# Optimization of Phase-Only Computer-Generated Holograms Based on the Gradient Descent Method
## 1 Introduction  
Holography [1; 2] is an ideal technique for producing three-dimensional (3D) images based on wave front reconstruction. An object wave and a reference wave are optically superimposed to generate an interference pattern, which is then recorded on a photosensitive film to produce a hologram. Computer-generated holograms (CGHs) [3; 4; 5; 6] are produced as the interference pattern which is calculated by a computer and recorded on a film. The CGH technique enables the generation of 3D images of virtual objects stored on a computer. When CGHs are displayed on spatial light modulators (SLMs), moving 3D images can be produced. The holography technique transforms the amplitude and phase distributions of an object wave into the intensity distribution of an interference pattern. When a reconstruction wave illuminates the interference pattern, the amplitude and phase distributions of the object wave are reconstructed. Phase-only holograms, which modulate the phase distribution but not the amplitude distribution of the reconstruction wave, were developed to obtain bright reconstructed images. Optically recorded holograms can be bleached to produce phase-only holograms [7; 8; 9]. Because the amplitude information is lost, the reconstructed images are degraded. When phase-only holograms are produced using the CGH technique, the phase distributions can be optimized using a computer to reduce the degradation of the reconstructed images. The interference patterns do not need to be calculated because the shape of the light wavefront can be directly controlled by the phase distributions. The complex-conjugate image does not appear because the interference pattern is not recorded. Because of their high light efficiency, phase-only CGHs havebeen used for a wide range of applications, such as optical tweezers [10], optical interconnections [11], and optogenetic techniques [12]. Because neither the amplitude nor the phase distributions of the reconstructed object wave can be fully controlled using only phase distributions of the holograms, phase-only CGHs are generally designed to control the intensity distributions of reconstructed images. Because the phase distributions of phase-only holograms cannot be directly calculated from the intensity distributions of target images, a number of optimization techniques have been developed. Iterative algorithms, such as simulated annealing [13], genetic algorithms [14; 15; 16], and direct binary searches [17; 18; 19], have been developed to optimize phase-only CGHs. As the CGH resolution increases, the number of iterations required for these algorithms increases rapidly. Although genetic algorithms need few iterations, the large quantity of calculations is required at each iteration step. The Gerchberg-Saxton (GS) algorithm [20], which iteratively performs Fourier transforms and inverse Fourier transforms, can more rapidly optimize phase-only CGHs than the above algorithms. In particular, the number of iterations required for the GS algorithm does not increase substantially as the CGH resolution increases. Several variants of the GS algorithm have also been developed. The mixed-region amplitude freedom algorithm [21] reduced the optimization time; however, it suffered from a low diffraction efficiency. The offset mixed-region amplitude freedom algorithm [22] provided highly accurate reconstructed images, but the black region could not be controlled efficiently. [PERSON] et al. [23] proposed yet another revised GS algorithm, which yielded high-quality reconstructed images in the preset area, nevertheless, it left the remaining regions filled with speckles. A modified GS algorithm (MGSA) [24] has also been proposed, in which the Fourier transform is replaced by the Fresnel transform in order to generate reconstructed images at any depth position. The MGSA has been used to generate multiple reconstructed images at different depth positions to create 3D images with a wide depth range [7; 25].