## Abstract

Extraordinary optical transmission (EOT) through dielectric screens periodically loaded with sub-wavelength 1D discontinuities, such as apertures or metallic insets is analyzed. The results of the analysis and computational electromagnetic simulations show that the transmission is higher for for metallic inclusions than for empty slits. This effect confirms that EOT is a quite general property of weakly transparent periodic diffraction screens and opens the door to optically induced EOT in photo-conductive semiconductor screens.

©2011 Optical Society of America

## 1. Introduction

The first reports on extraordinary optical transmission (EOT) through metallic screens periodically perforated by sub-wavelength holes or slits [1, 2] stimulated many subsequent experimental and theoretical studies of this effect. The reader is referred to the excellent reviews by C. Genet et al. [3], F. J. García de Abajo [4] or F.J. García Vidal et al. [5] in order to have a complete overview of the topic. Soon after the seminal work of Ebbesen was published, it was reported that EOT also appears in periodically corrugated thin metal films without apertures [6, 7], and in non-metallic dielectric screens with apertures [8, 9]. Therefore, it seems that nor the metallic nature of the screen nor the presence of apertures are essential for the presence of EOT. On the contrary, it seems that EOT will appear, for the appropriate light polarization, when: (i) There is some periodic modulation of the electrical properties of the screen and (ii) there is some weak coupling mechanism between both surfaces of the screen.

Following this *ansatz*, we will study EOT through high permittivity thin dielectric screens, which will provide the required weak coupling between both sides of the screen. Our analysis and numerical simulations will show that dielectric screens periodically loaded either with apertures or with metallic inclusions may present EOT in the vicinity of Wood’s anomaly. Surprisingly, transmittance will be higher for metallic inclusions than for apertures.

## 2. Theory

In order to simplify the analysis we will consider EOT of p-polarized light through an array of 1D empty slits or parallel metallic inclusions in a dielectric screen. One-dimensional configurations have been previously analyzed in order to simplify the computational problem, see for example reference [2], and subsequently they have taken on a life of its own. The unit cell of this structure is shown in Fig. 1. In our analysis we will take advantage of the equivalence between the studied problem and the equivalent problem of a symmetric discontinuity in a parallel plate waveguide, as it is illustrated in Fig. 1. First of all, the electric and magnetic fields at both sides of the screen are expanded in series of TM waveguide modes:

*T*and

*R*are the transmission and reflection coefficients,

*T*and

_{n}*R*are the amplitudes of the different modes excited in the screen, ${Y}_{0}\hspace{0.17em}=\hspace{0.17em}\sqrt{{\varepsilon}_{0}/{\mu}_{0}}$ is the admittance of free space and

_{n}*Y*are the modal admittances of the empty waveguide. It is now assumed that the empty slits or metallic inclusions are small (

_{n}*b*≪

*a*), so that the modal components of the electric and magnetic fields at both sides of the screen can be linked through the matrix surface impedance of the dielectric slab:

*k*and

_{z,n}*Y*

_{d,n}are the wave-vector and the modal admittances of the TM modes of the dielectric filled waveguide, respectively. From Eq. (5) we can define the quantities $A\hspace{0.17em}=\hspace{0.17em}{E}_{y,n}^{+}\hspace{0.17em}+\hspace{0.17em}{E}_{y,n}^{-}\hspace{0.17em}-\hspace{0.17em}{Z}_{n}^{(1)}\hspace{0.17em}\left({H}_{x,n}^{+}\hspace{0.17em}-\hspace{0.17em}{H}_{x,n}^{-}\right)$ and $B\hspace{0.17em}=\hspace{0.17em}{E}_{y,n}^{+}\hspace{0.17em}-\hspace{0.17em}{E}_{y,n}^{-}\hspace{0.17em}-\hspace{0.17em}{Z}_{n}^{(2)}\hspace{0.17em}\left({H}_{x,n}^{+}\hspace{0.17em}+\hspace{0.17em}{H}_{x,n}^{-}\right)$ which must vanish in the intervals

*a*/2 ≥ |

*y*| ≥

*b*/2. Therefore, for

*b ≪ a*

*B*, where

*m*is an integer in the interval 1 ≤

*m*≲

*a/b*. After substitution of Eqs. (1)–(4) into the above relations, we find

*T*and

_{n}*R*as functions of the transmission and reflections coefficients

_{n}*T*and

*R*. These coefficients can now be determined if the surface impedances associated with the inclusions ${Z}_{s}^{(1)}$ and ${Z}_{s}^{(2)}$ are known (for instance, for perfect conductors it is ${Z}_{s}^{(1)}\hspace{0.17em}=\hspace{0.17em}{Z}_{s}^{(2)}\hspace{0.17em}=\hspace{0.17em}0$). In order to determine

