## 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

## 1. Introduction

Resonance of electromagnetic(EM) waves arise in various forms in subwavelength size metallic apertures. In contrast to the well-known plasmonic resonance in a periodic array of holes or slits [1]–[5], a single aperture can support dipole type resonance since it presents a complementary structure of optical dipole antenna [6]. It was pointed out that a rectangular hole in a highly conducting metal enhances transmission resonantly when the half wavelength *λ*/2 is equal to the long side *a* of a rectangle, and the transmitted energy becomes nearly independent of the short side *b* of a rectangular aperture [7]. These features have been confirmed experimentally using the terahertz EM wave [8]. Experimental results, however, indicate that the resonance condition seems to be modified by *λ*/2≈*n _{s}a* in the presence of a substrate with the refractive index

*n*. This is rather surprising since the aperture, filled with air as illustrated in Fig. 1, possesses the lowest resonant mode of wavelength 2

_{s}*a*independently of a substrate. If true, this will make an important application as we can measure the refractive index of a substrate in a spatially highly localized way. Nevertheless, the substrate effect so far has remained a curiosity without a clear physical understanding.

In this paper, we present a theoretical explanation for the substrate effect under the perfectly conducting metal approximation. The perfect conductor assumption is strictly valid only in the terahertz and microwave frequency regimes. In the optical regime, we nevertheless find that our model shows a good qualitative agreement with a rigorous FDTD (Finite Difference Time Domain) result taking into account of the optical metal dispersion. We show that the energy transmission through an aperture in a metal film is governed by the coupling strength of EM waves to the internal mode at the air-aperture (*W _{a}*) and the aperture-substrate (

*W*) interfaces. We find that if the thickness

_{s}*h*of metal film is very thin (

*h*≪

*λ*), the transmitted energy is inversely proportional to the absolute square of the average coupling strength, i.e., ∝1/|

*W*+

_{s}*W*|

_{a}^{2}where

*W*is dependent on the refractive index of a substrate. We show that the effect of a substrate in general is to shift the resonance peak to

_{s}*λ*/2=

*n*where the refractive index

_{res}a*n*depends critically on

_{res}*h*and lies in between 1 and

*n*. A simple diffraction theory calculation [9] is provided to account for these predictions which shows an excellent agreement with a rigorous but time consuming FDTD result. We discuss the substrate effect on aperture resonance in terms of destructive interference among evanescent modes or impedance mismatching.

_{s}## 2. Diffraction theory calculation

Consider a TM polarized light incident vertically onto an aperture of size *a*×*b* in a metal film (-*h*/2<*z*<*h*/2) as illustrated in Fig. 1. We denote the incident and reflected EM waves in region I and the transmitted EM waves in region III by

where Θ=*k _{x}*(

*x*-

*a*/2)+

*k*(

_{y}*y*-

*b*/2), ${k}_{\mathrm{mz}}=\pm \sqrt{{\epsilon}_{m}{k}_{0}^{2}-{k}_{x}^{2}-{k}_{y}^{2}}$

*m*= 1, 3 and

*ε*is a dielectric constant. Here, a harmonic time factor

_{m}*e*

^{-iωt}is suppressed and the sign of

*k*is chosen such that the imaginary part of

_{mz}*k*is positive. The electric field vector

_{mz}*E*⃗ is determined through the Maxwell’s equation

Inside the aperture, we make a single mode approximation by assuming that only the *TE*
_{10} mode becomes dominantly coupled to the incident wave. The validity of this assumption will be justified later. Explicitly, the *TE*
_{10} mode inside a rectangular aperture is described by

where *β*
_{2}=*k*
^{2}
_{0}-(*π*/*a*)^{2}. The divergence free condition of *H*⃗ (∇·*H*⃗=0 =*k*⃗·*g*⃗) and the vanishing of *E _{x}* at

*z*=

*h*/2 gives rise to

Similar relations hold for *f*⃗ with k1z replaced by -*k*
_{3z}. All undetermined coefficients are fixed through the boundary condition such as the continuity of *E _{y}* and

