Abstract

An optical setup to obtain all the fractional correlations of a one-dimensional input in a single display is implemented. The system works as a multichannel parallel correlator for a continuous set of fractional orders and presents a variable shift variance. Some experimental results together with computer simulations are performed to illustrate the performance of our proposal.

© 2001 Optical Society of America

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References

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2000

1997

1996

A. W. Lohmann, Z. Zalevsky, D. Mendlovic, “Synthesis of pattern recognition filters for fractional Fourier processing,” Opt. Commun. 128, 199–204 (1996).
[CrossRef]

1995

1994

1993

Andrés, P.

Arizaga, R.

Bitran, Y.

Dorsch, R. G.

Furlan, W. D.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw Hill, New York, 1996).

Granieri, S.

Hua, J.

Li D, G.

Liu, L.

Lohmann, A. W.

Lohmann, W.

Mendlovic, D.

Ozaktas, H. M.

Pellat-Finet, P.

Saavedra, G.

Sicre, E.

Sicre, E. E.

Tajahuerce, E.

Vander Lugt, A.

A. Vander Lugt, Optical Signal Processing (Wiley, New York, 1992).

Zalevsky, Z.

A. W. Lohmann, Z. Zalevsky, D. Mendlovic, “Synthesis of pattern recognition filters for fractional Fourier processing,” Opt. Commun. 128, 199–204 (1996).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

A. W. Lohmann, Z. Zalevsky, D. Mendlovic, “Synthesis of pattern recognition filters for fractional Fourier processing,” Opt. Commun. 128, 199–204 (1996).
[CrossRef]

Opt. Lett.

Other

A. Vander Lugt, Optical Signal Processing (Wiley, New York, 1992).

J. W. Goodman, Introduction to Fourier Optics (McGraw Hill, New York, 1996).

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Figures (6)

Fig. 1
Fig. 1

Top view of the optical setups to achieve (a) the filter H, which stores the Radon–Wigner display of the 1-D input t(x 0) and (b) the parallel fractional correlation between t(x 0) and t(x 0 + Δ) at the output plane. M1 and M2, mirrors; BS, beam splitter; L, varifocal lens; LC, cylindrical lens.

Fig. 2
Fig. 2

Intensity patterns of the output plane of the correlator: (a) and (b) computer simulations for displacements Δ = 0.5 mm and Δ = 0.7 mm, respectively; (c) and (d) experimental results.

Fig. 3
Fig. 3

Normalized fractional correlations for several fractional orders p obtained from different slices of the intensity distributions shown in Figs. 2(a) and 2(c) for the case of Δ = 0.5 mm: (a) numerical results; (b) experimental results.

Fig. 4
Fig. 4

Normalized fractional correlations for several fractional orders p obtained from different slices of the intensity distributions shown in Figs. 2(b) and 2(d) for the case of Δ = 0.7 mm: (a) numerical results; (b) experimental results.

Fig. 5
Fig. 5

Amplitude transmittance of the nonsymmetric double slit.

Fig. 6
Fig. 6

Fractional correlations for several fractional orders p obtained at the output plane of the setup of Fig. 1. Intensity patterns as a gray level (left), and the corresponding three-dimensional plot (right). The filter is registered for the double slit of Fig. 5. The input is the same object but with two different shifts (see the main text).

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

pfx, y=i expipπ/2sinpπ/2expiπx2+y2tanpπ/2×- fx0, y0expiπx02+y02tanpπ/2×exp-i2πxx0+yy0sinpπ/2dx0dy0,
Uzp, Rpx, y=exp-iπx2+y2λz1-Mp-Rpz Mp2 Rp×ptx0Mp, y0Mp,
Rp=s2λ-1 tan pπ/21+s2λz-1 tan pπ/2,
Mp=1+tan pπ/2 tan pπ/41+s2λz-1 tan pπ/2.
Uhx=exp-iπ xh2λz 1-Mp2-Rpz Rp+1a+1apML2 ptx0,
MLp=a1-s2λz-1 tan pπ/2l1-s2λ-1 tan pπ/21z-1l,
Cpx=-1{ptx0 p* tx0},
tx0=rectx0/a,
tx0=tx0+Δ,

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