Hybrid Analytical/Numerical Coupled-Mode Modeling of Guided-Wave Devices

Abstract

A general version of coupled-mode-theory for frequency domain scattering problems in integrated optics is proposed. As a prerequisite a physically reasonable field template is required, that typically combines modes of the optical channels in the structure with coefficient functions of in principle arbitrary coordinates. Upon 1-D discretizations of these amplitude functions into finite elements, a Galerkin procedure reduces the problem to a system of linear equations in the element coefficients, where given input amplitudes are included. Smooth approximate solutions are obtained by solving the system in a least squares sense. The versatility of the approach is illustrated by means of a series of 2-D examples, including a perpendicular crossing of waveguides, and a grating-assisted rectangular resonator. As an appendix, we show that alternatively a similar procedure can be derived by variational means, i.e. by restricting a suitable functional representation of the full 2-D/3-D vectorial scattering problem (with transparent influx boundary conditions for inhomogeneous exterior) to the respective field templates.

Scattering on plasmonic nanostructures arrays modeled with a surface integral formulation

“…There is therefore an increasing demand for a modeling tool accurate even in these extreme conditions. A wide variety of methods is available for modeling the interaction of electromagnetic radiation with matter: couple-mode theory [8], finite-difference time-domain [9], plane-wave expansion [10], finite elements method [11] (FEM). The FEM is proven very accurate, flexible to handle realistic structures [12], and produces sparse matrices efficiently solved with specific algorithms.…”

Chains of coupled square dielectric optical microcavities

“…Then one proceeds along the hybrid CMT approach (HCMT) of Ref. Hammer (2007): By variational means (Vassallo 1991;Sopaheluwakan 2006;van Groesen and Molenaar 2007) one extracts a linear system of equations for the coefficients of the resonator fields, and for the amplitude functions of the bus modes, discretized in terms of finite elements, as unknowns. Section 5 compares the model with rigorous numerical calculations.…”

“…Hammer (2007), the unknown functions u and d are discretized by 1-D linear finite elements. Discrete coefficients u l , d l replace the functions, such that the template (2) assumes the abstract form…”

“…a k ∈ u l , d l , r j are the corresponding unknown coefficients, with an abstract index k that covers the bus mode propagation directions, the element numbers, and the indices of the individual cavities. Basis for the following procedure is then the variational representation of the Helmholtz problem on a suitable 2-D domain that covers the interesting region around the structure, with transparent-influx-boundary conditions that account for the outgoing half-infinite pieces of the bus waveguides in the exterior (Hammer 2007). By restricting the corresponding functional to the discretized field template (3) and looking for vectors of coefficients a k at which the restricted functional becomes stationary, a small to moderate sized system of linear equations for the a k can be obtained (only 1-D FEM discretizations are involved).…”

Optical resonances in multilayer structures

“…Provided that suitable QNM basis fields can be made available by analytical or numerical means, generalizations could be based on the functional representations of the frequency domain Maxwell equations for higher dimensions [84], [85]. …”