Abstract

The angular distribution of diffuse reflection is elucidated with greater understanding by studying a homogeneous turbid medium. We modeled the medium as an infinite slab and studied the reflection dependence on the following three parameters: the incident direction, optical depth, and asymmetry factor. The diffuse reflection is produced by incoherent multiple scattering and is solved through radiative transfer theory. At large optical depths, the angular distribution of the diffuse reflection with small incident angles is similar to that of a Lambertian surface, but, with incident angles larger than 60°, the angular distributions have a prominent reflection peak around the specular reflection angle. These reflection peaks are found originating from the scattering within one transport mean free path in the top layer of the medium. The maximum reflection angles for different incident angles are analyzed and can characterize the structure of angular distributions for different asymmetry factors and optical depths. The properties of the angular distribution can be applied to more complex systems for a better understanding of diffuse reflection.

© 2013 Optical Society of America

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2013 (1)

2011 (2)

M. Leonetti and C. López, “Measurement of transport mean-free path of light in thin systems,” Opt. Lett. 36, 2824–2826 (2011).
[CrossRef]

A. García-Valenzuela, F. L. S. Cuppo, and J. A. Olivares, “An assessment of Saunderson corrections to the diffuse reflectance of paint films,” J. Phys. Conf. Ser. 274, 012125 (2011).
[CrossRef]

2010 (1)

M. Janecek and W. W. Moses, “Simulating scintillator light collection using measured optical reflectance,” IEEE Trans. Nucl. Sci. 57, 964–970 (2010).

2008 (4)

M. I. Mishchenko, “Multiple scattering, radiative transfer, and weak localization in discrete random media: unified microphysical approach,” Rev. Geophys. 46, RG2003 (2008).
[CrossRef]

M. Janecek and W. W. Moses, “Optical reflectance measurements for commonly used reflectors,” IEEE Trans. Nucl. Sci. 55, 2432–2437 (2008).
[CrossRef]

P. W. Zhai, G. W. Kattawar, and P. Yang, “Impulse response solution to the three-dimensional vector radiative transfer equation in atmosphere-ocean systems. I. Monte Carlo method,” Appl. Opt. 47, 1037–1047 (2008).
[CrossRef]

G. Zonios and A. Dimou, “Melanin optical properties provide evidence for chemical and structural disorder in vivo,” Opt. Express 16, 8263–8268 (2008).
[CrossRef]

2007 (1)

2006 (2)

Z. Wang, M. A. Webster, A. M. Weiner, and K. J. Webb, “Polarized temporal impulse response for scattering media from third-order frequency correlations of speckle intensity patterns,” J. Opt. Soc. Am. A 23, 3045–3053 (2006).
[CrossRef]

S. C. Gebhart, W. C. Lin, and A. Mahadevan-Jansen, “In vitro determination of normal and neoplastic human brain tissue optical properties using inverse adding-doubling,” Phys. Med. Biol. 51, 2011–2027 (2006).
[CrossRef]

2003 (1)

M. D. Shawkey, A. M. Estes, L. M. Siefferman, and G. E. Hill, “Nanostructure predicts intraspecific variation in ultraviolet–blue plumage colour,” Proc. R. Soc. B 270, 1455–1460 (2003).
[CrossRef]

2001 (1)

R. B. A. Koelemeijer, P. Stammes, J. W. Hovenier, and J. F. de Haan, “A fast method for retrieval of cloud parameters using oxygen A-band measurements from the global ozone monitoring experiment,” J. Geophys. Res. Atmos. 106, 3475–3490 (2001).
[CrossRef]

1999 (2)

A. Springsteen, “Standards for the measurement of diffuse reflectance—an overview of available materials and measurement laboratories,” Anal. Chim. Acta 380, 379–390 (1999).
[CrossRef]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41 (1999).
[CrossRef]

1998 (1)

C. Sergent, C. Leroux, E. Pougatch, and F. Guirado, “Hemispherical-directional reflectance measurements of natural snows in the 0.9–1.45 μm spectral range: comparison with adding-doubling modelling,” Ann. Glaciol. 26, 59–63 (1998).

