Electromagnetically-induced transparency (EIT) results from a quantum interference effect induced by the interaction between laser beams and atom ensembles under a two-photon resonance condition [1, 2, 3]. This effect underlies many interesting ideas such as the transfer of quantum correlations [4], nonlinear optical processes at low light levels, and ultraslow light propagation [5, 6]. Compared to EIT in atomic systems, plasmonic EIT in metamaterials has the advantages of room-temperature manipulability, large bandwidth, and the ability to integrate with nanoplasmonic circuits. A great deal of attention has therefore been paid to the classical analogue of EIT based on mechanical oscillators, RLC circuits [7], optical resonators [8, 9, 10, 11, 12], optical dipole antennas [13, 14, 15, 16, 17], trapped-mode patterns [18, 19], split-ring resonators [20, 21, 22], and array of metallic nanoparticles [23].

Thanks to the merging of plasmonics and metamaterials, it is of great perspective to manipulate light by employing metal nanostructures in a unique way [13, 24]. The involvement of optical dipole antennas in the classical analogue of EIT is a specific example of this merging. Zhang et al. [13] first proposed the plasmon-induced transparency rendered by the coupling of bright and dark plasmonic modes, resembling a three-level atomic system. This was subsequently developed as a tripod system manifesting the classical analogue of quantum coherence swapping [14, 17]. Recently, Liu et al. [16] experimentally demonstrated plasmonic EIT at the Drude damping limit using a stacked optical metamaterial composed of an upper gold strip and a lower pair of gold strips with a dielectric spacer. It was found that the asymmetry is a prerequisite for the plasmonic EIT; in its absence, only a single absorption peak is visible, without any sign of an EIT-like effect, because of no evident coupling between bright and dark modes. Most of researchers also hold this view [16, 22, 25] explicitly or implicitly, since the dark mode is unlikely to be excited and the coupling is unavailable if the symmetry of the unit cell is unbroken.

In this work, we propose a scheme for the generation of plasmonic EIT even in symmetric structures. This scheme depends on a minor modification of the symmetric structure described in Ref. 16, in which EIT-like effect cannot be achieved. The underlying reasons are also elucidated in detail, based on the picture of magnetic plasmon resonance (MPR)-mediated plasmonic EIT.

 

Fig. 1. (a) Three-dimensional and (b) two-dimensional views of the unit cell. The geometric parameters are w = 80 nm, d = 100 nm, l ₁ = 346 nm, l ₂ = 790 nm, and s = 0. The vertical distance between the upper gold strip and the lower pair of gold strips is denoted by h and the thickness t of each strip is 40 nm. The periodicity is 870 nm in both the x and y directions. The incident plane waves is irradiated along the z direction, and its electric component, E, is parallel to the x direction.

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As described in Ref. 16 and illustrated in Fig. 1, each unit cell consists of an upper gold strip as a bright mode, a pair of lower gold strips as a dark mode, and a dielectric spacer. In particular, the parameter s represents lateral displacement; symmetric and asymmetric configurations therefore have s = 0 and s ≠ 0, respectively. Plasmonic EIT originates from the coupling of bright and dark modes when the symmetry is broken [16, 22, 29]. In essence, the former serves as an optical dipole antenna, and the latter as a quadrupole antenna, when the cell is illuminated perpendicularly and the electric field of the light is parallel to the upper strip. Here we do not restrict ourselves to the fundamental modes, such as the dipole and the quadrupole modes, and the higher order modes are also considered; in addition, our study is concentrated on the symmetric structure, and similar geometric parameters are used with two major differences. One difference from that in Ref. 16 is that, for simplicity, the dielectric spacer and the substrate are not taken into account; i.e., they are treated as air, which does not affect the EIT-like feature except for a blueshift and does not lead to any loss of generality in the discussion that follows. The other difference is that the pair of lower gold strips is elongated to 790 nm, about twice the length of the strips described in Ref. 16. The reason for this elongation is that the higher order modes could be excited as well as the fundamental mode. The numerical calculations are carried out using the finite integration package (CST Microwave Studio). The permittivity of gold is described by the Drude model, with a plasmon frequency ω_p of 2π × 2.175 × 10¹⁵ rad/s and a collision frequency ν_c of 1.225 × 10¹⁴ Hz, three times larger than that of bulk gold [16, 26, 27].

 

Fig. 2. (color online) (a) Transmission and (b) absorption spectra with various vertical distances h. The black curves in (a) and (b) are obtained with the same parameters as in Fig. 1 (h = 70 nm), except l ₂ = 315 nm.

