Wave-front sensing from defocused images by use of wave-front slopes

We describe a novel technique for deriving wave-front aberrations from two defocused intensity measurements. The intensity defines a probability density function, and the method is based on the evolution of the cumulative density function of the intensity with light propagation. In one dimension, the problem is easily solved with a histogram specification procedure, with a linear relationship between the wave-front slope and the difference in the abscissas of the histograms. In two dimensions, the method requires use of a Radon transform. Simulation results demonstrate that good reconstructions can be attained down to 100 photons in each detector. In addition, the method is insensitive to scintillation at the aperture.