## Abstract

In this paper we investigate the optical response of periodically structured metallic films constituted of sub-wavelength apertures. Our approach consists in studying the diffraction of transverse magnetic polarized electromagnetic waves by a one-dimensional grating. The method that we use is the Rigorous Coupled Waves Analysis allowing us to obtain an analytical model to calculate the diffraction efficiencies. The zero and first order terms allow determining the transmission, reflectivity and absorption of symmetric or asymmetric nanostructures surrounded either by identical or different dielectric media. For both type of nanostructures the spectral shape of the enhanced resonant transmission associated to surface plasmons displays a Fano profile. In the case of symmetric nanostructures, we study the conditions of formation of coupled surface plasmon-polaritons as well as their effect on the optical response of the modulated structure. For asymmetric nanostructures, we discuss the non-reciprocity of the reflectivity and we investigate the spectral dependency of the enhanced resonant transmission on the refractive index of the dielectric surrounding the metal film.

©2005 Optical Society of America

## 1. Introduction

Optical properties of periodically modulated metallic films have attracted a considerable interest during the past years after the observation of the enhanced transmission of light through metal films with periodic sub-wavelength hole arrays [1,2]. This effect has also been observed recently in the terahertz domain [3] and over the microwave spectral range [4]. The present common understanding of the mechanism responsible for this enhanced transmission is the excitation of surface plasmon polaritons (SPPs) at the interfaces of the structured film. Other mechanisms have been suggested such as : dynamical diffraction [5,6], resonant Wood anomalies[7], waveguiding modes in slits [8], localized surface plasmons [9,10]. Recent experimental measurements [11,12] and theoretical investigations [13–16] predict that the transmission enhancement can also occur in periodically structured continuous metal films. These works substantiate the fact that light tunneling via surface plasmon-polariton states plays a key role in the phenomenon of enhanced transmission.

Several numerical models have been developed to describe the light diffraction by 1D or 2D metallic gratings [8,13,14,17–20]. They are based on the resolution of Maxwell equations
with the appropriate boundary conditions and involve sophisticated computer calculations. For example Popov *et al*. [13,19] have used a Fourier model extended to crossed gratings to explain the extraordinary optical transmission of two-dimensional arrays of sub-wavelength holes designed in metallic films. They have also applied this method for continuous metallic films and they have shown that a resonant transmission can still occur, provided the film thickness is modulated. Salomon et al. [14,20] have used a differential method with R-matrix to model similar structures. In the case of a continuous metal film with a periodic structure of metal or dielectric ridges on one or both interfaces, they have investigated the dependencies of the enhanced transmission on the ridge width and height as well as the ridge arrangements on the opposite interfaces. Treacy [5,6] has used the dynamical diffraction theory to explain the light transmission anomalies observed for thin-metal film gratings. He pointed out that the surface plasmons are an intrinsic component of the diffracted wave field and play no independent causal role in the anomalies. Sarrazin *et al*. [7] have used a coupled method which combines a scattering matrix formalism with a plane wave representation of the fields to simulate the optical response (transmission, reflectivity and losses) of thin metallic films with bi-dimensional array of subwavelength holes. They concluded that the transmission properties of these systems could be explained to result from resonant Wood anomalies. Regarding the analytical treatment of optical properties of periodically structured metallic films, two models ought to be mentioned [15, 16]. In the first one [15] analytical equations describing the transmittance, the reflectance and the absorbance are derived and discussed. The transmission of a metal film is also investigated. This model works only for symmetric surrounding media. In the second model [16] an analytical calculation of the light transmitted and reflected by periodically structured metal films, due to a tunnelling of surface plasmon polariton modes, is presented. The authors have only discussed the transmission spectra and in their approach they consider the metal as a perfect conductor, the imaginary part of the dielectric function of the metal being neglected.

