Abstract

This tutorial is devoted to the Maxwell Garnett approximation and related theories. Topics covered in this first, introductory part of the tutorial include the Lorentz local field correction, the Clausius–Mossotti relation and its role in the modern numerical technique known as the discrete dipole approximation, the Maxwell Garnett mixing formula for isotropic and anisotropic media, multicomponent mixtures and the Bruggeman equation, the concept of smooth field, and Wiener and Bergman–Milton bounds.

© 2016 Optical Society of America

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References

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  1. J. C. M. Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London A 203, 385–420 (1904).
    [Crossref]
  2. J. C. M. Garnett, “Colours in metal glasses, in metallic films, and in metallic solutions II,” Philos. Trans. R. Soc. London 205, 237–288 (1906).
    [Crossref]
  3. R. Resta, “Macroscopic polarization in crystalline dielectrics: the geometrical phase approach,” Rev. Mod. Phys. 66, 899–915 (1994).
    [Crossref]
  4. R. Resta and D. Vanderbilt, “Theory of polarization: a modern approach,” in Physics of Ferroelectrics: A Modern Perspective (Springer, 2007), pp. 31–68.
  5. D. M. Wood and N. W. Ashcroft, “Effective medium theory of optical properties of small particle composites,” Philos. Mag. 35(2), 269–280 (1977).
    [Crossref]
  6. G. A. Niklasson, C. G. Granqvist, and O. Hunderi, “Effective medium models for the optical properties of inhomogeneous materials,” Appl. Opt. 20, 26–30 (1981).
    [Crossref]
  7. T. G. Mackay, A. Lakhtakia, and W. S. Weiglhofer, “Strong-property-fluctuation theory for homogenization of bianisotropic composites: formulation,” Phys. Rev. E 62, 6052–6064 (2000).
    [Crossref]
  8. W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
    [Crossref]
  9. R. Ruppin, “Evaluation of extended Maxwell–Garnett theories,” Opt. Commun. 182, 273–279 (2000).
    [Crossref]
  10. C. F. Bohren, “Do extended effective-medium formulas scale properly?” J. Nanophoton. 3, 039501 (2009).
    [Crossref]
  11. In a spherical system of coordinates (r,θ,φ), the angular average of cos2 θ is 1/3; that is, 〈cos2〉=(4π)−1∫02πdφ∫0πsin θdθ cos2 θ=1/3.
  12. Equation (8) can be obtained by using the expression ϕ(r)=∫[ρ(R)/|r−R|]d3R for the electrostatic potential ϕ(r), assuming that the charge density ρ(R) is localized around a point r′, using the expansion 1/|r−R|=1/|r−r′|+(R−r′)·∇r′(1/|r−r′|)+…, defining the dipole moment of the system as d=∫rρ(r)d3r, and, finally, computing the electric field according to E(r)=−∇rϕ(r).
  13. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [Crossref]
  14. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
    [Crossref]
  15. B. T. Draine and J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
    [Crossref]
  16. L. L. Foldy, “The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers,” Phys. Rev. 67, 107–119 (1945).
    [Crossref]
  17. M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. 23, 287–310 (1951).
    [Crossref]
  18. P. B. Allen, “Dipole interactions and electrical polarity in nanosystems: the Clausius-Mossotti and related models,” J. Chem. Phys. 120, 2951–2962 (2004).
    [Crossref]
  19. D. Vanzo, B. J. Topham, and Z. G. Soos, “Dipole-field sums, Lorentz factors and dielectric properties of organic molecular films modeled as crystalline arrays of polarizable points,” Adv. Funct. Mater. 25, 2004–2012 (2015).
    [Crossref]
  20. Of course, infinite media do not exist in nature. Here we mean a host medium that is so large that the field created by the dipole is negligible at its boundaries. In general, one should be very careful not to make a mathematical mistake when applying the concept of “infinite medium,” especially when wave propagation is involved.
  21. D. A. G. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. 416, 665–679 (1935).
    [Crossref]
  22. D. A. G. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. II. Dielektrizitätskonstanten und Leitfähigkeiten von Vielrkistallen der nichtregularen Systeme,” Ann. Phys. 417, 645–672 (1936).
    [Crossref]
  23. S. Berthier and J. Lafait, “Effective medium theory: mathematical determination of the physical solution for the dielectric constant,” Opt. Commun. 33, 303–306 (1980).
    [Crossref]
  24. R. Jansson and H. Arwin, “Selection of the physically correct solution in the n-media Bruggeman effective medium approximation,” Opt. Commun. 106, 133–138 (1994).
    [Crossref]
  25. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
  26. D. Schmidt and M. Schubert, “Anisotropic Bruggeman effective medium approaches for slanted columnar thin films,” J. Appl. Phys. 114, 083510 (2013).
    [Crossref]
  27. S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” J. Exp. Theor. Phys. 2, 466–475 (1956).
  28. This is true for sufficiently small h, as long as the phase shift of the wave propagating in the structure over one period is small compared to π. Homogenization is possible only if this condition holds.
  29. O. Wiener, “Die Theorie des Mischkörpers für das Feld der Stationären Strömung. Erste Abhandlung: Die Mittelwertsätze für Kraft, Polarisation und Energie,” Abhandlungen der Mathathematisch-Physikalischen Klasse der Königl. Sächsischen Gesellschaft der Wissenschaften 32, 507–604 (1912).
  30. Z. Hashin and S. Shtrikman, “A variational approach to the theory of the effective magnetic permeability of multiphase materials,” J. Appl. Phys. 33, 3125–3131 (1962).
    [Crossref]
  31. D. J. Bergman, “Exactly solvable microscopic geometries and rigorous bounds for the complex dielectric constant of a two-component composite material,” Phys. Rev. Lett. 44, 1285–1287 (1980).
    [Crossref]
  32. G. W. Milton, “Bounds on the complex dielectric constant of a composite material,” Appl. Phys. Lett. 37, 300–302 (1980).
    [Crossref]

