Abstract

A useful relationship between the fractional Fourier transform power spectra of a two-dimensional symmetric optical beam, on the one hand, and its Wigner distribution, on the other, is established. This relationship allows a significant simplification of the standard procedure for the reconstruction of the Wigner distribution from the field intensity distributions in the fractional Fourier domains. The Wigner distribution of a symmetric optical beam is analyzed, both in the coherent and in the partially coherent case.

© 2000 Optical Society of America

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References

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  1. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  2. M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
    [CrossRef]
  3. W. Mecklenbräuker, F. Hlawatsch, eds., The Wigner Distribution: Theory and Applications in Signal Processing (Elsevier, Amsterdam, 1997).
  4. M. G. Raymer, M. Beck, D. F. McAlister, “Complexwave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
    [CrossRef] [PubMed]
  5. D. F. McAlister, M. Beck, L. Clarke, A. Mayer, M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995).
    [CrossRef] [PubMed]
  6. Y. B. Karasik, “Expression of the kernel of a fractional Fourier transform in elementary functions,” Opt. Lett. 19, 769–770 (1994).
    [CrossRef] [PubMed]
  7. A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. 37, 2130–2141 (1998).
    [CrossRef]
  8. A. Sahin, M. A. Kutay, H. M. Ozaktas, “Nonseparable two-dimensional fractional Fourier transform,” Appl. Opt. 37, 5444–5453 (1998).
    [CrossRef]
  9. A. W. Lohmann, “Image rotation, Wigner rotation and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  10. M. J. Bastiaans, P. G. J. van de Mortel, “Wigner distribution function of a circular aperture,” J. Opt. Soc. Am. A 13, 1698–1703 (1996).
    [CrossRef]
  11. L. Yu, Y. Lu, X. Zeng, M. Huang, M. Chen, W. Huang, Z. Zhu, “Deriving the integral representation of a fractional Hankel transform from a fractional Fourier transform,” Opt. Lett. 23, 1158–1160 (1998).
    [CrossRef]
  12. E. Wolf, “New theory of partial coherence in the space-frequency domain. I. Spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
    [CrossRef]
  13. F. Gori, M. Santarsiero, G. Guattari, “Coherence and the spatial distribution of intensity,” J. Opt. Soc. Am. A 10, 673–679 (1993).
    [CrossRef]

1998

1996

1995

1994

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

Y. B. Karasik, “Expression of the kernel of a fractional Fourier transform in elementary functions,” Opt. Lett. 19, 769–770 (1994).
[CrossRef] [PubMed]

1993

1982

1978

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

1932

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, P. G. J. van de Mortel, “Wigner distribution function of a circular aperture,” J. Opt. Soc. Am. A 13, 1698–1703 (1996).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

Beck, M.

Chen, M.

Clarke, L.

Gori, F.

Guattari, G.

Huang, M.

Huang, W.

Karasik, Y. B.

Kutay, M. A.

Lohmann, A. W.

Lu, Y.

Mayer, A.

McAlister, D. F.

Mendlovic, D.

Ozaktas, H. M.

Raymer, M. G.

Sahin, A.

Santarsiero, M.

van de Mortel, P. G. J.

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Wolf, E.

Yu, L.

Zeng, X.

Zhu, Z.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

Opt. Lett.

Phys. Rev.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Phys. Rev. Lett.

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

Other

W. Mecklenbräuker, F. Hlawatsch, eds., The Wigner Distribution: Theory and Applications in Signal Processing (Elsevier, Amsterdam, 1997).

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Equations (52)

