Abstract

A volumetric display system using an optical imaging device consisting of numerous dihedral corner reflectors placed perpendicular to the surface of a metal plate is proposed. Image formation by the dihedral corner reflector array (DCRA) is free from distortion and focal length. In the proposed volumetric display system, a two-dimensional real image is moved by a mirror scanner to scan a three-dimensional (3D) space. Cross-sectional images of a 3D object are displayed in accordance with the position of the image plane. A volumetric image is observed as a stack of the cross-sectional images. The use of the DCRA brings compact system configuration and volumetric real image generation with very low distortion. An experimental volumetric display system including a DCRA, a galvanometer mirror, and a digital micro-mirror device was constructed to verify the proposed method. A volumetric image consisting of 1024×768×400 voxels was formed by the experimental system.

© 2012 Optical Society of America

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References

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  1. B. G. Blundell and A. J. Schwarz, Volumetric Three-Dimensional Display Systems (Wiley, 2000).
  2. J.-Y. Son, B. Javidi, and K.-D. Kwack, “Methods for displaying three-dimensional images,” Proc. IEEE 94, 502–523 (2006).
    [CrossRef]
  3. G. E. Favalora, J. Napoli, D. M. Hall, R. K. Dorval, M. G. Giovinco, M. J. Richmond, and W. S. Chun, “100 million-voxel volumetric display,” Proc. SPIE 4712, 300–312 (2002).
    [CrossRef]
  4. P. Soltan, M. Lasher, W. Dahlke, N. Acantilado, and M. McDonald, “Laser-projected 3-D volumetric displays,” Proc. SPIE 3057, 496–506 (1997).
    [CrossRef]
  5. S. Suyama, M. Date, and H. Takada, “Three-dimensional display system with dual-frequency liquid-crystal varifocal lens,” Jpn. J. Appl. Phys. 39, 480–484 (2000).
    [CrossRef]
  6. D. Miyazaki and K. Matsushita, “Volume scanning three-dimensional display that uses an inclined image plane,” Appl. Opt. 40, 3354–3358 (2001).
    [CrossRef]
  7. D. Miyazaki, K. Shiba, K. Sotsuka, and K. Matsushita, “Volumetric display system based on three-dimensional scanning of inclined optical image,” Opt. Express 14, 12760–12769 (2006).
    [CrossRef]
  8. D. Miyazaki, T. Honda, K. Ohno, and T. Mukai, “Three-dimensional user interface using a haptic device for volumetric display,” in Information Photonics 2008 (2008), pp. 198–199.
  9. D. Miyazaki, T. Honda, K. Ohno, and T. Mukai, “Volumetric display system using a digital micromirror device based on inclined-plane scanning,” J. Disp. Technol. 6, 548–552(2010).
    [CrossRef]
  10. D. Miyazaki, N. Hirano, Y. Maeda, K. Ohno, and S. Maekawa, “Volumetric display using a roof mirror grid array,” Proc. SPIE 7524, 75240N (2010).
    [CrossRef]
  11. D. Miyazaki, K. Ohno, and T. Mukai, “Real-time updatable volumetric display system based on inclined-image scanning,” in The Sixth International Conference on Intelligent Information Hiding and Multimedia Signal Processing (IIH-MSP) (2010), pp. 684–687.
  12. S. Maekawa, K. Nitta, and O. Matoba, “Transmissive optical imaging device with micromirror array,” Proc. SPIE 6392, 63920E (2006).
    [CrossRef]
  13. Y. Matsukura, K. Nitta, O. Matoba, and S. Maekawa, “Numerical analysis for an imaging optics with micromirror array,” in Information Photonics 2008 Technical Digest (Optical Society of Japan, 2008), pp. 206–207.
  14. S. Yokoyama, K. Nitta, O. Matoba, and S. Maekawa, “Imaging characteristics of array of dihedral corner reflectors by use of Gaussian beam decomposition,” in International Display Workshops ’10 (Institute of Image Information and Television Engineers/Society for Information Display2010), pp. 1249–1250.
  15. S. Maekawa, K. Nitta, and O. Matoba, “Advances in passive imaging elements with micromirror array,” Proc. SPIE 6803, 68030B (2008).
    [CrossRef]
  16. D. Dudley, W. M. Duncan, and J. Slaughter, “Emerging digital micro-mirror device (DMD) applications,” Proc. SPIE 498, 14–25 (2003).
    [CrossRef]
  17. I. E. McDowall, “Multi-use light engine-fast projection,” in IEEE Conference on CVPR2007 (IEEE, 2007), p. 1.

