Abstract

This study is devoted to the examination of scattering of waves by a slab containing randomly located cylinders. For the first time to our knowledge, the complete transmission problem has been solved numerically. We have compared the radiative transfer theory with a numerical solution of the wave equation. We discuss the coherent effects, such as forward-scattering dip and backscattering enhancement. It is seen that the radiative transfer equation can be used with great accuracy even for optically thin systems whose geometric thickness is comparable with the wavelength. We have also shown the presence of dependent scattering.

© 2001 Optical Society of America

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  2. K. D. Lathrop, “Use of discrete ordinates methods for the solution of neutron transport equations,” Nucl. Sci. Eng. 24, 381–388 (1966).
  3. A. Lagendijk, B. A. van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–216 (1996).
    [CrossRef]
  4. H. C. Hottel, A. F. Sarofim, I. A. Vasalos, W. H. Dalzell, “Multiple scatter: comparison of theory with experiment,” Trans. ASME, Ser. C J. Heat Transfer 92, 285–291 (1970).
    [CrossRef]
  5. A. K. Fung, Microwave Scattering and Emission Models and Their Applications (Artech House, Norwood, Mass., 1994).
  6. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).
  7. U. Frisch, “Wave propagation in random media,” in Probabilistic Methods in Applied Mathematics, A. Bharucha-Reid, ed. (Academic, New York, 1968), pp. 75–198.
  8. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [CrossRef]
  9. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).
  10. L. A. Apresyan, Y. A. Kravtsov, Radiation Transfer, Statistical and Wave Aspects (Gordon & Breach, Amsterdam, 1996).
  11. L. Ryzhik, G. Papanicolaou, J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
    [CrossRef]
  12. M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, “Multiple light scattering by polydispersions of randomly distributed, perfectly-aligned, infinite Mie cylinders illuminated perpendicularly to their axes,” J. Quant. Spectrosc. Radiat. Transf. 47, 401–410 (1992).
    [CrossRef]
  13. J. Keller, “Stochastic equations and wave propagation in random media,” in Stochastic Processes in Mathematical Physics and Engineering, Vol. 16 of Proceedings of Symposia in Applied Mathematics, R. Bellman, ed. (American Mathematical Society, Providence, R.I., 1964), pp. 145–170.
  14. L. Tsang, J. A. Kong, “Scattering of electromagnetic waves from random media with strong permittivity fluctuations,” Radio Sci. 16, 303–320 (1981).
    [CrossRef]
  15. R. West, D. Gibbs, L. Tsang, A. K. Fung, “Comparison of optical scattering experiments and the quasi-crystalline approximation for dense media,” J. Opt. Soc. Am. A 11, 1854–1858 (1994).
    [CrossRef]
  16. C. E. Mandt, Y. Kuga, L. Tsang, A. Ishimaru, “Microwave propagation and scattering in a dense distribution of non-tenuous spheres: experiment and theory,” Waves Random Media 2, 225–234 (1992).
    [CrossRef]
  17. A. Nashshibi, K. Sarabandi, “Experimental characterization of the effective propagation constant of dense random media,” IEEE Trans. Antennas Propag. 47, 1454–1462 (1999).
    [CrossRef]
  18. L. Tsang, C. E. Mandt, K. H. Ding, “Monte Carlo simulations of the extinction rate of dense media with randomly distributed dielectric spheres based on solution of Maxwell’s equations,” Opt. Lett. 17, 314–316 (1992).
    [CrossRef] [PubMed]
  19. L. M. Zurk, L. Tsang, K. H. Ding, D. P. Winnebrener, “Monte Carlo simulations of the extinction rate of densely packed spheres with clustered and nonclustered geometries,” J. Opt. Soc. Am. A 12, 1772–1781 (1995).
    [CrossRef]
  20. L. Tsang, K. H. Ding, S. Shih, J. Kong, “Scattering of electromagnetic waves from dense distributions of spheroidal particles based on Monte Carlo simulations,” J. Opt. Soc. Am. A 15, 2660–2669 (1998).
    [CrossRef]
  21. P. R. Siqueira, K. Sarabandi, “T-matrix determination of effective pemittivity for three-dimensional dense random media,” IEEE Trans. Antennas Propag. 48, 317–327 (2000).
    [CrossRef]
  22. P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Academic, San Diego, Calif., 1995).
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    [CrossRef]
  24. F. E. Nicodemus, “Reflectance nomenclature and directional reflectance and emissivity,” Appl. Opt. 9, 1474–1475 (1970).
    [CrossRef] [PubMed]
  25. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).
  26. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  27. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  28. Z. Jin, K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere–ocean system,” Appl. Opt. 33, 431–442 (1994).
    [CrossRef] [PubMed]
  29. J. Moore, H. Ling, C. S. Liang, “The scattering and absorption characteristics of material coated periodic gratings under oblique incidence,” IEEE Trans. Antennas Propag. 41, 1281–1288 (1993).
    [CrossRef]
  30. P. Mareschal, “Étude de la diffraction électromagnétique par des géométries bidimensionnelles infinies,” Ph.D. dissertation (Ecole Centrale Paris, Paris, 1996).
  31. A. Ishimaru, Y. Kuga, “Attenuation constant of a coherent field in a dense distribution of particles,” J. Opt. Soc. Am. 72, 1317–1320 (1982).
    [CrossRef]
  32. J. J. Greffet, M. Nieto-Vesperinas, “Field theory for generalized bidirectional reflectivity: derivation of Helmholtz’s reciprocity principle and Kirchhoff’s law,” J. Opt. Soc. Am. A 15, 2735–2744 (1998).
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  33. L. M. Zurk, L. Tsang, D. P. Winebrenner, “Scattering properties of dense media from Monte Carlo simulations with application to active remote sensing of snow,” Radio Sci. 31, 803–819 (1996).
    [CrossRef]
  34. K. Stamnes, R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387–399 (1981).
    [CrossRef]
  35. D. A. DeWolf, “Electromagnetic reflections from an extended turbulent medium: cumulative forward-scatter single-backscatter approximation,” IEEE Trans. Antennas Propag. 19, 254–262 (1971).
    [CrossRef]
  36. E. Wolf, G. Maret, “Weak localisation and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
    [CrossRef] [PubMed]
  37. M. P. V. van Albada, A. Lagendijk, “Observation of weak localisation of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
    [CrossRef] [PubMed]
  38. Y. Kuga, A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. A 1, 831–835 (1984).
    [CrossRef]
  39. Y. A. Kravtsov, L. A. Apresyan, “Radiative transfer: new aspects of the old theory,” in Progress in Optics, E. Wolf, ed. (Elsevier Science BV, Amsterdam, 1996), Vol. XXXVI, pp. 179–244.
  40. Ref. 7, Chap. 3.

