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Abstract
Cylindrical holography has attracted widespread attention for its 360° look-around property. However, commercial cylindrical spatial light modulators(SLM) are not available. Additionally, on account of the heavy burden of memory space and computing time, the wavelength in the terahertz band rather than the visible light is employed in experiments. These two constraints are a great challenge to optical realization of cylindrical holography. By contrast, although curved holography, as a segment of cylindrical holography, can be realized optically to increase the field of view, the angle of view is still limited due to its intrinsic limitations. In this paper, an optical realization of 360° cylindrical holography is proposed by using isophase surface transformation within visible light with a 45° conical mirror and a commercial planar SLM. The diffraction formula of the proposed method is derived theoretically, and the feasibility of the proposed method is further demonstrated by numerical simulations and optical experiments. This brand-new method is expected to have practical applications in the field of cylindrical holography in the future.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Holographic display can provide the required parallax and depth information for the human eyes, which is regarded as a promising display technique for truly realizing 3D displays [1]. Computer-generated hologram (CGH) is a key technique to realizing holographic display by recording a hologram of 3D virtual object digitally. Generally, holograms are loaded onto a planar SLM, meanwhile the field of view (FOV) is limited by the size and pixel pitch of SLM [2]. Curved holograms are an effective way to overcome the limitations of FOV [3], while computer-generated cylindrical holograms (CGCH) are considered ideal for extending FOV. This is because it has a 360° “look-around property”, providing a better immersive experience [4]. Immersion, where users can immerse themselves in the metaverse experience and ignore the real environment they are in, is one of the eight elements of the metaverse.
Up to now, researchers have devoted many efforts to the cylindrical holographic display. Sando et al. proposed a convolution algorithm using the Fast-Fourier Transform (FFT) to obtain cylindrical holograms [5]. Additionally, they used Bessel function expansions to reduce the calculation time and memory usage for CGCH [6]. Jackin et al. introduced a fast calculation method based on FFT, in which the angular spectrum diffraction formula of the cylindrical model was proposed and the transfer function was found [7], and this method was also used to simulate the reconstruction of 3D objects [8]. Zhao et al. accelerated computation of CGCH by introducing a wavefront recording plane [9]. Wang et al. proposed a unified and accurate diffraction calculation method based on FFT by analyzing the obliquity factor [10] and unifying the two propagation models of concentric cylinders [11]. Li et al. proposed a method for occlusion culling of CGCH using the horizontal optical-path-limit function [12]. Jin et al. used the cylindrical self-diffraction iteration algorithm to suppress speckle noise [13]. The cylindrical holographic algorithms proposed in the above articles are all based on cylindrical SLMs. However, current commerical SLMs are all planar, which leads to cylindrical holographic algorithms that can only be verified in numerical simulation, but cannot be carried out with optical experiment. Furthermore, in all the existing studies above, if the wavelength is taken in the visible region, the number of samples will be very large due to the sampling theorem, and the storage space and computing time are also a heavy burden, so that the wavelength has to be taken in the terahertz band. These two limitations pose a huge challenge to perform optical experiments and practical applications in visible region.
On the other hand, the curved holography, as a segment of cylindrical holography, is another way to increase the FOV of the holographic display. In recent years, some progresses on curved holography are reported. In 2019, a curved hologram generation method was proposed to expand the viewing angle of the holographic display [14]. In 2020, an acceleration algorithm based on approximate phase compensation were proposed for curved hologram generation [15]. In 2021, the two-step acceleration calculation method [16] and the look-up table method of curved holography [17] were proposed to further accelerate the generation of curved holograms. In 2022, the calculation method of color curved hologram based on angle multiplexing was proposed [18]. The above studies utilize a planar SLM to achieve the curved holographic display in visible light, increasing the viewing angle. However, due to its intrinsic limitations, the FOV of curved holographic display remains limited.
