Abstract

We consider the scattering and absorption of light in discrete random media of densely packed spherical particles. In what we term “radiative transfer with reciprocal transactions” (R2T2), we introduce a volume element of the random medium, derive its scattering and absorption characteristics using the superposition T-Matrix method (STMM), and compute its frequency-domain incoherent volume-element scattering characteristics. Using an order-of-scattering approach, we then compute a numerical Monte Carlo solution for the scattering problem with an exact treatment of the interaction between two volume elements. We compute both the direct and reciprocal contributions along a sequence of volume elements, allowing us to evaluate the coherent backscattering effects. We show that the R2T2 and exact STMM solutions are in mutual agreement for large finite systems of densely packed spherical particles. We conclude that the R2T2 method provides a viable numerical solution for scattering by asymptotically infinite systems of particles.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).
  2. L. Tsang and A. Ishimaru, J. Electromagn. Waves Appl. 1, 59 (1987).
    [Crossref]
  3. K. Muinonen, Waves Random Media 14, 365 (2004).
    [Crossref]
  4. K. Muinonen, M. I. Mishchenko, J. M. Dlugach, E. Zubko, A. Penttilä, and G. Videen, Astrophys. J. 760, 118 (2012).
    [Crossref]
  5. L. M. Zurk, L. Tsang, K. H. Ding, and D. P. Winebrenner, J. Opt. Soc. Am. A 12, 1772 (1995).
    [Crossref]
  6. C. C. Lu, W. C. Chew, and L. Tsang, Radio Sci. 30, 25 (1995).
    [Crossref]
  7. J. Markkanen and A. J. Yuffa, J. Quant. Spectrosc. Radiat. Transfer 189, 181 (2017).
    [Crossref]
  8. D. W. Mackowski and M. I. Mishchenko, J. Quant. Spectrosc. Radiat. Transfer 112, 2182 (2011).
    [Crossref]
  9. N. P. Barabashev, Astron. Nachr. 217, 445 (1922).
    [Crossref]
  10. B. Lyot, Recherches sur la polarisation de la lumière des planètes et de quelques substances terrestres (Observatoire de Paris, 1929), Vol. 8.
  11. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles (Cambridge University, 2006).

2017 (1)

J. Markkanen and A. J. Yuffa, J. Quant. Spectrosc. Radiat. Transfer 189, 181 (2017).
[Crossref]

2012 (1)

K. Muinonen, M. I. Mishchenko, J. M. Dlugach, E. Zubko, A. Penttilä, and G. Videen, Astrophys. J. 760, 118 (2012).
[Crossref]

2011 (1)

D. W. Mackowski and M. I. Mishchenko, J. Quant. Spectrosc. Radiat. Transfer 112, 2182 (2011).
[Crossref]

2004 (1)

K. Muinonen, Waves Random Media 14, 365 (2004).
[Crossref]

1995 (2)

1987 (1)

L. Tsang and A. Ishimaru, J. Electromagn. Waves Appl. 1, 59 (1987).
[Crossref]

1922 (1)

N. P. Barabashev, Astron. Nachr. 217, 445 (1922).
[Crossref]

Barabashev, N. P.

N. P. Barabashev, Astron. Nachr. 217, 445 (1922).
[Crossref]

Chew, W. C.

C. C. Lu, W. C. Chew, and L. Tsang, Radio Sci. 30, 25 (1995).
[Crossref]

Ding, K. H.

Dlugach, J. M.

K. Muinonen, M. I. Mishchenko, J. M. Dlugach, E. Zubko, A. Penttilä, and G. Videen, Astrophys. J. 760, 118 (2012).
[Crossref]

Ishimaru, A.

L. Tsang and A. Ishimaru, J. Electromagn. Waves Appl. 1, 59 (1987).
[Crossref]

Kong, J. A.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Lacis, A. A.

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

Lu, C. C.

C. C. Lu, W. C. Chew, and L. Tsang, Radio Sci. 30, 25 (1995).
[Crossref]

Lyot, B.

B. Lyot, Recherches sur la polarisation de la lumière des planètes et de quelques substances terrestres (Observatoire de Paris, 1929), Vol. 8.

Mackowski, D. W.

D. W. Mackowski and M. I. Mishchenko, J. Quant. Spectrosc. Radiat. Transfer 112, 2182 (2011).
[Crossref]

Markkanen, J.

J. Markkanen and A. J. Yuffa, J. Quant. Spectrosc. Radiat. Transfer 189, 181 (2017).
[Crossref]

Mishchenko, M. I.

K. Muinonen, M. I. Mishchenko, J. M. Dlugach, E. Zubko, A. Penttilä, and G. Videen, Astrophys. J. 760, 118 (2012).
[Crossref]

D. W. Mackowski and M. I. Mishchenko, J. Quant. Spectrosc. Radiat. Transfer 112, 2182 (2011).
[Crossref]

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

Muinonen, K.

K. Muinonen, M. I. Mishchenko, J. M. Dlugach, E. Zubko, A. Penttilä, and G. Videen, Astrophys. J. 760, 118 (2012).
[Crossref]

K. Muinonen, Waves Random Media 14, 365 (2004).
[Crossref]

Penttilä, A.

