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Analysis of reconstruction quality for computer-generated holograms using a model free of circular-convolution error

Xiaoting Wang, Zehao He, and Liangcai Cao

Open Access Open Access

Abstract

Continuous complex-amplitude computer-generated holograms (CGHs) are converted to discrete amplitude-only or phase-only ones in practical applications to cater for the characteristics of spatial light modulators (SLMs). To describe the influence of the discretization correctly, a refined model that eliminates the circular-convolution error is proposed to emulate the propagation of the wavefront during the formation and reconstruction of a CGH. The effects of several significant factors, including quantized amplitude and phase, zero-padding rate, random phase, resolution, reconstruction distance, wavelength, pixel pitch, phase modulation deviation and pixel-to-pixel interaction, are discussed. Based on evaluations, the optimal quantization for both available and future SLM devices is suggested.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Holography was first presented in 1948 by D. Gabor. The initial hologram was recorded on a physical medium through the interference of wavefront, hence it was called optical hologram. With the aid of electronic and computational technologies, computer-generated holography substitutes the generation of a hologram from an optical process to a simulated one [1]. The object used in computer-generated holography can be a fictional subject, as long as its mathematic model is available. Computer-generated holography can be applied in many fields, including optical communications [2], three-dimensional (3D) display [3,4], encryption [5], and laser processing [6].

In a practical system, the modulation of a computer-generated hologram (CGH) is often realized by a specific device called spatial light modulator (SLM). Current SLMs are basically classified into three major categories, which are digital micro-mirror device (DMD), liquid crystal on silicon (LCoS), and metasurface-based device. The metasurface-based device can modulate the complex-amplitude distribution, which is often employed to present a high-quality reconstruction [7,8]. However, the high cost in manufacturing and the difficulty in dynamic modulation limit its application currently. The DMD and LCoS are amplitude-only and phase-only devices, respectively. It is difficult for them to directly modulate both the amplitude and phase simultaneously [9,10]. There have been efforts in employing iterative algorithms to optimize both the amplitude and phase on the designated plane to achieve complex-amplitude modulation [11,12]. However, such algorithms suffer from huge demand of calculation, which are impractical in the reconstruction of large 3D scenes. Thus, a practical system often employs a phase-only or amplitude-only CGH that is calculated from the complex-amplitude distribution. The ignorance of the amplitude or the phase leads to a decline in image quality of the holographic reconstruction. Additionally, a typical SLM has a pixelated structure that requires a discrete modulation. Holographic algorithms need to adopt this feature by approximating each pixel of the CGH to the nearest available modulation state. The approximation significantly affected the quality of reconstruction.

The quantization errors of CGHs have been widely studied previously [13–17]. However, these researches were mostly to show various optimization methods, rather than quantitative analysis, for CGHs. They mainly studied on amplitude-only or phase-only CGHs, seldom on complex-amplitude CGHs. Our previous research has succeeded in describing a global and systematic approach to evaluate holographic reconstruction quality under various settings [18]. The effect of quantization on holographic reconstruction was analyzed by traversal comparisons, using the angular-spectrum model (ASM) as the propagation model and the peak signal-to-noise ratio (PSNR) as the evaluation index. However, in this research, the Fourier transform was approximately operated by the fast Fourier transform (FFT). The circular-convolution error was accordingly introduced. Since the circular-convolution error was not considered, the simulation of the wavefront propagation was not accurate enough. It could not describe the actual situation in some cases.

In this work, a refined ASM that eliminates the circular-convolution error is proposed. This model uses FFT algorithm to calculate the Fourier transform of the signal to ensure the calculation efficiency [19]. Zero-padding and image cropping are employed to avoid circular-convolution error. It can accurately evaluate the fidelity of the reconstructed image in wavefront propagation and is especially useful as available SLMs have limited performance on amplitude and phase modulation. Based on the refined model, the reconstruction quality of CGHs with continuous complex amplitude, quantized complex amplitude, quantized amplitude and quantized phase are all analyzed. Several significant factors including quantization bit depth, zero-padding rate, random phase, resolution, reconstruction distance, wavelength, pixel pitch and phase modulation deviation are also evaluated. Based on the analysis results, the optimal quantization bit depth for CGHs is suggested.

