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Optimal quantization for amplitude and phase in computer-generated holography

Zehao He, Xiaomeng Sui, Guofan Jin, Daping Chu, and Liangcai Cao

Open Access Open Access

Abstract

Owing to the characteristics of existing spatial light modulators (SLMs), the computer-generated hologram (CGH) with continuous complex-amplitude is conventionally converted to a quantized amplitude-only or phase-only CGH in practical applications. The quantization of CGH significantly affects the holographic reconstruction quality. In this work, we evaluated the influence of the quantization for both amplitude and phase on the quality of holographic reconstructions by traversing method. Furthermore, we considered several critical CGH parameters, including resolution, zero-padding size, reconstruction distance, wavelength, random phase, pixel pitch, bit depth, phase modulation deviation, and filling factor. Based on evaluations, the optimal quantization for both available and future SLM devices is suggested.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Holography is a type of technology that enables building mathematical connections between target objects and interference patterns. In digital holography, a captured interference pattern is often employed to reconstruct the corresponding target object [1,2]. It has been applied in numerous applications such as sonar [3,4], radar [5,6], and microscopy [7 –9]. In computer-generated holography, a target object is often employed to calculate the corresponding interference pattern, which is known as a computer-generated hologram (CGH). Such a calculated interference pattern can precisely reproduce the three-dimensional (3D) image of the target object. CGHs have been employed in many fields such as display [10 –13], optical manipulation [14,15], encryption [16,17], lithography [18], and beam shaping [19,20]. In recent years, the emerging of vortex light field also expand the applications for CGHs [21].

The primary goal in computer-generated holography is to achieve a high quality of reconstruction. The complex-amplitude CGH, with a continuous complex distribution, exhibits the highest enhancement in reconstruction quality because it records both the amplitude and phase of the target object without approximation. In practical computer-generated holographic systems, CGHs rely on wavefront modulation devices called spatial light modulators (SLMs). Currently, the available SLMs are digital micro-mirror devices (DMDs), liquid crystal on silicon (LCoS), and metasurfaces. Among them, a DMD is a type of amplitude-only device, and an LCoS is a type of phase-only device. It is not easy to directly modulate both the amplitude and phase by these devices simultaneously. Consequently, complicated encoding methods [22] or additional optical devices [23,24] have been introduced to realize complex-amplitude modulation, which dramatically increases the complexity of the computer-generated holographic system. A metasurface is a device that can present the complex-amplitude CGH and improve the image quality of the holographic reconstructions [25,26]. However, metasurfaces are not easy to manufacture and are incapable of performing dynamic modulation, and consequently, their applications in holographic display are limited.

Some iterative methods [27,28] have been proposed recently to achieve the control of both the amplitude and phase at a plane of interest to realize a complex-amplitude modulation. With the employment of iterations, heavy computational burdens are brought to holographic display systems. This issue would be significantly serious in large-size scenes and 3D scenes. Thus, the phase-only or amplitude-only CGH which is calculated from the complex-amplitude CGH by ignoring either the amplitude or the phase, is often employed for existing SLMs. However, ignoring the amplitude or the phase causes a severe decrease in the reconstruction quality.

Interestingly, SLMs have a pixelated structure with a series of discrete pixel values. Holographic algorithms, therefore, need to cater to this scenario. The obtained CGH is set to a specific resolution, while the pixel value of the CGH is approximated to the nearest available modulation state. This restrictive behavior significantly affects the quality of the holographic reconstruction. The quantization errors of CGHs have been extensively studied [29 –36], especially in the early stage of the computer-generated holography. However, the purpose of these researches was mostly to show optimization methods for CGHs, such as iteration-based methods. Results of quantitative analysis were rarely exhibited. These researches mainly focused on either the amplitude-only or the phase-only CGH. Complex-amplitude CGHs were seldom studied. Notably, the modulated result of the CGH is occasionally different from its desired pattern due to manufacturing defects of the SLM [36]. The look-up table (LUT) of the SLM should be updated to address this issue [37,38].

In this study, we described a global and systematic approach to gain insights on the reconstruction obtained from a CGH for a large parameter space, specifically in relation to our setting. An optimization model based on the angular-spectrum theory was employed to calculate the CGH of the target object. The peak signal-to-noise ratio (PSNR) was used as an index to evaluate the quality of the holographic reconstruction. We examined the influence of the quantization on the holographic reconstruction. Furthermore, we analyzed the impacts of parameters such as resolution, zero-padding, propagation distance, wavelength, random phase, and pixel pitch on the reconstruction quality. We extensively discussed the effects of bit depth, phase modulation deviation, filling factor, and optimal quantization on the reconstruction. The resolution and the refresh rate of CGHs determine the richness and the fluency of the displayed information, while the factors in our work determine the fidelity of the displayed information. This work is especially useful when one considers an advanced holographic display that is based on SLMs with a limited performance in terms of amplitude and phase modulation.