1996 (1)

H. van de Hulst, “Scaling laws in multiple light scattering under very small angles,” Rev. Mod. Astron. 9, 1–16 (1996).

1993 (3)

1992 (1)

J.-L. Roujean, M. Leroy, and P.-Y. Deschamps, “A bidirectional reflectance model of the Earth’s surface for the correction of remote sensing data,” J. Geophys. Res. 97, 20455–20468 (1992).
[CrossRef]

1991 (2)

A. Steyerl, S. S. Malik, and L. R. Iyengar, “Specular and diffuse reflection and refraction at surfaces,” Physica B 173, 47–64 (1991).
[CrossRef]

A. Morel and B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on sun angle as influenced by the molecular scattering contribution,” Appl. Opt. 30, 4427–4438 (1991).
[CrossRef]

1987 (2)

C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987).
[CrossRef]

J. F. Dehaan, P. B. Bosma, and J. W. Hovenier, “The adding method for multiple-scattering calculations of polarized-light,” Astron. Astrophys. 183, 371–391 (1987).

1977 (1)

W. J. Wiscombe, “The delta–m method: rapid yet accurate radiative flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. 34, 1408–1422 (1977).
[CrossRef]

1973 (1)

G. W. Kattawar and G. N. Plass, “Interior radiances in optically deep absorbing media. 1. Exact solutions for one-dimensional model,” J. Quant. Spectrosc. Radiat. Transfer 13, 1065–1080 (1973).
[CrossRef]

1967 (1)

1965 (1)

1942 (1)

1941 (1)

L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Arridge, S. R.

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41 (1999).
[CrossRef]

Bell, G. R. R.

L. M. Mäthger, S. L. Senft, M. Gao, S. Karaveli, G. R. R. Bell, R. Zia, A. M. Kuzirian, P. B. Dennis, W. J. Crookes-Goodson, R. R. Naik, G. W. Kattawar, and R. T. Hanlon, “Bright white scattering from protein spheres in color changing, flexible cuttlefish skin,” Adv. Funct. Mat. (to be published).
[CrossRef]

Bohren, C.

C. Bohren and E. Clothiaux, Fundamentals of Atmospheric Radiation (Wiley-VCH, 2006).

Bohren, C. F.

C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge University, 1999).

Bosma, P. B.

J. F. Dehaan, P. B. Bosma, and J. W. Hovenier, “The adding method for multiple-scattering calculations of polarized-light,” Astron. Astrophys. 183, 371–391 (1987).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover Publications, Inc., 1960).

Clothiaux, E.

C. Bohren and E. Clothiaux, Fundamentals of Atmospheric Radiation (Wiley-VCH, 2006).

Crookes-Goodson, W. J.

L. M. Mäthger, S. L. Senft, M. Gao, S. Karaveli, G. R. R. Bell, R. Zia, A. M. Kuzirian, P. B. Dennis, W. J. Crookes-Goodson, R. R. Naik, G. W. Kattawar, and R. T. Hanlon, “Bright white scattering from protein spheres in color changing, flexible cuttlefish skin,” Adv. Funct. Mat. (to be published).
[CrossRef]

Cuppo, F. L. S.

A. García-Valenzuela, F. L. S. Cuppo, and J. A. Olivares, “An assessment of Saunderson corrections to the diffuse reflectance of paint films,” J. Phys. Conf. Ser. 274, 012125 (2011).
[CrossRef]

de Haan, J. F.

R. B. A. Koelemeijer, P. Stammes, J. W. Hovenier, and J. F. de Haan, “A fast method for retrieval of cloud parameters using oxygen A-band measurements from the global ozone monitoring experiment,” J. Geophys. Res. Atmos. 106, 3475–3490 (2001).
[CrossRef]

Dehaan, J. F.

