Guiding mode expansion of a TE and TM transverse-mode integral equation for dielectric slab waveguides with an abrupt termination

Vectorial modal analysis of dielectric waveguides based on a coupled transverse-mode integral equation. I. Mathematical formulation

We propose a rigorous full-vector integral-equation formulation for analyzing modal characteristics of the complex, two-dimensional, rectangular-like dielectric waveguide that is divisible into vertical slices of one-dimensional layered structures. The entire electromagnetic mode field is completely determined by the y-component electric and magnetic field functions on the interfaces between slices. These interfacial functions are governed by a system of vector-coupled transverse-mode integral equations (VCTMIE) whose kernels are made of orthonormal sets of both TE-to-y and TM-to-y modes from each slice. To solve for the unknown functions, we construct sets of suitable expansion functions and turn VCTMIE into a nonlinear matrix equation via orthogonal projection. The eigenvectors of the matrix provide the mode field solutions of the complex dielectric waveguide.

Vectorial modal analysis of dielectric waveguides based on a coupled transverse-mode integral equation. II. Numerical analysis

We present numerical implementation and verification of a rigorous full-vector, integral-equation formulation suitable for analyzing modal characteristics of complex, two-dimensional (2D) rectangular-like dielectric waveguides. By dividing the waveguide into vertical slices, a system of integral equations we call vector-coupled transverse-mode integral equations (VCTMIE) is derived. The entire electromagnetic mode fields are completely determined by one-dimensional unknown field functions on the slice interfaces. To further reduce numerical computation, we expand these functions in terms of the guiding modes of a slab waveguide with a large normalized frequency. Through orthogonal projection the resulting nonlinear eigenvalue and eigenvector matrix formulation enables us to obtain the effective mode index with 10(-7) precision and to compute with high resolution the 2D vectorial mode field solutions of an open dielectric waveguide. We show stable and speedy convergence of our method as well as techniques to overcome the Gibbs phenomenon in the reconstruction of the transverse fields.