Transmission of electromagnetic waves through thick perfect conducting slabs perforated by one-dimensional arrays of rectangular holes was studied experimentally in the microwave frequency range. The observed thickness-dependent transmission clearly exhibits the evanescent and propagating nature of the involved electromagnetic excitations on the considered structures, which are effective surface plasmons and localized waveguide resonances, respectively. The 1D crystals showing transmission based on localized resonances further manifests the frequency-dependent effective refractive index depending on the filling ratio of the holes and accompanies resonant guided wave propagation.
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Since it was found that the light transmission through periodic subwavelength apertures can be extraordinarily enhanced compared to that through a single subwavelength hole in a thin metallic screen , this remarkable light transmission, known as extraordinary optical transmission (EOT), has invoked a lot of studies aiming to understand the underlying origin . Based on the analysis into the relationship between the resonant wavelength and the lattice period, the collective electromagnetic excitations on metal films such as surface plasmon polaritons (SPPs) [3,4] have been noted as the key player in such enhancement. However, later findings have elucidated the role of individual apertures, which is expressed as the shape dependent resonant wavelength and the Fabry-Perot resonances due to the waveguide modes in such apertures [5–7].
Along with the aforementioned achievements on two-dimensional (2D) hole arrays with high symmetry, there have been efforts to figure out the light transmission phenomena in minimal systems consisting of subwavelength holes with restricted symmetry such as one-dimensional (1D) chains of holes , orthogonal chains of holes , and 2D hole arrays with minimal number of holes [10–12]. Among these minimal systems, 1D chains of holes have been found to be the basic unit exhibiting EOT . Under certain conditions such as cutoff wavelength larger than the half-period and thickness comparable to the considered wavelength, these 1D plasmonic crystals have been reported to behave effectively as dielectric media with the frequency-dependent index of refraction [14,17]. Moreover, this type of structures have been noted for its capability of guiding electromagnetic waves confined on the surface , which is promising for plasmonic waveguide applications  and even as an interaction structure for micro-free-electron-lasers [15,16].
In this work, we study the transmission of electromagnetic waves through 1D arrays of rectangular holes in thick metal slabs using microwaves ranging from 6 to 12 GHz. To elucidate the different transmission mechanisms based on two types of collective electromagnetic excitations, effective surface plasmons [18–20] and localized resonances [14,17], the geometrical parameters such as period and thickness of the 1D plasmonic crystals were varied while maintaining the dimension of individual holes.
2. Enhanced transmission through 1D plasmonic crystals
The considered 1D plasmonic crystal is shown in Fig. 1 , which is a 1D array (period d) of rectangular holes of sides s and a, perforated on a metallic slab of thickness L. Each sample was formed out of two aluminum plates, one flat plate and the other with periodic grooves on it. The width and depth of each groove were given as a = 4 mm and s = 17 mm, respectively. Thereafter, these two aluminum plates were joined into one to form a metallic structure with rectangular holes as shown in Fig. 1. To measure the transmission of microwaves through the structures, microwaves were generated by an Anritsu Vector Network Analyzer (VNA) and a traveling wave tube amplifier (TWTA) in the frequency range between 6 and 12 GHz . The generated microwaves were radiated through a horn antenna and impinged on the sample with its electric field vector parallel to that of TE mode of the hole-waveguide. The transmitted waves were collected by another horn antenna behind the sample and fed back into the VNA. The whole measurement was conducted inside an anechoic chamber to prevent spurious effects due to diffraction and reflection.
For 2D arrays of rectangular holes perforated in metallic films, the 2D analog of the considered 1D systems in optical regime, it has been theoretically shown that the transmission of light exhibits two types of transmission spectra based on different electromagnetic excitations, which are SPPs and localized resonances . The involvement of these different excitations is related to the relative magnitude of the period of the array, d, to the cutoff wavelength of the hole waveguide, λc. For d>λc, the SPP-based resonant transmission occurs at wavelengths adjacent to the period of the array. For d<λc, the transmission shows resonances at wavelengths near λc mediated by the localized resonances of each hole waveguide.
In microwave frequency range where metals behave as perfect conductors, SPPs are no longer strongly bound to the surface. However, achieving strongly confined electromagnetic modes at such low frequencies is still possible by structuring the perfect conductors with subwavelength apertures [18,19]. These strongly confined electromagnetic modes, effective surface plasmons, have been known to be responsible for EOT in 2D arrays of holes in perfect conductors . To clarify the involvement of these effective surface plasmons together with the localized resonances in the considered 1D systems at microwave frequency range, the period and thickness of the array were varied around the cutoff wavelength λc, which was fixed to be 34 mm. The period d was varied to be 27, 30, and 35 mm while the thickness was ranging from 1 mm to 200 mm. The transmittance through each sample was obtained by normalizing the transmission spectra by the reference spectra measured without loading samples. Figure 2 shows the transmission spectra (normalized to the total area of the holes) of normally incident microwaves for the samples with d = 35 mm (> λc) and d = 27 mm (< λc), while varying the thickness: 1, 10, and 50 mm. Each transmission spectrum is compared with those obtained from numerical simulation (solid curves) using CST Microwave Studio .
For d = 35 mm (> λc), the transmission through the structure is expected to be associated with the effective surface plasmons and thus several features of EOT in 2D hole arrays based on SPPs  are observed in the experimental results shown in Fig. 2(a). The transmission resonances occur at wavelengths close to the period of the array and the resonance peak is blue-shifted from λ/d = 1.115 to λ/d = 1.007 as the thickness is increased from 1 mm to 50 mm, respectively. Meanwhile, the peak height decays as the thickness increases, which is attributed to the evanescent nature of the involved excitation, effective surface plasmons. The incident wave coupled to the surface mode on the top surface decays along the z-direction in the hole before it is coupled to the same mode on the bottom surface to finally couple out to free space.
