Abstract

Constructive interference between coherent waves traveling time-reversed paths in a random medium gives rise to the enhancement of light scattering observed in directions close to backscattering. This phenomenon is known as enhanced backscattering (EBS). According to diffusion theory, the angular width of an EBS cone is proportional to the ratio of the wavelength of light λ to the transport mean-free-path length ls* of a random medium. In biological media a large ls*0.52  mm  λ results in an extremely small (0.001°) angular width of the EBS cone, making the experimental observation of such narrow peaks difficult. Recently, the feasibility of observing EBS under low spatial coherence illumination (spatial coherence length Lscls*) was demonstrated. Low spatial coherence behaves as a spatial filter rejecting longer path lengths and thus resulting in an increase of more than 100 times in the angular width of low coherence EBS (LEBS) cones. However, a conventional diffusion approximation-based model of EBS has not been able to explain such a dramatic increase in LEBS width. We present a photon random walk model of LEBS by using Monte Carlo simulation to elucidate the mechanism accounting for the unprecedented broadening of the LEBS peaks. Typically, the exit angles of the scattered photons are not considered in modeling EBS in the diffusion regime. We show that small exit angles are highly sensitive to low-order scattering, which is crucial for accurate modeling of LEBS. Our results show that the predictions of the model are in excellent agreement with the experimental data.

© 2006 Optical Society of America

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2005 (3)

2004 (4)

2003 (2)

Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, M. J. Goldberg, A. K. Kromin, K. Chen, and V. Backman, "Simultaneous measurement of angular and spectral properties of light scattering for characterization of tissue microarchitecture and its alteration in early precancer," IEEE J. Sel. Top. Quantum Electron. 9, 243-256 (2003).
[CrossRef]

G. Labeyrie, D. Delande, C. A. Muller, C. Miniatura, and R. Kaiser, "Coherent backscattering of light by an inhomogeneous cloud of cold atoms," Phys. Rev. A 67, 033814 (2003).
[CrossRef]

2002 (3)

R. Lenke, R. Tweer, and G. Maret, "Coherent backscattering of turbid samples containing large Mie spheres," J. Opt. A , Pure Appl. Opt. 4, 293-298 (2002).
[CrossRef]

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, "Imaging skin pathology with polarized light," J. Biomed. Opt. 7, 329-340 (2002).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

2000 (3)

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, S. Shapzhay, T. Valdez, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, J. Van Dam, and M. S. Feld, "Detection of preinvasive cancer cells," Nature 406, 35-36 (2000).
[CrossRef] [PubMed]

A. Wax, S. Bali, and J. E. Thomas, "Time-resolved phase-space distributions for light backscattered from a disordered medium," Phys. Rev. Lett. 85, 66-69 (2000).
[CrossRef] [PubMed]

R. Lenke and G. Maret, "Magnetic field effects on coherent backscattering of light," Eur. Phys. J. B 17, 171-185 (2000).
[CrossRef]

1999 (2)

G. Labeyrie, F. de Tomasi, J. C. Bernard, C. A. Muller, C. Miniatura, and R. Kaiser, "Coherent backscattering of light by cold atoms," Phys. Rev. Lett. 83, 5266-5269 (1999).
[CrossRef]

K. Sokolov, R. A. Drezek, K. Gossage, and R. R. Richards-Kortum, "Reflectance spectroscopy with polarized light: is it sensitive to cellular and nuclear morphology?" Opt. Express 5, 302-317 (1999).
[CrossRef] [PubMed]

1998 (1)

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, and M. S. Feld, "Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution," Phys. Rev. Lett. 80, 627-630 (1998).
[CrossRef]

1997 (2)

1995 (4)

M. H. Eddowes, T. N. Mills, and D. T. Delpy, "Monte-Carlo simulations of coherent backscatter for identification of the optical coefficients of biological tissue in vivo," Appl. Opt. 34, 2261-2267 (1995).
[CrossRef] [PubMed]

L. H. Wang, S. L. Jacques, and L. Q. Zheng, "MCML--Monte Carlo modeling of photon transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

