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SECTION 1

Introduction

The phenomenon of extraordinary optical transmission (EOT) through periodically arranged subwavelength holes in metal films has attracted considerable interest over the past few decades since the pioneering work of Ebbesen et al. [1]. The main mechanisms responsible for this phenomenon have been widely believed to be the excitation of surface plasmon polaritons (SPPs) and Wood's anomalies (WAs) set up by periodically arranged holes [2], [3], [4], [5]. In addition, several works have shown that the shape of the aperture has a strong effect on the transmission spectra [4], [5], [6], [7], [8], [9], [10], [11]. For example, Koerkamp et al. [4] showed that shape resonances (SRs), which are in effect localized modes, play an important role in explaining the different transmission spectra between the circular and rectangular hole arrays. Ruan and Qiu [5] calculated the normalized transmission spectra and the band structure of a rectangular-shaped-hole array drilled in a perfect-electric-conductor (PEC) film to distinguish surface plasmon resonances and localized resonances. They also found that the localized modes are almost independent of the periodicity by calculating the transmission spectra of random arrays of holes. Moreover, Lee et al. [6] experimentally demonstrated the resonant peak dominantly depends on the hole length, which confirmed the theoretical predictions of Garcia-Vidal et al. [8] and Ruan and Qiu [5]. Mary et al. [9] analyzed theoretically how SPPs and localized resonances evolve and mix when the period of the array is varied.

In order to figure out more detailed properties of extraordinary transmission, some structures with more complex-shaped apertures have been studied. For example, a range of arrays composing of double holes that slightly overlap provides an additional local field enhancement [12], [13], [14], which could be potentially applied to nanolithography, biochemical sensing, or nonlinear processes. Arrays with C- or U-shaped apertures [15], [16] have been designed for the purpose to redshift the shape resonant wavelength. In particular, Sun et al. [17] experimentally showed the transmission properties of H-shaped arrays, and Liu et al. [18] theoretically investigated the same structure by analyzing the transmission spectra and the near-field amplitude and phase distributions. In this paper, we study the optical properties of three similar but differently shaped apertures, which are basically composed of two horizontal slits, named Formula$x$-arm slits, and one vertical slit, named the Formula$y$-arm slit. The difference between them lies in the horizontal position of the Formula$y$-arm silt, which is shifted from the alignment with the left of the two Formula$x$-arm silts to one forth length and finally to the middle. Fig. 1(a) shows the top view and structural parameters of samples 1 (U-shaped) to 3 (H-shaped). The experimental observation of such structures has been reported [19]. The SR wavelengths could be roughly estimated by the modified cutoff wavelengths of the rectangular waveguide Formula$\lambda_{res} = 2n_{eff}L_{res}/m$, where Formula$n_{eff} = \sqrt{(n_{substrate}^{2} + n_{air}^{2})/2}$ is the effective refractive index, Formula$L_{res}$ is resonant length, and Formula$m$ is an integer [10]. However, when the shape of the aperture becomes more complex and higher order modes coexist, Formula$L_{res}$ would have some ambiguities to be determined. We thus use our in-house developed 3-D finite-difference time-domain (FDTD) program [20] to simulate transmission spectra and near-field distributions at several particular wavelengths to gain more understanding of the physical characteristics of the resonant modes.

Figure 1
Fig. 1. (a) Top view and structural parameters of samples 1–3. (b) The unit cell and near-field plane cuts in our simulation. In-plane cut: perpendicular to the direction of propagation; perpendicular-plane cut: parallel to the direction of propagation.

