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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 6 — Mar. 12, 2012
  • pp: 6521–6526
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Slot antenna as a bound charge oscillator

Jong-Ho Choe, Ji-Hun Kang, Dai-Sik Kim, and Q-Han Park  »View Author Affiliations


Optics Express, Vol. 20, Issue 6, pp. 6521-6526 (2012)
http://dx.doi.org/10.1364/OE.20.006521


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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.

© 2012 OSA

1. Introduction

The coupling of light to a subwavelength size circular hole in an infinitely thin metal was first studied rigorously by Bethe in 1944 who showed that light transmission is inversely proportional to the fourth power of wavelength in the long wavelength limit [1

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

]. This was later revised to include higher order corrections [2

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

] and also extended to the cases of finitely thick metal or finite size holes [3

3. A. Roberts, “Electromagnetic theory of diffraction by a circular aperture in a thick, perfectly conducting screen,” J. Opt. Soc. Am. A 4, 1970–1983 (1987) [CrossRef]

]. Recently, Bethe’s work received a big attention in the wake of the extraordinary optical transmission phenomenon arising in a periodic array of small holes [4

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]

]. Resonantly enhanced transmission arises even for a single hole of finite size. If the size of a hole is near the cutoff wavelength, transmission resonance arises due to the resonant coupling of an incident light to localized surface plasmons [5

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

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 Nano 2, 25–32. (2008). [CrossRef]

]. A rectangular shape hole has a simple, yet more controllable resonance feature compared to a circular hole as the ratio of two sides of rectangle is an additional parameter to choose. For a normally incident light polarized along the short side of the rectangle, it has been shown that transmission peak develops near the cutoff wavelength twice the long side, and as the ratio of two sides increases, linewidth decreases with increasing maximum transmittance [7

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]

] while the total transmitted power remains constant [7

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

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]

].

The long wavelength behavior of Bethe’s hole is also a characteristic of the Rayleigh scattering cross section. In fact, Babinet’s principle relating a hole with a complementary disk asserts that the transmission of light through a hole can be identified as the scattering of light by a disk [9

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

] which acts as a dipole scatterer in the long wavelength limit. Inclusion of resonance features in the Rayleigh scattering can be made by considering the scattering of light by a bound charge oscillator. The bound charge oscillator manifests resonant scattering and contains the Rayleigh scattering as the long wavelength limit and the Thomson scattering of free electrons as the short wavelength limit [9

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

]. As demonstrated through the slot antenna applications [10

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

], the rectangular hole is an efficient resonant scatterer. This raises an important question as to whether one can associate the resonant scattering of a rectangular hole, regarded as an optical slot antenna, with the scattering of light by a bound charge oscillator.

2. Resonance in slot antenna

We consider a slot antenna formed from a narrow rectangular hole of size a × b (ba) in a metal film patched on a dielectric substrate of refractive index n as shown in Fig. 1(a). When light is incident upon a narrow slot with polarization normal to the long side of rectangle, transmission becomes strongly enhanced due to the capacitative coupling of light to the metallic slit structure [12

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. Photonics 3, 152–156 (2009). [CrossRef]

]. Enhanced transmission has been also confirmed through rigorous calculations of the energy flow through the rectangular hole [7

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]

, 11

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. Express 17, 15652–15658 (2009). [CrossRef] [PubMed]

]. In particular, using the single-mode approximation inside the hole region and matching electromagnetic fields at the air-hole and the hole-substrate boundaries, the Poynting vector representing energy flux has been found. For a highly thin perfect electric conductor, the Poynting vector component Sznorm normalized to the incident plane wave is given by [11

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. Express 17, 15652–15658 (2009). [CrossRef] [PubMed]

]
Sznorm32π2Re[Wsubs]|Wair+Wsubs|2
(1)
where the coupling factor Wm (m = air, substrate and εair = 1, εsubs = n2) is
Wm=ab8π2dkxdkyεmk02kx2k0εmk02kx2ky2sinc2bky2(sincπ+akx2+sincπakx2)2.
(2)
Though Wm has a closed-form expression, due to the complexity of integration only numerical studies have been made previously [11

