6.05 × 10 [rad/s] results. Fig. 2.4 illustrates the result of the SCP test for the circular trajectory. Figs. 2.4a, 2.4b, and 2.4c illustrate the trajectory of the SCP for Meshes 1, 2, and 3, respectively.

Figs. 2.4d, 2.4e, and 2.4f show the RPDs for four test particles on each mesh. It is seen that RPDs for the analytic test particle is very large (several orders of magnitude) compared to the RPDs of the EM-PIC simulation with SPAI-based explicit ﬁeld solver for k = 2, 4, and 6. We note again that the RPD for the analytic test particle arises from space and time discretization errors, whereas the other RPDs are due solely to the inverse approximation error. Therefore, these results indicate that inverse approximation error is negligible compared to the other inherent numerical errors.

We also note, as expected, that the RPD due to the discretization error decreases as the mesh is progressively reﬁned (curve with in Figs. 2.4d, 2.4e, and 2.4f). On the other hand, the RPD due to the inverse approximation error remains fairly constant across the diﬀerent meshes

(curves with +, ×, and in Figs. 2.4d, 2.4e, and 2.4f). Examining these ﬁgures, it is also observed that the error decreases as the parameter k increases.

Fig. 2.4g, 2.4h, and 2.4i show the RPD bands normalized by the analytic test particle’s RPD (i.e. setting the RPD of the analytical result to unity radius in the plot).

In all cases, the normalized RPD bands rotate around the origin (LU-decomposition implicit solution) around nearly circular orbits. Such normalized RPD bands for test particles 2, 3, and 4 become larger as mesh is reﬁned since the space and time discretization errors decrease, as noted above.

-2 Analytic

10 SPAI w/ k=2 SPAI w/ k=4 Analytic SPAI w/ k=6 -6 10 SPAI w/ k=2 SPAI w/ k=4 SPAI w/ k=6 -10 10 RPD [a.u.] RPD

-14 10

-18 10 0 1 2 3 -7 time [sec] x 10 (a) (b) (c)

Figure 2.5: Results for a trajectory with drift. (a) (b) (c) Particle trajectory history. (d) (e) (f) RPDs versus time for the four test particles. (g) (h) (i) Normalized RPD bands for the four test particles.

E × B drift motion

ext −3 2 ext 3 In this case, we set Bz = 5.085×10 [Wb/m ] and Ey = −5×10 [V/m]. This add a drift motion to the trajectory of the particle, as seen in Fig. 2.5a. We consider

mesh 3 result only, for brevity. The RPD data is shown in Fig. 2.5b and Fig. 2.5c.

Similar to the pure circular trajectory case, the RPDs for diﬀerent k are very small

compared to analytic RPD. It is again seen that the bands converge to the center of

the circle, which stands for the position of the standard particle, as k increases.

2.3.2 Plasma ball expansion

In the next example, we consider the simulation of an expanding plasma ball.

We consider 5 × 104 superparticles, each representing 200 electrons, initially placed uniformly within a circle of 0.5 [m] radius centered at the origin. At t = 0 positive and negative charged particles overlap, with net zero charge everywhere. Negative particles are initialized with Maxwellian distribution with thermal velocity |vth| =

0.1×c [m/s]. Positive charged are assumed with zero velocity at all times. The initial

] 0 2

/m -0.2

A/m] = 9,000 µ µA -0.4 𝑡𝑡 ∆𝑡𝑡

-0.6

-0.8 SPAI-PIC w/ k=2 -1 SPAI-PIC w/ k=4

radial currentradial density [ SPAI-PIC w/ k=6 -1.2 LUD-PIC radial current density [

0 1 2 3 4 5 radiusradius[m] [m]

Figure 2.6: Radial current versus radius coordinate for the expanding plasma at time step n = 9×104 using the LU-based implicit ﬁelds solver and the SPAI-based explicit ﬁeld solver with k = 2, 4, and 6.

4 −3 density of particles is n ≈ 6.37 × 10 [m ] and the Debye length is λD ≈ 0.663 [m],

5 resulting on a plasma parameter Λ = 4πnλd ≈ 2.34 × 10 . The unstructured mesh used in this simulation has 1880 faces, 2884 edges, and 1005 nodes. A PML is used to truncate the solution domain. A time step increment ∆t = 5 [ps] is used, and the simulation is terminated at 10 [ns].

Fig. 2.6 shows the radial current density from the plasma expansion at t = 9 ×

103∆t as a function of the radial coordinate computed by implicit LU-based and explicit SPAI-based ﬁeld solvers with k = 2, 4, and 6. The picture in the inset of

Fig. 2.6 shows a snapshot of the particle distribution at t = 9 × 103∆t. There is no discernible diﬀerence in the current density proﬁle among the results shown in

Fig. 2.6.

28 nodal index, i nodal index, i (a) (b) (c)

Figure 2.7: (a) Normalized residuals of the discrete continuity equation for the plasma ball expansion example using diﬀerent ﬁeld solvers, at t = 2 × 104∆t. (b) Similar results for the discrete Gauss’ law. (c) Averaged normalized residuals for the discrete Gauss’ law versus time step index.

In order to check charge conservation, we plot in Fig. 2.7a the normalized residual

(NR) for DCE (2.8) and DGL (2.14). These residuals are evaluated for each time

1 step n + 2 or n and node i, and deﬁned as

1 n+1 n n+ 2 Qi − Qi NRDCE = 1 + (2.20) i h i n+ 1 PN1 ˜ 2 ∆t j=1 Ddiv [J]j i,j n n Qi NRDGLi = 1 − h i (2.21) PN0 ˜ PN1 n j=1 Ddiv k=1 [?]aj,k [E]k i,j

1 n+ 2 where N0 denotes the total number of nodes in the mesh. Fig. 2.7a shows |NRDCEi | 1 n+ 2 at n=20, 000 versus the nodal index for diﬀerent solvers. It is seen |NRDCEi | is fairly low, about 10−13, in all cases. The small noise above the double-precision ﬂoor 10−15

can be attributed from arithmetic round-oﬀ errors in the scatter process. Fig. 2.7b

n shows a similar plot now for |NRDGLi |, which is very close to the double-precision ﬂoor. In order to verify that residual levels of the DGL are maintained by (2.16)

n during the time-update, we also plot |NRDGLi | averaged across all nodes of the mesh,

29 n PN0 n i.e. |NRDGL|ave = i=1 NRDGLi /N0 as a function of the time step n in Fig. 2.7c. As

n seen, |NRDGL|ave has nearly constant values close to the double-precision ﬂoor, with only a very small increase due to cumulative round-oﬀ error.

2.3.3 Electron beam in a vacuum diode

In order to further verify charge conservation and stability for long-time simulations, we simulate next an electron beam accelerated by a vacuum diode. The domain

Ω = {(x, y) ∈ [0, 1]2} has lateral walls representing anode and cathode surfaces with potential diﬀerence set as 1.5×105 [V]. The top and bottom boundaries of the domain are truncated by a PML. The unstructured mesh has 2301 faces, 3524 edges, and 1224 nodes. The time step interval is set to ∆t = 270 [ps], and the simulation is run up to

16.2 [µs]. Each superparticle used in the simulation represents 50×106 electrons. For the thermionic emission of electrons from the cathode at the left boundary, a slow initial mean velocity of 104 [m/s] is assumed for the electrons. Fig. 2.8 presents snapshots of the particle distribution and the self-field (electric) profile. Fig. 2.8a and Fig. 2.8d show the field and particle distribution for the charge-conserving EM-PIC algorithm with LU-based implicit field solver. Fig. 2.8b and Fig. 2.8e show the field and particle distribution for the charge-conserving EM-PIC algorithm with SPAI-based (k = 2) explicit field solver. Finally, Fig. 2.8c and Fig. 2.8f show the field and particle distribution for an EM-PIC with LU-based implicit field solver and conventional gather step (non-charge-conserving on an unstructured grid) where edge currents are obtained from the straightforward projection of the instantaneous product qv, summed

(1) over all particles, onto the edge element Wj , i.e.

30 (a) (b) (c)

(d) (e) (f)

Figure 2.8: Results for the accelerated electron beam at t = 6 × 104∆t. (a) (b) Particle distribution snapshot from charge-conserving EM-PIC algorithms using an LU-based implicit solver and a SPAI-based (k = 2) explicit solver, respectively . (c) Particle distribution snapshot from a conventional (non-charge conserving on the unstructured grid) EM-PIC algorithm with an LU-based implicit solver. (d) (e) (f) Corresponding electric-ﬁeld proﬁle distributions.

1 1 1 n+ 2 X n+ 2 (1) n+ 2 Jj = qpvp · Wj xp (2.22) p

1 n+ 2 n+1 n where xp = (xp + xp )/2. In the latter case, violation of the continuity equation produced spurious bunching of the charges into strips of higher density. In addition, the self ﬁeld is highly asymmetric and randomly oriented near the beam center. These

31 0 10 1 = 3,000 SPAI w/ k=2 SPAI w/ k=4 𝑡𝑡 ∆𝑡𝑡 SPAI w/ k=6 0.75 𝑐𝑐 LUD −3 m -1 10 0.5 density

0.25 average velocity/

-2 10 0 -0.5 0 0.5

𝑥𝑥 𝑚𝑚 Figure 2.9: Number density and average velocity of particles across a transversal section of the electron beam at t = 3 × 103∆t, after steady-state has been reached.

spurious eﬀects are not present in either the implicit and explicit charge-conserving

simulations.

Fig. 2.9 shows the average particle density and the average velocity of particles

across a transverse section of the beam versus the longitudinal direction x along the beam at time step n = 3000, for the charge-conserving algorithm with LU-based implicit solver and with SPAI-based explicit solver using k = 2, 4, 6. As expected, the number density of particles monotonically decreases as the average velocity of particles increases, keeping a uniform current ﬂow in steady-state across x. There is an excellent agreement among all these cases, indicating the robustness of the

SPAI-based explicit solver.

32 2.3.4 Electron Bernstein waves

Electron Bernstein waves are instrumental for many applications such as plasma heating, driving plasma currents, and temperature measurement diagnostics [2]. Such waves are present in over-dense plasmas otherwise inaccessible to electromagnetic

(EM) electron cyclotron waves. Because electron Bernstein wave propagation is only possible inside the magnetized warm plasma, mode conversion from EM waves including ordinary (O) or extraordinary (X) modes [2] should be performed.

Here, we analyze dispersion characteristics of electron Bernstein waves propagating in the magnetized warm plasma by using the proposed FETD-PIC algorithm on irregular grids. It is shown that the use of non-charge-conserving scatter algorithms in

FETD-PIC simulations induces a spurious static (self-)ﬁeld due to charge deposition on the grid and, as a result, produces more noisy spectral bands. In contrast, the proposed charge-conserving FETD-PIC solver [1,51,56] is shown to produce sharper spectral bands with less noise.

Magnetized warm plasmas can support two types of waves both propagating and polarized in a direction perpendicular to the stationary magnetic ﬁeld: (i) X mode and (ii) electron Bernstein waves. In what follows, we compare the dispersion relations for the X mode and electron Bernstein waves obtained analytically and numerically by means of FETD-PIC simulations.

We assume a z-directed stationary magnetic ﬁeld and electron Bernstein wave propagation along x, with the same conditions as used in [87]. Consider a magne-

20 −3 tized warm plasma with electron density ne = 2.4 × 10 [m ] and static applied magnetic ﬁeld B~ = 5.13ˆz [T]. The electrons have initial random distribution over

[0.0005, 0.012] × [0, 0.000025] and Maxwellian distribution for the thermal velocity

33 Figure 2.10: Simulated ω × k dispersion diagram for the X mode propagation and for electron Bernstein waves in a magnetized warm plasma. Here ωpe is the plasma frequency and ∆x is the grid spacing, chosen uniform. The analytical results are indicated by the red dots in the diagram. Note that the use of a charge-conserving scatter step in PIC algorithm as described in [1] reduces the numerical noise and yields cleaner spectral bands in the numerically generated band diagrams. In addition, a charge-conserving scatter step mitigates the spurious DC ﬁeld cause by spurious charge accumulation on the grid nodes, as observed at the bottom of the zoomed plots. Overall, a very good agreement is observed between the numerical and the analytical results.

with |~vth| = 0.07c. We set the total number of (macro-)particles in the simulation equal to 13,800, corresponding to a scaling factor of 5 × 109. The motion of ions (of mass mi) is neglected since mi/me ≈ 1, 838, where me is the electron mass. Also, we

11 11 have ωpe = 8.7 × 10 [rad/s] and ωce = 9.0 × 10 [rad/s], where ωpe is the plasma frequency [rad/s] and ωce is the gyrofrequency [rad/s]. Using the FETD-PIC simulation data, we perform a Fourier analysis of electric ﬁeld sampled in space and time to obtain the dispersion relation ω(kx) for the X mode and for the electron Bernstein wave.

Fig. 2.10 shows the dispersion relations computed analytically and numerically.

The reference analytical result for this problem is obtained from [2]. It can be seen

34 that the X mode is dominant for small kx, but as kx increases the electron Bern- stein wave becomes dominant. The two close-in views compare results from charge- conserving and non charge-conserving EM-PIC simulations. In the latter case, a strong spurious static (self-)ﬁeld is produced, which perturbs the particle trajectories and is evidenced by the noisy spectral bands. The absolute spectral resolution is aﬀected by the time step increments employed in the EM-PIC simulation. With this in mind, we have chosen identical increments for both simulations.

2.4 Conclusion

We have developed a EM-PIC algorithm suited for unstructured grids that com- bines a local explicit field solver with a charge-conserving scatter-gather scheme. A sparse approximate inverse is pre-computed to obviate the need for a linear solver at each time step and to retain the local nature of the algorithm. Excellent agreement was verified between EM-PIC simulations utilizing the proposed field solver and a conventional (implicit) field solver based on a LU-solver. The explicit and local nature of the proposed EM-PIC algorithm makes it suitable for integration with asynchronous time stepping techniques as well.

35 Chapter 3: Relativitic Extension of Particle-Pusher

Particle-in-cell (PIC) algorithms [11–13, 88, 89] have been a very successful tool in many scientiﬁc and engineering applications such as electron accelerators [59, 60,

90], laser-plasma interactions [88, 91–94], astrophysics [95, 96], vacuum electronic devices [56, 97, 98] and semiconductor devices [99–102]. In many cases, the particles of interest are often in the relativistic regime and the relevant physical phenomena need to be described by taking into account fully relativistic eﬀects. Relativistic PIC algorithms can be found in a variety of references [90–94,99,103–106].

In this chapter, the previously developed charge conserving FETD PIC algorithm developed in [1, 51] for time-dependent Maxwell-Vlasov equations is extended to the relativistic regime. In particular, we integrate Boris [107], Vay [38], and Higuera-

Cary [108] relativistic pushers in the conservative PIC-FETD algorithm for solving time-dependent Maxwell-Vlasov equations and provide a brief comparison among them. Several examples such as particle cyclotron motion, harmonic particle oscillation in the Lorentz-boosted frame, and relativistic Bernstein modes in magnetized charge-neutral (pair) plasmas are presented for validation. We adopt MKS units throughout this work.

36 3.1 Particle-pushers in the relativistic regime

In the particle update step, the particle mass is modiﬁed to account for relativistic eﬀects such that

dr u p = p , (3.1) dt γp dup q = [E (rp, t) + vp × B (rp, t)] , (3.2) dt m0 where up = γpvp, vp is the velocity of the p-th particle, and γp is its relativistic factor

−2 2 2 deﬁned as γp = 1 − |vp| /c . Using the central-diﬀerences to approximate the time derivatives, Eqs. (3.1) and (3.2) are discretized as

1 n+1 n n+ 2 rp − rp up = 1 , (3.3) ∆t n+ 2 γp n+ 1 n− 1 2 2 up − up q n n q n u¯p n = Ep + v¯p × Bp = Ep + × Bp , (3.4) ∆t m0 m0 γ¯p

1 where v¯p is the mean particle velocity between the n ± 2 time steps, which can also be approximated as u¯p/γ¯p with u¯p =γ ¯pv¯p.

In the non-relativistic case, γp → 1, v¯p can be chosen based on the midpoint rule, 1 1 n n n+ 2 n− 2 viz. v¯p = vp = vp + vp /2, to obtain updated phase coordinates explicitly. In this case, the (non-relativistic) Boris algorithm is typically used not only due to its computationally-eﬃcient velocity update obtained by separating irrotational (electric) and rotational (magnetic) forces but also because of its long-term numerical stability.

The latter property essentially means that, in spite of not being symplectic, the non- relativistic Boris algorithm preserves phase-space volume such that it provides energy conservation bounded within a ﬁnite interval. Note that every symplectic integrator guarantees phase-space volume-preservation but not vice-versa. In contrast, in the

37 relativistic regime v¯p should be carefully determined to accurately model the kinetics of high-energy particles. Next, we examine in detail three diﬀerent relativistic pushers proposed by Boris, Vay, and Higuera-Cary.

3.1.1 Relativistic Boris pusher

The main tenet of the relativistic Boris pusher is basically similar to the non- relativistic-Boris-pusher, viz. separation of irrotational and rotational forces [89].

Importantly, it averages v¯p as

1 1 n+ 2 n n− 2 n v up − p + v up + p ¯v = (3.5) p,B 2 q 2 2 n n where v (u) = u/ 1 + |u| /c , p = αEp , and α = q∆t/2m0. The particle velocity update in the relativistic Boris pusher follows the procedure below [89]

1 − n− 2 n uB = up + p , (3.6)

0 − − uB = uB + uB × tB, (3.7)

+ − 0 uB = uB + uB × sB, (3.8)

1 n+ 2 + n up = uB + p , (3.9)

− 0 + where uB, uB, and uB are auxiliary vectors and the subscript B refers to the Boris

n 2 n n algorithm. In addition, tB = βp /γ¯p,B, sB = 2tB/ (1 + |tB| ), and βp = αBp . The

factorγ ¯p,B is computed as

q q − 2 2 + 2 2 γ¯p,B = 1 + |uB| /c = 1 + |uB| /c , (3.10)

n and to obtain Bp , we set:

1 n+ 1 n− 1 Bn = B 2 + B 2 . (3.11) p 2 p p

38 Note that the relativistic Boris pusher has two variants: with and without correc-

tion. The relativistic Boris pusher without correction uses tB as deﬁned above. The

n relativistic Boris pusher with correction uses tB = (βp/|βp|) tan βp /γ¯p,B instead. The separation of two diﬀerent forces can be easily observed by substituting Eqs.

(3.6) and (3.9) into Eq. (3.4), which results in + − + − uB + uB n uB − uB = α × Bp . (3.12) γ¯p,B

n In Eq. (3.12), the eﬀect of Ep is completely removed, so that only magnetic rotation is eﬀected.

The relativistic Boris pusher preserves volumes in the phase-space because the 1 1 n n n− 2 n+1 n+ 2 determinant of Jacobian for time-update map, ψB : rp , up → rp , up equals to one [107, 108]. To verify that, we express determinant of the Jacobian for

n ψB as 1 n n+1 n n+1 n− 2 ∂ψB ∂rp /∂rp ∂rp /∂up = 1 1 1 , (3.13) n− 1 n+ n+ n− n 2 ∂u 2 /∂rn ∂u 2 /∂u 2 ∂ rp , up p p p p

n+ 1 n+1 n ¯ 2 n ¯ where from (3.3) we have that ∂rp /∂rp = I + ∂up /∂xp , with I being the 3 × 3 identity matrix. If we assume that electromagnetic ﬁelds to be uniform along the

1 n+ 2 n particle trajectory during one time step, ∂up /∂xp = 0. In addition, it is clear that n+ 1 n+1 n ¯ 2 ∂xp /∂xp = I. Substituting up from (3.4) into (3.3) and taking a derivative w.r.t. 1 1 1 1 n− 2 n+1 n− 2 n+ 2 n− 2 ∂up of the resulting equation, we obtain ∂xp /∂up = ∆t ∂up /∂up . As n+ 1 n− 1 2 2 a result, the determinant in (3.13) simply takes the form of ∂up /∂up . This derivative can be computed by splitting one time-update into two half-time-updates

and evaluating each serially as follows n+ 1 n+ 1 2 1 1 2 ∂up /∂u¯p n+ 2 n− 2 ∂up /∂u¯p ∂up /∂up = = . (3.14) n− 1 n− 1 2 2 ∂up /∂u¯p ∂up /∂u¯p

39 But since [108]

n 2 n 2 n+ 1 n− 1 β + β · u¯p,B ∂u 2 /∂u¯ = ∂u 2 /∂u¯ = 1 + p p , (3.15) p p,B p p,B 4 γ¯p,B

n+ 1 n− 1 2 2 it follows that ∂up /∂up = 1 and therefore the relativistic-Boris-pusher is phase- space volume-preserving. This means that energy conservation is attained for long

time simulations down to machine precision accuracy (with residual error stemming

from round-oﬀ errors). However, the main disadvantage of the relativistic-Boris-

pusher is that it cannot accurately capture correct the particle acceleration by electric

forces. This is because magnetic rotation fails to consider the varying relativistic

factor due to electric ﬁeld eﬀects during the magnetic rotation.

3.1.2 Vay pusher

The Vay pusher corrects trajectories of relativistic particles experiencing electric

ﬁelds by averaging v¯p as 1 1 n+ 2 n− 2 v up + v up ¯v = . (3.16) p,V 2

The particle velocity update follows the procedure below [38]

1 1 n n− 2 n n− 2 n up = up + p + vp × βp , (3.17)

0 n n uV = up + p , (3.18)

1 h 0 0 0 i n+ 2 up = |sV | uB + uV · tV tV + uV × tV , (3.19) s r n+ 1 2 2 2 n ∗2 γp = 0.5σ + 0.5 σ + 4 × βp + uV , (3.20)

02 n 2 σ = γV − βp , (3.21)

1 n+ 0 0 q 02 n 2 2 ∗ n 0 2 where tV = βp /γp , sV = 2tV / (1 + |tV | ), uV = uV ·βp /c, and γV = 1 + uV /c . The Vay pusher correctly models energetic particle motion under electric forces with

40 Lorentz (relativistic) invariance. In other words, the relation between the particle’s trajectory observed in a (relativistic) moving and laboratory frames satisfy Lorentz transformation.

3.1.3 Higuera-Cary pusher

The Higuera-Cary pusher provides an accurate treatment of electric forces by approximating the average velocity as [108]

n+ 1 n− 1 ! u 2 + u 2 ¯v = v p p . (3.22) p,H 2

The particle velocity update is basically similar to Boris algorithm except for the relativistic factor as " 1 2 n 2 γ¯p,H = γ¯ − β + 2 p,B p # r n 2 n 2 n − 2 γ¯p,B − βp + 4 βp + βp · uB . (3.23)

Therefore, the Higuera-Cary pusher is also phase-space volume-preserving.

3.2 Numerical results

3.2.1 Synchrocyclotron

To validate the relativistic PIC formulation, we ﬁrst examine a synchrocyclotron example. As electrons usually have velocities near to the speed of light in this case, progressively more energy needs to be delivered to accelerate them due to relativistic eﬀects. The relativistic mass increase results in a lower orbital (cyclotron) frequency.

Therefore, the driving RF electric ﬁeld should have variable frequencies matching this relativistic cyclotron frequency. Fig. 3.1b illustrates the computational mesh used for the simulation where centripetal force from external magnets is present in the red

41 (a) (b)

Figure 3.1: (a) Cyclotron configuration. (b) Computational domain, where the blue vertical strip indicates the region where an external longitudinal RF electric field is applied. The DC magnetic field is applied in the whole computational region except for the RF acceleration gap (red).

region leading to a circular motion of electrons and a longitudinal RF electric ﬁeld is present in the vertical blue strip leading to a periodic electron acceleration.

Fig. 3.2a shows the electron cyclotron motion in a non-relativistic regime, where the relativistic factor is assumed to be one. An electron is injected at (x, y) =

7 (0.52, 0.5) m with an initial velocity of |v0| = v0 = 1 × 10 m/s. The static magnetic

−3 force is determined to be Bz = m0v0/qr = 2.84281 × 10 T for an initial orbital

5 radius r of 0.02 m and the RF electric force is set to be Ex = 2 × 10 V/m. The thickness of the vertical strip in which the longitudinal electric ﬁeld is present is 0.02 m. As can be seen, the spacing of two adjacent orbits becomes successively smaller due to the increasing velocity. Fig. 3.2b shows the trajectory of the electron with

42 Region of acceleration Non-relativistic case relativisticrelativistic case, case, unsynched unsynched relativisticrelativistic case, case, synched synched

Settings

# of time steps : 5000 Time increment (Δt) : 4e-11 sec.

Initial velocity (v0) : 1e7 m/s Initial position : (0.52, 0.5) Initial radius of circular trajectory : 0.02 m Gap between dees : 0.02 m

Magnetic force (Bz) = mv/(rq) = 2.84281e-3 Wb/m2 Electric force (Ex) = 2e5 V/m

(a) (b) (c)

6/29/2015 8 6/29/2015 8 Figure6/29/2015 3.2: Electron trajectories on a cyclotron: (a) Non-relativistic,7 (b) Relativistic, unsynchronized, and (c) Relativistic, synchronized.

same initial conditions. For this example, we use the PIC formulation with the rel-

ativistic Boris pusher with correction discussed in the previous section. In this case,

the frequency of the RF electric force is set to be constant (79.6 MHz), which results

in an unsynchronized phase between particle velocity and electric force and a mixed

trajectory. In Fig. 3.2c, the frequency of the RF electric ﬁeld is matched to the orbital

frequency of the electron, the synchrocyclotron frequency given by f = qB/2πγm0.

This frequency is shown, together with the relativistic factor, as a function of the number of time steps of the simulation, in Fig. 3.3. Therefore, the in-phase acceleration is maintained at all times and a circular trajectory is observed at higher energies.

Note that the total distance over which an electron moves in this case is shorter than that for the non-relativistic case because of its smaller speed caused by the relativistic mass. Fig. 3.4 shows the electron velocity magnitudes in the three cases. It is clearly observed that the deceleration occurs near 4000 time steps for the second case.

43 Figure 3.3: Orbital frequency and relativistic factor for the case shown in Fig. 3.2c.

Also, the third case shows slightly smaller magnitudes than the ﬁrst one due to the

relativistic mass.

Tables 3.1, 3.2, and 3.3 provide a veriﬁcation of Gauss’ law. The amount of

charge on arbitrarily selected mesh nodes is recorded at diﬀerent time steps. As

the rightmost columns in these tables show, the normalized residual associated to

the discrete version of Gauss’ law is very small and near the double precision ﬂoor

(< 10−15).

3.2.2 Harmonic oscillations in Lorentz-boosted frame

In order to compare how accurately the three diﬀerent kinds of relativistic particle

pushers capture relativistic E × B drift motions, we consider a harmonic oscillatory

motion of a positron in the Lorentz-boosted frame with γf = 2 such as in [38]. Initial

parameters of the harmonic motion are transformed via the Lorentz transformation

into the moving frame alongy ˆ and PIC simulations are performed in the moving frame. At the end of simulations, we re-transform the phase coordinates from the

44 Velocity comparison

9 10

3x108

8 10 |v|

7 10

non-relativistic relativistic, unsynched

6 relativistic, synched 10 0 1000 2000 3000 4000 5000 time steps

Figure 3.4: Comparison of electron velocity magnitudes of the three cases shown in Fig. 3.2.

