## Abstract

A theory is presented for the transmission of transverse magnetic waves through a finite number of subwavelength slits in metal film. While a single slit achieves the single channel limit on resonance, multiple slits show super-transmission exceeding the single channel limit. The phenomenon of super-transmission is revealed as a result of cross-coupling of modes and confirmed by simulations. The influence of finite permittivity in the IR and microwave regime is included by perturbative corrections to the theory. The theory agrees quantitatively with past experiments and finite-difference time-domain simulations. By considering two or more modes in the slit region, our theory provides an approach to the analysis of cross-coupling among slits, which allows for super-transmission and features of a Fano resonance.

© 2014 Optical Society of America

## 1. Introduction

Since the discovery of extraordinary optical transmission (EOT) through subwavelength hole arrays in metal films [1], the study of the interaction between light and subwavelength apertures has become an active area of optics [2–4]. Compared with two-dimensional (2D) hole arrays [5,6], the system of slits is simpler and therefore allows for an analytic understanding of the underlying physics [7–9]. On the other hand, such slits exhibit no cut-off for the transverse magnetic (TM) mode [10], which is attractive for resonantly confining and squeezing light to the nanoscale. The slit system plays a fundamental role in understanding the mechanisms of the interaction of light with subwavelength apertures, including transmission resonances [7], near-field interference [11,12], surface plasmon generation [13], superscattering [14] and Fano resonances [15]. It also has practical significance for applications, such as biosensing [16], surface enhanced Raman scattering (SERS) [17] and terahertz (THz) field enhancement [18,19].

In 1999, transmission resonances of narrow slits in metallic films were demonstrated [7]. In 2001, a Fabry-Pérot-like model was introduced, providing an analytic expression of resonance wavelength for single slit in perfect conductor (PEC) [8]. That model was experimentally investigated at the microwave frequencies, and the influence of finite conductivity of metal was reported [20,21]. In 2007, a general angle-dependent analytic formulation for a single slit was developed [9]. Recently, the concept of single channel limit [22], that states the maximum scattering cross section of an atom cannot exceed $3{\lambda}^{2}/2\pi $ in 3D or $2\lambda /\pi $in 2D, has been explored in nano-optics systems. A deep subwavelength double-slit was carefully designed to surpass the single channel limit by the interaction of different channels and dielectric loading of one of the slits [14]. Resonant transmission and field enhancement by one-nanometer-scale slits in gold and silver films has been experimentally demonstrated in the near-infrared and THz regimes [19]. Although subwavelength slit systems have been intensely investigated [2], few works have departed from investigating only single slit or periodically arranged slits [23–25], and the rich physics of multi-slit system requires further investigation.

In this paper, we introduce a theory for multi-slit transmission. The theory is formulated for a perfect electric conductor (PEC), but modified perturbatively to include the effects of finite permittivity (in the visible-IR regime) and finite conductivity (in the microwave regime). Super-transmission through a subwavelength arrangement of multiple slits in a metal film is demonstrated. Not only does this multiple slit system surpasses the single channel limit, it also shows transmission that is considerably larger than one through single slit with the slit width equivalent to the total width of the multi-slit system. The Fano resonance invoked by the interference of different channels is observed. Comprehensive numerical simulations are carried out to verify the theory introduced in this work, and excellent agreement is found.

## 2. Mode matching theory for a multiple slit system

Figure 1 shows an example of subwavelength multi-slit system. It includes three subwavelength slits with width $a$, separated by $d$ in an infinitely wide metallic film of the thickness $l$. The dimensions are normalized by the incident wavelength. The even eigenmodes of this multiple slit can be written as

## 3. Quantitative comparison with numerical simulations

Figure 2 shows the transmission through PEC subwavelength slits calculated by the mode matching theory in Section 2. The transmission cross section ${\sigma}_{\text{T}}$ is defined as the ratio between the amount of transmitted power and the intensity of the incident wave. To verify the theoretical calculations, a comprehensive numerical method of finite difference time domain (FDTD) is used as simulations. Excellent agreement is found for both the single slit and three slit systems. The single slit shows Fabry-Pérot resonances, which have been demonstrated from past works with the same geometry [8].

The three slits system shows similar Fabry-Pérot resoannces, but also a Fano-like behavior due to the interaction between multiple transmission channels. Around the Fano resonance, the transmission substantially increases then abruptly drops to the minimum as will be discussed in more detail later.

## 4. Influence of finite permittivity/conductance in a real metal

In the case of real metal, the propagation of TM electromagnetic wave in a narrow slit of metallic film may be governed by the gap mode [28]:

Figure 3 shows the influence of finite conductance in real metal. In microwave regime, obvious resonant frequency shift emerges only in very deep subwavelength slits while in near-infrared regime substantially frequency shift happens in deep subwavelength slit. Comparing with experimental data (Fig. 3(a)) or simulation (Fig. 3(b)), it is clear that this perturbation can be applied to our theoretical model to capture the influence of the real metal’s response with high accuracy. It is clear from Fig. 3(a) that good accuracy for single mode-matching, within 1% for the resonance frequency, is obtained for slit widths up to 20% of the wavelength.

Our approach in this perturbation formulation is to assume that the mode shape is well-represented by the PEC case (e.g., Eqs. (1) and (2)), and that only the propagation constant within the slit changes, but otherwise the theory is identical to the PEC case. This approximation is clearly not valid if the slit skin-depth becomes comparable to the slit-width. As we can see from Fig. 3(b), a small departure from the comprehensive simulations (~1%) is seen only for the 100 nm slit width, and here the skin depth is about 10 nm. So, as long as the mode-shape does not vary by more than 10% of the PEC case, good accuracy is expected from this perturbative approach.