In this paper, an optical realization of 360° cylindrical holography is proposed by using isophase surface transformation within visible light with a commercial planar SLM. In the proposed method, a brand-new diffraction model is proposed to convert the isophase surface of the diffracted field from a plane to a cylinder by using a 45° conical mirror. In order to derive the diffraction formula of the model, the whole diffraction process is divided into three stages: plane-to-plane diffraction, plane-to-cylinder diffraction, and cylinder-to-cylinder diffraction. For the stage of plane-to-cylinder diffraction, since the precise diffraction calculation by the Point Source (PS) method is time-consuming, an approximate approach is proposed by planar diffraction, and the feasibility of the approximation is fully demonstrated. Also, the sampling conditions of the method are discussed. The feasibility of the proposed method is verified through numerical simulations and optical experiments. Therefore, a 360° cylindrical holography within visible light is optically realized successfully.
2. Theoretical derivation
By using a 45° conical mirror, the holographic display from the hologram plane to the cylindrical object surface can be achieved, so that a large viewing angle of 360° can be obtained. The schematic of the proposed model is shown in Fig. 1, in which the entire diffraction process can be regarded as being completed in three stages. Firstly, light diffracts from the hologram plane to the middle plane. The light is then redirected from the middle plane to the inner cylindrical surface after being reflected by the conical mirror. Finally, the light propagates from inner cylindrical surface to outer object surface. In the proposed model, hologram plane is a circle with a radius a. Middle plane is a virtual circular plane with vertex V as the center and radius a. The height of the conical mirror and the radius of the bottom circle which is also called refocusing plane, are both a. The height and radius of inner cylindrical surface are also a. It should be noted that the proposed model has excellent rotational symmetry. Therefore, the theoretical derivation section will be carried out in the cylindrical coordinate system.
Fig. 1. (a) Schematic of proposed holographic diffraction model. (b) Three stages of diffraction process.
2.1 Diffraction from inner cylindrical surface to object surface
In this section, the derivation process of the diffraction formula of two concentric cylindrical surfaces will be discussed. As shown in Fig. 2, $Q({{r_s},{\theta_s},{z_s}} )$ is a source point on the inner cylindrical surface, while $Q^{\prime}({{r_d},{\theta_d},{z_d}} )$ is a destination point on the outer cylindrical object surface. Point Q emits a conical beam within the diffraction angle $2\beta $, and point ${Q_0}$ is the center point of the beam on the object surface.
The distributions of diffraction on the inner surface and outer surface are represented by ${u_s}({{r_s},{\theta_s}} )$ and ${u_d}({{r_d},{\theta_d}} )$, respectively. In cylindrical coordinates, The Rayleigh–Sommerfeld integral equation [12] can be written as
When the object surface is diffracted to the inner cylindrical surface, since the object surface is loaded with a grid pattern, it can be known from the grating equation that the diffraction angle of a pixel is at most $2\beta $. Similarly, each pixel on the inner cylindrical surface will also have a diffraction angle when the diffraction process is reversed. Hence, the light beam emitted from point Q is a light cone with a divergence angle of $2\beta $. Approximately, the half-divergence angle $\beta $ can be expressed by the maximum diffraction angle of gratings [19]
where p represents the sampling pitch. The diffractive region on the object surface is a ellipse where the light cone meets the cylindrical object surface. Usually, $\beta $ is a small angle, which can approximate the curved diffraction area on the cylindrical object surface as a flat area. Thus the corss-section is a circular area. Any destination point whose propagation distance d less than $\sqrt {{r_d}^2 + {r_s}^2 - 2{r_d}{r_s}cos\beta + {{[{({{r_d} - {r_s}} )\times tan\beta } ]}^2}} $ is within this circular area. The maximum frequency of the circular-shaped PSF is $1/2p$, which will map to a circle with radius $1/2p$ in Fourier space. Since the inner cylindrical surface is sampled with a square lattice, its actual bandwidth is a square with side length $1/2p$. Therefore, only the frequencies within the inscribed circle of that square are used, constituting only of $\pi /4 \approx 78.5\%$ of the total available bandwidth [20]. In order to fully utilize the total bandwidth of the inner cylindrical surface, we establish a rectangular filtering function $g({\theta ,z} )$ embedded in the PSF.As mentioned, $g({\theta ,z} )$ is used to determine whether to retain the light ray or not, and it should be multiplied by the PSF $h({\theta ,z} )$. Therefore, the new PSF is $h^{\prime}({\theta ,z} )$ as follows
2.2 Diffraction from middle plane to inner cylindrical surface
2.2.1 Derivation of diffraction formula
Diffraction from middle plane to inner cylindrical surface is an important part that needs to be discussed in detail since it is the key step in converting planar diffraction into cylinder diffraction. A schematic diagram of the model is shown in Fig. 3. The height of the 45° conical mirror and the radius of its bottom base are both a. V is the vertex of conical mirror, and O is the midpoint of its base. Middle plane is a circle with V as center and a as radius. Inner cylindrical surface is a cylinder of radius and height a. $P - M - Q$ is a possible light path. $\overrightarrow {{n_m}} $ is the normal vector at the point M.
