Abstract

In the second part of this tutorial, we consider several advanced topics related to the Maxwell Garnett approximation.

© 2016 Optical Society of America

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References

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  1. V. A. Markel, “Introduction to the Maxwell Garnett approximation: tutorial,” J. Opt. Soc. Am. A 33, 1244–1256 (2016).
    [Crossref]
  2. A. Chipouline, C. Simovski, and S. Tretyakov, “Basics of averaging of the Maxwell equations for bulk materials,” Metamaterials 6, 77–120 (2012).
    [Crossref]
  3. A. P. Vinogradov and A. M. Merzlikin, “Comment on ‘Basics of averaging of the Maxwell equations for bulk materials’,” Metamaterials 6, 121–125 (2012).
    [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. A notable exception is the Aharonov–Bohm effect, which cannot be understood within the classical electromagnetic theory.
  6. V. Agranovich and V. Ginzburg, Spatial Dispersion in Crystal Optics and the Theory of Excitons (Wiley-Interscience, 1966).
  7. L. D. Landau and L. P. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1984), Sect. 6.
  8. H. G. Tompkins and W. A. McGahan, eds., Spectroscopic Ellipsometry and Reflectometry: A User’s Guide (Wiley, 1999).
  9. R. D. King-Smith and D. Vanderbilt, “Theory of polarization of crystalline solids,” Phys. Rev. B 47, 1651–1654 (1993).
    [Crossref]
  10. R. Resta, “Macroscopic electric polarization as a geometric quantum phase,” Europhys. Lett. 22, 133–138 (1993).
    [Crossref]
  11. D. Vanderbilt and R. D. King-Smith, “Electric polarization as a bulk quantity and its relation to surface charge,” Phys. Rev. B 48, 4442–4455 (1993).
    [Crossref]
  12. R. Resta, “Macroscopic polarization in crystalline dielectrics: the geometrical phase approach,” Rev. Mod. Phys. 66, 899–915 (1994).
    [Crossref]
  13. N. A. Spaldin, “A beginner guide to the modern theory of polarization,” J. Sol. St. Chem. 195, 2–10 (2012).
    [Crossref]
  14. R. Resta, “Electrical polarization and orbital magnetization: the modern theories,” J. Phys. Condens. Matter 22, 123201 (2010).
    [Crossref]
  15. R. Bianco and R. Resta, “Orbital magnetization as a local property,” Phys. Rev. Lett. 110, 087202 (2013).
    [Crossref]
  16. G. D. Mahan and G. Obermair, “Polaritons at surfaces,” Phys. Rev. 183, 834–841 (1969).
    [Crossref]
  17. J. E. Sipe and J. Van Kranendonk, “Macroscopic electromagnetic theory of resonant dielectrics,” Phys. Rev. A 9, 1806–1822 (1974).
    [Crossref]
  18. W. Lamb, D. M. Wood, and N. W. Ashcroft, “Long-wavelength electromagnetic propagation in heterogeneous media,” Phys. Rev. B 21, 2248–2266 (1980).
    [Crossref]
  19. 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]
  20. P. A. Belov and C. R. Simovski, “Homogenization of electromagnetic crystals formed by uniaxial resonant scatterers,” Phys. Rev. E 72, 026615 (2005).
    [Crossref]
  21. F. J. G. Abajo, “Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79, 1267–1290 (2007).
    [Crossref]
  22. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
    [Crossref]
  23. V. A. Markel, “Scattering of light from two interacting spherical particles,” J. Mod. Opt. 39, 853–861 (1992).
    [Crossref]
  24. M. A. Yurkin, “Computational approaches for plasmonics,” in Handbook of Molecular Plasmonics (Pan Stanford, 2013), pp. 83–135.
  25. P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447–466 (1998).
    [Crossref]
  26. B. T. Draine, “The discrete dipole approximation for light scattering by irregular targets,” in Light Scattering by Nonspherical Particles (Academic, 2000), pp. 226–273.
  27. J. M. Borwein, M. L. Glasser, R. C. McPhedran, J. G. Wan, and I. J. Zucker, Lattice Sums Then and Now (Cambridge, 2013).
  28. V. M. Agranovich and Y. N. Gartstein, “Electrodynamics of metamaterials and the Landau–Lifshitz approach to the magnetic permeability,” Metamaterials 3, 1–9 (2009).
    [Crossref]
  29. V. A. Markel and I. Tsukerman, “Current-driven homogenization and effective medium parameters for finite samples,” Phys. Rev. B 88, 125131 (2013).
    [Crossref]
  30. V. A. Markel and J. C. Schotland, “Homogenization of Maxwell’s equations in periodic composites,” Phys. Rev. E 85, 066603 (2012).
    [Crossref]
  31. U. C. Hasar, G. Buldu, M. Bute, J. J. Barroso, T. Karacali, and M. Ertugrul, “Determination of constitutive parameters of homogeneous metamaterial slabs by a novel calibration-independent method,” AIP Adv. 4, 107116 (2014).
    [Crossref]
  32. T. D. Karamanos, S. D. Assimonis, A. I. Dimitriadis, and N. V. Kantartzis, “Effective parameter extraction of 3D metamaterial arrays via first-principle homogenization theory,” Photon. Nanostruct. 12, 291–297 (2014).
    [Crossref]
  33. N. C. J. Clausen, S. Arslanagic, and O. Breinbjerg, “Comparison of spatial harmonics in infinite and finite Bragg stacks for metamaterial homogenization,” Photon. Nanostruct. 12, 419–428 (2014).
    [Crossref]
  34. A. Ciattoni and C. Rizza, “Nonlocal homogenization theory in metamaterials: effective electromagnetic spatial dispersion and artificial chirality,” Phys. Rev. B 91, 184207 (2015).
    [Crossref]
  35. V. Sozio, A. Vallecchi, M. Albani, and F. Capolino, “Generalized Lorentz–Lorenz homogenization formulas for binary lattice metamaterials,” Phys. Rev. B 91, 205127 (2015).
    [Crossref]
  36. A. A. Shcherbakov and A. V. Tishchenko, “3D periodic dielectric composite homogenization based on the generalized source method,” J. Opt. 17, 065101 (2015).
    [Crossref]
  37. A. A. Krokhin, J. Arriaga, L. N. Gumen, and V. P. Drachev, “High-frequency homogenization for layered hyperbolic metamaterials,” Phys. Rev. B 93, 075418 (2016).
    [Crossref]
  38. M. Caleap and W. Drinkwater, “Metamaterials: supra-classical dynamic homogenization,” New J. Phys. 17, 123022 (2016).
    [Crossref]
  39. A. P. Vinogradov, Electrodynamics of Composite Materials (URSS, 2001) (in Russian).

