Abstract

A detailed review of the theory of effective permittivity for one- and two-dimensional periodic structures shows its limited validity for metal-dielectric structures in the visible and near infra-red if the feature dimensions are comparable with the metal skin depth. We propose a phenomenological correction to the static formulae using a realistic assumption for the electric field behavior inside the metal features. This approach allows us to obtain analytical expressions for the effective permittivity in the case when the electric field is not sufficiently homogeneous within the unit cell of the gratings. A comparison with the numerical results of the Fourier modal method demonstrates the validity of the analytical formulae. Additional study is made on the impedance approximation at the outer boundaries of the periodical structure in order to propose analytical formulae for the reflection coefficient that permits better correspondence with the numerical results. The link between the values of effective permittivity and permeability defined as the ratios between the averaged fields, and the metamaterial permittivity and permeability is discussed.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. J. C. Maxwell-Garnett, “Colors in metal glasses and in metallic films,” Philos. Trans. R. Soc. London Ser. A 203(359–371), 385– 420 (1904).
  2. I. Barzilay, M. L. Myers, L. B. Cooper, and G. N. Graser, “Mechanical and chemical retention of laboratory cured composite to metal surfaces,” J. Prosthet. Dent. 59(2), 131–137 (1988).
    [Crossref] [PubMed]
  3. P. Vukusic, J. R. Sambles, C. R. Lawrence, and R. J. Wootton, “Quantified interference and diffraction in single Morpho butterfly scales,” Proceedings: Biological Sciences, The Royal Society of London 266, 1403–1411 (1999).
  4. H. Tada, S. Mann, I. Miaoulis, and P. Wong, “Effects of a butterfly scale microstructure on the iridescent color observed at different angles,” Opt. Express 5(4), 87–92 (1999).
    [Crossref] [PubMed]
  5. S. Tian and H. Brill Robert, Ancient Glass Research along the Silk Road (World Scientific, 2009).
  6. G. W. Milton, The Theory of Composites (Cambridge University, 2002).
  7. G. W. Milton and K. Golden, “Representations for the Conductivity Functions of Multicomponent Composites,” Commun. Pure Appl. Math. 43(5), 647–671 (1990).
    [Crossref]
  8. R. Clausius, Die Mechanische Behandlung der Electricität (Vieweg + Teubner Verlag, 1879).
  9. O. F. Mossotti, “Discussione analitica sull’influenza che l’azione di un mezzo dielettrico ha sulla distribuzione dell’elettricità alla superficie di più corpi elettrici disseminati in esso,” Memorie di Mathematica e di Fisica della Società Italiana della Scienza Residente in Modena 24, 49–74 (1850).
  10. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68(2), 248–263 (1980).
    [Crossref]
  11. P. Lalanne and D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43(10), 2063–2085 (1996).
    [Crossref]
  12. N. A. Nicorovici and R. C. McPhedran, “Transport properties of arrays of elliptical cylinders,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(2), 1945–1957 (1996).
    [Crossref] [PubMed]
  13. F. L. Galeener, “Submicroscopic-Void Resonance: The Effect of Internal Roughness on Optical Absorption, Phys. Rev. Lett. 27, 421-423 (1971), “Erratum,” ibid, p.769 (1971).
  14. S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).
  15. P. Lalanne, “Effective medium theory applied to photonic crystals composed of cubic or square cylinders,” Appl. Opt. 35(27), 5369–5380 (1996).
    [Crossref] [PubMed]
  16. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14(10), 2758–2767 (1997).
    [Crossref]
  17. G. Gao, C. Torres-Verdin, and T. M. Habashy, “Analytical Techniques to evaluate the integrals of 3D and 2D spatial Dyadic Green’s functions,” PIERS 52, 47–80 (2005).
    [Crossref]
  18. D. E. Gray, ed., American Institute of Physics Handbook (McGraw-Hill, 1957), sec.6.
  19. S. R. Coriell and J. L. Jackson, “Bounds on transport coefficients of two-phase materials,” J. Appl. Phys. 39(10), 4733–4736 (1968).
    [Crossref]
  20. T. Weiss, G. Granet, N. A. Gippius, S. G. Tikhodeev, and H. Giessen, “Matched coordinates and adaptive spatial resolution in the Fourier modal method,” Opt. Express 17(10), 8051–8061 (2009).
    [Crossref] [PubMed]
  21. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta (Lond.) 28(3), 413–428 (1981).
    [Crossref]
  22. J. Y. Suratteau, M. Cadilhac, and R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).
    [Crossref]
  23. K. B. Dossou, C. G. Poulton, and L. C. Botten, “Effective impedance modeling of metamaterial structures,” J. Opt. Soc. Am. A 33(3), 361–372 (2016).
    [Crossref] [PubMed]
  24. D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).
    [Crossref]
  25. J. Yang, C. Sauvan, T. Paul, C. Rockstuhl, F. Lederer, and P. Lalanne, “Retrieving the effective parameters of metamaterials from the single interface scattering problem,” Appl. Phys. Lett. 97(6), 061102 (2010).
    [Crossref]

