Abstract

We propose a technique for determining the pair-correlation function of a quasi-homogeneous medium. The method uses the variation of the spatial-coherence properties of the incident beam to generate two separate volumes of coherence where the field is correlated. Using this specially prepared beam, we reconstruct experimentally the correlation function of a scattering potential by recording the scattered intensity in only one direction.

© 2004 Optical Society of America

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References

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  1. E. Wolf, “Principles of development of diffraction tomography,” in Trends in Optics, A. Consortini, ed. (Academic, San Diego, Calif., 1996), Vol. 3, pp. 83–110.
  2. D. G. Fisher, E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133, 17–21 (1997).
    [CrossRef]
  3. P. M. Voyles, J. M. Gibson, M. M. J. Treacy, “Fluctuation microscopy: a probe of atomic correlations in disordered materials,” J. Electron Microsc. 49, 259–266 (2000).
    [CrossRef]
  4. J. Rosen, M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. 39, 4107–4111 (2000).
    [CrossRef]
  5. W. Wang, H. Kozaki, J. Rosen, M. Takeda, “Synthesis of longitudinal coherence functions by spatial modulation of an extended light source: a new interpretation and experimental verifications,” Appl. Opt. 41, 1962–1971 (2002).
    [CrossRef] [PubMed]
  6. W. H. Carter, E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85–90 (1988).
    [CrossRef]
  7. C. Iaconis, I. A. Walmsley, “Direct measurement of the two-point field correlation function,” Opt. Lett. 21, 1783–1785 (1996).
    [CrossRef] [PubMed]
  8. 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]
  9. J. Rosen, A. Yariv, “Longitudinal partial coherence of optical radiation,” Opt. Commun. 117, 8–12 (1995).
    [CrossRef]
  10. A. Zarubin, “Three-dimensional generalization of the Van Cittert–Zernike theorem to wave and particle scattering,” Opt. Commun. 100, 491–507 (1993).
    [CrossRef]
  11. E. Baleine, A. Dogariu, “Variable coherence tomography,” Opt. Lett. 29, 1233–1235 (2004).
    [CrossRef] [PubMed]
  12. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Chap. 3, pp. 97–102.

2004 (1)

2002 (1)

2000 (2)

J. Rosen, M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. 39, 4107–4111 (2000).
[CrossRef]

P. M. Voyles, J. M. Gibson, M. M. J. Treacy, “Fluctuation microscopy: a probe of atomic correlations in disordered materials,” J. Electron Microsc. 49, 259–266 (2000).
[CrossRef]

1997 (1)

D. G. Fisher, E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133, 17–21 (1997).
[CrossRef]

1996 (1)

1995 (1)

J. Rosen, A. Yariv, “Longitudinal partial coherence of optical radiation,” Opt. Commun. 117, 8–12 (1995).
[CrossRef]

1994 (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]

1993 (1)

A. Zarubin, “Three-dimensional generalization of the Van Cittert–Zernike theorem to wave and particle scattering,” Opt. Commun. 100, 491–507 (1993).
[CrossRef]

1988 (1)

W. H. Carter, E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85–90 (1988).
[CrossRef]

Baleine, E.

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]

Carter, W. H.

W. H. Carter, E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85–90 (1988).
[CrossRef]

Dogariu, A.

Fisher, D. G.

D. G. Fisher, E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133, 17–21 (1997).
[CrossRef]

Gibson, J. M.

P. M. Voyles, J. M. Gibson, M. M. J. Treacy, “Fluctuation microscopy: a probe of atomic correlations in disordered materials,” J. Electron Microsc. 49, 259–266 (2000).
[CrossRef]

Iaconis, C.

Kozaki, H.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Chap. 3, pp. 97–102.

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]

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]

Rosen, J.

Takeda, M.

Treacy, M. M. J.

P. M. Voyles, J. M. Gibson, M. M. J. Treacy, “Fluctuation microscopy: a probe of atomic correlations in disordered materials,” J. Electron Microsc. 49, 259–266 (2000).
[CrossRef]

Voyles, P. M.

P. M. Voyles, J. M. Gibson, M. M. J. Treacy, “Fluctuation microscopy: a probe of atomic correlations in disordered materials,” J. Electron Microsc. 49, 259–266 (2000).
[CrossRef]

Walmsley, I. A.

Wang, W.

Wolf, E.

D. G. Fisher, E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133, 17–21 (1997).
[CrossRef]

W. H. Carter, E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85–90 (1988).
[CrossRef]

E. Wolf, “Principles of development of diffraction tomography,” in Trends in Optics, A. Consortini, ed. (Academic, San Diego, Calif., 1996), Vol. 3, pp. 83–110.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Chap. 3, pp. 97–102.

Yariv, A.

J. Rosen, A. Yariv, “Longitudinal partial coherence of optical radiation,” Opt. Commun. 117, 8–12 (1995).
[CrossRef]

Zarubin, A.

A. Zarubin, “Three-dimensional generalization of the Van Cittert–Zernike theorem to wave and particle scattering,” Opt. Commun. 100, 491–507 (1993).
[CrossRef]

Appl. Opt. (2)

J. Electron Microsc. (1)

P. M. Voyles, J. M. Gibson, M. M. J. Treacy, “Fluctuation microscopy: a probe of atomic correlations in disordered materials,” J. Electron Microsc. 49, 259–266 (2000).
[CrossRef]

Opt. Commun. (4)

D. G. Fisher, E. Wolf, “Theory of diffraction tomography for quasi-homogeneous random objects,” Opt. Commun. 133, 17–21 (1997).
[CrossRef]

J. Rosen, A. Yariv, “Longitudinal partial coherence of optical radiation,” Opt. Commun. 117, 8–12 (1995).
[CrossRef]

A. Zarubin, “Three-dimensional generalization of the Van Cittert–Zernike theorem to wave and particle scattering,” Opt. Commun. 100, 491–507 (1993).
[CrossRef]

W. H. Carter, E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85–90 (1988).
[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)

E. Wolf, “Principles of development of diffraction tomography,” in Trends in Optics, A. Consortini, ed. (Academic, San Diego, Calif., 1996), Vol. 3, pp. 83–110.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Chap. 3, pp. 97–102.

