Non-iterative 3D computer-generated hologram based on single full-support optimized random phase and phase compensation

The main problem faced by traditional three-dimensional (3D) holographic displays is the time-consuming and poor flexibility of the hologram generation process. To address this issue, this paper proposes a non-iterative 3D computer-generated hologram (SFS-ORAP-PC-3D) method based on single full-support optimized random phase and phase compensation. Combining the full-support optimized random phase (FS-ORAP) method and the 3D layer-based idea to efficiently and non-iteratively generate the phase-only hologram of a 3D object with arbitrary positions and sizes using single FS-ORAP, thus overcoming the limitations of the original ORAP method in target position and size. Meanwhile, using a Fresnel lens for phase compensation allows for free selection of reconstruction planes. Numerical and optical experiments validate the feasibility of our proposed method.


Introduction
Three-dimensional (3D) holograms enables to reconstruct the complete optical field of a 3D scene [1], providing the depth cues that are observed by the human eye [2].The progress in spatial light modulators (SLMs) and computational technology has made it possible to dynamically reconstruct 3D scenes using computer-generated holograms (CGH) [3].CGH uses a computer to simulate the recording process The CGH is loaded onto a SLM and then illuminated by coherent light to achieve reconstruction.Compared with optical holography, CGH can not only record real objects but also virtual objects without the need for complex optical systems, and can achieve dynamic holographic 3D display [4].Application areas for CGH include holographic displays and holographic projections, or use as a variety of other diffractive optical elements such as lenses and diffraction gratings [5].
In 3D CGH, iterative methods are commonly employed to generate the phase-only holograms [6].Based on the classical Gerchberg-Saxton (GS) [7] algorithm, two prominent 3D CGH methods are the 3D sequential GS algorithm [8] and the 3D global GS algorithm [9].The former utilizes the last plane as a constraint and lacks the flexibility to choose a reconstruction plane.Moreover, both methods require iteration for specific targets, resulting in time-consuming hologram generation.In 2018, Alejandro Velez Zea et al proposed the optimized random phase (ORAP) method [10] for non-iterative generation of phase-only holograms of different targets with the fixed support as the target window.However, a new ORAP is required to be regenerated if the size and position of the target changed.
To address the time-consuming drawback of traditional GS methods and the limited flexibility of ORAP methods, this paper introduces a non-iterative 3D CGH method based on single full-support optimized random phase and phase compensation (SFS-ORAP-PC-3D).Our SFS-ORAP-PC-3D method can be directly applied to produce holograms for 3D holographic displays, surpassing the limitations of fixed support size and position in the spatial domain encountered by the original ORAP method.The proposed method utilizes a single full-support ORAP to generate phase holograms for each layer without requiring iteration, thereby significantly accelerating the generation of phase holograms for 3D objects.Numerical and optical experiments confirm the effectiveness of the SFS-ORAP-PC-3D method.
The article is structured as follows: the first chapter briefly introduces the importance and application of the CGH method as well as the traditional 3DCGH method and presents the methodology of this paper.Chapter 2 describes the basic CGH methods, including the principle of CGH, the 3D-GS method and the FS-ORAP method.Chapter 3 describes in detail the principles and steps of the proposed method in this paper.Chapter 4 verifies the performance of the proposed method in this paper through simulation and real experiments.Chapter 5 summarizes this paper.

3D computer-generated holograms
The process of holographic display can be divided into two primary steps: wavefront recording and wavefront reconstruction [11].During wavefront recording, an interference pattern is generated between the wavefront of the object and a reference wavefront.This interference pattern is then captured and recorded as a hologram.When illuminated by light waves, the hologram functions as a diffraction screen, faithfully reproducing the original wavefront of the object.
In order to generate holograms of 3D targets, a commonly used approach is the layer-based method [12], which devides the 3D scene into multiple 2D layers according to their depth information, followed by generating phase holograms for each layer.Subsequently, these individual holograms are superimposed and the superimposed holograms are loaded onto SLM for reconstruction.Figure 1 illustrates a schematic diagram depicting the layer-based method for generating 3D holograms.