*H*at the interfaces

_{x}*z*=±

*h*/2. To solve explicitly for momentum variables

*f*⃗,

*g*⃗,

*A*and

*B*, we take appropriate inverse Fourier transforms of boundary matching equations and find after a straightforward calculation that

$${g}_{x}=\delta \left({k}_{x}\right)\delta \left({k}_{y}\right)-\genfrac{}{}{0.1ex}{}{{k}_{y}^{2}+{\left({k}_{1z}\right)}^{2}}{{k}_{0}{k}_{3z}}\genfrac{}{}{0.1ex}{}{\mathrm{abJ}}{{\pi}^{3}D}\left[\genfrac{}{}{0.1ex}{}{\beta}{{k}_{0}}\mathrm{cos}\left(\mathrm{\beta h}\right)-{\mathrm{iW}}_{3}\mathrm{sin}\left(\mathrm{\beta h}\right)\right]$$

where the denominator *D* and the coupling coefficients *J* and *W _{m}*,

*m*=1,3 are defined by

$$J=\genfrac{}{}{0.1ex}{}{2}{\mathrm{ab}}{\int}_{0}^{a}\mathrm{dx}{\int}_{0}^{b}\mathrm{dy}\mathrm{sin}\left(\genfrac{}{}{0.1ex}{}{\mathrm{\pi x}}{a}\right){e}^{i\Theta}=\mathrm{sinc}\left(\genfrac{}{}{0.1ex}{}{{\mathrm{bk}}_{y}}{2}\right)\left[\mathrm{sinc}(\genfrac{}{}{0.1ex}{}{\pi}{2}+\genfrac{}{}{0.1ex}{}{{\mathrm{ak}}_{x}}{2})+\mathrm{sinc}(\genfrac{}{}{0.1ex}{}{\pi}{2}-\genfrac{}{}{0.1ex}{}{{\mathrm{ak}}_{x}}{2})\right]$$

Thus, the electromagnetic field components of *TE*
_{10} mode inside the aperture are

$${H}_{x}=\genfrac{}{}{0.1ex}{}{8}{\mathrm{\pi D}}\sqrt{\genfrac{}{}{0.1ex}{}{{\epsilon}_{0}}{{\mu}_{0}}}\mathrm{sin}\left(\genfrac{}{}{0.1ex}{}{\mathrm{\pi x}}{a}\right)\left[-i\genfrac{}{}{0.1ex}{}{{\beta}^{2}}{{k}_{0}^{2}}\mathrm{sin}\left[\beta \right(z+\genfrac{}{}{0.1ex}{}{h}{2}\left)\right]+\genfrac{}{}{0.1ex}{}{\beta}{{k}_{0}}{W}_{3}\mathrm{cos}\left[\beta (z+\genfrac{}{}{0.1ex}{}{h}{2})\right]\right].$$

The resulting energy flux is described by the time averaged Poynting vector component *S _{z}*,

$${S}_{z}^{\mathrm{norm}}=\genfrac{}{}{0.1ex}{}{32{\mid \beta \mid}^{2}}{{\pi}^{2}{k}_{0}^{2}}\genfrac{}{}{0.1ex}{}{\mathrm{Re}\left({W}_{3}\right)}{{\mid D\mid}^{2}},$$

where *S ^{norm}_{z}* is

*S*normalized by the incident energy flux of plane wave impinging on the aperture region.

_{z}## 3. Substrate effect

A closed expression of field amplitudes and energy flux as given in Eqs. (7) and (8) allows us to analyze the substrate effect on aperture resonance. Firstly, we consider a thin metal film satisfying the condition *βh*≪1. In this case, the denominator factor *D* in Eq. (6) becomes approximately *D*≈(*β*/*k*
_{0})(*W*
_{1}+*W*
_{3}) so that the normalized energy flow *S ^{norm}_{z}* simplifies to

Thus, the transmission is completely determined by the coupling strengths *W*
_{1}(=*W _{a}*) and

*W*

_{3}(=

*W*) that depends on the aperture shape parameters

_{s}*a*,

*b*and the dielectric constants

*ε*

_{1}=

*ε*=1,

_{a}*ε*

_{3}=

*ε*. To illustrate the nature of

_{s}*W*, we take for example

*a*=0.5,

*b*=0.05,

*ε*=1.7

_{s}^{2}and evaluate

*W*explicitly as described in Fig. 2. Since the present system possesses scale symmetry, we choose 2

*a*as the unit length without loss of generality. While the positive real part of

*W*varies little as wavelength increases, the imaginary part of

_{s}*W*passes zero at a critical wavelength that is implicitly determined by the refractive index of a substrate. This is a general feature of

_{s}*W*and Eq. (9) indicates that the maximum of

*S*occurs around the zero of imaginary part of averaged

^{norm}_{z}*W*=(

_{ave}*Wa*+

*W*)/2. We also note that the imaginary part of

_{s}*W*results from the contribution of evanescent modes over the

_{m}*k*- and the

_{x}*k*- integration. The integrand changes sign as

_{y}*k*becomes larger than √

_{x}*ε*

_{m}k_{0}. Thus, we may conclude that the aperture resonance, or alternatively the maximum of

*S*, arises when the evanescent modes contributing to the coupling minimize their effect through the destructive interference.

