Abstract

We consider the coherent reflection and transmission of electromagnetic waves from a slab of a dilute system of randomly located, polarizable, spherical particles. We focus our attention on the case where the size of the spheres is comparable to the wavelength of the incident radiation. First, using wave-scattering and Mie theories, we derive expressions for the coherent fields that are transmitted and reflected by a very thin slab. Then we find the effective-current distribution that would act as a source of these fields. We conclude that if the effective currents were induced in an effective medium, this medium must possess, besides an effective electric permittivity, also an effective magnetic permeability. We find that both of these optical coefficients become functions of the angle of incidence and the polarization of the incident wave. Then we calculate the reflection coefficient of a half-space by considering a semi-infinite pile of thin slabs and compare the result with Fresnel relations. Numerical results are presented for the optical coefficients as well as for the half-space reflectance as a function of several parameters. The reflectance is compared with that obtained without considering the magnetic response. Finally, we discuss the relevance and the physics behind our results and indicate as well the measurements that could be performed to obtain an experimental verification of our theory.

© 2003 Optical Society of America

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  1. Lord Rayleigh, “On the transmission of light through an atmosphere containing small particles in suspension, and on the origin of the blue of the sky,” Philos. Mag. 47, 375–384 (1899).
  2. R. Ruppin, “Evaluation of extended Maxwell Garnett theories,” Opt. Commun. 182, 273–279 (2000).
    [CrossRef]
  3. J. C. Maxwell Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London 203, 385–420 (1904).
    [CrossRef]
  4. See, for example, the historical review of R. Landauer, “Electrical conductivity in inhomogeneous media,” in Proceedings of the First Conference on Electrical Transport and Optical Properties of Inhomogeneous Media, J. C. Garland, D. B. Tanner, eds., American Institute of Physics Conf. Proc. No. 40 (American Institute of Physics, New York, 1978), pp. 2–45.
  5. J. E. Gubernatis, “Scattering theory and effective medium approximation to heterogeneous materials,” in Proceedings of the First Conference on Electrical Transport and Optical Properties of Inhomogeneous Media, J. C. Garland, D. B. Tanner, eds., American Institute of Physics Conf. Proc. No. 40 (American Institute of Physics, New York, 1978), pp. 84–97.
  6. R. Fuchs, “Optical properties of small particle composites,” in Proceedings of the First Conference on Electrical Transport and Optical Properties of Inhomogeneous Media, J. C. Garland, D. B. Tanner, eds., American Institute of Physics Conf. Proc. No. 40 (American Institute of Physics, New York, 1978), pp. 276–281.
  7. For more recent references, see, for example, Proceedings of the Fifth International Conference on Electrical Transport and Optical Properties of Inhomogeneous Media (ETOPIM 5), P. M. Hui, P. Sheng, L.-H. Tang, eds., Physica B279, Nos. 1–3 (2000).
  8. D. A. G. Bruggeman, “Berechnung verschiedener physikalicher Konstanten von heterogenen Substanzen. I. Dielektrizitätkonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. (Leipzig) 24, 636–679 (1935).
    [CrossRef]
  9. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  10. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  11. D. Stroud, F. P. Pan, “Self-consistent approach to electromagnetic wave propagation in composite media: application to model granular metals,” Phys. Rev. B 17, 1602–1610 (1978).
    [CrossRef]
  12. P. Chýlek, V. Srivastava, “Dielectric constant of a composite inhomogeneous medium,” Phys. Rev. B 27, 5098–5106 (1983).
    [CrossRef]
  13. W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
    [CrossRef]
  14. A. Wachniewski, H. B. McClung, “New Approach to effective medium for composite materials: application to electromagnetic properties,” Phys. Rev. B 33, 8053–8059 (1986).
    [CrossRef]
  15. C. A. Grimes, D. M. Grimes, “Permeability and permittivity spectra of granular materials,” Phys. Rev. B 43, 10780–10788 (1991).
    [CrossRef]
  16. C. F. Bohren, “Applicability of effective medium theories to problems of scattering and absorption by nonhomogeneous atmospheric particles,” J. Atmos. Sci. 43, 468–475 (1986).
    [CrossRef]
  17. P. Chýlek, G. Videen, D. J. W. Geldart, J. S. Dobbie, H. C. W. Tso, “Effective medium approximations for heterogeneous particles,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, New York, 2000), Chap. 9.
  18. See for example, L. Tsang, J. A. Kong, R. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).
  19. L. Tsang, J. A. Kong, “Multiple scattering of electromagnetic waves by random distribution of discrete scatterers with coherent potential and quantum mechanical formalism,” J. Appl. Phys. 51, 3465–3485 (1980).
    [CrossRef]
  20. L. Tsang, J. A. Kong, “Effective propagation constants for coherent electromagnetic waves propagating in media embedded with dielectric scatterers,” J. Appl. Phys. 53, 7162–7173 (1982).
    [CrossRef]
  21. Y. Kuga, D. Rice, R. D. West, “Propagation constant and the velocity of the coherent wave in a dense strongly scattering medium,” IEEE Trans. Antennas Propag. 44, 326–332 (1996).
    [CrossRef]
  22. V. A. Loiko, V. P. Dick, A. P. Ivanov, “Features in coherent transmittance of a monolayer of particles,” J. Opt. Soc. Am. A 17, 2040–2045 (2000).
    [CrossRef]
  23. L. L. Foldy, “The multiple scattering of waves,” Phys. Rev. 67, 107–119 (1945).
    [CrossRef]
  24. M. Lax, “Multiple scattering of waves II. The effective field in dense systems,” Phys. Rev. 85, 621–629 (1952).
    [CrossRef]
  25. M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. 23, 287–310 (1951).
    [CrossRef]
  26. L. Tsang, J. A. Kong, Scattering of Electromagnetic Waves; Advanced Topics (Wiley, New York, 2001), Chap. 3, pp. 128–130.
  27. A. Garcı́a-Valenzuela, R. G. Barrera, “Electromagnetic response of a random half-space of Mie scatterers within the effective medium approximation and the determination of the effective optical coefficients,” J. Quant. Spectrosc. Radiat. Transfer (to be published).
  28. C. Yang, A. Wax, M. S. Feld, “Measurement of the anomalous phase velocity of ballistic light in a random me-dium by use of a novel interferometer,” Opt. Lett. 26, 235–237 (2001).
    [CrossRef]
  29. G. H. Meeten, A. N. North, “Refractive index measurement of absorbing and turbid fluids by reflection near the critical angle,” Meas. Sci. Technol. 6, 214–221 (1995).
    [CrossRef]
  30. M. Mohammadi, “Colloidal refractometry: meaning and measurement of refractive index for dispersions; the science that time forgot,” Adv. Colloid Interface Sci. 62, 17–29 (1995).
    [CrossRef]
  31. A. Garcı́a-Valenzuela, M. Peña-Gomar, C. Fajardo-Lira, “Measuring and sensing a complex index of refraction by laser reflection near the critical angle,” Opt. Eng. 41, 1704–1716 (2002).
    [CrossRef]
  32. W. E. Vargas, G. A. Niklasson, “Applicability conditions of the Kubelka–Munk theory,” Appl. Opt. 36, 5580–5586 (1997).
    [CrossRef] [PubMed]
  33. J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors, and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
    [CrossRef]

