Phasing rectangular apertures

Several techniques have been developed to phase apertures in the context of astronomical telescopes with segmented mirrors. Phasing multiple apertures, however, is important in a wide range of optical applications. The application of primary interest in this paper is the phasing of multiple short pulse laser beams for fast ignition fusion experiments. In this paper analytic expressions are derived for parameters such as the far-field distribution, a line-integrated form of the far-field distribution that could be fit to measured data, enclosed energy or energy-in-a-bucket and center-of-mass that can then be used to phase two rectangular apertures. Experimental data is taken with a MEMS device to simulate the two apertures and comparisons are made between the analytic parameters and those derived from the measurements. Two methods, fitting the measured far-field distribution to the theoretical distribution and measuring the ensquared energy in the far-field, produced overall phase variance between the 100 measurements of less than 0.005 rad(2) or an RMS displacement of less than 12 nm.