J. R. Mourant, I. J. Bigio, J. Boyer, R. L. Conn, T. Johnson, and T. Shimada, "Spectroscopic diagnosis of bladder cancer with elastic light scattering," Lasers Surg. Med. 17, 350-357 (1995).
[CrossRef] [PubMed]

D. S. Wiersma, M. P. van Albada, and A. Lagendijk, "Coherent backscattering of light from amplifying random media," Phys. Rev. Lett. 75, 1739-1742 (1995).
[CrossRef] [PubMed]

1993 (1)

1992 (1)

A. Dogariu, J. Uozumi, and T. Asakura, "Enhancement of the backscattered intensity from fractal aggregates," Waves Random Media 2, 259-263 (1992).
[CrossRef]

1991 (1)

M. Tomita and H. Ikari, "Influence of finite coherence length of incoming light on enhanced backscattering," Phys. Rev. B 43, 3716-3719 (1991).
[CrossRef]

1990 (2)

1988 (2)

M. B. van der Mark, M. P. van Albada, and A. Lagendijk, "Light scattering in strongly scattering media: multiple scattering and weak localization," Phys. Rev. B 37, 3575-3592 (1988).
[CrossRef]

P. E. Wolf, G. Maret, E. Akkermans, and R. Maynard, "Optical coherent backscattering by random media: an experimental study," J. Phys. (Paris) 49, 63-75 (1988).
[CrossRef]

1987 (1)

S. Eternad, R. Thompson, and M. J. Andrejco, "Weak localization of photons: termination of coherent random walks by absorption and confined geometry," Phys. Rev. Lett. 59, 1420-1423 (1987).
[CrossRef]

1986 (1)

E. Akkermans, P. E. Wolf, and R. Maynard, "Coherent backscattering of light by disordered media: analysis of peak line shape," Phys. Rev. Lett. 56, 1471-1474 (1986).
[CrossRef] [PubMed]

1985 (1)

J. S. Hendricks and T. E. Booth, "MCNP variance reduction overview," Lect. Notes Phys. 240, 83-92 (1985).
[CrossRef]

1984 (1)

1983 (1)

B. C. Wilson and G. Adam, "A Monte Carlo model for the absorption and flux distributions of light in tissue," Med. Phys. 10, 824-830 (1983).
[CrossRef] [PubMed]

1941 (1)

L. G. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 93, 70-83 (1941).
[CrossRef]

Adam, G.

B. C. Wilson and G. Adam, "A Monte Carlo model for the absorption and flux distributions of light in tissue," Med. Phys. 10, 824-830 (1983).
[CrossRef] [PubMed]

Akkermans, E.

P. E. Wolf, G. Maret, E. Akkermans, and R. Maynard, "Optical coherent backscattering by random media: an experimental study," J. Phys. (Paris) 49, 63-75 (1988).
[CrossRef]

E. Akkermans, P. E. Wolf, and R. Maynard, "Coherent backscattering of light by disordered media: analysis of peak line shape," Phys. Rev. Lett. 56, 1471-1474 (1986).
[CrossRef] [PubMed]

Alfano, R. R.

Andrejco, M. J.

S. Eternad, R. Thompson, and M. J. Andrejco, "Weak localization of photons: termination of coherent random walks by absorption and confined geometry," Phys. Rev. Lett. 59, 1420-1423 (1987).
[CrossRef]

Arendt, J. T.

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, S. Shapzhay, T. Valdez, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, J. Van Dam, and M. S. Feld, "Detection of preinvasive cancer cells," Nature 406, 35-36 (2000).
[CrossRef] [PubMed]

Asakura, T.

A. Dogariu, J. Uozumi, and T. Asakura, "Enhancement of the backscattered intensity from fractal aggregates," Waves Random Media 2, 259-263 (1992).
[CrossRef]

Backman, V.

Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, and V. Backman, "Low-coherent backscattering spectroscopy for tissue characterization," Appl. Opt. 44, 366-377 (2005).
[CrossRef] [PubMed]

Y. L. Kim, Y. Liu, V. M. Turzhitsky, R. Wali, H. Roy, and V. Backman, "Depth-resolved low-coherent backscattering in tissue," Opt. Lett. 30, 741-743 (2005).
[CrossRef] [PubMed]

Y. L. Kim, Y. Liu, V. M. Turzhitsky, H. K. Roy, R. K. Wali, and V. Backman, "Coherent backscattering spectroscopy," Opt. Lett. 29, 1906-1908 (2004).
[CrossRef] [PubMed]

Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, M. J. Goldberg, A. K. Kromin, K. Chen, and V. Backman, "Simultaneous measurement of angular and spectral properties of light scattering for characterization of tissue microarchitecture and its alteration in early precancer," IEEE J. Sel. Top. Quantum Electron. 9, 243-256 (2003).
[CrossRef]

A. Wax, C. H. Yang, V. Backman, K. Badizadegan, C. W. Boone, R. R. Dasari, and M. S. Feld, "Cellular organization and substructure measured using angle-resolved low-coherence interferometry," Biophys. J. 82, 2256-2264 (2002).
[CrossRef] [PubMed]

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, S. Shapzhay, T. Valdez, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, J. Van Dam, and M. S. Feld, "Detection of preinvasive cancer cells," Nature 406, 35-36 (2000).
[CrossRef] [PubMed]

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, and M. S. Feld, "Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution," Phys. Rev. Lett. 80, 627-630 (1998).
[CrossRef]

Badizadegan, K.

A. Wax, C. H. Yang, V. Backman, K. Badizadegan, C. W. Boone, R. R. Dasari, and M. S. Feld, "Cellular organization and substructure measured using angle-resolved low-coherence interferometry," Biophys. J. 82, 2256-2264 (2002).
[CrossRef] [PubMed]

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, S. Shapzhay, T. Valdez, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, J. Van Dam, and M. S. Feld, "Detection of preinvasive cancer cells," Nature 406, 35-36 (2000).
[CrossRef] [PubMed]

Bali, S.

A. Wax, S. Bali, and J. E. Thomas, "Time-resolved phase-space distributions for light backscattered from a disordered medium," Phys. Rev. Lett. 85, 66-69 (2000).
[CrossRef] [PubMed]

Bernard, J. C.

G. Labeyrie, F. de Tomasi, J. C. Bernard, C. A. Muller, C. Miniatura, and R. Kaiser, "Coherent backscattering of light by cold atoms," Phys. Rev. Lett. 83, 5266-5269 (1999).
[CrossRef]

Berrocal, E.

Bigio, I. J.

J. R. Mourant, I. J. Bigio, J. Boyer, R. L. Conn, T. Johnson, and T. Shimada, "Spectroscopic diagnosis of bladder cancer with elastic light scattering," Lasers Surg. Med. 17, 350-357 (1995).
[CrossRef] [PubMed]

Boone, C. W.

A. Wax, C. H. Yang, V. Backman, K. Badizadegan, C. W. Boone, R. R. Dasari, and M. S. Feld, "Cellular organization and substructure measured using angle-resolved low-coherence interferometry," Biophys. J. 82, 2256-2264 (2002).
[CrossRef] [PubMed]

Booth, T. E.

J. S. Hendricks and T. E. Booth, "MCNP variance reduction overview," Lect. Notes Phys. 240, 83-92 (1985).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge U. Press, 1999), pp. 572-580.

Boyer, J.

J. R. Mourant, I. J. Bigio, J. Boyer, R. L. Conn, T. Johnson, and T. Shimada, "Spectroscopic diagnosis of bladder cancer with elastic light scattering," Lasers Surg. Med. 17, 350-357 (1995).
[CrossRef] [PubMed]

Chen, K.

Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, M. J. Goldberg, A. K. Kromin, K. Chen, and V. Backman, "Simultaneous measurement of angular and spectral properties of light scattering for characterization of tissue microarchitecture and its alteration in early precancer," IEEE J. Sel. Top. Quantum Electron. 9, 243-256 (2003).
[CrossRef]

Chen, W. R.

Cheung, C.

