Resonant-state expansion applied to planar open optical systems

Abstract

The resonant state expansion (RSE), a novel perturbation theory of Brillouin-Wigner type developed in electrodynamics [Muljarov, Langbein, and Zimmermann, Europhys. Lett., 92, 50010 (2010)], is applied to planar, effectively one-dimensional optical systems, such as layered dielectric slabs and Bragg reflector microcavities. It is demonstrated that the RSE converges with a power law in the basis size. Algorithms for error estimation and their reduction by extrapolation are presented and evaluated. Complex eigenfrequencies, electro-magnetic fields, and the Green's function of a selection of optical systems are calculated, as well as the observable transmission spectra. In particular we find that for a Bragg-mirror microcavity, which has sharp resonances in the spectrum, the transmission calculated using the resonant state expansion reproduces the result of the transfer/scattering matrix method.

Resonant-state expansion of light propagation in nonuniform waveguides

“…12 versus basis size N and computational time on a CPU Intel Core i7-5830K. We find a convergence of the relative error scaling with N −2.5 , which is close to the N −3 scaling of the RSE [21,22]. The somewhat slower convergence can be related to the residual role on the continuum represented by the cut, which does not allow for a natural discretization.…”

“…Based on the concept of RSs, a rigorous approach in electrodynamics called resonant-state expansion (RSE) has recently been developed [21], enabling accurate calculation of RSs in photonic systems [22][23][24][25][26]. The RSE calculates RSs of a given optical system using RSs of a basis system which is typically analytically treatable, as a basis for expansion, and maps Maxwell's wave equation onto a linear matrix eigenvalue problem.…”

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.…”

“…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].…”

“…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:…”