Abstract

Recently two optical interpretations of the fractional Fourier transform operator were introduced. We address implementation issues of the fractional-Fourier-transform operation. We show that the original bulk-optics configuration for performing the fractional-Fourier-transform operation [J. Opt. Soc. Am. A 10, 2181 (1993)] provides a scaled output using a fixed lens. For obtaining a non-scaled output, an asymmetrical setup is suggested and tested. For comparison, computer simulations were performed. A good agreement between computer simulations and experimental results was obtained.

© 1995 Optical Society of America

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References

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  1. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation. Part I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  2. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transformations and their optical implementation. Part II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [CrossRef]
  3. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  4. A. Yariv, Optical Electronics, 3rd ed. (Holt, New York, 1985).
  5. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994)
    [CrossRef] [PubMed]

1994 (1)

1993 (3)

Lohmann, A. W.

Mendlovic, D.

Ozaktas, H. M.

Yariv, A.

A. Yariv, Optical Electronics, 3rd ed. (Holt, New York, 1985).

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Figures (5)

Fig. 1
Fig. 1

Setup for performing a two-dimensional fractional Fourier transform according to the WDF definition.

Fig. 2
Fig. 2

Setup for performing the fixed-scale asymmetrical FRT.

Fig. 3
Fig. 3

Input function (a) cross section and (b) two-dimensional, separable function obtained with a pair of different one-dimensional functions in outer-product form.

Fig. 4
Fig. 4

FRT experimental result obtained with the optical symmetrical setup of Fig. 1 with the parameters of Table 1. The FRT order is (a) a = 0.25, (b) a = 0.5, (c) a = 0.75, and (d) a = 1.

Fig. 5
Fig. 5

FRT experimental result obtained with the asymmetrical optical setup of Fig. 2 with the parameters of Table 2. The FRT order is (a) a = 0.25, (b) a = 0.5, (c) a = 0.75 and (d) a = 1.

Tables (2)

Tables Icon

Table 1 Parameters a Used for Performing the Symmetrical Setup Experiments

Tables Icon

Table 2 Parameters a Used for Performing the Asymmetrical Setup Experiments

Equations (11)

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Ψ m ( x ) = H m ( 2 x / ω ) exp ( x 2 / ω 2 ) ,
a [ u ( x ) ] = m A m Ψ m ( x ) exp ( i β m a L ) ,
f = f 1 / Q , z = f 1 R ,
R = tan ( ϕ / 2 ) , Q = sin ( ϕ ) , ϕ = a ( π / 2 ) .
a [ u ( x ) ] = C 1 u ( x 0 ) exp ( i π x 0 2 + x 2 λ f 1 tan ϕ ) × exp ( i 2 π x x 0 λ f 1 sin ϕ ) d x 0 ,
ω 2 = λ f 1 / π .
u ( x ) = u 0 ( x 0 ) exp [ i π λ f 1 ( x 0 2 1 Q B B + A Q A B + x 2 1 Q A B + A Q A B 2 x x 0 B + A Q A B ) ] d x 0 .
| u ( x ) | = | a [ u 0 ( x 0 ) ] | ,
A = sin ϕ ( 1 cos ϕ ) / Q cos ϕ , B = ( 1 cos ϕ ) / Q .
u a = a [ u 0 ] = Ψ β a Ψ 1 u 0 ,
z = f tan ϕ / 2 sin ϕ ,

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