Abstract

This article presents an analytical solution of the effective index of the fundamental waveguide mode of one- dimensional (1D) metallodielectric grating for TM polarization. In contrast to the existing numerical solution involving a transcendental equation, it is shown that the square of the effective index (nEff) of the fundamental waveguide mode of 1D grating is inversely proportional to the slit width (w) and the refractive index (nm) of the ridge material and varies linearly with the incident wavelength (λ). Further, it has also been demonstrated that the dependence of nEff on the grating period (P) and the incidence angle (θ) is minimal. Agreement between the results obtained using the solution presented in this article and published data is excellent.

© 2011 Optical Society of America

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References

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  1. P. 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 2, 48–51 (2000).
    [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. Y. Pang, C. Genet, and T. Ebbesen, “Optical transmission through subwavelength slit apertures in metallic films,” Opt. Commun. 280, 10–15 (2007).
    [CrossRef]
  4. T. J. Kim, T. Thio, T. W. Ebbesen, D. E. Grupp, and H. J. Lezec, “Control of optical transmission through metals perforated with subwavelength hole arrays,” Opt. Lett. 24, 256–258 (1999).
    [CrossRef]
  5. 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]
  6. Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88, 057403 (2002).
    [CrossRef] [PubMed]
  7. P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916(1982).
    [CrossRef]
  8. A. V. Tishchenko, “Phenomenological representation of deep and high contrast lamellar gratings by means of the modal method,” Opt. Quantum Electron. 37, 309–330 (2005).
    [CrossRef]
  9. Y. Takakura, “Optical resonance in a narrow slit in a thick metallic screen,” Phys. Rev. Lett. 86, 5601–5603 (2001).
    [CrossRef] [PubMed]
  10. F. J. García-Vidal and L. Martín-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metal,” Phys. Rev. B 66, 155412 (2002).
    [CrossRef]
  11. T. Gaylord and M. Moharam, “Analysis and applications of optical diffraction by gratings,” in Proceedings of the IEEE (IEEE, 1985), pp. 894–937.
    [CrossRef]
  12. T. Clausnitzer, T. Kämpfe, E. B. Kley, A. Tünnermann, U. Peschel, A. V. Tishchenko, and O. Parriaux, “An intelligible explanation of highly-efficient diffraction in deep dielectric rectangular transmission gratings,” Opt. Express 13, 10448–10456(2005).
    [CrossRef] [PubMed]
  13. K. R. Catchpole, “A conceptual model of the diffuse transmittance of lamellar diffraction gratings on solar cells,” J. Appl. Phys. 102, 013102 (2007).
    [CrossRef]
  14. N. M. Lyndin, O. Parriaux, and A. V. Tishchenko, “Modal analysis and suppression of the Fourier modal method instabilities in highly conductive gratings,” J. Opt. Soc. Am. A 24, 3781–3788(2007).
    [CrossRef]
  15. M. Foresti, L. Menez, and A. V. Tishchenko, “Modal method in deep metal–dielectric gratings: the decisive role of hidden modes,” J. Opt. Soc. Am. A 23, 2501–2509 (2006).
    [CrossRef]
  16. R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Pearson Addison-Wesley, 2006), Vol.  2.
  17. A. T. M. A. Rahman, K. Vasilev, and P. Majewski, “Designing 1D grating for extraordinary optical transmission for TM polarization,” Photon. Nanostr. Fundam. Appl., doi:10.1016/j.photonics.2011.08.002 (posted 17 August 2011, in press).
    [CrossRef]
  18. E.D.Palik, ed., Handbook of Optical Constants of Solids(Academic, 1985).

2007 (3)

Y. Pang, C. Genet, and T. Ebbesen, “Optical transmission through subwavelength slit apertures in metallic films,” Opt. Commun. 280, 10–15 (2007).
[CrossRef]

K. R. Catchpole, “A conceptual model of the diffuse transmittance of lamellar diffraction gratings on solar cells,” J. Appl. Phys. 102, 013102 (2007).
[CrossRef]

N. M. Lyndin, O. Parriaux, and A. V. Tishchenko, “Modal analysis and suppression of the Fourier modal method instabilities in highly conductive gratings,” J. Opt. Soc. Am. A 24, 3781–3788(2007).
[CrossRef]

2006 (1)

2005 (2)

2002 (2)

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

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

2001 (1)

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

2000 (2)

P. 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 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]

1999 (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]

T. J. Kim, T. Thio, T. W. Ebbesen, D. E. Grupp, and H. J. Lezec, “Control of optical transmission through metals perforated with subwavelength hole arrays,” Opt. Lett. 24, 256–258 (1999).
[CrossRef]

1982 (1)

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916(1982).
[CrossRef]

Astilean, S.

P. 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 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]

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, 057403 (2002).
[CrossRef] [PubMed]

Catchpole, K. R.

K. R. Catchpole, “A conceptual model of the diffuse transmittance of lamellar diffraction gratings on solar cells,” J. Appl. Phys. 102, 013102 (2007).
[CrossRef]

Clausnitzer, T.

