# Vectoral Modal Analysis for Anisotropic Waveguides

## Introduction

Modal analysis for optical waveguides is one of the most important area in modeling and simulation of guided wave photonic devices. The problem of the numerical boundary condition is expected to become much more accurate for the leaky modes, as the modal fields at the edges of the computation window are traveling waves and the…

## Appendix I

E-Formulation The full vectorial wave equation is given by: Considering  into Equation 1, we get: The term  is null. We can separate Equation 2 into two equations, one for longitudinal terms and another for transversal terms: for longitudinal terms and for the transversal terms. By substituting Equation 3 into the first right hand term of Equation 4,…

## Full H-vector Formulation

The FD-VBPM based on the E and H fields are equivalent and yield almost identical results . Similar to the vector wave equation for the electric field, the equation for the magnetic field considering transversely scaled version of PML is: here  . The double-curl Equation 1 involves three vector components of the magnetic field, while strictly…

## 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:…

## Appendix II

H-Formulation The full vectorial wave equation for H is given by: By substituting  into Equation 1 we get: The term  is null. We can separate Equation 2 into one for longitudinal terms and another one for transversal terms: for longitudinal terms, and By substituting into the first right hand term of Equation 4, we get: By substituting Equation…

## References

           [Berenger, 1994] J. P. Berenger, “A Perfectly Matched Layer for the Absorption of Electromagnetic Waves,” J. Comput. Phys., No. 114, 1994, pp. 185-200.            [Teixeira, 1998] F. L. Teixeira and W. C. Chew, ” Systematic Derivation of Anisotropic PML Absorbing Media in Cylindrical and Spherical Coordinates”, IEEE Microwave and Guided Lett., vol. 8, No. 6,…