## Abstract

We present an intuitive reasoning and derivation leading to an approximated, simple closed-form model for predicting and explaining the general emergence of enhanced transmission resonances through rectangular, optically thick metallic gratings in various configurations and polarizations. This model is based on an effective index approximation and it unifies in a simple way the underlying mechanism of enhanced transmission as emerging from standing wave resonances of the different diffraction orders of periodic structures. The model correctly predicts the conditions for the enhanced transmission resonances in various geometrical configurations, for both TE and TM polarizations, and in both the subwavelength and non-subwavelength spectral regimes, using the same underlying mechanism and one simple closed-form equation, and does not require explicitly invoking specific polarization dependent mechanisms. The known excitation of surface plasmons polaritons or slit cavity modes, emerge as limiting cases of a more general condition. This equation can be used to easily design and analyze the optical properties of a wide range of rectangular metallic transmission gratings.

© 2011 OSA

## 1. Introduction

The electromagnetic response of subwavelength metallic corrugated structures gives rise to many physically interesting, potentially useful, and sometime initially counterintuitive phenomena. Perhaps the most widely known is that of resonant ”extraordinary optical transmission” or more generally enhanced transmission (ET) in arrayed subwavelength holes or slits in metallic sheets [1]. In subwavelength slit arrays, ET is defined as light transmission superseding the geometrical ratio of the open, transparent slits area and the opaque, metal surface area. Carefully engineered subwavelength slit arrays were shown to exhibit other interesting qualities, such as beaming and focusing of light [2–5], and large local field amplifications [6, 7] among others. Such structures therefore has been suggested as functional parts for various device realizations [5, 8, 9].

While current numerical calculations and semi-analytical models accurately predicted the emergence of ET in 1D arrays of subwavelength metallic gratings [10–13], the underlying physical mechanism has been under scientific debate for several years [11, 14–16]. In 2D hole arrays, it is quite accepted that surface waves, or more specifically surface plasmon polaritons (SPP) play an instrumental part in transporting the electromagnetic energy (and thus the light) from one side of the film to the other ([17] and references therein). Therefore, most explanations for 1D slits described ET for incoming light in TM polarization (the magnetic field component of the incoming light is parallel to the slits) and invoked, in one way or the other, resonant excitation of modified forms of surface waves. This is since for 1D slits, TM polarized light always has an electric field component perpendicular to the slits, and can therefore resonantly excite surface waves, i.e., SPP [6]. These phenomena were thought to be essential for ET, though different mechanisms such as the full dynamical diffraction theory [11], and slit cavity-like resonances [15] were also suggested as possible explanations for the emergence of ET. Accordingly, several rigorous analytical models were introduced which derived a transmission function explaining ET in 1D slit arrays through the use of the SPP [18–20]. These models show a good fit with the numerical and experimental data for the TM polarization, where SPP can be excited by the incoming light. A very good literature review can be found in [17, 18].

In 2006, E. Moreno et al. [21] showed theoretically that ET is possible for TE polarization as well, where surface excitations are not allowed, and therefore such ET is essentially a plasmon-less ET. They showed that such plasmonless ET can be realized by laying a thin dielectric layer on top of the metallic grating. In their explanation, the reason for the emergence of the ET is based on a coupling of the incoming light to slab waveguide modes in the thin dielectric layer, which take the role of surface waves as the resonant mediator for the light transmission. Similar structures and related effects were studied in a different context by Rosenblatt et al. [22]: they analyzed, both theoretically and experimentally, corrugated dielectric gratings which they termed grating waveguide structures. Several later works have investigated various modifications of such grating waveguide structure designs [5, 7, 23, 24]. However, recent works have shown the emergence of ET in TE polarization without the extra dielectric layer [25, 26], with no apparent surface modes.

Various numerical and analytical models predict the ET emergence in these structures as well. Specifically the RCWA method [11, 27] and analytical transmission functions based on the same principles [10, 12, 15, 17, 28] are in very good agreement with the various experimental results in different configurations. One outstanding problem is that different configurations required different explanations and different models for the emergence of ET. While the slit cavity-like resonances explained very well the reason for ET in the TM polarization, it is difficult to explain other configurations, such as ET in the subwavelength regime in the TE polarization with the same model. This wide variety of models, coupled with the not fully understood contribution of SPP has caused some confusion in the literature.