Table 3.1: Veriﬁcation of discrete Gauss’ law for the non-relativistic case (Fig. 3.2a).

n n n Nodal Index qn S · [? ] · en Se·[?]·e −q e qn 1000 89 -7.62302 ×10−20 -7.62302 ×10−20 1.15269 ×10−14 2000 26 -1.29865 ×10−20 -1.29865 ×10−20 4.95884 ×10−13 3000 110 -3.39127 ×10−20 -3.39127 ×10−20 3.19447 ×10−15 4000 233 -1.10205 ×10−19 -1.10205 ×10−19 1.26699 ×10−14 5000 259 -1.98727 ×10−20 -1.98727 ×10−20 2.24868 ×10−13

45 Table 3.2: Veriﬁcation of discrete Gauss’ law for the relativistic case without syn- chronization (Fig. 3.2b).

n n n Nodal Index qn S · [? ] · en Se·[?]·e −q e qn 1000 235 -7.20396 ×10−20 -7.20396 ×10−20 7.10129 ×10−14 2000 247 -3.70748 ×10−20 -3.70748 ×10−20 5.20444 ×10−13 3000 83 -5.30434 ×10−20 -5.30434 ×10−20 1.44212 ×10−13 4000 39 -8.31747 ×10−20 -8.31747 ×10−20 2.76415 ×10−14 5000 143 -8.17890 ×10−20 -8.17890 ×10−20 1.83523 ×10−13

Table 3.3: Veriﬁcation of discrete Gauss’ law for the relativistic case with synchro- nization (Fig. 3.2c).

n n n Nodal Index qn S · [? ] · en Se·[?]·e −q e qn 1000 235 -9.02849 ×10−20 -9.02849 ×10−20 9.86590 ×10−15 2000 179 -4.05879 ×10−20 -4.05879 ×10−20 1.68005 ×10−13 3000 196 -1.75078 ×10−20 -1.75078 ×10−20 4.06155 ×10−13 4000 116 -7.70014 ×10−21 -7.70014 ×10−21 2.83334 ×10−14 5000 332 -8.90351 ×10−20 -8.90351 ×10−20 2.06847 ×10−14

46 Figure 3.5: Motion of harmonic oscillator of a single positron inverse-Lorentz- transformed into Laboratory frame.

moving frame into the laboratory frame by using the inverse Lorentz transformation.

We compare the resultant trajectories of the harmonic oscillators obtained with three

diﬀerent particle-pushers and analytic predictions in Fig. 3.5. As we discussed in Sec.

C, the relativistic Boris pusher (without or with correction) cannot capture correct

relativistic E × B drift motion; on the other hand, results obtained with the Vay pusher and Higuera-Cary pusher accurately match the analytic prediction.

3.2.3 Relativistic Bernstein Modes in Magnetized Pair-Plasma

The non-relativistic electron Bernstein mode [2, 109], which is a purely electrostatic plasma wave perpendicularly propagating to a stationary magnetic ﬁeld, has

47 Figure 3.6: Dispersion relations for classical (non-relativistic) electron Bernstein modes of PIC results (Parula colormap) and analytic predictions [2] (dashed red line).

been mainly explored in magnetic plasma conﬁnement fusion as a promising alter-

native to conventional electron cyclotron electromagnetic waves such as the ordinary

(O) or extraordinary (X) modes which have frequency cutoﬀs associated with plasma

density [110]. The Bernstein mode is free from the density cut-oﬀ, and as a re-

sult, it is able to reach the core of over-dense plasmas in Tokamak devices and heat

the plasma electrons eﬀectively. It is well known that conventional non-relativistic

Bernstein waves are present at harmonics of electron cyclotron resonances [2], as

shown in Fig. 3.6, which illustrates dispersion relations, in terms of the normalized ˆ frequencyω ˆ = ω/ωc and normalized transverse wavenumber is k⊥ = kxc/ωc, of non-

relativistic electron Bernstein and X modes based on PIC simulations (colormap) and

compare the simulation results with analytic predictions (dashed red line). In this

11 11 20 −3 case, ωc = 9.0 × 10 rad/s, B0 = 5.13ˆz T, ωp = 8.7 × 10 , n0 = 2.4 × 10 m , and the initial isotropic speed distribution (equilibrium state) obeys a Maxwellian

48 distribution with vth = 0.07c. The simulation parameters are chosen similar to [87].

Runaway electrons can cause plasma discharge disruptions in fusion devices. The interaction of runaway electron beams with plasma turbulence requires full electromagnetic treatment in relativistic regimes. Relativistic plasma waves propagating in energetic electron-positron pair-plasmas have also been of interest in astrophysics

(pulsar atmospheres). Therefore, it is desirable to extend the investigation of Bern- stein modes to the relativistic regime. Recently, analytic works have been done to characterize the behaviors of Bernstein modes in relativistic pair-plasmas [111–114].

There are several features distinguishing the classical Bernstein wave from the relativistic one: (1) The classical Maxwellian distribution (equilibrium state) is modi-

fied to the Maxwell-Boltzmann-Jüttnerdistribution (relativistic Maxwellian), (2) the mobility of positively-charged particles is identical to that of negatively-charged particles, and (3) the conventional dispersion relations are significantly transformed to undamped or damped closed curve shapes. In this example, we use the developed

FETD PIC algorithm to perform simulations of Bernstein modes propagating in relativistic magnetized pair-plasmas and compare the results with analytic predictions.

Analytic prediction

To derive analytic dispersion relations of magnetized plasma waves [2, 111–114], we should obtain a complex permittivity tensor, ¯ associated with plasma currents.

First of all, we consider a small perturbation imposed on equilibrium magnetized

49 pair-plasmas with parameters of

fs (r, p, t) = f0,s (p) + f1,s (r, p, t) , (3.24)

i(k·r−ωt) B = B0 + B1e , (3.25)

i(k·r−ωt) E = E1e , (3.26)

where fs is a distribution function represented in the phase space for the species s,

E is electric ﬁeld intensity, B is magnetic ﬂux density, and subscriptions of 0 and

1 denote equilibrium and perturbed quantities, respectively. Note that perturbed

electromagnetic ﬁelds are proportional to ei(k·r−ωt). Equilibrium relativistic electron-

positron pair-plasmas are typically described by Maxwell-Boltzmann-J¨uttner(rela-

tivistic Maxwellian) distribution which is given by

MBJ 1 η −ηγ f0 (p) = 2 3 e (3.27) 4πm0 c K2 (η)

m0c where η = , kB is the Boltzmann constant, T is the kinetic temperature, K2 (·) kB T p2 −1/2 is the modiﬁed Bessel function of the second kind, and γ = (1 + c2 ) . The evolution of the distribution function is governed by the Vlasov equation. Its ﬁrst-order

approximation takes the form of

df (r, p, t) ∂f (p) 1,s = −q (E + v × B ) ei(k·r−ωt) · 0,s . (3.28) dt s 1 1 ∂p

Substituting Eq. (3.27) into Eq. (3.28) and integrating Eq. (3.28) over time, solutions

for the perturbed distribution function, f1,s can be obtained. Then, plasma currents are calculated based on f1,s as

Z X qs X J = pf (r, p, t) d3p = σ¯ · E (3.29) m 1,s s 1 s s s

50 (a) (b)

Figure 3.7: An isotropic 2D Maxwell-Boltzmann-J¨uttnervelocity distribution, f0 (p) for η = 1/20: (a) Speed distribution and (b) relativistic velocity distribution.

where σ¯ is a conductivity tensor from which we obtain the complex permittivity associated with plasma currents as

¯ ¯ σ¯ ¯ = 0 I − . (3.30) iω0

We are interested in longitudinal electrostatic plasma waves propagating in the x-

direction. Thus, (ω, kx) curves yielding zeros of xx form the dispersion relations for

Bernstein modes in a magnetized relativistic pair-plasma. The expression for xx can be written as [111]

2( ) 2ˆω2η Z ∞ xx p η 2 −ηγ 1 3 ˆ2 = 1 − pˆ e × 2F3 , 1; , 1 − γω,ˆ 1 + γωˆ; −β dpˆ − 1 , 0 kˆ x K2 (η) 0 2 2 (3.31)

1 3 2 where 2F3 2 , 1; 2 , 1 − a, 1 + a; −b is a hypergeometric function, which is deﬁned as Z ∞ 1 3 2 1 πa 2F3 , 1; , 1 − a, 1 + a; −b = sin θJa (b sin θ) J−a (b sin θ) dθ, 2 2 2 0 sin (πa) (3.32)

51 Figure 3.8: Dispersion relations for plasma waves propagating in magnetized relativistic pair-plasma for η = 1/20: Comparison of PIC results and analytic prediction.

ˆ ˆ ωˆp = ωp/ωc,p ˆ = p/ (m0c) β = k⊥pˆ, and Jν (·) denotes the Bessel function of the

ﬁrst kind for ν. One can ﬁnd details in [111] on how to numerically compute the

integral in Eq. (3.31), which exhibits singularities at harmonics of the (rest) cyclotron

frequencies.

FETD PIC results

We consider the case ofω ˆp = 3 and η = 20. Other parameters are speciﬁed

11 12 as follows: B0 = 5ˆz T, ωc = 8.7941 × 10 rad/s, ωp = 2.6382 × 10 rad/s, ne =

21 −3 21 2.1870 × 10 m , and electron or positron density, n0 = 2.1870 × 10 . The Debye

−5 length, λD equals to 2.55 × 10 m and the characteristic (relativistic) gyroradius,

−5 rg becomes 7.92 × 10 m. An irregular mesh with triangular elements of size lx ×

52 (a) (b)

Figure 3.9: Normalized residuals versus nodal index for (a) discrete continuity equation (DCE) and (b) discrete Gauss law (DGL).

ly is constructed with ly = rg and lx = 1000 × rg and average mesh element size comparable to λD. The number of nodes, edges, and faces in the mesh are 7252,

18123, 10872, respectively. The left and right boundaries of the mesh are terminated by perfectly matched layers (PML) [74, 115] to mimic open boundaries. Periodic boundary conditions (PBC) are applied at the top and bottom boundaries. In order to obtain the dispersion relation in (kx, ω) for Bernstein waves propagating along x, we spatially sample the electric field along the x direction at each time-step and perform a Fourier transform on the resulting data set in (x, t). The PML treatment of the left and right boundaries not only reduces unwanted reflections but also avoids aliasing effects in the dispersion relation caused by PBC with low sampling rates and the presence of image sources. Whenever particles meet a PBC boundary wall, they are removed and reassigned the same relative position on the other PBC boundary wall and same momentum. The total number of superparticles for e and p species

10 representing 1.7222×10 electrons and positrons, respectively, is set to Nsp,e+Nsp,p =

5 4 × 10 . Note that Nsp,e and Nsp,p are identical, and our simulation is initialized

53 so that the average number of superparticles per grid element is around 40. This

number attempts to provide a good trade-oﬀ between simulation speed and the rate

of plasma self-heating. Initially, superparticles are uniformly distributed on the mesh

in a pairwise fashion. i.e. with each species-e superparticle collocated with another

species-p superparticle. This arrangement produces zero initial electric ﬁeld. Based

on an Maxwell-Boltzmann-J¨uttnerdistribution for the initial speed as depicted in Fig.

3.7a), superparticles for species e are then launched in random 2D directions. The corresponding p superparticles are simultaneously launched with same speed in the opposite direction. Consequently, both species of superparticles follow an isotropic

Maxwell-Boltzmann-J¨uttnervelocity distribution, see Fig. 3.7b. Fig. 3.7a compares the Maxwell-Boltzmann-J¨uttnerdistribution and classical Maxwellian distribution.

It can be seen that the classical Maxwellian velocity distribution starts to gradually deviate from the relativistic one above the most probable velocity, which is about v/c = 0.21. We simulate up to 100, 000 of time-steps and employ ∆t = 0.01 ps chosen with Courant factor of 0.2. Then, we perform space and time Fourier transforms of sampled data to obtain the dispersion relations.

Fig. 3.8 illustrates the dispersion relations for relativistic Bernstein waves with

η = 1/20, compared to analytic predictions. It is observed in Fig. 3.8 that there are solutions of curved shape between every two neighboring harmonics of the (rest) cyclotron frequency. Our PIC simulations capture this feature quite well, which dis- tinguishes the relativistic Bernstein wave from the classical one as predicted by theory.

It is also interesting to note that every upper curve of each solution is considerably weaker since, as pointed out in [113], the damping coeﬃcient for the upper curve is larger than that for the lower curve. In addition, there are two kinds of stationary

54 modes at aroundω ˆ = 0.9 andω ˆ = 4.1. As shown in [111], these stationary modes are nearly dependent on ωp and the gap and location frequency between two stationary modes becomes smaller as η decreases (ultra-relativistic). It should be noted that even though PIC simulations are somewhat noisy, some physical damping should be ˆ certainly present (especially in warm plasmas and for high kx). Introducing a spectral filtering scheme to reduce the noise may lead to heating and artifacts. It is highly desirable to compare numerical results with analytical predictions here since although there are many PIC codes available, there is little understanding of how well PIC codes describe classical plasma effects, like Bernstein modes. It is expected that the most efficient way to reduce the numerical noise is to use high-order particle gather/scatter procedures. We leave these issues to a future work.

In order to check charge conservation, Fig. 3.9 illustrates normalized residuals for the discrete continuity equation (DCE) (Fig. 7.15a) and the discrete Gauss law

(DGL) (Fig. 7.15b) across all mesh nodes at three diﬀerent time-steps, n = 10, 000,

20, 000, and 30, 000. It is observed in Fig. 7.15a and Fig. 7.15b that the normalized residuals are fairly low, near the double precision ﬂoor, which again indicates that no spurious charges are deposited at the nodes.

3.3 Conclusion

A finite element time-domain particle-in-cell algorithm is presented to simulate the relativistic Maxwell-Vlasov equation. The key feature of the algorithm is to combine a new charge-conserving scatter/gather scheme for irregular meshes with relativistic particle pushers for efficient plasma simulations. Three different relativistic particle

55 pushers are considered and brieﬂy compared. Several numerical examples are provided for illustration purposes.

56 Chapter 4: Multipactor

Resonant electron discharges from metallic or dielectric surface, also known as multipactor effects, are often observed in high-power radio frequency (RF) devices such as accelerators, vacuum tubes, and satellite payloads. This effect generally degrades and limits device performance [116,117]. On the other hand, discharge effects can be harnessed by various technologies including electron guns, plasma displays, and for energy dissipation to protect highly sensitive receivers [118]. As a bridge between the theoretical and experimental analysis, computer simulations have been employed in recent years to analyze multipactor effects via electromagnetic particle-in-cell (EM-

PIC) algorithms [11, 119, 120], which basically solve Maxwell-Vlasov equations for tracking the non-linear evolution of coarse-grained distribution of charged particles

(tenuous plasma) and its interaction with the RF ﬁeld.

In this chapter, we numerically investigate multipactor eﬀects using the novel

EM-PIC algorithm based on the ﬁnite-element time-domain (FETD) method implemented on unstructured (irregular) grids [1, 51]. The use of unstructured grids enables local mesh reﬁnement and simulation of complex geometries with minimal geometrical defeaturing, see Fig. 4.7 below. The present algorithm attains energy and charge conservation [51], a feature that has eluded previous EM-PIC algorithm implementations on irregular grids. In addition, the present algorithm implements

57 Figure 4.1: Schematic illustration of a typical SEE process in an irregular-grid-based EM- PIC simulation. Note that electric current densities by the primary or secondaries are deposited on red- or blue-highlighted edges, respectively.

the Furman-Pivi probabilistic model [40], based on a broad phenomenological fit to experiment data, to obtain accurate simulations of secondary electron emission (SEE) process (rather than a conventional monoenergetic one). We illustrate the proposed algorithm by examining multipactor effects taking place on waveguides with flat or corrugated (triangularly-grooved) walls. We contrast the evolution of the electron population in various cases and investigate the respective saturation process arising from self-field counterbalance effects.

We illustrate the proposed algorithm by examining multipactor effects taking place on waveguides with flat or corrugated (triangularly-grooved) walls. We contrast the evolution of the electron population in various cases and investigate the respective saturation process arising from self-field counterbalance effects.

58 (a) (b)

Figure 4.2: Comparison of simulation and experimental results for SEE on copper [(a) and (b)] and stainless steel [(c) and (d))] surfaces. Figures (a) and (c) illustrate SEY δ versus the primary incident energy. Figures (b) and (d) show the emitted-energy spectrum dδ/dE.

59 4.1 Irregular-Grid EM-PIC Algorithm integrated with Furman- Pivi model

We integrate the probabilistic Furman-Pivi SEE model into the EM-PIC algorithm. In this model [40], three types of (macroscopic) mechanisms producing secondary electrons are incorporated: (1) backscattered (BS) or almost elastic, (2) re- diﬀused (RD) or partially elastic, and (3) true-secondary (TS) or inelastic. A typical scenario for the SEE process during one time step ∆t (i.,e. from time-step n to n+1) in the EM-PIC algorithm is illustrated in Fig. 4.1. When the primary electron (red solid line) trajectory intersects a metallic surface, the SEE algorithm launches secondaries

(BS, RD, or TS, indicated by blue solid lines) via a stochastic process governed by the primary electron kinetic energy and incidence angle. Once all trajectories of the secondary electrons are obtained, the scatter step in the EM-PIC algorithm converts the electron trajectories into equivalent electric currents along edges of the irregular grid. In this process, the trajectory of the primary inside the metal (red dashed line) is discarded and the primary becomes dummy. As noted, the use of irregular grids enables a more accurate description of complex geometries, including curved and textured-surface treatments employed to suppress the multipactor effects [121] or found in electron gun technologies. In the present EM-PIC algorithm, implementation of electron emission process from curved boundaries is more natural than in regular-grid-based EM-PIC algorithms. This is because the latter necessitates the use of ad hoc cut-cell methods or conformal finite-difference approaches [21]. For the purpose of implementing the field boundary conditions, the metallic surfaces are assumed as perfect electric conductors (PEC) at RF frequencies. A summary description and a pseudocode implementing the SEE process are provided in Appendix II.

60 (a) (b)

Figure 4.3: Geometrical illustration of exact charge conservation on irregular grids for a primary impact (also applicable for secondary electrons emitted on the opposite way) at PEC surfaces during ∆t. Plot (a) depicts the charge variation rate at jth node. Plot (b) depicts the divergence of current on jth node, which is equal to the sum of ith and kth currents.

4.2 Charge-conserving scatter near conducting surface

Given a particle trajectory (motion) during one unit time interval ∆t, the scatter

step of a PIC algorithm assigns the consequent current onto the edges of the grid

around the metallic boundary. From Maxwell’s equations, the tangential electric and

magnetic ﬁeld components at a PEC surface are zero. As such, the present ﬁeld solver

enforces zero tangential ﬁelds at the metallic surfaces. A charged particle next to a

metallic surface will induce a surface charge distribution on the surface that can be

obtained from image theory assuming a charged particle suﬃciently close to a locally

planar surface. As the charged particle approaches the surface from the grid domain,

the ﬁelds due to the charged particle and its image will cancel each other. When

the particle hits the surface, the associated ﬁeld becomes zero and the charge is then

61 absorbed (discarded) by the EM-PIC algorithm. On the grid, charge conservation is

obtained by a proper balance between the variation of the node-based charges and

the edge-based currents that touch a given node. This is illustrated in Fig. 4.3: when

a single charged particle inside the kth face (triangle) crossed the metallic (PEC)

surfaces and leaves the problem domain (dashed black line), there is an associated

th th non-zero grid charge at the i node (green square) and associated currents at the j1

th (red) and j2 (blue) edges. From a geometrical viewpoint, the grid charge variation rate on ith node, which from the Whitney 0-form expansion is associated to the green- colored area divided by ∆t, is equal to the sum of the grid currents ﬂowing in/out

th th th of the i node, which is the sum of two grid currents at j1 and j2 edges. Form the Whitney 1-form expansion of the currents, the latter is equivalent to the sum of two areas colored in red and blue. A mathematical derivation of these results can be found in [1,51].

4.3 Furman-Pivi SEE model implementation

The basic steps implementing the probabilistic Furman-Pivi SEE model are summarized in the EM-PIC code are summarized in Algorithm 1. Initially, a vacant 1-D workspace of size Np,max × 1 is set for either dummy or eﬀective macroparticles. At each time-step, a for loop is performed with respect to the index p that checks whether or not the p-th particle is dummy by Dummy Effective Checker, yielding the integer

a: 1 (eﬀective) or 0 (dummy). If a = 1, Particle Acceleration and Particle Push

accelerate and push the p-th particle during one time-step, respectively, and yield

1 n+ 2 n+1 its updated velocity vp and position xp . Afterwards, Impact Checker tests the occurrence of the impact of p-th particle on metal surfaces, producing the integer b:

62 n n+1 1 (impact) or 0 (no impact). If b = 0, scatter uses xp and xp to compute grid

n+ 1 electric currents, j 2 . If b = 1, the SEE algorithm determines the impact position,

energy, and angle via Impact Position, Energy, and Angle, respectively. After

n imp the primary’s trajectory between xp to impact position xp is converted to an grid electric current, the primary particle is discarded (absorbed). In accordance with

the impact information obtained above, the number of the secondaries NSEY is com-

puted by Compute SEY based on the Furman-Pivi probabilistic SEE model. Another for loop is performed to compute the launching energy and angle for the secondaries.

This information is are saved in vacant bins found by Min Dummy Finder. Finally, the

new secondaries are launched and their current transferred to the grid in the scatter

step.

4.4 Numerical Results and Discussion

4.4.1 Veriﬁcation of SEE model in EM-PIC simulations

In order to validate the probabilistic SEE process for embedding in present EM-

PIC algorithm, we carried out impact simulations on copper and stainless steel sur-

faces without external ﬁelds. We obtained statistical averages of two important pa-

rameters: the secondary electron yield (SEY) δ and the emitted-energy spectrum

dδ/dE. We compare the simulation results against experimental data from [122].

Fig. 4.2a and Fig. 4.2c plot δ versus the incident electron energy [eV] for normal

5 incidence when the superparticle number Np is set equal to 5 × 10 , for copper and

stainless steel surfaces, respectively. Fig. 4.2b and Fig. 4.2d show dδ/dE versus the

secondary electron energy assuming primary impact energy E0 = 295 [eV] and E0 =

300 [eV], respectively, and normal incidence. The superparticle number Np in these

63 Algorithm 1: Basic steps for implementation of SEE in the EM-PIC algorithm.

for p ∈ [1, 2, ..., Np,max] do a ← Dummy Effective Checker(p); if a == 1 then 1 n+ 2 vp ← Particle Acceleration(p); n+1 xp ← Particle Push(p); b ← Impact Checker(p); if b == 1 then imp xp ← Impact Position(p); d ← Impact Energy(p); e ← Impact Angle(p); n+ 1 n imp j 2 ← Scatter xp , xp ; Delete Primary(p); NSEY ← Compute SEY(d,e); for q ∈ [1, 2, ..., NSEY ] do r ← Min Dummy Finder(); Compute Launch Energy(r); Compute Launch Angle(r); n+1 xr ← Particle Push(r); n+ 1 imp n+1 j 2 ← Scatter xp , xr ; end else n+ 1 n n+1 j 2 ← Scatter xp , xp ; end end end

simulations is 1 × 106. Overall, there is very good agreement between the present simulation results and the experimental data. We also tested the angular dependence of the SEE process for various primary incident angles. This is shown in Fig. 4.4, where it can be seen that grazing incidence generates more secondaries than the normal incidence, as expected. Note that present algorithm assumes the same angular distribution function (cosine-like) as introduced in [40].

64 Figure 4.4: Angular dependence of δ on a copper surface.

(a) (b) (c)

Figure 4.5: PIC results for probabilistic SEE model. (a) Superparticle population versus time (RF voltage periods). (b) and (c) Snapshots of particle’s trajectories for copper and stainless steel cases, respectively. These trajectory snapshots are taken during four successive half-periods of the RF signal, i.e.: t/TRF ∈ (0, 0.5), t/TRF ∈ (0.5, 1), t/TRF ∈ (1, 1.5), and t/TRF ∈ (1.5, 2), where TRF = 1/fRF = 0.96 [ns].

65 (a) t/TRF ∈ (0, 0.5), (b) t/TRF ∈ (0.5, 1), (c) t/TRF ∈ (1, 1.5), (d) t/TRF ∈ (1.5, 2), copper copper copper copper

(e) t/TRF ∈ (0, 0.5), (f) t/TRF ∈ (0.5, 1), (g) t/TRF ∈ (1, 1.5), (h) t/TRF ∈ (1.5, 2), stainless steel stainless steel stainless steel stainless steel

Figure 4.6: Particle trajectory snapshots on the phase space. The coordinate axes represent x/10 [m], y [m], and the normalized speed of the particles (|vp| /20c). Each plot corresponds to a half-period of the RF signal, as in Fig. 4.5.

4.4.2 Multipactor on copper versus stainless steel surfaces

Consider parallel metallic plates separated by a 2 mm gap. An external RF voltage

is applied to the plates as shown in Fig. 4.5b and Fig. 4.5c. We assume a RF voltage

a with amplitude VRF = 300 V and frequency fRF = 1.044 GHz. These parameters are chosen to meet the resonant condition for multipacting, see Eqn. (4.1) below.

We initially place 100 seed superparticles uniformly distributed along a line parallel

to and near the lower plate. Each superparticle (both as initial seeds and future

secondaries) in the simulation represents about 2.5 × 108 actual electrons. Here and

in what follows, the metallic surfaces are assumed as PEC surfaces for the purpose of

implementing the ﬁeld boundary conditions in the RF frequency regime. The left and

66 right ends of the grid are terminated by a perfectly matched layer (PML) [41]. Copper and stainless steel have diﬀerent δ proﬁles, cf. Fig. 4.2a and Fig. 4.2c in [40]. Except for primary impacts with very low incident energy, electron avalanches can occur since

δ is overall larger than unity. It is worth noting that in the copper plates most of the secondaries tend to be true-secondary. On the other hand, both backscattered and true-secondary electrons are prevalent in the stainless steel plates. Fig. 4.5 shows some EM-PIC simulation results that capture the distinct features in copper and stainless steel. In Fig. 4.5a, we compare the temporal growth of superparticle population between the plates over ﬁrst two periods of the RF voltage. A nearly stepwise increase of the population can be observed in the copper plate whenever primary electrons hit the walls since the most of SEE is produced by TS emission and they are regular accelerated along the retarding RF voltage. On the other hand, in the stainless steel plate the net superparticle population increases rather more gradually due to the balance between almost elastic and inelastic secondary emissions. This implies that roughly half of electrons, which are of TS and BS, feel a accelerating force (in-phase) from the RF ﬁeld while the remaining electrons are not in phase; however, this is enough for them to hit the wall so that net electron avalanches also occur. This is evidenced in Fig. 4.6, which illustrates the particles’ trajectory in the phase space for successive half cycles of the RF signal. In these plots, the particle trajectories are colored with respect to their speed (faster by red lines, slower by blue lines). The axes represent the spatial coordinates, x/10 m and y m, versus normalized speed of the particles |vp| /20c for each half of the RF voltage period, as indicated.

It can be observed that for the second half-period the stainless steel surface creates more energetic electrons at the moment of the emission than the copper surface. As a

67 (a)

(b) (c)

Figure 4.7: Multipactor in parallel plate waveguides. (a) Schematics of the problem geometry. (b) Flat surface waveguide meshing. (c) Triangular-grooved waveguide meshing.

result, they produce more primary impacts that are out of phase with the RF ﬁeld (in

addition to the regularly accelerated ones). This causes the total electron population

to increase gradually in the stainless steel case rather than the stepwise sense as in

the copper case, see also Fig. 4.5a.

