1. Introduction

Extraordinary transmission of electromagnetic waves through a thin metal film perforated with periodic arrays of subwavelength circular holes was investigated in 1998 by Ebbesen et. al. [1]. This was explained by the coupling between photon and surface plasmon polaritons (SPPs), the SPPs are the fluctuations in the electron density at the interface between metal and dielectric materials excited by photon, and then the resonant surface wave reconverts to photon and enhances the transmission light. The influence of holes shape on extraordinary transmission including the polarization-dependent transmission intensity [2] and the shift of the spectral position of the resonances [3] have attracted lots of attention. In particular, a large enhancement in transmission intensity has been shown to appear when the polarization of the incident light is perpendicular to long edge of rectangular holes. In subsequent research, the strong enhanced transmission through rectangular hole arrays were investigated both in experiments [4] and theory [5, 6]. Klein et. al. [7] proved that the shape resonance originated from the contribution of individual hole by arranging the rectangular holes in random distribution. Mary et. al. [8] theoretically demonstrated that SPPs and localized resonance modes are all present in 2-D array of rectangular holes. Chen et. al. [9] found that the cross shaped hole arrays gave rise to larger transmission of light than those perforated with square or rectangular hole. The transmission peaks red-shifted when the aspect ratio of the cross increased, whereas the transmission dip representing the Wood’s anomalies stayed at the same wavelength. It is interesting to know how the Wood’s anomaly affects the peak wavelength of the transmission spectrum of the shape resonance.

In this paper, the rectangular hole arrays were arranged in a rectangular lattice to tailor the position of Wood’s anomaly away from the peak of localized resonance. The real peak position of the localized waveguide resonance λ_res was observed which is different from that appears at the fundament shape resonance λ_res ≈ 2L in free standing structure, where L is the hole length. The simplified transmission-line model was utilized to fit the experimental values.

2. Experiment

Fig. 1(a) and 1(b) show the top and side views of the square array of rectangular holes, respectively. The 100 nm silver thin film was deposited on the periodic photoresist rod array and lifted off to form a silver film perforated with periodic hole array on top of a doubly-polished n-type silicon wafer. The period of the array along x and y directions, a_x and a_y, are fixed at 15 μm. The rectangular holes have different aspect ratios (L/W), i.e., R=2, 4, 6, 7, 9, 11, and ∞ (1-D grating), the L is hole length and W is hole width. All the patterns were designed by fixed the hole area the same (18 μm²). For 1-D grating, the width of the hole is 1.2 μm. It is well known that both of the length and width in the rectangle could affect the peak transmission wavelength due to the coupling between surface plasmons on the long edges of hole [10]. Therefore, the coupling mechanism on the geometrical shape will be first investigated. Fig. 1(c) shows the experimental setup and the sample lies in the xy plane with light incident in z direction while the polarized light was along y direction. A Bruker IFS 66 v/s system was used to measure transmission spectra. The dispersion relations along K_y direction were measured by rotating samples around x axis 1° per step form 0° to 50°.

 

Fig. 1. The schematic diagrams showing the (a) top and (b) side view of the rectangular hole array in a square lattice. (c) The measurement setup and the sample lies in the xy plane with light incident in z direction.

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3. Transmission results and discussions

For normal incident light, the free-space wavelength of surface plasmon polaritons (SPPs), λ_spp, in square lattice is given by [11]

For normal incident light, the free-space wavelength λ_WA that satisfies Wood- Rayleigh anomaly condition and results in a transmission minimum is given by

where i and j are integers corresponding to the specific order of the SPP and WA mode, a (=15 μm) is the lattice constant, ε_d (=11.7 at 52 μm) and ε_m (= -9.2∝10⁴ at 52 μm) are the dielectric constants of silicon and silver, respectively. According to Eq. (1) the (1, 0) Ag/Si SPP mode should exhibit a peak wavelength at 52 μm, whereas the Wood’s anomaly shows a minimum at 51.3 μm. Fig. 2 shows the zero-order transmission spectra of rectangular holes in square lattice array with different aspect ratios R = 2, 4, 6, 7, 9, 11, and ∞ (1-D) while the polarized light is along y direction.