The fact that various analytical models coming from the same fundamental equations, are able to calculate the transmission function in very good agreement with the experimental data, but are seemingly using different underlying mechanisms, hints that it might be possible to distill the basic reason for ET emergence common to all the models. Therefore, the purpose of this paper is to propose *the simplest analytical condition* which will still be able to predict the emergence of ET, *based on existing models*. Such a condition has to be able to explain the emergence of ET *in all of the above configurations* with the same underlying physical mechanism, and be able to give an intuitive understanding of the physics.

In this paper, we will start with the rigorous eigenvalue solution, and use the single propagating mode approximation which was already shown [10, 12] to provide an excellent agreement with the experimental results. These models showed that ET can be explained by slit cavity-like resonances inside the slits of the grating structure. To be able to explain on the same footing configurations which do not have this propagating mode (such as the subwavelength regime in the TE polarization) we will try to simplify the model and define the general condition for the emergence of ET, with which all the ET configurations above can be explained. We will also discuss the surface waves observed in some of the configurations. Furthermore, we will provide an approximated closed-form condition for ET emergence, and show that in these cases, the metallic grating can be treated as an *effective dielectric medium*, thus greatly simplifying the problem. We will then show that this effective model, which we term the Effective Bragg Cavity (EBC) model, provides excellent agreement with rigorous numerical simulations for various configurations.

## 2. Effective Bragg-cavity model: derivation

The structure geometry under consideration and all the relevant geometrical, optical, and material parameters are shown in Fig. 1. Here the incoming plane wave has a positive wave vector *k _{z}*,

*d*is the periodicity of the grating,

*a*the slit width,

*w*is the grating thickness.

*n*

_{1},

*n*

_{3}are the refractive index of the infinite dielectric layers before and after the grating, and

*n*is the refractive index inside the slits. In some of the configurations, an extra thin dielectric layer will be added, with the refractive index

_{s}*n*

_{2}, and thickness

*w*

_{2}that is of the same order of magnitude as the grating thickness

*w*.

We will now show how to derive a closed form solution for the ET maxima, based upon the single propagating mode approximation and the approximations made in Ref. [28]. This closed form solution is based on the condition of a standing wave in the subwavelength slit array for the first bragg order (the first diffraction order). This will lead to a mapping of this subwavelength slit array to a similar configuration but with a non corrugated homogeneous dielectric layer replacing the grating, which is termed here the effective bragg-cavity (EBC) model, and is depicted in Fig. 2.

We start our derivation from the source free Maxwell equations in an inhomogeneous medium,

*ɛ*is the dielectric constant and

*μ*is the relative permeability. The vacuum wave vector of the plane wave with a wavelength

*λ*, incident on the grating is

*k*= 2

*π*/

*λ*.

Using these equations, we can find the eigenvalues of Eq. (2) in each of the four layers depicted in Fig. 1 separately. In the homogeneous dielectric layers before and after the grating (indexed 1,2,3 in Fig. 1), Eqs. (1), (2) reduce to

Let us treat first the case where the incident plane wave is in the TM polarization, and *w*_{2} = 0 (i.e. no added thin dielectric layer). Assuming a one-dimensional array of slits, periodic along the x axis, with *μ*(*r*) = 1 everywhere, the eigenfunctions of Eq. (2) for the magnetic field inside the periodic metal grating will be in the form of Bloch waves, [11]:

*j*indexes the eigenmode,

*g*= 2

*π*/

*d*,

*k*is the same as that of the incident electromagnetic wave, and ${k}_{z}^{\left(j\right)}$ will be found from satisfying Eq. (2). The total magnetic field inside the grating layer is given by the sum of all Bloch-wave excitation

_{x}*ψ*

_{(j)}denotes the excitation of the j-th eigenmode inside the grating. In the dielectric layers 1,3, before and after the grating respectively, the eigenmode solutions are just

*k*given by ${k}_{z}^{1,3}=\sqrt{{\left({k}_{1,3}\right)}^{2}-{\left({k}_{1,3}\text{sin}\theta +gm\right)}^{2}}$,

_{z}*k*

_{1,3}=

*k*

_{0}

*n*

_{1,3},

*θ*is the incidence angle depicted in Fig. 1, and ${n}_{1,3}=\sqrt{{\varepsilon}_{1,3}}$ is the refractive index of the homogeneous layers before and after the metallic grating respectively.

