Vector Finite Difference (VFD) Mode Solver

Based on:

A. B. Fallahkhair, K. S. Li, and T. E. Murphy, “Vector Finite Difference Modesolver for Anisotropic Dielectric Waveguides”, Journal of Lightwave Technology, Vol. 26, No. 11, pp. 1423–1431, 2008. DOI: 10.1109/JLT.2008.923643

Introduction

This solver implements a full-vector finite-difference mode-solving algorithm for analyzing dielectric waveguides, based on the method presented by Fallahkhair et al. It is designed to compute the guided modes of anisotropic, lossy, and inhomogeneous dielectric structures in two dimensions (2D), making it ideal for modeling complex photonic components such as:

  • Integrated optical waveguides

  • Photonic crystal fibers

  • Nanophotonic waveguides

Unlike scalar or semi-vectorial approximations, this solver fully resolves the vectorial coupling of the electromagnetic field, supporting anisotropic dielectric permittivity and complex materials with loss.

Key Features

  • Full-vector field computation: Solves for all components (Ex, Ey, Ez) of the electric field.

  • Anisotropic permittivity support: Handles diagonal tensor permittivity (εx, εy, εz).

  • Complex eigenvalues: Supports lossy materials and leaky modes.

  • Finite difference discretization: Uses central-difference schemes on structured Cartesian grids.

  • TE, TM, and hybrid modes: Accurately identifies all types of guided modes.

  • Bend Mode Analysis: Accurately computes all types of bend modes via conformal mapping.

  • Perfectly Matched Layer (PML): Implements absorbing boundary conditions to prevent artificial reflections.

Mathematical Method

The method solves the frequency-domain curl-curl Maxwell equation:

\[\nabla \times \left( \frac{1}{\mu} \nabla \times \mathbf{E} \right) = \omega^2 \varepsilon \mathbf{E}\]

Using finite difference discretization, this becomes a large, sparse generalized eigenvalue problem of the form:

\[\mathbf{A} \mathbf{E} = \beta^2 \mathbf{B} \mathbf{E}\]

Where:

  • \(\mathbf{E}\) is the electric field eigenvector,

  • \(\beta\) is the complex propagation constant,

  • \(\mathbf{A}\) and \(\mathbf{B}\) are sparse matrices representing the differential operators.

The eigenvalues \(\beta^2\) are used to extract the effective refractive index:

\[n_{\text{eff}} = \frac{\beta}{k_0}\]

Applications

  • Silicon photonics and integrated optical waveguides

  • Photonic crystal fibers and birefringent fiber design

  • Electro-optic and magneto-optic materials

  • Design of structures with material loss or gain

  • Analysis of hybrid modes in high-index-contrast devices

  • Analysis of bend modes in waveguides with curvature