Abstract

We present a theoretical basis for calculation of the angular profile of the coherent backscattering intensity under low spatial coherence illumination. We take into account two contributions to the intensity, namely, the diffusion contribution and the contribution from the waves that experience the small-angle multiple scattering before and after single deflection in the backward direction. The latter contribution describes transport of light at subdiffusion length scales and is responsible for the wings of the backscattering angular profile. Our results are in good agreement with data of Monte-Carlo simulations and experiment.

© 2012 Optical Society of America

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  2. M. P. Van Albada and A. Lagendijk, Phys. Rev. Lett. 55, 2692 (1985).
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    [CrossRef]
  4. P. E. Wolf, G. Maret, E. Akkermans, and R. Maynard, J. Phys. France 49, 63 (1988).
  5. M. B. van der Mark, M. P. van Albada, and A. Lagendijk, Phys. Rev. B 37, 3575 (1988).
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    [CrossRef]
  7. R. Lenke, R. Tweer, and G. Maret, J. Opt. A 4, 293 (2002).
  8. R. Sapienza, S. Mujumdar, C. Cheung, A. G. Yodh, and D. Wiersma, Phys. Rev. Lett. 92, 033903 (2004).
    [CrossRef]
  9. Y. L. Kim, Y. Liu, V. M. Turzhitsky, H. K. Roy, R. K. Wali, and V. Backman, Opt. Lett. 29, 1906 (2004).
    [CrossRef]
  10. H. Subramanian, P. Pradhan, Y. L. Kim, Y. Liu, X. Li, and V. Backman, Appl. Opt. 45, 6292 (2006).
    [CrossRef]
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    [CrossRef]
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2011 (2)

V. V. Marinyuk and D. B. Rogozkin, Phys. Rev. E 83, 066604 (2011).
[CrossRef]

V. Turzhitsky, A. J. Radosevich, J. D. Rogers, N. N. Mutyal, and V. Backman, J. Biomed. Opt. 16, 067007 (2011).

2010 (1)

V. Turzhitsky, J. D. Rogers, N. N. Mutyal, H. K. Roy, and V. Backman, IEEE J. Sel. Top. Quantum Electron. 16, 619 (2010).
[CrossRef]

2009 (1)

2007 (1)

2006 (1)

2004 (2)

R. Sapienza, S. Mujumdar, C. Cheung, A. G. Yodh, and D. Wiersma, Phys. Rev. Lett. 92, 033903 (2004).
[CrossRef]

Y. L. Kim, Y. Liu, V. M. Turzhitsky, H. K. Roy, R. K. Wali, and V. Backman, Opt. Lett. 29, 1906 (2004).
[CrossRef]

2002 (1)

R. Lenke, R. Tweer, and G. Maret, J. Opt. A 4, 293 (2002).

1995 (1)

D. S. Wiersma, M. P. van Albada, B. A. van Tiggelen, and A. Lagendijk, Phys. Rev. Lett. 74, 4193 (1995).
[CrossRef]

1993 (1)

1988 (2)

P. E. Wolf, G. Maret, E. Akkermans, and R. Maynard, J. Phys. France 49, 63 (1988).

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, Phys. Rev. B 37, 3575 (1988).

1985 (2)

M. P. Van Albada and A. Lagendijk, Phys. Rev. Lett. 55, 2692 (1985).
[CrossRef]

P. E. Wolf and G. Maret, Phys. Rev. Lett. 55, 2696 (1985).
[CrossRef]

1984 (1)

1980 (1)

1973 (1)

Y. N. Barabanenkov, Radiophys. Quantum Electron. 16, 65 (1973).
[CrossRef]

1971 (1)

D. A. de Wolf, IEEE Trans. Antennas Propag. AP- 19, 254 (1971).

Akkermans, E.

P. E. Wolf, G. Maret, E. Akkermans, and R. Maynard, J. Phys. France 49, 63 (1988).

Alfano, R. R.

Backman, V.

V. Turzhitsky, A. J. Radosevich, J. D. Rogers, N. N. Mutyal, and V. Backman, J. Biomed. Opt. 16, 067007 (2011).

V. Turzhitsky, J. D. Rogers, N. N. Mutyal, H. K. Roy, and V. Backman, IEEE J. Sel. Top. Quantum Electron. 16, 619 (2010).
[CrossRef]

J. D. Rogers, I. R. Capoglu, and V. Backman, Opt. Lett. 34, 1891 (2009).
[CrossRef]

H. Subramanian, P. Pradhan, Y. L. Kim, Y. Liu, X. Li, and V. Backman, Appl. Opt. 45, 6292 (2006).
[CrossRef]

Y. L. Kim, Y. Liu, V. M. Turzhitsky, H. K. Roy, R. K. Wali, and V. Backman, Opt. Lett. 29, 1906 (2004).
[CrossRef]

Barabanenkov, Y. N.

Y. N. Barabanenkov, Radiophys. Quantum Electron. 16, 65 (1973).
[CrossRef]

Capoglu, I. R.

Cheung, C.

