Abstract

We propose a study of a generalized Airy-like formula of the transmittance through planar resonators. A complete and analytical analysis of total transmission conditions is given according to physical parameters. The homogeneous problem related to free oscillations of resonators, which leads to complex resonance frequencies and quality factors, is also resolved. After a quick validation on well-known parallel-plate dielectric layers behaving as Fabry–Perot resonators, our discussion is applied on subwavelength metallic lamellar gratings from an analytical modal theory assuming perfectly electric conductors.

© 2012 Optical Society of America

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References

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  1. F. J. Garcia-Vidal, L. Martin-Moreno, L. Kuipers, and T. W. Ebbesen, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010).
    [CrossRef]
  2. S. Astilean, P. Lalanne, and M. Palamaru, “Light transmission through metallic channels much smaller than the wavelength,” Opt. Commun. 175, 265–273 (2000).
    [CrossRef]
  3. N. Garcia and M. Nieto-Vesperinas, “Theory of electromagnetic wave transmission through metallic gratings of subwavelength slits,” J. Opt. A Pure Appl. Opt. 9, 490–495 (2007).
    [CrossRef]
  4. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  5. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  6. G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
    [CrossRef]
  7. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1019–1023(1996).
    [CrossRef]
  8. E. Popov, M. Nevière, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100–16108 (2000).
    [CrossRef]
  9. 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,” J. Opt. A Pure Appl. Opt. 86, 1114–1117 (2001).
  10. D. Maystre, A. L. Fehrembach, and E. Popov, “Plasmonic antiresonance through subwavelength hole arrays,” J. Opt. Soc. Am. A 28, 342–355 (2011).
    [CrossRef]
  11. J. R. Andrewartha, G. H. Derrick, and R. C. McPhredran, “A general modal theory for reflection gratings,” Opt. Acta 28, 1501–1516 (1981).
    [CrossRef]
  12. J. X. Chen, P. Wang, X. L. Wang, Y. H. Lu, R. S. Zheng, and H. Ming, “Analytical investigation of transmission properties of metallic gratings,” Chin. Phys. Lett. 25, 4385–4387 (2008).
    [CrossRef]
  13. A. Kobyakov, A. R. Zakharian, A. Mafi, and S. A. Darmanyan, “Semi-analytical method for light interaction with 1d-periodic nanoplasmonic structures,” Opt. Express 16, 8938–8957(2008).
    [CrossRef]
  14. P. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A Pure Appl. Opt. 2, 48–51 (2000).
    [CrossRef]
  15. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999).
    [CrossRef]
  16. J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. 94, 1–4 (2005).
    [CrossRef]
  17. H. Liu and P. Lalanne, “Comprehensive microscopic model of the extraordinary optical transmission,” J. Opt. Soc. Am. A 27, 2542–2550 (2010).
    [CrossRef]
  18. M. Guillaumee, L. A. Dunbar, and R. P. Stanley, “Description of the modes governing the optical transmission through metal gratings,” Opt. Express 19, 4740–4755 (2011).
    [CrossRef]
  19. A. A. Antonov, “Resonance on real and complex frequencies,” Eur. J. Sci. Res. 28, 193–204 (2009).
  20. A. A. Antonov, “New interpretation of resonance,” Int. J. Pure Appl. Sci. Technol. 2, 1–12 (2010).
  21. P. Boyer, G. Renversez, E. Popov, and M. Nevière, “Improved differential method for microstructured optical fibres,” J. Opt. A Pure Appl. Opt. 9, 728–740 (2007).
    [CrossRef]

2011 (2)

2010 (3)

F. J. Garcia-Vidal, L. Martin-Moreno, L. Kuipers, and T. W. Ebbesen, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010).
[CrossRef]

A. A. Antonov, “New interpretation of resonance,” Int. J. Pure Appl. Sci. Technol. 2, 1–12 (2010).

H. Liu and P. Lalanne, “Comprehensive microscopic model of the extraordinary optical transmission,” J. Opt. Soc. Am. A 27, 2542–2550 (2010).
[CrossRef]

2009 (1)

A. A. Antonov, “Resonance on real and complex frequencies,” Eur. J. Sci. Res. 28, 193–204 (2009).

2008 (2)

J. X. Chen, P. Wang, X. L. Wang, Y. H. Lu, R. S. Zheng, and H. Ming, “Analytical investigation of transmission properties of metallic gratings,” Chin. Phys. Lett. 25, 4385–4387 (2008).
[CrossRef]

