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1. CONVERGENCE ANALYSIS OF YEE-FDTD SCHEMES FOR 3D MAXWELL’S EQUATIONS IN LINEAR DISPERSIVE MEDIA

   Sakkaplangkul, P; Bokil, V. (January 2021, International journal of numerical analysis and modeling)

   null (Ed.)

   In this paper, we develop and analyze finite difference methods for the 3D Maxwell’s equations in the time domain in three different types of linear dispersive media described as Debye, Lorentz and cold plasma. These methods are constructed by extending the Yee-Finite Difference Time Domain (FDTD) method to linear dispersive materials. We analyze the stability criterion for the FDTD schemes by using the energy method. Based on energy identities for the continuous models, we derive discrete energy estimates for the FDTD schemes for the three dispersive models. We also prove the convergence of the FDTD schemes with perfect electric conducting boundary conditions, which describes the second order accuracy of the methods in both time and space. The discrete divergence-free conditions of the FDTD schemes are studied. Lastly, numerical examples are given to demonstrate and confirm our results. 

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2. Finite-difference time-domain methods

   https://doi.org/10.1038/s43586-023-00257-4  

   Teixeira, F. L.; Sarris, C.; Zhang, Y.; Na, D.-Y.; Berenger, J.-P.; Su, Y.; Okoniewski, M.; Chew, W. C.; Backman, V.; Simpson, J. J. (December 2023, Nature Reviews Methods Primers)

   The finite-difference time-domain (FDTD) method is a widespread numerical tool for full-wave analysis of electromagnetic fields in complex media and for detailed geometries. Applications of the FDTD method cover a range of time and spatial scales, extending from subatomic to galactic lengths and from classical to quantum physics. Technology areas that benefit from the FDTD method include biomedicine — bioimaging, biophotonics, bioelectronics and biosensors; geophysics — remote sensing, communications, space weather hazards and geolocation; metamaterials — sub-wavelength focusing lenses, electromagnetic cloaks and continuously scanning leaky-wave antennas; optics — diffractive optical elements, photonic bandgap structures, photonic crystal waveguides and ring-resonator devices; plasmonics — plasmonic waveguides and antennas; and quantum applications — quantum devices and quantum radar. This Primer summarizes the main features of the FDTD method, along with key extensions that enable accurate solutions to be obtained for different research questions. Additionally, hardware considerations are discussed, plus examples of how to extract magnitude and phase data, Brillouin diagrams and scattering parameters from the output of an FDTD model. The Primer ends with a discussion of ongoing challenges and opportunities to further enhance the FDTD method for current and future applications. 

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3. 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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4. Coupled-mode theory for plasmonic resonators integrated with silicon waveguides towards mid-infrared spectroscopic sensing

   https://doi.org/10.1364/OE.28.002020  

   Chen, Che; Oh, Sang-Hyun; Li, Mo (January 2020, Optics Express)

   Advances in mid-IR lasers, detectors, and nanofabrication technology have enabled new device architectures to implement on-chip sensing applications. In particular, direct integration of plasmonic resonators with a dielectric waveguide can generate an ultra-compact device architecture for biochemical sensing via surface-enhanced infrared absorption (SEIRA) spectroscopy. A theoretical investigation of such a hybrid architecture is imperative for its optimization. In this work, we investigate the coupling mechanism between a plasmonic resonator array and a waveguide using temporal coupled-mode theory and numerical simulation. The results conclude that the waveguide transmission extinction ratio reaches maxima when the resonator-waveguide coupling rate is maximal. Moreover, after introducing a model analyte in the form of an oscillator coupled with the plasmonics-waveguide system, the transmission curve with analyte absorption can be fitted successfully. We conclude that the extracted sensing signal can be maximized when analyte absorption frequency is the same as the transmission minima, which is different from the plasmonic resonance frequency. This conclusion is in contrast to the dielectric resonator scenario and provides an important guideline for design optimization and sensitivity improvement of future devices. 

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5. FDTD-Net: A Deep Learning technique to model the FDTD Method

   Alvarez, M; Arzuaga, E; Sierra, H (August 2023, 21 st LACCEI International Multi-Conference for Engineering, Education, and Technology: “Leadership in Education and Innovation in Engineering in the Framework of Global Transformations: Integration and Alliances for Integral Development”)

   The Finite-Difference Time-Domain (FDTD) method is a numerical modeling technique used by researchers as one of the most accurate methods to simulate the propagation of an electromagnetic wave through an object over time. Due to the nature of the method, FDTD can be computationally expensive when used in complex setting such as light propagation in highly heterogenous object such as the imaging process of tissues. In this paper, we explore a Deep Learning (DL) model that predicts the evolution of an electromagnetic field in a heterogeneous medium. In particular, modeling for propagation of a Gaussian beam in skin tissue layers. This is relevant for the characterization of microscopy imaging of tissues. Our proposed model named FDTD-net, is based on the U-net architecture, seems to perform the prediction of the electric field (EF) with good accuracy and faster when compared to the FDTD method. A dataset of different geometries was created to simulate the propagation of the electric field. The propagation of the electric field was initially generated using the traditional FDTD method. This data set was used for training and testing of the FDTD-net. The experiments show that the FDTD-net learns the physics related to the propagation of the source in the heterogeneous objects, and it can capture changes in the field due to changes in the object morphology. As a result, we present a DL model that can compute a propagated electric field in less time than the traditional method. 

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