# Finite Difference Mode Solver

## Introduction

This mode solver uses a magnetic field based formulation of the difference equations to convert the mode solving problem into a problem of finding eigenvectors of a large system of linear equations. The formulation is due to Lüsse . Once the linear system is found from this formulation, the modes can be found by finding eigenvectors, and…

## Magnetic Formulation

Inside a dielectric, the time independent Maxwell curl equations with a positive time convention ( ejωt ) are and the divergence equations are In the magnetic formulation, the electric field is eliminated from (1) by taking the curl of the second equation and substituting from the first. For regions of constant permittivity ε , there are no gradients of ε ,…

## Magnetic Finite Difference Equations

In order to reduce the partial differential equation into difference equations for solving, a mesh needs to be defined and applied. The mesh used is shown in Fig. 1 below. It shows a sample of the mesh at a point P that is not on one of the calculation window boundaries. The magnetic fields are defined…

## Implicitly Restarted Arnoldi Method (IRAM)

The repeated application of (10) for every node, and the application of a similar equation for hy leads to an eigenvector problem of a large system of linear equations. The Finite Difference Mode Solver uses the Implicitly Restarted Arnoldi Method as described in Ref.  to find the eigenvectors of this system, and thereby find the modes of the waveguide.…

## Transparent Boundary Condition (TBC)

The equation (10) applies to nodes inside the mesh. Nodes on the boundaries need a different formula because at least one of the N, S, E, or W nodes will be outside of the mesh and therefore not included in the calculation. The situation for a node on the South boundary is illustrated in Fig.…

## References

 P. Lüsse, P. Stuwe, J. Schüle, H-G Unger, “Analysis of Vectorial Mode Fields in Optical Waveguides by a New Finite Difference Method”, Journal of Lightwave Technology, 12(3), p 487-493 (1994)  R.B. Lehoucq, D.C. Sorensen, C. Yang, “ARPACK User’s Guide”, SIAM (1998)