Abstract

An analytical model based on a modal expansion method is developed to investigate the optical transmission through metal gratings. This model gives analytical expressions for the transmission as well as for the dispersion relations of the modes responsible for high transmission. These expressions are accurate even for real metals used in the visible – near-infrared wavelength range, where surface plasmon polaritons (SPP’s) are excited. The dispersion relations allow the nature of the modes to be assessed. We find that the transmission modes are hybrid between Fabry-Pérot like modes and SPP’s. It is also shown that it is important to consider different refractive indices above and below the gratings in order to determine the nature of the hybrid modes. These findings are important as they clarify the nature of the modes responsible for high transmission. It can also be useful as a design tool for metal gratings for various applications.

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2009

M. Guillaumée, L. A. Dunbar, C. Santschi, E. Grenet, R. Eckert, O. J. F. Martin, and R. P. Stanley, “Polarization sensitive silicon photodiodes using nanostructured metallic grids,” Appl. Phys. Lett. 94(19), 193503 (2009).
[CrossRef]

2008

L. Martín-Moreno and F. J. Garcia-Vidal, “Minimal model for optical transmission through holey metal films,” J. Phys. Condens. Matter 20(30), 304214 (2008).
[CrossRef]

F. de León-Pérez, G. Brucoli, F. J. García-Vidal, and L. Martín-Moreno, “Theory on the scattering of light and surface plasmon polaritons by arrays of holes and dimples in a metal film,” N. J. Phys. 10(10), 105017 (2008).
[CrossRef]

2007

S. Collin, F. Pardo, and J. L. Pelouard, “Waveguiding in nanoscale metallic apertures,” Opt. Express 15(7), 4310–4320 (2007).
[CrossRef] [PubMed]

N. Garcia and M. Nieto-Vesperinas, “Theory of electromagnetic wave transmission through metallic gratings of subwavelength slits,” J. Opt. A, Pure Appl. Opt. 9(5), 490–495 (2007).
[CrossRef]

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445(7123), 39–46 (2007).
[CrossRef] [PubMed]

2005

2003

P. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, “Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures,” Phys. Rev. B 68(12), 125404 (2003).
[CrossRef]

2002

Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88(5), 057403 (2002).
[CrossRef] [PubMed]

F. J. García-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66(15), 155412 (2002).
[CrossRef]

2001

S. Collin, F. Pardo, R. Teissier, and J. L. Pelouard, “Strong discontinuities in the complex photonic band structure of transmission metallic gratings,” Phys. Rev. B 63(3), 033107 (2001).
[CrossRef]

Y. Takakura, “Optical resonance in a narrow slit in a thick metallic screen,” Phys. Rev. Lett. 86(24), 5601–5603 (2001).
[CrossRef] [PubMed]

2000

S. Astilean, P. Lalanne, and M. Palamaru, “Light transmission through metallic channels much smaller than the wavelength,” Opt. Commun. 175(4-6), 265–273 (2000).
[CrossRef]

E. Popov, M. Neviere, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62(23), 16100–16108 (2000).
[CrossRef]

Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and airy-like formulae for one-dimensional metallic gratings,” J. Opt. A, Pure Appl. Opt. 2(1), 48–51 (2000).
[CrossRef]

1999

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83(14), 2845–2848 (1999).
[CrossRef]

1998

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[CrossRef]

1996

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B Condens. Matter 54(9), 6227–6244 (1996).
[CrossRef] [PubMed]

1993

1987

1973

C. C. Chen, “Transmission of microwave through perforated flat plates of finite thickness,” IEEE Trans. Microw. Theory Tech. 21(1), 1–6 (1973).
[CrossRef]

Astilean, S.

S. Astilean, P. Lalanne, and M. Palamaru, “Light transmission through metallic channels much smaller than the wavelength,” Opt. Commun. 175(4-6), 265–273 (2000).
[CrossRef]

Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and airy-like formulae for one-dimensional metallic gratings,” J. Opt. A, Pure Appl. Opt. 2(1), 48–51 (2000).
[CrossRef]

Barnes, W. L.

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B Condens. Matter 54(9), 6227–6244 (1996).
[CrossRef] [PubMed]

Brucoli, G.

F. de León-Pérez, G. Brucoli, F. J. García-Vidal, and L. Martín-Moreno, “Theory on the scattering of light and surface plasmon polaritons by arrays of holes and dimples in a metal film,” N. J. Phys. 10(10), 105017 (2008).
[CrossRef]

Cao, Q.

Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88(5), 057403 (2002).
[CrossRef] [PubMed]

Chavel, P.

P. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, “Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures,” Phys. Rev. B 68(12), 125404 (2003).
[CrossRef]

Chen, C. C.

C. C. Chen, “Transmission of microwave through perforated flat plates of finite thickness,” IEEE Trans. Microw. Theory Tech. 21(1), 1–6 (1973).
[CrossRef]

Collin, S.

Crouse, D.

de León-Pérez, F.

F. de León-Pérez, G. Brucoli, F. J. García-Vidal, and L. Martín-Moreno, “Theory on the scattering of light and surface plasmon polaritons by arrays of holes and dimples in a metal film,” N. J. Phys. 10(10), 105017 (2008).
[CrossRef]

Depine, R. A.

Dunbar, L. A.

M. Guillaumée, L. A. Dunbar, C. Santschi, E. Grenet, R. Eckert, O. J. F. Martin, and R. P. Stanley, “Polarization sensitive silicon photodiodes using nanostructured metallic grids,” Appl. Phys. Lett. 94(19), 193503 (2009).
[CrossRef]

Ebbesen, T. W.

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445(7123), 39–46 (2007).
[CrossRef] [PubMed]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[CrossRef]

Eckert, R.

M. Guillaumée, L. A. Dunbar, C. Santschi, E. Grenet, R. Eckert, O. J. F. Martin, and R. P. Stanley, “Polarization sensitive silicon photodiodes using nanostructured metallic grids,” Appl. Phys. Lett. 94(19), 193503 (2009).
[CrossRef]

Enoch, S.

E. Popov, M. Neviere, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62(23), 16100–16108 (2000).
[CrossRef]

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(5), 490–495 (2007).
[CrossRef]

Garcia-Vidal, F. J.

L. Martín-Moreno and F. J. Garcia-Vidal, “Minimal model for optical transmission through holey metal films,” J. Phys. Condens. Matter 20(30), 304214 (2008).
[CrossRef]

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83(14), 2845–2848 (1999).
[CrossRef]

García-Vidal, F. J.

F. de León-Pérez, G. Brucoli, F. J. García-Vidal, and L. Martín-Moreno, “Theory on the scattering of light and surface plasmon polaritons by arrays of holes and dimples in a metal film,” N. J. Phys. 10(10), 105017 (2008).
[CrossRef]

F. J. García-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66(15), 155412 (2002).
[CrossRef]

Genet, C.

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445(7123), 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-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[CrossRef]

Greffet, J. J.

Grenet, E.

M. Guillaumée, L. A. Dunbar, C. Santschi, E. Grenet, R. Eckert, O. J. F. Martin, and R. P. Stanley, “Polarization sensitive silicon photodiodes using nanostructured metallic grids,” Appl. Phys. Lett. 94(19), 193503 (2009).
[CrossRef]

Guillaumée, M.

M. Guillaumée, L. A. Dunbar, C. Santschi, E. Grenet, R. Eckert, O. J. F. Martin, and R. P. Stanley, “Polarization sensitive silicon photodiodes using nanostructured metallic grids,” Appl. Phys. Lett. 94(19), 193503 (2009).
[CrossRef]

Hugonin, J. P.

P. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, “Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures,” Phys. Rev. B 68(12), 125404 (2003).
[CrossRef]

Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and airy-like formulae for one-dimensional metallic gratings,” J. Opt. A, Pure Appl. Opt. 2(1), 48–51 (2000).
[CrossRef]

Keshavareddy, P.

Kitson, S. C.

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B Condens. Matter 54(9), 6227–6244 (1996).
[CrossRef] [PubMed]

Lalanne, P.

P. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, “Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures,” Phys. Rev. B 68(12), 125404 (2003).
[CrossRef]

Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88(5), 057403 (2002).
[CrossRef] [PubMed]

S. Astilean, P. Lalanne, and M. Palamaru, “Light transmission through metallic channels much smaller than the wavelength,” Opt. Commun. 175(4-6), 265–273 (2000).
[CrossRef]

Lalanne, Ph.

Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and airy-like formulae for one-dimensional metallic gratings,” J. Opt. A, Pure Appl. Opt. 2(1), 48–51 (2000).
[CrossRef]

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[CrossRef]

Lochbihler, H.

Mansuripur, M.

Marquier, F.

Martin, O. J. F.