2. Method

The CGH of the target object is calculated using the angular-spectrum theory. If the holographic plane is set as the original plane, where z = 0, the complex-amplitude distribution on the holographic plane EH (x, y, 0) can be expressed as in [39,40]:

$${\widetilde E_H}({x,y,0} )= {F^{ - 1}}\left\{ {F\{{{E_O}({x,y,{z_0}} )R({x,y,{z_0}} )} \}\exp \left[ {j\frac{{2\pi }}{\lambda }{z_0}\sqrt {1 - {{({\lambda u} )}^2} - {{({\lambda v} )}^2}} } \right]} \right\}, $$
where F-1 represents the inverse Fourier transform, F is the symbol of the Fourier transform, EO (x, y, z0) is the amplitude of the target object on the object plane, z0 is the distance between the holographic plane and the object plane, R (x, y, z0) is the random phase, λ is the wavelength of the recording wave, and u and v are the spatial frequencies in the x- and y-directions, respectively. If α and β are the angles from the direction of the incident wave to the x- and y-directions, respectively, u and v can then be expressed as
$$u = \frac{{\cos \alpha }}{\lambda },v = \frac{{\cos \beta }}{\lambda }. $$
In the angular-spectrum theory, there are two limitations to the distance between the holographic plane and the object plane. Based on Nyquist’s sampling theorem [41], aliasing errors are introduced into the sampled transfer function when the reconstruction distance z0 is too large. The transfer function Hz (u, v) in the angular-spectrum theory can be expressed as
$${H_z}({u,v} )= \exp \left[ {j\frac{{2\pi }}{\lambda }{z_0}\sqrt {1 - {{({\lambda u} )}^2} - {{({\lambda v} )}^2}} } \right]. $$
For simplicity, only the one-dimensional (1D) form of the transfer function Hz (u, v) is considered. The 1D transfer function Hz (u) can be expressed as
$${H_z}(u )= \exp \left[ {j\frac{{2\pi }}{\lambda }{z_0}\sqrt {1 - {{({\lambda u} )}^2}} } \right] = \exp [{j\phi (u )} ], $$
where ϕ (u) is the phase of the 1D transfer function. Furthermore, f (u) is the local frequency of ϕ (u), which can be expressed as
$$f(u )= \frac{1}{{2\pi }}\frac{\partial }{{\partial u}}\phi (u )={-} \frac{{{z_0}u}}{{\sqrt {{\lambda ^{ - 2}} - {u^2}} }}. $$
In the angular-spectrum theory, the width of the 1D target object is equal to that of the CGH, while the 1D target object has the same sampling number as the CGH. For the 1D target object, the width is L, the sampling number is M, and the sampling interval is d. According to Nyquist’s sampling theorem, the maximal sampling interval Δu should be (2L)-1. To avoid the aliasing error in the 1D sampled transfer function, the local frequency f (u) should satisfy the condition
$${({2\Delta u} )^{ - 1}}\textrm{ = }L \ge |{f(u )} |. $$
Here, f (u) is a monotonically increasing function. Assuming the maximal value of the spatial frequency umax = M / 2L, the reconstruction distance z0 satisfies the condition
$${z_0} \le {z_{0\max }} = \frac{{L\sqrt {4{\lambda ^{ - 2}}{L^2} - {M^2}} }}{M} = Md\sqrt {4{\lambda ^{ - 2}}{d^2} - 1}, $$
where L = md. z0max is the upper limit of the reconstruction distance z0, which is called the effective distance.

In addition to the effective distance z0max, there is also a lower limit z0min for the angular-spectrum theory to prevent aliasing errors caused by the reconstructed results of different diffraction orders. Zero-order diffraction is the desired result. The diffraction angle of the zero-order θ0 is equal to 0. Meanwhile, the diffraction angle of the first-order θ +1 can be expressed as

$${\theta _{\textrm{ + }1}} = \arcsin \left( {\frac{\lambda }{d}} \right). $$
To avoid the aliasing error between the zero-order and the first-order, the distance between the same points in the reconstructed results of different orders should be larger than the width of the target object. Thus, the lower limit z0min becomes
$$L = Md \le {z_0}\tan \left[ {\arcsin \left( {\frac{\lambda }{d}} \right)} \right],$$
$${z_0} \ge {z_{0\min }} = \frac{{Md}}{{\tan \left[ {\arcsin \left( {\frac{\lambda }{d}} \right)} \right]}}.$$
During the calculation of the CGH, the distance between the holographic plane and the object plane should be within the range [z0min, z0max].