2002 (1)

A. Garcı́a-Valenzuela, M. Peña-Gomar, C. Fajardo-Lira, “Measuring and sensing a complex index of refraction by laser reflection near the critical angle,” Opt. Eng. 41, 1704–1716 (2002).
[CrossRef]

2001 (1)

2000 (2)

1999 (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors, and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

1997 (1)

1996 (1)

Y. Kuga, D. Rice, R. D. West, “Propagation constant and the velocity of the coherent wave in a dense strongly scattering medium,” IEEE Trans. Antennas Propag. 44, 326–332 (1996).
[CrossRef]

1995 (2)

G. H. Meeten, A. N. North, “Refractive index measurement of absorbing and turbid fluids by reflection near the critical angle,” Meas. Sci. Technol. 6, 214–221 (1995).
[CrossRef]

M. Mohammadi, “Colloidal refractometry: meaning and measurement of refractive index for dispersions; the science that time forgot,” Adv. Colloid Interface Sci. 62, 17–29 (1995).
[CrossRef]

1991 (1)

C. A. Grimes, D. M. Grimes, “Permeability and permittivity spectra of granular materials,” Phys. Rev. B 43, 10780–10788 (1991).
[CrossRef]

1989 (1)

W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
[CrossRef]

1986 (2)

A. Wachniewski, H. B. McClung, “New Approach to effective medium for composite materials: application to electromagnetic properties,” Phys. Rev. B 33, 8053–8059 (1986).
[CrossRef]

C. F. Bohren, “Applicability of effective medium theories to problems of scattering and absorption by nonhomogeneous atmospheric particles,” J. Atmos. Sci. 43, 468–475 (1986).
[CrossRef]