4.4.3 Surface treatment eﬀects

Consider a waveguide with copper plates separated by a gap size Dpp m and longi-

tudinal length Lpp, as depicted in Fig. 4.7a. In this case, a transverse electromagnetic

(TEM) wave is injected from the input (left) port by exciting a line current source between the plates. The output voltage is measured at the right-end port. We denote

68 Table 4.1: Multipactor simulation parameters for the parallel waveguide in Fig. 4.7a.

a Dpp [mm] fRF [GHz] VRF [V] Lpp [mm] Lmp [mm] 2 2 1,143 150 30

fRF and VRF as the frequency and RF voltage amplitude of the input signal. Ini- tially, we place 1, 000 superparticle seeds uniformly distributed near the lower plate

and launch them with zero velocities (the electron cloud region in Fig. 4.7a). Each

superparticle represents 2 × 107 actual electrons. In order to prevent stray electrons

spreading much laterally during multipacting, it is assumed that only the central

section of length Lmp (blue-glowed-solid lines in Fig. 4.7a) are constituted by copper

yielding the secondaries. The other surfaces are assumed as collectors that absorb

electrons. The resonant condition for multipacting is given by

r 1 a qe fRF = √ VRF [Hz] (4.1) 2 πDpp me where qe and me charge and mass for an electron. All parameters are chosen to meet the resonant condition and represented in Table 4.4. We consider two types of surfaces: ﬂat and triangularly-grooved surfaces, as illustrated in Fig. 4.7b and

Fig. 4.7c, respectively. The width, depth, angle, and number of grooves are denoted as wg, hg, αg, and Ng, respectively, and their values are given in Table 4.2. Note that a triangularly-grooved surface might reduce δ depending on αg, as discussed

in [121]. The domain is discretized using an unstructured mesh. Table 5.1 lists

some of the mesh parameters: N0, N1, and N2 are the number of nodes, edges, and

faces, respectively. In addition, ∆tmax denotes the maximum time-step interval for

stable simulations according to the Courant-Friedrichs-Lewy (CFL) criterion. Both

69 Table 4.2: Triangularly-grooved surface parameters.

wg [mm] hg [mm] αg [deg.] Ng 0.2 0.5495 40 150

Table 4.3: Mesh parameters.

N0 N1 N2 ∆tmax [fs] ﬂat surface 2,913 7,823 4,911 135 grooved surface 11,394 31,476 20,083 75

simulations adopt ∆t = 50 [fs]. In order to accurately capture the behaviors of ﬁelds around the grooved surface, we apply a local mesh reﬁnement near the tips, as shown in Fig. 4.7c.

4.4.4 Multipactor susceptibility to RF voltage amplitude

In order to examine the multipactor susceptibility to the RF voltage amplitude, we observed the electron population multiplication during the first five RF voltage periods (initial build-up) and for different RF voltage amplitudes. Reference [123] noted that the rate of change in the number of stray electrons over time to provide a measure of multipactor susceptibility. Here, we introduce a gain factor g as

nf ns 1/[2fRF(nf −ns)∆t] g = Np,eﬀ/Np,eﬀ (4.2)

n th where Np,eﬀ denotes the number of superparticle ﬂying between plates at the n time-step, and the exponent represents the inverse of the total number of primary hits during the considered time interval (nf − ns)∆t. As noted, we choose nf = 2.5

70 Figure 4.8: RF voltage amplitude susceptibility at fRFDpp = 4 [GHz·mm] for ﬂat and grooved copper surfaces.

ns and ns = 0 here. Multipactor occurs when g > 1 and becomes stronger for larger g, akin to the conventional δ.

Fig. 4.8 illustrates RF voltage amplitude versus g for ﬂat and grooved copper

surfaces with fRF · Dpp = 4 GHz·mm. On the ﬂat case, the simulations indicated that multipactor is triggered for RF voltage amplitudes VRF in the range from about 200

V to 1, 600 V. These simulation results are in good agreement with range estimates

from the Hatch-Williams model [122], represented by the red and blue shaded regions

in Fig. 4.8. The peaks observed in the low (at about 0.35 kV) and high (at about 0.85

kV) voltage regimes result from third-order and ﬁrst-order multipactor, respectively.

In contrast to these results, the susceptibility band in the grooved case becomes

wider and moves toward higher voltages as seen in Fig. 4.8. This can be explained

by the fact that the eﬀective gap size of the grooved waveguide is larger than the

71 Figure 4.9: RF voltage cycle versus population ampliﬁcation, An for both surfaces at a VRF = 1, 143.16 V.

ﬂat surfaced waveguide. For a ﬁxed frequency, eqn. (4.1) predicts that the voltage amplitude should increase with an increase on the gap size.

4.4.5 Multipactor saturation eﬀects

An exponential growth in the population of stray electrons can be observed during the initial build-up of multipactor. After many RF voltage cycles, the electron population saturates due to two main mechanisms: (1) acceleration-phase-mismatching and (2) the fact that secondary electrons are pulled back towards the surface by increasingly strong space-charge self ﬁelds. space-charge eﬀects by image theory [124].

Some symptoms by multipactor saturation are output power loss and harmonic generation. In order to capture the saturation phenomenon, based on the reference setting

a (i.e., with VRF = 1, 143.16 V), we ran EM-PIC simulations for 100 RF periods and

72 Table 4.4: Spectral amplitude of output voltage signals for high-order harmonics.

ﬂat surface grooved surface 2 3 fRF = 2 [GHz] 9.6620 × 10 1.0803 × 10 2nd harmonic 1.8682 × 10−1 - 3rd harmonic 4.7680 × 100 3.8502 × 100 4th harmonic 1.9876 × 10−1 - 5th harmonic 1.3681 × 10−1 4.1402 × 10−1 6th harmonic 1.4994 × 10−1 - 7th harmonic 7.2536 × 10−2 2.2400 × 10−1 8th harmonic 2.2514 × 10−1 - 9th harmonic 1.1574 × 10−1 5.6164 × 10−2 10th harmonic 1.3934 × 10−1 - 11th harmonic 1.3888 × 10−1 3.8213 × 10−2

for both types of surfaces. Fig. 4.9 shows the log-scale plot of the electron popula-

n n 0 tion ampliﬁcation factor A = Np /Np , where Np is total number of superparticles ﬂying between two metallic plates at time step n, vesus the RF voltage cycle. The

number density increases at an exponential rate up to an intermediate stage close to

about six RF cycles, beyond which saturation is reached due to strong space-charge

self ﬁelds. During the intermediate stage, the amplitude of RF ﬁelds prevail over the

space-charge self ﬁeld, and most secondaries successfully escape from the emission

surface.

Fig. 4.10a shows the instantaneous RF voltage at the output port over time. As

expected, the output voltage amplitudes in both cases are smaller than the input volt-

age amplitude (green-dashed line, 1, 143.16 V). Their spectra are shown in Fig. 4.10b where it can be clearly seen that, in addition to the original 2 GHz signal, many frequency harmonics are generated in both cases.

73 (a)

(b)

Figure 4.10: Output signals for both surfaces (a) in time-domain and (b) frequency domain.

74 Table 4.4 compares the spectral amplitudes of each harmonic for both cases. It

is seen that the ﬂat case includes all harmonics (even and odd) but the grooved

case exhibits only odd harmonics. According to [125], the output voltage signal

should include only odd orders to the fundamental frequency f0,Nmp = fRF/Nmp, which depends on the order of multipactor Nmp (here, fRF = 2 GHz). However, Fig.

4.10b shows that, in the ﬂat surface case, both odd- and even-order harmonics are present. Similar results have been observed in [126]. The presence of even harmonics in the ﬂat case might be due to the presence of stronger horizontal (lateral) currents due to drifting electrons with oblique SEE, which is not incorporated by the model considered in [125]. Further work is needed to test this hypothesis.

Figs. 4.11, 4.12, 4.13, and4.14 show snapshots of particle and ﬁelds evolutions and phase plots (vertical and horizontal components of velocity), respectively, taken over a half RF period at the saturation regime for both surfaces. It is observed in

Fig. 4.11 that many electrons inside the grooves experience multiple impacts. This eﬀectively lowers down the average number of the secondaries launched to the surface.

The breakdown of the focusing effect, which is one of symptoms from multipactor saturation [124, 126], can be seen in Fig. 4.13 for both surfaces. Fig. 4.14 shows the influence of the external lateral electric field (horizontal component) present in the grooved geometry on the horizontal electron speed distribution.

4.5 Conclusion

We have described the integration of a charge-conserving and energy-conserving

ﬁnite-element-based EM-PIC algorithm implemented on unstructured (irregular) meshes

75 (a) t = 24.50 [ns] (b) t = 24.55 [ns] (c) t = 24.60 [ns]

(d) t = 24.65 [ns] (e) t = 24.70 [ns] (f) t = 24.75 [ns]

(g) t = 24.50 [ns] (h) t = 24.55 [ns] (i) t = 24.60 [ns]

(j) t = 24.65 [ns] (k) t = 24.70 [ns] (l) t = 24.75 [ns]

Figure 4.11: Particle position snapshots taken over a half RF period during the saturation regime. The RF voltage period is 0.5 ns. Plots (a)-(f) are for the ﬂat surface and plots (g)-(l) are for the grooved surfaces.

with the Furman-Pivi probabilistic model describing SEE processes on metallic surfaces. The proposed SEE/EM-PIC algorithm enables mesh reﬁnement and is better suited to model to complex geometries. The algorithm was validated by comparing simulation results with available prior data.The algorithm was applied to evaluate and compare multipactor eﬀects on copper and stainless steel parallel plates. In addition, the algorithm was employed to compare multipactor on copper waveguides with

flat or corrugated (triangularly-grooved) walls. The multipactor saturation process was examined by quantifying the output power loss and harmonic generation arising from acceleration phase mismatching and self-field counterbalance effects.

76 (a) t = 24.50 [ns] (b) t = 24.55 [ns] (c) t = 24.60 [ns]

(d) t = 24.65 [ns] (e) t = 24.70 [ns] (f) t = 24.75 [ns]

(g) t = 24.50 [ns] (h) t = 24.55 [ns] (i) t = 24.60 [ns]

(j) t = 24.65 [ns] (k) t = 24.70 [ns] (l) t = 24.75 [ns]

Figure 4.12: External-field and self-field snapshots taken over a half period during the saturation regime. The RF voltage period is 0.5 ns. Plots (a)-(f) are for the flat surface and plots (g)-(l) are for the grooved surfaces.

(a) t = 24.50 (b) t = 24.55 (c) t = 24.60 (d) t = 24.65 (e) t = 24.70 (f) t = 24.75 [ns] [ns] [ns] [ns] [ns] [ns]

(g) t = 24.50 (h) t = 24.55 (i) t = 24.60 (j) t = 24.65 (k) t = 24.70 (l) t = 24.75 [ns] [ns] [ns] [ns] [ns] [ns]

Figure 4.13: Snapshots of vy [m/s] versus y [m] taken over a half RF period during the saturation regime. The RF voltage period is 0.5 ns. Plots (a)-(f) are for the ﬂat surface and plots (g)-(l) are for the grooved surfaces.

77 (a) t = 24.50 (b) t = 24.55 (c) t = 24.60 (d) t = 24.65 (e) t = 24.70 (f) t = 24.75 [ns] [ns] [ns] [ns] [ns] [ns]

(g) t = 24.50 (h) t = 24.55 (i) t = 24.60 (j) t = 24.65 (k) t = 24.70 (l) t = 24.75 [ns] [ns] [ns] [ns] [ns] [ns]

Figure 4.14: Snapshots of vx [m/s] versus y [m] taken over a half RF period during the saturation regime. The RF voltage period is 0.5 ns. Plots (a)-(f) are for the ﬂat surface and plots (g)-(l) are for the grooved surfaces.

78 Chapter 5: Numerical Cherenkov Radiation and Grid Dispersion Eﬀects

Electromagnetic particle-in-cell (EM-PIC) simulations have become an important tool for the study of a wide variety of problems associated with collisionless plasmas, including high power vacuum electronic devices [46, 47, 56], laser-wakefield acceleration [127], and astrophysical phenomena [128], to name just a few. Despite their success, there exist a number of outstanding challenges that limit the accuracy and robustness of EM-PIC simulations. Among them, the numerical Cherenkov radiation (NCR) instability, first observed by Godfrey [129], has been recognized as an important detrimental factor in EM-PIC simulations involving high-energy (relativistic) charged particles (including Lorentz-boosted frames) [130, 131] and collisionless shocks [132]. On regular periodic meshes such as used by the finite-difference time- domain (FDTD) method, NCR results from the coupling between numerical electromagnetic modes and plasma beam resonances. This coupling may result from the slow down of poorly-resolved waves due to numerical (grid) dispersion [133, 134] or from aliasing mechanisms [135,136].

The study of the causes and behavior of NCR is of critical importance for devel- oping mitigation strategies [15, 20, 135–138]. NCR has been extensively studied for

FDTD-based and spectral-based EM-PIC algorithms based on regular meshes [20,

79 129–132, 135, 136]. For problems involving complex geometries however, it is often advantageous to employ more general meshes which can better conform to curved and/or irregular boundaries as well as support adaptive mesh reﬁnement capabilities.

In this chapter, we analyze NCR effects in EM-PIC simulations based on more general meshes. The finite element time-domain (FETD)-based EM-PIC algorithm discussed in [1, 51, 52] is employed for this purpose. This explicit algorithm includes a charge-conserving scatter algorithm [1] and the Higuera-Cary particle-pusher to fully take into account relativistic effects and yield an overall energy-conserving algorithm [39, 52] . The reason for adopting this FETD-based EM-PIC algorithm in this study is twofold: (1) Contrary to most past FE-based EM-PIC algorithms implemented on general meshes 2, the present algorithm attains both charge and energy- conservation from first principles. (2) The standard FDTD algorithm can be retrieved as a special version of this mixed FETD scheme implemented on a regular mesh with square elements, in which low-order quadrature rules are employed in the evaluation of the mass (Hodge) matrices elements to yield diagonal matrices and a fully explicit field update [140]. This facilitates a direct comparison of NCR effects arising in

FETD-based EM-PIC simulations with those in FDTD-based EM-PIC simulations.

As noted, NCR is closely related to numerical dispersion. Most past numerical dispersion studies in FE-based Maxwell ﬁeld solvers have been carried out in the frequency domain (or equivalently, with no time discretization errors included) [22,

141,142] and only a few in the time domain [143]. Key conclusions from these studies are as follows: (1) a good quality mesh (i.e. one having near equilateral elements) is best for minimizing local phase errors per wavelength and (2) the cumulative phase

2Some notable exceptions are the compatible FE-based formulations described in Refs. [77,78,139]

80 error can be smaller on highly unstructured grids due to cancellation effects. The numerical dispersion analysis carried out in these works have focused on well-resolved waves, which is the practical regime of interest to provide sufficiently accurate results for pure EM simulations. However, in order to analyze NCR in EM-PIC simulations, a complete numerical dispersion map over the first Brillouin zone (the periodic layout of the mesh elements) must be determined because, among other reasons, electric currents, mapped from moving charged particles to the mesh during the scatter step of EM-PIC algorithm, are not explicitly bandlimited in contrast to the typical sources present in particle-free EM simulations. In this work, complete dispersion diagrams over the first Brillouin zone are obtained for meshes with different element shapes and layouts and analytical predictions for NCR are compared with numerical results from EM-PIC simulations.

5.1 Numerical Cherenkov Radiation in the FDTD-based EM- PIC Algorithm

Owing to its ﬂexibility and robustness, the FDTD algorithm is arguably the most popular ﬁeld solver for time-domain Maxwell’s equations [19]. As such, FDTD-based

(Yee) EM-PIC simulations are widely employed in plasma physics applications. The

FDTD uses central-diﬀerence approximations for both space and time derivatives applied on a structured regular mesh. The numerical dispersion for the FDTD algorithm in 2-D takes the form [19,144]

1 κ˜ h2 1 κ˜ h2 1 ω∆t2 sin x + sin y = sin , (5.1) h 2 h 2 c∆t 2 where c is the speed of light in vacuum, ∆t is the time step interval, h is the edge length of a square unit cell in the structured mesh, and κ˜ =κ ˜xxˆ +κ ˜yyˆ is the 2-D

81 (a) (b)

Figure 5.1: Numerical grid dispersion of the 2-D Yee’s FDTD scheme on a structured mesh. (a) The red color surface represents the dispersion diagram of the normalized frequency ω∆t/π versus the normalized numerical wavenumber κ˜h in radians. The olive color surface represents the light cone. The contour levels at the bottom represent the normalized phase errors (with respect to the color bar). (b) Wavenumber magnitude versus frequency for diﬀerent wave propagation angles with respect to the o o x axis, φp ∈ [0 , 45 ].

wavenumber for plane waves propagating on the mesh. We use the tilde to indicate numerical wavenumber κ˜ (as modiﬁed by numerical dispersion) as opposed to exact wavenumber κ. Throughout this work, the time convention ejωt is adopted.

The dispersion diagram (ω, κ˜) can be plotted by solving (5.1) [?]. Consider an √ example with h = 1 m and ∆t = h/ 2c ≈ 2.35 ns, which corresponds to the max-

imum time step allowed by the Courant-Friedrichs-Lewy (CFL) stability condition.

In Fig. 5.1a, the numerical grid dispersion diagram ωGD (κ˜) is displayed by a red

surface, and the light cone is shown in the olive green color. Contour levels at the

bottom of the ﬁgure represent the normalized phase diﬀerences (errors) between the

light cone and the numerical grid dispersion, [c|κ˜| − ωGD (κ˜)] ∆t/π, indicative of how

much faster or slower numerical waves propagate compared to the speed of light. The

82 normalized wave frequencies are plotted versus the normalized κ˜ for various prop-

o o 3 agation directions φp ∈ [0 , 45 ] along the grid in Fig. 5.1b. These results show that the numerical phase velocity has an anisotropic behavior on the FDTD mesh and is always slower than the speed of light in vacuum [145]. Suppose that a cold plasma beam is relativistically drifting along the x-axis with a bulk beam velocity of vb = 0.9c. Its space-charge mode (or entropy mode) can be characterized in the dispersion diagram by a plane with inclination given by the beam velocity (beam plane).

NCR is emitted from the coupling region where the numerical grid dispersion surface and the beam plane intersect in the first Brillouin zone. Furthermore, because of spatial and temporal aliasing effects, NCR can also be produced by resonances in other beam planes originated from higher-order Brillouin zones as well [135, 136], called aliased beams. The dispersion relation for space-charge modes, including spatial and temporal aliasing effects, on a structured mesh is given by

2π 2π ω − v = v κ˜ − u (5.2) ∆t b x h

where u and v are integers and the fundamental resonance mode has u = v = 0.

Since the spatial aliasing eﬀect directly depends on the shape factors used for current deposition (scatter step) onto the FDTD grid, NCR can be mitigated to some extent by increasing the spline order of the shape factors [146].

Fig. 5.2 illustrates the fundamental and aliased beam planes in the green and transparent yellow colors respectively, and the numerical dispersion in red within the

ﬁrst Brillouin zone, in which κ˜ ∈ [−π/h, π/h] × [−π/h, π/h] and ω ∈ [−π/∆t, π/∆t].

As seen in Fig. 5.2a, the NCR, caused by the fundamental beam resonance, is present

3 o o Here, φp = 0 corresponds to a direction along the grid axis and φp = 45 corresponds to a o o direction along the grid diagonals. Due to the FDTD symmetry, the behavior in the φp ∈ [0 , 45 ] repeats periodically along the other directions.

83 (a) (b)

Figure 5.2: Analytic NCR predictions on a structured FDTD grid for a bulk beam velocity vb = 0.9c. (a) 3-D numerical dispersion diagrams (in red) and beam planes (fundamental plane in green and aliased beams in transparent yellow). (b) Trajecto- ries of NCR solutions projected onto the 2-D κ˜-space.

over a relatively narrow spectrum of wavenumbers due to the slower phase velocity of poorly-resolved waves for wavenumbers close to the grid cut-oﬀ. On the other hand,

NCR produced by aliased beam resonances are spread out throughout the κ˜-space and may occur regardless of whether numerical dispersion is corrected or not. The loci for

NCR solutions can be found by using root-ﬁnd-solvers. These solutions are visualized more clearly in the κ˜-space as depicted in Fig. 5.2b with u, v ∈ {−3, −2, ..., 2, 3}.

Again, the fundamental beam resonance is shown in green and aliased ones are shown in yellow.

84 5.2 Numerical Cherenkov Radiation in ﬁnite-element-based EM-PIC Algorithms

To analyze NCR on more general meshes, an EM-PIC algorithm [1, 51, 52] based

on a mixed FETD ﬁeld solver [33, 76] is adopted here for the reasons listed in the

Introduction. The present FETD ﬁeld solver is based on an expansion of electro-

magnetic ﬁelds as a linear combination of discrete diﬀerential forms (Whitney forms)

deﬁned over mesh elements and an explicit (leap-frog) discretization in time. The

most essential aspects of the ﬁeld solver are outlined in Chapter 1.

We ﬁrst consider four periodic meshes, each with diﬀerent element shapes and lay-

outs as depicted in Fig. 5.3. They are denoted as square (SQ), right-angle triangular

(RAT), isosceles triangular (ISOT), and highly-irregular triangular (HIGT). This sec-

tion provides an analytical study of numerical dispersion and consequent NCR on SQ,

RAT, and ISOT meshes. These three meshes have periodic arrangements of elements and hence are amenable to such analysis. Analytical predictions of this Section are compared against EM-PIC simulation results in the next Section. NCR eﬀects in

FETD-based EM-PIC simulations on HIGT meshes are also presented in the next

Section. The FETD-based EM-PIC algorithm is implemented on all four meshes, whereas the FDTD-based EM-PIC algorithm is implemented exclusively on the SQ mesh.

In order to analyze the numerical dispersion in the FETD algorithm implemented on SQ, RAT, and ISOT meshes, we consider a numerical plane wave expressed as

j(ωn∆t−κ˜·r) E0e . In the FETD algorithm, the discrete degrees of freedom (DoF) of the electric ﬁeld are associated with the edges of the mesh. Suppose that tˆj is the tangential unit vector along the jth edge of the mesh. The plane wave solutions can

85 (a) (b)

Figure 5.3: Schematic illustration of the four types of mesh considered in this study. (a) Square regular (SQ) elements in both FDTD and FETD, (b) right-angle triangular (RAT) elements in FETD, (c) isosceles triangular (ISOT) elements in FETD, and (d) highly-irregular triangular (HIGT) elements in FETD.

86 j(ωn∆t−κ˜·r) be projected onto the edges as (E0 · tˆj)e and the factor E0 · tˆj can be taken as the DoF associated with the jthedge. In addition, by using the superscript n to

th n represent the n time step, we may denote the discrete DoFs as ej . For a plane wave propagating on a periodic mesh, it is possible to express the field value on an arbitrary edge using only the field values on a few number of so-called characteristic edges −−→ −−→ through multiplication of a spatial offset factor of the form e−jκ˜·AA0 , where AA0 is the relative position vector between the non-characteristic edge A0 and its corresponding characteristic edge A. Because of this, the number of DoFs can be restricted to the number of the characteristic edges and the size of the matrices involved can be greatly reduced, see also [22]. Note that the SQ mesh has only two characteristic edges whereas the RAT and ISOT meshes have three characteristic edges.

The full-discrete vector wave equation for the electric ﬁeld can be written as [33,79]

n+1 2 T n n−1 [?] · e = 2 [?] − ∆t C · [?µ−1 ] · C · e − [?] · e . (5.3)

For a discrete plane wave with harmonic evolution of the form ejωn∆t, it is clear that en±1 = ene±jω∆t so that (5.3) becomes

X · en = 2 [cos (ω∆t) − 1] M + ∆t2S · en = 0, (5.4)

T where M = [?] (mass matrix) and S = C · [?µ−1 ] · C (stiﬀness matrix). Non-trivial solutions can be obtained by solving det (X) = 0 which determines the numerical dispersion relation (ω, κ˜) on the mesh. In what follows, NCR analysis is presented for SQ, RAT, and ISOT meshes. The numerical dispersion analysis is similar to [22] except that time-discretization is also included.

87 Figure 5.4: Schematic of SQ mesh. There are two characteristic edges (A and B) directed along the y and x and colored in red and blue, respectively.

5.2.1 SQ Mesh

The SQ mesh has two characteristic edges, y- and x-directed. Let these two edges

be denoted as A and B, colored in red and blue, respectively, in Fig. 5.4. Local

matrices for the three facets spanned by the support of the edge elements A and B

are ﬁrst computed. Then, the global mass and stiﬀness matrices can be constructed as

a sum of three local matrices attributed to each facet. The dashed red (for y-directed

edges) and dashed blue (for x-directed edges) lines in Fig. 5.4 depict the relative position vectors between characteristic and non-characteristic edges. As detailed in

Chapter 1, in the mixed FETD algorithm, the electric ﬁeld is represented as a linear

88 combination of Whitney 1-forms associated 1:1 with mesh element edges and the

magnetic ﬂux density by a linear combination of Whitney 2-forms associated 1:1 with

mesh element faces (note that Whitney 1- and 2-forms are also known as edge and

face elements in the ﬁnite element literature). The vector proxies of Whitney 1- and

2-forms for the jth edge and kth face are written as [23]

(1) ! (1) ! α − α β − β W(1) (r) =α ˆΠ j Λ j , (5.5) j h/2 h ! ! zˆ x − x(2) y − y(2) W(2) (r) = Π k Π k , (5.6) k h2 h/2 h/2

where j and k are edge and face indices, α and β stand for Cartesian coordinates x and

th (p) (p) y or vice-versa depending on the direction of the j edge, and the point xq , yq

is the center of the qth p-cell (p = 1: edge, p = 2: face). In addition, Π (·) and Λ (·) are scalar pulse and roof-top functions deﬁned as

1, |ζ| ≤ 1 Π(ζ) = , (5.7) 0, |ζ| > 1 1 − |ζ| , |ζ| ≤ 1 Λ(ζ) = . (5.8) 0, |ζ| > 1

The global mass and stiﬀness matrices for DoFs on the characteristic edges, namely

n n T [eA, eB] , can then be written as

M = M1 + M2 + M3, (5.9)

S = S1 + S2 + S3, (5.10)

89 with

" −→ # 1/3 + e−jκ˜·AA1,r /6 0 M1 = 0∆ −−→ , 0 1/3 + e−jκ˜·BB1,u /6 −→ 1/3 + e−jκ˜·AA2,l ∆/6 0 M = ∆ , 2 0 0 0 0 0 M3 = 0∆ −−→ , (5.11) 0 1/3 + e−jκ˜·BB3,d /6 " −→ −−→ # 1 1 − e−jκ˜·AA1,r −1 + e−jκ˜·BB1,u S = −→ −−→ , 1 −jκ˜·AA −jκ˜·BB µ0∆ −1 + e 1,r 1 − e 1,u −→ −−→ −−→ 1 1 − e−jκ˜·AA2,l e−jκ˜·BB2,d − e−jκ˜·BB2,u S2 = , µ0∆ 0 0 1 0 0 S3 = −→ −→ −−→ , (5.12) −jκ˜·AA −jκ˜·AA3,r −jκ˜·BB µ0∆ e 3,l − e , 1 − e 3,d where ∆ is the area of the SQ mesh unit cell and the relative position vectors are given −→ −→ −→ −→ −−→ by AA1,r = (h, 0), AA2,l = (−h, 0), AA3,l = (0, −h), AA3,r = (h, −h), BB1,u = (0, h), −−→ −−→ −−→ BB2,d = (−h, 0), BB2,u = (−h, h), and BB3,d = (0, −h). As noted before, the

numerical grid dispersion can then be found by solving det (X) = 0 for diﬀerent ω and obtaining κ˜ on the ﬁrst Brillouin zone.

Fig. 5.5 illustrates the numerical grid dispersion of the FETD algorithm on the

SQ mesh with h = 1 [m] and ∆t = 1.35 [ns], which is the maximum time step for

the stable ﬁeld-update according the CFL limit4. Unlike the FDTD case, it is clear

from Fig. 5.5a that numerical plane waves propagate slightly faster than light in

this case, regardless of the propagation direction, over the entire ﬁrst Brillouin zone.

Fig. 5.5b projects the numerical dispersion on the normalized wavenumber/frequency

plane, with diﬀerent propagation directions illustrated by diﬀerent colors in the φp ∈

[0o, 45o] range. As noted before, the standard FDTD algorithm can be recognized as

4The maximum time step in the FETD scheme can be obtained through an eigenvalue analysis on M−1 · S [33].

