## Abstract

In this paper, a curved composite hologram (CCH) generation method with suppressed speckle noise is proposed. The recorded 3D object is considered as many layers with different depths and the planar hologram (PH) for each layer is generated accordingly. Then the PH is transformed to curved hologram (CH). The CH of the recorded 3D object can be generated by superposing the CHs for all the layers. Also, the linear phase factor is superposed to the CH of the object. For different objects, the bending angle and linear phase factor of the hologram are different. The CCH is generated by superimposing the CHs of different objects. Finally, the CCH is encoded by using the error diffusion method and optimized by superposing of the digital lens. When the CCH is reproduced from different angles, the reconstructed images with suppressed speckle noise can be displayed. The experimental results verify the feasibility of the proposed method.

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

## 1. Introduction

Computer-generated holography (CGH) uses the theory of physical optics to calculate the complex amplitude distribution of the object on the interference pattern. Then, the hologram is loaded on the spatial light modulator (SLM) to obtain the reconstructed image [1–3]. However, there are many challenges for the CGH. For example, the spatial bandwidth of the SLM is not enough, and the information capability is small [4–5]. Besides, the speckle noise seriously affects the viewing experience [6–8].

The traditional hologram is planar hologram (PH), and the field of view (FOV) of the reconstructed image is limited [9–11]. The method by using multiple SLMs has been proposed to increase FOV of the reconstructed image [12], but the seamless splicing of the SLMs is a great challenge. In order to solve this problem, the circular holographic video display system has been built to reconstruct the image with the FOV of 24° [13]. Of course, other holographic display systems with multiple SLMs in circular configuration have also been proposed [14–19]. In 2019, the see-through display method based on the curved holographic optical elements was proposed, and the virtual images with the FOV of 240° can be reconstructed [20]. Some researchers work on the design and calculation of curved holograms (CHs), in the hope to reconstruct the image with the FOV of 360° [21–22]. Moreover, the researchers propose a curved multiplexing method for 3D holographic display [23], which can effectively improve the bandwidth of the SLM. Recently, the CH manufactured on the flexible display materials has been proposed with the development of the femtosecond laser direct writing technology [24–26]. In order to achieve a large viewing angle holographic display, CH has been used to improve the FOV of the reconstructed image. Though the above works can increase the bandwidth of the SLM, the speckle noise existed in the reconstructed images affects the quality of the holographic display [27–28]. Up to now, the related methods of suppressing speckle noise are often used for PH reproduction, such as the pixel separation method [29], time average method [30], and the light source optimizing [31]. However, the speckle noise suppression method for CH are rarely reported, which seriously affects the viewing experience and the further development of curved holography.

In this paper, different from the traditional method, a method of curved composite hologram (CCH) generation is proposed. Firstly, the CH of the object is generated by transformation from the PH. Besides, different linear phase factors are superposed to the CH of the object. The CCH is generated by superimposing the CHs of different curved angles, and it is encoded by using the error diffusion method and optimized by superposing the digital lens. Then, the reconstructed images with suppressed speckle noise can be displayed. Compared with the traditional method [23], the proposed method has the following advantages: 1) The quality of the single reconstructed image by using the proposed method is better than traditional method. 2) The speckle noise caused by the interference of different images in the CCH is suppressed effectively. 3) The proposed method has the same spatial bandwidth and the higher image quality without any complicated optical system. With these advantages, the proposed method can be used in the holographic AR display. In Section 4, the application of the proposed method is also verified. The results show that the proposed method can greatly promote the application of the holographic display.

## 2. Principle of the method

As shown in Fig. 1, the proposed method is composed of three processes. Firstly, the recorded 3D object is extracted by layered processing. The PH of each layer is calculated by the angular spectrum method, and it transformed into CH by analyzing the corresponding relationship between the PH and CH. The CH of the recorded 3D object can be generated by superimposing all the layers. Secondly, the linear phase factor is superposed to the CH of the object. For different 3D objects, the curved angle and linear phase factor of the CH are different. So, the CCH can be generated by superimposing the CHs with different curved angles. Thirdly, the CCH is encoded by using the error diffusion method and optimized by superposing of the digital lens. Finally, different images with suppressed speckle noise can be reconstructed by using the optimized CCH.

