Abstract

Computer-generated holograms are employed to design any desired intensity distribution along the propagation axis for a finite specified distance.

© 1994 Optical Society of America

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References

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  1. W.-H. Lee, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. XVI, Chap. 3, p. 119.
    [CrossRef]
  2. J. Durnin, J. J. Miceli, J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
    [CrossRef] [PubMed]
  3. A. Vasara, J. Turunen, A. T. Friberg, J. Opt. Soc. Am. A 6, 1748 (1989).
    [CrossRef] [PubMed]
  4. A. J. Cox, D. C. Dibble, Appl. Opt. 30, 1330 (1991).
    [CrossRef] [PubMed]
  5. N. Davidson, A. A. Friesem, E. Hasman, Opt. Commun. 88, 326 (1992).
    [CrossRef]
  6. J. Rosen, Opt. Lett. 19, 369 (1994).
    [CrossRef] [PubMed]
  7. J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, New York, 1968), Chap. 4, p. 60.
  8. H. Stark, ed., Image Recovery Theory and Application, 1st ed. (Academic, New York, 1987), Chap. 8, p. 277.
  9. J. R. Fienup, Appl. Opt. 21, 2758 (1982).
    [CrossRef] [PubMed]
  10. M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, Appl. Opt. 26, 2788 (1987).
    [CrossRef] [PubMed]

1994 (1)

1992 (1)

N. Davidson, A. A. Friesem, E. Hasman, Opt. Commun. 88, 326 (1992).
[CrossRef]

1991 (1)

1989 (1)

1987 (2)

1982 (1)

Allebach, J. P.

Cox, A. J.

Davidson, N.

N. Davidson, A. A. Friesem, E. Hasman, Opt. Commun. 88, 326 (1992).
[CrossRef]

Dibble, D. C.

Durnin, J.

J. Durnin, J. J. Miceli, J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Fienup, J. R.

Friberg, A. T.

Friesem, A. A.

N. Davidson, A. A. Friesem, E. Hasman, Opt. Commun. 88, 326 (1992).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, New York, 1968), Chap. 4, p. 60.

Hasman, E.

N. Davidson, A. A. Friesem, E. Hasman, Opt. Commun. 88, 326 (1992).
[CrossRef]

Lee, W.-H.

W.-H. Lee, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. XVI, Chap. 3, p. 119.
[CrossRef]

Miceli, J. J.

J. Durnin, J. J. Miceli, J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Rosen, J.

Seldowitz, M. A.

Sweeney, D. W.

Turunen, J.

Vasara, A.

Appl. Opt. (3)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

N. Davidson, A. A. Friesem, E. Hasman, Opt. Commun. 88, 326 (1992).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Other (3)

J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, New York, 1968), Chap. 4, p. 60.

H. Stark, ed., Image Recovery Theory and Application, 1st ed. (Academic, New York, 1987), Chap. 8, p. 277.

W.-H. Lee, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. XVI, Chap. 3, p. 119.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Optical system used to obtain the axial Fourier transform.

Fig. 2
Fig. 2

(a) Cross section of the phase distribution of the hologram obtained by the projected-onto-constraint-sets algorithm. (b) Axial intensity distribution simulated by the computer. (c) Transversal intensity cross section of the beam at the second focal plane.

Fig. 3
Fig. 3

(a) Hologram obtained by the direct-binary-search algorithm. (b) Axial intensity distribution obtained around the second focal length. The long-dashed curve is the desired profile, the solid curve is the simulated result, and the crosses connected by the short-dashed curve are the experimental measurements.

Equations (8)

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u ( x , y , z ) = exp ( j k z ) j λ z - - g ( x 2 , y 2 ) × exp [ - j k 2 f ( x 2 2 + y 2 2 ) ] × exp { j k 2 z [ ( x - x 2 ) 2 + ( y - y 2 ) 2 ] } d x 2 d y 2 ,
g ( x 2 , y 2 ) = exp ( j k f ) j λ f - - g ( x 1 , y 1 ) × exp { j k 2 f [ ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 ] } d x 1 d y 1 .
u ( x , y , z ) = exp [ j k ( z + f ) ] j λ f - - g ( x 1 , y 1 ) × exp [ - j k ( z - f ) 2 f 2 ( x 1 2 + y 1 2 ) - j k f ( x x 1 + y y 1 ) ] d x 1 d y 1 .
u ( z ) = exp [ j k ( z + f ] 2 j λ f 0 t ( ρ ) exp [ - j k ( z - f ) 2 f 2 ρ ] d ρ = exp [ j k ( z + f ) ] 2 j λ f T ( z - f 2 λ f 2 ) ,
t ( ρ ) = 0 2 π g ( ρ , θ ) d θ ,             ρ = r 2 .
P 2 [ T i ( f ρ ) ] = { I 0 ( f ρ ) exp [ j Ψ i ( f ρ ) ] f ρ Δ f ρ T i ( f ρ ) otherwise ,
e i = 1 Δ f ρ Δ f ρ T i ( f ρ ) - P 2 [ T i ( f ρ ) ] 2 d f ρ .
g ( x 1 , y 1 ) = g ( r , θ ) = rect ( θ 2 π ) 0 t ( ρ ) δ ( ρ - r 2 ) d ρ .

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