From the perspective of a diffraction process, the complex amplitude of inner cylindrical surface can be calculated precisely by PS method, which is given by
For a given point P and Q, the coordinates of the reflection point M need to be known. Therefore, there are only two unknowns that need to be solved, ${r_m}$ and ${\theta _m}$. According to the Law of Reflection, the angle of incidence is equal to the angle of reflection, that is
Three Eqs. ,(11), (12) and (13), are obtained from the Law of Reflection and Fermat's principle, but only ${r_m}$ and ${\theta _m}$ are unknown, so these three equations are not independent of each other. As long as two of the equations are solved, the third equation is automatically satisfied. In other words, the Law of Reflection can be derived from Fermat's principle, and any ${r_m}$, ${\theta _m}$ that satisfies Eqs. (12), (13) must satisfy Eq. (11).
The coordinates of the reflection point M are obtained by Eqs. (11), (12) and (13), and the propagation distance ${L_{pq}}$ is calculated, so that the diffraction distribution of the inner cylindrical surface is obtained by the PS method. However, it is worth noting that there is no analytical solution to the equation for solving the reflection point M. Therefore, a set of nonlinear equations must be solved for each pair of source and destination points to obtain the numerical solution of ${L_{pq}}$. In addition, it takes a long time to obtain the diffraction distribution by the PS method.
For computational time considerations, we propose an approximate method to quickly calculate this diffraction process. When ${\theta _p} = {\theta _q}$, point $\tilde{Q}(a - {z_q},{\theta _q},a - {r_q})$ on the bottom surface of cone is the equivalent point of the point Q, namely $P\tilde{Q} = PQ$. Combined with Eq. (9), it can be known that the diffraction distributions on the two surfaces are the same. Therefore, the bottom surface can be regarded as the equivalent surface of the inner cylindrical surface, and it can be called as refocusing plane. Hence, the plane-to-cylinder diffraction can be replaced by the planar diffraction. Although $P\tilde{Q} = PQ$ no longer holds when ${\theta _p} \ne {\theta _q}$, when the actual diffraction distance ${L_{pq}}$ and the approximated diffraction distance ${L_{p\tilde{q}}}$ satisfy the following formula, it can still be approximated by the planar diffraction formula.