In this work, we present an original method based on the Rigorous Coupled Waves Analysis (RCWA) [21, 22] which we use to extract analytical expressions of the diffraction efficiencies of TM polarized electromagnetic waves illuminating a one-dimensional grating. This model goes beyond the preceding analytical descriptions and allows to consider absorbing metals as well as asymmetric nanostructures. By calculating -in addition to the zero order transmission and the specular reflection- the first diffracted orders in transmission and reflection, it gives a better description of the absorption of the modulated film and it provides new insights into the underlying physics. In particular, the Fano spectral line shapes of the resonant transmission which has been previously suggested [7, 23], clearly shows up. Moreover, for symmetric nanostructures, i.e., for gratings surrounded by identical dielectrics, the line shape of the enhanced transmission is strongly influenced by the coupling between the two degenerate surface plasmon polaritonic modes associated to the two sides of the metal film. In addition, the non-reciprocity of the reflectivity, which occurs when illuminating an asymmetric nanostructure from both sides [24], is demonstrated. The advantage of our approach, as compared to numerical investigations, is that it does not suffer from convergence problems due to discontinuities in the nanostructure. In addition it consistently conserves the energy for structures without loss. In Section 2 we describe the model and give the analytical expressions allowing calculating the transmittance, reflectance and absorption of a sinusoidally modulated structure. In Section 3 we apply the model to silver nanostructures allowing us to reproduce several experimental features previously reported and to get some insights into the physical mechanisms underlying the enhanced optical response of arrays of subwavelength holes.

## 2. Theoretical approach

#### 2.1 Diffraction of TM polarized waves by a 1D metallic grating.

In this section we describe the diffraction of TM polarized electromagnetic waves by a uni-dimensional grating surrounded by two different dielectric media. It is based on the modal method by Fourier expansion, commonly referred to as the Rigorous Coupled Wave Analysis (RCWA) [21]. In this method the dielectric function and the electromagnetic fields inside the periodic structure are expanded as Fourier series and Bloch-wave modes respectively and thereby the boundary-value problem is reduced to an algebraic eigenvalue problem.

The structure under investigation is sketched in Fig. 1. It consists in a periodically modulated metal film with a finite thickness *h* and a period *a*_{0}
. The structure is bound by two homogeneous media (*I* and *III*) with dielectric constants *ε*
_{I} for *z* <0, and ε_{III} for *z* >*h*. In the modulated region *II* (0<*z*<*h*), the dielectric function can be expanded in a Fourier series as

where *n* is an integer, *g* = 2*π*/*a*
_{0} is the period of the reciprocal lattice. In general, the Fourier components *ε*
_{n} are complex quantities. In the case of a grating with alternated regions made of a metal (dielectric function *ε*
_{m}) and a dielectric (slits with a width *d* and dielectric constant *ε*
_{s}) the Fourier harmonics are given by

where *f* = *d* / *a*
_{0} is the filling factor and *ε*
_{0} is the average value of the dielectric function of the structure.

Let us assume that the modulated structure is illuminated from the medium *I* with a p-polarized (TM) electromagnetic plane wave incident at an angle θ with a maximum amplitude normalized to unity. The temporal phase factor exp(*iωt*) can be discarded from all expressions of the fields. Therefore, the magnetic field of the incident wave can be written as

with *k*_{Ix}
=*k*_{I}
sin *θ* = *k*
_{0}-√*ε*_{I}
sin *θ*; *k*_{Iz}
= *k*
_{0}√*ε*_{I}
cos *θ*

The multiple diffraction of the incident light by the modulated structure gives rise to a series of diffracted waves (reflected or transmitted). Their directions are given by Bragg law: the in-plane component *k _{xn}* of the n

^{th}diffracted wave differs from that of the incident wave

*k*by:

_{Ix}where *n* is an integer.

Some of these diffracted orders are evanescent. They insure energy and wave vector conservation to excite surface plasmon-polariton (spp) modes at one or the other interface

The reflected and transmitted fields *H*_{Iy}
and *H*_{IIIy}
in regions *I* and *III* respectively, can be expressed as Rayleigh expansions

with *k*
^{2}
_{(I,III)zn} =${k}_{(I,\mathit{\text{III}})}^{2}$ -*k*
^{2}
_{xn}

Each of the coefficients *r*_{n}
and *t*_{n}
represents the normalized magnetic-field amplitude of the *n*^{th}
backward-diffracted (reflected) wave or forward-diffracted (transmitted) wave.

The total electric field in the two regions *I* and *III* is then found from Maxwell’s equation

*ε*_{f}
being the permittivity of the free space.

Inside the modulated region, Maxwell equations can be written as:

Bloch’s theorem allows writing the tangential magnetic and electric field components as generalized Fourier expressions:

*H _{IIn}* (

*z*) and

*E*(

_{IIn}*z*) are the normalized amplitudes of the spatial harmonics of the considered fields. A similar equation stands also for the component E

_{IIz}.

