Abstract

We present a simple theoretical model to study the effect of a substrate on the resonance of an aperture in a thin metal film. The transmitted energy through an aperture is shown to be governed by the coupling of aperture waveguide mode to the incoming and the outgoing electromagnetic waves into the substrate region. Aperture resonance in the energy transmission thus depends critically on the refractive index of a substrate. We explain the substrate effect on aperture resonance in terms of destructive interference among evanescent modes or impedance mismatch. Our model shows an excellent agreement with a rigorous FDTD calculation and is consistent with previous experimental observations.

©2009 Optical Society of America

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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-wavelength hole arrays,” Nature 391, 667–669 (1998).
    [Crossref]
  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]
  3. M. M. J. Treacy, “Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings,” Phys. Rev. B 66, 195105 (2002).
    [Crossref]
  4. 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]
  5. K. G. Lee and Q. H. Park, “Coupling of surface plasmon polaritions and light in metallic nanoslits,” Phys. Rev. Lett. 95, 103902 (2005).
    [Crossref] [PubMed]
  6. Q. H. Park, “Optical antennas and plasmonics,” Contemporary Phys. 50, 407–423 (2009).
    [Crossref]
  7. F. J. Garcia-Vidal, E. Moreno, J. A. Porto, and L. Martin-Moreno, “Transmission of Light through a Single Rectangular Hole,” Phys. Rev. Lett. 95, 193901 (2005).
    [Crossref]
  8. M. A. Seo, A. J. L. Adam, J. H. Kang, J. W. Lee, K. J. Ahn, Q. H. Park, P. C. M. Planken, and D. S. Kim, “Near field imaging of terahertz focusing onto rectangular apertures,” Opt. Express 16, 20484–20489 (2008).
    [Crossref] [PubMed]
  9. Q. H. Park, J. H. Kang, J. W. Lee, and D. S. Kim, “Effective medium description of plasmonic metamaterials,” Opt. Express 15, 6994–6999 (2007).
    [Crossref] [PubMed]
  10. K. L. Shuford, S. K. Gray, M. A. Ratner, and G. C. Schatz, “Substrate effects on surface plasmons in single nanoholes,” Chem. Phys. Lett. 435123–126 (2007).
    [Crossref]
  11. M. A. Seo et al., “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nature Photonics 3, 152–156 (2009).
    [Crossref]
  12. J. H. Kang, D. S. Kim, and Q. H. Park, “Local capacitor model for plasmonic electric field enhancement,” Phys. Rev. Lett. 102, 093906 (2009).
    [Crossref] [PubMed]

2009 (3)

Q. H. Park, “Optical antennas and plasmonics,” Contemporary Phys. 50, 407–423 (2009).
[Crossref]

M. A. Seo et al., “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nature Photonics 3, 152–156 (2009).
[Crossref]

J. H. Kang, D. S. Kim, and Q. H. Park, “Local capacitor model for plasmonic electric field enhancement,” Phys. Rev. Lett. 102, 093906 (2009).
[Crossref] [PubMed]

2008 (1)

2007 (2)

Q. H. Park, J. H. Kang, J. W. Lee, and D. S. Kim, “Effective medium description of plasmonic metamaterials,” Opt. Express 15, 6994–6999 (2007).
[Crossref] [PubMed]

K. L. Shuford, S. K. Gray, M. A. Ratner, and G. C. Schatz, “Substrate effects on surface plasmons in single nanoholes,” Chem. Phys. Lett. 435123–126 (2007).
[Crossref]

2005 (2)

F. J. Garcia-Vidal, E. Moreno, J. A. Porto, and L. Martin-Moreno, “Transmission of Light through a Single Rectangular Hole,” Phys. Rev. Lett. 95, 193901 (2005).
[Crossref]

K. G. Lee and Q. H. Park, “Coupling of surface plasmon polaritions and light in metallic nanoslits,” Phys. Rev. Lett. 95, 103902 (2005).
[Crossref] [PubMed]

2002 (2)

M. M. J. Treacy, “Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings,” Phys. Rev. B 66, 195105 (2002).
[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]

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]

1998 (1)

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, 667–669 (1998).
[Crossref]

Adam, A. J. L.

Ahn, K. J.

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]

Ebbesen, T. W.

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, 667–669 (1998).
[Crossref]

Garcia-Vidal, F. J.

