1. Introduction

Electromagnetically induced transparency (EIT) in atomic medium is a fascinating quantum interference effect, giving rise to an ultra-narrow and steep transparency window within a broad frequency band [1,2]. This intriguing phenomenon promises a serious of applications in slow light [3–12], optical nonlinearity [13,14], and switches [15,16]. Unfortunately, the EIT effect in application is limited by harsh conditions which are usually desired to maintain quantum states coherence, such as low temperature, strong gas lasers and macroscopic devices [1,17]. These requirements are unsuitable for the implementation of chip-scale. To overcome this issue, the analogues of EIT realized in classical optical systems have attracted wide attention, such as photonic crystals [18,19], metamaterials [20,21], and waveguide-cavity configurations [22]. All-optical analogue of EIT photonic crystals (PhCs) with ultra-high quality factor demonstrates group time delay up to 20 ps in experiment due to its low intrinsic loss in the silicon-based cavity [18]. Although the electromagnetic modes can be confined within the wavelength scale, PhCs are still far distant to the goal for controlling and manipulating light in nanoscale.

Surface plasmon polaritons (SPPs), collective electronic excitation, are strongly confined in the metal-dielectric interface. These evanescent waves are usually adopted in subwavelength optical devices due to their wavelengths being far below classical diffraction limit [23,24]. Plasmonically induced transparency (PIT) can be realized in metallic structures via destructive interference between bright and dark plasmonic modes or phase-coupling between two space-separated resonators [25]. Although plasmonic nanostructures possess the unique characteristics of giant near-field enhancement and ultra-compact configuration, the moderate slow light is obtained due to the intrinsic losses of the metal [26]. Usually, the group refractive index is less than 10² [17]. Consequently, the realizations of enhanced nonlinearities, enlarged spectral sensitivity, and increased phase shifts can be achieved by reducing the bandwidth to a small value [27–31]. Recently, it is a feasible way to compensate for the intrinsic Ohmic loss and pave the way to metallic metamaterials by resorting to the gain medium [32–34]. For example, Ambati et al. demonstrated a experiment of stimulated emission of SPPs with erbium doped glass as a gain-assisted medium in 2008 [35]. After that, Dong et al. demonstrated a loss-compensated resonance enhancement by using the gain-assisted PIT in a metallic metamaterial, and the Q-factor is up to 666.67 [36]. On the other hand, a two-dimensional material, graphene, demonstrates many excellent properties such as strong confinement, low resistive damping, and optical transparency, which gain wide attention from researchers in devise fields [37–40]. Up to now, a large number of graphene plasmonic metamaterials have been demonstrated to achieve the PIT effect [41]. Some nanostructured graphene are used to achieve optical devices with high slow light, as the intrinsic loss of graphene is much smaller than that of metallic structures in the mid-infrared regimes. For example, Gao et al. proposed a terahertz graphene metasurface with a group index as high as 935 in the four graphene ribbons [42]. In 2021, Cui et al. realized dual-PIT effect with a higher group index of ∼2553 in a hexagonal graphene coupled metasurface [5]. Then, can we combine gain material with graphene to obtain higher slow light?

In this paper, we demonstrate a graphene metamaterial possessing a sharp EIT-like window with a Q-factor of ∼10⁴. The high-Q transparency window is accomplished by employing the high-order mode to minimize the radiative losses, and gain medium to partially compensate the intrinsic Ohmic losses. In addition, the tunability of EIT-like window can be investigated by varying the gain level in dielectric layer and the Fermi energy level of GNR. The transfer matrix method and temporal coupled-mode theory are adopted to quantitatively describe the observed EIT-like responses, and these responses are in good with the finite-difference time-domain simulations.