1983 (1)

P. Chýlek, V. Srivastava, “Dielectric constant of a composite inhomogeneous medium,” Phys. Rev. B 27, 5098–5106 (1983).
[CrossRef]

1982 (1)

L. Tsang, J. A. Kong, “Effective propagation constants for coherent electromagnetic waves propagating in media embedded with dielectric scatterers,” J. Appl. Phys. 53, 7162–7173 (1982).
[CrossRef]

1980 (1)

L. Tsang, J. A. Kong, “Multiple scattering of electromagnetic waves by random distribution of discrete scatterers with coherent potential and quantum mechanical formalism,” J. Appl. Phys. 51, 3465–3485 (1980).
[CrossRef]

1978 (1)

D. Stroud, F. P. Pan, “Self-consistent approach to electromagnetic wave propagation in composite media: application to model granular metals,” Phys. Rev. B 17, 1602–1610 (1978).
[CrossRef]

1952 (1)

M. Lax, “Multiple scattering of waves II. The effective field in dense systems,” Phys. Rev. 85, 621–629 (1952).
[CrossRef]

1951 (1)

M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. 23, 287–310 (1951).
[CrossRef]

1945 (1)

L. L. Foldy, “The multiple scattering of waves,” Phys. Rev. 67, 107–119 (1945).
[CrossRef]

1935 (1)

D. A. G. Bruggeman, “Berechnung verschiedener physikalicher Konstanten von heterogenen Substanzen. I. Dielektrizitätkonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. (Leipzig) 24, 636–679 (1935).
[CrossRef]

1904 (1)

J. C. Maxwell Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London 203, 385–420 (1904).
[CrossRef]

1899 (1)

Lord Rayleigh, “On the transmission of light through an atmosphere containing small particles in suspension, and on the origin of the blue of the sky,” Philos. Mag. 47, 375–384 (1899).

Barrera, R. G.

A. Garcı́a-Valenzuela, R. G. Barrera, “Electromagnetic response of a random half-space of Mie scatterers within the effective medium approximation and the determination of the effective optical coefficients,” J. Quant. Spectrosc. Radiat. Transfer (to be published).

Bohren, C. F.

C. F. Bohren, “Applicability of effective medium theories to problems of scattering and absorption by nonhomogeneous atmospheric particles,” J. Atmos. Sci. 43, 468–475 (1986).
[CrossRef]

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

Bruggeman, D. A. G.

D. A. G. Bruggeman, “Berechnung verschiedener physikalicher Konstanten von heterogenen Substanzen. I. Dielektrizitätkonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. (Leipzig) 24, 636–679 (1935).
[CrossRef]

Chýlek, P.

P. Chýlek, V. Srivastava, “Dielectric constant of a composite inhomogeneous medium,” Phys. Rev. B 27, 5098–5106 (1983).
[CrossRef]

P. Chýlek, G. Videen, D. J. W. Geldart, J. S. Dobbie, H. C. W. Tso, “Effective medium approximations for heterogeneous particles,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, New York, 2000), Chap. 9.

Dick, V. P.

Dobbie, J. S.

P. Chýlek, G. Videen, D. J. W. Geldart, J. S. Dobbie, H. C. W. Tso, “Effective medium approximations for heterogeneous particles,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, New York, 2000), Chap. 9.

Doyle, W. T.

W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
[CrossRef]

Fajardo-Lira, C.

A. Garcı́a-Valenzuela, M. Peña-Gomar, C. Fajardo-Lira, “Measuring and sensing a complex index of refraction by laser reflection near the critical angle,” Opt. Eng. 41, 1704–1716 (2002).
[CrossRef]

Feld, M. S.

Foldy, L. L.

L. L. Foldy, “The multiple scattering of waves,” Phys. Rev. 67, 107–119 (1945).
[CrossRef]

Fuchs, R.

R. Fuchs, “Optical properties of small particle composites,” in Proceedings of the First Conference on Electrical Transport and Optical Properties of Inhomogeneous Media, J. C. Garland, D. B. Tanner, eds., American Institute of Physics Conf. Proc. No. 40 (American Institute of Physics, New York, 1978), pp. 276–281.

Garci´a-Valenzuela, A.

A. Garcı́a-Valenzuela, M. Peña-Gomar, C. Fajardo-Lira, “Measuring and sensing a complex index of refraction by laser reflection near the critical angle,” Opt. Eng. 41, 1704–1716 (2002).
[CrossRef]

A. Garcı́a-Valenzuela, R. G. Barrera, “Electromagnetic response of a random half-space of Mie scatterers within the effective medium approximation and the determination of the effective optical coefficients,” J. Quant. Spectrosc. Radiat. Transfer (to be published).

