According to standard diffraction theories, the transmission of a subwavelength aperture of diameter d is very weak because light cannot propagate through it if its wavelength λ>2d [1]. It has been shown that this can be overcome by orders of magnitude by surrounding a single subwavelength aperture with a periodic structure such as a hole array or by a periodic corrugation [2–17]. This enhancement arises from the coupling of the incident light with surface plasmon polaritons (SPPs) through the surface grating. In the case of single apertures, it has been shown furthermore that a periodic texture on the exit side can give rise to a beam with a small angular divergence, very different from the expected quasi-isotropic diffraction of a small hole [18–21]. Many investigations have been performed, both theoretical and experimental, and it is now established that these phenomena have a more general meaning. For instance it has been shown that the same results apply in the microwave region [22,23], or for perfect conductors [9,19], because the presence of the indentations in the metal provides an effective refractive index allowing the launching of SPP like surface waves. Very recently, the principle of the beaming has been predicted and observed at the exit of a photonic crystal [24,25].

In order to further understand the details of the transmission process through single apertures surrounded by corrugations, we have analyzed the effect of the aperture depth in bull’s eye structures (single apertures surrounded by concentric periodic grooves) in various configurations. We show among other things that the transmission can be divided into three independent steps whose product determines the total transmission intensity. This has consequences for further use of these structures for practical applications.

The structures were milled by focused ion beam in optically thick Ag films using an FEI Strata DB 235 system. We worked with free-standing metal membranes, which allow us to pattern either one or both surfaces of the film. The samples were then optically characterized by measuring their far-field transmission spectra with an optical microscope coupled to a spectrometer. In what follows, we will present the results obtained in the case of a periodic array of concentric grooves with period P=600 nm, groove width 240 nm and groove thickness 50 nm. The aperture diameter chosen for this study is 270 nm.

 

Fig. 1. Transmission spectra of cylindrical apertures surrounded by 5 concentric grooves on the input side, for a range of hole depths h. The geometric parameters are detailed in the text.

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Fig. 2. Intensity as a function of h for bull’s eye structures with corrugations on the input side (diamonds), and on both sides (squares). The data are derived from Figs. 1 and 3 at three different wavelengths: λ=650 nm (black), λ=700 nm (red), λ=750 nm (blue).

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We began by investigating the role of the aperture depth h on the transmission of a bull’s eye structure with corrugations only on the input side. Fig. 1 shows the results for a range of depths between 800 nm and 200 nm. All spectra exhibit a huge transmission peak at λ=650 nm, resulting from the excitation of SPP on the input side. As the hole become shallower, the peak position redshifts. In addition, Fig. 2 shows that the intensity at a given wavelength follows an exponential decay, which is the expected signature of the transmission in the absence of propagative modes inside the hole. This is the same evolution as that predicted and observed for hole arrays with deep holes, where the SPPs excited on both sides of the film remain uncoupled [4,9].

The experiments were then repeated for double-sided structures, where SPPs are excited on both surfaces. According to Fig. 3, the spectra collected for various hole depths have a peak which is better defined than in the previous case. However, it should be noted that a strong beaming occurs at the exit side of the samples at the peak wavelength [18–21]. Our experimental setup collects the light in a cone of 40° normal to the surface, which means that the part of the light collected at the beaming wavelength is more important than that of the other wavelengths. This partially explains why the peaks are better defined, but we will see later that other reasons exist. Apart from this difference, the spectra of Fig. 3 follow a very similar behavior as those of Fig. 1, since the peak grows and goes to the longer wavelengths as h decreases. In addition, the intensity follows the very same exponential law, as one can see in Fig. 2, where we compared the intensity versus h for the single and double-sided structures. Hence this plot shows that the transmission regime of a hole surrounded by a bull’s eye structure on the input side is not modified if we pattern the output surface in the same array.

 

Fig. 3. Transmission spectra of cylindrical apertures surrounded by 5 concentric grooves on both sides, for a range of hole depths h.

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As we show now, these results suggest that the enhancement of the input side is not correlated with the enhancement and the beaming of the exit side. This can be understood if one assumes that the transmission process can be separated into three independent steps: coupling in, transmission and coupling out. Let us first consider the case of a corrugation only on the input side. According to the above hypothesis, the transmitted intensity I_ci can be written as:

where f_ci is the coupling function of the corrugated input side i, T is the cutoff function of the central aperture and f_e is the coupling out function of the aperture on the exit side e. Note that f_ci and f_e are assumed independent of h, and T represents the exponential decay of Fig. 2. The same remarks can be applied for a single hole without corrugations, except that the transmission I(λ, h) differs by the input function f_i(λ):

Hence, the ratio I_ci/I is independent of h:

We can verify this point by dividing each spectrum of Fig. 1 with that of an uncorrugated single hole with same dimensions. Indeed as can be seen in Fig. 4, in such a case all the curves are identical, hence confirming our hypothesis.

