Abstract

Synthesizing computer-generated holograms (CGHs) of a general three-dimensional (3D) object is usually a heavy computational task. We propose and demonstrate a new algorithm for computing CGHs of 3D objects. In our scheme, many different angular projections of computer-designed 3D objects are numerically processed to yield a single two-dimensional complex matrix. This matrix is equivalent to the complex amplitude of a wave front on the rear focal plane of a spherical lens when the object is located near the front focal point and illuminated by a plane wave. Therefore the computed matrix can be used as a CGH after it is encoded to a real positive-valued transparency. When such CGH is illuminated by a plane wave, a 3D real image of the objects is constructed. The number of computer operations are equivalent to those of a two-dimensional Fourier CGH. Computer and optical constructions of 3D objects, both of which show the feasibility of the proposed approach, are described.

© 2003 Optical Society of America

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References

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  1. T. Inoue, H. Ohzu, “Accommodative responses to stereoscopic three-dimensional display,” Appl. Opt. 36, 4509–4515 (1997).
    [CrossRef] [PubMed]
  2. K. Langhans, D. Bezecny, D. Homann, C. Vogt, C. Blohm, K.-H. Scharschmidt, “New portable FELIX 3D display,” in Projection Displays, M. H. Wu, ed., Proc. SPIE3296, 204–216 (1998).
    [CrossRef]
  3. S.-W. Min, S. Jung, J.-H. Park, B. Lee, “Three-dimensional display system based on computer-generated integral photography,” in Stereoscopic Displays and Virtual Reality Systems VIII, A. J. Woods, M. T. Bolas, J. O. Merritt, S. A. Benton, eds., Proc. SPIE4297, 187–195 (2001).
    [CrossRef]
  4. S. A. Benton, “Holographic displays: 1975-1980,” Opt. Eng. 19, 686–696 (1980).
    [CrossRef]
  5. E. N. Leith, J. Upatnieks, “Wavefront reconstructionwith diffused illumination and three-dimensional objects,” J. Opt. Soc. Am. 54, 1295–1301 (1964).
    [CrossRef]
  6. R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography, 2nd ed. (Academic, New York, 1971), Chap. 10, pp. 266–289.
  7. A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms generated by computer,” Appl. Opt. 6, 1739–1748 (1967).
    [CrossRef] [PubMed]
  8. O. Bryngdahl, F. Wyrowski, “Digital holography-computer-generated holograms,” in Progress in Optics, Vol. XXVIII, E. Wolf, ed. (North-Holland, Amsterdam, 1990), pp. 1–86.
  9. P. Hariharan, Optical Holography, 2nd ed. (Cambridge, New York, 1996), Chap. 10, pp. 163–177.
  10. J. P. Waters, “Holographic image synthesis utilizing theoretical methods,” Appl. Phys. Lett. 9, 405–407 (1966).
    [CrossRef]
  11. Y. Ichioka, M. Izumi, T. Suzuki, “Scanning halftone plotter and computer-generated continuous tone hologram,” Appl. Opt. 10, 403–411 (1971).
    [CrossRef] [PubMed]
  12. T. Yatagai, “Stereoscopic approach to 3-D display using computer-generated holograms,” Appl. Opt. 15, 2722–2729 (1976).
    [CrossRef] [PubMed]
  13. N. George, J. T. McCrickerd, “Holography and stereoscopy: the holographic stereogram,” in Modulation Transfer Function, R. R. Shannon, R. J. Wollensack, eds., Proc. SPIE13, 342–350 (1969).
  14. L. P. Yaroslavskii, N. S. Merzlyakov, Methods of Digital Holography (Consultants Bureau, Plenum, New York, 1980).
  15. C. Frère, D. Leseberg, O. Bryngdahl, “Computer-generated holograms of 3-D objects composed of line segments,” J. Opt. Soc. Am. A 3, 726–730 (1986).
    [CrossRef]
  16. D. Leseberg, C. Frère, “Computer-generated holograms of 3-D objects composed of tilted planar segments,” Appl. Opt. 27, 3020–3024 (1988).
    [CrossRef] [PubMed]
  17. T. Tommasi, B. Bianco, “Computer-generated holograms of tilted planes by a spatial-frequency approach,” J. Opt. Soc. Am. A 10, 299–305 (1993).
    [CrossRef]
  18. M. Lucente, “Diffraction Specific Fringe Computation for Electro-Holography,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1994).
  19. C. D. Cameron, D. A. Pain, M. Stanley, C. W. Slinger, “Computational challenges of emerging novel true 3D holographic displays,” in Critical Technologies for the Future of Computing, L. J. Irakliotis, ed., Proc. SPIE4109, 129–140 (2000).
    [CrossRef]
  20. Y. Li, D. Abookasis, J. Rosen, “Computer-generated holograms of three-dimensional realistic objects recorded without wave interference,” Appl. Opt. 40, 2864–2870 (2001).
    [CrossRef]
  21. J.-N. Gillet, Y. Sheng, “Multiplexed computer-generated holograms with irregular-shaped polygonal apertures and discrete phase levels,” J. Opt. Soc. Am. A 19, 2403–2413 (2002).
    [CrossRef]
  22. A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
  23. S. Savchenko, 3D Graphics Programming: Games and Beyond (SAMS Publishing, Indianapolis, Ind., 2000), Chap. 2, pp. 46–53.
  24. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 5, p. 104.
  25. J. J. Burch, “A computer algorithm for the synthesis of spatial frequency filter,” Proc. IEEE 55, 599–600 (1967).
    [CrossRef]
  26. F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A 7, 961–969 (1990).
    [CrossRef]
  27. R. Piestun, J. Shamir, B. Webkamp, O. Bryngdahl, “On-axis computer-generated holograms for three-dimensional display,” Opt. Lett. 22, 922–924 (1997).
    [CrossRef] [PubMed]
  28. M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987).
    [CrossRef] [PubMed]
  29. Y. Li, J. Rosen, “Object recognition using three-dimensional optical quasi-correlation,” J. Opt. Soc. Am. A 19, 1755–1762 (2002).
    [CrossRef]