2. Methodology

2.1 ASM refined with zero-padding

In the ASM, if the holographic plane is set as the original plane, where z = 0, the complex-amplitude distribution on this plane EH (x, y, 0) can be described as [20,21]:

$${\tilde{E}_H}(x,y,0) = {\textrm{F}^{ - 1}}\left\{ {\textrm{F}\{{{E_O}(x,y,{z_0})R(x,y,{z_0})} \}\textrm{exp} [\textrm{j}\frac{{2\pi }}{\lambda }{z_0}\sqrt {1 - {{(\lambda u)}^2} - {{(\lambda v)}^2}} ]} \right\}$$
where F represents the Fourier transform, F−1 represents the inverse Fourier transform, z0 is the distance between the holographic plane and the object plane, EO (x, y, z0) is the amplitude of the target object, R (x, y, z0) is a random phase, λ is the wavelength of the illumination, and u and v are the spatial frequencies in the x- and y- directions. If α and β are the angles from the direction of the incident light to the x- and y- directions, respectively, u and v can be expressed as:
$$u = \frac{{\cos \alpha }}{\lambda },v = \frac{{\cos \beta }}{\lambda }$$

For simplicity, only one-dimensional (1D) ASM is considered. Correspondingly, Eq. (1) can be rewritten as:

$${\tilde{E}_H}(x) = {\textrm{F}^{ - 1}}\{{\textrm{F}\{{E{^{\prime}_O}(x)} \}{H_z}(u )} \}$$
where E’O (x) = EO (x) R (x), Hz (u) is the transfer function of ASM, which can be expressed as:
$${H_z}(u )= \textrm{exp} [\textrm{j}\frac{{2\pi }}{\lambda }{z_0}\sqrt {1 - {{(\lambda u)}^2}} ]$$

Equation (3) is then rewritten with a convolution form:

$${\widetilde E_H}(x) = E{^{\prime}_O}(x) \ast {h_z}(x)$$
where * represents the linear convolution with respect to x, hz (x) is the differential form of the transfer function Hz (u). For numerical computation, E’O (x) and hz (x) are sampled into two discrete distributions E’O (n) and hz (n). The linear convolution between E’O (n) and hz (n) can be expressed as
$${\widetilde E_H}(n) = \sum\limits_{m = 0}^{N - 1} {E{^{\prime}_O}(m) \cdot {h_z}(n - m)}$$
where N is the sampling number on the object plane. After convolution, the sampling number is expanded to 2N – 1. When we use FFT to replace the Fourier transform in numerical computation, the aperiodic distributions E’O (n) and hz (n) are extended to become periodic distributions. The linear convolution in Eq. (6) is converted to a circular convolution. If E’O (n) and hz (n) are expended to two distributions E’’O (n) and h’’z (n) with the sampling number L, the circular convolution is,
$$\widetilde E^{\prime}{^{\prime}_H}(n )= E^{\prime}{^{\prime}_O}(n )\otimes h^{\prime}{^{\prime}_z}(n )= \left[ {\sum\limits_{m = 0}^{L - 1} {E^{\prime}{^{\prime}_O}(m )\cdot h^{\prime}{^{\prime}_z}{{({({n - m} )} )}_L}} } \right]{R_L}(n )$$
where $\otimes$ represents the circular convolution; ((·))L represents the remainder of L, and RL (n) is the periodic extension with a period L. If L is smaller than the length of the conventional convolution result 2N – 1, the beginning part of one signal will overlap with the end of its previous signal during the periodic extension, leading to an aliasing error called circular-convolution error. To avoid the circular-convolution error, E’’O (n) should be expanded to a length L of at least 2N – 1, while the pixel value of the padded area should be set as 0 to avoid additional noise:
$$E^{\prime}{^{\prime}_O}(n) = \left\{ \begin{array}{l} E{^{\prime}_O}(n - N/2,0),N/2 \le n \le 3N/2\\ 0,\textrm{else} \end{array} \right.$$

In this way, the complex-amplitude distribution can be obtained by using FFT and inverse FFT (IFFT) without circular-convolution error.

The aforementioned process is illustrated in Fig. 1. To better utilize the active area of the SLM, the calculated complex-amplitude on the holographic plane is cropped. Because the random phase is superposed on the object plane, the cropped-off area contains part of information about the object. Therefore, the image quality is degraded in this process.

 figure: Fig. 1.

Fig. 1. Numerical construction and optical reconstruction in computer-generated holography.

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In the digitalization process, the computer-generated hologram, which is a continuous complex-amplitude distribution, has to be extracted and converted into a quantized amplitude or phase distribution, which can be expressed as:

$${E_A}({m,n,0} )= \frac{1}{{{2^{{B_A}}}}} \times \textrm{round}\left\{ {\frac{{\textrm{abs}[{{{\widetilde E}_H}^{^{\prime\prime}}({m,n,0} )} ]+ {E_{A0}}}}{{\max \{{\max [{{{\widetilde E}_H}^{^{\prime\prime}}({m,n,0} )} ]+ {E_{A0}}} \}}}} \right\}$$
$${E_P}({m,n,0} )= \frac{{2\pi }}{{{2^{{B_P}}}}} \times \textrm{round}\left\{ {\arg [{{{\widetilde E}_H}^{^{\prime\prime}}({m,n,0} )} ]\times \frac{{{2^{{B_P}}}}}{{2\pi }}} \right\}$$
where EA (m, n, 0) is the quantized amplitude, EP (m, n, 0) is the quantized phase, EA0 is a constant amplitude value, abs represents taking absolute value, arg represents taking argument value, round represents approximating a decimal to its nearest integer, and BA and BP are the bit depths of the amplitude and phase, respectively.

