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The transmission of light through metallic films with periodic double nanoholes is studied using vectorial three-dimensional finite element method. Special emphasis is given on understanding different transmission resonances arising in gold and silver films with periodic sub-wavelength holes of different shapes. The spectral shift of the hole-shape resonance resulting from a variation of the hole refractive index is analyzed for a double nanohole geometry in the transmission mode using numerical simulations. Specifically, the role of field enhancement at the apexes of the double nanohole in the sensing of medium within the hole cavity is pointed out and discussed. The presence of sharp apexes within the double nanoholes significantly improves the resonance sensitivity as compared to rectangular holes of comparable area. Impact of possible manufacturing errors on the overall sensitivity is also characterized. Robustness and a relatively simple fabrication procedure make these kinds of refractive index sensors practically attractive.
©2010 Optical Society of America
1. Introduction
Ever since the discovery, by Ebbesen and associates [1], of enhanced transmittance of light through perforated metal films at certain wavelengths, a significant growth of experimental and theoretical research has been experienced in the field of nanophotonics related to subwavelength slits and holes in thin metallic layers [2–9]. Metal films (exhibiting plasmonic properties) with individual and periodic holes of different shapes and sizes were studied in the recent past [5,9]. Although the exact reason behind the enhanced transmission is still not fully clarified, a substantial amount of theoretical work suggests that surface plasmons (SP) on the metal-dielectric interface definitely contribute to the phenomenon. Surface plasmons are collective free electron surface waves that can be generated at the interface of a metal and a dielectric medium at specific excitation (resonance) conditions when the incident light couples to the free electrons on the metal surface. By changing the environment at the metal-dielectric interface, SPs can be effectively exploited in sensing the refractive index (RI) variation at the interface. Normally, the Kretschmann configuration or the Otto configuration is employed in commercial SP resonance based sensors, wherein the power of reflected light or the spectral shift of the SP resonance is calibrated to the refractive index of the adsorbing substance at the interface [10]. There has also been a surge in interest of using metallic films (supporting SPs) with periodic sub-wavelength holes for sensing chemical substances, but the understanding of the sensing mechanism reported is relatively general [11,12].
Recently, optical properties of gold films with a periodic array of double nanoholes (DNH) have been experimentally studied and certain properties have been confirmed by numerical computations [13–18]. A DNH is a composite geometry that is generated by overlapping two circular holes by a particular distance. Such nanostructures can be fabricated using focused ion beam milling lithography and they are reported to provide a field enhancement of four orders of magnitude as compared to the usual rectangular, elliptical and circular hole shapes [13]. This makes them suitable for applications like surface enhanced Raman scattering (SERS), second harmonic generation (SHG), and super continuum generation (SCG) [14–16]. Some experimental results demonstrating the sensing capability of the DNH geometry have been published [17,18]. In these works, the sensor was shown to detect organic and biological molecules in the transmission mode at normal incidence. However, no consistent study has been done in characterizing the DNH transmission spectrum and explaining the nature of resonances that sense foreign molecules on the metal film riddled with holes.
In this paper, we model the enhanced transmission from metallic films with periodic subwavelength DNH arrays and analyze the sensing aspect of the resonant 'waveguide' modes (hole-shape resonances). Simulations suggest that the DNH shape resonance is very sensitive to the refractive index variation of the intra-cavity medium due to its sharp apexes. The sensitivity is more pronounced than that of a rectangular hole with comparable area. Although it is observed that an increase in the apex-to-apex distance of the DNH leads to a degradation of sensitivity, the mere presence of apexes provides more sensitivity to the material inside the cavity than conventional hole shapes such as rectangular, circular, elliptical, etc. This positive feature makes the double circular hole geometry very attractive for sensing applications.