Garnett, J. C. Maxwell

J. C. Maxwell Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London 203, 385–420 (1904).
[CrossRef]

Geldart, D. J. W.

P. Chýlek, G. Videen, D. J. W. Geldart, J. S. Dobbie, H. C. W. Tso, “Effective medium approximations for heterogeneous particles,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, New York, 2000), Chap. 9.

Grimes, C. A.

C. A. Grimes, D. M. Grimes, “Permeability and permittivity spectra of granular materials,” Phys. Rev. B 43, 10780–10788 (1991).
[CrossRef]

Grimes, D. M.

C. A. Grimes, D. M. Grimes, “Permeability and permittivity spectra of granular materials,” Phys. Rev. B 43, 10780–10788 (1991).
[CrossRef]

Gubernatis, J. E.

J. E. Gubernatis, “Scattering theory and effective medium approximation to heterogeneous materials,” in Proceedings of the First Conference on Electrical Transport and Optical Properties of Inhomogeneous Media, J. C. Garland, D. B. Tanner, eds., American Institute of Physics Conf. Proc. No. 40 (American Institute of Physics, New York, 1978), pp. 84–97.

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors, and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

Huffman, D. R.

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

Ivanov, A. P.

Kong, J. A.

L. Tsang, J. A. Kong, “Effective propagation constants for coherent electromagnetic waves propagating in media embedded with dielectric scatterers,” J. Appl. Phys. 53, 7162–7173 (1982).
[CrossRef]

L. Tsang, J. A. Kong, “Multiple scattering of electromagnetic waves by random distribution of discrete scatterers with coherent potential and quantum mechanical formalism,” J. Appl. Phys. 51, 3465–3485 (1980).
[CrossRef]

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

L. Tsang, J. A. Kong, Scattering of Electromagnetic Waves; Advanced Topics (Wiley, New York, 2001), Chap. 3, pp. 128–130.

Kuga, Y.

Y. Kuga, D. Rice, R. D. West, “Propagation constant and the velocity of the coherent wave in a dense strongly scattering medium,” IEEE Trans. Antennas Propag. 44, 326–332 (1996).
[CrossRef]

Landauer, R.

See, for example, the historical review of R. Landauer, “Electrical conductivity in inhomogeneous media,” in Proceedings of the First Conference on Electrical Transport and Optical Properties of Inhomogeneous Media, J. C. Garland, D. B. Tanner, eds., American Institute of Physics Conf. Proc. No. 40 (American Institute of Physics, New York, 1978), pp. 2–45.

Lax, M.

M. Lax, “Multiple scattering of waves II. The effective field in dense systems,” Phys. Rev. 85, 621–629 (1952).
[CrossRef]

M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. 23, 287–310 (1951).
[CrossRef]

Loiko, V. A.

McClung, H. B.

A. Wachniewski, H. B. McClung, “New Approach to effective medium for composite materials: application to electromagnetic properties,” Phys. Rev. B 33, 8053–8059 (1986).
[CrossRef]

Meeten, G. H.

G. H. Meeten, A. N. North, “Refractive index measurement of absorbing and turbid fluids by reflection near the critical angle,” Meas. Sci. Technol. 6, 214–221 (1995).
[CrossRef]

Mohammadi, M.

M. Mohammadi, “Colloidal refractometry: meaning and measurement of refractive index for dispersions; the science that time forgot,” Adv. Colloid Interface Sci. 62, 17–29 (1995).
[CrossRef]

Niklasson, G. A.

North, A. N.

G. H. Meeten, A. N. North, “Refractive index measurement of absorbing and turbid fluids by reflection near the critical angle,” Meas. Sci. Technol. 6, 214–221 (1995).
[CrossRef]

Pan, F. P.

D. Stroud, F. P. Pan, “Self-consistent approach to electromagnetic wave propagation in composite media: application to model granular metals,” Phys. Rev. B 17, 1602–1610 (1978).
[CrossRef]

Peña-Gomar, M.

A. Garcı́a-Valenzuela, M. Peña-Gomar, C. Fajardo-Lira, “Measuring and sensing a complex index of refraction by laser reflection near the critical angle,” Opt. Eng. 41, 1704–1716 (2002).
[CrossRef]

Pendry, J. B.

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors, and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

Rayleigh, Lord

Lord Rayleigh, “On the transmission of light through an atmosphere containing small particles in suspension, and on the origin of the blue of the sky,” Philos. Mag. 47, 375–384 (1899).

