Resonant-state expansion applied to three-dimensional open optical systems

Abstract

The resonant-state expansion (RSE), a rigorous perturbative method in electrodynamics, is developed for three-dimensional open optical systems. Results are presented using the analytically solvable homogeneous dielectric sphere as unperturbed system. Since any perturbation which breaks the spherical symmetry mixes transverse electric (TE) and transverse magnetic (TM) modes, the RSE is extended here to include TM modes and a zero-frequency pole of the Green's function. We demonstrate the validity of the RSE for TM modes by verifying its convergence towards the exact result for a homogeneous perturbation of the sphere. We then apply the RSE to calculate the modes for a selection of perturbations sequentially reducing the remaining symmetry, given by a change of the dielectric constant of half-sphere and quarter-sphere shape. Since no exact solutions are known for these perturbations, we verify the RSE results by comparing them with the results of state of the art finite element method (FEM) and finite difference in time domain (FDTD) solvers. We find that for the selected perturbations, the RSE provides a significantly higher accuracy than the FEM and FDTD for a given computational effort, demonstrating its potential to supersede presently used methods. We furthermore show that in contrast to presently used methods, the RSE is able to determine the perturbation of a selected group of modes by using a limited basis local to these modes, which can further reduce the computational effort by orders of magnitude.

How to calculate the pole expansion of the optical scattering matrix from the resonant states

“…It should be noted that this formulation of the normalization is slightly different than in most of our previous works on the resonant state expansion [10,[15][16][17][18][19][20][21]25,27,28], which is valid also for magnetic and bianisotropic materials [26]. This formulation can be reduced to our previous results for nonmagnetic materials that are solely described by the electric field, the electric permittivity, and the electric current as a special case.…”

“…When normalizing the resonant states appropriately [10,[15][16][17][18][19][20][21]26,27], they can be used together with possible cuts to expand the Green's dyadic with outgoing boundary conditions as follows:…”

“…Calculating the derivative is trivial for the first term on the right-hand side. For the second term, we have to differentiate the analytical continuation of F n on the complex k plane, which depends on the geometry of interest [15][16][17]20,21,27].…”

“…(18), (19), (23), and (29), while the internal fields are given by Eq. (17). Note that the background term can be reduced to the scattering matrix of homogeneous and isotropic space for some highly symmetric geometries, as it is assumed in the examples of Ref.…”

“…Several approaches have been suggested for the normalization of resonant states [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. This includes approximate formulations for highquality modes [23][24][25], the utilization of perfectly matched layers [14] or, equivalently, complex coordinates [11] in the exterior of the system, as well as numerical approaches [20,22].…”