To quantitatively compare the reconstruction quality, PSNR is employed as the evaluation index to describe the difference between the target objects and reconstructions. PSNR can be mathematically expressed as in [27]:

$$PSNR = 10\log \left\{ {\frac{{ma{x_g}^2}}{{MSE}}} \right\},$$
$$MSE = \frac{1}{{MN}}{\sum\limits_M {\sum\limits_N {[{{I_O}({m,n} )- {I_r}({m,n} )} ]} } ^2},$$
where maxg is the maximal value of the grayscale, MSE is the mean square error, Io (m, n) and Ir (m, n) are the pixel intensities of the target object and the reconstruction, respectively, and M and N are the number of pixels in the horizontal and vertical directions, respectively. In an 8-bit picture, the maximal value of the grayscale maxg is 255.

The processes of holographic recording and reconstruction are illustrated in Fig. 1. Owing to the illumination caused by the collimated coherent light, the information of the target object travels through a pre-set distance z0, and then the continuous complex-amplitude distribution on the holographic plane can be recorded. In this work, the recording process is conducted by the computer. The propagation of the diffracted information is restricted by the angular-spectrum theory. Owing to the characteristics of existing SLMs, the continuous complex-amplitude distribution is difficult to modulate directly. Therefore, the continuous complex-amplitude distribution should be converted to a CGH with a quantized amplitude or phase as follows:

$${E_A}({x,y,0} )= \frac{1}{{{2^B}}} \times \textrm{floor}\left\{ {\frac{{\textrm{real}[{{{\widetilde E}_H}({x,y,0} )} ]+ {E_0}}}{{\max \{{\textrm{real}[{{{\widetilde E}_H}({x,y,0} )} ]+ {E_0}} \}}} \times {2^B}} \right\},$$
$${E_P}({x,y,0} )= \frac{1}{{{2^B}}} \times \textrm{floor}\left\{ {\frac{{\arg [{{{\widetilde E}_H}({x,y,0} )} ]}}{{2\pi }} \times {2^B}} \right\}.$$

 figure: Fig. 1.

Fig. 1. Holographic recording and reconstruction. Recording process is conducted by computer, and reconstruction is realized by optical installation.

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Here, EA (x, y, 0) is the quantized amplitude, EP (x, y, 0) is the quantized phase, real represents the real part of the complex-amplitude, E0 is a constant, arg represents the argument value, floor represents rounding down decimals to integers, and BA and BP are the bit depths of the amplitude and phase, respectively. In a holographic display system, the reconstruction is actually realized by the optical configuration.

As shown in Fig. 2, we present the solution space for different BA and BP such as 0-bit, 8- bit, and 16-bit, i.e., 1 grayscale, 256 grayscales, and 65536 grayscales. The solution space can be regarded as a polar coordinate system. The polar axis indicates the amplitude, which ranges from 0 to 1. The polar angle indicates the phase, which ranges from 0 to 2π. When a complex-amplitude is quantized into a distribution with a 0-bit amplitude and 0-bit phase, no position in the solution space is occupied, as shown in Fig. 2(a). For an amplitude-only CGH, its phase is quantized into 0-bit. It only occupies the positions of some points on the polar axis. The density of the points increases with BA, as shown in Fig. 2(b) and 2(c). Similarly, for a phase-only CGH, its amplitude is quantized into 0-bit. It occupies the positions of points on the circumference with a radius of 1, from 0 to 2π. The number of the occupied points increases with BP, as shown in Fig. 2(d) and 2(g). However, irrespective of how large BA and BP are, the points which the amplitude-only or phase-only CGHs are occupied in the solution space are limited. Such excessive quantization has significantly limited the image quality of holographic reconstructions. Thus, the complex-amplitude CGH with both the amplitude and the phase is essential for holographic reconstructions. As shown in Fig. 2(e), 2(f), 2(h) and 2(i), when the quantization of the amplitude grows together with that of the phase, the positions of the occupied points become increasingly abundant. The information loss in such quantized complex-amplitude CGHs is smaller than that in amplitude-only and phase-only CGHs. To achieve a superior reconstruction quality, the quantized complex-amplitude CGH is a suitable choice. Although amplitude-only or phase-only SLMs are widely employed in current researches, we hope to investigate an optimal quantization for future SLMs.

 figure: Fig. 2.

Fig. 2. (a) Distribution with a 0-bit amplitude and 0-bit phase; (b), (c) amplitude-only distribution with different quantization of amplitude; (d), (g) phase-only distribution with different quantization of phase; (e), (f), (h) and (i) complex-amplitude distribution with different quantization of amplitude and phase.

