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1. High Spatial Order Energy Stable FDTD Methods for Maxwell’s Equations in Nonlinear Optical Media in One Dimension

   Bokil, Vrushali A ; Cheng, Yingda ; Jiang, Yan ; Li, Fengyan ( April 2018 , Journal of scientific computing)

   In this paper, we consider electromagnetic (EM) wave propagation in nonlinear optical media in one spatial dimension. We model the EM wave propagation by the time- dependent Maxwell’s equations coupled with a system of nonlinear ordinary differential equations (ODEs) for the response of the medium to the EM waves. The nonlinearity in the ODEs describes the instantaneous electronic Kerr response and the residual Raman molecular vibrational response. The ODEs also include the single resonance linear Lorentz dispersion. For such model, we will design and analyze fully discrete finite difference time domain (FDTD) methods that have arbitrary (even) order in space and second order in time. It is challenging to achieve provable stability for fully discrete methods, and this depends on the choices of temporal discretizations of the nonlinear terms. In Bokil et al. (J Comput Phys 350:420–452, 2017), we proposed novel modifications of second-order leap-frog and trapezoidal temporal schemes in the context of discontinuous Galerkin methods to discretize the nonlinear terms in this Maxwell model. Here, we continue this work by developing similar time discretizations within the framework of FDTD methods. More specifically, we design fully discrete modified leap-frog FDTD methods which are proved to be stable under appropriate CFL conditions. These method can be viewed as an extension of the Yee-FDTD scheme to this nonlinear Maxwell model. We also design fully discrete trapezoidal FDTD methods which are proved to be unconditionally stable. The performance of the fully discrete FDTD methods are demonstrated through numerical experiments involving kink, antikink waves and third harmonic generation in soliton propagation. 

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2. Dispersion analysis of finite difference and discontinuous Galerkin schemes for Maxwell’s equations in linear Lorentz media

   Jiang, Yan ; Sakkaplangkul, Puttha ; Bokil, Vrushali A ; Cheng, Yingda ; Li, Fengyan ( May 2019 , Journal of computational physics)

   In this paper, we consider Maxwell’s equations in linear dispersive media described by a single-pole Lorentz model for electronic polarization. We study two classes of commonly used spatial discretizations: finite difference methods (FD) with arbitrary even order accuracy in space and high spatial order discontinuous Galerkin (DG) finite element methods. Both types of spatial discretizations are coupled with second order semi-implicit leap-frog and implicit trapezoidal temporal schemes. By performing detailed dispersion analysis for the semi-discrete and fully discrete schemes, we obtain rigorous quantification of the dispersion error for Lorentz dispersive dielectrics. In particular, comparisons of dispersion error can be made taking into account the model parameters, and mesh sizes in the design of the two types of schemes. This work is a continuation of our previous research on energy-stable numerical schemes for nonlinear dispersive optical media [6,7]. The results for the numerical dispersion analysis of the reduced linear model, considered in the present paper, can guide us in the optimal choice of discretization parameters for the more complicated and nonlinear models. The numerical dispersion analysis of the fully discrete FD and DG schemes, for the dispersive Maxwell model considered in this paper, clearly indicate the dependence of the numerical dispersion errors on spatial and temporal discretizations, their order of accuracy, mesh discretization parameters and model parameters. The results obtained here cannot be arrived at by considering discretizations of Maxwell’s equations in free space. In particular, our results contrast the advantages and disadvantages of using high order FD or DG schemes and leap-frog or trapezoidal time integrators over different frequency ranges using a variety of measures 

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3. Body-of-revolution finite-difference time-domain modeling of hybrid-plasmonic ring resonators

   Mirzaei-Ghormish, S ; Shahabadi, M ; Smalley, D.E. ( September 2022 , Optics express)

   Development of a computational technique for the analysis of quasi-normal modes in hybrid-plasmonic resonators is the main goal of this research. Because of the significant computational costs of this analysis, one has to take various symmetries of these resonators into account. In this research, we consider cylindrical symmetry of hybrid-plasmonic ring resonators and implement a body-of-revolution finite-difference time-domain (BOR-FDTD) technique to analyze these resonators. We extend the BOR-FDTD method by proposing two different sets of auxiliary fields to implement multi-term Drude-Lorentz and multi-term Lorentz models in BOR-FDTD. Moreover, we utilize the filter-diagonalization method to accurately compute the complex resonant frequencies of the resonators. This approach improves numerical accuracy and computational time compared to the Fourier transform method used in previous BOR-FDTD methods. Our numerical analysis is verified by a 2D axisymmetric solver in COMSOL Multiphysics. 

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4. Body-of-revolution finite-difference time-domain modeling of hybrid-plasmonic ring resonators

   https://doi.org/10.1364/OE.468596  

   Mirzaei-Ghormish, S. ; Shahabadi, M. ; Smalley, D. E. ( September 2022 , Optics Express)

   Development of a computational technique for the analysis of quasi-normal modes in hybrid-plasmonic resonators is the main goal of this research. Because of the significant computational costs of this analysis, one has to take various symmetries of these resonators into account. In this research, we consider cylindrical symmetry of hybrid-plasmonic ring resonators and implement a body-of-revolution finite-difference time-domain (BOR-FDTD) technique to analyze these resonators. We extend the BOR-FDTD method by proposing two different sets of auxiliary fields to implement multi-term Drude-Lorentz and multi-term Lorentz models in BOR-FDTD. Moreover, we utilize the filter-diagonalization method to accurately compute the complex resonant frequencies of the resonators. This approach improves numerical accuracy and computational time compared to the Fourier transform method used in previous BOR-FDTD methods. Our numerical analysis is verified by a 2D axisymmetric solver in COMSOL Multiphysics.

    

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5. Full-wave modeling of EMIC wave packets: ducted propagation and reflected waves

   https://doi.org/10.3389/fspas.2023.1251563  

   Hanzelka, Miroslav ; Li, Wen ; Ma, Qianli ; Qin, Murong ; Shen, Xiao-Chen ; Capannolo, Luisa ; Gan, Longzhi ( October 2023 , Frontiers in Astronomy and Space Sciences)

   Electromagnetic ion cyclotron (EMIC) waves can scatter radiation belt electrons with energies of a few hundred keV and higher. To accurately predict this scattering and the resulting precipitation of these relativistic electrons on short time scales, we need detailed knowledge of the wave field’s spatio-temporal evolution, which cannot be obtained from single spacecraft measurements. Our study presents EMIC wave models obtained from two-dimensional (2D) finite-difference time-domain (FDTD) simulations in the Earth’s dipole magnetic field. We study cases of hydrogen band and helium band wave propagation, rising-tone emissions, packets with amplitude modulations, and ducted waves. We analyze the wave propagation properties in the time domain, enabling comparison within situobservations. We show that cold plasma density gradients can keep the wave vector quasiparallel, guide the wave energy efficiently, and have a profound effect on mode conversion and reflections. The wave normal angle of unducted waves increases rapidly with latitude, resulting in reflection on the ion hybrid frequency, which prohibits propagation to low altitudes. The modeled wave fields can serve as an input for test-particle analysis of scattering and precipitation of relativistic electrons and energetic ions.

    

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