1. Introduction

Electromagnetically-induced transparency (EIT) [1], as a result of a quantum destructive interference between two pathways induced by a coupling field, can make an absorptive medium transparent to a resonant probe field. Generally, this is realized in three-level atomic system, where the probe and the coupling light must satisfy two-photon resonance, and the Rabi frequency of the coupling light must exceed the effective dephasing rate of medium. Recently, much attention has been paid to the classical analogue of EIT because of the underlying physics and the advantages in applications, such as room-temperature operation, wide bandwidth, and integration with nanophotonic circuit. The analogue of EIT is realized in classical linear systems, based on a variety of coupling mechanism, such as mechanical oscillators, RLC circuits [2], optical resonators [3–6], optical antennas [7–11], trapped-mode patterns [12–14], split-ring resonators [15–18], and array of metallic nanoparticles [19]. The scheme, first proposed by Zhang et al. [7], is of great interest for applications of EIT-like effect in nanoplasmonic circuit at optical frequencies, where the coupling of a “bright” and a “dark” modes, arising from surface plasmon polaritons (SPPs), makes the otherwise opaque metamaterial transparent. Since the coupling is induced inherently by the near-field interaction, a single probe field is needed, without the involvement of a pump field as in atom system, which is the case in the other classical schemes as well. Obviously, the advantage is that the complexity of the experimental realization can be reduced considerably, while the disadvantage is that the tunability is lost inevitably. It means that the structure parameters must be varied if we would like to tune the optical response (i.e., the passive manipulation). However, this is difficult to change the geometrical size of elements after fabrication. Therefore, the active manipulation over subwavelength optical fields is of great importance for optical communication, sensing, and quantum information technology [20–22].

In this work, we analyze the underlying physics of the plasmonic EIT in metamaterials, which helps us to specifically understand the coupling process of the bright and the dark modes occurring in the nanoscale volume. Based on it, we propose an active plasmonic switching without using coupling/control fields required in the conventional EIT scheme by simply adjusting the incident angle of the optical field. The tunability of plasmonic EIT in this scheme is exhibited sufficiently. Although the mechanical control (i.e., adjusting the incident angle) has the disadvantage for high-speed applications, it paves the way for active manipulation of plasmonic EIT and benefits the clarification of its mechanism.

2. Scheme of active manipulation

A typical unit cell comprises a single metal strip (i.e., working as an optical dipole antenna) and a parallel metal pair. The former is excited as the “bright mode” and the latter works as the “dark mode” [7, 8], depending on how strong an incident light from free space can be coupled into the plasmonic mode [23, 24]. The bright mode is based on the strong coupling between the external plane wave and the SPPs; the dark mode is formed by the near-field interaction with the fields of the bright mode, since it cannot be excited by the external field at normal incidence [25] because of its vanishing dipole moment and/or bianisotropy [26].

Based on this typical unit cell, one more bright element is introduced to form a symmetric structure. The schematic of the unit cell is illustrated in Fig. 1, which consists of two bright elements (the metal strips distributed on the both sides) and one dark element (the parallel metal pair in the middle). An optical field is incident within the yz plane and its electric component is along the x axis. The geometrical parameters, W ₁, L ₁, W ₂, L ₂, s and d_c, are taken to be 50, 124, 30, 114, 80 and 18 nm, respectively. The thickness of each metal strip, t_m, is 20 nm. The permittivity of metal, ε_m, is modeled as silver using the Drude formula, where the plasma frequency and the collision frequency are ω_p = 1.366 × 10¹⁶ rad/s and ν_c = 3.07 × 10¹³ Hz, respectively. The unit cells are arranged with the periods of 280 and 420 nm in the x and the y directions, respectively. These periodicities guarantee that only the zeroth-order transmittance is investigated free from diffraction at the frequencies of interest when the incident angle is smaller than 10.98° according to the grating equation [27, 28]. The numerical simulations were performed using the finite-integration package, CST Microwave Studio.

 

Fig. 1 Schematic of the unit cell consisting of two bright elements (metal strips distributed on the both sides) and a dark element (two parallel strips in the middle), whose geometrical parameters, W ₁, L ₁, W ₂, L ₂, s and d_c, are taken to be 50, 124, 30, 114, 80 and 18 nm, respectively. The thickness of each metal strip, t_m, is 20 nm. An optical field is incident within the yz plane and its electric component is along the x axis.

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Most reports indicated that asymmetry is crucial to plasmonic EIT [8, 17] when we restrict to the interaction of fundamental plasmonic modes [29] or single dark mode [10, 11], because the asymmetry allows the excitation of the otherwise forbidden dark mode. The structure itself that we present here is free from the asymmetry at normal incidence. Nevertheless, as far as the whole problem is concerned, the symmetry of incident light must be taken into account as well as the structure. Therefore, the asymmetry can be introduced at oblique incidence without the variation of the structure, which brings about the tunability of plasmonic EIT in transition from the symmetry to the asymmetry.