2016 (3)

A. A. Krokhin, J. Arriaga, L. N. Gumen, and V. P. Drachev, “High-frequency homogenization for layered hyperbolic metamaterials,” Phys. Rev. B 93, 075418 (2016).
[Crossref]

M. Caleap and W. Drinkwater, “Metamaterials: supra-classical dynamic homogenization,” New J. Phys. 17, 123022 (2016).
[Crossref]

V. A. Markel, “Introduction to the Maxwell Garnett approximation: tutorial,” J. Opt. Soc. Am. A 33, 1244–1256 (2016).
[Crossref]

2015 (3)

A. Ciattoni and C. Rizza, “Nonlocal homogenization theory in metamaterials: effective electromagnetic spatial dispersion and artificial chirality,” Phys. Rev. B 91, 184207 (2015).
[Crossref]

V. Sozio, A. Vallecchi, M. Albani, and F. Capolino, “Generalized Lorentz–Lorenz homogenization formulas for binary lattice metamaterials,” Phys. Rev. B 91, 205127 (2015).
[Crossref]

A. A. Shcherbakov and A. V. Tishchenko, “3D periodic dielectric composite homogenization based on the generalized source method,” J. Opt. 17, 065101 (2015).
[Crossref]

2014 (3)

U. C. Hasar, G. Buldu, M. Bute, J. J. Barroso, T. Karacali, and M. Ertugrul, “Determination of constitutive parameters of homogeneous metamaterial slabs by a novel calibration-independent method,” AIP Adv. 4, 107116 (2014).
[Crossref]

T. D. Karamanos, S. D. Assimonis, A. I. Dimitriadis, and N. V. Kantartzis, “Effective parameter extraction of 3D metamaterial arrays via first-principle homogenization theory,” Photon. Nanostruct. 12, 291–297 (2014).
[Crossref]

N. C. J. Clausen, S. Arslanagic, and O. Breinbjerg, “Comparison of spatial harmonics in infinite and finite Bragg stacks for metamaterial homogenization,” Photon. Nanostruct. 12, 419–428 (2014).
[Crossref]

2013 (2)

V. A. Markel and I. Tsukerman, “Current-driven homogenization and effective medium parameters for finite samples,” Phys. Rev. B 88, 125131 (2013).
[Crossref]

R. Bianco and R. Resta, “Orbital magnetization as a local property,” Phys. Rev. Lett. 110, 087202 (2013).
[Crossref]

2012 (4)

N. A. Spaldin, “A beginner guide to the modern theory of polarization,” J. Sol. St. Chem. 195, 2–10 (2012).
[Crossref]

A. Chipouline, C. Simovski, and S. Tretyakov, “Basics of averaging of the Maxwell equations for bulk materials,” Metamaterials 6, 77–120 (2012).
[Crossref]

A. P. Vinogradov and A. M. Merzlikin, “Comment on ‘Basics of averaging of the Maxwell equations for bulk materials’,” Metamaterials 6, 121–125 (2012).
[Crossref]

V. A. Markel and J. C. Schotland, “Homogenization of Maxwell’s equations in periodic composites,” Phys. Rev. E 85, 066603 (2012).
[Crossref]

2010 (1)

R. Resta, “Electrical polarization and orbital magnetization: the modern theories,” J. Phys. Condens. Matter 22, 123201 (2010).
[Crossref]

2009 (1)

V. M. Agranovich and Y. N. Gartstein, “Electrodynamics of metamaterials and the Landau–Lifshitz approach to the magnetic permeability,” Metamaterials 3, 1–9 (2009).
[Crossref]

2007 (1)

F. J. G. Abajo, “Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79, 1267–1290 (2007).
[Crossref]

2005 (1)

P. A. Belov and C. R. Simovski, “Homogenization of electromagnetic crystals formed by uniaxial resonant scatterers,” Phys. Rev. E 72, 026615 (2005).
[Crossref]

1998 (1)

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447–466 (1998).
[Crossref]

1994 (1)

