Abstract

The Fresnel diffraction of tilted planar objects is determined. It can be calculated by a Fourier transformation, a coordinate transformation, and a subsequent multiplication by a quadratic phase. Images of 3-D objects are composed by superimposing several planar segments. The resultant complex amplitude is coded and stored in a computer-generated hologram. Optical reconstructions of planar segments nearly parallel to the optical axis are demonstrated.

© 1988 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).
  2. D. Leseberg, “Computer Generated Holograms: Cylindrical, Conical, and Helical Waves,” Appl. Opt. 26, 4385 (1987).
    [CrossRef] [PubMed]
  3. D. Leseberg, “Computer-Generated Holograms: Display Using One-Dimensional Transforms,” J. Opt. Soc. Am. A 3, 1846 (1986).
    [CrossRef]
  4. S. A. Benton, “Survey of Holographic Stereograms,” Proc. Soc. Photo-Opt. Instrum. Eng. 367, 15 (1982).
  5. H. J. Rabal, N. Bolognini, E. E. Sicre, “Diffraction by a Tilted Aperture. Coherent and Partially Coherent Cases,” Opt. Acta 32, 1309 (1985).
    [CrossRef]
  6. K. Patorski, “Fraunhofer Diffraction Patterns of Tilted Planar Objects,” Opt. Acta 30, 673 (1982).
    [CrossRef]
  7. S. Ganci, “Fourier Diffraction Through a Tilted Slit,” Eur. J. Phys. 2, 158 (1981).
    [CrossRef]
  8. G. Harburn, J. K. Ranniko, R. P. Williams, “An Aspect of Phase in Fraunhofer Diffraction Patterns,” Optik 48, 321 (1977).
  9. D. Schreier, Synthetische Holografie (Physik-Verlag, Weinheim, 1984).
  10. W. J. Dallas, “Computer-Generated Holograms,” in The Computer in Optical Research, B. R. Frieden, Ed. (Springer-Verlag, New York, 1980), p. 291.
  11. L. P. Yaroslavskii, N. S. Merzlyakov, Methods of Digital Holography (Plenum, New York, 1980).
  12. W. H. Lee, “Computer-Generated Holograms: Techniques and Applications,” Prog. Opt. 16, 119 (1978).
    [CrossRef]
  13. F. Wyrowski, R. Hauck, O. Bryngdahl, “Computer-Generated Holography: Hologram Repetition and Phase Manipulations,” J. Opt. Soc. Am. A 4, 694 (1987).
    [CrossRef]
  14. F. Wyrowski, O. Bryngdahl, “Iterative Fourier Transform Algorithm Applied to Computer Holography,” J. Opt. Soc. Am. A5 (1988), to be published.
    [CrossRef]

1987 (2)

1986 (1)

1985 (1)

H. J. Rabal, N. Bolognini, E. E. Sicre, “Diffraction by a Tilted Aperture. Coherent and Partially Coherent Cases,” Opt. Acta 32, 1309 (1985).
[CrossRef]

1982 (2)

K. Patorski, “Fraunhofer Diffraction Patterns of Tilted Planar Objects,” Opt. Acta 30, 673 (1982).
[CrossRef]

S. A. Benton, “Survey of Holographic Stereograms,” Proc. Soc. Photo-Opt. Instrum. Eng. 367, 15 (1982).

1981 (1)

S. Ganci, “Fourier Diffraction Through a Tilted Slit,” Eur. J. Phys. 2, 158 (1981).
[CrossRef]

1978 (1)

W. H. Lee, “Computer-Generated Holograms: Techniques and Applications,” Prog. Opt. 16, 119 (1978).
[CrossRef]

1977 (1)

G. Harburn, J. K. Ranniko, R. P. Williams, “An Aspect of Phase in Fraunhofer Diffraction Patterns,” Optik 48, 321 (1977).

Benton, S. A.

S. A. Benton, “Survey of Holographic Stereograms,” Proc. Soc. Photo-Opt. Instrum. Eng. 367, 15 (1982).

Bolognini, N.

H. J. Rabal, N. Bolognini, E. E. Sicre, “Diffraction by a Tilted Aperture. Coherent and Partially Coherent Cases,” Opt. Acta 32, 1309 (1985).
[CrossRef]

Bryngdahl, O.

F. Wyrowski, R. Hauck, O. Bryngdahl, “Computer-Generated Holography: Hologram Repetition and Phase Manipulations,” J. Opt. Soc. Am. A 4, 694 (1987).
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Iterative Fourier Transform Algorithm Applied to Computer Holography,” J. Opt. Soc. Am. A5 (1988), to be published.
[CrossRef]

Dallas, W. J.

W. J. Dallas, “Computer-Generated Holograms,” in The Computer in Optical Research, B. R. Frieden, Ed. (Springer-Verlag, New York, 1980), p. 291.

Ganci, S.

S. Ganci, “Fourier Diffraction Through a Tilted Slit,” Eur. J. Phys. 2, 158 (1981).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

Harburn, G.

G. Harburn, J. K. Ranniko, R. P. Williams, “An Aspect of Phase in Fraunhofer Diffraction Patterns,” Optik 48, 321 (1977).

Hauck, R.

Lee, W. H.

W. H. Lee, “Computer-Generated Holograms: Techniques and Applications,” Prog. Opt. 16, 119 (1978).
[CrossRef]

Leseberg, D.

Merzlyakov, N. S.

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

Patorski, K.