Rice, D.

Y. Kuga, D. Rice, R. D. West, “Propagation constant and the velocity of the coherent wave in a dense strongly scattering medium,” IEEE Trans. Antennas Propag. 44, 326–332 (1996).
[CrossRef]

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors, and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

Ruppin, R.

R. Ruppin, “Evaluation of extended Maxwell Garnett theories,” Opt. Commun. 182, 273–279 (2000).
[CrossRef]

Shin, R.

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

Srivastava, V.

P. Chýlek, V. Srivastava, “Dielectric constant of a composite inhomogeneous medium,” Phys. Rev. B 27, 5098–5106 (1983).
[CrossRef]

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors, and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

Stroud, D.

D. Stroud, F. P. Pan, “Self-consistent approach to electromagnetic wave propagation in composite media: application to model granular metals,” Phys. Rev. B 17, 1602–1610 (1978).
[CrossRef]

Tsang, L.

L. Tsang, J. A. Kong, “Effective propagation constants for coherent electromagnetic waves propagating in media embedded with dielectric scatterers,” J. Appl. Phys. 53, 7162–7173 (1982).
[CrossRef]

L. Tsang, J. A. Kong, “Multiple scattering of electromagnetic waves by random distribution of discrete scatterers with coherent potential and quantum mechanical formalism,” J. Appl. Phys. 51, 3465–3485 (1980).
[CrossRef]

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

L. Tsang, J. A. Kong, Scattering of Electromagnetic Waves; Advanced Topics (Wiley, New York, 2001), Chap. 3, pp. 128–130.

Tso, H. C. W.

P. Chýlek, G. Videen, D. J. W. Geldart, J. S. Dobbie, H. C. W. Tso, “Effective medium approximations for heterogeneous particles,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, New York, 2000), Chap. 9.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Vargas, W. E.

Videen, G.

P. Chýlek, G. Videen, D. J. W. Geldart, J. S. Dobbie, H. C. W. Tso, “Effective medium approximations for heterogeneous particles,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, New York, 2000), Chap. 9.

Wachniewski, A.

A. Wachniewski, H. B. McClung, “New Approach to effective medium for composite materials: application to electromagnetic properties,” Phys. Rev. B 33, 8053–8059 (1986).
[CrossRef]

Wax, A.

West, R. D.

Y. Kuga, D. Rice, R. D. West, “Propagation constant and the velocity of the coherent wave in a dense strongly scattering medium,” IEEE Trans. Antennas Propag. 44, 326–332 (1996).
[CrossRef]

Yang, C.

Adv. Colloid Interface Sci. (1)

M. Mohammadi, “Colloidal refractometry: meaning and measurement of refractive index for dispersions; the science that time forgot,” Adv. Colloid Interface Sci. 62, 17–29 (1995).
[CrossRef]

Ann. Phys. (Leipzig) (1)

D. A. G. Bruggeman, “Berechnung verschiedener physikalicher Konstanten von heterogenen Substanzen. I. Dielektrizitätkonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. (Leipzig) 24, 636–679 (1935).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

Y. Kuga, D. Rice, R. D. West, “Propagation constant and the velocity of the coherent wave in a dense strongly scattering medium,” IEEE Trans. Antennas Propag. 44, 326–332 (1996).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors, and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

J. Appl. Phys. (2)

L. Tsang, J. A. Kong, “Multiple scattering of electromagnetic waves by random distribution of discrete scatterers with coherent potential and quantum mechanical formalism,” J. Appl. Phys. 51, 3465–3485 (1980).
[CrossRef]

L. Tsang, J. A. Kong, “Effective propagation constants for coherent electromagnetic waves propagating in media embedded with dielectric scatterers,” J. Appl. Phys. 53, 7162–7173 (1982).
[CrossRef]

J. Atmos. Sci. (1)

C. F. Bohren, “Applicability of effective medium theories to problems of scattering and absorption by nonhomogeneous atmospheric particles,” J. Atmos. Sci. 43, 468–475 (1986).
[CrossRef]

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

Meas. Sci. Technol. (1)

G. H. Meeten, A. N. North, “Refractive index measurement of absorbing and turbid fluids by reflection near the critical angle,” Meas. Sci. Technol. 6, 214–221 (1995).
[CrossRef]

Opt. Commun. (1)

R. Ruppin, “Evaluation of extended Maxwell Garnett theories,” Opt. Commun. 182, 273–279 (2000).
[CrossRef]

Opt. Eng. (1)

A. Garcı́a-Valenzuela, M. Peña-Gomar, C. Fajardo-Lira, “Measuring and sensing a complex index of refraction by laser reflection near the critical angle,” Opt. Eng. 41, 1704–1716 (2002).
[CrossRef]

Opt. Lett. (1)

Philos. Mag. (1)

Lord Rayleigh, “On the transmission of light through an atmosphere containing small particles in suspension, and on the origin of the blue of the sky,” Philos. Mag. 47, 375–384 (1899).

