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 GS-based algorithm that allows reconstruction of 3D samples is demonstrated. The GS-based algorithms recover a complex-valued wavefront using 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 of 3D objects, including phase objects, are provided. Resolution enhancement obtained by iterative extrapolation of holograms is also discussed.

中文翻译：

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数字全息术的迭代相位检索：教程。

本文提供了基于数字全息术中基于Gerchberg-Saxton（GS）算法的迭代相位检索算法的教程。此外，还展示了一种新颖的基于GS的算法，该算法可重建3D样本。基于GS的算法使用两个平面之间的约束叠加在两个平面之间的波阵面来回传播来恢复复数值波阵面。迭代相位检索允许从其在线全息图中定量地校正物体幅值和相位分布，并进行双图像校正。本工作得出了定量标准，该标准要求重建2或3D样本的复数值对象分布需要多少个全息图。结果表明，对于可以近似为2D样本的样本，单次在线全息图足以重建样品的吸收和相分布。以前，基于GS的算法已成功用于重建限于2D平面的样本。但是，现实的物理对象始终具有一定的厚度，因此是3D而不是2D对象。这项研究表明，可以从两个或多个全息图重建3D样本（包括3D相位对象）。结果表明，原则上，两个全息图足以恢复3D样本分布衍射的整个波前。在这种方法中，通过在测量强度的平面之间应用迭代相位检索来执行重建。恢复的复数值波前然后传播回样本平面，从而重建样本的3D分布。此方法可以应用于3D样本，例如粒子的3D分布，浓厚的生物样本和其他3D相对象。提供了包括相位对象的3D对象的重建的示例。还讨论了通过全息图的迭代外推获得的分辨率增强。