Mathematical and numerical analysis of dielectric waveguides by the integral equation method

The eigenvalue problems for generalized natural modes of an inhomogeneous dielectric waveguide without a sharp boundary and a step-index dielectric waveguide with a smooth boundary of cross-section are formulated as problems for the set of time-harmonic Maxwell equations with partial radiation conditions at infinity in the cross-sectional plane. The original problems are reduced by the integral equation method to nonlinear spectral problems with Fredholm integral operators. Properties of the spectrum are investigated. The Galerkin and collocation methods for the calculations of generalized natural modes are proposed and convergence of the methods is proved. Some results of numerical experiments are discussed

Parallel algorithm of solving the electromagnetic wave diffraction problem on the spherical screen

The electromagnetic wave diffraction problem on a thin conducting spherical screen is reduced to pair summatorial equation relative to unknown coefficients of expansion into a series of spherical waves. This equation can be transformed to a regular infinite set of linear algebraic equations by integral-summatorial identities method. For all stages of numerical algorithm of solving the problem the parallel calculating processes are possible. At first, if field traces of outside source at the sphere are decomposed onto magnetic and electric parts, then magnetic and electric parts of the unknown field can be found independently. Secondly, if coefficients of field conjugation conditions at the sphere do not depend on longitude coordinate, then calculations also can be fulfilled independently for every number of the series coefficients. Thirdly, if by reduction of infinite set the finite set of linear equations of large dimension is obtained, then it can be solved by one of parallel algorithms. But the most effect can be obtained just at the stage of calculating the auxiliary integrals over screen

Natural modes of weakly guiding optical fiber

The eigenvalue problem for generalized natural modes of an inhomogeneous optical fiber is formulated as a problem for the Helmholtz equation with Reichardt condition at infinity in the cross-sectional plane. The generalized eigenvalues of this problem are the complex propagation constants on a logarithmic Reimann surface. The original problem is reduced to a spectral problem with compact integral operator. Theorem on spectrum localization is proved, and then it is proved that the set of all eigenvalues of the original problem can only be a set of isolated points on the Reimann surface, and it also proved that each eigenvalue depends continuously on the frequency and can appear and disappear only at the boundary of the Reimann surface. The existence of the surface modes is proved. The Galerkin method for numerical calculation of the surface modes is proposed. Some results of the numerical experiments are presented. © 2010 IEEE

Integral equation methods in optical waveguide theory

The eigenvalue problems for generalized natural modes of an inhomogeneous dielectric waveguide without a sharp boundary and a step-index dielectric waveguide with smooth boundary of cross-section are formulated as problems for the set of time-harmonic Maxwell equations with partial radiation conditions (Sveshnikov radiation conditions) at infinity in the crosssectional plane. The original problems by integral equations method are reduced to nonlinear spectral problems with Fredholm integral operators. Theorems on spectrum localization are proved, and then it is proved that the sets of all eigenvalues of the original problems can only be some sets of isolated points on the Reimann surface, ant it also proved that each eigenvalue depends continuously on the frequency and dielectric permittivity and can appear and disappear only at the boundary of the Reimann surface. The Galerkin method for numerical calculations of the generalized natural modes are proposed, and the convergence of the method is proved. Some results of numerical experiments are discussed

Projection methods for computation of spectral characteristics of weakly guiding optical waveguides

The original problem on surface and leaky eigen-modes of a weakly guiding step-index optical waveguide is considered. The original problem is reduced to a nonlinear spectral problem for the set of weakly singular boundary integral equations. We approximate the integral operator by collocation and Galerkin methods. Their convergence and quality are proved by numerical experiments. © 2013 IEEE

Electromagnetic wave diffraction on the conducting thin screen placed on the isotropic and anisotropic media interface

The over-determined boundary value problem method in the diffraction electromagnetic waves theory is extended to the case of anisotropic media. The solvability conditions of the over-determined boundary value problems for Maxwell equations set in the anisotropic semi-space are obtained in the case of one-axis anisotropy. The representations of solutions of Maxwell equations set by traces of tangential components of the field on the boundary of domain are constructed. The problem on reflection and refraction of the electromagnetic wave on the isotropic and anisotropic media interface is considered. The integral equation is obtained to determine field perturbation of conducting thin screen placed at the media interface

Numerical method for calculation 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 Reichardt condition at infinity in the cross-sectional plane. The generalized eigenvalues of this problem are the complex propagation constants on a logarithmic Reimann surface. The original problem is reduced to a nonlinear spectral problem with Fredholm integral operator. Theorem on spectrum localization is proved, and then it is proved that the set of all eigenvalues of the original problem can only be a set of isolated points on the Reimann surface, ant it also proved that each eigenvalue depends continuously on the frequency and refraction index and can appear and disappear only at the boundary of the Reimann surface. The Galerkin method for numerical calculation of the generalized natural modes is proposed, and the convergence of the method is proved. © 2008 IEEE

Numerical method for calculation 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 Reichardt condition at infinity in the cross-sectional plane. The generalized eigenvalues of this problem are the complex propagation constants on a logarithmic Reimann surface. The original problem is reduced to a nonlinear spectral problem with Fredholm integral operator. Theorem on spectrum localization is proved, and then it is proved that the set of all eigenvalues of the original problem can only be a set of isolated points on the Reimann surface, ant it also proved that each eigenvalue depends continuously on the frequency and refraction index and can appear and disappear only at the boundary of the Reimann surface. The Galerkin method for numerical calculation of the generalized natural modes is proposed, and the convergence of the method is proved. © 2008 IEEE

La Patrie : journal quotidien, politique, commercial et littéraire

17 août 18941894/08/17 (A54)

Mathematical and numerical analysis of dielectric waveguides by the integral equation method

The eigenvalue problems for generalized natural modes of an inhomogeneous dielectric waveguide without a sharp boundary and a step-index dielectric waveguide with a smooth boundary of cross-section are formulated as problems for the set of time-harmonic Maxwell equations with partial radiation conditions at infinity in the cross-sectional plane. The original problems are reduced by the integral equation method to nonlinear spectral problems with Fredholm integral operators. Properties of the spectrum are investigated. The Galerkin and collocation methods for the calculations of generalized natural modes are proposed and convergence of the methods is proved. Some results of numerical experiments are discussed