Philos. Trans. R. Soc. London (1)

J. C. Maxwell Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London 203, 385–420 (1904).
[CrossRef]

Phys. Rev. (2)

L. L. Foldy, “The multiple scattering of waves,” Phys. Rev. 67, 107–119 (1945).
[CrossRef]

M. Lax, “Multiple scattering of waves II. The effective field in dense systems,” Phys. Rev. 85, 621–629 (1952).
[CrossRef]

Phys. Rev. B (5)

D. Stroud, F. P. Pan, “Self-consistent approach to electromagnetic wave propagation in composite media: application to model granular metals,” Phys. Rev. B 17, 1602–1610 (1978).
[CrossRef]

P. Chýlek, V. Srivastava, “Dielectric constant of a composite inhomogeneous medium,” Phys. Rev. B 27, 5098–5106 (1983).
[CrossRef]

W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
[CrossRef]

A. Wachniewski, H. B. McClung, “New Approach to effective medium for composite materials: application to electromagnetic properties,” Phys. Rev. B 33, 8053–8059 (1986).
[CrossRef]

C. A. Grimes, D. M. Grimes, “Permeability and permittivity spectra of granular materials,” Phys. Rev. B 43, 10780–10788 (1991).
[CrossRef]

Rev. Mod. Phys. (1)

M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. 23, 287–310 (1951).
[CrossRef]

Other (10)

L. Tsang, J. A. Kong, Scattering of Electromagnetic Waves; Advanced Topics (Wiley, New York, 2001), Chap. 3, pp. 128–130.

A. Garcı́a-Valenzuela, R. G. Barrera, “Electromagnetic response of a random half-space of Mie scatterers within the effective medium approximation and the determination of the effective optical coefficients,” J. Quant. Spectrosc. Radiat. Transfer (to be published).

P. Chýlek, G. Videen, D. J. W. Geldart, J. S. Dobbie, H. C. W. Tso, “Effective medium approximations for heterogeneous particles,” in Light Scattering by Nonspherical Particles, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, New York, 2000), Chap. 9.

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

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

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

See, for example, the historical review of R. Landauer, “Electrical conductivity in inhomogeneous media,” in Proceedings of the First Conference on Electrical Transport and Optical Properties of Inhomogeneous Media, J. C. Garland, D. B. Tanner, eds., American Institute of Physics Conf. Proc. No. 40 (American Institute of Physics, New York, 1978), pp. 2–45.

J. E. Gubernatis, “Scattering theory and effective medium approximation to heterogeneous materials,” in Proceedings of the First Conference on Electrical Transport and Optical Properties of Inhomogeneous Media, J. C. Garland, D. B. Tanner, eds., American Institute of Physics Conf. Proc. No. 40 (American Institute of Physics, New York, 1978), pp. 84–97.

R. Fuchs, “Optical properties of small particle composites,” in Proceedings of the First Conference on Electrical Transport and Optical Properties of Inhomogeneous Media, J. C. Garland, D. B. Tanner, eds., American Institute of Physics Conf. Proc. No. 40 (American Institute of Physics, New York, 1978), pp. 276–281.

For more recent references, see, for example, Proceedings of the Fifth International Conference on Electrical Transport and Optical Properties of Inhomogeneous Media (ETOPIM 5), P. M. Hui, P. Sheng, L.-H. Tang, eds., Physica B279, Nos. 1–3 (2000).

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Figures (7)

Fig. 1
Fig. 1

Slab of a dilute random system of spheres. The centers of the particles are within the planes z=-d/2 and z=d/2.

Fig. 2
Fig. 2

Plots of the normalized change in the real and imaginary part of the effective index of refraction [Eq. (47)] as a function of a/λ for a system of nonmagnetic glass spheres (np=1.50) in vacuum (n=1.00).

