## Abstract

A simple semi-analytical model for the comprehension of the extraordinary optical transmission of a 1D array of periodic subwavelength slit is proposed. In this single mode model, the mono layer of the perforated metal film is considered as a homogeneous medium. Therefore, the electromagnetic response of this structure to a plane wave excitation is equivalent to that of a slab with homogeneous equivalent permittivity. A versatile phase correction is added to this model in order to handle the contribution of surface waves in the EOT phenomenon. The proposed model leads to a cavity-like dispersion relation that allows accurate prediction of the resonance frequencies of the 1D structure.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. introduction

The extraordinary optical transmission (EOT) phenomenon consists in a enhancement of the transmission of light through a subwavelength perforated opaque metallic film. This optical phenomenon is described as extraordinary because from the theory of diffraction, an incident field shining a subwavelength aperture must be diffracted isotropically in all directions. Thus the transmission of the aperture must be minimal. The EOT anomaly was observed first by Ebbesen [1] by exploring the optical properties of perforated metal films with cylindrical micro-cavities. Ebbesen *et al*. discovered that this matrix of submicrometer cylindrical cavities in metallic films exhibits very unusual zero-order transmission spectra at wavelengths larger than the array period. They remarked, a strong dependence of the maximum of the transmission with the electrical properties of the metallic material and they also noticed that the spectral position of the minima and maxima depended on the period of the structure, and the angle of incidence of the incident plane wave. Based on these observations, they suggested that surface plasmons are at the origin of EOT. However this transmission enhancement could also be observed in the case of a metal of infinite conductivity where surface plasmons do not exist. Since their publication, many theoretical and experimental works [2,3,5–9] were provided in order to understand the physical origin of this transmission enhancement. In the case of perfectly conductor metal, analytical approaches have been developed to approximatively predict the EOT [3, 4]. In 2001, Moreno *et al*. [3] proposed a first theoretical model based on modal theories of electromagnetism. In their demonstration, the authors suggest that the EOT results from a tunneling energy transfer between the two faces of the metal film through the excitation of a super mode of the grating. By analyzing the Fabry-Perot-type transmission coefficient of the structure, the authors approximately predicted the positions of the resonances of the structure. As pointed out by Haito and Lalanne in [6], Moreno’s theory is very interesting but does not explicitly describe the nature of excited waves on both interfaces of the film. To answer this question, Haito and Lalanne propose a very sophisticated model that takes into account the contribution of the surface plasmon as well as the mode of the periodic matrix of holes. Their model provides more accurate prediction over much broader spectral range from the visible to the microwave spectral range. However, an array of 1D periodic subwavelength slits can also exhibit a non Lorentz-like transmission enhancement in far infrared frequencies. We assert that both phenomena are linked to the same mode of the grating. But to the best of our knowledge, a unique formalism describing both phenomena is not reported in the literature. We propose in this paper a simple and versatile model of this phenomenon involving a specific mode living in an equivalent homogeneous medium and a phase correction due to surface waves. We provide a highly simplified model which can be analytically worked out from visible to infrared frequencies range.

## 2. EOT phenomenon and PMM analysis

Consider in a Cartesian coordinates system (**e _{x}**,

**e**,

_{y}**e**) the structure of Fig. 1, which is consisted of a metal film perforated with a subwavelength periodic array of nano-slits. The relative permittivity of the slits material is denoted by

_{z}*ε*

^{(slit)}. This structure is shined, from the upper medium (with relative permittivity

*ε*

^{(1)}) by a TM polarized plane wave (the magnetic field is parallel to the slits). The propagation constant of the incident wave is denoted by ${\mathbf{K}}_{\mathbf{0}}={k}_{0}\left({\alpha}_{0}^{(1)}{\mathbf{e}}_{\mathbf{x}}+{\beta}_{0}^{(1)}{\mathbf{e}}_{\mathbf{y}}+{\gamma}_{0}^{(1)}{\mathbf{e}}_{\mathbf{z}}\right)$, where

*k*

_{0}= 2

*π*/

*λ*=

*ω*/

*c*denotes the wavenumber,

*λ*being the wavelength and

*c*the light speed in vacuum. In this paper we consider a slits array with period

*d*= 165

*nm*<<

*λ*and width

*s*= 15

*nm*. The film layer height

*h*is set to

*h*= 800

*nm*. The metal film is then thick enough so that, there is no transmitted field without the slits array. The relative permittivity of the lower region is denoted by

*ε*

^{(3)}. The reflection, transmission and absorption spectra of this structure is computed throughout the polynomial modal method (PMM) [10–12]. In this simulation, the dispersive relative permittivity function

*ε*

^{(metal)}of the metal is described by the Drude-Lorentz model [14]. Gold (Au) is considered in our simulations. In this model both intraband

