## Abstract

In this paper, a method of color curved hologram calculation based on angle multiplexing is proposed. The relationship between the wavelength, center angle and sampling interval of the curved holograms is analyzed for the first time by analyzing the reconstruction process of the curved holograms with different wavelengths. Based on this relationship, the color curved holograms are calculated by compensating phase to the complex amplitude distribution of the planar holograms. To eliminate the chromatic aberration, the curved holograms of different objects with the same color channel are respectively used for angle multiplexing and phase compensation, and then the color composed curved hologram is generated. Different color objects without chromatic aberration can be reconstructed by bending the composed curved hologram into different central angles. The experimental results verify the feasibility of the proposed method. Besides, the application of the proposed method in augmented reality display is also shown.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

## 1. Introduction

Holographic display technology is one of the ideal approaches for three-dimensional (3D) display, and it has important application value in many fields such as medical diagnosis, aerospace, industrial manufacturing, education and entertainment [1–4]. However, since the diffraction angle of the holographic display is limited by the pixel pitch of the spatial light modulator (SLM), the field of view (FOV) of the holographic display is too narrow to meet the viewing requirements [5,6].

To increase the FOV of the planar hologram, a number of methods were proposed. In 2012, a wide-angle holographic display system based on the spatiotemporal multiplexing method was proposed [7–9], and the effective space bandwidth product of the system data was increased to 50 megapixels. In 2013, a linear phase factor superimposition method was proposed to realize viewing angle enlargement [10]. The horizontal viewing angle with a single SLM can be increased to 3.6 times. In 2017, an increased-viewing-angle full-color holographic display was realized by using two tiled SLMs and a 4*f* concave mirrors system [11]. In 2018, a large resonant scanner and a galvanometer scanner were used in the holographic display system for FOV enlargement [12]. Then the FOV can be increased to 48°. The common feature of the methods of expanding FOV based on the planar hologram is that it needs to splice the SLM in the time domain or the space domain. Thus, the system is usually complicated. On the other hand, the curved hologram is an effective way to increase the FOV of the holographic display without splicing the SLM [13]. And with the development of flexible materials, more and more researchers have begun to study curved holograms [14,15]. In 2014, an acceleration method for computer generated spherical hologram of a real-existing object was proposed [16]. After that, the see-through display method based on curved holographic optical elements was proposed [17,18]. Then, the potential applications of curved metasurface holograms were demonstrated in imaging, sensing, and anti-counterfeiting [19]. Besides, some researchers proposed the multiplexing method of curved hologram to improve the information capacity and computation time [20,21]. In 2020, a holographic system for recording a curved digital hologram was demonstrated [22]. In our previous work, we studied the curved hologram calculation method for speckle noise suppression [23].

At present, the curved hologram is one of the important research directions in the fields of large viewing angle holographic display. Unfortunately, there are few reports on the color curved holographic display. In the planar holographic display, the method of color holographic display mainly includes time multiplexing and space multiplexing [24–28]. Among them, the space-multiplexing method requires three SLMs or the separation of a single SLM into three regions. The time multiplexing method requires an SLM with a high refresh rate and synchronization mechanism. Besides, some researchers proposed color-dispersion-compensated synthetic phase holograms to realize the color reproduction based on a single SLM [29]. However, in the color curved holographic display, there are some chromatic aberrations that are different from the traditional planar chromatic aberration. Therefore, the color-multiplexing method based on the planar hologram cannot be directly applied to the color curved hologram calculation, and it is necessary to study the specific method of generating the color curved hologram.

