Abstract

A method is presented to construct computer-generated holograms from objects composed of line segments. In reconstruction, lines of finite lengths are displayed in three dimensions. These lines are analytically deduced from cylindrical and conical waves. It is possible to reconstruct lines of arbitrary orientation (inclined up to 85° to the hologram plane).

© 1986 Optical Society of America

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References

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  1. T. Kasahara, Y. Kimura, M. Kawai, “3-D construction of imaginary objects by the method of holographic stereogram,” in Applications of Holography, E. S. Barrekette, ed. (Plenum, New York, 1971), pp. 19–34.
    [Crossref]
  2. M. C. King, A. M. Noll, D. H. Berry, “A new approach to computer-generated holograms,” Appl. Opt. 9, 471–475 (1970).
    [Crossref] [PubMed]
  3. L. P. Yaroslavskii, N. S. Merzlyakov, Methods of Digital Holography (Plenum, New York, 1980).
  4. T. Yatagai, “Stereoscopic approach to 3-D display using computer-generated holograms,” Appl. Opt. 15, 2722–2729 (1976).
    [Crossref] [PubMed]
  5. B. R. Brown, A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Dev. 13, 160–168 (1969).
    [Crossref]
  6. W. J. Dallas, A. W. Lohmann, “Phase quantization in holograms—depth effects,” Appl. Opt. 11, 192–193 (1972).
    [Crossref] [PubMed]
  7. L. B. Lesem, P. M. Hirsch, J. A. Jordan, “Computer synthesis of holograms for 3-D display,” Commun. ACM 12, 661–674 (1968).
    [Crossref]
  8. J. P. Waters, “Three-dimensional Fourier transform method for synthesizing binary holograms,”J. Opt. Soc. Am. 58, 1284–1288 (1968).
    [Crossref]
  9. E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).
  10. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1983).
  11. A. Sommerfeld, Optics (Academic, New York, 1954).
  12. P. Frank, R. V. Mises, Die Differential- und Integralgleichungen der Mechanik und Physik, II. Teil, Physikalischer Teil (Dover, New York, 1961).
  13. D. Schreier, Synthetische Holografie (Physik Verlag, Weinheim, 1984).
  14. A. W. Lohmann, N. Streibl, “On the fundamentals of 3-D display,” Proc. Soc. Photo-Opt. Instrum. Eng. 402, 6–12 (1983).

1983 (1)

A. W. Lohmann, N. Streibl, “On the fundamentals of 3-D display,” Proc. Soc. Photo-Opt. Instrum. Eng. 402, 6–12 (1983).

1976 (1)

1972 (1)

1970 (1)

1969 (1)

B. R. Brown, A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Dev. 13, 160–168 (1969).
[Crossref]

1968 (2)

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “Computer synthesis of holograms for 3-D display,” Commun. ACM 12, 661–674 (1968).
[Crossref]

J. P. Waters, “Three-dimensional Fourier transform method for synthesizing binary holograms,”J. Opt. Soc. Am. 58, 1284–1288 (1968).
[Crossref]

Berry, D. H.

Born, M.

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

Brigham, E. O.

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

Brown, B. R.

B. R. Brown, A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Dev. 13, 160–168 (1969).
[Crossref]

Dallas, W. J.

Frank, P.

P. Frank, R. V. Mises, Die Differential- und Integralgleichungen der Mechanik und Physik, II. Teil, Physikalischer Teil (Dover, New York, 1961).

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “Computer synthesis of holograms for 3-D display,” Commun. ACM 12, 661–674 (1968).
[Crossref]

Jordan, J. A.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “Computer synthesis of holograms for 3-D display,” Commun. ACM 12, 661–674 (1968).
[Crossref]

Kasahara, T.

T. Kasahara, Y. Kimura, M. Kawai, “3-D construction of imaginary objects by the method of holographic stereogram,” in Applications of Holography, E. S. Barrekette, ed. (Plenum, New York, 1971), pp. 19–34.
[Crossref]

Kawai, M.

T. Kasahara, Y. Kimura, M. Kawai, “3-D construction of imaginary objects by the method of holographic stereogram,” in Applications of Holography, E. S. Barrekette, ed. (Plenum, New York, 1971), pp. 19–34.
[Crossref]

Kimura, Y.

T. Kasahara, Y. Kimura, M. Kawai, “3-D construction of imaginary objects by the method of holographic stereogram,” in Applications of Holography, E. S. Barrekette, ed. (Plenum, New York, 1971), pp. 19–34.
[Crossref]

King, M. C.

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “Computer synthesis of holograms for 3-D display,” Commun. ACM 12, 661–674 (1968).
[Crossref]

Lohmann, A. W.

A. W. Lohmann, N. Streibl, “On the fundamentals of 3-D display,” Proc. Soc. Photo-Opt. Instrum. Eng. 402, 6–12 (1983).

W. J. Dallas, A. W. Lohmann, “Phase quantization in holograms—depth effects,” Appl. Opt. 11, 192–193 (1972).
[Crossref] [PubMed]

B. R. Brown, A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Dev. 13, 160–168 (1969).
[Crossref]

Merzlyakov, N. S.

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

Mises, R. V.

