Resonant-state expansion applied to planar waveguides

Abstract

The resonant state expansion, a recently developed method in electrodynamics, is generalized here to planar open optical systems with non-normal incidence of light. The method is illustrated and verified on exactly solvable examples, such as a dielectric slab and a Bragg reflector microcavity, for which explicit analytic formulas are developed. This comparison demonstrates the accuracy and convergence of the method. Interestingly, the spectral analysis of a dielectric slab in terms of resonant states reveals an influence of waveguide modes in the transmission. These modes, which on resonance do not couple to external light, surprisingly do couple to external light for off-resonant excitation.

Resonant-state expansion of light propagation in nonuniform waveguides

“…The formalism of RSs has been recently applied to a uniform planar WG, and all types of RSs, including WG, anti-WG, and FP modes were calculated for an infinitely extended dielectric slab surrounded by vacuum [2]. It has also been shown that in spite of their exponential growth outside the WG, unbound RSs naturally discretize the continuum of radiation modes and are suited for expansion of the EM field inside the WG.…”

“…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. This approach has been applied to uniform WGs [2] for the case of a fixed real in-plane propagation wavevector. For the description of propagation along a waveguide, we consider here instead RSs for a fixed real frequency, having in general complex in-plane wavevectors, and use them to formulate a fixed-frequency RSE for homogeneous parts of WGs.…”

“…2(a)) which divides the k-plane into two half-planes, one of them corresponding to the physical sheet. The k-plane contains all possible values k n of RSs of the BWG, which include WG (ik n < 0), anti-WG (ik n > 0) and FP (Re(k n ) = 0) modes [2]. For the chosen BWG they are roots of the secular equation…”

“…Interestingly, the normalization condition Eq. (A5) for RSs at a fixed frequency does not contain in the volume term the permittivity of the system ε(x) as a weight function, unlike RSs of a planar systems defined at a fixed in-plane wave vector [2,21].…”

“…These modes present a small, though significant subgroup of a larger class of resonant states (RSs) of an optical system, among which there are also unbound solutions, such as Fabry-Perot (FP) and anti-WG modes [2]. Formally, RSs are the eigenmodes of an open optical system, which satisfy either incoming or outgoing wave boundary conditions (BCs), and describe, with mathematical rigor, optical resonances of different linewidth which exist in the system.…”