2000 (1)

P. R. Siqueira, K. Sarabandi, “T-matrix determination of effective pemittivity for three-dimensional dense random media,” IEEE Trans. Antennas Propag. 48, 317–327 (2000).
[CrossRef]

1999 (1)

A. Nashshibi, K. Sarabandi, “Experimental characterization of the effective propagation constant of dense random media,” IEEE Trans. Antennas Propag. 47, 1454–1462 (1999).
[CrossRef]

1998 (2)

1996 (3)

L. M. Zurk, L. Tsang, D. P. Winebrenner, “Scattering properties of dense media from Monte Carlo simulations with application to active remote sensing of snow,” Radio Sci. 31, 803–819 (1996).
[CrossRef]

A. Lagendijk, B. A. van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–216 (1996).
[CrossRef]

L. Ryzhik, G. Papanicolaou, J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[CrossRef]

1995 (1)

1994 (2)

1993 (1)

J. Moore, H. Ling, C. S. Liang, “The scattering and absorption characteristics of material coated periodic gratings under oblique incidence,” IEEE Trans. Antennas Propag. 41, 1281–1288 (1993).
[CrossRef]

1992 (3)

C. E. Mandt, Y. Kuga, L. Tsang, A. Ishimaru, “Microwave propagation and scattering in a dense distribution of non-tenuous spheres: experiment and theory,” Waves Random Media 2, 225–234 (1992).
[CrossRef]

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, “Multiple light scattering by polydispersions of randomly distributed, perfectly-aligned, infinite Mie cylinders illuminated perpendicularly to their axes,” J. Quant. Spectrosc. Radiat. Transf. 47, 401–410 (1992).
[CrossRef]