Since the phasor form of electric field can be expressed as Formula${\bf E} = \vert E_{x}\vert e^{i\psi_{x}}\mathhat{x} + \vert E_{y}\vert e^{i\psi_{y}}\mathhat{y} + \vert E_{z}\vert e^{i \psi_{z}}\mathhat{z}$, we analyzed the distributions of electric field amplitude (Formula$\vert E_{x}\vert$, Formula$\vert E_{y}\vert$, and Formula$\vert E_{z}\vert$) and phase (Formula$\psi_{x}$, Formula$\psi_{y}$, and Formula$\psi_{z}$) simultaneously. The modulus of the electric field, Formula$\vert{\bf E}\vert$, is defined by Formula$\sqrt{\vert E_{x}\vert^{2} + \vert E_{y}\vert^{2} + \vert E_{z}\vert^{2}}$. We find three features are helpful in recognizing the effective SR lengths: the large magnitude of Formula$\vert{\bf E}\vert$ contours clear geometry of resonant length. Within the shape resonant paths, the distribution of the phase component parallel to the polarization of the incident light (e.g., Formula$\psi_{x}$ under Formula$x$-polarized light) is quite homogeneous, and the phase component along the propagation direction, Formula$\psi_{z}$, exhibits a 180° phase difference with respect to the center of the slit width. Therefore, by utilizing these thumb of rules and the symmetric requirements, we can clearly distinguish the SR lengths and also predict possible contours. For example, assigning the SR peaks of H-shaped apertures in the shorter wavelength range to be the second U-shaped resonances is not appropriate [19] since Formula$\psi_{z}$ needs to be with even symmetry with respect to the center of the Formula$x$-coordinate under Formula$y$-polarized light. Therefore, the resonant path will not pass the Formula$y$-arm slit, and we find that it is indeed related to the merely two Formula$x$-arm slits resonance. Besides, Schnell et al. [21], [22] recently experimentally demonstrated the antiphase of Formula$\psi_{z}$ at the metal rod center as a direct evidence of the dipolar near-field mode. This also gives us a hint for the counterpart relationship between SRs and the plasmonic resonances for metal islands.

On the other hand, many theoretical analyses of WAs and surface electromagnetic (EM) modes have revealed their origin and roles in EOT phenomenon [23], [24], [25], [26], but few have shown their corresponding features in near-field analyses. For example, WA phenomenon has been known to occur when a diffracted order becomes grazing and the in-phase multiple scattering accumulates [23]. We thus find the corresponding Formula$\psi_{z}$ distribution is quite homogeneous. Besides, the weak coupling of the incident field with surface EM modes near the apertures may contribute to the transmission peaks [23], [24] and be related to the more severe phase change near the hole. Therefore, except utilizing band diagrams or the far-field spectra corresponding to different changes of the structure to distinguish different modes, we can also employ the near-field features to gain more insight of each resonance.

SECTION 2

Far-Field Transmission Spectra and Dispersion Relations

Fig. 1(a) and (b) show the geometry of the subwavelength metallic structures we studied in this paper: the unit cell with lattice constants of Formula$a_{x} = a_{y} = 14\ \mu\hbox{m}$, where the length of each silt Formula$l_{x} = l_{y} = 8\ \mu\hbox{m}$, and the width of the silt Formula$w = 1\ \mu\hbox{m}$. The metal (silver) film with thickness 75 nm is coated on a silicon substrate. In addition, the relative permittivity of the silicon substrate is taken as 11.9, and that of the silver [27] is modeled by the Drude dispersion with a plasma frequency of Formula$\omega_{p} = 1.3709 \times 10^{16}\ \hbox{s}^{-1}$ and a collision frequency of Formula$\gamma = 3.2258 \times 10^{13}\ \hbox{s}^{-1}$. These structures are illuminated by Formula$x$-polarized and Formula$y$-polarized plane waves, respectively, under normal incidence. The calculated transmission spectra shown in Fig. 3 are found to agree well with the measured transmission data shown in Fig. 2. Fig. 2(a)(c) were obtained under the same experimental setup using a Bruker IFS 66 v/s Fourier-Transform Infrared (FTIR) spectrometer, as reported in [19], with Fig. 2(c) corresponding to Fig. 7(a) in [19]. In the experiment, samples 1 and 2 have the same material structures as that of sample 3, except the top view shapes differ, as shown in Fig. 1(a). The spectra exhibit very different features under Formula$x$-polarized and Formula$y$-polarized incident light because of the anisotropic geometry of the hole structure. The transmission minima are associated with the WAs which could happen when a diffracted order becomes tangent to the plane of the grating, and the free-space wavelength that satisfies WA condition under normal incidence is given by Formula TeX Source $$\lambda_{WA} = {\Lambda \over \sqrt{i^{2} + j^{2}}} \times \sqrt{\varepsilon_{d}}\eqno{\hbox{(1)}}$$ where Formula$i$ and Formula$j$ are integers referring to the specific orders of the WA modes in the Formula$x$ and Formula$y$ directions, respectively, Formula$\Lambda\ (= 14\ \mu\hbox{m})$ is the lattice constant, and Formula$\varepsilon_{d}\ (= 11.9)$ is the relative dielectric constants of silicon. Notice that only Ag/Si WAs, denoted as (Formula$i, j$) mode Ag/Si WAs, will be excited in the wavelength range of 20–140 Formula$\mu\hbox{m}$. Adjacent to the WA dips are some small peaks which are due to the coupling of the incident field with surface EM modes [23], [24], [25]. Since the property of silver is PEC-like in the far-infrared range, the surface EM modes dominantly result from the multiple scattering of the diffracted modes within the holey metal surface [24], [28]. We have examined the spectra with the silver film replaced by a perfect conductor which usually does not support surface plasmon waves. The spectra are almost the same with Fig. 3 (not shown), while the peak values corresponding to SRs increase because of the lossless material and the spectral positions of SRs are slightly blue shifted due to the lack of penetration depth for perfect conductor.