Finding the *z* components of the wave vector inside the grating,
${k}_{z}^{\left(j\right)}$ can be done using numerical calculations (e.g. RCWA solution [27]). An analytic solution can be achieved by noticing that in the subwavelength regime (*λ*/*n _{s}* > 2

*a*), there is only one propagating mode inside the slits of the grating. We will denote this mode by ${k}_{z}^{\mathit{prop}}$. Under the condition that the metallic grating thickness

*w*is very large compared to the metal skin depth, one can use the approximation that only the propagating mode is excited by the incoming wave (i.e. discarding the evanescent modes inside the grating) [6, 12, 29]. Equation (3) then becomes:

Considering the case where the structure is illuminated at normal incidence (i.e. *θ* = 0 in Fig. 1), *k _{x}* = 0 in Eq. (4) and
${k}_{x}^{m}=gm$, the excited Bloch mode becomes a superposition of wave functions with

*k*=

_{x}*gm*. Solving the boundary conditions between the different layers under the single mode approximation, with the added assumptions that the metal is perfectly conducting leads to the semi-analytical models for the transmission [10, 12].

However, instead of exactly solving the equations, one can intuitively identify the cause for the ET: Under the condition that the wavelength of the incoming light satisfies *λ*/*n*_{1,3} > *d* (which means *k*_{1,3} < *g*), there can be only one propagating mode outside of the grating, having
${k}_{x}^{m}=0$ (this is just the zero order transmission), while all the modes having
${k}_{x}^{m}=gm$ with *m* ≠ 0 are evanescent. However, inside the grating, viewing the excited Bloch mode as a superposition of plane waves with
${k}_{x}^{m}=gm$ (each plane wave corresponds to a different value of *m*), all these plane waves are propagating, even those with *m* ≠ 0, as is clearly seen in Eq. (4).

Hence, these modes, which are evanescent outside the grating, will be confined to the grating. Again, from Eq. (4) we see that all these plane waves have
${k}_{z}={k}_{z}^{\mathit{prop}}$, for all values of *m*, even though *k _{x}* =

*gm*≠ 0.

One can now ask what is the condition for a standing wave for the different bragg orders (*m* ≠ 0). The general condition for a standing wave including all orders of *m* can be solved analytically with the approximation of a perfectly conducting metal [29]. However, when the slit width *a* is of the same order of magnitude as the slit periodicity *d*, we can with good accuracy take only excited modes with *m* = ±1. That is because for ideal metals, for the Bloch mode inside the grating in Eq. (4), the relative amplitude *H _{m}* of each order of

*m*is proportional to the fourier transform of a rectangular box of width $\frac{a}{d}$. Since this fourier transform $\sim \mathit{sinc}\left(\frac{a}{d}m\right)$, we get a rapid decrease in the contributions of higher orders of

*m*for $\frac{a}{d}\sim 0.5$ (Even for a range real metals, this approximation holds reasonably well, as observed from calculating the values of

*H*using an RCWA numerical calculation for different real metals). In this case, a much simpler picture emerges:

_{m}*we can map this problem into a similar one by replacing the metallic grating with a dielectric material*whose refractive index is defined as

*n*=

*n*, surrounded by two lower refractive index dielectric layers,

_{eff}*n*

_{1},

*n*

_{3}. Thus, solving the standard slab waveguide transverse resonance condition: will give us the values of

*k*which produces the standing wave inside the grating layer for

*m*= 1. This resonant

*k*will be denoted by ${k}_{0}^{r}$. Here we have ${\varphi}_{12}={\text{tan}}^{-1}\left(\widehat{\gamma}/{k}_{z}^{\mathit{prop}}\right)$, ${\varphi}_{23}={\text{tan}}^{-1}\left(\widehat{\delta}/{k}_{z}^{\mathit{prop}}\right)$, $\widehat{\gamma}={\left({n}_{\mathit{eff}}/{n}_{1}\right)}^{2}\sqrt{{g}^{2}-{\left({n}_{1}{k}_{0}^{r}\right)}^{2}}$, $\widehat{\delta}={\left({n}_{\mathit{eff}}/{n}_{3}\right)}^{2}\sqrt{{g}^{2}-{\left({n}_{3}{k}_{0}^{r}\right)}^{2}}$, with