3D Gerchberg-Saxton iterative algorithms
There are two commonly used GS algorithms for 3D CGH: sequential GS and global GS.The sequential GS algorithm firstly propagates the light field generated by the initial random phase to each layered plane and replaces the amplitude on each plane with the target amplitude [13].As the propagation reaches the final layer, the complex field is propagated to the SLM plane, where the amplitude is discarded while preserving the phase.The next iteration begins using the phase hologram obtained.After multiple iterations, the phase in the hologram plane is referred to as a multi-plane phase hologram [12].The global GS algorithm propagates the light field generated by the initial random phase to all planes simultaneously, replacing the amplitude on each plane with the target amplitude.Then, propagate the complex field of each plane to the SLM plane, sum them up, and extract the phase.Start the next iteration using the acquired phase hologram.After multiple iterations, the phase hologram of the three-dimensional target is obtained.
The key difference between the 3D global GS algorithm and the 3D sequential GS algorithm lies in the propagation of phase holograms.In the global GS algorithm, phase holograms propagate simultaneously to all layers rather than sequentially, and the holograms of each layer are averaged during each iteration.
Both algorithms are based on traditional GS algorithms, but exhibit different performance.Sequential GS tends to diverge and use the last plane as a constraint, while global GS is more stable due to averaging the contributions of all planes simultaneously in each iteration.The above two GS algorithms are traded for high-quality reconstruction by sacrificing flexibility and increasing computational costs.

Full support optimized random phase method
The ORAP method, proposed by Alejandro Velez Zea et al is limited to generating holograms for specific sizes and positions of targets.Consequently, when applying the ORAP method to generate holograms for 3D objects, it requires an iterative process to generate an ORAP based on the size and position of each layer.However, this iterative approach results in longer times for a new ORAP generation and limited flexibility.To overcome the limitations of ORAP, this paper uses the Full Support Optimized Random Phase (FS-ORAP) method [14].The FS-ORAP method tackles the constraints of ORAP by replacing the fixed support with full support in the spatial domain, resulting in a single FS-ORAP.By using a single FS-ORAP, holograms for targets with arbitrary size and position can be efficiently generated, leading to time savings and enhanced flexibility.In the FS-ORAP method, the first step involves generating a full-support unit amplitude that matches the size of the SLM plane.This amplitude is then multiplied by a random phase, followed by an inverse Fourier transform of the product.The resulting complex amplitude undergoes multiple iterations of the standard GS loop, where the amplitude is replaced with the target amplitude at each iteration.The SLM plane uses a uniform amplitude of the same size as the SLM plane, while the reconstruction plane employs the previously created target window.These iterations ensure a satisfactory approximation between the amplitude of the reconstructed window and the target window.Once the iterations are completed, the FS-ORAP is generated.The FS-ORAP is multiplied by the amplitude target, and an inverse Fourier transform is performed on the product, replacing the amplitude of the result with a unit amplitude to obtain a phase-only hologram.Figure 2 provides an illustration of the process of generating phase-only holograms using the FS-ORAP method.

Proposed method
We propose a non-iterative 3D CGH method, referred to as the Single Full-Support Optimized Random Phase and Phase Compensation for 3D holograms (SFS-ORAP-PC-3D).This SFS-ORAP-PC-3D method combines the layer-based approach, FS-ORAP method, and phase compensation technique.
As depicted in figure 2, firstly, the initialization module and FS-ORAP generation module, are utilized to generate the FS-ORAP.Secondly, the 3D target is divided into layers based on the depth information.The amplitude of each layer is then multiplied by the FS-ORAP, followed by an inverse Fourier transform to generate phase-only holograms for each layer.The holographic generation modules for each layer can be observed in figure 3.
To incorporate depth information and facilitate the reconstruction of 3D targets, phase compensation is applied to the phase holograms of each layer.Since the individual layers do not possess depth information, they are unable to accurately reconstruct the 3D object and determine an appropriate reconstruction plane.The method proposed in this paper involves the superposition of Fresnel lens phase holograms to obtain a composite phase hologram [15].Multiple Fresnel lenses with different focal lengths f i (i = 1, 2, 3, 4) are combined with the phase holograms to generate new phase distributions [16] where φ i is the phase hologram with added Fresnel lens, φ i is the phase hologram of each layer, φ FL i is the phase of a Fresnel lens with a focal length of f i , φ final is the final generated phase-only hologram, λ is the wavelength and u, v is the coordinate.The composite phase holograms from each layer are converted into complex amplitudes and superposed, preserving the resulting phase.This process yields the phase hologram of the three-dimensional target.The phase compensation module is depicted in figure 4. By incorporating the Fresnel lens into the hologram generation, the phase holograms can be reconstructed using the Fresnel transform.The Fresnel diffraction integral formula is applied in this process: where k is the wave number, x, y are the coordinates, and z is the distance between the SLM plane and the observation plane.The overall flow chart of SFS-ORAP-PC-3D is presented in figure 5.By employing the non-iterative hologram generation process, our proposed SFS-ORAP-PC-3D method enables high-quality reproduction while enhances the computational speed and efficiency.