J. F. Dehaan, P. B. Bosma, and J. W. Hovenier, “The adding method for multiple-scattering calculations of polarized-light,” Astron. Astrophys. 183, 371–391 (1987).

Dennis, P. B.

L. M. Mäthger, S. L. Senft, M. Gao, S. Karaveli, G. R. R. Bell, R. Zia, A. M. Kuzirian, P. B. Dennis, W. J. Crookes-Goodson, R. R. Naik, G. W. Kattawar, and R. T. Hanlon, “Bright white scattering from protein spheres in color changing, flexible cuttlefish skin,” Adv. Funct. Mat. (to be published).
[CrossRef]

Deschamps, P.-Y.

J.-L. Roujean, M. Leroy, and P.-Y. Deschamps, “A bidirectional reflectance model of the Earth’s surface for the correction of remote sensing data,” J. Geophys. Res. 97, 20455–20468 (1992).
[CrossRef]

Dimou, A.

Domke, H.

J. W. Hovenier, C. van der Mee, and H. Domke, Transfer of Polarized Light in Planetary Atmospheres: Basic Concepts and Practical Methods (Kluwer Academic, 2004).

Estes, A. M.

M. D. Shawkey, A. M. Estes, L. M. Siefferman, and G. E. Hill, “Nanostructure predicts intraspecific variation in ultraviolet–blue plumage colour,” Proc. R. Soc. B 270, 1455–1460 (2003).
[CrossRef]

Gao, M.

L. M. Mäthger, S. L. Senft, M. Gao, S. Karaveli, G. R. R. Bell, R. Zia, A. M. Kuzirian, P. B. Dennis, W. J. Crookes-Goodson, R. R. Naik, G. W. Kattawar, and R. T. Hanlon, “Bright white scattering from protein spheres in color changing, flexible cuttlefish skin,” Adv. Funct. Mat. (to be published).
[CrossRef]

García-Valenzuela, A.

A. García-Valenzuela, F. L. S. Cuppo, and J. A. Olivares, “An assessment of Saunderson corrections to the diffuse reflectance of paint films,” J. Phys. Conf. Ser. 274, 012125 (2011).
[CrossRef]

Gebhart, S. C.

S. C. Gebhart, W. C. Lin, and A. Mahadevan-Jansen, “In vitro determination of normal and neoplastic human brain tissue optical properties using inverse adding-doubling,” Phys. Med. Biol. 51, 2011–2027 (2006).
[CrossRef]

Gentili, B.

Greenstein, J. L.

L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Guirado, F.

C. Sergent, C. Leroux, E. Pougatch, and F. Guirado, “Hemispherical-directional reflectance measurements of natural snows in the 0.9–1.45 μm spectral range: comparison with adding-doubling modelling,” Ann. Glaciol. 26, 59–63 (1998).

Haag, J. M.

Hanlon, R. T.

L. M. Mäthger, S. L. Senft, M. Gao, S. Karaveli, G. R. R. Bell, R. Zia, A. M. Kuzirian, P. B. Dennis, W. J. Crookes-Goodson, R. R. Naik, G. W. Kattawar, and R. T. Hanlon, “Bright white scattering from protein spheres in color changing, flexible cuttlefish skin,” Adv. Funct. Mat. (to be published).
[CrossRef]

Henyey, L. C.

L. C. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Hill, G. E.

M. D. Shawkey, A. M. Estes, L. M. Siefferman, and G. E. Hill, “Nanostructure predicts intraspecific variation in ultraviolet–blue plumage colour,” Proc. R. Soc. B 270, 1455–1460 (2003).
[CrossRef]

Hovenier, J. W.