## 5. The single channel limit and beyond

Figure 4 firstly shows that the Fabry-Pérot transmission resonances of a single slit achieve the single channel limit, which has not been widely noted in past works. It should be noted that the single channel limit for 2D transmission is$\lambda /\pi $, since there is also a reflected component with equal contribution.

Figure 4 also shows super-transmission exceeding the single channel limit through a three slit system in a PEC. There are two transmission resonances in one period in the three slit system with symmetric or asymmetric profiles. The symmetric transmission is basically restricted by the single channel limit. However, the asymmetric one exhibits larger transmission than that. The origin of the asymmetric resonance can be explained with Fano resonance [30]. The three slits support two different scattering channels that couple through the diffraction at the ends of the slits. It is well known that $\pi $ phase jump occurs around the resonance, and this exhibits in-phase and out-phase interference to produce enhanced and reduced transmission.

An interesting result can be found in Fig. 4 by noting that the total width of the three slit structure is equal to the width of single slit. The smaller throughout area of three slits transmits more light than that transmitted by bigger single slit.

In microwave and THz regimes, metals are well-approximated by a PEC and super-transmission, exceeding the single channel limit, should be readily observed. In the visible and near infrared regime, metal loss will broaden the sharp Fano resonances and reduce the strength. For example, Fig. 5 shows an example of super-transmission for three 40 nm slits separated by 40 nm in a 140 nm thick silver film, and the associated single slit transmission for Ag in the visible regime.

## 6. Discussion

The theory introduced in this work is an application of mode matching; however, it is not the typical mode-matching of a discrete set of modes on either side of the boundary, but mode-matching to a continuum, which considers the infinite modes in the semi-infinite region. This work moves forward from past works, by considering two or more modes in the slit region. This adds significantly complexity since each mode not only self-couples, but also cross-couples to the other modes. Interesting physics arises from this cross-coupling since it produces nearly degenerate resonances that exhibit Fano interference.

To clarify the effect of the cross-coupling, we plot the transmission of multi-slit again compared with that neglecting the cross-coupling terms. As shown in Fig. 6, the cross-coupling plays the key role in super-transmission for deeply subwavelength multi-slit system.

The rich physics of this system is that it allows for a Fano resonance in a subwavelength structure. This occurs precisely because it is one of the few physical systems that allows for degenerate resonances in a (laterally) subwavelength confined region. The constructive zone of the Fano resonance allows for superscattering beyond the single channel limit. Interestingly, it also allows for total suppression of radiation. Figure 7 shows the FDTD simulation results of the distributions of electric fields at points A and B indicated in Fig. 5. The intensity of the components in the direction normal to the slits is plotted in log scale. At point A, the fields of satellite slits are clearly enhanced and the collective effect is like two radiating dipoles. However, at point B, the fields of satellite slits are weaker than the central one, and this reduces the beam width and suppresses radiation.

The approach introduced here for three slits can be extended to multiple slits in a straightforward way. For example, Fig. 8 shows the application to five slits, where there are three even modes cross-coupling and the transmission curve exhibits two Fano resonances in one period since the modes are not entirely degenerate. Extending to a greater number of slits can be obtained with the same approach. It should be noted that others have investigated the infinite slit case, but for that case, it is a standard mode-matching response [8].

For two slits, only one even mode is excited for normal incidence, so it cannot present super-transmission. For four slits, super-transmission is again possible. Therefore, by symmetry arguments, three slits is the minimum required to obtain super-transmission for normal incidence. In Ref. 14, a dielectric-loaded double-slit structure shows super-transmission at angled incidence, where symmetry breaking is present. In that work, a coupled-mode theory was used that treats the modes as generic oscillators, which requires extraction of the oscillator parameters ab extra, whereas our theory is self-contained. Also, the dielectric loading of the slits introduces experimental complexity.

The dependence of Fano resonance on the thickness of the metal can be written as the usual Fabry-Pérot cavity condition $\beta l+\phi =m\pi $ [19], where$\beta $ is the propagation constant in the slit cavity, $l$ is the length of the slit, $m$is an integer and $\phi $ is the shifted phase due to the refection at the end of the slit. Figure 9 shows the example of the dependence for the three slit in a PEC film, where the propagation reduced as ${k}_{0}=2\pi /\lambda .$ The additional phase $\phi $ can be found as $0.992\pi $ at Fano resonances by $\phi =m\pi -2\pi l/{\lambda}_{res}$from Fig. 8 and $0.926\pi $ for the Fabry-Pérot resonance. The phase shift for the higher order mode is closer to π because it is not only extremely subwavelength, but also has a rapid transverse oscillation which makes the ${I}_{\nu \nu}$ integral of Eq. (6) predominantly imaginary. This difference comes from the distinction of the dominant modes and cross-coupling; that is, both modes play a role in the Fano resonance, while the dominant mode of Fabry-Pérot resonance is only the lowest order “bright” mode.

## 7. Conclusions

We presented a theory of transmission through multiple subwavelength slits in metal film. Taking the influence of finite conductance or permittivity into account, this theory shows excellent agreement with experiment and simulation for real metals over a broad range of frequencies, as well as for a PEC. We investigated the single channel limit in the single slit system and super-transmission from multiple slits, where the single-channel limit is exceeded through a Fano resonance through multiple channels. We provided an analytical approach to study the cross-coupling of slits cavity. The results presented are interesting to enhanced spectroscopy applications where squeezing light below the diffraction limit is required.

## Acknowledgments

Shuwen Chen acknowledges the funding support from the Chinese Scholarship Council. This work is supported by the NSERC (Canada) Discovery Grant.

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