Abstract

A method is presented for measuring the correlation between the optical fields scattered in different directions by an arbitrary three-dimensional turbid medium. The field angular correlation function is obtained by processing ensemble average intensity data, which are recorded experimentally by a single-channel Shack–Hartmann wave-front sensor. Some general properties of scattered light are expressed in terms of the field angular correlation function, and the correlation function is measured for transmission through a suspension of microspheres under nonballistic transport conditions.

© 1999 Optical Society of America

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References

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  1. J. A. Moon, P. R. Battle, M. Bashkansky, R. Mahon, M. D. Duncan, J. Reintjes, “Achievable spatial resolution of time-resolved transillumination imaging systems which utilize multiply scattered light,” Phys. Rev. E 53, 1142–1155 (1996).
    [CrossRef]
  2. See J. S. Preston, “Retro-reflexion by diffusing surfaces,” Nature (London) 213, 1007–1008 (1967) and references therein.
    [CrossRef]
  3. K. M. Watson, “Multiple scattering of electromagnetic waves in an underdense plasma,” J. Math. Phys. 10, 688–702 (1969).
    [CrossRef]
  4. D. Léger, J. C. Perrin, “Real-time measurement of surface roughness by correlation of speckle patterns,” J. Opt. Soc. Am. 66, 1210–1217 (1976).
    [CrossRef]
  5. T. R. Michel, K. A. O’Donnell, “Angular correlation functions of amplitudes scattered from a one-dimensional, perfectly conducting rough surface,” J. Opt. Soc. Am. A 9, 1374–1384 (1992).
    [CrossRef]
  6. Shechao Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988);Shechao Feng, “Novel correlations and fluctuations in speckle patterns,” in Scattering and Localization of Classical Waves in Random Media, Ping Sheng, ed., Vol. 8 of World Scientific Series on Directions in Condensed Matter Physics (World Scientific, Singapore, 1990).
    [CrossRef] [PubMed]
  7. E. N. Leith, B. G. Hoover, D. S. Dilworth, P. P. Naulleau, “Ensemble-averaged Shack–Hartmann wavefront sensing for imaging through turbid media,” Appl. Opt. 37, 3643–3650 (1998).
    [CrossRef]
  8. D. S. Wiersma, P. Bartolini, A. Lagendijk, R. Righini, “Localization of light in a disordered medium,” Nature (London) 390, 671–673 (1997).
    [CrossRef]
  9. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 5.
  10. E. W. Marchand, E. Wolf, “Angular correlation and the far-zone behavior of partially coherent fields,” J. Opt. Soc. Am. 62, 379–385 (1972).
    [CrossRef]
  11. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 10.

1998 (1)

1997 (1)

D. S. Wiersma, P. Bartolini, A. Lagendijk, R. Righini, “Localization of light in a disordered medium,” Nature (London) 390, 671–673 (1997).
[CrossRef]

1996 (1)

J. A. Moon, P. R. Battle, M. Bashkansky, R. Mahon, M. D. Duncan, J. Reintjes, “Achievable spatial resolution of time-resolved transillumination imaging systems which utilize multiply scattered light,” Phys. Rev. E 53, 1142–1155 (1996).
[CrossRef]

1992 (1)

1988 (1)

Shechao Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988);Shechao Feng, “Novel correlations and fluctuations in speckle patterns,” in Scattering and Localization of Classical Waves in Random Media, Ping Sheng, ed., Vol. 8 of World Scientific Series on Directions in Condensed Matter Physics (World Scientific, Singapore, 1990).
[CrossRef] [PubMed]

1976 (1)

1972 (1)

1969 (1)

K. M. Watson, “Multiple scattering of electromagnetic waves in an underdense plasma,” J. Math. Phys. 10, 688–702 (1969).
[CrossRef]

1967 (1)

See J. S. Preston, “Retro-reflexion by diffusing surfaces,” Nature (London) 213, 1007–1008 (1967) and references therein.
[CrossRef]

Bartolini, P.

D. S. Wiersma, P. Bartolini, A. Lagendijk, R. Righini, “Localization of light in a disordered medium,” Nature (London) 390, 671–673 (1997).
[CrossRef]

Bashkansky, M.

J. A. Moon, P. R. Battle, M. Bashkansky, R. Mahon, M. D. Duncan, J. Reintjes, “Achievable spatial resolution of time-resolved transillumination imaging systems which utilize multiply scattered light,” Phys. Rev. E 53, 1142–1155 (1996).
[CrossRef]

Battle, P. R.

