Iterative phase retrieval for digital holography: tutorial

Abstract

This paper provides a tutorial of iterative phase retrieval algorithms based on the Gerchberg-Saxton (GS) algorithm applied in digital holography. In addition, a novel GSbased algorithm that allows reconstruction of 3D samples is demonstrated. The GS-based algorithms recover complex-valued wavefront-by-wavefront back-and forth propagation between two planes with constraints superimposed in these two planes. Iterative phase retrieval allows quantitatively correct and twin-image-free reconstructions of object amplitude and phase distributions from its in-line hologram. The present work derives the quantitative criteria on how many holograms are required to reconstruct a complex-valued object distribution, be it a 2D or 3D sample. It is shown that for a sample that can be approximated as a 2D sample, a single-shot in-line hologram is sufficient to reconstruct the absorption and phase distributions of the sample. Previously, the GS-based algorithms have been successfully employed to reconstruct samples that are limited to a 2D plane. However, realistic physical objects always have some finite thickness and therefore are 3D rather than 2D objects. This study demonstrates that 3D samples, including 3D phase objects, can be reconstructed from two or more holograms. It is shown that in principle, two holograms are sufficient to recover the entire wavefront diffracted by a 3D sample distribution. In this method, the reconstruction is performed by applying iterative phase retrieval between the planes where intensity was measured. The recovered complex-valued wavefront is then propagated back to the sample planes, thus reconstructing the 3D distribution of the sample. This method can be applied for 3D samples such as 3D distribution of particles, thick biological samples, and other 3D phase objects. Examples of reconstructions of3Dobjects, including phase objects, are provided. Resolution enhancement obtained by iterative extrapolation of holograms is also discussed.

The numerics of phase retrieval

“…While holography leads to relatively simple algorithms for solving the phase retrieval problems, it does pose numerous challenges in the experimental practice. For a detailed discussion of various practical issues with holography, such as resolution limitations, see Duadi et al (2011), Latychevskaia and Fink (2015), Shechtman et al (2015), Saliba et al (2016) and Latychevskaia (2019).…”

Holography and Coherent Diffraction Imaging with Low-(30–250 eV) and High-(80–300 keV) Energy Electrons: History, Principles, and Recent Trends

“…The object distribution can be also reconstructed from a single in-line hologram (defocus image) by applying iterative reconstruction, as explained in detail in the literature [53,[72][73][74][75][76]. We provide only a brief summary of the reconstruction steps here.…”

“…In optical holography, the complete reconstruction of a wavefront from a sequence of intensity measurements using an iterative procedure has been demonstrated in series of studies between 2003 and 2006 [ 95 , 96 , 97 , 98 , 99 ]. It has been recently demonstrated that 3D samples, including 3D phase objects, can be reconstructed from two or more holograms recorded at different z -distances from the sample [ 75 ], as illustrated in Figure 24 . This reconstruction is performed by applying iterative phase retrieval only between the planes in which the intensity distributions were measured ( H 1 and H 2 in Figure 24 a), i.e., without involving any planes within the sample and hence with no need for constraints on the sample.…”

“…(b) Reconstructed amplitude distributions at four planes within the 3D sample distribution. Adapted from[75].…”

Phase Retrieval of On-Axis Digital Holography With Modified Coherent Diffraction Imaging

“…Several techniques [10]- [20] have been applied to remove the undesired terms in on-axis DH. Phase shifting methods [10]- [12] are commonly used to remove the undesired terms and reconstruct the sample image.…”