Dirichlet-to-Neumann Transparent Boundary Conditions for Photonic Crystal Waveguides

Klindworth, Dirk; Schmidt, Kersten

doi:10.1109/tmag.2013.2285412

Cited by 1 publication

(1 citation statement)

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“…Several other problems have been approached with artificial boundary conditions and NEPs; see, e.g., the NEP arising in the modeling of an electromagnetic cavity [35], the model of an optical fiber in [31] and bi-periodic slabs [48]. There are also several approaches leading to NEPs developed in the context of photonic crystals [46,19,32,16]. To our knowledge, none of the methods developed in those papers have been adapted to resonance problems of our type.…”

Section: Introductionmentioning

confidence: 99%

Efficient resonance computations for Helmholtz problems based on a Dirichlet-to-Neumann map

Araujo-Cabarcas

Engström

Jarlebring

2018

Journal of Computational and Applied Mathematics

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We present an efficient procedure for computing resonances and resonant modes of Helmholtz problems posed in exterior domains. The problem is formulated as a nonlinear eigenvalue problem (NEP), where the nonlinearity arises from the use of a Dirichlet-to-Neumann map, which accounts for modeling unbounded domains. We consider a variational formulation and show that the spectrum consists of isolated eigenvalues of finite multiplicity that only can accumulate at infinity. The proposed method is based on a high order finite element discretization combined with a specialization of the Tensor Infinite Arnoldi method. Using Toeplitz matrices, we show how to specialize this method to our specific structure. In particular we introduce a pole cancellation technique in order to increase the radius of convergence for computation of eigenvalues that lie close to the poles of the matrix-valued function. The solution scheme can be applied to multiple resonators with a varying refractive index that is not necessarily piecewise constant. We present two test cases to show stability, performance and numerical accuracy of the method. In particular the use of a high order finite element discretization together with TIAR results in an efficient and reliable method to compute resonances.

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