2.2 Range of propagation distance

In the ASM, there are two limitations for z0. Based on the Nyquist’s sampling theorem [22], to prevent aliasing errors in the sampled transfer function, z0 should have an upper limit. The local frequency f (u) of the phase in the 1D transfer function expressed by Eq. (4) can be derived as:

$$f(u) = \frac{1}{{2\pi }}\frac{\partial }{{\partial u}}\phi (u) = \frac{1}{{2\pi }}\frac{\partial }{{\partial u}}\arg [{{H_z}(u )} ]={-} \frac{{{z_0}u}}{{\sqrt {{\lambda ^{ - 2}} - {u^2}} }}$$

If L is the size of the target object, M is the sampling number, d is the sampling interval, the maximal sampling interval Δu should be (2 L)−1 according to Nyquist’s sampling theorem. To avoid the aliasing error, the local frequency f (u) should meet the following requirement:

$$\frac{1}{{\Delta u}} \ge 2\max (|{f(u)} |)$$

Since | f (u)| increases monotonously with u, it reaches the maximum value at umax = M / (2 L). Thus, the maximum propagation distance can be calculated:

$${z_0} \le {z_0}_{\max } = \frac{{L\sqrt {4{\lambda ^{ - 2}}{L^2} - {M^2}} }}{M} = Md\sqrt {4{\lambda ^{ - 2}}{d^2} - 1}$$

Here, z0max is called the effective distance.

In addition, there is also a lower limit z0min for the propagation distance in the ASM to prevent aliasing of different diffraction orders. Specifically, to separate the zero-order and the first-order of the diffraction, the distance between the same object points of adjacent orders on the reconstruction plane should be larger than the size of the target object L, which can be expressed as:

$$L \le {z_0}\tan {\theta _{ + 1}} = {z_0}\tan \left[ {\arcsin \left( {\frac{\lambda }{d}} \right)} \right]$$
where θ+1 is the diffraction angle of the first-order. Thus, the minimum propagation distance z0min can be calculated:
$${z_0} \ge {z_{0\min }} = \frac{{Md}}{{\tan \left[ {\arcsin (\frac{\lambda }{d})} \right]}}$$

In the following discussions, the distance between the holographic plane and the object plane is within the range of [z0min, z0max].

2.3 Evaluation of image quality

The peak signal-to-noise ratio (PSNR) is employed as the evaluation parameter to describe the difference between the target objects and corresponding holographic reconstructions, which can be expressed as [23]:

$${P_{SNR}} = 10{\log _{10}}\left\{ {\frac{{{g_{\max }}^2}}{{{M_{SE}}}}} \right\} = 10{\log _{10}}\left\{ {\frac{{MN \cdot {g_{\max }}^2}}{{\sum\limits_M {\sum\limits_N {{{[{{I_O}(m,n) - {I_R}(m,n)} ]}^2}} } }}} \right\}$$
where PSNR is the symbol of PSNR, MSE is the symbol of mean square error, gmax is the maximal grayscale value of the image, IO (m, n) and IR (m, n) are the pixel values of the target object and its corresponding reconstruction, respectively. The maximal grayscale value is related to the bit depth of the image. For an 8-bit image, the maximal grayscale value gmax is 255.

2.4 Random phase in calculation of CGH

In a holographic recording process, the illumination wave, which is a coherent plane wave, passes through the target object and then propagates to the recording plane. During the propagation, different frequency components of the target object have different diffusion angles, which can be expressed as:

$${\theta _{\textrm{diff}}} \propto \arcsin (\lambda f)$$
where θdiff is the diffusion angle, f is the frequency of the certain component. Therefore, the diffusion angle of the high-frequency component is larger than that of the low-frequency component. The low-frequency component is usually recorded in a small region of the CGH. Limited by the insufficient dynamic range of the pixel, the information of the low-frequency component on the holographic plane cannot be recorded completely. Accordingly, holographic reconstruction of low-frequency component has unacceptable image quality.

When a random phase is superposed on the target object, the low-frequency component can be recorded by more pixels because its diffusion angle is expanded [24]. Therefore, it has a possibility of being better reconstructed because the limited dynamic range of a certain pixel can be compensated by the adjacent pixels. However, the introduction of the random phase leads to speckle noise, which decreases the image quality of the reconstruction. Therefore, in the following discussions, holographic reconstructions both with and without random phase are carried out.