 

Fig. 4. Ratio between the transmission spectra of Fig. 1 and those of isolated apertures without corrugations of same dimensions.

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If we wish to make a similar analysis for the double sided structures, we must take care to divide the data by the right reference. In other words, because the diffraction pattern of an aperture surrounded by corrugations on the output side is very different from that of a simple aperture, the comparison here had to be done between double-sided structures and apertures with corrugation just on the output side. In such case, they both have the same diffraction pattern and therefore the transmission intensity collected by the microscope can be compared. Following the same reasoning as above (Eq. (1) and (2)), the transmitted intensity I_ce of a single aperture with a corrugation only on the exit side is given by the product :

where f_ce(λ) is the contribution from the corrugated exit surface. On the other hand the transmission I_cie(λ, h) of double sided structures is then:

If we now take the ratio between Eq. (5) and (4), one obtains the same ratio as in Eq. (3), namely:

The experimental ratios I_cie(λ, h)/I_ce(λ, h) in Fig. 5 shows that indeed they are again invariant with depth and furthermore that they are superimposible with the corresponding curves in Fig. 4. This confirms the hypotheses made in formulating Eq. (1)–(6).

In summary, the transmission mechanism through a single aperture can be divided into three independent steps, namely coupling in, transmission and coupling out. The transmitted intensity can be seen as the product of functions associated with each step. The coupling in function reflects the amount of incident light converted into evanescent waves at the entrance of the subwavelength aperture. The evanescent field is then exponentially attenuated by the cut-off function of the hole. At the exit side, the evanescent waves recombine into propagative waves with an efficiency given by the coupling out function.

 

Fig. 5. Ratio between the transmission spectra of Fig. 3 and those of holes of same dimensions surrounded by a bull’s eye structure on the exit side only.

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One consequence is that, for a given hole depth, shape, and lateral dimensions, the only way to modify the transmission properties of a subwavelength aperture is to act upon the coupling and decoupling of light. Indeed, the cut-off function follows in any case the exponential law of a single aperture without corrugations, whereas the coupling in and out functions can be controlled by patterning the metal surface. If the surface presents a periodic texture, the coupling process is mediated by exciting SPPs on the grating. In the case of an input corrugation, this gives rise to a strong enhancement of the transmission at wavelengths slightly higher than the periodicity. If a grating is patterned on the output surface, the light exiting the structure at a given SPP wavelength is mainly reemitted in a narrow beam. The input and the output coupling functions can be optimized as shown in other studies [16,17,19] by tailoring for instance the width and depth of the groove structures for a given transmission wavelength. It should be noted that the presence of groove modes plays a key role both for the beaming induced by the output corrugation and for the field enhancement on the input side [19].

It should also be pointed out that the coupling and decoupling process for a single subwavelength hole without corrugations also involves surface plasmons. These modes are different from SPPs launched on a grating, as they are localized surface plasmons on the aperture ridge and have no dispersion [26]. Their excitation leads to the presence of broad peaks in the transmission spectra of the holes. The boost in transmission is not as strong as the enhancement provided by SPPs launched along a corrugated surface, therefore its spectral effects can be hidden if the attenuation through the hole is too important. The presence of a secondary peak at λ=800 nm for the shallowest samples of Fig. 1 and Fig. 3 are in fact signatures of such localized modes.

For hole arrays, two transmission regimes were predicted and found depending on the hole depth [4,9]. For deep holes, the SPPs of the two surfaces remain uncoupled and the intensity is also exponentially sensitive to the aperture depth, while for shallow holes the SPP modes of both sides couple leading to a leveling off of the intensity. The results presented here show that both single and doubled sided bull’s eye structures display only the exponential regime on hole depth even for the shallowest samples. For the single sided structure, it is not surprising, however, one might have expected coupling between modes in the double sided structures. This is probably explained by the fact that in the case of single apertures surrounded by corrugations, the interaction area (i.e. the hole section) between the two surfaces is very small compared to the overall structure. In the case of hole arrays, each period provides an interaction channel.

One question that remains is whether the output side corrugation contributes to the transmission enhancement. In conditions similar to our experiments, a theoretical analysis indicates just a weak contribution [13]. Our results can not provide any information in this regard because we are not able to collect the total transmitted intensity over 2π. However the reciprocity principle requires that if everything is symmetrical, then it doesn’t matter which side a corrugation is in order for the enhancement to occur. In our experiments, it is not possible to obtain completely symmetric conditions since the wave-front is modified after passage through the aperture and we are unable to recreate such a complex wave-front in the reverse path. In this regard, it should be remembered that in a hole array, the wave-front of the transmitted light and the incident light are both planar and the transmission enhancement is independent on which side the SPP modes are located [2,5,6].