Noise propagation in wave-front sensing with phase diversity

The phase diversity technique is studied as a wave-front sensor to be imple mented with widely extended sources. The wave-front phase expanded on the Z ernike polynomials is estimated from a pair of images (in focus and out of focus) by use of a maximum-likelihood approach. The propagation of the phot on noise in the images on the estimated phase is derived from a theoretical analysis. The covariance matrix of the phase estimator is calculated, and the optimal distance between the observation planes that minimizes the nois e propagation is determined. The phase error is inversely proportional to t he number of photons in the images. The noise variance on the Zernike polyn omials increases with the order of the polynomial. These results are confir med with both numerical and experimental validations. The influence of the spectral bandwidth on the phase estimator is also studied with simulations. (C) 1999 Optical Society of America.