Applications of Fourier-synthesis methods to the analysis of specular reflectivity

The inversion of a real-space density profile from specular reflection data is a highly nonlinear transformation and, in the absence of phase information, a direct inversion is not possible. Fourier-synthesis techniques used in crystallography (where a similar phaseless problem is encountered) have been applied to the analysis of reflectivity data. The approach involves approximating the density profile by a Fourier series and then using the correspondence between the Bragg-peak intensities and the corresponding Fourier coefficients to fit the reflectivity data. Although the phase is not determined at each point, the overall shape of the peak provides a measure of the relative magnitudes of the sine and cosine coefficients and hence a partial measure of the relative phase. This technique provides an excellent tool for the inversion of reflectivity data from regular periodic structures and is a promising step towards the final goal of establishing the uniqueness of the inverted density profile. The Fourier method of analysis has been further applied to the analysis of reflectivity from diblock copolymer films.