Digital holography from shadowgraphic phase estimates

Digital phase front retrieval from inline, Gabor-type holograms has to overcome the challenge of separating the object wave from its conjugate by retrieving the phase of the optical field. Recently, the so-called 'twin image problem' has received revived interest, mainly in conjunction with lens-less digital holography applications in the XUV or X-ray bands. In this context, we propose to use a recently devised algorithm, the iterative shadowgraphy method (ISM), to solve the twin-image problem and use the retrieved phase front for digital holography applications. The algorithm is based on the principle that the measurement of phase gradients, which drive the diffraction process, enable the retrieval of the transverse phase profile of a field by observing its intensity distribution on different propagation planes. We have proven rigorously that for small phase modulated object waves, the algorithm converges to the correct object wavefront using just two snapshots of the propagated intensity field as input. Because the algorithm is akin to a deconvolution algorithm, experimental noise can destabilize the iteration scheme. In this work, we discuss the influence of noise in the ISM and apply a wavelet-based scheme to regularize the data. We show that the phase retrieved from two experimental, defocused pictures of a weakly absorbing, scattering object can be used to accurately reconstruct the object trough numerical back-propagation. Thus we prove that ISM is suitable for digital holography applications. We compare the ISM to various other schemes, such as direct backpropagation and the Gerchberg-Saxton algorithm and find that the ISM scheme gives a much improved reconstruction of the phase front.