## Abstract

A curved hologram can increase the view angle in a holographic display. The huge data processing and curved computer-generated hologram (CCGH) computation time is still a challenge for real-time display. Here, we propose two fast methods to accelerate the computation. The first one is a diffraction compensation (DC) method where the diffraction calculation is from the wave-front recording plane (WRP) to a CCGH. The other is an approximate compensation (AC) method that adds a phase difference distribution to the WRP to obtain the CCGH. Numerical simulations and optical experiments are performed, which demonstrate that the two methods are feasible and the computation time is dramatically reduced. The AC method can further reduce time significantly compared with the DC method. And the image quality for proposed methods is similar. It is expected that these fast methods can be combined with curved display screen and flexible display materials in the future.

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

## 1. Introduction

A holographic display is regarded as an ultimate three-dimensional (3D) display technology which is promising to achieve true 3D display without any wearable devices. A computer-generated hologram (CGH) is a key technique to realizing a holographic display by recording a hologram of 3D virtual object digitally. However, the field of view (FOV) of a plane hologram is limited and can’t meet the display requirements of a large view angle (FOV is defined as the field where one can see the whole reconstructed image [1]). A curved hologram is an effective way to overcome the constraint of FOV. Most reports of curved holograms are about cylindrical hologram [2–5]. The cylindrical hologram has a 360° look-around property and can be observed from any direction [6]. However, calculation of the cylindrical hologram encounters large computation time [3,4]. Some fast calculation methods are proposed to reduce the computing time of cylindrical hologram. In these reported works, the object usually needs to be set to a complicated multilayer cylinder and the sampling pitch is limited. Also, the cylindrical hologram is not easy to process and can’t be combined with the curved screen. A curved hologram as a part of the cylinder is easier to combine with the curved screen and flexible material compared with the cylindrical hologram [7]. The graphene-based material [8–10] and metasurface [11] can be combined with the CGH by write-once phase manipulation for a 3D holographic image with the potential of wide FOV. Previously, we proposed curved multiplexing based on a curved hologram to improve FOV and information capacity, where CCGH is calculated by point source (PS) method [1]. It is time-consuming and hard to realize real-time holographic displays. The wave-recording plane (WRP) is an effective way to reduce the computational complexity for plane CGH generation [12]. Hence the fast calculation method by using wave-recording surface (WRS) is proposed for cylindrical CGH generation [13].

In this paper, two fast calculation methods for a curved hologram based on WRP are proposed to reduce the computation time. One is the diffraction compensation (DC) method which consists of two steps. The first step is to calculate the distribution of WRP. The second step is to generate the CCGH from the WRP by a diffraction propagation. The other is the approximate compensation (AC) method, where the first step is the same as the DC method. The second step is to generate the CCGH by adding a phase difference distribution to the WRP. The calculation time of CCGH generation by these two methods is dramatically reduced compared with that of the PS method. Moreover, the AC method can further reduce the calculation time without affecting image quality compared with the DC method.

## 2. Methods

#### 2.1 Principle of DC and AC methods

The principle schematic of DC method is shown in Fig. 1(a). It consists of two steps. In the first step, a WRP is placed between the object and CCGH, and the complex amplitude distribution of WRP is calculated by the angular spectrum method. Here the angular spectrum method can be replaced by many fast calculation methods for the planar hologram [14–20]. In the second step, the complex amplitude distribution of CCGH is generated by Rayleigh-Sommerfeld diffraction propagation from the WRP. The principle schematic of the AC method is proposed, as shown in Fig. 1(b). The first step is the same with the DC method. In the second step, the CCGH is generated by adding a phase difference to the complex amplitude distribution of WRP where the phase difference distribution is caused by the optical path between the WRP and CCGH. The reconstruction process of the curved hologram is shown in Fig. 1(c). The image can be reconstructed when the curved hologram is illuminated by the cylindrical reference beam.

#### 2.2 CCGH generation

The geometric schematics of the DC and AC methods are shown in Fig. 2. The first step of these proposed methods is the same where the WRP is placed near the CCGH. The diffraction propagation between WRP and object is calculated by the angular spectrum method [21,22]. The wave-front decomposes into many plane waves with different spatial frequencies and propagates a distance z to a parallel WRP. Then they compose to obtain the diffraction field of the WRP. The process can be calculated by using fast Fourier transform (FFT). The complex amplitude distribution ${H_W}(x,y,{z_1})$ of WRP can be expressed as

where $O(u,v,{z_0})$ is the object wave distribution, $FFT[{\cdot} ]$ and $IFFT[{\cdot} ]$ are the Fourier and inverse Fourier operators, $({f_u},{f_v})$ is the spatial frequency. $T({f_u},{f_v})$ is the transfer function that is given by where k*=2π/λ*is wave number,

*λ*is the wavelength of the light wave,

*z = z*is the propagation distance between the object and WRP.

