1. Introduction

An optically thick metal film perforated with a periodic array of subwavelength holes transmits light remarkably efficiently at certain resonant wavelengths[1, 2], which is called extraordinary optical transmission (EOT). Such a phenomenon has evoked great scientific interest and attracted attention for its significant potential applications[3]. To understand it, both of one-dimensional (1D) periodic arrays of slits[4, 5] and two-dimensional (2D) subwavelength periodic hole arrays with different shapes[6, 7, 8] were studied intensively. It has been established that two main mechanisms are responsible for the enhanced transmission: surface plasmon resonance (SPR) and localized waveguide resonance[9, 10, 11]. The former mechanism relies on the excitation of surface plasmon polaritons (SPPs) on either side of the metal film, and the latter depends on the shape of single hole.

In previous works, much attention has been paid to the optical transmission of slit or hole arrays, consisting of only one hole or slit in each primitive cell. Recently, for 1D periodic structure, compound metallic gratings[12, 13, 14] formed by several grooves/slits in the period were proposed and a third kind of resonance known as phase resonance was found. It is characterized by the splitting of transmission peak, usually found in the waveguide resonance. For 2D periodic structure, the enhanced transmission through compound hole arrays composed of several holes in each unit cell, also has been demonstrated. For example, the rotation -symmetry of lattices[15] or Fourier coefficient of reciprocal vectors[16] was suggested to tune the transmission properties of light in the metallic nanostructures. However, unlike those reported 1D compound metallic gratings, the enhanced transmission in these compound 2D structures either is induced by symmetric arrangement of hole arrays or the involvement of localized surface plasmon[17], so the splitting of transmission peak was not observed. In the present work, by changing the geometrical arrangement of compound 2D periodic hole arrays, the symmetry of square hole arrays is broken and the sharp dips in the allowed transmission band are found, together with the suppression of SPR (0, 1) transmission peak. We attribute this phenomenon to the phase resonance of SPPs. Our results provide another effective method to tailor the enhanced optical transmission in metallic nanostructures.

 

Fig. 1. Schematic illustrations of the compound subwavelength arrays of square hole. (a) A unit cell of compound subwavelength hole arrays, consisting of two square holes width identical width. (b) corresponds to d_x = 0. (c) the general case. (d) the case d_y = 0. The width of square hole are a = 200 nm, filled with air, and the periodicity is p = 600 nm. The thickness of Ag film is 100 nm.

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2. Simulated model and method

In Fig. 1, we schematically show the 2D compound subwavelength hole arrays with different arrangement style, perforated on the Ag film deposited on a substrate assumed as quartz with dielectric constant ϕ_d = 2.16. Fig. 1(a) illustrates one unit cell of compound subwavelength hole arrays, consisting of two square holes with identical width a = 200 nm. The hole width a and period of compound hole arrays p_x = p_y = 600 nm are fixed for the whole paper. In the complex unit cell, the hole B is inserted to the position with both lateral and the longitudinal displacements with respect to the hole A, labeled as d_x and d_y , respectively. The interspace between two holes in a unit cell along the y direction is denoted as d_AB. Changing the d_x or d_y, different geometrical arrangement can be obtained. Figs. 1(b) and (d) depict the cases of d_x = 0 and d_y = 0, and the holes are aligned to the y and x direction, respectively. Also, the general case is demonstrated in Fig. 1(c). The three-dimensional Finite Difference Time Domain (3D-FDTD) method[18] was employed to simulate the interaction between metal sliver film and the incident wave. In our simulation, the spatial mesh steps were set Δx = Δy = Δz = 10 nm and the time step was set Δt = Δx/2c (c is the velocity of light). Also, perfectly matched layer (PML) absorbing boundary conditions were applied at either end of the computing space where periodic boundary conditions were used on other boundaries. The structure is illuminated by a p-polarized plane wave at normal incidence (i.e., the incoming electric field points along the y direction). The thickness of sliver film is h = 100 nm and the frequency-dependent permittivities of sliver are referred to Ref. [19].