 

Fig. 2. Zero-order transmission spectra of rectangular hole array in a square lattice with aspect ratios R = 2, 4, 6, 7, 9, 11, and ∞ (1-D grating). All the lattice constants are 15 μm

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It is clear that the transmission peaks shift from 52 μm to longer wavelength (76 μm) as the aspect ratio of the rectangular holes increase from 2 to 11, and to much longer wavelength (27360 μm) as R → ∞ (1-D grating). This phenomenon had been observed before [3, 7]. One of the reasons for red shift is due to the shift of the cut-off wavelength of rectangular waveguide to longer wavelength, but because the metal thickness is thin, the Fabry-Perot resonant mode may not be able to exist in this short waveguide [12–15]. It is also noticed that the wavelength of transition minima are almost the same for all samples which are attributed to the Wood’s anomalies [14].

The dispersion relations of rectangular hole arrays along K_y direction with different aspect ratios, i.e., R = 4, 7, 9, and ∞ (1-D) are shown in Figs. 3(a)–(d), respectively. The top views of the samples were shown in the inset of the figures. In Fig. 3(a), aspect ratio R = 4, the transmission maxima and minima are represented by the yellow and blue curves, respectively. It can be seen that the yellow curve is accompanied by the slightly higher energy blue curve representing the (0, -1) WA mode. A slightly flat dispersion curve was observed near low K_y values, and then the curve follows and mixes with the (0, -1) Ag/Si SPP mode at high incident angles, showing that a localized mode coexists with the SPP mode [4]. When the aspect ratio is increased to 7, the yellow curve becomes flat indicating a localized resonance mode. This confirms the previous reports [5, 16] that the transparency can only be realized through the shape resonance. To see the fine details, the transmission spectra of samples with aspect ratio of 6 and 7 with different incident angle (from 0° to 50°) are shown in Fig. 4(a) and 4(b), respectively. It can be seen definitely in Fig. 4(b) for R = 7 that the long wavelength tail of the transmission peak keeps the same profile when measured at different incident angles as compared to that shown in Fig. 4(a) for R=6. This is because the localized resonance dominates the transmission through the structure for R = 7 sample, deriving from the competition between propagating SPPs and the cut-off wavelength of the rectangular waveguide [8]. A Wood’s anomaly cannot shift resonances. It can make a transmission peak seem to be at a different position than the resonant position. For the sample with R = ∞ (1-D grating), only transmission minimum (Wood’s anomalies) was observed, the transmission maximum which is related to the cut-off wavelength of the waveguide and determined by the length of the long edge of the rectangle moves all the way to long wavelength outside the detection range due to the infinite long edge of the 1-D grating.

 

Fig. 3. The dispersion relations and transmission intensity of rectangular hole array as a function of photon energy and k_y⃑. The aspect ratios of rectangular holes are (a) 4, (b) 7, (c) 9 and (d) ∞ (1-D grating). All the lattice constants are 15 μm .

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Fig. 4 Zero-order transmission spectra of rectangular hole array in a square lattice with different incident angles. The lattice constants are 15 μm , and the aspect ratios R of rectangular holes are (a) 6 and (b) 7, respectively.

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To see the effect of Wood’s anomalies and cut-off condition on the spectral shape of the transmission spectra, the rectangular hole arrays in a rectangular lattice were designed and fabricated. The purpose was to tailor the position of Wood’s anomalies [17] away from the peak position of shape resonance. In Fig. 5, the transmission spectra for the specific rectangular holes with aspect ratio of 9, i.e., the length is 12.7 μm and the width is 1.4 μm, were arranged in a rectangular lattice with a fixed period of 15 μm along x axis and different periods from 3 to 18 μm along y axis. As the period along y axis a_y is decreased from 18 μm, the transmission peak gradually shifts to shorter wavelength until the period reaches 6 μm, the peak is then fixed at 52.5 μm. This is because Wood’s anomaly moves further to shorter wavelength, away from the shape resonance, its influence on the shape of the transmission peaks diminishs. The transmission peaks appear at the fundamental shape resonance λ_res ≈ 2L as observed in the free-standing structure [16], but in our sample this value should be multiplied by the effective refractive index n_eff since the metallic periodic structure is sandwiched between the Si substrate and air.

