Abstract

A simple semi-analytical model for the comprehension of the extraordinary optical transmission of a 1D array of periodic subwavelength slit is proposed. In this single mode model, the mono layer of the perforated metal film is considered as a homogeneous medium. Therefore, the electromagnetic response of this structure to a plane wave excitation is equivalent to that of a slab with homogeneous equivalent permittivity. A versatile phase correction is added to this model in order to handle the contribution of surface waves in the EOT phenomenon. The proposed model leads to a cavity-like dispersion relation that allows accurate prediction of the resonance frequencies of the 1D structure.

© 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. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelenght hole arrays,” Nature 391, 667–669 (1998).
    [Crossref]
  2. C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445, 39–46 (2007).
    [Crossref] [PubMed]
  3. L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
    [Crossref] [PubMed]
  4. F. J. Garcia de Abajo, R. Gomez-Medina, and J. J. Saenz, “Full transmission through perfect-conductor subwavelength hole arrays,” Phys. Rev. E. 72, 016608 (2005).
    [Crossref]
  5. L. Aigouy, P. Lalanne, J. P. Hugonin, G. Julie, V. Mathet, and M. Mortier, “Near field analysis of surface waves launched at nano-slit apertures,” Phys. Rev. Lett. 98, 153902 (2007).
    [Crossref]
  6. H. T. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452, 728–731 (2008).
    [Crossref] [PubMed]
  7. A. Y. Nikitin, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, “In the diffraction shadow: Norton waves versus surface plasmon polaritons in the optical region,” New J. Phys. 11, 123020 (2009).
    [Crossref]
  8. F. I. Baida, Y. Poujet, J. Salvi, D. van Labeke, and B. Guizal, “Extraordinary transmission beyond the cut-off through sub-lambda annular aperture arrays,” Opt. Commun. 282, 1463–1466 (2009).
    [Crossref]
  9. F. I. Baida, Y. Poujet, B. Guizal, and D. van Labeke, “New design for enhanced transmission and polarization control through near-field optical microscopy probes,” Opt. Commun. 256, 190–195 (2005).
    [Crossref]
  10. K. Edee, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings,” J. Opt. Soc. Am. A 28, 2006–2013 (2011).
    [Crossref]
  11. K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).
    [Crossref]
  12. K. Edee and J.-P. Plumey, “Numerical scheme for the modal method based on subsectional Gegenbauer polynomial expansion: application to biperiodic binary grating,” J. Opt. Soc. Am. A 32, 402–410 (2015).
    [Crossref]
  13. R. Brendel and D. Bormann, “An infrared dielectric function model for amorphous solids,” J. Appl. Phys 71, 1–6 (1992).
    [Crossref]
  14. A. D. Rakiá, A. B. Djurišić, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37, 5271–5283 (1998).
    [Crossref]
  15. B. Guizal and D. Felbacq, “Electromagnetic beam diffraction by a finite strip grating,” Opt. Commun. 165, 1–6 (1999).
    [Crossref]

2015 (1)

2013 (1)

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).
[Crossref]

2011 (1)

2009 (2)

A. Y. Nikitin, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, “In the diffraction shadow: Norton waves versus surface plasmon polaritons in the optical region,” New J. Phys. 11, 123020 (2009).
[Crossref]

F. I. Baida, Y. Poujet, J. Salvi, D. van Labeke, and B. Guizal, “Extraordinary transmission beyond the cut-off through sub-lambda annular aperture arrays,” Opt. Commun. 282, 1463–1466 (2009).
[Crossref]

2008 (1)

H. T. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452, 728–731 (2008).
[Crossref] [PubMed]

2007 (2)

L. Aigouy, P. Lalanne, J. P. Hugonin, G. Julie, V. Mathet, and M. Mortier, “Near field analysis of surface waves launched at nano-slit apertures,” Phys. Rev. Lett. 98, 153902 (2007).
[Crossref]

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445, 39–46 (2007).
[Crossref] [PubMed]

2005 (2)

F. J. Garcia de Abajo, R. Gomez-Medina, and J. J. Saenz, “Full transmission through perfect-conductor subwavelength hole arrays,” Phys. Rev. E. 72, 016608 (2005).
[Crossref]

