A treatise on cascaded computer generated holograms

In diffractive optics and computer holography, generally, a common and well‐studied scenario is using one hologram plane to generate a single image plane. As a continuation, in this work, cascaded phase only holograms are presented for single and multi‐plane image formation. Traditional design methods, such as deterministic iterative and stochastic simulated annealing algorithms, for the one planar hologram case have been expanded and enriched to handle cascaded computer generated holograms. This special arrangement of holograms demonstrates interesting attributes in the reconstruction of classical 2D images. When it comes to 3D images, on the other hand, the performance of these cascaded‐holograms is found to follow a trend and is even more remarkable: in order to facilitate the system, 3D images may be conceived to consist of successive 2D image planes; after application of the cascaded holograms for reconstruction, numerical results from computer experiments show that increasing the number of holograms for input, decreases the error in all output image planes for 3D setup. In addition, the cascaded holograms technique can be combined with the classical method of noise window which is achieved by planar extension of a single hologram by forfeiting some resolution. Thus when cascaded holograms are also expanded in the lateral domain, the results are even more promising. Specifically, it is observed that, while lateral extension of holograms as an application of the classical noise window concept is responsible for noise removal phenomena up to some degree in a general sense, longitudinal extension as a form of cascades may especially become a remedy for apparent incompatibility between image planes appearing in multi‐focal systems, which is another remarkable outcome. As a continuation of the work, instead of cascaded phase only holograms, amplitude modulation is taken into account and phase plus amplitude modulation with a certain distance between them, is analyzed. This configuration shows some interesting properties and some significant superiority with respect to traditional techniques although some limitations, such as requirement of larger pixel sizes which is on the boundary of diffractive optics, exist. ❧ The proposed schemes in this work may be applicable to a formation of a simple and light‐weight setup. Therefore, by using the techniques, results and algorithms here, especially on the cascaded phase only holograms, a functional static micro‐holographic 3D display may be realizable. In addition, phase and amplitude modulating setup can be used on applications requiring large pixel size, such as on devices with large area electronics. ❧ Additionally, methods to overcome possible hurdles in both computational design and physical application stages of the novel displays whose properties are mentioned above are briefly discussed. Although the work up to now is concentrated on algorithms and computation, experimental techniques for fabrication and error consideration coming from physical misalignment are also proposed. We believe that this work will be not only useful and functional in the analysis of cascaded phase elements but also a thorough examination on algorithms, noise removal or some other issues on computer holography and computational optics.

在衍射光学与计算全息领域，常规研究中普遍采用的经典方案是利用单个全息平面生成单一像平面。作为该方案的延伸，本文提出了可实现单平面与多平面成像的级联纯相位全息图（cascaded phase only holograms）。针对单平面全息场景的传统设计方法——如确定性迭代算法与随机模拟退火算法——已被拓展并完善，以适配级联计算生成全息图的设计需求。这种特殊的全息阵列结构在经典二维图像的重建过程中展现出独特的性能特性。
而就三维图像而言，这类级联全息图的表现更具规律且效果更为出色：为简化系统设计，三维图像可被视为由一系列连续二维像平面构成；通过级联全息图进行重建后，计算机仿真的数值结果表明，增加输入全息图的数量，可降低三维成像系统中所有输出像平面的重建误差。
此外，级联全息技术可与经典的噪声窗口（noise window）方法相结合，该方法通过牺牲部分分辨率来实现单全息图的平面扩展。当级联全息图在横向域进行拓展时，其重建效果将更为优异。具体而言，基于经典噪声窗口概念的全息图横向扩展，可在一定程度上实现噪声抑制；而作为级联形式的纵向扩展，则可专门解决多焦系统中不同像平面间的明显适配矛盾，这是另一项重要的研究成果。
作为本文工作的延伸，本文并未局限于级联纯相位全息图，还探讨了振幅调制方案，并对间距固定的相位-振幅联合调制结构展开了分析。相较于传统技术，该配置展现出了独特的性能与显著的优势，尽管其仍存在一定局限，例如需要更大的像素尺寸，而这已接近衍射光学的技术边界。
❧ 本文提出的方案可应用于构建轻量化简易系统。因此，借助本文所述的技术、研究结果与算法——尤其是级联纯相位全息图相关内容——可实现功能完备的静态微型全息三维显示器（static micro-holographic 3D display）。此外，相位-振幅联合调制结构可应用于对像素尺寸要求较高的场景，例如搭载大面积电子器件的设备。
❧ 此外，本文还简要讨论了克服上述新型显示系统在计算设计与物理实现阶段可能遇到的障碍的方法。尽管目前的研究主要聚焦于算法与计算层面，本文同时也提出了用于制备的实验技术，以及由物理对准误差带来的误差分析方法。我们认为，本文不仅可为级联相位元件的分析提供实用支撑，还可对计算全息与计算光学领域的算法研究、噪声抑制及其他相关问题展开全面的探讨。