This is because the PS method are used to calculate the diffraction distribution, the variation of ${L_{pq}}$ in Eq. (9) is much less than that in $exp(jk{L_{pq}})$ since the wavenumber k is quite a great value. Therefore, the approximation is reasonable as long as the difference between the approximated phase and the actual phase is small. The feasibility of the approximate approach will be discussed in detail in subsection 3.1.2.2.2 Coordinate transformation
The diffraction distribution formed by the middle plane on the inner cylindrical surface can be approximately equivalent to the diffraction distribution on the bottom surface of the cone. In other words, bottom surface is the refocusing plane. The function of the conical mirror is to perform the coordinate transformation between the cylindrical coordinate system and the plane polar coordinate system [21,22]. The point $\tilde{Q}({r,\theta } )$ in Fig. 4(a) is equivalent to the point $Q({r^{\prime},L} )$ in Fig. 4(b), and their coordinate relationship meet the following equation
where a is the height of the cone, and $\theta $ is in the range of $[{0,2\pi } ]$.However, it is worth noting that commercial SLMs are in the shape of a rectangular grid, and there is no SLM with a center-radiation distribution like Fig. 4(a). Therefore, it is necessary to convert the polar coordinate system of the refocusing plane to a Cartesian coordinate system
This step of coordinate system transformation is mathematically accurate, but in practice there will be approximation errors. This is because the pixel sizes of the SLMs of Fig. 4(a) are different, while the pixel sizes of the actual SLMs are same. As a result, it is not difficult to imagine that the closer the pixel is to the center of the circle, the greater the error caused by the change.2.3 Diffraction from hologram plane to middle plane
As shown in Fig. 5, the diffraction process from the hologram plane to the middle plane is the diffraction between two planes. Based on well-known scalar diffraction theory, there are many formulas to calculate this process, such as the angular spectrum formula, Rayleigh–Sommerfeld formula and the Fresnel–Kirchhoff formula. The angular spectrum method will be used here
The area that is useful in the middle plane, in other words the area that is actually reflected to the inner cylindrical surface, is a circle of radius a. However, since the object plane is loaded onto the SLM of a rectangular shape, the diffraction area of the middle plane is also a rectangle. Therefore, zero-padding is performed on the unused area on the rectangular middle plane. The reason for the zero-padding is simply because the SLM is a rectangle and not a circle. Since the zero-filled area will not participate in the next diffraction process, this operation will not have effect on the subsequent diffraction.
2.4 Sampling conditions
In order to reasonably carry out numerical simulations and optical experiments, both plane-to-plane diffraction and cylinder-to-cylinder diffraction must satisfy the sampling theorem. The minimum number of samples required in the x and y directions in plane-to-plane diffraction is
where L is the length of planes. In the concentric cylindrical model, the Nyquist theorem must be satisfied in both the azimuthal and the vertical directions. Since the spatial frequency of ${u_s}({{r_s},{\theta_s}} )$ is much smaller than $h^{\prime}({\theta ,z} )$, only the maximum value of the spatial frequency of $h^{\prime}({\theta ,z} )$ needs to be considered. Obviously the variation in $({\textrm{cos}\alpha /j\lambda d} )$ is much less than that in $\textrm{exp}({j\lambda d} )$ since k is quite a huge number. Therefore, the spatial frequencies ${f_\theta }({\theta ,z} )$ and ${f_z}({\theta ,z} )$ in azimuthal and vertical directions respectively are3. Numerical simulations
3.1 Feasibility of planar diffraction approximation
In Section 2.2.1, it is mentioned that the diffraction from middle plane to inner cylindrical surface can be approximately regarded as the planar diffraction from middle plane to refocusing plane. Since the diffraction from hologram plane to middle plane is an exact planar diffraction process, the plane-to-cylinder diffraction from hologram plane to inner cylindrical surface can be approximately equivalent to plane-to-plane diffraction. The feasibility of this approximation and the optimal position of the inner cylindrical surface will be discussed below.
In the simulation, the resolutions of hologram plane and inner cylindrical surface are both 64×64, so a total of ${64^4} \approx 17M$ rays are simulated. The light wavelength $\lambda $ is selected as 671 nm, the cone height a is fixed at 5 mm, the height of the hologram plane is ${Z_{p^{\prime}}} = Z$, and the radius of inner cylindrical surface is ${R_q} = R$. As shown in Fig. 6, the abscissa is the error rate $k|{{L_{pq}} - {L_{p\tilde{q}}}} |/2\pi $ (The phase errors are unified into the range $[{0,1} ]$), and the ordinate represents the cumulative percentage of rays that do not exceed this error rate to all 17M rays. For example, the point (0.1, 82.27%) in the blue polyline means that 82.27% of the 17M simulated rays have an error rate of less than 0.1. The graph shows that when $Z = 300mm,R = 5mm$, the proportion of rays with a ratio less than 0.05 accounts for 72.58% and the proportion of rays with a ratio less than 0.2 reaches 90.80%, which indicates that most of the light rays satisfy the approximate condition of Eq. (14). In the other three experimental conditions, the proportion of rays satisfying this approximation formula decreases at the same error rate. Therefore, the optimal position of the inner cylindrical surface is at ${R_q} = 5mm$.
