Abstract

We study the effective index of random media composed of two-layered spheres by using the energy-density coherent potential approximation method. As expected from the Ewald–Oseen extinction theorem, in the long-wavelength limit, the optical properties of a random medium composed of two-layered spheres are identical to those of a random medium composed of the corresponding pure spheres, while in the Mie-scattering region, the single-scattering resonances lead to an overall shift of the effective refractive index with the modified volume fraction.

© 2011 Optical Society of America

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  1. C. M. Aegerter and G. Maret, “Coherent backscattering and Anderson localization of light,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2009), Vol.  52, pp. 1–62.
    [CrossRef]
  2. B. A. van Tiggelen, A. Lagendijk, and D. S. Wiersma, “Reflection and transmission of waves near the localization threshold,” Phys. Rev. Lett. 84, 4333–4336 (2000).
    [CrossRef] [PubMed]
  3. A. L. Diederik, S. Wiersma, Paolo Bartolini, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
  4. A. Ioffe and A. Regel, “Non-crystalline, amorphous and liquid electronic semiconductors,” Prog. Semicond. 4, 237–291 (1960).
  5. M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near Anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
    [CrossRef] [PubMed]
  6. P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
    [CrossRef]
  7. M. Störzer, C. M. Aegerter, and G. Maret, “Reduced transport velocity of multiply scattered light due to resonant scattering,” Phys. Rev. E 73, 065602 (2006).
    [CrossRef]
  8. G. Labeyrie, E. Vaujour, C. A. Müller, D. Delande, C. Miniatura, D. Wilkowski, and R. Kaiser, “Slow diffusion of light in a cold atomic cloud,” Phys. Rev. Lett. 91, 223904 (2003).
    [CrossRef] [PubMed]
  9. R. Sapienza, P. D. García, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, “Observation of resonant behavior in the energy velocity of diffused light,” Phys. Rev. Lett. 99, 233902 (2007).
    [CrossRef]
  10. R. Tweer, “Vielfachstreuung von Licht in Systemen dicht gepackter Mie–Streuer: Auf dem Weg zur Anderson–Lokalisierung?” Ph.D. thesis (University of Konstanz, 2002).
  11. P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena, 2nd ed. (Springer, 2006).
  12. E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University Press, 2007).
    [CrossRef]
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    [CrossRef] [PubMed]
  14. T. C. Choy, Effective Medium Theory (Oxford University Press, 1999).
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    [CrossRef]
  16. C. M. Soukoulis, S. Datta, and E. N. Economou, “Propagation of classical waves in random media,” Phys. Rev. B 49, 3800–3810(1994).
    [CrossRef]
  17. K. Busch and C. M. Soukoulis, “Transport properties of random media: an energy-density CPA approach,” Phys. Rev. B 54, 893–899 (1996).
    [CrossRef]
  18. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).
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    [CrossRef]
  22. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Cambridge University Press, 1997).
    [PubMed]
  23. 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]
  24. R. G. Barrera, A. Reyes-Coronado, and A. García-Valenzuela, “Nonlocal nature of the electrodynamic response of colloidal systems,” Phys. Rev. B 75, 184202 (2007).
    [CrossRef]

2011 (1)

2009 (1)

C. M. Aegerter and G. Maret, “Coherent backscattering and Anderson localization of light,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2009), Vol.  52, pp. 1–62.
[CrossRef]

2007 (3)

R. Sapienza, P. D. García, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, “Observation of resonant behavior in the energy velocity of diffused light,” Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University Press, 2007).
[CrossRef]

R. G. Barrera, A. Reyes-Coronado, and A. García-Valenzuela, “Nonlocal nature of the electrodynamic response of colloidal systems,” Phys. Rev. B 75, 184202 (2007).
[CrossRef]

2006 (3)

P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena, 2nd ed. (Springer, 2006).