2. Modeling
The propagation of light through metal films with subwavelength sized holes is modeled by three-dimensional vectorial finite element method using COMSOL Multiphysics software package. Light is normally incident from air onto the metal film (Au or Ag) of thickness t, deposited on a glass substrate (RIglass=1.5). The metal film is perforated with periodic holes throughout its whole thickness. We consider two kinds of holes, namely rectangular holes and circular DNHs. The rectangular hole parameters and the periodicity (425 nm) are chosen in order to facilitate comparisons with the experimental results of [9]. The DNH geometry is created by varying the center-to-center distance between the two circular holes of diameter 112.5 nm. When the two circular holes are just touching each other, the effective width of the hole is 225 nm (i.e., equal to the width of the rectangular holes in [9]). Three cases of double nanoholes are studied in this manuscript: overlap factors (distances) of 5, 10, and 15 nm corresponding to the effective hole width of 220, 215, and 210 nm, respectively. The surface area of the rectangular hole is 1.71 times the DNH area with a 5 nm overlap. By varying the overlap factor, the apex-to-apex distance is also modified. Figure 1 shows the basic layout and the unit cell of our models. Periodic boundary conditions are applied at the unit cell boundaries to simulate a square array. We only consider the case of incident polarization along the shorter edge of the rectangular holes (and along the apexes of the DNHs) as the transmission is highly attenuated for the perpendicular polarization [9,13]. The RI of the metal film (Au and Ag) is obtained from literature [19]. In order to simulate the transmission from a single hole, the periodicity is set to 1.5 microns and the metal domain is covered with perfect matching layers except in the direction of propagation.
Fig. 1 Two types of metallic films with an infinite periodic array of: (a) double nanoholes, (b) rectangular holes of 425 nm periodicity, and (c) and (d) their respective unit cells. In (c), the diameter of the circular holes is 112.5 nm and they are overlapped by a distance of x nm to give the double nanohole with an effective width of (225 - x) nm. Incident light is polarized along the two apexes (shorter dimension) of the double nanohole (rectangular hole) as shown in (c) and (d).
3. Results and discussion
It is known that Bragg resonances arise from the periodicity of the array of subwavelength holes in a metal film whenever light is incident on it. At normal incidence, the resonances of a two dimensional subwavelength hole array are given by [13,17]:
where L is the periodicity of the array, i and j are the different Bragg diffraction orders, and εm and εd are the dielectric constants of the metal and dielectric (glass in our case) at the interface, respectively. Two sets of resonances are expected in the transmission spectra that correspond to the two interfaces in our models: the air-metal and the metal-glass interface. The air-metal resonance occurs at a wavelength approximately equal to the periodicity of holes, i.e., 425 nm in our case, and the metal-glass resonance at 425 nm x RIglass = 637.5 nm, for Au and Ag films (based on the theory given in [1]). Minimas can be found approximately at these wavelengths in the transmission spectrum of a metallic film of 100 nm thickness (t) with periodic rectangular holes (Fig. 2 ).fabricated in Au and Ag films of thickness t.It is observed that for Ag, the transmittance at 310 nm wavelength doubles as thickness (t) is reduced from 100 nm to 50 nm. Also, the transmission drops by a factor of 2, as the thickness of the metal film is doubled. This wavelength corresponds to the narrow bulk plasmon resonance of Ag; for Au it occurs at a wavelength of 137 nm [1]. We mainly focus our discussion on the longest wavelength resonance occurring at 810 nm (Ag) and 840 nm (Au) for t = 100 nm. These resonances are related to the shape and size of the hole and they correspond to the resonant localized ‘waveguide’ mode (‘RLW’ mode). The subwavelength holes in a metal layer can be viewed as low Q-factor cavities [7] operating at a specific resonance wavelength depending on their size and shape. A redshift of 60 nm in the RLW mode accompanied by broadening of the peak is observed, when the thickness of the film is decreased from 100 nm to 50 nm. The results in Fig. 2 suggest that the thickness t of the metal film has opposite effects on the magnitude and the bandwidth (or Q) of the RLW mode. When t increases, the peaks become slightly narrower (higher Q) but the magnitude of the transmission also decreases. Taking all these findings into account and considering that the plasmonic skin depth of Au/Ag is ~20 nm, we set the value of t equal to 100 nm for all practical purposes throughout the paper.