R. B. A. Koelemeijer, P. Stammes, J. W. Hovenier, and J. F. de Haan, “A fast method for retrieval of cloud parameters using oxygen A-band measurements from the global ozone monitoring experiment,” J. Geophys. Res. Atmos. 106, 3475–3490 (2001).
[CrossRef]

J. F. Dehaan, P. B. Bosma, and J. W. Hovenier, “The adding method for multiple-scattering calculations of polarized-light,” Astron. Astrophys. 183, 371–391 (1987).

J. W. Hovenier, C. van der Mee, and H. Domke, Transfer of Polarized Light in Planetary Atmospheres: Basic Concepts and Practical Methods (Kluwer Academic, 2004).

Humphreys, G.

M. Pharr and G. Humphreys, Physically Based Rendering: From Theory to Implementation (Morgan Kaufmann, 2010).

Iyengar, L. R.

A. Steyerl, S. S. Malik, and L. R. Iyengar, “Specular and diffuse reflection and refraction at surfaces,” Physica B 173, 47–64 (1991).
[CrossRef]

Jacques, S. L.

Jaffe, J. S.

Janecek, M.

M. Janecek and W. W. Moses, “Simulating scintillator light collection using measured optical reflectance,” IEEE Trans. Nucl. Sci. 57, 964–970 (2010).

M. Janecek and W. W. Moses, “Optical reflectance measurements for commonly used reflectors,” IEEE Trans. Nucl. Sci. 55, 2432–2437 (2008).
[CrossRef]

Karaveli, S.

L. M. Mäthger, S. L. Senft, M. Gao, S. Karaveli, G. R. R. Bell, R. Zia, A. M. Kuzirian, P. B. Dennis, W. J. Crookes-Goodson, R. R. Naik, G. W. Kattawar, and R. T. Hanlon, “Bright white scattering from protein spheres in color changing, flexible cuttlefish skin,” Adv. Funct. Mat. (to be published).
[CrossRef]

Kattawar, G. W.

P. W. Zhai, G. W. Kattawar, and P. Yang, “Impulse response solution to the three-dimensional vector radiative transfer equation in atmosphere-ocean systems. I. Monte Carlo method,” Appl. Opt. 47, 1037–1047 (2008).
[CrossRef]

G. W. Kattawar and G. N. Plass, “Interior radiances in optically deep absorbing media. 1. Exact solutions for one-dimensional model,” J. Quant. Spectrosc. Radiat. Transfer 13, 1065–1080 (1973).
[CrossRef]

L. M. Mäthger, S. L. Senft, M. Gao, S. Karaveli, G. R. R. Bell, R. Zia, A. M. Kuzirian, P. B. Dennis, W. J. Crookes-Goodson, R. R. Naik, G. W. Kattawar, and R. T. Hanlon, “Bright white scattering from protein spheres in color changing, flexible cuttlefish skin,” Adv. Funct. Mat. (to be published).
[CrossRef]

Koelemeijer, R. B. A.

R. B. A. Koelemeijer, P. Stammes, J. W. Hovenier, and J. F. de Haan, “A fast method for retrieval of cloud parameters using oxygen A-band measurements from the global ozone monitoring experiment,” J. Geophys. Res. Atmos. 106, 3475–3490 (2001).
[CrossRef]

Kuzirian, A. M.

L. M. Mäthger, S. L. Senft, M. Gao, S. Karaveli, G. R. R. Bell, R. Zia, A. M. Kuzirian, P. B. Dennis, W. J. Crookes-Goodson, R. R. Naik, G. W. Kattawar, and R. T. Hanlon, “Bright white scattering from protein spheres in color changing, flexible cuttlefish skin,” Adv. Funct. Mat. (to be published).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University, 2006).

Leonetti, M.

Leroux, C.

C. Sergent, C. Leroux, E. Pougatch, and F. Guirado, “Hemispherical-directional reflectance measurements of natural snows in the 0.9–1.45 μm spectral range: comparison with adding-doubling modelling,” Ann. Glaciol. 26, 59–63 (1998).

Leroy, M.