3. Results

3.1 Continuous complex-amplitude CGH

To analyze the effect of the quantization, holographic reconstructions of continuous complex-amplitude CGHs are firstly presented, which are used for comparison. The target objects employed in the analysis are USAF Resolution Target, Baboon and Otter, as shown in Fig. 2(a)–2(c). In the analysis, the original resolution of these target objects is 1000 × 1000, the recording wavelength is 532 nm, and the pixel pitch of the CGH is 3.7 µm. The reconstruction distance is set as 51 mm to satisfy the requirements of Eq. (13) and Eq. (15). When no random phase is superposed on the target object, the PSNRs of holographic reconstructions for USAF Resolution Target, Baboon and Otter are 12.36 dB, 19.67 dB and 28.20 dB, respectively. However, when random phase is applied, the PSNRs of holographic reconstructions for these target objects decreases dramatically to 9.81 dB, 12.80 dB and 13.22 dB, respectively. The results in Figs. 2(e)–2(i) indicates that even if the SLM is able to modulate continuous complex-amplitude CGHs, the reconstructed results still differ from the original ones. The processes of zero-padding and image cropping employed during the calculation of CGHs filter out high-frequency components of the target objects, resulting in the fuzziness of verges. Here, zero-padding means wrapping the original image with a black margin. When random phase is employed, the image quality of holographic reconstructions decreases sharply, overshadowing the benefit of compensating the dynamic range of a certain pixel by using more adjacent pixels.

 figure: Fig. 2.

Fig. 2. (a) USAF Resolution Target; (b) Baboon; (c) Otter; (d)-(f) Holographic reconstructions by continuous complex-amplitude CGHs (without random phase) of USAF Resolution Target, Baboon and Otter; (g)-(i) Holographic reconstructions by continuous complex-amplitude CGHs (with random phase) of USAF Resolution Target, Baboon and Otter.

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The impacts of parameters including resolution, zero-padding size, reconstruction distance, wavelength, random phase and pixel pitch on the image quality of holographic reconstructions have also been studied by employing the variable-controlling approach. As shown in Fig. 3, zero-padding can increase the reconstruction quality regardless of whether random phase is superposed on the target objects during the calculation. Here, the zero-padding size coefficient is defined as the result of dividing the size of the zero-padded object by that of the original object. As zero-padding size coefficient increases, holographic reconstruction yields higher accuracy. This phenomenon can be explained by regarding the zero-padding of the original image as increasing the size of the CGH. As the size of the CGH increases, more information from the original image can be recorded. Therefore, a better reconstruction quality can be also achieved. In addition, the average PSNR of results reconstructed from CGHs without random phase is 10 dB higher than that of reconstructions from CGHs with random phase. It proves the negative impact of random phase on the continuous complex-amplitude CGH.

 figure: Fig. 3.

Fig. 3. Impact of zero-padding on reconstruction quality.

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Except for zero-padding size and random phase, other parameters have few impacts on the image quality of reconstruction. For example, the relation between the reconstruction quality and the resolution is shown in Fig. 4. Here, no random phase is superposed on the target objects. The resolution changes from 1000 pixels to 2000 pixels, increasing by 200 pixels at each step in both x- and y- directions.

 figure: Fig. 4.

Fig. 4. Impact of resolution on reconstruction quality (without random phase).

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3.2 Complex-amplitude CGH with quantized amplitude

As mentioned in Section 1, a typical SLM often has a pixelated structure that requires a discrete modulation. Thus, the pixel value of a continuous CGH should be approximated to a nearest available state. To explore the influence of SLM’s discrete property on the image quality, the impact of pixel value approximation is worthy studying.

In this section, the impact of amplitude quantization is analyzed. The amplitude of each pixel is quantized into an 8-bit value, which has 255 possible states. With z0, λ and d unchanged, and no random phase is superposed on the target objects, the reconstructions of USAF Resolution Target, Baboon and Otter have the PSNRs of 12.36 dB, 19.66 dB and 28.20 dB, respectively. For comparison, when a certain random phase is superposed on the target objects, the reconstructions of USAF Resolution Target, Baboon and Otter have the PSNRs of 9.81 dB, 12.80 dB and 13.24 dB, respectively. Compared with the reconstruction of the continuous CGH, the impact caused by the quantization of amplitude is trivial.

The impacts of the resolution, pixel pitch and zero-padding on the reconstruction quality are also analyzed by adopting the variable-controlling approach. As shown in Fig. 5, zero-padding can also increase the reconstruction quality when CGHs with quantized amplitude are employed. Besides, the effect of changing the resolution and pixel pitch shares the same variation tendencies in Fig. 4, which has no obvious impact on the reconstruction quality.

 figure: Fig. 5.

Fig. 5. Impact of zero-padding on reconstruction quality when CGHs with quantized amplitude are employed.