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

1993 (4)

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]

R. D. King-Smith and D. Vanderbilt, “Theory of polarization of crystalline solids,” Phys. Rev. B 47, 1651–1654 (1993).
[Crossref]

R. Resta, “Macroscopic electric polarization as a geometric quantum phase,” Europhys. Lett. 22, 133–138 (1993).
[Crossref]

D. Vanderbilt and R. D. King-Smith, “Electric polarization as a bulk quantity and its relation to surface charge,” Phys. Rev. B 48, 4442–4455 (1993).
[Crossref]

1992 (1)

V. A. Markel, “Scattering of light from two interacting spherical particles,” J. Mod. Opt. 39, 853–861 (1992).
[Crossref]

1988 (1)

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

1980 (1)

W. Lamb, D. M. Wood, and N. W. Ashcroft, “Long-wavelength electromagnetic propagation in heterogeneous media,” Phys. Rev. B 21, 2248–2266 (1980).
[Crossref]

1974 (1)

J. E. Sipe and J. Van Kranendonk, “Macroscopic electromagnetic theory of resonant dielectrics,” Phys. Rev. A 9, 1806–1822 (1974).
[Crossref]

1969 (1)

G. D. Mahan and G. Obermair, “Polaritons at surfaces,” Phys. Rev. 183, 834–841 (1969).
[Crossref]

Abajo, F. J. G.

F. J. G. Abajo, “Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79, 1267–1290 (2007).
[Crossref]

Agranovich, V.

V. Agranovich and V. Ginzburg, Spatial Dispersion in Crystal Optics and the Theory of Excitons (Wiley-Interscience, 1966).

Agranovich, V. M.

V. M. Agranovich and Y. N. Gartstein, “Electrodynamics of metamaterials and the Landau–Lifshitz approach to the magnetic permeability,” Metamaterials 3, 1–9 (2009).
[Crossref]

Albani, M.

V. Sozio, A. Vallecchi, M. Albani, and F. Capolino, “Generalized Lorentz–Lorenz homogenization formulas for binary lattice metamaterials,” Phys. Rev. B 91, 205127 (2015).
[Crossref]

Arriaga, J.

A. A. Krokhin, J. Arriaga, L. N. Gumen, and V. P. Drachev, “High-frequency homogenization for layered hyperbolic metamaterials,” Phys. Rev. B 93, 075418 (2016).
[Crossref]

Arslanagic, S.

N. C. J. Clausen, S. Arslanagic, and O. Breinbjerg, “Comparison of spatial harmonics in infinite and finite Bragg stacks for metamaterial homogenization,” Photon. Nanostruct. 12, 419–428 (2014).
[Crossref]

Ashcroft, N. W.

W. Lamb, D. M. Wood, and N. W. Ashcroft, “Long-wavelength electromagnetic propagation in heterogeneous media,” Phys. Rev. B 21, 2248–2266 (1980).
[Crossref]

Assimonis, S. D.

T. D. Karamanos, S. D. Assimonis, A. I. Dimitriadis, and N. V. Kantartzis, “Effective parameter extraction of 3D metamaterial arrays via first-principle homogenization theory,” Photon. Nanostruct. 12, 291–297 (2014).
[Crossref]

Barroso, J. J.

U. C. Hasar, G. Buldu, M. Bute, J. J. Barroso, T. Karacali, and M. Ertugrul, “Determination of constitutive parameters of homogeneous metamaterial slabs by a novel calibration-independent method,” AIP Adv. 4, 107116 (2014).
[Crossref]

Belov, P. A.

P. A. Belov and C. R. Simovski, “Homogenization of electromagnetic crystals formed by uniaxial resonant scatterers,” Phys. Rev. E 72, 026615 (2005).
[Crossref]

Bianco, R.

R. Bianco and R. Resta, “Orbital magnetization as a local property,” Phys. Rev. Lett. 110, 087202 (2013).
[Crossref]

Borwein, J. M.

J. M. Borwein, M. L. Glasser, R. C. McPhedran, J. G. Wan, and I. J. Zucker, Lattice Sums Then and Now (Cambridge, 2013).

Breinbjerg, O.

N. C. J. Clausen, S. Arslanagic, and O. Breinbjerg, “Comparison of spatial harmonics in infinite and finite Bragg stacks for metamaterial homogenization,” Photon. Nanostruct. 12, 419–428 (2014).
[Crossref]

Buldu, G.

U. C. Hasar, G. Buldu, M. Bute, J. J. Barroso, T. Karacali, and M. Ertugrul, “Determination of constitutive parameters of homogeneous metamaterial slabs by a novel calibration-independent method,” AIP Adv. 4, 107116 (2014).
[Crossref]

Bute, M.

U. C. Hasar, G. Buldu, M. Bute, J. J. Barroso, T. Karacali, and M. Ertugrul, “Determination of constitutive parameters of homogeneous metamaterial slabs by a novel calibration-independent method,” AIP Adv. 4, 107116 (2014).
[Crossref]

Caleap, M.

M. Caleap and W. Drinkwater, “Metamaterials: supra-classical dynamic homogenization,” New J. Phys. 17, 123022 (2016).
[Crossref]

Capolino, F.

V. Sozio, A. Vallecchi, M. Albani, and F. Capolino, “Generalized Lorentz–Lorenz homogenization formulas for binary lattice metamaterials,” Phys. Rev. B 91, 205127 (2015).
[Crossref]

Chipouline, A.