3.1. Identification of resonances in double nanoholes (DNH)
It is essential to study the transmission spectrum of periodic DNHs as well as a single DNH to identify which resonances correspond to the DNHs, i.e., the RLW modes, and to know their dependencies. We present the case of an air filled DNH with a 5 nm overlap distance in a gold film. Figure 3 shows the transmission spectrum of one single DNH and periodic DNHs. One can see the presence of two distinct resonances marked as ‘X’ and ‘Y’ for both cases, which get stronger and narrower as the periodicity is reduced. These resonances are the RLW modes for the DNH geometry and their magnitudes can be tuned by varying the periodicity. The increase in transmission magnitude is not surprising since decreasing the periodicity (i.e., more number of holes per unit area) lead to more light being transmitted through the holes. It is also interesting to note that the RLW mode magnitude is quite comparable for both rectangular holes and DNHs (comparing Fig. 2 and Fig. 3(b)) for the same periodicity of 425 nm, even though the surface area of the rectangular holes is almost double the DNH. This suggests that the geometrical shape of the hole also plays a vital role along with the size in determining the net transmission magnitude at RLW mode condition. Another resonance that can be observed at 500 nm (also seen for rectangular holes in Fig. 2) does not depend on the shape of the hole. It is present even for a non-perforated thin Au film, but its magnitude changes by varying the periodicity of the holes. Similar effects are observed for Ag but the peak at 500 nm is weaker as compared to Au.
Fig. 3 Transmission for (a) a single DNH and (b) periodic DNHs with an overlap distance of 5 nm in a gold film. The legends in (b) are the different periodicity values. Markers ‘X’ and ‘Y’ show the shape resonances or RLW modes.
3.2. Analysis of ideal DNH shape resonances
The bulk refractive index (i.e., RI) of the hole is varied to determine the sensitivity of the RLW modes. The goal is to compare the sensing performance of ideal DNHs (with perfect sharp apexes) and rectangular holes, with 425 nm periodicity. Figure 4 shows the sensitivity of the RLW modes for varying RIs of the DNH cavity in Au and Ag films. Three cases of DNH geometry are considered: overlap factors of 5 nm (blue), 10 nm (red), and 15 nm (black), respectively. It is confirmed by our models that the DNH with an overlap factor of 5 nm (15 nm) provides the best (worst) field enhancement in the cases considered [13].
Fig. 4 Transmittance vs wavelength of an ideal periodic DNH structure with an overlap distance of 5 nm (blue), 10 nm (red), and 15 nm (black) in (a) Au film and (b) Ag film. Solid lines (-), dashed lines (--), and dotted lines (···) correspond to refractive index of 1, 1.05, and 1.1 of the DNH cavity, respectively. In (a), markers ‘X’ and ‘Y’ denote the shape resonance peaks of the DNH.
As the overlap factor increases, a blue shift in the RLW modes is observed thereby providing an option to spectrally tune the shape resonance peaks. Previous experimental and theoretical work shows that in the case of rectangular holes, as the aspect ratio (i.e., length/breadth ratio of the rectangular hole) increases, there is a redshift in the shape resonance [9]. Hence, as we increase the overlap factor, the effective hole width is reduced, thereby causing a subsequent blueshift in the RLW modes. The smaller peak ‘X’ (in Fig. 4(a)) undergoes an equivalent shift along with the dominant RLW mode ‘Y’ for both, Au and Ag. Peak ‘X’ depends on the sharpness of the apexes, apex-to-apex distance of the DNH, and the RI of the cavity. These dependencies will be discussed in more detail later in the paper. Overall, it can be observed that Ag yields higher transmittances than Au. The magnitude of the RLW mode ‘Y’ in an Ag film (~0.6) is also almost twice that of Au (~0.3) when the refractive index of the hole is 1 for a periodicity of 425 nm. There is also a gradual reduction in the magnitude of peak ‘Y’ (almost by a factor of 2 in the case of Ag), as the overlap factor increases from 5 nm to 15 nm. The calculated values of sensitivity for the dominant RLW mode ‘Y’ on varying the RI of DNHs (Fig. 4) is given in Table 1 below.