J. A. Moon, P. R. Battle, M. Bashkansky, R. Mahon, M. D. Duncan, J. Reintjes, “Achievable spatial resolution of time-resolved transillumination imaging systems which utilize multiply scattered light,” Phys. Rev. E 53, 1142–1155 (1996).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 10.

Dilworth, D. S.

Duncan, M. D.

J. A. Moon, P. R. Battle, M. Bashkansky, R. Mahon, M. D. Duncan, J. Reintjes, “Achievable spatial resolution of time-resolved transillumination imaging systems which utilize multiply scattered light,” Phys. Rev. E 53, 1142–1155 (1996).
[CrossRef]

Feng, Shechao

Shechao Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988);Shechao Feng, “Novel correlations and fluctuations in speckle patterns,” in Scattering and Localization of Classical Waves in Random Media, Ping Sheng, ed., Vol. 8 of World Scientific Series on Directions in Condensed Matter Physics (World Scientific, Singapore, 1990).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 5.

Hoover, B. G.

Kane, C.

Shechao Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988);Shechao Feng, “Novel correlations and fluctuations in speckle patterns,” in Scattering and Localization of Classical Waves in Random Media, Ping Sheng, ed., Vol. 8 of World Scientific Series on Directions in Condensed Matter Physics (World Scientific, Singapore, 1990).
[CrossRef] [PubMed]

Lagendijk, A.

D. S. Wiersma, P. Bartolini, A. Lagendijk, R. Righini, “Localization of light in a disordered medium,” Nature (London) 390, 671–673 (1997).
[CrossRef]

Lee, P. A.

Shechao Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988);Shechao Feng, “Novel correlations and fluctuations in speckle patterns,” in Scattering and Localization of Classical Waves in Random Media, Ping Sheng, ed., Vol. 8 of World Scientific Series on Directions in Condensed Matter Physics (World Scientific, Singapore, 1990).
[CrossRef] [PubMed]

Léger, D.

Leith, E. N.

Mahon, R.

J. A. Moon, P. R. Battle, M. Bashkansky, R. Mahon, M. D. Duncan, J. Reintjes, “Achievable spatial resolution of time-resolved transillumination imaging systems which utilize multiply scattered light,” Phys. Rev. E 53, 1142–1155 (1996).
[CrossRef]

Marchand, E. W.

Michel, T. R.

Moon, J. A.

J. A. Moon, P. R. Battle, M. Bashkansky, R. Mahon, M. D. Duncan, J. Reintjes, “Achievable spatial resolution of time-resolved transillumination imaging systems which utilize multiply scattered light,” Phys. Rev. E 53, 1142–1155 (1996).
[CrossRef]

Naulleau, P. P.

O’Donnell, K. A.

Perrin, J. C.

Preston, J. S.

See J. S. Preston, “Retro-reflexion by diffusing surfaces,” Nature (London) 213, 1007–1008 (1967) and references therein.
[CrossRef]

Reintjes, J.

J. A. Moon, P. R. Battle, M. Bashkansky, R. Mahon, M. D. Duncan, J. Reintjes, “Achievable spatial resolution of time-resolved transillumination imaging systems which utilize multiply scattered light,” Phys. Rev. E 53, 1142–1155 (1996).
[CrossRef]

Righini, R.

D. S. Wiersma, P. Bartolini, A. Lagendijk, R. Righini, “Localization of light in a disordered medium,” Nature (London) 390, 671–673 (1997).
[CrossRef]

Stone, A. D.

Shechao Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988);Shechao Feng, “Novel correlations and fluctuations in speckle patterns,” in Scattering and Localization of Classical Waves in Random Media, Ping Sheng, ed., Vol. 8 of World Scientific Series on Directions in Condensed Matter Physics (World Scientific, Singapore, 1990).
[CrossRef] [PubMed]

Watson, K. M.

K. M. Watson, “Multiple scattering of electromagnetic waves in an underdense plasma,” J. Math. Phys. 10, 688–702 (1969).
[CrossRef]

Wiersma, D. S.

D. S. Wiersma, P. Bartolini, A. Lagendijk, R. Righini, “Localization of light in a disordered medium,” Nature (London) 390, 671–673 (1997).
[CrossRef]

Wolf, E.

