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Mathematics
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Advanced Mathematical and Numerical Modeling in Electrodynamics and Photonics
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Advanced Mathematical and Numerical Modeling in Electrodynamics and Photonics

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: 30 April 2024 | Viewed by 10127

Special Issue Editors

Dr. André Nicolet

E-Mail Website
Guest Editor
Institut Fresnel, Aix-Marseille Université, 13013 Marseille, France
Interests: mathematical methods for electromagnetic theory; numerical methods for partial differential equations; computational electromagnetics and photonics
Dr. Guillaume Demésy

E-Mail Website
Guest Editor
Institut Fresnel, Aix-Marseille Université, 13013 Marseille, France
Interests: electromagnetism; nanophotonics; finite element

Special Issue Information

Dear Colleagues,

I am pleased to invite you to contribute to a Special Issue on “Advanced Mathematical and Numerical Modelling in Electrodynamics and Photonics”. In recent years, intensive numerical modeling has become an important player in electromagnetic wave and nanophotonics research. Technological advances have allowed the design of complex structures at the wavelength scale of visible light, providing innovative optical functions and opening new applications, including metamaterials, microstructured optical fibers, nonlinear optics, plasmonics, graphene and 2D material devices, topological photonics, integrated quantum photonics, etc. In parallel, numerical and applied mathematics have provided very efficient new techniques that have profited from the increase in available computer power to provide us with the powerful computational methods necessary for the design of complex new structures. The purpose of this issue is to present the current state of the art of this research field in electromagnetic wave modeling and its wide panel of mathematical and computational tools:  finite elements, boundary integrals, finite differences, discontinuous Galerkin, quasi-normal mode expansions, domain decomposition, absorbing boundary techniques, etc.

Dr. André Nicolet
Dr. Guillaume Demésy
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • electromagnetic waves
  • nanophotonics
  • numerical methods in electrodynamics
  • finite element, finite difference, discontinuous galerkin methods
  • integral methods
  • eigenvalue problems, quasi-normal mode expansions
  • domain decomposition methods
  • optimization

Published Papers (7 papers)