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3.3 Complex-amplitude CGH with quantized phase

Subsequently, the impact of phase quantization is analyzed. In this section, the phase of each pixel is quantized into an 8-bit value. When no random phase is superposed on the target objects, the reconstructions of USAF Resolution Target, Baboon and Otter have the PSNRs of 12.31 dB, 19.65 dB and 28.19 dB, respectively. While the random phases are employed, the reconstructions have the PSNRs of 9.82 dB, 12.80 dB and 13.24 dB, respectively. The image qualities by employing CGHs with continuous complex-amplitude, quantized amplitude and quantized phase are compared in Fig. 6. The impacts of both amplitude and phase quantization are minimal, while the reconstruction quality is slightly more sensitive to the phase quantization. Besides, it can be concluded from Fig. 6 that without careful selection of the random phase, the image quality of holographic reconstruction might be greatly damaged.

 figure: Fig. 6.

Fig. 6. Comparison of reconstruction qualities by employing CGHs with continuous complex-amplitude, quantized amplitude and quantized phase. The selected part on the left is enlarged and shown on the right.

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For generosity, when a constant random phase is employed during the calculation of the CGH, the impact of zero-padding on the reconstruction quality of complex-amplitude CGH with either quantized phase or quantized amplitude is presented in Fig. 7. With the increase in zero-padding size coefficient, the reconstruction quality of both amplitude-quantized CGH and phase-quantized CGH is improved, from around 9 dB to 18 dB. Therefore, the phase quantization and amplitude quantization have similar impacts on the reconstruction quality as the zero-padding is concerned.

 figure: Fig. 7.

Fig. 7. The impact of zero-padding on reconstruction quality when either amplitude-quantized CGH or phase-quantized CGH is employed.

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In addition, as parameters including resolution, reconstruction distance, wavelength and pixel pitch change, the reconstruction quality almost keeps unchanged. These factors have few effects on the reconstruction quality, only changing the PSNR by less than 1 dB. For the impact of these factors, please refer to the Supplement 1.

3.4 Phase-only CGH and amplitude-only CGH

As mentioned in Section 1, most current SLMs only modulates either the amplitude or phase. Thus, either the amplitude or phase should be ignored during the modulation. Such an ignorance has a significant impact on the reconstruction quality.

The ignorance of the amplitude is first analyzed in this section. When no random phase is superposed during the calculation of the phase-only CGHs, the reconstructions of USAF Resolution Target, Baboon and Otter have PSNRs of 12.82 dB, 16.94 dB and 18.41 dB, respectively. When the random phase is employed, the reconstructions have PSNRs of 9.1 dB, 12.6 dB and 13.1 dB, respectively. Evidently, with the ignorance of the amplitude, the reconstructions of the phase-only CGHs calculated without the random phase have a huge decrease in quality. However, when a random phase is employed during the calculation of the CGHs, the decline in PSNRs of the reconstructions is only 1 dB lower compared to the results in Section 3.2 and Section 3.3.

From the reconstructions, the employment of the random phase has some positive effects on the image quality. With the employment of the random phase, more pixels are involved during the holographic recording. Therefore, the low-frequency components of the USAF Resolution Target are better reconstructed, as shown in Fig. 8. Since the employment of the random phase brings the speckle noise and excessive diffusion, the high-frequency components are severely damaged during the reconstruction.

 figure: Fig. 8.

Fig. 8. (a) Reconstruction of USAF Resolution Target with random phase, partly enlarged in low-frequency and high-frequency components. (b) Reconstruction of USAF Resolution Target without random phase, partly enlarged in low-frequency and high-frequency components.

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The impact of zero-padding on reconstruction quality of the phase-only CGH is also studied. CGHs calculated with and without random phase exhibit different tendencies as zero-padding size coefficient increases. As expected, when the random phase is imposed, the reconstruction quality increases with the zero-padding size coefficient. Yet the positive effect of the zero-padding on the reconstruction quality of the phase-only CGH is not as significant as shown in that of the complex-amplitude CGH, which means that the benefit of the zero-padding is overshadowed by the damage caused by the ignorance of the amplitude.

However, when no random phase is imposed, the effect of the zero-padding seems to be extraordinary that cannot be readily explained. As shown in Fig. 9, when the zero-padding size coefficient increases, the reconstruction quality exhibits a clear tendency of decline. The PSNR under a zero-padding size coefficient of 2 is even lower than that under a zero-padding size coefficient of 1. To further verify this effect, structural similarity (SSIM) is also employed in the evaluation [25]. The SSIM shows a more obvious decline tendency, from 0.59 under a zero-padding size coefficient of 1 to 0.06 under a zero-padding size coefficient of 2.

 figure: Fig. 9.

Fig. 9. Impact of zero-padding on reconstruction quality of phase-only CGHs.

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Practically, the phase-only CGH uploaded on the SLM is a CGH with an 8-bit quantized phase. When the phase-only CGHs are calculated without random phase and an 8-bit quantization is applied on the phase, the reconstructions of USAF Resolution Target, Baboon and Otter have PSNRs of 12.82 dB, 16.94 dB and 18.41 dB, respectively. It suggests that the quantization of the phase is an unimportant influence factor on the reconstruction quality.

