1 Introduction

Due to electromagnetic coupling between atoms in a narrow transmission window, electromagnetically induced transparency renders an opaque medium transparent [1], which significantly tweaks the dispersion properties of the medium. EIT is an interference phenomenon between the super-radiant and sub-radiant modes, which are equal in strength but opposite in phase. Sub-radiant modes are inherently dipole-inactive and weakly coupled with the incident light. There are two ways to excite EIT: (1) excitation of super- and sub-radiant modes, which require a dipole-active resonator (super-radiant) whose resonance mode directly couples or can easily be excited and transferred to a nearby sub-radiant mode through near-field coupling [2, 3]. Once excited, the sub-radiant mode produces EIT by interacting with the super-radiant mode, which results in high transmission; (2) on the other hand, axial symmetric structures, which are not identical, produce anti-parallel currents due to inductive coupling that leads to EIT windows [4, 5]. EIT has enormous applications in self-enhanced nonlinearity [6], ultraslow group velocity, storage of light, quantum information and communication, classical optics, and photonics [7]. To achieve EIT effect in the conventional materials is very difficult.

Recently, it is observed that metamaterials (MM) can strongly confine electromagnetic fields and show sharp spectral resonance features [8, 9]. Several metamaterials have been demonstrated for EIT effect from microwave to ultraviolet spectral range [3, 10,11,12,13,14,15]. Most of the reported EIT metamaterials work for fixed wavelengths, which significantly limit its applications. MM structure with multiple EIT resonances can offer huge opportunities in multi-wavelength biosensing [16], slow light systems and storage over the multiple frequency bands [17]. To produce multi-band EIT peaks, a bull’s-eye-shaped structure is proposed by Kim et al [4] and a structure made of parallel cut wires of different sizes is proposed by Zhu et al [5]. Moreover, independent tunability of EIT windows further enhanced the prospects of designing tunable, and multi-spectral terahertz (THz) functional devices [17]. Several approaches have been reported to realize tunable multi-spectral EIT windows [18,19,20]. These methods mainly depend on the nonlinear properties of active materials, which inevitably results in low modulation depth and range. Moreover, most of the reported metastructures are polarization sensitive, complex and their spectral windows could not be tuned independently. To avoid any loss of information due to unpredictable change in polarization state of THz waves during its propagation, polarization insensitive EIT metasurface is highly desirable [21].

In this paper, we report a simple method to design a graphene-based, polarization-insensitive, THz metasurface for ‘n’ EIT spectral windows. We report our simulation results up to quint-EIT spectral windows that can be further extended to higher number. Quint-EIT resonances in transmittance spectra are obtained when three concentric CKSSRR are put together. Simulation and analytical results show that the proposed structure can achieve the multi-band slow light effects and possess multi-band EIT-like properties. Independent tunability of spectral windows is obtained by varying the relative dimensions of the resonators, which has potential applications in multi-wavelength selective switching, multi-wavelength biosensing, slow light systems, and light storage over multiple frequency bands. The simulation results are in good agreement with the analytical results. Our research can open new doors to a quick and accurate fabrication of the multi-band slow light devices.

Fig. 1
figure 1

Schematic of designed THz metamaterial structure; b the zoom view of the unit cell along with structural parameters; c dual-resonators as an example of axially symmetric multi-resonator metamaterial for multi-band EIT. The opposite direction of induced currents (dotted lines) indicates the formation of transparency windows

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2 Structural parameter and unit cell design

We design a metamaterial unit cell that consists of a metallic CKSSRR made of graphene, surrounded by two similar concentric metallic rings (Fig. 1). Due to negative permittivity of graphene in terahertz (THz) spectral region, each MM unit cell behaves like hyperbolic medium with large anisotropic permittivity [22, 23], which acts as a subwavelength resonator to realize optical cavity for emission enhancement [24]. The metallic patterns are designed on the top of preperm L450 substrate (permittivity = 4.5, loss tangent = 0.0005) with \(10 \upmu\)m thickness. Simulations are carried out using CST microwave studio suite 2018 in frequency domain solver. The moderate mesh accuracy with   \(\lambda /10\) (where \(\lambda\) is the wavelength of incident light) is adopted to make good trade-off between accuracy, memory requirements, and simulation time. The unit cell boundary conditions are used in x- and y-directions, whereas open add space is applied along the z-direction. Figure 1a shows a schematic illustration of unit cell arrays with an incident plane polarized THz wave on the metasurface along with the transmitted THz wave, which is used to compute the transmittance characteristics. The optimized parameters of metastructure (see Fig. 1b) are: periodicity \((P) = 50 \upmu\)m, the separation between first and second concentric resonators \((s_1) =3.89 \upmu\)m, the separation between second and third concentric resonators \((s_2) = 4.23 \upmu\)m, inner and outer lengths of the resonators (\(l_{in1}=8.48 \upmu \textrm{m}, l_{in2}=18.39 \upmu \textrm{m}, l_{in3}=29.7\mu m, l_{o1}=10.61 \upmu \textrm{m}, l_{o2}=20.51\,m, l_{o3}=31.82 \upmu \textrm{m})\), width of the resonators \((w) =1.06 \upmu \textrm{m}\), thickness of the resonator \((t_r)=0.2 \upmu \textrm{m}\) and thickness of the substrate \((t_s)=10 \upmu \textrm{m}\). Strong electromagnetic resonances are obtained for normally incident THz wave [25].