Figure 2
Fig. 2. Measured transmission spectra of (a) sample 1, (b) sample 2, and (c) sample 3. [(c) corresponds to [19, Fig. 7(a)].]
Figure 3
Fig. 3. Calculated transmission spectra with polarized incident light at (a) Formula$x$ polarization and (b) Formula$y$ polarization under normal incidence.

As mentioned in [19], the localized shape resonant path is mainly the U-shaped part of the hole in our case. The broadband peaks at 109 Formula$\mu\hbox{m}$, 96 Formula$\mu\hbox{m}$, and 87 Formula$\mu\hbox{m}$ for samples 1 to 3 under Formula$x$-polarized light correspond to the 1st U-shaped resonance. The blue-shifted trend of the spectral positions from samples 1 to 3 is due to the shrinkage of the effective resonant length of aperture referring to the green path indicated in Fig. 1(a). On the other hand, the peaks at 51 Formula$\mu\hbox{m}$ and 47 Formula$\mu\hbox{m}$ in Fig. 3(b) correspond to the second U-shaped resonances for samples 1 and 2 under Formula$y$-polarized light. The broadband resonance at 42 Formula$\mu \hbox{m}$ of sample 3 may seem to also be the second U-shaped resonance according to the trend. However, from the help of near-field analyses, this peak is related to the two Formula$x$-arm slits resonance.

SECTION 3

Near-Field Distributions

Next, we analyze the distributions of electric field amplitude and phase simultaneously to investigate the relation between transmission spectra and near-field features. The in-plane cut shown in Fig. 1(b) is along the silicon substrate side right after the back surface of thin metal film perforated with 2-D hole arrays, while the perpendicular-plane Formula$\perp_{x}\ (\perp_{y})$-plane cut is the Formula$x$Formula$z$ (Formula$y$Formula$z$) plane cut along the center of Formula$y\ (x)$-coordinate of the unit cell. According to Liu et al. [18], since the H-shaped hole pattern has twofold mirror symmetry, the components of electric field (Formula$E_{x}$, Formula$E_{y}$, and Formula$E_{z}$) under normal incidence could be classified into four symmetric categories, which is summarized in Table 1. An incident field will preserve its particular symmetry when transmitting through subwavelength hole structures drilled in metallic film. Therefore, for Formula$x$-polarized light, Formula$E_{x}$ has even–even symmetry, and Formula$E_{y}$ has odd–odd symmetry. On the other hand, for Formula$y$-polarized light, Formula$E_{x}$ has odd–odd symmetry and Formula$E_{y}$ has even–even symmetry. For example, Fig. 4(a) shows Formula$\psi_{x}$ and Formula$\psi_{y}$, respectively, at in-plane cut for (±2, 0) mode Ag/Si WA under Formula$x$-polarized light and Fig. 4(b) shows those for (0, ±2) mode Ag/Si WA under Formula$y$-polarized light. These figures demonstrate that Formula$E_{x}$ and Formula$E_{y}$ indeed possess specific symmetries for H-shaped hole arrays. However, for samples 1 and 2, mirror symmetry with respect to the center of the Formula$x$-coordinate no longer exists [see Fig. 4(c)(d)].

