Abstract

Computer-generated holograms of plane surfaces tilted and shifted with respect to the hologram plane are considered. The analysis is made in the spatial frequency domain by using the translation and rotation transformations of angular spectra. The frequency approach permits the use of the fast-Fourier-transform algorithm, which decreases the computation time and makes it possible to consider any position of the planes in space. Various configurations of tilted and shifted planes have been investigated, and computer-generated holograms of off-axis planes have been obtained. Computer and optical reconstructions, both of which show the feasibility of the proposed approach, have been carried out.

© 1993 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  2. L. P. Yaroslavskii, N. S. Merzlyakov, Methods of Digital Holography (Plenum, New York, 1980).
  3. T. Yatagai, “Stereoscopic approach to 3-D display using computer-generated holograms,” Appl. Opt. 15, 2722–2729 (1976).
    [CrossRef] [PubMed]
  4. T.S. Huang, “Digital holography,” Proc. IEEE 59, 1335–1346 (1971).
    [CrossRef]
  5. Ch. Frere, D. Leseberg, O. Bryngdahl, “Computer-generated holograms of three-dimensional objects composed of line segments,” J. Opt. Soc. Am. A 3, 726–730 (1986).
    [CrossRef]
  6. S. Bara, Ch. Frere, Z. Jaroszewicz, A. Kolodziejczyk, D. Leseberg, “Modulated on-axis circular zone plates for a generation of 3-D focal curves,” J. Mod. Opt.37, 1287–1295 (1990).
  7. Z. Jaroszewicz, A. Kolodziejczyk, D. Mouriz, S. Bara, “Analytic design of computer-generated Fourier-transform holograms for plane curves reconstruction,” J. Opt. Soc. Am. A 8, 559–565 (1991).
    [CrossRef]
  8. D. Leseberg, Ch. Frere, “Computer-generated holograms of 3-D objects composed of tilted planar segments,” Appl. Opt. 27, 3020–3024 (1988).
    [CrossRef] [PubMed]
  9. Ch. Frere, D. Leseberg, “Large objects reconstructed from computer-generated holograms,” Appl. Opt. 28, 2422–2425 (1989).
    [CrossRef] [PubMed]
  10. T. Tommasi, B. Bianco, “Frequency analysis of light diffraction between rotated planes,” Opt. Lett. 17, 556–558 (1992).
    [CrossRef] [PubMed]

1992 (1)

1991 (1)

1990 (1)

S. Bara, Ch. Frere, Z. Jaroszewicz, A. Kolodziejczyk, D. Leseberg, “Modulated on-axis circular zone plates for a generation of 3-D focal curves,” J. Mod. Opt.37, 1287–1295 (1990).

1989 (1)

1988 (1)

1986 (1)

1976 (1)

1971 (1)

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

Bara, S.

Z. Jaroszewicz, A. Kolodziejczyk, D. Mouriz, S. Bara, “Analytic design of computer-generated Fourier-transform holograms for plane curves reconstruction,” J. Opt. Soc. Am. A 8, 559–565 (1991).
[CrossRef]

S. Bara, Ch. Frere, Z. Jaroszewicz, A. Kolodziejczyk, D. Leseberg, “Modulated on-axis circular zone plates for a generation of 3-D focal curves,” J. Mod. Opt.37, 1287–1295 (1990).

Bianco, B.

Bryngdahl, O.

Frere, Ch.

Goodman, J. W.

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

Huang, T.S.

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

Jaroszewicz, Z.

Z. Jaroszewicz, A. Kolodziejczyk, D. Mouriz, S. Bara, “Analytic design of computer-generated Fourier-transform holograms for plane curves reconstruction,” J. Opt. Soc. Am. A 8, 559–565 (1991).
[CrossRef]

S. Bara, Ch. Frere, Z. Jaroszewicz, A. Kolodziejczyk, D. Leseberg, “Modulated on-axis circular zone plates for a generation of 3-D focal curves,” J. Mod. Opt.37, 1287–1295 (1990).

Kolodziejczyk, A.