A. Kobyakov, A. R. Zakharian, A. Mafi, and S. A. Darmanyan, “Semi-analytical method for light interaction with 1d-periodic nanoplasmonic structures,” Opt. Express 16, 8938–8957(2008).
[CrossRef]

2007 (2)

P. Boyer, G. Renversez, E. Popov, and M. Nevière, “Improved differential method for microstructured optical fibres,” J. Opt. A Pure Appl. Opt. 9, 728–740 (2007).
[CrossRef]

N. Garcia and M. Nieto-Vesperinas, “Theory of electromagnetic wave transmission through metallic gratings of subwavelength slits,” J. Opt. A Pure Appl. Opt. 9, 490–495 (2007).
[CrossRef]

2005 (1)

J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. 94, 1–4 (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,” J. Opt. A Pure Appl. Opt. 86, 1114–1117 (2001).

2000 (3)

P. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A Pure Appl. Opt. 2, 48–51 (2000).
[CrossRef]

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100–16108 (2000).
[CrossRef]

S. Astilean, P. Lalanne, and M. Palamaru, “Light transmission through metallic channels much smaller than the wavelength,” Opt. Commun. 175, 265–273 (2000).
[CrossRef]

1999 (1)

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999).
[CrossRef]

1996 (3)

1981 (2)

M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
[CrossRef]

J. R. Andrewartha, G. H. Derrick, and R. C. McPhredran, “A general modal theory for reflection gratings,” Opt. Acta 28, 1501–1516 (1981).
[CrossRef]

Andrewartha, J. R.

J. R. Andrewartha, G. H. Derrick, and R. C. McPhredran, “A general modal theory for reflection gratings,” Opt. Acta 28, 1501–1516 (1981).
[CrossRef]

Antonov, A. A.

A. A. Antonov, “New interpretation of resonance,” Int. J. Pure Appl. Sci. Technol. 2, 1–12 (2010).

A. A. Antonov, “Resonance on real and complex frequencies,” Eur. J. Sci. Res. 28, 193–204 (2009).

Astilean, S.

S. Astilean, P. Lalanne, and M. Palamaru, “Light transmission through metallic channels much smaller than the wavelength,” Opt. Commun. 175, 265–273 (2000).
[CrossRef]

P. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A Pure Appl. Opt. 2, 48–51 (2000).
[CrossRef]

Boyer, P.

P. Boyer, G. Renversez, E. Popov, and M. Nevière, “Improved differential method for microstructured optical fibres,” J. Opt. A Pure Appl. Opt. 9, 728–740 (2007).
[CrossRef]

Catrysse, P. B.

J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. 94, 1–4 (2005).
[CrossRef]

Chen, J. X.

J. X. Chen, P. Wang, X. L. Wang, Y. H. Lu, R. S. Zheng, and H. Ming, “Analytical investigation of transmission properties of metallic gratings,” Chin. Phys. Lett. 25, 4385–4387 (2008).
[CrossRef]

Darmanyan, S. A.

Derrick, G. H.

J. R. Andrewartha, G. H. Derrick, and R. C. McPhredran, “A general modal theory for reflection gratings,” Opt. Acta 28, 1501–1516 (1981).
[CrossRef]

Dunbar, L. A.

Ebbesen, T. W.

F. J. Garcia-Vidal, L. Martin-Moreno, L. Kuipers, and T. W. Ebbesen, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010).
[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,” J. Opt. A Pure Appl. Opt. 86, 1114–1117 (2001).

Enoch, S.

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100–16108 (2000).
[CrossRef]

Fan, S.

J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. 94, 1–4 (2005).
[CrossRef]

Fehrembach, A. L.

Garcia, N.

N. Garcia and M. Nieto-Vesperinas, “Theory of electromagnetic wave transmission through metallic gratings of subwavelength slits,” J. Opt. A Pure Appl. Opt. 9, 490–495 (2007).
[CrossRef]

Garcia-Vidal, F. J.

F. J. Garcia-Vidal, L. Martin-Moreno, L. Kuipers, and T. W. Ebbesen, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010).
[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,” J. Opt. A Pure Appl. Opt. 86, 1114–1117 (2001).

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999).
[CrossRef]

Gaylord, T. K.

Granet, G.

Guillaumee, M.

Guizal, B.

Hugonin, J. P.

P. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A Pure Appl. Opt. 2, 48–51 (2000).
[CrossRef]

Kobyakov, A.

Kuipers, L.

F. J. Garcia-Vidal, L. Martin-Moreno, L. Kuipers, and T. W. Ebbesen, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010).
[CrossRef]

Lalanne, P.