Ebbesen, T.

Y. Pang, C. Genet, and T. Ebbesen, “Optical transmission through subwavelength slit apertures in metallic films,” Opt. Commun. 280, 10–15 (2007).
[CrossRef]

Ebbesen, T. W.

Feynman, R. P.

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Pearson Addison-Wesley, 2006), Vol.  2.

Foresti, M.

Garcia-Vidal, F. J.

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]

García-Vidal, F. J.

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

Gaylord, T.

T. Gaylord and M. Moharam, “Analysis and applications of optical diffraction by gratings,” in Proceedings of the IEEE (IEEE, 1985), pp. 894–937.
[CrossRef]

Genet, C.

Y. Pang, C. Genet, and T. Ebbesen, “Optical transmission through subwavelength slit apertures in metallic films,” Opt. Commun. 280, 10–15 (2007).
[CrossRef]

Grupp, D. E.

Hugonin, J. P.

P. 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 2, 48–51 (2000).
[CrossRef]

Kämpfe, T.

Kim, T. J.

Kley, E. B.

Lalanne, P.

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

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. Moller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A 2, 48–51 (2000).
[CrossRef]

Leighton, R. B.

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Pearson Addison-Wesley, 2006), Vol.  2.

Lezec, H. J.

Lyndin, N. M.

Majewski, P.

A. T. M. A. Rahman, K. Vasilev, and P. Majewski, “Designing 1D grating for extraordinary optical transmission for TM polarization,” Photon. Nanostr. Fundam. Appl., doi:10.1016/j.photonics.2011.08.002 (posted 17 August 2011, in press).
[CrossRef]

Martín-Moreno, L.

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

Menez, L.

Moharam, M.

T. Gaylord and M. Moharam, “Analysis and applications of optical diffraction by gratings,” in Proceedings of the IEEE (IEEE, 1985), pp. 894–937.
[CrossRef]

Moller, K. D.

P. 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 2, 48–51 (2000).
[CrossRef]

Palamaru, M.

P. 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 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]

Pang, Y.

Y. Pang, C. Genet, and T. Ebbesen, “Optical transmission through subwavelength slit apertures in metallic films,” Opt. Commun. 280, 10–15 (2007).
[CrossRef]

Parriaux, O.

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, 2845–2848 (1999).
[CrossRef]

Peschel, U.

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]

Rahman, A. T. M. A.

A. T. M. A. Rahman, K. Vasilev, and P. Majewski, “Designing 1D grating for extraordinary optical transmission for TM polarization,” Photon. Nanostr. Fundam. Appl., doi:10.1016/j.photonics.2011.08.002 (posted 17 August 2011, in press).
[CrossRef]

Sanda, P. N.

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916(1982).
[CrossRef]

Sands, M.

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Pearson Addison-Wesley, 2006), Vol.  2.

Sheng, P.

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916(1982).
[CrossRef]

Stepleman, R. S.

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916(1982).
[CrossRef]

Takakura, Y.

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

Thio, T.

Tishchenko, A. V.

Tünnermann, A.

Vasilev, K.

A. T. M. A. Rahman, K. Vasilev, and P. Majewski, “Designing 1D grating for extraordinary optical transmission for TM polarization,” Photon. Nanostr. Fundam. Appl., doi:10.1016/j.photonics.2011.08.002 (posted 17 August 2011, in press).
[CrossRef]

J. Appl. Phys. (1)

K. R. Catchpole, “A conceptual model of the diffuse transmittance of lamellar diffraction gratings on solar cells,” J. Appl. Phys. 102, 013102 (2007).
[CrossRef]

J. Opt. A (1)

P. 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 2, 48–51 (2000).
[CrossRef]

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

Opt. Commun. (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]

Y. Pang, C. Genet, and T. Ebbesen, “Optical transmission through subwavelength slit apertures in metallic films,” Opt. Commun. 280, 10–15 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

A. V. Tishchenko, “Phenomenological representation of deep and high contrast lamellar gratings by means of the modal method,” Opt. Quantum Electron. 37, 309–330 (2005).
[CrossRef]

Phys. Rev. B (2)

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

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916(1982).
[CrossRef]

Phys. Rev. Lett. (3)

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

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]

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

Other (4)

T. Gaylord and M. Moharam, “Analysis and applications of optical diffraction by gratings,” in Proceedings of the IEEE (IEEE, 1985), pp. 894–937.
[CrossRef]

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Pearson Addison-Wesley, 2006), Vol.  2.

A. T. M. A. Rahman, K. Vasilev, and P. Majewski, “Designing 1D grating for extraordinary optical transmission for TM polarization,” Photon. Nanostr. Fundam. Appl., doi:10.1016/j.photonics.2011.08.002 (posted 17 August 2011, in press).
[CrossRef]

E.D.Palik, ed., Handbook of Optical Constants of Solids(Academic, 1985).

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Figures (7)

Fig. 1
Fig. 1

1D lamellar grating.

