Abstract

Atmospheric turbulence precludes the direct measurement of the spatial-spectral phase of a small stellar object; however, measurement of the gradient of the phase, followed by integration to obtain the phase function, has proved a viable procedure. Our phase-gradient (PG) algorithm is based on the Fourier differentiation theorem and requires the evaluation of the cross-correlation of the nth speckle image i_n with each of i_n, (xi_n), and (yi_n). Modulus information is obtained from conventional speckle interferometry. At very low light levels where the detector can deliver the image as a list of photon addresses, the cross-correlations are simply 2-D histograms of weighted address differences. Bias terms are eliminated by omitting all photon self-products. A photon-event version of the PG algorithm has been successfully demonstrated on computer-simulated data and on real stellar data obtained from Steward Observatory, Tucson. A detailed comparison with the well-known Knox-Thompson algorithm, operating in the photon-address mode, shows that the PG process gives lower rms errors, is less sensitive to defects in the data, and requires only half of the memory size and number of computing operations.

© 1987 Optical Society of America