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3. Results

3.1 Continuous complex-amplitude CGH

To analyze the effect of the quantization of the continuous complex-amplitude CGH on the quality of the holographic reconstruction, we first considered the perfectly reconstructed result of the target object without approximation. In this work, the target objects employed in the analysis were the USAF Resolution Target, Opera, and Pills, as shown in Fig. 3(a)–3(c). The reconstruction distance was set as 102 mm to satisfy the requirements of Eq. (7) and Eq. (9). The resolution of the target object was 2000 × 2000, the recording wavelength was 532 nm, and the pixel pitch of the CGH was 3.7 µm. The reconstruction of the USAF Resolution Target is shown in Fig. 3(d). The MSE of the reconstructed result was 0, which implied that the PSNR was infinite. Thus, the calculation of a continuous complex-amplitude CGH and its holographic reconstruction were perfect reciprocal processes without errors. Furthermore, the intensity curves in the USAF Resolution Target and its reconstructed result are shown in Fig. 3(e). It can be observed that the curves are completely coincident with each other. Reconstructions by continuous complex-amplitude CGHs for Opera and Pills exhibited similar results as that of the USAF Resolution Target.

 figure: Fig. 3.

Fig. 3. (a) USAF Resolution Target; (b) Opera; (c) Pills; (d) Holographic reconstruction by the continuous complex-amplitude CGH for USAF Resolution Target; (e) Intensity curves of USAF Resolution Target and its reconstructed result.

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To ensure a systematic research approach, the study extensively emphasized related parameters, such as resolution, zero-padding size, reconstruction distance, wavelength, random phase, and pixel pitch. To quantitatively analyze the effect of every variable, we employed a variable-controlling approach. The results showed that the MSEs of the reconstructed results were always 0 when continuous complex-amplitude CGHs were employed, regardless of the variations in the parameters. This conclusion again successfully verified the fact that the calculation of a continuous complex-amplitude CGH and its holographic reconstruction were perfect reciprocal processes.

Furtherly, when the zero-padding is operated on the target object, the image quality of the reconstruction by the continuous complex-amplitude CGH keeps unchanged. The operation of zero-padding is shown in Fig. 4. The resolution of the target object was 2000 × 2000. A black margin, whose pixel value is always 0, is placed outside the target object. The total resolution, including both the target object and the black margin, changes from 2000 × 2000 to 4000 × 4000. Because the margin contains no information, the continuous complex-amplitude CGH of the padded image can be regarded as a continuous complex-amplitude CGH of the original image but captured by a larger camera sensor. Spatial frequencies contained in the large CGH are more abundant than those in the small CGH. However, the MSEs of the reconstructions before and after zero-padding are both 0. It means that the reconstruction quality of the continuous complex-amplitude CGH is not determined by the contained spatial frequencies.

 figure: Fig. 4.

Fig. 4. Holographic recording (a) before zero-padding and (b) after zero-padding.

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3.2 Complex-amplitude CGH with quantized amplitude

Practically, the CGH was stored in a computer with the quantized amplitude and the quantized phase. We separately analyzed the complex-amplitude CGHs with the quantized amplitude and with the quantized phase to determine which quantization process was more critical to the holographic reconstruction.

We first studied the effect of the quantization of the amplitude. The amplitude was quantized into an 8-bit distribution, which had 255 grayscales. With the reconstruction distance, wavelength, random phase, and pixel pitch remaining unchanged, the PSNRs of the reconstructions for USAF Resolution Target, Opera, and Pills were 53.73 dB, 52.34 dB, and 52.03 dB, respectively. Thus, the 8-bit quantization of the amplitude for the continuous complex-amplitude CGH caused a decline in the reconstruction quality, as expected. In this case, the calculation of a CGH and its holographic reconstruction were not perfect reciprocal processes, but the error was minuscule.

Similarly, we adopted the variable-controlling approach in the analysis of the relationship between the reconstruction quality and the resolutions, while using the complex-amplitude CGH with an 8-bit quantized amplitude. Figure 5 shows the change in the reconstruction quality with the increase in the resolution. The resolution of the target object changed from 2000 × 2000 to 4000 × 4000, increasing by 200 pixels each time in both the x- and y-directions. The PSNRs of the reconstructions fluctuated near the average PSNR value, and the fluctuation range was generally smaller than 1 dB. We also studied the relationship between the reconstruction quality and other parameters, such as zero-padding, reconstruction distance, wavelength, random phase, and pixel pitch. With changes in these parameters, the PSNRs of the reconstructions shared the same variation tendencies as shown in Fig. 5.

 figure: Fig. 5.

Fig. 5. Relationship between the PSNR and the resolution when the complex-amplitude CGH with an 8-bit quantized amplitude is employed.

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3.3 Complex-amplitude CGH with quantized phase

Subsequently, we utilized the complex-amplitude CGH with an 8-bit quantized phase. Considering that the parameters were identical to those described in section 3.1, the PSNRs of the reconstructions for the USAF Resolution Target, Opera, and Pills were 49.49 dB, 46.42 dB, and 45.21 dB, respectively. The decline in the PSNR caused by the 8-bit quantization for the phase was more severe than that caused by the 8-bit quantization for the amplitude.

