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OPTICS LETTERS / Vol. 38, No. 23 / December 1, 2013
Fast calculation method for computer-generated cylindrical holograms based on the three-dimensional Fourier spectrum Yusuke Sando,1,2,3,* Daisuke Barada,1,2 Boaz Jessie Jackin,1 and Toyohiko Yatagai1 1
Center for Optical Research & Education, Utsunomiya University, 7-1-2 Yoto, Utsunomiya, Tochigi 321-8585, Japan 2
Graduate School of Engineering, Utsunomiya University, 7-1-2 Yoto, Utsunomiya, Tochigi 321-8585, Japan 3
Technology Research Institute of Osaka Prefecture, 2-7-1 Ayumino, Izumi, Osaka 594-1157, Japan *Corresponding author: [email protected]‑u.ac.jp Received September 23, 2013; revised October 22, 2013; accepted October 30, 2013; posted October 31, 2013 (Doc. ID 198206); published November 27, 2013
The relation between a three-dimensional (3D) object and its diffracted wavefront in the 3D Fourier space is discussed at first and then a rigorous diffraction formula onto cylindrical surfaces is derived. The azimuthal direction and the spatial frequency direction corresponding to height can be expressed with a one-dimensional (1D) convolution integral and a 1D inverse Fourier transform in the 3D Fourier space, respectively, and fast Fourier transforms are available for fast calculation. A numerical simulation of a diffracted wavefront on cylindrical surfaces is presented. An alternative optical experiment equivalent of the optical reconstruction from cylindrical holograms is also demonstrated. © 2013 Optical Society of America OCIS codes: (090.1760) Computer holography; (090.2870) Holographic display; (090.5694) Real-time holography. http://dx.doi.org/10.1364/OL.38.005172
Computer-generated holograms (CGHs), which are holograms synthesized in a computer by simulating the interference between a reference wave and an object wave, have considerable potential as ideal three-dimensional (3D) displays because they are fully compatible with physiological perception factors and can be used to reconstruct virtual objects [1] and moving 3D objects with a spatial light modulator (SLM) [2]. In a CGH, the narrow viewing angle is one of the key issues to be resolved. Methods incorporating a stereoscopic approach [3] and utilizing high-order diffraction [4] have been shown to enlarge the viewing angle. The observational area is, however, limited because the shape of common CGHs is planar and the reconstructed images cannot be observed from the opposite side of the CGHs. Thus, a cylindrical hologram could be the most effective shape for a 3D display since it offers a viewing angle of 360°, which enables full motion and binocular parallax [5,6]. However, even planar CGHs require substantial calculation time. In a cylindrical CGH having an area that is much larger than that of a planar CGH, much longer calculation time is inevitable, making the reduction of calculation time extremely important for synthesizing a cylindrical CGH [7]. The fast Fourier transform (FFT) is the most common technique for fast calculation and its effect is drastic. Algorithms using FFTs, in which the diffraction integral for a cylindrical surface is calculated through a convolution method [8] and through a spectral propagation method [9], have been proposed by defining a 3D object in the cylindrical coordinate system. Kashiwagi and Sakamoto proposed a new method based on the angular spectrum of a plane wave; in this method, a 3D object is defined in the Cartesian coordinate system [10]. However, although these methods using FFTs are very advantageous in terms of the calculation time, they suffer from a serious drawback in that hidden surface cannot be removed, and their removal is essential for realistic reconstruction. 0146-9592/13/235172-04$15.00/0