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near Anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
[CrossRef] [PubMed]

M. Störzer, C. M. Aegerter, and G. Maret, “Reduced transport velocity of multiply scattered light due to resonant scattering,” Phys. Rev. E 73, 065602 (2006).
[CrossRef]

2003 (1)

G. Labeyrie, E. Vaujour, C. A. Müller, D. Delande, C. Miniatura, D. Wilkowski, and R. Kaiser, “Slow diffusion of light in a cold atomic cloud,” Phys. Rev. Lett. 91, 223904 (2003).
[CrossRef] [PubMed]

2002 (1)

R. Tweer, “Vielfachstreuung von Licht in Systemen dicht gepackter Mie–Streuer: Auf dem Weg zur Anderson–Lokalisierung?” Ph.D. thesis (University of Konstanz, 2002).

2000 (1)

B. A. van Tiggelen, A. Lagendijk, and D. S. Wiersma, “Reflection and transmission of waves near the localization threshold,” Phys. Rev. Lett. 84, 4333–4336 (2000).
[CrossRef] [PubMed]

1999 (1)

T. C. Choy, Effective Medium Theory (Oxford University Press, 1999).

1998 (1)

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley , 1998).
[CrossRef]

1997 (2)

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Cambridge University Press, 1997).
[PubMed]

A. L. Diederik, S. Wiersma, Paolo Bartolini, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).

1996 (1)

K. Busch and C. M. Soukoulis, “Transport properties of random media: an energy-density CPA approach,” Phys. Rev. B 54, 893–899 (1996).
[CrossRef]

1995 (1)

K. Busch and C. M. Soukoulis, “Transport properties of random media: a new effective medium theory,” Phys. Rev. Lett. 75, 3442–3445 (1995).
[CrossRef] [PubMed]

1994 (2)

K. Busch, C. M. Soukoulis, and E. N. Economou, “Transport and scattering mean free paths of classical waves,” Phys. Rev. B 50, 93–98 (1994).
[CrossRef]

C. M. Soukoulis, S. Datta, and E. N. Economou, “Propagation of classical waves in random media,” Phys. Rev. B 49, 3800–3810(1994).
[CrossRef]

1986 (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]

1982 (1)

D. E. Aspnes, “Local-field effects and effective-medium theory: a microscopic perspective,” Am. J. Phys. 50, 704–709 (1982).
[CrossRef]

1981 (1)

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

1960 (1)

A. Ioffe and A. Regel, “Non-crystalline, amorphous and liquid electronic semiconductors,” Prog. Semicond. 4, 237–291 (1960).

1958 (1)

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
[CrossRef]

Aegerter, C. M.

C. M. Aegerter and G. Maret, “Coherent backscattering and Anderson localization of light,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2009), Vol.  52, pp. 1–62.
[CrossRef]

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near Anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
[CrossRef] [PubMed]

M. Störzer, C. M. Aegerter, and G. Maret, “Reduced transport velocity of multiply scattered light due to resonant scattering,” Phys. Rev. E 73, 065602 (2006).
[CrossRef]

Akkermans, E.

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University Press, 2007).
[CrossRef]

Anderson, P. W.

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes, “Local-field effects and effective-medium theory: a microscopic perspective,” Am. J. Phys. 50, 704–709 (1982).
[CrossRef]

Barrera, R. G.

R. G. Barrera, A. Reyes-Coronado, and A. García-Valenzuela, “Nonlocal nature of the electrodynamic response of colloidal systems,” Phys. Rev. B 75, 184202 (2007).
[CrossRef]

Bartolini, Paolo

A. L. Diederik, S. Wiersma, Paolo Bartolini, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).

Bertolotti, J.

R. Sapienza, P. D. García, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, “Observation of resonant behavior in the energy velocity of diffused light,” Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

Blanco, A.

R. Sapienza, P. D. García, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, “Observation of resonant behavior in the energy velocity of diffused light,” Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley , 1998).
[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]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Cambridge University Press, 1997).
[PubMed]

Busch, K.

K. Busch and C. M. Soukoulis, “Transport properties of random media: an energy-density CPA approach,” Phys. Rev. B 54, 893–899 (1996).
[CrossRef]

K. Busch and C. M. Soukoulis, “Transport properties of random media: a new effective medium theory,” Phys. Rev. Lett. 75, 3442–3445 (1995).
[CrossRef] [PubMed]

K. Busch, C. M. Soukoulis, and E. N. Economou, “Transport and scattering mean free paths of classical waves,” Phys. Rev. B 50, 93–98 (1994).
[CrossRef]

Choy, T. C.

