Deterministic phase retrieval from diffracted intensities speckle fields Giancarlo Pedrini * , Fucai Zhang, Wolfgang Osten Institut fu ¨ r Technische Optik, Universita ¨ t Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart, Germany Received 23 January 2007; received in revised form 20 April 2007; accepted 20 April 2007 Abstract We present a model where the phase of a wave front is obtained by processing the intensity of speckle field pattern. By using the light propagation equation and knowing the intensity of the wavefront it is possible to build a system of linear equations which can be solved to give the wanted phase information. Simulations are presented and the convergence of the solution is discussed. Ó 2007 Elsevier B.V. All rights reserved. 1. Introduction If the amplitude and the phase of a monochromatic wave front are known at a certain plane, it is possible by using the law of propagation to calculate the object wave front at a given distance from that plane. This allows one, e.g., to focus or defocus an image simply by computer simulation. Modern computers allow these kinds of recon- struction to be performed almost in real time. The problem that remains to be solved is how to get the complex ampli- tude of a wave field, because it is well known that detectors like films, photorefractive crystals, CCDs, CMOSs, etc., are not sensitive to the phase (the phase information is lost during the recording process). One way to get the complex amplitude of a wave front is to overlap to it a reference wave and to use a detector to record the interference produced by that two waves (holog- raphy, [1]). During the last 10 years we had an impressive development of techniques where the holograms are recorded on electronic devices (CCD, CMOS) and digitally reconstructed [2–5]. The technique has been used in the past years for different application in particular in the domain of interferometry; where two or more wave fronts recorded at different times need to be compared [4]. Configuring a separate reference beam from a single laser source, however, entails additional optical compo- nents and reduces the available laser power. Furthermore, configuring and recording the interference pattern formed by the superposition of the information-carrying beam and the reference beam involve a tedious and sometimes cumbersome process of optimizing the beam ratio and fringe spacing. Reconstruction methods without reference beam or commonly referred to as phase-retrieval methods have been utilized in optical microscopy of phase objects [6] and in the imaging of nanoparticles using electron micro-diffraction patterns [7]. A wave front reflected or transmitted by an object can be seen as a sum of waves originating from different point sources. These waves produce a 3D interference pattern that we may consider as a volume speckle field. Even without a reference wave, as in the case of holography, from the 3D intensity distribution inside this volume it is possible to get the information about the amplitude and phase. In the past many investigations have been made with the purpose to reconstruct amplitude and phase from an intensity pattern only. Gerchberg–Saxton [8,9] and Yang–Gu [10] algorithms are iterative methods which allow to get the phase information if the intensity is know at a certain plane and we have some additional information about the object wave front in another plane: e.g. pure amplitude object or pure phase object. Recently it was shown that by recording two of more intensity 0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.04.041 * Corresponding author. Tel.: +49 711 685 66078; fax: +49 711 685 66586. E-mail address: pedrini@ito.uni-stuttgart.de (G. Pedrini). www.elsevier.com/locate/optcom Optics Communications 277 (2007) 50–56