1. Introduction

When two antennas are close to each other, they become coupled electromagnetically, because each antenna affects the other through scattered electromagnetic fields. Coupling of dipole nanoantennas has been described by simple models such as hybridization [1–3] or oscillator [4] models that predict the occurrence of hybridized modes, e.g., the bright and the dark modes of two coupled dipoles. These hybridized modes exhibit richer spectral features than the fundamental mode of a single antenna. Particularly the dark mode, which features strongly suppressed radiation coupling, possesses a sharp spectral resonance peak of an exceptionally high quality factor [4–6] that can be used for sensing, filtering and energy transport [7,8]. Coupled antennas are also capable of generating diverse radiation patterns, such as frequency-dependent beam steering using two nanoparticle resonators [9]. It was shown that multimodes can be used to tailor scattering through overlapping the electric and magnetic modes of a resonator [10]. This raises the possibility of controlling radiation using hybridized dark and bright modes. However, the selective excitation of dark modes in a controlled way is difficult due to their suppressed coupling to external lights. Various methods have been used to excite dark modes, such as by using a localized source [11,12], a near field from another resonator [13], obliquely incident light [14,15], radially polarized light [16], and a spatial phase-shaped beam [17]. Coupled asymmetric resonators have been used to excite dark modes indirectly which interferes with the bright mode resulting in electromagnetically induced transparency [5,6,18–20]. Until now, the controlled and selective excitation of the dark and bright modes of coupled nanoantennas has been unachievable.

Here, we present a coupled nanoantenna system, consisting of two identical plasmonic slot resonators and additional phase modulating elements, which enables the selective excitation of bright and dark modes. We explain the resonance characteristics of bright and dark modes in terms of the coupling between two slot resonators. We confirm experimentally the coupling-dependent resonance enhancement of the bright mode in the IR range and those of bright and dark modes in the microwave range. We also demonstrate that by applying a phase modulating unit to each pair of coupled antennas, bright and dark modes can be selectively excited by changing the polarization of the incident microwave. In particular, for the first time we excite only the dark mode using a plane wave. We explain that the radiation patterns of the bright and the dark mode are those of a dipole and a quadruple and that the superposed radiation patterns vary depending on the polarization state the incident wave. In particular, leftward or rightward uni-directional radiations arise upon the injection of left or right circularly polarized waves. These features can be applied to design of metasurface devices possessing novel functionalities.

2. Results

3.1 Coupled resonators

Consider plasmonic nanoantennas made of two identical narrow rectangular slots of size $a\times b(b\ll a)$ in a metal film as shown in Fig. 1. The long side of the rectangle is oriented along the y-direction and the two rectangles are separated by distance $d$ along the x-direction. Upon the injection of x-polarized light, two slots coupled through multiple scattering support bright and dark modes as hybridized fundamental modes. Figure 1 illustrates the configuration of the induced charge and the x-components of the electric field when the bright and dark modes are separately excited. Resonance occurs when the wavelength is about $2a$ and the strength of the resonance depends on the distance separating the nanoantennas [4,21]. To further understand the resonance characteristics of the bright and dark modes, we calculate the scattering cross sections of the coupled antennas using the FDTD numerical method assuming the metal film is a perfect conductor. We have used the FDTD grid size 10 nm for the IR range and 0.5 mm for the microwave applications. Keeping the aspect ratio of the rectangle $a/b=10$ and varying the separation $d$, we calculate the quality factors (Q-factor) of bright and dark mode resonances with the results shown in Fig. 1. The Q-factor of the bright mode oscillates around 6.4, which is the value of the single slot resonance, and can be enhanced up to 10. Remarkably, the Q-factor of the dark mode resonance increases limitlessly as separation becomes zero.

 

Fig. 1 When two slot antennas are coupled electromagnetically, hybridized modes occur named as (b) bright mode (c) dark mode. The columns show, from left to right, the schematic of the hybridized modes, the calculated electric field map, and the Q-factor of each mode.

