Abstract

The modified unrestricted effective-medium refractive index is defined as one that yields accurate values of a representative set of far-field scattering characteristics (including the scattering matrix) for an object made of randomly heterogeneous materials. We validate the concept of the modified unrestricted effective-medium refractive index by comparing numerically exact superposition T-matrix results for a spherical host randomly filled with a large number of identical small inclusions and Lorenz–Mie results for a homogeneous spherical counterpart. A remarkable quantitative agreement between the superposition T-matrix and Lorenz–Mie scattering matrices over the entire range of scattering angles demonstrates unequivocally that the modified unrestricted effective-medium refractive index is a sound (albeit still phenomenological) concept provided that the size parameter of the inclusions is sufficiently small and their number is sufficiently large. Furthermore, it appears that in cases when the concept of the modified unrestricted effective-medium refractive index works, its actual value is close to that predicted by the Maxwell-Garnett mixing rule.

© 2014 Optical Society of America

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References

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  1. J. C. Maxwell-Garnett, Phil. Trans. R. Soc. A 203, 385 (1904).
    [CrossRef]
  2. D. A. G. Bruggeman, Ann. Phys. 416, 636 (1935).
    [CrossRef]
  3. P. Chýlek, G. Videen, D. J. W. Geldart, J. S. Dobbie, and H. C. W. Tso, in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic, 2000), p. 273.
  4. M. I. Mishchenko, J. W. Hovenier, W. J. Wiscombe, and L. D. Travis, in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic, 2000), p. 29.
  5. F. M. Kahnert, J. Quant. Spectrosc. Radiat. Transfer 79–80, 775 (2003).
    [CrossRef]
  6. A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles (Springer, 2006).
  7. C. F. Bohren, J. Atmos. Sci. 43, 468 (1986).
    [CrossRef]
  8. H. A. Lorentz, The Theory of Electrons (B. G. Teubner, Leipzig, 1916).
  9. L. Rosenfeld, Theory of Electrons (North-Holland, 1951).
  10. G. Russakoff, Am. J. Phys. 38, 1188 (1970).
    [CrossRef]
  11. J. Vlieger, Can. J. Phys. 49, 1384 (1971).
    [CrossRef]
  12. J. D. Jackson, Classical Electrodynamics (Wiley, 1999).
  13. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (Cambridge University, 2006).
  14. M. I. Mishchenko, Electromagnetic Scattering by Particles and Particle groups: An Introduction (Cambridge University, 2014).
  15. A. Doicu and T. Wriedt, J. Opt. A Pure Appl. Opt. 3, 204 (2001).
    [CrossRef]
  16. M. I. Mishchenko and A. Macke, J. Quant. Spectrosc. Radiat. Transfer 57, 767 (1997).
    [CrossRef]
  17. D. W. Mackowski, J. Quant. Spectrosc. Radiat. Transfer 133, 264 (2014).
    [CrossRef]
  18. P. R. Siqueira and K. Sarabandi, IEEE Trans. Antennas Propag. 48, 317 (2000).
    [CrossRef]
  19. C. Meiners, J. Psilopoulos, and A. F. Jacob, Microwave Opt. Technol. Lett. 40, 523 (2004).
  20. M. I. Mishchenko, L. Liu, and D. W. Mackowski, Opt. Lett. 39, 1701 (2014).
    [CrossRef]
  21. M. I. Mishchenko, L. Liu, B. Cairns, and D. W. Mackowski, Opt. Lett. 39, 2607 (2014).
    [CrossRef]
  22. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

2014 (3)

2004 (1)

C. Meiners, J. Psilopoulos, and A. F. Jacob, Microwave Opt. Technol. Lett. 40, 523 (2004).

2003 (1)

F. M. Kahnert, J. Quant. Spectrosc. Radiat. Transfer 79–80, 775 (2003).
[CrossRef]

2001 (1)

A. Doicu and T. Wriedt, J. Opt. A Pure Appl. Opt. 3, 204 (2001).
[CrossRef]

2000 (1)

P. R. Siqueira and K. Sarabandi, IEEE Trans. Antennas Propag. 48, 317 (2000).
[CrossRef]

1997 (1)

M. I. Mishchenko and A. Macke, J. Quant. Spectrosc. Radiat. Transfer 57, 767 (1997).
[CrossRef]

1986 (1)