Appl. Opt. (1)

J. Math. Phys. (1)

K. M. Watson, “Multiple scattering of electromagnetic waves in an underdense plasma,” J. Math. Phys. 10, 688–702 (1969).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Nature (London) (2)

D. S. Wiersma, P. Bartolini, A. Lagendijk, R. Righini, “Localization of light in a disordered medium,” Nature (London) 390, 671–673 (1997).
[CrossRef]

See J. S. Preston, “Retro-reflexion by diffusing surfaces,” Nature (London) 213, 1007–1008 (1967) and references therein.
[CrossRef]

Phys. Rev. E (1)

J. A. Moon, P. R. Battle, M. Bashkansky, R. Mahon, M. D. Duncan, J. Reintjes, “Achievable spatial resolution of time-resolved transillumination imaging systems which utilize multiply scattered light,” Phys. Rev. E 53, 1142–1155 (1996).
[CrossRef]

Phys. Rev. Lett. (1)

Shechao Feng, C. Kane, P. A. Lee, A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988);Shechao Feng, “Novel correlations and fluctuations in speckle patterns,” in Scattering and Localization of Classical Waves in Random Media, Ping Sheng, ed., Vol. 8 of World Scientific Series on Directions in Condensed Matter Physics (World Scientific, Singapore, 1990).
[CrossRef] [PubMed]

Other (2)

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 5.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 10.

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Figures (8)

Fig. 1
Fig. 1

Ray-optical schematic depicting two plane waves simultaneously scattered by an inhomogeneous medium.

Fig. 2
Fig. 2

Schematic of the general scattering problem, with an arbitrary two-dimensional (x, z) coherent input wave and a three-dimensional scattering medium.

Fig. 3
Fig. 3

Schematic arrangement for measuring the field angular correlation. The scattered wavefront and the Fourier intensity distribution are shown for an instantaneous observation. The elements to the right of the scattering medium constitute a single-channel wave-front sensor; those to the left are replaced in the laboratory by a Mach–Zehnder interferometer.

Fig. 4
Fig. 4

(a) Typical intensity distributions recorded by a wave-front sensor system at two sampling coordinates. In each plot the dashed curve indicates the incoherent component, which is determined by the routine described in the text. (b) The coherent component of each intensity distribution in (a).

Fig. 5
Fig. 5

Optical system for measurement of field angular correlation: TP, Thompson prism; ES, electronic shutter system; WP, half-wave plate; SA, sampling aperture; GT, Glan–Thompson polarizing prism.

Fig. 6
Fig. 6

Data polynomial vector projections onto their respective solution planes, for Λ=1.73 mm (crosses) and Λ=3.42 mm (circles). Elliptical loci are predicted by the theory.

Fig. 7
Fig. 7

Experimental results. (a) Real and (b) imaginary parts of the normalized field angular correlation function at three spatial periods Λ (mm) for transmission through a microsphere suspension. The mean incidence angle is small: f¯i=Δfi.

Fig. 8
Fig. 8

Geometrical construction for proving the necessary condition for nonzero field angular correlation.

Equations (38)