*ε*and interband

_{intra}*ε*contributions are taken into account:

_{inter}*ε*is described by the Drude model:

_{intra}*χ*(

_{j}*ω*) [Eq. (4)], applicable to an amorphous solid in the far-IR part of the spectrum. However, A. D. Rakic

*et al*. [14] have extended this model to describe optical properties of a wide range of materials including metals. In this model the interband contribution is expressed as

*k*is the number of Brendel and Bormann oscillators used to interpret the interband part of the spectrum ; and

*α*= [

_{j}*ω*(

*ω*+

*i*Γ

*)]*

_{j}^{1/2}, and

*w*(

*z*) =

*e*

^{−z2}

*w̃*(

*iz*) (

*Im*(

*z*) > 0);

*w̃*is the complementary error function: In the case of gold-metal, we use the following parameters: for

*ω*in

*ev*,

*f*

_{0}= 0.770, Γ

_{0}= 0.050,

*ω*= 9.03

_{p}*ev*,

*f*∈ [0.054, 0.050, 0.312, 0.719, 1.648], Γ

_{j}*∈ [0.074, 0.035, 0.083, 0.125, 0.179],*

_{j}*ω*∈ [0.218, 2.885, 4.069, 6.137, 27.97],

_{j}*σ*∈ [0.742, 0.349, 0.830, 1.246, 1.795].

_{j}The reflection, transmission and absorption spectrum of the array of subwavelength 1D nano-slits are plotted in Fig. 2. A Lorentz-like resonance corresponding to an EOT phenomenon occurs around *λ* = 3.37*μm*. The modulus of the magnetic field plotted in Fig. 3 at *λ* = 3.37*μm* supports the fact that this EOT phenomenon is linked to the resonance of the slit mode. At this frequency, both electric and magnetic fields are well confined in the slit cavity. Beyond the very near infrared range, another enhancement of the transmission can also be observed. Although this second transmission enhancement doesn’t seem to be a Lorentz-like resonance, we assert that all these enhancements would have in common the excitation of a particular eigenmode of the slit grating. To demonstrate our assertion, we provide an equivalent homogeneous dielectric model of the 1D periodic slit array and show that these different enhancement phenomena are caused by this mode which has very different confining properties depending on the frequency range. But, let us first briefly outline the main steps of the polynomial Modal Method (PMM) based on Gegenbauer polynomials ${G}_{n}^{\mathrm{\Lambda}}(x)$ expansion) [10–12]. In the PMM, the structure is divided into 3 intervals ${I}_{x}^{(k)}$, *k* = 1 : 3 in the (*Ox*) direction and 3 layers ${I}_{z}^{(k)}$, *k* = 1 : 3 in (*O*, *z*) direction. In each layer ${I}_{z}^{(k)}$, each component of the electromagnetic field is expanded on a set of eigenfunctions, i.e. ; solutions |*ψ _{q}*〉 of the eigenvalue equation, Eq. (6):

*k*the ${H}_{y}^{(k)}(x,z)$ component is then written as linear combination of the eigenfunctions

*ψ*of Eq. (6):

_{q}*k*= 1 : 3, the eigenvalues ${\gamma}_{q}^{(k)}$ corresponding to the effective indices of all modes living in these layers are obtained. In particular in layer 2, and because of the metal, these solutions are generally complex. We consider the most slowly decaying evanescent mode that will be denoted by ${\gamma}_{0}^{(2)}$. Figures 4 and 5 present the behavior of ${\gamma}_{0}^{(2)}$ with respect to different values of the wavelength

*λ*∈ [0.7, 500]

*μm*) for three values of the period

*d*and for

*s*= 15

*nm*. One can remark that the value of the effective index ${\gamma}_{0}^{(2)}$ hardly depends on the period in the visible and near-infrared ranges. For this frequency range, the mode behaves like the eigenmode of the isolated slits. At these frequencies, this mode is the gap-plasmon mode of the Au/slit/Au gap. On the other hand, in the case of large wavelengths, the effective index of the mode strongly depends on the grating periodicity. The mode decreases very slowly in the metal and thus it is no longer confined in the slits as shows Fig. (6). It behaves like a lattice mode. ${\gamma}_{0}^{(2)}$ is not the effective index of the fundamental guided mode living in a slit viewed as a perfectly conducting waveguide. Therefore ${\gamma}_{0}^{(2)}$ cannot be replaced by the refractive index $\sqrt{{\epsilon}^{(\mathit{slit})}}$ of the slit.