90 (a) (b)

Figure 5.5: Numerical grid dispersion for the FETD scheme on the SQ mesh. (a) The red color surface represents the dispersion diagram of the normalized frequency ω∆t/π versus the normalized numerical wavenumber κ˜h in radians. The olive color surface represents the light cone. The contour levels at the bottom represent the normalized phase errors (with respect to the color bar). Note that the normalized phase error is always negative in this case because of a slightly faster-than-light numerical phase velocity. (b) Projected dispersion curves for diﬀerent wave propagation angles with o o respect to the x axis φp ∈ [0 , 45 ].

a special version of the FETD on the SQ mesh in which low-order quadrature rules are employed in evaluating the mass (Hodge) matrices elements to yield diagonal matrices [140]. Conversely, the FETD algorithm on the SQ mesh can be regarded as a modified FDTD scheme in which non-diagonal coupling, present in the mass matrices, mimics an extended finite-difference stencil approximating spatial derivatives. As discussed in [20], the extended stencil makes numerical waves propagate faster than the speed of light. The latter effect can also be understood from the fact that the extended spatial stencil results in a stronger coupling of various degrees of freedom on the mesh. Because of faster numerical wave speeds, the maximum time step for a stable update in the FETD algorithm on a SQ mesh is smaller than the FDTD limit on the same mesh by a factor of 0.58, which agrees with [20].

91 (a) (b)

Figure 5.6: Analytic prediction of NCR for the FETD algorithm on the SQ mesh when vb = 0.9c. (a) 3-D dispersion diagram. (b) NCR solution contours projected onto the ﬁrst Brillouin zone in the κ˜-space.

Because the wave phase velocity of the FETD algorithm on the SQ mesh is larger

than c, it is expected that NCR will not arise from the wave resonance coupling to the fundamental beam plane. However, NCR is still expected in the solution domain due to the presence of spatially and temporally aliased beams, as illustrated in Fig.

5.6 with u ∈ {−5, −4, ..., 4, 5} and v ∈ {−3, −2, ..., 2, 3}.

5.2.2 Triangular-element-based FE meshes

On triangular-element-based FE meshes, we can again solve det (X) = 0 to de-

termine the numerical dispersion relation (ω, κ˜) on the mesh. However, the mass M and stiffness S matrices appearing on the expression for X should be modified. To derive these matrices, we first recall the expression for the vector proxies of Whitney

92 Figure 5.7: A periodically-arranged triangular grid. It has three characteristic edges denoted by A, B, and C. Labels inside circles denote global facet indexes and labels inside rectangles and pentagons denote local edge and node indexes, respectively.

1- and 2-forms on triangular meshes [147–149]

W(1) (r) = λ ∇λ − λ ∇λ (5.13) j [1]j [2]j [2]j [1]j

W(2) (r) = 2∇λ × ∇λ (on 2-dimensional meshes). (5.14) k {1}k {2}k

th th th at each j edge and k facet on the mesh, where [j]i and {j}i denote the j local node index for ith edge and the jth local node index for ith facet, respectively. In

th th addition, in what follows (j)i denotes the j local edge index for i facet. For an arbitrary point (x, y) inside a kth facet, the relationship between local nodal and barycentric coordinates is given by

x x x  λ  x {1}k {2}k {3}k {1}k y y y · λ = y . (5.15)  {1}k {2}k {3}k   {2}k    1 1 1 λ 1 {3}k

93 Consider a periodically-arranged triangular grid, as shown in Fig. 5.7. The local

mass matrix for the kth facet can be obtained by [23,150]

M loc M loc M loc  (1,1)k (1,2)k (1,3)k loc loc loc loc M = M(2,1) M(2,2) M(2,3)  , (5.16) k  k k k  M loc M loc M loc (3,1)k (3,2)k (3,3)k where ∇λ · ∇λ  {1}k {1}k loc M = 2Ak αi,j βi,j γi,j · ∇λ{1} · ∇λ{2} , (5.17) (i,j)k  k k  ∇λ · ∇λ {2}k {2}k

1 1 1 −1 −1 1  1 −1 −1 1 1 1 α = 1 3 1 , β = −1 3 3 , γ = −1 1 1 . (5.18) 12   12   12   1 3 1 1 3 3 −1 1 3

th and Ak is the area of the k facet. Each element index in the expression of the

local mass matrix above also denotes the local edge index of the kth facet. The local

loc locT loc loc stiﬀness matrix Sk = Ck · [?µ−1 ]k · Ck is given by Sloc Sloc Sloc  (1,1)k (1,2)k (1,3)k loc loc loc loc S th = S(2,1) S(2,2) S(2,3)  , (5.19) k  k k k  Sloc Sloc Sloc (3,1)k (3,2)k (3,3)k

loc i+j where S = (−1) /Ak. We deﬁne a function F ({j} , k) that yields one of (i,j)k k th the characteristic edges of the k facet according to the local edge index {j}k, i.e.,

−1 F ({j}k , k) = gk(= A, B, or C). Likewise, the inverse function F = H yields the

th th j local edge index of the k facet, {j}k, i.e. H (F ({j}k , k)) = {j}k. Projecting plane wave solutions into DoFs on edges, the relationship between the

DoFs for similar edges in diﬀerent facets kth and Kth can be written as

−jκ˜·g−−−→g −jϕ k K [gk→gK ] egK = egk e = e e (5.20) F ({j}k,k)

−−→ where gkgK is the relative position vector from the center of the edge gk to gK and

T ϕ[gk→gK ] is the corresponding phase delay. For e = [eA, eB, eC ] , the global mass and

94 stiﬀness matrices can be assembled by

4 ( loc M , for Gp = gp X (H(Gp),H(gp))p MG1,g1 = loc −jϕ[g →g ] , (5.21) M e 1 p , for Gp 6= gp p=1 (H(Gp),H(gp))p 4 ( loc S , for Gp = gp X (H(Gp),H(gp))p SG1,g1 = loc −jϕ[g →g ] , (5.22) S e 1 p , for Gp 6= gp p=1 (H(Gp),H(gp))p where Gp and gp are outputs of the F function for local edges in p facets (see Fig. 5.7).

As noted before, the numerical grid dispersion of the FETD schemes on periodically- arranged triangular grids can then be obtained by substituting the above mass and stiﬀness matrices into (5.4) and solving the characteristic equation det (X) = 0. We will examine next triangular meshes composed of RAT and ISOT elements.

Right-angle triangular-element (RAT) mesh

The unit cell of the RAT mesh corresponds to setting a = 0 and b = 1 (see

Fig. 5.7). The numerical dispersion diagram of the FETD-RAT scheme over the

ﬁrst spatial Brillouin zone is illustrated in Fig. 5.8. The ﬁrst spatial Brillouin zone of the RAT mesh in the κ˜-space is shaped as an inclined hexagon. As depicted in

Fig. 5.8a, red and blue surfaces correspond to the lower and upper grid dispersion of the mesh. The normalized phase errors are shown by means of contour maps at the bottom and top planes along the (vertical) frequency axis. Fig. 5.8b shows

o o the numerical dispersion curves for diﬀerent propagation directions φp ∈ [−45 , 45 ].

Dirac points denote the points where the lower and upper dispersion bands meet5.

It is observed that the numerical dispersion in the lower band is highly anisotropic.

5This nomenclature is borrowed from solid state physics, where it is used to describe attachment of valence and conduction energy bands.

95 (a) (b)

Figure 5.8: Numerical grid dispersion for the FETD scheme on the RAT mesh with the CFL number equal to one. Unlike the FDTD or FETD-SQ cases, this diagram exhibits an additional (upper) dispersion band. (a) The red (lower band) and blue (upper band) color surfaces represent the dispersion diagram of the normalized frequency ω∆t/π versus the normalized numerical wavenumber κ˜h in radians. The olive color surface represents the light cone. The contour levels at the bottom represent the normalized phase errors (with respect to the color bar). (b) Projected dispersion o o curves for diﬀerent wave propagation angles with respect to the x axis φp ∈ [−45 , 45 ].

Although in the upper band wave propagation is faster than light due to its inverse- like shape compared to the lower band, NCR may still be produced by intersection with aliased beams. The existence of an upper dispersion band on meshes based on RAT elements can be understood from the fact that the normal component of the (vector proxy of) Whitney 1-forms used to expand the electric ﬁeld on the mesh exhibit discontinuities at the edges of triangular meshes. Fig. 5.9 illustrates the normal discontinuity of Whitney 1-forms. This is in contrast to meshes based on SQ elements where Whitney 1-forms exhibit both tangential and normal continuity (due to zero normal components). Strictly speaking, Whitney 1-forms on triangular grids are only tangentially continuous [147, 149]. Indeed, the numerical grid dispersion behavior on meshes with periodically-arranged triangular elements is reminiscent of

96 (a) (b) (c)

−→ Figure 5.9: (a) The vector proxy of a Whitney 1-form associated with the edge AB on a triangular mesh. (b) Tangential component along edge. (c) Normal component to the edge direction.

that of photonic band gap structures in which the discontinuity of the normal ﬁeld

component is caused by periodic material interfaces [151,152].

Fig. 5.10 shows the analytic prediction of NCR for the FETD-based EM-PIC

scheme on the RAT mesh assuming a plasma beam with bulk velocity vb = 0.9c. Fig.

5.10a depicts the numerical dispersion diagram over the ﬁrst Brillouin zone superim-

posed to the fundamental and aliased beams. Set of NCR solutions in the κ˜-space

are shown in Fig. 5.10b with u ∈ {−5, −4, ..., 4, 5} and v ∈ {−3, −2, ..., 2, 3}.

Isosceles triangular-element (ISOT) mesh

The unit cell for the ISOT mesh corresponds to a = 0.5 and b = 1. Similar to the

RAT case, the numerical dispersion diagram exhibits both lower and upper dispersion

bands as shown in Fig. 5.11. Fig. 5.12 shows the NCR prediction for a beam bulk

97 (a) (b)

Figure 5.10: Analytic prediction of NCR for the FETD-based EM-PIC scheme on the RAT mesh assuming a plasma beam with bulk velocity vb = 0.9c. (a) Dispersion diagram. (b) NCR solution contours projected onto the ﬁrst Brillouin zone in the κ˜-space.

(a) (b)

Figure 5.11: Numerical grid dispersion for the FETD scheme on the ISOT mesh with CFL number equal to one. Unlike the FDTD or FETD-SQ cases, this diagram exhibits an additional (upper) dispersion band. (a) The red (lower band) and blue (upper band) color surfaces represent the dispersion diagram of the normalized frequency ω∆t/π versus the normalized numerical wavenumber κ˜h in radians. The olive color surface represents the light cone. The contour levels at the bottom and top represent the normalized phase errors (with respect to the color bar). (b) Projected dispersion curves for diﬀerent wave propagation angles with respect to the x axis o o φp ∈ [26.57 , 90 ].

98 (a) (b)

Figure 5.12: Analytic prediction of NCR for the FETD-based EM-PIC scheme on the ISOT mesh assuming a plasma beam with bulk velocity vb = 0.9c. (a) Dispersion diagram. (b) NCR solution contours projected onto the ﬁrst Brillouin zone in the κ˜-space.

velocity vb = 0.9c. There is no NCR caused by the fundamental beam resonances in this case; however, NCR is still excited by waves coupling to aliased beam resonances as illustrated in Fig. 5.12b, where u ∈ {−5, −4, ..., 4, 5} and v ∈ {−3, −2, ..., 2, 3}.

Highly-irregular triangular-element (HIGT) mesh

In contrast to the SQ, RAT, and ISOT meshes considered above, the HIGT mesh does not have a periodic layout of elements. The aperiodic layout of the elements in the HIGT mesh precludes a similar type of analytical study of NCR as made before for the other meshes. As a result, NCR eﬀects in HIGT are investigated exclusively by means of numerical simulations in the next Section. Note that the phase errors due to numerical dispersion on the meshes with periodic triangular elements, as seen in Figs. 5.8 and 5.11, can be positive or negative (i.e. with dispersion curves above or below the light line) depending on angle of propagation relative

99 to the (local) element orientation. Therefore, it is expected that for an unstructured

(aperiodic) mesh composed of triangular elements of diﬀerent shapes and orientations, the cummulative phase error may be reduced due to some cancellation eﬀects. This result has been numerically observed before in [22].

5.3 Numerical Experiments

In this Section, EM-PIC simulations are conducted to verify the analytic predictions made in the previous Section. In addition, FETD-based EM-PIC simulations on the HIGT mesh (for which no analytical prediction is available) are also included for comparison. Two basic scenarios are considered here: a relativistically-drifting

(electron-positron) pair-plasma and a single electron-positron pair moving in the relativistic regime.

First, consider a relativistic pair-plasma drifting along the x-axis with the velocity vb = 0.9c (equivalent to a Lorentz factor of γb ≈ 2.3). The electron plasma

5 8 −2 frequency is set to ωpe ≈ 4 × 10 rad/s and the electron density to ne = 1 × 10 m

(same for positrons). Superparticles representing 2.5 × 106 charged particles are used for each species (electrons and positrons). The average number of superparticle per cell is set to 40 for the SQ mesh and 20 for the RAT, ISOT, HIGT meshes 6. In all cases the problem domain Ω = (x, y) ∈ [0, 128]2 m2 is terminated by periodic boundary conditions for both ﬁelds and particles. Initially, all superparticles are uniformly distributed over the entire simulation domain and each pair of superparticles is placed at the same position to produce zero net initial ﬁelds. The initial velocity distribution in the beam rest frame vbf for both electrons and positrons is Maxwellian

6Note that for similar edge sizes, SQ mesh elements are about twice the size of the triangular mesh elements

100 (a) (b) (c) (d)

Figure 5.13: Initial velocity distributions for a relativistic pair plasma beam with bulk velocity vb = 0.9c (γb ≈ 2.3). (a) Phase space in the beam rest frame. (b) Velocity distribution in the beam rest frame. (c) Phase space in the laboratory frame. (d) Velocity distribution in the laboratory frame.

with thermal velocity vth = 0.005c (see Fig. 5.13a and 5.13b). The resulting Debye

bf p 2 2 length equals to λD = 0vthme/ (neqe ) ≈ 2.657 m. The initial velocity distribution in the laboratory frame vlf is shown in Fig. 5.13c and Fig. 5.13d. The relation-

lf bf ship between vx and vx can be obtained by applying the Lorentz velocity transfor- q lf bf bf 2 lf bf 2 bf 2 mation vx = vx + vb / 1 + vbvx /c and vy = vy 1 − (vb/c) / 1 + vbvx /c . Due to length contraction, the Debye length in the laboratory frame reduces to

lf bf λD = λD/γb ≈ 1.158 m. EM-PIC simulations are performed based on the following setups: (1-a) FDTD-

based solver on the SQ mesh, (1-b) FETD-based solver on the SQ mesh, (2) FETD-

based solver on the RAT mesh, (3) FETD-based solver on the ISOT mesh, and (4)

FETD-based solver on the HIGT mesh. The HIGT mesh is shown in Fig. 5.14. Note

again that the SQ, RAT, and ISOT meshes have periodic layouts of elements, whereas

HIGT has an aperiodic layout. To ensure a good mesh quality in the latter case, the

angles of triangular elements are enforced to be no less by 30o. The average angle

is near 60o. The angle distribution (histogram) is shown in Fig. 5.14. All meshes

101 (a) (b) (c)

Figure 5.14: (a) HIGT mesh. (b) Histogram of the edge lengths. (c) Histogram of the triangular element angles.

lf are designed so that the average edge length lavg is comparable to λD to mitigate self-heating eﬀects. Table 5.1 lists the basic properties of the four types of mesh.

Table 5.1: Basic meshes properties.

parameters simulation type: (1-a) (1-b) (2) (3) (4) h [m] 1.00 1.00 1.00 1.00 - lavg [m] 1.00 1.00 1.14 1.08 1.24 N0 (# nodes) 16,641 16,641 16,641 16,641 13,239 N1 (# edges) 33,024 33,024 49,408 49,408 39,202 N2 (# faces) 16,384 16,384 32,768 32,768 25,964 ∆tmax [ns] 2.35 1.35 1.11 1.03 1.02

In the deep relativistic regime (i.e. very large γb) for 2-D FDTD-based EM-PIC

simulations, the optimal time step ∆tmag for the lowest rate of NCR production √ has been determined [135, 136] to be ∆topt ≈ 0.9192∆tmax,2D = 0.9192h/ 2c where

∆tmax,2D is the maximum time step for stability, as dictated by the CFL condition.

On the other hand, the NCR growth rate in the mildly relativistic regime has been

102 observed to monotonically decrease as ∆t increases, that is ∆topt → ∆tmax,2D. In

either case, ∆topt does not diﬀer substantially from ∆tmax,2D. The present FDTD-

based EM-PIC simulations adopt ∆tmax,2D as a reference for comparison.

To obtain NCR dispersion maps in the κ˜-space representation, the z component of

the B ﬁeld is measured across the mesh at the end of the simulation (t = 47 µs). A 2-

D fast Fourier transform (FFT) is then performed on the sampled data to obtain B in the κ˜-space representation. Fig. 5.15 shows contour plots of the amplitude of B in log scale over the ﬁrst Brillouin zone in the κ˜-space from FDTD-based and FETD-based

EM-PIC simulations on the SQ meshe (cases (1-a) and (1-b)). Fig. 5.16 illustrates the same for FETD-based EM-PIC simulations on the RAT and ISOT meshes (cases

(2) and (3)). Analytic prediction curves are superimposed on the simulation results in

Figs. 5.15b, 5.15d, 5.16b and 5.16d. The black and gray colors denote fundamental and aliased beams, respectively. A very good agreement is observed between the numerical results and the analytic predictions in all cases.

It is observed that on periodic triangular grids, NCR can be purely transverse and longitudinal unlike the SQ case (FDTD or FETD) as discussed in [136]. On the other hand, propagation along certain directions (‘characteristics edges’) is prohibited. It is evident also that the NCR distribution in the κ˜-space is strongly dependent on the mesh element shapes. The existence of the upper grid dispersion bands is also conﬁrmed from their contributions to the NCR solutions in the κ˜-space. Compared to the FDTD-based simulation and to the FETD-based simulations on the RAT and

ISOT meshes, the NCR observed in FETD-based EM-PIC simulation on the SQ mesh exhibits weaker amplitudes. Also of note is that even though there is no NCR

103 (a) (b)

Figure 5.15: B ﬁeld amplitude distribution (log scale) over the ﬁrst Brillouin zone in the κ˜-space as measured from EM-PIC simulation snapshots at 47 µs. (a) and (c) plots correspond to FDTD- and FETD-based EM-PIC simulations on the SQ mesh, respectively. In (b) and (d), the analytical predictions are superimposed to the numerical results.

104 (a) (b)

Figure 5.16: B ﬁeld amplitude distribution (log scale) over the ﬁrst Brillouin zone in the κ˜-space as measured from EM-PIC simulation snapshots at 47 µs. (a) and (c) plots correspond to FETD-based EM-PIC simulations on the RAT and ISOT meshes, respectively. In (b) and (d), the analytical predictions are superimposed to the numerical results.

105 produced by the fundamental beam in FETD-ISOT case (3), the NCR caused by

from aliasing beams shows up stronger than those in FETD-SQ case (1-b).

The NCR amplitude distribution in the κ˜-space for the FETD-based EM-PIC simulation on the HIGT mesh is shown in Fig. 5.17. Unlike the previous cases, the aperiodic layout of mesh elements of the HIGT mesh precludes spatial coherency.

Instead, a diﬀusive-like (spatially incoherent) pattern in the κ˜-space is present instead.

Fig. 5.18 shows a quantitative comparison of the B ﬁeld amplitude distribution in the κ˜-space between the FDTD and FETD-HIGT cases. Fig. 5.18a depicts the amplitude of B versusκ ˜yh at some ﬁxed values ofκ ˜xh and vice-versa in Fig. 5.18b.

This corresponds to vertical and horizontal cuts, respectively, on Figs. 5.15a and

5.17. It can be seen that the peak spectral amplitude in the FDTD case is about two orders of magnitude larger than that in the FETD-HIGT case. The peaks in the

FDTD result correspond to spatially coherent NCR modes. In the FETD-HIGT case, on the other hand, NCR is more evenly spread in the κ˜-space.

As also observed in [146,153], these results above confirm that the observed magnetic field originates from NCR. Similar to the analysis done in [146, 153] we next compare the growth rate of the NCR-induced magnetic field by evaluating the total magnetic field energy on the mesh given by [34]:

1 1 1 n+ 2 1 n+ n+ W = b 2 · [? −1 ] · b 2 . (5.23) m 2 µ

The above expression is computed as a function of time for the various types of mesh

considered above. These results are shown in Fig. 5.19. Among all cases, the FETD-

based EM-PIC simulation on the SQ mesh exhibits the smallest growth rate at earlier

times. More importantly, the magnetic energy produced by NCR in the SQ, RAT,

and ISOT meshes (periodic layouts) reach saturation levels which are at least one

106 Figure 5.17: B ﬁeld amplitude distribution (log scale) over the ﬁrst Brillouin zone in the κ˜-space as measured from FETD-based EM-PIC simulation snapshots at 47 µs on the HIGT mesh.

(a) (b)

Figure 5.18: The qualitative comparison of the B ﬁeld amplitude distribution (log scale) on the κ˜-space between FDTD and FETD-HIGT cases. (a) shows the spectral amplitude of B versusκ ˜yh at some ﬁxed values ofκ ˜yh and vice-versa in (b).

107 Figure 5.19: Evolution of the magnetic energy Wm due to NCR on various meshes.

order of magnitude above that in the HIGT mesh (aperiodic layout). This could be attributed to the fact that, as noted above, the latter type of mesh does not support spatially coherent NCR modes. These features could be explored to devise possible strategies for NCR mitigation such as for example, use of hybrid meshes composed of SQ and HIGT elements in diﬀerent subdomains.

In order to further illustrate the distinct NCR behavior across various meshes, we consider the simulation of a single electron-positron pair moving at relativistic velocity. Although strictly speaking an EM-PIC simulation of a single particle pair may not describe very precisely the underlying physics due to the ﬁnite mesh resolution, it is nevertheless useful for unveiling NCR patterns. We assume an electron (e) and

5 a positron (p) are launched with ve,p = vbxˆ ± 1.7 × 10 yˆ m/s, respectively, where vb = 0.9c m/s. We observe the resulting magnetic ﬁeld on the very same meshes as considered before. Fig. 5.20 shows snapshots of magnetic ﬁeld on each mesh at three

108 time instants, as indicated. It can be seen that in the case of meshes with periodic layouts, NCR have preferential directions of propagation according to the intersection points in the ﬁrst Brillouin zone. In contrast, the NCR pattern on the HIGT mesh has a diﬀusive-like shape originating from the particle trail.

5.4 Conclusion

We analyzed numerical Cherenkov radiation (NCR) effects arising in finite-element- based EM-PIC algorithms on different types of mesh. Complete dispersion diagrams over the first Brillouin zone were derived for periodic meshes with different element shapes and layouts. Analytical NCR predictions were compared against numerical results from EM-PIC simulations. Considering a relativistic plasma beam simulation, it was observed that the mesh element shape and mesh layout have a marked influence on the ensuing NCR properties. In particular, it was also observed that

EM-PIC simulations on an unstructured mesh (with irregular triangular elements) does not support spatially coherent NCR modes due to the aperiodic nature of the mesh layout. In this case, a diﬀusive-like behavior is observed for the NCR in the spatial domain. Importantly, it was observed that the spurious energy produced by

NCR on the unstructured mesh reaches saturation levels that are considerably lower than those on meshes based on periodic layout of (rectangular or triangular) elements.

For simplicity, the analysis was carried out here in 2-D but is is expected that similar conclusions apply to 3-D as well.

109 (a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

(m) (n) (o)

Figure 5.20: Snaphots of the magnetic ﬁeld distribution resulting from EM-PIC simulations of a single electron-positron pair moving relativistically. The snapshots are taken at 75.2 ns, 112.8 ns, and 150.4 ns, as indicated. The results correspond to: (a-c) FDTD-based EM-PIC simulation on SQ mesh , (d-f) FETD-based EM-PIC simulation on SQ mesh, (g-i) FETD-based EM-PIC simulation on the RAT mesh, (j-l) FETD-based EM-PIC simulation on ISOT mesh, (m-o) FETD-based EM-PIC simulation on HIGT mesh.

110 Chapter 6: Finite-Element Time-Domain Body-of-Revolution Maxwell-Solver

The solution of Maxwell’s equations in circularly symmetric or body-of-revolution

(BOR) geometries is important for a plethora of applications involving analysis and design of microwave devices (e.g. cavity resonators, coaxial cables, waveguides, antennas, high-power ampliﬁers, etc.) [23,56,154–160], electromagnetic scattering [161–

164], metamaterials [165], and exploration geophysics [3, 166–171], to name a few.

Azimuthal field variations in BOR problems can be described by Fourier modal decomposition, with the modal field solutions reduced to a two-dimensional (2-D) problem in the meridian ρz-plane. Frequency-domain finite element (FE) Maxwell solvers

for BOR problems have been developed in the past by discretizing the second-order

vector wave equation using edge elements for either the electric or the magnetic ﬁeld

[158,159,163,165,172] which avoids some of the pitfalls encountered when using scalar

elements [161].

It is highly desirable to develop BOR FE solvers in the time domain as well. Time-

domain FE solvers are better suited for simulating broadband problems, for capturing

transient processes such as those involved in beam-wave interactions [1, 27, 51], and

for handling non-linear problems. However, the use of the second-order vector wave

111 equation as a starting point for a time-domain FE formulation, as done in frequency- domain Maxwell FE solvers, is inadequate. As mentioned in Chapter 2, this is because the vector wave equation admits solutions of the form t∇φ, which are not original solutions of Maxwell’s equations and, even if not excited by (properly set) initial conditions, may emerge in the course of the simulation due to round-oﬀ errors and pollute the results for long integration times [173]. To avoid this problem, a mixed

(basis) FE solver based directly on the ﬁrst-order should be adopted in the time domain [25,28,74,75].

In this chapter, we present a mixed FE BOR solver for time-domain Maxwell’s curl equations based on transformation optics (TO) [26, 41–45] and discretization principles based on the discrete exterior calculus (DEC) of diﬀerential forms [27–

36]. We explore TO principles to map the original three-dimensional (3-D) BOR problem to an equivalent problem on the 2-D meridian plane where the resulting metric is not the cylindrical one but instead the Cartesian one (i.e., with no radial factors present). The cylindrical metric becomes fully embedded into the constitutive properties of an effective (artificial) inhomogeneous anisotropic medium that fills the entire domain. In this way, a Cartesian 2-D FE code can be retrofitted to this problem with no modifications necessary except to accommodate the presence of anisotropic media. Similar ideas have been explored in the past but restricted to the frequency- domain finite-difference (FD) context and to structured grids only [174]. In the FE context considered here, DEC principles are used to discretize Maxwell’s equations on unstructured (irregular) grids using discrete differential (Whitney) forms [24, 26,

29, 32, 76]. Unstructured grids permits a more ﬂexible representation of irregular geometries and reduce the need for geometrical defeaturing. In addition to the above

112 Figure 6.1: Depiction of an axisymmetric structure.

advantages, the proposed formalism facilitates treatment of the coordinate singularity on the axis of symmetry (z axis) because it does not require any modiﬁcation of the basis functions for ρ = 0 (otherwise necessary in prior BOR FE solvers [158,163,172]).

As detailed in the Appendix, the DEC formalism also facilitates implementation of perfectly matched layers (PML) to truncate the outer boundaries. We validate the algorithm against analytical solutions for resonant ﬁelds in cylindrical cavities and against pseudo-analytical solutions for the radiated ﬁelds by cylindrically symmetric antennas in layered media.

6.1 Formulation

6.1.1 Exploration of transformation optics (TO) concepts

Consider a BOR object with symmetry axis along z, such as the waveguide structure depicted in Fig. 6.1. It is well known that the vector operators (gradient, curl, and divergence) in cylindrical coordinates have additional metric scaling factors not present in Cartesian coordinates. However, by exploiting TO concepts [30,41,42], we

113 can map the cylindrical-system Maxwell’s curl equations to a Cartesian-like equations where the metric factors are embedded into artificial constitutive tensors. For convenience we denote these calculations under the generic banner of TO but some of these ideas actually predate TO per se. They can be traced to earlier applications involving Maxwell’s equations in BOR geometries and to Weitzenbock identities involving differential forms of different degrees [175] in cylindrical (polar) coordinates.