2010 (2)

D. Miyazaki, T. Honda, K. Ohno, and T. Mukai, “Volumetric display system using a digital micromirror device based on inclined-plane scanning,” J. Disp. Technol. 6, 548–552(2010).
[CrossRef]

D. Miyazaki, N. Hirano, Y. Maeda, K. Ohno, and S. Maekawa, “Volumetric display using a roof mirror grid array,” Proc. SPIE 7524, 75240N (2010).
[CrossRef]

2008 (1)

S. Maekawa, K. Nitta, and O. Matoba, “Advances in passive imaging elements with micromirror array,” Proc. SPIE 6803, 68030B (2008).
[CrossRef]

2006 (3)

D. Miyazaki, K. Shiba, K. Sotsuka, and K. Matsushita, “Volumetric display system based on three-dimensional scanning of inclined optical image,” Opt. Express 14, 12760–12769 (2006).
[CrossRef]

S. Maekawa, K. Nitta, and O. Matoba, “Transmissive optical imaging device with micromirror array,” Proc. SPIE 6392, 63920E (2006).
[CrossRef]

J.-Y. Son, B. Javidi, and K.-D. Kwack, “Methods for displaying three-dimensional images,” Proc. IEEE 94, 502–523 (2006).
[CrossRef]

2003 (1)

D. Dudley, W. M. Duncan, and J. Slaughter, “Emerging digital micro-mirror device (DMD) applications,” Proc. SPIE 498, 14–25 (2003).
[CrossRef]

2002 (1)

G. E. Favalora, J. Napoli, D. M. Hall, R. K. Dorval, M. G. Giovinco, M. J. Richmond, and W. S. Chun, “100 million-voxel volumetric display,” Proc. SPIE 4712, 300–312 (2002).
[CrossRef]

2001 (1)

2000 (1)

S. Suyama, M. Date, and H. Takada, “Three-dimensional display system with dual-frequency liquid-crystal varifocal lens,” Jpn. J. Appl. Phys. 39, 480–484 (2000).
[CrossRef]

1997 (1)

P. Soltan, M. Lasher, W. Dahlke, N. Acantilado, and M. McDonald, “Laser-projected 3-D volumetric displays,” Proc. SPIE 3057, 496–506 (1997).
[CrossRef]

Acantilado, N.

P. Soltan, M. Lasher, W. Dahlke, N. Acantilado, and M. McDonald, “Laser-projected 3-D volumetric displays,” Proc. SPIE 3057, 496–506 (1997).
[CrossRef]

Blundell, B. G.

B. G. Blundell and A. J. Schwarz, Volumetric Three-Dimensional Display Systems (Wiley, 2000).

Chun, W. S.

G. E. Favalora, J. Napoli, D. M. Hall, R. K. Dorval, M. G. Giovinco, M. J. Richmond, and W. S. Chun, “100 million-voxel volumetric display,” Proc. SPIE 4712, 300–312 (2002).
[CrossRef]

Dahlke, W.

P. Soltan, M. Lasher, W. Dahlke, N. Acantilado, and M. McDonald, “Laser-projected 3-D volumetric displays,” Proc. SPIE 3057, 496–506 (1997).
[CrossRef]

Date, M.

S. Suyama, M. Date, and H. Takada, “Three-dimensional display system with dual-frequency liquid-crystal varifocal lens,” Jpn. J. Appl. Phys. 39, 480–484 (2000).
[CrossRef]

Dorval, R. K.

G. E. Favalora, J. Napoli, D. M. Hall, R. K. Dorval, M. G. Giovinco, M. J. Richmond, and W. S. Chun, “100 million-voxel volumetric display,” Proc. SPIE 4712, 300–312 (2002).
[CrossRef]

Dudley, D.

D. Dudley, W. M. Duncan, and J. Slaughter, “Emerging digital micro-mirror device (DMD) applications,” Proc. SPIE 498, 14–25 (2003).
[CrossRef]

Duncan, W. M.