In the first step, the CH of the recorded object is generated by the PH. In the proposed method, the PH is calculated based on angular spectrum method. The 3D object is regarded as *L* layers with different depths, where *L* = 0, 1, 2, 3…*l*… *L*. The amplitude of each layer can be extracted from the rendered image, and the uniform phase is added to each layer. In the space Cartesian coordinates system, the light field of the layer is *h*_{p} (*x*, *y*, *z*_{h}). According to the angular spectrum method, the *h*_{p} (*x*, *y*, *z*_{h}) can be calculated as follows:

*λ*is the wavelength,

*A*(

*f*

_{x},

*f*

_{y};

*l*) is the angular spectrum distribution of the

*l*

^{th}layer, and Δ

*z*is the distance between the PH and the

*l*

^{th}layer of the object.

Then, the PH is transformed into CH by analyzing the corresponding relationship between the PH and the CH. Figure 2 shows the top-view of the conversion processes from the PH to CH. The sampling points of the PH and CH are drawn as yellow and red points, respectively. *w* is the sampling interval and *R* is the curvature radius of the CH. The multiply of the sampling points and the sampling interval is *w _{h}*.

*β*is the central angle of the CH. In curved holographic display, the curvature radius of the CH that can be derived is also an important parameter. But for convenience, the curved hologram is generally described by the central angle.

The diffraction boundary of the sampling point is drawn as the green line. *α* is the diffraction angle. The sampling interval of the PH is the same as that of the CH. The diffraction angle of each pixel on the PH depends on the pixel of the SLM. When the diffraction angle only covers the corresponding single pixel on the CH, the calculation is the simple. The maximum distance between the corresponding pixel points of the PH and the CH needs to be smaller than the ratio of the pixel interval to the tangent of the maximum diffraction angle. It means that the phase retardation caused by the diffraction of each yellow point is only added to a single linked red point. The complex amplitude of the CH of the layer can be given by the following equation:

*h*

_{c}(

*x*,

*y*,

*z*

_{(x, y)}) is the complex amplitude distribution of the CH of one layer, and it can be regarded as sub-CH of the object.

*T*(

*x*,

*y*;

*β*) is the conversion phase factor.

*Δz*

_{(x, y; β)}=

*z*

_{(x, y)}-

*z*is the propagation distance between yellow point and red point, which can be expressed as follows: where

_{h}*z*

_{c}is the maximum of the

*Δz*

_{(x, y)}. Then the CH of object can be given by Eq. (4):

*and CH*

_{l}*represent the complex amplitude of the*

_{l}*l*

^{th}layer on the planar and sub-CH, respectively. Therefore, the CH of the object with central angle

*β*can be obtained by superposing all the sub-CHs.

In the second step, in order to suppress the speckle noise caused by the interference between different images, the reconstructed image can be moved away from each other by superposing a linear phase factor to the CH of the object. As shown in Fig. 3, when one image is reconstructed based on the CCH, the other images will disturb in the form of the speckle noise. The image can be separated from the other images by superimposing the linear phase factor. The angle of the linear phase factor is *θ*. The CH of the object can be expressed as follows:

*k*= 2

*π*/

*λ*is the wave number,

*N*is the order number of the CH, and

*w*is the sampling interval. Different linear phase factors are corresponding to the CHs of different objects so that the interference can be eliminated, and speckle noise can be suppressed.

Besides, in order to improve the spatial bandwidth, the CCH is generated by superimposing the CHs with different curved angles. Figure 4 shows the schematic diagram of the CCH. The letters ‘BH’ and the letters ‘3D’ are regarded as two recorded objects. Two CHs can be generated according to the previous steps. Not only the angle of the linear phase factor but also the central angle of the CH is different for different images. Take two objects as an example, the complex amplitude of the CCH can be expressed as follows:

In the third step, the CCH is encoded by using the error diffusion method and optimized by superposing of the digital lens, as shown in Fig. 4. The complex amplitude of the CCH contains not only phase information but also the amplitude information. However, in the traditional method, only the phase information is extracted because the amplitude and the phase information cannot be modulated at the same time by using the usual SLM. For each pixel (*x*, *y*) of the CCH, the complex amplitude can also be expressed as follows:

*A*(

*x*,

*y*) represents the amplitude and

*φ*(

*x*,

*y*) is the phase value. The phase value is extracted as the pure phase value. The error between the phase information and the complex amplitude may cause the speckle noise of the reconstructed image [28]. The error can be expressed as follows: Then the complex amplitude CCH is scanned in pixels from the first pixel, and it can be optimized as follows: where

*H*is the complex amplitude of the CCH,

*H*’ is the updated complex amplitude, and

*a*is the error scan coefficient. The error is spread to the adjacent pixels

*H*(

*x*,

*y*+1),

*H*(

*x*+1,

*y*-1),

*H*(

*x+*1,

*y*),

*H*(

*x*+1,

*y*+1) in the ratio of 7:3:5:1. The sum of the error scan coefficient is set to be 1. After all pixels of the CCH have been scanned, the phase of the digital lens is superposed to that of the CCH in order to optimize CCH further. The optimized CCH can be generated by Eq. (10): where

*f*is focal length of the digital lens. The digital lens can separate the reconstructed images from the undesirable light.