F. I. Baida, Y. Poujet, B. Guizal, and D. van Labeke, “New design for enhanced transmission and polarization control through near-field optical microscopy probes,” Opt. Commun. 256, 190–195 (2005).
[Crossref]

2001 (1)

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[Crossref] [PubMed]

1999 (1)

B. Guizal and D. Felbacq, “Electromagnetic beam diffraction by a finite strip grating,” Opt. Commun. 165, 1–6 (1999).
[Crossref]

1998 (2)

A. D. Rakiá, A. B. Djurišić, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37, 5271–5283 (1998).
[Crossref]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelenght hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

1992 (1)

R. Brendel and D. Bormann, “An infrared dielectric function model for amorphous solids,” J. Appl. Phys 71, 1–6 (1992).
[Crossref]

Aigouy, L.

L. Aigouy, P. Lalanne, J. P. Hugonin, G. Julie, V. Mathet, and M. Mortier, “Near field analysis of surface waves launched at nano-slit apertures,” Phys. Rev. Lett. 98, 153902 (2007).
[Crossref]

Baida, F. I.

F. I. Baida, Y. Poujet, J. Salvi, D. van Labeke, and B. Guizal, “Extraordinary transmission beyond the cut-off through sub-lambda annular aperture arrays,” Opt. Commun. 282, 1463–1466 (2009).
[Crossref]

F. I. Baida, Y. Poujet, B. Guizal, and D. van Labeke, “New design for enhanced transmission and polarization control through near-field optical microscopy probes,” Opt. Commun. 256, 190–195 (2005).
[Crossref]

Bormann, D.

R. Brendel and D. Bormann, “An infrared dielectric function model for amorphous solids,” J. Appl. Phys 71, 1–6 (1992).
[Crossref]

Brendel, R.

R. Brendel and D. Bormann, “An infrared dielectric function model for amorphous solids,” J. Appl. Phys 71, 1–6 (1992).
[Crossref]

Djurišic, A. B.

Ebbesen, T. W.

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445, 39–46 (2007).
[Crossref] [PubMed]

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[Crossref] [PubMed]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelenght hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

Edee, K.

Elazar, J. M.

Felbacq, D.

B. Guizal and D. Felbacq, “Electromagnetic beam diffraction by a finite strip grating,” Opt. Commun. 165, 1–6 (1999).
[Crossref]

Fenniche, I.

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).
[Crossref]

Garcia de Abajo, F. J.

F. J. Garcia de Abajo, R. Gomez-Medina, and J. J. Saenz, “Full transmission through perfect-conductor subwavelength hole arrays,” Phys. Rev. E. 72, 016608 (2005).
[Crossref]

Garcia-Vidal, F. J.

A. Y. Nikitin, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, “In the diffraction shadow: Norton waves versus surface plasmon polaritons in the optical region,” New J. Phys. 11, 123020 (2009).
[Crossref]

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[Crossref] [PubMed]

Genet, C.

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445, 39–46 (2007).
[Crossref] [PubMed]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelenght hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

Gomez-Medina, R.

F. J. Garcia de Abajo, R. Gomez-Medina, and J. J. Saenz, “Full transmission through perfect-conductor subwavelength hole arrays,” Phys. Rev. E. 72, 016608 (2005).
[Crossref]

Granet, G.

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).
[Crossref]

Guizal, B.

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).
[Crossref]

F. I. Baida, Y. Poujet, J. Salvi, D. van Labeke, and B. Guizal, “Extraordinary transmission beyond the cut-off through sub-lambda annular aperture arrays,” Opt. Commun. 282, 1463–1466 (2009).
[Crossref]

F. I. Baida, Y. Poujet, B. Guizal, and D. van Labeke, “New design for enhanced transmission and polarization control through near-field optical microscopy probes,” Opt. Commun. 256, 190–195 (2005).
[Crossref]

B. Guizal and D. Felbacq, “Electromagnetic beam diffraction by a finite strip grating,” Opt. Commun. 165, 1–6 (1999).
[Crossref]

Hugonin, J. P.