A. Chipouline, C. Simovski, and S. Tretyakov, “Basics of averaging of the Maxwell equations for bulk materials,” Metamaterials 6, 77–120 (2012).
[Crossref]

Ciattoni, A.

A. Ciattoni and C. Rizza, “Nonlocal homogenization theory in metamaterials: effective electromagnetic spatial dispersion and artificial chirality,” Phys. Rev. B 91, 184207 (2015).
[Crossref]

Clausen, N. C. J.

N. C. J. Clausen, S. Arslanagic, and O. Breinbjerg, “Comparison of spatial harmonics in infinite and finite Bragg stacks for metamaterial homogenization,” Photon. Nanostruct. 12, 419–428 (2014).
[Crossref]

de Vries, P.

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447–466 (1998).
[Crossref]

Dimitriadis, A. I.

T. D. Karamanos, S. D. Assimonis, A. I. Dimitriadis, and N. V. Kantartzis, “Effective parameter extraction of 3D metamaterial arrays via first-principle homogenization theory,” Photon. Nanostruct. 12, 291–297 (2014).
[Crossref]

Drachev, V. P.

A. A. Krokhin, J. Arriaga, L. N. Gumen, and V. P. Drachev, “High-frequency homogenization for layered hyperbolic metamaterials,” Phys. Rev. B 93, 075418 (2016).
[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]

B. T. Draine, “The discrete dipole approximation for light scattering by irregular targets,” in Light Scattering by Nonspherical Particles (Academic, 2000), pp. 226–273.

Drinkwater, W.

M. Caleap and W. Drinkwater, “Metamaterials: supra-classical dynamic homogenization,” New J. Phys. 17, 123022 (2016).
[Crossref]

Ertugrul, M.

U. C. Hasar, G. Buldu, M. Bute, J. J. Barroso, T. Karacali, and M. Ertugrul, “Determination of constitutive parameters of homogeneous metamaterial slabs by a novel calibration-independent method,” AIP Adv. 4, 107116 (2014).
[Crossref]

Gartstein, Y. N.

V. M. Agranovich and Y. N. Gartstein, “Electrodynamics of metamaterials and the Landau–Lifshitz approach to the magnetic permeability,” Metamaterials 3, 1–9 (2009).
[Crossref]

Ginzburg, V.

V. Agranovich and V. Ginzburg, Spatial Dispersion in Crystal Optics and the Theory of Excitons (Wiley-Interscience, 1966).

Glasser, M. L.

J. M. Borwein, M. L. Glasser, R. C. McPhedran, J. G. Wan, and I. J. Zucker, Lattice Sums Then and Now (Cambridge, 2013).

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]

Gumen, L. N.

A. A. Krokhin, J. Arriaga, L. N. Gumen, and V. P. Drachev, “High-frequency homogenization for layered hyperbolic metamaterials,” Phys. Rev. B 93, 075418 (2016).
[Crossref]

Hasar, U. C.

U. C. Hasar, G. Buldu, M. Bute, J. J. Barroso, T. Karacali, and M. Ertugrul, “Determination of constitutive parameters of homogeneous metamaterial slabs by a novel calibration-independent method,” AIP Adv. 4, 107116 (2014).
[Crossref]

Kantartzis, N. V.

T. D. Karamanos, S. D. Assimonis, A. I. Dimitriadis, and N. V. Kantartzis, “Effective parameter extraction of 3D metamaterial arrays via first-principle homogenization theory,” Photon. Nanostruct. 12, 291–297 (2014).
[Crossref]

Karacali, T.

U. C. Hasar, G. Buldu, M. Bute, J. J. Barroso, T. Karacali, and M. Ertugrul, “Determination of constitutive parameters of homogeneous metamaterial slabs by a novel calibration-independent method,” AIP Adv. 4, 107116 (2014).
[Crossref]

Karamanos, T. D.

T. D. Karamanos, S. D. Assimonis, A. I. Dimitriadis, and N. V. Kantartzis, “Effective parameter extraction of 3D metamaterial arrays via first-principle homogenization theory,” Photon. Nanostruct. 12, 291–297 (2014).
[Crossref]

King-Smith, R. D.

R. D. King-Smith and D. Vanderbilt, “Theory of polarization of crystalline solids,” Phys. Rev. B 47, 1651–1654 (1993).
[Crossref]

D. Vanderbilt and R. D. King-Smith, “Electric polarization as a bulk quantity and its relation to surface charge,” Phys. Rev. B 48, 4442–4455 (1993).
[Crossref]

Krokhin, A. A.

A. A. Krokhin, J. Arriaga, L. N. Gumen, and V. P. Drachev, “High-frequency homogenization for layered hyperbolic metamaterials,” Phys. Rev. B 93, 075418 (2016).
[Crossref]

Lagendijk, A.

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447–466 (1998).
[Crossref]

Lamb, W.

W. Lamb, D. M. Wood, and N. W. Ashcroft, “Long-wavelength electromagnetic propagation in heterogeneous media,” Phys. Rev. B 21, 2248–2266 (1980).
[Crossref]

Landau, L. D.

L. D. Landau and L. P. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1984), Sect. 6.

Lifshitz, L. P.

L. D. Landau and L. P. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1984), Sect. 6.

Mahan, G. D.

