  Laterally periodic nanostructures were investigated with grazing incidence
small angle X-ray scattering (GISAXS) by using the diffraction patterns to
reconstruct the surface shape. To model visible light scattering, rigorous
calculations of the near and far field by numerically solving Maxwell's
equations with a finite-element method are well established. The application of
this technique to X-rays is still challenging, due to the discrepancy between
incident wavelength and finite-element size. This drawback vanishes for GISAXS
due to the small angles of incidence, the conical scattering geometry and the
periodicity of the surface structures, which allows a rigorous computation of
the diffraction efficiencies with sufficient numerical precision. To develop
dimensional metrology tools based on GISAXS, lamellar gratings with line widths
down to 55 nm were produced by state-of-the-art e-beam lithography and then
etched into silicon. The high surface sensitivity of GISAXS in conjunction with
a Maxwell solver allows a detailed reconstruction of the grating line shape
also for thick, non-homogeneous substrates. The reconstructed geometrical line
shape models are statistically validated by applying a Markov chain Monte Carlo
(MCMC) sampling technique which reveals that GISAXS is able to reconstruct
critical parameters like the widths of the lines with sub-nm uncertainty.