*n*given by Eq. (5), and

_{eff}*l*is a non negative integer. For the specific case where

*n*=

_{s}*n*

_{1}=

*n*

_{3}, we get an even simpler form:

When finding the condition for a standing wave for the bragg orders for incoming light in the TE polarization, the derivation is similar, though one should start from Eq. (1) instead of Eq. (2). Using a similar procedure we get an equation similar to Eq. (6) - the TE slab waveguide transverse resonance, by setting $\widehat{\gamma}=\sqrt{{g}^{2}-{\left({n}_{1}{k}_{0}^{r}\right)}^{2}}$, $\widehat{\delta}=\sqrt{{g}^{2}-{\left({n}_{3}{k}_{0}^{r}\right)}^{2}}$ and ${k}_{z}^{\mathit{prop}}$ is calculated as will be explained in section 2.1. Therefore, Eq. (6) is general and applies for both polarizations.

So far we have derived the resonant standing wave condition for the bragg diffractions. Our claim is that the condition on the wavelength of an ET resonance, in both TM and TE polarization approximately reduces to the effective waveguide confinement analytic condition of Eq. (6): the crucial point is that, as in a Fabry-Perot cavity, these standing wave conditions will have a visible effect on the transmission. Essentially these standing waves for the higher bragg orders (*m* ≠ 0) will cause the forward transmission to be at maximum value, because of constructive interference with the propagating *m* = 0 mode (zero order transmission), similar to a Fabry-Perot cavity. The argument is similar in spirit to the one proposed by Rosenblatt et al. [22] for reflection resonances in dielectric grating waveguide structures. Since generally the waveguide condition is transcendental, it is difficult to show analytically that the standing wave condition (Eq. (6)) and the ET condition are identical. However, we will show that for a range of different configurations and for incoming light *in either* TE or TM polarization, the wavelength *λ*_{0} for which an ET maxima occurs in rigorous numerical calculations matches well the solution for the analytical equation for the standing wave condition (i.e. Eq. (6)). Therefore, our simplified model is that *for there to be ET, there has to be a standing wave in the ẑ direction inside the system for the bragg modes having m* ≠ 0*. The ET resonance condition is therefore approximately given by Eq. 6. This, in essence, is the Effective Bragg-Cavity (EBC) Model.* Importantly, this simple model predicts correctly the emergence of ET in a vast variety of 1D configurations, i.e, for both TE and TM polarizations, and for the subwavelength and non-subwavelength spectral regimes, *using the same analytical condition* (Eq. (6)). The only difference between these different configurations is the wave-number of the propagating mode in grating, and the effective region where this standing wave occurs, as will be explained next. (It is also important to note that while taking higher orders of *m* into consideration makes it difficult to map the problem to the dielectric picture, for a perfectly conducting metal it is still possible to find a closed form solution using all orders of *m*, see Ref. [29] and references therein).

Figure 2 shows a schematic representation of the EBC model. Three different standing wave configurations can be achieved. Figure 2(a)(1–3) shows the numerically calculated near field intensity for an incoming *λ* at an ET maximum in the different configurations. These configurations are summarized in Table. 1. Figure 2(b)(1–3) shows the EBC model mapping, with the standing wave which corresponds to each of the configurations ((*λ*/*n*_{1}), (*λ*/*n*_{1}) < *d* for all cases):

- corresponds to an incident plane wave in either the TM polarization, in which case both (
*λ*/*n*) ≥ 2_{s}*a*and (*λ*/*n*) < 2_{s}*a*are valid, or in the TE polarization, for*w*_{2}= 0 (no thin dielectric layer) for (*λ*/*n*) < 2_{s}*a*(here both the dielectric materials*n*_{1},*n*_{3}are approximated as having infinite thickness). This is the usual scenario of a bare grating discussed in the literature (specifically, the near field calculation shown coincides with TM polarized light resonance). This standing wave also corresponds to TM polarized incoming light, with an added thin dielectric layer*n*_{2}, for*λ*/*n*_{2}>*d*. - corresponds to the case where a thin dielectric layer
*n*_{2}with a finite thickness is added and*λ*/*n*_{2}<*d*, in both polarizations. In the case of incoming light in the TE polarization, an extra condition*λ*/*n*< 2_{s}*a*applies. - corresponds to an incoming plane wave in the TE polarization, with a thin dielectric layer
*n*_{2}, (*λ*/*n*) > 2_{s}*a*, and (*λ*/*n*_{2}) <*d*.

