# Weighted Constraint Iterative Algorithm for Phase Hologram Generation
## 2 Method  
An iterative optimization algorithm is an effective technique to design a phase hologram for converting the incident plane wave in the hologram plane into the target intensity distribution in the image plane. In this work, we focus on the generation of Fourier phase holograms, and an optical Fourier transform system is shown as Figure 1. The hologram plane is located in the front focal plane of the Fourier lens, and the image plane is located in the back focal plane of the Fourier lens. The incident plane wave in the hologram plane is modulated by the phase hologram and then converted into the target intensity distribution in the image plane by the optical Fourier transform system. A conventional iterative algorithm, such as the GS algorithm, is a stable algorithm to design phase holograms. However, the GS algorithm gives the same weight to all sampling points in the image plane, leading a slow and stagnant convergence. Our method would effectively improve the visual quality of the reconstructed image by addressing different constraint strategies at the corresponding regions in the image plane. The image plane is partitioned into two regions according to the amplitude distribution of the target image. The signal region is the area where the signal pattern is located, and the non-signal region is the area where there is no designed signal. Figure 2 illustrates two typical examples of the target images. The left one is a grayscale image where the non-signal region is set at the surrounding area. The right one shows a target image where the signal region is located in the annulus with uniform intensity, which is often used in beam shaping applications. In this case, the signal region is not rectangular, and the non-signal region is set at the black area which is separated by the annulus. The working principle of our proposed algorithm is shown in Figure 3. The initial random phase \(\varphi\) is applied to the target image as \(A_{0}=|A_{\text{target}}|\text{exp}(i\varphi)\), where \(|A_{\text{target}}|\) is the amplitude distribution of the target image, and the resulting complex amplitude \(A_{0}\) is set as the initial input of the iteration process. Complex amplitude \(a_{k}\) in the hologram plane is calculated by the inverse Fourier transform of the input  
Figure 1: Schematic geometry for optical Fourier transform system: the hologram plane is located in the front focal plane of the Fourier lens, and the image plane is located in the back focal plane of the Fourier lens. Figure 2: The partition strategy of the weighted constraint iterative algorithm (WCIA): The signal region is the area where the signal pattern is located, and the non-signal region is the area where there is no designed signal. complex amplitude \(A_{k}\) in the image plane. Then, the phase hologram \(a^{\prime}{}_{k}\) is generated from the complex amplitude \(a_{k}\) after a phase extraction and amplitude regularization. Subsequently, the reconstructed complex amplitude distribution \(A^{\prime}{}_{k}\) in the image plane is calculated by the Fourier transform of the phase hologram \(a^{\prime}{}_{k}\). The input complex amplitude \(A_{k+1}\) for the next iteration is generated after the amplitude constraint in the image plane. In the amplitude constraint of the image plane, only the amplitude distribution of the signal region is constrained using an adaptive over-compensation method. The over-compensation method is always utilized to speed up the convergence of the iteration process [23]. However, the convergence of the conventional over-compensation method would be easily degraded if used alone. Here, we modify the over-compensation method by introducing adaptive parameters which can change with the iteration progress. During the iteration process, the enforced amplitude constraint for the next iteration in the image plane is  
\[|A_{\text{con}}|=\left\{ \begin{array}{ccc}|A_{k}|\Big{(}|A_{\text{target}}| \Big{)}\Big{|}A^{\prime}_{k}\Big{|}^{\beta_{k}}&,&(u,v)\in S\\ &\Big{|}A^{\prime}_{k}\Big{|}&,&(u,v)\
otin S\end{array} \right.