Abstract

A synthetic generation of three-dimensional image holograms is suggested. The complex amplitude in the hologram is calculated by using the spectrum of plane waves. Specifically, three-dimensional images are composed of planar and inclined two-dimensional distributions. In display applications, large hologram apertures can be achieved by introducing a remote exit pupil. A synthetic three-dimensional image is optically reconstructed.

© 1992 Optical Society of America

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References

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  1. R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).
  2. W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).
  3. J. B. DeVelis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley, Reading, Mass., 1967).
  4. H. J. Caulfield, S. Lu, The Applications of Holography (Wiley, New York, 1970).
  5. L. Rosen, “Focused-image holography with extended sources,” Appl. Phys. Lett. 9, 337–339 (1966).
    [CrossRef]
  6. A. W. Lohmann, D. P. Paris, “Binary image holograms,” J. Opt. Soc. Am 56, 537 (A) (1966).
  7. W.-H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. 16, pp. 119–232.
    [CrossRef]
  8. W.-J. Dallas, “Computer-generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed. (Springer-Verlag, Berlin, 1980), pp. 291–366.
  9. T. S. Huang, “Digital holography,” Proc. IEEE 59, 1335–1346 (1971).
    [CrossRef]
  10. L. P. Yaroslavskii, N. S. Merzlyakov, Methods of Digital Holography (Consultants Bureau, New York, 1980).
  11. O. Bryngdahl, F. Wyrowski, “Digital holography/computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1990), Vol. 28, pp. 1–86.
    [CrossRef]
  12. O. Bryngdahl, “Image-transfer characteristics of carrier-frequency photography,” J. Opt. Soc. Am 62, 807–813 (1972).
    [CrossRef]
  13. B. B. Baker, E. T. Copson, The Mathematical Theory of Huygens’ Principle (Clarendon, Oxford, 1969).
  14. P. C. Clemmow, The Plane Waυe Spectrum Representation of Electromagnetic Fields (Pergamon, Oxford, 1966).
  15. I. N. Sneddon, Fourier Transforms (McGraw-Hill, New York, 1951).
  16. K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).
  17. D. C. Champeney, Fourier Transforms and Their Physical Applications (Academic, London, 1973).
  18. P. P. Ewald, “Zur Theorie der Interferenzen der Röntgen-strahlen in Kristallen,” Phys. Z. 14, 465–472 (1913).
  19. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1983).
  20. D. Leseberg, C. Frère, “Computer-generated holograms of 3-D objects composed of tilted planar segments,” Appl. Opt. 27, 3020–3024 (1988).
    [CrossRef] [PubMed]
  21. C. Frère, D. Leseberg, “Large objects reconstructed from computer-generated holograms,” Appl. Opt. 28, 2422–2425 (1989).
    [CrossRef] [PubMed]
  22. J. W. Goodman, An Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  23. E. O. Brigham, The Fast Fourier Transformation (Prentice-Hall, Englewood Cliffs, N.J., 1974).
  24. A. W. Lohmann, A. S. Marathay, “About periodicities in 3-D wavefields,” Appl. Opt. 28, 4419–4423 (1989).
    [CrossRef] [PubMed]
  25. D. C. Champeney, A Handbook of Fourier Theorems (Cambridge U. Press, Cambridge, 1987).
    [CrossRef]
  26. S. A. Benton, “Hologram reconstruction with extended incoherent sources,” J. Opt. Soc. Am. 59, 1545(A) (1969).
  27. D. Leseberg, O. Bryngdahl, “Computer-generated rainbow holograms,” Appl. Opt. 23, 2441–2447 (1984).
    [CrossRef] [PubMed]

1989 (2)

1988 (1)

1984 (1)

1972 (1)

O. Bryngdahl, “Image-transfer characteristics of carrier-frequency photography,” J. Opt. Soc. Am 62, 807–813 (1972).
[CrossRef]

1971 (1)

T. S. Huang, “Digital holography,” Proc. IEEE 59, 1335–1346 (1971).
[CrossRef]

1969 (1)

S. A. Benton, “Hologram reconstruction with extended incoherent sources,” J. Opt. Soc. Am. 59, 1545(A) (1969).

1966 (2)

L. Rosen, “Focused-image holography with extended sources,” Appl. Phys. Lett. 9, 337–339 (1966).
[CrossRef]

A. W. Lohmann, D. P. Paris, “Binary image holograms,” J. Opt. Soc. Am 56, 537 (A) (1966).

1913 (1)

P. P. Ewald, “Zur Theorie der Interferenzen der Röntgen-strahlen in Kristallen,” Phys. Z. 14, 465–472 (1913).

Baker, B. B.

B. B. Baker, E. T. Copson, The Mathematical Theory of Huygens’ Principle (Clarendon, Oxford, 1969).

Benton, S. A.

S. A. Benton, “Hologram reconstruction with extended incoherent sources,” J. Opt. Soc. Am. 59, 1545(A) (1969).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1983).

Brigham, E. O.