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Wf(r, q)=-f(r+12r)f*(r-12r)×exp(-i2πq·r)dr
Wf(r, q)=-Fπ/2(q+12q)Fπ/2*(q-12q)×exp(i2πq·r)dq,
whereFπ/2(q)=-f(r)exp(-i2πq·r)dr.
Wf(r, 0)=Wf(x1, x2, 0, 0)=--Fπ/2(12u1,12u2)×Fπ/2*(-12u1,-12u2)×exp[i2π(u1x1+u2x2)]du1du2.
Fα1,α2(u1, u2)=Rα1,α2[f(x1, x2)](u1, u2)=--K(α1, x1, u1)K(α2, x2, u2)×f(x1, x2)dx1dx2,
K(αn, xn, un)=×exp(iαn)/2i sin αnexpiπ(xn2+un2)cos αn-2unxnsin αn.
x1=R1 cos β1,x2=R2 cos β2,
u1=R1 sin β1,u2=R2 sin β2.
f(x1, x2)W˜f(R1, β1, R2, β2),
fractionalFTrotationofWD
Rα1,α2[f]W˜Rα1,α2[f](R1, β1, R2, β2)
=W˜f(R1, β1+α1, R2, β2+α2),
W˜f(R1, α1, R2, α2)=W˜Rα1,α2[f](R1, 0, R2, 0)=W˜Fα1,α2(R1, 0, R2, 0)=WFα1,α2(x1, x2, 0, 0).
W˜f(R1, α1, R2, α2)
=--Fπ/2+α1,π/2+α2(12u1,12u2)×Fπ/2+α1,π/2+α2*(-12u1,-12u2)×exp[i2π(u1R1+u2R2)]du1du2.
f(-r)=±f(r),
Wf(-r,-q)=±Wf(r,q).
Fα1,α2(-u1,-u2)=±Fα1,α2(u1, u2).
W˜f(R1, α1, R2, α2)
=±--|Fπ/2+α1, π/2+α2(12u1,12u2)|2×exp[i2π(u1R1+u2R2)]du1du2.
f(r)=f(|r|)=f(ρx),
Fπ/2(u1, u2)=Hπ/2(ρu)=0f(ρx)J0(2πρuρx)ρxdρx,
Wˆf(ρx, θx, ρu, θu)=0ρxdρx02πdθxf[rx(+)]f*[rx(-)]×exp[-i2πρuρx cos(θu-θx)],
Wˆf(ρx, θx, ρu, θu)=0ρudρu02πdθuFπ/2[ru(+)]Fπ/2*[ru(-)]×exp[i2πρxρu cos(θx-θu)],
rx(+)=[ρx2+14ρx2+ρxρx cos(θx-θx)]1/2,
rx(-)=[ρx2+14ρx2-ρxρx cos(θx-θx)]1/2,
ru(+)=[ρu2+14ρu2+ρuρu cos(θu-θu)]1/2,
ru(-)=[ρu2+14ρu2-ρuρu cos(θu-θu)]1/2,
x1=ρx cos θx,x1=ρx cos θx,
x2=ρx sin θx,x2=ρx sin θx,
u1=ρu cos θu,u1=ρu cos θu,
u2=ρu sin θu,u2=ρu sin θu.
ρx cos θx=x1=R1 cos α1,
ρx sin θx=x2=R2 cos α2,
ρu cos θu=u1=R1 sin α1,
ρu sin θu=u2=R2 sin α2.
Wˆf(ρx, θx-γ, ρu, θu)=Wˆf(ρx, θx, ρu, θu+γ),
Wˆf(ρx, θx-θu, ρu, 0)=Wˆf(ρx, θx, ρu, θu)=Wˆf(ρx, 0, ρu, θu-θx).
Wˆf(ρx, θ, ρu, θ)=Wˆf(ρx, 0, ρu, 0).
W˜f(R1, α1, R2, 0)=--|Fπ/2+α1,π/2(12u1,12u2)|2×exp[i2π(u1R1+u2R2)]du1du2,
W˜f(R1, 0, R2, 0)=--|Fπ/2,π/2(12u1,12u2)|2 ×exp[i2π(u1R1+u2R2)]du1du2=0|Hπ/2(12ρu)|2J0(2πRρu)ρudρu=Wˆf(R, 0, 0, 0),
Hα(ρu)=Rα[f(ρx)](ρu)=exp(iα)i sin α0exp[iπ(ρx2+ρu2)cot α]×J02πρxρusin αf(ρx)ρxdρx.
W˜f(R1, α, R2, α)=-|Fπ/2+α,π/2+α(12u1,12u2)|2×exp[i2π(u1R1+u2R2)]du1du2=0|Hπ/2+α(12ρu)|2J0(2πρuR)ρudρu.
Wˆf(ρx, θ, ρu, θ)=Wˆf(R cos α, θ, R sin α, θ)=0|Hπ/2+α(12ρ)|2J0(2πρR)ρdρ=Wˆf(ρx, 0, ρu, 0),
R2=ρx2+ρu2,
tan α=ρu/ρx.
G(r1, r2)=m=0λmgm(r1)gm*(r2).
W(r, q)=-G(r+12r, r-12r)×exp(-i2πq·r)dr
W(r, q)=m=0λmwm(r, q),
wm(r, q)=-gm(r+12r)gm*(r-12r)×exp(-i2πq·r)dr.
G(-r1,-r2)=±G(r1, r2)
G(r1, r2)=G(|r1|, |r2|),

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