2015 (1)

D. Vanzo, B. J. Topham, and Z. G. Soos, “Dipole-field sums, Lorentz factors and dielectric properties of organic molecular films modeled as crystalline arrays of polarizable points,” Adv. Funct. Mater. 25, 2004–2012 (2015).
[Crossref]

2013 (1)

D. Schmidt and M. Schubert, “Anisotropic Bruggeman effective medium approaches for slanted columnar thin films,” J. Appl. Phys. 114, 083510 (2013).
[Crossref]

2009 (1)

C. F. Bohren, “Do extended effective-medium formulas scale properly?” J. Nanophoton. 3, 039501 (2009).
[Crossref]

2004 (1)

P. B. Allen, “Dipole interactions and electrical polarity in nanosystems: the Clausius-Mossotti and related models,” J. Chem. Phys. 120, 2951–2962 (2004).
[Crossref]

2000 (2)

T. G. Mackay, A. Lakhtakia, and W. S. Weiglhofer, “Strong-property-fluctuation theory for homogenization of bianisotropic composites: formulation,” Phys. Rev. E 62, 6052–6064 (2000).
[Crossref]

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

1994 (2)

R. Jansson and H. Arwin, “Selection of the physically correct solution in the n-media Bruggeman effective medium approximation,” Opt. Commun. 106, 133–138 (1994).
[Crossref]

R. Resta, “Macroscopic polarization in crystalline dielectrics: the geometrical phase approach,” Rev. Mod. Phys. 66, 899–915 (1994).
[Crossref]

1993 (1)

B. T. Draine and J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
[Crossref]

1989 (1)

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

1988 (1)

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[Crossref]

1981 (1)

1980 (3)

D. J. Bergman, “Exactly solvable microscopic geometries and rigorous bounds for the complex dielectric constant of a two-component composite material,” Phys. Rev. Lett. 44, 1285–1287 (1980).
[Crossref]