K. Muinonen, M. I. Mishchenko, J. M. Dlugach, E. Zubko, A. Penttilä, and G. Videen, Astrophys. J. 760, 118 (2012).
[Crossref]

Shin, R. T.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Travis, L. D.

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

Tsang, L.

L. M. Zurk, L. Tsang, K. H. Ding, and D. P. Winebrenner, J. Opt. Soc. Am. A 12, 1772 (1995).
[Crossref]

C. C. Lu, W. C. Chew, and L. Tsang, Radio Sci. 30, 25 (1995).
[Crossref]

L. Tsang and A. Ishimaru, J. Electromagn. Waves Appl. 1, 59 (1987).
[Crossref]

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

Videen, G.

K. Muinonen, M. I. Mishchenko, J. M. Dlugach, E. Zubko, A. Penttilä, and G. Videen, Astrophys. J. 760, 118 (2012).
[Crossref]

Winebrenner, D. P.

Yuffa, A. J.

J. Markkanen and A. J. Yuffa, J. Quant. Spectrosc. Radiat. Transfer 189, 181 (2017).
[Crossref]

Zubko, E.

K. Muinonen, M. I. Mishchenko, J. M. Dlugach, E. Zubko, A. Penttilä, and G. Videen, Astrophys. J. 760, 118 (2012).
[Crossref]

Zurk, L. M.

Astron. Nachr. (1)

N. P. Barabashev, Astron. Nachr. 217, 445 (1922).
[Crossref]

Astrophys. J. (1)

K. Muinonen, M. I. Mishchenko, J. M. Dlugach, E. Zubko, A. Penttilä, and G. Videen, Astrophys. J. 760, 118 (2012).
[Crossref]

J. Electromagn. Waves Appl. (1)

L. Tsang and A. Ishimaru, J. Electromagn. Waves Appl. 1, 59 (1987).
[Crossref]

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

J. Quant. Spectrosc. Radiat. Transfer (2)

J. Markkanen and A. J. Yuffa, J. Quant. Spectrosc. Radiat. Transfer 189, 181 (2017).
[Crossref]

D. W. Mackowski and M. I. Mishchenko, J. Quant. Spectrosc. Radiat. Transfer 112, 2182 (2011).
[Crossref]

Radio Sci. (1)

C. C. Lu, W. C. Chew, and L. Tsang, Radio Sci. 30, 25 (1995).
[Crossref]

Waves Random Media (1)

K. Muinonen, Waves Random Media 14, 365 (2004).
[Crossref]

Other (3)

B. Lyot, Recherches sur la polarisation de la lumière des planètes et de quelques substances terrestres (Observatoire de Paris, 1929), Vol. 8.

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

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley, 1985).

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Figures (5)

Fig. 1.
Fig. 1. Discrete spherical random medium of equal-sized spherical particles. The phase angle α denotes the angle between the source of illumination (in the direction n inc ) and the observer ( n sca ) as seen from the object. The size parameters of the random medium and the particles are k R and k a , respectively. Finally, k = 2 π / λ , where λ is the wavelength.
Fig. 2.
Fig. 2. Ensemble-averaged incoherent scattering matrix element S 11 (left) and the degree of linear polarization for unpolarized incident light S 21 / S 11 (right) as a function of the scattering angle θ = π α for a volume element of spherical particles with size parameter k a = 2 and refractive index m = 1.31 . The volume densities are v = 0.125 (blue) and v = 0.25 (red).
Fig. 3.
Fig. 3. Scattering phase function S 11 (left) and the degree of linear polarization for unpolarized incident light S 21 / S 11 (right) for a spherical discrete random medium (size parameter k R = 100 ) of spherical particles ( k a = 2 and m = 1.31 ). We show the asymptotically exact results from the STMM (red dashed line), the results from radiative transfer with reciprocal transactions (blue solid line), and the results from radiative transfer and coherent backscattering (blue dashed–dotted line). The volume fraction of particles is v = 0.125 .
Fig. 4.
Fig. 4. As in Fig. 3 for v = 0.25 .
Fig. 5.
Fig. 5. Multiple scattering by a large medium of spherical particles ( k R = 10000 , k a = 2 , and m = 1.31 + i 10 4 ). We depict the scattering phase function S 11 (left) and the degree of linear polarization for unpolarized incident light S 21 / S 11 (right) for two volume fractions of v = 0.125 (blue) and v = 0.25 (red).

Equations (4)

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E ( r ) = E inc ( r ) + E s ( r ) = E inc ( r ) + i = 1 N E i 0 ( r ) + i = 1 N j = 1 , j i N E i j ( r ) + i = 1 N j = 1 , j i N k = 1 , k j N E i j k ( r ) + ,
E s , c ( r ) = E s ( r ) = lim n 1 n i = 1 n E i s ( r ) ,
E i s , ic ( r ) = E i s ( r ) E s , c ( r ) .
E s , ic ( r ) = 0 , | E s , ic ( r ) | 2 = | E s ( r ) | 2 | E s , c ( r ) | 2 .

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