Phase retrieval from intensity measurements in one-parameter canonical-transform systems

Martin J. Bastiaans and Kurt Bernardo Wolf

Phase retrieval and local frequency estimation of a signal from intensity profiles are important problems in radio location, optical signal processing, quantum mechanics, and other fields. Several successful iterative algorithms for phase reconstruction from the squared modulus of the signal and its power spectrum, or its Fresnel spectrum, have been proposed recently, and related techniques are applied in various regions of the electromagnetic spectrum and in quantum mechanics. The development of non-iterative procedures for generic systems remains an attractive research topic.

A non-iterative approach for phase retrieval, based on the so-called transport-of-intensity equation in optics, was proposed by Teague [1] and then further developed by others. It was shown that the longitudinal derivative of the Fresnel spectrum is proportional to the transversal derivative of the product of the instantaneous power and the instantaneous frequency of the signal. A similar procedure was proposed for the fractional Fourier transform [2].

In this paper we show that a non-iterative formulation applies for general one-parameter canonical transforms [3]. We show that the local frequency (the first derivative of the phase of the signal) is directly related to the derivative of the squared modulus of the one-parameter canonical transform with respect to the parameter, and given by the evolution Hamiltonian of the optical medium. From this relationship we conclude that the phase of the signal can be reconstructed by letting it propagate in such systems, and measuring the intensity profiles of the signal for two close values of the parameter.

  1. M. R. Teague, "Deterministic phase retrieval: a Green function solution," J. Opt. Soc. Am., vol. 73, pp. 1434-1441, 1983.
  2. T. Alieva, M. J. Bastiaans, and LJ. Stankovic, "Signal reconstruction from two close fractional Fourier power spectra," IEEE Trans. Signal Process., vol. 51, pp. 112-123, 2003.
  3. K. B. Wolf, Integral Transforms in Science and Engineering, Chap. 9, Plenum Press, New York, 1979.

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