1. Introduction

Propagation of electromagnetic waves through arrays of subwavelength apertures in metallic films has attracted considerable attention following the initial demonstration of enhanced optical transmission [1]. Apparently simple structures have been shown to exhibit a range of complex electromagnetic interactions that lead to rich transmission and reflection spectra. The initial report was followed by a large number of studies seeking to elucidate the role that surface plasmon polaritons (SPPs), localized surface plasmons (LSPs), material properties and device geometry play in the optical response of these hole arrays (for reviews see [2, 3]). The shape of the apertures has been shown to be particularly important since the excitation of LSPs can produce resonances accompanied by transmission maxima even though the apertures are arranged randomly [4, 5].

The emphasis of most theoretical and computational studies into the electromagnetic behavior of hole arrays has been on plane wave illumination. It has been noted that some features in the transmission spectra, most notably those associated with SPP excitation, are highly dependent on angle of incidence and polarization, whereas other features, such as those associated with LSP resonances, are robust to incidence parameters [2]. There have been some interesting investigations into the behaviour of metallic structures when illuminated with more complex beams. These include rigorous modeling of scattering of a focused spot by a grating [6], the excitation of surface plasmons on the surfaces of metallic films by focused [7] and optical vortex [8] beams and surface plasmon focusing by illuminating a coaxial aperture with radially polarized light [9].

Here we look at the specific case of transmission through a periodic array of annular apertures in a perfectly conducting screen [10, 11]. This structure replicates the rich resonant behavior of nanometric plasmonic devices and provides a useful platform for investigating the relative contributions of planar surface wave and localized resonances [12]. By studying the interaction of off-normal incidence plane waves with the array, we show that it is possible to filter the angular spectrum of certain beams. In particular, linearly polarized Gaussian-like and radially and azimuthally polarized Gauss-Laguerre-like beams are considered. We see that the capacity of the incident beam to excite different electromagnetic modes of the system significantly influences transmission of these beams.

2. Theoretical and computational background

The structure under consideration is shown in Fig. 1 . The perfectly conducting screen has a thickness h and is perforated with coaxial apertures arranged in a square grid with period d. The apertures have an outer radius of a and an inner radius b. Calculations of transmission of electromagnetic plane waves through this structure are performed using the modal method [11]. The incident field is taken to be a monochromatic plane wave with wavelength λ. The electric and magnetic fields above and below the grid structure are written as a Rayleigh expansion in propagating and evanescent linearly polarized plane waves, while the fields within the apertures are written as a superposition of coaxial waveguide modes. Mode and field amplitudes are determined using boundary conditions at the upper and lower surfaces of the structure.

 

Fig. 1 Periodic array of annular (coaxial) apertures in a metallic screen.

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These calculations can be extended to the examination of transmission by arbitrary propagating fields by writing these as an expansion in plane waves. Specifically, the electric field can be written in the form:

In the case of an arbitrary field, evanescent components with $k_x^2+k_y^2>k^2$need to be included in the range of integration in Eq. (1). In many practical situations, however, the distance from the source of the illumination is such that evanescent contributions to the field will be negligible. In this case, the integrals in Eq. (1) are restricted to a range for which $k_x^2+k_y^2\le k^2$ [13]. Furthermore, the illumination system provides an additional restriction on the range of spatial frequencies interacting with a sample of interest. This can be incorporated into calculations by restricting the angular range implicit in Eq. (1) to a cone described by the numerical aperture (NA) of the system. Given the direct relationship between k_x, k_y, the angle of incidence and the polarization of the incident field, it is possible, with knowledge of plane wave transmission through the structure over the relevant range of incidence angles and polarizations, to determine the transmitted electric field using (1) and the magnetic field using an analogous expression.

Here the incident field is defined in terms of its angular spectrum decomposition (1) rather than its spatial distribution in a plane. Three different types of illumination are considered:

3. Results and discussion

In order to determine the transfer function of the hole array, the transmission spectrum as a function of polarization and angle of incidence was calculated. Figure 2 shows plots of the transmission through arrays of apertures as a function of angle of incidence for both TE (Fig. 2(a)) and TM (Fig. 2(b)) polarization. In both cases the ratio of the outer radius of the rings, a, to the period of the arrays, d, a/d is 0.45, the ratio of inner radius to period, b/d, is 0.40 and the ratio of screen thickness to period, h/d is 0.40. No substrate was included in calculations. A high-aspect ratio screen was chosen to clearly isolate a resonance of the TEM mode that has a wavelength approximately equal to twice the thickness of the screen.

 

Fig. 2 Transmission through coaxial hole arrays as a function of angle of incidence and wavelength for TE (a) and TM (b) polarization. The apertures are separated by a distance d, have outer radius 0.45 d, inner radius 0.4d and are located in a screen of thickness 1.5d.

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Figure 2 exhibits the rich spectral response of these structures and the results are consistent with those of other studies [14]. The locations of peaks in the spectra are a consequence of the excitation of surface waves near Wood anomalies, localized aperture resonances or a combination of the two. At normal incidence, the longest wavelength Wood anomaly (±1,0), (0,±1) occurs at a wavelength of λ = d. The maximum at 2.65d is associated with the cutoff wavelength of the coaxial waveguide TE₁₁ mode. Note that this resonance is a zeroth order Fabry-Perot resonance and, unlike surface wave resonances, has a position that is independent of the thickness of the grid and angle of incidence. In the case of TM polarization (Fig. 2(b)), the excitation of the TEM mode at a wavelength of 3.18d at off-normal angles of incidence can be seen. This mode cannot be excited with normally incident plane waves nor off-normally incident TE waves due to its radial symmetry.