It is clear from Fig. 2(b) that the model mapping is essentially the same for all three configurations, and the only difference is the area which confines the standing wave at the ET resonances.

As the model predicts the appearance of ET resonances also in TE polarization, which is less well known and less discussed in previous literature, in the following we will specifically elaborate on these cases.

#### 2.1. ET in TE polarization - no thin dielectric layer

As was noted before, the EBC model predicts ET in TE polarization as well. To show the generality of this simple picture, let us explain the emergence of ET in TE polarization in the framework of the EBC model. As noted earlier, for incoming plane waves in the TE polarization, Eq. (6) gives the ET resonance condition. The one major difference from an incoming light in the TM polarization, is that
${k}_{z}^{\mathit{prop}}$ in the TE polarization behaves differently than in the TM polarization: approximating the grating slits to infinite metallic slab waveguides (with a correction to the width *a* in case of non-ideal metal to account for skin depth),
${k}_{z}^{\mathit{prop}}$ is then given by the equation

*k*found numerically in rigorous numerical calculations (RCWA) with the one given by Eq. (8). The approximation holds remarkably well as long as the imaginary part of the propagating

_{z}*k*is small, which is valid as long as

_{z}*λ*/

*n*< 2

_{s}*a*. With this approximation of ${k}_{z}^{\mathit{prop}}$, Eq. (6) modified for TE polarization indeed predicts well the ET resonances for a bare grating (

*w*

_{2}= 0) for the case

*λ*/

*n*< 2

_{s}*a*.

From Eq. (8) it is clear that there is a cutoff wavelength. Hence, in the subwavelength regime ((*λ*/*n _{s}*) ≥ 2

*a*) there are

**no propagating modes**inside the grating. Therefore, the ET in TE polarization, which was both observed in wire grating experiments [26] and calculated by numerical simulations [25, 26], is expected to appear only in the non-subwavelength regime. We note that while ET in this configuration was previously explained using models that differ from models for ET in TM polarization [25, 26] (with good results), this ET can now be explained by the EBC model, with the same underlying mechanism as the ET in the TM polarization.

#### 2.2. ET in TE polarization - with thin dielectric layer

ET in the TE polarization cannot trivially occur in the subwavelength regime ((*λ*/*n _{s}*) > 2

*a*), because of the lack of a propagating wave inside the grating, as seen from the cutoff of ${k}_{z}^{\mathit{prop}}$ in Eq. (8). Therefore, the system has to be configured differently for ET emergence in this regime. An addition of a thin dielectric layer on top of the grating allows for a standing wave in the system, even for

*λ*/

*n*> 2

_{s}*a*(given that

*λ*/

*n*

_{2}<

*d*) [21, 30]. This is because under these conditions, while there is no longer a propagating mode inside the grating, the thin dielectric layer (with refractive index

*n*

_{2}) can still support one, allowing for a standing wave in the thin dielectric layer. In this case, the grating acts as one of the boundaries.

Therefore, in this case there are two regimes in which a standing wave can occur in the system: (I) For the case where there is a propagating mode for the first bragg order (*m* = 1) for both the grating and the thin dielectric layer *n*_{2} ((*λ*/*n _{s}*) < 2

*a*), the standing wave will be in both these layers, and is given by the equation for a two layer dielectric waveguide (with

*n*

_{2}and

*n*for the grating layer), corresponding to Fig. 2(b)(2).