G. W. Milton, “Bounds on the complex dielectric constant of a composite material,” Appl. Phys. Lett. 37, 300–302 (1980).
[Crossref]

S. Berthier and J. Lafait, “Effective medium theory: mathematical determination of the physical solution for the dielectric constant,” Opt. Commun. 33, 303–306 (1980).
[Crossref]

1977 (1)

D. M. Wood and N. W. Ashcroft, “Effective medium theory of optical properties of small particle composites,” Philos. Mag. 35(2), 269–280 (1977).
[Crossref]

1973 (1)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

1962 (1)

Z. Hashin and S. Shtrikman, “A variational approach to the theory of the effective magnetic permeability of multiphase materials,” J. Appl. Phys. 33, 3125–3131 (1962).
[Crossref]

1956 (1)

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” J. Exp. Theor. Phys. 2, 466–475 (1956).

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. I. General theory of isotropic scattering by randomly distributed scatterers,” Phys. Rev. 67, 107–119 (1945).
[Crossref]

1936 (1)

D. A. G. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. II. Dielektrizitätskonstanten und Leitfähigkeiten von Vielrkistallen der nichtregularen Systeme,” Ann. Phys. 417, 645–672 (1936).
[Crossref]

1935 (1)

D. A. G. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. 416, 665–679 (1935).
[Crossref]

1912 (1)

O. Wiener, “Die Theorie des Mischkörpers für das Feld der Stationären Strömung. Erste Abhandlung: Die Mittelwertsätze für Kraft, Polarisation und Energie,” Abhandlungen der Mathathematisch-Physikalischen Klasse der Königl. Sächsischen Gesellschaft der Wissenschaften 32, 507–604 (1912).

1906 (1)

J. C. M. Garnett, “Colours in metal glasses, in metallic films, and in metallic solutions II,” Philos. Trans. R. Soc. London 205, 237–288 (1906).
[Crossref]

1904 (1)

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

Allen, P. B.

P. B. Allen, “Dipole interactions and electrical polarity in nanosystems: the Clausius-Mossotti and related models,” J. Chem. Phys. 120, 2951–2962 (2004).
[Crossref]

Arwin, H.

R. Jansson and H. Arwin, “Selection of the physically correct solution in the n-media Bruggeman effective medium approximation,” Opt. Commun. 106, 133–138 (1994).
[Crossref]

Ashcroft, N. W.

D. M. Wood and N. W. Ashcroft, “Effective medium theory of optical properties of small particle composites,” Philos. Mag. 35(2), 269–280 (1977).
[Crossref]

Bergman, D. J.

D. J. Bergman, “Exactly solvable microscopic geometries and rigorous bounds for the complex dielectric constant of a two-component composite material,” Phys. Rev. Lett. 44, 1285–1287 (1980).
[Crossref]

Berthier, S.

S. Berthier and J. Lafait, “Effective medium theory: mathematical determination of the physical solution for the dielectric constant,” Opt. Commun. 33, 303–306 (1980).
[Crossref]

Bohren, C. F.

C. F. Bohren, “Do extended effective-medium formulas scale properly?” J. Nanophoton. 3, 039501 (2009).
[Crossref]

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

Bruggeman, D. A. G.

D. A. G. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. II. Dielektrizitätskonstanten und Leitfähigkeiten von Vielrkistallen der nichtregularen Systeme,” Ann. Phys. 417, 645–672 (1936).
[Crossref]

D. A. G. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. 416, 665–679 (1935).
[Crossref]

Doyle, W. T.

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

Draine, B. T.

B. T. Draine and J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
[Crossref]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[Crossref]

Foldy, L. L.

L. L. Foldy, “The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers,” Phys. Rev. 67, 107–119 (1945).
[Crossref]

Garnett, J. C. M.

J. C. M. Garnett, “Colours in metal glasses, in metallic films, and in metallic solutions II,” Philos. Trans. R. Soc. London 205, 237–288 (1906).
[Crossref]

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

Goodman, J.

