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.
KEYWORDS: Fourier transforms, Optical engineering, Numerical integration, Digital filtering, Convolution, Data integration, Statistical analysis, Point spread functions, Deflectometry, Linear filtering
The objective of profilometry is to obtain the topography of a surface. Some methods are based on the measurement of the slope of the test surface. Then, by integration the profile of a surface can be determined. The slope is measured in a given set of points and from these data it is necessary to obtain the profile with the highest possible accuracy. Most frequently, the integration is carried out by numerical integration methods that assume different kinds of polynomial approximation of data between sampling points. We propose the integration of the function by means of processing in the Fourier domain. The analysis of the different integration methods in the Fourier domain enables us to easily study and compare their performance.
A compact scanning deflectometer is presented for the fast topography measurement of semiconductor wafers. The technique, however, is equally well suited for any flat or slightly curved specular reflective surface. The measurement principle is based on the 2D measurement of the local slope vector by means of a narrow Laser beam scanning rapidly across the sample surface. The fast linear scanning is combined with sample rotation to measure the complete surface of circular samples. There is no physical contact to the measured surface. The topography of the sample is derived from the slope data by a novel 2D integration method, which is robust with respect to noise in the slope signals. We present the full-size topography of unpatterned and patterned wafers of different polishing quality.
In VanderLugt type correlators, the input scene and the filter could be implemented onto twisted liquid crystal displays (LCD's). The modulator used to display the scene and the elements placed before the filter usually introduce phase aberrations. These aberrations have an important influence in the final correlation plane. We propose a new method to evaluate and correct <i>in situ</i> these aberrations by using the correlator as a point diffraction interferometer. In this work, the wave front phase distribution evaluation is performed by means of the phase shift interferometry (PSI) technique. We present the theory on which the method is based and the experimental results obtained by applying it in a convergent correlator.
In some measurement techniques the profile f(x) of a function should be obtained from the measured slope f'(x) data by integration. The slope is measured in a given set of points x<sub>n</sub> = nΔx, n = 1, N[f'(x<sub>n</sub>)=f'n] and from these data we should obtain the profile with the highest possible accuracy. In general, all integration methods perform some kind of interpolation between the data: Newton-Cotes quadrature expressions (trapezoidal rule, Simpson rule, ...), spline interpolation, etc. We propose here the integration of the function in the Fourier Domain then the most accurate interpolation is automatically carried out. In section 2 we present the classical methods. In section 3 our proposal is described. Finally, in section 4, a comparison between the different methods is carried out.