A Spherical FDTD Numerical Dispersion Relation Based on Elemental Spherical Wave Functions

The numerical dispersion relation for the finite-difference time-domain method in three-dimensional spherical coordinates is derived. Derivation is based on matching spherical wave functions and results in a relation that is equally valid in the entire spherical space. It also matches the sensitivity of the under-lying method to singular solutions behavior near the origin and the z-axis. Derived relation is confirmed to converge in the far field to the corresponding relation for Cartesian space. It also converges to the appropriate continuous space limit when all discrete steps approach zero. Detailed sensitivity analysis to mesh parameters and absolute position within spherical space is also presented.