R. Sapienza, S. Mujumdar, C. Cheung, A. G. Yodh, and D. Wiersma, Phys. Rev. Lett. 92, 033903 (2004).
[CrossRef]

Das, B. B.

de Wolf, D. A.

D. A. de Wolf, IEEE Trans. Antennas Propag. AP- 19, 254 (1971).

Ishimaru, A.

Y. Kuga and A. Ishimaru, J. Opt. Soc. Am. A 1, 831 (1984).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

Kim, Y. L.

Kuga, Y.

Lagendijk, A.

D. S. Wiersma, M. P. van Albada, B. A. van Tiggelen, and A. Lagendijk, Phys. Rev. Lett. 74, 4193 (1995).
[CrossRef]

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, Phys. Rev. B 37, 3575 (1988).

M. P. Van Albada and A. Lagendijk, Phys. Rev. Lett. 55, 2692 (1985).
[CrossRef]

Lenke, R.

R. Lenke, R. Tweer, and G. Maret, J. Opt. A 4, 293 (2002).

Li, X.

Liu, Y.

Maret, G.

R. Lenke, R. Tweer, and G. Maret, J. Opt. A 4, 293 (2002).

P. E. Wolf, G. Maret, E. Akkermans, and R. Maynard, J. Phys. France 49, 63 (1988).

P. E. Wolf and G. Maret, Phys. Rev. Lett. 55, 2696 (1985).
[CrossRef]

Marinyuk, V. V.

V. V. Marinyuk and D. B. Rogozkin, Phys. Rev. E 83, 066604 (2011).
[CrossRef]

Maynard, R.

P. E. Wolf, G. Maret, E. Akkermans, and R. Maynard, J. Phys. France 49, 63 (1988).

McCormick, N. J.

Mujumdar, S.

R. Sapienza, S. Mujumdar, C. Cheung, A. G. Yodh, and D. Wiersma, Phys. Rev. Lett. 92, 033903 (2004).
[CrossRef]

Mutyal, N. N.

V. Turzhitsky, A. J. Radosevich, J. D. Rogers, N. N. Mutyal, and V. Backman, J. Biomed. Opt. 16, 067007 (2011).

V. Turzhitsky, J. D. Rogers, N. N. Mutyal, H. K. Roy, and V. Backman, IEEE J. Sel. Top. Quantum Electron. 16, 619 (2010).
[CrossRef]

Pradhan, P.

Radosevich, A. J.

V. Turzhitsky, A. J. Radosevich, J. D. Rogers, N. N. Mutyal, and V. Backman, J. Biomed. Opt. 16, 067007 (2011).

Reynolds, L.

Rogers, J. D.

V. Turzhitsky, A. J. Radosevich, J. D. Rogers, N. N. Mutyal, and V. Backman, J. Biomed. Opt. 16, 067007 (2011).

V. Turzhitsky, J. D. Rogers, N. N. Mutyal, H. K. Roy, and V. Backman, IEEE J. Sel. Top. Quantum Electron. 16, 619 (2010).
[CrossRef]

J. D. Rogers, I. R. Capoglu, and V. Backman, Opt. Lett. 34, 1891 (2009).
[CrossRef]

Rogozkin, D. B.

V. V. Marinyuk and D. B. Rogozkin, Phys. Rev. E 83, 066604 (2011).
[CrossRef]

Roy, H. K.

V. Turzhitsky, J. D. Rogers, N. N. Mutyal, H. K. Roy, and V. Backman, IEEE J. Sel. Top. Quantum Electron. 16, 619 (2010).
[CrossRef]

Y. L. Kim, Y. Liu, V. M. Turzhitsky, H. K. Roy, R. K. Wali, and V. Backman, Opt. Lett. 29, 1906 (2004).
[CrossRef]

Sapienza, R.

R. Sapienza, S. Mujumdar, C. Cheung, A. G. Yodh, and D. Wiersma, Phys. Rev. Lett. 92, 033903 (2004).
[CrossRef]

Sheppard, C. J. R.

Subramanian, H.

Turzhitsky, V.

V. Turzhitsky, A. J. Radosevich, J. D. Rogers, N. N. Mutyal, and V. Backman, J. Biomed. Opt. 16, 067007 (2011).

V. Turzhitsky, J. D. Rogers, N. N. Mutyal, H. K. Roy, and V. Backman, IEEE J. Sel. Top. Quantum Electron. 16, 619 (2010).
[CrossRef]

Turzhitsky, V. M.

Tweer, R.

R. Lenke, R. Tweer, and G. Maret, J. Opt. A 4, 293 (2002).

van Albada, M. P.

D. S. Wiersma, M. P. van Albada, B. A. van Tiggelen, and A. Lagendijk, Phys. Rev. Lett. 74, 4193 (1995).
[CrossRef]

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, Phys. Rev. B 37, 3575 (1988).

M. P. Van Albada and A. Lagendijk, Phys. Rev. Lett. 55, 2692 (1985).
[CrossRef]

van der Mark, M. B.

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, Phys. Rev. B 37, 3575 (1988).

van Tiggelen, B. A.