In this paper, a color curved hologram calculation method based on angle multiplexing is proposed. The relationship between the wavelength, center angle and sampling interval of the curved holograms is analyzed for the first time. By analysis, we find that the holographic reproduction of the color curved hologram contains distortion chromatic aberration and crosstalk chromatic aberration. To eliminate the chromatic aberration, the recorded 3D color object is divided into three RGB channels firstly, and the corresponding three planar holograms are generated by using the angular spectrum method. Then, three curved holograms of the recorded 3D color object are generated by compensating phase to the complex amplitude distribution of the three planar holograms. For the different recorded 3D color objects, the central angle of the curved hologram is different. The phase compensation is added to the three color holograms of different objects to eliminate the crosstalk chromatic aberration. Finally, the RGB three channels are arranged in a parallel manner, and the curved holograms of different objects in one channel are composited by using angle multiplexing to avoid the distortion chromatic aberration. When the full color composed curved hologram (CCH) is bent into a curved hologram with different central angles, the different 3D reproduced images are reconstructed. The proposed method realizes the multi-angle holographic reconstruction of the color curved hologram successfully. Additionally, the reason why the color-multiplexing method cannot be directly applied to the full color CCH calculation is analyzed. Moreover, the application of the proposed method in augmented reality (AR) display has also been verified.

## 2. Principle of the method

The principle schematic of the proposed method is shown in Fig. 1 and the method consists of three steps. For two different 3D objects, we record them as 3D object 1 and 3D object 2. Firstly, the information of the two 3D objects is extracted in red, green and blue channels, respectively. By analyzing the relationship between the wavelength, center angle and sampling interval of the curved holograms, the curved holograms of three colors are calculated respectively. The red, green and blue curved holograms of 3D object 1 are recorded as R_CH_{1}, G_CH_{1} and B_CH_{1}, respectively. The red, green and blue curved holograms of 3D object 2 are recorded as R_CH_{2}, G_CH_{2} and B_CH_{2}, respectively. For different objects, the center angles of the holograms are different accordingly. Secondly, the linear phase factor is added to separate the crosstalk chromatic aberration between the two 3D objects. Taking the red channel as an example, different phase factors are loaded on R_CH_{1} and R_CH_{2} respectively. Then the red curved hologram of object 1 and the red curved hologram of object 2 are superposed to generate the red CCH. In this way, the three color CCHs can be calculated by using the angle multiplexing method accordingly. Thirdly, the final full color CCH is generated by splicing the three color CCHs. In the process of holographic reconstruction, the reconstructed light of the three colors illuminates the CCH of the corresponding color region respectively. When the full color CCH is bent to different angles, the corresponding color objects can be reproduced.

#### 2.1 Step 1: process of curved hologram generation

The red, green, and blue scene information for a color 3D object is processed separately. For each color channel, the 3D object is divided into *L* layers with different depths by using the layer-based method [30]. *L _{l}* represents the

*l*th layer,

*l*= 1, 2, 3…

*L*. For each layer, the amplitude information is extracted from the rendered image, and the phase information is preset to a uniform value. Then, the complex amplitude distribution of the wavefront recording plane (WRP) is generated by using the angular spectrum method (ASM) [31]:

*U*

_{o}(

*x*,

*y*) and

*U*(

_{p}*x*,

*y*) represent the complex amplitude distributions of the object and the WRP, respectively.

*A*

_{0}and

*φ*

_{0}are the amplitude and phase distribution of each depth layer, respectively. FFT[•] and IFFT[•] are the fast Fourier transform (FFT) and inverse FFT operators, respectively.

*H*(

_{f}*f*,

_{x}*f*) is the transfer function, which can be expressed as follows:

_{y}*k*represents the wavenumber,

*λ*is the wavelength, and Δ

*z*is the distance between the layer of the object and the WRP. For different color channel, the wavelength is different accordingly.

After the WRP of the object in each channel is generated, the curved hologram of the object in each channel can be transformed from the WRP by analyzing the phase difference distribution caused by the optical path between the WRP and curved hologram. The conversion process from WRP to the curved hologram is shown in Fig. 2, where CH represents the curved hologram plane. The yellow points and red points represent the sampling points of the WRP and CH, respectively. *w* is the sampling interval and *w _{h}* is the size of the WRP.

*R*is the curvature radius of the CH, which is derived from the central angle

*β*.