D. Dudley, W. M. Duncan, and J. Slaughter, “Emerging digital micro-mirror device (DMD) applications,” Proc. SPIE 498, 14–25 (2003).
[CrossRef]

Favalora, G. E.

G. E. Favalora, J. Napoli, D. M. Hall, R. K. Dorval, M. G. Giovinco, M. J. Richmond, and W. S. Chun, “100 million-voxel volumetric display,” Proc. SPIE 4712, 300–312 (2002).
[CrossRef]

Giovinco, M. G.

G. E. Favalora, J. Napoli, D. M. Hall, R. K. Dorval, M. G. Giovinco, M. J. Richmond, and W. S. Chun, “100 million-voxel volumetric display,” Proc. SPIE 4712, 300–312 (2002).
[CrossRef]

Hall, D. M.

G. E. Favalora, J. Napoli, D. M. Hall, R. K. Dorval, M. G. Giovinco, M. J. Richmond, and W. S. Chun, “100 million-voxel volumetric display,” Proc. SPIE 4712, 300–312 (2002).
[CrossRef]

Hirano, N.

D. Miyazaki, N. Hirano, Y. Maeda, K. Ohno, and S. Maekawa, “Volumetric display using a roof mirror grid array,” Proc. SPIE 7524, 75240N (2010).
[CrossRef]

Honda, T.

D. Miyazaki, T. Honda, K. Ohno, and T. Mukai, “Volumetric display system using a digital micromirror device based on inclined-plane scanning,” J. Disp. Technol. 6, 548–552(2010).
[CrossRef]

D. Miyazaki, T. Honda, K. Ohno, and T. Mukai, “Three-dimensional user interface using a haptic device for volumetric display,” in Information Photonics 2008 (2008), pp. 198–199.

Javidi, B.

J.-Y. Son, B. Javidi, and K.-D. Kwack, “Methods for displaying three-dimensional images,” Proc. IEEE 94, 502–523 (2006).
[CrossRef]

Kwack, K.-D.

J.-Y. Son, B. Javidi, and K.-D. Kwack, “Methods for displaying three-dimensional images,” Proc. IEEE 94, 502–523 (2006).
[CrossRef]

Lasher, M.

P. Soltan, M. Lasher, W. Dahlke, N. Acantilado, and M. McDonald, “Laser-projected 3-D volumetric displays,” Proc. SPIE 3057, 496–506 (1997).
[CrossRef]

Maeda, Y.

D. Miyazaki, N. Hirano, Y. Maeda, K. Ohno, and S. Maekawa, “Volumetric display using a roof mirror grid array,” Proc. SPIE 7524, 75240N (2010).
[CrossRef]

Maekawa, S.

D. Miyazaki, N. Hirano, Y. Maeda, K. Ohno, and S. Maekawa, “Volumetric display using a roof mirror grid array,” Proc. SPIE 7524, 75240N (2010).
[CrossRef]

S. Maekawa, K. Nitta, and O. Matoba, “Advances in passive imaging elements with micromirror array,” Proc. SPIE 6803, 68030B (2008).
[CrossRef]

S. Maekawa, K. Nitta, and O. Matoba, “Transmissive optical imaging device with micromirror array,” Proc. SPIE 6392, 63920E (2006).
[CrossRef]

Y. Matsukura, K. Nitta, O. Matoba, and S. Maekawa, “Numerical analysis for an imaging optics with micromirror array,” in Information Photonics 2008 Technical Digest (Optical Society of Japan, 2008), pp. 206–207.

S. Yokoyama, K. Nitta, O. Matoba, and S. Maekawa, “Imaging characteristics of array of dihedral corner reflectors by use of Gaussian beam decomposition,” in International Display Workshops ’10 (Institute of Image Information and Television Engineers/Society for Information Display2010), pp. 1249–1250.

Matoba, O.

S. Maekawa, K. Nitta, and O. Matoba, “Advances in passive imaging elements with micromirror array,” Proc. SPIE 6803, 68030B (2008).
[CrossRef]

S. Maekawa, K. Nitta, and O. Matoba, “Transmissive optical imaging device with micromirror array,” Proc. SPIE 6392, 63920E (2006).
[CrossRef]

Y. Matsukura, K. Nitta, O. Matoba, and S. Maekawa, “Numerical analysis for an imaging optics with micromirror array,” in Information Photonics 2008 Technical Digest (Optical Society of Japan, 2008), pp. 206–207.