J.-L. Roujean, M. Leroy, and P.-Y. Deschamps, “A bidirectional reflectance model of the Earth’s surface for the correction of remote sensing data,” J. Geophys. Res. 97, 20455–20468 (1992).
[CrossRef]

Lin, W. C.

S. C. Gebhart, W. C. Lin, and A. Mahadevan-Jansen, “In vitro determination of normal and neoplastic human brain tissue optical properties using inverse adding-doubling,” Phys. Med. Biol. 51, 2011–2027 (2006).
[CrossRef]

Liou, K. N.

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, 2002).

López, C.

Mahadevan-Jansen, A.

S. C. Gebhart, W. C. Lin, and A. Mahadevan-Jansen, “In vitro determination of normal and neoplastic human brain tissue optical properties using inverse adding-doubling,” Phys. Med. Biol. 51, 2011–2027 (2006).
[CrossRef]

Malik, S. S.

A. Steyerl, S. S. Malik, and L. R. Iyengar, “Specular and diffuse reflection and refraction at surfaces,” Physica B 173, 47–64 (1991).
[CrossRef]

Mäthger, L. M.

L. M. Mäthger, S. L. Senft, M. Gao, S. Karaveli, G. R. R. Bell, R. Zia, A. M. Kuzirian, P. B. Dennis, W. J. Crookes-Goodson, R. R. Naik, G. W. Kattawar, and R. T. Hanlon, “Bright white scattering from protein spheres in color changing, flexible cuttlefish skin,” Adv. Funct. Mat. (to be published).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, “Multiple scattering, radiative transfer, and weak localization in discrete random media: unified microphysical approach,” Rev. Geophys. 46, RG2003 (2008).
[CrossRef]

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University, 2006).

Mobley, C.

C. Mobley, Light and Water-Radiative Transfer in Natural Waters (Academic, 1994).

Morel, A.

Moses, W. W.

M. Janecek and W. W. Moses, “Simulating scintillator light collection using measured optical reflectance,” IEEE Trans. Nucl. Sci. 57, 964–970 (2010).

M. Janecek and W. W. Moses, “Optical reflectance measurements for commonly used reflectors,” IEEE Trans. Nucl. Sci. 55, 2432–2437 (2008).
[CrossRef]

Naik, R. R.

L. M. Mäthger, S. L. Senft, M. Gao, S. Karaveli, G. R. R. Bell, R. Zia, A. M. Kuzirian, P. B. Dennis, W. J. Crookes-Goodson, R. R. Naik, G. W. Kattawar, and R. T. Hanlon, “Bright white scattering from protein spheres in color changing, flexible cuttlefish skin,” Adv. Funct. Mat. (to be published).
[CrossRef]

Nicodemus, F. E.

Olivares, J. A.

A. García-Valenzuela, F. L. S. Cuppo, and J. A. Olivares, “An assessment of Saunderson corrections to the diffuse reflectance of paint films,” J. Phys. Conf. Ser. 274, 012125 (2011).
[CrossRef]

Petty, G. W.

G. W. Petty, A First Course in Atmospheric Radiation, 2nd ed. (Sundog Publishing, 2006).

Pharr, M.

M. Pharr and G. Humphreys, Physically Based Rendering: From Theory to Implementation (Morgan Kaufmann, 2010).

Plass, G. N.

G. W. Kattawar and G. N. Plass, “Interior radiances in optically deep absorbing media. 1. Exact solutions for one-dimensional model,” J. Quant. Spectrosc. Radiat. Transfer 13, 1065–1080 (1973).
[CrossRef]

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C. Sergent, C. Leroux, E. Pougatch, and F. Guirado, “Hemispherical-directional reflectance measurements of natural snows in the 0.9–1.45 μm spectral range: comparison with adding-doubling modelling,” Ann. Glaciol. 26, 59–63 (1998).

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[CrossRef]

Saunderson, J. L.