Figure 6(a) shows the change in the reconstruction quality with the increase in the resolution, when we used the complex-amplitude CGH with an 8-bit quantized phase. The results exhibited a trend similar to that in Fig. 5, but the average value of the PSNR was lower by more than 7 dB. As other parameters including the reconstruction distance, wavelength, random phase, and pixel pitch changed separately, the PSNRs also shared the same variation tendencies, as shown in Fig. 6(a). However, conducting zero-padding on target objects made a distinctive difference. The total resolution, including both the target object and the black margin, changed from 2000 × 2000 to 4000 × 4000, increasing by 200 pixels each time in both the x- and y-directions. As shown in Fig. 6(b), the PSNRs of the reconstructed results had a very obvious growth trajectory with the expansion of the zero-padding area. The PSNR of the reconstruction with a total resolution of 4000 × 4000 was more than 5 dB higher than that with a total resolution of 2000 × 2000. Therefore, when we used a complex-amplitude CGH with an 8-bit phase quantization, zero-padding was an effective method to improve the quality of the reconstruction.

 figure: Fig. 6.

Fig. 6. (a) Relationship between PSNRs of reconstructed results and resolutions; (b) Relationship between PSNRs of reconstructed results and zero-paddings.

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In addition to improving the reconstruction quality, zero-padding can also be utilized to increase the effective distance of the CGH. In the angular-spectrum theory, the resolution of the CGH is equal to the total resolution, including both the target object and the black margin. If the resolution of an original target object is M × M, while the total resolution of the target object together with the black margin is 2M × 2M, the effective distance of the new CGH is 2z0max. Meanwhile, the overlap of the zero-padding area does not cause an aliasing error between the effective reconstructed areas of different diffraction orders. Zero-padding does not change the lower limit of the reconstruction distance.

3.4 Phase-only CGH and amplitude-only CGH

The phase-only CGH is one of the most employed CGH in the holographic display. Some optimization methods [42,43] have been proposed to realize the control of both amplitude and phase by a phase-only CGH. Iteration-based methods are hardware intensive and time-consuming, especially in the display of large-size scenes and 3D scenes. Iteration-free optimization methods generally rely on additional mathematical strategies and optical elements, increasing the cost of the holographic display system. Thus, the method by flattening the amplitude from the complex-amplitude is still a widely employed method to calculate phase-only CGHs.

Phase-only CGHs could be uploaded on phase-only SLMs directly. Keeping the parameters mentioned in section 3.1 unchanged, we analyzed the effect of ignoring the amplitude on the quality of the reconstruction, as shown in Fig. 7(a)–7(c). When continuous phase-only CGHs were applied, the PSNRs of the USAF Resolution Target, Opera, and Pills were 16.27 dB, 12.95 dB, and 12.02 dB, respectively. As shown in Fig. 7(d), the reconstructed result of the USAF Resolution Target was locally enlarged. An extraordinarily negative reconstruction quality was obtained in this case. The intensity curves in the target object and its reconstruction by the continuous phase-only CGH are shown in Fig. 7(e). The maximal intensity of the reconstructed result was approximately 25% lower than that of the target object. All of the evidence reveals that ignoring the amplitude was detrimental to the quality of the reconstruction, compared to other factors such as amplitude quantization and phase quantization.

 figure: Fig. 7.

Fig. 7. (a) Phase-only reconstruction of USAF Resolution Target; (b) Phase-only reconstruction of Opera; (c) Phase-only reconstruction of Pills; (d) Locally enlarged pattern from the reconstruction of USAF Resolution Target; (e) intensity curves of USAF Resolution Target and its reconstructed result by continuous phase-only CGH.

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Additionally, we observed a prominent fluctuation in the intensity curve of the reconstructed result, which implied that the speckle noise was introduced during the reconstruction by the continuous phase-only CGH. In incoherent display, the speckle noise can be reduced by using an incoherent illumination or a rotating diffuser. However, in SLM-based holographic display systems, CGHs are mainly designed based on coherent properties. With the decrease of the coherence, the reconstruction quality decreases as well. Eliminating speckle noise through subsequent designs for the phase-only CGH is an important issue which should be studied continuously.

The relationship between the reconstruction quality and the resolution of the target object was remarkably similar to that shown in Fig. 6(a). The average PSNR value of the reconstructions by phase-only CGHs decreased to 10–15 dB, which was far smaller than that of the complex-amplitude CGH. Other parameters including reconstruction distance, wavelength, random phase, and pixel pitch also had similar influences on the reconstructed results as the resolution. The effect of the zero-padding operation on the reconstruction quality was strikingly similar to the result shown in Fig. 6(b). With the expansion of the zero-padding area, the PSNR of the reconstructed result had an obvious growth trajectory. However, the maximal PSNR value in phase-only holography was only 21 dB, which is nearly 35 dB lower than that shown in Fig. 6(b).