G. D. Mahan and G. Obermair, “Polaritons at surfaces,” Phys. Rev. 183, 834–841 (1969).
[Crossref]

Markel, V. A.

V. A. Markel, “Introduction to the Maxwell Garnett approximation: tutorial,” J. Opt. Soc. Am. A 33, 1244–1256 (2016).
[Crossref]

V. A. Markel and I. Tsukerman, “Current-driven homogenization and effective medium parameters for finite samples,” Phys. Rev. B 88, 125131 (2013).
[Crossref]

V. A. Markel and J. C. Schotland, “Homogenization of Maxwell’s equations in periodic composites,” Phys. Rev. E 85, 066603 (2012).
[Crossref]

V. A. Markel, “Scattering of light from two interacting spherical particles,” J. Mod. Opt. 39, 853–861 (1992).
[Crossref]

McPhedran, R. C.

J. M. Borwein, M. L. Glasser, R. C. McPhedran, J. G. Wan, and I. J. Zucker, Lattice Sums Then and Now (Cambridge, 2013).

Merzlikin, A. M.

A. P. Vinogradov and A. M. Merzlikin, “Comment on ‘Basics of averaging of the Maxwell equations for bulk materials’,” Metamaterials 6, 121–125 (2012).
[Crossref]

Obermair, G.

G. D. Mahan and G. Obermair, “Polaritons at surfaces,” Phys. Rev. 183, 834–841 (1969).
[Crossref]

Resta, R.

R. Bianco and R. Resta, “Orbital magnetization as a local property,” Phys. Rev. Lett. 110, 087202 (2013).
[Crossref]

R. Resta, “Electrical polarization and orbital magnetization: the modern theories,” J. Phys. Condens. Matter 22, 123201 (2010).
[Crossref]

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

R. Resta, “Macroscopic electric polarization as a geometric quantum phase,” Europhys. Lett. 22, 133–138 (1993).
[Crossref]

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

Rizza, C.

A. Ciattoni and C. Rizza, “Nonlocal homogenization theory in metamaterials: effective electromagnetic spatial dispersion and artificial chirality,” Phys. Rev. B 91, 184207 (2015).
[Crossref]

Schotland, J. C.

V. A. Markel and J. C. Schotland, “Homogenization of Maxwell’s equations in periodic composites,” Phys. Rev. E 85, 066603 (2012).
[Crossref]

Shcherbakov, A. A.

A. A. Shcherbakov and A. V. Tishchenko, “3D periodic dielectric composite homogenization based on the generalized source method,” J. Opt. 17, 065101 (2015).
[Crossref]

Simovski, C.

A. Chipouline, C. Simovski, and S. Tretyakov, “Basics of averaging of the Maxwell equations for bulk materials,” Metamaterials 6, 77–120 (2012).
[Crossref]

Simovski, C. R.

P. A. Belov and C. R. Simovski, “Homogenization of electromagnetic crystals formed by uniaxial resonant scatterers,” Phys. Rev. E 72, 026615 (2005).
[Crossref]

Sipe, J. E.

J. E. Sipe and J. Van Kranendonk, “Macroscopic electromagnetic theory of resonant dielectrics,” Phys. Rev. A 9, 1806–1822 (1974).
[Crossref]

Sozio, V.

V. Sozio, A. Vallecchi, M. Albani, and F. Capolino, “Generalized Lorentz–Lorenz homogenization formulas for binary lattice metamaterials,” Phys. Rev. B 91, 205127 (2015).
[Crossref]

Spaldin, N. A.

N. A. Spaldin, “A beginner guide to the modern theory of polarization,” J. Sol. St. Chem. 195, 2–10 (2012).
[Crossref]

Tishchenko, A. V.

A. A. Shcherbakov and A. V. Tishchenko, “3D periodic dielectric composite homogenization based on the generalized source method,” J. Opt. 17, 065101 (2015).
[Crossref]

Tretyakov, S.

A. Chipouline, C. Simovski, and S. Tretyakov, “Basics of averaging of the Maxwell equations for bulk materials,” Metamaterials 6, 77–120 (2012).
[Crossref]

Tsukerman, I.

V. A. Markel and I. Tsukerman, “Current-driven homogenization and effective medium parameters for finite samples,” Phys. Rev. B 88, 125131 (2013).
[Crossref]

Vallecchi, A.

V. Sozio, A. Vallecchi, M. Albani, and F. Capolino, “Generalized Lorentz–Lorenz homogenization formulas for binary lattice metamaterials,” Phys. Rev. B 91, 205127 (2015).
[Crossref]

van Coevorden, D. V.

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447–466 (1998).
[Crossref]

Van Kranendonk, J.

J. E. Sipe and J. Van Kranendonk, “Macroscopic electromagnetic theory of resonant dielectrics,” Phys. Rev. A 9, 1806–1822 (1974).
[Crossref]

Vanderbilt, D.

D. Vanderbilt and R. D. King-Smith, “Electric polarization as a bulk quantity and its relation to surface charge,” Phys. Rev. B 48, 4442–4455 (1993).
[Crossref]

R. D. King-Smith and D. Vanderbilt, “Theory of polarization of crystalline solids,” Phys. Rev. B 47, 1651–1654 (1993).
[Crossref]

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

Vinogradov, A. P.