L. Aigouy, P. Lalanne, J. P. Hugonin, G. Julie, V. Mathet, and M. Mortier, “Near field analysis of surface waves launched at nano-slit apertures,” Phys. Rev. Lett. 98, 153902 (2007).
[Crossref]

Julie, G.

L. Aigouy, P. Lalanne, J. P. Hugonin, G. Julie, V. Mathet, and M. Mortier, “Near field analysis of surface waves launched at nano-slit apertures,” Phys. Rev. Lett. 98, 153902 (2007).
[Crossref]

Lalanne, P.

H. T. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452, 728–731 (2008).
[Crossref] [PubMed]

L. Aigouy, P. Lalanne, J. P. Hugonin, G. Julie, V. Mathet, and M. Mortier, “Near field analysis of surface waves launched at nano-slit apertures,” Phys. Rev. Lett. 98, 153902 (2007).
[Crossref]

Lezec, H. J.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[Crossref] [PubMed]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelenght hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

Liu, H. T.

H. T. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452, 728–731 (2008).
[Crossref] [PubMed]

Majewski, M. L.

Martin-Moreno, L.

A. Y. Nikitin, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, “In the diffraction shadow: Norton waves versus surface plasmon polaritons in the optical region,” New J. Phys. 11, 123020 (2009).
[Crossref]

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[Crossref] [PubMed]

Mathet, V.

L. Aigouy, P. Lalanne, J. P. Hugonin, G. Julie, V. Mathet, and M. Mortier, “Near field analysis of surface waves launched at nano-slit apertures,” Phys. Rev. Lett. 98, 153902 (2007).
[Crossref]

Mortier, M.

L. Aigouy, P. Lalanne, J. P. Hugonin, G. Julie, V. Mathet, and M. Mortier, “Near field analysis of surface waves launched at nano-slit apertures,” Phys. Rev. Lett. 98, 153902 (2007).
[Crossref]

Nikitin, A. Y.

A. Y. Nikitin, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, “In the diffraction shadow: Norton waves versus surface plasmon polaritons in the optical region,” New J. Phys. 11, 123020 (2009).
[Crossref]

Pellerin, K. M.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[Crossref] [PubMed]

Pendry, J. B.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[Crossref] [PubMed]

Plumey, J.-P.

Poujet, Y.

F. I. Baida, Y. Poujet, J. Salvi, D. van Labeke, and B. Guizal, “Extraordinary transmission beyond the cut-off through sub-lambda annular aperture arrays,” Opt. Commun. 282, 1463–1466 (2009).
[Crossref]

F. I. Baida, Y. Poujet, B. Guizal, and D. van Labeke, “New design for enhanced transmission and polarization control through near-field optical microscopy probes,” Opt. Commun. 256, 190–195 (2005).
[Crossref]

Rakiá, A. D.

Rodrigo, S. G.

A. Y. Nikitin, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, “In the diffraction shadow: Norton waves versus surface plasmon polaritons in the optical region,” New J. Phys. 11, 123020 (2009).
[Crossref]

Saenz, J. J.

F. J. Garcia de Abajo, R. Gomez-Medina, and J. J. Saenz, “Full transmission through perfect-conductor subwavelength hole arrays,” Phys. Rev. E. 72, 016608 (2005).
[Crossref]

Salvi, J.

F. I. Baida, Y. Poujet, J. Salvi, D. van Labeke, and B. Guizal, “Extraordinary transmission beyond the cut-off through sub-lambda annular aperture arrays,” Opt. Commun. 282, 1463–1466 (2009).
[Crossref]

Thio, T.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[Crossref] [PubMed]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelenght hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

van Labeke, D.

