Since the publication six years ago of a paper [1] demonstrating resonant transmission by a film with a periodic array of holes, over 100 papers have been devoted to analyse the mechanisms of this phenomena both experimentally and theoretically. Whereas the enhanced transmission due to a perfectly conducting structure has been known for some time and used to produce grid polarizers in the infrared or frequency selective surfaces in the microwave domain, the physics involved in the optical regime for noble metals is less simple. The presence of surface-plasmon polaritons which are excitations of a flat bare interface and the presence of losses introduce significant differences. Many papers have put forward the role of surface plasmons [2, 3, 4, 5, 6, 7]. More recently, it has been argued that excitation of surface-plasmon polaritons entails nearly zero transmission [8]. A very recent paper has shown that surface plasmons yield a peak of absorption that coexists with a peak of transmission in periodic arrays of subwavelength holes in a metal film [9]. Many authors have studied the enhanced transmission by a periodic array of slits. For p-polarization, surface waves can be excited but there is no cut-off frequency for the modes guided in the slits. It has been argued that the key mechanism is the excitation of resonances in the slits [10, 11], a mechanism that cannot be invoked in the case of holes due to the presence of a cut-off frequency. It has also been argued that a resonant transmission is a manifestation of the excitation of a resonance of a diffracting structure independently of the presence of surface-plasmon polaritons [8, 12]. This extremely rapid review of the litterature indicates that the resonant transmission is explained in terms of excitation of modes in the slits for the case of slits, in terms of excitation of surface plasmons for both slits and holes. The relative role of these excitations is not clearly established, the role of losses seems to have been clarified. Finally, we note that a problem is still pending. No simple explanation has been put forward to explain the large transmission. In this paper, we focus on the problem of propagation through a silver film with a periodic array of parallel slits. We will argue that the enhanced transmission is associated with the excitation of modes of the whole structure. We will show that these modes can be described as coupled modes mixing both cavity modes and surface modes. We will finally discuss the origin of the large value of the transmission.

In order to analyse the resonant transmission, we have extensively studied the transmission and the absorption of a free standing silver film in air with a periodic array of slits, period of 500 nm, thickness of 400 nm, slits of 50 nm-width for wavelengths in the range [0.5,2] µm. We use a Drude model for the dielectric constant ε(ω)=1-${\omega }_p^2$ /(ω ²+iγω) with ω_p =1.29.10¹⁶ rad.s^-1 and γ=1.14.10¹⁴ rad.s^-1. The numerical results obtained with the method described in ref. [13] are plotted in Figure 1 in a (ω,k) plane.

 

Fig. 1. Transmission efficiency (a) and Absorption efficiency (b) in the plane (ω, k). Period of the grating : Λ=500 nm, Filling factor : F=0.9 and thickness h=400 nm. The brighter the region, the larger the transmission or absorption.

Download Full Size | PDF

Large transmission peaks are observed with a transmission that is above 0.6–0.8. By looking at Fig. 1(b), we observe that the transmission peaks coincide with the absorption peaks. This result agrees with recent experimental data reported in ref. [9]. It indicates that both the enhanced transmission and the enhanced absorption can be attributed to the resonant excitation of a mode of the structure. It also appears that the transmission is limited by losses. We will come back to this point later.

We now study the resonances. It is clear from Fig. 1 (a,b) that the loci of maxima yields the dispersion relation of modes of the structure. The peaks follow closely the dispersion relation of the surface-plasmon polariton of a flat metallic surface in many parts of the (ω,k) space. This is particularly true close to the light line ω=ck and the corresponding folded branches. Yet, we also observe horizontal lines that do not match the usual dispersion relation for surface-plasmon polaritons. A closer look reveals that the position of the resonant frequencies of these horizontal modes are multiple of the same fundamental frequency. This suggests that organ-pipe type modes, hereafter called cavity modes, are excited in the slits. The frequency of these modes should be given by the condition k2d+2Φ=n2π where d stands for the thickness of the film, k for the wavector of the mode, Φ is the phase change upon the reflection at the end of the cavity and n is an integer. By assuming that the mode has a constant phase velocity v and neglecting the phase change at reflection [14], we find ω=nπv/d. In order to further confirm the existence of such a cavity mode, we have studied the electromagnetic field in the structure. Fig. 2(a) shows the field at point B (see Fig. 1). It is clearly seen that there are cavity modes with two nodes excited in the cavity. It is also seen that the field is enhanced along the interfaces. The existence of cavity modes and their contribution to modes with flat dispersion relation has already been discussed in different contexts previously [15, 16, 17].

 

Fig. 2. Near-field intensity of the electric field in logarithmic scale at point B (a), and at point C (b).

Download Full Size | PDF

We now examine the near field structure of the field for the point C close to the light line (see Fig. 2(b)). It is seen that a cavity mode is still excited in the cavity, but the energy is lower than at point B. Conversely, surface modes are excited on both sides of the film. In order to further prove that surface modes of the surface polariton type are excited along the interface, we have studied the amplitude of the order -1 of the grating which is an evanescent mode. We have found that the amplitude of this mode as a function of frequency for a fixed wavevector has resonances (not shown) for the same frequencies than the transmission. This indicates that the surface wave is indeed excited when the transmission is enhanced.

Having established the character of cavity modes or surface modes of the structure modes depending on the position in the (ω,k) plane, we now propose to analyse the data using the concept of coupled mode. We plot in Fig. 3 a schematic view of the dispersion relation of a surface-plasmon polariton in the reduced zone scheme.

