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Rational eigenvalue problems and applications to photonic crystals

We establish new analytic results for a general class of rational spectral problems. They arise e.g. in modelling photonic crystals whose capability to control the flow of light depends on specific features of the eigenvalues. Our results comprise a complete spectral analysis including variational principles and two-sided bounds for all eigenvalues, as well as numerical implementations. They apply

Efficient and reliable hp-FEM estimates for quadratic eigenvalue problems and photonic crystal applications

We present a-posteriori analysis of higher order finite element approximations (hp-FEM) for quadratic Fredholm-valued operator functions. Residual estimates for approximations of the algebraic eigenspaces are derived and we reduce the analysis of the estimator to the analysis of an associated boundary value problem. For the reasons of robustness we also consider approximations of the associated in

A Subspace Iteration Algorithm for Fredholm Valued Functions

We present an algorithm for approximating an eigensubspace of a spectral component of an analytic Fredholm valued function. Our approach is based on numerical contour integration and the analytic Fredholm theorem. The presented method can be seen as a variant of the FEAST algorithm for infinite dimensional nonlinear eigenvalue problems. Numerical experiments illustrate the performance of the algor

Topology and shape optimization of plasmonic nano-antennas

Metallic nano-antennas are devices used to concentrate the energy in light into regions that are much smaller than the wavelength. These structures are currently used to develop new measurement and printing techniques, such as optical microscopy with sub-wavelength resolution, and high-resolution lithography. Here, we analyze and design a nano-antenna in a two-dimensional setting with the source b

Spectral approximation of quadratic operator polynomials arising in photonic band structure calculations

Galerkin spectral approximation theory for non-self-adjoint quadratic operator polynomials with periodic coefficients is considered. The main applications are complex band structure calculations in metallic photonic crystals, periodic waveguides, and metamaterials. We show that the spectrum of the considered operator polynomials consists of isolated eigenvalues of finite multiplicity with a nonzer

High-order absorbing boundary conditions for the meshless radial point interpolation method in the frequency domain

The meshless radial point interpolation method (RPIM) in frequency domain for electromagnetic scattering problems is presented. This method promises high accuracy in a simple collocation approach using radial basis functions. The treatment of high-order non-reflecting boundary conditions for open waveguides is discussed and implemented up to fourth-order. RPIM allows the direct calculation of high

Recent developments of the meshless radial point interpolation method for time-domain electromagnetics

Meshless methods are a promising new field in computational electromagnetics. Instead of relying on an explicit mesh topology, a numerical solution is computed on an unstructured set of collocation nodes. This allows to model fine geometrical details with high accuracy and facilitates the adaptation of node distributions for optimization or refinement purposes. The radial point interpolation metho

Linearization techniques for band structure calculations in absorbing photonic crystals

Band structure calculations for photonic crystals require the numerical solution of eigenvalue problems. In this paper, we consider crystals composed of lossy materials with frequency-dependent permittivities. Often, these frequency dependencies are modeled by rational functions, such as the Lorentz model, in which case the eigenvalue problems are rational in the eigenvalue parameter. After spatia

On high-order FEM applied to canonical scattering problems in plasmonics

In this paper a high-order finite element method with curvilinear elements is proposed for the simulation of plasmonic structures. Most finite element packages use low order basis functions and non-curved elements, which is very costly for demanding problems such as the simulation of nanoantennas. To enhance the performance of finite elements, we use curvilinear quadrilateral elements to calculate

Adaptive meshless methods in electromagnetic modeling : A gradient-based refinement strategy

Meshless methods are numerical methods that have the advantage of high accuracy without the need of an explicitly described mesh topology. In this class of methods, the Radial Point Interpolation Method (RPIM) is a promising collocation method where the application of radial basis functions yields high interpolation accuracy for even strongly unstructured node distributions. For electromagnetic si

A comparison of three meshless algorithms : Radial point interpolation, non-symmetric and symmetric Kansa method

Three different meshless methods based on radial basis functions are investigated for the numerical solution of electromagnetic eigenvalue problems. The three algorithms, the non-symmetric Kansa approach, the symmetric Kansa method and the radial point interpolation method, are first described putting emphasis on the influence of their formalism on practical implementation. The convergence rate of

Characterization of an adaptive refinement algorithm for a meshless eigenvalue solver based on radial basis functions

A meshless method based on a radial basis collocation approach is presented to calculate eigenvalues for the second-order wave equation. Instead of an explicit mesh topology only a node distribution is required to calculate electric fields, thus facilitating dynamic alteration of the discretization of an electromagnetic problem. An algorithm is presented that automatically adapts an initially very

Residual-based adaptive refinement for meshless eigenvalue solvers

The concept of an adaptive meshless eigenvalue solver is presented and implemented for two-dimensional structures. Based on radial basis functions, eigenmodes are calculated in a collocation approach for the second-order wave equation. This type of meshless method promises highly accurate results with the simplicity of a node-based collocation approach. Thus, when changing the discrete representat

Eigenvalue analysis and longtime stability of resonant structures for the meshless radial point interpolation method in time domain

A meshless collocation method based on radial basis function (RBF) interpolation is presented for the numerical solution of Maxwell's equations. RBFs have attractive properties such as theoretical exponential convergence for increasingly dense node distributions. Although the primary interest resides in the time domain, an eigenvalue solver is used in this paper to investigate convergence properti

Complex dispersion relation calculations with the symmetric interior penalty method

A high-order discontinuous Galerkin method for calculations of complex dispersion relations of two-dimensional photonic crystals is presented. The medium is characterized by a complex-valued permittivity and we relate for this absorptive system the spectral parameter to the time frequency. We transform the non-linear eigenvalue problem for a Lorentz material in air into a non-Hermitian linear eige

On the spectrum of a holomorphic operator-valued function with applications to absorptive photonic crystals

We study electromagnetic wave propagation in a periodic and frequency dependent material characterized by a space- and frequency-dependent complex-valued permittivity. The spectral parameter relates to the time-frequency, leading to spectral analysis of a holomorphic operator-valued function. We apply the Floquet transform and show for a fixed quasi-momentum that the resulting family of spectral p

On a method to reduce uncertainties in bulk property measurements of two-component composites

For two-component composites, we address the inverse problem of estimating the structural parameters and decrease measurement errors in bulk property measurements. A measurement of the effective permittivity at one frequency gives microstructural information about the composite that is used in cross-property bounds to estimate the effective permittivity at other frequencies. We use this informatio

Meshless eigenvalue analysis for resonant structures based on the radial point interpolation method

Meshless methods are a promising field of numerical methods recently introduced to computational electromagnetics. The potential of conformal and multi-scale modeling and the possibility of dynamic grid refinements are very attractive features that appear more naturally in meshless methods than in classical methods. The Radial Point Interpolation Method (RPIM) uses radial basis functions for the a

On the spectrum of an operator pencil with applications to wave propagation in periodic and frequency dependent materials

We study wave propagation in periodic and frequency dependent materials when the medium in a frequency interval is characterized by a real-valued permittivity. The spectral parameter relates to the quasi momentum, which leads to spectral analysis of a quadratic operator pencil where frequency is a parameter. We show that the underlying operator has a discrete spectrum, where the eigenvalues are sy