_{eff}(II) For the subwavelength regime, where there is no propagating mode in the grating ((*λ*/*n _{s}*)

*>*2

*a*), the thin dielectric layer can still support a propagating mode for the first bragg order (given that

*λ*/

*n*

_{2}<

*d*). Even though there is no propagating mode in the grating, there will still be an evanescent eigenmode with a relatively small imaginary wave vector (as long as (

*λ*/

*n*) is only slightly larger than 2

_{s}*a*), which will be denoted by ${k}_{z}^{ev}$. ${k}_{z}^{ev}$ can be either estimated or calculated numerically (using RCWA for example). Thus for thin gratings, an evanescent coupling of the first bragg diffraction to the waveguide mode in the thin dielectric layer will still be possible [21]. We can then consider only the first bragg order, and find the solution for a standing wave inside the dielectric layer

*n*

_{2}: mapping the grating into a homogeneous dielectric layer lets us use Eq. (6) for finding the standing wave condition, with the changes ${k}_{z}^{\mathit{prop}}=\sqrt{{\left({n}_{2}{k}_{0}^{r}\right)}^{2}-{g}^{2}}$, $\widehat{\gamma}=\sqrt{{g}^{2}-{\left({n}_{1}{k}_{0}^{r}\right)}^{2}}$ and $\widehat{\delta}=\mathit{Im}\left({k}_{z}^{\mathit{ev}}\right)$. Surprisingly, as will be shown in the next section, the maximum transmission peaks observed in this configuration indeed satisfy the EBC condition of Eq. (6) (with the standing wave occurring in the dielectric layer), corresponding to the configuration in Fig. 2(b)(3).

#### 2.3. ET in TM polarization - with thin dielectric layer

The case of the configuration of incoming light in the TM polarization, with an added thin dielectric layer does not contain new physical insight, but is added here for completeness. In this configuration there are two regimes where ET can occur:

The first regime corresponds to the case where the first bragg order (*m* = 1) is evanescent in all the homogeneous dielectric layers (*λ*/*n _{i}* >

*d*, for

*i*∈ {1,2,3}). In this case the ET corresponds to a standing wave inside the grating layer, with layers

*n*

_{1},

*n*

_{2}acting as the boundaries. This is given by Eq. (6), with $\widehat{\gamma}={\left({n}_{\mathit{eff}}/{n}_{2}\right)}^{2}\sqrt{{g}^{2}-{\left({n}_{2}{k}_{0}^{r}\right)}^{2}}$. Note that in the case where the dielectric layer

*n*

_{2}is very thin, or where the the mode is only slightly evanescent in the thin dielectric layer, one needs to take into account also the layer

*n*

_{1}, leading to a slightly more complicated standing wave condition. This case corresponds to Fig. 2(b)(1).

The second regime where ET can occur is when the first bragg order (*m* = 1) is propagating in the thin dielectric layer *n*_{2}, but evanescent in layers 1,3. In this case, as in the non-subwavelengh regime for incoming light in the TE polarization with an extra thin dielectric layer (section 2.2), the ET will correspond to standing wave in both the grating and the thin dielectric layer. This will be given by the equation for a two layer dielectric waveguide (with *n*_{2} for the thin dielectric layer and *n _{eff}* for the grating layer), and correspond to Fig. 2(b)(2).

To summarize this section, we have shown how the various observed ET resonances are intuitively explained by the same EBC model, as arising from various ways of fulfilling the waveguide condition of Eq. (6). Therefore we claim that all these distinct effects have the same underlying mechanism. In the next section, we will show explicitly that the predictions of the EBC model are in a very good agreement with exact numerical calculations, confirming the validity of the above picture.

## 3. Comparison to numerical calculations

To check the validity of our model predictions for the spectral position of the ET maxima, we compare them to a full numerical calculation using an RCWA method [11,27]. We specifically compare three different configurations: incoming light in the TM polarization with no added thin dielectric layer, in the TE polarization with no added thin dielectric layer, and in the TE polarization with a thin dielectric layer. We show that in all these configurations, the EBC model correctly predicts all occurrences of ET.

#### 3.1. TM polarization

Figure 3(a) shows the numerically calculated zero order transmission intensity for different wavelengths and grating thickness, corresponding to the configuration depicted in Fig. 2(b)(1). The numerical calculation was done for a perfectly conducting metal, *d* = 0.9 *μm*, *a* = 0.35 *μm*, and (*n*_{1} = *n*_{3} = *n _{s}* = 1) on both sides of the grating and inside the slits. The predictions of the ET maxima, given by Eq. (6) are plotted in Fig. 3(a), by the white dotted lines. A very good agreement between the numerically calculated transmission maxima and the EBC model is clearly seen, with no fitting parameters.

