# Weighted Constraint Iterative Algorithm for Phase Hologram Generation
## 2 Method
\tag{1}\]  
where \(|A_{k}|\) is the amplitude distribution of the input in \(k\)th iteration, \(|A^{\prime}{}_{k}|\) is the amplitude distribution of the reconstructed image in \(k\)th iteration, \((u,v)\) are the coordinates in the image plane, and \(S\) denotes the signal region. Adaptive parameters \(\beta_{k}\) are used here to enhance convergence of the iterative algorithm. In the numerical calculation, a small number \(\sim\)\(10^{-10}\) should be added on \(|A^{\prime}{}_{k}|\) for avoiding the division by zero in the amplitude constraint of the signal region. The amplitude constraint strategy in the image plane improves the reconstruction quality in the signal region by relaxing the amplitude constraint in the non-signal region. The enforced amplitude constraint changes during the iteration process, so that the difference between \(|A_{k+1}|\) and \(|A^{\prime}{}_{k}|\) is reduced. In this way, when the complex amplitude \(a_{k+1}\) in the hologram plane is calculated by the inverse Fourier transform of the image plane, it has better amplitude uniformity than the previous iteration, which helps to improve the reconstruction quality. Furthermore, the amplitude distribution of the signal region is constrained by the adaptive over-compensation method. From Equation (1), we can see that in \(k\)th iteration that if amplitude distribution \(|A^{\prime}{}_{k}|\) at some pixels of the signal region is smaller than the target amplitude \(|A_{\text{target}}|\), the enforced amplitude \(|A_{\text{con}}|\) at the corresponding pixels will be enlarged in \((k+1)\)th iteration, which helps to enlarge the amplitude \(|A^{\prime}{}_{k+1}|\) at the corresponding pixels and make it closer to the target amplitude. After several iterations, the difference between the amplitude distribution of the reconstructed image and the amplitude distribution of the target image is reduced. It should be noted that the convergence of the conventional over-compensation method would be easily degraded if it is used alone to design phase holograms [23]. Hence, it is always used as a final step to optimize further results obtained by some stable iterative algorithms, such as the GS  
Figure 3: The block diagram of WCIA. algorithm. Here, we modify the conventional over-compensation method by introducing adaptive parameters \(\beta_{k}\) into the iteration process. The adaptive parameters \(\beta_{k}\) are given by  
\[\beta_{k}=\sqrt{\beta_{k-1}} \tag{2}\]  
where the initial value \(\beta_{0}\) is a small number, \(\sim\)\(10^{-8}\). \(\beta_{k}\) would increase and approach to one after a certain number of iterations. When \(\beta_{k}\) is relatively small, the constraint strategy in the signal region is similar to the GS algorithm so that the convergence of the iteration process is effective and stable. After several iterations, \(\beta_{k}\) would be close to one, then the constraint strategy in the signal region is similar to the conventional over-compensation method, and a further optimized result can be obtained. In order to suppress the noise in the non-signal region, total energy in the hologram plane is controlled based on the energy conservation principle. In the Fourier transform system, the energy in the image plane is equal to the energy in the hologram plane:  
\[\iint\big{|}A^{\prime}_{k}(u,v)\big{|}^{2}dudv=\iint\big{|}A_{\text{holo}}(x,y) \big{|}^{2}dxdy \tag{3}\]  
By controlling the total energy in the hologram plane, the total energy in the image plane is constrained.