2016 (1)

2010 (1)

J. Yang, C. Sauvan, T. Paul, C. Rockstuhl, F. Lederer, and P. Lalanne, “Retrieving the effective parameters of metamaterials from the single interface scattering problem,” Appl. Phys. Lett. 97(6), 061102 (2010).
[Crossref]

2009 (1)

2005 (1)

G. Gao, C. Torres-Verdin, and T. M. Habashy, “Analytical Techniques to evaluate the integrals of 3D and 2D spatial Dyadic Green’s functions,” PIERS 52, 47–80 (2005).
[Crossref]

2002 (1)

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).
[Crossref]

1999 (2)

P. Vukusic, J. R. Sambles, C. R. Lawrence, and R. J. Wootton, “Quantified interference and diffraction in single Morpho butterfly scales,” Proceedings: Biological Sciences, The Royal Society of London 266, 1403–1411 (1999).

H. Tada, S. Mann, I. Miaoulis, and P. Wong, “Effects of a butterfly scale microstructure on the iridescent color observed at different angles,” Opt. Express 5(4), 87–92 (1999).
[Crossref] [PubMed]

1997 (1)

1996 (3)

P. Lalanne, “Effective medium theory applied to photonic crystals composed of cubic or square cylinders,” Appl. Opt. 35(27), 5369–5380 (1996).
[Crossref] [PubMed]

P. Lalanne and D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43(10), 2063–2085 (1996).
[Crossref]

N. A. Nicorovici and R. C. McPhedran, “Transport properties of arrays of elliptical cylinders,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(2), 1945–1957 (1996).
[Crossref] [PubMed]

1990 (1)

G. W. Milton and K. Golden, “Representations for the Conductivity Functions of Multicomponent Composites,” Commun. Pure Appl. Math. 43(5), 647–671 (1990).
[Crossref]

1988 (1)

I. Barzilay, M. L. Myers, L. B. Cooper, and G. N. Graser, “Mechanical and chemical retention of laboratory cured composite to metal surfaces,” J. Prosthet. Dent. 59(2), 131–137 (1988).
[Crossref] [PubMed]

1983 (1)

J. Y. Suratteau, M. Cadilhac, and R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).
[Crossref]

1981 (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta (Lond.) 28(3), 413–428 (1981).
[Crossref]

1980 (1)

D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68(2), 248–263 (1980).
[Crossref]

1968 (1)

S. R. Coriell and J. L. Jackson, “Bounds on transport coefficients of two-phase materials,” J. Appl. Phys. 39(10), 4733–4736 (1968).
[Crossref]

1956 (1)

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

1904 (1)

J. C. Maxwell-Garnett, “Colors in metal glasses and in metallic films,” Philos. Trans. R. Soc. London Ser. A 203(359–371), 385– 420 (1904).

1850 (1)

O. F. Mossotti, “Discussione analitica sull’influenza che l’azione di un mezzo dielettrico ha sulla distribuzione dell’elettricità alla superficie di più corpi elettrici disseminati in esso,” Memorie di Mathematica e di Fisica della Società Italiana della Scienza Residente in Modena 24, 49–74 (1850).

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta (Lond.) 28(3), 413–428 (1981).
[Crossref]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta (Lond.) 28(3), 413–428 (1981).
[Crossref]

Barzilay, I.

I. Barzilay, M. L. Myers, L. B. Cooper, and G. N. Graser, “Mechanical and chemical retention of laboratory cured composite to metal surfaces,” J. Prosthet. Dent. 59(2), 131–137 (1988).
[Crossref] [PubMed]

Botten, L. C.