It is apparent from Fig. 3(a) that changing the grating thickness changes the wavelength for which the transmission maximum occurs [6] as expected from a cavity-like behavior. Furthermore, in region (1) in Fig. 3(a) we see a linear dependence of the resonant wavelength on the cavity width
$w=\frac{{\lambda}_{0}l}{2{n}_{s}}$ (*l* being an integer), similar to Fabry-Perot cavities. This region is termed the slit cavity-like ET (which also exhibits a high intensity of the electromagnetic field inside the slits). There is also a smooth transition between this region and the region marked by (2) in Fig. 3(a), which deviates from the linear slope. This region is termed SPP-like ET (with the local field intensities centralized inside the slits and outside them as well). As seen, this transition is also predicted by the EBC model. Indeed, since in this configuration we have *n*_{1} = *n*_{3} = *n _{s}*, in the limit where
$\frac{\lambda}{{n}_{1}}\gg d$, we get

*n*≫

_{eff}*n*

_{1,3}and

*χ*≫ 1. Therefore, in this limit, Eq. (7) becomes

*ϕ*

_{12}=

*ϕ*

_{23}≈

*tan*

^{−1}(

*χ*

^{3}) ≈

*π*/2 and Eq. (6) reduces to 2

*n*

_{s}k_{0}

*w*= 2

*πl*which is the pure metallic slab waveguide condition, or

*l*∈ ℕ, as in a typical Fabry-Perot resonance in the metallic grating. This means that the standing wave is confined exactly inside the metallic grating, corresponding to the slit cavity-like ET maxima in Ref. [10], and to region (1). The other limit, where $\frac{\lambda}{{n}_{1}}\to d$ gives us a confined mode with an effective length in the

*ẑ*direction which is much larger than the grating width

*w*. This limit corresponds to the SPP-like ET in Ref. [10]: due to the slow spatial decay of the field intensity of the confined mode into the surrounding dielectric layers in this limit, the near field intensity distribution of the standing wave resembles a surface plasmon polariton-like mode. However, it does not need to correspond to actual surface plasmon excitations, which explains the appearance of ET in this limit in perfectly conducting metals. In this sense, the cavity-like modes and the SPP-like modes are just two limits of the general EBC model.

In Fig. 3(b) it can be seen that treating the metal as a real metal - Ag (approximated by the Drude model, but with an additional high loss put in artificially to separate it further from a perfectly conducting metal), changes the zero order transmission image. However, the EBC model still correctly predicts the wavelengths corresponding to ET with no change to the model, with a slight deviation. Note that for a thick grating, because of the added loss of the metal, the absolute value of the transmission decreases. This fact is not predicted by the EBC model, which calculates the wavelength of the maximum transmission, but not its value. Therefore we conclude that the EBC model works quite well also for cases of real metal, and not only for perfectly conducting ones.

#### 3.2. TE polarization - no thin dielectric layer

In Fig. 4 we see a similar graph for the TE polarization, with no added thin dielectric layer, and with the parameters *d* = 0.9 *μm* and *a* = 0.55 *μm*. we can see that for (*λ*/*n _{s}*) <

*d*the transmission is not dependent on the grating thickness, and no ET resonance is observed. However, since

*d*< 2

*a*we see ET in this polarization when

*d*< (

*λ*/

*n*) < 2

_{s}*a*, again in good agreement with the EBC model. As can be seen, the behavior of the ET lines differs from the TM polarization. This difference is largely explained by the difference in the metallic slab waveguide equations (and thus ${k}_{z}^{\mathit{prop}}$) between TE and TM.

#### 3.3. TE polarization - with thin dielectric layer

Figure 5 shows the zero order transmission for the case where a thin dielectric layer was added on top of the grating (configuration (c) in Fig. 2). The relevant parameters are *d* = 0.9 *μm*, *a* = 0.35 *μm*, *w*_{2} = 0.93 *μm*, *n*_{1} = *n*_{3} = 1 and *n*_{2} = 1.52.