## 3. Semi-analytical and single mode model

At this stage, we affirm that this mode should allow to describe, at least partially the EOT mechanisms that occur in the system. Since the transversal geometrical parameters of the grating are very smaller than the incident field wavelength *λ* (*d* << *λ*), the wave functions describing the components of the electromagnetic fields and densities only slightly vary spatially at the scale of the period. Under these hypothesis, we derive the average parameters viewed by the incident plane wave. Since *d* << *λ*, the electromagnetic field quantities, namely normal components of electric density flux and tangential components of electric field, that are continuous at the interfaces slit/metal, can be supposed to be constant. From constitutive relation **D** = *ε***E**, it follows for *E _{x}* component

*D*is supposed to be constant and equal to

_{x}*D*

_{0}. Therefore

*ϕ*(

*x*)〉 denotes the averaged value of the quantity

*ϕ*over the period. The averaged permittivity associated with the equivalent effective medium is defined throughout the following relation:

*d*<<

*λ*), is then equivalent to that of a slab with equivalent permittivity

*ε*

^{(2)}and height

*h*. See Fig. 7. Its reflection and transmission coefficients

*r*and

*t*are then given by and where ${\varphi}_{1}={\varphi}_{3}={e}^{-i{k}_{0}{\gamma}_{0}^{(2)}h}$,

*r*and

_{i}*t*,

_{i}*i*= 1, 2 are the Fresnel coefficients at the interface

*ε*

^{(i)}/

*ε*

^{(i+1)}in TM polarization:

*r*behavior Eq. (12), it is possible to provide an interpretation of the curve shapes of Fig. 2. The zeros of the reflection coefficient

*r*Eq. (12) approximatively give the resonance frequencies. For example, as soon as

*ϕ*

_{1}and

*ϕ*

_{3}as follows:

*γ′*,

*γ″*,

*α′*

_{1,3},

*α″*

_{1,3}∈ ℝ

^{+}. We distinguish two cases:

- First case: We remark from numerical simulations that the real and imaginary parts of the modes effective indices are bounded when the wavelength of the incident wave
*λ*becomes larger and larger. See Figs. 4 and 5. Therefore, the arguments of*ϕ*_{1}and*ϕ*_{3}, tend to zero when*λ*becomes larger. Thus the terms*ϕ*_{1}and*ϕ*_{3}tend to 1; the reflection is minimal and the transmission is enhanced - Second case: the real parts of the effective indices are comparable to the wavelength. In this case the transmission enhancement is driven by the phase condition which implies that$${k}_{0}\left[2{\gamma}^{\prime}h+{\alpha}_{1}^{\prime}{a}^{(1)}+{\alpha}_{3}^{\prime}{a}^{(3)}\right]=2\pi p,\hspace{0.17em}\hspace{0.17em}p\in \mathbb{N}.$$The resonance wavelengths
*λ*are then solutions of where_{p}

Using Eq. (19) we provide some numerical simulations in different cases by modifying the angle of incidence, the relative permittivity of the slit, of the substrate, and that of the superstrate. See Figs. 12 and 13. In Fig. 12 we compare the resonance frequencies obtained with dispersion relation of Eq. (19) with reflection curves computed with HSM (dotted line), PMM (solid line), HHSM (dashed line) for the following numerical parameters: *ε*^{(1)} = *ε*^{(3)} = 1.54^{2}, *ε*^{(slit)} = 1, incidence angle= 50°, *h* = 800*nm*, *d* = 165*nm*, *s* = 15*nm*. And in Fig. 13, the following arbitrary numerical parameters are used: *ε*^{(1)} = *ε*^{(slit)} = 1.54^{2}, *ε*^{(3)} = 1, incidence angle= 20°, *h* = 800*nm*, *d* = 165*nm*, *s* = 15*nm*. In all these results, the resonance wavelengths are accurately predicted by Eq. (19).

## 4. Conclusion

In conclusion, we have proposed a simple model, of the extraordinary optical transmission through a 1D array of periodic subwavelength slits. This model is based on a single mode approach with surface waves phase correction. The mode of the structure is efficiently expanded on a series sum of Gegenbauer polynomials and its effective index is computed as eigenfunctions of a boundary value problem. This polynomial modal method is very stable, accurate and converges very rapidly. The effective index is then introduced in an equivalent homogeneous slab model. We provide some numerical simulations by modifying the incidence angle, permittivity of the slits, the substrate and the superstrate. These simulations show that the proposed model which involves a single mode in an homogeneous medium and the surface waves phase correction is valid and robust. Finally, this model could be extended to finite non periodic structures as well [15].

## Funding

French government research program “Investissements d’Avenir,” IDEX-ISITE initiative (16-IDEX-0001) (CAP 20-25).

## Acknowledgement

The author is grateful to B. Guizal, G. Granet, and E. Centeno for the attention they paid to his work.

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