In the calculation of the curved hologram, the phase retardation caused by the diffraction from the WRP to the curved hologram can be regarded as an approximate compensation. Then the complex amplitude of the curved hologram can be generated by the geometric optical path difference of each point in the WRP. The complex amplitude of the curved hologram *U _{c}* (

*x*,

*y*) is expressed by the following equation:

*T*(

*x*,

*y*;

*β*) is the conversion phase factor. Δ

*z*is the propagation distance between the point on the WRP and the corresponding point on the CH. The origin of the

*x*-axis is set on the

*z*-axis. Δ

*z*can be expressed as follows: where

*z*represents the maximum of the Δ

_{c}*z*. Only the distance

*z*

_{c}is small enough according to the limited hologram size and the central angle of curved hologram, the phase difference distribution can be regarded as an approximate compensation generated by the geometric optical path difference. Therefore, the complex amplitude of the curved hologram with central angle

*β*can be generated by transformation of the WRP.

In planar holographic display, only the same reference beam as the recording light can accurately reconstruct the recorded object. When the wavelength is different, the position of the reproduced image is correspondingly different, then the chromatic aberration occurs. According to the diffraction principle of the planar holographic display, the complex distribution of the reconstructed image *I*(*x*, *y*) can be expressed as follows:

*U*(

_{o}*x, y*) represents the complex amplitude distribution of the recorded object, |

*r*(

*x, y*)|

*exp*(•) represents the reference beam in the recording process and |

*r*’(

*x, y*)|

*exp*(•) represents the reference beam in the reconstruction process. From Eq. (6) and Eq. (7) we can see that when the reference beam in the reconstruction process is consistent with the reference beam in the recording process, the two parts of the reference beam can be counteracted. When the reference beam does not match, the reconstructed image will be multiplied by a phase factor. Therefore, the redundancy caused by the wavelength mismatch of the reference beam will cause chromatic aberration in the reproduced image.

As shown in Fig. 3, when the green channel planar hologram (G_PH) is irradiated with three color reference beams simultaneously, the red, blue and green reconstructed images do not overlap. *z*_{0} is the reconstructed distance. Δ*y*_{1} and Δ*z*_{1} represent the chromatic aberration caused by blue light. Δ*y*_{2} and Δ*z*_{2} represent the chromatic aberration caused by the red light.

However, in the reconstruction process of the color curved hologram, the distortion chromatic aberration exists when the wavelength of the reference beam does not match the recorded light. The generation of the curved hologram is mainly transformed from the WRP according to Eq. (4). One of the important parameters is the conversion phase factor *T* (*x*, *y*; *β*). It can be found that the conversion phase factor is wavelength dependent, which means that the WRPs of different color channels correspond to different conversion phase factors. The reconstruction process based on the curved hologram is as follows: The first is the inverse phase conversion of the curved hologram to WRP, and the second is to use WRP for diffraction reconstruction. The reversal transformation from the curved hologram to WRP is equivalent to multiplying the reversal transformation phase factor in the curved hologram light field. The inverse conversion phase factor depends on the wavelength and phase distribution of the reference beam in the reconstruction process. Therefore, only when the reference beam in the reconstruction process is the same as the reference beam in the recording process, the inverse conversion phase factor and the conversion phase factor can be canceled. Then the image of the recorded object can be reconstructed correctly. If the wavelength of the reference beam is mismatched in the reconstruction process, the reconstructed image will be stretched or squeezed, which is the distorted chromatic aberration in the curved holographic display.

As shown in Fig. 4, when the green channel curved hologram (G_CH) is irradiated with three color reference beams simultaneously, the positions of the reconstructed images are different, and the shape of the reconstructed images is deformed due to the distortion chromatic aberration. Among them, the green reconstructed image is normal, the blue reconstructed image is squeezed, and the red reconstructed image is stretched. After verification, we find that the reconstructed image will be stretched if the wavelength of the reference beam of the reconstruction process is greater than that of the recording process. Otherwise, it will be squeezed.

In addition, when the green reference beam is used to illuminate different color curved holograms, multiple images will be also reconstructed, as shown in Fig. 5. The distortion chromatic aberration also exists. The stretched image is reconstructed at the position of the blue dotted line. Similarly, the squeezed image is reconstructed at the position of the red dotted line. It can be seen that, due to the existence of distorted chromatic aberration, the traditional planar color multiplexing method cannot be directly applied to color CCH calculation.