3. Results and discussion

At normal incidence, the position of transmission peak due to the excitation SPPs is given approximately by the surface plasmon dispersion for a smooth film[20],

where i, j are integers defining the different diffraction orders, ϕ_d and ϕ_m are the dielectric constant of the interface medium and metal, respectively. Fig. 2(a) shows the zero-transmission spectra of the compound hole arrays shown as in Fig. 1(b). For comparison, we also give the transmission spectrum of square hole arrays with only one hole in each cell (dashed line). According to Eq. (1), these transmission peaks can be indexed as Q (0, 1) at 930 nm, Q (1, 1) at 760 nm, and A (0, 1) at 710 nm, A (1, 1) at 590 nm as shown in Figs. 2(a) and (b). Here, Q denotes the SPPs at the quartz-silver interface and A represents the SPPs at air-sliver interface, respectively. With respect to the physics origin of these peaks, it has been demonstrated that the reflected SPPs Bloch waves from the holes produce a multiple beam interference pattern of the symmetry determined by the orientation of the hole rows[21].

 

Fig. 2. Transmission spectra for different arrangement of compound subwavelength hole arrays. (a)-(b) Transmission spectra correspond to the arrangement style shown as in Figs. 1(b) and (c). The transmission for simple cell arrays with the same width and periodicity as the compound structures is also plotted (dashed line), shown in the inset as “simple”.

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We first consider the effect of the distance d_AB on the transmission spectrum for compound hole arrays. When d_AB =100 nm, the interspace between every two holes in the y direction is equal, which actually becomes rectangular arrays of square hole with p_y = p_x/2 = 300 nm. In Fig. 2(a), two distinct peaks are observed in the transmission spectra (line with circles). However, if we change the distance d_AB, both of the two transmission bands split and two sharp dips are formed, one at wavelength λ = 620 nm and the other at λ = 870 nm (the arrow denoted). Moreover, as the distance d_AB changes from 80 nm to 60 nm, the dip of A (1, 1) peak at λ = 620 nm is further deepened while the other dip at λ = 870 nm is not so sensitive to d_AB. Comparing the transmission spectra with square hole arrays consisting of simple unit cell(dashed line), another interesting feature is the suppression of two (0, 1) peaks (A (0, 1) and Q (0, 1)) in the transmission spectra of compound hole arrays. The SPR peak arises from the constructive interference of SPPs, which is determined by the phase-match condition[22]: Re(k_Spp)p_x(y)+arg(τ) = 0 modulo 2π. Here, $k_{\mathrm{spp}}=2\pi /\lambda \sqrt{\frac{{\epsilon }_d{\epsilon }_m}{{\epsilon }_d+{\epsilon }_m}}$ is the SPPs wave vector, and τ is the transmittance coefficient of the SPPs mode. As pointed out in Ref. [21], the SPR (0, 1) peak is mainly induced by the Bragg scattering from the holes of y direction. When the square hole arrays B are inserted along to the y direction, the additional phase difference is introduced and the condition of constructive interference for the SPPs mode is broken, leading to the suppression of two (0,1) peaks. This phenomenon also agrees well with the experimental results in Ref. [16], where the suppression of transmission minima and maxima for compound graphite arrays and rectangular arrays were observed. While for the two (1, 1) peaks (A (1, 1) and Q (1, 1)), the asymmetrical compound hole arrays can only be regarded as rectangular arrays of square hole with structural defects, for the SPPs (1,1) modes are mainly scattered by the holes along the diagonal of the alternating arrays. As a result, one or more forbidden channels are formed within the allowed transmission bands, resulting in sudden variations of diffracted efficiency at a certain wavelength. On the other hand, due to the different dispersion relation of SPPs on two interfaces, the phase difference accumulated by the variation of d_AB is different for two dips at λ = 620 nm and λ = 870 nm, which gives rise to their different sensitivity to d_AB.

 

Fig. 3. Transmission spectra for various d_x, corresponds to the compound subwavelength square hole arrays illustrated in Fig. 1(d).