L. Tsang, C. E. Mandt, K. H. Ding, “Monte Carlo simulations of the extinction rate of dense media with randomly distributed dielectric spheres based on solution of Maxwell’s equations,” Opt. Lett. 17, 314–316 (1992).
[CrossRef] [PubMed]

1985 (2)

E. Wolf, G. Maret, “Weak localisation and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

M. P. V. van Albada, A. Lagendijk, “Observation of weak localisation of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
[CrossRef] [PubMed]

1984 (1)

1982 (1)

1981 (2)

L. Tsang, J. A. Kong, “Scattering of electromagnetic waves from random media with strong permittivity fluctuations,” Radio Sci. 16, 303–320 (1981).
[CrossRef]

K. Stamnes, R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387–399 (1981).
[CrossRef]

1971 (1)

D. A. DeWolf, “Electromagnetic reflections from an extended turbulent medium: cumulative forward-scatter single-backscatter approximation,” IEEE Trans. Antennas Propag. 19, 254–262 (1971).
[CrossRef]

1970 (2)

H. C. Hottel, A. F. Sarofim, I. A. Vasalos, W. H. Dalzell, “Multiple scatter: comparison of theory with experiment,” Trans. ASME, Ser. C J. Heat Transfer 92, 285–291 (1970).
[CrossRef]

F. E. Nicodemus, “Reflectance nomenclature and directional reflectance and emissivity,” Appl. Opt. 9, 1474–1475 (1970).
[CrossRef] [PubMed]

1968 (1)

1966 (1)

K. D. Lathrop, “Use of discrete ordinates methods for the solution of neutron transport equations,” Nucl. Sci. Eng. 24, 381–388 (1966).

1965 (1)

Apresyan, L. A.

Y. A. Kravtsov, L. A. Apresyan, “Radiative transfer: new aspects of the old theory,” in Progress in Optics, E. Wolf, ed. (Elsevier Science BV, Amsterdam, 1996), Vol. XXXVI, pp. 179–244.

L. A. Apresyan, Y. A. Kravtsov, Radiation Transfer, Statistical and Wave Aspects (Gordon & Breach, Amsterdam, 1996).

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Dalzell, W. H.

H. C. Hottel, A. F. Sarofim, I. A. Vasalos, W. H. Dalzell, “Multiple scatter: comparison of theory with experiment,” Trans. ASME, Ser. C J. Heat Transfer 92, 285–291 (1970).
[CrossRef]

DeWolf, D. A.

D. A. DeWolf, “Electromagnetic reflections from an extended turbulent medium: cumulative forward-scatter single-backscatter approximation,” IEEE Trans. Antennas Propag. 19, 254–262 (1971).
[CrossRef]

Ding, K. H.

Dlugach, J. M.

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, “Multiple light scattering by polydispersions of randomly distributed, perfectly-aligned, infinite Mie cylinders illuminated perpendicularly to their axes,” J. Quant. Spectrosc. Radiat. Transf. 47, 401–410 (1992).
[CrossRef]

Frisch, U.

U. Frisch, “Wave propagation in random media,” in Probabilistic Methods in Applied Mathematics, A. Bharucha-Reid, ed. (Academic, New York, 1968), pp. 75–198.

Fung, A. K.

Gibbs, D.

Greffet, J. J.

Hottel, H. C.

H. C. Hottel, A. F. Sarofim, I. A. Vasalos, W. H. Dalzell, “Multiple scatter: comparison of theory with experiment,” Trans. ASME, Ser. C J. Heat Transfer 92, 285–291 (1970).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983).

Ishimaru, A.

C. E. Mandt, Y. Kuga, L. Tsang, A. Ishimaru, “Microwave propagation and scattering in a dense distribution of non-tenuous spheres: experiment and theory,” Waves Random Media 2, 225–234 (1992).
[CrossRef]

Y. Kuga, A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. A 1, 831–835 (1984).
[CrossRef]

A. Ishimaru, Y. Kuga, “Attenuation constant of a coherent field in a dense distribution of particles,” J. Opt. Soc. Am. 72, 1317–1320 (1982).
[CrossRef]

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

Jin, Z.

Keller, J.

