Model-based autofocus for near-field phase retrieval

The phase problem is a well known ill-posed reconstruction problem of coherent lens-less microscopic imaging, where only the intensities of a complex wave-field are measured by the detector and the phase information is lost. For the reconstruction of sharp images from holograms in a near-field experimental setting, it is crucial to solve the autofocus problem, i.e., to precisely estimate the Fresnel number of the forward model. Otherwise, blurred out-of focus images that also can contain artifacts are the result. In general, a simple distance measurement at the experiment is not sufficiently accurate, thus the fine-tuning of the Fresnel number has to be done prior to the actual reconstructions. This can be done manually or automatically by an estimation algorithm. To automatize the process, as needed, e.g., for <i>in-situ/operando</i> experiments, different focus criteria have been widely studied in literature but are subjected to certain restrictions. The methods often rely on image analysis of the reconstructed image, making them sensitive to image noise and also neglecting algorithmic properties of the applied phase retrieval. In this paper, we propose a novel criterion, based on a model-matching approach, which improves autofocusing by also taking the underlying reconstruction algorithm, the forward model and the measured hologram into account. We derive a common autofocusing framework, based on a recent phase-retrieval approach and a downhill-simplex method for the automatic optimization of the Fresnel number. We further demonstrate the robustness of the framework on different data sets obtained at the nano imaging endstation of P05 at PETRA III (DESY, Hamburg) operated by Helmholtz-Zentrum Hereon.