_{1}-z_{0}We process the second step of the DC method as shown in Fig. 2(a). In a cylindrical coordinate system, the complex amplitude distributions of the WRP and CCGH are ${H_W}(x,y,{z_1})$ and ${H_C}(x,y,{z_2})$, respectively. z_{1} is a constant and z_{2} can be expressed as *Rcosθ.* The diffraction propagation between WRP and CCGH is calculated by the Rayleigh-Sommerfeld diffraction. And the complex amplitude distribution ${H_C}(x,y,{z_2})$ of CCGH can be described as

*N*is the pixel number of the WRP, $\cos (\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n} ,{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _W})$ is a direction factor, $\vec{n}$ is the normal vector of the plane hologram,, ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _W}$ is the direction vector between the point A on the WRP and the point B on the CCGH. When the two diffraction planes are very close, it can’t be regarded as the paraxial approximation system. Therefore, the direction factor $\cos (\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n} ,{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _W})$ remained according to the Rayleigh-Sommerfeld diffraction equation. ${r_W}$ is the distance between the two points of the WRP and CCGH. It can be given by According to the maximum diffraction angle theory, diffraction areas of different points can be calculated by the distance. We regard both shapes of the diffraction area and the pixel approximately as a square. The diffraction area is given by where

*D*is side length of the diffraction area,

_{C}*d*is vertical distance of the point,

_{c}*β*is the maximum diffraction angle of a pixel of WRP [23]. It is calculated as where

_{max}*p*is the pixel pitch. The diffraction areas of all the points are small due to the close distance between WRP and CCGH. Therefore, the computational amount is small according to the Eq. (3).

The second step of the AC method is shown in Fig. 2(b). The CCGH is generated by add a phase difference distribution to the complex amplitude distribution of WRP. The complex amplitude distribution $H{^{\prime}_C}(x,y,{z_2})$ of the CCGH is given by

where*z*is the perpendicular distance between WRP and CCGH. It varies with different locations.$\exp [jk({z_2} - {z_1})]$ is phase difference distribution. Since the distance between the WRP and the CCGH is small enough according to the limited hologram size and the central angle of CCGH, the phase difference distribution can be regard as an approximate compensation generated by the geometric optical path difference. The coordinate relationship between point A$({x_W},{y_W},{z_1})$ on WRP and point B$({x_C},{y_C},{z_2})$ on the CCGH can be given by Point A on the WRP is the vertical projection of point B on the CCGH. Every point on the CCGH has a corresponding vertical projection point on the WRP. Hence, the pixel number of them needs to be set to the same in the AC method. There is no such constraint in the DC method.

_{2}–z_{1}#### 2.3 Curved multiplexing based on AC method

The curved multiplexing based on a curved hologram can improve FOV and information capacity [1]. However, the computation time is huge because multiple holograms need be calculated in the process. AC method is combined with the curved multiplexing to reduce the computation time. The flow chart is shown in Fig. 3. Firstly, three original objects are calculated to generate three WRPs. The distances between the objects and WRPs are all same. Secondly, the three CCGHs of different central angles are generated from the WRPs by the AC method, respectively. The CCGHs have same size and pixel number. Finally, the complex amplitude distribution of three CCGHs are added and synthetized into a composite hologram. The complex amplitude distribution of the composite hologram is given by

Where*i*is the number of the CCGHs. When the composite hologram is bended into the different central angles and illuminated with the corresponding cylindrical reference beam, the different objects can be reconstructed one by one at the same position in the reconstruction process.

## 3. Numerical simulations and optical experiments

#### 3.1 Results of DC and AC methods

We carry out numerical simulations to demonstrate our proposed methods. All the simulations were performed on the platform of MATLAB R2016b, Xeon, 2.4 GHz, and 128G RAM. These simulations are according to the physical propagation process of the optical experiment.