T. C. Choy, Effective Medium Theory (Oxford University Press, 1999).

Datta, S.

C. M. Soukoulis, S. Datta, and E. N. Economou, “Propagation of classical waves in random media,” Phys. Rev. B 49, 3800–3810(1994).
[CrossRef]

Delande, D.

G. Labeyrie, E. Vaujour, C. A. Müller, D. Delande, C. Miniatura, D. Wilkowski, and R. Kaiser, “Slow diffusion of light in a cold atomic cloud,” Phys. Rev. Lett. 91, 223904 (2003).
[CrossRef] [PubMed]

Diederik, A. L.

A. L. Diederik, S. Wiersma, Paolo Bartolini, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).

Economou, E. N.

K. Busch, C. M. Soukoulis, and E. N. Economou, “Transport and scattering mean free paths of classical waves,” Phys. Rev. B 50, 93–98 (1994).
[CrossRef]

C. M. Soukoulis, S. Datta, and E. N. Economou, “Propagation of classical waves in random media,” Phys. Rev. B 49, 3800–3810(1994).
[CrossRef]

García, P. D.

R. Sapienza, P. D. García, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, “Observation of resonant behavior in the energy velocity of diffused light,” Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

García-Valenzuela, A.

R. G. Barrera, A. Reyes-Coronado, and A. García-Valenzuela, “Nonlocal nature of the electrodynamic response of colloidal systems,” Phys. Rev. B 75, 184202 (2007).
[CrossRef]

Gross, P.

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near Anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
[CrossRef] [PubMed]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley , 1998).
[CrossRef]

Ioffe, A.

A. Ioffe and A. Regel, “Non-crystalline, amorphous and liquid electronic semiconductors,” Prog. Semicond. 4, 237–291 (1960).

Kaiser, R.

G. Labeyrie, E. Vaujour, C. A. Müller, D. Delande, C. Miniatura, D. Wilkowski, and R. Kaiser, “Slow diffusion of light in a cold atomic cloud,” Phys. Rev. Lett. 91, 223904 (2003).
[CrossRef] [PubMed]

Labeyrie, G.

G. Labeyrie, E. Vaujour, C. A. Müller, D. Delande, C. Miniatura, D. Wilkowski, and R. Kaiser, “Slow diffusion of light in a cold atomic cloud,” Phys. Rev. Lett. 91, 223904 (2003).
[CrossRef] [PubMed]

Lagendijk, A.

B. A. van Tiggelen, A. Lagendijk, and D. S. Wiersma, “Reflection and transmission of waves near the localization threshold,” Phys. Rev. Lett. 84, 4333–4336 (2000).
[CrossRef] [PubMed]

López, C.

R. Sapienza, P. D. García, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, “Observation of resonant behavior in the energy velocity of diffused light,” Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

Maret, G.

C. M. Aegerter and G. Maret, “Coherent backscattering and Anderson localization of light,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2009), Vol.  52, pp. 1–62.
[CrossRef]

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near Anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
[CrossRef] [PubMed]

M. Störzer, C. M. Aegerter, and G. Maret, “Reduced transport velocity of multiply scattered light due to resonant scattering,” Phys. Rev. E 73, 065602 (2006).
[CrossRef]

Martín, M. D.

R. Sapienza, P. D. García, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, “Observation of resonant behavior in the energy velocity of diffused light,” Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

Miniatura, C.

G. Labeyrie, E. Vaujour, C. A. Müller, D. Delande, C. Miniatura, D. Wilkowski, and R. Kaiser, “Slow diffusion of light in a cold atomic cloud,” Phys. Rev. Lett. 91, 223904 (2003).
[CrossRef] [PubMed]

Montambaux, G.

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University Press, 2007).
[CrossRef]

Müller, C. A.

G. Labeyrie, E. Vaujour, C. A. Müller, D. Delande, C. Miniatura, D. Wilkowski, and R. Kaiser, “Slow diffusion of light in a cold atomic cloud,” Phys. Rev. Lett. 91, 223904 (2003).
[CrossRef] [PubMed]

Regel, A.

