An overview of spatial spectral methods with complex-plane deformations for the representation of waves in homogeneous and layered media without absorbing boundary conditions

The prevention of reflections from the edge of the computational domain is a challenge incomputational electromagnetics. Although ways exist to absorb/negate such reflections, we recently proposed an entirely different strategy. Based on a representation in the spectral domain, we analytically represent waves on the entirety of space, but with accuracy focused only on a certain region. Therefore, we can employ formulations without worrying about boundary conditions. We show several examples of this technique, including simulationsin layered media

A note on Gabor coefficient computing with Taylor series expansion

We present an improvement on a previously proposed method for computing Gabor coefficients of characteristic functions with polygonal cross sections, based on a Taylor series expansion and Olver's algorithm. Several requirements are proposed to make the method more robust. Numerical evidence is given to show a convergent solution can be obtained based on a sufficiently high truncation number and working precision

Inverse scattering with a parametrized spatial spectral volume integral equation for finite scatterers

In wafer metrology, the knowledge of the photomask together with the deposition process only reveals the approximate geometry and material properties of the structures on a wafer as a priori information. With this prior information and a parametrized description of the scatterers, we demonstrate the performance of the Gauss-Newton method for the precise and noise-robust reconstruction of the actual structures, without further regularization of the inverse problem. The structures are modeled as three-dimensional finite dielectric scatterers with a uniform polygonal cross-section along their height, embedded in a planarly layered medium. A continuous parametrization in terms of the homogeneous permittivity and the vertex coordinates of the polygons is employed. By combining the global Gabor frame in the spatial spectral Maxwell solver with the consistent parametrization of the structures, the underlying linear system of the Maxwell solver inherits all the continuity properties of the parametrization. Two synthetically generated test cases demonstrate the noise-robust reconstruction of the parameters by surpassing the reconstruction capabilities of traditional imaging methods at signal-to-noise ratios up to -3 dB with geometrical errors below λ/7, where λ is the illumination wavelength. For signal-to-noise ratios of 10 dB, the geometrical parameters are reconstructed with errors of approximately λ/60 and the material properties are reconstructed with an error of around 0.03%. The continuity properties of the Maxwell solver and the use of prior information are key contributors to these results.In wafer metrology, the knowledge of the photomask together with the deposition process only reveals the approximate geometry and material properties of the structures on a wafer as a priori information. With this prior information and a parametrized description of the scatterers, we demonstrate the performance of the Gauss–Newton method for the precise and noise-robust reconstruction of the actual structures, without further regularization of the inverse problem. The structures are modeled as 3D finite dielectric scatterers with a uniform polygonal cross-section along their height, embedded in a planarly layered medium. A continuous parametrization in terms of the homogeneous permittivity and the vertex coordinates of the polygons is employed. By combining the global Gabor frame in the spatial spectral Maxwell solver with the consistent parametrization of the structures, the underlying linear system of the Maxwell solver inherits all the continuity properties of the parametrization. Two synthetically generated test cases demonstrate the noise-robust reconstruction of the parameters by surpassing the reconstruction capabilities of traditional imaging methods at signal-to-noise ratios up to −3dB with geometrical errors below 𝜆/7, where 𝜆 is the illumination wavelength. For signal-to-noise ratios of 10 dB, the geometrical parameters are reconstructed with errors of approximately 𝜆/60, and the material properties are reconstructed with errors of around 0.03%. The continuity properties of the Maxwell solver and the use of prior information are key contributors to these results

A Parallel 3D Spatial Spectral Volume Integral Equation Method for Electromagnetic Scattering from Finite Scatterers

Parallel computing for the three-dimensional spatial spectral volume integral equation method is presented for the computation of electromagnetic scattering by finite dielectric scatterers in a layered medium. The first part exploits the Gabor-frame expansion to compute the Gabor coefficients of scatterers in a parellel manner. The second part concerns the decomposition and restructuring of the matrix-vector product of this spatial spectral volume integral equation into (partially) independent components to enable parallel computing. Both capitalize on the hardware to reduce the computation time by shared-memory parallelism. Numerical experiments in the form of solving electrically large scattering problems, namely volumes up to 1300 cubic wavelengths, in combination with a large number of finite scatterers show a significant reduction in wall-clock time owing to parallel computing, while maintaining accuracy.Parallel computing for the three-dimensional spatial spectral volume integral equation method is presented for the computation of electromagnetic scattering by finite dielectric scatterers in a layered medium. The first part exploits the Gabor-frame expansion to compute the Gabor coefficients of scatterers in a parellel manner. The second part concerns the decomposition and restructuring of the matrix-vector product of this spatial spectral volume integral equation into (partially) independent components to enable parallel computing. Both capitalize on the hardware to reduce the computation time by shared-memory parallelism. Numerical experiments in the form of solving electrically large scattering problems, namely volumes up to 1300 cubic wavelengths, in combination with a large number of finite scatterers show a significant reduction in wall-clock time owing to parallel computing, while maintaining accuracy

THE GABOR FRAME AS A DISCRETIZATION FOR THE 2D TRANSVERSE-ELECTRIC SCATTERING-PROBLEM DOMAIN INTEGRAL EQUATION

We apply the Gabor frame as a projection method to numerically solve a 2D ransverse-electric-polarized domain-integral equation for a homogeneous medium. Since the Gabor frame is spatially as well as spectrally very well convergent, it is convenient to use for solving a domain integral equation. The mixed spatial and spectral nature of the Gabor frame creates a natural and fast way to Fourier transform a function. In the spectral domain we employ a coordinate scaling to smoothen the branchcut found in the Green function. We have developed algorithms to perform multiplication and convolution efficiently, scaling as O(N log N ) on the number of Gabor coefficients, yielding an overall algorithm that also scales as O(N log N )

Local normal vector field formulation for polygonal building blocks in a Gabor representation

The usage of a spatial spectral domain integralequation solver for electromagnetic scattering from dielectricobjects provides a means to execute scattering simulationsfor lithography.We consider the extension of the localnormal vector field formulation to support polygonal buildingblocks in a Gabor series representation of functions.The usage of a spatial spectral domain integral equation solver for electromagnetic scattering from dielectric objects provides a means to execute scattering simulations for lithography. We consider the extension of the local normal vector field formulation to support polygonal building blocks in a Gabor series representation of functions