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Figures (7)

Fig. 1
Fig. 1

Typical scattering configuration for variable-coherence tomography.

Fig. 2
Fig. 2

Intensity pattern in the plane (ξ, η) of the incoherent source shown in Fig. 1.

Fig. 3
Fig. 3

The degree of spatial coherence μ(i)(Δx,Δy,Δz) of the incident field produced by the source shown in Fig. 2 is plotted as a function of the separation of the two points Δy and Δz and for Δx=0.

Fig. 4
Fig. 4

Scattering medium.

Fig. 5
Fig. 5

Experimental setup for variable-coherence tomography.

Fig. 6
Fig. 6

Oscillating part of the scattered intensity for x0=0.2 mm and three different values of y0. From top to bottom y0=-3 mm, -3.3 mm, -2.6 mm. The dashed curve is the intensity envelope.

Fig. 7
Fig. 7

Envelopes of the scattered intensity for x0=0.2 mm and y0=-3 mm, -3.3 mm, -2.6 mm.

Equations (29)

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U(s)(r, ω)=VF(r, ω)U(i)(r, ω) exp(ik|r-r|)|r-r|d3r.
F(r, ω)=14π k2[n2(r, ω)-1],
W(α)(r1, r2, ω)=U(α)*(r1, ω)U(α)(r2, ω)U,
W(i)(r1, r2)=I(i)r1+r22μ(i)(Δr),
CF(r1, r2, ω)=F*(r1, ω)F(r2, ω)FSFr1+r22, ωμF(Δr, ω),
W(s)(r1, r2, ω)=VVCF(r1, r2, ω)×W(i)(r1, r2, ω)×exp(-ik|r1-r1|)|r1-r1|×exp(ik|r2-r2|)|r2-r2|d3r1d3r2.
exp(ik|ru-r|)|ru-r|=exp(ikr)rexp(-ikur),
I(s)(ru)=1r2VVSFr1+r22μF(Δr)×I(i)r1+r22μ(i)(Δr)×exp(ikuΔr)d3r1d3r2.
μ(i)(Δr)=exp(-ikΔz) I0(ξ, η)expi 2πλf (ξΔx+ηΔy)+i πΔzλf2 (ξ2+η2)dξdηI0(ξ, η)dξdη,
I0(ξ, η)=121+m cos2π (ξ-x0)2+(η-y0)2α2-βDisk(ξ2+η2)1/2R.
μ(i)(Δr)=exp(-ikΔz)g(Δr)+m2exp{i[ϕ(Δr0)-kΔz]}g(Δr+Δr0)+m2exp{-i[ϕ(Δr0)+kΔz]}g(Δr-Δr0),
Δr0=2λfα2 (-x0x-y0y+fz)
g(Δr)=12Disk(ξ2+η2)1/2Rexpi 2πλf (ξΔx+ηΔy)+i πΔzλf2 (ξ2+η2)dξdηI0(ξ, η)dξdη.
I(s)(ru, Δr0)
=DIμF(Δr)g(Δr)exp[ik(u-z)Δr]d3Δr+m2exp[iϕ(Δr0)]μF(Δr)g(Δr+Δr0)×exp[ik(u-z)Δr]d3Δr+m2×exp[-iϕ(Δr0)]μF(Δr)g(Δr-Δr0)×exp[ik(u-z)Δr]d3Δr,
DI=1r2VSF(r2)I(i)(r2)d3r2.
I(s)(ru, Δr0)
=DIG(ku, 0)1+mG(ku, Δr0)G(ku, 0)cos[k(u-z)Δr0-ϕ(Δr0)-ϕG(ku, Δr0)+ϕG(ku, 0)].
G(ku, Δr0)=μF(r-Δr0)g(r)exp[ik(u-z)r]d3r,
μF(r)=δ(r)+mFδ(r+Δrscat)+mFδ(r-Δrscat),
G(ku, Δr0)
=g(Δr0)exp[ik(u-z)Δr0]+mFg(Δr0-Δrscat)×exp[ik(u-z)(Δr0-Δrscat)]+mFg×(Δr0+Δrscat)exp[ik(u-z)(Δr0+Δrscat)].
G(ku, Δr0)mFg(Δr0-Δrscat)×exp[ik(u-z)(Δr0-Δrscat)].
I(s)(ru, Δr0)=DI{1+mFm|g(Δr0-Δrscat)|×cos[k(u-z)Δr0-ϕ(Δr0)-ϕG(ku, Δr0)+ϕG(ku, 0)]}.
F(r)=iδ(r-ri)+δ(r-ri-Δrscat),
CF(r1, r2)δ(Δr)+12 δ(Δr+Δrscat)+12 δ(Δr-Δrscat),
IA(x, y)=121+cos2πλ ΔL (x-xA)2+(y-yA)2fc2-4πλ ΔL,
-2 fsΔL(χfc)2 x0=Δxscat,-2 fsΔL(χfc)2 y0=Δyscat,
2 fs2ΔL(χfc)2=Δzscat.

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