#### 2.2 Step 2: process of CCH generation

In order to reproduce multiple objects, the angle multiplexing method is used to calculate the curved holograms of different objects. The two WRPs of the objects are generated by using the ASM. The diffraction distances between the WRPs and the objects are the same. Then, the curved holograms of the two objects can be transformed from two WRPs by adding the phase retardations. It should be noted that the curved holograms have the same pixel number and sampling interval, but the center angles of the curved holograms for different objects are different. The complex amplitude distribution of the CCH can be generated by adding all the curved holograms.

where*n*is the number of curved holograms. In the reconstruction process, different reconstructed images can be displayed in sequence by bending the CCH into different central angles.

Unfortunately, the angle multiplexing method of the curved holograms will produce additional chromatic aberrations. Since the angle multiplexing method is to superimpose curved holograms with different central angles, different conversion phase factors correspond to different curved hologram central angles according to Eq. (4). Therefore, the inverse conversion phase factor is also different. Figure 6 is the reconstruction of the color CCH illuminated with the color reference beam. When the CCH is bent into a different central angle, the image corresponding to the recorded central angle can be correctly reconstructed, while the other center angles cannot reproduce the correct reconstructed image since the inverse conversion phase factor cannot be counteracted. The distorted image will exist in the background of the correct reconstructed image, which will affect the viewing effect, as shown in Figs. 6(a)-6(b).

Due to the crosstalk between the curved holograms, the quality of the reconstructed image of the CCH is affected. To suppress the crosstalk chromatic aberration, the linear phase factor is superimposed on the curved hologram of the recorded object. Before generating the CCH using the angle multiplexing method, the same linear phase factor is added to each color curved hologram of the same recorded object. The curved hologram of the object is expressed as follows:

where*θ*is the angle of the linear phase factor. Since there are no curved SLMs currently available, we have to perform phase compensation on the planar SLM for experimental verification [32]. Then, the phase distribution

*H*'(

*x*,

*y*) loaded on the SLM is expressed as follows:

By loading different linear phase factors on the curved holograms of different objects, the images of different objects can be separated, and the crosstalk chromatic aberration caused by angle multiplexing can be eliminated, as shown in Figs. 6(c)-(d).

#### 2.3 Step 3: process of holographic reconstruction

In the reconstruction process, we use the method of spatial multiplexing to splice the curved holograms of three colors into a color curved hologram, and control the reproduced light of the three colors to illuminate the hologram of the corresponding color areas, thereby avoiding distortion chromatic aberration. The phase compensation is added to the three color holograms of different objects to eliminate the crosstalk chromatic aberration. The final full color CCH is generated by splicing the three color CCHs and the corresponding area of the final full-color CCH is illuminated with the RGB reference beam simultaneously. The reconstruction process of the final full-color CCH is shown in Fig. 7. When the final full-color CCH is bent into different central angle, the different 3D reproduced images without chromatic aberration can be seen.

## 3. Simulation, experiments and results

To verify the feasibility of the proposed method, a color holographic display system is built, as shown in Fig. 8. The red, green, and blue lasers are used as the color reference beam. The wavelengths for the three colors are 671nm, 532nm, and 473nm, respectively. After passing through the prism and beam splitter (BS), the three color lasers illuminate one third of the SLM area respectively. The SLM used in the system has a pixel pitch of 6.4µm and a resolution of 1920×1080. The full color CCH is loaded onto the SLM, and the loading area of each color CCH is the same as the irradiation position of the corresponding color laser. The size of the BS is 25.4mm×25.4mm×25.4mm. The 4*f* system is composed of two lenses and an aperture to eliminate the zero-order light. The SLM locates at the front focal plane. The reconstruction position of the 3D object is determined by the diffraction distance set in the experiment. When the reproduced image passes through the aperture, by adjusting the aperture size, it can be ensured that only the first-order diffraction image passes through, thereby avoiding the interference of high-order diffraction images and zero-order light. After the reconstructed light passes through the BS and 4*f* system, the reconstructed images of three color CCHs can be spatially overlapped, then the ideal holographic reconstructed image can be captured by the CCD.