Fig. 6. Cumulative percentage of number of rays that do not exceed Error rate to total number of rays in different experimental conditions.
In order to further illustrate the feasibility of the approximation, an image with a resolution of 64×64 is selected on the inner cylindrical surface. The hologram iplaced at Z = 300/295/285/265mm, respectively. The inner cylindrical surface, reconstruction plane, is placed at different distances with the radius R = 5/10/20/40mm, respectively. The hologram is obtained by plane-to-plane diffraction method at a propagation distance of $l = Z - R + a = 300mm$. Then, the images are reconstructed by plane-to-cylinder diffraction method under four different experimental conditions, and the results are shown in Fig. 7(c),(d),(e),(f). The results are as expected, the higher the proportion of rays that satisfy the approximate formula, the higher the quality of the reconstructed image. Therefore, Fig. 7(c) has the highest quality, and the planar diffraction approximation works best when $Z = 300mm,R = 5mm$.
Fig. 7. (a) The object on Inner Cylindrical Surface. (b) is the Hologram Plane. (c),(d),(e),(f) are reconstructed results of object when Z = 300/295/285/265mm, R = 5/10/20/40mm.
3.2 Diffraction pattern
To verify the effectiveness of the proposed model, we conduct the following numerical simulation. The parameters are as follows: ${Z_{p^{\prime}}} = 300mm,{R_q} = 5mm,{R_{q^{\prime}}} = 10mm$., the resolution of the object surface is 3840×540(Aspect ratio 64:9), and the hologram plane is 1080×1080, the sampling interval of the SLM is 8 um×8 um, the wavelength of the laser is 671 nm. Due to the limitation of sampling theorem, $\beta $ is taken to be 0.020 rad, not 0.041 rad calculated by the grating equation. As shown in Fig. 8, letters (a)-1, vehicles (b)-1, and coast panoramas (c)-1 are used as test images on the Object Surface, respectively. After a cylinder-to-cylinder diffraction, the diffraction patterns on the inner cylindrical surface become (a)-2, (b)-2, (c)-2. It can be seen that the diffraction effect produced by the distance is weak, which is caused by the short diffraction distance and the small divergence angle. If the former is too large, images on the object surface will be stretched and deformed seriously in the lateral direction, and if the latter is too large, the sampling theorem will not be satisfied and the diffraction patterns will be aliased. After the reflection of the conical mirror, the optical axis of the diffraction field changes, and the coordinate system changes. Correspondingly, the diffraction patterns on the Refocusing Plane are (a)-3, (b)-3, (c)-3. The hologram(d) is obtained by final plane-to-plane diffraction.
Fig. 8. (a)-1,(b)-1,(c)-1 are the images on the Object Surface. (a)-2,(b)-2,(c)-2 are the diffraction patterns on the Inner Cylindrical Surface. (a)-3,(b)-3,(c)-3 are the diffraction patterns on the Refocusing Plane. (d) is a typical hologram on the Hologram Plane.
In the above results, the images are flat expanded view. To simulate the scene realistically, we present the results of the reconstructed images from four different angles, as shown in Fig. 9, which proves that the reconstructed images from the hologram is viewable from all perspectives after using the conical mirror.
4. Optical experiments
In this section, the effectiveness of the proposed model is verified by optical experiments. The schematic of optical reconstructed system is shown in Fig. 10. The wavelength of the He-Ne laser is 671 nm. An 8-bit reflective phase-only SLM is used with the sampling interval of 8 um×8 um and the resolution of 1920×1080. A 4-f filter system was used to eliminate multi-order diffraction images. The diffraction field will be imaged on the cylindrical Object Surface after being deflected by the conical mirror. All other parameters are the same as those selected in the simulation.