Table 1
TABLE 1 Categories of symmetry of electric field components for H-shaped holes. For samples 1 and 2, they only preserve the symmetry with respect to the center of the Formula$y$-coordinate of the unit cell
Figure 4
Fig. 4. For sample 3 within twofold mirror symmetry, Formula$E_{x}$ possesses even–even symmetry, and Formula$E_{y}$ possesses odd–odd symmetry under Formula$x$-polarized light excitation, while Formula$E_{x}$ possesses odd–odd symmetry, and Formula$E_{y}$ possesses even–even symmetry under Formula$y$-polarized light. For samples 1 and 2, mirror symmetry with respect to the center of the Formula$x$-coordinate no longer exists. For example, the Formula$\psi_{x}$ and Formula$\psi_{y}$ distributions for (±2, 0) and (0, ±2) Ag/Si WA are shown in (a) and (b) for sample 3 and in (c) and (d) for sample 1, respectively.

3.1. Formula$x$-Polarized Light

First, we analyze the near-field distributions corresponding to different orders of WA modes for samples 1 to 3 under Formula$x$-polarized light, and we find that they all exhibit some similar features. For example, Fig. 5(a) shows the modulus of the electric field Formula$\vert{\bf E}\vert$ at in-plane cut and the modulus of the electric field amplitude and the phase along the propagation direction, Formula$\vert E_{z}\vert$ and Formula$\psi_{z}$, at Formula$\perp_{x}$-plane cut for Formula$(\pm 1, \pm 1)$ Ag/Si WA modes at 35 Formula$\mu\hbox{m}$, and Fig. 5(b) shows those for the adjacent surface EM modes at 34 Formula$\mu\hbox{m}$ of sample 3. Both Formula$\vert{\bf E}\vert$ and Formula$\vert E_{z}\vert$ display distinct field patterns at the Ag/Si interface for WAs and surface EM modes. By comparing the Formula$\psi_{z}$ distributions in Fig. 5(a) and (b), we find that Formula$\psi_{z}$ is quite homogeneous for WAs, while Formula$\psi_{z}$ has changed severely near the hole in the substrate for surface EM modes. According to Abajo et al. [23], the transmittance of the metallic hole array could be derived from the reflectance of the complementary metallic disk array by Babinet's principle. Therefore, the transmission for the metallic hole array can be expressed as Formula TeX Source $$T = {1 \over 1 + \left({A \over 2\pi \times k}{\rm Re}\left[{1 \over \alpha_{E}} - G_{E}\right]\right)^{2}}\eqno{\hbox{(2)}}$$ where Formula$A$ is the area of the unit cell, Formula$k$ is the light momentum in free space, Formula$\alpha_{E}$ is the electric polarizability of the disks, and Formula$G_{E} = \sum_{R\neq 0}(k^{2} + \partial_{xx}^{2})e^{ikR}/R$ is the summation of the dipole-dipole interaction dyadic under normal incidence. Formula$G_{E}$ diverges when one of the diffracted orders becomes grazing, which is the same condition that produces WAs and results in transmission minimum according to (2). The divergence of Formula$G_{E}$ is related to the accumulation of in-phase long-distance multiple scattering. This may be associated with the homogeneity of Formula$\psi_{z}$ for WAs in our near-field analyses. On the other hand, two channels are responsible for the EOT phenomenon as described by Fano [25]: One is simply the direct scattering of the field through the apertures and can be represented by a continuum of states, and the other is via a discrete resonant state (surface EM mode) and then coupled to the continuum of states. Besides, it is near those divergences associated with WAs that surface EM modes can exist [26]. Surface EM modes are confined to the surface and partially coupled to a continuum near the apertures, and thus resulting in the transmission peaks and the severe change of Formula$\psi_{z}$. For samples 1 and 2, the Formula$\psi_{z}$ distributions are a little bit disordered due to the broken symmetry with respect to the center of the Formula$x$-coordinate, but the distinct features of Formula$\psi_{z}$ for WAs and surface EM modes still reserve, as shown in Fig. 6.