F. J. Garcia-Vidal, E. Moreno, J. A. Porto, and L. Martin-Moreno, “Transmission of Light through a Single Rectangular Hole,” Phys. Rev. Lett. 95, 193901 (2005).
[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, 2845–2848 (1999).
[Crossref]

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, 667–669 (1998).
[Crossref]

Gray, S. K.

K. L. Shuford, S. K. Gray, M. A. Ratner, and G. C. Schatz, “Substrate effects on surface plasmons in single nanoholes,” Chem. Phys. Lett. 435123–126 (2007).
[Crossref]

Kang, J. H.

Kim, D. S.

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]

Lee, J. W.

Lee, K. G.

K. G. Lee and Q. H. Park, “Coupling of surface plasmon polaritions and light in metallic nanoslits,” Phys. Rev. Lett. 95, 103902 (2005).
[Crossref] [PubMed]

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, 667–669 (1998).
[Crossref]

Martin-Moreno, L.

F. J. Garcia-Vidal, E. Moreno, J. A. Porto, and L. Martin-Moreno, “Transmission of Light through a Single Rectangular Hole,” Phys. Rev. Lett. 95, 193901 (2005).
[Crossref]

Moreno, E.

F. J. Garcia-Vidal, E. Moreno, J. A. Porto, and L. Martin-Moreno, “Transmission of Light through a Single Rectangular Hole,” Phys. Rev. Lett. 95, 193901 (2005).
[Crossref]

Park, Q. H.

Q. H. Park, “Optical antennas and plasmonics,” Contemporary Phys. 50, 407–423 (2009).
[Crossref]

J. H. Kang, D. S. Kim, and Q. H. Park, “Local capacitor model for plasmonic electric field enhancement,” Phys. Rev. Lett. 102, 093906 (2009).
[Crossref] [PubMed]

M. A. Seo, A. J. L. Adam, J. H. Kang, J. W. Lee, K. J. Ahn, Q. H. Park, P. C. M. Planken, and D. S. Kim, “Near field imaging of terahertz focusing onto rectangular apertures,” Opt. Express 16, 20484–20489 (2008).
[Crossref] [PubMed]

Q. H. Park, J. H. Kang, J. W. Lee, and D. S. Kim, “Effective medium description of plasmonic metamaterials,” Opt. Express 15, 6994–6999 (2007).
[Crossref] [PubMed]

K. G. Lee and Q. H. Park, “Coupling of surface plasmon polaritions and light in metallic nanoslits,” Phys. Rev. Lett. 95, 103902 (2005).
[Crossref] [PubMed]

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]

Planken, P. C. M.

Porto, J. A.

F. J. Garcia-Vidal, E. Moreno, J. A. Porto, and L. Martin-Moreno, “Transmission of Light through a Single Rectangular Hole,” Phys. Rev. Lett. 95, 193901 (2005).
[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, 2845–2848 (1999).
[Crossref]

Ratner, M. A.

K. L. Shuford, S. K. Gray, M. A. Ratner, and G. C. Schatz, “Substrate effects on surface plasmons in single nanoholes,” Chem. Phys. Lett. 435123–126 (2007).
[Crossref]

Schatz, G. C.

K. L. Shuford, S. K. Gray, M. A. Ratner, and G. C. Schatz, “Substrate effects on surface plasmons in single nanoholes,” Chem. Phys. Lett. 435123–126 (2007).
[Crossref]

Seo, M. A.

Shuford, K. L.

K. L. Shuford, S. K. Gray, M. A. Ratner, and G. C. Schatz, “Substrate effects on surface plasmons in single nanoholes,” Chem. Phys. Lett. 435123–126 (2007).
[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, 667–669 (1998).
[Crossref]

Treacy, M. M. J.