Finally, different images with suppressed speckle noise can be reconstructed using only an optimized CCH, as shown in Fig. 5. When the optimized CCH is bent into different central angles, the corresponding image with suppressed speckle noise can be displayed on the receiving screen. In the optical experiment, the optimized CCH is compensated for curved holographic reconstruction according to the central angle of the CH. The phase distribution loaded on the SLM can be expressed as follows:

where*H*(

_{SLM}*x*,

*y*) is the phase distribution loaded on the SLM.

Besides, we record the speckle contrast (SC) of the reconstructed image as *C* to calculate the quality of the reconstructed image. *C* satisfies the following equation:

*σ*represents the standard deviation of intensity and

*Ī*is the average of intensity.

## 3. Experiments and results

In order to verify the feasibility of the proposed method, the holographic display system is built, as shown in Fig. 6. The system consists of a laser, an aperture, an SLM, two lenses, a beam splitter (BS), a filter and a receiving screen. The laser with the wavelength of 532 nm, an aperture and lens 1 compose the collimated light source. The BS is located behind the SLM. The resolution and pixel pitch of the SLM are 1920×1080 and 6.4 µm, respectively. The SLM has the frame rates of 60 Hz. The filter and lens 2 are used to eliminate the undesirable light, and the filter is located at the focal plane of lens 2. When the optimized CCH is loaded on the SLM, the image with suppressed speckle noise can be reconstructed on the receiving screen.

In the traditional curved multiplexing CH method, the speckle noise exists in the reconstructed images of experimental results due to the interference between the different CHs [23]. In proposed method, the linear phase factor is used to separate the multiple reconstructed images, thereby avoiding the interference among all the CHs. In order to demonstrate the effect of the linear phase factor, a verification experiment has been carried out. As the recorded objects, the resolution of the letters ‘EF’ and ‘3D’ is 800×800. The central angles of the CHs are 15° and 30°, respectively. The angles of the linear phase factors are 1° and 2°, respectively. The focal length of the digital lens is 20mm. The diffraction distance of the object is 150mm. The reproduced images using the proposed method are shown in Figs. 7(a)-(b). In contrast, when the optimized CCH without the linear phase factor is loaded on the SLM, the reproduced images are shown in Figs. 7(c)-(d). The SC of Figs. 7(a)–7(d) is 0.3157, 0.4081, 0.7532 and 0.6953, respectively. The result of using the linear phase factor is better than that without the linear phase factor.

Besides, a verification experiment has been carried out in order to demonstrate the effect of the digital lens. Due to the pixel structure of the SLM, the zero-order diffraction light and high-order diffraction images exist in the reconstructed image. These undesirable light and images can not only occupy most of the energy, but also affect the quality of the reconstructed image. Therefore, the digital lens and filter are used to improve the quality of the reconstructed image. As the recorded objects, the resolution of the letters ‘EF’ and ‘3D’ is 800×800. Then two CHs are generated. The central angles of the CHs for ‘EF’ and ‘3D’ are 15° and 30°, respectively. Two linear phase factors loaded on the CH are 1° and 2°, respectively. The focal length of the digital lens is 20mm. The reproduced images by using the proposed method are shown in Figs. 8(a)-(b). In contrast, when the optimized CCH without the digital lens is loaded on the SLM, the reproduced images are shown in Figs. 8(c)-(d). The SC of Figs. 8(a)–8(d) is 0.2914, 0.3022, 0.5768 and 0.5082, respectively. Since the digital lens can separate the reconstructed image from the rest of the undesirable light and the unnecessary interference can be avoided. So, the result of using the digital lens is better than that without the digital lens.