E. O. Brigham, The Fast Fourier Transformation (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Bryngdahl, O.

D. Leseberg, O. Bryngdahl, “Computer-generated rainbow holograms,” Appl. Opt. 23, 2441–2447 (1984).
[CrossRef] [PubMed]

O. Bryngdahl, “Image-transfer characteristics of carrier-frequency photography,” J. Opt. Soc. Am 62, 807–813 (1972).
[CrossRef]

O. Bryngdahl, F. Wyrowski, “Digital holography/computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1990), Vol. 28, pp. 1–86.
[CrossRef]

Burckhardt, C. B.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

Cathey, W. T.

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).

Caulfield, H. J.

H. J. Caulfield, S. Lu, The Applications of Holography (Wiley, New York, 1970).

Champeney, D. C.

D. C. Champeney, A Handbook of Fourier Theorems (Cambridge U. Press, Cambridge, 1987).
[CrossRef]

D. C. Champeney, Fourier Transforms and Their Physical Applications (Academic, London, 1973).

Clemmow, P. C.

P. C. Clemmow, The Plane Waυe Spectrum Representation of Electromagnetic Fields (Pergamon, Oxford, 1966).

Collier, R. J.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

Copson, E. T.

B. B. Baker, E. T. Copson, The Mathematical Theory of Huygens’ Principle (Clarendon, Oxford, 1969).

Dallas, W.-J.

W.-J. Dallas, “Computer-generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed. (Springer-Verlag, Berlin, 1980), pp. 291–366.

DeVelis, J. B.

J. B. DeVelis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley, Reading, Mass., 1967).

Ewald, P. P.

P. P. Ewald, “Zur Theorie der Interferenzen der Röntgen-strahlen in Kristallen,” Phys. Z. 14, 465–472 (1913).

Frère, C.

Goodman, J. W.

J. W. Goodman, An Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Huang, T. S.

T. S. Huang, “Digital holography,” Proc. IEEE 59, 1335–1346 (1971).
[CrossRef]

Lee, W.-H.

W.-H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. 16, pp. 119–232.
[CrossRef]

Leseberg, D.

Lin, L. H.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

Lohmann, A. W.

A. W. Lohmann, A. S. Marathay, “About periodicities in 3-D wavefields,” Appl. Opt. 28, 4419–4423 (1989).
[CrossRef] [PubMed]

A. W. Lohmann, D. P. Paris, “Binary image holograms,” J. Opt. Soc. Am 56, 537 (A) (1966).

Lu, S.

H. J. Caulfield, S. Lu, The Applications of Holography (Wiley, New York, 1970).

Marathay, A. S.

Merzlyakov, N. S.

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

Paris, D. P.

A. W. Lohmann, D. P. Paris, “Binary image holograms,” J. Opt. Soc. Am 56, 537 (A) (1966).

Reynolds, G. O.

J. B. DeVelis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley, Reading, Mass., 1967).

Rosen, L.

L. Rosen, “Focused-image holography with extended sources,” Appl. Phys. Lett. 9, 337–339 (1966).
[CrossRef]

Sneddon, I. N.

I. N. Sneddon, Fourier Transforms (McGraw-Hill, New York, 1951).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1983).

Wolf, K. B.

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).

Wyrowski, F.

O. Bryngdahl, F. Wyrowski, “Digital holography/computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1990), Vol. 28, pp. 1–86.
[CrossRef]

Yaroslavskii, L. P.

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

Appl. Opt. (4)

Appl. Phys. Lett. (1)

L. Rosen, “Focused-image holography with extended sources,” Appl. Phys. Lett. 9, 337–339 (1966).
[CrossRef]

J. Opt. Soc. Am (2)

A. W. Lohmann, D. P. Paris, “Binary image holograms,” J. Opt. Soc. Am 56, 537 (A) (1966).

O. Bryngdahl, “Image-transfer characteristics of carrier-frequency photography,” J. Opt. Soc. Am 62, 807–813 (1972).
[CrossRef]

J. Opt. Soc. Am. (1)

S. A. Benton, “Hologram reconstruction with extended incoherent sources,” J. Opt. Soc. Am. 59, 1545(A) (1969).

Phys. Z. (1)

P. P. Ewald, “Zur Theorie der Interferenzen der Röntgen-strahlen in Kristallen,” Phys. Z. 14, 465–472 (1913).

Proc. IEEE (1)

T. S. Huang, “Digital holography,” Proc. IEEE 59, 1335–1346 (1971).
[CrossRef]

Other (17)

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

O. Bryngdahl, F. Wyrowski, “Digital holography/computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1990), Vol. 28, pp. 1–86.
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1983).

B. B. Baker, E. T. Copson, The Mathematical Theory of Huygens’ Principle (Clarendon, Oxford, 1969).

P. C. Clemmow, The Plane Waυe Spectrum Representation of Electromagnetic Fields (Pergamon, Oxford, 1966).

I. N. Sneddon, Fourier Transforms (McGraw-Hill, New York, 1951).

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979).

D. C. Champeney, Fourier Transforms and Their Physical Applications (Academic, London, 1973).

W.-H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. 16, pp. 119–232.
[CrossRef]

W.-J. Dallas, “Computer-generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed. (Springer-Verlag, Berlin, 1980), pp. 291–366.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).