P. Frank, R. V. Mises, Die Differential- und Integralgleichungen der Mechanik und Physik, II. Teil, Physikalischer Teil (Dover, New York, 1961).

Noll, A. M.

Schreier, D.

D. Schreier, Synthetische Holografie (Physik Verlag, Weinheim, 1984).

Sommerfeld, A.

A. Sommerfeld, Optics (Academic, New York, 1954).

Streibl, N.

A. W. Lohmann, N. Streibl, “On the fundamentals of 3-D display,” Proc. Soc. Photo-Opt. Instrum. Eng. 402, 6–12 (1983).

Waters, J. P.

Wolf, E.

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

Yaroslavskii, L. P.

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

Yatagai, T.

Appl. Opt. (3)

Commun. ACM (1)

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “Computer synthesis of holograms for 3-D display,” Commun. ACM 12, 661–674 (1968).
[Crossref]

IBM J. Res. Dev. (1)

B. R. Brown, A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Dev. 13, 160–168 (1969).
[Crossref]

J. Opt. Soc. Am. (1)

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

A. W. Lohmann, N. Streibl, “On the fundamentals of 3-D display,” Proc. Soc. Photo-Opt. Instrum. Eng. 402, 6–12 (1983).

Other (7)

T. Kasahara, Y. Kimura, M. Kawai, “3-D construction of imaginary objects by the method of holographic stereogram,” in Applications of Holography, E. S. Barrekette, ed. (Plenum, New York, 1971), pp. 19–34.
[Crossref]

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

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

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

A. Sommerfeld, Optics (Academic, New York, 1954).

P. Frank, R. V. Mises, Die Differential- und Integralgleichungen der Mechanik und Physik, II. Teil, Physikalischer Teil (Dover, New York, 1961).

D. Schreier, Synthetische Holografie (Physik Verlag, Weinheim, 1984).

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Figures (4)

Fig. 1
Fig. 1

Geometrical relationship between hologram and reconstructed line segment.

Fig. 2
Fig. 2

Hologram portions of a–c, cylindrical waves and d–f, conical waves. The reconstructed line segments are a, parallel to the x axis; b, in the xy plane inclined at a = 25° to the x axis; c, in the xz plane tilted at γ = 1°; d, in the xz plane tilted at γ = 85°; e, same as in d but inclined at α = 25°; f, same as in e but shifted in the x and y directions.

Fig. 3
Fig. 3

Hologram of object F tilted at y = 80° to the hologram plane.

Fig. 4
Fig. 4

Optical reconstructions of the object F: a, screen parallel to hologram at z = Rmin; b, screen parallel to hologram at z = Rmax; c, screen tilted at γ = 80° (reduction 2.5 ×); d, same as in c but adjusted lengths of horizontal bars.

Equations (17)

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u ( x , y , 0 ) = exp ( i π λ R y 2 ) ,
u ( x , y , 0 ) = exp ( i π λ R y 2 ) rect ( x 2 a ) ,
rect ( x 2 a ) = { 1 for - a x a 0 elsewhere .
x = X cos α - Y sin α , y = X sin α + Y cos α ,
u ( x , y , 0 ) = exp [ i π λ R ( X sin α + Y cos α ) 2 ] × rect ( X cos α - Y sin α 2 a ) = exp ( i π λ R y 2 ) rect ( x 2 a ) .
R ( x , γ ) = R 0 - x tan γ ,
exp ( i 2 π sin γ λ x ) .
u ( x , y , 0 ) = exp ( i 2 π sin γ λ x ) exp [ i π λ y 2 cos γ R ( x , γ ) ] rect ( x 2 a ) .
exp ( - i 2 π sin γ λ x ) .
u ( x , y , 0 ) = exp [ i π λ y 2 R ( x , γ ) ] rect ( x 2 a ) .
exp [ i 2 π λ R 0 ( x 0 x + y 0 y ) ] ,
u ( x , y , 0 ) = exp [ i 2 π λ R 0 ( x 0 x + y 0 y ) ] exp [ i π λ y 2 R ( x , γ ) ] × rect ( x 2 a ) .
t ( x , y ) = { 1 + cos [ 2 π ν 0 x + 2 π λ R 0 ( x 0 x + y 0 y ) + π λ y 2 R ( x , γ ) ] } rect ( x 2 a ) .
u ( x , y , z ) = - + - + u ( x , y , 0 ) exp { i π λ z [ ( x - x ) 2 + ( y - y ) 2 ] } d x d y .
u [ x , y , - R ( x , γ ) ] = - + - + exp [ i 2 π λ R 0 ( x 0 x + y 0 y ) ] × exp [ i π λ y 2 R ( x , γ ) ] exp { - i π λ 1 R ( x , γ ) × [ ( x - x ) 2 + ( y - y ) 2 ] } rect ( x 2 a , y 2 b ) d x d y ,
1 R 0 - x tan γ 1 R 0             for γ 85 °
u [ x , y , - R ( x , γ ) ] = exp ( - i π λ y 2 R 0 ) exp ( i π λ x 0 2 R 0 ) × exp ( i 2 π λ R 0 x 0 x ) 2 b sinc [ 2 b λ R 0 ( y + y 0 ) ] × - a + a exp { - i π λ R 0 [ x - ( x 0 + x ) ] 2 } d x .

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