Here we discuss the influence of a substrate on transmission through subwavelength holes in metallic films. In particular, we show that in the case of transmission maxima associated with localized resonances of the apertures, that the wavelength at which this maximum occurs are strongly influenced by the presence of a substrate and the aspect ratio of the structure. Furthermore, we show that removing a shallow region of the substrate immediately below the apertures leads to blue-shifting of the resonance and increased transmission compared to that in the presence of a homogeneous substrate.
© 2011 OSA
Considerable interest in the transmission of light through subwavelength apertures in metallic films has persisted since unexpectedly high transmission was first seen in periodic arrangements of apertures . Applications utilizing this phenomenon have been demonstrated and envisaged in optical sensing  novel compact filters  and in wavefront control devices [4–6]. A large number of issues associated with these structures have been investigated: the arrangement of the apertures, the geometry of the holes and the nature of the illumination as well as the optical properties of the metal, the super- and substrate and the material within the apertures. Substrates are commonly used to lend robustness to a thin metallic film and provide support for isolated metallic elements. Furthermore, understanding the response of plasmonic systems to variations in the dielectric environment is central to the development of novel sensing applications . It is well-known that in the case of apertures in metallic films [2,7–10] as well as metallic nanoparticles [11,12] that the presence and optical properties of a substrate can significantly modify the transmission from that in its absence. In the case of high transmission through structures where the dominant mechanism underlying high transmission relies on surface wave excitation [2,8,9] the peak wavelength is red-shifted. This is apparent in the dispersion relation for the excitation of surface plasmon polaritons (SPPs) at a metal-dielectric interface. The case of high transmission emerging from the excitation of localized resonances within the apertures, however, is less clear [13–21] since fields that occur at resonance are tightly confined within the aperture and do not extend substantially into the substrate.
Here we investigate transmission through arrays of annular (or coaxial) apertures in metallic films (Fig. 1 ). In particular, we look at the role played by the refractive index and structure of the substrate and the thickness of the metal layer. The investigation of arrays of apertures with the distinct resonances seen in these structures has the potential to provide insight not only into the mechanisms underpinning changes in transmission in these particular structures introduced by a substrate, but may also inform the design of novel sensors and metamaterials. Furthermore, we show that selective removal of a thin layer of the substrate immediately below the apertures leads to a blue-shifting of the resonance compared with that in the presence of a homogeneous substrate.
2. Substrate effects in frequency selective surfaces, aperture arrays and metamaterials
Localized resonances within apertures in metal films (also known as Localized Surface Plasmon (LSP) resonances) are cavity Fabry-Perot resonances where the cavity is defined by the aperture walls and the upper and lower boundaries of the metal film. The location of each resonance is influenced by phase shifts at the ends of the cavity . By modifying the refractive index of the superstrate, the substrate or the material filling the apertures, this phase is modified leading to a shift in the resonant frequency and corresponding free-space wavelength. This phenomenon can also be considered as arising from changes in the capacitance of the structure . Arrays of apertures with slot [7,24], cross-shaped [20,25] or annular [13–19] geometries with small gaps possess relatively high capacitances and distinct aperture resonances can be designed to be well-separated from any surface wave resonant excitations. When a dielectric substrate is introduced, the capacitance is increased producing a red-shift in the resonance.
The topic of the influence of a dielectric substrate on transmission through lossless inductive and capacitive grids , originally developed as frequency-selective surfaces, has been extensively investigated [26,27]. By invoking a monomodal approximation and other arguments, the performance of inductive and capacitive grids can be described using an equivalent circuit model where, in the case of lossless media, the grid can be ascribed an effective capacitance, C0, and inductance, L0. It has been shown  that in the presence of a semi-infinit substrate with refractive index, n, the capacitance of an infinitesimally thin, perfectly conducting capacitive grid is given by:
Resonances of metamaterials, have also been shown to be extremely sensitive to the presence of a substrate or dielectric overlayer [28,29] and the thickness of the metal [28,29]. In particular, it has been shown [28,29] that increasing the thickness of a metamaterial leads to a shift in the resonant frequency and that this is accompanied by a decrease in sensitivity to the presence of a substrate . As in the case of frequency-selective surfaces, these effects can similarly be attributed to changes in the capacitance of the structure. The loss of sensitivity to the substrate would, therefore, be anticipated since by increasing the thickness of the structure, the contribution to its total capacitance by the substrate would be reduced.