2002 (2)

2001 (1)

1997 (2)

1993 (1)

1990 (1)

1988 (1)

1987 (1)

1986 (1)

1980 (1)

S. A. Benton, “Holographic displays: 1975-1980,” Opt. Eng. 19, 686–696 (1980).
[CrossRef]

1976 (1)

1971 (1)

1967 (2)

A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms generated by computer,” Appl. Opt. 6, 1739–1748 (1967).
[CrossRef] [PubMed]

J. J. Burch, “A computer algorithm for the synthesis of spatial frequency filter,” Proc. IEEE 55, 599–600 (1967).
[CrossRef]

1966 (1)

J. P. Waters, “Holographic image synthesis utilizing theoretical methods,” Appl. Phys. Lett. 9, 405–407 (1966).
[CrossRef]

1964 (2)

E. N. Leith, J. Upatnieks, “Wavefront reconstructionwith diffused illumination and three-dimensional objects,” J. Opt. Soc. Am. 54, 1295–1301 (1964).
[CrossRef]

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

Abookasis, D.

Allebach, J. P.

Benton, S. A.

S. A. Benton, “Holographic displays: 1975-1980,” Opt. Eng. 19, 686–696 (1980).
[CrossRef]

Bezecny, D.

K. Langhans, D. Bezecny, D. Homann, C. Vogt, C. Blohm, K.-H. Scharschmidt, “New portable FELIX 3D display,” in Projection Displays, M. H. Wu, ed., Proc. SPIE3296, 204–216 (1998).
[CrossRef]

Bianco, B.

Blohm, C.

K. Langhans, D. Bezecny, D. Homann, C. Vogt, C. Blohm, K.-H. Scharschmidt, “New portable FELIX 3D display,” in Projection Displays, M. H. Wu, ed., Proc. SPIE3296, 204–216 (1998).
[CrossRef]

Bryngdahl, O.

Burch, J. J.

J. J. Burch, “A computer algorithm for the synthesis of spatial frequency filter,” Proc. IEEE 55, 599–600 (1967).
[CrossRef]

Burckhardt, C. B.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography, 2nd ed. (Academic, New York, 1971), Chap. 10, pp. 266–289.

Cameron, C. D.

