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Sparsifying preconditioner for the time-harmonic Maxwell's equations
摘要： This paper presents the sparsifying preconditioner for the time-harmonic Maxwell’s equations in the integral formulation. Following the work on sparsifying preconditioner for the Lippmann–Schwinger equation, this paper generalizes that approach from the scalar wave case to the vector case. The key idea is to construct a sparse approximation to the dense system by minimizing the non-local interactions in the integral equation, which allows for applying sparse linear solvers to reduce the computational cost. When combined with the standard GMRES solver, the number of preconditioned iterations remains small and essentially independent of the frequency. This suggests that, when the sparsifying preconditioner is adopted, solving the dense integral system can be done as efficiently as solving the sparse system from PDE discretization.
关键词： Electromagnetic scattering,Preconditioner,Maxwell’s equations,Sparse linear algebra
更新于2025-09-23 15:23:52
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Variational source condition for ill-posed backward nonlinear Maxwell's equations
摘要： This paper examines the mathematical analysis of an electromagnetic inverse problem governed by nonlinear evolutionary Maxwell’s equations. The aim of the inverse problem is to recover electromagnetic ?elds at the past time by noisy measurement data at the present time. We consider the Tikhonov regularization method to cope with the ill-posedness of the governing backward nonlinear Maxwell’s equations. By means of the semigroup theory, we study its convergence analysis and derive optimality conditions through a rigorous ?rst-order analysis and adjoint calculus. The ?nal part of the paper is focused on the convergence rate analysis of the Tikhonov regularization method under a variational source condition (VSC), which leads to power-type convergence rates. Employing the spectral theory, the complex interpolation theory and fractional Sobolev spaces, we validate the proposed VSC on account of an appropriate regularity assumption on the exact initial data and the material parameters.
关键词： convergence rate analysis,electromagnetic inverse problem,variational source condition,nonlinear Maxwell’s equations,Tikhonov regularization
更新于2025-09-23 15:21:21
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Nonperturbative nonlinear effects in the dispersion relations for TE and TM plasmons on two-dimensional materials
摘要： We analytically obtain the dispersion relations for transverse-electric (TE) and transverse-magnetic (TM) surface plasmon-polaritons in a nonlinear two-dimensional (2D) conducting material with inversion symmetry lying between two Kerr-type dielectric media. To this end, we use Maxwell’s equations within the quasielectrostatic, weakly dissipative regime. We show that the wavelength and propagation distance of surface plasmons decrease due to the nonlinearity of the surrounding dielectric. In contrast, the effect of the nonlinearity of the 2D material depends on the signs of the real and imaginary parts of the third-order conductivity. Notably, the dispersion relations obtained by naively replacing the permittivity of the dielectric medium by its nonlinear counterpart in the respective dispersion relations of the linear regime are not accurate. We apply our analysis to the case of doped graphene and make predictions for the surface plasmon wavelength and propagation distance.
关键词： Maxwell’s equations,quasielectrostatic regime,weakly dissipative regime,nonlinear two-dimensional materials,surface plasmon-polaritons,doped graphene,Kerr-type dielectric media
更新于2025-09-23 15:21:21
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[Differential-Algebraic Equations Forum] || Systems of Differential Algebraic Equations in Computational Electromagnetics
摘要： Starting from space-discretisation of Maxwell’s equations, various classical formulations are proposed for the simulation of electromagnetic fields. They differ in the phenomena considered as well as in the variables chosen for discretisation. This contribution presents a literature survey of the most common approximations and formulations with a focus on their structural properties. The differential-algebraic character is discussed and quantified by the differential index concept.
关键词： Quasistatic approximations,Maxwell’s equations,DAE index
更新于2025-09-23 15:21:21
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Homogenization of time-harmonic Maxwella??s equations in nonhomogeneous plasmonic structures
摘要： We carry out the homogenization of time-harmonic Maxwell’s equations in a periodic, layered structure made of two-dimensional (2D) metallic sheets immersed in a heterogeneous and in principle anisotropic dielectric medium. In this setting, the tangential magnetic field exhibits a jump across each sheet. Our goal is the rigorous derivation of the effective dielectric permittivity of the system from the solution of a local cell problem via suitable averages. Each sheet has a fine-scale, inhomogeneous and possibly anisotropic surface conductivity that scales linearly with the microstructure scale, d. Starting with the weak formulation of the requisite boundary value problem, we prove the convergence of its solution to a homogenization limit as d approaches zero. The effective permittivity and cell problem express a bulk average from the host dielectric and a surface average germane to the 2D material (metallic layer). We discuss implications of this analysis in the modeling of plasmonic crystals.