Firstly, we simulate the distortion chromatic aberration in the reconstructed image. When red, green and blue lasers are used to illuminate the green curved hologram, the simulation results are shown in Fig. 9. The center angle of the curved hologram is 10°. It can be seen that the correct image can only be reproduced when the green curved hologram is illuminated by a green laser. When the green curved hologram is illuminated with red and blue lasers, the reconstructed image is distorted. So, in the color curved holographic reproduction, we need to keep the three colors of light illuminating the hologram of the corresponding color for the purpose of distortion chromatic aberration elimination.

Then, the 3D object is used for verification. The letters ‘3’ and ‘D’ are located on different depth planes to verify the 3D reproduction effect. The diffraction distances of the two depth objects are 15cm and 17cm, respectively. The resolution of the 3D object is 800×600. When generating three-color curved holograms, we spatially splice them together to form a color curved hologram. The resolution of the monochromatic curved hologram is 640×1080. The resolution and curved angle of the color curved hologram are 1920×1080 and 10°, respectively. Then the recorded full color curved hologram of the 3D object is loaded onto the SLM according to the calculation. In the simulation experiment, we do not convert the curved hologram into a plane by phase compensation, but directly reconstructed the curved hologram. The results are shown in Fig. 10. Figure 11 shows the result when the diffraction distance is 17cm. It can be seen that the reconstructed image ‘D’ is in focus at this time, while the reconstructed image ‘3’ is blurred. When the CCD is located at a distance of 15cm, the results are shown in Fig. 12. It can be seen clearly that the reconstructed image ‘3’ is in focus at this time, while the reconstructed image ‘D’ is blurred. So, the color curved holographic display can be realized without chromatic aberration.

In order to verify the color display effect of the CCH, two different objects ‘H’ and ‘W’ are recorded respectively. The resolutions of the two objects are 320×240. The resolutions of the curved holograms and full color CCH are 640×1080 and 1920×1080, respectively. The diffraction distances of the two objects are 15cm. Taking ‘H’ as an example, ‘H’ is firstly separated for color processing and three color curved holograms of the object ‘H’ are generated respectively with the center angles of 10°. Similarly, the object ‘W’ is separated into three colors and three color curved holograms of the object ‘W’ are generated respectively with the center angles of 30°. The green curved holograms of ‘H’ and ‘W’ are superimposed to generate a green CCH, the red curved holograms of ‘H’ and ‘W’ are superimposed to generate a red CCH, and the blue curved hologram of ‘H’ and ‘W’ are superimposed to generate a blue CCH. The three color CCHs are spliced into a full color CCH and loaded on the SLM. When the bending angle of the full color CCH is set to 10°, the results of the color reconstructed image are shown in Figs. 13(a)-13(c). It can be seen that the object ‘H’ is reproduced, but the reproduced image is disturbed by another angled object ‘W’. When the bending angle of the full color CCH is set to 30°, the reconstructed images of the object ‘W’ are shown in Figs. 13(d)-13(f). It can be seen that the reproduced image is disturbed by the object ‘H’. Therefore, when the full color CCH is used for reproduction, the reconstructed image will be disturbed by the chromatic aberration of other curved angle images.

In order to eliminate this crosstalk chromatic aberration, different phase factors are added to the curved holograms of ‘H’ and ‘W’ before the CCHs are generated according to Eq. (9). The angles of the linear phase factors added on the CH of the object are 1° and 3°, respectively. The center angles of the color curved holograms are 10° and 30° respectively. The reconstructed results of the object ‘H’ are shown in Fig. 14. Figures 14(a)-14(c) are the blue, green, and red reconstructed images respectively. Figure 14(d) is the color reconstructed image. It can be seen clearly that the chromatic aberration and the crosstalk have been eliminated. When the bending angle of the full color CCH is set to 30°, the color reconstructed images of the object ‘W’ are shown in Fig. 15. Hence, by using the proposed method, the reconstructed images of color curved holograms at different angles can be achieved without chromatic aberration. When the bending angle is different, the object of the corresponding angle can be displayed. This angle multiplexing can also increase the information capacity in the holographic display.

