Spectral Approximation of a Boundary Condition for an Eigenvalue Problem

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Abstract

To compute the guided modes of an optical fiber, the authors use a scalar approximation of Maxwell’s equations. This formulation leads to a bidimensional eigenvalue problem set in an unbounded domain. An equivalent formulation set in a bounded domain is derived. The boundary condition involves a Fourier series expansion. For the numerical treatment, only a finite number N of terms of the series is retained. The authors prove that the error on the eigenvalues and the eigenfunctions decreases faster than any power of ${1 / N}$.

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