3. Simulations and discussion
Here we present simulations showing the influence of both the thickness of the metal film and the refractive index of the substrate on the resonant wavelength of arrays of coaxial apertures in perfectly electrically conducting (PEC) and silver films where the optical constants of the material are explicitly included . In the former case, a rigorous modal method  is utilized. This technique provides a relatively computationally efficient method to explore the principles influencing the properties of interest. We also use the Finite Element Method (FEM) implemented in COMSOL Multiphysics to investigate the performance of materials in the visible and near-infrared.
3.1. PEC - rigorous modal method
The modal method involves expanding the electric and magnetic fields within the apertures in terms of coaxial waveguide modes. The fields above the metal (PEC) and in the substrate below the structure are expanded in Rayleigh modes. By enforcing boundary conditions at the upper and lower surfaces of the metal, the various modal amplitudes and, hence, the transmitted power, can be determined. The transmission through square arrays of coaxial apertures with periodicity d in PEC of thickness h was calculated. In all cases the outer radius, a, of the apertures was set to be 0.45d and the inner radius, b, was 0.40d. Normally incident plane wave illumination and a semi-infinite substrate were assumed. Figure 2 shows the transmission through these grids for different PEC thicknesses and substrate indices. In Figs. 2(a) and 2(b), the transmission as a function of both normalized wavelength, λ/d and substrate index, n, is shown for two different PEC thicknesses. In Fig. 2(a), the thickness of the PEC is equal to 0.1d, whereas in Fig. 2(b) it is 0.4d. In both cases, a clear long-wavelength maximum that occurs at a wavelength, λres, that increases with the substrate index is apparent. The maximum transmission also depends on the substrate index in line with the Fresnel transmission coefficient. In Figs. 2(c) and 2(d), the transmission through the structure as a function of the thickness of the PEC film for substrate refractive indices of n = 1 and n = 1.52 are shown. In both cases, there is a shift in λ res as h is increased. When n = 1, λres is shifted to longer wavelengths, whereas if n = 1.52 (corresponding to BK7), it is blue-shifted. As the thickness of the metal film increases, as has been observed elsewhere [22,32], higher-order Fabry-Perot harmonics of the aperture resonance appear at shorter wavelengths in the plotted spectrum and shift to longer wavelengths as the thickness of the metal increases. In both cases, it can also be seen that as the screen becomes thicker, the resonance decreases in width and the Q increases.
In Fig. 3 , the position of λres for the geometries shown in Fig. 2 is plotted. In Fig. 3(a) the dependence on substrate index is shown for both thicknesses shown in Figs. 2(a) and 2(b) as well as that for h = 0.001d. The expected dependence given by Eq. (2) (i.e. that for h = 0) is also shown as a dashed line. It is apparent that the resonant wavelength increases monotonically with increasing substrate index and that its dependence on n approaches that given by Eq. (2) as h → 0. As discussed earlier in the context of metamaterials, the origins of this red-shift lie in the increased capacitance of the structure which leads to a decrease in the resonant frequency of the equivalent circuit and, hence, a red-shifting of the wavelength. As the thickness of the metal film increases, however, the sensitivity of λres to the substrate index decreases. This is expected since as the metal becomes thicker, the relative contribution to the total capacitance from the ends of the cavity would decrease.
Figure 3(b) shows that in the absence of the substrate (n = 1), the location of the maximum transmission shifts to longer wavelengths as the thickness, h, of the metal film increases, whereas if n = 1.52, the location shifts to shorter wavelengths. For very thick films, the resonance is expected to approach the cutoff wavelength of the TE11 coaxial waveguide mode (shown as a dashed line in the figure) which, for the parameters considered here, is 2.66d. In the case of films supported by a substrate, the resonant wavelength shifts to shorter wavelengths and approaches that of the free-standing film.
It is clear that the dependence of the resonant wavelength on the refractive index of the substrate and the thickness are consistent with those seen in  for THz metamaterials and with analytic results for thin screens.
3.2. Silver films – finite element method
In order to accommodate the optical properties of a real metal in the visible and near-infrared regions of the electromagnetic spectrum, the transmission properties of the device were modeled using the Finite Element Method (FEM) implemented in COMSOL Multiphysics 4.1a. Floquet boundary conditions were used to simulate the periodicity of the array and perfectly matched layers (PML) were used at the upper and lower surfaces. Optical properties of Ag were taken from  and the refractive index of the substrate (when included) was taken to be 1.52, to correspond to that of BK7 glass. The transmitted power was calculated by integrating the z-component of the Poynting flux over a plane below the metal film and above the PML. This was normalized to the same value calculated in the absence of the metal film and any substrate.