J. Keller, “Stochastic equations and wave propagation in random media,” in Stochastic Processes in Mathematical Physics and Engineering, Vol. 16 of Proceedings of Symposia in Applied Mathematics, R. Bellman, ed. (American Mathematical Society, Providence, R.I., 1964), pp. 145–170.

Keller, J. B.

L. Ryzhik, G. Papanicolaou, J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[CrossRef]

Kong, J.

Kong, J. A.

L. Tsang, J. A. Kong, “Scattering of electromagnetic waves from random media with strong permittivity fluctuations,” Radio Sci. 16, 303–320 (1981).
[CrossRef]

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

Kravtsov, Y. A.

L. A. Apresyan, Y. A. Kravtsov, Radiation Transfer, Statistical and Wave Aspects (Gordon & Breach, Amsterdam, 1996).

Y. A. Kravtsov, L. A. Apresyan, “Radiative transfer: new aspects of the old theory,” in Progress in Optics, E. Wolf, ed. (Elsevier Science BV, Amsterdam, 1996), Vol. XXXVI, pp. 179–244.

Kuga, Y.

Lagendijk, A.

A. Lagendijk, B. A. van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–216 (1996).
[CrossRef]

M. P. V. van Albada, A. Lagendijk, “Observation of weak localisation of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
[CrossRef] [PubMed]

Lathrop, K. D.

K. D. Lathrop, “Use of discrete ordinates methods for the solution of neutron transport equations,” Nucl. Sci. Eng. 24, 381–388 (1966).

Liang, C. S.

J. Moore, H. Ling, C. S. Liang, “The scattering and absorption characteristics of material coated periodic gratings under oblique incidence,” IEEE Trans. Antennas Propag. 41, 1281–1288 (1993).
[CrossRef]

Ling, H.

J. Moore, H. Ling, C. S. Liang, “The scattering and absorption characteristics of material coated periodic gratings under oblique incidence,” IEEE Trans. Antennas Propag. 41, 1281–1288 (1993).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).

Mandt, C. E.

C. E. Mandt, Y. Kuga, L. Tsang, A. Ishimaru, “Microwave propagation and scattering in a dense distribution of non-tenuous spheres: experiment and theory,” Waves Random Media 2, 225–234 (1992).
[CrossRef]

L. Tsang, C. E. Mandt, K. H. Ding, “Monte Carlo simulations of the extinction rate of dense media with randomly distributed dielectric spheres based on solution of Maxwell’s equations,” Opt. Lett. 17, 314–316 (1992).
[CrossRef] [PubMed]

Mareschal, P.

P. Mareschal, “Étude de la diffraction électromagnétique par des géométries bidimensionnelles infinies,” Ph.D. dissertation (Ecole Centrale Paris, Paris, 1996).

Maret, G.

E. Wolf, G. Maret, “Weak localisation and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

Mishchenko, M. I.

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, “Multiple light scattering by polydispersions of randomly distributed, perfectly-aligned, infinite Mie cylinders illuminated perpendicularly to their axes,” J. Quant. Spectrosc. Radiat. Transf. 47, 401–410 (1992).
[CrossRef]

Moore, J.

J. Moore, H. Ling, C. S. Liang, “The scattering and absorption characteristics of material coated periodic gratings under oblique incidence,” IEEE Trans. Antennas Propag. 41, 1281–1288 (1993).
[CrossRef]

Nashshibi, A.

A. Nashshibi, K. Sarabandi, “Experimental characterization of the effective propagation constant of dense random media,” IEEE Trans. Antennas Propag. 47, 1454–1462 (1999).
[CrossRef]

Nicodemus, F. E.

Nieto-Vesperinas, M.

Papanicolaou, G.

L. Ryzhik, G. Papanicolaou, J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[CrossRef]

Ryzhik, L.

L. Ryzhik, G. Papanicolaou, J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[CrossRef]

Sarabandi, K.

P. R. Siqueira, K. Sarabandi, “T-matrix determination of effective pemittivity for three-dimensional dense random media,” IEEE Trans. Antennas Propag. 48, 317–327 (2000).
[CrossRef]

A. Nashshibi, K. Sarabandi, “Experimental characterization of the effective propagation constant of dense random media,” IEEE Trans. Antennas Propag. 47, 1454–1462 (1999).
[CrossRef]

Sarofim, A. F.

