Abstract

In digital holography the first step is the determination of the complex amplitude in the hologram. Large objects can be synthesized from tilted planar segments. In the calculation step, configurations of inclined object and inclined hologram plane are considered. The Fresnel transformation is generalized, and its implementation is discussed. Optical reconstructions show the feasibility.

© 1989 Optical Society of America

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References

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  1. D. Leseberg, C. Frère, “Computer-Generated Holograms of 3-D Objects Composed of Tilted Planar Segments,” Appl. Opt. 27, 3020–3024 (1988).
    [CrossRef] [PubMed]
  2. K. Patorski, “Fraunhofer Diffraction Patterns of Tilted Planar Objects,” Opt. Acta 30, 673 (1982).
    [CrossRef]
  3. J. J. Burch, “A Computer Algorithm for the Synthesis of Spatial Frequency Filter,” Proc. IEEE 55, 599 (1967).
    [CrossRef]
  4. W. J. Dallas, “Computer-Generated Holograms,” in The Computer in Optical Research, B. R. Frieden, Ed. (Springer-Verlag, New York, 1980), p. 291.
  5. K. Biedermann, O. Holmgren, “Large-Size Distortion-Free Computer-Generated Holograms in Photoresist,” Appl. Opt. 16, 2014–2016 (1977).
    [CrossRef] [PubMed]
  6. E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, NJ, 1974).

1988 (1)

1982 (1)

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

1977 (1)

1967 (1)

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

Biedermann, K.

Brigham, E. O.

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

Burch, J. J.

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

Frère, C.

Holmgren, O.

Leseberg, D.

Patorski, K.

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

Appl. Opt. (2)

Opt. Acta (1)

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

Proc. IEEE (1)

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

Other (2)

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

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

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Figures (3)

Fig. 1
Fig. 1

Geometry for the determination of the complex amplitude in the hologram plane H. The object plane O is inclined by γ to the auxiliary optical axis za, which is tilted by α to the principal optical axis zp.

Fig. 2
Fig. 2

Optical reconstruction outside the Fresnel region of the principal optical axis zp for an inclination of α = 2.5°. The reconstruction wave is on-axis, and the image I is tilted (γ = 45°).

Fig. 3
Fig. 3

Optical reconstruction for an inclination of α = 45° with a reconstruction wave on the auxiliary optical axis za; i.e., the hologram is inclined by α = 45° to za. The image I is inclined −45° to zp (γ = 0).

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

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