We have proposed a fast calculation method using FFTs for diffracted wavefronts traveling in arbitrary directions on the basis of the 3D Fourier spectrum [11]. Moreover, hidden surface removal can be incorporated in this method by adding a geometrical selection of light rays [12]. However, CGHs synthesized by using these methods are limited to a planar shape and it is impossible to synthesize a cylindrical CGH. In this Letter, we propose a fast calculation algorithm for cylindrical CGHs by applying the above 3D Fourier spectrum approaches. As shown in Fig. 1, a 3D object and its diffracted wavefront propagating in the z direction on the z R plane are defined as ox; y; z and f 0 x0 ; y in the respective coordinate system. By defining Ou; v; w as the 3D Fourier spectrum of ox; y; z and λ as a wavelength of light, the diffracted wavefront f 0 x0 ; y is given by [11] p Ou; v; 1∕λ2 − u2 − v2 p 1∕λ2 − u2 − v2 q × exp i2πR 1∕λ2 − u2 − v2
i f 0 x0 ; y 4π
ZZ
× expi2πux0 vydudv: y
x R x
y L0
R
z 3-D object o(x, y, z)
z=R
y L
(1)
f (x , y) f (0, y)=f c ( , y) x0 f 0 (x 0 , y)
f 0 (0, y) =f c (0, y)
Fig. 1. Schematic of the virtual optical system for observation of diffracted wavefronts. © 2013 Optical Society of America
December 1, 2013 / Vol. 38, No. 23 / OPTICS LETTERS
p Ou; v; 1∕λ2 − u2 − v2 in Eq. (1) represents the spectrum on the hemisphere with radius 1∕λ in 3D Fourier space and can be treated with two independent variables. Here, this hemispherical spectrum O is handled with variables θ; v defined as follows: r 1 sin θ u λ2 − v2 v v;
(2)
where θ is the azimuthal angle from the w axis shown in Fig. 2. The integration variables of Eq. (1) are converted into θ; v from u; v according to Eq. (2). Moreover, although the diffracted wavefront on the z R plane is considered in Eq. (1), we shall focus simply on the lineal wavefront f 0 0; y on the line L0 at x0 0 shown in Fig. 1 from here on. By using this treatment, the wavefront f 0 0; y on the line L0 is given by i f 0 0; y 4π
ZZ
π∕2 −π∕2
q Os θ; v exp i2πR cos θ 1∕λ2 − v2
× expi2πvydθdv:
(3)
Os θ; v is another representation of the hemispherical p spectrum Ou; v; 1∕λ2 − u2 − v2 in the coordinate system θ; v. Next, a rect function is introduced to extend the integral range of θ from −π∕2 ≤ θ < π∕2 to −π ≤ θ < π as follows: f 0 0; y
q Os θ; v exp i2πR cos θ 1∕λ2 − v2 −π θ expi2πvydθdv: (4) × rect π i 4π
ZZ
π
The range of the definition of Os θ; v is also extended from the hemisphere to the sphere by using a rect function. As described above, Eq. (4) gives the wavefront propagating in the z direction on the line L0 . Here, as shown in Fig. 1, the wavefront f ϕ xϕ ; y propagating in the direction rotated by ϕ around the y axis is considered and its lineal wavefront f ϕ 0; y on the line Lϕ at xϕ 0 is redefined as f c ϕ; y. The distribution f c ϕ; y, which is to be obtained in this Letter, is the diffracted wavefront on the cylindrical observation surface of radius R. According to [11], when the direction of propagation is rotated by ϕ around the y axis, the position of the required hemispherical spectrum is also v 1/ v
O(u,v, 1/ -u2-v2 ) =Os( , v) u w Spectrum on the hemispherical surface
Fig. 2. Coordinate system θ; v for representing the spectrum on the spherical surface.