Figure 5
Fig. 5. Formula$\vert{\bf E}\vert$ distribution at in-plane cut and Formula$\vert E_{z}\vert$ and Formula$\psi_{z}$ distributions at Formula$\perp_{x}$-plane cut of sample 3 at (a) 35 Formula$\mu\hbox{m}$ for Formula$(\pm 1, \pm 1)$ WA and (b) 34 Formula$\mu\hbox{m}$ for the adjacent surface EM mode.
Figure 6
Fig. 6. Formula$\psi_{z}$ distributions at Formula$\perp_{x}$-plane cut of sample 1 at (a) 35 Formula$\mu\hbox{m}$ for Formula$(\pm 1, \pm 1)$ WA and (b) 34 Formula$\mu\hbox{m}$ for the adjacent surface EM mode. (Those for sample 2 demonstrate similar features.)

Next, we focus on the brand-band peaks in the spectra of Fig. 3(a) corresponding to the first U-shaped resonances for samples 1 to 3. As shown in Fig. 7, the large magnitude of Formula$\vert{\bf E}\vert$ and the homogeneous distribution of Formula$\psi_{x}$ under Formula$x$-polarized light within the shape resonant paths may result in the strong transmission. Formula$\vert{\bf E}\vert$ also contours clear geometrical patterns of the resonant modes and Formula$\vert E_{z}\vert$ at Formula$\perp_{y}$-plane cut is strongly localized near the edge of the slits for SRs. Formula$\vert E_{z}\vert$ for samples 2 and 3 are not shown since they demonstrate similar features. The most interesting part is that Formula$\psi_{z}$ exhibits a 180° phase difference with respect to the center of the slit width along the shape resonant path. Therefore, the effective resonant lengths for SRs could be clearly determined by analyzing both Formula$\vert{\bf E}\vert$ and Formula$\psi_{z}$. For example, by analyzing the Formula$\psi_{z}$ distribution of sample 2, which could be divided into two possible U-shaped paths as shown in Fig. 1(a), we can clearly distinguish that the dominant resonant pattern at 96 Formula$\mu\hbox{m}$ is along the green path since Formula$\psi_{z}$ only processes 180° phase difference within that path. Moreover, for sample 3, Formula$\psi_{z}$ fulfills the requirement in two symmetric U-shaped holes as shown in Fig. 7(c), and thus, both paths would contribute to the resonance. Furthermore, since both measured and simulation transmission spectra exhibit peaks at 51 Formula$\mu\hbox{m}$ for sample 1 and at 52 Formula$\mu\hbox{m}$ for sample 2, we thus analyze the near-field features and find that the peak at 51 Formula$\mu\hbox{m}$ is related to the first Formula$y$-arm slit SR, while that of sample 2 corresponds to the red-path [referring to Fig. 1(a)] U-shaped resonance. As shown in Fig. 8, Formula$\vert{\bf E}\vert$ displays both features for surface EM modes and SRs: distinct field patterns at the Ag/Si interface indicating the characteristic of surface EM modes as well as contouring clear geometric patterns at the boundary of the hole for SRs. Besides, Formula$\psi_{x}$ and Formula$\psi_{z}$ exhibit the properties of homogeneity and the 180 ° phase difference within the shape resonant path, respectively, showing good agreement with the mentioned features for SRs.

Figure 7
Fig. 7. Formula$\vert{\bf E}\vert$, Formula$\psi_{x}$, and Formula$\psi_{z}$ distributions at in-plane cut and Formula$\vert E_{z}\vert$ distribution at Formula$\perp_{y}$-plane cut corresponding to the first LSR for (a) sample 1 at 109 Formula$\mu\hbox{m}$, (b) sample 2 at 96 Formula$\mu\hbox{m}$, and (c) sample 3 at 87 Formula$\mu\hbox{m}$.
Figure 8
Fig. 8. Formula$\vert{\bf E}\vert$, Formula$\psi_{x}$, and Formula$\psi_{z}$ distributions at in-plane cut for (a) sample 1 at 51 Formula$\mu \hbox{m}$ and (b) sample 2 at 52 Formula$\mu\hbox{m}$.