A. P. Vinogradov and A. M. Merzlikin, “Comment on ‘Basics of averaging of the Maxwell equations for bulk materials’,” Metamaterials 6, 121–125 (2012).
[Crossref]

A. P. Vinogradov, Electrodynamics of Composite Materials (URSS, 2001) (in Russian).

Wan, J. G.

J. M. Borwein, M. L. Glasser, R. C. McPhedran, J. G. Wan, and I. J. Zucker, Lattice Sums Then and Now (Cambridge, 2013).

Wood, D. M.

W. Lamb, D. M. Wood, and N. W. Ashcroft, “Long-wavelength electromagnetic propagation in heterogeneous media,” Phys. Rev. B 21, 2248–2266 (1980).
[Crossref]

Yurkin, M. A.

M. A. Yurkin, “Computational approaches for plasmonics,” in Handbook of Molecular Plasmonics (Pan Stanford, 2013), pp. 83–135.

Zucker, I. J.

J. M. Borwein, M. L. Glasser, R. C. McPhedran, J. G. Wan, and I. J. Zucker, Lattice Sums Then and Now (Cambridge, 2013).

AIP Adv. (1)

U. C. Hasar, G. Buldu, M. Bute, J. J. Barroso, T. Karacali, and M. Ertugrul, “Determination of constitutive parameters of homogeneous metamaterial slabs by a novel calibration-independent method,” AIP Adv. 4, 107116 (2014).
[Crossref]

Astrophys. J. (2)

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]

Europhys. Lett. (1)

R. Resta, “Macroscopic electric polarization as a geometric quantum phase,” Europhys. Lett. 22, 133–138 (1993).
[Crossref]

J. Mod. Opt. (1)

V. A. Markel, “Scattering of light from two interacting spherical particles,” J. Mod. Opt. 39, 853–861 (1992).
[Crossref]

J. Opt. (1)

A. A. Shcherbakov and A. V. Tishchenko, “3D periodic dielectric composite homogenization based on the generalized source method,” J. Opt. 17, 065101 (2015).
[Crossref]

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

J. Phys. Condens. Matter (1)

R. Resta, “Electrical polarization and orbital magnetization: the modern theories,” J. Phys. Condens. Matter 22, 123201 (2010).
[Crossref]

J. Sol. St. Chem. (1)

N. A. Spaldin, “A beginner guide to the modern theory of polarization,” J. Sol. St. Chem. 195, 2–10 (2012).
[Crossref]

Metamaterials (3)

A. Chipouline, C. Simovski, and S. Tretyakov, “Basics of averaging of the Maxwell equations for bulk materials,” Metamaterials 6, 77–120 (2012).
[Crossref]

A. P. Vinogradov and A. M. Merzlikin, “Comment on ‘Basics of averaging of the Maxwell equations for bulk materials’,” Metamaterials 6, 121–125 (2012).
[Crossref]

V. M. Agranovich and Y. N. Gartstein, “Electrodynamics of metamaterials and the Landau–Lifshitz approach to the magnetic permeability,” Metamaterials 3, 1–9 (2009).
[Crossref]

New J. Phys. (1)

M. Caleap and W. Drinkwater, “Metamaterials: supra-classical dynamic homogenization,” New J. Phys. 17, 123022 (2016).
[Crossref]

Photon. Nanostruct. (2)

T. D. Karamanos, S. D. Assimonis, A. I. Dimitriadis, and N. V. Kantartzis, “Effective parameter extraction of 3D metamaterial arrays via first-principle homogenization theory,” Photon. Nanostruct. 12, 291–297 (2014).
[Crossref]

N. C. J. Clausen, S. Arslanagic, and O. Breinbjerg, “Comparison of spatial harmonics in infinite and finite Bragg stacks for metamaterial homogenization,” Photon. Nanostruct. 12, 419–428 (2014).
[Crossref]

Phys. Rev. (1)

G. D. Mahan and G. Obermair, “Polaritons at surfaces,” Phys. Rev. 183, 834–841 (1969).
[Crossref]

Phys. Rev. A (1)

J. E. Sipe and J. Van Kranendonk, “Macroscopic electromagnetic theory of resonant dielectrics,” Phys. Rev. A 9, 1806–1822 (1974).
[Crossref]

Phys. Rev. B (7)

W. Lamb, D. M. Wood, and N. W. Ashcroft, “Long-wavelength electromagnetic propagation in heterogeneous media,” Phys. Rev. B 21, 2248–2266 (1980).
[Crossref]

R. D. King-Smith and D. Vanderbilt, “Theory of polarization of crystalline solids,” Phys. Rev. B 47, 1651–1654 (1993).
[Crossref]

D. Vanderbilt and R. D. King-Smith, “Electric polarization as a bulk quantity and its relation to surface charge,” Phys. Rev. B 48, 4442–4455 (1993).
[Crossref]

A. Ciattoni and C. Rizza, “Nonlocal homogenization theory in metamaterials: effective electromagnetic spatial dispersion and artificial chirality,” Phys. Rev. B 91, 184207 (2015).
[Crossref]

V. Sozio, A. Vallecchi, M. Albani, and F. Capolino, “Generalized Lorentz–Lorenz homogenization formulas for binary lattice metamaterials,” Phys. Rev. B 91, 205127 (2015).
[Crossref]

A. A. Krokhin, J. Arriaga, L. N. Gumen, and V. P. Drachev, “High-frequency homogenization for layered hyperbolic metamaterials,” Phys. Rev. B 93, 075418 (2016).
[Crossref]