5173
rotated by ϕ around the v axis in 3D Fourier space. Thus, by generalizing the extractive position of the hemispherical spectrum with respect to the azimuthal angle in Eq. (4), the wavefront f c ϕ; y can be expressed by i f c ϕ; y 4π
ZZ
π
Os θ; v q × exp i2πR cosθ − ϕ 1∕λ2 − v2 −π
× rect
θ−ϕ expi2πvydθdv: π
(5)
Then, the following spectral kernel function is introduced: q θ 2 2 hθ; v exp i2πR cos θ 1∕λ − v rect : π
(6)
By using the function hθ; v, Eq. (5) is simplified as follows: i f c ϕ; y 4π
Z
Os ϕ; vϕ hϕ; v expi2πvydv;
(7)
where the symbol ϕ indicates the one-dimensional (1D) convolution integral with respect to the ϕ direction. In Eq. (7), the calculations with respect to the azimuthal direction ϕ and the spatial frequency direction v can be regarded as the 1D convolution integral and 1D inverse Fourier transform, respectively. Therefore, the fast calculation of Eq. (7) becomes possible by using FFTs in both directions. In addition, it is advantageous that hidden surface removal can be easily incorporated in Eq. (7) if the spectrum Os ϕ; v on the spherical surface is obtained with the geometrical selection according to [12]. However, in the calculation method of [10], in which a formula similar to Eq. (7) is used, hidden surfaces cannot be removed because this method basically calculates the diffraction from a planar surface to a cylindrical surface. Numerical simulations have been demonstrated to verify the calculating formula of Eq. (7). The 3D object used for the simulation is a sphere, on which the world map is spherically mapped, and the positional relation between the object and a cylindrical observation surface is also indicated in Fig. 3. The size of the 3D computational area is 10.15 mm × 10.15 mm × 10.15 mm, and the sampling number is 256 × 256 × 256. A globe of radius D 3.5 mm is placed inside this area. The number of sampling points of the observation surface is 131072 × 256. The wavelength λ is set to 632 nm. Since the center of the cylindrical observation surface coincides with the origin, its radius R is considered as the diffraction distance. Figures 4(a)–4(c) show the diffracted wavefronts at different diffraction distances calculated by using our proposal. In Fig. 4(a), the vertically central area is defocused and the top and bottom areas are in focus locally because the diffraction distance is smaller than the radius of the globe. In contrast, the vertically central area is in focus extensively in Fig. 4(b). This is because the diffraction distance is the same as the radius. In Fig. 4(c), the diffraction distance is larger than the
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y R Object o
x
Observation Plane x
R
z
y
z (a)
(b)
Fig. 3. Schematic of the virtual optical system for calculation of wavefronts on the cylindrical surface: (a) perspective view and (b) top view.
radius and no area is in focus. These results are consistent with the geometrical configuration of Fig. 3, and verify that the calculating formula of Eq. (7) is correct. Moreover, it can be seen that the defocus effect in the ϕ direction is stronger than that in the y direction. This is because the spectral bandwidth of the height is strictly limited by the sampling number of the object, which is 256 in this simulation as described above. Figures 4(a)– 4(c) are calculated from the spherical spectrum allowing for hidden surface removal according to [12]. A result without hidden surface removal is also shown in Fig. 4(d) just for reference. It is found that the hidden surface removal functions effectively in Figs. 4(a)–4(c) compared with Fig. 4(d). The calculation time is about 60 s. If the total calculation time is proportional to the product of the sampling numbers of the object and the observation surface, our method would be approximately 100 times faster than the method proposed in [7], in which FFTs are not used. The other reported methods [8–10] use FFTs and have speeds comparable to the proposed method. However, the other methods suffer from a serious drawback in that hidden surface cannot be removed. The above simulations are for the diffracted wavefront on the cylindrical surface, not the reconstruction from cylindrical CGHs. Here, optical reconstruction is
in focus
in focus