Numerical experiments
To validate the feasibility of the proposed SFS-ORAP-PC-3D method for generating phase holograms of 3D objects, four sets of comparative experiments are conducted.In Experiment 1, the feasibility of generating phase holograms of 3D targets using the SFS-ORAP-PC-3D method is tested by varying the interval between different layers.In Experiment 2, the phase-only holograms of the 'dragon' model with accurate depth precision and a wide depth range is numerically demonstrated.In Experiment 3, the performance of the sequential GS method, the global GS method, and the SFS-ORAP-PC-3D method are compared in terms of reconstruction quality and computational cost of 3D CGH generation.In Experiment 4, the feasibility of 3D CGH of dynamic 3D targets are tested using both the SFS-ORAP-PC-3D method and the 3D-ORAP method.The reconstruction quality are evaluated using the peak signal-to-noise ratio (PSNR, in dB) and correlation coefficient (CC): PSNR = 20log 10 where I R is the reconstructed target, I O is the original target, ĪR and ĪO are its mean values, M × N is the target resolution, MAX is the maximum pixel value of the image, and [n, m] are pixel coordinates.The hardware environment used in these experiments includes a Core i5-12450 H processor, 16GB of memory, and the Windows 11 operating system.All programs are implemented in MATLAB R2019a.
Experiment 1 aimed to examine the feasibility of generating 3D phase holograms using SFS-ORAP-PC-3D.The SFS-ORAP-PC-3D method allowed for the selection of reconstruction planes, with 'A,' 'H,' and 'U' representing the three layers of the 3D target.Each layer and the FS-ORAP size were set to 1024 × 1024, with 20 iterations, and the distance between the nearest plane and the holographic plane was set to 500 mm.The distance between each layer was modified to generate the respective hologram and perform reconstruction.Figure 6 depicts the numerical reconstruction results of this comparative experiment.The experiment on complex 3D targets can be viewed in appendices A and B.
As depicted in figure 6, it can be observed that when the distances between layers are relatively close, the projected images of other planes can be seen on a specific plane.As the distance increases, the projected images of other planes become more blurred.This phenomenon occurs because when the spacing between layers is very small, each layer can be considered to be on the same plane.Consequently, the images of all layers can be clearly displayed on that plane, leading to significant crosstalk.Conversely, when the distance between layers is very large, the target can be treated as a single plane, and only the image of that particular layer can be observed, crosstalk is noticeably minimal.
The objective of Experiment 2 is to validate the advantages of the SFS-ORAP-PC-3D method proposed in this paper in comparison to the sequential GS method and the global GS method.Figure 7 presents the reconstruction results obtained from the sequential GS, global GS, and SFS-ORAP-PC-3D methods.Table 1 displays the time taken for hologram generation using the sequential GS, global GS, and SFS-ORAP-PC-3D methods.
As illustrated in figure 7, the sequential GS method is limited in that it cannot select a specific reconstruction plane and can only reconstruct the image displaying the last layer.On the other hand, although the global GS method is capable of reconstructing images of each plane, the reconstructed image exhibits uneven amplitude distribution, resulting in the presence of bright and dark spots and the ability to observe crosstalk images from other planes.In contrast, the SFS-ORAP-PC-3D method proposed in this paper showcases favorable numerical reconstruction results.
According to table 1, the sequential GS method and the global GS method require more computational time for hologram generation since they iterate on the target amplitude and generate holograms separately.When faced with different target amplitudes, the sequential GS method and the global GS method will take longer.The SFS-ORAP-PC-3D method, on the other hand, primarily spends time on the generation process of each layer of FS-ORAP, while the generation of holograms from each layer of FS-ORAP takes less time.The SFS-ORAP-PC-3D method exhibits the advantage of flexibility.