The proposed method can be used for holographic AR display and multi-plane display, etc. In order to verify the application of color curved holographic display, two real objects are placed in front of a camera with an imaging lens to obtain the holographic AR display effect. Here, a ‘car’ and an ‘angel’ at two different depth planes are used as the real objects. The object ‘H’ and ‘3D’ are used as the two recorded objects. The resolution of the two objects is 320×240. The resolutions of the curved hologram and full color CCH are 640×1080 and 1920×1080. In the generation process of the CCH, the center angles of ‘H’ and ‘3D’ are set to 10° and 30°, respectively. The angle of the linear phase factors added on the CH of the object is 1° and 2°, respectively. For the object ‘3D’, the letters ‘3’ and ‘D’ are at different depth planes. The diffraction distances of ‘H’ and ‘3’ are the same, and their reconstructed images are on the same plane as the real object ‘car’. The reconstructed image of ‘D’ is on the same plane as the real object ‘angel’. When the bending angle of the full color CCH is set to 10°, the color reconstructed images of the object ‘H’ can be captured by the camera. At the same time, the real object ‘car’ can be seen clearly, as shown in Fig. 16(a). When the bending angle of the full color CCH is set to 30°, the color reconstructed images of the object ‘3D’ can be captured by the camera. When the diffraction distance is 15cm, the real object ‘car’ and the reconstructed image of ‘3’ can be clearly seen in Fig. 16(b), while ‘angel’ and the reconstructed image of ‘D’ are blurred. When the diffraction distance is 25cm, the real object ‘angel’ and the reconstructed image of ‘D’ are in focus, while ‘car’ and the reconstructed image of ‘3’ are blurred, as shown in Fig. 16(c). It can be seen from the result that when the CCH is bent at different angles, the color reproduced image at the corresponding angle and the real object can be seen simultaneously. Compared with the traditional holographic AR display, the proposed method not only realizes the color reproduction but also increases the information content of the holographic display.

There are different ways to generate the curved hologram. One way is to calculate 3D object directly. This curved hologram can enlarge the FOV of holographic reproduction, but it takes a long time to directly calculate the curved hologram from 3D object. In order to improve the calculation speed, some researchers obtain the complex amplitude distribution of the curved hologram by approximate solution of planar hologram. In this case, the parameters such as pixel pitch of curved hologram are affected by the planar hologram. The main reason for the increase of the viewing angle of curved hologram is that its curvature increases the viewing angle. Affected by the pixel pitch of the SLM, the limited diffraction angle of the reproduced image is restricted by the maximum diffraction angle of the SLM, which is a small fixed value. When the SLM changes from a planar surface to a curved surface, there is a curvature change, which artificially introduces an additional perspective. Due to the limitation of experimental conditions, we use a planar modulator for equivalent experimental verification. In this case, the increase of FOV can not be verified. Of course, we are also studying curved modulation elements based on micro-nano materials. When the curved hologram is loaded on the curved modulator, the curved modulator has a larger diffraction angle than that of the planar modulator, Then the FOV can be increased. Besides, although the angle multiplexing method can improve the information capacity of the hologram, the number of the curved holograms with different central angle is limited by the crosstalk. When the number of composites increases, the crosstalk of the reproduced image will increase accordingly. The minimum angular pitch are important topics to be addressed. When the bending angle becomes large, the crosstalk between different central angle curved holograms will lead to the degradation of the reconstructed image quality.

## 4. Conclusion

In this paper, a color curved hologram calculation method based on angle multiplexing is proposed. By dividing the recorded 3D color object into three RGB channels, the corresponding three color planar holograms are generated. Then, three curved holograms can be generated by compensating phase to the complex amplitude distribution of three planar holograms. The chromatic aberration in the curved hologram reproduction is analyzed and compensated. For the different recorded 3D color objects, the central angle of the curved hologram is different. Finally, the curved holograms of different objects in the same color channel are composited by using angle multiplexing to generate the full color CCH. The different color objects can be reproduced when the full color CCH is bent into different central angles. Besides, the application of the proposed method in holographic AR display is also verified. We believe that the proposed method can promote the holographic display application.

## Funding

National Natural Science Foundation of China (62020106010, 62011540406); National Research Foundation of Korea (2020K2A9A2A06038623).