H. Liu and P. Lalanne, “Comprehensive microscopic model of the extraordinary optical transmission,” J. Opt. Soc. Am. A 27, 2542–2550 (2010).
[CrossRef]

P. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A Pure Appl. Opt. 2, 48–51 (2000).
[CrossRef]

S. Astilean, P. Lalanne, and M. Palamaru, “Light transmission through metallic channels much smaller than the wavelength,” Opt. Commun. 175, 265–273 (2000).
[CrossRef]

P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
[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,” J. Opt. A Pure Appl. Opt. 86, 1114–1117 (2001).

Li, L.

Liu, H.

Lu, Y. H.

J. X. Chen, P. Wang, X. L. Wang, Y. H. Lu, R. S. Zheng, and H. Ming, “Analytical investigation of transmission properties of metallic gratings,” Chin. Phys. Lett. 25, 4385–4387 (2008).
[CrossRef]

Mafi, A.

Martin-Moreno, L.

F. J. Garcia-Vidal, L. Martin-Moreno, L. Kuipers, and T. W. Ebbesen, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010).
[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,” J. Opt. A Pure Appl. Opt. 86, 1114–1117 (2001).

Maystre, D.

McPhredran, R. C.

J. R. Andrewartha, G. H. Derrick, and R. C. McPhredran, “A general modal theory for reflection gratings,” Opt. Acta 28, 1501–1516 (1981).
[CrossRef]

Ming, H.

J. X. Chen, P. Wang, X. L. Wang, Y. H. Lu, R. S. Zheng, and H. Ming, “Analytical investigation of transmission properties of metallic gratings,” Chin. Phys. Lett. 25, 4385–4387 (2008).
[CrossRef]

Moharam, M. G.

Möller, K. D.

P. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A Pure Appl. Opt. 2, 48–51 (2000).
[CrossRef]

Morris, G. M.

Nevière, M.

P. Boyer, G. Renversez, E. Popov, and M. Nevière, “Improved differential method for microstructured optical fibres,” J. Opt. A Pure Appl. Opt. 9, 728–740 (2007).
[CrossRef]

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100–16108 (2000).
[CrossRef]

Nieto-Vesperinas, M.

N. Garcia and M. Nieto-Vesperinas, “Theory of electromagnetic wave transmission through metallic gratings of subwavelength slits,” J. Opt. A Pure Appl. Opt. 9, 490–495 (2007).
[CrossRef]

Palamaru, M.

P. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A Pure Appl. Opt. 2, 48–51 (2000).
[CrossRef]

S. Astilean, P. Lalanne, and M. Palamaru, “Light transmission through metallic channels much smaller than the wavelength,” Opt. Commun. 175, 265–273 (2000).
[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,” J. Opt. A Pure Appl. Opt. 86, 1114–1117 (2001).

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,” J. Opt. A Pure Appl. Opt. 86, 1114–1117 (2001).

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999).
[CrossRef]

Popov, E.

D. Maystre, A. L. Fehrembach, and E. Popov, “Plasmonic antiresonance through subwavelength hole arrays,” J. Opt. Soc. Am. A 28, 342–355 (2011).
[CrossRef]

P. Boyer, G. Renversez, E. Popov, and M. Nevière, “Improved differential method for microstructured optical fibres,” J. Opt. A Pure Appl. Opt. 9, 728–740 (2007).
[CrossRef]

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100–16108 (2000).
[CrossRef]

Porto, J. A.

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999).
[CrossRef]

Reinisch, R.

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100–16108 (2000).
[CrossRef]

Renversez, G.

P. Boyer, G. Renversez, E. Popov, and M. Nevière, “Improved differential method for microstructured optical fibres,” J. Opt. A Pure Appl. Opt. 9, 728–740 (2007).
[CrossRef]

Shen, J. T.

J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. 94, 1–4 (2005).
[CrossRef]

Stanley, R. P.

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,” J. Opt. A Pure Appl. Opt. 86, 1114–1117 (2001).

Wang, P.

J. X. Chen, P. Wang, X. L. Wang, Y. H. Lu, R. S. Zheng, and H. Ming, “Analytical investigation of transmission properties of metallic gratings,” Chin. Phys. Lett. 25, 4385–4387 (2008).
[CrossRef]

Wang, X. L.

J. X. Chen, P. Wang, X. L. Wang, Y. H. Lu, R. S. Zheng, and H. Ming, “Analytical investigation of transmission properties of metallic gratings,” Chin. Phys. Lett. 25, 4385–4387 (2008).
[CrossRef]

Zakharian, A. R.