K. B. Dossou, C. G. Poulton, and L. C. Botten, “Effective impedance modeling of metamaterial structures,” J. Opt. Soc. Am. A 33(3), 361–372 (2016).
[Crossref] [PubMed]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta (Lond.) 28(3), 413–428 (1981).
[Crossref]

Cadilhac, M.

J. Y. Suratteau, M. Cadilhac, and R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).
[Crossref]

Cooper, L. B.

I. Barzilay, M. L. Myers, L. B. Cooper, and G. N. Graser, “Mechanical and chemical retention of laboratory cured composite to metal surfaces,” J. Prosthet. Dent. 59(2), 131–137 (1988).
[Crossref] [PubMed]

Coriell, S. R.

S. R. Coriell and J. L. Jackson, “Bounds on transport coefficients of two-phase materials,” J. Appl. Phys. 39(10), 4733–4736 (1968).
[Crossref]

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta (Lond.) 28(3), 413–428 (1981).
[Crossref]

Dossou, K. B.

Gao, G.

G. Gao, C. Torres-Verdin, and T. M. Habashy, “Analytical Techniques to evaluate the integrals of 3D and 2D spatial Dyadic Green’s functions,” PIERS 52, 47–80 (2005).
[Crossref]

Giessen, H.

Gippius, N. A.

Golden, K.

G. W. Milton and K. Golden, “Representations for the Conductivity Functions of Multicomponent Composites,” Commun. Pure Appl. Math. 43(5), 647–671 (1990).
[Crossref]

Granet, G.

Graser, G. N.

I. Barzilay, M. L. Myers, L. B. Cooper, and G. N. Graser, “Mechanical and chemical retention of laboratory cured composite to metal surfaces,” J. Prosthet. Dent. 59(2), 131–137 (1988).
[Crossref] [PubMed]

Habashy, T. M.

G. Gao, C. Torres-Verdin, and T. M. Habashy, “Analytical Techniques to evaluate the integrals of 3D and 2D spatial Dyadic Green’s functions,” PIERS 52, 47–80 (2005).
[Crossref]

Jackson, J. L.

S. R. Coriell and J. L. Jackson, “Bounds on transport coefficients of two-phase materials,” J. Appl. Phys. 39(10), 4733–4736 (1968).
[Crossref]

Lalanne, P.

J. Yang, C. Sauvan, T. Paul, C. Rockstuhl, F. Lederer, and P. Lalanne, “Retrieving the effective parameters of metamaterials from the single interface scattering problem,” Appl. Phys. Lett. 97(6), 061102 (2010).
[Crossref]

P. Lalanne, “Effective medium theory applied to photonic crystals composed of cubic or square cylinders,” Appl. Opt. 35(27), 5369–5380 (1996).
[Crossref] [PubMed]

P. Lalanne and D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43(10), 2063–2085 (1996).
[Crossref]

Lawrence, C. R.

P. Vukusic, J. R. Sambles, C. R. Lawrence, and R. J. Wootton, “Quantified interference and diffraction in single Morpho butterfly scales,” Proceedings: Biological Sciences, The Royal Society of London 266, 1403–1411 (1999).

Lederer, F.

J. Yang, C. Sauvan, T. Paul, C. Rockstuhl, F. Lederer, and P. Lalanne, “Retrieving the effective parameters of metamaterials from the single interface scattering problem,” Appl. Phys. Lett. 97(6), 061102 (2010).
[Crossref]

Lemercier-Lalanne, D.

P. Lalanne and D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43(10), 2063–2085 (1996).
[Crossref]

Li, L.

Mann, S.

Markoš, P.

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).
[Crossref]

Maxwell-Garnett, J. C.

J. C. Maxwell-Garnett, “Colors in metal glasses and in metallic films,” Philos. Trans. R. Soc. London Ser. A 203(359–371), 385– 420 (1904).

McPhedran, R. C.

N. A. Nicorovici and R. C. McPhedran, “Transport properties of arrays of elliptical cylinders,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(2), 1945–1957 (1996).
[Crossref] [PubMed]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta (Lond.) 28(3), 413–428 (1981).
[Crossref]

Miaoulis, I.