C. D. Cameron, D. A. Pain, M. Stanley, C. W. Slinger, “Computational challenges of emerging novel true 3D holographic displays,” in Critical Technologies for the Future of Computing, L. J. Irakliotis, ed., Proc. SPIE4109, 129–140 (2000).
[CrossRef]

Collier, R. J.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography, 2nd ed. (Academic, New York, 1971), Chap. 10, pp. 266–289.

Frère, C.

George, N.

N. George, J. T. McCrickerd, “Holography and stereoscopy: the holographic stereogram,” in Modulation Transfer Function, R. R. Shannon, R. J. Wollensack, eds., Proc. SPIE13, 342–350 (1969).

Gillet, J.-N.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 5, p. 104.

Hariharan, P.

P. Hariharan, Optical Holography, 2nd ed. (Cambridge, New York, 1996), Chap. 10, pp. 163–177.

Homann, D.

K. Langhans, D. Bezecny, D. Homann, C. Vogt, C. Blohm, K.-H. Scharschmidt, “New portable FELIX 3D display,” in Projection Displays, M. H. Wu, ed., Proc. SPIE3296, 204–216 (1998).
[CrossRef]

Ichioka, Y.

Inoue, T.

Izumi, M.

Jung, S.

S.-W. Min, S. Jung, J.-H. Park, B. Lee, “Three-dimensional display system based on computer-generated integral photography,” in Stereoscopic Displays and Virtual Reality Systems VIII, A. J. Woods, M. T. Bolas, J. O. Merritt, S. A. Benton, eds., Proc. SPIE4297, 187–195 (2001).
[CrossRef]

Langhans, K.

K. Langhans, D. Bezecny, D. Homann, C. Vogt, C. Blohm, K.-H. Scharschmidt, “New portable FELIX 3D display,” in Projection Displays, M. H. Wu, ed., Proc. SPIE3296, 204–216 (1998).
[CrossRef]

Lee, B.

S.-W. Min, S. Jung, J.-H. Park, B. Lee, “Three-dimensional display system based on computer-generated integral photography,” in Stereoscopic Displays and Virtual Reality Systems VIII, A. J. Woods, M. T. Bolas, J. O. Merritt, S. A. Benton, eds., Proc. SPIE4297, 187–195 (2001).
[CrossRef]

Leith, E. N.

Leseberg, D.

Li, Y.

Lin, L. H.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography, 2nd ed. (Academic, New York, 1971), Chap. 10, pp. 266–289.

Lohmann, A. W.

Lucente, M.

M. Lucente, “Diffraction Specific Fringe Computation for Electro-Holography,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1994).

McCrickerd, J. T.

N. George, J. T. McCrickerd, “Holography and stereoscopy: the holographic stereogram,” in Modulation Transfer Function, R. R. Shannon, R. J. Wollensack, eds., Proc. SPIE13, 342–350 (1969).

Merzlyakov, N. S.

L. P. Yaroslavskii, N. S. Merzlyakov, Methods of Digital Holography (Consultants Bureau, Plenum, New York, 1980).

Min, S.-W.

S.-W. Min, S. Jung, J.-H. Park, B. Lee, “Three-dimensional display system based on computer-generated integral photography,” in Stereoscopic Displays and Virtual Reality Systems VIII, A. J. Woods, M. T. Bolas, J. O. Merritt, S. A. Benton, eds., Proc. SPIE4297, 187–195 (2001).
[CrossRef]

Ohzu, H.

Pain, D. A.

C. D. Cameron, D. A. Pain, M. Stanley, C. W. Slinger, “Computational challenges of emerging novel true 3D holographic displays,” in Critical Technologies for the Future of Computing, L. J. Irakliotis, ed., Proc. SPIE4109, 129–140 (2000).
[CrossRef]

Paris, D. P.

Park, J.-H.

S.-W. Min, S. Jung, J.-H. Park, B. Lee, “Three-dimensional display system based on computer-generated integral photography,” in Stereoscopic Displays and Virtual Reality Systems VIII, A. J. Woods, M. T. Bolas, J. O. Merritt, S. A. Benton, eds., Proc. SPIE4297, 187–195 (2001).
[CrossRef]

Piestun, R.

Rosen, J.

Savchenko, S.

S. Savchenko, 3D Graphics Programming: Games and Beyond (SAMS Publishing, Indianapolis, Ind., 2000), Chap. 2, pp. 46–53.

Scharschmidt, K.-H.