Fig. 3
Fig. 3

Plots of the normalized change in the real (a), (c), and imaginary (b), (d) part of the optical coefficients [Eqs. (39) – (44)] as a function of the particle radius a divided by the wavelength λ, for two different angles of incidence, 30° (a), (b) and 70° (c), (d). The subindex eff in the optical coefficients was removed here for clarity. The plots are for a system of nonmagnetic glass spheres (np=1.50) in vacuum (n=1.00); dotted curves are for TE, dashed-dotted curves for TM, solid curves for μTE, and dashed curves for μTM.

Fig. 4
Fig. 4

Plots of the normalized change in the real (a) and imaginary (b) part of the optical coefficients [Eqs. (39) –(44)] as a function of the angle of incidence for particles of radius a=0.5λ. The plots are for a system of nonmagnetic glass spheres (np=1.50) in vacuum (n=1.00). The subindex eff in the optical coefficients was removed here for clarity. Dotted curves are for TE, dashed-dotted curves for TM, solid curves for μTE, and dashed curves for μTM.

Fig. 5
Fig. 5

Model of a half-space as a semi-infinite stack of 2D sheets. The sheets are located at z=zn=nd with n=0,1,2,3,  . The fields are calculated at the intermediate planes z=ζn=(n+1/2)d.

Fig. 6
Fig. 6

Plot of the coherent reflectance R of unpolarized light [average of Eqs. (55), (56)] for a system of nonmagnetic glass spheres (np=1.50) in vacuum (n=1.00) with a filling fraction of f=0.1; (a) as a function of the angle of incidence and for several values of the radius a of the particles, (b) as a function of the particle radius a divided by the wavelength λ for an angle of incidence θi=85°. For comparison we also plot the reflectance ignoring the effective magnetic susceptibility [Rnm from Eq. (57)].

Fig. 7
Fig. 7

Interesting features of the coherent reflectance R as a function of the angle of incidence for a system of nonmagnetic glass spheres (np=1.50) in vacuum (n=1.00) with a filling fraction of f=0.1. (a) Brewster’s angle for TM polarization [Eq. (56)] for particles of radius a=0.2λ, (b) Brewster’s angle for TE polarization for particles of radius a=1.5λ. For comparison we also plot the reflectance ignoring the effective magnetic susceptibility [Rn-m from Eq. (57)].

Equations (76)