demonstrated through an optical experiment with CGHs synthesized using our proposal. However, because cylindrical CGHs are difficult to fabricate, we perform an alternative optical experiment that is optically equivalent to the reconstruction from cylindrical CGHs under the limited conditions. First, a segment cut out of a cylindrical CGH is considered, as shown in Fig. 5(a). When a point light source is placed at the origin as a reference light, both the reference wave and the object wave propagate radially. Thus, when the radius R of the observation surface is much larger than the size of the segmented CGH and the 3D object, the phase difference between the reference wave and the object wave stays mostly unchanged within the small region indicated in Fig. 5(a). Generally, because interference patterns depend strongly on the phase difference between two such waves, the CGH pattern remains unchanged within the small limited region where the phase difference does not change. This means that the CGH pattern designed for a cylindrical surface is the same as that designed for a planar or even arbitrary surface within the region. So, under these limited conditions, if a segment of a cylindrical CGH is radially projected onto the tangent plane and displayed on an SLM as the planar CGH shown in Fig. 5(b), the same images as the original cylindrical CGH would be reconstructed. This is the alternative optical experiment demonstrated in this Letter. Although the parameters for calculating a cylindrical CGH are the same as those for the above simulation, the 3D object used for this experiment is the north hemisphere of the globe to avoid overlap with the conjugate image. The diffraction distance R is also changed to 206 mm. A segmented CGH with 1280 × 256 pixels is cut out of the cylindrical CGH synthesized under these conditions. As an example, Fig. 6 shows a segmented CGH. Since the object is the north hemisphere, its diffracted wavefront mainly propagates in the upper half space and hence the bottom of Fig. 6 looks dark. For calculating the spherical spectrum, the spectral periodicity of the discretized object is utilized with respect to the azimuthal angle according to [11]. Thus, the results of
Cylindrical CGH Object o
y
Small region Segmenting direction Cutout x z
in focus (a)
Radial projection
(b) (a) Planar SLM
(c)
(d)
Fig. 4. Simulation results. The horizontal and vertical axes represent the azimuth angle ϕ and the height y, respectively. (a), (b), and (c) are calculated under the conditions of R 0.8D, 1.0D, and 1.2D, respectively. (d) is calculated without hidden surface removal for R 1.0D.
Digital camera
Point light source R=206 mm (b) Fig. 5. Schematics of (a) a cutout of a segmented CGH and the radial projection onto the tangent plane and (b) the optical reconstruction system from the segmented CGH.
December 1, 2013 / Vol. 38, No. 23 / OPTICS LETTERS
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Fig. 6. Segment of the cylindrical CGH.
its convolution also shows the quasi-local periodicity and the fringe pattern is repeated horizontally, which is about 4 times as shown in Fig. 6. The SLM used for this experiment is a transmissive liquid crystal display removed from a commercially available projector. The pixel number is 1280 × 800, the SLM is not uniform. To adjust to these specifications, the segmented CGH is interpolated in the y direction so that the pixel number becomes 800 and is then binarized. This CGH is displayed on the transmissive planar SLM. The optically reconstructed images are photographed with a digital camera, as shown in Fig. 5(b), and photographs are shown in Fig. 7. As can be seen from Fig. 7, the first-order diffracted image and its conjugate image are located on the top and bottom, respectively, and they change properly in accordance with the observation direction, namely, the segmenting direction of the cylindrical CGH. Therefore, these results have verified the validity of our proposal and we conclude that a 3D display with a viewing angle of 360° is realized if a CGH synthesized according to Eq. (7) is displayed cylindrically. In conclusion, we have proposed a fast calculation method for cylindrical CGHs on the basis of 3D Fourier spectrum. In this method, the calculation formula with respect to the azimuthal angle ϕ and the spatial frequency v is expressed with a convolution integral and an inverse Fourier transform, respectively. This enables fast calculation by using FFTs. It is very easy to incorporate a hidden surface removal process into our proposal. Both numerical simulations and an alternative optical experiment have been demonstrated to verify that. Our method is very effective for calculating large cylindrical CGHs.
Fig. 7. Optically reconstructed images captured from the different segmented CGHs corresponding to ϕ (a) 0°, (b) 45°, (c) 135°, and (d) −90° , respectively.