B. T. Draine and J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
[Crossref]

Granqvist, C. G.

Hashin, Z.

Z. Hashin and S. Shtrikman, “A variational approach to the theory of the effective magnetic permeability of multiphase materials,” J. Appl. Phys. 33, 3125–3131 (1962).
[Crossref]

Huffman, D. R.

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

Hunderi, O.

Jansson, R.

R. Jansson and H. Arwin, “Selection of the physically correct solution in the n-media Bruggeman effective medium approximation,” Opt. Commun. 106, 133–138 (1994).
[Crossref]

Lafait, J.

S. Berthier and J. Lafait, “Effective medium theory: mathematical determination of the physical solution for the dielectric constant,” Opt. Commun. 33, 303–306 (1980).
[Crossref]

Lakhtakia, A.

T. G. Mackay, A. Lakhtakia, and W. S. Weiglhofer, “Strong-property-fluctuation theory for homogenization of bianisotropic composites: formulation,” Phys. Rev. E 62, 6052–6064 (2000).
[Crossref]

Lax, M.

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

Mackay, T. G.

T. G. Mackay, A. Lakhtakia, and W. S. Weiglhofer, “Strong-property-fluctuation theory for homogenization of bianisotropic composites: formulation,” Phys. Rev. E 62, 6052–6064 (2000).
[Crossref]

Milton, G. W.

G. W. Milton, “Bounds on the complex dielectric constant of a composite material,” Appl. Phys. Lett. 37, 300–302 (1980).
[Crossref]

Niklasson, G. A.

Pennypacker, C. R.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Purcell, E. M.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

Resta, R.

R. Resta, “Macroscopic polarization in crystalline dielectrics: the geometrical phase approach,” Rev. Mod. Phys. 66, 899–915 (1994).
[Crossref]

R. Resta and D. Vanderbilt, “Theory of polarization: a modern approach,” in Physics of Ferroelectrics: A Modern Perspective (Springer, 2007), pp. 31–68.

Ruppin, R.

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

Rytov, S. M.

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” J. Exp. Theor. Phys. 2, 466–475 (1956).

Schmidt, D.

D. Schmidt and M. Schubert, “Anisotropic Bruggeman effective medium approaches for slanted columnar thin films,” J. Appl. Phys. 114, 083510 (2013).
[Crossref]

Schubert, M.

D. Schmidt and M. Schubert, “Anisotropic Bruggeman effective medium approaches for slanted columnar thin films,” J. Appl. Phys. 114, 083510 (2013).
[Crossref]

Shtrikman, S.

Z. Hashin and S. Shtrikman, “A variational approach to the theory of the effective magnetic permeability of multiphase materials,” J. Appl. Phys. 33, 3125–3131 (1962).
[Crossref]

Soos, Z. G.

D. Vanzo, B. J. Topham, and Z. G. Soos, “Dipole-field sums, Lorentz factors and dielectric properties of organic molecular films modeled as crystalline arrays of polarizable points,” Adv. Funct. Mater. 25, 2004–2012 (2015).
[Crossref]

Topham, B. J.

D. Vanzo, B. J. Topham, and Z. G. Soos, “Dipole-field sums, Lorentz factors and dielectric properties of organic molecular films modeled as crystalline arrays of polarizable points,” Adv. Funct. Mater. 25, 2004–2012 (2015).
[Crossref]

Vanderbilt, D.

R. Resta and D. Vanderbilt, “Theory of polarization: a modern approach,” in Physics of Ferroelectrics: A Modern Perspective (Springer, 2007), pp. 31–68.

Vanzo, D.

D. Vanzo, B. J. Topham, and Z. G. Soos, “Dipole-field sums, Lorentz factors and dielectric properties of organic molecular films modeled as crystalline arrays of polarizable points,” Adv. Funct. Mater. 25, 2004–2012 (2015).
[Crossref]

Weiglhofer, W. S.