H. C. Hottel, A. F. Sarofim, I. A. Vasalos, W. H. Dalzell, “Multiple scatter: comparison of theory with experiment,” Trans. ASME, Ser. C J. Heat Transfer 92, 285–291 (1970).
[CrossRef]

Sheng, P.

P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Academic, San Diego, Calif., 1995).

Shih, S.

Shin, R. T.

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

Siqueira, P. R.

P. R. Siqueira, K. Sarabandi, “T-matrix determination of effective pemittivity for three-dimensional dense random media,” IEEE Trans. Antennas Propag. 48, 317–327 (2000).
[CrossRef]

Stamnes, K.

Z. Jin, K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere–ocean system,” Appl. Opt. 33, 431–442 (1994).
[CrossRef] [PubMed]

K. Stamnes, R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387–399 (1981).
[CrossRef]

Swanson, R. A.

K. Stamnes, R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387–399 (1981).
[CrossRef]

Tsang, L.

L. Tsang, K. H. Ding, S. Shih, J. Kong, “Scattering of electromagnetic waves from dense distributions of spheroidal particles based on Monte Carlo simulations,” J. Opt. Soc. Am. A 15, 2660–2669 (1998).
[CrossRef]

L. M. Zurk, L. Tsang, D. P. Winebrenner, “Scattering properties of dense media from Monte Carlo simulations with application to active remote sensing of snow,” Radio Sci. 31, 803–819 (1996).
[CrossRef]

L. M. Zurk, L. Tsang, K. H. Ding, D. P. Winnebrener, “Monte Carlo simulations of the extinction rate of densely packed spheres with clustered and nonclustered geometries,” J. Opt. Soc. Am. A 12, 1772–1781 (1995).
[CrossRef]

R. West, D. Gibbs, L. Tsang, A. K. Fung, “Comparison of optical scattering experiments and the quasi-crystalline approximation for dense media,” J. Opt. Soc. Am. A 11, 1854–1858 (1994).
[CrossRef]

C. E. Mandt, Y. Kuga, L. Tsang, A. Ishimaru, “Microwave propagation and scattering in a dense distribution of non-tenuous spheres: experiment and theory,” Waves Random Media 2, 225–234 (1992).
[CrossRef]

L. Tsang, C. E. Mandt, K. H. Ding, “Monte Carlo simulations of the extinction rate of dense media with randomly distributed dielectric spheres based on solution of Maxwell’s equations,” Opt. Lett. 17, 314–316 (1992).
[CrossRef] [PubMed]

L. Tsang, J. A. Kong, “Scattering of electromagnetic waves from random media with strong permittivity fluctuations,” Radio Sci. 16, 303–320 (1981).
[CrossRef]

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

van Albada, M. P. V.

M. P. V. van Albada, A. Lagendijk, “Observation of weak localisation of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
[CrossRef] [PubMed]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

van Tiggelen, B. A.

A. Lagendijk, B. A. van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–216 (1996).
[CrossRef]

Vasalos, I. A.

H. C. Hottel, A. F. Sarofim, I. A. Vasalos, W. H. Dalzell, “Multiple scatter: comparison of theory with experiment,” Trans. ASME, Ser. C J. Heat Transfer 92, 285–291 (1970).
[CrossRef]

Walther, A.

West, R.

Winebrenner, D. P.

L. M. Zurk, L. Tsang, D. P. Winebrenner, “Scattering properties of dense media from Monte Carlo simulations with application to active remote sensing of snow,” Radio Sci. 31, 803–819 (1996).
[CrossRef]

Winnebrener, D. P.

Wolf, E.

E. Wolf, G. Maret, “Weak localisation and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).

Yanovitskij, E. G.

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, “Multiple light scattering by polydispersions of randomly distributed, perfectly-aligned, infinite Mie cylinders illuminated perpendicularly to their axes,” J. Quant. Spectrosc. Radiat. Transf. 47, 401–410 (1992).
[CrossRef]

Zurk, L. M.