M. M. J. Treacy, “Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings,” Phys. Rev. B 66, 195105 (2002).
[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, 667–669 (1998).
[Crossref]

Chem. Phys. Lett. (1)

K. L. Shuford, S. K. Gray, M. A. Ratner, and G. C. Schatz, “Substrate effects on surface plasmons in single nanoholes,” Chem. Phys. Lett. 435123–126 (2007).
[Crossref]

Contemporary Phys. (1)

Q. H. Park, “Optical antennas and plasmonics,” Contemporary Phys. 50, 407–423 (2009).
[Crossref]

Nature (1)

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, 667–669 (1998).
[Crossref]

Nature Photonics (1)

M. A. Seo et al., “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nature Photonics 3, 152–156 (2009).
[Crossref]

Opt. Express (2)

Phys. Rev. B (1)

M. M. J. Treacy, “Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings,” Phys. Rev. B 66, 195105 (2002).
[Crossref]

Phys. Rev. Lett. (5)

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]

K. G. Lee and Q. H. Park, “Coupling of surface plasmon polaritions and light in metallic nanoslits,” Phys. Rev. Lett. 95, 103902 (2005).
[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]

F. J. Garcia-Vidal, E. Moreno, J. A. Porto, and L. Martin-Moreno, “Transmission of Light through a Single Rectangular Hole,” Phys. Rev. Lett. 95, 193901 (2005).
[Crossref]

J. H. Kang, D. S. Kim, and Q. H. Park, “Local capacitor model for plasmonic electric field enhancement,” Phys. Rev. Lett. 102, 093906 (2009).
[Crossref] [PubMed]

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Figures (4)
Fig. 1.
Fig. 1. Rectangular aperture of size a×b in a metal film of thickness h deposited on an substrate.

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Fig. 2.
Fig. 2. Plot of coupling strengths Wa and Ws with aperture size a=0.5, b=0.05. We choose 2a as the unit length and the refractive index ns =1.7. It is interesting to note that the zero of imaginary part of Wave indeed occurs around the maximum of Snorm z in Fig. 3.

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Fig. 3.
Fig. 3. (a) Aperture resonance in the normalized energy flow calculated by a diffraction theory with a=0.5, b=0.05, ns =1.7. (b), (c) and (d) are the FDTD results for the gold film case with aperture sizes, 200µm×20µm,400nm×40nm,200nm×40nm respectively.

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Fig. 4.
Fig. 4. FDTD calculation of the resonant mode electric and magnetic field maps in the plane y=b/2 parallel to the xz-plane. Each field components are measured through the time averaged magnitude without (air) and with (sub) a substrate. Cross cuts of field profiles are along the horizontal center line and the metal thickness h=0.05 and a=0.5, b=0.05.

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Equations (14)

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H→I = ε0μ0 ∫dkxdky[x^δ(kx)δ(ky)e−ik1z(z–h/2)+g→(kx,ky)eiΘ+ik1z(z–h/2)
H→III = ε0μ0 ∫dkxdkyeiΘf→(kx,ky)e−ik3z(z+h⁄2),
∇×H→=∂D→∂t=−iωε0E→=−ik0μ0ε0E→.
ExII=0,EyII=sin(πxa)[Ae−iβz+Biβz],EzII=0,HxII=βk0ε0μ0sin(πxa)[Ae−iβz−Beiβz],HyII=0,HzII=−iπk0aε0μ0cos(πxa)[Ae−iβz+Beiβz],
gz = − 1k1z (kxgx+kygy)=−kxk1zky2+k1z2gx,gy=−kxkyky2+k1z2gx.
A = 4πD e−iβh⁄2 (βk0+W3),B=4πDeiβh⁄2(βk0−W3),fx=ky2+(k3z)2k0k3zabJπ3βk0D
gx = δ (kx)δ(ky)− ky2+(k1z)2k0k3z abJπ3D [βk0cos(βh)−iW3sin(βh)]
D = − i sin (βh)(β2k02+W1W3)+βk0(W1+W3)cos(βh);Wm=∫dkxdkyεmk02−kx2k0kmzab8π2J2.
J = 2ab ∫0adx ∫0bdy sin (πxa) eiΘ = sinc(bky2)[sinc(π2+akx2)+sinc(π2−akx2)]
Ey = 8πD sin (πxa) [βk0cos[β(z+h2)]−iW3sin[β(z+h2)]] ,
Hx = 8πD ε0μ0 sin (πxa) [−iβ2k02sin[β(z+h2)]+βk0W3cos[β(z+h2)]] .
Sz = 12 Re (ExHy*−EyHx*)=−12Re(EyHx*)=−32∣β∣2π2k02ε0μ0sin2(πxa)Re(W3)∣D∣2,
Sznorm = 32∣β∣2π2k02 Re(W3)∣D∣2 ,
Sznorm = 32π2 Re(W3)∣W1+W3∣2 .

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