2. Structure and method

The designed graphene metamaterials schematically show in Fig. 1, which consist of several layers of vertically separated graphene nanoribbon (GNR) array embedded into gain-assisted dielectric. The surface conductivity of graphene is described by σ = (e²/πħ²) iE_Fj/ (ω + iτ ⁻¹), where τ = μE_Fj / eν_F² is the intrinsic inelastic lifetime (μ = 2.5 × 10⁴ cm² V⁻¹s⁻¹, and ν_F = 10⁶ ms⁻¹, see Ref. [29,43]). In the FDTD simulation, the single-layer graphene is modeled as an effective dielectric layer with a thickness (H) of 1 nm with a dielectric constant ε_r = 1 + iσ/(ωε₀H). The dielectric constant of single-layer graphene can also be dynamically adjusted by changing its Fermi energy E_Fj through electrostatic doping [44]. Here, according to a direct experimental evidence of stimulated emission of surface plasmon polaritons (SPPs) with gain medium [35], the Er³⁺ ions are doped in the dielectric spacer as a gain-assisted medium when the gain is needed, and the complex permittivity of this medium is ε = ε` + iε``. The real part of the permittivity is taken to be ε` = 2 and the gain is introduced through the imaginary part ε``. Since the gain as a function of wavelength can be approximately depicted as a Gaussian profile, the maximum gain coefficient α₀ is located at 2800 nm wavelength and the gain bandwidth is γ₀ = 0.03ω₀ [45].

 

Fig. 1. (a) The schematic illustration of the graphene metamaterials consisting of several layers of longitudinally separated graphene nanoribbon (GNR) array embedded into gain-assisted dielectric. W_j and h_j represent the width of GNRj (j = 1,2,…) and the separation between GNRj and GNRj+1, respectively. P is the period. (b) Cross section of the proposed one unit cell with j-layer GNRs in the metamaterials. The dashed lines are reference planes of GNRs.

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As shown schematically in Fig. 1(a), when a linearly polarized light propagates along the + y-axis direction, the plasmon resonances at a specific wavelength will be excited in GNRs. For a unit cell of graphene metamaterials in Fig. 1(b), the E^(j)_d,in and E^(j)_d,out (j = 1, 2, …) are represent the amplitudes of the incident and outgoing lights of the GNRs. At GRAj (“j” represents the jth nanoribbon), the subscripts d = “+” or “−” indicate that the wave propagates in the orientations of the + y-axis and the −y-axis and the incident and outgoing waves are denoted by the subscripts “in” and “out”, respectively. When the incident light with a frequency ω_0j is inputted only from the upper (E^(j)_-,in = 0), the temporal normalized mode amplitude of j-th graphene nanoribbon a_j can be written as

Combined with the above Eqs. (1), (2), and (3), the complex transmission and reflection coefficients t_j and r_j of the single-layer GNRs structure can be described as

As shown in Fig. 1(b), the transport characteristics of the proposed F-P microcavity with two spatially separated GNRs (i.e., GRA1 and GRA2) can be described by the transfer matrix model. The transfer equation can be expressed as

Matrice S_j represents the propagate wave in the h_j-thickness dielectric interlayer and M_j represents in the graphene interface. These matrices are given by

Here, β(ω) stands for the propagation constant. When the gain medium is introduced, the imaginary part of β(ω) will be a negative value (i.e., β = β` + iβ`` and β``< 0). Considering the dual-layer GNRs system with the dielectric interlayer, the transmission can be obtained by

Then, Eq. (4) is substituted into Eq. (7), we have,

In addition, the numerical transmission spectra can be obtained by using the two dimensional (2D) finite-difference time-domain (FDTD) method. And perfectly matched layers (PML) boundaries are used in the y-axis directions, as well as periodic boundary conditions (P = 283.6 nm) are set at x-axis directions. The thickness of single-layer GNR is taken to 1 nm, and the maximum mesh size is set as Δx = 0.4 nm and Δy = 0.2 nm, which ensure the convergence of numerical results.

3. Results and discussions

Figure 2(a) demonstrates the transmission spectra of the dual-layer GNRs system at δE_F = 0.3 meV (i.e., E_F2 - E_F1 = 0.3 meV, E_F1 and E_F2 represent the Fermi energy levels of two layer ribbons) for different gain coefficients α₀. In the numerical calculations, at the expance of efficient pump power, the imaginary part of permittivity ε`` will be a negative value. When the pump power is increased, the value of -ε`` will rise and the medium will eventually show the significant gain effect. Here, the gain medium is embedded into the graphene nanoribbon array, for which the active characteristic can be described by the gain coefficient α₀ = −(2π/λ)Im((ε` + iε``)^1/2) [36,46–48]. It is clearly shows that compare with the lower value of α₀, the PIT window is more prominent, and the bandwidth is narrower with the higher value of α₀ because increasing the value of α₀ decreases the intrinsic Ohmic loss. Noteworthily, a gain level within 11.54-13.34 cm⁻¹ can realize a sufficiently large transmittance in the transparency window. In Fig. 2(b), the values of radiative and intrinsic damping rates related to the graphene material are obtained by theoretical fittings. The radiative damping rates κ_e1 and κ_e2 do not vary significantly with increasing α₀, whereas the damping rate of the intrinsic loss κ_o1 and κ_o2 decrease markedly from 8.98 cm⁻¹. In addition, due to the compensation of the gain medium for the intrinsic losses, the damping rate is gradually dominated by the radiative damping. On the other hand, with the increase of α₀, the value of intrinsic loss κ_o1 and κ_o2 are getting closer and the value of intrinsic loss κ_o1 is always smaller than κ_o2 because the Fermi energy level of GNR1 is smaller than GNR2.