Starting from Maxwell’s equations in cylindrical coordinates, and considering ar- tiﬁcial anisotropic permittivity and permeability tensors ¯0 and µ¯0 of the form

ρ 0 0 0 ¯ −1 ¯ = ¯· R = ¯· 0 ρ 0 , (6.1) 0 0 ρ ρ−1 0 0  0 ¯ µ¯ = µ¯ · Rµ = µ¯ ·  0 ρ 0  , (6.2) 0 0 ρ−1 where the constitutive parameters of the original medium are given by     ρ 0 0 µρ 0 0 ¯ =  0 φ 0  , µ¯ =  0 µφ 0  . 0 0 z 0 0 µz and using the following rescaling for the ﬁelds

1 0 0 0 ¯ E = RE · E = 0 ρ 0 · E, (6.3) 0 0 1 ρ 0 0 0 ¯ D = RD · D = 0 1 0 · D, (6.4) 0 0 ρ ρ 0 0 0 ¯ B = RB · B = 0 1 0 · B, (6.5) 0 0 ρ 1 0 0 0 ¯ H = RH · H = 0 ρ 0 · H, (6.6) 0 0 1

114 we can rewrite the resulting Maxwell’s curl equations as

∂B0 ∇0 × E0 = − , (6.7) ∂t ∂D0 ∇0 × H0 = , (6.8) ∂t D0 = ¯0 · E0, (6.9)

B0 = µ¯0 · H0, (6.10)

with

ρˆ φˆ zˆ 0 0 ∂ ∂ ∂ ∇ × A = ∂ρ ∂φ ∂z . (6.11) 0 0 0 Aρ Aφ Az The modiﬁed curl operator in the equivalent (primed) system seen in (6.11) is devoid

of any radial scaling and thus locally isomorphic to the Cartesian curl operator.

6.1.2 Field decomposition

We decompose the ﬁelds into two sets: TEφ- and TMφ-polarized ﬁelds, corre-

0 0 0 0 0 0 sponding to {E ρ,E z,B φ} and {E φ,B ρ,B z}, respectively. In what follows, we use superscripts k or ⊥ to denote ﬁelds transverse or normal to the 2-D meridian plane.

The TEφ ﬁeld components can be expressed as E0k and B0⊥ and the TMφ as E0⊥ and

B0k. In the DEC context, the electric field intensity, the magnetic flux density, the electric flux density, and the magnetic field intensity are likewise represented as 1-, 2-,

2-, and 1-forms7 on the 3-D Euclidean space, respectively [30]. For present analysis based on the meridian plane (a 2-D manifold), E k is transverse to the plane and still is represented as a 1-form. On the other hand, E ⊥ should be represented as a 0-form

since it is a point-based quantity on this manifold. Likewise, although B⊥ is a 2-form

in 3-D, Bk is represented as a 1-form on the 2-D meridian plane.

71- and 2-forms correspond to physical quantities naturally associated to line and surface integrals, respectively.

115 (a) (b)

Figure 6.2: (2+1) setup for ﬁelds on (a) primal and (b) dual meshes at the meridian plane. The vertical axis is ρ and the horizontal axis is z.

6.1.3 Mixed FE time-domain BOR solver

We factor the transverse (i.e. ρ and z) and normal (i.e. φ) variations of the polarization-decomposed Maxwell ﬁelds on the 2-D meridian plane as

Mφ Mφ 0 X 0k X 0⊥ E (ρ, φ, z, t) = E m (ρ, z, t)Φm (φ) + E m (ρ, z, t)Ψm (φ) , (6.12)

m=−Mφ m=−Mφ

Mφ Mφ 0 X 0⊥ X 0k B (ρ, φ, z, t) = B m (ρ, z, t)Φm (φ) + B m (ρ, z, t)Ψm (φ) , (6.13)

m=−Mφ m=−Mφ where Mφ is the maximum order of the Fourier harmonics considered and  cos (mφ) , for m < 0 Φm (φ) = 1, for m = 0 , (6.14) sin (mφ) , for m > 0  sin (mφ) , for m < 0 Ψm (φ) = 1, for m = 0 . (6.15) cos (mφ) , for m > 0

116 Substituting (6.12) and (6.13) into (6.7), by using the orthogonality between modes, i.e.

Z 2π Φm (φ)Φn (φ) dφ = Cmδmn, (6.16) 0 Z 2π Ψm (φ)Ψn (φ) dφ = Cmδmn, (6.17) 0 where Cm = π for m 6= 0 and C0 = 2π, we obtain the modal Faraday’s law as

∂B0⊥ (ρ, z, t) ∇0k × E0k (ρ, z, t) = − m , (6.18) m ∂t ∂B0k (ρ, z, t) ∇0k × E0⊥ (ρ, z, t) = − m + |m| E0k (ρ, z, t) × φ,ˆ (6.19) m ∂t m

0k for m = −Mφ, ..., Mφ, where ∇ =ρ∂/∂ρ ˆ +z∂/∂z ˆ .

We discretize (6.18) and (6.19) on the meridian plane using an unstructured mesh based on simplicial (triangular) cells and by expanding the fields in a mixed basis as scalar or vector proxies of discrete differential forms (Whitney forms) [28, 30, 35]. In particular, the TEφ field is expanded as

N1 0k X k (1) E m (ρ, z, t) = Ej,m (t) Wj (ρ, z) , (6.20) j=1

N2 0⊥ X ⊥ (2) B m (ρ, z, t) = Bk,m (t) Wk (ρ, z) , (6.21) k=1 (p) (p) where Wq is the vector proxy of a Whitney p-form wq [1] associated with the q-th p-cell (p = 0, 1, 2 for nodes, edges, and facets, respectively) on the grid, and Np is the total number of p-cells on the grid. The expressions for the Whitney forms and their proxies are provided in Appendix C. Likewise, the TMφ ﬁeld is represented as

N0 0⊥ X ⊥ ˆ (0) E m (ρ, z, t) = Ei,m (t) φ Wi (ρ, z) , (6.22) i=1

N1 0k X k (RWG) B m (ρ, z, t) = Bj,m (t) Wj (ρ, z) . (6.23) j=1

117 (1) ˆ (RWG) In what follows, we denote Wj × φ = Wj , since this expression recovers the so-called Rao-Wilton-Glisson (RWG) functions [176,177]8. Note that we use dummy

index subscripts i, j, and k to indicate the i-th node, j-th edge, and k-th face, respectively. The various basis functions above are depicted in Fig. 6.3, see also [147,

178].

By substituting (6.20) and (6.21) into (6.18), and (6.22) and (6.23) into (6.19), we obtain the following equations

N1 N2 X k k (1) ∂ X (2) (t) ∇0 × W = − ⊥ (t) W (6.24) Ej,m j ∂t Bk,m k j=1 k=1

N0 N1 N1 X k (0) ∂ X k (1) X k (1) ⊥ (t) ∇0 W = − (t) W + |m| (t) W , (6.25) Ei,m i ∂t Bj,m j Ej,m j i=1 j=1 j=1 0k ˆ (0) for m = −Mφ, ..., Mφ and where we have used the fact that ∇ × φWi = 0k (0) ˆ ∇ Wi × φ. The equations above can be recast using the exterior calculus of diﬀerential forms as

N1 N2 X k k (1) ∂ X (2) (t) d0 w = − ⊥ (t) w , (6.26) Ej,m j ∂t Bk,m k j=1 k=1

N0 N1 N1 X k (0) ∂ X k (1) X k (1) ⊥ (t) d0 w = − (t) w + |m| (t) w , (6.27) Ei,m i ∂t Bj,m j Ej,m j i=1 j=1 j=1

where d0k = dρ ∂/∂ρ + dz ∂/∂z is the exterior derivative on the meridian plane.

Applying DEC principles, (6.26) can be paired to the 2-cells of the mesh and (6.27)

to the 1-cells of the mesh (see Appendix C) so that, by invoking the generalized Stokes’

theorem [28, 30, 32, 35, 36] (see Appendix C), the exterior derivative can be replaced

by incidence operators on the mesh (see also Appendix C). Next, by discretizing

the time derivatives using central-diﬀerences in a staggered manner (leap-frog time

8 (RWG) (1) In other words, Wj is the Hodge dual of Wj in 2-D [32,35, 85].

118 (a) (b)

(1) Figure 6.3: Vector proxies of various degrees of Whitney forms on the mesh: (a) Wj , (2) (0) (RWG) (b) Wk , (c) Wi , and (d) Wj . Note that tj is a unit vector tangential to j−th edge and parallel to its direction and nk is a unit vector normal to k−th face.

119 discretization) we obtain the following update equations for Faraday’s law

n+ 1 n− 1 n ⊥ 2 ⊥ 2 k Bm = Bm − ∆t [Dcurl] · Em , (6.28)

n+ 1 n− 1 n k 2 k 2 ⊥ n k Bm = Bm − ∆t [Dgrad] · Em − |m| Em , (6.29) where ∆t is a time step increment and the superscript n indicates the time-step index. [Dcurl] and [Dgrad] are N2 × N1 and N1 × N0 incidence matrices, respectively, that encode the curl and the gradient operators on the FE mesh with elements in the set {−1, 0, 1} (see C.5). The ﬁeld unknowns are represented by the column vectors h i h iT h i h iT ⊥ = ⊥ , ..., ⊥ T , k = k , ..., k , k = k , ..., k , Bm Bm,1 Bm,N2 Em Em,1 Em,N1 Bm Bm,1 Bm,N1

⊥ ⊥ ⊥ T and Em = Em,1, ..., Em,N0 . We proceed along similar lines for Ampere’s law by expressing the D0 and H0 ﬁelds as

Mφ Mφ 0 X 0k X 0⊥ D (ρ, φ, z, t) = D m (ρ, z, t)Φm (φ) + D m (ρ, z, t)Ψm (φ) , (6.30) m=0 m=0

Mφ Mφ 0 X 0⊥ X 0k H (ρ, φ, z, t) = H m (ρ, z, t)Φm (φ) + H m (ρ, z, t)Ψm (φ) . (6.31) m=0 m=0

After substituting (6.30) and (6.31) to (6.8), applying trigonometric orthogonality to the resulting equations, and matching the ﬁeld components, we arrive at

∂D0⊥ (ρ, z, t) ∇0k × H0k (ρ, z, t) = m , (6.32) m ∂t ∂D0k (ρ, z, t) ∇0k × H0⊥ (ρ, z, t) = m − |m| H0k (ρ, z, t) × φ.ˆ (6.33) m ∂t m

As before, we discretize (6.32) and (6.33) on the 2-D meridian plane, the important diﬀerence being that the discretization for D0 and H0 is on the dual mesh [26,30,35,85],

120 as opposed to the FE (primal) mesh as done for E0 and B0. In this way, we obtain

N˜1 0k X k ˜ (RWG) D m (ρ, z, t) = Dj,m (t) Wj (ρ, z) , (6.34) j=1

N˜0 0⊥ X ⊥ ˆ ˜ (0) H m (ρ, z, t) = Hi,m (t) φWi (ρ, z) , (6.35) i=1

N˜2 0⊥ X ⊥ ˜ (2) D m (ρ, z, t) = Dk,m (t) Wk (ρ, z) , (6.36) k=1

N˜1 0k X k ˜ (1) H m (ρ, z, t) = Hj,m (t) Wj (ρ, z) . (6.37) j=1 where we use the tilde˜to denote quantities associated with the dual mesh. Similar to the discrete counterparts of Faraday’s law, by substituting (6.34) and (6.35) into (6.32) and (6.36) and (6.37) into (6.33) and by applying DEC principles and a leap-frog time discretization to the resulting equations, we obtain the discrete representations of Ampere’s law as

n+1 n h i n+ 1 ⊥ ⊥ ˜ k 2 Dm = Dm + ∆t Dcurl · Hm , (6.38)

n+1 n h i n+ 1 n+ 1 k k ˜ ⊥ 2 k 2 Dm = Dm + ∆t Dgrad · Hm − |m| Hm , (6.39)

h ˜ i h ˜ i ˜ ˜ where Dcurl and Dgrad are incidence matrices on the dual mesh, with sizes N2 ×N1 h i h i ˜ ˜ ⊥ k k ⊥ and N1×N0, respectively. As before, Hm , Dm , Hm , and Dm are column vectors containing the degrees of freedom of the modal ﬁelds.

121 We use the (discrete) Hodge star operator [26, 30, 35, 85] to convert the discrete

Ampere’s law from the dual mesh to the primal mesh. In this way,

0→0 ⊥ n+1 0→0 ⊥ n [?] · Em = [?] · Em

n+ 1 T 1→1 k 2 + ∆t [Dgrad] · [?µ−1 ] · Bm , (6.40)

1→1 k n+1 1→1 k n [?] · Em = [?] · Em

n+ 1 n+ 1 T 2→2 ⊥ 2 1→1 k 2 + ∆t [Dcurl] · [?µ−1 ] · Bm − |m| [?µ−1 ] · Bm , (6.41)

h ˜ i T h ˜ i T where Dcurl = [Dgrad] , Dgrad = [Dcurl] and the discrete Hodge matrix elements are given by [30,35,76] Z Z 1→1 (1) (1) (1) (1) [?]J,j = (0ρ) wJ ∧ ? wj = (0ρ) WJ · Wj dV , (6.42) Ω Ω | {z } vector proxy representation Z Z 2→2 −1 (2) (2) −1 (2) (2) [?µ−1 ]K,k = µ0 ρ wK ∧ ? wk = µ0 ρ WK · Wk dV , (6.43) Ω Ω | {z } vector proxy rep. Z Z 0→0 −1 (0) (0) −1 h (0) î h (0) î [?]I,i = 0ρ wI ∧ ? wi = 0ρ WI φ · Wi φ dV , (6.44) Ω Ω | {z } vector proxy rep. Z 1→1 −1 (RWG) (RWG) [?µ−1 ]J,j = (µ0ρ) wJ ∧ ? wj Ω Z −1 h (1) î h (1) î = (µ0ρ) WJ × φ · Wj × φ dV , (6.45) Ω | {z } vector proxy rep. where Ω is the (compact) spatial support of the Whitney forms, and the ρ, ρ−1

factors result from the use of the TO in the mapping, as discussed before, where

they enter as modiﬁers of constitutive properties rather than diﬀerential operator

factors. The discrete Hodge matrices deﬁned in (6.42), (6.43), (6.44), and (6.45)

are instantiations of the (discrete) Galerkin-Hodge operator. It should be emphasized

that the Galerkin-Hodge operator is not a natural consequence of DEC. The Galerkin-

Hodge operator was originally proposed in [179] and it satisﬁes a number of built-in

122 properties for stability in arbitrary simplicial meshes as discussed in, for example:

[35], [180], [181], [182].

The ﬁeld updates in (6.40) and (6.41) call for sparse linear solvers due to the

0→0 1→1 presence of the matrices [?] and [?] . From (6.42) and (6.44), it is seen that

0→0 1→1 [?] and [?] are diagonally dominant and symmetric positive deﬁnite matrices;

consequently, the linear solve can be performed very quickly. Nevertheless, this needs

to be repeated at every time step. The linear solve can be obviated by computing

0→0 1→1 a sparse approximate inverse (SPAI) of [?] and [?] prior to the start of the

time updating procedure. This strategy is discussed in [51] and [33]. The present

algorithm is explicit and hence conditionally stable. The stability conditions are

discussed in E.

6.1.4 Symmetry axis singularity treatment

For BOR problems where the line ρ = 0 (symmetry axis) is part of the solution domain (for example, in hollow waveguides), it becomes necessary to treat the ﬁeld behavior there by means of appropriate boundary conditions. The boundary conditions at ρ = 0 are mode-dependent and should account for the cylindrical coordinate system singularity and the related degeneracy of theρ ˆ and φˆ unit vectors there. When

m = 0, there is no ﬁeld variation along azimuth and, in the absence of charges at

ρ = 0, both azimuthal and radial ﬁeld components are zero at ρ = 0. On the other

hand, the axial ﬁeld component should be zero for m 6= 0 [183] since the axial di-

rection is invariant with respect to φ and a ﬁeld dependency of the form cos (mφ) or

sin (mφ) with m 6= 0 would imply a multivalued result at ρ = 0 due to the coordinate

degeneracy there. As a result, when m = 0, the boundary ρ = 0 can be represented

123 (a)

(b)

(c)

(d)

Figure 6.4: Field boundary conditions on the primal mesh for the TEφ field with (a) perfect magnetic conductor (m = 0) and (b) perfect electric conductor (m 6= 0) and for the TMφ field with (c) perfect magnetic conductor (m 6= 0) and (d) perfect electric conductor (m = 0). Dashed lines indicate Dirichlet boundary condition, for example edges on the z axis representing a perfect electric conductor boundary for TEφ field in (b), or nodes on the z axis representing a perfect electric conductor boundary for the TMφ field in (d).

124 as a perfect electric conductor for the TEφ ﬁeld and as a perfect magnetic conductor

for the TMφ field. Conversely, when m 6= 0, the ρ = 0 boundary can be represented as a perfect magnetic conductor for the TEφ field and as a perfect electric conductor for the TMφ field. A homogeneous Neumann boundary condition for the electric field can be used to represent the perfect magnetic conductor case and a homogeneous

Dirichlet boundary condition for the perfect electric conductor case. Implementation of such boundary conditions on the primal mesh is illustrated in Fig. 6.4. Dashed lines in Fig. 6.4b and 6.4d denote the Dirichlet boundary implementation: along the z axis, the perfect electric conductor condition is enforced on grid edges for the TEφ case and on grid nodes for the TMφ case. Likewise, Fig. 6.4a and 6.4c illustrate application of the Neumann boundary condition: along the z axis, the perfect magnetic

conductor condition is enforced on grid edges for the TEφ case and on grid nodes for

the TMφ case.

Using the boundary conditions described above, the present FETD-BOR Maxwell

solver does not require any modiﬁcations in the basis functions on the grid cells

adjacent to the z axis, unlike prior FE-BOR Maxwell solvers.

6.2 Numerical Examples

In order to validate present FETD-BOR Maxwell solver, we ﬁrst consider a cylin-

drical cavity and compare the resonance frequency results to the analytical predic-

tions. Then, we illustrate two practical examples of devices based on BOR geome-

tries: logging-while-drilling sensors used for Earth formation resistivity proﬁling in

geophysical exploration and relativistic BWO for high-power microwave applications.

125 6.2.1 Cylindrical cavity

We simulate the eigenfrequencies of a hollow cylindrical cavity with metallic walls

using the present FETD-BOR Maxwell solver, and compare the results to analytic

predictions. The cavity has radius a = 0.5 m and height h = 1 m, as depicted in

Fig. 6.5. Magnetic and electric dipole current sources M (r, t) and J (r, t) oriented

along φ and excited by broadband Gaussian-modulated pulses are placed at arbitrary

locations inside the cavity rs = (ρs, φs, zs), so that

ˆ M (r, t) , J (r, t) = φ G(t) δ (r − rs) =

 Mφ  ˆ k k X = φ G(t) δ r − rs π + 2π cos [m (φ − φs)] (6.46) m=1

2 −[(t−tg)/(2σg)] where G(t) = e sin [2πfg (t − tg)] with tg = 20 ns, σg = 1.9 ns, and

k fg = 300 MHz, and r = ρρˆ + zzˆ. We use Fourier series expansion to describe

δ (φ − φs) in (6.46) in order to match the modal ﬁeld expansion used before. A total of four dipole sources (electric and magnetic currents) are used to excite a rich gamut of eigenmodes, as illustrated in Fig. 6.5. The meridian plane of the cylindrical cavity is discretized by an unstructured grid with 4, 045 nodes, 11, 939 edges, and 7, 895 faces

(seen as the ρz plane for φ = 180o in Fig. 6.8). The metallic boundaries are treated as perfect electric conductors. In this case, the maximum azimuthal modal order Mφ was set equal to 4 to investigate the ﬁeld solution up to this order. Higher order modes can be included by simply increasing Mφ. This is straightforward since azimuthal modal

ﬁelds with diﬀerent orders are orthogonal to each other. From the stability analysis in E, the maximum time-step intervals for various cases are presented in Table 6.1.

Here we chose ∆t = 1 ps for the simulations and used a total of 1 × 107 time steps to provide suﬃciently narrow resonance peaks. By recording the time history of the

126 Figure 6.5: Schematic view of the simulated cylindrical cavity with perfect electric conductor (PEC) walls. The cavity dimensions are a = 0.5 m and h = 1 m.

Table 6.1: Maximum time-step intervals for various cases in the simulation of cylindrical metallic cavity.

m = 0 m 6= 0 TEφ-pol. TMφ-pol. m = 1 m = 2 m = 3 m = 4

∆tmax [ps] 10.009 10.249 10.009 6.4792 4.5545 3.4843

127 electric ﬁeld values at arbitrary locations inside the cavity and performing a Fourier transform, we obtain the eigenfrequencies as peaks in the Fourier spectrum. Fig. 6.6 shows the normalized spectral amplitude as a function of frequency. The black solid line is the result obtained by using present FETD-BOR Maxwell solver. The red dashed and blue solid lines indicate analytic predictions for the eigenfrequencies of the TEmnp and TMmnp modes in this cavity, respectively. The analytic expressions for the eigenfrequencies are given by

r 2c pπ 2 f = χ02 + , TEmnp π mn h for m = 0, 1, ..., n = 1, 2, ..., p = 1, 2, ... , (6.47) r 2c pπ 2 f = χ2 + , TMmnp π mn h for m = 0, 1, ..., n = 1, 2, ..., p = 0, 1, ... , (6.48)

0 where c is speed of light, χmn and χ mn are the roots of the equations Jm (aχmn) = 0

0 0 and J m (aχ mn) = 0, respectively, with Jm (·) being the Bessel function of ﬁrst kind

0 and J m (·) its derivative with respect to the argument. It is clear from Fig. 6.6 that there is a great agreement between the simulated and analytic eigenfrequencies. Table

7.2 shows the relative error between the simulated fs and analytical fa frequencies.

The relative error is below 0.03 % in all cases, indicating the accuracy of the proposed

ﬁeld solver.

To illustrate the field behavior, Fig. 6.8 shows snapshots for electric field intensity and magnetic flux density distribution inside the cavity on four ρz planes with φ = 0o,

φ = 90o, 180o, 270o and two ρφ planes with z = 0.2 m and 0.8 m, at two time instants:

1.0024 µs, 1.0028 µs, 1.0032 µs, and 1.0036 µs. Due to the location of the dipole sources, the transient ﬁelds produced include many eigenmodes, and are basically

128 Figure 6.6: Normalized spectral amplitude for E, showing the eigenfrequencies of the cavity. Black solid lines correspond to the present FETD-BOR result. Red solid and blue dashed lines are analytic predictions for the TEmnp and TMmnp eigenfrequencies, respectively.

asymmetric. It can be seen that the (tangential or normal) boundary conditions on the outer perfect electric conductor walls for electric field intensity and magnetic flux density are well satisfied. Moreover, the correct field distribution along the symmetry axis is well reproduced by the chosen boundary conditions at ρ = 0, without any spurious artifacts.

6.2.2 Logging-while-drilling sensor simulation

Logging-while-drilling sensors have BOR geometries and are routinely used for hydrocarbon exploration [3,167–171]. As the drilling process is performed, these sensors record logs obtained by the measurements of ﬁelds produced by loop (multi-coil) antennas present in the sensor and reﬂected from the surrounding geological formation. Logging-while-drilling sensors are typically equipped with a series of transmitter

129 (a) (b)

Figure 6.7: Transient snapshots for Ez inside the cylindrical cavity at (a) 1.0024 [µs], (b) 1.0028 [µs], (c) 1.0032 [µs], and (d) 1.0036 [µs].

130 (a) (b)

Figure 6.8: Transient snapshots for Bz inside the cylindrical cavity at (a) 1.0024 [µs], (b) 1.0028 [µs], (c) 1.0032 [µs], and (d) 1.0036 [µs].

131 Table 6.2: Eigenfrequencies for the cylindrical cavity and normalized errors between numerical and analytic results.

Resonant modes fa [MHz] |fa − fs| /fa × 100 [%] −2 TM010 229.6369 1.1854 × 10 −4 TE111 231.1104 8.0278 × 10 −2 TM011 274.2865 2.4558 × 10 −2 TE211 327.9619 1.0503 × 10 −2 TE112 347.7241 1.7614 × 10 −2 TM110 365.8931 2.8110 × 10 −3 TM012 377.8003 7.0851 × 10 −3 TE011, TM111 395.4463 1.5709 × 10 −3 TE212 418.4005 9.3816 × 10 −3 TE311 428.3025 6.0946 × 10 −2 TE012, TM112 473.1572 1.2629 × 10 −3 TE113 483.1273 4.4680 × 10 −3 TM210 490.4134 5.2154 × 10 −2 TE312 500.9421 1.0443 × 10 −3 TM013 505.2060 2.0998 × 10 −3 TM211 512.8404 8.3352 × 10 −2 TM020 527.1202 2.3910 × 10 −3 TE411 529.4750 6.8899 × 10 −2 TE121 530.7481 2.5411 × 10 −3 TE213 536.2453 5.4133 × 10 −2 TM021 548.0472 2.9989 × 10

and receiver loop antennas that are wrapped around the outer diameter of a metallic mandrel attached to the bit drill [184–189]. Fields produced by the transmitter coil(s) interact with the adjacent well-bore environment and are detected by a pair (or more) of receiver coils along the logging-while-drilling sensor at same axial distance from the transmitter(s). Two types of measurements are typically used to determine the resistivity proﬁles of the adjacent formation. The ﬁrst is the amplitude ratio

(AR) between the electromotive force (e.m.f.) excited at the two receiver coils and

132 Figure 6.9: Logging-while-drilling sensor problem geometry (from inner to outer features): metallic mandrel, transmit (Tx) and receive (Rx) coil antennas, mud-ﬁlled borehole, and adjacent geological formation.

the second is their phase difference (PD). In this section, we consider a prototypical concentric logging-while-drilling sensor generating a TMφ field distribution in the formation with m = 09. The logging-while-drilling sensor depicted in Fig. 6.9 consists of a metallic cylindrical mandrel modeled as a perfect electric conductor inside a concentric cylindrical borehole. Three loop antennas are used: one as transmitter and two as receivers. The borehole created by the drilling process is filled with a lubricant

ﬂuid (mud). The three coil antennas are moving downward in tandem as the drilling process occur.

We consider two scenarios for the adjacent Earth formation, as shown in Fig.

6.10. In the ﬁrst scenario, the borehole is ﬁlled with a low conductive (oil-based)

9Not only the geometry but also the ﬁeld excitation is axisymmetric in this case.

133 fluid (mud) having σ = 0.0005 S/m and surrounded by geological formations with different conductivities. We compute the AR and PD as a function of the formation conductivity. In the second scenario, the borehole is filled with a high conductive

(water-based) ﬂuid having σ = 2 S/m, and the formation has three horizontal layers with diﬀerent conductivities as shown. We compute the AR and PD as the set of coil antennas (sensor) moves downward. In both cases, the relative permittivity and permeability are assumed equal to one everywhere, and the transmitter coil radiates a 2 MHz signal. In the time domain, this is implemented through a current signal along the transmitter coil given by ITx(t) = r(t) sin (ωt), where  0, t < 0   ωt r(t) = 0.5 1 − cos , 0 t < αT (6.49) 2α 6   1, t > αT, is a raised-cosine ramp function, T = 2π/ω is the signal period, and α is the number of sine wave cycles during the ramp duration αT . The use of ramp function mitigates high frequency components otherwise produced by an abrupt turn-on at t = 0, and yields faster convergence of AR and PD (after approximately one time period T ) [3].