K. Patorski, “Fraunhofer Diffraction Patterns of Tilted Planar Objects,” Opt. Acta 30, 673 (1982).
[CrossRef]

Rabal, H. J.

H. J. Rabal, N. Bolognini, E. E. Sicre, “Diffraction by a Tilted Aperture. Coherent and Partially Coherent Cases,” Opt. Acta 32, 1309 (1985).
[CrossRef]

Ranniko, J. K.

G. Harburn, J. K. Ranniko, R. P. Williams, “An Aspect of Phase in Fraunhofer Diffraction Patterns,” Optik 48, 321 (1977).

Schreier, D.

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

Sicre, E. E.

H. J. Rabal, N. Bolognini, E. E. Sicre, “Diffraction by a Tilted Aperture. Coherent and Partially Coherent Cases,” Opt. Acta 32, 1309 (1985).
[CrossRef]

Williams, R. P.

G. Harburn, J. K. Ranniko, R. P. Williams, “An Aspect of Phase in Fraunhofer Diffraction Patterns,” Optik 48, 321 (1977).

Wyrowski, F.

F. Wyrowski, R. Hauck, O. Bryngdahl, “Computer-Generated Holography: Hologram Repetition and Phase Manipulations,” J. Opt. Soc. Am. A 4, 694 (1987).
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Iterative Fourier Transform Algorithm Applied to Computer Holography,” J. Opt. Soc. Am. A5 (1988), to be published.
[CrossRef]

Yaroslavskii, L. P.

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

Appl. Opt. (1)

Eur. J. Phys. (1)

S. Ganci, “Fourier Diffraction Through a Tilted Slit,” Eur. J. Phys. 2, 158 (1981).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Acta (2)

H. J. Rabal, N. Bolognini, E. E. Sicre, “Diffraction by a Tilted Aperture. Coherent and Partially Coherent Cases,” Opt. Acta 32, 1309 (1985).
[CrossRef]

K. Patorski, “Fraunhofer Diffraction Patterns of Tilted Planar Objects,” Opt. Acta 30, 673 (1982).
[CrossRef]

Optik (1)

G. Harburn, J. K. Ranniko, R. P. Williams, “An Aspect of Phase in Fraunhofer Diffraction Patterns,” Optik 48, 321 (1977).

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

S. A. Benton, “Survey of Holographic Stereograms,” Proc. Soc. Photo-Opt. Instrum. Eng. 367, 15 (1982).

Prog. Opt. (1)

W. H. Lee, “Computer-Generated Holograms: Techniques and Applications,” Prog. Opt. 16, 119 (1978).
[CrossRef]

Other (5)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

F. Wyrowski, O. Bryngdahl, “Iterative Fourier Transform Algorithm Applied to Computer Holography,” J. Opt. Soc. Am. A5 (1988), to be published.
[CrossRef]

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

W. J. Dallas, “Computer-Generated Holograms,” in The Computer in Optical Research, B. R. Frieden, Ed. (Springer-Verlag, New York, 1980), p. 291.

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

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Figures (3)

Fig. 1
Fig. 1

Geometry for determination of the complex amplitude in the xy hologram plane due to the plane object which lies in the inclined xy′ plane.

Fig. 2
Fig. 2

Distortion of a square νμ raster by an inclination of the input plane to the y″ axis.

Fig. 3
Fig. 3

Optical reconstruction of two surfaces of a die: (a) geometry; (b), (c) surfaces inclined at an angle of −10° and 80°, respectively.

Equations (12)

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U h ( x , y ) = 1 i λ - + u 0 ( x , y ) exp ( i 2 π λ y sin ψ ) × exp [ i 2 π λ r ( x , y , x , y ) ] r ( x , y , x , y ) χ ( x , y , x , y ) d x d y .
r = ( z 0 - y sin ψ ) 2 + ( x - x ) 2 + ( y - y cos ψ ) 2 .
r 0 ( x , y , z 0 ) = z 0 2 + x 2 + y 2
U h ( x , y ) = exp ( i 2 π λ r 0 ) - u 0 ( x , y ) exp [ i π λ r 0 ( x 2 + y 2 ) ] × exp { - i 2 π λ r 0 [ x x + y y cos ψ + ( z 0 - r 0 ) y sin ψ ] } d x d y .
exp [ i π λ r 0 ( x , y , z 0 ) ( x 2 + y 2 ) ]
exp [ i π λ z 0 ( x 2 + y 2 ) ] .
ν ( x , y ) = 1 λ · x r 0 , μ ( x , y ) = cos ψ λ · y r 0 + sin ψ λ · z 0 - r 0 r 0 .
U h ( x , y ) = exp [ i 2 π λ r 0 ( x , y , z 0 ) ] U [ ν ( x , y ) , μ ( x , y ) ] ,
U [ ν ( x , y ) , μ ( x , y ) ] = - + U ( x , y ) × exp [ - i 2 π ( ν x + μ y ) ] d x d y ,
u ( x , y ) = u 0 ( x , y ) exp [ i π λ z 0 ( x 2 + y 2 ) ] .
U h ( x , y ) = exp [ i 2 π λ r 0 ( x , y , z 0 ) ] × [ U ( ν , μ ) n , m = - + δ ( ν - n Δ ν , μ - m Δ μ ) ] ,
t ( x , y ) = ½ [ 1 + U ¯ h ( x , y ) cos { 2 π ( α 0 x + β 0 y ) + arg [ U h ( x , y ) ] } ] ,

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