Figure 6 shows the transmission spectra of the rectangular hole arrays with the aspect ratio of 9 on doubly-polished Si and Ge substrates. There are two different periods along y axis a_y, i.e., 4 and 6 μm, and a fixed period of a_x = 15 μm along x axis. Regardless of the a_y, the spectra show that the transmission peaks only correlates with the wafer type, the peak shifts from 52.5 μm to the longer wavelength of 61 μm when the Si substrate was replaced by Ge substrate. To estimate the peak wavelength, a transmission-line model that describes the effective index variation with different substrates was applied [18–19]. For a capactive strip grating, the electric field direction is perpendicular to the long edge of rectangular hole, the equivalent circuit is a capacitor with capacitance (n ² ₁ + n ² ₂)/2 times of the same grating in free standing structure, and its reactance X_c is given by

 

Fig. 5. The transmission spectra of the rectangular hole array with aspect ratio of 9, i.e., the length is 12.7 μm and the width is 1.4 μm , in a rectangular lattice with the fixed period of 15 μm along x axis and the different periods from 3 to 18 μm m along y axis.

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Fig. 6. The transmission spectra of the rectangular hole arrays on Si or Ge substrate with aspect ratio of 9 in a rectangular lattice. The periods are 15 μm along x axis for all four samples, the periods along y axis a_y are either 4 or 6 μm .

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Where n₁ and n₂ are the refraction indexes of the two adjacent media, i.e. air and Si (Ge) substrate, a is one half of the space between two adjacent holes, and g is the period of the periodic structure. The ω′₀ is the resonant frequency of the periodic metal arrays sandwiched between two dielectric medium with n₁ and n₂ refraction indexes, respectively. ω′₀ is related to the resonant frequency of the same metallic arrays in free standing structure ω ₀ by

When the periodic metal structure deposited on Si or Ge substrate, the refraction index n₁ of air is unity, the refraction index of substrates n₂ (Si) and n₂ (Ge) are 3.35 and 4 at the peak wavelengths, respectively. From Eq. (4), the cut-off resonance wavelength $({\lambda }_{\mathrm{res}}\approx 2n_{\mathrm{eff}}L=2\sqrt{\frac{\left(n_1^2+n_2^2\right)}{2}L})$ will be modified by effective refraction index of 2.47 (Si) and 2.92 (Ge). The real aperture of the rectangular hole with aspect ratio of 9 has a length of 11.5 μm and width of 1.3 μm. The theoretical resonance peaks were estimated to be 56.8 μm and 67.2 μm for Si and Ge substrates, respectively. The measured values, i.e., 52.5 μm for Si substrate and 61 μm for Ge substrate, are slightly smaller than the theoretical value.

4. Conclusions

In conclusion, the extraordinary transmission through the periodic metal arrays perforated with subwavelength rectangular holes was investigated. When the aspect ratio of rectangular holes exceeds 7, the propagating SPP modes transform to the localized resonant modes that was attributed to the cut-off wavelength of the rectangular waveguide beyond the wavelength of SPP mode. It was found that the Wood’s anomalies play an important role in shaping the spectral profile of the transmission peak. When the Wood’s anomalies are moved far apart form the shape resonance by using rectangular lattice, the real peak position of the localized resonance mode appears and stays at the same wavelength irrespective of the period of the lattice. The peak position of the localized or shape resonance was demonstrated to be relevant to the substrate type and can be explained by a transmission-line model.