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. Express 17, 15652–15658 (2009). [CrossRef] [PubMed]

] thereby restricting further analytic understanding of the system. Here, we carry out integration explicitly by separately evaluating the real and imaginary parts of Wm. The real part of Wm results from the integration over the domain kx2+ky2εmk0 which can be evaluated readily if we keep only the leading order term in b/λ. This approximation is valid for the narrow slot case (bλ). The imaginary part of Wm, with the integral domain εmk0<kx2+ky2, admits a direct dky-integration while the remaining dkx-integration can be carried out through the contour integral with a contour as in Fig. 1(b) (detailed derivations are suppressed). The final result is
Re[Wm]=bλ(εmλ24a2)[Si(Δ+)Si(Δ)]+bπλ(εm+λ24a2)[Ci(Δ)Ci(Δ+)+ln(Δ+Δ)]2bεm1/2πacos2(πaεm1/2λ)+𝒪(b2/λ2);Δ±π±2aπεm1/2λ
Im[Wm]=12(εmλ24a2)1/2G2,42,1(12,112,12,12,0|π2b2λ2(εmλ24a2)),λ<2aεm1/2=12π(λ24a2εm)1/2G2,42,2(12,112,12,12,0|π2b2λ2(λ24a2εm)),2aεm1/2λ
(3)
where Ci, Si and G are cosine-integral, sine-integral and the Meijer G function respectively. To help understand the spectral behavior of Wm, we also keep only the leading order term in b/λ for the imaginary part of Wm. For the real part of Wm, we make an additional approximation by keeping the leading order term in a/λ which is valid since ba < λ. This brings Wm into a simple form
Wm32εm3/2ab3πλ2+ibλ(εmλ24a2)[ln(π2b2λ2|εmλ24a2|)+2γ3]
(4)
where γ (≈ 0.577) is the Euler-Gamma constant. It was noted that resonance occurs around the zero of the sum of imaginary part of W such that [11

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. Express 17, 15652–15658 (2009). [CrossRef] [PubMed]

]
Im[Wair(λres)+Wsubs(λres)]=0.
(5)
The virtue of closed-form expression Wm in Eq. (4) is that now we can find the resonance condition explicitly in terms of geometrical parameters. Solving Eq. (5) for λres, we obtain the resonance condition,
λres=2(n2+1)a.
(6)
Figure 2(a) shows that the resonance condition in Eq. (6) agrees nicely with rigorous numerical results of the coupled-mode theory. In particular, resonance wavelength is proportional to the long side of the slot and depends on the refractive index of the substrate. If the substrate is absent (n = 1), resonant wavelength becomes two times of the long side of the slot consistent with the resonance condition in rectangular waveguide [9

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

, 10

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

].

Fig. 1 (a) Schematic of a slot antenna. (b) Contours used for the dkx-integral. Wavy lines denote branch cuts emanating from ±εmk0 to infinity.
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.

3. Bound Charge Oscillator

To understand the nature of radiation from slot antenna and also for the sake of simplicity, we consider the free standing case without a substrate (n = 1). In terms of angular frequency ω (= 2πc/λ), the total scattering cross section of free standing slot antenna is obtained from Eq. (1) and Eq. (4) such that
σtotalslot=Sznorm×ab=3πc2T2ω4(ω2ω02)2Δ2(ω2)+T2ω6
(7)
where
ω0=2πcλres=πca,T=16a3π2candΔ(ω2)=ln|b24c2(ω2ω02)|+2γ3.
(8)
The logarithmic factor Δ(ω2) is a slowly varying function of ω if ωω0. The quasi-Lorentzian total cross section in Eq. (7) may be compared to the case of light scattering by a bound charge oscillator. If radiative effects are small, the total cross section of scattering of radiation by a bound charge oscillator can be written [9