3 Quint band EIT window development

The unit cell of multi-band EIT is shown in Fig. 1b. It consists of three axially symmetric CKSSRR with different structural parameters in the same plane. Due to different structural parameters, they possess different resonance frequencies. The incident THz wave is perpendicular to the plane of the metasurface with an electric field along the x-direction. Under the excitation of a linearly polarized incident THz wave, CKSSRR behaves as a dipole that induced electrical currents along the polarization direction. Since the two CKSSRRs as shown in Fig. 1c are not identical, the outer resonator will experience a stronger coupling to the incident excitation compared with the inner resonator. Hence, these two resonators will produce anti-parallel currents due to inductive coupling between the two resonators. Such a current distribution leads to destructive interference of scattering fields and induces a pronounced transparency window [26, 27]. A similar process will also be followed by others resonators, as a result, new additional transparency windows are produced without affecting the original ones. As illustrated in Fig. 1b, additional resonators fitted within the inner resonator produce more EIT windows at higher frequencies, which extends the bandwidth of EIT. These results are in agreement with the previous reports [4, 5].

4 Results and discussion

The transmission curves of the proposed MM structure are shown in Fig. 2a, b. Figure 2a reveals that with an increase in the number of CKSSRR, number of EIT windows towards the lower frequency end increases, which greatly extend the EIT bandwidth. We restrict our investigations to quint-EIT windows to avoid the complexity of simulations and also to reduce the simulation time (see Figs. 1b and 2a) although this method can be used for a larger number of EIT windows. The axial symmetry of the CKSSRR ensures that the proposed structure is polarization insensitive. Transmission spectra obtain for incident THz radiation is the same for both TE and TM modes (see Fig. 2b), which also confirms the polarization insensitivity of the proposed device.

Fig. 2
figure 2

a Transmission spectra for different resonators. b Transmission spectra for TE and TM mode

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Independent modulation of transparency windows is realized by changing the outer and inner lengths of the CKSSRR (see Fig. 3a–e). First transparency window 1.90-\(-\)2.64 THz shifts through 1.61-\(-\)2.64 THz while other windows remain unaffected with the change in inner length \((l_{in3})\) of third resonator from 26.82 \(\upmu\)m to 29.7\(\upmu\)m keeping other parameters unchanged. Similarly, second transparency window 2.94-\(-\)3.70 THz shifts through 2.73-\(-\)3.69 THz with change in inner length \((l_{in2})\) of second resonator from 15.51\(\upmu\)m to 17.67\(\upmu\)m; third window 3.66–4.51THz shifts through 3.53-\(-\)4.49 THz with change in outer length \((l_{o3})\) of third resonator from 31.82\(\upmu\)m to 34.7 \(\upmu\)m; fourth transparency window 4.51–5.11THz revamps through 4.42–4.95THz with change in inner length \((l_{in1})\) of first resonator from 4.88\(\upmu\)m to 8.48\(\upmu\)m. It is also observed that second, third, fourth, and fifth transparency windows can be modulated simultaneously by changing \(l_{o1}\). We observed that independent and simultaneous tuning of EIT windows depend on the relative separation, width and outer–inner lengths of CKSSRR. The EIT tuning occurs due to change in relative resonant coupling. Thus, all transparency windows can be revamped independently by varying one specific parameter of the resonator and keeping other parameters fixed. This is very useful in accomplishing selective tunable response of the multi-band EIT effect for the application in selective switching.

Fig. 3
figure 3

Independent tunability of EIT with variation of inner and outer length of KSSRR

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Fig. 4
figure 4

Electric field distribution at all resonance frequencies

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We examine electric field distributions at different resonant frequencies (1.61, 2.66, 3.66, 4.50, 5.1, 5.52THz) for deep understanding (Fig. 3) of the physical process. Color ramp in Fig. 4 indicates the intensity of the electric field, where formation of multi-band EIT windows is clearly observed. In Fig. 4a, the electric field at the edges of the outer resonator is much stronger than the inner resonators before the onset of the EIT windows. As we increase the frequency of THz, EIT window starts occurring at the overlap of the absorption band of the metal resonators (Fig. 4b). In Fig. 4b, we notice that the electric fields of the middle resonator are much stronger than outer and inner resonators. The electric field of the outer resonator is suppressed due to the existence of the middle resonator. The destructive inferences between these concentric CKSSRR result in pronounced transparency windows between 1.90-\(-\)2.64 THz frequencies. With further increase in the frequency, the prominent electric fields appear on inner resonators. Thus, EIT windows occurring at higher frequencies repeat the same process as shown in Fig. 4c–e.