Senft, S. L.

L. M. Mäthger, S. L. Senft, M. Gao, S. Karaveli, G. R. R. Bell, R. Zia, A. M. Kuzirian, P. B. Dennis, W. J. Crookes-Goodson, R. R. Naik, G. W. Kattawar, and R. T. Hanlon, “Bright white scattering from protein spheres in color changing, flexible cuttlefish skin,” Adv. Funct. Mat. (to be published).
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C. Sergent, C. Leroux, E. Pougatch, and F. Guirado, “Hemispherical-directional reflectance measurements of natural snows in the 0.9–1.45 μm spectral range: comparison with adding-doubling modelling,” Ann. Glaciol. 26, 59–63 (1998).

Shawkey, M. D.

M. D. Shawkey, A. M. Estes, L. M. Siefferman, and G. E. Hill, “Nanostructure predicts intraspecific variation in ultraviolet–blue plumage colour,” Proc. R. Soc. B 270, 1455–1460 (2003).
[CrossRef]

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M. D. Shawkey, A. M. Estes, L. M. Siefferman, and G. E. Hill, “Nanostructure predicts intraspecific variation in ultraviolet–blue plumage colour,” Proc. R. Soc. B 270, 1455–1460 (2003).
[CrossRef]

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R. B. A. Koelemeijer, P. Stammes, J. W. Hovenier, and J. F. de Haan, “A fast method for retrieval of cloud parameters using oxygen A-band measurements from the global ozone monitoring experiment,” J. Geophys. Res. Atmos. 106, 3475–3490 (2001).
[CrossRef]

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[CrossRef]

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H. van de Hulst, “Scaling laws in multiple light scattering under very small angles,” Rev. Mod. Astron. 9, 1–16 (1996).

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J. W. Hovenier, C. van der Mee, and H. Domke, Transfer of Polarized Light in Planetary Atmospheres: Basic Concepts and Practical Methods (Kluwer Academic, 2004).

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L. V. Wang and H.-I. Wu, Biomedical Optics: Principles and Imaging (Wiley, 2007).

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M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge University, 1999).

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L. V. Wang and H.-I. Wu, Biomedical Optics: Principles and Imaging (Wiley, 2007).

Xia, J.

Yang, P.

Yao, G.

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Zia, R.

L. M. Mäthger, S. L. Senft, M. Gao, S. Karaveli, G. R. R. Bell, R. Zia, A. M. Kuzirian, P. B. Dennis, W. J. Crookes-Goodson, R. R. Naik, G. W. Kattawar, and R. T. Hanlon, “Bright white scattering from protein spheres in color changing, flexible cuttlefish skin,” Adv. Funct. Mat. (to be published).
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A. Springsteen, “Standards for the measurement of diffuse reflectance—an overview of available materials and measurement laboratories,” Anal. Chim. Acta 380, 379–390 (1999).
[CrossRef]

Ann. Glaciol. (1)

C. Sergent, C. Leroux, E. Pougatch, and F. Guirado, “Hemispherical-directional reflectance measurements of natural snows in the 0.9–1.45 μm spectral range: comparison with adding-doubling modelling,” Ann. Glaciol. 26, 59–63 (1998).

Appl. Opt. (6)

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S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41 (1999).
[CrossRef]

J. Atmos. Sci. (1)

W. J. Wiscombe, “The delta–m method: rapid yet accurate radiative flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. 34, 1408–1422 (1977).
[CrossRef]

J. Geophys. Res. (1)

J.-L. Roujean, M. Leroy, and P.-Y. Deschamps, “A bidirectional reflectance model of the Earth’s surface for the correction of remote sensing data,” J. Geophys. Res. 97, 20455–20468 (1992).
[CrossRef]

J. Geophys. Res. Atmos. (1)

R. B. A. Koelemeijer, P. Stammes, J. W. Hovenier, and J. F. de Haan, “A fast method for retrieval of cloud parameters using oxygen A-band measurements from the global ozone monitoring experiment,” J. Geophys. Res. Atmos. 106, 3475–3490 (2001).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