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f=sin θλ
Usα(fs, t)=[Uiα(fi)hαα(fs, t; fi)+Uiα¯(fi)hα¯α(fs, t; fi)]dfi.
Jα(fs1, fs2)=Usα(fs1, t)Usα*(fs2, t)=Uiα(fi1)hαα(fs1, t; fi1)dfi1×Uiα*(fi2)hαα*(fs2, t; fi2)dfi2=Uiα(fi1)Uiα*(fi2)hαα(fs1, t; fi1)×hαα*(fs2, t; fi2)dfi1dfi2.
fs1-fi1=fs2-fi2.
Jααα(f1, f2; f, f)hαα(f1, t; f)hαα*(f2, t; f)
Is(f)=Isα(f)+Isα¯(f),
Isα(f)=Jα(f, f)={|Uiα(fi)|2|hαα(f, t; fi)|2+|Uiα¯(fi)|2|hα¯α(f, t; fi)|2+2 Re[Uiα(fi)Uiα¯*(fi)Jα¯αα(f, f; fi, fi)]}dfi,
Jα(x1, x2)=Jα(f1, f2)×exp[2πj(f1x1-f2x2)]df1df2.
hαα(f, t; f|x)
=rectf2 fc[hαα(f, t;f)*w sinc(wf)exp(-2πjfx)],
Isα(f|x)=Jα(f, f|x)=w2 rectf2 fcUiα(fi1)Uiα*(fi2)×hαα(fs1, t; fi1)sinc[w(f-fs1)]×exp[-2πj(f-fs1)x]dfs1×hαα*(fs2, t; fi2)sinc[w(f-fs2)]×exp[2πj(f-fs2)x]dfs2dfi1dfi2=w2 rectf2 fc{Uiα(fi1)Uiα*(fi2)×Jααα(fs1, fs2; fi1, fi2)sinc[w(f-fs1)]×sinc[w(f-fs2)]exp[2πj(fs1-fs2)x]}×dfs1dfs2dfi1dfi2.
Isαinc(f|x)=w2 rectf2 fc[sinc2(wf) * |Uiα(fi)|2|hαα(f, t; fi)|2dfi].
Isα(f|x)=Isαinc(f|x)+Isαc(f|x).
Isαc(f|x)=2w2 rectf2 fcRefi1fi2>fi1fs1fs2fs1×{Uiα(fi1)Uiα*(fi2)Jααα(fs1, fs2; fi1, fi2)×sinc[w(f-fs1)]sinc[w(f-fs2)]×exp[2πj(fs1-fs2)x]dfs1dfs2}dfi1dfi2.
Δfi=fi2-fi12,Δfs=fs2-fs12,
f¯i=fi2+fi12,f¯s=fs2+fs12.
Jααα(fs1, fs2; fi1, fi2)=Jααα(Δfi, f¯i, f¯s)δ(Δfs-Δfi),
Isαc(f|x)=8w2 rectf2 fcReΔfi>0f¯if¯sUiα(f¯i-Δfi)×Uiα*(f¯i+Δfi)Jααα(Δfi, f¯i, f¯s)×sinc[w(f-f¯s+Δfi)]×sinc[w(f-f¯s-Δfi)]×exp(-4πjΔfix)df¯sdf¯idΔfi
Uiα(f¯i-Δfi)Uiα*(f¯i+Δfi)
=δ(f¯i-Δfi-fi1)δ(f¯i+Δfi-fi2)
Isαc(f|x)=8w2 rectf2 fcReexp(-4πjΔfix)×f¯sJααα(Δfi, f¯i, f¯s)sinc[w(f-f¯s+Δfi)]×sinc[w(f-f¯s-Δfi)]df¯s,
L(f)sinc[w(f+Δfi)]sinc[w(f-Δfi)],
Isαc(f|x)=8w2 rectf2 fcRe{exp(-4πjΔfix)×[L(f) * Jααα(Δfi, f¯i, f)]}.
Isαc(f|x)=8w2 rectf2 fcL(f) * [Jααα(Δfi, f¯i, f)×cos(4πΔfix)+Jααα(Δfi, f¯i, f)sin(4πΔfix)].
Isαinc(f|x)=w2 rectf2 fcsinc2(wf) * [|hαα(f, t; fi1)|2+|hαα(f, t; fi2)|2].
μααα(Δfi, f¯i, f¯s)Jααα(Δfi, f¯i, f¯s)sinc2(wΔfi)×(|hαα(fs1, t; fi1)|2|hαα(fs2, t; fi2)|2)1/2.
V(Δfi, f¯i, f¯s)=[μ(Δfi, f¯i, f¯s)2+μ(Δfi, f¯i, f¯s)2]1/2.
E=1+exp(jφ),
φ=k×OPD=k(sin θi-sin θs)Δx=2π(fi-fs)Δx,
J(fs1, fs2; fi1, fi2)=E1E2*=[1+exp(jφ1)][1+exp(-jφ2)].
ddΔxtan-1Im(E1E2*)Re(E1E2*)=0,
ddΔxIm(E1E2*)Re(E1E2*)
=ddΔxsin φ1-sin φ2+sin(φ1-φ2)1+cos φ1+cos φ2+cos(φ1-φ2)=0.
dφ1dΔx=dφ2dΔx,
fs1-fi1=fs2-fi2,
Jα(f1, f2)Usα(f1, t)Usα*(f2, t)=usα(s1, t)usα*(s2, t)×exp[2πj(f2s2-f1s1)]ds1ds2,
Jα(f1, f2)exp[2πj(f1x1-f2x2)]df1df2
=(usα(s1, t)usα*(s2, t)exp{2πj[(s2-x2)f2-(s1-x1)f1]})df1df2ds1ds2=usα(s1, t)usα*(s2, t)δ(s2-x2)δ(s1-x1)ds1ds2=usα(x1, t)usα*(x2, t)=Jα(x1, x2).

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