3. Results and discussion

The metal strips, parallel to the electric component of incidence, can be electrically polarized and form the bright plasmonic modes, where a very strong charge displacement is formed and the localized electric and magnetic fields are induced. This magnetic field is possible to activate the neighboring dark mode. Of course, there is another possibility that the excitation of the dark mode results from both the electric and magnetic fields. Since the bright mode is considered as an electric dipole [30–32], we suppose that the magnetic component of the dipole field is normal to the plane of the parallel metal pair so as to excite the dark mode based on the magnetic plasmon resonance (MPR) [33, 34] regardless of its small amplitude [35]. For instance, it has theorectically proved that plasmonic EIT is accessible even in the symmetric structures based on the second-order MPR [29]. Here, we would not like to exclude the contribution from the electric component of the dipole field, since it might points, e.g., right at the position of the upper one in the parallel metal pair while it points left at the position of the lower one. This distribution of the electric field might have the same effects on the parallel metal pair as the MPR. The reason that we concentrate on MPR is that the electric component of the plane wave has taken part in the excitation of the bright mode, while the magnetic component is still untapped. In order to seek an additional degree of freedom to manipulate the optical response, the utilization of MPR is a better candidate. Finally, the active manipulation by varying the angle of incidence, illustrated in the following, convinces us that MPR plays a crucial role in plasmonic EIT.

Figure 2(a) shows the transmittance spectrum at normal incidence, where the EIT-like effect vanishes and only a dip can be seen. Explicitly, if the second-order MPR [29] or the double dark resonance [10, 11] does not take part in the coupling process, the preservation of symmetry, both the structure and the incident light, definitely gives rise to the disappearance of plasmonic EIT, which was observed in experiment as well [8]. Based on the picture of MPR, it can be explained that the magnetic field of the bright mode on the left has the same amplitude, but the opposite direction, with respect to that on the right and these magnetic fields induce the opposite circular currents, thereby canceling out each other. As a result, the MPR is suppressed and the dark mode is inactivated, as shown in Fig. 2(b), where the electromagnetic fields are concentrated on the both wings and vanishes in the middle (i.e., only two bright modes are activated). Consequently, the transmission at the dip is considerably suppressed becase the metal strips have a larger effective electric-dipole moment than that of the single strip [36].

 

Fig. 2 (a) Transmittance spectrum at normal incidence. (b) Amplitude distribution of the magnetic field at the transmission dip.

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In order to achieve the plasmonic EIT, the circular currents must be reestablished, which depends on the introduction of imbalance between the two magnetic fields from the both wings. However, since the metal strips on the both wings are equivalently excited by the incident light, an additional influence is required. This is the vertical magnetic component H _0z of plane wave and it is available at oblique incidence. Under such circumstances, although the symmetry of the structure itself is still preserved, that of the whole system is broken, allowing for both the structure and the incidence. As a result, the whole problem turns into the asymmetry from the symmetry. Figure 3(a) shows the angle-dependent spectra of transmittance, where the incident angle is varied from 2 to 10° with an increment of 2°. It is found that the plasmonic EIT becomes more evident with the increase of angle because of the excitation of MPR, arising from the imbalance of the dipole fields on the both wings, as shown in Fig. 3(b). Here, the enhanced magnetic field is generated near the parallel metal pair. Besides, one bright mode is effectively suppressed and the magnetic field is not much enhanced around it, while the other is not. Consequently, the otherwise opaque metamaterial becomes somewhat transparent due to this partial suppression of the bright modes. The moderate transmittance, compared with that reported in Ref. [7], is ascribed as follows: (1) a relatively small net magnetic moment due to two partially counteractive magnetic moments; (2) a coupling between the net magnetic moment and the magnetic component of the incident light, which does not occur in Ref. [7] because of the orthogonality between the magnetic moment and the external magnetic component. In our scheme, the numerical calculations show that the transmittance increases to be 34% at an incident angle of 10° with a large quality factor of 77, as shown in Fig. 3(c). The quality factor is evaluated by Q = f ₀/Δf, where f ₀ is the frequency of transmission peak and Δf is the full width at half maximum. Despite of the relatively moderate transmittance, it still brings about a large modulation of transmittance, the maximum ΔT/T ≃ 96%, at a frequency of 456.9 THz through mechanical tuning, accompanied by slow-light effect, as shown in Fig. 3(d). The group index is roughly estimated according to n_g = c ₀ dk/dω = (c ₀/t_m)dϕ/dω [37], where c ₀ is the speed of light in vacuum, ϕ is the phase, and ω is the angular frequency.

 

Fig. 3 (a) Transmittance spectra at various angles from 2 to 10° with an increment of 2° in the yz plane, as shown in the inset on the right. The region of the transmittance around the frequencies of plasmonic EIT is zoomed in, as illustrated in the inset on the left. (b) Amplitude distribution of the magnetic field at the frequency of plasmonic EIT of 456.9 THz and at an incident angle of 10°. Here the component of the magnetic field normal to the plane of the symmetric structure can be estimated by |H_0z| = |H₀|sinθ_i ≃ 4.7×10⁻⁴ A/m provided that the electric field of incidence |E₀| = 1 V/m and the incident angle θ_i = 10°. (c) Dependence of both the amplitude and the quality factor of the transmission peak on the angles of incidence from 4 to 10°. (d) Phase change and group index across the symmetric structure as a function of frequency at an incident angle of 10°.

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4. Conclusions

We propose an active control of plasmonic EIT in the metamaterial based on the excitation of MPR such that the role of MPR is further revealed. The bright and the dark modes can be either coupled or uncoupled, depending on the angle of incidence in the symmetric structure. Therefore, the active control of plasmonic EIT can be implemented even using only one optical field by controlling the incident angle. Even though the mechanical control has the disadvantage for high-speed applications, it paves the way for active manipulation of plasmonic EIT and benefits the clarification of its mechanism.