J. Phys. Conf. Ser. (1)

A. García-Valenzuela, F. L. S. Cuppo, and J. A. Olivares, “An assessment of Saunderson corrections to the diffuse reflectance of paint films,” J. Phys. Conf. Ser. 274, 012125 (2011).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

G. W. Kattawar and G. N. Plass, “Interior radiances in optically deep absorbing media. 1. Exact solutions for one-dimensional model,” J. Quant. Spectrosc. Radiat. Transfer 13, 1065–1080 (1973).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Med. Biol. (1)

S. C. Gebhart, W. C. Lin, and A. Mahadevan-Jansen, “In vitro determination of normal and neoplastic human brain tissue optical properties using inverse adding-doubling,” Phys. Med. Biol. 51, 2011–2027 (2006).
[CrossRef]

Physica B (1)

A. Steyerl, S. S. Malik, and L. R. Iyengar, “Specular and diffuse reflection and refraction at surfaces,” Physica B 173, 47–64 (1991).
[CrossRef]

Proc. R. Soc. B (1)

M. D. Shawkey, A. M. Estes, L. M. Siefferman, and G. E. Hill, “Nanostructure predicts intraspecific variation in ultraviolet–blue plumage colour,” Proc. R. Soc. B 270, 1455–1460 (2003).
[CrossRef]

Rev. Geophys. (1)

M. I. Mishchenko, “Multiple scattering, radiative transfer, and weak localization in discrete random media: unified microphysical approach,” Rev. Geophys. 46, RG2003 (2008).
[CrossRef]

Rev. Mod. Astron. (1)

H. van de Hulst, “Scaling laws in multiple light scattering under very small angles,” Rev. Mod. Astron. 9, 1–16 (1996).

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G. W. Petty, A First Course in Atmospheric Radiation, 2nd ed. (Sundog Publishing, 2006).

J. W. Hovenier, C. van der Mee, and H. Domke, Transfer of Polarized Light in Planetary Atmospheres: Basic Concepts and Practical Methods (Kluwer Academic, 2004).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University, 2006).

L. M. Mäthger, S. L. Senft, M. Gao, S. Karaveli, G. R. R. Bell, R. Zia, A. M. Kuzirian, P. B. Dennis, W. J. Crookes-Goodson, R. R. Naik, G. W. Kattawar, and R. T. Hanlon, “Bright white scattering from protein spheres in color changing, flexible cuttlefish skin,” Adv. Funct. Mat. (to be published).
[CrossRef]

M. Pharr and G. Humphreys, Physically Based Rendering: From Theory to Implementation (Morgan Kaufmann, 2010).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge University, 1999).

S. Chandrasekhar, Radiative Transfer (Dover Publications, Inc., 1960).

C. Mobley, Light and Water-Radiative Transfer in Natural Waters (Academic, 1994).

L. V. Wang and H.-I. Wu, Biomedical Optics: Principles and Imaging (Wiley, 2007).

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, 2002).

C. Bohren and E. Clothiaux, Fundamentals of Atmospheric Radiation (Wiley-VCH, 2006).

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Figures (13)

Fig. 1.
Fig. 1.

HG phase function p(θs) with asymmetry factor g=0, 0.5, 0.8, and 0.9.

Fig. 2.
Fig. 2.

(a) Incident direction is in the xz plane along direction (180θi,ϕi), where θi is the incident angle and ϕi is the incident azimuthal angle. (b) The reflection zenith angle θ[90°,90°] and azimuthal angle ϕ[90°,90°].

Fig. 3.
Fig. 3.

Angular reflectance versus both zenith angle θ and azimuthal angle ϕ. The system has an adjusted optical depth τ*=10, asymmetry factor g=0.9, and incident angle (a) θi=0° and (b) θi=70°.