R. Sapienza, S. Mujumdar, C. Cheung, A. G. Yodh, and D. Wiersma, "Anisotropic weak localization of light," Phys. Rev. Lett. 92, 033903 (2004).
[CrossRef] [PubMed]

Churmakov, D. Y.

Conn, R. L.

J. R. Mourant, I. J. Bigio, J. Boyer, R. L. Conn, T. Johnson, and T. Shimada, "Spectroscopic diagnosis of bladder cancer with elastic light scattering," Lasers Surg. Med. 17, 350-357 (1995).
[CrossRef] [PubMed]

Crawford, J. M.

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, S. Shapzhay, T. Valdez, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, J. Van Dam, and M. S. Feld, "Detection of preinvasive cancer cells," Nature 406, 35-36 (2000).
[CrossRef] [PubMed]

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, and M. S. Feld, "Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution," Phys. Rev. Lett. 80, 627-630 (1998).
[CrossRef]

Dasari, R. R.

A. Wax, C. H. Yang, V. Backman, K. Badizadegan, C. W. Boone, R. R. Dasari, and M. S. Feld, "Cellular organization and substructure measured using angle-resolved low-coherence interferometry," Biophys. J. 82, 2256-2264 (2002).
[CrossRef] [PubMed]

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, S. Shapzhay, T. Valdez, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, J. Van Dam, and M. S. Feld, "Detection of preinvasive cancer cells," Nature 406, 35-36 (2000).
[CrossRef] [PubMed]

de Tomasi, F.

G. Labeyrie, F. de Tomasi, J. C. Bernard, C. A. Muller, C. Miniatura, and R. Kaiser, "Coherent backscattering of light by cold atoms," Phys. Rev. Lett. 83, 5266-5269 (1999).
[CrossRef]

Delande, D.

G. Labeyrie, D. Delande, C. A. Muller, C. Miniatura, and R. Kaiser, "Coherent backscattering of light by an inhomogeneous cloud of cold atoms," Phys. Rev. A 67, 033814 (2003).
[CrossRef]

Delpy, D. T.

Dogariu, A.

A. Dogariu, J. Uozumi, and T. Asakura, "Enhancement of the backscattered intensity from fractal aggregates," Waves Random Media 2, 259-263 (1992).
[CrossRef]

Drezek, R. A.

Eddowes, M. H.

Eternad, S.

S. Eternad, R. Thompson, and M. J. Andrejco, "Weak localization of photons: termination of coherent random walks by absorption and confined geometry," Phys. Rev. Lett. 59, 1420-1423 (1987).
[CrossRef]

Feld, M. S.

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R. Sapienza, S. Mujumdar, C. Cheung, A. G. Yodh, and D. Wiersma, "Anisotropic weak localization of light," Phys. Rev. Lett. 92, 033903 (2004).
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E. Akkermans, P. E. Wolf, and R. Maynard, "Coherent backscattering of light by disordered media: analysis of peak line shape," Phys. Rev. Lett. 56, 1471-1474 (1986).
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V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, S. Shapzhay, T. Valdez, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, J. Van Dam, and M. S. Feld, "Detection of preinvasive cancer cells," Nature 406, 35-36 (2000).
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L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, and M. S. Feld, "Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution," Phys. Rev. Lett. 80, 627-630 (1998).
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L. G. Henyey and J. L. Greenstein, "Diffuse radiation in the galaxy," Astrophys. J. 93, 70-83 (1941).
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A. Wax, C. H. Yang, V. Backman, K. Badizadegan, C. W. Boone, R. R. Dasari, and M. S. Feld, "Cellular organization and substructure measured using angle-resolved low-coherence interferometry," Biophys. J. 82, 2256-2264 (2002).
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L. H. Wang, S. L. Jacques, and L. Q. Zheng, "MCML--Monte Carlo modeling of photon transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
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R. Lenke and G. Maret, "Magnetic field effects on coherent backscattering of light," Eur. Phys. J. B 17, 171-185 (2000).
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S. L. Jacques, J. C. Ramella-Roman, and K. Lee, "Imaging skin pathology with polarized light," J. Biomed. Opt. 7, 329-340 (2002).
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R. Lenke, R. Tweer, and G. Maret, "Coherent backscattering of turbid samples containing large Mie spheres," J. Opt. A , Pure Appl. Opt. 4, 293-298 (2002).
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Nature (1)