Milton, G. W.

G. W. Milton and K. Golden, “Representations for the Conductivity Functions of Multicomponent Composites,” Commun. Pure Appl. Math. 43(5), 647–671 (1990).
[Crossref]

Mossotti, O. F.

O. F. Mossotti, “Discussione analitica sull’influenza che l’azione di un mezzo dielettrico ha sulla distribuzione dell’elettricità alla superficie di più corpi elettrici disseminati in esso,” Memorie di Mathematica e di Fisica della Società Italiana della Scienza Residente in Modena 24, 49–74 (1850).

Myers, M. L.

I. Barzilay, M. L. Myers, L. B. Cooper, and G. N. Graser, “Mechanical and chemical retention of laboratory cured composite to metal surfaces,” J. Prosthet. Dent. 59(2), 131–137 (1988).
[Crossref] [PubMed]

Nicorovici, N. A.

N. A. Nicorovici and R. C. McPhedran, “Transport properties of arrays of elliptical cylinders,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(2), 1945–1957 (1996).
[Crossref] [PubMed]

Paul, T.

J. Yang, C. Sauvan, T. Paul, C. Rockstuhl, F. Lederer, and P. Lalanne, “Retrieving the effective parameters of metamaterials from the single interface scattering problem,” Appl. Phys. Lett. 97(6), 061102 (2010).
[Crossref]

Petit, R.

J. Y. Suratteau, M. Cadilhac, and R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).
[Crossref]

Poulton, C. G.

Rockstuhl, C.

J. Yang, C. Sauvan, T. Paul, C. Rockstuhl, F. Lederer, and P. Lalanne, “Retrieving the effective parameters of metamaterials from the single interface scattering problem,” Appl. Phys. Lett. 97(6), 061102 (2010).
[Crossref]

Rytov, S. M.

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

Sambles, J. R.

P. Vukusic, J. R. Sambles, C. R. Lawrence, and R. J. Wootton, “Quantified interference and diffraction in single Morpho butterfly scales,” Proceedings: Biological Sciences, The Royal Society of London 266, 1403–1411 (1999).

Sauvan, C.

J. Yang, C. Sauvan, T. Paul, C. Rockstuhl, F. Lederer, and P. Lalanne, “Retrieving the effective parameters of metamaterials from the single interface scattering problem,” Appl. Phys. Lett. 97(6), 061102 (2010).
[Crossref]

Schultz, S.

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).
[Crossref]

Smith, D. R.

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).
[Crossref]

Soukoulis, C. M.

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).
[Crossref]

Suratteau, J. Y.

J. Y. Suratteau, M. Cadilhac, and R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).
[Crossref]

Tada, H.

Tikhodeev, S. G.

Torres-Verdin, C.

G. Gao, C. Torres-Verdin, and T. M. Habashy, “Analytical Techniques to evaluate the integrals of 3D and 2D spatial Dyadic Green’s functions,” PIERS 52, 47–80 (2005).
[Crossref]

Vukusic, P.

P. Vukusic, J. R. Sambles, C. R. Lawrence, and R. J. Wootton, “Quantified interference and diffraction in single Morpho butterfly scales,” Proceedings: Biological Sciences, The Royal Society of London 266, 1403–1411 (1999).

Weiss, T.

Wong, P.

Wootton, R. J.

P. Vukusic, J. R. Sambles, C. R. Lawrence, and R. J. Wootton, “Quantified interference and diffraction in single Morpho butterfly scales,” Proceedings: Biological Sciences, The Royal Society of London 266, 1403–1411 (1999).

Yaghjian, D.

D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68(2), 248–263 (1980).
[Crossref]

Yang, J.

