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.