V. Backman, M. B. Wallace, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, S. Shapzhay, T. Valdez, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, J. Van Dam, and M. S. Feld, "Detection of preinvasive cancer cells," Nature 406, 35-36 (2000).
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Opt. Express (1)

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[CrossRef]

D. S. Wiersma, M. P. van Albada, and A. Lagendijk, "Coherent backscattering of light from amplifying random media," Phys. Rev. Lett. 75, 1739-1742 (1995).
[CrossRef] [PubMed]

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[CrossRef]

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[CrossRef] [PubMed]

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, and M. S. Feld, "Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution," Phys. Rev. Lett. 80, 627-630 (1998).
[CrossRef]

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B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991), pp. 351-352.

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Figures (10)

Fig. 1
Fig. 1

Normalized intensity at different exit angles, θ c i in the diffusive multiple scattering regime ( L sc l s * ) is plotted as a function of radial distance r. Intensities are calculated by using MC simulation from a medium with l s * = 2   mm , g = 0.9 (at λ = 520   nm ) and L sc = 50   mm . Intensity profiles for different exit angles remain constant in the diffusive multiple scattering regime.

Fig. 2
Fig. 2

Normalized intensity at different exit angles, θ c i in the low-order scattering regime ( L sc l s * ) is plotted as a function of radial distance r. Intensities are calculated using by MC simulation ( l s * = 2   mm , g = 0.9 at λ = 520   nm ) with L sc = 600   μm . When the number of scattering events is restricted due to the finite spatial coherence area by using a low spatial coherence illumination, the intensity profile over r becomes broader as θ c i increases from 1.5° to 80°.

Fig. 3
Fig. 3

r ( θ c i , L sc ) as a function of exit angle θ ci for four different L sc . r ( θ c i , L sc ) is simulated for a sample with l s * = 2   mm and g = 0.9 (at λ = 520   nm ) for different exit angles θ c i varying from 1° to 90°. r ( θ c i , L sc ) is insensitive to the exit angle θ c i when L sc l s * , while r ( θ c i , L sc ) increases with the increase in the exit angle in the low order scattering regime (L scl s *).

Fig. 4
Fig. 4

(a) r ( θ c i , L sc ) as a function of angle θ c i obtained for four different L sc . (b) r ( θ c i , L sc ) as a function of L sc obtained for four different θ c i . r ( θ c i , L sc ) is calculated by using MC simulation from a medium with l s * = 2   mm and g = 0.9 (at λ = 520   nm for θ c i varying from 1° to 90° and different L sc varying between 35 and 140   μm . r ( θ c i , L sc ) is proportional to both θ c i and L sc .

Fig. 5
Fig. 5

r ( θ c i , L sc ) as a function of r and depth of penetration of scattered photons z for a fixed θ c i = 45 ° and a fixed L sc = 600   μm . r ( θ c i , L sc ) is calculated from a medium with l s * = 2   mm and g = 0.9 (at λ = 520   nm ) by using MC simulation. r ( θ c i , L sc ) is proportional to the penetration depth z.

Fig. 6
Fig. 6

Probability of exit angle P ex ( θ c i ) as a function of θ c i obtained for low-order scattering regime. P ex ( θ c i ) is calculated by using MC simulation ( l s * = 2   mm , g = 0.9 at λ = 520   nm ) for a fixed L sc = 200   μm by varying θ c i between 1° and 90°. P ex ( θ c i ) converges at small angles of approximately 1 ° 3 ° when L sc l s * .

