Phaseless inverse scattering with a parametrized spatial spectral volume integral equation for finite scatterers in the soft x-ray regime

Soft x-ray wafer-metrology experiments are characterized by low signal-to-noise ratios and lack phase information, which both cause difficulties with the accurate three-dimensional profiling of small geometrical features of structures on a wafer. To this end, we extend an existing phase-based inverse-scattering method to demonstrate a sub-nanometer and noise-robust reconstruction of the targets by synthetic soft x-ray scatterometry experiments. The targets are modeled as three-dimensional finite dielectric scatterers embedded in a planarly layered medium, where a scatterer's geometry and spatial permittivity distribution are described by a uniform polygonal cross section along its height. Each cross section is continuously parametrized by its vertices and homogeneous permittivity. The combination of this parametrization of the scatterers and the employed Gabor frames ensures that the underlying linear system of the spatial spectral Maxwell solver is continuously differentiable with respect to the parameters for phaseless inverse-scattering problems. In synthetic demonstrations, we demonstrate the accurate and noise-robust reconstruction of the parameters without any regularization term. Most of the vertex parameters are retrieved with an error of less than <i>λ</i>/13 with <i>λ</i>=13.5<i>n</i> <i>m</i>, when the ideal sensor model with shot noise detects at least five photons per sensor pixel. This corresponds to a signal-to-noise ratio of 3.5 dB. These vertex parameters are retrieved with an accuracy of <i>λ</i>/90 when the signal-to-noise ratio is increased to 10 dB, or approximately 100 photons per pixel. The material parameters are retrieved with errors ranging from 0.05% to 5% for signal-to-noise ratios between 10 dB and 3.5 dB.