Computational aspects of a spatial-spectral domain integral equation for scattering by objects of large longitudinal extent

With a 3D spatial spectral integral-equation method for EM scattering from finite objects, a significant part of the computation time is spent on a middle region around the origin of the spectral domain. Especially when the scatterer extends to more than a wavelength in the stratification direction, a fine discretization on this region is required, consuming much computation time in the transformation to the spatial domain. Numerical evidence is shown that the information in the middle region of the spectral domain is largely linearly dependent. Therefore, a truncated singular-value decomposition is proposed to make the computation time largely independent of the discretization on this middle region. For a practical example the increased computational efficiency and the approximation error of the singular-value decomposition are shown

A domain integral equation approach for simulating two dimensional transverse electric scattering in a layered medium with a Gabor frame discretization.

We solve the 2D transverse-electrically polarized domain-integral equation in a layered background medium by applying a Gabor frame as a projection method. This algorithm employs both a spatial and a spectral discretization of the electric field and the contrast current in the direction of the layer extent. In the spectral domain we use a representation on the complex plane that avoids the poles and branchcuts found in the Green function. Because of the special choice of the complex-plane path in the spectral domain and because of the choice to use a Gabor frame to represent functions on this path, fast algorithms based on FFTs are available to transform to and from the spectral domain, yielding an O(Nlog⁡N) scaling in computation time

An efficient spatial spectral integral-equation method for EM scattering from finite objects in layered media

We propose a mixed spatial spectral method aimed directly at aperiodic, finite scatterers in a layered medium. By using a Gabor frame to discretize the problem a straightforward and fast way to Fourier transform is available. The poles and branchcuts in the spectral-domain Green function can be avoided by representing the induced currents and Green function on a path deformed into the complex spectral domain

A Hermite-interpolation discretization and a uniform path deformation for the spatial spectral domain integral equation method in multilayered media for TE polarization

Two alternative approaches to the spatial spectral integral equation method are proposed. The first enhancement comprises a Hermite interpolation as the set of basis functions instead of the Gabor frame. The continuity, differentiability, equidistant spacing, and small support of these basis functions allows for an efficient and accurate numerical implementation. The second approach encompasses a method to transform between spatial domain and the deformed path in the complex-plane spectral domain. This method allows for more general path shapes, removing the need to decompose the complex-plane spectral domain path into distinct straight sections. Both enhancements are implemented for the case of TE polarization, and the results are validated against the finite element method and rigorous coupled-wave analysis

Computing Gabor Coefficients for a scattering problem: super exponential converging accuracy and a more localized representation

Optical scatterometry is a non-destructive inspection technique to evaluate possible production deformations for integrated circuits. A key component of optical scatterometry is a Maxwell solver to obtain accurate information regarding the electromagnetic scattering from the structures of interest. This Maxwell solver also needs to be capable to quickly adapt the geometrical details of these structures to take various possible deformities into consideration for the fast computation of electromagnetic scattering. A fitting Maxwell solver [1] takes the form of a spatial-spectral domain integral equation such that its computational domain is comparable to the volume of the scattering objects of interest, e.g. parts of an integrated circuit. The spatial-spectral representation in terms of Gabor frames allows to analytically and efficiently incorporate the response of a planarly layered background medium, since the setting of optical scatterometry for integrated circuits is characterized by such a background description. The Gabor frames as in [1] make it possible to describe the electromagnetic fields and scattering objects as weighted sums of shifted and modulated Gaussian window functions, where the so-called Gabor coefficients are the weighting coefficients. We want to make more efficient use of the Gabor frames, which requires the computation of Gabor coefficients, such that the Maxwell solver [1] becomes faster and more suitable for optical scatterometry applications

2D TM scattering problem for finite dielectric objects in a dielectric stratified medium employing Gabor frames in a domain integral equation

We present a method to simulate two-dimensional scattering by dielectric objects embedded in a dielectric layered medium with transverse magnetic polarization through a domain integral equation formulation. A mixed spatial-spectral discretization is employed with both a spatial and a spectral representation along the direction of the layer interfaces. In the spectral domain, a discretization on a path through the complex plane is used on which the Green function is well behaved. To calculate the field-material interaction in the spatial domain, an auxiliary field is employed similar to the Li factorization rules. Numerical results show that this auxiliary-field formulation significantly improves accuracy, compared to a formulation that directly employs the electric field

A 3D spatial spectral integral equation method for electromagnetic scattering from finite objects in a layered medium

The generalization of a two-dimensional spatial spectral volume integral equation to a three-dimensional spatial spectral integral equation formulation for electromagnetic scattering from dielectric objects in a stratified dielectric medium is explained. In the spectral domain, the Green function, contrast current density, and scattered electric field are represented on a complex integration manifold that evades the poles and branch cuts that are present in the Green function. In the spatial domain, the field-material interactions are reformulated by a normal-vector field approach, which obeys the Li factorization rules. Numerical evidence is shown that the computation time of this method scales as O(Nlog N) on the number of unknowns. The accuracy of the method for three numerical examples is compared to a finite element method reference