Noise propagation in wave-front sensing with phase diversity.
The phase diversity technique is studied as a wave-front sensor to be implemented with widely extended sources. The wave-front phase expanded on the Zernike 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 photon 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 noise propagation is determined. The phase error is inversely proportional to the number of photons in the images. The noise variance on the Zernike polynomials increases with the order of the polynomial. These results are confirmed with both numerical and experimental validations. The influence of the spectral bandwidth on the phase estimator is also studied with simulations.