S. Yokoyama, K. Nitta, O. Matoba, and S. Maekawa, “Imaging characteristics of array of dihedral corner reflectors by use of Gaussian beam decomposition,” in International Display Workshops ’10 (Institute of Image Information and Television Engineers/Society for Information Display2010), pp. 1249–1250.

Matsukura, Y.

Y. Matsukura, K. Nitta, O. Matoba, and S. Maekawa, “Numerical analysis for an imaging optics with micromirror array,” in Information Photonics 2008 Technical Digest (Optical Society of Japan, 2008), pp. 206–207.

Matsushita, K.

McDonald, M.

P. Soltan, M. Lasher, W. Dahlke, N. Acantilado, and M. McDonald, “Laser-projected 3-D volumetric displays,” Proc. SPIE 3057, 496–506 (1997).
[CrossRef]

McDowall, I. E.

I. E. McDowall, “Multi-use light engine-fast projection,” in IEEE Conference on CVPR2007 (IEEE, 2007), p. 1.

Miyazaki, D.

D. Miyazaki, T. Honda, K. Ohno, and T. Mukai, “Volumetric display system using a digital micromirror device based on inclined-plane scanning,” J. Disp. Technol. 6, 548–552(2010).
[CrossRef]

D. Miyazaki, N. Hirano, Y. Maeda, K. Ohno, and S. Maekawa, “Volumetric display using a roof mirror grid array,” Proc. SPIE 7524, 75240N (2010).
[CrossRef]

D. Miyazaki, K. Shiba, K. Sotsuka, and K. Matsushita, “Volumetric display system based on three-dimensional scanning of inclined optical image,” Opt. Express 14, 12760–12769 (2006).
[CrossRef]

D. Miyazaki and K. Matsushita, “Volume scanning three-dimensional display that uses an inclined image plane,” Appl. Opt. 40, 3354–3358 (2001).
[CrossRef]

D. Miyazaki, T. Honda, K. Ohno, and T. Mukai, “Three-dimensional user interface using a haptic device for volumetric display,” in Information Photonics 2008 (2008), pp. 198–199.

D. Miyazaki, K. Ohno, and T. Mukai, “Real-time updatable volumetric display system based on inclined-image scanning,” in The Sixth International Conference on Intelligent Information Hiding and Multimedia Signal Processing (IIH-MSP) (2010), pp. 684–687.

Mukai, T.

D. Miyazaki, T. Honda, K. Ohno, and T. Mukai, “Volumetric display system using a digital micromirror device based on inclined-plane scanning,” J. Disp. Technol. 6, 548–552(2010).
[CrossRef]

D. Miyazaki, T. Honda, K. Ohno, and T. Mukai, “Three-dimensional user interface using a haptic device for volumetric display,” in Information Photonics 2008 (2008), pp. 198–199.

D. Miyazaki, K. Ohno, and T. Mukai, “Real-time updatable volumetric display system based on inclined-image scanning,” in The Sixth International Conference on Intelligent Information Hiding and Multimedia Signal Processing (IIH-MSP) (2010), pp. 684–687.

Napoli, J.

G. E. Favalora, J. Napoli, D. M. Hall, R. K. Dorval, M. G. Giovinco, M. J. Richmond, and W. S. Chun, “100 million-voxel volumetric display,” Proc. SPIE 4712, 300–312 (2002).
[CrossRef]

Nitta, K.

S. Maekawa, K. Nitta, and O. Matoba, “Advances in passive imaging elements with micromirror array,” Proc. SPIE 6803, 68030B (2008).
[CrossRef]

S. Maekawa, K. Nitta, and O. Matoba, “Transmissive optical imaging device with micromirror array,” Proc. SPIE 6392, 63920E (2006).
[CrossRef]

Y. Matsukura, K. Nitta, O. Matoba, and S. Maekawa, “Numerical analysis for an imaging optics with micromirror array,” in Information Photonics 2008 Technical Digest (Optical Society of Japan, 2008), pp. 206–207.

S. Yokoyama, K. Nitta, O. Matoba, and S. Maekawa, “Imaging characteristics of array of dihedral corner reflectors by use of Gaussian beam decomposition,” in International Display Workshops ’10 (Institute of Image Information and Television Engineers/Society for Information Display2010), pp. 1249–1250.

Ohno, K.