T. G. Mackay, A. Lakhtakia, and W. S. Weiglhofer, “Strong-property-fluctuation theory for homogenization of bianisotropic composites: formulation,” Phys. Rev. E 62, 6052–6064 (2000).
[Crossref]

Wiener, O.

O. Wiener, “Die Theorie des Mischkörpers für das Feld der Stationären Strömung. Erste Abhandlung: Die Mittelwertsätze für Kraft, Polarisation und Energie,” Abhandlungen der Mathathematisch-Physikalischen Klasse der Königl. Sächsischen Gesellschaft der Wissenschaften 32, 507–604 (1912).

Wood, D. M.

D. M. Wood and N. W. Ashcroft, “Effective medium theory of optical properties of small particle composites,” Philos. Mag. 35(2), 269–280 (1977).
[Crossref]

Abhandlungen der Mathathematisch-Physikalischen Klasse der Königl. Sächsischen Gesellschaft der Wissenschaften (1)

O. Wiener, “Die Theorie des Mischkörpers für das Feld der Stationären Strömung. Erste Abhandlung: Die Mittelwertsätze für Kraft, Polarisation und Energie,” Abhandlungen der Mathathematisch-Physikalischen Klasse der Königl. Sächsischen Gesellschaft der Wissenschaften 32, 507–604 (1912).

Adv. Funct. Mater. (1)

D. Vanzo, B. J. Topham, and Z. G. Soos, “Dipole-field sums, Lorentz factors and dielectric properties of organic molecular films modeled as crystalline arrays of polarizable points,” Adv. Funct. Mater. 25, 2004–2012 (2015).
[Crossref]

Ann. Phys. (2)

D. A. G. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen,” Ann. Phys. 416, 665–679 (1935).
[Crossref]

D. A. G. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. II. Dielektrizitätskonstanten und Leitfähigkeiten von Vielrkistallen der nichtregularen Systeme,” Ann. Phys. 417, 645–672 (1936).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

G. W. Milton, “Bounds on the complex dielectric constant of a composite material,” Appl. Phys. Lett. 37, 300–302 (1980).
[Crossref]

Astrophys. J. (3)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[Crossref]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[Crossref]

B. T. Draine and J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
[Crossref]

J. Appl. Phys. (2)

Z. Hashin and S. Shtrikman, “A variational approach to the theory of the effective magnetic permeability of multiphase materials,” J. Appl. Phys. 33, 3125–3131 (1962).
[Crossref]

D. Schmidt and M. Schubert, “Anisotropic Bruggeman effective medium approaches for slanted columnar thin films,” J. Appl. Phys. 114, 083510 (2013).
[Crossref]

J. Chem. Phys. (1)

P. B. Allen, “Dipole interactions and electrical polarity in nanosystems: the Clausius-Mossotti and related models,” J. Chem. Phys. 120, 2951–2962 (2004).
[Crossref]

J. Exp. Theor. Phys. (1)

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” J. Exp. Theor. Phys. 2, 466–475 (1956).

J. Nanophoton. (1)

C. F. Bohren, “Do extended effective-medium formulas scale properly?” J. Nanophoton. 3, 039501 (2009).
[Crossref]

Opt. Commun. (3)

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

S. Berthier and J. Lafait, “Effective medium theory: mathematical determination of the physical solution for the dielectric constant,” Opt. Commun. 33, 303–306 (1980).
[Crossref]

R. Jansson and H. Arwin, “Selection of the physically correct solution in the n-media Bruggeman effective medium approximation,” Opt. Commun. 106, 133–138 (1994).
[Crossref]

Philos. Mag. (1)

D. M. Wood and N. W. Ashcroft, “Effective medium theory of optical properties of small particle composites,” Philos. Mag. 35(2), 269–280 (1977).
[Crossref]

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

J. C. M. Garnett, “Colours in metal glasses, in metallic films, and in metallic solutions II,” Philos. Trans. R. Soc. London 205, 237–288 (1906).
[Crossref]

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

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

Phys. Rev. (1)