关键词： Jump condition on hypersurface,Time-harmonic Maxwell’s equations,Layered structures,Two-scale convergence,Homogenization
更新于2025-09-23 15:21:01
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Effect of nonlinear cap layer on TM-polarized surface waves in a graphene-based photonic crystal
摘要： In this paper, we investigated the effect of nonlinear cap layer and the chemical potential of the graphene on the p-polarized surface waves (SWs) in a one-dimensional graphene-based photonic crystal (GBPC). Three different cases were considered for the dielectric tensor of the nonlinear cap layer: (1) perpendicular uniaxial approximation, (2) parallel uniaxial approximation and (3) generic case. The results of the study indicated that the effect of nonlinear cap layer on TM SWs may be neglected due to the perpendicular approximation. On the other hand, in parallel approximation as well as generic form of dielectric function, the frequency of the TM SWs within the graphene-induced band gap (GIPBG) may be controlled by the nonlinear cap layer and the chemical potential of the graphene. Moreover, according to the results, in the GIPBG of GBPC with self-defocusing cap layer, enhancing chemical potential and optical nonlinearity shift the frequency of the SWs in the same direction. However, with respect to the self-focusing cap layer, enhancing chemical potential and optical nonlinearity shift the frequency of the SWs in the opposite direction. Consequently, by adjusting the chemical potential in the self-defocusing cap layer, the tunability range of the SW frequency may be extended and in the case of self-focusing, optical nonlinear effects may be eliminated.
关键词： Graphene,First integral,Nonlinear optics,Surface waves,Maxwell’s equations,Photonic crystal
更新于2025-09-23 15:21:01
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[IEEE 2019 PhotonIcs & Electromagnetics Research Symposium - Spring (PIERS-Spring) - Rome, Italy (2019.6.17-2019.6.20)] 2019 PhotonIcs & Electromagnetics Research Symposium - Spring (PIERS-Spring) - Stability of Least Squares for the Solution of an Ill-posed Inverse Problem of Reconstructing Real Value of the Pcrrnittivity of a Dielectric Layer in a Rectangular Waveguide
摘要： We use the solution to Maxwell’s equations in a rectangular single-mode waveguide with multi-mode boundary conditions at its flanges to determine the parameters of the inclusion. The well-posedness of the inverse problem is studied using explicit expressions for the S-parameters of the waveguide when the inclusion is a plane-parallel dielectric slab. The methods of reconstructing permittivity from the measured (experimental) data must be stable and well-conditioned. The importance of these demands is discussed taking as an example the determination of real permittivity from noisy measurement data of the transmission coefficient of the principal waveguide mode. Generally, this problem is unsolvable because the range of the function to be inverted forms a set of measure zero on the complex plane. In addition, the occurrence of self-intersections of the parametric curve leads to non-uniqueness of the solution to the inverse problem. Therefore, approximate methods of the permittivity reconstruction in the vicinity of such points may be unstable or ill-conditioned. In our study, we present several examples of such algorithms. We demonstrate that the method of least squares applied for reconstructing permittivity of the inclusion from multi-frequency measurement data is a stable algorithm for the solution to this inverse problem. This approach does not use a priori estimates for the sought parameter and information about the location of singularities of the parametric curve. We determine the interval of variation of the condition number for the method of least squares and show that this quantity decreases as one of the following parameters increases: the width of the dielectric layer, the measurement frequency band, or its distance to the lower cutoff value. Using these results, we estimate the rate of convergence of the approximate solution to the exact value of the sought parameter when the quality of the measurement data is improved and show how to choose optimal parameters of the experiment and the measurement setup.
关键词： inverse problem,permittivity,Maxwell’s equations,waveguide,least squares method
更新于2025-09-23 15:21:01
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Nonlinear Optics || Wave-Equation Description of Nonlinear Optical Interactions
摘要： We have seen in the last chapter how nonlinearity in the response of a material system to an intense laser field can cause the polarization of the medium to develop new frequency components not present in the incident radiation field. These new frequency components of the nonlinear polarization act as sources of new frequency components of the electromagnetic field. In the present chapter, we examine how Maxwell’s equations describe the generation of these new components, and more generally we see how the various frequency components of the field become coupled by the nonlinear interaction.
关键词： wave equation,electromagnetic field,frequency components,Maxwell’s equations,nonlinear optics,polarization
更新于2025-09-23 15:21:01
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Propagation of Electromagnetic Waves in an Open Planar Dielectric Waveguide Filled with a Nonlinear Medium II: TM Waves
摘要： A nonlinear eigenvalue problem on an interval is considered. The nonlinearity in the equation is specified by a nonnegative monotonically increasing function, and the boundary conditions nonlinearly depend both on the sought-for functions and on the spectral parameter. The discrete eigenvalues are defined using an additional (local) condition at one end of the interval. This problem describes the propagation of monochromatic (polarized) electromagnetic TM waves in a planar dielectric waveguide filled with a nonlinear medium. The nonlinearity covers a wide range of laws of nonlinear optics corresponding to self-interaction effects. Results on the solvability of the problem and the properties of the eigenvalues are obtained.
关键词： Maxwell’s equations,nonlinear eigenvalue problem,nonlinear permittivity,nonlinear Sturm–Liouville problem,planar dielectric waveguide,asymptotics of eigenvalues,comparison theorem
更新于2025-09-23 15:21:01
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Plasma and Plasmonics || 9 Numerical Methods for Electromagnetics
摘要： In this chapter, we will discuss some of the basic numerical methods that can be used to perform computer simulations of Maxwell’s equations in various conditions. Computational electromagnetics is a very broad field with too many different methods available for different situations. But we will only give a brief introduction to this topic in this book and encourage the reader to refer to other sources [31] for more details.
关键词： Maxwell’s equations,Electromagnetics,Numerical Methods,Computational Electromagnetics
更新于2025-09-23 15:19:57