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Research

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46 pages, 6674 KiB  
Article
Polyharmonic Representation of the Electromagnetic Field Generated by an Oscillating Particle near a Dispersive Bulk
by Mauricio Garcia-Vergara, Guillaume Demésy, André Nicolet and Frédéric Zolla
Mathematics 2024, 12(2), 321; https://0-doi-org.brum.beds.ac.uk/10.3390/math12020321 - 18 Jan 2024
Viewed by 606
Abstract
This study introduces a polyharmonic framework for analyzing the electromagnetic (EM) field generated by an oscillating point charge near a dispersive bulk of size comparable to the wavelength under study. We critically evaluate traditional approaches such as Liénard-Wiechert, Landau, and Raimond, and propose [...] Read more.
This study introduces a polyharmonic framework for analyzing the electromagnetic (EM) field generated by an oscillating point charge near a dispersive bulk of size comparable to the wavelength under study. We critically evaluate traditional approaches such as Liénard-Wiechert, Landau, and Raimond, and propose a Fourier representation of sources that simplifies numerical implementation and enhances analytical clarity. Our method effectively addresses the limitations of conventional models and is applicable to both relativistic and non-relativistic scenarios. It includes the oscillating point dipole fields, providing a comprehensive understanding of the EM field behavior. The Finite Element Method (FEM) is employed for numerical analysis, demonstrating the method’s adaptability to complex geometries. While offering significant insights, this study acknowledges certain limitations and outlines directions for future research. Full article
(This article belongs to the Special Issue Advanced Mathematical and Numerical Modeling in Electrodynamics and Photonics)
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16 pages, 2172 KiB  
Article
TE-Polarized Electromagnetic Wave Diffraction by a Circular Slotted Cylinder
by Garnik V. Abgaryan and Yury V. Shestopalov
Mathematics 2023, 11(9), 1991; https://0-doi-org.brum.beds.ac.uk/10.3390/math11091991 - 23 Apr 2023
Cited by 1 | Viewed by 1022
Abstract
The problem of diffraction of a TE-polarized electromagnetic wave by a circular slotted cylinder is investigated. The boundary value problem in question for the Helmholtz equation is reduced to an infinite system of linear algebraic equations of the second kind (SLAE-II) using integral [...] Read more.
The problem of diffraction of a TE-polarized electromagnetic wave by a circular slotted cylinder is investigated. The boundary value problem in question for the Helmholtz equation is reduced to an infinite system of linear algebraic equations of the second kind (SLAE-II) using integral summation identities (ISI). A detailed study of the matrix operator of the problem is performed and its Fredholm property in the weighted Hilbert space of infinite sequences is proven. The convergence of the truncation method constructed in the paper for the numerical solution of SLAE-II is justified and the results of computations are presented and discussed, specifically considering the determination of resonance modes. Full article
(This article belongs to the Special Issue Advanced Mathematical and Numerical Modeling in Electrodynamics and Photonics)
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15 pages, 6076 KiB  
Article
Numerical Analysis and Parametric Optimization of T-Shaped Symmetrical Metasurface with Broad Bandwidth for Solar Absorber Application Based on Graphene Material
by Meshari Alsharari, Ammar Armghan and Khaled Aliqab
Mathematics 2023, 11(4), 971; https://0-doi-org.brum.beds.ac.uk/10.3390/math11040971 - 14 Feb 2023
Cited by 4 | Viewed by 1346
Abstract
Solar energy is an essential renewable energy source among all the other renewable energy sources. It is possible to improve the efficiency of the solar energy absorber by increasing the solar energy absorber’s capacity for absorption, which can help in building better solar-based [...] Read more.
Solar energy is an essential renewable energy source among all the other renewable energy sources. It is possible to improve the efficiency of the solar energy absorber by increasing the solar energy absorber’s capacity for absorption, which can help in building better solar-based renewable energy devices. The need of covering the whole solar spectrum led us to design this T-shaped metasurface solar absorber which is based on graphene material. The T-shaped absorber gives 90, 88 and 57% absorption in the visible, infrared and UV regions, respectively. This symmetrical structure is also periodic with respect to x-axis and y-axis. This solar absorber demonstrates better efficiency compared to many other existing solar absorbers. The solar absorber is also compared with two other square-1 and square-2 designs to show the improvement in solar energy absorption. The parametric optimization method is applied to optimize the design. The parameters, such as the length and width of the substrate and the thicknesses of the T-shaped metasurface and substrate, are varied to find out the optimized design for maximum solar energy absorption. The optimized parameters obtained from the optimization are 1000, 2500, 3000 and 3000 nm, for resonator thickness, substrate thickness, substrate length and substrate width, respectively. The design results for graphene material and its potential variation are also observed. The design also shows good absorption for a wide-angle of incidence of about 0 to 50°. The increased efficiency of this design can be applied in future solar absorber devices. Full article
(This article belongs to the Special Issue Advanced Mathematical and Numerical Modeling in Electrodynamics and Photonics)
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16 pages, 1896 KiB  
Article
Solution of an Electrodynamic Problem for a Homogeneous Equivalent Segment of a Coaxial Load, Considering Heat Losses in the Conductors
by Polina V. Tatarenko and Alexander S. Tatarenko
Mathematics 2022, 10(24), 4732; https://0-doi-org.brum.beds.ac.uk/10.3390/math10244732 - 13 Dec 2022
Viewed by 929
Abstract