In order to test the correctness and reconstruction capability of the DC and AC methods, the proposed two methods are used for recording process and the PS method is used for the reconstruction process in the simulations. The vegetable image is used to calculate the CCGH for the 15° and 90° central angles, respectively. The parameters used in the numerical simulation are given as following: The resolutions of original image and CCGH are 1024×1024 pixels and 1920×1080 pixels, respectively. The sampling interval is 8×8µm. The wavelength of the light is 532 nm. The distance between the object and WRP is 200 mm. The reconstructed results of 15° and 90° CCGHs generated by the DC and AC are shown in the Figs. 4(a)–4(d), respectively. Usually we use the peak signal-to-noise ratio (PSNR) and speckle contract (SC) to evaluate the quality of reconstructed images. PSNR and SC of the reconstructed images are 8.96 dB, 9.01 dB, 8.86 dB, 8.94 dB and 0.515, 0.506, 0.506, 0.517, respectively. The CCGH generated by the proposed two methods can reconstruct the images by PS method successfully. It is demonstrated that both AC and the DC methods are correct and feasible for the CCGH generation. The reconstructed result of the AC method for the 90° central angle has little distortion in Fig. 4(d). The information of vertical propagation remains and the other directional information loses in the AC method. The loss information increases when the distance between the WRP and CCGH increases with the augmented central angle. Hence the distortion is caused by the loss information. Normally, the FOV will be cut off due to the limitation of the diffraction angle when the central angle become larger [1]. Hence, the central angle of CCGH needn’t set to 90°. Here, the CCGH of central angle 90° is calculated to test feasibility of the proposed methods for wide range of central angles. The PS method is a conventional algorithm that generates the CCGH from the object directly. Both the recording and reconstruction processes by PS method are calculated to compare with the proposed methods. The reconstructed results are shown in the Figs. 4(e) and 4(f). The PSNR and SC are 10.20 dB, 10.32 dB and 0.427, 0.428, respectively.

We compare the computation time among the DC, AC, and PS methods, as listed in Table 1. The computation time of the DC and AC methods both consist of two parts for the two steps. The total time for CCGHs of 15° and 90° central angles by the DC method is 81.53 and 346.53s. Different central angles correspond to different computational amount. The relationship between the central angle and computation time is shown in Fig. 5. We can see that the computation time increases as the central angle becomes larger. When the central angle increases, the distance between the WRP and the CCGH increases. Thus, the diffraction area and computational amount becomes larger according to Eq. (3) and Eq. (5). The computation time of 15° and 90° CCGHs generated by PS method is 55149s and 55153s. It is obvious that the DC method can dramatically reduce the calculation time compared with the PS method. AC method takes only totally 0.684s and 0.674s for the two CCGHs of the 15° and 90° central angles. It can further accelerate the computation compared with the DC method. The computation time for different central angles is similar by AC method. The DC method is more precise in theory and has no constraint of pixel number. But computation time of AC method is much less. Though the PSNR and SC of reconstructed images by PS method is a little higher than that of proposed methods. We prefer to use the proposed methods to calculate CCGH considering the computation time.

#### 3.2 Optical experiment setup

The schematic of the optical experimental setup for reconstruction is shown in Fig. 6. The green laser with wavelength 532 nm is collimated by the collimator which consists of spatial filter and collimating lens. The flat reflection-type phase-only spatial light modulator (SLM) and plane reference wave are used in the optical experiment. Hence, the phase distribution of CCGH should be remained and pre-compensated before it is loaded on the SLM [1]. The pixel pitch and resolution of SLM is 8um and 1920×1080, respectively. It is addressed with 256 phase modulation gray-scale levels. The 4f system that consisted of two Fourier lens and a filter is used to eliminate the impact of the zero order beam introduced by SLM on the reconstructed image [24]. The focal lengths of L1 and L2 are 350 mm. The reconstructed image is recorded by the CCD.