Zheng, R. S.

J. X. Chen, P. Wang, X. L. Wang, Y. H. Lu, R. S. Zheng, and H. Ming, “Analytical investigation of transmission properties of metallic gratings,” Chin. Phys. Lett. 25, 4385–4387 (2008).
[CrossRef]

Chin. Phys. Lett. (1)

J. X. Chen, P. Wang, X. L. Wang, Y. H. Lu, R. S. Zheng, and H. Ming, “Analytical investigation of transmission properties of metallic gratings,” Chin. Phys. Lett. 25, 4385–4387 (2008).
[CrossRef]

Eur. J. Sci. Res. (1)

A. A. Antonov, “Resonance on real and complex frequencies,” Eur. J. Sci. Res. 28, 193–204 (2009).

Int. J. Pure Appl. Sci. Technol. (1)

A. A. Antonov, “New interpretation of resonance,” Int. J. Pure Appl. Sci. Technol. 2, 1–12 (2010).

J. Opt. A Pure Appl. Opt. (4)

P. Boyer, G. Renversez, E. Popov, and M. Nevière, “Improved differential method for microstructured optical fibres,” J. Opt. A Pure Appl. Opt. 9, 728–740 (2007).
[CrossRef]

N. Garcia and M. Nieto-Vesperinas, “Theory of electromagnetic wave transmission through metallic gratings of subwavelength slits,” J. Opt. A Pure Appl. Opt. 9, 490–495 (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,” J. Opt. A Pure Appl. Opt. 86, 1114–1117 (2001).

P. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Möller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A Pure Appl. Opt. 2, 48–51 (2000).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Opt. Acta (1)

J. R. Andrewartha, G. H. Derrick, and R. C. McPhredran, “A general modal theory for reflection gratings,” Opt. Acta 28, 1501–1516 (1981).
[CrossRef]

Opt. Commun. (1)

S. Astilean, P. Lalanne, and M. Palamaru, “Light transmission through metallic channels much smaller than the wavelength,” Opt. Commun. 175, 265–273 (2000).
[CrossRef]

Opt. Express (2)

Phys. Rev. B (1)

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100–16108 (2000).
[CrossRef]

Phys. Rev. Lett. (2)

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999).
[CrossRef]

J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. 94, 1–4 (2005).
[CrossRef]

Rev. Mod. Phys. (1)

F. J. Garcia-Vidal, L. Martin-Moreno, L. Kuipers, and T. W. Ebbesen, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82, 729–787 (2010).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

General profile of the studied planar resonators.

Fig. 2.
Fig. 2.

Profile and optogeometrical parameters of the studied LMG.

Fig. 3.
Fig. 3.

Transmittance and Fs,c,± functions versus wavelength for the lamellar metallic grating of Fig. (2) under normal incidence. The parameters are ϵinf=ϵsup=ϵcav, d=0.9μm, h=1.8μm, and w=90nm. The ambient medium is air. Values of resonance order k are reported over each peak.

Fig. 4.
Fig. 4.

Map of quality factor deduced from resonance frequencies as solutions of Eq. (12) for k=1, versus |b1|/|b2| and arg(b2)arg(b1) (complex plane of b1/b2). The dashed line shows the trajectory for the CFPR resonator when ηcav changes. All reported points refer to the LMG’s resonance when h varies (see Table 4).

Fig. 5.
Fig. 5.

Schematic representation of waves inside and outside the resonator. The curves with arrows are related to propagating waves and the remaining curves to the evanescent ones. No evanescent waves exist in the case of a CFPR. On the other hand, the evanescent waves surrounding the LMG reinforce the field, confining and minimize radiative losses.

Fig. 6.
Fig. 6.

S^ given in Eq. (3) and coupling coefficient Y˜ versus resonance wavelength λ^max varying according to h and at total transmission. All other optogeometrical parameters are the same as in Fig. 3.

Tables (4)

Tables Icon

Table 1. Pair of Functions Fs,± and Fc,± Needed to Find λ^max According to the Sign of P^

Tables Icon

Table 2. Substitutions of Parameters (α, β, γ) in Eq. (2) When the Unknown Quantity ψ^max Is Replaced by e{ψr} or m{ψr}

Tables Icon

Table 3. Wavelengths and Angular Frequencies at Peak Maxima of LMG Transmittance Shown in Fig. 3: Complex Resonance Wavelengths and Angular Frequencies, and Corresponding Quality Factors

Tables Icon

Table 4. Complex Resonance Wavelengths and Angular Frequencies According to h When k=1 and for the Same LMG Studied in Fig. 3