L. M. Zurk, L. Tsang, D. P. Winebrenner, “Scattering properties of dense media from Monte Carlo simulations with application to active remote sensing of snow,” Radio Sci. 31, 803–819 (1996).
[CrossRef]

L. M. Zurk, L. Tsang, K. H. Ding, D. P. Winnebrener, “Monte Carlo simulations of the extinction rate of densely packed spheres with clustered and nonclustered geometries,” J. Opt. Soc. Am. A 12, 1772–1781 (1995).
[CrossRef]

Appl. Opt. (3)

IEEE Trans. Antennas Propag. (4)

J. Moore, H. Ling, C. S. Liang, “The scattering and absorption characteristics of material coated periodic gratings under oblique incidence,” IEEE Trans. Antennas Propag. 41, 1281–1288 (1993).
[CrossRef]

D. A. DeWolf, “Electromagnetic reflections from an extended turbulent medium: cumulative forward-scatter single-backscatter approximation,” IEEE Trans. Antennas Propag. 19, 254–262 (1971).
[CrossRef]

A. Nashshibi, K. Sarabandi, “Experimental characterization of the effective propagation constant of dense random media,” IEEE Trans. Antennas Propag. 47, 1454–1462 (1999).
[CrossRef]

P. R. Siqueira, K. Sarabandi, “T-matrix determination of effective pemittivity for three-dimensional dense random media,” IEEE Trans. Antennas Propag. 48, 317–327 (2000).
[CrossRef]

J. Atmos. Sci. (1)

K. Stamnes, R. A. Swanson, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres,” J. Atmos. Sci. 38, 387–399 (1981).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

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

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, “Multiple light scattering by polydispersions of randomly distributed, perfectly-aligned, infinite Mie cylinders illuminated perpendicularly to their axes,” J. Quant. Spectrosc. Radiat. Transf. 47, 401–410 (1992).
[CrossRef]

Nucl. Sci. Eng. (1)

K. D. Lathrop, “Use of discrete ordinates methods for the solution of neutron transport equations,” Nucl. Sci. Eng. 24, 381–388 (1966).

Opt. Lett. (1)

Phys. Rep. (1)

A. Lagendijk, B. A. van Tiggelen, “Resonant multiple scattering of light,” Phys. Rep. 270, 143–216 (1996).
[CrossRef]

Phys. Rev. Lett. (2)

E. Wolf, G. Maret, “Weak localisation and coherent backscattering of photons in disordered media,” Phys. Rev. Lett. 55, 2696–2699 (1985).
[CrossRef] [PubMed]

M. P. V. van Albada, A. Lagendijk, “Observation of weak localisation of light in a random medium,” Phys. Rev. Lett. 55, 2692–2695 (1985).
[CrossRef] [PubMed]

Radio Sci. (2)

L. Tsang, J. A. Kong, “Scattering of electromagnetic waves from random media with strong permittivity fluctuations,” Radio Sci. 16, 303–320 (1981).
[CrossRef]

L. M. Zurk, L. Tsang, D. P. Winebrenner, “Scattering properties of dense media from Monte Carlo simulations with application to active remote sensing of snow,” Radio Sci. 31, 803–819 (1996).
[CrossRef]

Trans. ASME, Ser. C J. Heat Transfer (1)

H. C. Hottel, A. F. Sarofim, I. A. Vasalos, W. H. Dalzell, “Multiple scatter: comparison of theory with experiment,” Trans. ASME, Ser. C J. Heat Transfer 92, 285–291 (1970).
[CrossRef]

Wave Motion (1)

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[CrossRef]

Waves Random Media (1)

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[CrossRef]

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Ref. 7, Chap. 3.

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Figures (13)

Fig. 1
Fig. 1

2D scattering system.

Fig. 2
Fig. 2

2D geometry numerically simulated by the moment method.

Fig. 3
Fig. 3

Directional–hemispherical reflectance and transmittance (total energy flux) in polarization V for three concentrations: (a) 1.5%, (b) 3%, and (c) 15% by the MM and RT methods. d/λ=1.5.

Fig. 4
Fig. 4

Collimated reflectance and transmittance for three concentrations: (a) 1.5%, (b) 3%, and (c) 15%: coherent (MM) and collimated (RT) energy. d/λ=1.5.

Fig. 5
Fig. 5

Diffuse directional–hemispherical reflectance and transmittance for three concentrations: (a) 1.5%, (b) 3%, and (c) 15%: incoherent (MM) and diffuse (RT) energy. d/λ=1.5.