As can be seen, in the non-subwavelength regime ((*λ*/*n _{s}*) < 2

*a*), corresponding to the region beneath the black dashed line in Fig. 5, the maximum transmission lines behave similarly to the configuration without an added dielectric layer (see Fig. 4 for comparison and the discussion in Sec. 3.2). However, the effective thickness is different - the cavity in this case is the grating layer + dielectric layer

*n*

_{2}. It can also be seen that ET is observed in the subwavelength regime (above the dashed black line) as predicted by the EBC model in Sec. 2.2. This is due to the evanescent coupling discussed earlier. The extra observed features of transmission minima lines closely correspond to the waveguide condition in the thin dielectric layer

*n*

_{2}, taking the grating as a homogeneous metallic slab [21], and accordingly these minima do not change with the metal width. Note that in the subwavelength regime, the transmission maximum does not change spectrally with the metal thickness. This is to be expected, because there is no propagating mode in the grating, as already explained.

As we have shown in Sec. 2.2, one can use the EBC model for the subwavelength regime, mapping the metallic grating to a homogeneous dielectric layer, in which the standing wave is evanescent (it propagates only inside the thin dielectric layer). In order to check the prediction of the model, we want to vary a parameter which will change the wavelength for which the ET occurs in the subwavelength regime. We note that by changing the slit width *a*, we can change the properties of the evanescent wave in the slits. This change is manifested by a change in
${k}_{z}^{ev}$ when *a* is changed. In Fig. 6 the zero order transmission maxima is extracted from the numerical model for different values of the slit width *a* in the subwavelength regime, and is compared to the predicted value given by the EBC model (the full RCWA transmission map makes the maximum hard to see, and so is not included). As can be seen, there is a good correspondence between the two. Note that the whole wavelength spectral range shown in Fig. 6 is only 20 *nm*, so we do not expect a perfect fit on this spectral scale, as was explained before. However it is clear that the trend of both lines is the same.

It is important to note some limitations of the EBC model. We note that in both polarizations, the EBC model predicts quite accurately the behavior of the ET, but with a small deviation from the actual maximum. This can mostly be explained by the approximations in the derivations of the model. The fact that our approximation took into account only the first bragg order causes a slight inaccuracy (which increases the farther away $\frac{2a}{d}$ is from 1). The yellow line in Fig. 3 is the EBC model calculated first transmission maximum when all the bragg orders of the waveguide condition [29] are taken into account. As can be seen, this improves the accuracy of the model’s prediction. The second approximation, of taking only the propagating mode inside the grating into account causes discrepancies when the grating is not optically thick (for thin gratings) [12]. Finally, as can be seen, the EBC model works best for the region $\frac{\lambda}{d}\sim 1$. For $\frac{\lambda}{d}\gg 1$ a different mapping of the grating to a homogeneous dielectric layer is possible for perfectly conducting metals [29]. Also, the model works quite well for real metals, however, an estimation of the effective slit width due to the finite skin depth has to be done properly, to improve its accuracy in these cases.

We have also compared the EBC model to numerical calculations in which *n*_{1} ≠ *n*_{3}. In the spectral regimes where all the higher bragg diffraction orders are evanescent in both infinite dielectric layers (layers 1,3), the agreement with our analytical model was just as good. When one of the dielectric layers starts supporting a propagating mode with *m* ≠ 0, no real ET is apparent as is expected.

## 4. Conclusions

In this paper we presented a simple model for explaining the emergence of ET through 1D metallic gratings. An approximation of this model, valid for the case in which *a* and *d* are of the same order of magnitude, allowed a derivation of a simple analytical closed form condition (Eq. (6)) for the spectral position of the ET maxima and of the EBC model: a mapping of the problem to a much simpler one of a standing wave condition with an effective homogeneous dielectric layer instead of the metal grating. We have analyzed various different configuration in which ET emerges, and explained them all as being realizations of the same standing wave model, with the standing wave being confined in different layers for different configurations. Finally, we compared the model predictions with rigorous numerical calculations and have shown a very good agreement between the two.

It is still an open question whether one can extend this model to explain ET through 2D arrays of holes [1], in which no propagating mode exists in the metallic layer.

## Acknowledgments

This research was partially funded by the FP7 Marie Curry IRG grant and the BIKURA program of the Israel Science Foundation. We acknowledge the generous support of the Wolfson Foundation. IS would like to thanks Itamar Rosenberg for the technical assistance.

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