Measurement of surface position by focusing an interference pattern of multiple apertures with a confocal system.

We propose a method to measure the position of a flat mirror based on the identification of the best focused interferogram on the mirror. The interferogram is generated using a wavefront-splitting interferometer in a confocal configuration. The wavefront is sampled with an array of identical circular apertures. We analyze experimentally and theoretically the diffraction pattern for one, two, and three apertures when they are focused on the mirror. For the theoretical analysis we solve numerically the Rayleigh-Sommerfeld integral, and for the experimental analysis we measure the axial irradiance and the square power of the interferogram as a function of the axial displacement of the mirror. The maximum of these measurements corresponds to the best focused interferogram. The experimental and theoretical results are in good agreement. We found that the uncertainty associated with the axial position of the mirror is reduced significantly (about 161 times) when we use the interferogram generated by three apertures in comparison with the diffraction pattern for one aperture.

Extrapolation, interpolation, and identification of spots in Hartmann patterns.

Based on Fourier analysis, we propose a simple method to extrapolate spots outside an aperture that bounds a Hartmann pattern, and interpolate spots in those regions where some spots are lost or damaged. It allows us to smoothen or remove the discontinuities of the wavefront slopes at the border of the aperture or in those inner regions where some spots are lost. The method changes the spot pattern by a fringe pattern. The extrapolated or interpolated spots are obtained from the intersection of the fringe maxima. The fringe pattern is also used to identify the spots, and it is particularly useful in highly distorted Hartmann patterns. Experimental results show how the method works.

Power maps and wavefront for progressive addition lenses in eyeglass frames.

PURPOSE: To evaluate a method for measuring the cylinder, sphere, and wavefront of progressive addition lenses (PALs) in eyeglass frames. METHOD: We examine the contour maps of cylinder, sphere, and wavefront of a PAL assembled in an eyeglass frame using an optical system based on a Hartmann test. To reduce the data noise, particularly in the border of the eyeglass frame, we implement a method based on the Fourier analysis to extrapolate spots outside the eyeglass frame. The spots are extrapolated up to a circular pupil that circumscribes the eyeglass frame and compared with data obtained from a circular uncut PAL. RESULTS: By using the Fourier analysis to extrapolate spots outside the eyeglass frame, we can remove the edge artifacts of the PAL within its frame and implement the modal method to fit wavefront data with Zernike polynomials within a circular aperture that circumscribes the frame. The extrapolated modal maps from framed PALs accurately reflect maps obtained from uncut PALs and provide smoothed maps for the cylinder and sphere inside the eyeglass frame. CONCLUSIONS: The proposed method for extrapolating spots outside the eyeglass frame removes edge artifacts of the contour maps (wavefront, cylinder, and sphere), which may be useful to facilitate measurements such as the length and width of the progressive corridor for a PAL in its frame. The method can be applied to any shape of eyeglass frame.

Improvement in the measurement of focal length using spot patterns and spherical aberration.

Using an optical setup that includes a square array of 3×3 holes, we used nine meridional rays to measure the effective focal length of a lens. We observed the selected meridional rays as a spot pattern on a diffuse screen. First, we generated a regular square spot pattern (reference pattern) without a lens to test, and then we generated two spot patterns in two different axial positions when the lens being tested refracts the rays. By selecting two sets of four rays of each spot pattern, we were able to measure the difference of the longitudinal (primary) spherical aberration in two positions. With this difference we were able to improve the calculation of the effective focal length. To determine the method's precision, we first simulated the relative error in the effective focal length considering the error in the measurement of the ray heights. Then we determined the experimental relative error by means of the standard deviation of the focal lengths obtained for each spot (in the image of reference and for the images at the two different locations) for both sets of four spots. The experimental results agree very well with the simulation. The error analysis allows us to establish under what conditions it is possible to obtain relative errors of less than 1% in the effective focal length.

Nonredundant array of apertures to measure the spatial coherence in two dimensions with only one interferogram.

We propose to use a mask with a nonredundant array (NRA) of multiple apertures to measure spatial coherence in two dimensions. The spatial distribution of the apertures in the mask is made in such a way that we obtain a quasi-uniform sampling in the coherence domain. The spatial coherence is obtained by Fourier transform of the interferogram generated by the mask when it is illuminated by the light field under analysis.

Corneal topographer based on the Hartmann test.

PURPOSE: The purpose of this article is to show the performance of a topographer based on the Hartmann test for convex surfaces of F/# approximately 1. This topographer, called "Hartmann Test topographer (HT topographer)," is a prototype developed in the Physics Department of the Universidad Nacional de Colombia. METHODS: From the Hartmann pattern generated by the surface under test, and by the Fourier analysis and the optical aberration theory we obtain the sagitta (elevation map) of the surface. Then, taking the first and the second derivatives of the sagitta in the radial direction we obtain the meridional curvature map. The method is illustrated with an example. RESULTS: To check the performance of the HT topographer a toric surface, a revolution aspherical surface, and two human corneas were measured. Our results are compared with those obtained with a Placido ring topographer (Tomey TMS-4 videokeratoscope), and we show that our curvature maps are similar to those obtained with the Placido ring topographer. CONCLUSIONS: The HT topographer is able to reconstruct the corneal topography potentially eradicating the skew ray problem, therefore, corneal defects can be visualized more. The results are presented by elevation and meridional curvature maps.