Christian Engström

A Floquet--Bloch Decomposition of Maxwell's Equations Applied to Homogenization

Using Bloch waves to represent the full solution of Maxwell’s equations in periodic media, we study the limit where the material’s period becomes much smaller than the wavelength. It is seen that for steady-state fields, only a few of the Bloch waves contribute to the full solution. Effective material parameters can be explicitly represented in terms of dyadic products of the mean values of the non-vanishing Bloch waves, providing a new means of homogenization. The representation is valid for an arbitrary wave vector in the first Brillouin zone.

On two numerical methods for homogenization of Maxwell's equations

When the wavelength is much larger than the typical scale of the microstructure in a material, it is possible to define effective or homogenized material coefficients. The classical way of determination of the homogenized coefficients consists of solving an elliptic problem in a unit cell. This method and the Floquet-Bloch method, where an eigenvalue problem is solved, are numerically compared with respect to accuracy and contrast sensitivity. Moreover, we provide numerical bounds on the effective permittivity. The Floquet-Bloch method is shown to be a good alternative to the classical homogenization method, when the contrast is modest.

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 symmetrically placed with respect to the real and imaginary axis. Moreover, we discretize the operator pencil with finite elements and use a Krylov space method to compute eigenvalues of the resulting large sparse matrix pencil.

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 representation of a physical model, only node locations have to be adapted, hence avoiding the numerical overhead of handling an explicit mesh topology. The accuracy of the method comes at a cost of dealing with poorly-conditioned matrices. This is circumvented by applying a leave-one-out-cross-validation optimization algorithm to get stable results. A node adaptivity algorithm is presented to efficiently refine an initially coarse discretization. The convergence is evaluated in two numerical examples with analytical solutions. The most relevant parameter of the adaptation algorithm is numerically investigated and its influence on the convergence rate examined.

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 problems has a spectrum consisting of at most countably many isolated eigenvalues of finite multiplicity. These eigenvalues depend continuously on the quasi-momentum and no nonzero real eigenvalue exists when the material is absorptive. Moreover, we reformulate the special case of a rational operator-valued function in terms of a polynomial operator pencil and study two-component dispersive and absorptive crystals in detail.

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 nonzero imaginary part. The spectral problem is equivalent to a non-compact block operator matrix and norm convergence is shown for a block operator matrix having the same generalized eigenvectors as the original operator. Convergence rates of finite element discretizations are considered and numerical experiments with the

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

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

p-version and the