Mathematical analysis of the generalized natural modes of an inhomogeneous optical fiber

The eigenvalue problem for generalized natural modes of an inhomogeneous optical fiber without a sharp boundary is formulated as a problem for the set of time-harmonic Maxwell equations with the Reichardt condition at infinity in the cross-sectional plane. The generalized eigenvalues (including, as subsets, the well-known guided and leaky modes) of this problem are the complex propagation constants on a logarithmic Riemann surface. A theorem on spectrum localization is proved, and then the original problem is reduced to a nonlinear spectral problem with a compact integral operator. It is proved that the set of all eigenvalues of the original problem can only be a set of isolated points on the Riemann surface, and it is also proved that each eigenvalue depends continuously on the frequency and refraction index and can appear and disappear only at the boundary of the Riemann surface. © 2005 Society for Industrial and Applied Mathematics

Numerical Modeling of Optical Fibers Using the Finite Element Method and an Exact Non-reflecting Boundary Condition

© 2018 Walter de Gruyter GmbH, Berlin/Boston. The original problem for eigenwaves of weakly guiding optical fibers formulated on the plane is reduced to a convenient for numerical solution linear parametric eigenvalue problem posed in a disk. The study of the solvability of this problem is based on the spectral theory of compact self-adjoint operators. Properties of dispersion curves are investigated for the new formulation of the problem. An efficient numerical method based on FEM approximations is developed. Error estimates for approximate solutions are derived. The rate of convergence for the presented algorithm is investigated numerically

Halfspace Matching: a Domain Decomposition Method for Scattering by 2D Open Waveguides

We study a scattering problem for the Helmholtz equation in 2D, which involves non-parallel open waveguides, by means of the halfspace matching method. This method has formerly been applied to periodic media and homogeneous anisotropic media, and we extend it to open waveguides. It allows the reformulation of the Helmholtz equation in an exterior domain to a set of equations for particular traces of the solution, reducing the overall dimension of the problem by 1, making it accessible for numerical discretisation. We show the well-posedness of the halfspace matching method for a model problem in the exterior of a triangular domain, assuming the presence of absorption. Furthermore, we introduce a numerical discretisation which allows the realisation of transparent boundary conditions by a system of coupled integral equations. To illustrate the practicality of this method, we study a number of optimisation examples involving junctions of open waveguides by means of material optimisation

Двумерные сингулярные и слабо сингулярные интегральные уравнения в теории диэлектрических волноводов

На основе метода интегральных уравнений исследован широкий круг задач теории диэлектрических волноводов. Для научных работников в области математического моделирования и численных методов решения задач математической физики.20

Mathematical analysis of the generalized natural modes of an inhomogeneous optical fiber

The eigenvalue problem for generalized natural modes of an inhomogeneous optical fiber without a sharp boundary is formulated as a problem for the set of time-harmonic Maxwell equations with the Reichardt condition at infinity in the cross-sectional plane. The generalized eigenvalues (including, as subsets, the well-known guided and leaky modes) of this problem are the complex propagation constants on a logarithmic Riemann surface. A theorem on spectrum localization is proved, and then the original problem is reduced to a nonlinear spectral problem with a compact integral operator. It is proved that the set of all eigenvalues of the original problem can only be a set of isolated points on the Riemann surface, and it is also proved that each eigenvalue depends continuously on the frequency and refraction index and can appear and disappear only at the boundary of the Riemann surface. © 2005 Society for Industrial and Applied Mathematics

Mathematical analysis of the generalized natural modes of an inhomogeneous optical fiber

The eigenvalue problem for generalized natural modes of an inhomogeneous optical fiber without a sharp boundary is formulated as a problem for the set of time-harmonic Maxwell equations with the Reichardt condition at infinity in the cross-sectional plane. The generalized eigenvalues (including, as subsets, the well-known guided and leaky modes) of this problem are the complex propagation constants on a logarithmic Riemann surface. A theorem on spectrum localization is proved, and then the original problem is reduced to a nonlinear spectral problem with a compact integral operator. It is proved that the set of all eigenvalues of the original problem can only be a set of isolated points on the Riemann surface, and it is also proved that each eigenvalue depends continuously on the frequency and refraction index and can appear and disappear only at the boundary of the Riemann surface. © 2005 Society for Industrial and Applied Mathematics

Mathematical analysis of the generalized natural modes of an inhomogeneous optical fiber

The eigenvalue problem for generalized natural modes of an inhomogeneous optical fiber without a sharp boundary is formulated as a problem for the set of time-harmonic Maxwell equations with the Reichardt condition at infinity in the cross-sectional plane. The generalized eigenvalues (including, as subsets, the well-known guided and leaky modes) of this problem are the complex propagation constants on a logarithmic Riemann surface. A theorem on spectrum localization is proved, and then the original problem is reduced to a nonlinear spectral problem with a compact integral operator. It is proved that the set of all eigenvalues of the original problem can only be a set of isolated points on the Riemann surface, and it is also proved that each eigenvalue depends continuously on the frequency and refraction index and can appear and disappear only at the boundary of the Riemann surface. © 2005 Society for Industrial and Applied Mathematics