Abstract

In holographic applications in which the phase is a free parameter, computer-generated holograms offer attractive possibilities. Lateral repetition of the hologram structure is a natural way to increase the space–bandwidth product of a Fourier hologram. To circumvent unwanted effects, two methods of phase manipulations of repeated holograms are presented. Improvement of the diffraction efficiency and an increase in the number of reconstructed image points are discussed. Experimental verifications are shown in the form of optical reconstructions of computer-generated holograms.

© 1987 Optical Society of America

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References

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  1. W. J. Dallas, “Computer-generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed., Vol. 41 of Topics in Applied Physics (Springer-Verlag, Berlin, 1980), pp. 291–366.
    [CrossRef]
  2. W.-H. Lee, “Computer-generated holograms: techniques and applications,” Prog. Opt. 16, 119–232 (1978).
    [CrossRef]
  3. L. B. Lesem, P. M. Hirsch, J. A. Jordan, Commun. ACM 11, 661–674 (1968).
    [CrossRef]
  4. A. Papoulis, Systems and Transformations with Applications in Optics (McGraw-Hill, New York, 1968); E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).
  5. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  6. J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980); “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]

1980 (1)

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980); “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

1978 (1)

W.-H. Lee, “Computer-generated holograms: techniques and applications,” Prog. Opt. 16, 119–232 (1978).
[CrossRef]

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1968 (1)

L. B. Lesem, P. M. Hirsch, J. A. Jordan, Commun. ACM 11, 661–674 (1968).
[CrossRef]

Dallas, W. J.

W. J. Dallas, “Computer-generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed., Vol. 41 of Topics in Applied Physics (Springer-Verlag, Berlin, 1980), pp. 291–366.
[CrossRef]

Fienup, J. R.

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980); “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, Commun. ACM 11, 661–674 (1968).
[CrossRef]

Jordan, J. A.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, Commun. ACM 11, 661–674 (1968).
[CrossRef]

Lee, W.-H.

W.-H. Lee, “Computer-generated holograms: techniques and applications,” Prog. Opt. 16, 119–232 (1978).
[CrossRef]

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, Commun. ACM 11, 661–674 (1968).
[CrossRef]

Papoulis, A.

A. Papoulis, Systems and Transformations with Applications in Optics (McGraw-Hill, New York, 1968); E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Commun. ACM (1)

L. B. Lesem, P. M. Hirsch, J. A. Jordan, Commun. ACM 11, 661–674 (1968).
[CrossRef]

Opt. Eng. (1)

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980); “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

Optik (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Prog. Opt. (1)

W.-H. Lee, “Computer-generated holograms: techniques and applications,” Prog. Opt. 16, 119–232 (1978).
[CrossRef]

Other (2)

W. J. Dallas, “Computer-generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed., Vol. 41 of Topics in Applied Physics (Springer-Verlag, Berlin, 1980), pp. 291–366.
[CrossRef]

A. Papoulis, Systems and Transformations with Applications in Optics (McGraw-Hill, New York, 1968); E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).

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Figures (7)

Fig. 1
Fig. 1

Illustration of how the spectrum is modified by sampling an object. In (a) the spectrum ũ(ν) of the continuous object u(x) in (b) is shown, and in (c) the spectrum p ˜(ν) of the sampled object p(x) in (d) is shown.

Fig. 2
Fig. 2

One period of the phase distribution of Eq. (6) superposed onto the periodic spectrum of Fig. 1(c) is given in (a), and the same portion of the reconstruction as in Fig. 1(d) is given in (b).

Fig. 3
Fig. 3

One period of the phase distribution of Eq. (10) superposed onto the periodic spectrum of Fig. 1(c) is given in (a), and the same portion of the reconstruction as in Fig. 1(d) is given in (b).

Fig. 4
Fig. 4

Simulated reconstructions of a subhologram constructed with a random phase (a) and an iterated phase (b).

Fig. 5
Fig. 5

Optical reconstruction of a CGH of an object sampled in 322 points constructed by a mere repetition of the subholograms.

Fig. 6
Fig. 6

Optical reconstruction of a CGH of an object sampled in 322 points constructed by repetition of phase-manipulated (P = 4, r = 1) subholograms.

Fig. 7
Fig. 7

Optical reconstruction of a CGH of an object sampled in 322 points constructed by repetition of phase-manipulated (P2 = r = 4) subholograms.

Equations (20)

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u ˜ ( ν ) = FT [ u ( x ) ] = - + u ( x ) exp ( - i 2 π ν x ) d x ,
p ˜ ( ν ) = u ˜ ( ν ) * comb ( ν T Δ ν ) ,
p ( x ) = u ( x ) comb ( x δ x ) ,
p ˜ ( ν ) = j = 0 P - 1 p ˜ j ( ν ) ,
p j ( x ) = u ( x ) comb ( x δ x / P ) exp ( i 2 π j x / δ x ) .
v ˜ a ( ν ) = j = 0 P - 1 u ˜ ( ν ) * { comb ( ν - j T Δ ν P T Δ ν ) exp [ i 2 π ( a j ν + b j ) ] } ,
v a ( x ) = u ( x ) j = 0 P - 1 comb ( x - a j δ x / P ) exp [ i 2 π ( j x δ x + b j ) ] .
a j = j δ x P 2 = j P 2 T Δ ν ,
v a ( x ) = u ( x ) comb ( x δ x / P 2 ) e i ϕ ( x ) .
v ˜ b ( ν ) = j = 0 P - 1 [ u ˜ ( ν ) * comb ( ν - j T Δ ν P T Δ ν ) ] exp [ i 2 π ( a j ν + b j ) ] ,
v b ( x ) = j = 0 P - 1 u ( x - a j ) comb ( x - a j δ x / P ) × exp { i 2 π [ j ( x - a j ) δ x + b j ] } = j = 0 P - 1 p j ( x - a j ) exp ( i 2 π b j ) .
v ˜ t ( ν ) = v ˜ ( ν ) rect ( ν R T Δ ν ) .
v t ( x ) = v ( x ) * sinc ( x δ x / R ) .
u ( x ) = m = - u m sinc ( m - x Δ ν ) ,
u ( x ) = m = 0 M - 1 u m sinc ( m - x Δ ν )
u ˜ k = m = 0 M - 1 u m exp ( - i 2 π m k M ) ,
u m = w m exp ( i ϕ m ) .
u ( x ) = m = 0 M - 1 w m exp ( i ϕ m ) sinc ( m - x Δ ν ) .
R = r P 2 ,
h k l = b + u ˜ k l cos ( π ( k + l ) 2 + ϕ ˜ k l ) ,

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