M. Guillaumée, L. A. Dunbar, C. Santschi, E. Grenet, R. Eckert, O. J. F. Martin, and R. P. Stanley, “Polarization sensitive silicon photodiodes using nanostructured metallic grids,” Appl. Phys. Lett. 94(19), 193503 (2009).
[CrossRef]

Martin-Moreno, L.

F. J. García-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66(15), 155412 (2002).
[CrossRef]

Martín-Moreno, L.

F. de León-Pérez, G. Brucoli, F. J. García-Vidal, and L. Martín-Moreno, “Theory on the scattering of light and surface plasmon polaritons by arrays of holes and dimples in a metal film,” N. J. Phys. 10(10), 105017 (2008).
[CrossRef]

L. Martín-Moreno and F. J. Garcia-Vidal, “Minimal model for optical transmission through holey metal films,” J. Phys. Condens. Matter 20(30), 304214 (2008).
[CrossRef]

Moller, K. D.

Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and airy-like formulae for one-dimensional metallic gratings,” J. Opt. A, Pure Appl. Opt. 2(1), 48–51 (2000).
[CrossRef]

Moloney, J.

Neviere, M.

E. Popov, M. Neviere, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62(23), 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(5), 490–495 (2007).
[CrossRef]

Palamaru, M.

Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and airy-like formulae for one-dimensional metallic gratings,” J. Opt. A, Pure Appl. Opt. 2(1), 48–51 (2000).
[CrossRef]

S. Astilean, P. Lalanne, and M. Palamaru, “Light transmission through metallic channels much smaller than the wavelength,” Opt. Commun. 175(4-6), 265–273 (2000).
[CrossRef]

Pardo, F.

Pelouard, J. L.

Pendry, J. B.

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83(14), 2845–2848 (1999).
[CrossRef]

Popov, E.

E. Popov, M. Neviere, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62(23), 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(14), 2845–2848 (1999).
[CrossRef]

Preist, T. W.

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B Condens. Matter 54(9), 6227–6244 (1996).
[CrossRef] [PubMed]

Reinisch, R.

E. Popov, M. Neviere, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62(23), 16100–16108 (2000).
[CrossRef]

Rodier, J. C.

P. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, “Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures,” Phys. Rev. B 68(12), 125404 (2003).
[CrossRef]

Sambles, J. R.

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B Condens. Matter 54(9), 6227–6244 (1996).
[CrossRef] [PubMed]

Santschi, C.

M. Guillaumée, L. A. Dunbar, C. Santschi, E. Grenet, R. Eckert, O. J. F. Martin, and R. P. Stanley, “Polarization sensitive silicon photodiodes using nanostructured metallic grids,” Appl. Phys. Lett. 94(19), 193503 (2009).
[CrossRef]

Sauvan, C.

P. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, “Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures,” Phys. Rev. B 68(12), 125404 (2003).
[CrossRef]

Stanley, R. P.

M. Guillaumée, L. A. Dunbar, C. Santschi, E. Grenet, R. Eckert, O. J. F. Martin, and R. P. Stanley, “Polarization sensitive silicon photodiodes using nanostructured metallic grids,” Appl. Phys. Lett. 94(19), 193503 (2009).
[CrossRef]

Takakura, Y.

Y. Takakura, “Optical resonance in a narrow slit in a thick metallic screen,” Phys. Rev. Lett. 86(24), 5601–5603 (2001).
[CrossRef] [PubMed]

Teissier, R.

S. Collin, F. Pardo, R. Teissier, and J. L. Pelouard, “Strong discontinuities in the complex photonic band structure of transmission metallic gratings,” Phys. Rev. B 63(3), 033107 (2001).
[CrossRef]

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[CrossRef]

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[CrossRef]

Xie, Y.

Zakharian, A.

Appl. Opt.

Appl. Phys. Lett.

M. Guillaumée, L. A. Dunbar, C. Santschi, E. Grenet, R. Eckert, O. J. F. Martin, and R. P. Stanley, “Polarization sensitive silicon photodiodes using nanostructured metallic grids,” Appl. Phys. Lett. 94(19), 193503 (2009).
[CrossRef]

IEEE Trans. Microw. Theory Tech.