The impact of ignoring the phase on the image quality of holographic reconstruction is studied. When no random phase is superposed during the calculation of the amplitude-only CGHs, the reconstructions of USAF Resolution Target, Baboon and Otter have PSNRs of 11.33 dB, 20.67 dB and 22.22 dB, respectively. It indicates that ignoring amplitude has larger influence on reconstruction quality. The reconstructions quality of amplitude-only CGHs calculated with the random phase are severely misshaped, indicating that the random phase might not be suitable in the calculation of the amplitude-only CGHs. The effect of zero-padding is examined, while it has little effect on the reconstruction quality. As padding size coefficient changes from 1 to 2, the PSNR only changes by less than 0.5 dB, indicating that in amplitude-only CGHs, the loss of information is so severe that even zero-padding can be of little assistance.

4. Discussions

4.1 Quantized phase-only and amplitude-only CGH with different bit depths

In some applications such as the high-resolution holographic video display, the data volume is of great significance to ensure speedy data transmission. Thus, compressing the bit depth of the phase-only CGH is a profitable option. As shown in Fig. 10, the reconstruction quality increases with the bit depth in the beginning and remains stable after a certain bit depth. The optimal bit depth for the phase-only CGH calculated with random phase is 3-bit, while for that without random phase is 5-bit.

 figure: Fig. 10.

Fig. 10. Relationship between the PSNR and bit depth of the phase-only CGH.

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The impact of quantization for amplitude-only CGH is also studied. The reconstruction quality under different bit-depths of amplitude shares a similar tendency with that shown in Fig. 10. The optimal bit depth for the amplitude-only CGH calculated without random phase is 3-bit, while the optimal bit depth calculated with random phase is 2-bit.

The reconstructions of phase-only CGHs with different bit depths which are uploaded on an LCoS are presented in Fig. 11. It can be seen from the figures that the reconstruction quality reaches its maximum at around 5-bit without random phase imposed and at around 3-bit with random phase imposed. These results are in conformity with the simulations, indicating that the information redundancy exists in an 8-bit phase-only CGH.

 figure: Fig. 11.

Fig. 11. Reconstructions by using an LCoS. (a) to (c): Reconstructions of 3-bit, 5-bit and 8-bit CGHs calculated without random phase imposed. (d) to (e): Reconstructions of 1-bit, 3-bit and 8-bit CGHs with random phase imposed.

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4.2 Phase modulation deviation

The LUT bridges the grayscale value of the CGH and the phase change of a phase-only SLM. The grayscale value of an 8-bit phase-only CGH ranges from 0 to 255, corresponding to the ideal modulation range of [0, 2π] for an ideal phase-only SLM. However, the realistic phase modulation range of an SLM often differs from the designed LUT owing to manufacturing defects, which disrupts the desired phase modulation.

To make the trend easy to be recognized, the SSIM is employed as the evaluation parameter. As shown in Fig. 12, with the random phase imposed, as the phase modulation range changes from [0, π] to [0, 3π] with a step of π/8, the reconstruction quality increases as the maximum phase value changes from π to 2π, and decreases as the maximum phase value changes from 2π to 3π. The variation of SSIM as the maximum phase value locates in the range [1.75π, 2.25π] is relatively smaller than that as the maximum phase value locates in the range [0, 1.75π] and [2.25π, 3π]. It indicates that in some specific applications such as anti-counterfeiting, rough calibration on the LUT can achieve acceptable results.

 figure: Fig. 12.

Fig. 12. Effect of phase modulation deviation on reconstruction quality

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4.3 Pixel-to-pixel interaction

In a current phase-only type SLM, the phase is modulated by imposing voltage on each pixel of the SLM panel. Correspondingly, the effective refractive index of each pixel is changed, resulting in changing the optical path of incident light. However, due to the small pixel size, the voltage imposed on one pixel may affect its adjacent pixels. The mutual interference caused by adjacent pixels affects the phase modulation, leading to a decline in reconstruction quality.

In this work, the effect of pixel-to-pixel interaction is simplified. Assuming that the ideal phase modulation of one selected pixel is φ0, and ideal phase modulations of its four adjacent pixels are φ1, φ2, φ3 and φ4 respectively, the phase modulation of this selected pixel after considering the mutual interference can be expressed as:

$${\varphi _0}^{\prime} = {\alpha _0}{\varphi _0} + {\alpha _1}{\varphi _1} + {\alpha _2}{\varphi _2} + {\alpha _3}{\varphi _3} + {\alpha _4}{\varphi _4}$$
where α0, α1, α2, α3 and α4 are different weight coefficients, respectively. In this work, the value of α0 is set as one. It is assumed that a maximum leakage coefficient C0 represents the largest possible value for α1, α2, α3 and α4. Thus, the value of these coefficients can be expressed as:
$${\varphi _i} = {C_0} \times \mathrm{{\cal N}}(0,1)\textrm{ }i = 1,2,3,4$$
where $\cal {N}$ (0, 1) represents a normally distributed random number between 0 and 1.