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Ei(r, t)=E0 exp[i(ki·r-ωt)]eˆi,
ES(r)=p=1Nd3rd3rG¯¯0(r, r)·T¯¯(r-rp, r-rp)·Ei(r),
G¯¯0(r, r)=i8π2dkxsdkys1kzs(1¯¯-kˆ±skˆ±s)exp[ik±s·(r-r)]
T¯¯(r-rp, r-rp)=1(2π)6d3pd3p exp[ip·(r-rp)]T¯¯(p, p)×exp[-ip·(r-rp)],
ES(r)=i8π2E0p=1Ndkxsdkys(1¯¯-kˆ±skˆ±s)kzs·T¯¯(k±s, ki)·eˆi exp[-i(k±s-ki)·rp]×exp(ik±s·r).
ES(r)slab=E+S exp(iki·r)forz>d/2E-S exp(ikr·r)forz<-d/2 ,
E+S=iE02ρ(1¯¯-kˆikˆi)kzi·T¯¯(ki, ki)·eˆid,
E-S=iE02ρ(1¯¯-kˆrkˆr)kzi·T¯¯(kr, ki)·eˆisin kzidkzi.
EfarS(r)=E0exp(ikr)rF¯¯(kˆs, kˆi)·eˆi,
(1¯¯-kˆikˆi)·T¯¯(ki,ki)=4πF¯¯(kˆi,kˆi),
(1¯¯-kˆrkˆr)·T¯¯(kr,ki)=4π(kˆr, kˆi).
EfarSEfarS=exp(ikr)-ikrS2(θ)S4(θ)S3(θ)S1(θ)EiEi,
E+S=-E0γkdcos θiS(0)eˆi,
E-S=-E0γkcos θisin kzidkzi×[-(cos θiaˆy+sin θiaˆz)(cos θiaˆy-sin θiaˆz)×S2(π-2θi)+aˆxaˆxS1(π-2θi)]·eˆi,
E+S=-E0γkdcos θiS(0)aˆx,
E-S=-E0γkcos θisin kzidkziS1(π-2θi)aˆx.
Ei(r, t)=E0 exp[i(kyiy+kziz)]aˆx,
J=j0xδ(z)exp(ikyiy)aˆx,
EJ=E+J exp(iki·r)forz>0E-J exp(ikr·r)forz<0,
E±xJ=-12μ0j0xωkzi,
JC=limε0 jC[δ(z+ε/2)-δ(z-ε/2)]exp(ikyy)aˆx=JCδ(z)exp(ikyiy)aˆx,
J=×M,
M=m0yδ(z)exp(ikyiy)eˆy,
E±xJ=±i2ωμ0m0y,
E±xJ=12μ0ω-j0xkzi±im0y,
M=-m0zδ(z)exp(ikyiy)eˆz.
E±xJ=i2μ0ωm0zkyikzi.
E±xJ=12μ0ω-j0xkzi±im0y+im0zkyikzi.
J=Pt-iωP,
j0x=-iω0χSEE0,
m0y=χSHH0 cos θi=χSHkωμ0E0 cos θi,
m0z=χSBB0 sin θi=χSBkωE0 sin θi,
E±xJ=i2kχSEcos θi±χSH cos θi+μ0χSBsin2 θicos θiE0.
χSEχEd,
χSHχHd,
χSBχHdμχHdμ0.
χE+χH cos2 θi+χH sin2 θi=2iγS(0),
χE-χH cos2 θi+χH sin2 θi=2iγS1(π-2θi)sin kzidkzid,
μ˜effTE(θi)=1+iγS-(1)(θi)cos2 θi,
˜effTE(θi)=1+iγ[2S+(1)(θi)-S-(1)(θi)tan2 (θi)],
S+(m)(θi)12[S(0)+Sm(π-2θi)],
S-(m)(θi)S(0)-Sm(π-2θi),
˜effTM(θi)=1+iγS-(2)(θi)cos2 θi,
μ˜effTM(θi)=1+iγ[2S+(2)(θi)-S-(2)(θi)tan2(θi)].
neff(m)(θi)=[˜eff(m)(θi)μ˜eff(m)(θi)]1/2
=1+2iγS(0)-γ2cos2 θi[S(0)2-Sm(π-2θi)2]1/2,
neff1+iγS(0),
S1(θi)-ix3β,
S2(θi)-ix3β cos θi,
μ˜effTE(θi)=μ˜effTM(θi)μ˜eff=1,
˜effTE(θi)=˜effTM(θi)˜eff=1+3βf.
μ˜effTE(0)=μ˜effTM(0)μ˜eff(0)=1+iγ[S(0)-S1(π)],
˜effTE(0)=˜effTM(0)˜eff(0)=1+iγ[S(0)+S1(π)],
rhsTE=γS1(π-2θi)/cos θii{cos θi+[cos2 θi+2iγS(0)]1/2}-γS(0)/cos θi.
rhsTE=μ˜effTE(θi)kzi-kzeffμ˜effTE(θi)kzi+kzeff,
rhsTE=˜effTM(θi)kzi-kzeff˜effTM(θi)kzi+kzeff,
rnmTE=kzi-kzeffkzi+kzeff,rnmTM=neff2kzi-kzeffneff2kzi+kzeff,
rh=rhs[1-exp(2ikzeffh)]1-rhs2 exp(2ikzeffh),
th=1-rhs21-rhs2 exp(2ikzeffh)×exp[-i(kzi-kzeff)h].
rcm=rm+rhs exp(2iknm cos θmg)1+rmrhs exp(2iknm cos θmg),
rhs=γSm(π-2θi)/cos θii(cos θi+{cos2 θi+2iγS(0)-(γ2/cos2 θi)[S(0)2-Sm(π-2θi)2]}1/2)-γS(0)/cos θi,
α=-γkcos θiS(0),
β=-γkcos θiS1(π-2θi).
En=(En+eˆi+En-eˆr)exp(ikxix+ikyiy),
En+=Eni+m=0n(βEm-+αEm-1+)exp[ikzi(ζn-zm)]d,
En-=m=n(βEm++αEm+1-)exp[-ikzi(ζn-zm+1)]d,
E+(z)=Ei(z)+0z[βE-(z)+αE+(z)]×exp[ikzi(z-z)]dz,
E-(z)=z[βE+(z)+αE-(z)]×exp[-ikzi(z-z)]dz.
E+(z)=E0 exp(ikzeffz),
E-(z)=rhsE0 exp(ikzeffz),
rhs=i(kzeff-kzi)-αβ,
rhs=-βi(kzeff+kzi)+α.
kzeff=[(kzi)2-2ikziα+β2-α2]1/2,
kzeff=k[cos2 θi+2iγS(0)]1/2
rhsTE=γS1(π-2θi)/cos θii(cos θi+[cos2 θi+2iγS(0)]1/2)-γS(0)/cos θi,
neff=[1+2iγS(0)]1/21+iγS(0),

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