Practically, the phase-only CGH uploaded on the phase-only SLM was the CGH with an 8-bit quantized phase. We also examined the effect of phase quantization for the phase-only CGH. The PSNRs of the USAF Resolution Target, Opera, and Pills were 16.19 dB, 12.88 dB, and 11.97 dB, respectively. The negative influence of the phase quantization was almost overshadowed by the influence of ignoring the amplitude.

Furtherly, we studied the effect of ignoring the phase on the image quality of the reconstruction. The amplitude-only CGH was obtained by flattening the phase from the complex-amplitude in this work. When continuous amplitude-only CGHs were applied, the PSNRs of the USAF Resolution Target, Opera, and Pills were 12.17 dB, 9.31 dB, and 9.16 dB, respectively. The decline in the PSNR caused by ignoring the phase was more severe than that caused by ignoring the amplitude. As parameters including the resolution, reconstruction distance, wavelength, random phase, and pixel pitch changed separately, the PSNRs shared the same variation tendencies as those in the reconstructions of phase-only CGHs. Meanwhile, the effect of the zero-padding operation on the reconstruction quality was similar to the result in the reconstruction by the phase-only CGH. In addition, the negative influence of the amplitude quantization in the amplitude-only CGH was also almost overshadowed when ignoring the phase.

Practically, the reconstruction of the amplitude-only CGH was realized by the amplitude-only SLM such the DMD. The DMD is a gray scale modulation device. As the gray scale is always positive, the practical amplitude-only CGH EA_CGH could be expressed as:

$${E_{A\_CGH}}({x,y,0} )= \textrm{real}({{{\widetilde E}_H}({x,y,0} )} )+ {E_0}.$$
Here, E0 is a constant, whose function is to make EA_CGH always positive. Physically, E0 is the bright zero-order image, which reduces the contrast of the reconstruction. Thus, amplitude-only CGHs are encoded by the off-axis method in most applications. However, for both on-axis and off-axis amplitude-only CGHs, the existence of E0 limits the diffraction efficiency and the image quality. From this standpoint, the phase-only CGH has a preferable performance compared to the amplitude-only CGH.

4. Discussions

4.1 Quantized phase-only CGH with different bit depths

In some specific applications, such as high-resolution holographic video display, high storage capacity is a critical requirement because of the ultra-high resolution and real-time refresh rate. To reduce the demand for data storage, compressing the bit depth of the phase-only CGH is a necessary objective. The PSNRs of the reconstructions by phase-only CGHs with different bit depths are shown in Fig. 8. The holographic reconstructions were obtained by using a Holoeye GAEA-2 phase-only SLM. When the bit depth of the phase-only CGH changed from 1-bit to 5-bit, the PSNR of the reconstruction increased gradually. The larger the bit depth, the slower the rising speed of the PSNR. As the bit depth exceeded 5-bit, the PSNR remained almost unchanged. Therefore, information redundancy existed in the CGH with the bit depth larger than 5-bit. The phase-only CGH with 5-bit quantization could be an optimal choice for applications with extremely high data storage requirements.

 figure: Fig. 8.

Fig. 8. (a) Relationship between PSNR and bit depth of the phase-only CGH. The reconstructions by the CGHs with the bit depths of 1 bit, 3 bit and 5 bit are shown in (b), (c) and (d).

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4.2 Phase modulation deviation

The LUT is designed to establish the relationship between the grayscale values of the CGH and the phase changes on the phase-only SLM panel. The grayscale value of an 8-bit phase-only CGH ranges from 0 to 255. As the pixel value of the CGH changes by one grayscale, the phase value of the wavefront, which was modulated by the phase-only SLM, should change by π/128. As a result, the ideal modulation range of a phase-only SLM is [0, 2π]. However, the relationship between the grayscale value of the CGH and the phase modulation on the SLM panel often differs from the designed LUT owing to some manufacturing defects. For example, if the realistic modulation range of a phase-only SLM is [0, 4π], the change of one grayscale in the CGH represents a π/64 phase change on the SLM panel. In this case, the desired phase modulation is disrupted.

As shown in Fig. 9, when the modulation range of a phase-only SLM changes from [0, π/8] to [0, 4π] with a step of π/8, the consequent PSNRs of the holographic reconstructed results change. When the modulation range changes from [0, π/8] to [0, 2π], the PSNR increases with the increase in the modulation range. When the modulation range changes from [0, 2π] to [0, 4π], the PSNR decreases with the increase in the modulation range. As the modulation range is located between [0, π] and [0, 3π], the reconstructed results can be recognizable to human vision. Thus, in some specific applications, such as anti-counterfeiting, which has a lower requirement for the reconstruction quality, rough calibration on the LUT can achieve approximate results.

 figure: Fig. 9.

Fig. 9. (a) Relationship between PSNR of holographic reconstruction and phase modulation deviation of the phase-only CGH. The reconstructions by the CGHs with the maximal random phase ranges of 0.125π, π, 2π, 3π, and 4π are shown in (b)–(f).