Table 1. Wavelength sensitivity of the RLW mode for different overlap distances of circular holes in ideal double nanohole structures.
The sensitivity of the DNH structure with an overlap factor of 5 nm is the maximum among the cases we considered: a wavelength shift of 20 nm for every 0.05 change in the hole RI. On the other hand, for rectangular and circular holes, the calculated sensitivity is 8 nm when the RI of the hole is varied from 1 to 1.05 (not shown here). It is seen that as the overlap factor increases from 5 nm to 15 nm, the sensitivity to the change in RI of the nanohole degrades but it is still better than for the conventional rectangular and circular holes. This suggests an active role of the field confinement and geometry of the DNH cavity in enhanced sensitivity (more than double) to the medium in the nanohole. Generally, it is expected that by increasing the number of pointed features (e.g., apexes in the DNH) in the nanoholes that perform the role of confining the field (or surface plasmons), the Q-factor of the cavity is modified. So, any change in the medium within the cavity will definitely affect the resonance condition, thereby shifting the RLW modes.
In a real life scenario, a sample of liquid whose RI is to be measured is dropped on the surface of the perforated metal film. The thickness of the liquid layer on the surface could be a few millimeters, and our RI sensor is operating in the visible-NIR range. If the thickness of the liquid layer is much larger than the wavelength of the RLW mode, a feasible approach is to vary the RI of the medium of incidence (which is air normally) along with that of the hole in our model. One can observe that for DNH with a 5 nm overlap factor, the RLW mode sensitivity changes to 45 nm (48 nm) for Au (Ag) for 0.1 RI change. It is a slight increase with respect to the previous case when the medium of incidence was set to air (RI = 1): spectral sensitivity was 40 nm per 0.1 RI change. Figure 5 shows a comparison between the two situations when the RI of the incident medium is varied/not varied along with the hole, for Au and Ag films. Thus, the minimum sensitivity for any given thickness of the liquid layer on top of the film is expected to be 40 nm per 0.1 RI change for an ideal DNH structure with an overlap factor of 5 nm.
Fig. 5 Shift of RLW mode when the medium of incidence is varied together with the medium in the DNH cavity for Au and Ag films of thickness t.
3.3. Effect of blunt apexes on DNH shape resonances
To investigate the effects of imperfect apexes, as would be expected in real DNH samples due to fabrication errors, simulations similar to Fig. 4 were performed for blunt apexes. The apexes were blunted by chopping off a portion of the apexes at a particular height h with respect to the original ideal apex points. It is important to note that the apex-to-apex distance of the DNH is also inadvertently modified as the apexes are blunted. In particular, the DNH with an overlap factor of 5 nm is considered for further analysis as it provides maximum sensitivity. Figure 6 shows the behavior of RLW mode for blunt apexes with h = 2 nm and 4 nm, in the case of Au when the refractive index of the DNH cavity is varied from 1 to 1.1. It is interesting to note that the smaller peak (‘X’ in Fig. 4 (a)) disappears when the apexes are blunted and the main RLW mode gets blue-shifted by 50 nm as h changes from 0 (ideal DNH) to 2 nm. This reveals that the smaller resonance is related to the sharpness of the apexes in the DNH. The blue shift of the RLW modes can be explained by the inadvertent variation in the apex-to-apex distance as the apexes are blunted (similar to blue-shifts by changing overlap factors in Fig. 4). No shift in the RLW mode wavelengths would be expected had the apex-to-apex distance remained constant. The disappearance of the smaller resonance is accompanied by almost 33% increase in the magnitude of the main RLW mode for blunt apexes. For a RI change of 0.1, the spectral sensitivity degrades to 38 nm (h = 2 nm) and 32 nm (h = 4 nm) as compared to 40 nm in the case of ideal apexes. However, these values are still much greater than that for rectangular holes (16 nm) of equal hole width. Similar results are obtained for Ag and it does not provide any further novelty.