Substituting eqs. (11) and (12) into eqs. (9) and (10) and by doing some algebraic operations we find that the spatial harmonics *H _{IIn}* (

*z*) and

*E*(

_{IIn}*z*) are related by a set of coupled equations which can be written in the following matrix forms

Finally, the system can be written in one of the compact forms:

or

with

and

where [*ε*] denotes the Toeplitz matrix generated by the Fourier coefficients of *ε*(*x*) such that its (*n*, *m*) element is *ε*
_{n-m}. *I*_{d}
is the identity matrix and *K*_{x}
is a diagonal matrix which elements are *k*_{xn}
/*k*_{0}
. *M*, *M*’ and *K*_{x}
are (*N*×*N*) matrices, *N* being the number of the harmonics retained in the field expansion. [H_{II}] and [E_{II}] are the field vectors with components *H _{IIn}* (

*z*) and

*E*(

_{IIn}*z*).

Previous works concerning the RCWA [21, 22] solve numerically the set of Eqs. (16) for the magnetic field H_{II} rather than for the electric field, by using a matrix *M*’ given by Eq. (18) [21]. This approach suffers from convergence problems due to the discontinuities inside the modulated nanostructure. As discussed by Li [25], one can bypass these convergence problems by using a modified matrix M’. As shown in the following, we found that an analytical expression can be extracted from Eq. (15). It gives a stable solution. In addition, it always conserves the energy for structures without loss.

The resolution of the differential Eq. (15) requires the determination of the eigenvalues and eigenvectors of the matrix *M*. In these conditions, the spatial harmonics *E _{IIn}* (

*z*) can be represented by a sum of plane waves:

*q*_{m}
being the positive square root of the eigenvalues of the matrix *M* and *E*_{n,m}
being the elements of the corresponding eigenvector matrix [*E*] . Using the Eqs. (9) and (19), the spatial harmonics *H _{IIn}* (

*z*) can be written as

*H*_{n,m}
are the elements of the product matrix [*H*]= [*E*] [*ε*][*Q*], where [*Q*] is a diagonal matrix with diagonal elements -1/k_{0}
*q*_{m}
.

The unknown Bloch excitations (${\psi}_{m}^{+}$,${\psi}_{m}^{-}$) and amplitudes of the diffracted fields (*r*_{n}
, *t*_{n}
) are determined by matching the tangential components of the electric and the magnetic fields at the boundaries.

At the input boundary (*z*=0)

And at the output boundary (*z* = *h*)

By eliminating *r*_{n}
and *t*_{n}
from Eqs. (22) and (24), we get the following set of coupled equations

$$\sum _{m=1}^{N}\left[\frac{{{k}_{\mathit{IIIz}}}_{n}}{{\epsilon}_{\mathit{III}}}{H}_{n,m}+i{E}_{n,m}\right]\mathrm{exp}\left(-{k}_{0}{q}_{m}h\right){\psi}_{m}^{+}+\sum _{m=1}^{N}\left[-\frac{{{k}_{\mathit{IIIz}}}_{n}}{{\epsilon}_{\mathit{III}}}{H}_{n,m}+i{E}_{n,m}\right]{\psi}_{m}^{-}=0$$

or in matrix form

*X*, *K*_{IZ}
and *K*_{IIIZ}
are diagonal matrices with elements: exp(-*k*
_{0}
*q*_{m}
*h*), $\frac{{{k}_{\mathit{Iz}}}_{n}}{{\epsilon}_{I}}$ and $\frac{{{k}_{\mathit{IIIz}}}_{n}}{{\epsilon}_{\mathit{III}}}$ respectively. *INC* is a vector with components: $\frac{2{k}_{I}}{{\epsilon}_{I}}{\delta}_{n0}$.

The diffraction efficiencies are defined as:

*n*_{I}
and *n*_{III}
are the indices of refraction of the media *I* and *III* respectively.