A. Ioffe and A. Regel, “Non-crystalline, amorphous and liquid electronic semiconductors,” Prog. Semicond. 4, 237–291 (1960).

Reyes-Coronado, A.

R. G. Barrera, A. Reyes-Coronado, and A. García-Valenzuela, “Nonlocal nature of the electrodynamic response of colloidal systems,” Phys. Rev. B 75, 184202 (2007).
[CrossRef]

Righini, R.

A. L. Diederik, S. Wiersma, Paolo Bartolini, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).

Sapienza, R.

R. Sapienza, P. D. García, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, “Observation of resonant behavior in the energy velocity of diffused light,” Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

Sheng, P.

P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena, 2nd ed. (Springer, 2006).

Soukoulis, C. M.

K. Busch and C. M. Soukoulis, “Transport properties of random media: an energy-density CPA approach,” Phys. Rev. B 54, 893–899 (1996).
[CrossRef]

K. Busch and C. M. Soukoulis, “Transport properties of random media: a new effective medium theory,” Phys. Rev. Lett. 75, 3442–3445 (1995).
[CrossRef] [PubMed]

K. Busch, C. M. Soukoulis, and E. N. Economou, “Transport and scattering mean free paths of classical waves,” Phys. Rev. B 50, 93–98 (1994).
[CrossRef]

C. M. Soukoulis, S. Datta, and E. N. Economou, “Propagation of classical waves in random media,” Phys. Rev. B 49, 3800–3810(1994).
[CrossRef]

Störzer, M.

M. Störzer, C. M. Aegerter, and G. Maret, “Reduced transport velocity of multiply scattered light due to resonant scattering,” Phys. Rev. E 73, 065602 (2006).
[CrossRef]

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near Anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
[CrossRef] [PubMed]

Tweer, R.

R. Tweer, “Vielfachstreuung von Licht in Systemen dicht gepackter Mie–Streuer: Auf dem Weg zur Anderson–Lokalisierung?” Ph.D. thesis (University of Konstanz, 2002).

van de Hulst, H. C.

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

van Tiggelen, B. A.

B. A. van Tiggelen, A. Lagendijk, and D. S. Wiersma, “Reflection and transmission of waves near the localization threshold,” Phys. Rev. Lett. 84, 4333–4336 (2000).
[CrossRef] [PubMed]

Vaujour, E.

G. Labeyrie, E. Vaujour, C. A. Müller, D. Delande, C. Miniatura, D. Wilkowski, and R. Kaiser, “Slow diffusion of light in a cold atomic cloud,” Phys. Rev. Lett. 91, 223904 (2003).
[CrossRef] [PubMed]

Viña, L.

R. Sapienza, P. D. García, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, “Observation of resonant behavior in the energy velocity of diffused light,” Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

Wiersma, D. S.

R. Sapienza, P. D. García, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, “Observation of resonant behavior in the energy velocity of diffused light,” Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

B. A. van Tiggelen, A. Lagendijk, and D. S. Wiersma, “Reflection and transmission of waves near the localization threshold,” Phys. Rev. Lett. 84, 4333–4336 (2000).
[CrossRef] [PubMed]

Wiersma, S.

A. L. Diederik, S. Wiersma, Paolo Bartolini, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).

Wilkowski, D.

G. Labeyrie, E. Vaujour, C. A. Müller, D. Delande, C. Miniatura, D. Wilkowski, and R. Kaiser, “Slow diffusion of light in a cold atomic cloud,” Phys. Rev. Lett. 91, 223904 (2003).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Cambridge University Press, 1997).
[PubMed]

Xu, M.

Zhang, H.

Zhu, H.