Z. Jaroszewicz, A. Kolodziejczyk, D. Mouriz, S. Bara, “Analytic design of computer-generated Fourier-transform holograms for plane curves reconstruction,” J. Opt. Soc. Am. A 8, 559–565 (1991).
[CrossRef]

S. Bara, Ch. Frere, Z. Jaroszewicz, A. Kolodziejczyk, D. Leseberg, “Modulated on-axis circular zone plates for a generation of 3-D focal curves,” J. Mod. Opt.37, 1287–1295 (1990).

Leseberg, D.

Merzlyakov, N. S.

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

Mouriz, D.

Tommasi, T.

Yaroslavskii, L. P.

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

Yatagai, T.

Appl. Opt. (3)

J. Mod. Opt. (1)

S. Bara, Ch. Frere, Z. Jaroszewicz, A. Kolodziejczyk, D. Leseberg, “Modulated on-axis circular zone plates for a generation of 3-D focal curves,” J. Mod. Opt.37, 1287–1295 (1990).

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

Opt. Lett. (1)

Proc. IEEE (1)

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

Other (2)

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

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

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Figures (10)

Fig. 1
Fig. 1

Geometry for the determination of the light diffracted by plane I onto hologram plane H. The angular spectra of virtual planes A and B are considered for this calculation.

Fig. 2
Fig. 2

Geometry for the generation of the hologram of an on-axis tilted plane A, illuminated by a plane wave u.

Fig. 3
Fig. 3

Hologram of the letters CGH tilted by 45 deg with respect to the hologram plane.

Fig. 4
Fig. 4

Hologram of the word OPTICAL tilted by 80 deg with respect to the hologram plane.

Fig. 5
Fig. 5

Optical reconstructions of the on-axis tilted letters CGH. A tilted screen to reconstruct the original pattern (top) and a screen parallel to the hologram plane (bottom) are used.

Fig. 6
Fig. 6

Optical reconstructions of the tilted word OPTICAL. A screen tilted by 80 deg (top) and a screen parallel to the hologram (bottom) are used. Two different focus positions are shown, which bring into focus the initial (right) and the final (left) letters of the word.

Fig. 7
Fig. 7

Geometry for the generation of the hologram of an off-axis plane tilted by an angle θ = 45 deg.

Fig. 8
Fig. 8

Holograms and related computer reconstructions for the configuration shown in Fig. 7. From left to right and from top to bottom: the hologram generated by the frequency approach and by the Rayleigh–Sommerfeld integral; the corresponding reconstructions performed by using the Rayleigh–Sommerfeld equation.

Fig. 9
Fig. 9

Holograms and reconstructions analogous to those shown in Fig. 8 but related to the configuration in Fig. 1.

Fig. 10
Fig. 10

Computer reconstructions showing a higher diffraction order. The reconstructions have been obtained by applying the Rayleigh–Sommerfeld integral to holograms generated by the frequency approach (right) and by the Rayleigh–Sommerfeld integral (left).

Equations (12)

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α = λ f x , β = λ f y , γ = 1 α 2 β 2
A 1 ( f x , f y , z ) = A 0 ( f x , f y ) exp [ 2 π j z ( 1 / λ 2 f x 2 f y 2 ) 1 / 2 ] .
( x , y , z ) t = M ( ξ , η , ζ ) t ,
( f ξ , f η , f ζ ) t = M t ( f x , f y , f z ) t ,
f z = ( 1 / λ 2 f x 2 f y 2 ) 1 / 2 .
A 2 ( f ξ , f η , 0 ) = { 0 outside C A 1 ( f x , f y , 0 ) f z / f ζ inside C ,
f z > 0 , f ζ > 0 .
U ( x , y ) = T ( x , y ) exp ( 2 π j y cos θ / λ ) ,
i = n / 2 + D f x + 1 , j = n / 2 + D f y + 1 , ( i , j = 1 n ) .
j = c 1 j j + c 2 ( D 2 i i 2 j j 2 ) 1 / 2 , i = i i ,
w = D β / λ n / 2
w = D β / λ + n / 2

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