J. Yang, C. Sauvan, T. Paul, C. Rockstuhl, F. Lederer, and P. Lalanne, “Retrieving the effective parameters of metamaterials from the single interface scattering problem,” Appl. Phys. Lett. 97(6), 061102 (2010).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

J. Yang, C. Sauvan, T. Paul, C. Rockstuhl, F. Lederer, and P. Lalanne, “Retrieving the effective parameters of metamaterials from the single interface scattering problem,” Appl. Phys. Lett. 97(6), 061102 (2010).
[Crossref]

Commun. Pure Appl. Math. (1)

G. W. Milton and K. Golden, “Representations for the Conductivity Functions of Multicomponent Composites,” Commun. Pure Appl. Math. 43(5), 647–671 (1990).
[Crossref]

J. Appl. Phys. (1)

S. R. Coriell and J. L. Jackson, “Bounds on transport coefficients of two-phase materials,” J. Appl. Phys. 39(10), 4733–4736 (1968).
[Crossref]

J. Mod. Opt. (1)

P. Lalanne and D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43(10), 2063–2085 (1996).
[Crossref]

J. Opt. (Paris) (1)

J. Y. Suratteau, M. Cadilhac, and R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Prosthet. Dent. (1)

I. Barzilay, M. L. Myers, L. B. Cooper, and G. N. Graser, “Mechanical and chemical retention of laboratory cured composite to metal surfaces,” J. Prosthet. Dent. 59(2), 131–137 (1988).
[Crossref] [PubMed]

Memorie di Mathematica e di Fisica della Società Italiana della Scienza Residente in Modena (1)

O. F. Mossotti, “Discussione analitica sull’influenza che l’azione di un mezzo dielettrico ha sulla distribuzione dell’elettricità alla superficie di più corpi elettrici disseminati in esso,” Memorie di Mathematica e di Fisica della Società Italiana della Scienza Residente in Modena 24, 49–74 (1850).

Opt. Acta (Lond.) (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta (Lond.) 28(3), 413–428 (1981).
[Crossref]

Opt. Express (2)

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

J. C. Maxwell-Garnett, “Colors in metal glasses and in metallic films,” Philos. Trans. R. Soc. London Ser. A 203(359–371), 385– 420 (1904).

Phys. Rev. B (1)

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).
[Crossref]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

N. A. Nicorovici and R. C. McPhedran, “Transport properties of arrays of elliptical cylinders,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(2), 1945–1957 (1996).
[Crossref] [PubMed]

PIERS (1)

G. Gao, C. Torres-Verdin, and T. M. Habashy, “Analytical Techniques to evaluate the integrals of 3D and 2D spatial Dyadic Green’s functions,” PIERS 52, 47–80 (2005).
[Crossref]

Proc. IEEE (1)

D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68(2), 248–263 (1980).
[Crossref]

Proceedings: Biological Sciences, The Royal Society of London (1)

P. Vukusic, J. R. Sambles, C. R. Lawrence, and R. J. Wootton, “Quantified interference and diffraction in single Morpho butterfly scales,” Proceedings: Biological Sciences, The Royal Society of London 266, 1403–1411 (1999).

Sov. Phys. JETP (1)

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

Other (5)

D. E. Gray, ed., American Institute of Physics Handbook (McGraw-Hill, 1957), sec.6.

F. L. Galeener, “Submicroscopic-Void Resonance: The Effect of Internal Roughness on Optical Absorption, Phys. Rev. Lett. 27, 421-423 (1971), “Erratum,” ibid, p.769 (1971).

R. Clausius, Die Mechanische Behandlung der Electricität (Vieweg + Teubner Verlag, 1879).

S. Tian and H. Brill Robert, Ancient Glass Research along the Silk Road (World Scientific, 2009).

G. W. Milton, The Theory of Composites (Cambridge University, 2002).

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Figures (16)