In order to verify the effect of the error diffusion encoding method and the reconstructed image with suppressed speckle noise for the single CH is better than that of traditional CH algorithms, the single CH of the object is calculated by using the proposed method without curved multiplexing. In the traditional method [23], the diffraction propagation between the CH and object is calculated by using the point source method. Then the CCH is generated by using curved multiplexing method. The traditional method is added with the random phase, and the iteration is not used in the algorithm. So, it is time-consuming and difficult to optimize the quality of the reconstructed image. As shown in Fig. 9, (a) ‘frog’ image of 800×693 pixels is regarded as the recorded object. Then, the CH of 1024×1024 pixels is generated with a central angle of 5°. The focal length of the digital lens is 20 mm. The diffraction distance of the objects is 100 mm, and the linear phase factor is 1.5°. The result based on the traditional method without curved multiplexing is also given as the comparative experiment. The SC of the result in Fig. 9(a) and Fig. 9(b) are 0.2693 and 0.7354, respectively. The intensity distribution of the reconstructed image is also analyzed. It can be seen that the speckle noise is suppressed greatly according to the enlarged part of the intensity distribution.

The next experiment demonstrates that the different images with suppressed speckle noise can be displayed by the optimized CCH, as shown in Fig. 10. Firstly, two CHs of 1024×1024 pixels are generated with the central angle of 15° and 30°, respectively. The focal length of the digital lens is 30 mm. The diffraction distance of the objects is 100 mm. The two linear phase factors loaded on the CH are 1.5° and 3°, respectively. Among them, the effect for the linear phase factor of 3° is equivalent to the linear phase of -1.76°. Then, the optimized CCH is calculated by the superposing two CHs. Figures 10(a)–10(b) are the CH of the letters ‘AB’ and ‘CD’, respectively. Figure 10(c) is the optimized CCH. Figures 10(d)–10(e) are the reconstructed image of the letters ‘AB’ and ‘CD’. The SC of the reconstructed images is 0.2658 and 0.3313, respectively. Therefore, it is obvious that the quality of the reconstructed image by the proposed method is high.

Through all the experiments, we can see that the high-quality reconstructed image is the result of the combined effect of the linear phase factor, the digital lens and the error diffusion encoding method.

In addition, two 3D objects are used for holographic reproduction. One object is the letters ‘3D’, and the other is the letters ‘BH’. The two 3D objects are divided into two layers with different depths. The letter ‘3’ and the letter ‘B’ are in the same depth *Z*_{1}. The letter ‘D’ and the letter ‘H’ are in the same depth *Z*_{2}. The CH with the central angle 15° is calculated for the letters ‘3D’. And the CH with the central angle 30° is calculated for the letters ‘BH’. The resolutions of all the objects are both 800×800, and the CHs are 1024×1024. The reconstructed depth *Z*_{1} and *Z*_{2} are 150 mm and 200 mm, respectively. The focal length of the digital lens is 40 mm.

Then the optimized CCH can be generated based on the proposed method. The experimental results are shown in Fig. 11. The result of using the traditional method is also given as the comparative experiment. The SC of Figs. 10(a)–10(h) are 0.3878, 0.4084, 0.4605, 0.4544, 0.9721, 0.8732, 0.9513 and 0.8654, respectively. In the holographic display, the focus and defocus of the 3D object are used to simulate the object in different depths. It is obvious that the quality of the proposed method is greatly improved.

Besides, the comparative experiment has been carried out to prove the differences of the CH and the PH. As the recorded objects, the resolution of the object ‘cube’ and the letter ‘2D’ is 800×800. The central angles of the CHs are 15° and 30°, respectively. Two linear phase factors loaded on the CH are 1° and 2°, respectively. The focal length of the digital lens is 20mm. Then the optimized CCH is calculated by the proposed method. Two PHs can be generated by using proposed method without the transformation process. However, since the PH cannot use curved multiplexing, the number of the PHs is more than CCH. The reproduced images are shown in Figs. 12(a)-(d). The SC of Figs. 12(a)–12(d) is 0.4035, 0.3952, 0.4503 and 0.3714, respectively. They are almost the same, but the CCH can display more information than the PH.

In summary of the above experiments, the proposed method has better quality not only in the reconstructed image based on a single CH, but also in the reconstructed images based on the optimized CCH. The proposed method not only optimizes the phase distribution of the CH, but also reduces the interference between different images. Therefore, the proposed method has wide spatial bandwidth and suppressed speckle noise at the same time.

