Abstract

A numerical algorithm based on a single fast Fourier transform is proposed. Its precision and calculation efficiency show better performance than those of previously published algorithms. It is also shown that if specific conditions are met, the numerical calculations of two successive fractional Fourier transforms produce results that are similar to the analytical solution.

© 1998 Optical Society of America

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References

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  1. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation. II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [CrossRef]
  2. Adolf Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  3. Y. Bitran, D. Mendlovic, R. Dorsch, A. Lohmann, H. Ozaktas, “Fractional Fourier transform: simulations and experimental results,” Appl. Opt. 34, 1329–1332 (1995).
    [CrossRef] [PubMed]
  4. J. Garcia, D. Mas, R. Dorsch, “Fractional-Fourier-transform calculation through the fast-Fourier-transform algorithm,” Appl. Opt. 35, 7013–7018 (1996).
    [CrossRef] [PubMed]
  5. J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
    [CrossRef]
  6. D. Mendlovic, H. M. Ozaktas, A. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
    [CrossRef] [PubMed]
  7. E. O. Brigham, The Fast Fourier Transform and Its Applications (Prentice-Hall, Englewood Cliffs, N.J., 1988).
  8. J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), pp. 16–19.

1996 (1)

1995 (2)

1993 (2)

1965 (1)

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Bitran, Y.

Brigham, E. O.

E. O. Brigham, The Fast Fourier Transform and Its Applications (Prentice-Hall, Englewood Cliffs, N.J., 1988).

Cooley, J. W.

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Dorsch, R.

Garcia, J.

Goodman, J.

J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), pp. 16–19.

Lohmann, A.

Lohmann, Adolf

Mas, D.

Mendlovic, D.

Ozaktas, H.

Ozaktas, H. M.

Tukey, J. W.

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Plots of the FRT with p=0.5, N=512, and s2=0.0633 mm2, obtained with the S-FFT, for a rectangle with width a=0.5 mm: (a) normalized amplitude distribution, (b) phase distribution, (c) computed amplitude values versus exact analytic amplitude values, (d) computed phase values versus exact analytic phase values.

Fig. 2
Fig. 2

Zoomed plots of the region [0, 0.05] of (a) Fig. 1(c) and (b) Fig. 3(c).

Fig. 3
Fig. 3

Plots of the FRT with p=0.5, N=512 and s2=0.0633 mm2, obtained with the D-FFT, for a rectangle with width a=0.5 mm: (a) normalized amplitude distribution, (b) phase distribution, (c) computed amplitude values versus exact analytic amplitude values, (d) computed phase values versus exact analytic phase values.

Fig. 4
Fig. 4

Results of the application of the S-FFT algorithm to the calculation of the cascade of two FRT’s of orders p=0.4 and p2=0.6, when the input is a rectangle function, with width a=0.5 mm. N=256, and s2=0.0633 mm2: (a) normalized amplitude distribution, (b) phase distribution.

Equations (28)

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F(p)[u0(x0)]=u0(x0)exp[iπ(xp2+x02)/(s2 tan ϕ)]×exp[-i2πxpx0/(s2 sin ϕ)]dx0,
F(p)[u0(sx sin ϕ)]
=U(ξ)=s sin ϕu0(sx sin ϕ)×exp[iπ(ξ2+x2 sin2 ϕ)/tan ϕ]×exp(-i2πξx)dx=s sin ϕ exp(iπξ2/tan ϕ)u(x)×exp(iπx2 sin ϕ cos ϕ)exp(-i2πξx)dx,
ξ=xps,x=x0s sin ϕ,
u(x)=u0(sx sin ϕ).
U(kδξ)=sδx exp(iπk2δξ2/tan ϕ)sin ϕ×n=-N/2(N/2)-1u(nδx)exp(-i2πnkδxδξ+iπn2δx2 sin ϕ cos ϕ),
x=nδx,ξ=kδξ,
Uk=sΔx sin ϕNexp[iπk2/(Δx2 tan ϕ)]×n=-N/2N/2-1u˜n exp(-i2πnk/N)=sΔx sin ϕNexp[iπk2/(Δx2 tan ϕ)]DFT{u˜n},
u˜n=u(nΔx/N)exp[iπn2Δx2 sin 2ϕ/(2N2)]
Uk=Δx0Nexp[iπk2s2 sin 2ϕ/(2Δx02)]×n=-N/2N/2-1u0(nΔx0/N)×exp[iπn2Δx02/(N2s2 tan ϕ)]×exp(-i2πnk/N).
δx0s2 tan ϕΔx0.
δxps2 tan ϕΔxp.
Δxp=Ns2 sin ϕΔx0,δxp=s2 sin ϕΔx0,
Δx0=Nδx0,
Δx02s2 tan ϕN2Δx02s2 sin 2ϕ.
Ns2 sin 2ϕ21/2Δx0(Ns2 tan ϕ)1/2.
up(xp)=(-1)5/4s(tan ϕ)1/22exp(-iπxp2 tan ϕ/s2)×erf(-1)3/4π(-2xp+a cos ϕ)2s(cos ϕ sin ϕ)1/2+erf(-1)3/4π(2xp+a cos ϕ)2s(cos ϕ sin ϕ)1/2,
erf(x)=2π0x exp(-t2)dt,
T0=1νmax=2s2 tan ϕ1Δx0.
Δx0=k0T0.
Δxp1=Ns2 sin ϕ1/Δx0.
exp[iπxp12/(s2 tan ϕ1)]exp[iπxp12/(s2 tan ϕ2)].
T1=2s2/[Δxp1(1/tan ϕ1+1/tan ϕ2)].
T1=2s2 sin ϕ1 cos ϕ1Δxp1.
Δxp1=k1T1,
Δxp2=k2T2
Cp1(x)=F(1)(F(p)[u0(x0)]{F(p)[v0(x0)]}*),
Cp1,p2(x)=F(p2)(F(p1)[u0(x0)]{F(p1)[v0(x0)]}*),

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