Fig. 1
Fig. 1 Schematic presentation of the elementary cell of a 1D (a) or 2D (b) and (c) periodical system with notations.
Fig. 2
Fig. 2 Spectral dependence of Re( ε eff,xx ) for a 1D lamellar grating (the medium 2 in Fig. 1(a) is silver. Medium 1 is air, d = 30 nm, w = 15 nm). Long (a) and short (b) wavelength regions. Numerical values (Fourier modal method) are presented with dots, harmonic mean from Eq. (16) in blue, Eq. (20) in red, and the correction by using Eq. (25) in black.
Fig. 3
Fig. 3 Same as in Fig. 2, but d = 300 nm and w = 150 nm.
Fig. 4
Fig. 4 Re( ε eff,yy ) for the case presented in Fig. 3 as a function of the filling factor f = w/d. λ = 1µm.
Fig. 5
Fig. 5 Same as in Fig. 4 but for TM polarization.
Fig. 6
Fig. 6 Spectral dependence of the real part of silver relative permittivity [18].
Fig. 7
Fig. 7 Spectral dependence of Re( ε eff,xx ) = Re( ε eff,yy ) for an array of silver circular cylinders in air, Fig. 1(c). Square cell with d = 3 nm and R = 0.75 nm. Numeric values presented with dots, static limit, Eq. (13) in red, lower and upper limits, Eq. (27) in blue and rose. Long (a) and short (b) wavelength domains.
Fig. 8
Fig. 8 Same as in Fig. 7, but with d = 300 nm and R = 75 nm. The black curve is made by using the correction in Eq. (29).
Fig. 9
Fig. 9 Same as in Fig. 8 but for shorter wavelengths.
Fig. 10
Fig. 10 The dependence of Re( ε eff,xx,yy ) on the filling factor f for an array of circular or square cylinders. Dots present the numerical results (red squares with d = 3 nm, black for squares, and green for circles with d = 300 nm). Red curve Eq. (13), lower and upper limits, Eq. (27) in blue and rose, and the correction from Eq. (29) in black.
Fig. 11
Fig. 11 Convergence of the relative error of the real part of εeff and of the reflectivity as a function of the number of Fourier harmonics [-N, N] for a lamellar diffraction grating made of silver with d = 0.3 µm, w = 0.15 µm suspended in air in normal incidence and TM polarization, λ = 1 µm, h = 0.25 µm.
Fig. 12
Fig. 12 Reflection by a 1D lamellar silver grating suspended in air in normal incidence and TM polarization at λ = 1 µm. (a) dependence on the grating thickness h. In red, d = 3 nm, w = 1.5 nm, with points, numerical results, and with a line, Eq. (35) using Eqs. (16) and (36). d = 300 nm and w = 150 nm, in black, numerical results, in blue, modal method using Eq. (42) for ns. (b) dependence of the first maximum of the reflectivity on the lamellae width w for d = 300 nm (n2 = 0.129 + i 6.83 [18]).
Fig. 13
Fig. 13 (a) Reflectivity of a 2D array of silver circular cylinders (Fig. 1(c)) in air as a function of the grating height h. Normal incidence, λ = 1 µm. Red points, numerical values with d = 3 nm, R = 1 nm, red curve, the static limit. Black curve, numerical results for d = 300 nm, R = 100 nm, blue line, correction from Eqs. (42), (48), and (49). (b) the dependence of the maximum of the reflectivity on the cylinder diameter for d = 300 nm.
Fig. 14
Fig. 14 Spectral dependence of the real parts of the metamaterial permittivity εM and permeability µM, calculated using the values of neff and ns from Sec. 2 and 4 for a lamellar Ag grating with 50% filling ratio, and having two different periods, (a) d = 3 nm in red and (b) d = 300 nm in black.
Fig. 15
Fig. 15 1D lamellar Ag grating in air with d = 0.15 µm, h = 0.24 µm, and w = 0.1244 µm in TM polarization. (a) Reflectivity as a function of the wavelength, numerical results in black, modal correction in blue, the static results, red line. The numerical results for d = 1.5 nm and w = 1.244 nm is presented with red circles. (b) The spectral dependence of the metamaterial parameters εM and µM, calculated using Eq. (55) with the modal correction for neff and ns from Sec. 2 and 4.
Fig. 16
Fig. 16 Schematical view of a rectangular cross-section with notations.

Equations (83)