L. L. Foldy, “The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers,” Phys. Rev. 67, 107–119 (1945).
[Crossref]

Phys. Rev. B (1)

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

Phys. Rev. E (1)

T. G. Mackay, A. Lakhtakia, and W. S. Weiglhofer, “Strong-property-fluctuation theory for homogenization of bianisotropic composites: formulation,” Phys. Rev. E 62, 6052–6064 (2000).
[Crossref]

Phys. Rev. Lett. (1)

D. J. Bergman, “Exactly solvable microscopic geometries and rigorous bounds for the complex dielectric constant of a two-component composite material,” Phys. Rev. Lett. 44, 1285–1287 (1980).
[Crossref]

Rev. Mod. Phys. (2)

R. Resta, “Macroscopic polarization in crystalline dielectrics: the geometrical phase approach,” Rev. Mod. Phys. 66, 899–915 (1994).
[Crossref]

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

Other (6)

In a spherical system of coordinates (r,θ,φ), the angular average of cos2 θ is 1/3; that is, 〈cos2〉=(4π)−1∫02πdφ∫0πsin θdθ cos2 θ=1/3.

Equation (8) can be obtained by using the expression ϕ(r)=∫[ρ(R)/|r−R|]d3R for the electrostatic potential ϕ(r), assuming that the charge density ρ(R) is localized around a point r′, using the expansion 1/|r−R|=1/|r−r′|+(R−r′)·∇r′(1/|r−r′|)+…, defining the dipole moment of the system as d=∫rρ(r)d3r, and, finally, computing the electric field according to E(r)=−∇rϕ(r).

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

Of course, infinite media do not exist in nature. Here we mean a host medium that is so large that the field created by the dipole is negligible at its boundaries. In general, one should be very careful not to make a mathematical mistake when applying the concept of “infinite medium,” especially when wave propagation is involved.

R. Resta and D. Vanderbilt, “Theory of polarization: a modern approach,” in Physics of Ferroelectrics: A Modern Perspective (Springer, 2007), pp. 31–68.

This is true for sufficiently small h, as long as the phase shift of the wave propagating in the structure over one period is small compared to π. Homogenization is possible only if this condition holds.

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Figures (3)

Fig. 1.
Fig. 1.

Illustration of the setup used in the derivation (i) of Eq. (2). The dipole moment of the system of two charges in the Z direction is d z = q h . The electric field in the midpoint P is E z ( P ) = 8 q / β 3 h 2 . The oval shows the region of space where the strong field is supported; its volume scales as β 3 h 3 . Note that rigorous integration of the electric field of a truly point charge is not possible due to a divergence. In this figure, the charges are shown to be of finite size. In this case, a small spherical region around each charge gives a zero contribution to the integral.

Fig. 2.
Fig. 2.

Collection of dipoles in an external field. The particles are distributed inside a spherical volume either randomly (as shown) or periodically. It is assumed, however, that the macroscopic density of particles is constant inside the sphere and equal to v 1 = N / V . Here v is the specific volume per one particle.

Fig. 3.
Fig. 3.

Illustration of the Wiener and Bergman–Milton bounds for a two-component mixture with ϵ 1 = 1.5 + 1.0 i and ϵ 2 = 4.0 + 2.5 i . Curves MG1, MG2, and BG are parametric plots of the complex functions ( ϵ MG ) p [formula (34)] and ϵ BG [formula (30)] as functions of f for 0 f 1 and three different values of ν p , as labeled. Arcs ACB and ADB are parametric plots of ( ϵ MG ) p as a function of ν p for fixed f and different choice of the host medium (hence two different arcs). Notations: LWB—linear Wiener bound; CWB—circular Wiener bound; MG1—Maxwell Garnett mixing formula in which ϵ h = ϵ 1 , ϵ i = ϵ 2 and f = f 2 ; MG2—same but for ϵ h = ϵ 2 , ϵ i = ϵ 1 and f = f 1 ; BG—symmetric Bruggeman mixing formula; A, B, C, D, E mark the points on the CWB, LWB, MG1, MG2, BG curves for which f 1 = 0.7 and f 2 = 0.3 . Point E and curve BG are shown in panel (a) only. Only the curves MG1 and MG2 depend on ν p . The region Ω (delineated by LWB and CWB) is the locus of all points ( ϵ eff ) p that are attainable for the two-component mixture regardless of f 1 and f 2 ; Ω (between the two arcs ACB and ADB) is the locus of all points that are attainable for this mixture and f 1 = 0.7 , f 2 = 0.3 .

