Abstract

The use of a single off-axis holographic lens for Fourier transformation results in phase errors that degrade its performance. A configuration of two identical off-axis holographic elements is proposed for performing Fourier transformation without phase errors. Such a configuration can be readily folded to form a compact and cascadable element that can be conveniently incorporated into optical correlators. The grating functions of the holographic elements needed for performing the desired transformation were used to record the planar configuration, which was then experimentally evaluated. The results reveal that phase errors were indeed eliminated, in close agreement with the calculations.

© 1995 Optical Society of America

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References

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  1. R. C. Fairchild, R. J. Fienup, Opt. Eng. 21, 133 (1982).
  2. J. Kedmi, A. A. Friesem, J. Opt. Soc. Am. A 3, 2011 (1986).
    [Crossref]
  3. E. Hasman, A. A. Friesem, J. Opt. Soc. Am. A 6, 62 (1989).
    [Crossref]
  4. Y. Amitai, J. W. Goodman, Opt. Lett. 16, 952 (1991).
    [Crossref] [PubMed]
  5. R. Blank, A. A. Friesem, Opt. Eng. 31, 544 (1992).
    [Crossref]
  6. J. Jahns, A. Huang, Opt. Commun. 76, 313 (1989).
    [Crossref]
  7. Y. Amitai, J. W. Goodman, Appl. Opt. 30, 2376 (1991).
    [Crossref] [PubMed]
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5, p. 80.

1992 (1)

R. Blank, A. A. Friesem, Opt. Eng. 31, 544 (1992).
[Crossref]

1991 (2)

1989 (2)

1986 (1)

1982 (1)

R. C. Fairchild, R. J. Fienup, Opt. Eng. 21, 133 (1982).

Amitai, Y.

Blank, R.

R. Blank, A. A. Friesem, Opt. Eng. 31, 544 (1992).
[Crossref]

Fairchild, R. C.

R. C. Fairchild, R. J. Fienup, Opt. Eng. 21, 133 (1982).

Fienup, R. J.

R. C. Fairchild, R. J. Fienup, Opt. Eng. 21, 133 (1982).

Friesem, A. A.

Goodman, J. W.

Hasman, E.

Huang, A.

J. Jahns, A. Huang, Opt. Commun. 76, 313 (1989).
[Crossref]

Jahns, J.

J. Jahns, A. Huang, Opt. Commun. 76, 313 (1989).
[Crossref]

Kedmi, J.

Appl. Opt. (1)

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

Opt. Commun. (1)

J. Jahns, A. Huang, Opt. Commun. 76, 313 (1989).
[Crossref]

Opt. Eng. (2)

R. C. Fairchild, R. J. Fienup, Opt. Eng. 21, 133 (1982).

R. Blank, A. A. Friesem, Opt. Eng. 31, 544 (1992).
[Crossref]

Opt. Lett. (1)

Other (1)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5, p. 80.

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Figures (5)

Fig. 1
Fig. 1

On-axis optical Fourier transformation with two refractive lenses.

Fig. 2
Fig. 2

Folded optical Fourier-transformation scheme with a holographic doublet.

Fig. 3
Fig. 3

Unfolded configuration of the doublet. HOE’s, holographic optical elements.

Fig. 4
Fig. 4

Experimental readout arrangement.

Fig. 5
Fig. 5

Experimental results: (a) input, (b) output from the unfolded configuration, (c) output from the folded planar configuration.

Equations (11)

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ϕ h = - 2 π λ ( x 2 + y 2 2 f ) ,
ϕ h = - 2 π λ ( x 2 cos 2 θ o + y 2 2 F - x sin θ o ) ,
U 1 ( x i , y i ) = t ( x i , y i ) exp [ - i 2 π λ × ( x i 2 cos 2 θ o + y i 2 2 F - x i sin θ o ) ] = t ( x i , y i ) exp ( + i 2 π λ x i sin θ o ) × exp [ - i 2 π λ ( x i 2 cos 2 θ o + y i 2 2 F ) ] ,
U 1 ( x i , y i ) = A 0 ( f x , f y ) exp { + i 2 π λ [ ( λ f x + sin θ o ) x i + λ f y y i ] } d f x d f y ,
A 0 ( f x , f y ) = t ( x i , y i ) exp ( - i 2 π λ x i 2 cos 2 θ o + y i 2 2 F ) × exp [ - i 2 π λ ( λ f x x i + λ f y y i ) ] d x i d y i .
α = sin θ o + f x λ ,             β = f y λ ,             γ = [ 1 - α 2 - β 2 ] 1 / 2 .
γ cos θ o [ 1 - λ f x sin θ o cos 2 θ o - λ 2 ( f x 2 + f y 2 ) 2 cos 2 θ o - 1 2 λ 2 f x 2 sin 2 θ o cos 4 θ o ] .
x i = x 0 - F sin θ o ,             y i = y o ,
U 2 ( x o , y o ) = A 0 ( f x , f y ) × exp [ + i 2 π λ ( x 0 2 cos 2 θ o + y o 2 2 F ) ] × exp [ i 2 π λ ( λ f x x o + λ f y y o - F λ 2 2 × 1 cos 2 θ o f x 2 - F λ 2 2 f y 2 ) ] d f x d f y .
U 2 ( x o , y o ) = C t ( x i , y i ) exp [ - i 2 π ( x i x o cos 2 θ o λ F + y i y o λ F ) ] d x i d y i ,
f ˜ x = x o cos 2 θ o λ F ,             f ˜ y = y o λ F .

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