Mathematical aspects of solving an electrodynamic problem in the field of designing coaxial devices in the microwave range are considered. The solution of the electrodynamic problem for a homogeneous equivalent segment of a coaxial load in the single-mode approximation, considering the heat losses [...] Read more.
Mathematical aspects of solving an electrodynamic problem in the field of designing coaxial devices in the microwave range are considered. The solution of the electrodynamic problem for a homogeneous equivalent segment of a coaxial load in the single-mode approximation, considering the heat losses in the central and outer conductors, was obtained. A mathematical model of the microwave load, linking the high-frequency and design-technological parameters of the device, was built. To refine the model, we consider second-order effects associated with considering inhomogeneities that occur in places where the cross-section of the coaxial structure changes. The design of the 50-Ω load and the results of its experimental investigation are presented for comparison with theoretical calculations. Full article
(This article belongs to the Special Issue Advanced Mathematical and Numerical Modeling in Electrodynamics and Photonics)
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18 pages, 1640 KiB  
Article
Open-Source Computational Photonics with Auto Differentiable Topology Optimization
by Benjamin Vial and Yang Hao
Mathematics 2022, 10(20), 3912; https://0-doi-org.brum.beds.ac.uk/10.3390/math10203912 - 21 Oct 2022
Cited by 3 | Viewed by 1955
Abstract
In recent years, technological advances in nanofabrication have opened up new applications in the field of nanophotonics. To engineer and develop novel functionalities, rigorous and efficient numerical methods are required. In parallel, tremendous advances in algorithmic differentiation, in part pushed by the intensive [...] Read more.
In recent years, technological advances in nanofabrication have opened up new applications in the field of nanophotonics. To engineer and develop novel functionalities, rigorous and efficient numerical methods are required. In parallel, tremendous advances in algorithmic differentiation, in part pushed by the intensive development of machine learning and artificial intelligence, has made possible large-scale optimization of devices with a few extra modifications of the underlying code. We present here our development of three different software libraries for solving Maxwell’s equations in various contexts: a finite element code with a high-level interface for problems commonly encountered in photonics, an implementation of the Fourier modal method for multilayered bi-periodic metasurfaces and a plane wave expansion method for the calculation of band diagrams in two-dimensional photonic crystals. All of them are endowed with automatic differentiation capabilities and we present typical inverse design examples. Full article
(This article belongs to the Special Issue Advanced Mathematical and Numerical Modeling in Electrodynamics and Photonics)
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11 pages, 1357 KiB  
Article
Role of Static Modes in Quasinormal Modes Expansions: When and How to Take Them into Account?
by Mondher Besbes and Christophe Sauvan
Mathematics 2022, 10(19), 3542; https://0-doi-org.brum.beds.ac.uk/10.3390/math10193542 - 28 Sep 2022
Cited by 1 | Viewed by 1177
Abstract
The scattering of electromagnetic waves by a resonator is determined by the excitation of the eigenmodes of the system. In the case of open resonators made of absorbing materials, the system is non-Hermitian, and the eigenmodes are quasinormal modes. Among the whole set [...] Read more.
The scattering of electromagnetic waves by a resonator is determined by the excitation of the eigenmodes of the system. In the case of open resonators made of absorbing materials, the system is non-Hermitian, and the eigenmodes are quasinormal modes. Among the whole set of quasinormal modes, static modes (modes with a zero eigenfrequency) occupy a specific place. We study the role of static modes in quasinormal modes expansions calculated with a numerical solver implemented with the finite-element method. We show that, in the case of a dielectric permittivity described by a Lorentz model, static modes markedly contribute to the electromagnetic field reconstruction but are incorrectly calculated with a solver designed to compute modes with non-zero eigenfrequencies. We propose to solve this issue by adding to the solver a separate, specific computation of the static modes. Full article
(This article belongs to the Special Issue Advanced Mathematical and Numerical Modeling in Electrodynamics and Photonics)
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Review

Jump to: Research

19 pages, 437 KiB  
Review
The Discrete Dipole Approximation: A Review
by Patrick Christian Chaumet
Mathematics 2022, 10(17), 3049; https://0-doi-org.brum.beds.ac.uk/10.3390/math10173049 - 24 Aug 2022
Cited by 14 | Viewed by 2225
Abstract
There are many methods for rigorously calculating electromagnetic diffraction by objects of arbitrary shape and permittivity. In this article, we will detail the discrete dipole approximation (DDA) which belongs to the class of volume integral methods. Starting from Maxwell’s equations, we will first [...] Read more.
There are many methods for rigorously calculating electromagnetic diffraction by objects of arbitrary shape and permittivity. In this article, we will detail the discrete dipole approximation (DDA) which belongs to the class of volume integral methods. Starting from Maxwell’s equations, we will first present the principle of DDA as well as its theoretical and numerical aspects. Then, we will discuss the many developments that this method has undergone over time and the numerous applications that have been developed to transform DDA in a very versatile method. We conclude with a discussion of the strengths and weaknesses of the DDA and a description of the freely available DDA-based electromagnetic diffraction codes. Full article
(This article belongs to the Special Issue Advanced Mathematical and Numerical Modeling in Electrodynamics and Photonics)
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