Am. J. Phys. (1)

D. E. Aspnes, “Local-field effects and effective-medium theory: a microscopic perspective,” Am. J. Phys. 50, 704–709 (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]

Nature (1)

A. L. Diederik, S. Wiersma, Paolo Bartolini, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).

Opt. Express (1)

Phys. Rev. (1)

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
[CrossRef]

Phys. Rev. B (4)

K. Busch, C. M. Soukoulis, and E. N. Economou, “Transport and scattering mean free paths of classical waves,” Phys. Rev. B 50, 93–98 (1994).
[CrossRef]

C. M. Soukoulis, S. Datta, and E. N. Economou, “Propagation of classical waves in random media,” Phys. Rev. B 49, 3800–3810(1994).
[CrossRef]

K. Busch and C. M. Soukoulis, “Transport properties of random media: an energy-density CPA approach,” Phys. Rev. B 54, 893–899 (1996).
[CrossRef]

R. G. Barrera, A. Reyes-Coronado, and A. García-Valenzuela, “Nonlocal nature of the electrodynamic response of colloidal systems,” Phys. Rev. B 75, 184202 (2007).
[CrossRef]

Phys. Rev. E (1)

M. Störzer, C. M. Aegerter, and G. Maret, “Reduced transport velocity of multiply scattered light due to resonant scattering,” Phys. Rev. E 73, 065602 (2006).
[CrossRef]

Phys. Rev. Lett. (5)

G. Labeyrie, E. Vaujour, C. A. Müller, D. Delande, C. Miniatura, D. Wilkowski, and R. Kaiser, “Slow diffusion of light in a cold atomic cloud,” Phys. Rev. Lett. 91, 223904 (2003).
[CrossRef] [PubMed]

R. Sapienza, P. D. García, J. Bertolotti, M. D. Martín, A. Blanco, L. Viña, C. López, and D. S. Wiersma, “Observation of resonant behavior in the energy velocity of diffused light,” Phys. Rev. Lett. 99, 233902 (2007).
[CrossRef]

B. A. van Tiggelen, A. Lagendijk, and D. S. Wiersma, “Reflection and transmission of waves near the localization threshold,” Phys. Rev. Lett. 84, 4333–4336 (2000).
[CrossRef] [PubMed]

K. Busch and C. M. Soukoulis, “Transport properties of random media: a new effective medium theory,” Phys. Rev. Lett. 75, 3442–3445 (1995).
[CrossRef] [PubMed]

M. Störzer, P. Gross, C. M. Aegerter, and G. Maret, “Observation of the critical regime near Anderson localization of light,” Phys. Rev. Lett. 96, 063904 (2006).
[CrossRef] [PubMed]

Prog. Semicond. (1)

A. Ioffe and A. Regel, “Non-crystalline, amorphous and liquid electronic semiconductors,” Prog. Semicond. 4, 237–291 (1960).

Other (8)

C. M. Aegerter and G. Maret, “Coherent backscattering and Anderson localization of light,” in Progress in Optics, E.Wolf, ed. (Elsevier, 2009), Vol.  52, pp. 1–62.
[CrossRef]

R. Tweer, “Vielfachstreuung von Licht in Systemen dicht gepackter Mie–Streuer: Auf dem Weg zur Anderson–Lokalisierung?” Ph.D. thesis (University of Konstanz, 2002).

P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena, 2nd ed. (Springer, 2006).

E. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University Press, 2007).
[CrossRef]

T. C. Choy, Effective Medium Theory (Oxford University Press, 1999).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6th ed. (Cambridge University Press, 1997).
[PubMed]

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

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley , 1998).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Scheme of the ECPA method for random media composed of pure spheres. After ensemble averaging, the electromagnetic energy stored in the core–mantle sphere in (a) is equal to that in the core–mantle sphere in (b).

Fig. 2
Fig. 2

Calculated effective refractive indices n e for random media composed of two-layered spheres with r 1 = 500 nm , r 2 = 600 nm in the long-wavelength limit. The squares and upper line show the results obtained by the ECPA method and the Maxwell– Garnett theory, respectively. The lower line illustrates the results obtained by the ECPA method for the case in which two-layered-sphere systems are approximated by pure-sphere systems with r = 600 nm and ϵ .

Fig. 3
Fig. 3

Numerical comparison for the formulas of ϵ respectively defined by Eqs. (11, 16). ϕ = ( r 1 / r 2 ) 3 .