 

Fig. 2. (a) Transmission spectra of the dual-layer GNRs system based on theoretical fittings (solid lines) and numerical calculations (scattered open dots) for different gain coefficients α₀. The other structural parameters are h₁ = 1.98 µm, W₁ = W₂ = 141.8 nm, and the Fermi energy levels (E_F1 and E_F2) of the dual-layer GNRs are set as 799.85 and 800.15 meV. (b) Damping rates κ_o1(2) and κ_e1(2) are the parameters used in the transmission fitting of the graphene metamaterial under different gain coefficients α₀.

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As shown in Fig. 3(a), under the gain level α₀ = 12.83 cm⁻¹, the tunability of EIT-like response can be investigated by adjusting the δE_F, which can be realized by alkaline surface doping in experiment. A trade-off between peak transmission and the full width at half maximum (FWHM) at an EIT-like window is apparent for the dual-layer GNRs system. The transmission spectrum has a small split due to the absence of δE_F in an almost lossless environment. As δE_F gradually increases from 0.2 meV to 0.5 meV, the transmission undergoes a large modulation, with an increase in the transmission peak. Ultimately, an EIT-like window with a transmission amplitude of 77% is obtained at 2800 nm when δE_F reaches 0.5 meV. Significantly reducing the bandwidth of the transparency window is valuable for slow light applications. Because sufficient gain level is chosen in the system, the bandwidth of EIT-like window remains narrow and the Q-factor originally 5.9×10³ at δE_F = 0.5 meV, reaches the maximum value of 2×10⁴. To demonstrate the physical picture of the dual-layer GNRs system, we plot the x-component of the electric field distribution (i.e., E_x) with δE_F = 0.3 meV in Figs. 3(b)–3(d). When the incoming lights pass through the system, as depicted in Figs. 3(b) and 3(d), strong higher-order resonances indicate the effective excitation of surface plasmon in single GNR at the transmission dips λ₁ and λ₂. As illustrated in Fig. 3(c), both two-layer GNRs are excited at the central wavelength λ₁₂ of the transmission window, leading to Fabry-Perot (F-P) resonance between the two nanoribbons. According to the F-P model, the transparency peak with the wavelength detuning δλ = |λ₂ - λ₁| appears at the resonance wavelength λ₁₂ = (λ₁+λ₂)/2 when the phase difference β`h₁ equals nπ (n = 1, 2, 3…..). In our investigation, the wavelengths of transmission dips are λ₁ = 2799.71 nm and λ₂ = 2800.12 nm, and the resonance wavelength λ₁₂ =_{2799.92 nm.}

 

Fig. 3. (a) Transmission spectra of the dual-layer GNRs system by adjusting the difference of Fermi energy level (i.e., δE_F). δE_F = 0 meV, δE_F = 0.2 meV, δE_F = 0.3 meV, δE_F = 0.4 meV, and δE_F = 0.5 meV. The red solid curves are the theoretical fitting results and the blue spheres are the simulation results calculated by FDTD method. The Fermi energy levels (E_F1 and E_F2) of two-layer GNRs are set as 800 and 800 meV, 799.9 and 800.1 meV, 799.85 and 800.15 meV, 799.8 and 800.2 meV, 799.75 and 800.25 meV. And the other parameters are h₁ = 1.98 µm and W₁ = W₂ = 141.8 nm. (b)-(d) Numerical field distributions (E_x) with the incident wavelengths at λ₁ = 2799.71 nm, λ₁₂ = 2799.92 nm, and λ₂ = 2800.12 nm. The inset (black dash squares) shows the field distribution around the 2nd-layer GNR.