#### 3.3 Results of curved multiplexing based on AC method

The AC method is used to combine the curved multiplexing to minimize the computation time. At the same time, this method also has the effect of improving image quality. The numerical and experimental results are shown in Fig. 7. Vegetable, baboon, cameraman are corresponding to the CCGHs of the central angle 0°, 8°, 15°, respectively. The resolutions of images and CCGHs are 1024×1024 pixels and 1920×1080 pixels, respectively. The sampling interval is 8×8µm. The light wavelength is 532 nm. The distance between the object and WRP is 200 mm. The reconstructed results of the curved multiplexing based on the PS method is shown in the Figs. 7(a)–7(f). The PSNR and SC of the numerical reconstructed images are 9.57 dB, 9.77 dB, 5.64 dB and 0.517, 0.522, 0.529. It is obvious that the image quality is low due to the crosstalk among the different CCGHs when they are synthesized into a composite hologram. The total computational time is 165447s that is three times a single CCGH generation. Figures 7(g)–7(l) are the reconstructed results of the curved multiplexing based on the proposed AC method. The PSNR and SC are 9.89 dB, 10.41 dB, 7.04 dB and 0.436, 0.481, 0.392. The total computation time is 2.052s. It is seen that the image quality by the AC method is better and the computation time is shorter. The interferences among points cause more speckle noise in the calculation process of PS method. The crosstalk of curved multiplexing combined with the PS method is more when the three CCGHs are composed to a composite hologram. Hence, the AC method perform better than PS method in the simulation and optical experiment. In order to further improve the reconstructed image quality, the Fidoc algorithm [24] and gradient-limited random phase (GL-RP) methods [25] are used to combine with the curved multiplexing based on the AC method. The reconstructed images are shown in the Figs. 7(m)–7(r). The PSNR and SC are 13.66 dB, 13.85 dB, 9.51 dB and 0.238, 0.286, 0.228, respectively. It is seen that the image quality is significantly improved in this way. Both crosstalk and background noise of the reconstructed images are reduced by combining the Fidoc and GL-RP methods. The optical experimental results are in good agreement with numerical simulations.

We reconstruct s 3D scene to verify the feasibility of curved multiplexing based on the AC method. The 3D objects are calculated to generate a hologram of 1920×1080 pixels composed of two CCGHs with the central angles 0° and 8°. The 3D scene is divided into multiple 2D slices of 400×200 pixels located at different distances. The first 0° CCGH is calculated for the 3D object which are letter ‘A’ and ‘C’ focused on the distance z1 = 200 mm and z2 = 250 mm. The second 8° CCGH is corresponding to the letter ‘B’ and ‘D’ focused on the distance z1 = 200 mm and z2 = 250 mm. The numerical and optical reconstructed 3D images are shown in the Fig. 8. In the reconstructed process of the CCGH 0°, the letter ‘A’ turns from in-focus to blurred, while the letter ‘C’ turns from blurred to in-focus. By that analogy, the letter ‘B’ and ‘D’ are displayed in the different distance for the 8° CCGH. It is obvious that the 3D scene with the depth information can be successfully reconstructed from a composite hologram generated by the curved multiplexing based on the AC method. The crosstalk in the reconstructed images appears during composing where the transverse fringe crosstalk due to the horizontal bending direction of a curved hologram.

## 4. Discussion

Usually the direction factor is omitted in the PS method due to the paraxial approximation. since the diffraction distance between the WRP and CCGH is closed that the system can’t meet the condition of paraxial approximation. The direction factor is retained in DC method in the section 2 of this paper. We compared the two situations with and without direction factor. The simulation results of CCGH with 90° central angle are shown in Fig. 9. The PSNR and SC are 8.868 dB, 8.868 dB, 0.513, 0.513, respectively. It is seen that image quality is very similar. That is to say the reconstructed results are less affected by direction factor in the DC method. The maximum diffraction angle *β _{max}* of a pixel on the WRP is limited just 3.8 degree due to the pixel pitch, the diffraction area is very small according to Eq. (5). Hence the maximum direction factor $\cos (\vec{n},{\vec{r}_{cp}})$ is 0.9994. Since it is very close to cos(0°) = 1, the reconstructed results with and without the direction factor have not much differences. The direction factor can be also omitted in the DC method. The computational amount can be further reduced without the direction factor.

## 5. Conclusion

We propose a DC and AC methods to accelerate the curved hologram computation. The two methods significantly reduce the huge amount of calculation time compared with the PS method. The DC method is more precise in theory and the AC method takes much less time. And the image quality of the proposed methods is similar. We combined it with the curved multiplexing to reduce the computation time and improved the image quality at the same time. The numerical and experimental results indicate that the 2D and 3D objects can be reconstructed by the proposed methods correctly and rapidly. The curved holographic display based on the AC method is expected to achieve real-time computing by implementing on the high-performance CPU and GPU. It has a promising prospect by combining with the curved display screen and flexible display materials in the future.

## Funding

National Key Research and Development Program of China Stem Cell and Translational Research (2017YFB1002900); National Natural Science Foundation of China (61975014); Newton Fund.

## Disclosures

The authors declare no conflicts of interest.

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