V. A. Markel and I. Tsukerman, “Current-driven homogenization and effective medium parameters for finite samples,” Phys. Rev. B 88, 125131 (2013).
[Crossref]

Phys. Rev. E (2)

V. A. Markel and J. C. Schotland, “Homogenization of Maxwell’s equations in periodic composites,” Phys. Rev. E 85, 066603 (2012).
[Crossref]

P. A. Belov and C. R. Simovski, “Homogenization of electromagnetic crystals formed by uniaxial resonant scatterers,” Phys. Rev. E 72, 026615 (2005).
[Crossref]

Phys. Rev. Lett. (1)

R. Bianco and R. Resta, “Orbital magnetization as a local property,” Phys. Rev. Lett. 110, 087202 (2013).
[Crossref]

Rev. Mod. Phys. (3)

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

F. J. G. Abajo, “Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79, 1267–1290 (2007).
[Crossref]

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70, 447–466 (1998).
[Crossref]

Other (9)

B. T. Draine, “The discrete dipole approximation for light scattering by irregular targets,” in Light Scattering by Nonspherical Particles (Academic, 2000), pp. 226–273.

J. M. Borwein, M. L. Glasser, R. C. McPhedran, J. G. Wan, and I. J. Zucker, Lattice Sums Then and Now (Cambridge, 2013).

M. A. Yurkin, “Computational approaches for plasmonics,” in Handbook of Molecular Plasmonics (Pan Stanford, 2013), pp. 83–135.

A. P. Vinogradov, Electrodynamics of Composite Materials (URSS, 2001) (in Russian).

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

A notable exception is the Aharonov–Bohm effect, which cannot be understood within the classical electromagnetic theory.

V. Agranovich and V. Ginzburg, Spatial Dispersion in Crystal Optics and the Theory of Excitons (Wiley-Interscience, 1966).

L. D. Landau and L. P. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1984), Sect. 6.

H. G. Tompkins and W. A. McGahan, eds., Spectroscopic Ellipsometry and Reflectometry: A User’s Guide (Wiley, 1999).

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Figures (5)

Fig. 1.
Fig. 1. One-dimensional illustration of the ambiguity of defining the dipole moment of an element of volume. The idea of the figure is borrowed from Ref. [13].
Fig. 2.
Fig. 2. Illustration of the charge density given by Eq. (18) for L=10h. Here Lorentzians of the width γ=0.01h are plotted instead of true delta functions.
Fig. 3.
Fig. 3. A schematic illustration of a two-component photonic crystal. Identical inclusions are periodically arranged in a host medium. For simplicity, we assume that the host medium is vacuum. The pattern is three-dimensional.
Fig. 4.
Fig. 4. Illustration of the anisotropic inclusions and polarization labels used in the simulations shown in Fig. 5(c) below.
Fig. 5.
Fig. 5. Numerical and analytical results for the effective permittivity of different composites. The inclusions are (a) spherical, (b) cubic, and (c) parallelepipedal. The shapes of the parallelepipeds and the corresponding polarization labels (directions of the fundamental Bloch harmonic P0), as shown in Fig. 4. The volume fraction of the inclusions is varied from zero to the maximum value fmax allowed by the geometry. In particular, (a) fmax=π/6, (b) fmax=1, and (c) fmax=0.5, Inclusion A, and fmax=0.25, Inclusion B. Results are shown for both cases when εh=ε1, εi=ε2 (dielectric host) and εh=ε2, εi=ε1 (metal host). The labeling of various curves and data sets is of the form X or XY or XvY. Here X takes the values X=MG (isotropic Maxwell Garnett mixing formula), X=BG (isotropic Bruggeman’s mixing formula), X=S [Eq. (80) for spherical inclusions], X=C [Eq. (80) for cubic inclusions], and X=A,B [Eq. (80) for parallelepipedal inclusions A or B shown in Fig. 4]. The polarization label takes two values v=s or v=p and is applicable only to X=A,B (see Fig. 4). Finally, the label Y takes two values: Y=DH for the dielectric host and Y=MH for the metal host. LWB and CWB are the linear and circular Wiener bounds, respectitvely. Note that some data points shown as circles were computed for the values of f slightly different from the exact limiting values (for example, f=0.01 instead of f=0) to avoid complete overlap and obscuration of two or more data points.

Equations (88)