3.2. Formula$y$-Polarized Light

Fig. 9 shows the Formula$\vert{\bf E} \vert$ distribution at in-plane cut and Formula$\vert E_{z}\vert$ and Formula$\psi_{z}$ distributions at Formula$\perp_{y}$-plane cut for Formula$(\pm 1, \pm 1)$ Ag/Si WA modes at 35 Formula$\mu$ m and those for the adjacent surface EM modes at 34 Formula$\mu\hbox{m}$ of sample 1 under Formula$y$-polarized light. Sample 2 demonstrates similar near-field features (not shown). Both Formula$\vert{\bf E}\vert$ and Formula$\vert E_{z}\vert$ display distinct field patterns at the Ag/Si interface for WAs and surface EM modes as in the case under Formula$x$-polarized light. We again find the homogeneity of Formula$\psi_{z}$ for WAs and the severer phase change for surface EM modes, thus showing the consistency of the observation. Moreover, in order to check whether these features are general to WAs and adjacent surface EM modes, we have varied the structural parameters to examine their near-field distributions in the optical range. The same profiles have been observed, thus providing evidence that the different Formula$\psi_{z}$ distributions near the apertures are meaningful to the mentioned physical characteristics of WAs and adjacent surface EM modes.

Figure 9
Fig. 9. Formula$\vert{\bf E}\vert$ distribution at in-plane cut and Formula$\vert E_{z}\vert$ and Formula$\psi_{z}$ distributions at Formula$\perp_{y}$-plane cut of sample 1 at (a) 35 Formula$\mu\hbox{m}$ for Formula$(\pm 1, \pm 1)$ WA and (b) 34 Formula$\mu\hbox{m}$ for the adjacent surface EM mode.

Finally, we focus on the broadband peaks in the spectra of Fig. 3(b). As shown in Fig. 10(a) and (b), the Formula$\vert{\bf E}\vert$ distributions, which are concentrated near the boundaries of the hole with a node in the middle of the Formula$y$-arm slit, demonstrate geometric resonant patterns of 2nd U-shaped resonances and Formula$\psi_{y}$ is quite homogenous under Formula$y$-polarized light. Moreover, the Formula$\vert E_{z}\vert$ distributions at Formula$\perp_{y}$-plane cut are strongly localized near the edges of the slits and the Formula$\psi_{z}$ distributions exhibit a 180 ° phase difference with respect to the center of the slit width within the shape resonant paths, which all show determinative features for SRs. Notice that in Fig. 10(c), the Formula$\psi_{z}$ distribution owns the 180° phase difference with respect to the center of two Formula$x$-arm slits' width only. The lack of phase difference in Formula$y$-arm slit is due to the requirement of even symmetry with respect to the center of the Formula$x$-coordinate for H-shaped holes under Formula$y$-polarized light (see Table 1). For samples 1 and 2, there is no such symmetry requirement for their Formula$y$-arm slits since the mirror symmetry with respect to the center of Formula$x$-coordinate no longer exists. Besides, the SR has the priority to follow the longest possible path, thus exciting the second U-shaped resonance for samples 1 and 2, as well as the two Formula$x$-arm slits resonance for sample 3. In order to validate that this resonance is associated with the two Formula$x$-arm slits resonance rather than the second U-shaped resonance for sample 3, we enlarge the difference of effective resonant lengths between these two resonances by changing the separation distances Formula$l_{y}$ between the two Formula$x$-arm slits from 8 Formula$\mu\hbox{m}$ to 6 Formula$\mu \hbox{m}$ and 4 Formula$\mu\hbox{m}$, respectively. The effective lengths of the 2nd U-shaped resonance shrink from 15 Formula$\mu\hbox{m}$ to 13 Formula$\mu\hbox{m}$ and 11 Formula$\mu\hbox{m}$, respectively, while those of the two Formula$x$-arm slits resonance will not be affected by different Formula$l_{y}$'s and keeps at 8 Formula$\mu \hbox{m}$. However, due to the great effects of WAs in shaping the spectral profiles [10], we could hardly tell the response of resonant wavelengths corresponding to different separation distances. We thus simulate the same condition for merely the two Formula$x$-arm slits aperture, as shown in Fig. 11, and find that the spectral profiles can almost overlap with each other under the same separation distance for the H-shaped and two Formula$x$-arm slits apertures, therefore giving evidence to the fact that the peak at 42 Formula$\mu \hbox{m}$ is associated with the two Formula$x$-arm slits resonance.