D. Miyazaki, N. Hirano, Y. Maeda, K. Ohno, and S. Maekawa, “Volumetric display using a roof mirror grid array,” Proc. SPIE 7524, 75240N (2010).
[CrossRef]

D. Miyazaki, T. Honda, K. Ohno, and T. Mukai, “Volumetric display system using a digital micromirror device based on inclined-plane scanning,” J. Disp. Technol. 6, 548–552(2010).
[CrossRef]

D. Miyazaki, T. Honda, K. Ohno, and T. Mukai, “Three-dimensional user interface using a haptic device for volumetric display,” in Information Photonics 2008 (2008), pp. 198–199.

D. Miyazaki, K. Ohno, and T. Mukai, “Real-time updatable volumetric display system based on inclined-image scanning,” in The Sixth International Conference on Intelligent Information Hiding and Multimedia Signal Processing (IIH-MSP) (2010), pp. 684–687.

Richmond, M. J.

G. E. Favalora, J. Napoli, D. M. Hall, R. K. Dorval, M. G. Giovinco, M. J. Richmond, and W. S. Chun, “100 million-voxel volumetric display,” Proc. SPIE 4712, 300–312 (2002).
[CrossRef]

Schwarz, A. J.

B. G. Blundell and A. J. Schwarz, Volumetric Three-Dimensional Display Systems (Wiley, 2000).

Shiba, K.

Slaughter, J.

D. Dudley, W. M. Duncan, and J. Slaughter, “Emerging digital micro-mirror device (DMD) applications,” Proc. SPIE 498, 14–25 (2003).
[CrossRef]

Soltan, P.

P. Soltan, M. Lasher, W. Dahlke, N. Acantilado, and M. McDonald, “Laser-projected 3-D volumetric displays,” Proc. SPIE 3057, 496–506 (1997).
[CrossRef]

Son, J.-Y.

J.-Y. Son, B. Javidi, and K.-D. Kwack, “Methods for displaying three-dimensional images,” Proc. IEEE 94, 502–523 (2006).
[CrossRef]

Sotsuka, K.

Suyama, S.

S. Suyama, M. Date, and H. Takada, “Three-dimensional display system with dual-frequency liquid-crystal varifocal lens,” Jpn. J. Appl. Phys. 39, 480–484 (2000).
[CrossRef]

Takada, H.

S. Suyama, M. Date, and H. Takada, “Three-dimensional display system with dual-frequency liquid-crystal varifocal lens,” Jpn. J. Appl. Phys. 39, 480–484 (2000).
[CrossRef]

Yokoyama, S.

S. Yokoyama, K. Nitta, O. Matoba, and S. Maekawa, “Imaging characteristics of array of dihedral corner reflectors by use of Gaussian beam decomposition,” in International Display Workshops ’10 (Institute of Image Information and Television Engineers/Society for Information Display2010), pp. 1249–1250.

Appl. Opt. (1)

J. Disp. Technol. (1)

D. Miyazaki, T. Honda, K. Ohno, and T. Mukai, “Volumetric display system using a digital micromirror device based on inclined-plane scanning,” J. Disp. Technol. 6, 548–552(2010).
[CrossRef]

Jpn. J. Appl. Phys. (1)

S. Suyama, M. Date, and H. Takada, “Three-dimensional display system with dual-frequency liquid-crystal varifocal lens,” Jpn. J. Appl. Phys. 39, 480–484 (2000).
[CrossRef]

Opt. Express (1)

Proc. IEEE (1)

J.-Y. Son, B. Javidi, and K.-D. Kwack, “Methods for displaying three-dimensional images,” Proc. IEEE 94, 502–523 (2006).
[CrossRef]

Proc. SPIE (6)

G. E. Favalora, J. Napoli, D. M. Hall, R. K. Dorval, M. G. Giovinco, M. J. Richmond, and W. S. Chun, “100 million-voxel volumetric display,” Proc. SPIE 4712, 300–312 (2002).
[CrossRef]

P. Soltan, M. Lasher, W. Dahlke, N. Acantilado, and M. McDonald, “Laser-projected 3-D volumetric displays,” Proc. SPIE 3057, 496–506 (1997).
[CrossRef]

D. Miyazaki, N. Hirano, Y. Maeda, K. Ohno, and S. Maekawa, “Volumetric display using a roof mirror grid array,” Proc. SPIE 7524, 75240N (2010).
[CrossRef]