C. F. Bohren, J. Atmos. Sci. 43, 468 (1986).
[CrossRef]

1971 (1)

J. Vlieger, Can. J. Phys. 49, 1384 (1971).
[CrossRef]

1970 (1)

G. Russakoff, Am. J. Phys. 38, 1188 (1970).
[CrossRef]

1935 (1)

D. A. G. Bruggeman, Ann. Phys. 416, 636 (1935).
[CrossRef]

1904 (1)

J. C. Maxwell-Garnett, Phil. Trans. R. Soc. A 203, 385 (1904).
[CrossRef]

Bohren, C. F.

C. F. Bohren, J. Atmos. Sci. 43, 468 (1986).
[CrossRef]

Bruggeman, D. A. G.

D. A. G. Bruggeman, Ann. Phys. 416, 636 (1935).
[CrossRef]

Cairns, B.

Chýlek, P.

P. Chýlek, G. Videen, D. J. W. Geldart, J. S. Dobbie, and H. C. W. Tso, in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic, 2000), p. 273.

Dobbie, J. S.

P. Chýlek, G. Videen, D. J. W. Geldart, J. S. Dobbie, and H. C. W. Tso, in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic, 2000), p. 273.

Doicu, A.

A. Doicu and T. Wriedt, J. Opt. A Pure Appl. Opt. 3, 204 (2001).
[CrossRef]

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles (Springer, 2006).

Eremin, Y. A.

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles (Springer, 2006).

Geldart, D. J. W.

P. Chýlek, G. Videen, D. J. W. Geldart, J. S. Dobbie, and H. C. W. Tso, in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic, 2000), p. 273.

Hovenier, J. W.

M. I. Mishchenko, J. W. Hovenier, W. J. Wiscombe, and L. D. Travis, in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic, 2000), p. 29.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

Jacob, A. F.

C. Meiners, J. Psilopoulos, and A. F. Jacob, Microwave Opt. Technol. Lett. 40, 523 (2004).

Kahnert, F. M.

F. M. Kahnert, J. Quant. Spectrosc. Radiat. Transfer 79–80, 775 (2003).
[CrossRef]

Lacis, A. A.

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

Liu, L.

Lorentz, H. A.

H. A. Lorentz, The Theory of Electrons (B. G. Teubner, Leipzig, 1916).

Macke, A.

M. I. Mishchenko and A. Macke, J. Quant. Spectrosc. Radiat. Transfer 57, 767 (1997).
[CrossRef]

Mackowski, D. W.

Maxwell-Garnett, J. C.

J. C. Maxwell-Garnett, Phil. Trans. R. Soc. A 203, 385 (1904).
[CrossRef]

Meiners, C.

C. Meiners, J. Psilopoulos, and A. F. Jacob, Microwave Opt. Technol. Lett. 40, 523 (2004).

Mishchenko, M. I.

M. I. Mishchenko, L. Liu, and D. W. Mackowski, Opt. Lett. 39, 1701 (2014).
[CrossRef]

M. I. Mishchenko, L. Liu, B. Cairns, and D. W. Mackowski, Opt. Lett. 39, 2607 (2014).
[CrossRef]

M. I. Mishchenko and A. Macke, J. Quant. Spectrosc. Radiat. Transfer 57, 767 (1997).
[CrossRef]

M. I. Mishchenko, Electromagnetic Scattering by Particles and Particle groups: An Introduction (Cambridge University, 2014).

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

M. I. Mishchenko, J. W. Hovenier, W. J. Wiscombe, and L. D. Travis, in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic, 2000), p. 29.

Psilopoulos, J.

C. Meiners, J. Psilopoulos, and A. F. Jacob, Microwave Opt. Technol. Lett. 40, 523 (2004).

Rosenfeld, L.

L. Rosenfeld, Theory of Electrons (North-Holland, 1951).

Russakoff, G.

G. Russakoff, Am. J. Phys. 38, 1188 (1970).
[CrossRef]

Sarabandi, K.

P. R. Siqueira and K. Sarabandi, IEEE Trans. Antennas Propag. 48, 317 (2000).
[CrossRef]

Siqueira, P. R.

