Abstract

The determination of the complex amplitude in the hologram is one major step in the realization of computer generated holograms. Usually the input distribution is assumed to be in a plane parallel to the hologram plane. However, sometimes a more flexible geometry is of interest. Here, in particular an arbitrarily shaped computer generated hologram of a plane input distribution is considered. The calculation of the complex amplitude in the hologram mainly consists of a Fourier transformation and a nonlinear coordinate transformation. The feasibility of this procedure is demonstrated by optical reconstructions of cylindrically shaped computer generated holograms in arbitrarily oriented planes.

© 1990 Optical Society of America

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References

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  1. E. O. Brigham, The Fast Fourier Transform, (Prentice Hall, Englewood Cliffs, 1974).
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw Hill, San Francisco, 1968).
  3. D. Leseberg, C. Frere, “Computer-Generated Holograms of 3-D Objects Composed of Tilted Planar Segments,” Appl. Opt. 27, 3020–3024 (1988).
    [CrossRef] [PubMed]
  4. C. Frère, D. Leseberg, “Large Objects Reconstructed from Computer-Generated Holograms,” Appl. Opt. 28, 2422–2425 (1989).
    [CrossRef] [PubMed]
  5. K. Patorski, “Fraunhofer Diffraction Patterns of Tilted Planar Objects,” Opt. Acta 30, 673–679 (1982).
    [CrossRef]
  6. J. J. Burch, “A Computer Algorithm for the Synthesis of Spatial Frequency Filter,” Proc. IEEE 55, 599–601 (1967).
    [CrossRef]
  7. O. Bryngdahl, F. Wyrowski, ”Digital Holography/Computer-Generated Holograms,” Prog. Opt.28, E. Wolf ed. North Holland, Amsterdam, to appear (1990).
    [CrossRef]

1989

1988

1982

K. Patorski, “Fraunhofer Diffraction Patterns of Tilted Planar Objects,” Opt. Acta 30, 673–679 (1982).
[CrossRef]

1967

J. J. Burch, “A Computer Algorithm for the Synthesis of Spatial Frequency Filter,” Proc. IEEE 55, 599–601 (1967).
[CrossRef]

Brigham, E. O.

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

Bryngdahl, O.

O. Bryngdahl, F. Wyrowski, ”Digital Holography/Computer-Generated Holograms,” Prog. Opt.28, E. Wolf ed. North Holland, Amsterdam, to appear (1990).
[CrossRef]

Burch, J. J.

J. J. Burch, “A Computer Algorithm for the Synthesis of Spatial Frequency Filter,” Proc. IEEE 55, 599–601 (1967).
[CrossRef]

Frere, C.

Frère, C.

Goodman, J. W.

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

Leseberg, D.

Patorski, K.

K. Patorski, “Fraunhofer Diffraction Patterns of Tilted Planar Objects,” Opt. Acta 30, 673–679 (1982).
[CrossRef]

Wyrowski, F.

O. Bryngdahl, F. Wyrowski, ”Digital Holography/Computer-Generated Holograms,” Prog. Opt.28, E. Wolf ed. North Holland, Amsterdam, to appear (1990).
[CrossRef]

Appl. Opt.

Opt. Acta

K. Patorski, “Fraunhofer Diffraction Patterns of Tilted Planar Objects,” Opt. Acta 30, 673–679 (1982).
[CrossRef]

Proc. IEEE

J. J. Burch, “A Computer Algorithm for the Synthesis of Spatial Frequency Filter,” Proc. IEEE 55, 599–601 (1967).
[CrossRef]

Other

O. Bryngdahl, F. Wyrowski, ”Digital Holography/Computer-Generated Holograms,” Prog. Opt.28, E. Wolf ed. North Holland, Amsterdam, to appear (1990).
[CrossRef]

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

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

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Figures (5)

Fig. 1
Fig. 1

Geometry to determine the complex amplitude in an arbitrary shaped hologram (H) while the input plane (IP) is inclined by 90° − γ to the optical axis z.

Fig. 2
Fig. 2

Geometry to determine the complex amplitude in the cylindrically shaped hologram (H).

Fig. 3
Fig. 3

Distortion of an equidistant raster (a) by a cylindrically shaped hologram and an input plane perpendicular (b) and inclined (c) to the optical axis.

Fig. 4
Fig. 4

Amplitude and phase of a part of the repeated spectrum before (a) and after (b) the coordinate transformation.

Fig. 5
Fig. 5

Optical reconstruction of computer generated cylindrically shaped holograms for an inclination of the image plane by γ = 0° (a) and γ = 45° (b).

Equations (12)

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u o ( x , y ) exp [ i 2 π λ y sin γ ]
u h ( x , y ) = 1 i λ i + u o ( x y ) exp [ i 2 π λ y sin γ ] × z o - y sin γ + Δ z r 1 r exp [ i 2 π λ r ] d x d y ,
r = ( z 0 - y sin γ + Δ z ) 2 + ( Δ x - x ) 2 + ( Δ y - y cos γ ) 2 .
r o = r ( 0 , 0 , x , y ) = ( z o + Δ z ) 2 + Δ x 2 + Δ y 2 ,
u ( x , y ) = exp [ i 2 π λ r o ] - + u o ( x , y ) exp [ - i 2 π ( ν x + μ y ) ] × exp [ i π λ r o ( x 2 + y 2 ) ] exp [ - i π λ t o 3 z o 2 y 2 sin 2 γ ] × exp [ i π λ r o 3 { x 2 + y 2 - 2 [ Δ x x + ( Δ z sin γ + Δ y cos γ ) y ] } z o y sin γ ] d x d y
ν ( x , y ) = 1 λ r o Δ x μ ( x , y ) = 1 λ r o [ Δ y cos γ + ( z o + Δ z - r 0 ) sin γ ] ,
exp [ i π λ r o ( x 2 + y 2 ) ] exp [ - i π λ r o 3 z o 2 y 2 sin 2 γ ] exp [ i π λ z o ( x 2 + y 2 cos 2 γ ) ] exp [ i π λ r o 3 ( x 2 + y 2 - 2 ) [ Δ x x + ( Δ z sin γ + Δ y cos γ ) y ] } z o y sin γ ] exp [ i π λ r o 2 ( x 2 + y 2 ) y sin γ ] .
2 [ Δ x x + ( Δ z sin γ + Δ y cos γ ) y ] y sin γ + [ x 2 + y 2 ( 1 - 3 sin 2 γ ) ] Δ z λ z o 2 ,
Δ x = x Δ y = R sin y R Δ z = R cos y R .
u h ( x , y ) = exp [ i 2 π λ r o ] - + u o ( x , y ) exp [ i π λ z o ( x 2 + y 2 cos 2 γ ) ] × exp [ i π λ z o 2 ( x 2 + y 2 ) y sin γ ] exp [ - i 2 π ( ν x + μ y ) ] d x d y ,
ν ( x , y ) = 1 λ r o x μ ( x , y ) = 1 λ r o [ R sin ( y R - γ ) + ( z o - r o ) sin γ ] .
t h ( x , y ) = ½ + ½ u ¯ h ( x , y ) cos { arg [ u h ( x , y ) ] } ,

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