As the value of C0 ranges from 0 to 0.08, the relationship between the reconstruction quality and the value of maximum leakage coefficient is shown in Fig. 13. With a random phase imposed, the PSNR declines as the maximum leakage coefficient increases, but the effect is not obvious. This implies that under the conditions of this simulation, the pixel-to-pixel interaction has limited effect on reconstruction quality. However, this conclusion is obtained under a relatively simplified model. The actual effect of pixel-to-pixel interaction remains to be studied.

 figure: Fig. 13.

Fig. 13. Reconstruction quality with pixel-to-pixel interaction effect under different maximum leakage coefficients (with random phase)

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4.3 Optimized quantization in complex-amplitude CGH

When no random phase is employed, the contour map of PSNR when the amplitude and phase of a complex-amplitude CGH are quantized into different bit depths is presented in Fig. 14(a). Specifically, when the bit depth of the phase is fixed, the relationship between the PSNR and the quantization of the amplitude is shown in Fig. 14(b). It can be seen from the results that when the bit depth of the phase is fixed, the PSNR initially increases with the bit depth of the amplitude and quickly reaches a turning point. As the bit depth of the phase has a small value, the maximal value of the PSNR is relatively small, no matter how large the bit depth of the amplitude is. These two factors have a coupled impact on the reconstruction quality.

 figure: Fig. 14.

Fig. 14. (a) Contour map of PSNR when the amplitude and phase of a complex-amplitude CGH are quantized into different bit depths. (b) Relationship between the PSNR and the quantization of the amplitude when the bit depth of the phase is fixed.

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It can be seen from Fig. 14(b) that when the phase is quantized to less than 4-bit, the turning point of the PSNR curve appears when the bit depth of the amplitude is 3-bit. When the bit depth of the phase reaches 4-bit, the turning point appears when the bit depth of the amplitude is 3-bit. As the bit depth of the phase continues to increase, the turning point reaches 5-bit, but the improvement in PSNR is trivial.

The results have showed that the bit depths of both the amplitude and phase are 4-bit might be an optimal combination for complex-amplitude SLMs. Currently, some devices such as metasurfaces can modulate both the amplitude and phase simultaneously [26]. However, their application is limited by the balance between the reconstruction quality and the manufacturing cost [27]. The results indicate that during the design of these devices, it is desired to consider the quantization of the amplitude and phase simultaneously, rather than only improving one of them. Through an appropriate combination of quantization for both the amplitude and phase, high image quality can be even achieved under an affordable manufacturing cost.

5. Conclusion

In this work, a refined model without circular-convolution error for computer-generated holography has been proposed. By employing the refined model, the impacts of a wide variety of parameters on the image quality of holographic reconstructions have been analyzed. The quantization of both the amplitude and phase can reduce the reconstruction quality, while the impact of phase quantization is more significant. For amplitude-only and phase-only CGHs, ignoring either the phase or amplitude has an extremely significant influence on the reconstruction quality. In most cases, zero-padding is an effective method to improve the reconstruction quality because it makes CGH suffer less from the information loss. Random phase has negative influence on the reconstruction quality in most situations. However, for phase-only CGHs, random phase improves the recognizability of the holographic reconstruction because it can diffuse low-frequency components, avoiding the negative influence caused by the insufficient dynamic range of a single pixel in a phase-only CGH. For a phase-only CGH, the influence of the bit depth is studied. The results show that when no random phase is employed, 5-bit phase is an optimal choice for applications that have demanding storage requirements. While the random phase is imposed, the optimal bit depth can be set as 3-bit. For a complex-amplitude CGH, the influence of the amplitude quantization and phase quantization on the reconstruction quality is correlated. Increasing one of them dramatically while leaving the other one unmatched can be of little help in increasing image quality. The bit depths of both the amplitude and phase are 4-bit might be an optimal combination for complex-amplitude SLMs. The results in this work can be employed in the design of different types of devices, which have potentials for optical communications, 3D display, encryption, and laser processing.

Funding

National Natural Science Foundation of China (62035003, 62205173); China Postdoctoral Science Foundation (BX2021140).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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Supplementary Material (1)

NameDescription
Supplement 1       Effects of factors including resolution, reconstruction distance, wavelength and pixel pitch on reconstruction quality

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (14)
Fig. 1.
Fig. 1. Numerical construction and optical reconstruction in computer-generated holography.

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Fig. 2.
Fig. 2. (a) USAF Resolution Target; (b) Baboon; (c) Otter; (d)-(f) Holographic reconstructions by continuous complex-amplitude CGHs (without random phase) of USAF Resolution Target, Baboon and Otter; (g)-(i) Holographic reconstructions by continuous complex-amplitude CGHs (with random phase) of USAF Resolution Target, Baboon and Otter.