## Acknowledgement

We would like to thank Nanofabrication facility in Beihang Nano for technique consultation.

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

## 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.

## References

**1. **L. Shi, B. Li, C. Kim, P. Kellnhofer, and W. Matusik, “Towards real-time photorealistic 3D holography with deep neural networks,” Nature **591**(7849), 234–239 (2021). [CrossRef]

**2. **T. Zhan, J. H. Xiong, J. Y. Zhou, and S. T. Wu, “Multifocal displays: review and prospect,” PhotoniX **1**(1), 10 (2020). [CrossRef]

**3. **D. Wang, C. Liu, C. Shen, Y. Xing, and Q. H. Wang, “Holographic capture and projection system of real object based on tunable zoom lens,” PhotoniX **1**(1), 6 (2020). [CrossRef]

**4. **K. Wakunami, P. Y. Hsieh, R. Oi, T. Senoh, H. Sasaki, Y. Ichihashi, M. Okui, Y. P. Huang, and K. Yamamoto, “Projection-type see-through holographic three-dimensional display,” Nat. Commun. **7**(1), 12954 (2016). [CrossRef]

**5. **X. Ding, Z. Wang, G. Hu, J. Liu, K. Zhang, H. Li, B. Ratni, S. N. Burokur, Q. Wu, J. Tan, and C. W. Qiu, “Metasurface holographic image projection based on mathematical properties of Fourier transform,” PhotoniX **1**(1), 16 (2020). [CrossRef]

**6. **Z. Wang, G. Lv, Q. Feng, A. Wang, and H. Ming, “Simple and fast calculation algorithm for computer-generated hologram based on integral imaging using look-up table,” Opt. Express **26**(10), 13322–13330 (2018). [CrossRef]

**7. **T. Kozacki, G. Finke, P. Garbat, W. Zaperty, and M. Kujawińska, “Wide angle holographic display system with spatiotemporal multiplexing,” Opt. Express **20**(25), 27473–27481 (2012). [CrossRef]

**8. **T. Kozacki, M. Kujawinska, G. Finke, B. Hennelly, and N. Pandey, “Extended viewing angle holographic display system with tilted SLMs in a circular configuration,” Appl. Opt. **51**(11), 1771–1780 (2012). [CrossRef]

**9. **Z. Zhang, C. P. Chen, Y. Li, B. Yu, L. Zhou, and Y. Wu, “Angular multiplexing of holographic display using tunable multi-stage gratings,” Mol. Cryst. Liq. Cryst. **657**(1), 102–106 (2017). [CrossRef]

**10. **Y. Z. Liu, X. N. Pang, S. Jiang, and J. W. Dong, “Viewing-angle enlargement in holographic augmented reality using time division and spatial tiling,” Opt. Express **21**(10), 12068–12076 (2013). [CrossRef]

**11. **Z. Zeng, H. Zheng, Y. Yu, A. K. Asundi, and S. Valyukh, “Full-color holographic display with increased-viewing-angle,” Appl. Opt. **56**(13), F112–F120 (2017). [CrossRef]

**12. **J. Li, Q. Smithwick, and D. Chu, “Full bandwidth dynamic coarse integral holograohic displays with large field of view using a large resonant scanner and a galvanometer scanner,” Opt. Express **26**(13), 17459–17476 (2018). [CrossRef]

**13. **Y. Sando, M. Itoh, and T. Yatagai, “Fast calculation method for cylindrical computer-generated holograms,” Opt. Express **13**(5), 1418–1423 (2005). [CrossRef]

**14. **S. Q. Li, X. Xu, R. M. Veetil, V. Valuckas, R. P. Domínguez, and A. I. Kuznetsov, “Phase-only transmissive spatial lightmodulator based on tunable dielectric metasurface,” Science **364**(6445), 1087–1090 (2019). [CrossRef]

**15. **Q. S. Wei, B. Sain, Y. T. Wang, B. Reineke, X. W. Li, L. L. Huang, and T. Zentgraf, “Simultaneous spectral and spatial modulation for color printing and holography using all-dielectric metasurfacces,” Nano Lett. **19**(12), 8964–8971 (2019). [CrossRef]