Fig. 4
Fig. 4

Effective refractive indices n e of random media composed of pure and two-layered spheres in the long-wavelength limit. The radius of the mantle of the sphere, r 2 = 600 nm , and the radius of the core is changed from r 1 = 100 to 500 nm . The numerical results for random media composed of two-layered spheres are denoted by r 1 i , and the results for random media composed of pure spheres with ϵ are denoted by r 1 i .

Fig. 5
Fig. 5

Effective refractive indices n e ( n e = ϵ e ) of random media composed of two-layered spheres with the volume fraction Φ 1 = 10 % (red solid line), Φ 2 = 40 % (black solid line) and random media composed of pure spheres with r = 600 nm , n = ϵ , Φ 1 = 10 % (cyan dashed line), Φ 2 = 40 % (green dashed line). The sizes and refractive indices of the two-layered spheres are (a)  r 1 = 100 nm , r 2 = 600 nm , n 1 = 1.44 , n 2 = 2.7 and (b)  r 1 = 300 nm , r 2 = 600 nm , n 1 = 1.44 , n 2 = 2.7 , respectively.

Fig. 6
Fig. 6

Effective refractive indices n e ( n e = ϵ e ) of random media composed of two-layered spheres with n 1 = 1.44 , n 2 = 2.7 and r 1 = 300 nm , r 2 = 600 nm . The volume fraction f of the system is changed from Φ = 10 % to Φ = 60 % .

Equations (16)

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V c ρ 1 d V + V m ρ 2 d V = V c + V m ρ ¯ e d V ,
ρ ( r ) = 1 2 [ ϵ ( r ) | E ( r ) | 2 + μ ( r ) | H ( r ) | 2 ] .
l = 1 N V l ρ l d V = V l ρ e ¯ d V .
E l = n = 1 E n ( f n ( l ) M o 1 n ( 1 ) i g n ( l ) N e 1 n ( 1 ) + v n ( l ) M o 1 n ( 2 ) i w n ( l ) N e 1 n ( 2 ) ) ,
H l = k l ω μ l n = 1 E n ( g n ( l ) M e 1 n ( 1 ) + i f n ( l ) N o 1 n ( 1 ) + w n ( l ) M e 1 n ( 2 ) + i v n ( l ) N o 1 n ( 2 ) ) ,
E n = E 0 i n 2 n + 1 n ( n + 1 ) .
Ω = l = 1 N Ω l ,
Ω l = 1 2 · ω 2 c 2 ϵ l n = 1 1 k l 3 k l r l 1 k l r l ρ 2 d ρ [ ( | f n ( l ) | 2 + | g n ( l ) | 2 ) W n ( j n , j n ; ρ ) + ( | v n ( l ) | 2 + | w n ( l ) | 2 ) W n ( n n , n n ; ρ ) + 2 ( | f n ( l ) · v n ( l ) | + | g n ( l ) · w n ( l ) | ) W n ( j n , n n ; ρ ) ] ,
W n ( z n , z ¯ n ; r ) = ( 2 n + 1 ) z n ( r ) z ¯ n ( r ) + ( n + 1 ) z n 1 ( r ) z ¯ n 1 ( r ) + n z n + 1 ( r ) z ¯ n + 1 ( r ) ,
ϵ e = ϵ 2 ( 1 + 3 Φ β 1 Φ β ) ,
ϵ = [ n 2 ϕ ( n 2 n 1 ) ] 2 ,
α = 4 π r 2 3 A ,
A = ( ϵ 2 ϵ m ) ( ϵ 1 + 2 ϵ 2 ) + ϕ ( ϵ 1 ϵ 2 ) ( ϵ m + 2 ϵ 2 ) ( ϵ 2 + 2 ϵ m ) ( ϵ 1 + 2 ϵ 2 ) + ϕ ( 2 ϵ 2 2 ϵ m ) ( ϵ 1 ϵ 2 ) .
ϵ e ϵ m ϵ e + 2 ϵ m = 1 3 α ,
ϵ e ϵ m ϵ e + 2 ϵ m = Φ ϵ ϵ m ϵ + 2 ϵ m .
ϵ = 3 ϵ m 1 A 2 ϵ m .

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