Equations (53)

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E d ( r ) = 3 r ^ ( r ^ · d ) d r 3 ,
E d ( r ) = 3 r ^ ( r ^ · d ) d r 3 4 π 3 δ ( r ) d ,
α = a 3 ϵ 1 ϵ + 2 .
E int = 3 ϵ + 2 E ext .
E dep = ϵ 1 ϵ + 2 E ext = 1 a 3 d .
r < a E dep d 3 r = 4 π 3 d .
r < R ( E E ext ) d 3 r = r < R E d d 3 r = 4 π 3 d .
G ( r , r ) = r r 1 | r r | .
d tot = N d = N α E ext .
d tot = V P = V ϵ 1 4 π E .
E = E ext + n E n ( r ) , r V .
E n ( r ) = 1 V V E d ( r r n ) d 3 r 4 π 3 d V ,
E = E ext + n E n = E ext N 4 π 3 d V = ( 1 4 π 3 α v ) E ext .
α v = ϵ 1 4 π ( 1 4 π 3 α v ) .
ϵ = 1 + 4 π ( α / v ) 1 ( 4 π / 3 ) ( α / v ) = 1 + ( 8 π / 3 ) ( α / v ) 1 ( 4 π / 3 ) ( α / v ) .
α v = 3 4 π ϵ 1 ϵ + 2 .
ϵ MG = 1 + 2 f ϵ 1 ϵ + 2 1 f ϵ 1 ϵ + 2 = 1 + 1 + 2 f 3 ( ϵ 1 ) 1 + 1 f 3 ( ϵ 1 ) .
ϵ MG = ϵ h 1 + 2 f ϵ i ϵ h ϵ i + 2 ϵ h 1 f ϵ i ϵ h ϵ i + 2 ϵ h = ϵ h ϵ h + 1 + 2 f 3 ( ϵ i ϵ h ) ϵ h + 1 f 3 ( ϵ i ϵ h ) .
E d ( r ) = 1 ϵ h [ 3 r ^ ( r ^ · d ) d r 3 4 π 3 δ ( r ) d ] .
P h ( r ) = ϵ h 1 4 π E ( r ) , P i ( r ) = ϵ ( r ) ϵ h 4 π E ( r ) .
· ϵ h E ( r ) = · [ ϵ ( r ) ϵ h ] E ( r ) = 4 π ρ i ( r ) ,
α = a 3 ϵ h ϵ i ϵ h ϵ i + 2 ϵ h [ compare to ( 3 ) ] ,
E dep = ϵ i ϵ h ϵ i + 2 ϵ h E ext = d ϵ h a 3 [ compare to ( 5 ) ] .
r < R ( E E ext ) d 3 r = r < R E d d 3 r = 4 π 3 ϵ h d .
E = ( 1 4 π 3 ϵ h α v ) E ext ,
ϵ MG = ϵ h + 4 π ( α / v ) 1 ( 4 π / 3 ϵ h ) ( α / v ) .
ϵ MG ϵ h ϵ MG + 2 ϵ h = f ϵ n ϵ h ϵ n + 2 ϵ h ,
ϵ MG ϵ h ϵ MG + 2 ϵ h = n = 1 N f n ϵ n ϵ h ϵ n + 2 ϵ h ,
ϵ n ϵ n and f n f m , 1 n , m N .
ϵ n ϵ h and f n f h , 1 n N .
n = 1 N f n ϵ n ϵ BG ϵ n + 2 ϵ BG = 0 , where n = 1 N f n = 1 .