K. Langhans, D. Bezecny, D. Homann, C. Vogt, C. Blohm, K.-H. Scharschmidt, “New portable FELIX 3D display,” in Projection Displays, M. H. Wu, ed., Proc. SPIE3296, 204–216 (1998).
[CrossRef]

Seldowitz, M. A.

Shamir, J.

Sheng, Y.

Slinger, C. W.

C. D. Cameron, D. A. Pain, M. Stanley, C. W. Slinger, “Computational challenges of emerging novel true 3D holographic displays,” in Critical Technologies for the Future of Computing, L. J. Irakliotis, ed., Proc. SPIE4109, 129–140 (2000).
[CrossRef]

Stanley, M.

C. D. Cameron, D. A. Pain, M. Stanley, C. W. Slinger, “Computational challenges of emerging novel true 3D holographic displays,” in Critical Technologies for the Future of Computing, L. J. Irakliotis, ed., Proc. SPIE4109, 129–140 (2000).
[CrossRef]

Suzuki, T.

Sweeney, D. W.

Tommasi, T.

Upatnieks, J.

VanderLugt, A.

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

Vogt, C.

K. Langhans, D. Bezecny, D. Homann, C. Vogt, C. Blohm, K.-H. Scharschmidt, “New portable FELIX 3D display,” in Projection Displays, M. H. Wu, ed., Proc. SPIE3296, 204–216 (1998).
[CrossRef]

Waters, J. P.

J. P. Waters, “Holographic image synthesis utilizing theoretical methods,” Appl. Phys. Lett. 9, 405–407 (1966).
[CrossRef]

Webkamp, B.

Wyrowski, F.

F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A 7, 961–969 (1990).
[CrossRef]

O. Bryngdahl, F. Wyrowski, “Digital holography-computer-generated holograms,” in Progress in Optics, Vol. XXVIII, E. Wolf, ed. (North-Holland, Amsterdam, 1990), pp. 1–86.

Yaroslavskii, L. P.

L. P. Yaroslavskii, N. S. Merzlyakov, Methods of Digital Holography (Consultants Bureau, Plenum, New York, 1980).

Yatagai, T.

Appl. Opt. (7)

Appl. Phys. Lett. (1)

J. P. Waters, “Holographic image synthesis utilizing theoretical methods,” Appl. Phys. Lett. 9, 405–407 (1966).
[CrossRef]

IEEE Trans. Inf. Theory (1)

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Opt. Eng. (1)

S. A. Benton, “Holographic displays: 1975-1980,” Opt. Eng. 19, 686–696 (1980).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (1)

J. J. Burch, “A computer algorithm for the synthesis of spatial frequency filter,” Proc. IEEE 55, 599–600 (1967).
[CrossRef]

Other (11)

S. Savchenko, 3D Graphics Programming: Games and Beyond (SAMS Publishing, Indianapolis, Ind., 2000), Chap. 2, pp. 46–53.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 5, p. 104.

O. Bryngdahl, F. Wyrowski, “Digital holography-computer-generated holograms,” in Progress in Optics, Vol. XXVIII, E. Wolf, ed. (North-Holland, Amsterdam, 1990), pp. 1–86.

P. Hariharan, Optical Holography, 2nd ed. (Cambridge, New York, 1996), Chap. 10, pp. 163–177.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography, 2nd ed. (Academic, New York, 1971), Chap. 10, pp. 266–289.

K. Langhans, D. Bezecny, D. Homann, C. Vogt, C. Blohm, K.-H. Scharschmidt, “New portable FELIX 3D display,” in Projection Displays, M. H. Wu, ed., Proc. SPIE3296, 204–216 (1998).
[CrossRef]

S.-W. Min, S. Jung, J.-H. Park, B. Lee, “Three-dimensional display system based on computer-generated integral photography,” in Stereoscopic Displays and Virtual Reality Systems VIII, A. J. Woods, M. T. Bolas, J. O. Merritt, S. A. Benton, eds., Proc. SPIE4297, 187–195 (2001).
[CrossRef]

M. Lucente, “Diffraction Specific Fringe Computation for Electro-Holography,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1994).