Equations (47)

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

T=a|b1cos(ψ)+b2sin(ψ)|2,
α=βsin2(ψ^max)+γcos(ψ^max)sin(ψ^max)with{α=a|b1|2β=|b2|2|b1|2γ=2(b1·b2),
S^=γ2+2βα±γΔ2(β2+γ2),
C^=γ2+2β(βα)γΔ2(β2+γ2),
Δ=γ2+4α(βα)=4[(b1·b2)2+(a|b1|2)(|b2|2a)].
Fs,±=sin(2πλ^maxn˜h)±S^=0,
Fc,±=cos(2πλ^maxn˜h)±C^=0.
min{a|b1|22(b1·b2);a|b2|22(b1·b2)}P^max{a|b1|22(b1·b2);a|b2|22(b1·b2)}.
Δ0,
Δ=γ2+2βαγΔ,
Δ=γ2+2β(βα)±γΔ,
G=tan(ψr)+G=0,
G=b1b2=|b2|2[b1·b2i.det(b1,b2)],
Q=e{λr}2m{λr}=e{ωr}2m{ωr}.
ψr=kπatan(|b1b2|),
tan(x+iy)=sin(x)cos(x)+i.sinh(y)cosh(y)cos2(x)cosh2(y)+sin2(x)sinh2(y).
ψr=kπ+δ+i.atanh(|b1b2|ξ)with{δ=0andξ=1orδ=π2andξ=1.
S^=a|b1|2|b2|2|b1|2.
b1=a=1,
b2=i(Wcav+{Wcav}1)/2,
Q=kπatanh[2(Wcav+{Wcav}1)1].
Et(j)=exeiωtdp=+eiαpx[Ap(j)eiγp(j)(zhj)+Bp(j)eiγp(j)(zhj)],
Ht(j)=H(j)=eyη0eiωtdp=+eiαpxηp(j)[Ap(j)eiγp(j)(zhj)Bp(j)eiγp(j)(zhj)],
Et(cav)=exeiωtm=0+g˜m(x)[A˜meiγ˜mz+B˜meiγ˜m(zh)],
Ht(cav)=H(cav)=eyη0eiωtm=0+g˜m(x)η˜m[A˜meiγ˜mzB˜meiγ˜m(zh)],
xΩ,p=+eiαpxηp(inf)(δp0rp)=m=0+g˜m(x)η˜m(A˜mB˜mum),
xΩ,p=+eiαpxηp(sup)tp=m=0+g˜m(x)η˜m(A˜mumB˜m),
p=+eiαpx(δp0+rp)={m=0+g˜m(x)(A˜m+B˜mum),xΩ0,xRΩ,
p=+eiαpxtp={m=0+g˜m(x)(A˜mum+B˜m),xΩ0,xRΩ.
p=+ηp(inf)(δp0rp)g˜m,p*=η˜m(A˜mB˜mum),mN,
p=+ηp(sup)tpg˜m,p*=η˜m(A˜mumB˜m),mN.
g˜m,p=1dw/2w/2g˜m(x)eiαpxdx.
δp0+rp=m=0+(A˜m+B˜mum)g˜m,p,pZ,
tp=m=0+(A˜mum+B˜m)g˜m,p,pZ.
tp=(A˜u+B˜)g˜p,pZ,
rp=δp0+(A˜+B˜u)g˜p,pZ,
p=+ηp(inf)(δp0rp)g˜p*=η˜(A˜B˜u),
p=+ηp(sup)tpg˜p*=η˜(A˜uB˜),
([C˜(sup)η˜]uη˜+C˜(sup)η˜+C˜(inf)[C˜(inf)η˜]u)(A˜B˜)=(02η0(inf)g˜0*),
C˜(j)=p=+ηp(j)|g˜p|2.
tp=4uη0(inf)η˜g˜pg˜0*[η˜+C˜(inf)][η˜+C˜(sup)]u2[η˜C˜(inf)][η˜C˜(sup)],
rp=δp0+2η0(inf)g˜pg˜0*[η˜+C˜(sup)]+u2[η˜C˜(inf)][η˜+C˜(inf)][η˜+C˜(sup)]u2[η˜C˜(inf)][η˜C˜(sup)],
b1=1+iY˜,
b2=W˜Y˜i2(W˜+W˜1)+i2W˜Y˜2.
Y˜=1iη0|g˜0|2pZ*ηp|g˜p|2.
ΔY˜2=Δ|b2|21=2(Y˜2+|b2|21).
S^=|b1|21|b1|2+|b2|22,

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