We investigate phase shift and group delays of the proposed metasurface for slow light device applications that can trap photons inside the metasurface for long time due to strong dispersion and as a result enhances light–matter interactions. The plots for phase shift and group delays are shown in Fig. 4. We use \(\tau _{gd}=-\frac{d\phi }{d\omega }\)(where \(\phi\) and \(\omega\) are the effective phase and angular frequency of the terahertz transmission spectrum, respectively) [28] to calculate the group delays. Positive group delay in the transparency window is a phenomenal characteristic of the EIT effect. The value of obtained group delay is 6.05 ps. Delay-bandwidth product (DBP) is also one of the important features of slow optical devices. We obtain DBP using \(DBP=\tau _{gd}*f\), where f is the full-width at half-maxima. Maximum value of group delay (i.e., 6.05 ps) is obtained close to 5.50THz having FWHM as 0.041THz (see Fig. 5a, b), which reflects strong dispersion characteristics of our metastructure. The obtained DBP at 5.50THz is 0.248.

Fig. 5
figure 5

Phase change and the group delay of designed THz metamaterial EIT device as a function of frequency

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We develop a general theoretical model to design a device for ‘n’ EIT windows for wide range of EM spectrum. The model is also capable to describe the physical mechanism behind the active EIT behavior. In our theoretical model, we extend the basic principles of transmission line theory [29], which we call extended transmission line theory. Expressions for resonator impedance and net impedance including the substrate of our extended transmission theory model for ‘n’ EIT windows are obtained as:

$$\begin{aligned} Z(\omega )= & {} \sum _{i=1}^{n-1}\frac{(Z_i Z_{i+1}+\omega ^2 M_i^2)}{ (Z_i+Z_{i+1}-2j\omega M_i)},\\ Z_T= & {} \sum _ {i=1}^ {n-1}\frac{Z_s(Z_i Z_{i+1}+\omega ^2 M_i^2)}{Z_s [(Z_i+Z_{i+1}-2j\omega M_i)+(Z_i Z_{i+1}+\omega ^2 M_i^2)]}, \end{aligned}$$

where \(Z_i=R+j\omega L_i+1/j\omega C_i\), \(Z_T=Z_r \parallel Z_s\), \(M_i\) is mutual inductance, \(\omega\) is angular frequency of incident radiation and \(R_i, L_i, C_i\) are the resistance, inductance and capacitance of \(i^{th}\) frequency, respectively.

We verify our proposed model using our simulation data for THz radiation. The calculated value of \(Z_s\) is 177. The normalized transmission amplitude of the MM unit cell is calculated using \(T= \frac{2Z_T}{Z_0+Z_T}\). The values of analytical parameters for our metasurface are \(L_1=80\,\textrm{pH}, L_2=50\,\textrm{pH}, L_3=48\,\textrm{pH}, L_4=35\,\textrm{pH}, L_5=30\,\textrm{pH}, L_6=25\,\textrm{pH}, C_1=0.95\,\textrm{fF}, C_2=0.085\,\textrm{fF}, C_3=0.040\,\textrm{fF}, C_4=0.032\,\textrm{fF}, C_5=0.026\,\textrm{fF}, C_6=0.023\,\textrm{fF}, M_1=6\,\textrm{pH}, M_2=8\,\textrm{pH}, M_3=7\,\textrm{pH}, M_4=5\,\textrm{pH}, M_5=7\,\textrm{pH}\). We also find that calculated transmittance from the analytical model is in good agreement with the simulation results as shown in Fig. 6. Our novel method can be very useful to design a THz meta-device with a large number of EIT bands, which can significantly enhance the operational spectral range of the device.

Fig. 6
figure 6

Comparison of simulation and analytical results

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5 Conclusion

In summary, we demonstrate a novel method for independent tuning of a large number of EIT windows by varying the inner and outer lengths of the resonators. We carry out simulations to investigate the electromagnetic response of the concentric CKSSRR for quint-EIT bands. The normalized transmission spectra reveal that the addition of more concentric CKSSRR increases the number of transmission windows and consequently the operational bandwidth of the device. We obtain 6.05-ps group delay and 0.248 delay-bandwidth product, which reflect strong dispersion characteristics of our metastructure. The proposed metastructure has magnificent advantages like polarization insensitivity, independent tunability, \(\tau _{gd}=6.05\) ps, and \(DBP=0.248\), which are highly desirable, particularly in multi-wavelength switching, slow-light and optical storage devices. The transmission curves of the proposed MM structure are shown in Fig. 2a, b reveal that number of EIT windows towards the lower frequency end can be increased with an increase in the number of CKSSRR, which greatly extend the EIT bandwidth. Transmission spectra obtained for incident THz radiation are same for both TE and TM modes, which confirm the polarization insensitivity of the proposed metasurface. Although we restrict our investigations to quint-EIT windows to avoid the complexity and simulation time, our novel method can be used for a large number of EIT windows for wider spectral range.