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To test the Q-factor enhancement of the bright mode experimentally, we directly measured the scattering cross section of the slot antennas in the IR region. Double slots, sized $180\text{nm}\times 1300\text{nm}$ and separated by $d=1500\text{nm}$, were fabricated on a silver film of thickness 300nm over a glass substrate as shown in Fig. 2(a). Slots are fabricated using the FIB etching after the deposition of silver film using thermal evaporators. Figure 2(b) shows the calculated Q-factor of the bright mode of the double slots (black line) and that of a single slot (red line). Figure 2(c) and 2(d) show both the numerical and experimental results for the transmission spectra of the single and double slots, respectively. Compared to the numerical results, the measured Q-factors are slightly reduced due to imperfections of fabrication. Nevertheless, overall the Q-factor was enhanced by $4.9/3.1\approx 1.6$ experimentally when using the double slot, a figure in good agreement with the numerically predicted value, $6.9/4.1\approx 1.7$.

 

Fig. 2 (a) Ag slot antennas on a glass substrate designed so that the resonance wavelength is 3.5 μm. (b) Dependence of the Q-factor of the bright mode on the separation between slot antennas. (c) Resonance spectra of the single slot antenna. (d) Resonance spectra of the double slot antennas when the Q-factor is maximized. The columns show, from left to right, the calculated spectrum, the SEM image and the measured spectrum.

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3.2 Excitation of the dark mode

Generally, exciting a dark mode is difficult. Due to the anti-symmetric field configuration of a dark mode as shown in Fig. 1(b), an incident plane wave can excite the dark mode component only when the plane wave is incident obliquely to break the symmetry or when the two slots are not identical. Even with these broken symmetries, excitation of a dark mode is usually accompanied by the excitation of a bright mode. In the numerical FDTD calculations, we excited the dark mode alone by placing numerical wave sources inside the waveguide of each slot with phases differing by 180 degree. Here, we present a method of exciting only the dark mode using an incident plane wave. To do so, we locate C-shaped split ring resonators (SRRs) in front of a double slot antenna, rotated ± 45° about the y–axis as shown in Fig. 3(a), generating a phase difference of up to 180 degree [22,23]. These SRR structures generate a phase difference between the waves entering each slot, through a subwavelength scale scattering process. The orientation of the SRRs and double slot structures are fixed specifically as shown in Fig. 3(a). Then, given an incoming y-polarized planewave, the x-component of the transmitted field generated by SRR scattering has a 180 degree phase difference in front of each slot while the y-component of the transmitted field maintains the same phase. This rule of phase change also applies to an incoming x-polarized planewave with x- and y- interchanged. Applying the rule, we can excite the dark and bright modes selectively since only the electric field component $E_x$, orthogonal to the long side of a rectangular slot, contributes to transmission through the slot. Specifically, according to the rule, while a normally incident x-polarized wave generates an in-phase electric field component $E_x$ in front of each slot the y-polarized wave generates an out-of-phase electric field component $E_x$. Consequently, the x- and y-polarized waves excite only the bright and the dark modes respectively. This is demonstrated numerically using the FDTD method by the calculated field maps for in front of and inside each slot, as shown in Fig. 3(b).

 

Fig. 3 (a) To selectively excite the dark and the bright modes of the double slot antennas, arrays of circular SRRs are placed for phase control of the incident electric field. (b) Depending on the polarization of the incident wave, the phase of the $E_x$ field is modulated as shown in the field maps, exciting either the dark or the bright mode. (c) The measured phase difference between slot antennas depending on the excited mode. Separation between slot antennas is 5 mm (black), 3 mm (red), 2 mm (blue). (d) Measured and calculated near field spectra of the dark mode with the near field spectrum of the single slot antenna (green).

