We present a phase retrieval technique for the recovery of complex-valued wave-fields from multiple near-field diffraction measurements. The proposed method does neither rely on any a priori knowledge about the sample nor on knowledge about an external reference wave, but instead uses multiple self-referencing object exit surface waves that are iteratively recovered. The key ingredient to our approach is a system of relaxed coupled waves that allow for the incorporation of holographic data. We use diffraction measurements of multiple exit surface waves as well as their holograms at multiple sample-detector distances to provide sufficient data redundancy to successfully reconstruct the complex-valued wave field. Parameters for stable performance are investigated. Numerical reconstruction is shown by simulation and experiment to be robust against systematic errors such as position uncertainty and noise. The method proposed is realizable at low cost with instrumentation available in typical optical laboratories.
During the past decades the optical imaging community witnessed a rapid emergence of novel imaging modalities such as coherent diffraction imaging (CDI), propagation-based imaging and ptychography. These methods have been demonstrated to recover complex-valued scalar wave fields from redundant data without the need for refractive or diffractive optical elements. This renders these techniques suitable for imaging experiments with EUV and x-ray radiation, where the use of lenses is complicated by fabrication, photon efficiency and cost. However, decoherence effects can have detrimental effects on the reconstruction quality of the numerical algorithms involved. Here we demonstrate propagation-based optical phase retrieval from multiple near-field intensities with decoherence effects such as partially coherent illumination, detector point spread, binning and position uncertainties of the detector. Methods for overcoming these systematic experimental errors - based on the decomposition of the data into mutually incoherent modes - are proposed and numerically tested. We believe that the results presented here open up novel algorithmic methods to accelerate detector readout rates and enable subpixel resolution in propagation-based phase retrieval. Further the techniques are straightforward to be extended to methods such as CDI, ptychography and holography.