Abstract

A fast algorithm to compute a scattering pattern produced by a two-dimensional object is presented. Rather than by use of an approximation, this algorithm is derived by decomposition of the spherical nature of a radiation pattern into the sum of plane waves. Thus it precludes the phase errors that exist in approximation-based Fresnel holograms. Moreover, the amount of computation required in this method is shown to be significantly smaller than that of a well-known ray-tracing method; excellent performance, as good as one could obtain with the ray-tracing method, is achieved. Implementation results as well as sampling constraints are also presented.

© 1996 Optical Society of America

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References

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  1. E. Wolf, Progress in Optics (North-Holland, Amsterdam, 1990), Vol. 28, pp. 3–86.
    [CrossRef]
  2. D. Leseberg, C. Frere, Appl. Opt. 27, 3020 (1988).
    [CrossRef] [PubMed]
  3. D. Leseberg, Appl. Opt. 31, 223 (1992).
    [CrossRef] [PubMed]
  4. M. Lucente, Proc. SPIE 1667, 32 (1992).
    [CrossRef]
  5. D. Leseberg, O. Bryngdahl, Appl. Opt. 23, 2441 (1984).
    [CrossRef] [PubMed]
  6. P. Morse, H. Feshback, Methods of Theoretical Physics (McGraw-Hill, New York, 1968).
  7. J. W. Goodman, Introduction of Fourier Optics (McGraw-Hill, New York, 1968), pp. 48–54.
  8. H. Yang, M. Soumekh, IEEE Trans. Image Process. 2, 80 (1993).
    [CrossRef] [PubMed]

1993 (1)

H. Yang, M. Soumekh, IEEE Trans. Image Process. 2, 80 (1993).
[CrossRef] [PubMed]

1992 (2)

1988 (1)

1984 (1)

Bryngdahl, O.

Feshback, H.

P. Morse, H. Feshback, Methods of Theoretical Physics (McGraw-Hill, New York, 1968).

Frere, C.

Goodman, J. W.

J. W. Goodman, Introduction of Fourier Optics (McGraw-Hill, New York, 1968), pp. 48–54.

Leseberg, D.

Lucente, M.

M. Lucente, Proc. SPIE 1667, 32 (1992).
[CrossRef]

Morse, P.

P. Morse, H. Feshback, Methods of Theoretical Physics (McGraw-Hill, New York, 1968).

Soumekh, M.

H. Yang, M. Soumekh, IEEE Trans. Image Process. 2, 80 (1993).
[CrossRef] [PubMed]

Wolf, E.

E. Wolf, Progress in Optics (North-Holland, Amsterdam, 1990), Vol. 28, pp. 3–86.
[CrossRef]

Yang, H.

H. Yang, M. Soumekh, IEEE Trans. Image Process. 2, 80 (1993).
[CrossRef] [PubMed]

Appl. Opt. (3)

IEEE Trans. Image Process. (1)

H. Yang, M. Soumekh, IEEE Trans. Image Process. 2, 80 (1993).
[CrossRef] [PubMed]

Proc. SPIE (1)

M. Lucente, Proc. SPIE 1667, 32 (1992).
[CrossRef]

Other (3)

P. Morse, H. Feshback, Methods of Theoretical Physics (McGraw-Hill, New York, 1968).

J. W. Goodman, Introduction of Fourier Optics (McGraw-Hill, New York, 1968), pp. 48–54.

E. Wolf, Progress in Optics (North-Holland, Amsterdam, 1990), Vol. 28, pp. 3–86.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Holographic system geometry.

Fig. 2
Fig. 2

Original one-dimensional object.

Fig. 3
Fig. 3

Object reconstructed without zero padding.

Fig. 4
Fig. 4

Object reconstructed with zero padding.

Fig. 5
Fig. 5

Object reconstructed with the ray-tracing method.

Equations (10)

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g ( u , v = Y 1 ) = g 1 ( u ) = f ( x ; y = Y 1 ) exp [ j k r ( x , u ) ] r ( x , u ) d x ,
r ( x , u ) = [ ( x - u ) 2 + Z 1 2 ] 1 / 2 .
g 1 ( u ) 1 Z 1 f ( x , y = Y 1 ) exp { j k [ ( x - u ) 2 + Z 1 2 ] 1 / 2 } d x .
exp { j k [ ( x - u ) 2 + Z 1 2 } = exp { j [ k 2 - α 2 Z 1 + ( x - u ) α ] } k 2 - α 2 d α .
G 1 ( k u ) = f ( x , y = Y 1 ) × exp [ j ( k 2 - α 2 Z 1 + α x ) ] k 2 - α 2 δ ( k u + α ) d α d x ,
G 1 ( k u ) = H ( k u ) F x ( k u , y = Y 1 ) ,
- k D + L Z 1 k u k D + L Z 1 .
Δ u Z 1 λ 2 ( D + L ) ,
Δ k u < 2 π 2 ( D + λ Z 1 2 Δ u ) .
2 D z > 2 D + λ Z 1 Δ u ,

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