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Figure 2 displays the simulated transmission and absorption spectra for symmetric structures with varying vertical distances h. The absorption spectra are calculated using formula A = 1 − T − R, where T and R denote the transmission and the reflection, respectively. Astonishingly, the EIT-like feature completely disappears when the length of the lower pair is relatively small, l ₂ = 315 nm, without varying the other parameters. In contrast, this feature clearly manifests itself at approximately 240 THz when the lower pair is elongated to 790 nm. Obviously, the question of whether the plasmonic EIT can be excited depends on the length of the lower pair, as well as the structural asymmetry. There is a seeming inconsistency with the conclusion in Ref. 16, in which it was believed that the coupling wanes due to the structural symmetry. To answer this question then, the nature of the coupling must be deciphered. Lu et al. [15] provided a physical picture for plasmonic EIT in which the phenomenon is considered as a result of plasmonic coherent interference in the near-field zone based on the excitation of surface plasmon polaritons (SPPs) and MPR. The former occurs on the upper strip and behaves as an optical dipole antenna [28], while the latter is induced by the magnetic component of the dipole fields. According to this picture, the disappearance of the EIT-like effect can be explained by the fact that, if the structure is symmetric, the magnetic components have the same magnitudes but in opposite directions on both sides of the upper strip, and thus the induced currents cancel each other out. Despite their opposite directions, however, in the absence of symmetry, the two magnetic components cannot be equal and thus produce a current or quadrupole in the lower pair. This means that the pivot of the plasmonic EIT is determined by the excitement of the lower pair (i.e., dark mode).

The point of importance is that the quadrupole cannot in general be excited by normal incidence because of its vanishing dipole moment (i.e., because it is dark). Thus, in order for the quadrupole to become activated, highly angled illumination must be employed [13, 29]. As shown in Fig. 3, two dark modes are magnetically excited at 120 and 240 THz when the plane wave irradiates the elongated pair (790 nm) along the -y direction, without broken symmetry in either the incident field or the structure itself. The latter frequency is twice as high as the former, which can be ascribed to be second-order MPR [30]. The second resonant peak at 240 THz coincided with the plasmonic EIT peak, as shown in Fig. 2. Therefore, on the basis of this physical picture, it can be concluded that the lower pair is excited at 240 THz.

 

Fig. 3. (a) Schematics for the incident plane wave on the lower pair of gold strips (l₂ = 790 nm), in which the wave is parallel to the strips and its electric field along the x direction. The arrow is an E_y probe placed 10 nm away from the center of the end facet. (b) Spectral response of the E_y probe.

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Taking a closer look at the z-component distribution of the magnetic field at the frequency of plasmonic EIT, it is unambiguous that the magnetic fields point in and out of xy plane on both sides of the upper strip [see Fig. 4(a)]. Since there is the fact that the magnetic field is π/2 out of phase with the electric field [15, 31], the y-component distribution of the electric field is illustrated in Fig. 4(b) with a π/2 phase difference compared to the magnetic field; here, the dark mode is activated and two antisymmetric quadrupolar resonances are induced on each side of the upper strip by the two antisymmetric magnetic fields of the dipole fields. Due to the excitations of the two circular currents, plasmonic EIT is also achieved in the symmetric structures. Because the lower pair is elongated, more room is provided to accommodate the two circular currents, thereby avoiding cancelation. Consequently, the second-order MPR is excited so that plasmonic EIT can appear even in the symmetric structures (see Fig. 2). This possibility was previously not even considered as a candidate for plasmonic EIT.

Furthermore, as shown in Fig. 3, the intensity of the second-order MPR is larger than that of the fundamental mode, broadening the plasmonic EIT peak compared to the peak described in Ref. 16. It is generally believed that strong coupling induces peak broadening [20, 21]. On the other hand, the strength of the coupling can be tuned by adjusting the vertical distance h. The larger is the vertical distance, the smaller is the peak width; of course, this narrowing reduces transmission, as shown in Fig. 2.

In conclusion, it is possible for plasmonic EIT to be realized even in symmetric structures depending on the possibility of the high-order MPR to be excited. This clarifies the essence of coupling and strongly validates the physical picture of MPR-mediated plasmonic EIT. Essentially, MPR, besides symmetry or asymmetry, is the crux of the plasmonic analogue of EIT. At the time of this writing, corresponding modifications were not carried out on the upper strip because the frequency scaling law for SPPs is more complicated than that for MPR arising from phase shifts [32]. The point of great interest is that the optical response is invariant, whereas the restrictions on size are partially relaxed, making fabrication easier.

 

Fig. 4. (a) The z-component distribution of the magnetic field at a frequency of plasmonic EIT with h = 70 nm, where the phase is 150°. (b) The y-component distribution of the electric field at the same frequency, but with a phase of 60°.

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Yuehui Lu would like to express his gratitude to Nguyen Thanh Tung for valuable discussions. This work was supported by the MEST/NRF through the Quantum Photonic Science Research Center, Korea.