Figure 3 shows the computed zeroth-order angular spectrum intensity, for the incident (Figs. 3(a), (e) and (i)) and transmitted (Figs. 3(b-d), (f-h), (j-l)) spectrum for the three different beams under consideration at illumination wavelengths of 1.02d, 2.65d and 3.18d. These wavelengths correspond to normal incidence excitation of surface waves (1.02d), excitation of the TE₁₁ mode (2.65d) and excitation of the TEM mode at off-normal incidence for TM polarization (3.18d). Note that at 2.65d and 3.18d only the zeroth transmitted diffracted order is propagating for the range of angles contributing to the angular spectrum. It can be seen that at a wavelength of 2.65d (Figs. 3(c), (g) and (k)) the angular spectrum is transmitted with high fidelity. Not only is the shape of the angular spectrum transmitted with no apparent modification, but the normalized transmission of the beam is close to 100%. This wavelength corresponds to a localized aperture resonance and would be analogous to excitation of localized surface plasmons inside nanoscale apertures in real metals [4]. At 1.02d and 3.18d, however, the transmission process is more complex since there is a dependence of transmission of plane waves on angle of incidence. At a wavelength that corresponds to high transmission at normal incidence due to the excitation of surface waves, but relatively low transmission at other angles of incidence, (λ = 1.02d) (Figs. 3(b), (f) and (j)) the spectrum of the x-polarized (Fig. 3(b)) beam is narrowed along the polarization direction. The changes in spectral shape seen at this wavelength are consistent with the ‘beaming’ effect seen in other structures where surface wave or surface plasmon polariton effects are responsible [15]. Different mechanisms are at play at a wavelength of 3.18d. As has been discussed, this wavelength corresponds to the excitation of the TEM mode, but for only off-normal incidence and for TM polarization. Hence, the low-spatial frequency components of the x-polarized beam (Fig. 3(d)) are suppressed relative to higher order contributions so that the spectrum is dominated by two lobes and the total transmission is relatively low (less than 5%). The radially polarized beam (Fig. 3(h)), however, is transmitted with an efficiency of ~35% and it can be seen that low spatial frequencies are suppressed relative to higher spatial frequencies as expected. The shape of the spectrum of the transmitted azimuthally polarized beam (Fig. 3(l)) is largely unchanged from the incident field. The normalized transmission, however, is extremely low (<1%).

 

Fig. 3 Transmitted zeroth order angular spectrum intensity, ${\left|A\left(k_x,k_y\right)\right|}^2$, for linearly (a-d), radially (e-h) and azimuthally polarized (i-l) beams at wavelengths of 1.02d ((b), (f) and (j)), 2.67d ((c), (g) and (k)) and 3.18d ((d), (h) and (l)). All plots are normalized.

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Figure 4 shows the incident spatial intensity profile (Figs. 4(a), (e) and (f)) as well as the near-zone intensity, a distance 0.05d below the lower surface of the structure. The beam and wavelength labeling are consistent with Fig. 3. It should be noted that the angular spectrum was kept fixed and, hence, a broadening of the incident spatial profile with increasing wavelength occurs. At a wavelength of 1.02d (Figs. 4 (b), (f) and (i)) the near-zone fields are quite different in shape from the incident fields due to the excitation of surface waves. At 2.65d, however, where the angular spectrum is transmitted relatively intact, there is excellent agreement between the general shape of the incident spatial intensity profile for all beams (Figs. 4(c), (g) and (k)). The role played by the TE₁₁ mode is obvious on close inspection since the intensity profile of this mode, with maxima in intensity parallel to the direction of polarization, is apparent in the near-fields adjacent to the apertures. At 3.18d (Figs. 4(d), (h) and (l)), there are again significant differences between the near-field profiles and those of the incident field. The excitation of the TEM mode in both the x-polarized and radially polarized beam is apparent. In particular, the near-field intensity profile of the radially polarized beam is narrower than that incident on the structure as expected from the suppression of low spatial frequency components seen in Fig. 3(h). In the case of the azimuthally polarized beam (Fig. 4(l)), the TEM mode cannot be excited and the means of transmission is still via the TE₁₁ mode. Since this is off-resonance, however, the transmission is (as indicated earlier) very low.

 

Fig. 4 Incident and near-field (a distance 0.05d below the lower surface of the array described in the Fig. 2) intensity patterns. The intensity patterns correspond to the beams and wavelengths of Fig. 3. All values are normalized to the maximum of each quantity.

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4. Conclusion

The angular spectra and near-field intensities of beams transmitted through periodic arrays of coaxial apertures in perfectly conducting screens have been considered. Transmission was effected by the excitation of either surface waves or aperture resonances. In the case of wavelengths close to a Wood anomaly, the transmitted angular spectrum was a complex function of spatial frequency. Since the excitation of the TE₁₁ mode is relatively insensitive to the angle of incidence and the polarization, any beam illuminating the sample at a wavelength corresponding to this localized resonance is transmitted with high fidelity with little modification of its angular spectrum. In the case of wavelengths corresponding to resonances of the TEM mode, however, low spatial frequencies are suppressed and only TM-polarized angular spectrum components are transmitted. The particular examples of linearly, radially and azimuthally polarized beams were presented. It is anticipated that these conclusions would extend to hole arrays being designed for used in the visible and near-infrared regions of the electromagnetic spectrum and may provide a novel method for wavefield modification, spatial filtering and interrogating plasmonic devices.