Optics experiments
The feasibility and advantages of the SFS-ORAP-PC-3D method proposed in this paper were verified by generating phase-only holograms of different types of 3D targets using sequential GS, global GS, and SFS-ORAP-PC-3D methods and performing optical reconstruction.The LCOS used in this experiment is MD1280 produced by Three Fivesystems, with a resolution of 1280 × 1024, with a pixel spacing of 12 µm, used for loading phase holograms.The focus length of the lens used in this experiment is 30 cm, and the camera uses Canon EOS 600D.The laser wavelength used in the laboratory is 532 nm.In order to reduce the influence of zero order light and multi-level diffraction image [17] on the experimental results, the spherical focusing wave and Blazed grating are superimposed on the phase-only hologram.Then, the superimposed phase-only hologram is loaded onto the LCOS, projected onto the camera, and the imaging results are captured by the camera.The experimental optical path is shown in figure 8.
In Experiment, a phase-only hologram of the 'dragon' model was generated using the SFS-ORAP-PC-3D method and reconstructed.The 'dragon' model was layered according to depth, with a distance of 200 mm  between the nearest layer and the holographic plane.The optical reconstruction results are shown in figure 9.
Real experiments comparing with other methods can be found in appendix C.
From figure 9, it can be seen from the optical reconstruction results of the 'Dragon' model using the SFS-ORAP-PC-3D method that the reconstruction results can restore the shape and contour of the object, verifying the feasibility of generating phase-only holograms of complex 3D targets using the SFS-ORAP-PC-3D method.

Conclusion
This article presents the SFS-ORAP-PC-3D method, which enables the non-iterative generation of phase holograms for 3D objects and can be directly applied to hologram generation and reconstruction.In comparison to the 3D ORAP method, SFS-ORAP-PC-3D exhibits advantages in terms of accommodating targets of varying sizes and positions, showcasing high flexibility.Additionally, SFS-ORAP-PC-3D not only offers faster processing speeds compared to the sequential GS and global GS methods but also addresses the limitation of the sequential GS method's inability to select reconstruction planes.In summary, the proposed SFS-ORAP-PC-3D method strikes a balance between reproduction quality and computational cost, meeting specific quality requirements in practical applications.The SFS-ORAP-PC-3D method proposed in this paper allows for the utilization of FS-ORAP in hologram generation, regardless of changes in the size or position of the target layer.In contrast, when employing the 3D-ORAP method to generate three-dimensional holograms, regenerating the ORAP becomes highly time-consuming whenever there are changes in the size or position of the target layer.This demonstrates that FS-ORAP offers higher flexibility, and this advantage becomes even more prominent when the size and distance of the target layer undergo multiple changes.

Figure 1 .
Figure 1.Schematic diagram of layer-based generation of 3D holograms.SLM stands for spatial light modulator, and zn represents the distance from the nth layer of the 3D target to the SLM plane.

Figure 6 .
Figure 6.Simulation results using SFS-ORAP-PC-3D method at different intervals.(a) Target 3D intensity map.(b) Reconstruction results using SFS-ORAP-PC-3D method with an interval of 0.001 m.(c) Reconstruction results using SFS-ORAP-PC-3D method with an interval of 0.01 m.(d) Reconstruction results using SFS-ORAP-PC-3D method with an interval of 0.1 m.

Figure 7 .
Figure 7. Simulation results of SFS-ORAP-PC-3D compared with existing algorithms.(a) The target three-dimensional intensity map contains 5 letters (0.1 m apart) on a uniformly spaced depth plane.(b) The reconstruction results of the sequential GS method.(c) The reconstruction results of the global GS method.(d) The reconstruction results of the SFS-ORAP-PC-3D method.

Figure 9 .
Figure 9. Phase-only holograms (first row) of the three-dimensional target 'Dragon' at different angles (a)-(d) and their corresponding reconstruction results (second row).