Abstract

The problem of fitting a wave-front distortion estimate to a (single-instant) set of phase-difference measurements has been formulated as an unweighted least-square problem. The least-square equations have been developed as a set of simultaneous equations for a square array of phase-difference sensors, with phase estimates at the corner of each measurement element. (This corresponds to the standard Hartmann configuration and to one version of a shearing interferometer of a predetection compensation wave-front sensor.) The noise dependence in the solution of the simultaneous equations is found to be expressible in terms of the solution to a particular version of the measurement inputs to the simultaneous equation, a sort of “Green’s-function” solution. The noise version of the simultaneous equations is solved using relaxation techniques for array sizes from 4 × 4 to 40 × 40 phase estimation points, and the mean-square wave-front error calculated as a function of the mean-square phase-difference measurement error. It is found that the results can be approximated within a fraction of a percent accuracy by $〈{(\delta \mathrm{\Phi })}^2〉=0.6558[1+0.2444\hspace{0.17em}\text{ln}(N^2)]{\sigma }_{pd}^2$, where 〈(δΦ)²〉 is the mean-square error (rad²) in the estimation of the wave-front distortion, for a square array consisting of N² square subaperture elements over which two phase-difference measurements are made—one phase difference across the x dimension and the other difference across the y dimension. Here ${\sigma }_{pd}^2$ is the mean-square error (rad²) in each phase-difference measurement.

© 1977 Optical Society of America

Least-squares wave front errors of minimum norm

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