References 1. A. W. Lohmann and D. P. Paris, Appl. Opt. 6, 1739 (1967). 2. N. Hashimoto and S. Morokawa, J. Electron. Imaging 2, 93 (1993). 3. T. Yatagai, Appl. Opt. 15, 2722 (1976). 4. T. Mishina, M. Okui, and F. Okano, Appl. Opt. 41, 1489 (2002). 5. T. H. Geong, J. Opt. Soc. Am. 57, 1396 (1967). 6. O. D. D. Soares and J. C. A. Fernandes, Appl. Opt. 21, 3194 (1982). 7. T. Yamaguchi, T. Fujii, and H. Yoshikawa, Appl. Opt. 47, D63 (2008). 8. Y. Sando, M. Itoh, and T. Yatagai, Opt. Express 13, 1418 (2005). 9. B. J. Jackin and T. Yatagai, Opt. Express 18, 25546 (2010). 10. A. Kashiwagi and Y. Sakamoto, Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2007), paper DWB7. 11. Y. Sando, D. Barada, and T. Yatagai, Opt. Express 20, 20962 (2012). 12. Y. Sando, D. Barada, and T. Yatagai, Appl. Opt. 52, 4871 (2013).
OPTICS LETTERS / Vol. 38, No. 23 / December 1, 2013
Fast calculation method for computer-generated cylindrical holograms based on the three-dimensional Fourier spectrum Yusuke Sando,1,2,3,* Daisuke Barada,1,2 Boaz Jessie Jackin,1 and Toyohiko Yatagai1 1
Center for Optical Research & Education, Utsunomiya University, 7-1-2 Yoto, Utsunomiya, Tochigi 321-8585, Japan 2
Graduate School of Engineering, Utsunomiya University, 7-1-2 Yoto, Utsunomiya, Tochigi 321-8585, Japan 3
Technology Research Institute of Osaka Prefecture, 2-7-1 Ayumino, Izumi, Osaka 594-1157, Japan *Corresponding author: [email protected]‑u.ac.jp Received September 23, 2013; revised October 22, 2013; accepted October 30, 2013; posted October 31, 2013 (Doc. ID 198206); published November 27, 2013
The relation between a three-dimensional (3D) object and its diffracted wavefront in the 3D Fourier space is discussed at first and then a rigorous diffraction formula onto cylindrical surfaces is derived. The azimuthal direction and the spatial frequency direction corresponding to height can be expressed with a one-dimensional (1D) convolution integral and a 1D inverse Fourier transform in the 3D Fourier space, respectively, and fast Fourier transforms are available for fast calculation. A numerical simulation of a diffracted wavefront on cylindrical surfaces is presented. An alternative optical experiment equivalent of the optical reconstruction from cylindrical holograms is also demonstrated. © 2013 Optical Society of America OCIS codes: (090.1760) Computer holography; (090.2870) Holographic display; (090.5694) Real-time holography. http://dx.doi.org/10.1364/OL.38.005172
Computer-generated holograms (CGHs), which are holograms synthesized in a computer by simulating the interference between a reference wave and an object wave, have considerable potential as ideal three-dimensional (3D) displays because they are fully compatible with physiological perception factors and can be used to reconstruct virtual objects [1] and moving 3D objects with a spatial light modulator (SLM) [2]. In a CGH, the narrow viewing angle is one of the key issues to be resolved. Methods incorporating a stereoscopic approach [3] and utilizing high-order diffraction [4] have been shown to enlarge the viewing angle. The observational area is, however, limited because the shape of common CGHs is planar and the reconstructed images cannot be observed from the opposite side of the CGHs. Thus, a cylindrical hologram could be the most effective shape for a 3D display since it offers a viewing angle of 360°, which enables full motion and binocular parallax [5,6]. However, even planar CGHs require substantial calculation time. In a cylindrical CGH having an area that is much larger than that of a planar CGH, much longer calculation time is inevitable, making the reduction of calculation time extremely important for synthesizing a cylindrical CGH [7]. The fast Fourier transform (FFT) is the most common technique for fast calculation and its effect is drastic. Algorithms using FFTs, in which the diffraction integral