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D = ε 0 ε eff E ,
D V = 1 V V D dV = V 1 V 1 V 1 V 1 D 1 dV + V 2 V 1 V 2 V 2 D 2 dV =(1f) D 1 V 1 +f D 2 V 2 ,
E V =f E 2 ( r ) V 2 +(1f) E 1 ( r ) V 1 .
D 2 V 2 = ε 0 ε 2 E 2 V 2 . D 1 V 1 = ε 0 ε 1 E 1 V 1
f( ε eff ε 2 ) E 2 V 2 =(1f)( ε eff ε 1 ) E 1 V 1 .
E 2 V 2 = Q m E 1 V 1 .
ε eff =[ 1(1f) ε 1 +f Q m ε 2 ] [ 1f(1 Q m ) ] 1
ε eff,ii = (1f) ε 1 +f Q m,ii ε 2 1f(1 Q m,ii ) ,i=x,y,z
Q m =Q .
E 2 ( r ) 3 ε 1 2 ε 1 + ε 2 E 1 ( r )
ε eff,ii = ε 1 2(1f) ε 1 +(1+2f) ε 2 (2+f) ε 1 +(1f) ε 2 ,i=x,y,z
E 2 ( r )( 2 ε 1 /( ε 1 + ε 2 ) 0 0 0 2 ε 1 /( ε 1 + ε 2 ) 0 0 0 1 ) E 1 ( r ),
ε eff,ii = ε 1 (1f) ε 1 +(1+f) ε 2 (1+f) ε 1 +(1f) ε 2 ,i=x,y. ε eff,zz =(1f) ε 1 +f ε 2
E 2x = ε 1 ε 2 E 1x E 2y = E 1y . E 2z = E 1z
ε eff,yy = ε eff,zz =f ε 2 +(1f) ε 1 ε ¯ TE .
ε eff,xx = ε 1 ε 2 f ε 1 +(1f) ε 2 ε ¯ TM .
ε eff,xx = ε 1 (1f) .
ε eff,xx ε 1
ε eff,yy ε ¯ TE + ( d λ ) 2 [ f(1f)( ε 2 ε 1 ) ] 2 π 2 3 .
ε eff,xx ε ¯ TM + ( d λ ) 2 [ f(1f) ( ε 2 ε 1 ) ε 1 ε 2 ] 2 π 2 3 ε ¯ TM 3 ε ¯ TE 3 .
E 2 (x) E 2,0 cos( k 0 n 2 x) cos( k 0 n 2 w/2) , w 2 x w 2 , E 1 (x) E 1 (w/2)
E 2 q E 2,0
q= tan( k 0 n 2 w 2 ) k 0 n 2 w 2 .
Q m =qQ
Q m,xx = ε 1 ε 2 tan( k 0 n 2 w/2) k 0 n 2 w/2 .
Q m,yy = tan( k 0 n 2 w/2) k 0 n 2 w/2 .
Re[ f y ε 1 ε 2 f x ε 1 +(1 f x ) ε 2 +(1 f y ) ε 1 ]Re( ε eff,xx )Re [ f x f y ε 2 +(1 f y ) ε 1 + (1 f x ) ε 1 ] 1 .
E 2 (ρ,φ)= E 2,0 (ρ,φ) J 0 ( k 0 n 2 ρ) J 0 ( k 0 n 2 R) .
E 2 1 π R 2 0 2π E 2,0 (R,φ)dφ 0 R J 0 ( k 0 n 2 ρ) J 0 ( k 0 n 2 R) ρdρ= E 2,0 2 J 1 ( k 0 n 2 R) k 0 n 2 R J 0 ( k 0 n 2 R) q E 2,0 .
E 2 = E 2,0 (ρ,φ) J 0 ( k 0 n 2 ρ ˜ ) J 0 ( k 0 n 2 R ) ,
ρ ˜ 2 = R 2 ( x 2 a 2 + y 2 b 2 )=ρ R 2 ( cos 2 φ a 2 + sin 2 φ b 2 ),
Rab= ab .
E 2 1 π R 2 0 2π E 2,0 dφ 0 ρ ˜ =R J 0 ( k 0 n 2 ρ ˜ ) J 0 ( k 0 n 2 R) ρdρ= E 2,0 2 J 1 ( k 0 n 2 R) k 0 n 2 R J 0 ( k 0 n 2 R) ,
r cl = n cl n s n cl + n s ,
r= r cl + r sub exp(2i k 0 n eff h) 1+ r cl r sub exp(2i k 0 n eff h) ,
n s = n eff ε eff,xx .
H cl = H i exp(i k 0 n cl x)+ H r exp(i k 0 n cl x), H gr = H 0 exp(i k 0 n eff,xx z)ψ(x)
k jx = k 0 ε j ε eff,xx . ψ(x)={ cos[ k 1x (| x |d/2)] cos[ k 1x (dw)/2] in region 1 cos( k 2x x) cos( k 2x w/2) in region 2