#### 2.2 Analytical expressions of the diffraction efficiencies

In order to get a tractable expression of the diffraction efficiencies *R*_{n}
and *T*_{n}
, let us consider the case of a structure illuminated at normal incidence (θ=0). We truncate the expansions by considering only the first three components in the Fourier series of the periodic dielectric function of the structure in Eq. (1) and of the fields in the different media in Eq. (6, 7, 11 and 12). This truncature allows to obtain only one surface plasmon-polariton resonance, determined by Eq. (5), at each interface of the structure. The higher order of the expansion would give the other SPP modes. Let us also notice that with such truncation, the higher order diffraction efficiencies *R*_{n}
and *T*_{n}
are rigorously equal to zero. Therefore the normalization condition $\sum _{n}\left({R}_{n}+{T}_{n}\right)=1$, valid for a nanostructure without loss, applies only to the zero and ±1 diffraction orders. The development of the dielectric function (1) is then given by:

Since by symmetry consideration in the plane of the structure *ε*
_{-1} = *ε*
_{1}, one has:

with *α* = *ε*
_{1} / *ε*
_{0}. Within this approximation, the initial structure sketched in Fig. 1 is reduced to a sinusoidally modulated metallic film.

The matrices *K*_{x}
and [*ε*] can be written in this case as

Thus the matrix *M* is given by

This matrix has three eigenvalues, which are given by

$${q}_{2}^{2}=\frac{1}{2{k}_{0}^{2}}\left(-2{\epsilon}_{0}{k}_{0}^{2}+{g}^{2}-\sqrt{{g}^{4}+8{\epsilon}_{0}{k}_{0}^{2}{\alpha}^{2}\left({\epsilon}_{0}{k}_{0}^{2}-{g}^{2}\right)}\right)$$

$${q}_{3}^{2}=\frac{1}{{k}_{0}^{2}}\left(-{\epsilon}_{0}{k}_{0}^{2}+\frac{{g}^{2}}{1-2{\alpha}^{2}}\right)$$

The corresponding eigenvector matrix can be written as

with

$${E}_{2}={k}_{0}^{2}\frac{{\epsilon}_{0}+{q}_{1}^{2}}{\alpha \left({\epsilon}_{0}{k}_{0}^{2}-{g}^{2}\right)}$$

Therefore the matrix [*H*] is given by

where

$${H}_{3}=-\frac{{\epsilon}_{0}}{{k}_{0}{q}_{1}}\left({E}_{1}+2\alpha \right)\phantom{\rule{1.2em}{0ex}}{H}_{4}=-\frac{{\epsilon}_{0}}{{k}_{0}{q}_{2}}\left({E}_{2}+2\alpha \right)$$

The set of boundary conditions (26) can be then reduced to a set of 6×6 coupled equations with the unknown quantities (${\psi}_{1}^{\pm}$,${\psi}_{2}^{\pm}$,${\psi}_{3}^{\pm}$), which can be easily reduced to the following 4×4 matrix equation

where *A* is given by

with

We define Δ = det[*A*] as the determinant of the matrix *A* and Δ_{1,2,3,4} as the co-determinants of the matrices obtained by replacing the corresponding column of the matrix *A* with the vector defined in the right hand-side of Eq. (39). Therefore, the unknown Bloch excitations can be written as: ${\psi}_{\mathrm{3,1}}^{+}=\frac{{\Delta}_{\mathrm{1,2}}}{\Delta}$, and ${\psi}_{\mathrm{3,1}}^{-}=\frac{{\Delta}_{\mathrm{3,4}}}{\Delta}$. Then, by using Eq. (21) and (23), one can calculate the amplitudes of the different diffracted orders at normal incidence as

$${t}_{1}=\frac{1}{\Delta}\left[-{\Delta}_{1}{H}_{1}\mathrm{exp}\left(-{k}_{0}{q}_{1}h\right)-{\Delta}_{2}{H}_{2}\mathrm{exp}\left(-{k}_{0}{q}_{2}h\right)+{\Delta}_{3}{H}_{1}+{\Delta}_{4}{H}_{2}\right]$$

$${r}_{0}=\frac{1}{\Delta}\left[-{\Delta}_{1}{H}_{3}-{\Delta}_{2}{H}_{4}+{\Delta}_{3}{H}_{3}\mathrm{exp}\left(-{k}_{0}{q}_{1}h\right)+{\Delta}_{4}{H}_{4}\mathrm{exp}\right(-{k}_{0}{q}_{2}h\left)\right]-1$$

$${r}_{1}=\frac{1}{\Delta}\left[-{\Delta}_{1}{H}_{1}-{\Delta}_{2}{H}_{2}+{\Delta}_{3}{H}_{1}\mathrm{exp}\left(-{k}_{0}{q}_{1}h\right)+{\Delta}_{4}{H}_{2}\mathrm{exp}\right(-{k}_{0}{q}_{2}h\left)\right]$$