We choose α = 0.5 to yield a continuous ﬁrst-order derivative and no DC (zero- frequency) component for the signal. From the time-domain signals computed at the two receivers, we extract the corresponding phases θ and amplitudes A using

q sin (ωt ) − q sin (ωt ) θ = tan−1 2 1 1 2 , (6.50) q1 cos (ωt2) − q2 cos (ωt1)

q1 A = , (6.51) sin (ωt1 + θ)

134 (a) (b)

Figure 6.10: Logging-while-drilling sensor responses. (a) First scenario: the conductivity of the adjacent geological formation is varied. (b) Second scenario: the sensor moves downward through a borehole surrounded by a geological formation with three horizontal layers.

where q1 and q2 are signals computed at times t1 and t2, respectively [3]. Next, the

AR and PD are calculated as

AR = ARx2 /ARx1 , (6.52)

PD = θRx2 − θRx1 . (6.53)

The azimuthal electric current along the transmitter coil is modeled as a nodal current density on the meridian plane and the metallic mandrel is regarded as perfect electric conductor. The FE domain is truncated by a PML to mimic an open domain.

We use 8 layers for the PML to yield a reﬂectance below −50 dB [74].

135 (a) (b)

Figure 6.11: Computed (a) AR and (b) PD (in deg.) by a logging-while-drilling sensor surrounded by homogeneous geological formations with diﬀerent conductivities. This corresponds to the ﬁrst scenario in Fig. 6.10. The results from the present algorithm are compared against FDTD and NMM results [3] (see more details in the main text).

Fig. 6.11 shows results for the behavior of AR and PD versus the conductivity

on a homogeneous formation. The results are compared against previous results ob-

tained by the ﬁnite-diﬀerence time-domain (FDTD) and the numerical mode matching

(NMM) methods [3]. There is excellent agreement between the results. Results for

the second scenario are shown in Fig. 6.12, where PD is plotted as a function of the

z position of the transmitter, zTx, and compared against previous results obtained by

the FDTD and NMM methods [3]. Again, an excellent agreement is obtained. As

expected, the PD is higher when the coil antennas are within high attenuation (high

conductivity) layer and vice versa. The conductance proﬁle and the corresponding

axial extension of each formation is shown in green color in Fig. 6.12. Fig. 6.13a−Fig.

6.13f show snapshots of the electric field distributions for different zTx to illustrate the field behavior.

136 Figure 6.12: Computed PD (deg.) between the two receivers of the logging-while- drilling sensor versus the z position of the transmitter coil antenna. This corresponds to the second scenario in Fig. 6.10. The results from the present algorithm are compared against FDTD and NMM results [3] (see more details in the main text).

6.3 Conclusion

We presented a new finite-element time-domain (FETD) Maxwell solver for the analysis of body-of-revolution (BOR) geometries. The proposed solver is based on discrete exterior calculus (DEC) and transformation optics (TO) concepts. We explored TO principles to map the original 3-D problem from a cylindrical coordinate system to an equivalent problem on a 2-D (Cartesian-like) meridian ρz plane, where the cylindrical metric is factored out from the differential operators and embedded on an effective (artificial) inhomogeneous and anisotropic medium that fills the domain.

This enables the use of Cartesian 2-D FE code with no modiﬁcations necessary except to accommodate the presence of anisotropic media. The spatial discretization is done on an unstructured mesh on the 2-D meridian plane and eﬀected by decomposing the

137 (a) (b)

(e) (f)

Figure 6.13: Electric ﬁeld distribution during the half period for zTx = (a) −50 inch, (b) −25 inch, (c) 5 inch, (d) 25 inch, (e) 50, and (f) 70 inch. Note that zTx = 0 at the interface between ﬁrst (5 S/m) and second (0.0005 S/m) formations.

138 ﬁelds into their TEφ and TMφ components and expanding each eigenmode into an appropriate set of (vector or scalar) basis functions (Whitney forms) based on DEC principles. A leap-frog (symplectic) time-integrator is applied to the semi-discrete

Maxwell curl equations and used to obtain a fully discrete, marching-on-time evolution algorithm. Unlike prior solvers, the present FETD-BOR Maxwell solver does not require any modiﬁcations on the basis functions adjacent to the symmetry axis.

Rather, the ﬁeld behavior on the symmetry axis can be simply implemented through properly selected homogeneous Dirichlet and Neumann applied to the eigenmodal expansion.

139 Chapter 7: Axisymmetric Electromagnetic Particle-in-Cell Algorithm: Application to Microwave Vacuum Electronic Devices

Historically the need for high-power electromagnetic (EM) radiation sources in the gigahertz and terahertz frequency ranges has triggered signiﬁcant technical advances in vacuum electronic devices (VED) [46–50], such as the gyrotron, free electron

Laser, and traveling wave tube (TWT). These devices serve as a basis for a variety of applications in radar and communications systems, plasma heating for fusion, and radio-frequency (RF) accelerators [63,190].

Amplification of RF signals is usually obtained by exploiting resonance Cerenkov interactions between an electron beam and the modal field supported by a slow-wave structure (SWS) [47,191–194]. SWSs are often made by imposing periodic ripples on the conducting wall of cylindrically symmetric waveguides, as illustrated in Fig. 7.1a, so that the phase velocity of the modal field becomes slower than the speed of light in vacuum due to the Bragg scattering [195].

According to the dispersion relations associated with the geometry of SWSs, resultant Cerenkov interactions can amplify forward or backward waves10. Similarly

10Along the direction of the group velocity w. r. t. the beam velocity.

140 to plasma instabilities, the evolution of forward and backward waves can be characterized by convective instabilities that grow over time while traveling away from the location of initial disturbance and absolute instabilities that propagate a local initial disturbance throughout the whole device volume [47]. Traveling-wave tube ampliﬁers

(TWTA) and backward-wave oscillators (BWO) are two practical examples utilizing convective and absolute instabilities, respectively.

Recent studies have shown that a particular SWS geometry may significantly en- hance the system performance of TWTs. For example, nonuniform (locally periodic) ripples used in BWOs may improve mode conversion efficiency [196,197], and tapering ripples may reduce reflections at the output of TWTA and prevent internal oscillations [191,198]. More importantly, smooth device edges are preferred for high-output power applications in order to mitigate pulse shortening, which is a major bottleneck for increasing output powers beyond the gigawatts range [199, 200]. This is because extremely strong field singularities, which accumulate on the sharp edges, may create interfering plasmas that terminate the output signal at an earlier time. Sinusoidally corrugated slow wave structures (SCSWS) have been increasingly adopted in many modern high-power BWO systems to combat this problem [?]. In addition, a variety of micro-machining fabrication techniques have been developed to enable better device performances by using much tighter tolerances. These technological advances have allowed the production of devices operating at higher frequencies, including the

THz regime.

Computational experiments for VEDs employ electromagnetic particle-in-cell (EM-

PIC) algorithms [11, 13, 57, 201], which numerically solve the Maxwell-Vlasov equations describing weakly coupled (collision-less) systems, where the collective behavior

141 y x z

(a) (b)

Figure 7.1: Schematics of two examples of axisymmetric vacuum electronic devices. (a) Backward-wave oscillator producing bunching eﬀects on an electron beam. Wall ripples are designed to support slow-wave modes in the device. (b) Space-charge- limited cylindrical vacuum diode.

of charged particles prevails over their binary collisions [11–13]. A typical PIC al-

gorithm tracks the temporal evolution of macro-particles seeded in a coarse-grained

six-dimensional (6D) phase space11. A typical PIC algorithm consists of four basic steps, viz. the field solver, the field gather, the particle push, and the particle charge/current scatter, which are repeated at every time iteration. This provides a self-consistent update of particles and field states in time.

As a field solver, most EM-PIC simulations employ the celebrated Yee’s finite- difference time-domain (FDTD) method for regular grids, due to its simplicity. There is a plethora of FDTD-based EM-PIC codes such as UNIPIC, MAGIC, TWOQUICK, KARAT,

VORPAL, and others [97, 202, 203]. However, the relatively poor grid-dispersion properties of this algorithm [19] causes spurious numerical Cerenkov radiation [20]. More- over, in complex geometries such as those of modern VEDs, “staircase” (step-cell)

11That is, a ﬁnite-size ensemble of physical particles with positions and momenta.

142 effects present a critical challenge. Using FDTD for an accurate analysis of geometrically complex devices, which typically have curved boundaries or very fine geometrical features, may require excessive mesh refinement and therefore result in a waste of computational resources. Many studies have been done to mitigate the staircasing errors in finite-difference (FD) methods, in particular through using conformal FD discretizations [21,98].

On the other hand, the finite-element time-domain (FETD) method [27, 70] fun- damentally eliminates the undesired staircase effects since it is naturally based on unstructured (irregular) grids, which can more easily be made conformal to complex geometries and can be augmented by powerful mesh refinement algorithms. Unfor- tunately, conventional FETD-based PIC algorithms have historically faced numerical challenges that result from a lack of exact charge conservation on unstructured grids.

This gives rise to the accumulation of spurious charges which must be removed by applying costly a posteriori corrections [14,71]. In addition, implicit time updates used in conventional FETD require repeated linear solves at each time-step [27]. Recently, a novel charge-conserving scatter scheme for unstructured grids, inspired by diﬀerential- geometric ideas and the exterior calculus of diﬀerential forms [26,30,35,178,204,205], has been proposed in [1]. Other charge-conservative EM-PIC algorithms for unstructured grids were also developed under similar tenets in [77, 78, 206]. In addition, an charge-conserving EM-PIC algorithm with explicit time-update that is both local (i.e. preserves sparsity) and obviates the need for linear solvers at each time step has been described in [25,33,51] based on the sparse approximate inverse (SPAI) of the discrete

Hodge operator (the ﬁnite-element “mass” matrix).

143 These recent advances have made possible the present work, which is motivated by the demand to accurately capture realistic physics of beam-SWS interactions in complex geometry devices. In this chapter, we present a charge-conservative EM-PIC algorithm based on unstructured grids and optimized for the analysis and design of axisymmetric VEDs. Since conventional SWSs are cylindrically axisymmetric (invariant along φ), SWS studies can be best done with algorithms that explore this symmetry, so that significant computational resources can be saved and the algorithm may be feasibly implemented as a forward engine in a design loop [98]. We show that under the assumption of cylindrical symmetry of fields and sources one can reduce the original 3D geometry to a 2.5D setup by introducing an artificial inhomogeneous medium12, considering TEφ-polarized fields in the meridian (ρz) plane.

Note that this is a special version of the BOR-FETD scheme, viz. only accounting for only the zeroth azimuthal eigenmode, combined with the Cartesian EM-PIC algorithm. We validate our algorithm using analytical results for a cylindrical cavity and previously obtained results for a space-charge-limited (SCL) vacuum diode (see

Fig. 7.1b). We include a micro-machined SWS-based BWO example, designed to harness particle bunching eﬀects from coherent Cerenkov beam-wave interaction, to demonstrate the advantages of utilizing unstructured grids without staircasing error to predict the device performance.

12In a manner akin to the “transformation optics” technique [30,41,42, 207].

144 7.1 Spatial dimensionality reduction

In this section we describe a numerical model for 3D VEDs with cylindrical ax-

isymmetry based on an equivalent 2D model discretized on an unstructured (irregular)

grid in the meridian plane.

7.1.1 Exterior calculus representation of Maxwell’s equations

We represent Maxwell’s equations using the exterior calculus of diﬀerential forms [35,

77,178,205,208] as

∂B dE = − , (7.1) ∂t ∂D dH = + J , (7.2) ∂t dD = Q, (7.3)

dB = 0, (7.4)

where E and H are 1-forms for the electric and magnetic ﬁeld intensity, D and B are

2-forms for the electric and magnetic ﬂux density, J is 2-form for the electric current

density, Q is 3-form for the electric charge density, and operator d is the exterior

derivative encompassing conventional curl and divergence operators [34,147,209,210].

These 1-, 2-, and 3-forms can be expressed using a set of non-orthonormal-basis in

a cylindrical coordinate system (dρ, dφ, dz) [41, 209]. For instance, E (1-form) is ~ expressed as E = Eρdρ + Eφdφ + Ezdz; then, its vector proxy can be written as E =

Eφ ˆ Eρρˆ+ ρ φ+Ezzˆ. Similarly, B (2-form) is given by B = Bρdφ∧dz+Bφdz∧dρ+Bzdρ∧dφ (where ∧ is the exterior or wedge product [204, 205, 210]) and its vector proxy takes

~ Bρ ˆ Bz the form of B = ρ ρˆ + Bφφ + ρ zˆ [209].

145 (a)

(b)

Figure 7.2: A charged ring travels inside an axisymmetric object bounded by PEC: (a) a 3D view, (b) the meridian plane.

7.1.2 Cylindrical axisymmetry constraints

Consider a charged ring with constant density along azimuth that travels inside

of a cylindrically axisymmetric tube bounded by a perfect electric conductor (PEC)

with a radial boundary proﬁle of ∂Ω = R (z) where Ω is the computational domain and R is the wall radius which only depends on z, as shown in Fig. 7.2a. Cylindrical

axisymmetry, used here, implies that there is no variation along φ (∂/∂φ = 0) in the

device geometry, ﬁelds, and sources. It should be noted that axisymmetric sources in

the meridian plane are represented as charge rings (see Fig. 7.2b). There exist two

146 ∂ useful constraints that simplify the original 3D problem: (i) elimination of the dφ ∂φ

∂ ∂ term in the exterior derivative d, viz. d = dρ ∂ρ + dz ∂z and (ii) retainment of only transverse magnetic (TM) eigenmodes with m = 013. The ﬁrst constraint enables the

same calculus in the meridian plane as in the 2D Cartesian coordinate system with

the cylindrical metric factor embedded into the constitutive relations, as discussed

below. The second constraint simpliﬁes expressions for ﬁelds and sources as

E = Eρdρ + Ezdz, (7.5)

B = Bφdz ∧ dρ, (7.6)

D = Dρdφ ∧ dz + Dzdρ ∧ dφ, (7.7)

H = Hφdφ. (7.8)

From equations of (7.5), (7.6), (7.7), (7.8), it is straightfoward to show that a 3D

problem with cylindrical axisymmetry can be represented by a 2D problem describing

TEφ-polarized ﬁelds on the meridian plane (see also Fig. 7.3).

7.1.3 Modiﬁed Hodge star operator

The Hodge star operator ? map p-forms into (n − p)-forms in n-dimensional space14 [26,34,35,85,209,211]. In our case, we have

−1 H = µ0 ? B, (7.9)

D = 0 ? E. (7.10)

13The index m is used here to denote azimuthal harmonics. 14The Hodge star operator can be understood geometrically as yielding the orthogonal complement of a given diﬀerential form to the volume form in n-space. In the 3D case for example, ?(dx) = dy ∧ dz, ?(dz ∧ dρ) = ρdφ, ?(1) = dx ∧ dy ∧ dz = ρ dρ ∧ dφ ∧ dz and so forth [209]. For an arbitrary p-form A in n-space, we have A ∧ (?A) = |A|2dV , where is dV is the volume element in n-space. As such, A ∧ (?A) provide the L2 ( or“energy”) element norm of A.

147 The Hodge operators incorporate the metrical properties of the system, which in the

cylindrical case are expressed by a metric tensor diag(1, ρ2, 1). For the magnetic ﬁeld

and ﬂux density, substituting (7.8) into the left-hand side term of (7.9) gives

H = Hφdφ, (7.11)

and substituting (7.6) into the right-hand side term of (7.9) yields

−1 −1 −1 µ0 ? B = µ0 ? (Bφdz ∧ dρ) = µ0 Bφ ? (dz ∧ dρ)

−1 −1 −1 = µ0 Bφρdφ = µ0 ρ Bφdφ = µ (ρ) Bφdφ. (7.12)

Note that ? only acts on the diﬀerentials such as dz, dρ, and dφ. By comparing (7.11)

−1 and (7.12) and introducing an artificial magnetic permeability, µ (ρ) = µ0ρ , we can extract the radial factor ρ from the Hodge star operator. As a result, we can reuse simple Cartesian space calculus with metric tensor diag(1, 1, 1) and a Hodge operator devoid of additional metric factors. For the electric field and flux density, substituting

(7.7) into the left-hand side term of (7.10) yields

D = Dρdφ ∧ dz + Dzdρ ∧ dφ, (7.13) and substituting (7.5) into the right-hand side term of (7.10) gives

0 ? E = 0 ? (Eρdρ + Ezdz) = 0 (Eρ ? dρ + Ez ? dz)

= 0 (Eρρdφ ∧ dz + Ezdρ ∧ ρdφ) = (0ρ)(Eρdφ ∧ dz + Ezdρ ∧ dφ)

= (ρ)(Eρdφ ∧ dz + Ezdρ ∧ dφ) . (7.14)

In a similar fashion, an artiﬁcial electrical permittivity takes the form of (ρ) = 0ρ.

Essentially, an original 3D problem with cylindrical axisymmetry is replaced by an equivalent 2D problem with TEφ-polarized ﬁelds immersed in Cartesian space with an

148 Figure 7.3: The original problem shown in Fig. 7.2 is replaced by an equivalent 2D problem in the meridian plane as depicted above, which considers TEφ-polarized EM fields on Cartesian space with an artificial inhomogeneous medium. The variable coloring serves to stress the dependency of the artificial medium parameters on ρ.

Table 7.1: Estimation of the run time of EM-PIC simulations based on FETD and FDTD at each time-update.

Each time-update Field-solver Gather Particle-push Scatter

FETD (T1) 5N1 3Np Np 3Np FDTD (T2) 4N1 4Np Np 4Np

inhomogeneous medium with artiﬁcial permittivity and permeability, as illustrated schematically in Fig. 7.3. As a result, present ﬁeld-solver borrowing the concept of TO are free from the axial issues associated with the ρ factor present in ∇×~ in cylindrical coordinate systems.

We can estimate the run time of FETD and FDTD at each time-update. Each step will be proportional to some factors, as follows: where N1 is the number of edges, and Np is the number of superparticles, and T1 and T2 are the run time of

149 FETD and FDTD during one time-update. Then, T1 ≈ 5N1 + 3Np + Np + 3Np and

T2 ≈ 4N1 + 4Np + Np + 4Np. In addition, we need to consider the time to track which element the particle is and FDTD can track particles much easier than FETD due to the use of structured grids. However, for the curved boundary, FDTD may need more edges to accurately model the boundary. Then, the total time of FETD will be nT1 + SPAI solver time and FDTDs is nT2 with n is the total number of time-steps for a PIC simulation. As a results, the run time is comparable to each other.

7.2 Validation

In this section, we provide validation examples. First, we consider a metallic cylindrical cavity and compare the resonant frequencies of the TM0np cavity modes obtained by the present ﬁeld solver with the exact (analytic) results. Second, we model a space-charge-limited cylindrical diode with a ﬁnite-length emitter and compare the maximum injection currents for divergent and convergent electron beam

ﬂows against previously published results.

7.2.1 Metallic hollow cylindrical cavity

Assume a hollow cylindrical cavity with metallic walls, radius a = 0.5 m and height h = 1 m (see Fig. 7.4). Two magnetic point sources M1 (ρ, z, t) and M2 (ρ, z, t) excited by Gaussian-modulated pulses of broadband spectrum are placed at arbitrary positions given as

2 t−tg/2 − 2σ M1 (ρ = 0.37, z = 0.2, t) =0.7e g cos [ωg (t − tg/2)] (7.15)

2 t−tg − 2σ M2 (ρ = 0.08, z = 0.7, t) =0.5e g cos [ωg (t − tg)] (7.16)

150 Figure 7.4: Snapshots for electric ﬁeld distribution at 2 µs. Note that RGB colors and white arrows indicate magnitudes and vectors of the electric ﬁelds, respectively.

9 with tg = 40 ns, σg = 1 ns, and ωg = π × 10 rad/s. The source locations are denoted by Tx in Fig. 7.4. These sources excite resonant modes inside the cavity. The 2D meridian plane of the cylindrical cavity is discretized by an unstructured grid with

8, 095 nodes, 23, 928 edges , and 15, 834 faces. The three lateral metallic boundaries

are assumed as perfect electric conductors (PEC). The remaining boudary is the z- axis (axisymmetric boundary). The time step interval ∆t is chosen as 1 ps, and the simulation runs over a total of 2 × 106 time steps. Fig. 7.4 shows a snapshot for electric ﬁeld distribution in the cavity at 2 µs. The RGB colormap and the white

arrows indicate magnitudes and vectors of the electric ﬁelds, respectively. The time

signal are detected at (ρ = 0.71, z = 0.39) (Rx in Fig. 7.4) and a Fourier analysis

is performed to obtain a spectrum of the signal. The resulting spectrum shows the

resonant cavity modes from 1 MHz to 1 GHz in Fig. 7.5, where blue-solid lines

are axisymmetric FETD ﬁeld-solver results and red-dashed lines are analytic results.

An excellent agreement can be observed. In addtion, Table. 7.2 shows the resonant

151 Figure 7.5: Spectrum for resonant cavity modes from 1 MHz to 1 GHz.

frequencies for TM cavity modes15 and the normalized error deﬁned as |fs−fa| mnp fa where fs and fa are numerical and analytic resonant frequencies, respectively. It is seen that all resonant cavity modes are axisymmetric (m = 0) and the normalized errors are fairly low.

7.2.2 Space-charge-limited (SCL) cylindrical diode

As a second validation example, we test the accuracy of present EM-PIC algorithm by modeling a SCL cylindrical diode with finite-length emitter. By applying an external voltage to the cathode, a fast rise in the number of electrons emitted from the cathode initially occurs; however, in the steady-state the injection current density eventually becomes saturated due to space charge effects. For an infinitely long cylindrical diode and electrodes, Langmuir-Blodgett’s law describes the SCL current

15m, n, and p are associated with eigenmode orders along azimuthal (φ), radial (ρ), and longitudinal directions (z), respectively.

152 Table 7.2: Resonant frequencies for axisymmetric cavity modes and normalized errors between numerical and analytic works.

|fa−fs| |fa−fs| (m, n, p) fa [GHz] × 100 [%] (m, n, p) fa [GHz] × 100 [%] fa fa

(0, 1, 0) 0.230 0.083 (0, 1, 5) 0.784 0.0035 (0, 1, 1) 0.274 0.040 (0, 2, 4) 0.799 0.0048 (0, 1, 2) 0.378 0.040 (0, 3, 0) 0.826 0.0014 (0, 1, 3) 0.505 0.047 (0, 3, 1) 0.840 0.019 (0, 2, 0) 0.527 0.042 (0, 3, 1) 0.879 0.017 (0, 2, 1) 0.548 0.031 (0, 3, 1) 0.917 0.027 (0, 2, 2) 0.607 0.0030 (0, 2, 5) 0.929 0.027 (0, 1, 4) 0.642 0.023 (0, 3, 3) 0.941 0.016 (0, 2, 3) 0.693 0.034 − − −

per unit length J1D,LB as

8π r2e V 3/2 J ≡ 0 (7.17) 1D,LB 9 m ρβ

where V is the external voltage, e and m are charge and mass of electrons, ρ is a

radial coordinate, and

2µ2 11µ3 47µ4 β = µ − + − + ..., (7.18) 5 120 3300

with µ = ln (ρ/ρc) and where ρc denotes the radius of the cathode. Here, we assume

the emitting electrode with axial length Le. The ﬁnite element solution domain is given by Ω = {(ρ, z) ∈ [ρi, ρo] × [0,Lz]} with ρi = 5 mm, ρo = 15 mm, and Lz = 100

mm. The domain has the horizontal wall segments representing electrode surfaces

(cathode or anode), as shown in Fig. 7.6. If the inner conductor is chosen as a

cathode, the electron ﬂow becomes divergent, on the other hand, it is convergent.

The left and right boundaries of the domain are truncated by a perfectly matched

layer (PML) [115, 212]. The unstructured mesh has 7, 313 nodes, 21, 334 edges, and

14, 022 faces. We choose ∆t = 0.15 ps and the simulation runs up to a total 30, 000 time-steps.

153 (a)

(b)

Figure 7.6: Schematics for divergent and convergent ﬂows in the cylindrical diode.

Fig. 7.6 illustrates snapshots for electron beam distribution with Le = 10 mm and

V = 1 kV. In order to determine J2D,max injection, we ﬁrst ﬁx the superparticle scaling factor Csp, indicating the number of electrons for each superparticle, and gradually increase the superparticle injection rate until the virtual cathode starts to form. We simulate a total pf 8 cases, including Le/ρo = 0.4, 1, 2.425, and 4 for each divergent

and convergent electron ﬂow, and compare the maximum injection current density

without formation of a virtual cathode, J2D,max injection to the previous results ob-

tained by [4] with KARAT, which is a FDTD-based EM-PIC algorithm. Fig. 7.7 shows

J2D,max injection versus Lz/ρo for divergent and convergent electron ﬂows. Red-solid

(divergent) and -dashed (convergent) lines are KARAT results and blue-markers with

upper ranges are obtained from present EM-PIC simulations. The upper ranges on

the blue markers stand for an interval where an exact solutions for the current density

154 Figure 7.7: Space-charge-limited current density for various Lz/ρo and comparison between present EM-PIC simulations and KARAT by [4].

may exist. The marker indicates maximum current density without the virtual cathode formation and upper horizontal line is minimum injection current density with virtual cathode formation. Two markers are used because of the stepwise increases in the current density by the assumed superparticle number in our EM-PIC model.

Decreasing superparticle scaling factor or increasing superparticle injection rate can yield higher resolution. Fig. 7.7 shows very good agreement between the results of the present EM-PIC algorithm and those of KARAT, for both divergent and convergent

ﬂows. In addition, it is seen that, as Le/ρo increases, the current density converges to the limit of Langmuir-Blodgett’s law as expected.

Fig. 7.8 shows the magnitude of the electrical self-field and external field at the instant of virtual cathode formation. Since the external field is stronger as ρ decreases, a divergent flow produces more charges on the cathode surface so that their self-field

155 (a)

(b)

Figure 7.8: Electric ﬁeld intensity of self- and external ﬁelds at the instant of virtual cathode formation.

may cancel the external field. As a result, the current density of divergent flows is larger than that of convergent flow.

7.3 Numerical examples

In this section, we present simulations of a relativistic BWO device operating at

π-point by using the proposed EM-PIC algorithm. First we consider a SWS with sinusoidal ripples on a cylindrical waveguide section and determine its characteristics by performing a “cold” test (i.e. without the presence of an electron beam). Then, we perform “hot” tests (with electron beams) of BWO to check the reliability and validity of our axisymmetric EM-PIC algorithm. In particular, we are interested in investigating eﬀects of staircasing errors on the predicted behavior of the BWO system.

156 Figure 7.9: Shematics of backward-wave oscillator with an instant particle distribution snapshots at t = 21.50 ns.

Figure 7.10: Electric potential distribution (contour plots) and corresponding electric ﬁelds (vector plots) between the cathode and the anode.

157 7.3.1 Relativistic backward-wave oscillator (BWO)

Consider a BWO system composed of the cathode-anode, the SCSWS region,

the beam collector, the output port, and the beam focusing system. In order to

produce the relativistic electron beam, we apply an external voltage diﬀerence V0

between cathode and anode. We choose −550 kV which produces the electron beam

with mean axial velocity vbeam = 0.877c with the width of 2 mm. Fig. 7.10 illustrates

electric potential distribution (contour plots) and corresponding electric ﬁelds (vector

plots) by solving Poisson equations.

