## 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 Optical Society of America

## 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]. This was later revised to include higher order corrections [2] and also extended to the cases of finitely thick metal or finite size holes [3]. 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]. 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, 6]. 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] while the total transmitted power remains constant [7, 8].

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] 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]. As demonstrated through the slot antenna applications [10], 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.

In this paper, we demonstrate that the slot antenna can indeed be regarded as a bound charge oscillator within a reasonable approximation scheme. In particular, we show that the transmission spectrum of a rectangular hole obtained by a rigorous modal expansion method can be approximated to a simple expression of the scattering spectrum of a bound charge oscillator. From this, a closed form of the resonance condition that reveals the effect of a dielectric substrate [11] is obtained. The spectral width of scattering resonance is also obtained in terms of structural parameters and the refractive index of a substrate. We show that spectral width is determined by a radiative damping caused by the Abraham-Lorentz force acting on a bound charge. The bound charge oscillator model presents a simple physical interpretation of an optical slot antenna and shows good agreements with more rigorous numerical calculations using the finite-difference time-domain method.

## 2. Resonance in slot antenna

We consider a slot antenna formed from a narrow rectangular hole of size *a* × *b* (*b* ≪ *a*) 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]. Enhanced transmission has been also confirmed through rigorous calculations of the energy flow through the rectangular hole [7, 11]. 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
${S}_{z}^{\mathit{norm}}$ normalized to the incident plane wave is given by [11]

*W*(

_{m}*m*= air, substrate and

*ε*

_{air}= 1,

*ε*

_{subs}=

*n*

^{2}) is

*W*has a closed-form expression, due to the complexity of integration only numerical studies have been made previously [11] thereby restricting further analytic understanding of the system. Here, we carry out integration explicitly by separately evaluating the real and imaginary parts of

_{m}*W*. The real part of

_{m}*W*results from the integration over the domain ${k}_{x}^{2}+{k}_{y}^{2}\le {\epsilon}_{m}{k}_{0}$ which can be evaluated readily if we keep only the leading order term in

_{m}*b/λ*. This approximation is valid for the narrow slot case (

*b*≪

*λ*). The imaginary part of

*W*, with the integral domain ${\epsilon}_{m}{k}_{0}<{k}_{x}^{2}+{k}_{y}^{2}$, admits a direct d

_{m}*k*-integration while the remaining d

_{y}*k*-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

_{x}*G*are cosine-integral, sine-integral and the Meijer G function respectively. To help understand the spectral behavior of

*W*, we also keep only the leading order term in

_{m}*b/λ*for the imaginary part of

*W*. For the real part of

_{m}*W*, we make an additional approximation by keeping the leading order term in

_{m}*a/λ*which is valid since

*b*≪

*a*<

*λ*. This brings

*W*into a simple form

_{m}*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]

*W*in Eq. (4) is that now we can find the resonance condition explicitly in terms of geometrical parameters. Solving Eq. (5) for

_{m}*λ*

_{res}, we obtain the resonance condition, 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, 10].

By putting Eq. (4) into Eq. (1), we get normalized transmission spectrum
${S}_{z}^{\text{norm}}$ in a closed form. This Lorentzian shape spectrum and the resonance condition in Eq. (6) characterize the nature of slot antenna resonance. Figure 2 describes spectral properties of slot antenna resonance. It shows that both the resonant wavelength and the quality factor depend on the refractive index *n* of substrate while the dependence is much weaker for the quality factor. As expected, the quality factor increases as the slot becomes narrower. In Fig. 2(c), Lorentzian shape spectrums calculated for four distinct cases of *n* using three different methods are compared. Dashed lines representing results of the coupled-mode theory show that resonance peaks are slightly blue-shifted compared to the rigorous numerical results using the Finite Difference Time Domain(FDTD) method. This is caused by the single-mode approximation adopted in the coupled-mode theory calculation which does not apply well for thin layer holes. Solid lines representing closed form of
${S}_{z}^{\text{norm}}$ obtained by substituting Eq. (4) into Eq. (1) show reasonable agreements with other results. This confirms once again the validity of our narrow slot approximation. The real importance, however, lies in the fact that the explicit closed form of
${S}_{z}^{\text{norm}}$ admits a simple physical interpretation of slot antenna as a bound charge oscillator as explained in next section.

## 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* (= 2*πc/λ*), the total scattering cross section of free standing slot antenna is obtained from Eq. (1) and Eq. (4) such that

*ω*

^{2}) is a slowly varying function of

*ω*if

*ω*≠

*ω*

_{0}. 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]

*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*

_{t}*ω*

^{6}-dependence of the line broadening term

*T*

^{2}

*ω*

^{6}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],

*τ*(=

*μ*

_{0}

*e*

^{2}/

*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.

## 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

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