Direct retrieval of a complex wave from its diffraction pattern

A direct, non-iterative method of recovering the complex wave in the exit surface plane of an object from its diffraction pattern is presented. The unknown scattered wave may be uniquely recovered via the solution of a set of linear equations, provided the illuminating wave is well-characterised and meets certain geometrical requirements. These conditions are satisfied by a general class of illumination functions, examples of which are those with a Gaussian profile or with a compact support. The linear equations are obtained by analyzing the autocorrelation function of the exit surface wave, which is constructed by inverse Fourier transforming the diffraction pattern. This method has potential applications to imaging nanoparticles and single biological molecules.