Fig. 4.
Fig. 4.

Simulated flux reflectance versus incident angle θi with asymmetry factor g=0, 0.5, and 0.9, and adjusted optical depth τ*=0.1, 1, and 10 (dashed lines). The red lines are the two stream results for the same adjusted optical depth τ*=0.1, 1, and 10.

Fig. 5.
Fig. 5.

Simulated results (solid lines) and the single-scattering approximation (dashed lines) for the angular reflectance versus zenith angle θ with asymmetry factor g=0.9. (a) Incident angle θi=0° and adjusted optical depths τ*=0.0001, 0.001, 0.01, and 0.1, and (b) τ*=0.001 and θi from 0° to 80° with an interval of 10°.

Fig. 6.
Fig. 6.

Angular reflectance versus zenith angle θ for isotropic scattering (g=0). The solid lines are simulated results for an adjusted optical depth τ*=1000; the discrete points are theoretical results.

Fig. 7.
Fig. 7.

Angular reflectance and transmittance versus zenith angle θ at zenith angle ϕ=0° for asymmetry factor g=0.9, adjusted optical depth τ*=0.01 (upper panels), τ*=0.1 (lower panels), and incident angle θi from 0° to 80° with an interval of 10°.

Fig. 8.
Fig. 8.

Same as Fig. 7 but for τ*=1 (upper panels) and τ*=10 (lower panels).

Fig. 9.
Fig. 9.

Maximum reflection angle (θr) versus incident angle (θi) for adjusted optical depths τ*=0.5, 1, and 2, and asymmetry factor g from 0 to 0.9 with an interval of 0.1. The red solid lines indicate θr=θi. The solid curved lines are the results for g=0.9.

Fig. 10.
Fig. 10.

Maximum reflection angles (θr) versus incident angles (θi) for asymmetry factor g=0.9 and adjusted optical depth τ*. τ* for the dashed lines from upper to lower, respectively, are 0.001, 0.01, 0.1 to 1.5 (with an interval of 0.1), 2, 3, 5, 10, and 100. The red solid line indicates θr=θi, and the solid curve indicates τ*=1.2, which has the minimum deviation from the red line.

Fig. 11.
Fig. 11.

Maximum reflection angles (θr) versus incident angles (θi) for asymmetry factor g=0.9 and adjusted optical depth τ*=103 to 105.

Fig. 12.
Fig. 12.

Non-Lambertian reflection ratio (ρ) versus incident angle θi for adjusted optical depth τ*=0.1, 1, 10, and 100, and asymmetry factor g=0.9.

Fig. 13.
Fig. 13.

(a) Difference between the angular reflectance for adjusted optical depth τ*=1 and τ*=10 at incident angles θi=0° to 80° with steps of 10°. (b) The angular reflectance difference between τ*=1 and τ*=1, 1.1 to 1.5 (with steps of 0.1), 2, 3, 5, 10, 100, and 1000 at incident angle θi=80°. Asymmetry factor g=0.9 is considered.

Equations (11)

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p(μ)=14π1g2(1+g22gμ)3/2,
g=4πp(μ)μdΩ.
R(μ,ϕ;μi,ϕi)=μI(μ,ϕ)μiF0,
f(μ,ϕ;μi,ϕi)=I(μ,ϕ)μiF0.
R(μi,ϕi)=02πdϕ01dμR(μ,ϕ;μi,ϕi).
Rdiff=τ*2μi+τ*.
R1(μ,ϕ;μi,ϕi)=μμ+μi[1exp(μ+μiμμiτ)]p(μ),
μ=μiμ+1μi21μ2cos(ϕiϕ).
R(μ,ϕ;μi,ϕi)=14πμμ+μiH(μi)H(μ),
ΔR2=02πdϕ01dμ[R(μ,ϕ;μi,ϕi)RL(μ,ϕ)]2.
ρ=ΔR2R,

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