.. _vfd-intro:

Vector Finite Difference (VFD) Mode Solver
==========================================


.. contents::
   :local:
   :depth: 2
   :backlinks: top

**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 <https://doi.org/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 (ε\ :sub:`x`\, ε\ :sub:`y`\, ε\ :sub:`z`\).
- **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.

Mathematical Method
-------------------

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

.. math::

   \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:

.. math::

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

Where:

- :math:`\mathbf{E}` is the electric field eigenvector,
- :math:`\beta` is the complex propagation constant,
- :math:`\mathbf{A}` and :math:`\mathbf{B}` are sparse matrices representing the differential operators.

The eigenvalues :math:`\beta^2` are used to extract the effective refractive index:

.. math::

   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