Quantitative evaluation of an off-axis parabolic mirror by using a tilted null screen

Abstract

We report the testing of a fast off-axis surface based on the null screen principles. Here we design a tilted null screen with drop shaped spots drawn on it in such a way that its image, which is formed by reflection on the test surface, becomes an exact square array of circular spots if the surface is perfect. Any departure from this geometry is indicative of defects on the surface. Here the whole surface is tested at once. The test surface has a radius of curvature of r = 20.4 mm (F/0.206). The surface departures from the best surface fit are shown; in addition, we show that the errors in the surface shape are below 0.4 mum when the errors in the determination of the coordinates of the centroids of the reflected images are less than 1 pixel, and the errors in the coordinates of the spots of the null screen are less than 0.5 mm.

Surface shape evaluation with a corneal topographer based on a conical null-screen with a novel radial point distribution

“…It is important to mention that the proposed technique takes into account the fact that the cornea is a highly asymmetric and irregular surface, because according to Eq. (14), that gives the shape of the surface, is an exact expression, and is not limited to any model of eye. As pointed out, the conical null-screen is designed in such way that it gives us an ordered array of points if the surface is a perfect conic, any departure from this geometry is indicative of defects on the evaluated surface.…”

“…The shape of the test surface can be obtained from measurements of the positions of the incident points on the CCD plane through the formula [19] z − z i − Z P f P i n x n z dx n y n z dy ; (14) where n x , n y , and n z are the Cartesian components of the normal vector N to the test surface, and z i is the sagitta for one point of the surface that must be known in advance. This expression is exact; evaluating the normals and performing the numerical integration, however, are approximate, so they introduce some errors that must be reduced.…”

“…The next step is the numerical evaluation of Eq. (14). A simple method used for the discrete evaluation of the integral is the trapezoidal rule for nonequally spaced data [20],…”

“…With the calculated normals, the shape of the surface is obtained using Eq. (14) with the trapezoidal rule as the integration procedure [Eq. (15)].…”

“…In previous works [11][12][13][14], we proposed the null-screen method to test fast aspheric convex, concave, and off-axis surfaces. The method consists of drawing a set of curved lines or spots (similar to ellipses) on a cylinder or a plane in such a way that by reflection on the test surface, the image consists of a perfect grid or an array of circular spots.…”