Generalized modes of optical fiber

Abstract

The eigenvalue problem for generalized natural modes of an inhomogeneous optical fiber is formulated as a problem for 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, ant 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 collcation method for numerical calculation of the surface and leaky modes is proposed. The convergence of this method is investigated. Some results of the numerical experiments are presented.

Methods of analytical regularization in the spectral theory of open waveguides

“…The collocation method for numerical approximation of weakly singular domain integral operators associated with problem (15), (16) is proposed in [34] and is developed in [35], [36]. The statement similar to Theorem 3 concerning convergence of the collocation method is proved in [37].…”