Fig. 2
Fig. 2

Real part of the effective index of the fundamental mode as a function of slit width corresponding to n d = 1 , θ = 0 ° , P = 900 nm , λ = 1433 nm for silver gratings.

Fig. 3
Fig. 3

Real part of the effective index of the fundamental mode as a function of slit width corresponding to n d = 1 , θ = 0 ° , P = 900 nm , λ = 1433 nm for silver gratings.

Fig. 4
Fig. 4

(a) Real and (b) imaginary components of the effective index of the fundamental mode as a function of Im ( n m ) while keeping Re ( n m ) constant corresponding to n d = 1 , θ = 0 ° , w = 21 nm , P = 150 nm , λ = 1500 nm . Grating parameters have been taken from [17].

Fig. 5
Fig. 5

(a) Real and (b) imaginary components of the effective index of the fundamental mode as a function of λ, where n d = 1 , θ = 0 ° , w = 21 nm , and P = 150 nm . n m = 0.530 + 9.5070 i is that of gold at λ = 1500 nm . Grating parameters have been taken from [17].

Fig. 6
Fig. 6

Re ( n Eff ) as a function of (a) P and (b) incidence angle θ corresponding to n d = 1 , w = 21 nm , and λ = 1500 nm . For (b), P is equal to 150 nm .

Fig. 7
Fig. 7

Zeroth-order transmittance corresponding to P = 3.50 μm , w = 0.50 μm , h = 4.00 μm , n d = 1 , and θ = 0 ° [6]. n m is that of gold [18]. Transmission efficiency is based on the model of [1, 6], where n Eff is needed to complete the calculation. In [6], n Eff has been found using a technique called method of line, while, in plotting this graph, we have used Eq. (12).

Tables (1)

Tables Icon

Table 1 Effective Index Corresponding to the Fundamental Mode with n d = 1 a

Equations (12)

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cos ( k 0 P sin θ ) cos ( β r P ) cos ( α f P ) + 1 2 [ ϵ m α β + β ϵ m α ] sin ( β r P ) sin ( α f P ) = 0 ,
cos ( k 0 P sin θ ) cos ( k 0 n m r P ) cos ( k 0 ρ f P ) + 1 2 [ ϵ m α β + β ϵ m α ] sin ( k 0 n m r P ) sin ( k 0 ρ f P ) = 0.
cos ( k 0 P sin θ ) cos q cos p + 1 2 n m ρ sin q sin p = 0 ,
1 2 n m ρ ( p p 3 3 ! + p 5 5 ! ) sin q ( 1 p 2 2 ! + p 4 4 ! p 6 6 ! + ) cos q + cos ( k 0 P sin θ ) = 0 , n m 4 π f γ ( p 2 p 4 3 ! + p 6 5 ! ) sin q ( 1 p 2 2 ! + p 4 4 ! p 6 6 ! + ) cos q + cos ( k 0 P sin θ ) = 0.
A p 4 B p 2 + C = 0 ,
p 2 = B ± B 2 4 A C 2 A , n Eff 2 = 1 3 λ 2 4 π 2 w 2 [ 1 + 1 D ± 1 [ 2 cos q 8 cos ( k 0 P sin θ ) 3 cos q ] 1 D + 1 D 2 ] ,
n Eff 2 = 1 3 λ 2 4 π 2 w 2 [ 1 + π w π w + i λ n m ± [ 1 [ 2 3 8 cos ( k 0 P sin θ ) exp { i k 0 ( P w ) η } 3 cosh { k 0 ( P w ) κ } ] π w π w + i λ n m + π 2 w 2 ( π w + i λ n m ) 2 ] 1 / 2 ] .
n Eff 2 = 1 3 λ 2 4 π 2 w 2 [ 1 + π w π w + i λ n m ± [ 1 [ 1 3 4 cos ( k 0 P sin θ ) exp { i k 0 ( P w ) η } 3 cosh { k 0 ( P w ) κ } ] π w π w + i λ n m + π 2 w 2 2 ( π w + i λ n m ) 2 ] ] .
n Eff 2 = 1 [ 1 cos ( k 0 P sin θ ) exp { i k 0 ( P w ) η } cosh { k 0 ( P w ) κ } ] λ 2 π w ( π w + i λ n m ) + 3 λ 2 8 ( π w + i λ n m ) 2 = 1 + i [ 1 cos ( k 0 P sin θ ) exp { i k 0 ( P w ) η } cosh { k 0 ( P w ) κ } ] λ π n m w ( 1 i π w λ n m ) 3 8 n m 2 ( 1 i π w λ n m ) 2 .
n Eff 2 = 1 [ 11 8 n m 2 i λ π w n m ] + [ 1 n m 2 i λ π w n m ] cos ( k 0 P sin θ ) cosh { k 0 ( P w ) κ } exp { i k 0 ( P w ) η } .
n Eff 2 = 1 [ 11 8 n m 2 i λ π w n m ] .
n Eff 2 = 1 + i λ π w n m .

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