C. C. Chen, “Transmission of microwave through perforated flat plates of finite thickness,” IEEE Trans. Microw. Theory Tech. 21(1), 1–6 (1973).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

Ph. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and airy-like formulae for one-dimensional metallic gratings,” J. Opt. A, Pure Appl. Opt. 2(1), 48–51 (2000).
[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(5), 490–495 (2007).
[CrossRef]

J. Phys. Condens. Matter

L. Martín-Moreno and F. J. Garcia-Vidal, “Minimal model for optical transmission through holey metal films,” J. Phys. Condens. Matter 20(30), 304214 (2008).
[CrossRef]

N. J. Phys.

F. de León-Pérez, G. Brucoli, F. J. García-Vidal, and L. Martín-Moreno, “Theory on the scattering of light and surface plasmon polaritons by arrays of holes and dimples in a metal film,” N. J. Phys. 10(10), 105017 (2008).
[CrossRef]

Nature

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998).
[CrossRef]

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445(7123), 39–46 (2007).
[CrossRef] [PubMed]

Opt. Commun.

S. Astilean, P. Lalanne, and M. Palamaru, “Light transmission through metallic channels much smaller than the wavelength,” Opt. Commun. 175(4-6), 265–273 (2000).
[CrossRef]

Opt. Express

Phys. Rev. B

P. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, “Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures,” Phys. Rev. B 68(12), 125404 (2003).
[CrossRef]

S. Collin, F. Pardo, R. Teissier, and J. L. Pelouard, “Strong discontinuities in the complex photonic band structure of transmission metallic gratings,” Phys. Rev. B 63(3), 033107 (2001).
[CrossRef]

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[CrossRef]

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Other

The commercially available software GSolver has been used. More information can be found at http://www.gsolver.com/ (2010).

Data may be retrieved at http://www.sopra-sa.com (20010).

The commercially available software Omnisim has been used. More information can be found at http://www.photond.com/products/omnisim.htm (2010).

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, (Springer-Verlag, 1988).

A. Yariv, Optical electronics in modern communications (Oxford University Press, 2007).

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Figures (12)

Fig. 1
Fig. 1

A schematic of the studied structure and definitions used in this article for the structure dimensions and incident field directions.

Fig. 2
Fig. 2

Transmission spectra of a gold grating with p = 1000 nm, w = 50 nm, h = 600 nm, ε 1 = ε 2 = ε 3 = 1 and normal incidence obtained from: modal expansion method described in Ref. [15]. (red solid curve), RCWA (green dashed curve) and Eq. (12) (blue dotted curve).

Fig. 3
Fig. 3

(a) Transmission of a gold grating for w = 50 nm, h = 600 nm, ε 1 = ε 2 = 1, ε 3 = 2.25 and normal incidence in function of p and λ. The brighter the region, the larger is the transmission. (b) Dispersion relations of the modes obtained from the solutions of Eq. (13) (blue and red dashed curves) for the same set of parameters as in Fig. 3(a). The cyan and green solid lines correspond to the dispersion relations of SPP modes excited at the top and bottom of the grating respectively.

Fig. 4
Fig. 4

Real (blue curves) and imaginary part (red curves) of the terms (a) Y 1,1; (b) ΣY 1,n (solid lines) and Y 2 (dashed lines) for a gold grating for w = 50 nm, h = 600 nm, p = 1000, ε 1 = ε 2 = 1 and normal incidence. In panel (a) λR 1,1 and λSPP 1,1 are represented by vertical dashed lines and the phase of Y 1,1 is represented by the violet curve and the right hand axis corresponds to the phase of Y 1.

Fig. 5
Fig. 5

Schematic of the coupling mechanism between SPP and FP modes in metal gratings. In panel (a) and (b) ε 1 < ε 3 and the difference between ε 1 and ε 3 is reduced from (a) to (b). In panel (c), ε 1 = ε 3.

Fig. 6
Fig. 6

(a) Transmission of a gold grating for w = 50 nm, h = 600 nm, ε 1 = ε 2 = ε 3 = 1 and normal incidence in function of p and λ. The brighter the region, the larger is the transmission. (b) Dispersion relations of the modes obtained from the solutions of Eq. (15) (blue dotted curves) and Eq. (16) (red dashed curves) for the same set of parameters as in Fig. 6(a). The green solid curves correspond to the SPP dispersion relations.

Fig. 7
Fig. 7

Map of the magnetic field intensity (a) - (c) and the magnetic field amplitude (d) – (f) plotted in the (x, y) plane over one period (p = 500 nm) in the x direction for different wavelengths. Each figure is obtained from the simplified model. Grey rectangles represent the metal regions.

Fig. 8
Fig. 8

Map of the magnetic field intensity (a), (c) and the magnetic field amplitude (b), (d) plotted as in Fig. 7 for λ = 3000 nm and two different periods.