J. B. DeVelis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley, Reading, Mass., 1967).

H. J. Caulfield, S. Lu, The Applications of Holography (Wiley, New York, 1970).

D. C. Champeney, A Handbook of Fourier Theorems (Cambridge U. Press, Cambridge, 1987).
[CrossRef]

J. W. Goodman, An Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

E. O. Brigham, The Fast Fourier Transformation (Prentice-Hall, Englewood Cliffs, N.J., 1974).

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Figures (5)

Fig. 1
Fig. 1

Calculation of complex amplitudes in hologram plane H. (a) In the space domain the object parts can be parallel to the hologram (O1) or of an arbitrary orientation (O2). For O2, an auxiliary coordinate system with axis z′ is introduced. (b) The spatial frequency spectrum of a monochromatic wave has nonvanishing amplitudes on the Ewald sphere only. The contributions of object part O2 are determined in an auxiliary coordinate system with axis k2 first and then transformed to the main coordinate system.

Fig. 2
Fig. 2

Optical reconstruction of a synthetic image hologram. The objects, six circles, are sampled in 2562 arrays and lie on six planes in depth. The smallest circle is farthest away from the hologram plane, the largest is almost in the hologram plane: a–d are optical reconstructions in laser light; e and fare optical reconstructions in collimated white light. Focusing is achieved to the outermost circle in a, c, and e, and to the innermost circle in b, d, and f. Two different perspective views are shown (a, b, and c, d), which were taken by displacing the camera horizontally. In f the outermost circle is defocused and the innermost circle is broadened due to dispersed light.

Fig. 3
Fig. 3

Calculated spectrum restricted to a portion of the Ewald sphere. This is approximately a horizontal slit, which then becomes the exit pupil of the hologram.

Fig. 4
Fig. 4

Optical reconstruction setup used for synthetic image holograms without a vertical diffusing element. Plane wave P is transformed to a convergent spherical wave by lens L that focuses at F on axis. Thus, z = zF is the Fourier plane of hologram H. The diffracted light of +1 order converges to E, which becomes the exit pupil. E is a scaled version of the calculated portion of the spectrum (see Fig. 3).

Fig. 5
Fig. 5

Offsets used to avoid overlap of spectrum U and twin spectrum U*. (a) A vertical offset has the disadvantage of alternately occurring U and U* such that an inaccuracy in vertical positioning mixes the desired image and the longitudinally reversed image. (b) A horizontal offset shows an unambiguous reconstruction at the expense of a horizontally enlarged transform array. Therefore, the horizontal offset is preferable.

Equations (21)

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U ( k ¯ ) = w 3 + u ( r ¯ ) exp [ i ( k ¯ · r ¯ ) ] d x d y d z ,
u ( r ¯ ) = w 3 U ( k ¯ ) exp [ i ( k ¯ · r ¯ ) ] d k x d k y d k z .
U ( k ¯ ) = 1 w [ U r ( k x , k y ) δ ( k z k z ) + U l ( k x , k y ) δ ( k z + k z ) ] ,
k z = ( k 0 2 k x 2 k y 2 ) 1 / 2 .
u r ( r ¯ ) = w 2 + U r ( k x , k y ) exp [ i z ( k 0 2 k x 2 k y 2 ) 1 / 2 ] × exp [ i ( x k x + y k y ) ] d k x d k y .
U ( k ¯ ) = U r ( k x , k y ) ,
U r ( k x , k y ) = w 2 u r ( x , y ; 0 ) exp [ i ( k x x + k y y ) ] d x d y .
U r ( k ¯ ) = w δ ( k z k z ) u r ( x , y ; z 0 ) δ ( z z 0 ) × exp [ i ( k ¯ · r ¯ ) ] d x d y d z ,
U r ( k ¯ ) = w 2 δ ( k z k z ) exp ( i k z z 0 ) u r ( x , y ; z 0 ) × exp [ i ( k x x + k y y ) ] d x d y .
U ( k ¯ ) = U ( R 1 k ¯ ) .
u h ( x , y ) = u r ( x , y ; z h ) .
u h ( x , y ) = w 2 U 0 ( k x , k y ) exp [ i ( x k x + y k y ) ] × exp [ i ( z 0 z h ) ( k 0 2 k x 2 k y 2 ) 1 / 2 ] d k x d k y ,
U 0 ( k x , k y ) = w 2 u 0 ( x , y ) exp [ i ( x k x + y k y ) ] d x d y .
Δ x h = M δ x h , Δ y h = N δ y h
δ k x = 2 π Δ x h , δ k y = 2 π Δ y h .
exp [ i 2 π z λ ( 1 λ 2 Δ x h 2 m 2 λ 2 Δ y h 2 n 2 ) 1 / 2 ] ,
Δ k x h = 2 π δ x h , Δ k y h = 2 π δ y h .
u h ( x , y ) = i u i ( x , y ; z h ) .
δ y h = 2 π Δ k y = Δ y h N
Δ y k = λ f Δ k y = λ f π N Δ y h .
F 2 - D { U ( k x , k y , k z ) + U * ( k x , k y , k z ) } = 2 Re { u ( x , y , 0 ) } .

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