F. I. Baida, Y. Poujet, J. Salvi, D. van Labeke, and B. Guizal, “Extraordinary transmission beyond the cut-off through sub-lambda annular aperture arrays,” Opt. Commun. 282, 1463–1466 (2009).
[Crossref]

F. I. Baida, Y. Poujet, B. Guizal, and D. van Labeke, “New design for enhanced transmission and polarization control through near-field optical microscopy probes,” Opt. Commun. 256, 190–195 (2005).
[Crossref]

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelenght hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

Appl. Opt. (1)

J. Appl. Phys (1)

R. Brendel and D. Bormann, “An infrared dielectric function model for amorphous solids,” J. Appl. Phys 71, 1–6 (1992).
[Crossref]

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

Nature (3)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelenght hole arrays,” Nature 391, 667–669 (1998).
[Crossref]

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445, 39–46 (2007).
[Crossref] [PubMed]

H. T. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452, 728–731 (2008).
[Crossref] [PubMed]

New J. Phys. (1)

A. Y. Nikitin, S. G. Rodrigo, F. J. Garcia-Vidal, and L. Martin-Moreno, “In the diffraction shadow: Norton waves versus surface plasmon polaritons in the optical region,” New J. Phys. 11, 123020 (2009).
[Crossref]

Opt. Commun. (3)

F. I. Baida, Y. Poujet, J. Salvi, D. van Labeke, and B. Guizal, “Extraordinary transmission beyond the cut-off through sub-lambda annular aperture arrays,” Opt. Commun. 282, 1463–1466 (2009).
[Crossref]

F. I. Baida, Y. Poujet, B. Guizal, and D. van Labeke, “New design for enhanced transmission and polarization control through near-field optical microscopy probes,” Opt. Commun. 256, 190–195 (2005).
[Crossref]

B. Guizal and D. Felbacq, “Electromagnetic beam diffraction by a finite strip grating,” Opt. Commun. 165, 1–6 (1999).
[Crossref]

Phys. Rev. E. (1)

F. J. Garcia de Abajo, R. Gomez-Medina, and J. J. Saenz, “Full transmission through perfect-conductor subwavelength hole arrays,” Phys. Rev. E. 72, 016608 (2005).
[Crossref]

Phys. Rev. Lett. (2)

L. Aigouy, P. Lalanne, J. P. Hugonin, G. Julie, V. Mathet, and M. Mortier, “Near field analysis of surface waves launched at nano-slit apertures,” Phys. Rev. Lett. 98, 153902 (2007).
[Crossref]

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[Crossref] [PubMed]

Prog. Electromagn. Res. (1)

K. Edee, I. Fenniche, G. Granet, and B. Guizal, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: weighting function, convergence and stability,” Prog. Electromagn. Res. 133, 17–35 (2013).
[Crossref]

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Figures (13)

Fig. 1
Fig. 1 Schematic of dispersive metal film perforated with a subwavelength periodic array of 1D nano-slits.

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Fig. 2
Fig. 2 Reflection, transmission and absorption spectrum of the array of subwavelength 1D nano-slits. Illustration of EOT phenomenon on a metal film perforated with a subwavelength periodic array of 1D nano-slits. Parameters: ε(1) = ε(3) = ε(slit) = 1, incidence angle= 0°, h = 800nm, d = 165nm, s = 15nm.

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Fig. 3
Fig. 3 Modulus of the magnetic field Hx(x, z) at λ = 3.37μm. Parameters: ε1 = ε3 = εslit = 1, incidence angle= 0°, h = 800nm, d = 165nm, s = 15nm.

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Fig. 4
Fig. 4 Spectrum of the real part of the most slowly decaying evanescent mode of the subwavelength periodic periodic structure. Parameters: ε(slit) = 1, incidence angle= 0°, s = 15nm.

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Fig. 5
Fig. 5 Spectrum of the imaginary part of the most slowly decaying evanescent mode of the subwavelength periodic structure. Parameters: ε(slit) = 1, incidence angle= 0°, s = 15nm.

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Fig. 6
Fig. 6 Illustration of the confinement properties of Hy(x) for different values of the wavelength and for d = 165nm. Parameters: ε(slit) = 1, incidence angle= 0°, s = 15nm.

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Fig. 7
Fig. 7 Schematic of dispersive metal film perforated with a subwavelength periodic array of 1D nano-slits and its equivalent homogeneous slab with effective medium permittivity ε(2) in static limit.

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Fig. 8
Fig. 8 Reflection spectrum for λ ∈ [0.6, 3]μm. Parameters: ε(1) = ε(3) = ε(slit) = 1, incidence angle= 0°.