P. R. Siqueira and K. Sarabandi, IEEE Trans. Antennas Propag. 48, 317 (2000).
[CrossRef]

Travis, L. D.

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

M. I. Mishchenko, J. W. Hovenier, W. J. Wiscombe, and L. D. Travis, in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic, 2000), p. 29.

Tso, H. C. W.

P. Chýlek, G. Videen, D. J. W. Geldart, J. S. Dobbie, and H. C. W. Tso, in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic, 2000), p. 273.

van de Hulst, H. C.

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

Videen, G.

P. Chýlek, G. Videen, D. J. W. Geldart, J. S. Dobbie, and H. C. W. Tso, in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic, 2000), p. 273.

Vlieger, J.

J. Vlieger, Can. J. Phys. 49, 1384 (1971).
[CrossRef]

Wiscombe, W. J.

M. I. Mishchenko, J. W. Hovenier, W. J. Wiscombe, and L. D. Travis, in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic, 2000), p. 29.

Wriedt, T.

A. Doicu and T. Wriedt, J. Opt. A Pure Appl. Opt. 3, 204 (2001).
[CrossRef]

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles (Springer, 2006).

Am. J. Phys. (1)

G. Russakoff, Am. J. Phys. 38, 1188 (1970).
[CrossRef]

Ann. Phys. (1)

D. A. G. Bruggeman, Ann. Phys. 416, 636 (1935).
[CrossRef]

Can. J. Phys. (1)

J. Vlieger, Can. J. Phys. 49, 1384 (1971).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

P. R. Siqueira and K. Sarabandi, IEEE Trans. Antennas Propag. 48, 317 (2000).
[CrossRef]

J. Atmos. Sci. (1)

C. F. Bohren, J. Atmos. Sci. 43, 468 (1986).
[CrossRef]

J. Opt. A Pure Appl. Opt. (1)

A. Doicu and T. Wriedt, J. Opt. A Pure Appl. Opt. 3, 204 (2001).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (3)

M. I. Mishchenko and A. Macke, J. Quant. Spectrosc. Radiat. Transfer 57, 767 (1997).
[CrossRef]

D. W. Mackowski, J. Quant. Spectrosc. Radiat. Transfer 133, 264 (2014).
[CrossRef]

F. M. Kahnert, J. Quant. Spectrosc. Radiat. Transfer 79–80, 775 (2003).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

C. Meiners, J. Psilopoulos, and A. F. Jacob, Microwave Opt. Technol. Lett. 40, 523 (2004).

Opt. Lett. (2)

Phil. Trans. R. Soc. A (1)

J. C. Maxwell-Garnett, Phil. Trans. R. Soc. A 203, 385 (1904).
[CrossRef]

Other (9)

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

H. A. Lorentz, The Theory of Electrons (B. G. Teubner, Leipzig, 1916).

L. Rosenfeld, Theory of Electrons (North-Holland, 1951).

A. Doicu, T. Wriedt, and Y. A. Eremin, Light Scattering by Systems of Particles (Springer, 2006).

P. Chýlek, G. Videen, D. J. W. Geldart, J. S. Dobbie, and H. C. W. Tso, in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic, 2000), p. 273.

M. I. Mishchenko, J. W. Hovenier, W. J. Wiscombe, and L. D. Travis, in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, eds. (Academic, 2000), p. 29.

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

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

M. I. Mishchenko, Electromagnetic Scattering by Particles and Particle groups: An Introduction (Cambridge University, 2014).

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Figures (1)

Fig. 1.
Fig. 1.

Elements of the normalized scattering matrix (2) for randomly heterogeneous as well as homogeneous spherical particles with a fixed size parameter X = 10 . The color legend in panel (b) applies to panels (a)–(f).

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

1 T t t + T d t exp ( i ω t ) = T 2 π / ω 0 .
F ˜ ( Θ ) = [ a 1 ( Θ ) b 1 ( Θ ) 0 0 b 1 ( Θ ) a 2 ( Θ ) 0 0 0 0 a 3 ( Θ ) b 2 ( Θ ) 0 0 b 2 ( Θ ) a 4 ( Θ ) ]
1 2 0 π d Θ sin Θ a 1 ( Θ ) = 1 .
a 2 ( Θ ) / a 1 ( Θ ) 1 .

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