**16. **G. Li, K. Hong, J. Yeom, N. Chen, J. H. Park, N. Kim, and B. Lee, “Acceleration method for computer generated spherical hologram calculation of real objects using graphics processing unit,” Chin. Opt. Lett. **12**(6), 060016 (2014). [CrossRef]

**17. **K. Bang, C. Jang, and B. Lee, “Curved holographic optical elements and applications for curved see-through displays,” J. Inf. Disp. **20**(1), 9–23 (2019). [CrossRef]

**18. **J. Burch and A. D. Falco, “Holography using curved metasurfaces,” Photonics **6**(1), 8 (2019). [CrossRef]

**19. **L. Zhou, C. P. Chen, Y. Wu, Z. Zhang, K. Wang, B. Yu, and Y. Li, “See-through near-eye displays enabling vision correction,” Opt. Express **25**(3), 2130–2142 (2017). [CrossRef]

**20. **R. Kang, J. Liu, G. Xue, X. Li, D. Pi, and Y. Wang, “Curved multiplexing computer-generated hologram for 3D holographic display,” Opt. Express **27**(10), 14369–14380 (2019). [CrossRef]

**21. **R. Kang, J. Liu, D. Pi, and X. Duan, “Fast method for calculating a curved hologram in a holographic display,” Opt. Express **28**(8), 11290–11300 (2020). [CrossRef]

**22. **J. P. Liu, W. T. Chen, H. H. Wen, and T. C. Poon, “Recording of a curved digital hologram for orthoscopic real image reconstruction,” Opt. Lett. **45**(15), 4353–4356 (2020). [CrossRef]

**23. **N. N. Li, D. Wang, Y. L. Li, and Q. H. Wang, “Method of curved composite hologram generation with suppressed speckle noise,” Opt. Express **28**(23), 34378–34389 (2020). [CrossRef]

**24. **M. Oikawa, T. Shimobaba, T. Yoda, H. Nakayama, A. Shhiraki, N. Masuda, and T. Ito, “Time-division color electroholography using one-chip RGB LED and synchronizing controller,” Opt. Express **19**(13), 12008–12013 (2011). [CrossRef]

**25. **T. Shimobaba, T. Takahashi, N. Masuda, and T. Ito, “Numerical study of color holographic projection using space-division method,” Opt. Express **19**(11), 10287–10292 (2011). [CrossRef]

**26. **Y. Zhao, K. C. Kwon, Y. L. Piao, S. H. Jeon, and N. Kim, “Depth-layer weighted prediction method for a full-color polygon-based holographic system with real objects,” Opt. Lett. **42**(13), 2599–2602 (2017). [CrossRef]

**27. **S. J. Liu, N. T. Ma, P. P. Li, and D. Wang, “Holographic near-eye 3D display method based on large-size hologram,” Front. Mater. **8**, 739449 (2021). [CrossRef]

**28. **S. F. Lin, P. Gentet, D. Wang, S. H. Lee, E. S. Kim, and Q. H. Wang, “Simply structured full-color holographic three-dimensional display using angular-compensating holographic optical element,” Opt. Laser. Eng. **138**, 106404 (2021). [CrossRef]

**29. **K. Choi, H. Kim, and B. Lee, “Full-color autostereoscopic 3D display system using color-dispersion-compensated synthetic phase holograms,” Opt. Express **12**(21), 5229–5236 (2004). [CrossRef]

**30. **Y. Zhao, L. Cao, H. Zhang, D. Kong, and G. Jin, “Accurate calculation of computer-generated holograms using angular-spectrum layer-oriented method,” Opt. Express **23**(20), 25440–25449 (2015). [CrossRef]

**31. **T. Shimobaba, K. Matsushim a, T. Kakue, N. Masuda, and T. Ito, “Scaled angular spectrum method,” Opt. Lett. **37**(19), 4128–4130 (2012). [CrossRef]

**32. **D. Wang, N. N. Li, Y. L. Li, Y. W. Zheng, and Q. H. Wang, “Curved hologram generation method for speckle noise suppression based on stochastic gradient descent algorithm,” Opt. Express **29**(26), 42650–42662 (2021). [CrossRef]