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Based on these findings, we expect the similar results would be found when we keep the distance d_AB in the y direction fixed and shift the hole arrays B a little along the x direction, the arrangement style shown as in Fig. 1(c). Indeed, the dips also appear at two transmission bands as illustrated in Fig. 2(b). When compared to that in Fig. 2(a), the dip in the second transmission band (840 nm) is more obvious. Moreover, further decrease of d_AB, the dip at 840 nm become deeper and the SPR (0, 1) peak appears gradually. As a matter of fact, the A (0, 1) peak and Q (1, 1) peak combine together in Fig. 2(b). This can attributed to the complex coupling behavior of electromagnetic field between the adjacent holes. Clearly, the phase difference for SPPs on both the metal-air and metal-quartz interface will decrease as the distance d_AB reduce, which lead the destructive interference for SPPs to be damaged and the SPR (0, 1) peak appear again. According to Figs. 2(a) and (b), the splitting of two (1, 1) peaks and the suppression of two (0, 1) peaks originate from the different SPPs interference behavior, which is governed by their phase difference related to the distance d_AB between two holes in a complex unit cell arranged in the SPPs propagation direction.

Since the SPPs modes mainly propagate along the y direction, the condition of SPPs resonance almost would not be affected if the two holes in every unit cell are arranged along the x axis shown as in Fig. 1(d). Thus, the above mentioned phenomena will not exist. This prediction is also verified by our FDTD simulation, as the transmission spectra shown in Fig. 3. We can find that all the transmission peaks for square hole arrays with simple unit cell appear again, except the Q (1, 1) peak and A (0, 1) peak combine together in some cases. Also, the Q (0, 1) peak for compound hole arrays exhibits redshift compared to the transmission for hole arrays with simple unit cell , which is caused by the variation of dispersion of SPPs by the additional holes. We also have calculated other cases for different d_x (not shown here), which all indicate that the transverse distance d_x is of minor importance to the transmission spectra for the compound hole arrays when illuminated by p-polarized incident light. This can be attribute to the fact that SPPs does not propagate along the x direction, confirming the above mentioned splitting of transmission peak and the formation of dips are caused by phase resonance of SPPs.

To further verify the explanation about the above results, we also give the field distribution at resonant and off-resonant wavelength in Fig. 4. The field distribution at wavelength λ = 610 nm, d_AB = 100 nm and λ= 620 nm, d_AB = 60 nm are depicted in Figs. 4(a) and (b), respectively. From Fig. 4(a), the hole arrays are distributed symmetrically, the SPPs Bloch wave interferes constructively at the edge of hole, leading to the transmission peak in Fig. 2(a). However, as to the asymmetrical case, the SPPs Bloch wave interferes destructively (Fig. 4(b)), which give rise to a dip in the transmission shown in Fig. 2(a). In order to better understand the suppression of two SPR (0, 1) peaks at metal surface, we also plot the field distribution at wavelength λ = 970 nm in Figs. 4(c) and (d), corresponding to the structure displayed in Figs. 1(d) and (b), respectively. The electric field distribution confirms that the interference behavior of SPPs is modulated by the arrangement of compound hole arrays.

 

Fig. 4. The calculated time-averaged density distribution of the electric field at dielectric-metal interfaces, the xy cross section. (a) corresponds to the case of d_AB = 100 nm, λ= 610 nm at the air-metal interface. (b) corresponds to d_AB = 60 nm, λ = 620 nm at the air-metal interface. (c)-(d) the electric field distribution at quartz-metal interface for Q (0, 1) peak at wavelength λ = 970 nm, corresponding to the arrangement of Figs. 1(d) and (b), respectively. The dotted lines mark the position of the holes.

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4. Conclusion

In summary, we have shown that the enhanced transmission can be tailored by changing the geometrical arrangement of compound subwavelength hole arrays. When the symmetry of hole arrays along the direction of polarized incident light is broken, the transmission bands show splitting behavior and the depth of the dip depends on the dielectric contacted with metal, as well as the distance between two holes. Additionally, as the symmetry of hole arrays changes, the two SPR (0, 1) peaks are suppressed, if the phase difference of SPPs resulting from the scattering by the holes is appropriated. However, the transmission spectra are not sensitive to the symmetry in the direction of perpendicular to the polarized light. We ascribe these phenomena to the phase resonance of SPPs, produced by the broking of symmetry of hole arrays. The proposed method to tailor enhanced transmission spectra is expected to have applications such as frequency selector or filter.