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Fig. 3.
Fig. 3. Impact of zero-padding on reconstruction quality.

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Fig. 4.
Fig. 4. Impact of resolution on reconstruction quality (without random phase).

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Fig. 5.
Fig. 5. Impact of zero-padding on reconstruction quality when CGHs with quantized amplitude are employed.

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Fig. 6.
Fig. 6. Comparison of reconstruction qualities by employing CGHs with continuous complex-amplitude, quantized amplitude and quantized phase. The selected part on the left is enlarged and shown on the right.

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Fig. 7.
Fig. 7. The impact of zero-padding on reconstruction quality when either amplitude-quantized CGH or phase-quantized CGH is employed.

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Fig. 8.
Fig. 8. (a) Reconstruction of USAF Resolution Target with random phase, partly enlarged in low-frequency and high-frequency components. (b) Reconstruction of USAF Resolution Target without random phase, partly enlarged in low-frequency and high-frequency components.

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Fig. 9.
Fig. 9. Impact of zero-padding on reconstruction quality of phase-only CGHs.

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Fig. 10.
Fig. 10. Relationship between the PSNR and bit depth of the phase-only CGH.

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Fig. 11.
Fig. 11. Reconstructions by using an LCoS. (a) to (c): Reconstructions of 3-bit, 5-bit and 8-bit CGHs calculated without random phase imposed. (d) to (e): Reconstructions of 1-bit, 3-bit and 8-bit CGHs with random phase imposed.

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Fig. 12.
Fig. 12. Effect of phase modulation deviation on reconstruction quality

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Fig. 13.
Fig. 13. Reconstruction quality with pixel-to-pixel interaction effect under different maximum leakage coefficients (with random phase)

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Fig. 14.
Fig. 14. (a) Contour map of PSNR when the amplitude and phase of a complex-amplitude CGH are quantized into different bit depths. (b) Relationship between the PSNR and the quantization of the amplitude when the bit depth of the phase is fixed.

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Equations (19)

Equations on this page are rendered with MathJax. Learn more.

E ~ H ( x , y , 0 ) = F − 1 { F { E O ( x , y , z 0 ) R ( x , y , z 0 ) } exp [ j 2 π λ z 0 1 − ( λ u ) 2 − ( λ v ) 2 ] }
u = cos ⁡ α λ , v = cos ⁡ β λ
E ~ H ( x ) = F − 1 { F { E O ′ ( x ) } H z ( u ) }
H z ( u ) = exp [ j 2 π λ z 0 1 − ( λ u ) 2 ]
E ~ H ( x ) = E O ′ ( x ) ∗ h z ( x )
E ~ H ( n ) = ∑ m = 0 N − 1 E O ′ ( m ) ⋅ h z ( n − m )
E ~ ′ H ′ ( n ) = E ′ O ′ ( n ) ⊗ h ′ z ′ ( n ) = [ ∑ m = 0 L − 1 E ′ O ′ ( m ) ⋅ h ′ z ′ ( ( n − m ) ) L ] R L ( n )
E ′ O ′ ( n ) = { E O ′ ( n − N / 2 , 0 ) , N / 2 ≤ n ≤ 3 N / 2 0 , else
E A ( m , n , 0 ) = 1 2 B A × round { abs [ E ~ H ′ ′ ( m , n , 0 ) ] + E A 0 max { max [ E ~ H ′ ′ ( m , n , 0 ) ] + E A 0 } }
E P ( m , n , 0 ) = 2 π 2 B P × round { arg ⁡ [ E ~ H ′ ′ ( m , n , 0 ) ] × 2 B P 2 π }
f ( u ) = 1 2 π ∂ ∂ u ϕ ( u ) = 1 2 π ∂ ∂ u arg ⁡ [ H z ( u ) ] = − z 0 u λ − 2 − u 2
1 Δ u ≥ 2 max ( | f ( u ) | )
z 0 ≤ z 0 max = L 4 λ − 2 L 2 − M 2 M = M d 4 λ − 2 d 2 − 1
L ≤ z 0 tan ⁡ θ + 1 = z 0 tan ⁡ [ arcsin ⁡ ( λ d ) ]
z 0 ≥ z 0 min = M d tan ⁡ [ arcsin ⁡ ( λ d ) ]
P S N R = 10 log 10 { g max 2 M S E } = 10 log 10 { M N ⋅ g max 2 ∑ M ∑ N [ I O ( m , n ) − I R ( m , n ) ] 2 }
θ diff ∝ arcsin ⁡ ( λ f )
φ 0 ′ = α 0 φ 0 + α 1 φ 1 + α 2 φ 2 + α 3 φ 3 + α 4 φ 4
φ i = C 0 × N ( 0 , 1 )   i = 1 , 2 , 3 , 4
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