We have also added the dispersion relation of the cavity modes which are horizontal as we discussed above. Since the dispersion curves have crossing points, the modes must couple producing new modes with a higher and a lower frequency. This allow to draw schematically a dispersion relation that has the features observed in Fig. 1. We conclude that the enhanced transmission of a periodic array of slits can be understood by introducing the concept of coupled modes. We argue that away from the crossing point, the modes have either a well-defined cavity-mode or surface-mode character. Conversely, close to the crossing point, the modes are in the strong coupling regime and cannot be called either surface-modes nor cavity-modes. A more detailed analysis should take into account the fact that surface plasmons polaritons of both interfaces are coupled through the cavity producing symmetric and antisymmetric modes with slightly different frequencies.

In order to quantitatively check this concept, we introduce a measure of the surface-mode character. Let us denote by U_s the energy of the surface mode component whereas U_c is the energy of the cavity-mode component of a given coupled mode. U_c is obtained by integrating the electromagnetic energy in the cavity (i.e. in the slit), U_s is obtained by integrating the electromagnetic energy of all the evanescent waves, on both interfaces. We now define the surface plasmon character of a mode by a normalized quantity that we will denote as surface-cavity ratio SC=U_s /(U_s +U_c ).

 

Fig. 3. Dispersion relation of surface plasmons (long-dashed lines) and cavity modes (dotted lines). The plain lines shows the coupling between both modes.

Download Full Size | PDF

 

Fig. 4. Branch B-C of the dispersion relation (a) and Surface-cavity ratio SC for this branch (b).

Download Full Size | PDF

We have plotted in Fig. 4 the result of our computations. It is seen that we can clearly define a region where the mode has a well-defined cavity type and there are regions where the mode has a well-defined surface-wave type.

So far, we have argued that the resonant transmission can be attributed to the excitation of modes of the structure. We have proposed to analyse these modes in terms of the coupling between the modes of the bare surface and the modes of the cavity. Yet, we have only studied the transmission and absorption resonances. We now proceed to a direct analysis of the modes themselves. To this aim, we have searched the complex frequencies that are poles of the scattering matrix following the approach outlined in ref. [10]. The outcome is displayed in Fig. 5 (a).

 

Fig. 5. Exact dispersion relation of the structure (a), and associated quality factor (b).

Download Full Size | PDF

We see that the exact dispersion relation is in close agreement with the loci of the enhanced transmission. An additional interesting information can be gained from this study. Since we look for complex eigenfrequencies of the structure, we obtain the decay rate γ=Im(ω) of each mode. Using this information, we have plotted the quality factor Re(ω)/2γ (ω) in Fig. 5(b). From this result, we learn that cavity modes usually have a low quality factor. Conversely, surface modes may have a large quality factor. By comparing Fig. 1 and Fig. 5(b), it is seen that the quality factor of the modes is the factor that governs the width of the transmission peaks, as discussed by Collin et al. in ref. [18].

We complete our discussion of the enhanced transmission by analysing the height of the transmission peaks. The question still open is to understand the origin of the large transmission factor. We first use a fundamental result derived many years ago [19] showing that a non-lossy resonant structure has a zero reflection and therefore a unity transmission if it is symetric with respect to the plane (y,z) and if there is a single propagating order. Owing to this property, which is a fundamental consequence of symmetry, unitarity and reciprocity, one expects a transmission peak of 1. This is a property often observed in the microwave or far infrared regime where losses are practically negligible [20, 21, 22]. When using noble metals in the optical spectrum, losses cannot be neglected and are responsible for the finite value of the transmission factor as discussed in ref. [9]. It is thus useful to introduce the radiative decay rate γ_R and the non-radiative decay rate γ_NR of the modes. The quantity η=γ_R /(γ_R +γ_NR ), analog to the quantum yield of a molecule, is the probability that the mode releases its energy by emitting a photon. We will refer to it as radiative yield. A high transmission is expected if η is large. It turns out that the transmission peak value can be shown to be η². Indeed, owing to the existence of a resonance, the transmission factor can be cast in the form :

where f(ω) is an unknown analytic function. A similar form was introduced in ref. [23]. In this paper, we make no assumption regarding f(ω). In the presence of ohmic losses, γ=γ_R +γ_NR . We know from ref. [19] that without losses (γ_NR =0), |t(ω ₀)|²=1 so that |f(ω ₀)|=γ_R . It follows that in the presence of losses, the transmission factor takes the peak value [γ_R /(γ_R +γ_NR )]². In figure 6, we plot the value of the peak transmission derived from a scattering calculation [13] and η² derived from the complex frequencies of the mode of the structure [10]. It is seen that η² follows quantitatively the transmission peak.

 

Fig. 6. Zero-order transmission efficiency and square of the quantum yield η² along the branch B-C of the dispersion relation.

Download Full Size | PDF

In conclusion, we have introduced the concept of coupled mode to analyse the physical origin of the enhanced transmission phenomenon. This concept allows to reconcile the different points of view put forward in the litterature so far to explain this phenomenon. It also allows to quantitatively explain the observations. We have found numerically the dispersion relations of these modes and shown that they follow the loci of the transmission peaks. We showed that three dimensionless numbers characterize the coupled modes : the surface-cavity ratio SC, the quality factor Q and the radiative yield η. Depending on the distance to the crossing point between a surface mode and a cavity mode, a structure mode may be either cavity-mode type or surface-mode type. The quality factor controls the width of the transmission peaks. We have emphasized the role of symmetry and resonance to explain the large values observed for the transmission factors. We have shown that the value of the transmission peak is limited by the losses and is simply given by the square of the radiative yield. Although the comparison of the transmission of the periodic array of slits with the transmission by a single slit suggests that the transmission is enhanced, the previous discussion shows that it is more apropriate to call it resonant transmission. We believe that these concepts simplify the analysis of the rich physics of the microstructured metallic films. They should allow to engineer the local density of states [24] in order to design resonant fluorescence or emitting light devices. They could aso be used to produce coherent thermal sources [25, 26, 27].