By using Eqs. (27) and (28) the different diffracted efficiencies can be calculated as follows: the zero order transmission coefficient ${T}_{0}=\frac{{n}_{I}}{{n}_{\mathit{III}}}{\mid {t}_{0}\mid}^{2}$,the first order transmission coefficient ${T}_{1}=\frac{{n}_{I}}{{n}_{\mathit{III}}}{\mid {t}_{1}\mid}^{2}\mathrm{Re}\left(\frac{{n}_{I}{k}_{\mathit{IIIz}}}{{k}_{0}{\epsilon}_{\mathit{III}}}\right)$, the specular reflection coefficient *R*
_{0} = ∣*r*
_{0}∣^{2} and the first order reflection coefficient ${R}_{1}={\mid {r}_{1}\mid}^{2}\mathrm{Re}\left(\frac{{k}_{\mathrm{Iz}}}{{k}_{0}{n}_{I}}\right)$. The absorption coefficient of the structure is then defined by

The frequencies of the surface plasmon polariton resonances *ω*_{spp}
are given by the relation Δ = 0. As Δ is a complex function of the frequency, its roots are generally complex and can be written as *ω*_{spp}
= Ω
_{spp}
-*i*Γ_{spp}. The finite linewidth Γ_{spp} of the SPP resonances is due to their decay into elementary excitations (coupling to the quasi-particle continuum) and to their damping by radiation. Thus, one cannot find a real value Ω of the frequency for which Δ(Ω)=0. In practice, the spectral positions Ω_{spp} correspond to the minima of the
quantity ∣Δ∣ as displayed in Fig. 2.

Finally, to test the accuracy and the consistency of our analytical model, we have checked the energy conservation for different modulated structures without loss (absorption *A* = 0) when varying different parameters (period, film thickness, slit width…). We found that this condition is always fulfilled within accuracy better than 10^{-13}. This was not the case when deriving from Eq. (16) an analytical expression of the diffraction efficiencies with the procedure described above. In addition, let us remark that the approximation of a sinusoidal profile of the dielectric function (Eq. 29) is not important in the spectral domain investigated in the following. Indeed, even when such truncation is not assumed, the higher diffraction orders (using for example a numerical approach) do not contribute to the diffraction efficiencies when *λ* ≥ *a*
_{0}√*ε*_{I}
∣*n*∣ for *R*_{n}
and *λ* ≥ *a*
_{0}√*ε*_{III}
∣*n*∣ for *T*_{n}
. In most practical cases, with realistic values of *λ*, *a*_{0}
and *ε*_{I,III}
these conditions are fulfilled only when ∣*n*∣ ≥ 2.

## 3. Diffraction efficiencies of structured metal films: line shape and coupling of Plasmon resonances

In this section we consider the structure sketched in Fig. 1. It consists in a silver grating deposited on a glass substrate, with the following parameters: *a*_{0}
= 700 nm, *d* = 200 nm and *h* = 120 nm. The dielectric function of silver has been taken from tabulated experimental data [26].

#### 3.1 Line shape of the enhanced transmission resonances: Fano profiles

Figure 3 displays the calculated zero order transmission spectrum of the structure. The enhanced transmission exhibits two resonances, located at 731 nm (resonance R_{I}) and 1097 nm (resonance R_{III}) respectively. Their spectral shapes are strongly asymmetric. They display the characteristic Fano profile reported in other works [7, 23]. Fano resonances are a general characteristic of systems where two transmission pathways interfere, a resonant and a non-resonant one. Depending on the relative phase and amplitude of these two pathways different line shapes occur, parameterized by a constant *q* called the asymmetry parameter. A study of this effect has been made recently for arrays of holes assuming that the Fano profile results from the interference of the resonant SPP and the non-resonant direct scattering from the metallic structure. The Fano line shape clearly shows up, from our model, for the two SPP resonances R_{I} and R_{III} associated to the metal/dielectric interfaces. As seen in the inset of Fig. 3, for R_{I}, the maximum of transmission is slightly red-shifted with respect to the position of the SPP resonance given by min ∣Δ(Ω)∣.

