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Transparent boundary conditions for Maxwell's equations: Numerical concepts beyond the PML method
(2009)
Optical technologies are ubiquitously used in hi-tech devices. As a common feature of such devices one finds structures with dimensions in the order of the wavelength of the used light. To design and produce such devices, the wave nature of light must be taken into account. Accordingly, robust simulation tools are required which are based on rigorously solving Maxwell's equations, the governing equations of light propagation within macroscopic media. This thesis contributes to the modeling and the numerical computation of light scattering problems: Light scattering problems are typically posed on the entire space. The Perfectly-Matched -Layer method (PML) is widely used to restrict the simulation problem onto a bounded computational domain. We propose an adaptive PML method which exhibits a good convergence even for critical problems where standard PML implementations fail. Besides the computation of the near field, that is the electromagnetic field within the computational domain, it is of major interest to evaluate the electromagnetic field in the exterior domain and to compute the far field. So far, this was numerically only possible for simple geometries such as homogeneous exterior domains or layered media. To deal with more complicated devices, for example with waveguide inhomogeneities, we develop an evaluation formula based on the PML solution which allows for an exterior domain field evaluation in a half space above the device. Finally, we generalize the PML method to problems with multiply structured exterior domains. The term “multiply structured exterior domain” is defined in this thesis and means that the exterior domain exhibits several half-infinite structures. Mathematically, this gives rise to various complications. For example, no analytical solutions to Maxwell's equations for standard light sources are available in the exterior domain, which are needed to describe the incoming field in a light scattering problem. To tackle this we propose a new light scattering problem formulation which fits well into the PML method framework and which may be regarded as an extension of classical contributions by Sommerfeld, Wiener and Hopf. An exterior domain evaluation formula for multiply structured exterior domains with an extended illumination is derived as well.
We present a domain decomposition approach for the computation of the electromagnetic field within periodic structures. We use a Schwarz method with transparent boundary conditions at the interfaces of the domains. Transparent boundary conditions are approximated by the perfectly matched layer method (PML). To cope with Wood anomalies appearing in periodic structures an adaptive strategy to determine optimal PML parameters is developed. We focus on the application to typical EUV lithography line masks. Light propagation within the multi-layer stack of the EUV mask is treated analytically. This results in a drastic reduction of the computational costs and allows for the simulation of next generation lithography masks on a standard personal computer.
Adaptive Multigrid Methods for the Vectorial Maxwell Eigenvalue Problem for Optical Waveguide Design
(2000)
This paper has been motivated by the need for a fast robust adaptive multigrid method to solve the vectorial Maxwell eigenvalue problem arising from the design of optical chips. Our nonlinear multigrid methods are based on a previous method for the scalar Helmholtz equation, which must be modified to cope with the null space of the Maxwell operator due to the divergence condition. We present two different approaches. First, we present a multigrid algorithm based on an edge element discretization of time-harmonic Maxwell's equations, including the divergence condition. Second, an explicit elimination of longitudinal magnetic components leads to a nodal discretization known to avoid discrete \emph{spurious modes} also and a vectorial eigenvalue problem, for which we present a multigrid solver. Numerical examples show that the edge element discretization clearly outperforms the nodal element approach.
Our focus is on Maxwell's equations in the low frequency range; two specific applications we aim at are time-stepping schemes for eddy current computations and the stationary double-curl equation for time-harmonic fields. We assume that the computational domain is discretized by triangles or tetrahedrons; for the finite element approximation we choose N\'{e}d\'{e}lec's $H(curl)$-conforming edge elements of the lowest order. For the solution of the arising linear equation systems we devise an algebraic multigrid preconditioner based on a spatial component splitting of the field. Mesh coarsening takes place in an auxiliary subspace, which is constructed with the aid of a nodal vector basis. Within this subspace coarse grids are created by exploiting the matrix graphs. Additionally, we have to cope with the kernel of the $curl$-operator, which comprises a considerable part of the spectral modes on the grid. Fortunately, the kernel modes are accessible via a discrete Helmholtz decomposition of the fields; they are smoothed by additional algebraic multigrid cycles. Numerical experiments are included in order to assess the efficacy of the proposed algorithms.
The paper surveys recent progress in a joint mathematical-medical project on cancer therapy planning. Within so-called regional hyperthermia the computational task is to tune a set of coupled radiofrequency antennas such that a carefully measured tumor is locally heated, but any outside hot spots are avoided. A mathematical model of the whole clinical system -- air, applicator with antennas, water bolus, individual patient body -- involves Maxwell's equations in inhomogeneous media and a parabolic bioheat transfer equation, which represents a simplified model of heat transfer in the human body (ignoring strong blood vessel heat transport). Both PDEs need to be computed fast and to medical reliability (!) on a workstation within a clinical environment. This requirement triggered a series of new algorithmic developments to be reported here, among which is an adaptive multilevel FEM for Maxwell's equations, which dominates the numerical simulation time. In total, however, the main bulk of computation time (see Table 3 in Section 4 below) still goes into segmentation -- a necessary preprocessing step in the construction a 3D virtual patient from the input of a stack of 2D computed tomograms (left out here).