Fig. 9
Fig. 9

Transmission spectrum of a gold grating for w = 50 nm, h = 600 nm, λ = 3000 nm, ε 1 = ε 2 = ε 3 = 1 and normal incidence in function of p. λR 1,1 and λSPP 1,1 are represented by vertical dashed lines. The inset shows in detail the shape of the λ(1,0) transmission peak.

Fig. 10
Fig. 10

Map of the magnetic field intensity plotted as in Fig. 7 for λ = 1700 nm and p = 1693 nm.

Fig. 11
Fig. 11

(a) Transmission spectrum of a gold grating for w = 600 nm, h = 600 nm, λ = 1750 nm, ε 1 = ε 2 = ε 3 = 1 and normal incidence. The magnetic field amplitudes at λSPP + ,1 = 1750.5 nm and λSPP - ,1 = 1751.9 nm are plotted in panel (b) and (c) respectively.

Fig. 12
Fig. 12

Transmission of a gold grating for p = 1750 nm, ε 1 = ε 2 = ε 3 = 1 and normal incidence in function of w and λ. The film thickness is (a) h = 600 nm; (b) h = 200 nm. The brighter the region, the larger is the transmission.

Equations (22)

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H z ( x , y ) = exp { i [ k x , 0 x k y 1 , 0 ( y h / 2 ) ] } + n = r n exp { i [ k x , n x + k y 1 , n ( y h / 2 ) ] }  for  y h / 2 ,
H z ( x , y ) = n = t n exp { i [ k x , n x k y 3 , n ( y + h / 2 ) ] }  for  y h / 2 ,
H z ( x , y ) = m = 0 X m ( x ) [ a m exp ( i β m y ) + b m exp ( i β m y ) ] ,
I j [ a j exp ( i β j h / 2 ) + b j exp ( i β j h / 2 ) ] = n = ( r n + δ n , 0 ) K j , n ,
I j [ a j exp ( i β j h / 2 ) + b j exp ( i β j h / 2 ) ] = n = t n K j , n ,
I j = 0 w X j ( x ) X j ( x ) d x = [ 1 + ( η 2 μ j ) 2 ] w 2 + η 2 μ j 2 ,
K j , n = 0 w exp ( i k x , n x ) X j ( x ) d x .
i k y 1 , q ( r q δ q , 0 ) = i ε 1 ε 2 m = 0 β m [ a m exp ( i β m h / 2 ) b m exp ( i β m h / 2 ) ] J q , m + η 1 n = ( r n + δ n , 0 ) Q q , n ,
i k y 3 , q t q = i ε 3 ε 2 m = 0 β m [ a m exp ( i β m h / 2 ) b m exp ( i β m h / 2 ) ] J q , m + η 3 n = t n Q q , n ,
J q , m = 1 / p 0 w exp ( i k x , q x ) X m ( x ) d x ,
Q q , n = 1 / p 0 w exp [ i ( k x , n k x , q ) x ] d x .
t q = 4 ( ε 3 / ε 2 ) J q , 0 K 0 , 0 ( k y 1 , 0 + k 0 ε 1 / 2 ) 1 ( k y 3 , q + k 0 ε 1 / 2 ) 1 k y 1 , 0 Y 2 ( n = Y 3 , n + Y 2 ) ( n = Y 1 , n + Y 2 ) e i β 0 h ( n = Y 3 , n Y 2 ) ( n = Y 1 , n Y 2 ) e i β 0 h ,
( n = Y 3 , n + Y 2 ) ( n = Y 1 , n + Y 2 ) e i β 0 h ( n = Y 3 , n Y 2 ) ( n = Y 1 , n Y 2 ) e i β 0 h = 0.
tan ( β 0 h ) = i β 0 G 1 + G 3 β 0 2 + G 1 G 3 ,
n = Y 1 , n + i Y 2 cot ( β 0 h / 2 ) = 0 ,
n = Y 1 , n i Y 2 tan ( β 0 h / 2 ) = 0.
β 0 tan ( β 0 h / 2 ) = i G 1 ,
β 0 cot ( β 0 h / 2 ) = i G 1 .
Y d , n = ε d ε 2 J n , 0 K 0 , n ( k y d , n + ε d k 0 ε 1 / 2 ) .
λ R d , n = p n ε d .
λ S P P d , n = p n ε d ε ε d ε .
β 0 h = π l ,

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