Resonant-state expansion of light propagation in nonuniform waveguides

Abstract

A new rigorous approach for precise and efficient calculation of light propagation along nonuniform waveguides is presented. Resonant states of a uniform waveguide, which satisfy outgoingwave boundary conditions, form a natural basis for expansion of the local electromagnetic field. Using such an expansion at fixed frequency, we convert the wave equation for light propagation in a non-uniform waveguide into an ordinary second-order matrix differential equation for the expansion coefficients depending on the coordinate along the waveguide. We illustrate the method on several examples of non-uniform planar waveguides and evaluate its efficiency compared to the aperiodic Fourier modal method and the finite element method, showing improvements of one to four orders of magnitude. A similar improvement can be expected also for applications in other fields of physics showing wave phenomena, such as acoustics and quantum mechanics.

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.…”

“…Note that we do not distinguish explicitly between pole and cut contributions, since they can be formally written in the same way and replaced by a finite number of cut poles in numerical calculations [15,28]. Furthermore, it has to be emphasized that the pole expansion of the Green's dyadic in Eq.…”

“…A fully analytical form of normalization has been derived in Ref. [10] and extended to various geometries and materials [15][16][17][18][19][20][21][26][27][28].…”

Analytic Mode Normalization for the Kerr Nonlinearity Parameter: Prediction of Nonlinear Gain for Leaky Modes

“…An analytic expression for the mode normalization in waveguide geometries can be found for slab waveguides in Ref. [26] and for optical fibers in Ref. [27].…”

“…[27]. Based on the correct mode normalization, it is possible to set up the so-called resonant-state expansion [22][23][24][25][26][27][28], which is a rigorous perturbative method that uses the normalized resonant states of a reference system for describing the optical properties of a perturbed system.…”