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Fig. 9
Fig. 9 Transmission spectrum for λ ∈ [0.6, 3]μm. Parameters: ε(1) = ε(3) = ε(slit) = 1, incidence angle= 0°.

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Fig. 10
Fig. 10 Reflection spectrum for λ ∈ [3, 100]μm. Parameters: ε(1) = ε(3) = ε(slit) = 1, incidence angle= 0°.

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Fig. 11
Fig. 11 Transmission spectrum for λ ∈ [3, 100]μm. Parameters: ε(1) = ε(3) = ε(slit) = 1, incidence angle= 0°.

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Fig. 12
Fig. 12 Comparison of resonance frequencies obtained with dispersion equation with reflection curves computed with HSM (dotted line), PMM (solid line), HHSM (dashed line). Parameters: ε(1) = ε(3) = 1.542, ε(slit) = 1, incidence angle= 50°.

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Fig. 13
Fig. 13 Comparison of resonance frequencies obtained with dispersion equation with reflection curves computed with HSM (dotted line), PMM (solid line), HHSM (dashed line). Parameters: ε(1) = ε(slit) = 1.542, ε(3) = 1, incidence angle= 20°.

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Equations (20)

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

ε ( metal ) ( ω ) = ε intra ( ω ) + ε inter ( ω ) .
ε intra ( ω ) = 1 − ω p 2 f 0 ω ( ω − i Γ 0 ) .
ε inter ( ω ) = i π ω p 2 2 2 ∑ j = 1 k χ j ( ω )
χ j ( ω ) = f j α j σ j [ w ( α j − ω j 2 σ j ) + w ( α j + ω j 2 σ j ) ] ,
w ˜ ( t ) = 2 π ∫ z ∞ e − t 2 d t .
ℒ ( ω ) | ψ q ( ω ) 〉 = γ q 2 ( ω ) | ψ q ( ω ) 〉
ℒ ( x , ω ) = ( c ω ) 2 ε ( x , ω ) ∂ x 1 ε ( x , ω ) ∂ x + ε ( x , ω ) .
H y ( k ) ( x , z , ω ) = ∑ q A q ( k ) ( ω ) e − i k 0 ( ω ) γ q ( k ) ( ω ) z ψ q ( x , ω ) .
E x ( x ) = 1 ε ( x ) D x ( x ) .
〈 E x 〉 = 〈 1 / ε ( x ) D x 〉 ≃ 〈 1 / ε ( x ) 〉 〈 D ( x ) 〉 = 〈 1 / ε ( x ) 〉 D 0
〈 D x 〉 = 〈 1 / ε ( x ) 〉 − 1 〈 E x 〉 = ε ( 2 ) 〈 E x 〉 .
r = r 1 + ϕ 1 r 2 ϕ 3 1 + r 1 ϕ 1 r 2 ϕ 3
t = t 1 t 2 ϕ 3 1 + r 1 ϕ 1 r 2 ϕ 3
r i = γ 0 ( i ) / ε ( i ) − γ 0 ( i + 1 ) / ε ( i + 1 ) γ 0 ( i ) / ε ( i ) + γ 0 ( i + 1 ) / ε ( i + 1 ) ,
t i = 2 γ 0 ( i ) / ε ( i ) γ 0 ( i ) / ε ( i ) + γ 0 ( i + 1 ) / ε ( i + 1 ) .
ϕ 1 r 2 ϕ 3 ≃ − r 1         and       1 + r 1 ϕ 1 r 2 ϕ 3 ≠ 0 ,
ϕ 1 = e − i k 0 γ 0 ( 2 ) h e − i k 0 α s p ( 1 ) a ( 1 ) ,     ϕ 3 = e − i k 0 γ 0 ( 2 ) h e − i k 0 α s p ( 3 ) a ( 3 )
k 0 [ 2 γ ′ h + α 1 ′ a ( 1 ) + α 3 ′ a ( 3 ) ] = 2 π p ,     p ∈ ℕ .
f p ( λ ) = λ
f p ( λ ) = 1 p [ 2 γ ′ ( λ ) h + α 1 ′ ( λ ) a ( 1 ) + α 3 ′ ( λ ) a ( 3 ) ] .

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