Abstract
Perturbation theory formulation of Maxwell's equations gives a theoretically elegant and computationally efficient way of describing small imperfections and weak interactions in electro-magnetic systems. It is generally appreciated that due to the discontinuous field boundary conditions in the systems employing high dielectric contrast profiles standard perturbation formulations fail when applied to the problem of shifted material boundaries. In this paper we developed a novel coupled mode and perturbation theory formulations for treating generic non-uniform (varying along the direction of propagation) perturbations of a waveguide cross-section based on Hamiltonian formulation of Maxwell equations in curvilinear coordinates. We show that our formulation is accurate and rapidly converges to an exact result when used in a coupled mode theory framework even for the high index-contrast discontinuous dielectric profiles. Among others, our formulation allows for an efficient numerical evaluation of induced PMD due to a generic distortion of a waveguide profile, analysis of mode filters, mode converters and other optical elements such as strong Bragg gratings, tapers, bends etc., and arbitrary combinations of thereof. To our knowledge, this is the first time perturbation and coupled mode theories are developed to deal with arbitrary non-uniform profile variations in high index-contrast waveguides.
Highlights
Standard perturbation and coupled mode theory formulations are known to fail or exhibit a very slow convergence [1,2,3,4,5,6] when applied to the analysis of geometrical variations in the structure of high index-contrast fibers.In a uniform coupled mode theory framework, eigenvalues of the matrix of coupling elements approximate the values of the propagation constants of a uniform waveguide of perturbed cross-section
We focus on the geometrical variation in fibers, geometric variations in generic waveguides can be readily described by the same theory
We presented a novel form of the coupled mode and perturbation theories to treat general geometric variations of a waveguide profile with an arbitrary dielectric index contrast
Summary
Standard perturbation and coupled mode theory formulations are known to fail or exhibit a very slow convergence [1,2,3,4,5,6] when applied to the analysis of geometrical variations in the structure of high index-contrast fibers. Ing perturbed electromaghetic modes in the waveguides with shifted high index-contrast dielectric boundaries It was demonstrated (see [4], for example) that for a uniform geometric perturbation of a fiber profile with abrupt high index-contrast dielectric interfaces, expansion of the perturbed modes into an increasing number of the modes of an unperturbed system does not converge to a correct solution when standard form of the coupling elements [7, 8] is used. We focus on the geometrical variation in fibers, geometric variations in generic waveguides can be readily described by the same theory (see [3] for planar waveguides, for example)
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