Fig. 10. Schematic of optical imaging system. CL, collimation lens; BS, beam splitter; L1-L2, 4f system; F, filter; phase pattern Φ, loaded on SLM.
The experimental results are shown in Fig. 11 and Fig. 12. In Fig. 11, the diffraction patterns on the object surface are first imaged on a white screen, and then photographed with the camera(NIKON D810). The pictures in Fig. 12 are captured by CMOS(SONY IMX 305). The imaging patterns can be viewed from all angles of 360°. The figures show the diffraction patterns from four angles of 90°, 180°, 270°, 360°. In addition to the expected images on the white screen can be seen clearly in Fig. 11, the diffraction patterns propagating to the conical mirror can also be seen. For example, the second picture in the first row is viewed at 180°. The letters “EN” are displayed on the white screen, and the distorted patterns “FA” are displayed on the conical mirror, while letters “FA” are exactly observed at 360°, which indicates “EN” and “FA” are opposite. The size of the images in Fig. 12 is scaled down in accordance with the actual images. Since the radius of the inner cylindrical surface is 5 mm and the radius of the object surface is 10 mm, the length of the latter in the lateral direction is twice that of the former, making images appear flatter. The optical experimental results agree well with the simulation results and agree with the theoretical expectations. The diffraction patterns of the letters “CHENYIFA” and the four modes of transportation: bus, train, speedboat, and airplane, are well-recognized, but some specific details, such as the train's chimney, are a little blurred. This blurring is mainly caused by the approximation error produced by the transformation of the 2.B.2 coordinate system. This approximation error is not caused by theoretical derivation, but caused by the fact that there is no commercial SLM with a center-radiation distribution, and the SLM in the shape of a rectangular grid has to be used. It is conceivable that the details in the upper half of the object surface are more blurred, and the details in the lower half can be well represented. As for the 360° dynamic optical demonstration videos can be found in Visualization 1 and Visualization 2.
Fig. 12. Pictures captured by CMOS. Lines 1 and 2 are diffraction patterns on inner cylindrical surface. Lines 3,4,5,6 are images on object surface. Line 7 is diffraction pattern on refocusing plane.
5. Conclusion
In this paper, an optical realization of 360° cylindrical holography is proposed within visible light only using a planar SLM. In our proposed method, placing a 45° conical mirror in the optical path can realize isophase surface transformation, thereby achieving the purpose of 360° cylindrical holographic display. Furthermore, the diffraction calculations can be processed in several separated stages by assuming virtual surfaces Middle Plane and Inner Cylindrical Surface in the optical path. Although the existing cylindrical holography can realize a 360° surround view, cylindrical SLM is required, which is not available in commerce. Furthermore, realizing cylindrical holographic displays within visible light remains a challenge due to heavy burden of memory space and computing time. Meanwhile, although the existing curved holography can utilize a planar SLM to display in visible light, the viewing angle is still limited. Therefore, our proposed method has unique advantages compared to cylindrical and curved holography. In the future, our proposed method is expected to have practical applications in the field of cylindrical holography.
Funding
Joint Fund of Civil Aviation Research (U1933132); Chengdu Science and Technology Program (2019-GH02-00070-HZ); Sichuan Province Science and Technology Support Program (2019YFG0205).