Each super-particle represents 1.495 × 107 electrons so that the total injection

current is around 1.5 kA. On average, 102 macro-particles are assigned to each grid

cell. The Debye length λD in our simulations is 10.7258 mm and much larger than

the average grid (edge) length, lav = 1.1123 mm. This avoids artiﬁcial heating of

the electron beam. The electron beam is emitted from the cathode and eventually

1 absorbed at the collector. We consider a SCSWS with radial proﬁle R(z) = 2 (A+B)+

1 2π 2 (A − B) cos C z , where A and B are maximum and minimum radii, respectively, and C is the corrugation period. The total number of corrugations along the structure

16 is denoted as Ncrg. Based on a eigenmode analysis (see the double-reﬁned case

in Fig. 7.20), the SCSWS was designed to have A = 17.1 mm, B = 12.9 mm,

C = 7.5 mm, and Ncrg = 14.5 for Ku-band operation. We terminate the output ports of the BWO system by inserting a PML. All left vertical walls except for the cathode are truncated by PML to avoid spurious reflections. By using a PML with thickness equals to 0.2λ0 where λ0 is wavelength of the center frequency, the PML reflection coefficient, defined in [74], is as low as −92 dB. In the beam focusing system, a static

16Corresponding to the absence of an electron beam or a so-called ‘cold test’.

158 Figure 7.11: A zoomed-in region of four rightmost corrugations of Fig. 7.9 with RGB color scales reﬂecting particle velocities.

axial magnetic ﬁeld is applied over the SWS region. We enforce axisymmetric ﬁelds

at the z-axis, by applying a perfect magnetic conductor (PMC) boundary condition

there (note that only the Ez component is present on axis). For the particles incident

on the axis we use a perfectly reﬂecting boundary condition.

System performance

Fig. 7.9 illustrates a snapshot of the electron beam at t = 21.50 ns. In Fig. 7.11

we plot a zoomed-in beam picture that shows the four rightmost corrugations, where

RGB colors indicate variations in normalized particle velocities vb/c. The periodic particle beam bunching is a result of particles being accelerated or decelerated, which means that the beam electrons synchronously lose and recover their kinetic energy.

The net energy is transferred from the beam to the waves as seen in Fig. 7.12, which illustrates a phase space distribution of the particle beam at 24.00 ns. The particles decelerated from the initial velocity (0.877c m/s) dominate the accelerated particles, so that a net loss of the beam kinetic energy leads to ampliﬁcation of the TM01 mode. Fig. 7.13 presents a vector plot of self-ﬁelds generated by the electron beam at t = 76.00 ns (in steady state). Coherent Cerenkov beam-wave interactions give rise

159 Figure 7.12: Phase-space plot at 24.00 ns.

Figure 7.13: A snapshot of steady-state self-ﬁelds (76.00 ns).

to a strong TM01 mode that may be observed within the SWS region. The self-ﬁelds at the output port were analyzed in time and frequency domains, as shown in Fig.

7.14a and Fig. 7.14b, respectively. It can be seen in Fig. 7.14a that although initially there are no oscillations, the output signal starts to oscillate simultaneously with the onset of beam bunching. This signal keeps evolving and eventually approaches a steady state at about 35 ns. By performing the Fourier analysis of steady-state output signals, we show that their spectrum shows a good degree of single-mode

160 purity at fosc = 15.575 GHz, as shown in Fig. 7.14b. In order to verify charge

(a) (b)

Figure 7.14: Output signal analysis in (a) time and (b) frequency domains.

conservation, we plot the normalized residuals (NR) versus the nodal indices for the discrete continuity equation (DCE) and discrete Gauss law (DGL) [51] in Figs. 7.15a and Fig. 7.15b, respectively. Fig. 7.15a shows that NR levels for DCE remain fairly low at all nodes and very close to the double precision ﬂoor (below 10−15). Sparsely observed peaks are due to the fact that the electron-beam edges, where electrons occasionally travel through extremely small cell fractions, generate numerical noise during the scatter step. However, these errors still remain well below 10−10. NR levels for DGL are also distributed around the double precision ﬂoor as shown in Fig.

7.15b, which means that charge conservation is maintained within round-oﬀ errors.

The transverse dynamics of an electron beam in the cross-section plane are sensitive to gyroradius value ρg which depends on the BFS magnetic ﬁeld strength. Fig. 7.16 shows 3D electron velocity plots for a weaker BFS magnetic ﬁeld reduced from 5 T

161 (a) (b)

Figure 7.15: Veriﬁcation of charge conservation at nodes along time (at time-steps of 7.5 × 104, 9 × 104, 12 × 104) by testing NR levels of (a) DCE and (b) DGL.

2.55

2.52 푣 p ×

2.49 10 8

2.46 [m/s]

2.43

2.40

Figure 7.16: 3D velocity plots for an electron beam with the BFS magnetic ﬁeld of 0.5 T.

to 0.5 T. RGB colors again indicate particle velocity magnitudes. The weaker axial magnetic ﬁeld yields a relatively larger ρg in the polar (ρφ) plane, which makes the electron beam to gradually expand radially.

Staircasing error analysis

We now examine staircasing errors resulting from a discrete approximation of the

SCSWS boundary. We consider eight cases comprising the same BWO geometry modeled by: (i) a coarse mesh, (ii) a double-reﬁned mesh, (iii) a quadruple-reﬁned

162 Figure 7.17: SCSWS boundary proﬁles for all cases.

Table 7.3: Mesh information for diﬀerent SCSWS cases

Case N0 N1 N2 ave ledge [mm]

coarse 4,564 12,978 8,415 2.50 double-refined 5,801 16,545 10,745 1.67 quadruple-refined 8,771 25,236 16,466 1.11 octuple-refined 15,333 44,542 29,210 0.74 coarse staircased 4,593 13,005 8,413 2.50 double-refined staircased 5,909 16,766 10,858 1.67 quadruple-refined staircased 8,939 25,547 16,610 1.11 octuple-refined staircased 15,543 44,904 29,362 0.74

mesh, (iv) a octuple-refined mesh, (v) a coarse mesh with staircased boundary, (vi) a double-refined mesh with staircased boundary, (vii) a quadruple-refined mesh with staircased boundary, and (viii) a octuple-refined mesh with staircased boundary.

Fig. 7.17 depicts one period of the SCSWS boundary as rendered by the diﬀerent meshes. All cases employ unstructured meshes. Although unstructured meshes do not necessitate staircased boundaries, the latter are enforced to reproduce the staircasing that would be present on structured meshes. At the same time, this setup allows for a detailed study that eﬀectively isolates staircasing error from grid-dispersion errors.

The mesh information for all cases is shown in Table 7.3. We have performed hot tests for all cases, keeping the number of superparticle per each cell by 106 on average.

163 (a) (b)

Figure 7.18: Field signal at the output port in (a) SCSWS and (b) staircased SCSWS in the time domain.

Field output signals in time and oscillation frequencies are displayed in Fig. 7.18 and Fig. 7.19. The results based on meshes devoid of staricasing error presented in

Fig. 7.18a converge much faster that those of meshes with staircased boundaries in

Fig. 7.18b The oscillation frequency values in Fig. 7.19 also shows the fast convergent rate of present EM-PIC simulations. In paricular, it is interesting to see that the results from the BWO with a staircased SCSWS are unable to capture mcuh RF oscillation at all in the case of coarse and double-reﬁned meshes (case (v) and (vi)).

The RF oscillation is more visible on quadruple- or octuple-reﬁned meshes (case (vii) and (viii)). This is because underestimation of the π-point frequency17 in the modal dispersion causes a slow-charge mode driven by the electron beam [213] to falls into the forward wave region and, as a result, the system does not act as an oscillator

17π-point denotes the solution of the modal ﬁeld which have maximum frequency in the passband.

164 Figure 7.19: Normalized spectral amplitude at the output port in SCSWS and staircased SCSWS.

anymore. Note that the BWO system here is designed to operate at π-point of the modal ﬁelds. The underestimation of the π-point frequency in the staircased boundary is shown by Fig. 7.20, which depicts the dispersion diagrams for the TM01 mode of each SWS. The analytic dispersion relation for the TM01 mode is obtained based on

Floquet’s theory by considering harmonics up to 6th order [214]. It is clearly seen that the π-point frequency decreases as the staircasing error become signiﬁcant. On the other hand, the cases without staircasing errors rapidly converge to the analytic prediction.

165 Figure 7.20: Dispersion relations from “cold tests”.

7.4 Conclusion

We introduced a new axisymmetric charge-conservative EM-PIC algorithm on

unstructured grids for the analysis and design of micromachined VEDs with cylin-

drical axisymmetry. We demonstrated that cylindrical symmetry in device geometry,

ﬁelds, and sources enables reduction of the 3D problem to a 2D one through intro-

ducing an artiﬁcial inhomogeneous medium, considering TEφ-polarized ﬁelds in the

meridian (ρz) plane. As a result, computational resources are signiﬁcantly reduced.

The unstructured-grid spatial discretization is achieved by using Whitney forms in the meridian plane. Using leapfrog time integration, we obtained space- and time- discretized Maxwell’s equations which form a so-called mixed E − B FETD scheme.

A local explicit time update is made possible by employing the SPAI approach [51].

166 Interpolation pf the ﬁeld values to the particles’ positions is also performed by Whit- ney forms and next used for solving the Newton-Lorentz equations of motion of each particle. Relativistic particles are accelerated and pushed in space with a corrected

Boris algorithm. In the particle scatter step, we utilized a Cartesian charge-conserving scatter scheme for unstructured grids [1]. The algorithm was validated considering cylindrical cavity and space-charge-limited (SCL) cylindrical diode problems. We also illustrated the advantages of the present algorithm in the analysis of a BWO system including a slow-wave waveguide structure with complex geometry.

167 Appendix A: Basics of Plasmas

In this chapter, the basics of plasmas are covered including fundamental parameters, its distinct feature (quasi-neutrality and collective interaction), and motion of a single charged particle under electromagnetic ﬁelds.

A.1 Fundamental parameters

Charge is denoted as q measured in Coulomb [C], for example, qe = −1.60217662×

10−1 [C] (negatively charged) is of a single electron with a subscription e. Note that, in what follows, subscriptions i, p, and n are for ion (proton), positron, and neutron,

−1 respectively. qi = qp = 1.60217662 × 10 [C] (positively charged).

Mass (at rest) are measured in kilogram [kg] denoted as m, then, me = mp =

−31 −27 −27 9.10938356×10 [kg] whereas mi = 1.672623×10 [kg] and mn = 1.674929×10

[kg]. Since ion is much heavier than electron, some of our simulations later will assume

that ions are stationary while electrons moving.

Number density, n is how many (charged) particles are present in unit volume,

−3 −3 typically measured in [cm or m ]. Usually it refers to the electron density ne,

for example, 107 [m3] in the solar wind (1AU), 1020 [m3] in Tokamak for magnetic

conﬁnement fusion, 1012 [m3] in the ionosphere etc.

168 Temperature is a quantity measured with thermometer, but here, it is to be related to the molecular kinetic energy to characterize thermal or random motions in gases.

Consider a gas composed of many molecules, then, the ideal gas law18 has states that

PV = NkBT (A.1)

where P [Pa or N/m2] is a pressure exerted by the gas, V [m−3] is the volume it

occupies, N is the number of the molecules, T [K] is its absolute temperature, and kB

is the Boltzmann constant given by kB = 1.3810×10−23 [J/K]. It can be equivalently

associated with kinetic parameters by examining momentum changes in the impaction

of a molecule to a rigid wall, as follows

1 PV = Nm v2 (A.2) 3

where v [m/s] is the speed of the molecules and h·i is the average operator. Note that

1/3 factor ahead of Eqn. (A.2) comes from the space dimensionality of 3. Since PV

in Eqn. (A.2) canceled (divided) by N and 2/3 equals to the average kinetic energy of the gas, a new relationship between temperature and kinetic energy can be obtained as

1 3 hKEi = m v2 = k T. (A.3) 2 2 B

Rearranging the above equation in terms of velocity results in r 3k T v ≡ phv2i = B , (A.4) th m

so called thermal velocity. As expected, the hotter gas, the higher thermal velocity.

It basically describes thermal (or random) motion of particles in the gas (at its rest)

18The relationship was deduced from experimental measurements of Charles’ law and Boyle’s law. It describes the behaviors of real gases under most conditions.

169 and a measure of the temperature. Another choice for thermal velocity could be the

root-mean-sqaure (rms) of the magnitude of the velocity in any one dimension as r k T v = B . (A.5) th m

Note that the kinetic temperature

1 T ≡ m v2 (A.6) 3

is measured through typically electron-volts [eV] unit and 1 [eV] is equivalent to

1.60217662 × 10−19 [J] (same as a single electron charge) or 1 [J] is equivalent to

6.24150913 × 1018 [eV].

Maxwell-Boltzmann distribution naturally models the thermal equilibrium state of gases. There are two speeds to characterize the Maxwell-Boltzmann distribution which are most probable velocity at the maximum probability and rms velocity same

to the thermal velocity.

A.2 Quasi-neutrality in plasma

Consider a plasma composed of an equal number of electrons and ions freely

moving, i.e. n ≡ ne ≈ ni. Although the plasma is almost neutral in a macroscopic

scale, the charge neutrality breaks down in shorter scales, hence, plasma is usually

called quasi-neutral. If we place a test positively-charged point particle into the quasi-

neutral plasma, its electric potential will create forces attracting electrons nearby,

forming a cloud around the test charge to neutralize it. As a consequence, the modiﬁed

potential with the consideration of both the test charge and the electron clouds will

170 take the form of19 √ ! q 2r φ (x) = T exp − (A.7) 4π0r λD where r is a radial distance from the test charge and

2 kBT X vth,sms λ2 = = . (A.8) D 0 P n q2 0 n q2 s 0,s s s 0,s s

As seen in Eqn. (A.7), the potential is exponentially decreasing, therefore, negligible enough since other particles feel it far away from the test charge. in other words, electrostatic fields the test charge is screened out and we usually this screening effect is effective beyond λD, called Debye length. Total number of charged particles per

Debye shpere is the measure of plasma parameter denoted as

4 Λ = πnλ3 . (A.9) 3 D

Usually, in collisionless plasmas, the plasmas parameter greater than 1.

A.3 Plasma oscillation

Here, we investigate dynamic response of a quasi-neutral plasma to a small perturbation, viz. a restoring force. When some electrons are slightly displaced, electric

ﬁelds will be immediately created between the separated gap and it tends to remain light electrons back to original position. Due to their inertia, electrons will overshoot and then vice versa again based on total energy conservation (from electrostatic to kinetic energy or oppositely) as of a pendulum motion. Such a oscillation will continue around the equilibrium position with a speciﬁc frequency depending on the property

19The original electric potential by the test charge is inversely proportional to r2.

171 of the plasma. This frequency is referred as plasma frequency given by s 2 nsqs ωp,s = (A.10) 0ms

for s species where 0 is permittivity in a vacuum. This is the most fundamental

timescale to characterize plasmas and usually the electrons’ is of interest in plasma physics.

A.4 Collisions in plasmas

One of distinct features in plasmas against neutral gases is the collective interaction among charged particles through long-range Coulomb force instead of direct binary collisions of which individual even causes a large deflection in the trajectory of the particle. A collision between two molecules in a neutral gas is basically rigid body collisions (billiard) resulting in large angle of deflection. This can be thought as direct, binary, and strong interaction each other. On the other hand, collisions in a plama are predominantly by Coulomb force which is long-range, so that individual particle will interact all nearby particles. Moreover, Coulomb force becomes rapidly week, most of charged particles will have minor deflections. Plasma can be divided into collisional and collisionless type. Collisional plasmas further can be decomposed into (i) fully ionized and (ii) partially or weakly ionized. In the latter, dominant collisions occur between charged particles and neutral atoms or molecules. For such a collision, we can introduce a key quantity cross sectional area σc to estimate how

often collisions occur. For binary collisions, simply

2 σc ≡ πd0 (A.11)

172 where d0 denotes the radius of the particle. collision frequency is given by

νn ≡ nnσc hvi (A.12)

Then, the average mean free path which is

hvi 1 lmpf ≡ = (A.13) νn nnσc

As expected, the more direct binary collisions happen by increase of velocity, number density, and large cross-section area.

On contrary, Coulomb collisions between charged particles are not straightforward.

This is because deflection by Coulomb collisions are not significant compared to the direct binary collisions. Alternatively, for Coulomb collisions, we define the collision frequency and mean free path is the measure of the particle trajectory is deflected by

90o angle by successive Coulomb interactions.

Being collisionless does not mean that charged particles never have any interactions but interactions dominates their behaviors. In collisionless plasmas, we take into account their collective behaviors, i.e. it is sufficient to consider the effect of the average EM fields on the particles instead of individual collision.

173 Appendix B: Kinetic Plasma Description

B.1 Plasma kinetic equation

In a vacuum, consider a fully ionized plasma composed of a huge number (denoted

as N) of charged particles such as electrons and ions. When individual charged

particle can be modeled as a point-like particle in 3-dimensional space moving with

a speciﬁc velocity, the plasma can be described by summing up all contributions

of which each would be a product between spatial and velocity delta distribution

function as

N m X f (x, v, t) = δ (x − xi (t)) δ (v − vi (t)) (B.1) i=1 where f m is called the microscopic phase space distribution function and the subscription m stands for its microscopic quantity. In other words, f m is the number

of the charged particles over an inﬁnitesimal volume in 6-dimensional phase space,

i.e. number density, therefore, its unit becomes [#/m6s−3]. It is straightforward to

check that, based on conservation of the number of particles (i.e. in the Hamiltonian

dynamical system), Z f m (x, v, t) dv = nm (x, t) , (B.2) 3 Rv Z Z f m (x, v, t) dvdx = N, (B.3) 3 3 Rx Rv

174 3 3 where Rx and Rv are 3-dimensional position and velocity spaces, respectively, and nm (x, t) denotes the microscopic number density. Then, trajectories of all charged paticles can be tracked from a set of

dv m i = q [Em (x , t) + v × Bm (x , t)] , for i = 1, 2, ..., N, (B.4) i dt i i i i

th m m where for the i particle, E (xi, t) B (xi, t) are instantaneous electric ﬁeld intensity

and magnetic ﬂux density, respectively, and mi and qi are mass and charge, respectively. These microscopic electromagnetic ﬁelds, which are very jumpy, are governed by Maxwell’s equation writing

∂Bm ∇ × Em = − , (B.5) ∂t m m 1 ∂E m ∇ × B = 2 + µ0J , (B.6) c0 ∂t m m ∇ · E = −ρ /0, (B.7)

∇ · Bm = 0, (B.8)

m m where ρ and J are electric charge and current densities, respectively, c0 is the speed

of light in a vacuum, and 0 and µ0 are permittivity and permeanbility, respectively.

The microscopic charge density and current densities can be obtained through zeroth and ﬁrst moments for f m over the velocity space, respectively, as follows

N Z N m X m X ρ (x, t) ≡ qi f (x, v, t) dv = qiδ (x − xi (t)) , (B.9) 3 i=1 Rv i N Z N m X m X J (x, t) ≡ qi vf (x, v, t) dv = qivi (t) δ (x − xi (t)) . (B.10) 3 i=1 Rv i

Note that solutions of the microscopic Maxwell’s equations driven by above sources are directly associated with Coulomb forces between two charged particles. Althgouh a system of such microscopic equations yields a complete and exact description of the

175 plasma evolution, it is impossible to solve it due to the (almost) inﬁnite number of

charged particles in usual plasmas (ranging from 1016 to 1024) beyond current high

performance computing capabilities. Alternatively, we can relax the complexity by

using an average procedure to macroscopically model the plasma but still staying

in the kinetic description. In order to develop the average procedure, let us ﬁrst

introduce a single evolution equation embedding the N number of equations of motion

by evaluating the total derivative of f m with respect to time such as

df m ∂f m dx ∂f m dv ∂f m ≡ + · + · dt ∂t dt ∂x dt ∂v ∂f m ∂f m q ∂f m = + v · + [Em + v × Bm] = 0. (B.11) ∂t ∂x m ∂v

This equation is called Klimontovich equation and the reason for which the total time derivative equals to zero can be deduced from df m/dt plugged with (B.1) as

m N df X ∂ dxi ∂ dvi ∂ = + · + · δ (x − x (t)) δ (v − v (t)) , (B.12) dt ∂t dt ∂x dt ∂v i i i=1 with the following equalities

(∂/∂t) δ (x − xi) = − (dxi/dt) · (∂/∂x) δ (x − xi) , (B.13)

(∂/∂t) δ (v − vi) = − (dvi/dt) · (∂/∂v) δ (v − vi) . (B.14)

As a ﬁrst step to average the Klimontovich equation, consider a small volume in the phase space, sized by ∆V = ∆Vx∆Vv where ∆Vx = ∆x∆y∆z and ∆Vv =

∆vx∆vy∆vz, containing N∆V number of charged particles. A carfule selection for the size of the volume is required, satisfying a following condition as

−1/3 n < ∆x < λD (B.15)

where n is the number density and λD is Debye length. This simply means that the average process should have the eﬀective relaxation to the complexity (for the smaller

176 statistical ﬂucuation) as well as capture the fundamental properties of the plasma.

As a consequence, one can ﬁnd the collective behaviors of the plasma present in the

3 2 order of the Debye length scale guarantieeng N∆V (nλD) 1. Let us decompose the microscopic distribution function in terms of the ensemble averaged one and an error term. Taking the ensemble average operator to the distribution function, denoted as hf mi, yields the macroscopic number density with respect to the small volume ∆V given by

N hf m (x, v, t)i ≡ lim ∆V . (B.16) −1/3 n <∆x<λD ∆V and the error term can be written as

δf m ≡ f m − hf mi (B.17) with a zero ensemble average, namely, hδf mi = 0. Specifically, hf mi represents the smoothened properties of the plasma in a scale larger than ∆x while δf m implies the discrete particle (very jumpy and spiky) effects of individual charged particles hidden under the scale (smaller than ∆x). The ensemble average operator should be taken also to the Maxwell dynamic variables, for the example of electric field intensity,

Em = hEmi + δEm. (B.18)

In terms of the macroscopic quantities, we can rewrite Klimontovich equation as

∂ hf mi ∂ hf mi q ∂ hf mi + v · + [hEmi + v × hBmi] ∂t ∂x m ∂v q ∂δf m = [δEm + v × δBm] . (B.19) m ∂v

Terms in the left hand side are in relation to the smoothened response of the plasma whereas the right represents binary Coulomb collisions within the Debye length scale.

177 It is important to note that there are many terminologies referring these two distinct

properties

Not only for its convenience but also due to the fact that the averaged proper-

ties are only of our main interest, we cancel the notations (1) bracket h·i and (2)

superscript m from all the ensemble averaged microscopic quantities, for example,

f (x, v, t) instead of hf m (x, v, t)i, such that

∂f ∂f q ∂f + v · + [E + v × B] = C (f) (B.20) ∂t ∂x m ∂v where C (f) refers a (binary) Coulomb collisional operator again coming from discreteness eﬀects (strong correlations between two charged particles via Coulomb force).

This is the fundamental set of equations decomposed into collisionless and collisional effects and provides the complete kinetic description of a plasma. It should be mentioned that it does not mean the Coulomb collisional operator can collectively take into account all collisional effects but in some cases such as magnetized plasmas the discreteness effect will be reaching and significant beyond the Debye length scale. Nevertheless, this discussion is still useful and intuitive enough at least for this manuscript intensively discussing about particle-in-cell algorithm later on.

B.2 Vlasov equation for collisionless plasmas

There are plasmas where contributions of the Coulomb collisional operator may be negligible, called collisionless plasmas, i.e. C (f) = 0, such that

∂f ∂f q ∂f + v · + [E + v × B] = 0, (B.21) ∂t ∂x m ∂v

which is called Vlasov equation. Following two conditions are typically used to ex-

amine whether or not plasmas are collisionless as (1) ωinterest νc and (2) λD lmfp

178 where ωinterest is a (angular) frequency of our interest, usually chosen as the plasma frequency ωp that is characteristic time scale of plasma, νc is the collisional frequency illustrating how often collisions occur, and lmfp denotes the mean free path that is

a average distance for charged particles to be deﬂected up to 90o. Here, much de-

tails about parameters related to collisional eﬀects are not covered. In a word, in

collisionless plasmas its characteristic time and distance scales are too short for col-

lisions to take place in. Consequently, in addition to the conservation of the number

of particles f becomes incompressible, meaning that their velocity cannot be drifted nor diﬀused20. Such incompressibility allows us to coarse-grain the phase space dis-

tribution function, identically known as superparticle, so as to model the collisionless plasmas more eﬃciently. This is an underlying feature on which particle-in-cell algorithm is based and enjoys for its realization.

B.3 Superparticle: Coarse-grained f (x, v, t)

Once and for all our main focus will be on finding the smoothened properties in the plasmas without any consideration of collisional effects. Let us first decompose the phase distribution function f (x, v, t) for a collisionless plasma with respect to species s as

e,i X f (x, v, t) = fs (x, v, t) . (B.22) s=1

Then, each species function can be represented by the superposition of its coarse- grained (segmentized) versions with an index p (Np in total), centered on the speciﬁc

20Coulomb collisions may lead to velocity drift or diﬀusion such that the phase space distribution function may have compression or expansion.

179 phase space coordinates (xp, vp) as

Np Np X X fs (x, v, t) ≈ fp (x, v, t) = Sx (x − xp) Sv (v − vp) (B.23) p=1 p=1 where fp is called superparticle (also called macroparticle or computational particle by one’s preference) used in particle-in-cell algorithms and S function is a shape function macroscopically describing the number density of actual charged particles at once instead of adding up many delta particles. Note that xp and vp are representing the center phase space coordinates of the superparticle. Each superparticle usually represent millions of actual charged particles that results in the huge reduction the computational costs for plasma kinetic simulations. The shape function is represented by a tensor product 3 Y Sζ (ζ − ζp) = Sζi ζi − ζip (B.24) i=1 where ζi denotes one of components in a generalized coordinate system, ζ = (ζ1, ζ3, ζ3) representing vectors in either Euclidean or velocity spaces. It should have following conditions:

(1) Compact support21

(2) Normalization Z Sζi ζi − ζip dζi = 1 (B.25) 1 Rζi

(3) Symmetry Sζi ζi − ζip = S ζip − ζi (B.26)

In particular, for the velocity coarse-graining, a delta function is usually preferred so as to avoid the dispersion of spatial shape function over the 3-dimensional Euclidean

21Each superparticle taking a small portion of the phase space is supposed not to share its support with others’. For example, exponential functions are not allowable.

180 space, marching on time, where our analysis is mainly done. For the spatial coarse-

graining, here two popular choices are introduced: one is again using a delta function

and the other one is with b-spline. We are going to discuss pros and cons of these

later more in detail.

B.4 Maxwell-Vlasov or Poisson-Vlasov systems

It is still diﬃcult to explicitly ﬁnd solutions of Vlasov equation due to its large

domain dimensionality (i.e. 6-dimensional phase space). Alternatively, equivalent

solutions for the evolution of collisionless plasmas can be obtained through the relax-

ation of Vlasov system which are either Poisson-Vlasov or Maxwell-Vlasov systems.

Plugging the superparticle expression (B.23) to the Vlasov equation (B.21) and eval-

uating zeroth and ﬁrst (with position and velocity vectors) orders of moments for the

resultant equation for all superparticles of s species turn out to be

dN p = 0, (B.27) dt dx p = v , (B.28) dt p dvp qs = [Ep + vp × Bp] , (B.29) dt ms

th for p = 1, 2, ..., Np where all quantities with subscription p are for p superparticle.

Note that electric and magnetic forces, associated with Ep and Bp in the above, are forces directly acting on pth superparticle and can be obtained through mediating

(averaging) electromagnetic ﬁelds by the spatial shape function as ZZZ Ep ≡ E (xp, t) = E (x, t) Sx (x − xp) dx, (B.30)

3 Rx ZZZ Bp ≡ B (xp, t) = B (x, t) Sx (x − xp) dx. (B.31)

3 Rx

181 For the example of the delta shape function, it is going to be direct evaluation of

electromagnetic ﬁelds at the location of superparticles.

Equation (B.27) is about conservation of the number of superparticle, meaning

that describes that any superparticle should not be disappeared nor created and

Newton’s law of motion and Lorentz force can be found in (B.28) and (B.29). Above

equations coupled with Maxwell’s (curl) equations is called Maxwell-Vlasov system

(multiphysical) while dealing with full electromagnetic eﬀects.