Fig. 6 Comparison between the transmission spectra of ideal and blunted DNH structures in an Au film. Solid lines (-) and dashed lines (--) correspond to the refractive index of 1 and 1.1 of the DNH cavity, respectively. Parameter h denotes the blunting factor.
3.4. Electric field distribution inside the DNH cavity
In order to explore the connection between the change of sensitivity for blunted DNHs and the near E-field distribution, we compare the magnitude of total electrical field |E| in the cavity for different h values. The |E| plots are calculated over the plane corresponding to t = 50 nm at RLW mode conditions (Fig. 7 ). The color scale comparatively depicts the magnitude of the total electric field in the different regions of the DNH cavity: red is the maximum and blue is the minimum. All the plots correspond to DNHs with 5 nm overlap factor and 425 nm periodicity.
Fig. 7 Magnitude of the E-field distribution for ideal and blunted DNH structures (half section of a symmetric DNH with 5 nm overlap factor) in a gold film at different hole-shape resonance (RLW) conditions.
The RLW mode condition occurs at 950 nm (ideal DNH), 900 nm (h = 2 nm), and 890 nm (h = 4 nm) for air filled cavities. The extra smaller resonance that only exists for ideal DNH occurs at 810 nm. It can be clearly observed that most of the energy is localized at and around the apexes for ideal and blunt DNHs. For ideal DNHs, the |E| maxima at 950 nm reduce by a factor of 2 at 810 nm, and the field enhancement (defined by the ratio of |E| maxima to |E| minima) is about 104 in both cases. The color scale bar is not shown in Fig. 7 but the field enhancement value is consistent to four orders, as reported in [13]. The reduction in |E| maxima can partially explain the lower transmittance at 810 nm. As h increases from 0 to 4 nm, the field distribution at RLW mode wavelengths change around the apexes but the field enhancement in the cavity still remains at four orders of magnitude (color coded in Fig. 7). For blunt apexes, the energy is distributed over a larger area but the total confined energy in the cavity still is comparable with the ideal DNH case. Hence, there is an evidence of the dependence of spectral sensitivity of the RLW mode on the E-field distribution of the hole.
4. Conclusion
A detailed study of the sensing properties of the RLW modes (shape resonances of the double nanohole) was done. Although the dominant RLW mode is relatively broad as compared to conventional SPR based sensors, the design and fabrication aspect of a double nanohole sensor is fairly non-complicated. The characterized resolution for sensing intra-cavity refractive index variation in the case of periodic double nanohole structures is two times better than that of rectangular holes in the transmission mode, for both Au and Ag films. Even with possible variation of the hole overlapping factor and blunting of apexes due to manufacturing limitations, the mere presence of apexes is enough to obtain a higher sensing spectral response than that of conventional hole shapes such as circular, elliptical, or rectangular holes of comparable surface areas. Imperfections in the apexes change the electric field distribution in the hole and this slightly affects the sensing response. It is also found that by controlling the size of the double nanoholes and varying the thickness of the metal film, one can tune the wavelength and bandwidth of the RLW modes efficiently, in order to design a simple and robust sensor operating within a specified range of wavelengths.
Acknowledgements
This work was financially supported by the Swedish Foundation for Strategic Research (SSF) and the Academy of Finland (SA). The authors wish to thank the anonymous reviewers for their constructive suggestions.
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