Methods to obtain the waveform profile from slope measurements

Abstract

There are many optical metrological techniques to determine the profile of a surface or a wave-front. A group of them are based on the measurements of the profile slopes, like deflectometry or wave-front sensors. In both sensors, the profile is then obtained by integrating the gradient information provided by the measurements. The used integration method influences the quality of the obtained results. In this work we compare the performance of different bi-dimensional integration methods to obtain the profile from the slopes, and we propose some new methods. The first kind of methods is based on a path integral, in which the profile in a given point (x,y) is obtained by a 1D integral from (0,0) to (x,0) followed by a 1D integral from (x,0) to (x,y). The second kind of methods is based on finite differences, where the profile in a point is related with the profile in the neighbor points and the slopes of those points. On these methods different interpolations can be used. Finally, the third kind of methods is based on Fourier domain integration. Several simulation results are obtained to study the influence of several parameters: spatial frequency of the signal, local slope errors, random noise, and edge effects. Fourier domain methods could be considered as the gold standard, they suffer from edge effects because the signals are not periodic. Moreover they can only be applied when regular Cartesian sampling is used. Path integral methods create artifacts along the integration paths, when local errors are present. Finite difference methods are more versatile, and their accuracy depends on the used interpolation methods.