Acknowledgments
Authors thank Dr. Xu GOU for his help with the experimental equipment and Dr. Dahai LI for his professional guidance, from the School of Electronics and Information Engineering, Sichuan University.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this Letter are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. J. Park, H. Kang, E. Stoykova, Y. Kim, S. Hong, Y. Choi, Y. Kim, S. Kwon, and S. Lee, “Numerical reconstruction of a full parallax holographic stereogram with radial distortion,” Opt. Express 22(17), 20776–20788 (2014). [CrossRef]
2. X. Zhang, G. Lv, Z. Wang, P. Dai, D. Li, M. Guo, H. Xiao, and Q. Feng, “Holographic display system with enhanced viewing angle using boundary folding mirrors,” Opt. Commun. 482, 126580 (2021). [CrossRef]
3. L. Xu, C. Chang, S. Feng, C. Yuan, and S. Nie, “Calculation of computer-generated hologram (CGH) from 3D object of arbitrary size and viewing angle,” Opt. Commun. 402, 211–215 (2017). [CrossRef]
4. A. Goncharsky and S. Durlevich, “Cylindrical computer-generated hologram for displaying 3D images,” Opt. Express 26(17), 22160–22167 (2018). [CrossRef]
5. Y. Sando, M. Itoh, and T. Yatagai, “Fast calculation method for cylindrical computer-generated holograms,” Opt. Express 13(5), 1418–1423 (2005). [CrossRef]
6. Y. Sando, D. Barada, B. J. Jackin, and T. Yatagai, “Bessel function expansion to reduce the calculation time and memory usage for cylindrical computer-generated holograms,” Appl. Opt. 56(20), 5775–5780 (2017). [CrossRef]
7. B. J. Jackin and T. Yatagai, “Fast calculation method for computer-generated cylindrical hologram based on wave propagation in spectral domain,” Opt. Express 18(25), 25546–25555 (2010). [CrossRef]
8. B. J. Jackin and T. Yatagai, “360° reconstruction of a 3D object using cylindrical computer generated holography,” Appl. Opt. 50(34), H147–H152 (2011). [CrossRef]
9. 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]
10. J. Wang, Q. H. Wang, and Y. Hu, “Fast diffraction calculation of cylindrical computer generated hologram based on outside-in propagation model,” Opt. Commun. 403, 296–303 (2017). [CrossRef]
11. J. Wang, Q. H. Wang, and Y. Hu, “Unified and accurate diffraction calculation between two concentric cylindrical surfaces,” J. Opt. Soc. Am. A 35(1), A45–A52 (2018). [CrossRef]
12. Y. Li, J. Wang, C. Chen, B. Li, R. Yang, and N. Chen, “Occlusion culling for computer-generated cylindrical holograms based on a horizontal optical-path-limit function,” Opt. Express 28(12), 18516–18528 (2020). [CrossRef]
13. F. M. Jin, Y. Wu, J. Wang, C. Chen, C.-J. Liu, and Y.-H. Hu, “Speckle suppression using cylindrical self-diffraction for cylindrical phase-only hologram,” IEEE Photonics J. 13(1), 1–11 (2021). [CrossRef]
14. 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]
15. 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]
16. D. Pi, J. Liu, and S. Yu, “Two-step acceleration calculation method to generate curved holograms using the intermediate plane in a three-dimensional holographic display,” Appl. Opt. 60(25), 7640–7647 (2021). [CrossRef]
17. D. Pi, J. Liu, R. Kang, Y. Han, and S. Yu, “Accelerating calculation method for curved computer-generated hologram using look-up table in holographic display,” Opt. Commun. 485, 126750 (2021). [CrossRef]
18. D. Wang, N. N. Li, Z. S. Li, C. Chen, B. Lee, and Q. H. Wang, “Color curved hologram calculation method based on angle multiplexing,” Opt. Express 30(2), 3157 (2022). [CrossRef]
19. H. K. Cao, S. F. Lin, and E. S. Kim, “Accelerated generation of holographic videos of 3-D objects in rotational motion using a curved hologram-based rotational-motion compensation method,” Opt. Express 26(16), 21279–21300 (2018). [CrossRef]
20. D. Blinder, T. Birnbaum, and P. Schelkens, “Pincushion point-spread function for computer-generated holography,” Opt. Lett. 47(8), 2077–2080 (2022). [CrossRef]
21. M. Leon Rodriguez, J. A. Rayas, A. Martinez Garcia, A. Martinez Gonzalez, A. Tellez Quinones, and R. Porras Aguilar, “Panoramic reconstruction of quasi-cylindrical objects with digital holography and a conical mirror,” Opt. Lett. 46(19), 4749–4752 (2021). [CrossRef]
22. A. Albertazzi and M. Viotti, “Full-field optical metrology in polar and cylindrical coordinates,” JPhys Photonics 3(4), 042001 (2021). [CrossRef]
