Computation of scattering resonances in absorptive and dispersive media with applications to metal-dielectric nano-structures

Abstract

In this paper we consider scattering resonance computations in optics when the resonators consist of frequency dependent and lossy materials, such as metals at optical frequencies. The proposed computational approach combines a novel hp-FEM strategy, based on dispersion analysis for complex frequencies, with a fast implementation of the nonlinear eigenvalue solver NLEIGS. Numerical computations illustrate that the pre-asymptotic phase is significantly reduced compared to standard uniform h and p strategies. Moreover, the efficiency grows with the refractive index contrast, which makes the new strategy highly attractive for metal-dielectric structures. The hp-refinement strategy together with the efficient parallel code result in highly accurate approximations and short runtimes on multi processor platforms.

Shape optimization for the strong directional scattering of dielectric nanorods

“…This, from the accuracy of the solutions and performance of the solver. 18,19 These conclusions can also be drawn for the scattering problems treated in this work. We emphasize this point in Section 3.5.1 where formula (62) gives a pre asymptotic estimate for controlling the pollution error.…”

“…So, if r 1 < R( n + p n , ) < r 2 ∀ we use n = 1 as initial guess, else we set the initial n to the largest constant c ∈ (0, 1) such that r 1 < R( n + cp n , ) < r 2 . In our setting the feasibility check is computationally inexpensive compared to solving the governing equation (19). For the Armijo condition (32) we use the recommended 34 value c 1 = 10 −4 .…”

“…The advantage of high-order FEM for wave scattering problems with smooth interfaces is greatly experienced when comparing dispersive pollution error of low-order versus high-order FE. Pre asymptotic error estimation predicts that increasing the polynomial degree p is more efficient than decreasing the mesh size h. The works by Araújo et al 18,19 present results for scattering resonance computations by comparing the resulting errors obtained by refining p and h, respectively. Particularly, h refinements consists on fixing p and decreasing h uniformly over the mesh, and p refinements consists of increasing p uniformly and keeping h fixed.…”

“…It is possible to improve the performance of the numerical optimization scheme and to lessen the memory requirements by implementing meshes designed a priori by using different FE approximation properties depending on the refractive index function. The outcome of the strategy is to shorten the preasymptotic phase of the FE error for Helmholtz problems with constant coefficients as described in Araújo et al 19 For clarity of the exposition, and simplicity of the proposed numerical optimization strategy we do not incorporate these ideas in the current work.…”

Refined isogeometric analysis for generalized Hermitian eigenproblems

“…We also use SLEPc, the scalable library for eigenvalue problem computations [49], for performing the eigenanalysis, allowing us to apply the shift-and-invert spectral transformation. SLEPc, used to solve different eigenproblems (see, e.g., [44,[65][66][67][68][69]), employs the Krylov-Schur algorithm by default, which is equal to the thick-restart Lanczos algorithm in the case of generalized Hermitian eigenproblems. SLEPc calculates almost the same number of eigenpairs for each shift, allowing us to estimate the computational costs efficiently.…”