Figure 10
Fig. 10. Formula$\vert{\bf E}\vert$, Formula$\psi_{y}$, and Formula$\psi_{z}$ distributions at in-plane cut and Formula$\vert E_{z}\vert$ distribution at Formula$\perp_{y}$-plane cut corresponding to the second U-shaped resonances for (a) sample 1 at 51 Formula$\mu\hbox{m}$, (b) sample 2 at 47 Formula$\mu\hbox{m}$, and the two Formula$x$-arm slits resonance for (c) sample 3 at 42 Formula$\mu\hbox{m}$.
Figure 11
Fig. 11. Simulated transmission spectra for (a) H-shaped apertures and (b) two Formula$x$-arm slits apertures under different separation distance Formula$l_{y}$ between the two Formula$x$-arm slits.
SECTION 4

Conclusion

We find that by analyzing the distributions of the electric field amplitude and phase simultaneously, the unique near-field features associated with specific resonant modes can be summarized as follows. 1) For SRs, the large magnitude of Formula$\vert{\bf E}\vert$ and the homogeneity of the phase component parallel to the polarization of the incident light within the shape resonant paths may result in transmission peaks. In order to determine the effective resonant lengths of SRs, we could use the properties of Formula$\vert{\bf E}\vert$ and Formula$\psi_{z}$: Formula$\vert{\bf E}\vert$ contours a clear geometry pattern of the resonant mode, and Formula$\psi_{z}$ exhibits a 180° phase difference with respect to the center of the slit width along the shape resonant path. 2) Formula$\vert E_{z}\vert$ is strongly localized near the edges of the slits for SRs, while Formula$\vert E_{z} \vert$ display distinct field patterns at the Ag/Si interface for WAs or surface EM modes. 3) For WAs, Formula$\psi_{z}$ is quite homogeneous due to the accumulation of the in-phase multiple scattering; the weak coupling of surface EM modes to a continuum cause the transmission peaks and severe phase variation near the hole in the substrate.

We thus use such information to investigate the SR at the wavelength of 42 Formula$\mu\hbox{m}$ under Formula$y$-polarized light because Formula$\psi_{z}$ exhibits 180° phase difference within the two Formula$x$-arm slits only due to the requirement of even symmetry with respect to the center of the Formula$x$-coordinate for H-shaped holes. The overlapped spectral profiles under the same separation distances between H-shaped and two Formula$x$-arm slits apertures give us a clue that this resonance would be attributed to the two Formula$x$-arm slits resonance.

Footnotes

This work was supported in part by the National Science Council of the Republic of China under Grant NSC99-2628-M-002-008 and Grant NSC98-2221-E-002-025-MY2, in part by the Excellent Research Projects of National Taiwan University under Grant 99R80306, and in part by the Ministry of Education of the Republic of China under “The Aim of Top University Plan” Grant. Corresponding author: H.-C. Chang (e-mail: hcchang@cc.ee.ntu.edu.tw).

H.-H. Hsiao is with the Graduate Institute of Photonics and Optoelectronics, National Taiwan University, Taipei 10617, Taiwan.

H.-F. Huang is with the Graduate Institute of Electronics Engineering, National Taiwan University, Taipei 10617, Taiwan.

S.-C. Lee is with the Graduate Institute of Electronics Engineering and Department of Electrical Engineering, National Taiwan University, Taipei 10617, Taiwan.

H.-C. Chang is with the Department of Electrical Engineering, Graduate Institute of Photonics and Optoelectronics, and Graduate Institute of Communication Engineering, National Taiwan University, Taipei 10617, Taiwan.

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Hui-Hsin Hsiao

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Hao-Fu Huang

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Si-Chen Lee

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Hung-Chun Chang

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