Abstract

We study the scattering properties of an optical slot antenna formed from a narrow rectangular hole in a metal film. We show that slot antennas can be modeled as bound charge oscillators mediating resonant light scattering. A simple closed-form expression for the scattering spectrum of a slot antenna is obtained that reveals the nature of a bound charge oscillator and also the effect of a substrate. We find that the spectral width of scattering resonance is dominated by a radiative damping caused by the Abraham-Lorentz force acting on a bound charge. The bound charge oscillator model provides not only an intuitive physical picture for the scattering of an optical slot antenna but also reasonable numerical agreements with rigorous calculations using the finite-difference time-domain method.

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References

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  1. H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev.66, 163–182 (1944).
    [CrossRef]
  2. C. J. Bouwkamp, “On Bethe’s theory of diffraction by small holes,” Philips Res. Rep.5, 321–332 (1950).
  3. A. Roberts, “Electromagnetic theory of diffraction by a circular aperture in a thick, perfectly conducting screen,” J. Opt. Soc. Am. A4, 1970–1983 (1987)
    [CrossRef]
  4. For a review, see for example, F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys.82, 729–787 (2010).
    [CrossRef]
  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. T.-H. Park, N. Mirin, J. B. Lassiter, C. L. Nehl, N. J. Halas, and P. Nordlander, “Optical properties of a nanosized hole in a thin metallic film,” ACS Nano2, 25–32. (2008).
    [CrossRef]
  7. F. J. Garcia-Vidal, E. Moreno, J. A. Porto, and L. Martin-Moreno, “Transmission of light through a single hole,” Phys. Rev. Lett.95, 103901 (2005).
    [CrossRef] [PubMed]
  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. Express16, 20484–20489 (2008).
    [CrossRef] [PubMed]
  9. J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 2001).
  10. J. D. Kraus and R. J. Marhefka, Antennas For All Applications, 3rd ed. (McGraw-Hill, 2002).
  11. J. H. Kang, J.-H. Choe, D. S. Kim, and Q-H. Park, “Substrate effect on aperture resonances in a thin metal film,” Opt. Express17, 15652–15658 (2009).
    [CrossRef] [PubMed]
  12. M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q-H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics3, 152–156 (2009).
    [CrossRef]
  13. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

2010

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

2009

J. H. Kang, J.-H. Choe, D. S. Kim, and Q-H. Park, “Substrate effect on aperture resonances in a thin metal film,” Opt. Express17, 15652–15658 (2009).
[CrossRef] [PubMed]

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q-H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics3, 152–156 (2009).
[CrossRef]

2008

T.-H. Park, N. Mirin, J. B. Lassiter, C. L. Nehl, N. J. Halas, and P. Nordlander, “Optical properties of a nanosized hole in a thin metallic film,” ACS Nano2, 25–32. (2008).
[CrossRef]

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. Express16, 20484–20489 (2008).
[CrossRef] [PubMed]

2005

F. J. Garcia-Vidal, E. Moreno, J. A. Porto, and L. Martin-Moreno, “Transmission of light through a single hole,” Phys. Rev. Lett.95, 103901 (2005).
[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]

1987

1950

C. J. Bouwkamp, “On Bethe’s theory of diffraction by small holes,” Philips Res. Rep.5, 321–332 (1950).

1944

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev.66, 163–182 (1944).
[CrossRef]

Adam, A. J. L.

Ahn, K. J.

Bethe, H. A.

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev.66, 163–182 (1944).
[CrossRef]

Bouwkamp, C. J.

C. J. Bouwkamp, “On Bethe’s theory of diffraction by small holes,” Philips Res. Rep.5, 321–332 (1950).

Choe, J.-H.

Choi, S. S.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q-H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics3, 152–156 (2009).
[CrossRef]

Ebbesen, T. W.

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

Garcia-Vidal, F. J.

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

F. J. Garcia-Vidal, E. Moreno, J. A. Porto, and L. Martin-Moreno, “Transmission of light through a single hole,” Phys. Rev. Lett.95, 103901 (2005).
[CrossRef] [PubMed]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Halas, N. J.

T.-H. Park, N. Mirin, J. B. Lassiter, C. L. Nehl, N. J. Halas, and P. Nordlander, “Optical properties of a nanosized hole in a thin metallic film,” ACS Nano2, 25–32. (2008).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 2001).

Kang, J. H.