## 4. Application

Due to the high quality of reproduced images, the proposed method has unique advantages in AR display. The feasibility is verified by the following experiments. In the AR display, the receiving screen is replaced by the BS. The reconstructed image can be reflected when it passes through the BS. The distance between the SLM and BS is 200mm. Besides, as a real reference, a little ‘toy’ is placed aside. The distance between the ‘toy’ and the reflection plane of BS is 100mm. The central angle of the CH is 5°. Then, the video with suppressed speckle noise can be captured by the camera. As shown in Fig. 13, (a) ‘teapot’ is rotating in the hand of the ‘toy’. The video is given in Visualization 1. Since the speckle noise is suppressed, the quality of the reconstructed image is good enough to satisfy the viewing requirement. Besides, not only the angular spectrum method but also other algorithms are applicable for the generation of the PH by using the proposed method. That means the proposed method can be combined with other fast algorithms to realize the fast calculation of the CH.

In addition, the proposed method has wide spatial bandwidth and suppressed speckle noise at the same time, which means the proposed method can be used in AR display with more information. In this verification experiment, the distance between SLM and the BS is 150mm. As the real references, the ‘toy’ and ‘car’ are placed at different depths. The ‘toy’ is placed appressed to the BS prism, and the distance between ‘toy’ and ‘car’ is 50mm. The optimized CCH with the 3D letters ‘BH’ and the 3D object ‘cloud’ is used to load on the SLM. As shown in Fig. 14, the letter ‘B’, the bigger ‘cloud’, and the ‘toy’ have the same depth, while the letter ‘H’, the smaller ‘cloud’, and the ‘car’ have the same depth. When the bigger ‘cloud’ and the letter ‘B’ are focused, the ‘toy’ can be seen clearly. When the smaller ‘cloud’ and the letter ‘H’ are focused, the ‘car’ can be seen clearly. It can be clearly seen that more information can be displayed by using the proposed method.

Compared with the traditional method, the calculation speed of the proposed method is faster. The calculation time of the proposed method is only about 1% of that based on the traditional method under the same conditions. Besides, the proposed method has the following advantages: 1) At present, the speckle noise is existed in the reconstructed image by using the traditional method, which affects the development of the holographic display based on the CH. The quality of the single reconstructed image by using the proposed method is better than the traditional CH algorithm. 2) Besides, the proposed method can suppress the speckle noise caused by the interference between different images. 3) The proposed method has wide spatial bandwidth and suppressed speckle noise at the same time. Moreover, there are no complicated optical systems or devices in the proposed method. Those features can be used in the holographic AR display in the future. Due to the characteristics of the large information capacity and suppressed speckle noise, the proposed method also has potential application value. Based on our current experimental conditions, a single SLM is used to verify the effect of the experiment. In the future, we will try to realize the curved multiplexing holographic display through the curved array of the SLMs, or looking for new bendable holographic recording materials. We will continue to research and make contribute to the development of holographic display. Besides, in general, by using the error diffusion method in CGH encoding, it will introduce some additional speckle noise. For example, some frequency domain information may be lost. In the future, we will further suppress the speckle noise by using the ringing algorithms or new iterative algorithm, and continue to solve this problem.

## 5. Conclusions

In this paper, a CCH generation method with suppressed speckle noise is proposed. The PH of the recorded 3D object is transformed to the CH. Secondly, the linear phase factor is superposed on the CH of the recorded 3D object. For different objects, the curved angle and linear phase factor of the CH are different. Then, in order to increase the information capacity, the CCH is generated by superimposing all the object CHs with different curved angles. Finally, the CCH is encoded by using the error diffusion method and optimized by superposing of the digital lens. When the CCH is loaded on the SLM, the reconstructed images with suppressed speckle noise can be display. The proposed method has wide spatial bandwidth and suppressed speckle noise at the same time. Experimental results verify the feasibility of the proposed method.

## Funding

National Natural Science Foundation of China (61927809, 62020106010); China Postdoctoral Science Foundation (2020T130039).