S. Maekawa, K. Nitta, and O. Matoba, “Transmissive optical imaging device with micromirror array,” Proc. SPIE 6392, 63920E (2006).
[CrossRef]

S. Maekawa, K. Nitta, and O. Matoba, “Advances in passive imaging elements with micromirror array,” Proc. SPIE 6803, 68030B (2008).
[CrossRef]

D. Dudley, W. M. Duncan, and J. Slaughter, “Emerging digital micro-mirror device (DMD) applications,” Proc. SPIE 498, 14–25 (2003).
[CrossRef]

Other (6)

I. E. McDowall, “Multi-use light engine-fast projection,” in IEEE Conference on CVPR2007 (IEEE, 2007), p. 1.

Y. Matsukura, K. Nitta, O. Matoba, and S. Maekawa, “Numerical analysis for an imaging optics with micromirror array,” in Information Photonics 2008 Technical Digest (Optical Society of Japan, 2008), pp. 206–207.

S. Yokoyama, K. Nitta, O. Matoba, and S. Maekawa, “Imaging characteristics of array of dihedral corner reflectors by use of Gaussian beam decomposition,” in International Display Workshops ’10 (Institute of Image Information and Television Engineers/Society for Information Display2010), pp. 1249–1250.

D. Miyazaki, K. Ohno, and T. Mukai, “Real-time updatable volumetric display system based on inclined-image scanning,” in The Sixth International Conference on Intelligent Information Hiding and Multimedia Signal Processing (IIH-MSP) (2010), pp. 684–687.

D. Miyazaki, T. Honda, K. Ohno, and T. Mukai, “Three-dimensional user interface using a haptic device for volumetric display,” in Information Photonics 2008 (2008), pp. 198–199.

B. G. Blundell and A. J. Schwarz, Volumetric Three-Dimensional Display Systems (Wiley, 2000).

Supplementary Material (1)

» Media 1: MOV (3192 KB)     

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Figures (13)

Fig. 1.
Fig. 1.

Schematic diagram of light rays passing through a DCRA: (a) top view, (b) side view, and (c) angular view.

Fig. 2.
Fig. 2.

Schematic diagram of incident light with optimum incident angle θop to maximize the quantity of light passing through a mirror hole with double reflection.

Fig. 3.
Fig. 3.

Schematic diagram of incident light with minimum incident angle for double reflection in a mirror hole on a DCRA. Incident light with an angle less than θmin cannot reflect twice on the adjacent walls in a mirror hole.

Fig. 4.
Fig. 4.

Light spread caused by diffraction effect at the aperture or a mirror hole on a DCRA.

Fig. 5.
Fig. 5.

Stripe images formed by a DCRA for several image distance and spatial frequencies: (a) 50 mm, 1.4lp/mm; (b) 80 mm, 1.4lp/mm; (c) 100 mm, 1.4lp/mm; and (d) 80 mm, 0.71lp/mm, respectively.

Fig. 6.
Fig. 6.

Plots of contrasts of stripe patterns formed at various distances from the DCRA with various spatial frequencies.

Fig. 7.
Fig. 7.

Volumetric display that uses an inclined 2D display and a mirror scanner.

Fig. 8.
Fig. 8.

Experimental results of volumetric image formation of a cube by a display system based on inclined-plane scanning using concave mirrors, reported in Miyazaki et al. [11]. The images were taken from different angles.

Fig. 9.
Fig. 9.

Experimental volumetric display system using a DCRA: (a) side view and (b) top view.

Fig. 10.
Fig. 10.

Experimental results of the volumetric display. A floating 3D image was captured from different viewing points.

Fig. 11.
Fig. 11.

Single-frame excerpts from video recordings of volumetric display system and a 3D image of a human skull (Media 1).

Fig. 12.
Fig. 12.

Pictures of a volumetric image of a wireframe cube taken from different viewing angles, whose difference was about 15 deg.

Fig. 13.
Fig. 13.

Light distribution on a diffusing plate placed at different depths around a 3D image consisting of two small disks formed by the experimental system. Diffusing plate was placed at the position of (a) upper disk and (b) lower disk, respectively.

Equations (3)

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tanθop=d2a.
tanθmin=d22a.
l=2sλa,

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