Objective Obtaining high-quality reconstruction is desirable in computer-generated holography. Continuous complex-amplitude computer-generated holograms (CGHs) can present the most enhanced reconstruction quality because accurate amplitude and phase values rather than approximate values are obtained. However, in a practical system, CGHs need to be uploaded on the spatial light modulator (SLM). The most commonly used SLMs can only modulate either amplitude or phase. In addition, SLMs generally have pixelated structures with limited value ranges. It is necessary to sample the continuous distribution into a two-dimensional matrix with specific resolution and discrete pixel values in practical applications. This characteristic may harm the holographic reconstruction quality. Therefore, an optimization method based on parameter space traversal is proposed in this study to evaluate the effect of quantization on the holographic reconstruction quality. Various related parameters are considered in the evaluation. Proper quantization in some specific applications is also suggested.
Methods The CGH of a target object is calculated using the angular-spectrum model. In this model, when the reconstruction distance is too large, an aliasing error in the transfer function will be introduced. The maximum reconstruction distance, also called the effective distance, is determined by the Shannon-Nyquist sampling theorem. Meanwhile, when the reconstruction distance is too small, different diffraction orders on the reconstruction plane will interfere with each other. The minimum reconstruction distance is determined by the grating function. To quantitatively evaluate the reconstruction quality, the peak signal-to-noise ratio (PSNR) is used as the index to measure the difference between the reconstructed and target objects. Moreover, a traversal method is used to quantitatively evaluate the influence of quantization. Considering the pixelated structure and discrete value ranges of current SLMs, the continuous complex-amplitude distribution is converted into quantized amplitude- or phase-only distribution by rounding down decimals to integers.
Results and Discussions PSNRs of reconstructions via continuous complex-amplitude CGHs are infinite (Fig. 3). No matter how many related parameters, such as resolution, zero-padding area, reconstruction distance, reconstruction wavelength, and pixel pitch change, this conclusion remains unchanged. The calculation and reconstruction of continuous complex-amplitude CGHs were inverse processes. The 8-bit quantization of amplitude in complex-amplitude CGHs induced the degradation of reconstruction quality. The calculation and reconstruction of CGHs were not perfect inverse processes in this situation. However, the difference is negligible (Fig. 4). Compared with results by complex-amplitude CGHs with 8-bit quantized amplitude, results by complex-amplitude CGHs with 8-bit quantized phase presented a worse reconstruction quality. In addition, a zero-padding operation could improve the quality of the reconstruction by CGHs with 8-bit quantized phase. When the size of the target objects was doubled via the zero-padding operation, the PSNRs of reconstructions increased by 6.32 dB (Fig. 5). Phase-only CGHs were obtained by neglecting the amplitude of the complex-amplitude. The neglect of the amplitude had an extremely negative impact on reconstruction quality. PSNRs of reconstruction by phase-only CGHs decreased by 34.77 dB compared with those by complex-amplitude CGHs with 8-bit quantized amplitude. In some specific applications, quantization parameters could be selected appropriately. Phase-only CGHs with 5-bit quantization were proved to be suitable for the applications of dynamic holographic displays. Practically, a look-up table (LUT) often deviates from the designed one. However, a small phase modulation deviation had little effect on the reconstruction quality. In the application of anticounterfeiting, rough calibration for LUT could also be effective (Fig. 6). The reconstruction quality was affected by the quantization of both amplitude and phase. A small increase in the quantization of both amplitude and phase induced a better effect than a huge increase in the quantization of only amplitude or phase (Fig. 8). This conclusion was also applicable when the pixel pitch was less than 1 μm, which would provide guidance for designing meta-surface devices.
Conclusions Because of the modulation characteristics of available SLMs, complex-amplitude CGHs with continuous values usually need to be converted to amplitude- or phase-only CGHs with discrete values. The quantization process of approximating continuous values to discrete values has a significant influence on the holographic reconstruction quality. In this study, a traversal method is used to quantitatively evaluate the influence of quantization. Various parameters, such as resolution, zero-padding area, reconstruction distance, reconstruction wavelength, random phase, and pixel pitch are considered. For phase-only CGHs, neglecting the amplitude has an extremely negative impact on reconstruction quality. The PSNRs of reconstruction by phase-only CGHs decrease by 34.77 dB compared with those by complex-amplitude CGHs with 8-bit quantized amplitude. In some specific applications, quantization parameters can be selected appropriately. Dynamic holographic display, holographic anticounterfeiting, and the design of meta-surface devices are discussed specifically. We hope this study will provide a guideline for designing CGH-based systems.