Figure 4 shows the transmission through an array of periodicity d = 400 nm of annular apertures of outer radius, a, of 125 nm and inner radius, b, of 75 nm in a silver film. Figure 4(a) shows the transmission through films ranging in thickness from 50 – 300 nm for a free-standing film, while Fig. 4(b) shows the equivalent results assuming a semi-infinite substrate with refractive index n = 1.52. In Fig. 5 , the resonant wavelength as a function of film thickness is plotted in both cases with dashed curves being exponential fits included to guide the eye. We see that the general trends observed in Figs. 2 and 3 are also apparent when the optical constants of real metals in the near-infrared are considered. These are also consistent with the general trends seen in similar research involving THz metamaterials [28,29]. It can also be seen that, as in the case of apertures in PEC films, as the thickness of the Ag film increases, the Q of the resonance increases.
Consider now an array of apertures in a metallic film located on a substrate where the region in the substrate immediately below the apertures has been selectively removed to a depth of t (Fig. 1(c)). It is possible to produce this structure using focused ion beam (FIB) milling, for example, since the material could be removed during the same process through which the metal film is processed. It is anticipated that this structure would possess a lower capacitance and, hence, shorter resonant wavelength than that of a structure with a homogeneous substrate. Given that many applications of plasmonic devices lie in the visible or near-infrared, the ability to blue-tune the structure while maintaining the mechanical benefits of a substrate using a simple process compatible with other fabrication techniques is highly desirable. In particular, there are resonant structures of interest which are not self-supporting, for example the arrays of coaxial apertures investigated here, where the ability to maintain a substrate while decreasing the resonant wavelength is extremely useful.
The transmission spectra for different values of t in a substrate of index 1.52 for a structure in a Ag film of thickness 150 nm are shown in Fig. 6(a) , while the location of the transmission maximum in each case is shown in Fig. 6(b). It can be seen that the location of the resonance is extremely sensitive to the removal of even a small amount of the substrate and saturates once the depth of the structure in the substrate reaches approximately 50 nm. At this depth, the fields, protruding only a short distance below the structure, do not interact significantly with the substrate.
This is illustrated in Fig. 7 where the magnitude of the electric field around these nanostructures at resonance is shown. The transverse field pattern is shown in Fig. 7(a), while the fields in meridional planes through the free-standing structure and the milled structure are shown in Figs. 7(b) and 7(c) respectively. The strong localization of the fields within the aperture and their limited extent into the region below the metal film are apparent. In Fig. 7(d), a plot of |E| at a distance (a + b)/2 from the centre of the aperture is plotted as a function of distance below the lower surface of the film. The plot line is indicated by the cross in Fig. 7(a) and the dashed line in Fig. 7(b). |E| is plotted for this structure in the presence and absence of the substrate and when the substrate has been modified. It is apparent that the field in each case decreases rapidly with distance from the lower surface of the metal with a characteristic 1/e distance of approximately 30-35 nm in each case. It is, therefore, not surprising that the resonant wavelength is extremely sensitive to milling depth. Overall, a shift in resonant wavelength of approximately 50 nm can be achieved by tailoring the substrate in this way. Furthermore, the structuring of the substrate improves the impedance match between the array and the substrate leading to higher transmission than that seen in the presence of a unpatterned semi-infinite substrate. It should also be pointed out that the corresponding shift in the location of the Rayleigh-Wood anomaly near 650 nm shifts by only approximately 15 nm as the substrate is modified.
We have shown that, in general, transmission through arrays of apertures where the dominant mechanism facilitating high transmission is the excitation of localized aperture resonances that both the substrate refractive index and the thickness of the metal film significantly influences the location and magnitude of the transmission maximum. The sensitivity of the transmission to the refractive index decreases as the film increases in thickness.
We have also shown that the influence of the substrate can be substantially moderated by selective removal of the substrate so that there is a coaxial gap in the substrate immediately below the film. The location of the resonant wavelength is sensitive to the removal of even small amounts of the substrate below the apertures. It is expected that since the resonances are a property of the apertures, rather than their arrangement, that the conclusions drawn here would also be true of an isolated aperture, or a random arrangement of apertures [18,33]. In the examples presented here, the geometry of the apertures was specifically chosen so that the localized aperture resonance occurs at a wavelength longer than any strong Rayleigh-Wood or other surface wave excitations. If the apertures were smaller, or their geometries tuned to produce resonances at shorter wavelengths, the phenomena discussed here would still exist, but may be masked by order-dependent excitations. The research lays the foundation for the development of new three-dimensional plasmonic structures and metamaterials and suggests new strategies for tuning the electromagnetic response of these devices. It also has consequences for new strategies for sensing based on localized aperture resonances.
This research was supported under the Australian Research Council's Discovery Projects funding scheme (project number DP0878268). The authors would like to acknowledge useful conversations with Dr Tim Davis.
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