## 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.

## References

**1. **H. M. Chen, J. P. Yang, H. T. Yen, Z. N. Hsu, Y. Huang, and S. T. Wu, “Pursuing high quality phase-only liquid crystal on silicon (LCoS) devices,” Appl. Sci. **8**(11), 2323 (2018). [CrossRef]

**2. **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]

**3. **H. Gao, Y. Wang, X. Fan, B. Jiao, T. Li, C. Shang, C. Zeng, L. Deng, W. Xiong, J. Xia, and M. Hong, “Dynamic 3D meta-holography in visible range with large frame number and high frame rate,” Sci. Adv. **6**(28), eaba8595 (2020). [CrossRef]

**4. **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]

**5. **D. Wang, C. Liu, N. N. Li, and Q. H. Wang, “Holographic zoom system with large focal depth based on adjustable lens,” IEEE Access **8**, 85784–85792 (2020). [CrossRef]

**6. **G. Yang, S. Jiao, J. P. Liu, T. Lei, and X. Yuan, “Error diffusion method with optimized weighting coefficients for binary hologram generation,” Appl. Opt. **58**(20), 5547–5555 (2019). [CrossRef]

**7. **D. S. Mehta, D. N. Naik, R. K. Singh, and M. Takeda, “Laser speckle reduction by multimode optical fiber bundle with combined temporal spatial and angular diversity,” Appl. Opt. **51**(12), 1894–1904 (2012). [CrossRef]

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

**9. **Y. Takaki and M. Nakaoka, “Scalable screen-size enlargement by multi-channel viewing-zone scanning holography,” Opt. Express **24**(16), 18772–18787 (2016). [CrossRef]

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

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

**12. **H. Sasaki, K. Yamamoto, K. Wakunami, Y. Lchihashi, R. Oi, and T. Senoh, “Large size three-dimensional video by electronic holography using multiple spatial light modulators,” Sci. Rep. **4**(1), 6177 (2015). [CrossRef]

**13. **F. Yaraş, H. Kang, and L. Onural, “Circular holographic video display system,” Opt. Express **19**(10), 9147–9156 (2011). [CrossRef]

**14. **J. Hahn, H. Kim, Y. Lim, G. Park, and B. Lee, “Wide viewing angle dynamic holographic stereogram with a curved array of spatial light modulators,” Opt. Express **16**(16), 12372–12386 (2008). [CrossRef]

**15. **T. Kozacki, M. Kujawinska, G. Finke, W. Zaperty, 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]

**16. **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]

**17. **A. Goncharsky and S. Durlevich, “Cylindrical computer-generated hologram for display 3D images,” Opt. Express **26**(17), 22160–22167 (2018). [CrossRef]

**18. **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]

**19. **H. K. Cao, S. F. Lin, and E. S. Kim, “Accelerated generation of holographic videos of 3D objects in rotational motion using a curved hologram-based rotational-motion compensation method,” Opt. Express **26**(16), 21279–21300 (2018). [CrossRef]

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

**21. **Y. Sando, D. Barada, B. J. Jackin, and T. Yatagai, “Fast calculation method for computer-generated cylindrical holograms based on the three-dimensional Fourier spectrum,” Opt. Lett. **38**(23), 5172–5175 (2013). [CrossRef]

**22. **Y. Zhao, M. L. Piao, G. Li, and N. Kim, “Fast calculation method of computer-generated cylindrical hologram using wave-front recording surface,” Opt. Lett. **40**(13), 3017–3020 (2015). [CrossRef]

**23. **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]

**24. **G. Xue, Q. Zhang, J. Liu, Y. Wang, and M. Gu, “Flexible holographic 3D display with wide viewing angle,” in Frontiers in Optics 2016, OSA Technical Digest (online) (Optical Society of America, 2016), paper FTu5A.3.

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

**26. **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]

**27. **S.-J. Liu, D. Wang, and Q.-H. Wang, “Speckle noise suppression method in holographic display using time multiplexing technique,” Opt. Comm. **436**, 253 (2019). [CrossRef]

**28. **P. W. M. Tsang, A. S. M. Jiao, and T. C. Poon, “Fast conversion of digital Fresnel hologram to phase-only hologram based on localized error diffusion and redistribution,” Opt. Express **22**(5), 5060–5066 (2014). [CrossRef]

**29. **M. Makowski, “Minimized speckle noise in lens-less holographic projection by pixel separation,” Opt. Express **21**(24), 29205–29216 (2013). [CrossRef]

**30. **Y. Takaki and M. Yokouchi, “Speckle-free and grayscale hologram reconstruction using time-multiplexing technique,” Opt. Express **19**(8), 7567–7579 (2011). [CrossRef]

**31. **D. Lee, C. Jang, K. Bang, S. Moon, G. Li, and B. Lee, “Speckle reduction for holographic display using optical path difference and random phase generator,” IEEE Trans. Ind. Inf. **15**(11), 6170–6178 (2019). [CrossRef]