J. H. Kang, J.-H. Choe, D. S. Kim, and Q-H. Park, “Substrate effect on aperture resonances in a thin metal film,” Opt. Express17, 15652–15658 (2009).
[CrossRef] [PubMed]

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q-H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics3, 152–156 (2009).
[CrossRef]

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. Express16, 20484–20489 (2008).
[CrossRef] [PubMed]

Kim, D. S.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q-H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics3, 152–156 (2009).
[CrossRef]

J. H. Kang, J.-H. Choe, D. S. Kim, and Q-H. Park, “Substrate effect on aperture resonances in a thin metal film,” Opt. Express17, 15652–15658 (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. Express16, 20484–20489 (2008).
[CrossRef] [PubMed]

Koo, S. M.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q-H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics3, 152–156 (2009).
[CrossRef]

Kraus, J. D.

J. D. Kraus and R. J. Marhefka, Antennas For All Applications, 3rd ed. (McGraw-Hill, 2002).

Kuipers, L.

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

Lassiter, J. B.

T.-H. Park, N. Mirin, J. B. Lassiter, C. L. Nehl, N. J. Halas, and P. Nordlander, “Optical properties of a nanosized hole in a thin metallic film,” ACS Nano2, 25–32. (2008).
[CrossRef]

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]

Marhefka, R. J.

J. D. Kraus and R. J. Marhefka, Antennas For All Applications, 3rd ed. (McGraw-Hill, 2002).

Martin-Moreno, L.

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

F. J. Garcia-Vidal, E. Moreno, J. A. Porto, and L. Martin-Moreno, “Transmission of light through a single hole,” Phys. Rev. Lett.95, 103901 (2005).
[CrossRef] [PubMed]

Mirin, N.

T.-H. Park, N. Mirin, J. B. Lassiter, C. L. Nehl, N. J. Halas, and P. Nordlander, “Optical properties of a nanosized hole in a thin metallic film,” ACS Nano2, 25–32. (2008).
[CrossRef]

Moreno, E.

F. J. Garcia-Vidal, E. Moreno, J. A. Porto, and L. Martin-Moreno, “Transmission of light through a single hole,” Phys. Rev. Lett.95, 103901 (2005).
[CrossRef] [PubMed]

Nehl, C. L.

T.-H. Park, N. Mirin, J. B. Lassiter, C. L. Nehl, N. J. Halas, and P. Nordlander, “Optical properties of a nanosized hole in a thin metallic film,” ACS Nano2, 25–32. (2008).
[CrossRef]

Nordlander, P.

T.-H. Park, N. Mirin, J. B. Lassiter, C. L. Nehl, N. J. Halas, and P. Nordlander, “Optical properties of a nanosized hole in a thin metallic film,” ACS Nano2, 25–32. (2008).
[CrossRef]

Park, D. J.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q-H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics3, 152–156 (2009).
[CrossRef]

Park, G. S.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q-H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics3, 152–156 (2009).
[CrossRef]

Park, H. R.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q-H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics3, 152–156 (2009).
[CrossRef]

Park, N. K.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q-H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics3, 152–156 (2009).
[CrossRef]

Park, Q-H.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q-H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics3, 152–156 (2009).
[CrossRef]

J. H. Kang, J.-H. Choe, D. S. Kim, and Q-H. Park, “Substrate effect on aperture resonances in a thin metal film,” Opt. Express17, 15652–15658 (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. Express16, 20484–20489 (2008).
[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]

Park, T.-H.

T.-H. Park, N. Mirin, J. B. Lassiter, C. L. Nehl, N. J. Halas, and P. Nordlander, “Optical properties of a nanosized hole in a thin metallic film,” ACS Nano2, 25–32. (2008).
[CrossRef]

Planken, P. C. M.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q-H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics3, 152–156 (2009).
[CrossRef]

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. Express16, 20484–20489 (2008).
[CrossRef] [PubMed]

Porto, J. A.

F. J. Garcia-Vidal, E. Moreno, J. A. Porto, and L. Martin-Moreno, “Transmission of light through a single hole,” Phys. Rev. Lett.95, 103901 (2005).
[CrossRef] [PubMed]

Roberts, A.

Seo, M. A.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q-H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics3, 152–156 (2009).
[CrossRef]

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. Express16, 20484–20489 (2008).
[CrossRef] [PubMed]

Suwal, O. K.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q-H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics3, 152–156 (2009).
[CrossRef]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

ACS Nano

T.-H. Park, N. Mirin, J. B. Lassiter, C. L. Nehl, N. J. Halas, and P. Nordlander, “Optical properties of a nanosized hole in a thin metallic film,” ACS Nano2, 25–32. (2008).
[CrossRef]

J. Opt. Soc. Am. A

Nat. Photonics

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q-H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics3, 152–156 (2009).
[CrossRef]

Opt. Express

Philips Res. Rep.