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It is well known that the EIT-like effect also supports slow light in plasmonic system. The slow-light effect can be described by group index n_g, which is expressed as

Here, v_g is the group velocity in the system. τ_g and φ(ω) stand for the group delay time and transmission phase shift. L is the length of the system. Figure 4 shows group indices are extracted from transmission spectra. As shown in Fig. 4(a), the slow light in the transparency peak is numerically investigated with E_F1 = 800.1 meV, E_F2 = 799.9 meV, W₁ = W₂ = 141.8 nm, and h₁ = 1.98 µm. It indicates that the decreasing phase is steepest at the position of the transparency peak. In Eq. (9), the value n_g is determined by the phase slope. This system exhibits a maximum group index over 10⁴ around the transparency-peak wavelength, as shown in Fig. 4(a). As δE_F varies, it can be predicted from Fig. 4(b) that the maximum group index will occur near δE_F = 0.15 meV. And the maximum value of n_g is about 3.4×10⁴, which corresponding optical delay of the transparency peak resonance reaches ∼230 ps.

 

Fig. 4. (a) Transmission phase shifts (black dashed line) and group indices (blue solid line) in the system with h₁ = 1.98 µm. E_F1 = 799.9 meV, E_F2 = 800.1 meV and W₁ = W₂ = 141.8 nm. (b) Group indices are extracted from transmission spectra for different δE_F.

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As presented in Fig. 5(a), a periodic relation between the optical delay of transmission peak and vertical separation is theoretically calculated. The maximum optical delay is about 180 ps can be obtained at vertical separations equal to 0.99 µm and 1.98 µm, which is agree well with the consequence exhibited by the spectrum. In addition, the evolution of the transmission spectrum with separation h is as detailed in Fig. 5(b). The result shows that the transparency peaks obviously display and redshift around h₁ = 0.99 µm and 1.98 µm. The periodic evolution of transmission characteristics can be explained by the F-P model. Finally, to verify the F-P model can be extended to multi-layer GNRs system, Fig. 6 reveals the transmission properties in triple-layer GNRs system with different vertical separations h_j (j = 1, 2). When j = 3, the transmission efficiency can be obtained by

 

Fig. 5. (a) Variation of optical delay of transmission peak with the separation h changed. (b) Evolution of transmission spectra as a function of separation h, and the other parameters are E_F1 = 799.9 meV, E_F2 = 800.1 meV, and W₁ = W₂ = 141.8 nm.

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Fig. 6. (a) Simulation (blue spheres) and theoretical (red solid curve) transmission spectra in the triple-layer-GNRs system with W₁ = W₂ = W₃ = 141.8 nm, h₁ = 1.98 µm, h₂ = 1.97 µm, and δE_F = 0.3 meV. (b) Optical delay (blue line) in this system. (c) and (d) Numerical field distributions (E_x) at the two transparency-resonance wavelengths: λ₁₁ = 2799.5 nm and λ₂₂ = 2800.11 nm.

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As shown in Fig. 6(a), the theoretical fittings based on Eq. (10) show good agreement with the numerical simulations. In addition, as shown in Fig. 6(b), the maximal optical delay reach to 96 ps and group index up to 1.44×10⁴. In this system, the separation between the 2nd and 3rd-layer GNR can be set as 1979.96 nm and the width of 3rd-layer GNR is W₃ = 141.8 nm. The Fig. 6(a) shows that there are two transparency windows at λ₁₁ = 2799.5 nm and λ₂₂ = 2800.11 nm, and the line and the dot are fit well, namely, the theoretical model is in a good agreement with the simulation results. As shown in Figs. 6(c) and 6(d), similar to a dual-layer GNRs system, the electric field distributions show that two F-P cavities at the wavelength of EIT-like peaks. Thus, it can be conjectured that the multi-layer GNRs system can also be implemented to achieve the graphene-based multiple EIT-like responses.

4. Conclusions

In summary, ultra-narrowband EIT-like effect in a gain-assisted graphene plasmonic metamaterial has been theoretically and numerically investigated. The EIT window can be tuned by varying the gain level in medium layer and altering the Fermi energy level in graphene. Moreover, based on the gain medium and high-order graphene plasmonic resonances, radiative and non-radiative loss can be reduced by the proposed system, and the metamaterials can be used to achieve drastic phase transition and ultra-high group index up to 10⁴. It should be possible, therefore, to achieve efficient slow light effects for better optical buffers and other nonlinear applications.