*T*and

*R*we simply integrate the quantities $C\hspace{0.17em}=\hspace{0.17em}{E}_{y,n}^{+}\hspace{0.17em}+\hspace{0.17em}{E}_{y,n}^{-}\hspace{0.17em}-\hspace{0.17em}{Z}_{s}^{(1)}\hspace{0.17em}\left({H}_{x,n}^{+}\hspace{0.17em}-\hspace{0.17em}{H}_{x,n}^{-}\right)$ and $D\hspace{0.17em}=\hspace{0.17em}{E}_{y,n}^{+}\hspace{0.17em}-\hspace{0.17em}{E}_{y,n}^{-}\hspace{0.17em}-\hspace{0.17em}{Z}_{s}^{(2)}\hspace{0.17em}\left({H}_{x,n}^{+}\hspace{0.17em}+\hspace{0.17em}{H}_{x,n}^{-}\right)$ along the aperture (i.e. in the interval –

*b*/2 ≤

*y*≤

*b*/2) and obtain

*x*) ≡ sin(

*x*)/

*x*. It is worth to mention here that this analysis is also valid for empty slits, which can be characterized by the surface impedances corresponding to the fundamental quasi-TEM mode in the slit, as well as for realistic metallic inclusions, which can be characterized by the appropriate surface impedances [10].

## 3. Results

EOT through low-loss high permittivity dielectric screens will now be considered. Transmittance through these screens is low (except at Fabry-Perot resonances) due to the high reflectance associated to the jump in the dielectric constant at the interfaces. However, if the screen is periodically perforated by an array of periodic parallel slits (Figs. 1(a) and 1(b)), an extraordinary increase of transmittance appears near Wood’s anomaly, just as in EOT through metallic screens. This effect is shown in Fig. 2 for a periodic array of slits perforated in a slab of zirconium-tin-titanate with *ε* = 92.7(1 + 0.005*i*) [11]. The results provided by the model reported in the previous Section are compared with numerical simulations carried out using the electromagnetic solver CST Microwave Studio. Electromagnetic simulations considered a parallel plate host waveguide with a length equal to four periods, and with 3 modes (the fundamental TEM mode and two evanescent modes) at the input and output ports. Lateral perfect magnetic walls, with an adaptive tetrahedral mesh were used for numerical computations. Computation time was about 1 min for each frequency point in CST, whereas computation time using the analytical model was negligible (less than 1 micro-second). As it can be seen in Fig. 2 there is a reasonable qualitative and quantitative agreement between the results of our analytical model and the numerical simulations.

In the THz range of frequencies most metals behave as PEC. Thus, transmittances through the same diffraction screen, but with the slits filled with a PEC, was analyzed. The results of this analysis, also shown in Fig. 2, illustrate how EOT can be induced by metallic inclusions in low-loss, high permittivity, thin dielectric screens operating in the THz frequency range and below. Surprisingly, transmittance is higher with metallic inclusions than with apertures.

Similar results are shown at infrared frequencies in Fig. 3 for screens of amorphous silicon (a-Si), periodically loaded with silver inclussions. In this analysis, the metallic inclusions were modeled using standard Drude theory with the constants already used in [13] for silver. The results of Fig. 3 show that the increase of transmittance in a-Si screens loaded with metallic insets is much higher than the corresponding increase of transmittance due to the presence of open slits in the screen, which is actually very weak.

It is well known that EOT can be associated to the excitation of weakly coupled leaky surface waves at both sides of the screen. In standard EOT, these surface waves are coupled through the apertures perforated in the screen. In low-loss dielectric screens, this coupling is ensured by the screen itself, and the role of the metallic inclusions is merely to allow for the excitation of the surface waves. In order to support this interpretation, we show in Fig. 4 the electric field distribution at both sides of the screen in the configuration of Fig. 3 at the frequency of maximum transmission. A clear standing wave pattern corresponding to the simultaneous excitation of two surface waves traveling at opposite directions along each side of the screen can be appreciated in both cases. These field distributions are very similar to those found in simulations of metallic screens at optical frequencies, thus confirming that the excitation of coupled surface waves with wavelengths close to the periodicity is the physical mechanism behind EOT in dielectric screens with conductive insets.

## 4. Conclusion

An analytical model for EOT through high permittivity 1D dielectric screens periodically loaded with open slits or metallic inclusions has been provided and validated with full wave electromagnetic simulations. Analytical and numerical results show that EOT through high permittivity dielectric screens loaded with a periodic distribution of metallic inclusions is higher than for the corresponding screens with empty slits.

Regions with a high density of free electrons can be induced in most semiconductors by using lasers of the appropriate wavelength. Thus, we feel that the reported effect opens the door to the design of photo-induced EOT in semiconductor screens at THz and infrared frequencies.

## Acknowledgments

This work has been supported by the Spanish Ministerio de Educación y Ciencia and European Union FEDER funds (project No. CSD2008-00066), by the Czech Grant Agency (project No. 102/09/0314), and by the Czech Technical University in Prague (project No. SGS10/271/OHK3/3T/13).

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