for a cylindrical surface is calculated through a convolution method [8] and through a spectral propagation method [9], have been proposed by defining a 3D object in the cylindrical coordinate system. Kashiwagi and Sakamoto proposed a new method based on the angular spectrum of a plane wave; in this method, a 3D object is defined in the Cartesian coordinate system [10]. However, although these methods using FFTs are very advantageous in terms of the calculation time, they suffer from a serious drawback in that hidden surface cannot be removed, and their removal is essential for realistic reconstruction. 0146-9592/13/235172-04$15.00/0
We have proposed a fast calculation method using FFTs for diffracted wavefronts traveling in arbitrary directions on the basis of the 3D Fourier spectrum [11]. Moreover, hidden surface removal can be incorporated in this method by adding a geometrical selection of light rays [12]. However, CGHs synthesized by using these methods are limited to a planar shape and it is impossible to synthesize a cylindrical CGH. In this Letter, we propose a fast calculation algorithm for cylindrical CGHs by applying the above 3D Fourier spectrum approaches. As shown in Fig. 1, a 3D object and its diffracted wavefront propagating in the z direction on the z R plane are defined as ox; y; z and f 0 x0 ; y in the respective coordinate system. By defining Ou; v; w as the 3D Fourier spectrum of ox; y; z and λ as a wavelength of light, the diffracted wavefront f 0 x0 ; y is given by [11] p Ou; v; 1∕λ2 − u2 − v2 p 1∕λ2 − u2 − v2 q × exp i2πR 1∕λ2 − u2 − v2
i f 0 x0 ; y 4π
ZZ
× expi2πux0 vydudv: y
x R x
y L0
R
z 3-D object o(x, y, z)
z=R
y L
(1)
f (x , y) f (0, y)=f c ( , y) x0 f 0 (x 0 , y)
f 0 (0, y) =f c (0, y)
Fig. 1. Schematic of the virtual optical system for observation of diffracted wavefronts. © 2013 Optical Society of America
December 1, 2013 / Vol. 38, No. 23 / OPTICS LETTERS
p Ou; v; 1∕λ2 − u2 − v2 in Eq. (1) represents the spectrum on the hemisphere with radius 1∕λ in 3D Fourier space and can be treated with two independent variables. Here, this hemispherical spectrum O is handled with variables θ; v defined as follows: r 1 sin θ u λ2 − v2 v v;
(2)
where θ is the azimuthal angle from the w axis shown in Fig. 2. The integration variables of Eq. (1) are converted into θ; v from u; v according to Eq. (2). Moreover, although the diffracted wavefront on the z R plane is considered in Eq. (1), we shall focus simply on the lineal wavefront f 0 0; y on the line L0 at x0 0 shown in Fig. 1 from here on. By using this treatment, the wavefront f 0 0; y on the line L0 is given by i f 0 0; y 4π
ZZ
π∕2 −π∕2
q Os θ; v exp i2πR cos θ 1∕λ2 − v2
× expi2πvydθdv:
(3)
Os θ; v is another representation of the hemispherical p spectrum Ou; v; 1∕λ2 − u2 − v2 in the coordinate system θ; v. Next, a rect function is introduced to extend the integral range of θ from −π∕2 ≤ θ < π∕2 to −π ≤ θ < π as follows: f 0 0; y
q Os θ; v exp i2πR cos θ 1∕λ2 − v2 −π θ expi2πvydθdv: (4) × rect π i 4π
ZZ
π
The range of the definition of Os θ; v is also extended from the hemisphere to the sphere by using a rect function. As described above, Eq. (4) gives the wavefront propagating in the z direction on the line L0 . Here, as shown in Fig. 1, the wavefront f ϕ xϕ ; y propagating in the direction rotated by ϕ around the y axis is considered and its lineal wavefront f ϕ 0; y on the line Lϕ at xϕ 0 is redefined as f c ϕ; y. The distribution f c ϕ; y, which is to be obtained in this Letter, is the diffracted wavefront on the cylindrical observation surface of radius R. According to [11], when the direction of propagation is rotated by ϕ around the y axis, the position of the required hemispherical spectrum is also v 1/ v
O(u,v, 1/ -u2-v2 ) =Os( , v) u w Spectrum on the hemispherical surface
Fig. 2. Coordinate system θ; v for representing the spectrum on the spherical surface.