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4.3 Optimized filling factor for phase-only CGH

In most applications, SLMs have a pixelated structure. Taking the phase-only CGH as an example, the realistic CGH EP’ (x, y, 0) with the consideration of the characteristic of SLMs can be expressed by:

$${E_P}^{\prime}({x,y,0} )= \textrm{rect}\left( {\frac{x}{{{d_x}}},\frac{y}{{{d_y}}}} \right) \ast \left\{ {\frac{1}{{{p_x}{p_y}}}\textrm{comb}\left( {\frac{x}{{{p_x}}},\frac{y}{{{p_y}}}} \right) \times {E_P}({x,y,0} )} \right\}. $$
Here, dx and dy are the pixel sizes, px and py are the sizes of the active area in each pixel, comb is the comb function which presents the periodic structure of SLMs, and * represents the convolution operation. In this work, we assume that dx = dy = d, and px = py  = p, the filling factor of the pixel µ can be defined as µ = a 2 / d 2. The relationship between the PSNR of the reconstruction and the filling factor of the pixel is shown in Fig. 10. For holographic display, a higher filling factor of CGH makes a better image quality of holographic reconstruction.

 figure: Fig. 10.

Fig. 10. (a) Relationship between PSNR of holographic reconstruction and filling factor of pixel. The reconstructions by the CGHs with the filling factors of 0.2, 0.4, 0.6, 0.8, and 1 are shown in (b)–(f).

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4.4 Optimized quantization in complex-amplitude CGH

Figure 11(a) presents the contour map of the PSNR when the amplitude and phase of the complex-amplitude CGH were quantized into different bit depths. When the bit depth of the phase was small, the upper limit of the PSNR was also significantly low, regardless of the bit depth of the amplitude. Similarly, when the bit depth of the amplitude was small, the upper limit of the PSNR also remained significantly low. The optimal quantization for the amplitude is related to that for the phase. These two factors are not independent of each other.

 figure: Fig. 11.

Fig. 11. (a) Contour map of the PSNR value when amplitude and phase in complex-amplitude CGH are quantized into different bit depths. (b) Relationship between the PSNR and the quantization of amplitude when the bit depth of the phase is fixed.

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The relationship between the PSNRs and the quantization of the amplitude is shown in Fig. 11(b). When the bit depth of the phase was fixed, the PSNRs initially increased with the growth of the quantization of the amplitude. When the quantization reached a critical value, the PSNRs remained unchanged with the growth of the amplitude quantization. When the phase of the CGH had a large bit depth, the critical value of the amplitude quantization was high as well. In this case, the PSNR of the reconstructed result had a significantly high upper limit. In the simulations, the pixel pitch was also changed to 1 µm on some occasions. The results had a regular pattern similar to that with a pixel pitch of 3.7 µm. Currently, the complex-amplitude modulation can be realized by some devices, such as metasurface-based devices. The target object can be reconstructed preeminently with such devices which have high quantization for both amplitude and phase. However, the difficulty of manufacture restricts the freewill quantization. The cost of manufacture rises dramatically with the increase of the quantization. Properly quantized amplitude and phase should be considered emphatically. The reconstruction quality vastly improved after slightly increasing the bit depths of both the amplitude and the phase, rather than improving one of them dramatically.

5. Conclusion

In this work a quantitative analysis in a larger parameter space was presented. The calculation of a continuous complex-amplitude CGH and its corresponding holographic reconstruction were perfect reciprocal processes without errors, regardless of the changes in the parameters. The reconstruction quality of the continuous complex-amplitude CGH was not determined by the contained spatial frequencies. The quantization of either the amplitude or the phase in continuous complex-amplitude CGH led to a decline in the reconstruction quality. The influence of the quantization of the phase was more severe than that of the amplitude. Without consideration of optimized designs, the phase-only CGH was calculated from the complex-amplitude CGH by ignoring its amplitude. Ignoring the amplitude had the most significant influence on the reconstruction quality. The operation of zero-padding was found to be a selective choice in some cases to improve the reconstruction quality. For the phase-only CGH, when the bit depth of the phase was larger than 5-bit, the PSNR of the reconstruction remained almost unchanged. The phase-only CGH with 5-bit quantization was verified to be an optimal choice for applications with extremely high data storage requirements. Small phase modulation deviation had a low influence on the reconstruction quality. Rough calibration on modulation devices achieved objectives in some specific applications that had a low requirement for the reconstruction quality. A higher filling factor of the CGH made a better image quality of the holographic reconstruction. The image quality of the holographic reconstruction was related to the quantization of both the amplitude and the phase. These two factors were not independent of each other. Complex-amplitude CGH with properly quantized amplitude and quantized phase significantly improved the quality. In the simulations, the results with a pixel pitch of 1 µm had a similar pattern similar to that with a pixel pitch of 3.7 µm, thereby providing a guideline for designing metasurface-based devices as well. We expect our research to provide optimal quantization guidelines for both future SLM devices and their currently available counterparts.

