Abstract

Dammann gratings are binary-phase Fourier holograms that are used to generate an array of point sources. Using symmetry and separability arguments, we extend Dammann’s method for generating one-dimensional symmetric arrays to general two-dimensional responses. Computer-generated holograms are used to verify the modified design method.

© 1989 Optical Society of America

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References

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  1. H. Dammann, K. Görtler, Opt. Commun. 3, 312 (1971).
    [CrossRef]
  2. H. Dammann, E. Klotz, Opt. Acta 24, 404 (1977).
  3. U. Killat, G. Rabe, W. Rave, Fiber Integ. Opt. 4, 159 (1982).
    [CrossRef]
  4. W. B. Veldkamp, J. R. Leger, G. J. Swanson, Opt. Lett. 11, 303 (1986).
    [CrossRef] [PubMed]
  5. J. Jahns, M. E. Prise, M. M. Downs, S. J. Walker, N. Streibl, J. Opt. Soc. Am. A 4(13), P69 (1987).
    [CrossRef]
  6. J. N. Mait, K.-H. Brenner, Appl. Opt. 27, 1692 (1988).
    [CrossRef] [PubMed]
  7. K.-H. Brenner, A. Huang, N. Streibl, Appl. Opt. 25, 3053 (1986).
  8. K.-H. Brenner, Appl. Opt. 25, 3061 (1986).
    [CrossRef] [PubMed]

1988

1987

J. Jahns, M. E. Prise, M. M. Downs, S. J. Walker, N. Streibl, J. Opt. Soc. Am. A 4(13), P69 (1987).
[CrossRef]

1986

1982

U. Killat, G. Rabe, W. Rave, Fiber Integ. Opt. 4, 159 (1982).
[CrossRef]

1977

H. Dammann, E. Klotz, Opt. Acta 24, 404 (1977).

1971

H. Dammann, K. Görtler, Opt. Commun. 3, 312 (1971).
[CrossRef]

Brenner, K.-H.

Dammann, H.

H. Dammann, E. Klotz, Opt. Acta 24, 404 (1977).

H. Dammann, K. Görtler, Opt. Commun. 3, 312 (1971).
[CrossRef]

Downs, M. M.

J. Jahns, M. E. Prise, M. M. Downs, S. J. Walker, N. Streibl, J. Opt. Soc. Am. A 4(13), P69 (1987).
[CrossRef]

Görtler, K.

H. Dammann, K. Görtler, Opt. Commun. 3, 312 (1971).
[CrossRef]

Huang, A.

K.-H. Brenner, A. Huang, N. Streibl, Appl. Opt. 25, 3053 (1986).

Jahns, J.

J. Jahns, M. E. Prise, M. M. Downs, S. J. Walker, N. Streibl, J. Opt. Soc. Am. A 4(13), P69 (1987).
[CrossRef]

Killat, U.

U. Killat, G. Rabe, W. Rave, Fiber Integ. Opt. 4, 159 (1982).
[CrossRef]

Klotz, E.

H. Dammann, E. Klotz, Opt. Acta 24, 404 (1977).

Leger, J. R.

Mait, J. N.

Prise, M. E.

J. Jahns, M. E. Prise, M. M. Downs, S. J. Walker, N. Streibl, J. Opt. Soc. Am. A 4(13), P69 (1987).
[CrossRef]

Rabe, G.

U. Killat, G. Rabe, W. Rave, Fiber Integ. Opt. 4, 159 (1982).
[CrossRef]

Rave, W.

U. Killat, G. Rabe, W. Rave, Fiber Integ. Opt. 4, 159 (1982).
[CrossRef]

Streibl, N.

J. Jahns, M. E. Prise, M. M. Downs, S. J. Walker, N. Streibl, J. Opt. Soc. Am. A 4(13), P69 (1987).
[CrossRef]

K.-H. Brenner, A. Huang, N. Streibl, Appl. Opt. 25, 3053 (1986).

Swanson, G. J.

Veldkamp, W. B.

Walker, S. J.

J. Jahns, M. E. Prise, M. M. Downs, S. J. Walker, N. Streibl, J. Opt. Soc. Am. A 4(13), P69 (1987).
[CrossRef]

Appl. Opt.

Fiber Integ. Opt.

U. Killat, G. Rabe, W. Rave, Fiber Integ. Opt. 4, 159 (1982).
[CrossRef]

J. Opt. Soc. Am. A

J. Jahns, M. E. Prise, M. M. Downs, S. J. Walker, N. Streibl, J. Opt. Soc. Am. A 4(13), P69 (1987).
[CrossRef]

Opt. Acta

H. Dammann, E. Klotz, Opt. Acta 24, 404 (1977).

Opt. Commun.

H. Dammann, K. Görtler, Opt. Commun. 3, 312 (1971).
[CrossRef]

Opt. Lett.

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Figures (6)

Fig. 1
Fig. 1

(a) Representation of a single period P ˜ e ( u ) of an even Dammann grating Pe(u); (b) representation of a single period P ˜ o ( u ) of an odd Dammann grating Po(u).

Fig. 2
Fig. 2

Results of a one-dimensional asymmetric design showing relative amplitudes of the diffracted orders. The dashed lines indicate the optic axis. (a) The desired response; (b) even component of the desired response; (c) odd component of the desired response; (d) predicted response for the second-order solution; (e) predicted response for the fourth-order solution; (f) predicted response for the zero-quantized fourth-order solution.

Fig. 3
Fig. 3

Experimental results for the zero-quantized fourth-order solution. The two spots in the center of the figure correspond to the two sources in Fig. 2(a) that lie to the right of the optic axis. The third desired source lies to their left. The brighter spot to the right corresponds to the 5 order in Fig. 2(f).

Fig. 4
Fig. 4

Two-dimensional Dammann gratings for generating a 3 × 3 array of impulses. The grating structure is shown on the left, and the resulting impulse array is shown on the right. Impulses whose amplitudes can be independently specified are shown within dashed boxes. A centrosymmetric separable response is represented in (a), centrosymmetric nonseparable responses are represented in (b) and (c), and an asymmetric nonseparable response is represented in (d).

Fig. 5
Fig. 5

Results of a two-dimensional nonseparable design showing the grating spatial response. The size of the spot indicates the relative amplitude. (a) The desired response and (b) the predicted response assuming that the impulse outside the dashed box in Fig. 4(c) has zero amplitude and is located at (u, υ) = (2, 0).

Fig. 6
Fig. 6

Experimental results for a two-dimensional design [compare with Fig. 5(b)].

Equations (5)

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p e ( x ) = n = p ˜ e ( n ) δ ( x n ) .
p ˜ e ( n ) = α q ˜ e ( n ) ,
p ( x ) = 1 2 [ p ( x ) + p ( x ) ] + 1 2 [ p ( x ) p ( x ) ] = p e ( x ) + p o ( x ) .
P ( u ) = P e ( u ) j P o ( u ) .
p ˜ o ( n ) = α q ˜ o ( n ) ,

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