Phase retrieval from intensity-only data by relative entropy minimization

A recursive algorithm, which appears to be new, is presented for estimating the amplitude and phase of a wave field from intensity-only measurements on two or more scan planes at different axial positions. The problem is framed as a nonlinear optimization, in which the angular spectrum of the complex field model is adjusted in order to minimize the relative entropy, or Kullback-Leibler divergence, between the measured and reconstructed intensities. The most common approach to this so-called phase retrieval problem is a variation of the well-known Gerchberg-Saxton algorithm devised by Misell (J. Phys. D6, L6, 1973), which is efficient and extremely simple to implement. The new algorithm has a computational structure that is very similar to Misell's approach, despite the fundamental difference in the optimization criteria used for each. Based upon results from noisy simulated data, the new algorithm appears to be more robust than Misell's approach and to produce better results from low signal-to-noise ratio data. The convergence of the new algorithm is examined.