In the electrostatic limit, one may be able to neglect the magnetic ﬂux density

term in the Lorentz force such that

dvp qs ≈ Ep. (B.32) dt ms

In this case, Poisson equation is to be associated with (B.28) and (B.32) instead of

Faraday’s and Ampere’s law, referred as Poisson-Vlasov system. Most earlier versions of plasma kinetic simulations were solving Poisson-Vlasov system, on the other hand, this manuscript will focus on the Maxwell-Vlasov system including the solution space of Poisson-Vlasov system.

182 Appendix C: Discrete Exterior Caclulus (DEC)

C.1 Whitney forms

Whitney p-forms are canonical interpolants of discrete diﬀerential p-forms [215].

As explained below, Whitney p-forms are naturally paired to the p-cells of the mesh, where p refers to the dimensionality, i.e. p = 0 refers to nodes, p = 1 to edges, p = 1 to facets and so on [30]. On simplices (e.g. on triangular cells in 2-D or tetrahedral cells in 3-D), Whitney 0-, 1-, and 2-forms are expressed as [30,147,215]

(0) wi = λi, (C.1)

(1) wi = λia dλib − λib dλia , (C.2)

(2) wi = 2 (λia dλib ∧ dλic + λib dλic ∧ dλia + λic dλia ∧ dλib ) , (C.3)

where d is the exterior derivative, ∧ is the exterior product, ia, ib, and ic denote the grid nodes belonging to the i-th p-cell for p = 1 or 2, and λ denotes the barycentric coordinate associated to a given node.

The corresponding vector proxies for Whitney 0-, 1-, and 2-forms write as [1,30]

(0) Wi = λi, (C.4)

(1) Wi = λia ∇λib − λib ∇λia , (C.5)

(2) Wi = 2 (λia ∇λib × ∇λic + λib ∇λic × ∇λia + λic ∇λia × ∇λib ) . (C.6)

183 C.2 Pairing operation

One of the key properties of Whitney p-forms is that they admit a natural “pairing” with the p-cells of the mesh [30]. Computationally, the pairing operation between

i (p) an i-th p-cell of the grid σ(p) and a Whitney form wj associated with the j-th p-cell is eﬀected by the integral below and yields [30,35] Z D i (p)E (p) σ(p), wj = wj = δi,j, (C.7) i σ(p)

where δi,j is the Kronecker delta, for p = 0,..., 3 in 3-D space.

C.3 Generalized Stokes’ theorem

The generalized Stokes’ theorem recovers Stokes’ and Gauss’ theorems of vector

calculus for p = 1, 2, respectively, and the fundamental theorem of calculus for p = 0.

The generalized Stokes’ theorem of exterior calculus for a Whitney p-form [30,32,35,

216,217] states

D (p)E D (p)E σ(p+1), dwj = ∂σ(p+1) (p) , wj (C.8)

where ∂ is the boundary operator that maps an (oriented) p-cell on the grid to the

set of (oriented) (p − 1)-cells comprising its boundary. Note that ∂2 = 0 and hence

d2 = 0 from (C.8). This latter identity is the exterior calculus counterpart of the

vector calculus identities ∇ × ∇ = 0 and ∇ · ∇× = 0.

C.4 Discretization of Maxwell’s equation

C.4.1 Cartesian coordinates case

Using the pairing operation and generalized Stokes’ theorem, we can obtain dis-

crete Maxwell’s equation on a irregular lattice (unstructured grid). For example,

184 applying pairing for the K-th 2-cell into Faraday’s law on the primal mesh gives

∂B σK , dE = σK , , (C.9) (2) (2) ∂t and applying generalized Stokes’ theorem into the left-hand side term of (C.17) yields

∂ ∂σK , E = σK , B . (C.10) (2) ∂t (2)

Substituting (7.5) and (7.6) into (C.18) and using

N1 K X j ∂σ(2) = CK,jσ(1) (C.11) j=1 where CK,j is an element in an incidence matrix which takes a value in the set of

{−1, 0, 1} [30,35,218,219], we obtain

* N1 N1 + * N2 + X X (1) ∂ X (2) C σj , (t) w = σK , (t) w . (C.12) K,j (1) Ej j ∂t (2) Bj k j=1 j=1 k=1 By using (C.7), (C.19) can be rewritten as

N X1 ∂ C (t) = (t) , (C.13) K,jEj ∂tBK j=1 for K = 1, ..., N2. (C.13) represents the discrete representation of Faraday’s law as written in (2.1). Discrete Ampere’s law can be obtained by a similar procedure on the dual mesh.

C.4.2 Body-of-revolution case

By pairing Faraday’s law for the TEφ ﬁeld with the m-th azimuthal eigenmode

k set in (6.26) with K-th 2-cell σ(2) of the FE grid (primal mesh) and applying the generalized Stokes’ theorem, we obtain

* N1 + * N2 + X k h k (1)i ∂ X (2) σK , (t) d0 w = − σK , ⊥ (t) w , (C.14) (2) Ej,m j (2) ∂t Bk,m k j=1 k=1

185 * N1 + * N2 + X k (1) ∂ X (2) σK , (t) w = − σK , ⊥ (t) w . (C.15) (2) Ej,m j (2) ∂t Bk,m k j=1 k=1

K PN1 j Using ∂σ(2) = j=1 CK,jσ(1), where CK,j is the incidence matrix associated to the exterior derivative applied to 1-forms (curl operator on the mesh), see (C.5), we obtain [30,35,218,219]

N1 X k ∂ C (t) = − ⊥ (t) , (C.16) K,jEj,m ∂tBK,m j=1

for m = −Mφ, ..., Mφ. The elements of the incidence matrix take values in the set of

{−1, 0, 1},

J Likewise, pairing (6.27) with J-th 1-cells σ(1) of the primal mesh gives

* N0 + * N1 + J X ⊥ h 0k (0)i J X k (1) σ(1), Ei,m (t) d wi − σ(1), |m| Ej,m (t) wj i=1 j=1

* N1 + ∂ X k (1) = − σJ , (t) w , (C.17) (1) ∂t Bj,m j j=1 and applying generalized Stokes’ theorem to the left-hand side of (C.17) yields

* N0 + * N1 + J X ⊥ (0) J X k (1) ∂σ(1), Ei,m (t) wi − σ(1), |m| Ej,m (t) wj i=1 j=1

* N1 + ∂ X k (1) = − σJ , (t) w , (C.18) (1) ∂t Bj,m j j=1

J PN0 i Similarly to before, we can write ∂σ(1) = i=1 GJ,iσ(1), where GJ,i is the incidence matrix associated to the exterior derivative applied to 0-forms (gradient operator on

the mesh), and obtain

N0 X k ∂ k G ⊥ (t) − |m| (t) = − (t) , (C.19) J,iEi,m EJ,m ∂tBJ,m i=1

for J = 1, ..., N1. An analogous procedure can be used to obtain the discrete rendering

of Ampere’s law for on the dual mesh.

186 Figure C.1: Example (primal) unstructured mesh.

C.5 Incidence Matrices

Frequently used in graph theory, in mathematics an incidence matrix is a matrix providing oriented connectivity information between two classes of objects, for example between nodes and edges, in the incident context. Incidence matrices can be used to represent on a mesh the discrete exterior derivative or, equivalently, the grad, curl, and div operators distilled from their metric structure [30,35,217]. Since, from (C.8), the discrete exterior derivative can be seen as the dual of the boundary operator, incidence matrices encode the relationship between each oriented p-cell of the mesh and its boundary oriented (p−1)-cells (say, between an edge and its boundary nodes, a face element and its boundary edges, and so on). To provide a concrete example, we consider a small mesh with perfect magnetic conductor (or free edges) boundaries

187 (a)

(b)

Figure C.2: Incidence matrices for (a) curl [Dcurl] and (b) gradient [Dgrad] operators for the mesh in Fig. C.1.

188 as depicted in Fig. C.1. Red-colored numbers denote the nodal indices, black-colored numbers the edge indices, and blue-colored numbers the face indices. Intrinsic edge orientation is deﬁned by ascending index order of the two nodes associated with any given edge. For example, if we consider [Dcurl], of size N2 × N1, there are three edges wrapping face number 6: edges 8, 9, and 20. As a result, [Dcurl]6,8 = 1, [Dcurl]6,9 = −1, and [Dcurl]6,9 = 1. The sign is determined by comparing the intrinsic orientation of each edge with the curl in Fig. C.1: if they are opposite, the element is −1, otherwise it is +1. Furthermore, [Dcurl]6,j = 0 for all other j−th edges. This is represented in

Fig. C.2a, which shows the entire [Dcurl] for this mesh. A curl orientation on each face is supposed to follow the intrinsic orientation of the ﬁrst local edge (i.e. an edge with the smallest index among three edges for the face). Likewise, if we consider

[Dgrad], of size N1 × N0, there are two nodes connected to edge 10: nodes 4 and 5.

The corresponding elements are [Dgrad]10,4 = −1 and [Dgrad]10,5 = 1. The element for the diverging node with the gradient (the intrinsic edge orientation) in Fig. C.1 is

−1, otherwise it is +1.

C.6 Discrete Hodge matrix

A (discrete) Hodge star operator encodes all the metric information and is used to transfer information between the primal and dual meshes [30,32,76,85,180]. Here, we use a Galerkin-Hodge construction [32,33,179,180], which leads to symmetric positive deﬁnite matrices and enables energy-conserving discretizations in arbitrary simplicial meshes [35]. As noted before in Section 2, the Galerkin-Hodge operator is not a natural consequence of DEC [182]. The Hodge operator also incorporates the constitutive properties (permittivity and permeability) of the background medium [74].

189 Inhomogeneous and anisotropic media can be easily dealt with by incorporating piecewise constant permittivity and permeability over each cell, for example. In the present FETD-BOR solver, the elements of the Hodge matrices including the radial scaling factor from the cylindrical metric are assembled by adding the contributions from all cells as:

N2 Z 1→1 X (1) (1) [?]J,j = (kρk) WJ · Wj dV, (C.20) k=1 Ωk N2 Z 2→2 X −1 (2) (2) [?µ−1 ]K,k = µk ρk WK · Wk dV, (C.21) k=1 Ωk N2 Z 0→0 X −1 h (0) î h (0) î [?]I,i = kρk WI φ · Wi φ dV, (C.22) k=1 Ωk N2 Z 1→1 X −1 −1 h (1) î h (1) î [?µ−1 ]J,j = µk ρk WJ × φ · Wj × φ dV, (C.23) k=1 Ωk

P3 where Ωk is the area of the k−th cell, and ρk = i=1 ρki /3 where ρki is ρ coordinate of i−th node touching k−th face and for simplicity we have assumed isotropic media assuming permittivity and permeability values k and µk, resp., on cell k. Since

Whitney forms have compact support, we can express the global discrete Hodge matrix as a sum of local matrices (excluding element-wise permittivity and permeability information) for the K-th face as

 1/6 1/12 1/12 0→0 [T ]K = ∆K 1/12 1/6 1/12 , (C.24) 1/12 1/12 1/6  1→1 1→1 1→1 T11 T12 T13 1→1 1→1 1→1 1→1 [T ]K = ∆K T21 T22 T23  , (C.25) 1→1 1→1 1→1 T31 T32 T33 2→2 ˆ [T ]K = 4∆K (∇λ1 × ∇λ2) · φ, (C.26)

190 where ∆K is the area of K-th face and

∇λ · ∇λ ∇λ · ∇λ ∇λ · ∇λ T 1→1 = 1 1 + 2 2 − 1 2 , (C.27) 1,1 6 6 6 ∇λ · ∇λ ∇λ · ∇λ ∇λ · ∇λ T 1→1 = 1 1 − 2 2 − 1 2 , (C.28) 1,2 6 6 6 ∇λ · ∇λ ∇λ · ∇λ ∇λ · ∇λ T 1→1 = 1 1 − 2 2 + 1 2 , (C.29) 1,3 6 6 6 1→1 1→1 T2,1 = T12 , (C.30) ∇λ · ∇λ ∇λ · ∇λ ∇λ · ∇λ T 1→1 = 1 1 + 2 2 + 1 2 , (C.31) 2,2 2 6 2 ∇λ · ∇λ ∇λ · ∇λ ∇λ · ∇λ T 1→1 = 1 1 + 2 2 + 1 2 , (C.32) 2,3 6 6 2 1→1 1→1 T3,1 = T13 , (C.33)

1→1 1→1 T3,2 = T23 , (C.34) ∇λ · ∇λ ∇λ · ∇λ ∇λ · ∇λ T 1→1 = 1 1 + 2 2 + 1 2 . (C.35) 3,3 6 2 2

Due to the local support of the Whitney forms, the above Hodge matrices are very sparse (and diagonally dominant). Their sparsity patterns for the mesh in Fig. C.1 are provided in Fig. C.3. The number of non-zero elements per row (or column) in these Hodge matrices is invariant with respect to the mesh size, so the sparsity increases for larger meshes.

For the axisymmetric EM-PIC code, its discrete Hodge matrices are identical to

1→1 −12→2 [?] and ?µ , respectively. In this case, it can be also analytically evaluated

[56] rather than using the elementwise ρk.

In the Cartesian coordinate system, its discrete Hodge matrices are also identical

1→1 −12→2 to [?] and ?µ , respectively, with ρk = 1.

191 (a) (b)

Figure C.3: Sparsity patterns for discrete Hodge matrices corresponding to the toy 0→0 1→1 −11→1 2→2 mesh depicted in Fig. C.1: (a) [?] , (b) [?] , (c) ?µ , and (d) [?µ−1 ] .

192 C.7 Barycentric dual lattice relations

The barycentric dual lattice [35,210] has a similar contraction identity to (C.7) in

n-dimensional space, given by Z D ˜i (p)E (p) σ˜(n−p), ?wj = ?wj = δ˜i,j. (C.36) ˜i σ˜(n−p) Note that p is the (primal) grid element dimension and a quantity with a tilde is of the dual mesh. Since we consider Hodge duals of electric current and charge densities in the primal mesh, we need to express them in dual formulation to be used in discrete

Ampere’s law. The electric current density can be expressed by

N1 ! N1 X (1) X (1) J = ? (J?) = ? J?,jwj = J?,j ? wj . (C.37) j=1 j=1 In order to obtain the discrete representation for Ampere’s law, we combine the D (K˜ ) E ˜ pairing, i.e. σ˜2 , (·) (at a K-th face in the dual mesh) with Ampere’s law. The electric current density term become

* N1 + N1 N1 (K˜ ) X (1) X D (K˜ ) (1)E X σ˜2 , J?,j ? wj = J?,j σ˜1 , ?wj = δK,j˜ J?,j. (C.38) j=1 j=1 j=1 Therefore, we can write above in matrix form as

[J] = [I] · [J?] (C.39)

where [J] is a column vector with all Dofs for J expanded in terms of Whitney 1- forms on the dual mesh, and [I] is the identity matrix. A similar procedure can be

done for the electric charge density in Gauss’ law:

[Q] = [I] · [Q?] (C.40)

where [Q] is a column vector with all Dofs for Q expanded in terms of Whitney 2-forms on the dual mesh. It is worth mentioning that our charge conserving scatter scheme

193 is a natural result of Galerkin projection of (ambient) current and charge densities,

which are basically to be associated with dual grids, via (known) Whitney forms in

the primal grids. Strictly speaking, the results of the Galerkin projection for the

sources are DoFs (J and Q) on the dual mesh likewise parameters in the constitutive

relations D and H. However, since discrete Hodge matrices for the sources are a tricky identity matrix as the above, such that we do not need to explicitly distinguish them.

194 Appendix D: Cartesian-like PML implementation

A perfectly matched layer (PML) is used to absorb outgoing waves in FE simulations, enabling analysis of open-domain problems [220, 221]. As described before, in the present FETD-BOR the spatial discretization is performed in the meridian plane mapped onto a Cartesian domain with the cylindrical metric factor transferred to the constitutive relations. The resulting constitutive relations correspond to a medium that is inhomogeneous and doubly anisotropic. As such, a Cartesian PML implementation extended to such media can be used. Such formulation exists [212] and is adapted here to the FETD-BOR case as follows.

In the 2-D Cartesian plane, the PML can be eﬀected as an analytic continuation on

R u 0 0 the spatial variables to complex space [212,221], given by u → u˜ = 0 su (u ) du where

0 su (u ) is a complex stretching variable and u stands for ρ or z. This transformation can also be expressed as

r0k → ˜r0k = Γ¯ · r0k, (D.1) where Γ¯ =ρ ˆρˆ(˜ρ/ρ)+ˆzzˆ(˜z/z). As before, the apostrophe 0 in r0k denotes the transverse coordinates on the 2-D meridian plane. The modiﬁed nabla operator (posterior to the TO-based transformation and hence devoid of the 1/ρ factor in the φ derivative)

195 following such analytical continuation is given by

1 ∂ ∂ 1 ∂ ∇0 → ∇˜ 0 =ρ ˆ + φˆ +z ˆ , (D.2) sρ ∂ρ ∂φ sz ∂z or simply

∇˜ 0 = S¯ · ∇0, (D.3)

¯ ˆˆ 0 where S =ρ ˆρˆ(1/sρ) + φφ (1) +z ˆzˆ(1/sz). Following [212], since su (u) and ∂/∂u commute when u 6= u0 and S¯ is a diagonal tensor, the following identity holds for any vector a in the Cartesian-like 2-D meridian plane:

−1 ∇0 × S¯−1 · a = detS¯ S¯ · S¯ · ∇0 × a. (D.4)

Applying this analytic continuation to (6.18), (6.19), (6.32), and (6.33) in the

Fourier domain (with time convention of ejωt) yields the modiﬁed Maxwell’s equations for each mode m as

0 0 ˜ 0k 0kc k 0⊥c k ∇ × E m ˜r = −jωB m ˜r , (D.5)

0 0 0 ˜ 0k 0⊥c k 0kc k 0kc k ˆ ∇ × E m ˜r = −jωB m ˜r + |m| E m ˜r × φ, (D.6)

0 0 ˜ 0k 0kc k 0⊥c k ∇ × H m ˜r = jωD m ˜r , (D.7)

0 0 0 ˜ 0k 0⊥c k 0kc k 0kc k ˆ ∇ × H m ˜r = jωD m ˜r − |m| H m ˜r × φ, (D.8) with constitutive relations in analytic-continued complex space as

0c 0k 0 0c 0k D m ˜r = ¯ (ω) · E m ˜r , (D.9)

0c 0k 0 0c 0k B m ˜r = µ¯ (ω) · H m ˜r , (D.10) where the superscript c denotes non-Maxwellian (complex space) ﬁelds and ¯0 and

µ¯0 indicates constitutive parameters of the original medium incorporating the radial

196 scaling factors from the TO mapping. Next, using (D.1) and (D.3), we can revert

(D.5)−(D.8) back to a real-valued spatial domain by writing

0 0 ¯ 0k 0kc ¯ k 0⊥c ¯ k S · ∇ × E m Γ · r = −jωB m Γ · r , (D.11)

0 0 0 ¯ 0k 0⊥c ¯ k 0kc ¯ k ˆ 0kc ¯ k S · ∇ × E m Γ · r = −jωB m Γ · r − |m| φ × E m Γ · r , (D.12)

0 0 ¯ 0k 0kc ¯ k 0⊥c ¯ k S · ∇ × H m Γ · r = jωD m Γ · r , (D.13)

0 0 0 ¯ 0k 0⊥c ¯ k 0kc ¯ k ˆ 0kc ¯ k S · ∇ × H m Γ · r = jωD m Γ · r + |m| φ × H m Γ · r . (D.14)

Using the identity (D.4), we can rewrite (D.11)−(D.14) as

h 0 i −1 0 0k ¯−1 0kc ¯ k ¯ ¯ 0⊥c ¯ k ∇ × S · E m Γ · r = −jω detS S · B m Γ · r , (D.15) h 0 i −1 0 0k ¯−1 0⊥c ¯ k ¯ ¯ 0kc ¯ k ∇ × S · E m Γ · r = −jω detS S · B m Γ · r −1 n 0 o ¯ ¯ ˆ 0kc ¯ k − |m| detS S · φ × E m Γ · r , (D.16) h 0 i −1 0 0k ¯−1 0kc ¯ k ¯ ¯ 0⊥c ¯ k ∇ × S · H m Γ · r = jω detS S · D m Γ · r , (D.17) h 0 i −1 0 0k ¯−1 0⊥c ¯ k ¯ ¯ 0kc ¯ k ∇ × S · H m Γ · r = jω detS S · D m Γ · r −1 n 0 o ¯ ¯ ˆ 0kc ¯ k + |m| detS S · φ × H m Γ · r . (D.18)

We can further verify the identity below

−1 n 0 o h 0 i ¯ ¯ ˆ 0kc ¯ k ˆ ¯−1 0kc ¯ k detS S · φ × E m Γ · r = φ × S · E m Γ · r , (D.19)

−1 n 0 o h 0 i ¯ ¯ ˆ 0kc ¯ k ˆ ¯−1 0⊥c ¯ k detS S · φ × H m Γ · r = φ × S · H m Γ · r . (D.20)

197 and introduce a new set of ﬁelds deﬁned as

0 0 0a k ¯−1 0c ¯ k E m r = S · E m Γ · r , (D.21)

0 0 0a k ¯−1 0c ¯ k H m r = S · H m Γ · r , (D.22)

0 −1 0 0a k ¯ ¯ 0c ¯ k D m r = detS S · D m Γ · r , (D.23)

0 −1 0 0a k ¯ ¯ 0c ¯ k B m r = detS S · B m Γ · r , (D.24) so that, by substituting (D.21)−(D.24) back into (D.15)−(D.18), and utilizing the identities (D.19) and (D.20), we ﬁnally obtain

0k 0ka 0k 0⊥a 0k ∇ × E m r = −jωB m r , (D.25)

0k 0⊥a 0k 0ka 0k 0ka 0k ˆ ∇ × E m r = −jωB m r + |m| E m r × φ, (D.26)

0k 0ka 0k 0⊥a 0k ∇ × H m r = jωD m r , (D.27)

0k 0⊥a 0k 0ka 0k 0ka 0k ˆ ∇ × H m r = jωD m r − |m| H m r × φ. (D.28) with

0 −1 n o 0 0a k ¯ ¯ 0 ¯ 0a k D m r = detS S · ¯ (ω) · S · E m r , (D.29) 0 −1 n o 0 0a k ¯ ¯ 0 ¯ 0a k B m r = detS S · µ¯ (ω) · S · H m r . (D.30)

0a 0a 0a 0a The above expressions show that E m, H m, D m, and B m obey Maxwell’s equations in an equivalent PML medium with constitutive parameters given by

−1 n o ¯PML = detS¯ S¯ · ¯0 (ω) · S¯ , (D.31)

−1 n o µ¯PML = detS¯ S¯ · µ¯0 (ω) · S¯ . (D.32)

198 As an example, consider a background medium with   ρ (ω) 0 0 ¯(ω) =  0 φ (ω) 0  , (D.33) 0 0 z (ω)   µρ (ω) 0 0 µ¯ (ω) =  0 µφ (ω) 0  , (D.34) 0 0 µz (ω) σm with ρ (ω) = φ (ω) = z (ω) = 1 + , corresponding to a lossy, isotropic, homo- jω0 geneous medium. After the TO-based mapping, we obtain

ρ (ω) ρ 0 0  ¯0 ¯ ¯ φ(ω) ¯ (ω) = ¯(ω) · R =  0 ρ 0  , (D.35) 0 0 z (ω) ρ

µρ (ω) ρ 0 0  ¯0 ¯ ¯ µφ(ω) µ¯ (ω) = µ¯ (ω) · Rµ =  0 ρ 0  , (D.36) 0 0 µz (ω) ρ As a result, by using (D.31) and (D.32), the elements of the resulting PML constitutive

tensor write as:

PML PML σm jω0 + σρ ρ (ω) = 0 1 + PML , (D.37) jω0 (jω0 + σz ) σ (jω )2 PML (ω) = 1 + m 0 , (D.38) φ 0 PML PML jω0 jω0 + σρ (jω0 + σz ) σ jω + σPML PML (ω) = 1 + m 0 z , (D.39) z 0 PML jω0 jω0 + σρ PML PML jω0 + σρ µρ (ω) = µ0 PML , (D.40) (jω0 + σz ) (jω )2 µPML (ω) = µ 0 , (D.41) φ 0 PML PML jω0 + σρ (jω0 + σz ) jω + σPML µPML (ω) = µ 0 z . (D.42) z 0 PML jω0 + σρ

PML PML where σρ and σz are the artiﬁcial PML conductivities along ρ and z respectively. The presence of jω factors in the above Fourier-domain elements produce modiﬁca-

tions in the corresponding ﬁeld equations in the time-domain. These modiﬁcations

199 are implemented using an auxiliary diﬀerential equation (ADE) approach as described in, e.g., [74,75].

200 Appendix E: Stability Conditions

To determine the stability conditions, we express the ﬁeld update in matrix form as

w¯ n+1 = G¯ · w¯ n = ¯I + T¯ · w¯ n (E.1) with

 1   n+ 1  ⊥ n− 2 ⊥ 2 Bm Bm 1 1 n− h in+ 2  h k i 2   k  n  Bm  n+1  Bm  w¯ =   , w¯ =   , (E.2)  ⊥ n   ⊥ n+1   Em  Em  n   n+1  h k i h k i  Em Em and  ¯ ¯ ¯  0N2×N2 , 0N2×N1 , 0N2×N0 , −∆t [Dcurl]  0¯ , 0¯ , −∆t [D ] , ∆t |m| ¯I  ¯  N1×N2 N1×N1 grad N1×N1  T = ¯ ¯ 2 ¯ 2 ¯ ,  0¯ , ∆tX φ , −∆t X¯ φ · [D ], ∆t |m| X φ   N0×N2 TM TM grad TM  ¯ ¯ 2 ¯ 2 ¯ 2 2 ¯ ∆tXTEφ , −∆t |m| A, −∆t |m| A · [Dgrad] , −∆t XTEφ · [Dcurl] − ∆t |m| A (E.3) where

¯ 0→0−1 T −11→1 XTMφ = [?] · [Dgrad] · ?µ , (E.4)

¯ 1→1−1 T −12→2 XTEφ = [?] · [Dcurl] · ?µ , (E.5)

¯ 1→1−1 −11→1 A = [?] · ?µ . (E.6)

201 ¯ A necessary condition for stability is |λG¯ | ≤ 1 for all eigenvalues λG¯ of G [222]. When m = 0, the ﬁeld update equation becomes decoupled into two independent

numerical integrators for TEφ and TMφ ﬁelds. In this case, following [33], we can

easily obtain the stability criteria for both polarizations in closed form as

2 ∆t φ ≤ , (E.7) TE ,m=0 r max λ XTEφ ·[Dcurl] 2 ∆t φ ≤ , (E.8) TM ,m=0 r max λ XTMφ ·[Dgrad]

φ φ where λX φ ·[Dcurl] and λ denote the eigenvalues of XTE ·[Dcurl] and XTM · TE XTMφ ·[Dgrad]

[Dgrad] respectively.

When m 6= 0, we can simply represent G¯ using 2 × 2 block matrices X¯ and [D] as

¯I , −∆t [D] G¯ = (N2+N1)×(N2+N1) (E.9) ¯ ¯ 2 ¯ ∆tX, I(N0+N1)×(N0+N1) − ∆t X · [D] where

¯ ¯ 0¯ , X¯ φ ¯ N0×N2 TM X = ¯ ¯ , (E.10) XTEφ , − |m| A and

¯ 0N2×N0 , [Dcurl] [D] = ¯ . (E.11) [Dgrad] − |m| IN1×N1

Therefore, the stability condition is similarly obtained as

2 ∆t ≤ (E.12) m6=0 r max λX¯ ·[D]

¯ where λX¯ ·[D] are the eigenvalues of X · [D]. Note that in this case the maximum time step depends on the modal index magnitude |m|.

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