C. J. Bouwkamp, “On Bethe’s theory of diffraction by small holes,” Philips Res. Rep.5, 321–332 (1950).

Phys. Rev.

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev.66, 163–182 (1944).
[CrossRef]

Phys. Rev. Lett.

F. J. Garcia-Vidal, E. Moreno, J. A. Porto, and L. Martin-Moreno, “Transmission of light through a single hole,” Phys. Rev. Lett.95, 103901 (2005).
[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]

Rev. Mod. Phys.

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

Other

J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 2001).

J. D. Kraus and R. J. Marhefka, Antennas For All Applications, 3rd ed. (McGraw-Hill, 2002).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

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Figures (3)

Fig. 1
Fig. 1

(a) Schematic of a slot antenna. (b) Contours used for the dkx-integral. Wavy lines denote branch cuts emanating from ± ε m k 0 to infinity.

Fig. 2
Fig. 2

Spectral properties of a slot antenna. (a) resonance wavelength and (b) the quality factor Q vs. substrate refractive index n. (c) Scattering spectrum for four cases of slot antenna. Filled dots denote rigorous FDTD results with a = 500 μm and h = 2 μm. Dashed and solid lines represent results using the coupled-mode theory and our approximations (oscillator model). We have chosen 2a for a unit length.

Fig. 3
Fig. 3

Radiation pattern of a slot antenna in comparision with the radiation pattern of an Abraham-Lorentz bound charge oscillator. Radiation pattern of a slot antenna is obtained using the near-to-far field transformation on the FDTD result [13] of the near field at the slot interface. In FDTD calculation, we assumed a free standing slot antenna on a perfect conductor with width b = 0.01, thickness h = 0.002 and wavelength λ = 1.03 in a length unit 2a. (a) XZ cross cut and (b) YZ cross cut of radiation patterns are shown.

Equations (13)

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S z norm 32 π 2 Re [ W subs ] | W air + W subs | 2
W m = a b 8 π 2 d k x d k y ε m k 0 2 k x 2 k 0 ε m k 0 2 k x 2 k y 2 sinc 2 b k y 2 ( sinc π + a k x 2 + sinc π a k x 2 ) 2 .
Re [ W m ] = b λ ( ε m λ 2 4 a 2 ) [ Si ( Δ + ) Si ( Δ ) ] + b π λ ( ε m + λ 2 4 a 2 ) [ Ci ( Δ ) Ci ( Δ + ) + ln ( Δ + Δ ) ] 2 b ε m 1 / 2 π a cos 2 ( π a ε m 1 / 2 λ ) + 𝒪 ( b 2 / λ 2 ) ; Δ ± π ± 2 a π ε m 1 / 2 λ
Im [ W m ] = 1 2 ( ε m λ 2 4 a 2 ) 1 / 2 G 2 , 4 2 , 1 ( 1 2 , 1 1 2 , 1 2 , 1 2 , 0 | π 2 b 2 λ 2 ( ε m λ 2 4 a 2 ) ) , λ < 2 a ε m 1 / 2 = 1 2 π ( λ 2 4 a 2 ε m ) 1 / 2 G 2 , 4 2 , 2 ( 1 2 , 1 1 2 , 1 2 , 1 2 , 0 | π 2 b 2 λ 2 ( λ 2 4 a 2 ε m ) ) , 2 a ε m 1 / 2 λ
W m 32 ε m 3 / 2 a b 3 π λ 2 + i b λ ( ε m λ 2 4 a 2 ) [ ln ( π 2 b 2 λ 2 | ε m λ 2 4 a 2 | ) + 2 γ 3 ]
Im [ W air ( λ res ) + W subs ( λ res ) ] = 0.
λ res = 2 ( n 2 + 1 ) a .
σ total slot = S z norm × a b = 3 π c 2 T 2 ω 4 ( ω 2 ω 0 2 ) 2 Δ 2 ( ω 2 ) + T 2 ω 6
ω 0 = 2 π c λ res = π c a , T = 16 a 3 π 2 c and Δ ( ω 2 ) = ln | b 2 4 c 2 ( ω 2 ω 0 2 ) | + 2 γ 3.
σ total B.C. = 6 π c 2 ω 0 4 Γ 2 ω 4 ( ω 2 ω 0 2 ) 2 + Γ t 2 ω 2
m x ¨ = m ω 0 2 x + e E 0 e i k x i ω t + m τ x
x ( t ) = e E 0 m ( 1 ω 0 2 ω 2 i τ ω 3 ) e i ω t
σ total B.C.A.L. = 6 π c 2 τ 2 ω 4 ( ω 0 2 ω 2 ) 2 + τ 2 ω 6 ,

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