C. D. Cameron, D. A. Pain, M. Stanley, C. W. Slinger, “Computational challenges of emerging novel true 3D holographic displays,” in Critical Technologies for the Future of Computing, L. J. Irakliotis, ed., Proc. SPIE4109, 129–140 (2000).
[CrossRef]

N. George, J. T. McCrickerd, “Holography and stereoscopy: the holographic stereogram,” in Modulation Transfer Function, R. R. Shannon, R. J. Wollensack, eds., Proc. SPIE13, 342–350 (1969).

L. P. Yaroslavskii, N. S. Merzlyakov, Methods of Digital Holography (Consultants Bureau, Plenum, New York, 1980).

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Figures (8)

Fig. 1
Fig. 1

Schematic of the computational process of the CGH and of the image construction from the CGH. fm,n=exp[-j2πb(xp sin ϕm+yp sin θn)].

Fig. 2
Fig. 2

Equivalent optical system for the CGH computation.

Fig. 3
Fig. 3

Nine projections of the 3D objects projected from various viewpoints.

Fig. 4
Fig. 4

Enlarged portion (100×100 pixels of 201×201) of (a) the magnitude and (b) the phase angle of the noncoded CGH generated by the algorithm shown in Fig. 1.

Fig. 5
Fig. 5

Simulation results from the hologram shown in Fig. 4 at the vicinity of the back focal plane of lens L for three successive transverse planes along the z axis.

Fig. 6
Fig. 6

Central part (300×300 pixels of 640×640) of the CGH computed by Eq. (13) from the complex function shown in Fig. 4.

Fig. 7
Fig. 7

Experimental results of the first diffraction order obtained from the CGH shown in Fig. 6 at the vicinity of the back focal plane of L for three transverse planes at z1=715 mm, z2=750 mm, and z3=781 mm.

Fig. 8
Fig. 8

Construction results from the gray-scale CGH of Fig. 6, including the three central diffraction orders.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

xp=xscos ϕm-zssin ϕm,
yp=yscos θn-zssin θncos ϕm-xssin ϕmsin θn.
s(m, n)=pmn(xp, yp)exp[-j2πb(xpsin ϕm+ypsin θn)]dxpdyp,
s(m, n)=t(xs, ys, zs)(ΔxsΔysΔzs)δ(x¯p-xp, y¯p-yp)exp[-j2πb(x¯psin ϕm+y¯psin θn)]dx¯pdy¯p=t(xs, ys, zs)exp[-j2πb(xpsin ϕm+ypsin θn)](ΔxsΔysΔzs),
s(m, n)=t(xs, ys, zs)exp{-j2πb[sin ϕm(xscos ϕm-zssin ϕm)+sin θn(yscos θn-zssin θncos ϕm-xssin ϕmsin θn)]}ΔxsΔysΔzs.
s(m, n)=s(m, n)dxsdysdzs=t(xs, ys, zs)×exp{-j2πb[sin ϕm(xscos ϕm-zssin ϕm)+sin θn(yscos θn-zssin θncos ϕm-xssin θnsin ϕm)]}dxsdysdzs.
s(m, n)t(xs, ys, zs)exp{-j2πb[xssin ϕm+yssin θn-zs(sin2 ϕm+sin2 θn)-xssin2 θnsin ϕm]}dxsdysdzs.
s(u, v)=t(xs, ys, zs)exp-j4πb sin ϕmaxΔuuxs+vys-zs2 sin ϕmaxΔu (u2+v2)dxsdysdzs.
g(u, v)=Ct(xs, ys, zs)exp-j 2πλfuxs+vys-zsu2+v22 fdxsdysdzs,
Mxx¯ox¯s=2λfb sin ϕmaxΔu,
Myy¯oy¯s=2λfbsin θmaxΔv=Mx,
Mzz¯oz¯s=8λbf sin ϕmaxΔu2,
MzMx=4f sin ϕmaxΔu=MzMy.
b sin ϕmaxBx,b sin θmaxBy,
b(sin2 ϕmax+sin2 θmax)Bz,
(Δϕ, Δθ)(Δ sin ϕ, Δ sin θ)1bWx, 1bWy,
T(u, v)=121+Res(u, v)expj 2πλf (dxu+dyv),
T(u, v)=12|s(u, v)|+Res(u, v)expj 2πλf (dxu+dyv).

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