5173
rotated by ϕ around the v axis in 3D Fourier space. Thus, by generalizing the extractive position of the hemispherical spectrum with respect to the azimuthal angle in Eq. (4), the wavefront f c ϕ; y can be expressed by i f c ϕ; y 4π
ZZ
π
Os θ; v q × exp i2πR cosθ − ϕ 1∕λ2 − v2 −π
× rect
θ−ϕ expi2πvydθdv: π
(5)
Then, the following spectral kernel function is introduced: q θ 2 2 hθ; v exp i2πR cos θ 1∕λ − v rect : π
(6)
By using the function hθ; v, Eq. (5) is simplified as follows: i f c ϕ; y 4π
Z
Os ϕ; vϕ hϕ; v expi2πvydv;
(7)
where the symbol ϕ indicates the one-dimensional (1D) convolution integral with respect to the ϕ direction. In Eq. (7), the calculations with respect to the azimuthal direction ϕ and the spatial frequency direction v can be regarded as the 1D convolution integral and 1D inverse Fourier transform, respectively. Therefore, the fast calculation of Eq. (7) becomes possible by using FFTs in both directions. In addition, it is advantageous that hidden surface removal can be easily incorporated in Eq. (7) if the spectrum Os ϕ; v on the spherical surface is obtained with the geometrical selection according to [12]. However, in the calculation method of [10], in which a formula similar to Eq. (7) is used, hidden surfaces cannot be removed because this method basically calculates the diffraction from a planar surface to a cylindrical surface. Numerical simulations have been demonstrated to verify the calculating formula of Eq. (7). The 3D object used for the simulation is a sphere, on which the world map is spherically mapped, and the positional relation between the object and a cylindrical observation surface is also indicated in Fig. 3. The size of the 3D computational area is 10.15 mm × 10.15 mm × 10.15 mm, and the sampling number is 256 × 256 × 256. A globe of radius D 3.5 mm is placed inside this area. The number of sampling points of the observation surface is 131072 × 256. The wavelength λ is set to 632 nm. Since the center of the cylindrical observation surface coincides with the origin, its radius R is considered as the diffraction distance. Figures 4(a)–4(c) show the diffracted wavefronts at different diffraction distances calculated by using our proposal. In Fig. 4(a), the vertically central area is defocused and the top and bottom areas are in focus locally because the diffraction distance is smaller than the radius of the globe. In contrast, the vertically central area is in focus extensively in Fig. 4(b). This is because the diffraction distance is the same as the radius. In Fig. 4(c), the diffraction distance is larger than the
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y R Object o
x
Observation Plane x
R
z
y
z (a)
(b)
Fig. 3. Schematic of the virtual optical system for calculation of wavefronts on the cylindrical surface: (a) perspective view and (b) top view.
radius and no area is in focus. These results are consistent with the geometrical configuration of Fig. 3, and verify that the calculating formula of Eq. (7) is correct. Moreover, it can be seen that the defocus effect in the ϕ direction is stronger than that in the y direction. This is because the spectral bandwidth of the height is strictly limited by the sampling number of the object, which is 256 in this simulation as described above. Figures 4(a)– 4(c) are calculated from the spherical spectrum allowing for hidden surface removal according to [12]. A result without hidden surface removal is also shown in Fig. 4(d) just for reference. It is found that the hidden surface removal functions effectively in Figs. 4(a)–4(c) compared with Fig. 4(d). The calculation time is about 60 s. If the total calculation time is proportional to the product of the sampling numbers of the object and the observation surface, our method would be approximately 100 times faster than the method proposed in [7], in which FFTs are not used. The other reported methods [8–10] use FFTs and have speeds comparable to the proposed method. However, the other methods suffer from a serious drawback in that hidden surface cannot be removed. The above simulations are for the diffracted wavefront on the cylindrical surface, not the reconstruction from cylindrical CGHs. Here, optical reconstruction is
in focus
in focus
