Abstract

An analytical theory for coherent backscattering (CBS) of low-coherence light is presented. An expression linking the CBS profile to the radial distribution of the incoherent backscattered light is derived when the incident light is partially spatially coherent. The backscattered snake light, which has experienced exactly two large-angle scatterings, is taken into account together with the diffuse light in the analysis. Monte Carlo simulations demonstrate that the model describes well the CBS profile as long as the spatial coherence length, Lc, of the incident beam is larger than the scattering mean free path of light in the medium. The intensity of the enhanced backscattered light in the exact backscattering direction and the width of the CBS cone are found to be proportional to Lc and Lc1, respectively, in the limit of small Lc.

© 2008 Optical Society of America

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2006 (1)

Y. L. Kim, V. M. Turzhitsky, Y. Liu, H. K. Roy, R. K. Wali, H. Subramanian, P. Pradhan, and V. Backman, J. Biomed. Opt. 11, 041125 (2006).
[CrossRef] [PubMed]

2005 (1)

2004 (2)

1996 (1)

1994 (1)

1991 (2)

J. X. Zhu, D. J. Pine, and D. A. Weitz, Phys. Rev. A 44, 3948 (1991).
[CrossRef] [PubMed]

M. Tomita and H. Ikari, Phys. Rev. B 43, 3716 (1991).
[CrossRef]

1986 (1)

E. Akkermans, P. E. Wolf, and R. Maynard, Phys. Rev. Lett. 56, 1471 (1986).
[CrossRef] [PubMed]

1985 (1)

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

1984 (1)

Akkermans, E.

E. Akkermans, P. E. Wolf, and R. Maynard, Phys. Rev. Lett. 56, 1471 (1986).
[CrossRef] [PubMed]

Alfano, R. R.

Asakura, T.

Backman, V.

Y. L. Kim, V. M. Turzhitsky, Y. Liu, H. K. Roy, R. K. Wali, H. Subramanian, P. Pradhan, and V. Backman, J. Biomed. Opt. 11, 041125 (2006).
[CrossRef] [PubMed]

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

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Pergamon, 2002), p. 572.

Gayen, S. K.

Ikari, H.

M. Tomita and H. Ikari, Phys. Rev. B 43, 3716 (1991).
[CrossRef]

Ishimaru, A.

Kim, Y. L.

Y. L. Kim, V. M. Turzhitsky, Y. Liu, H. K. Roy, R. K. Wali, H. Subramanian, P. Pradhan, and V. Backman, J. Biomed. Opt. 11, 041125 (2006).
[CrossRef] [PubMed]

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

Kuga, Y.

Liu, Y.

Y. L. Kim, V. M. Turzhitsky, Y. Liu, H. K. Roy, R. K. Wali, H. Subramanian, P. Pradhan, and V. Backman, J. Biomed. Opt. 11, 041125 (2006).
[CrossRef] [PubMed]

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

Maret, G.

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

Maynard, R.

E. Akkermans, P. E. Wolf, and R. Maynard, Phys. Rev. Lett. 56, 1471 (1986).
[CrossRef] [PubMed]

Okamoto, T.

Phillips, K. G.

Pine, D. J.

J. X. Zhu, D. J. Pine, and D. A. Weitz, Phys. Rev. A 44, 3948 (1991).
[CrossRef] [PubMed]

Pradhan, P.

Y. L. Kim, V. M. Turzhitsky, Y. Liu, H. K. Roy, R. K. Wali, H. Subramanian, P. Pradhan, and V. Backman, J. Biomed. Opt. 11, 041125 (2006).
[CrossRef] [PubMed]

Roy, H. K.

Y. L. Kim, V. M. Turzhitsky, Y. Liu, H. K. Roy, R. K. Wali, H. Subramanian, P. Pradhan, and V. Backman, J. Biomed. Opt. 11, 041125 (2006).
[CrossRef] [PubMed]

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

Subramanian, H.

Y. L. Kim, V. M. Turzhitsky, Y. Liu, H. K. Roy, R. K. Wali, H. Subramanian, P. Pradhan, and V. Backman, J. Biomed. Opt. 11, 041125 (2006).
[CrossRef] [PubMed]

Tomita, M.

M. Tomita and H. Ikari, Phys. Rev. B 43, 3716 (1991).
[CrossRef]

Turzhitsky, V. M.

Y. L. Kim, V. M. Turzhitsky, Y. Liu, H. K. Roy, R. K. Wali, H. Subramanian, P. Pradhan, and V. Backman, J. Biomed. Opt. 11, 041125 (2006).
[CrossRef] [PubMed]

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

Wali, R. K.