^{norm}_{z}For a relatively thick metal film, we note that in the denominator *D* the coupling coefficients *W* is combined with the waveguide momentum *β* whose magnitude has a minimum at *k*
_{0}=*π*/*a* or λ=2*a*. As the thickness of a metal film increases, this effectively shifts the aperture resonance from the minimum condition of coupling *W* toward the minimum condition of the waveguide momentum. All these features are explicitly demonstrated in Fig. 3. Figure 3(a) describes the normalized time-averaged Poynting vector component Sz calculated by the formula in Eq. (8) whereas Figs. 3(b)–3(d) are rigorous results for the gold film with various aperture sizes calculated by the FDTD method adopting nonuniform grids and the Drude type metal dispersion fitted to the experimental values for gold. Aperture sizes for Figs. 3(b), 3(c) and 3(d) are chosen to make resonance peaks to be located in the terahertz, IR and optical frequency domains. It is remarkable that the single mode approximation in Eq. (8) agrees nicely with the FDTD result in Fig. 3(b) even for small *h*. As explained before, in the presence of a substrate these results show that the resonance peak moves from the value 1.55 minimizing *W* toward the free-standing resonance value near 1 minimizing *β* as the thickness *h* increases. The earlier experimental observation [8] that substrate shifts the resonance peaks to *λ*=*n _{s}*×2

*a*=1.7 turns out to be inaccurate though the general trend is correct. Figures 3(c) and 3(d) describe FDTD results covering the IR and the optical ranges. Except for the overall strong red shift of resonance peaks [10], they show a remarkable qualitative agreement with our theoretical prediction in Fig. 3(a). thereby strengthening the validity of our simple approach. Figure 3 shows that the resonance peak decreases as the metal film becomes thicker. This agrees with the fact that the thicker slit has weaker capacitative field enhancement [11, 12].

In order to help understanding the substrate effect, we have computed the electric and mag- netic field profiles of the resonant mode *TE*
_{10} using the FDTD method with results in Fig. 4. Since the FDTD method solves the Maxwell’s equation directly, it includes contributions not only from the *TE*
_{10} mode but also from all higher order modes. Nevertheless, we note from Fig. 4 that *TE*
_{10} is dominant even for a thin metal film with thickness *h*=0.05. We find that components *E _{x}*,

*H*are extremely weak compared to

_{y}*E*,

_{y}*H*(note the difference in color scales showing the relative strength). The dominance of

_{x}*TE*

_{10}mode, together with the result in Fig.3 showing the remarkable agreement between two different methods, justifies the use of single mode approximation in our analysis [9].

The horizontal cut clearly shows that electric field component *E _{y}* reduces significantly in the presence of a substrate whereas magnetic field component behaves exactly in the opposite way. This is due to the impedance mismatch at the substrate surface, where the reflected wave reduces (increases) the total electric (magnetic) field by reversing (keeping) direction upon reflection. On the other hand, the Poynting vector component

*S*as a product of these two components does not change significantly unless the refractive index of a substrate is too big. Figure 4 also shows that field components are moved toward the substrate region. This is an outcome of the impedance mismatch and the continuity of electromagnetic fields at the substrate surface. Thus the increased occupation of EM field in the substrate region makes the aperture resonance to depend on the refractive index of a substrate.

_{z}In summary, we have shown that the presence of a substrate affects the resonance condition of an aperture in a metal film. In a thin metal film, the condition of aperture resonance is determined by the minimization of coupling between the waveguide mode and external evanescent modes that depends on the substrate. We obtained an explicit theoretical formulation of the resonance condition and offered a heuristic physical account for the substrate effect on resonance in terms of destructive interference and impedance mismatch. Due to the complexity of coupling function *W* and the multiple reflections involved in resonance, this physical account is only a qualitative one. A simpler, yet quantitative physical account of the substrate effect thus remains an open problem. More importantly, the dependence of resonance on a substrate can be utilized to measure the refractive index of a thin film in a spatially highly confined way. This could lead to a practical application of terahertz waves to the measurement of much smaller nano scale systems.

## Acknowledgements

We thank F. J. Garcia-Vidal for discussion. This work is supported in part by KOSEF, KRF, MKE and the Seoul R & BD program.

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