For d = 27 mm(< λc), the transmission resonances occur near the cutoff wavelength of the hole waveguide, λ/d = 1.26, as shown in Fig. 2(b). However, the peak height does not decay as the thickness increases in contrary to the case for d = 35 mm. Instead, Fabry-Perot resonances appear with the increase of the thickness, which is clearly seen for L = 50 mm where three Fabry-Perot peaks are observed above λ/d = 1. Each peak corresponds to the fundamental waveguide mode with the different phase variations along the hole-waveguide .
To corroborate the involvement of the evanescent and propagating electromagnetic modes in these two types of enhanced transmissions, the electric field profiles at the resonances for L = 50 mm were also obtained from numerical simulation. The two contour plots in Fig. 3 show the electric field distribution in the y-z plane at the transmission peaks λ/d = 1.007 and 1.261 for d = 35 and 27 mm, respectively. The evanescent field profile of the effective surface plasmons excited on the top surface due to the incident waves is clearly seen in Fig. 3(a). On the other hand, Fig. 3(b) shows the excited fundamental waveguide mode at the localized resonance.
3. Frequency-dependent index of refraction
The similarity between the Fabry-Perot resonances in Fig. 2(b) and the transmission spectra of ordinary dielectric slabs suggests the homogenization of the considered periodic structure into an effective dielectric medium. In our previous work , its homogenization into an isotropic dielectric medium has been made by adopting one-mode approximation for the TE-waveguide mode in the holes . From the comparison of the calculated transmittance of the periodic structure with the transmittance of an ordinary dielectric slab with isotropic index of refraction, the effective refractive index has been found to exhibit frequency dependence in the following form :25], which is well-known as a metamaterial with geometrically controllable index of refraction. The difference of the effective refractive index in Eq. (1) from n0 arises from the different amount of phase variation in the propagation direction, z-direction, between the two systems. Since the fundamental TE-waveguide mode of each hole governs the transmission of the incident waves through the considered 1D hole arrays, the phase accumulation along the sample thickness, L, is given by qL. Meanwhile, the phase accumulation in the cut-through slits is given by α0L since the transmission is governed by TEM mode. As a result, for the considered structure, the effective refractive index of 1D cut-through slits, n = d/a, is modified with the additional term, q/α0 as in Eq. (1).
According to this homogenization, the effective refractive index, n, increases as the filling ratio, d/a, increases, which would be exhibited as more densely located Fabry-Perot resonances above the cutoff frequency. Since the number of resonances increases for thicker samples according to the basic relation of Fabry-Perot resonance, kL = mπ, transmission spectra were recorded for thicker samples with L = 200mm to clearly see the density change of resonances. Figure 4 shows the transmission spectra of samples having filling ratio of 6.75 (d = 27 mm), 7.5 (d = 30 mm), and 8.75 (d = 35 mm). As expected, Fabry-Perot resonances for higher filling ratio are more densely packed as shown in Fig. 4. However, due to diffraction, the frequency dependence is observed in a finite range of frequency above the cutoff frequency and below the Rayleigh minimum, c/d. At normal incidence, samples with d = 27, 30, and 35 mm experience the first order diffraction at the normalized frequencies 1.259, 1.133, and 0.971, respectively.
Another important point in our previous work  is that the enhanced transmission occurs at the frequency where the eigenmode population of the effective dielectric slab is maximized. To verify this numerical prediction, we measured the propagation of electromagnetic waves on the considered 1D structures which support effective dielectric slab modes [13,14,17]. These effective guided modes were excited on the structures by coupling fundamental waveguide mode into the far-left hole from the bottom using a waveguide adapter. When the coupled waveguide mode arrives at the top boundary, some are coupled into the nearest neighbour hole whereas some are reflected back or diffracted into the far-field. The amount of guided propagation spectrum can be measured by recording the amount of coupled out waveguide mode from the far-right hole using another waveguide adapter.
The measured guided propagation spectra for d = 27 mm with different thicknesses are depicted in Fig. 5(a) , where the resonances correspond to the frequency where the eigenmode population of effective dielectric slab modes is maximized . As L is increased from 50 mm to 200 mm, the number of resonance peaks in the given frequency range increases. This increase in the number of peaks corresponds to the appearance of successive higher order modes with different phase variation in the z-direction [14,17]. The comparison of peak frequencies of farfield transmission and guided wave propagation spectra for 200 mm thick sample shows close proximity between the two spectra as shown in Fig. 5(b). This is in agreement with the explanation given in the previous numerical study  that the far-field transmission resonances occur where the eigenmode population is maximized.
In conclusion, we showed that enhanced transmission of electromagnetic waves still occurs in 1D plasmonic crystals consisting of arrays of rectangular holes in thick perfect conducting slabs using microwaves ranging from 6 to 12 GHz. The observed thickness-dependent transmission spectra clearly exhibits the evanescent and propagating nature of the involved electromagnetic excitations, which are effective surface plasmons and localized resonances, for d>λc and d<λc, respectively. Although limited to a finite frequency range, for d<λc, this 1D structure shows frequency dependence in its effective dielectric response, which is observed as density change of Fabry-Perot resonances. This dielectric response is further corroborated by the concomitant existence of guided modes.
This work was supported by National Research Foundation of Korea Grant funded by the Korean Goverment(grant code: 2009-0083512)
References and links
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