Y. L. Kim, V. M. Turzhitsky, Y. Liu, H. K. Roy, R. K. Wali, H. Subramanian, P. Pradhan, and V. Backman, J. Biomed. Opt. 11, 041125 (2006).
[CrossRef] [PubMed]

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

Weitz, D. A.

J. X. Zhu, D. J. Pine, and D. A. Weitz, Phys. Rev. A 44, 3948 (1991).
[CrossRef] [PubMed]

Wolf, E.

E. Wolf, Opt. Lett. 19, 2024 (1994).
[CrossRef] [PubMed]

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Pergamon, 2002), p. 572.

Wolf, P. E.

E. Akkermans, P. E. Wolf, and R. Maynard, Phys. Rev. Lett. 56, 1471 (1986).
[CrossRef] [PubMed]

Wolf, P.-E.

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

Xu, M.

Zhu, J. X.

J. X. Zhu, D. J. Pine, and D. A. Weitz, Phys. Rev. A 44, 3948 (1991).
[CrossRef] [PubMed]

J. Biomed. Opt. (1)

Y. L. Kim, V. M. Turzhitsky, Y. Liu, H. K. Roy, R. K. Wali, H. Subramanian, P. Pradhan, and V. Backman, J. Biomed. Opt. 11, 041125 (2006).
[CrossRef] [PubMed]

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

Opt. Express (2)

Opt. Lett. (3)

Phys. Rev. A (1)

J. X. Zhu, D. J. Pine, and D. A. Weitz, Phys. Rev. A 44, 3948 (1991).
[CrossRef] [PubMed]

Phys. Rev. B (1)

M. Tomita and H. Ikari, Phys. Rev. B 43, 3716 (1991).
[CrossRef]

Phys. Rev. Lett. (2)

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

E. Akkermans, P. E. Wolf, and R. Maynard, Phys. Rev. Lett. 56, 1471 (1986).
[CrossRef] [PubMed]

Other (1)

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Pergamon, 2002), p. 572.

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Figures (4)

Fig. 1
Fig. 1

Pair of time-reversal trajectories (solid and dashed lines) for a beam incident in the s i direction and escaping in the s f direction. A low-coherent source introduces an additional phase correlation between the light fields at the illumination points P 1 and P 2 .

Fig. 2
Fig. 2

Profiles of the backscattered incoherent light intensity I ( ρ ) versus the normalized radial distance ρ l t from MC simulations compared with theoretical predictions for Rayleigh and Mie scatterers ( g = 0.92 ) .

Fig. 3
Fig. 3

CBS profiles obtained from Monte Carlo simulations compared to the theoretical predictions for Rayleigh and Mie scatterers at L c l t = 0.1 (left) and 1 (right). The enhancement factor is underestimated by the theory for the Rayleigh scatterer with L c = 0.1 l t as L c = 0.1 l s is much less than the scattering mean free path.

Fig. 4
Fig. 4

(Left) Normalized intensity I CBS ( 0 ) I base and (right) normalized inverse width ( q 1 2 l t ) 1 of LEBS versus L c l t .

Equations (6)

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I CBS ( q ) = 1 A d ρ 1 d ρ 2 I ( ρ 2 ρ 1 ) exp [ i q ( ρ 2 ρ 1 ) ] J ( ρ 1 , ρ 2 ) .
J ( ρ 1 , ρ 2 ) = V 1 * V 2 = 2 J 1 ( ρ 1 ρ 2 L c ) ρ 1 ρ 2 L c exp ( i ρ 2 2 ρ 1 2 2 a L c ) ,
I CBS ( q ) = d ρ I ( ρ ) exp ( i q ρ ) γ ( ρ ) exp ( i ρ 2 2 a L c ) [ 2 J 1 ( ρ L c ) ρ L c ] 2 ,
I CBS ( q ) = 2 π 0 + ρ d ρ I ( ρ ) J 0 ( q ρ ) [ 2 J 1 ( ρ L c ) ρ L c ] 2 ,
A ( s , s ) = k 2 ( 2 π ) 2 d ρ [ 2 J 1 ( ρ L c ) ρ L c ] 2 exp [ i k ( s s 0 ) ρ ] ,
I ( ρ ) = 1 4 π l t 2 0 + d z 0 + d z exp ( z + z l t ) × [ G ( snake ) ( R ) + G ( diffuse ) ( r , r ) ] ,

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