Funding

National Natural Science Foundation of China (61775117, 62035003); Tsinghua University Initiative Scientific Research Program (20193080075); Tsinghua-Cambridge Joint Research Initiative.

Disclosures

The authors declare no conflicts of interest.

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Figures (11)
Fig. 1.
Fig. 1. Holographic recording and reconstruction. Recording process is conducted by computer, and reconstruction is realized by optical installation.

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Fig. 2.
Fig. 2. (a) Distribution with a 0-bit amplitude and 0-bit phase; (b), (c) amplitude-only distribution with different quantization of amplitude; (d), (g) phase-only distribution with different quantization of phase; (e), (f), (h) and (i) complex-amplitude distribution with different quantization of amplitude and phase.

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Fig. 3.
Fig. 3. (a) USAF Resolution Target; (b) Opera; (c) Pills; (d) Holographic reconstruction by the continuous complex-amplitude CGH for USAF Resolution Target; (e) Intensity curves of USAF Resolution Target and its reconstructed result.

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Fig. 4.
Fig. 4. Holographic recording (a) before zero-padding and (b) after zero-padding.

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Fig. 5.
Fig. 5. Relationship between the PSNR and the resolution when the complex-amplitude CGH with an 8-bit quantized amplitude is employed.

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Fig. 6.
Fig. 6. (a) Relationship between PSNRs of reconstructed results and resolutions; (b) Relationship between PSNRs of reconstructed results and zero-paddings.

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Fig. 7.
Fig. 7. (a) Phase-only reconstruction of USAF Resolution Target; (b) Phase-only reconstruction of Opera; (c) Phase-only reconstruction of Pills; (d) Locally enlarged pattern from the reconstruction of USAF Resolution Target; (e) intensity curves of USAF Resolution Target and its reconstructed result by continuous phase-only CGH.

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Fig. 8.
Fig. 8. (a) Relationship between PSNR and bit depth of the phase-only CGH. The reconstructions by the CGHs with the bit depths of 1 bit, 3 bit and 5 bit are shown in (b), (c) and (d).

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Fig. 9.
Fig. 9. (a) Relationship between PSNR of holographic reconstruction and phase modulation deviation of the phase-only CGH. The reconstructions by the CGHs with the maximal random phase ranges of 0.125π, π, 2π, 3π, and 4π are shown in (b)–(f).

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Fig. 10.
Fig. 10. (a) Relationship between PSNR of holographic reconstruction and filling factor of pixel. The reconstructions by the CGHs with the filling factors of 0.2, 0.4, 0.6, 0.8, and 1 are shown in (b)–(f).

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Fig. 11.
Fig. 11. (a) Contour map of the PSNR value when amplitude and phase in complex-amplitude CGH are quantized into different bit depths. (b) Relationship between the PSNR and the quantization of amplitude when the bit depth of the phase is fixed.

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Equations (16)

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 .
H z ( u , v ) = exp ⁡ [ j 2 π λ z 0 1 − ( λ u ) 2 − ( λ v ) 2 ] .
H z ( u ) = exp ⁡ [ j 2 π λ z 0 1 − ( λ u ) 2 ] = exp ⁡ [ j ϕ ( u ) ] ,
f ( u ) = 1 2 π ∂ ∂ u ϕ ( u ) = − z 0 u λ − 2 − u 2 .
( 2 Δ u ) − 1 = L ≥ | f ( u ) | .
z 0 ≤ z 0 max = L 4 λ − 2 L 2 − M 2 M = M d 4 λ − 2 d 2 − 1 ,
θ + 1 = arcsin ⁡ ( λ d ) .
L = M d ≤ z 0 tan ⁡ [ arcsin ⁡ ( λ d ) ] ,
z 0 ≥ z 0 min = M d tan ⁡ [ arcsin ⁡ ( λ d ) ] .
P S N R = 10 log ⁡ { m a x g 2 M S E } ,
M S E = 1 M N ∑ M ∑ N [ I O ( m , n ) − I r ( m , n ) ] 2 ,
E A ( x , y , 0 ) = 1 2 B × floor { real [ E ~ H ( x , y , 0 ) ] + E 0 max { real [ E ~ H ( x , y , 0 ) ] + E 0 } × 2 B } ,
E P ( x , y , 0 ) = 1 2 B × floor { arg ⁡ [ E ~ H ( x , y , 0 ) ] 2 π × 2 B } .
E A _ C G H ( x , y , 0 ) = real ( E ~ H ( x , y , 0 ) ) + E 0 .
E P ′ ( x , y , 0 ) = rect ( x d x , y d y ) ∗ { 1 p x p y comb ( x p x , y p y ) × E P ( x , y , 0 ) } .
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