Fig. 7
Fig. 7

Comparison of the EBS profile from MC simulation with that from analytical formulation (Ref. 40). Profile of the EBS peak from MC simulation I EBS ( θ ) , is calculated from a medium with l s * = 2   mm and g = 0.9 (at λ = 520   nm ) for L sc = 50   mm . Results from the simulation are in excellent agreement with the analytical results. Also, the LEBS simulation for θ c i = 1.5 ° agrees well with the results obtained from θ c i = 80 ° as the width of the EBS peak is insensitive to the θ c i in the diffusive multiple scattering regime ( L sc l s * ) .

Fig. 8
Fig. 8

Normalized intensity profile of the LEBS peak as a function of θ from MC simulation is compared with that of the experimental result from white paint under low spatial coherence illumination (Xenon lamp, λ = 520   nm , L sc = 160   μm ). L LEBS ( θ ) , is calculated by using MC simulation from a medium with l s * = 4   mm and g = 0.9 (at λ = 520   nm ) for L sc = 160   μm . Simulation results agree well with experimentally observed LEBS peak when L sc = 160   μm . Also, the EBS peak from L sc = 50   mm and LEBS peak from L sc = 160   μm are completely indistinguishable as the peak width is completely determined by the small l s * of the medium and the spatial coherence length plays an insignificant role in this regime.

Fig. 9
Fig. 9

Normalized intensity profile of the LEBS peak as a function of angle θ from MC simulation is compared with that of the aqueous suspensions of polystyrene microspheres (diameter = 0.89   μm ) under low spatial coherence illumination (Xenon lamp, λ = 520   nm , L sc = 48   μm ). L LEBS ( θ ) , is simulated for a medium with l s * = 2   mm , g = 0.9 (at λ = 520   nm ), and L sc = 48   μm . LEBS peak simulated by MC simulation at θ c i = 1.5 ° matches well with the LEBS peak profile recorded in the experiment. On the contrary, the LEBS peaks from θ c i = 80 ° are three times narrower than those obtained from the experiment and the large angles do not accurately predict the LEBS peak as they are insensitive to low orders of scattering.

Fig. 10
Fig. 10

Comparison of angular width W of the LEBS peaks obtained from MC simulation with that of LEBS peaks from experiment (bead diameter = 0.89   μm ) under low spatial coherence illumination (Xenon lamp, λ = 520   nm ). The width of LEBS peak from simulation ( l s * = 2   mm , g = 0.9 at λ = 520   nm ) is calculated for eight different L sc varying between 30 and 220   μm at a fixed θ c i = 1.5 ° . The error bars in the curves are the standard errors. The widths of the LEBS peaks predicted by MC simulation are in excellent agreement with those determined from the experiment.

Equations (16)

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ω hm = λ / ( 3 π l s * ) .
I EBS ( q ) = P ( r ) exp ( i q . r ) d 2 r ,
I EBS ( q ) r P ( r ) exp ( i q . r ) d r ,
C L sc ( r ) = 2 J 1 ( r / L sc ) / ( r / L sc ) ,
I LEBS ( θ ) r P ( r ) . C L sc ( r ) exp ( i 2 π sin θ λ r ) d r .
P HG ( cos θ ) = ( 1 g 2 ) / [ 2 ( 1 + g 2 2 g cos θ ) 3 / 2 ] ,
W = ( 0 I LEBS ( q ) d q ) 2 / [ I LEBS 2 ( q ) d q ] .
r = 0 θ = 0 π / 2 p ( r , θ ) d r d θ = N ,
r = 0 θ = 0 θci p ( r , θ ) d r d θ = r = 0 p ¯ θ c i ( r ) d r = N , c i .
0 P θ c i ( r ) d r = 1 .
r = 0 L sc θ =0 π / 2 p ( r , θ ) d r d θ = N L sc ,
0 L sc 0 θci p ( r , θ ) d r d θ = 0 L sc p ¯ θ ci ( r ) d r = N L sc , ci .
0 L sc P θ c i ( r ) d r = 1 .
0 L sc / 10 P θ c 1 ( r ) d r 0 L sc / 10 P θ c 2 ( r ) d r .
r ( θ c i , L sc ) = 0 L sc r P θ c 1 ( r ) d r L sc .
r ( θ c i , L sc ) = 0 L scj r P θ c i ( r ) d r ,

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