D. S. Wiersma, M. P. van Albada, B. A. van Tiggelen, and A. Lagendijk, Phys. Rev. Lett. 74, 4193 (1995).
[CrossRef]

Wali, R. K.

Wiersma, D.

R. Sapienza, S. Mujumdar, C. Cheung, A. G. Yodh, and D. Wiersma, Phys. Rev. Lett. 92, 033903 (2004).
[CrossRef]

Wiersma, D. S.

D. S. Wiersma, M. P. van Albada, B. A. van Tiggelen, and A. Lagendijk, Phys. Rev. Lett. 74, 4193 (1995).
[CrossRef]

Wolf, P. E.

P. E. Wolf, G. Maret, E. Akkermans, and R. Maynard, J. Phys. France 49, 63 (1988).

P. E. Wolf and G. Maret, Phys. Rev. Lett. 55, 2696 (1985).
[CrossRef]

Yodh, A. G.

R. Sapienza, S. Mujumdar, C. Cheung, A. G. Yodh, and D. Wiersma, Phys. Rev. Lett. 92, 033903 (2004).
[CrossRef]

Yoo, K. M.

Appl. Opt. (1)

IEEE J. Sel. Top. Quantum Electron. (1)

V. Turzhitsky, J. D. Rogers, N. N. Mutyal, H. K. Roy, and V. Backman, IEEE J. Sel. Top. Quantum Electron. 16, 619 (2010).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

D. A. de Wolf, IEEE Trans. Antennas Propag. AP- 19, 254 (1971).

J. Biomed. Opt. (1)

V. Turzhitsky, A. J. Radosevich, J. D. Rogers, N. N. Mutyal, and V. Backman, J. Biomed. Opt. 16, 067007 (2011).

J. Opt. A (1)

R. Lenke, R. Tweer, and G. Maret, J. Opt. A 4, 293 (2002).

J. Opt. Soc. Am. (1)

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

J. Phys. France (1)

P. E. Wolf, G. Maret, E. Akkermans, and R. Maynard, J. Phys. France 49, 63 (1988).

Opt. Lett. (4)

Phys. Rev. B (1)

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, Phys. Rev. B 37, 3575 (1988).

Phys. Rev. E (1)

V. V. Marinyuk and D. B. Rogozkin, Phys. Rev. E 83, 066604 (2011).
[CrossRef]

Phys. Rev. Lett. (4)

D. S. Wiersma, M. P. van Albada, B. A. van Tiggelen, and A. Lagendijk, Phys. Rev. Lett. 74, 4193 (1995).
[CrossRef]

M. P. Van Albada and A. Lagendijk, Phys. Rev. Lett. 55, 2692 (1985).
[CrossRef]

P. E. Wolf and G. Maret, Phys. Rev. Lett. 55, 2696 (1985).
[CrossRef]

R. Sapienza, S. Mujumdar, C. Cheung, A. G. Yodh, and D. Wiersma, Phys. Rev. Lett. 92, 033903 (2004).
[CrossRef]

Radiophys. Quantum Electron. (1)

Y. N. Barabanenkov, Radiophys. Quantum Electron. 16, 65 (1973).
[CrossRef]

Other (1)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

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Figures (3)

Fig. 1.
Fig. 1.

Radial profile of backscattering intensity I(ρ) from Monte-Carlo simulation data [10] (circles) and calculated theoretically (solid curve). The curve is normalized to the simulation data at ρ/ltr=0.4. The dotted and dashed curves are the diffusion and FSSB contributions, respectively. The Henyey–Greenstein phase function, g=0.9.

Fig. 2.
Fig. 2.

LEBS angular profile calculated theoretically (solid curve) and measured experimentally [18] (circles). The dotted and dashed curves are the diffusion and FSSB contributions, respectively. Spatial coherence length L=173μm, ltr=668μm, λ=680nm.

Fig. 3.
Fig. 3.

Dependence of the inverse LEBS peak width on ratio ltr/L. The solid curve is the result of our calculations. Circles are data of Monte-Carlo simulation [11]. The Henyey–Greenstein phase function, g=0.86.

Equations (9)

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p(ϑ)=α22πϑ0α2(ϑ02+ϑ2)α/2,(α>2),
JcLEBS(θ)=0I(ρ)C(ρ)J0(k0ρθ)2πρdρ,
Idiff(ρ)=316π2ltr2{ln(ltr/ρ)0.22,ρltr5.85·(ltr/ρ)3,ρltr.
IFSSB(ρ)=pb2π0qdqJ0(qρ)0dτ[|Iq(τ)|2e2τ],
Iq(τ)=exp(0τdτ[1p^(qlτ)]),
p^(ω)=0p(ϑ)J0(ωϑ)2πϑdϑ
IFSSB(ρ)=aα2πpbϑ0α2ltr2(ltrρ)αα1,
aα=2α(4α)8(α2)Γ(2ε)Γ(ε)Γ(1ε)[(α1)(α2)Γ(α/2)Γ(3α/2)]1α1,
IFSSB(ρ)=Γ((α1)/2)2πΓ(α/21)pbϑ0lρ

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