# Weighted Constraint Iterative Algorithm for Phase Hologram Generation  
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random phase and optimized phase modulation methods [13; 14; 15; 16; 17]. These methods possess the merit of low computation load; whereas the generalization of these methods would be affected due to the lack of further optimization. Iterative algorithms can optimize the phase profile of the hologram according to target images, which are widely used for designing phase holograms [18]. Iterative algorithms try to optimize the phase profile for improving the reconstruction quality by utilization of phase freedom in the image plane. The majority of iterative algorithms were developed on the basis of the error-reduction (ER) algorithm, which is also referred to as the Gerchberg-Saxton (GS) algorithm [19; 20; 21]. The GS algorithm has the property of non-increasing error under increasing quantity of iterations, leading effective convergence of the iteration process. Input-output algorithms were introduced to speed up convergence by introducing feedback calculation during the iterations [22; 23; 24]. However, the GS algorithm and input-output algorithms often stagnate into the local minimum. Instead of imposing amplitude constraint to the entire image plane, some iterative algorithms have been proposed to improve reconstruction quality by introducing amplitude freedom into the image plane [25; 26; 27; 28]. In these methods, the image plane was partitioned into a signal region and non-signal region, and only the amplitude distribution of the signal region was directly constrained. Recently, double-constraint iterative algorithms were introduced to suppress the speckle noise, in which the desired amplitude and smooth phase distribution were constrained in the signal region during the iteration process [29; 30]. These methods improve the reconstruction quality of the signal region by relaxing the amplitude constraint of the non-signal region. However, the reconstruction in the non-signal region would be less controlled, causing severe noise and low optical efficiency. Although the noise in the non-signal region could be reduced by introducing a noise suppression parameter [26], it would sacrifice the optimization space of the iterative algorithm and degrade the reconstruction quality in the signal region. It is difficult to obtain a satisfactory reconstruction while maintaining a high optical efficiency in these methods. In addition, the noise suppression parameter should be chosen manually, and the optimal value is dependent on different target images, which would be inconvenient when generating phase holograms with different target images. In our previous work [31], the noise in the non-signal region was suppressed by controlling the total energy in the hologram plane, and the amplitude distribution of the signal region was constrained by a two-step constraint strategy. This method can effectively reduce the noise in the non-signal region; however, the amplitude constraint strategy in the signal region is not stable and might degrade the convergence of the iteration process. In this study, a weighted constraint iterative algorithm (WCIA) is presented to calculate phase holograms with quality reconstruction and effective convergence. Specifically, the image plane is partitioned into signal region and non-signal region 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. Different constraint strategies are addressed on the corresponding regions during the iteration process. The amplitude distribution of the signal region is constrained directly according to the target image using an adaptive over-compensation method, which can change the constraint parameter in the image plane with the iteration progress for effective and fast convergence.