H i + H r = H 0 I ψ ,
n cl ( H i H r ) n ψ ¯ = n eff,xx H 0 I | ψ | 2 ,
I ψ = 1 d d/2 d/2 ψ(x)dx . I | ψ | 2 = 1 d d/2 d/2 | ψ(x) | 2 dx
n s = n eff,xx I | ψ | 2 | I ψ | 2 .
I | ψ | 2 1f+ f 2 [ sin( k 2x w) k 2x w + sh( k 2x w) k 2x w ]/ | cos( k 2x w 2 ) | 2 , I ψ 1f+f tan( k 2x w) k 2x w
H x,cl = H x,i exp(i k 0 n cl z)+ H x,r exp(i k 0 n cl z).
H x,gr = H x,0 (φ) ψ 2D (ρ)exp(i k 0 n eff,xx z)
ψ 2D ={ 1, ρ>R J 0 ( k 2ρ ρ)/ J 0 ( k 2ρ R), ρR .
H x,i + H x,r = H x,0 I 2D,ψ n cl ( H x,i H x,r ) I 2D, ψ ¯ = n eff,xx H x,0 I 2D, | ψ | 2
I 2D,ψ =1f+f 2 J 1 ( k 2ρ R) k 2ρ RJ 0 ( k 2ρ R)
I 2D, | ψ | 2 =1f+f[ 1+ | J 1 ( k 2ρ R) J 0 ( k 2ρ R) | 2 ]
B = μ 0 µ eff H
B = μ 0 H B = 1 V V μ 0 H dV = μ 0 H
µ eff 1
ε M μ M = n M μ M / ε M =(1+ r cl )/(1 r cl )
n M = n eff = ε eff,xx
ε M = n eff n s μ M = n eff n s
n eff = n s = ε eff,xx
E p ( r )= E unp ( r )+ k 0 2 ε 1 V 2 G( r r )( ε 2 ε 1 1 ) E p ( r )d r ,
G( r r )= P vG ( r r ) 1 k 0 2 ε 1 Lδ( r r ),
E p ( r ) E 2 ( r ) E unp (( r )L( ε 2 ε 1 1 ) E p (( r ),in V 2
E p ( r ) E 1 ( r ) E unp (( r ),in V 1 .
E 2 ( r S )Q E 1 ( r S )
Q= [ 1 + L( ε 2 ε 1 1 ) ] 1 .
Trace(L)=1.
rotrotG k 2 G=1δ( r r ).
k 2 G=1δG= 1 k 2 δ.
GΔG k 2 G=1δ(r r )
1 k 2 δΔG k 2 G=1δΔG+ k 2 G=( 1+ 1 k 2 )δ.
Δg+ k 2 g=δ( r r )
G( r r 0 )=( 1+ 1 k 2 )g( r r 0 ),
g(x x 0 )= i 2k exp( ik| x x 0 | ) d 2 d z 2 g(x x 0 )=δ(x x 0 ) k 2 g(x x 0 ) G(x x 0 )=( 1 k 0 2 ε 1 δ(x x 0 ) 0 0 0 g(x x 0 ) 0 0 0 g(x x 0 ) )
1 4π| r r 0 |
L= V 2 1 4π| r r 0 | S 2 n ^ 1 4π| r r 0 | dS= S 2 n ^ r r 0 4π | r r 0 | 3 dS,
L= C 2 n ρ 2π ρ 2 dc .
L= ( 1/3 0 0 0 1/3 0 0 0 1/3 ).
L=( 1/2 0 0 0 1/2 0 0 0 0 ).
L=( 1 0 0 0 0 0 0 0 0 ).
L=( 0 0 0 0 0 0 0 0 1 ).
E 2 ( r )( 1 0 0 0 1 0 0 0 ε 1 / ε 2 ) E 1 ( r ).
on c 1 :{ x= a 2 n ^ = x ^ ρ=x x ^ +y y ^ d c 1 =dy
L xx = 2 2π c 1 xdy x 2 + y 2 .
L xx = 1 2π c 1 a 2 4 cos 2 φ dφ a 2 4 + a 2 4 t g 2 φ = 1 π c 1 dφ = 2 π arctg b a
ρ =acosφ x ^ +bsinφ y ^ n ^ =gradρ/| gradρ | dc=| dρ dφ |
L xx = 1 2π c ab cos 2 φdφ a 2 cos 2 φ+ b 2 sin 2 φ = b a+b L yy = 1 2π c ab sin 2 φdφ a 2 cos 2 φ+ b 2 sin 2 φ = a a+b

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