#### 3.2 Reflectivity and absorption: SPP, Wood anomalies and non-reciprocity

Figure 4 shows the calculated zero order transmission (4a) and the specular reflection and absorption (4b) spectra of the structure when it is illuminated from both sides. For clarity only the spectral regions around the two SPP resonances are displayed. To each transmission maximum corresponds a reflectivity minimum and a maximum of absorption. This is in agreement with the recent experimental results of Barnes et al. [27]. A detailed analysis of the spectra shows that the spectral positions of the reflection and absorption extrema are slightly different from the corresponding transmission maxima. The absorption resonances are an additional indication of the role played by SPP. Indeed, as we have outlined above there are two possible channels for SPP decay: either by radiation and reemission of the photons, a process which enhances the transmission, or by absorption in the metal via its decay into quasi-particles. These two processes are indeed at the origin of the enhanced resonances both in transmission and absorption. Let us stress that, although the light coupling to the SPPs gives rise to a strong exaltation of the transmission, the large part of the transferred energy from the incident wave to the SPPs is dissipated in the metal via its decay into elementary excitations. This explains why the phenomenon of transmission enhancement strongly depends on the imaginary part of the dielectric function of the metal constituting the structure.

The Wood-Rayleigh anomalies given by *k*
_{(I,III)z} = 0, where *k*
_{(I, III)z} are defined in Eq. (41), are located at *λ* = 700 nm and *λ* = 1057 nm. They show up in the transmission, reflection and absorption spectra as discontinuities and they are not the cause of the transmission minima as it is usually assumed [1, 2].

It is important to notice that our modeling of the silver structure is consistent with the experimental results of Altewischer et al. [24] who studied arrays of holes in silver films with the following parameters: array period 700 nm, holes diameter 200 nm and film thickness 200 nm. Their reflection spectra show three peaks at the wavelengths 750 nm, 810 nm and 1100 nm. They assigned the first and the third peaks to the excitation of (±1,0) and (0,±1) SPPs on metal/air and metal/glass respectively. They correspond to the two resonances R_{I} and R_{III} found in our calculation, with very close wavelengths (731 and 1097 nm) in spite of the fact that we use a one-dimensional structure. However, the second peak at 810 nm, which in the experimental work is associated to the excitation of (±1,±1) SPP from the metal/air interface, cannot be reproduced by our analytical model based on a truncation to the first three orders of diffraction.

Figure 4 also shows that the transmission spectra are unchanged under reversal of the modulated structure in agreement with the Helmholtz reciprocity principle. However, there is a significant difference between the reflection and absorption spectra calculated from the air side and the glass side. This non-reciprocity of the reflection and absorption spectra contains important information about the optical response of these structures. When illuminating from the air-side the peak R_{I} is more pronounced than R_{III} in reflection and absorption, while from the glass-side, it is the reverse the peak R_{III} being larger. This feature can be explained qualitatively as follows. When illuminating an interface the efficient coupling of the incident light with the associated SPP is important. It leads to an important decrease of the reflectivity at the corresponding wavelength and a strong enhancement of the electromagnetic field at this wavelength. When exciting from the other interface the SPP resonance at this same wavelength is excited with a weaker field which has been attenuated after propagation through the structure. In that case, it leads to a weaker decrease of the reflectivity at this wavelength. The reverse process takes place for the other SPP resonance. For the transmission, the occurrence of the two mechanisms, the SPP generation and the absorption during propagation, is a commutative sequence. As already mentioned by Altewischer et al. [24] the non-reciprocity of the reflectivity under the reversal of the direction of illumination has practical consequences. For example, it is a powerful diagnostic to assign the SPP resonances to their associated interfaces.

#### 3.3 Coupled surface plasmon polaritons

In the symmetric case, the SPPs created at each interface are degenerated and have the same frequency. For very thick films they don’t interact with each other, but when the film thickness becomes smaller the enhanced electromagnetic fields associated with the two SPPs overlap. Thus, the SPPs on the two interfaces couple together to form two new coupled SPPs [16, 28]: the short range SPP (SRSPP) with lower frequency and symmetric charge density (electromagnetic field) distribution and the long range SPP (LRSPP) with higher frequency and antisymmetric charge density distribution. It is important to note that each of the two plasmons SRSPP and LRSPP propagates on both interfaces simultaneously. Our model reproduces these features quite nicely.