Fig. 6
Fig. 6

Comparison of the BRDF and the BTDF obtained by the MM and RT methods for a concentration c=1.5% and a thickness d/λ=1.5.

Fig. 7
Fig. 7

Single scattering is coherent in the strictly forward direction (Δϕ=0) and incoherent in all other directions (Δϕ0).

Fig. 8
Fig. 8

Finite-size random system of infinite cylinders under plane-wave incidence in polarization V (electric field parallel to cylinders’ axis). Parameters are a/λ=0.25,s=16.0.

Fig. 9
Fig. 9

Incoherent transmittivity near the forward direction. The influence of the system size is shown.

Fig. 10
Fig. 10

Comparison of the BRDF and the BTDF obtained by the MM and RT methods for a concentration of 3% with a thickness d/λ=6.

Fig. 11
Fig. 11

Coefficient of reflection in the backward direction as a function of incidence for c=3% and d/λ=1.5. Results obtained from MM, RT, and corrected RT are shown. The backscattering peak is recovered with the correction introduced in the RT model.

Fig. 12
Fig. 12

Same as Fig. 6, but for a concentration of 15%.

Fig. 13
Fig. 13

Dependent and independent attenuation constants.

Tables (1)

Tables Icon

Table 1 Characteristic Quantities of the System in the Independent-Scattering Approximation for Polarization V a

Equations (38)

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s  x=sin θ;sy=cos θ.
Is(θout)=BSDF(θout, θin)Iin(θin)cos θindθin,
ρ=0πρ(θout; θin)cos θoutdθout,
τ=0πτ(θout; θin)cos θoutdθout.
ρ+τ=1-α,
κext=nCext=Cexts c,κsca=nCsca=Cscas c,
cos θ dI(τ, θ)dτ=I(τ, θ)-S(τ, θ),
ρc(θ; θin)=Ic(0, θr)Iin |cos θin| δ(sin θ-sin θr),
τc(θ; θin)=Ic(κextd, θt)Iin|cos θin| δ(sin θ-sin θt).
S(τ, θs)=ϖ2π02πp(θs; θs)Id(τ, θs)dθs,+ϖ2π p(θs; θiII)Ic(τ, θiII),+ϖ2π p(θs; π-θiII)Ic(τ, π-θiII).
ρd(θsI; θin)=Id(0, θsI)Iin|cos θin|,
τd(θsIII; θin)=Id(κextd, θsIII)Iin|cos θin|.
ρ=j=1Nρd(θjI; θin)|cos θjI|wj+ρc(θr; θin)|cos θr|,
τ=j=1Nτd(θjIII; θin)|cos θjIII|wj+τc(θt; θin)|cos θt|.
EI(r)=Ein(r)+p=-+EpIexp(-jkpIr),
HI(r)=Hin(r)+p=-+HpIexp(-jkpIr).
EIII(r)=p=-+EpIIIexp(-jkpIIIr),
HIII(r)=p=-+HpIIIexp(-jkpIIIr).
kpz=0,
kpx=kinx+p 2πL,
kpy,I=(kI2-kpx2)1/2,
kpy,III=-(kIII2-kpx2)1/2,
PI=p{propag.}12ηI |Epz,I|2cos θpI.
p0,EpI=0.
PcohI=12ηI |E0z,I|2.
ρcoh(θs, θin)=PcohIPin δ(θs-θr).
P˜pI=12ηI (|Epz,I|2-|Epz,I|2).
P˜I=p{propag.}P˜pIcos θpI.
P˜IPin=-π/2+π/2ρ(θout, θin)cos θoutdθout,
Iin(θ)cos θ=Pinδ(θ-θin).
kp+1x-kpx=2πL=kIcos θpIΔθpI,
P˜IPinp{propag}p(θpI, θin)cos θpIΔθpI.
ρ(θpI, θin)=P˜pIPinkIL2πcos θpI,
ρ(ss, si)=ρ(-si, -ss)=ρ(si, ss).
Imultiple(θt)=IRTE(θt)-Isingle(θt).
Ibackward=Isingle(θin+π)+2Imultiple(θin+π)
=2IRTE(θin+π)-Isingle(θin+π).
Γ=4πmλ.

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