Constructing the scattering matrix for optical microcavities as a nonlocal boundary value problem

Abstract

We develop a numerical scheme to construct the scattering ($S$) matrix for optical microcavities, including the special cases with parity-time and other non-Hermitian symmetries. This scheme incorporates the explicit form of a nonlocal boundary condition, with the incident light represented by an inhomogeneous term. This approach resolves the artifact of a discontinuous normal derivative typically found in the $\mathcal{R}$-matrix method. In addition, we show that by excluding the aforementioned inhomogeneous term, the non-Hermitian Hamiltonian in our approach also determines the Periels-Kapur states, and it constitutes an alternative approach to derive the standard $\mathcal{R}$-matrix result in this basis. Therefore, our scheme provides a convenient framework to explore the benefits of both approaches. We illustrate this boundary value problem using one-dimensional and two-dimensional scalar Helmholtz equations. The eigenvalues and poles of the $S$ matrix calculated using our approach show good agreement with results obtained by other means.

10 pages, 5 figures

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Cited by in corpus (4)

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• Chiral symmetry in non-Hermitian systems: product rule and Clifford algebra
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• Insight of the Green's function as a defect state in a boundary value problem