demonstrated through an optical experiment with CGHs synthesized using our proposal. However, because cylindrical CGHs are difficult to fabricate, we perform an alternative optical experiment that is optically equivalent to the reconstruction from cylindrical CGHs under the limited conditions. First, a segment cut out of a cylindrical CGH is considered, as shown in Fig. 5(a). When a point light source is placed at the origin as a reference light, both the reference wave and the object wave propagate radially. Thus, when the radius R of the observation surface is much larger than the size of the segmented CGH and the 3D object, the phase difference between the reference wave and the object wave stays mostly unchanged within the small region indicated in Fig. 5(a). Generally, because interference patterns depend strongly on the phase difference between two such waves, the CGH pattern remains unchanged within the small limited region where the phase difference does not change. This means that the CGH pattern designed for a cylindrical surface is the same as that designed for a planar or even arbitrary surface within the region. So, under these limited conditions, if a segment of a cylindrical CGH is radially projected onto the tangent plane and displayed on an SLM as the planar CGH shown in Fig. 5(b), the same images as the original cylindrical CGH would be reconstructed. This is the alternative optical experiment demonstrated in this Letter. Although the parameters for calculating a cylindrical CGH are the same as those for the above simulation, the 3D object used for this experiment is the north hemisphere of the globe to avoid overlap with the conjugate image. The diffraction distance R is also changed to 206 mm. A segmented CGH with 1280 × 256 pixels is cut out of the cylindrical CGH synthesized under these conditions. As an example, Fig. 6 shows a segmented CGH. Since the object is the north hemisphere, its diffracted wavefront mainly propagates in the upper half space and hence the bottom of Fig. 6 looks dark. For calculating the spherical spectrum, the spectral periodicity of the discretized object is utilized with respect to the azimuthal angle according to [11]. Thus, the results of
Cylindrical CGH Object o
y
Small region Segmenting direction Cutout x z
in focus (a)
Radial projection
(b) (a) Planar SLM
(c)
(d)
Fig. 4. Simulation results. The horizontal and vertical axes represent the azimuth angle ϕ and the height y, respectively. (a), (b), and (c) are calculated under the conditions of R 0.8D, 1.0D, and 1.2D, respectively. (d) is calculated without hidden surface removal for R 1.0D.
Digital camera
Point light source R=206 mm (b) Fig. 5. Schematics of (a) a cutout of a segmented CGH and the radial projection onto the tangent plane and (b) the optical reconstruction system from the segmented CGH.
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5175
Fig. 6. Segment of the cylindrical CGH.
its convolution also shows the quasi-local periodicity and the fringe pattern is repeated horizontally, which is about 4 times as shown in Fig. 6. The SLM used for this experiment is a transmissive liquid crystal display removed from a commercially available projector. The pixel number is 1280 × 800, the SLM is not uniform. To adjust to these specifications, the segmented CGH is interpolated in the y direction so that the pixel number becomes 800 and is then binarized. This CGH is displayed on the transmissive planar SLM. The optically reconstructed images are photographed with a digital camera, as shown in Fig. 5(b), and photographs are shown in Fig. 7. As can be seen from Fig. 7, the first-order diffracted image and its conjugate image are located on the top and bottom, respectively, and they change properly in accordance with the observation direction, namely, the segmenting direction of the cylindrical CGH. Therefore, these results have verified the validity of our proposal and we conclude that a 3D display with a viewing angle of 360° is realized if a CGH synthesized according to Eq. (7) is displayed cylindrically. In conclusion, we have proposed a fast calculation method for cylindrical CGHs on the basis of 3D Fourier spectrum. In this method, the calculation formula with respect to the azimuthal angle ϕ and the spatial frequency v is expressed with a convolution integral and an inverse Fourier transform, respectively. This enables fast calculation by using FFTs. It is very easy to incorporate a hidden surface removal process into our proposal. Both numerical simulations and an alternative optical experiment have been demonstrated to verify that. Our method is very effective for calculating large cylindrical CGHs.
Fig. 7. Optically reconstructed images captured from the different segmented CGHs corresponding to ϕ (a) 0°, (b) 45°, (c) 135°, and (d) −90° , respectively.
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