Figure 5 represents the transmission (5a) and the reflection (5b) spectra of structures with the parameters *a*_{0}
= 700 nm, *d* = 200 nm, *ε*_{I}
=*ε*_{III}
=1 and a variable film thickness *h*. These spectra are calculated by neglecting losses by absorption in the metal i.e. by neglecting the imaginary part of the dielectric function of the metal. For smaller values of *h* the SPP resonance R_{I} is split in two peaks which shift in opposite directions when decreasing the film thickness. The transmission of the film approaches unity. As expected the peak of the LRSPP is narrower than that of the SRSPP. When increasing the thickness *h* the two resonances shift closer to each other until they form a unique resonance which amplitude decreases. The transmission of 100% means that there is a complete formation of SRSPP and LRSPP. The energy transfer from one side to the other is resonant provided that the lifetime of the SPPs is sufficiently long. In the case of thicker films the coupling between the SPPs of the two sides is weaker. They have the characteristics of isolated SPPs rather than coupled SPPs and the transmission is achieved by simple tunneling followed by a reemission on the other interface. This process is less efficient than the resonant tunneling. This situation is similar to the carrier transport in semiconductor heterostructures (more precisely to a three-barrier two-wells structure [18]).

When the losses by absorption are taken into account, using the complex dielectric function of silver, we obtain the spectra of Fig. 6. Clearly, both resonances are broadened and become undistinguishable for thick films. In addition, the amplitude of the two peaks in transmission is drastically reduced. In contrast to Fig. 5, where the absorption is absent, in Fig. 6(c) the spectral evolution of the absorption with decreasing film thickness (asymmetry, peak splitting and amplitude variation) also reflects the coupling between the two SPPs.

For comparison, let us mention the work of Gérard et al. [29] who investigated numerically the SPP coupling in the weak and strong regimes for a symmetric silver nanostructure modulated on both sides of the metal film. In addition, their study was performed for different incident angles. In the strong coupling regime associated to thin films, these authors found that at normal incidence the spectral resonance of the SPP modes become distinguishable, in agreement with our analytical results displayed in Figs. 5 and 6.

#### 3.4 Influence of the surrounding dielectric media

In the asymmetric case, the frequencies of the two SPP modes in the opposite interfaces are different. The resonant tunneling conditions are not satisfied. The transmission becomes smaller in both resonances. In this case, the transmission peaks are related to the direct photon tunneling via the SPP states.

Let us now study the influence of the dielectric function on the line shape of the surface plasmon resonances. This influence has been investigated in the case of gold structures by Krishnan et al [30]. The structure used in our simulation consists in a modulated silver film deposited on a quartz substrate *ε*_{I}
= 2.31 with the following parameters: *a*_{0}
= 600 nm, *d* = 200 nm and *h* = 150 nm. The dielectric constant of the medium I is varied from 1 to 3.24, simulating the effects of an increasing liquid index. Figure 7 represents the evolution of transmission spectra as *ε*_{I}
is increased. The model reproduces the main features of Krishnan’s experiment: the intensity of the SPP resonance *R*_{III}
increases and reaches its maximum value in the symmetric case, i.e., when *ε*_{I}
=*ε*_{III}
= 2.31. Simultaneously its spectral position slightly shifts to the red while the resonance *R*_{I}
drastically shifts towards *R*_{III}
until the two resonances become degenerate (Fig. 7(a)). Then, when *ε*_{I}
>*ε*_{III}
(Fig. 7(b)) the resonance *R*_{I}
decreases and continues shifting to the red while *R*_{III}
decreases until it completely vanishes when its spectral position coincides with the minimum of *R*_{I}
. Then, *R*_{III}
reappears and increases again when both resonances get far apart.

## 4. Conclusion

In this work, we have investigated the theoretical optical response of one dimensional periodically structured metallic films surrounded by different dielectric media. A tractable analytical model is obtained by truncating the Fourier series of the dielectric function and of the electromagnetic field. It gives simplified expressions of the diffraction efficiencies which allow us to obtain straightforwardly the transmission, reflectivity and absorption spectra of the nanostructure. Several important features associated to the surface plasmon polaritons excited on either side of the metallic film, which have been previously reported in various experimental studies, can be traced back by varying the parameters of the structure. In particular the Fano line shape profiles, the coupling between the two surface plasmons as well as the non-reciprocity of the reflectivity upon excitation from each side are well reproduced. This analytical model provides an easy way to investigate more complex situations in particular to compute the dynamical response of these nanostructures when they are excited by ultrashort light pulses. Further work in this direction is under investigation.

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