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

]
σtotalB.C.=6πc2ω04Γ2ω4(ω2ω02)2+Γt2ω2
(9)
where Γ and Γt are the radiative decay and total decay constants. Note that the total cross section in Eq. (7) has a similar Lorentzian shape as in Eq. (9) but with a different line broadening factor. The ω6-dependence of the line broadening term T2ω6 comes from the fact that the energy loss of a slot antenna is due to the radiation damping. If radiation is the dominant mechanism in the energy loss of the bound charge oscillator, the motion of bound charge driven by external light field is governed by the Abraham-Lorentz equation of motion [9

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

],
mx¨=mω02x+eE0eikxiωt+mτx
(10)
where the first two terms on the r.h.s. represent the restoring and external forces respectively. The last term is a radiative reaction force responsible for the radiative energy loss. From the steady-state solution of the Abraham-Lorentz equation
x(t)=eE0m(1ω02ω2iτω3)eiωt
(11)
we may calculate the radiation field and subsequently the total scattering cross section,
σtotalB.C.A.L.=6πc2τ2ω4(ω02ω2)2+τ2ω6,
(12)
where τ (=μ0e2/6πmc) is the characteristic time of Abraham-Lorentz force. Comparing Eqs. (7) and (12) and associating τ with T/Δ, we find that the total scattering cross section of a slot antenna is indeed that of a bound charge oscillator with a factor 1/2 because we observe antenna radiation only from one side. The characteristic time τ of electron is of the order of time taken for light to travel across nucleus (10−24 second), which is too small to have any significant change on non-relativistic motions. On the contrary, radiation is the dominant damping mechanism of a slot antenna so that a slot antenna is an efficient low Q radiator as in Fig. 2(b). The characteristic time (T/Δ) of a slot antenna depends on ω but only weakly away from resonance. It is of the order of time taken for light to travel the short side of rectangle and significant radiative loss arises during time T/Δ.

To further support the oscillator model, we have compared in Fig. 3 the far-field radiation pattern of slot antenna from the FDTD calculation with the dipole radiation pattern of a bound charge oscillator. We point out that polarization directions of incident light for each cases are orthogonal (y-polarized for slot antenna and x-polarized for bound charge oscillator), thus the direction of charge oscillations are also orthogonal. They result in nearly the same radiation pattern despite the opposite polarization of incident light according to the Babinet’s principle. This strengthens our interpretation of slot antenna as a bound charge oscillator.

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.

4. Discussion

So far, we have shown that the scattering problem of a slot antenna can be identified with that of a bound charge oscillator within a certain approximation of a coupled-mode theory. Total scattering cross sections and resonance conditions are found explicitly and the radiative damping of a slot antenna is explained in terms of the Abraham-Lorentz force acting on a bound charge oscillator. The bound charge oscillator presents a simple intuitive picture for the optical slot antenna which are usually fabricated on a substrate and driven by incident light. In this regard, it is an important open question whether we could extend the oscillator model to cope with interaction between separate slots and also to include different shape slots possessing an inductive nature.

Acknowledgments

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government ( MEST) (No. 2010-0000175: 2010-0028713: 2010-0019171), and a Korea University Grant.

References and links

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. A 4, 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 Nano 2, 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. Express 16, 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. Express 17, 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. Photonics 3, 152–156 (2009). [CrossRef]

13.

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

OCIS Codes
(050.1220) Diffraction and gratings : Apertures
(050.1960) Diffraction and gratings : Diffraction theory
(050.5745) Diffraction and gratings : Resonance domain
(290.5825) Scattering : Scattering theory
(050.6624) Diffraction and gratings : Subwavelength structures

ToC Category:
Diffraction and Gratings

History
Original Manuscript: August 22, 2011
Revised Manuscript: October 17, 2011
Manuscript Accepted: February 27, 2012
Published: March 6, 2012

Citation
Jong-Ho Choe, Ji-Hun Kang, Dai-Sik Kim, and Q-Han Park, "Slot antenna as a bound charge oscillator," Opt. Express 20, 6521-6526 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-6-6521


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References

  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).

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