H-Vectorial Modal Analysis for Anisotropic Waveguide

Assuming ∂ ⁄ ∂z = 0 and regarding n₀  as an effective refractive index, Equation 23 is reduced to a basic equation for the guided-mode analysis of anisotropic optical waveguides. Hence, from (9) we get the following Helmholtz equation:

One can cast the Helmholtz equation into the following matrix form:

where

and

The differential operators are defined as:

It is noted that in Equations [28] – [31], both the material and the geometrical properties of the waveguides contribute to the polarization dependence,

A_xx ≠ A_yy , and coupling:

A_xy ≠ 0 and A_yx ≠ 0 .

Like in E-Formulation Equation 25 can be solved by an efficient Arnodi method that takes into account the sparsity under consideration and computes just a few number of eigenvalues and eigenvectors that represent the propagation constants and fields distributions for the wanted modes.