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For an experimental demonstration of bright and dark mode excitations, we made a microwave measurement in the frequency range 2.6-3.9 GHz near the slot. Although our scheme works equally well for the slot antennas in the IR region, we have made microwave measurements for ease of fabrication and measurement. Slots are perforated in a stainless steel plate of thickness 1 mm and the same steel plate is used to fabricate the SRRs. The size of a slot is $5\text{mm}\times 50\text{mm}$ and the C-shaped SRRs have outer radii of 14 mm, inner radii of 2.5 mm, and split opens spanning the angular region from 39.5° to 50.5 ^o or 129.5 ^o to 140.5 ^o. The SRRs are arranged periodically in Styrofoam with 30 mm spacings as in Fig. 3(a) and the distance between the plane of the SRR array and the double slot antenna is 30 mm. We measured near fields using a hairpin antenna located at the edge of a slot in order to minimize disturbance of the local electromagnetic fields. The resulting differences between the phases measured at each slot are shown in Fig. 3(c), and confirm the out-of-phase and in-phase nature of the dark and bright modes. We also measured the spectrum of a dark mode excited by a y-polarized incident wave. The spectra measured with each of the three slot separation cases (d = 2,3,5 mm) and their corresponding FDTD numerical results are compared in Fig. 3(d). As expected, the dark mode exhibits a sharp spectral resonance with a highly enhanced Q-factor. In the case of smallest separation, 2 mm, the Q-factor reaches 64, which represents a more than a ten-fold enhancement over the single slot case.

3.3 Radiation characteristics of bright and dark modes

The radiation pattern of light scattered by a thin rectangular slot is known to resemble that of a dipole oriented along the long side of the slot [21]. Thus we may postulate that the radiation pattern of double slots is that of two dipoles which are parallel for the bright mode and anti-parallel for the dark mode. When the dipole at the origin is aligned along the y-direction, in the radiation zone the electromagnetic field of a single dipole $p$ and the radiated power per solid angle are:

 

Fig. 4 (a) Radiation pattern of a parallel dipole pair and the bright mode of coupled slot antennas. (b) Radiation pattern of anti-parallel dipole pair and the dark mode. Radiation pattern of dipole pair with (c) $\pi /2$ and (d) $-\pi /2$ phase difference between dipoles. Corresponding radiation patterns of slot antennas excited under (c) right and (d) left circularly polarized incident light. Three-dimensional maps are the radiation patterns of dipole pairs when the dipoles are separated in the x-direction. Two-dimensional graphs are the calculated radiation patterns of the slot antennas.

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Radiation patterns from a dipole pair can be further controlled by changing the relative phase between the two dipoles. In particular, if the phase difference is $\pm \pi /2$, the magnetic field and radiated power become

To experimentally confirm the radiation characteristics of bright and dark modes, we measured radiated field intensities away from the double slot and the resulting intensity maps of the bright and the dark modes are shown in Fig. 5(a) and 5(b). The first and second columns are numerically calculated intensity maps corresponding to xz- and xy-plane cross-cuts and the third column are the measured intensities in the xy-plane cross-cuts. Dotted lines in the first column indicate the position of the xy-cut in the second column, while dotted squares in the second column indicate the regions of the measured positions in the third column. Measurements are made with 5 mm spacing on the xy-plane, 130 mm away from the slots along the z-axis. In the figures the grey regions indicate slot antennas and circular SRRs. The radiation patterns measured in the bright and the dark modes confirm that they are indeed patterns of dipolar and quadrupolar radiation.

 

Fig. 5 Electric intensity maps of (a) the bright mode and (b) the dark mode. The first and second columns are calculated maps, and the last is the measurements taken during the microwave experiment. The intensity on the xy-plane is calculated and measured 130 mm away from slots in z-axis.

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3. Summary

We have the presented spectral and radiation characteristics of examples of double slot coupled resonators. Notably, the achievement of the excitation of a dark mode alone and resonance of an unusually high Q-factor have been demonstrated both theoretically and experimentally. We have also shown that various novel spectra and radiation pattern features arise depending on the polarization state of the incident wave. All these features suggest that coupled antennas, with associated additional modulating structures, can be versatile unit elements for metasurfaces and metamaterials leading to potential new applications of nanonantenna.