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×B=1cEt+4πcJ,×E=1cBt,
ρt+·J=0,
F=qE+qcv×B.
J(r,t)=Jext(r,t)+Jind(r,t).
f(r)=1DDf(r+s)d3s,
Jind=Pt+c×M,
P(r,t)=0fe(r,τ)E(r,tτ)dτ,
M(r,t)=0fm(r,τ)B(r,tτ)dτ.
ρind(r,t)=·P(r,t).
×H=1cDt+4πcJext,×E=1cBt,
D=E+4πP,H=B4πM.
Jext(r,t)=Re[Jext(r)eiωt],E(r,t)=Re[E(r)eiωt],
D(r)=ε(r)E(r),B(r)=μ(r)H(r),
ε(r)=1+4π0χe(r,τ)exp(iωτ)dτ,
μ1(r)=1+4π0χm(r,τ)exp(iωτ)dτ.
×H(r)=ikε(r)E(r)+4πcJext(r),
×E(r)=ikμ(r)H(r),
k=ω/c+i0.
d(t)Drρind(r,t)d3r=Dr[·P(r,t)]d3r=DP(r,t)d3r+Sr[P(r,t)·dS].
dtot(t)=VP(r,t)d3r.
VP(r,t)d3r=Vrϱ(r,t)d3r=dtot.
ϱ(z)=ϱ0{cos(2πz+Δh)+h2πsin(2πΔh)[δ(z)δ(zL)]},0zL.
Pz=ϱ0h2πsin(2πΔh)Δ/h0ϱ0Δ.
m=12cDr×Jindd3r,
m1=12cDr×Ptd3r,m2=12D(r××M)d3r.
m2=12[S(r·M)dSDMd3rD(r·)Md3r].
D(r·)Md3r=SM(r·dS)3DMd3r.
m2=DMd3r12Sr×M×dS.
mtot=VMd3r+12cVr×Ptd3r.
mtotmtot+12ca×tVPd3r=mtot+12ca×dtott.
Im(1/α)2k3/3.
1/α=1/α˜2ik3/3,
[(××)k2]E(r)=4πk2P(r)+4πikcJext(r).
E(r)=Eext(r)+G^(r,r)P(r)d3r,
[(××)k2]Eext(r)=4πikcJext(r),
[(××)k2]G^(r,r)=4πk2I^δ(rr),
Eext(r)=1iωG^(r,r)Jext(r)d3r.
Escatt(r)=G^(r,r)P(r)d3r.
Escatt(r)G^(r,r0)VP(r)d3r=G^(r,r0)d,
G^(r,r)=F^(rr)=K^(p)eip·(rr)d3p(2π)3.
[(p×p×)+k2]K^(p)=4πk2I^.
K^(p)=4πk2I^ppp2k2.
G^(r,r)=(k2I^rr)exp(ik|rr|)|rr|,
F^(r)=(k2I^+)exp(ikr)r.
F^(r)=4π3δ(r)+F^R(r),
|r|<aF^R(r)d3ra00.
F^R(r)=[(k2r+ikr21r3)I^+(k2r3ikr2+3r3)rrr2]eikr.
ReF^R(r)=(1r3+k22r)I^+(3r3+k22r)rrr2+O(r),
ImF^R(r)=(2  k332  k5r215)I^+k5r215rrr2+O(r4).
|r|<aF^R(r)d3r=8π3I^[(1ika)eika1]ka04π3I^[(ka)2+i2(ka)33+].
K^(p)=(4π/3)I^+K^R(p),
K^R(p)=4π3(2  k2+p2)I^3ppp2k2.
dn=α[Eext(rn)+mnG^R(rn,rm)dm].
1αd=S^(q)d,
S^(q)=rn0F^R(rn)eiq·rn
S^(q)=d3p(2π)3K^R(p)rn0ei(pq)·rn.
S^(q)=1h3nK^R(q+gn)d3p(2π)3K^R(p),
neip·rn=(2πh)3nδ(pgn),gn=2πh(nx,ny,nz).
d3p(2π)3K^R(p)=F^R(0)34πa3|r|<aF^R(r)d3r(k2a+i2a33)I^,
(1α˜+k2a)d=1h3nKR(q+gn)d.
1αLLd=1h3K^R(q)d+1h3gn0K^R(q+gn)d.
K^R(q+gn)kh,|qp|h04π3Q^n,
Q^n=I^3gngngn2,gn0.
gn0Q^n=limLgnB(L)Q^n,
1αLLd=4π3  h3(2  k2+q2)I^3qqq2k2d.
vαLL=4π3εeff+2εeff1,
εeff1εeff+2=fε1ε+2,
εeffεhεeff+2εh=fεiεhεi+2εh.
E(r)=4π3P(r)+F^R(rr)P(r)d3r.
P(r)=3χ4πmrVmF^R(rr)P(r)d3r,rVn,
χ=ε1ε+2
P˜(R)=3χ4πRV0W^(R,R)P˜(R)d3R,RV0,
W^(R,R)=mF^R(rnrm+RR)eiq·(rnrm+RR).
W^(R,R)=1h3nK^R(q+gn)eign·(RR).
P˜(R)=nPneign·R.
Pn=f3χ4πK^R(q+gn)mM(gngm)Pm,
M(g)=1VRV0eig·Rd3R.
P0=f3χ4πK^R(q)[P0+n0M(gn)Pn],
Pn=fχQ^n[M(gn)P0+m0M(gngm)Pm],n0.
n0M(gn)Pn=Σ^P0,
Σ^=n0M(gn)T^nM(gn).
[I^fχ(2  k2+q2)I^3qqq2k2(I^+Σ^)]P0=0,
1fχ2  k2+q2q2k2(1+Σ)=0.
εeff1εeff+2=fε1ε+2(1+Σ).
εeffεhεeff+2εh=fεiεhεi+2εh(1+Σ).
(εeff)pεh(εeff)p+2εh=fεiεhεi+2εh(1+Σp),p=x,y,z.
(εeff)p=εh1+2fεiεhεi+2εh(1+Σp)1fεiεhεi+2εh(1+Σp)=εhεh+1+2f(1+Σp)3(εiεh)εh+1f(1+Σp)3(εiεh).
νp=13(1Σp1+Σp1χ),where  χ=εiεhεi+2εh.

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