Abstract

It is shown how a coherent optical signal that contains only a finite number of Hermite–Gauss modes can be reconstructed from the knowledge of its Radon–Wigner transform—associated with the intensity distribution in a fractional-Fourier-transform optical system—at only two transversal points. The proposed method can be generalized to any fractional system whose generator transform has a complete orthogonal set of eigenfunctions.

© 2002 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  9. H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic Press, San Diego, Calif., 1999), Vol. 106, pp. 239–291.
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    [CrossRef]
  12. L. Yu, W. D. Huang, M. C. Huang, Z. Z. Zhu, X. M. Zeng, W. Ji, “The Laguerre–Gaussian series representation of 2-dimensional fractional Fourier transform,” J. Phys. A Math. Gen. 31, 9353–9357 (1998).
    [CrossRef]

2000 (2)

1999 (1)

1998 (3)

1996 (1)

1994 (2)

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

1983 (1)

Alieva, T.

Almeida, L. B.

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

Bastiaans, M. J.

Beck, M.

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Calvo, M. L.

Chen, N. X.

Cong, W. X.

Dorsch, R. G.

Gu, B. Y.

Huang, M. C.

L. Yu, W. D. Huang, M. C. Huang, Z. Z. Zhu, X. M. Zeng, W. Ji, “The Laguerre–Gaussian series representation of 2-dimensional fractional Fourier transform,” J. Phys. A Math. Gen. 31, 9353–9357 (1998).
[CrossRef]

Huang, W. D.

L. Yu, W. D. Huang, M. C. Huang, Z. Z. Zhu, X. M. Zeng, W. Ji, “The Laguerre–Gaussian series representation of 2-dimensional fractional Fourier transform,” J. Phys. A Math. Gen. 31, 9353–9357 (1998).
[CrossRef]

Ji, W.

L. Yu, W. D. Huang, M. C. Huang, Z. Z. Zhu, X. M. Zeng, W. Ji, “The Laguerre–Gaussian series representation of 2-dimensional fractional Fourier transform,” J. Phys. A Math. Gen. 31, 9353–9357 (1998).
[CrossRef]

Kutay, M. A.

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic Press, San Diego, Calif., 1999), Vol. 106, pp. 239–291.

Lohmann, A. W.

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformation in optics,” in Progress in Optics XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1998), pp. 263–342.

McAlister, D. F.

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Mendlovic, D.

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformation in optics,” in Progress in Optics XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1998), pp. 263–342.

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic Press, San Diego, Calif., 1999), Vol. 106, pp. 239–291.

Ozaktas, H. M.

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic Press, San Diego, Calif., 1999), Vol. 106, pp. 239–291.

Raymer, M. G.

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Tamura, S.

Teague, M. R.

Tu, J.

Wolf, K. B.

Yu, L.

L. Yu, W. D. Huang, M. C. Huang, Z. Z. Zhu, X. M. Zeng, W. Ji, “The Laguerre–Gaussian series representation of 2-dimensional fractional Fourier transform,” J. Phys. A Math. Gen. 31, 9353–9357 (1998).
[CrossRef]

Zalevsky, Z.

Z. Zalevsky, R. G. Dorsch, “Gerchberg–Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
[CrossRef] [PubMed]

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformation in optics,” in Progress in Optics XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1998), pp. 263–342.

Zeng, X. M.

L. Yu, W. D. Huang, M. C. Huang, Z. Z. Zhu, X. M. Zeng, W. Ji, “The Laguerre–Gaussian series representation of 2-dimensional fractional Fourier transform,” J. Phys. A Math. Gen. 31, 9353–9357 (1998).
[CrossRef]

Zhu, Z. Z.

L. Yu, W. D. Huang, M. C. Huang, Z. Z. Zhu, X. M. Zeng, W. Ji, “The Laguerre–Gaussian series representation of 2-dimensional fractional Fourier transform,” J. Phys. A Math. Gen. 31, 9353–9357 (1998).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Signal Process. (1)

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

J. Phys. A Math. Gen. (1)

L. Yu, W. D. Huang, M. C. Huang, Z. Z. Zhu, X. M. Zeng, W. Ji, “The Laguerre–Gaussian series representation of 2-dimensional fractional Fourier transform,” J. Phys. A Math. Gen. 31, 9353–9357 (1998).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (1)

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Other (2)

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformation in optics,” in Progress in Optics XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1998), pp. 263–342.

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic Press, San Diego, Calif., 1999), Vol. 106, pp. 239–291.

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Figures (2)

Fig. 1
Fig. 1

Amplitude of f(x).

Fig. 2
Fig. 2

Phase of f(x).

Equations (23)

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Rα[f(x)](u)=Fα(u)=-K(α, x, u)f(x)dx,
K(α, x, u)=exp[i(α/2)]i2π sin αexpi(x2+u2)cos α-2ux2 sin α.
|Fα(u)|2=n=-Qn(u)exp(inα),
Qn(u)=12π02π|Fα(u)|2 exp(-inα)dα.
Ψn(x)=1(2nn!π)1/2exp-x22Hn(x),
f(x)=n=0fnΨn(x).
K(α,x,u)=n=0Ψn*(x)Ψn(u)exp(-inα),
Fα(u)=n=0fn exp(-inα)Ψn(u).
f(x)=n=LL+NfnΨn(x)
Fα(u)=n=LL+Nfn exp(-inα)Ψn(u),
Qn(u)=Q-n*(u)=m=LL+N-nfmfm+n*Ψm(u)Ψm+n*(u)
(n=0,, N),
ξm=fL+mfL,ηm=fL+N-mfL+N(m=1,, N)
qm(u)=QN-m(u)QN(u),ψm(u)=ΨL+m(u)ΨL(u),
ϕm(u)=ΨL+N-m*(u)ΨL+N*(u)(m=1,, N).
ξkψk(u)+ηk*ϕk(u)=qk(u)-ck(u),(k=1,, N)
c1(u)=0,
ck(u)=m=1k-1ξmηk-m*ψm(u)ϕk-m(u)(k=2,, N).
ψk(u1)ϕk(u1)ψk(u2)ϕk(u2)ξkηk*=qk(u1)-ck(u1)qk(u2)-ck(u2)
(k=1,, N),
QN(u)ΨL(u)ΨL+N*(u)=|fL|2ξN*,
f(x)=0.54x4 exp(-x2/2)(1+ix23),
K(α, x, u)=n=0Φn*(x)Φn(u)exp(inαl),

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