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 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 -3d B 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.

Describing discontinuous finite 3D scattering objects in Gabor coefficients: fast and accurate methods.

In relation to the computation of electromagnetic scattering in layered media by the Gabor-frame-based spatial spectral Maxwell solver, we present two methods to compute the Gabor coefficients of the transverse cross section of three-dimensional scattering objects with high accuracy and efficiency. The first method employs the analytically obtained two-dimensional Fourier transform of the cross section of a scattering object, which we describe by two-dimensional characteristic functions, in combination with the traditional discrete Gabor transform method for computing the Gabor coefficients. The second method concerns the expansion of the so-called dual window function to compute the Gabor coefficients by employing the divergence theorem. Both methods utilize (semi)-analytical approaches to overcome the heavy oversampling requirement of the traditional discrete Gabor transform method in the case of discontinuous functions. Numerical results show significant improvement in terms of accuracy and computation time for these two methods against the traditional discrete Gabor transform method.