ϵ BG = b ± 8 ϵ 1 ϵ 2 + b 2 4 , b = ( 2 f 1 f 2 ) ϵ 1 + ( 2 f 2 f 1 ) ϵ 2 ,
ϵ MG ϵ h = 1 + 3 ϵ i ϵ h ϵ i + 2 ϵ h f + 3 ( ϵ i ϵ h ) 2 ( ϵ i + 2 ϵ h ) 2 f 2 + ,
ϵ BG ϵ h = 1 + 3 ϵ i ϵ h ϵ i + 2 ϵ h f + 9 ϵ i ( ϵ i ϵ h ) 2 ( ϵ i + 2 ϵ h ) 3 f 2 + .
α p = a x a y a z 3 ϵ h ( ϵ i ϵ h ) ϵ h + ν p ( ϵ i ϵ h ) , p = x , y , z .
( ϵ MG ) p ϵ h ϵ h + ν p [ ( ϵ MG ) p ϵ h ] = f ϵ i ϵ h ϵ h + ν p ( ϵ i ϵ h ) .
( ϵ MG ) p = ϵ h 1 + ( 1 ν p ) f ϵ i ϵ h ϵ h + ν p ( ϵ i ϵ h ) 1 ν p f ϵ i ϵ h ϵ h + ν p ( ϵ i ϵ h ) = ϵ h ϵ h + [ ν p ( 1 f ) + f ] ( ϵ i ϵ h ) ϵ h + ν p ( 1 f ) ( ϵ i ϵ h ) .
E = ( 1 ν ^ 4 π ϵ h α ^ v ) E ext .
E ( E E ext ) d 3 r = E E d d 3 r = ν ^ 4 π ϵ h d .
( ϵ ^ MG ) p = ϵ h 1 + 2 f 3 ϵ i ϵ h ϵ h + ν p ( ϵ i ϵ h ) 1 f 3 ϵ i ϵ h ϵ h + ν p ( ϵ i ϵ h ) = ϵ h ϵ h + ( ν p + 2 f / 3 ) ( ϵ i ϵ h ) ϵ h + ( ν p f / 3 ) ( ϵ i ϵ h ) .
ϵ ^ MG , ϵ ^ MG = ϵ h ( 1 + f ϵ i ϵ h ϵ h + ν p ( ϵ i ϵ h ) ) + O ( f 2 ) .
n = 1 N f n ϵ n ( ϵ BG ) p ( ϵ BG ) p + ν n p [ ϵ n ( ϵ BG ) p ] = 0 .
S ( r ) F ( r ) = S ( r ) F ( r ) ,
( ϵ eff ) x , y = ϵ ( z ) = n = 1 N f n ϵ n ,
( ϵ eff ) z = ϵ 1 ( z ) 1 = [ n = 1 N f n ϵ n ] 1 .
D x , y ( z ) = ϵ ( z ) E x , y ( z ) = ϵ ( z ) E x , y ( z ) .
E z ( z ) = ϵ 1 ( z ) D z ( z ) = ϵ 1 ( z ) D z ( z ) .
E p ( r ) = S p ( r ) [ β p + ( 1 β p ) ϵ ( r ) ] 1 ,
D p ( r ) = S p ( r ) ϵ ( r ) [ β p + ( 1 β p ) ϵ ( r ) ] 1 .
( ϵ eff ) p = ϵ ( r ) [ ϵ ( r ) + β p / ( 1 β p ) ] 1 [ ϵ ( r ) + β p / ( 1 β p ) ] 1 .
( ϵ MG ) p = ϵ ( r ) [ ϵ ( r ) + ( 1 / ν p 1 ) ϵ h ] 1 [ ϵ ( r ) + ( 1 / ν p 1 ) ϵ h ] 1 .
β p = ( 1 ν p ) ϵ h ν p + ( 1 ν p ) ϵ h .
ϵ 1 1 ( ϵ eff ) p ϵ , if    ϵ n > 0 .

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