1. Introduction

Slow light effect has become increasing interested in scientific research because of its various applications such as optical delay, optical buffers and various optical circuits [1,2]. Many approaches of achieving slow light have been proposed, including electromagnetically induced transparency (EIT) [3], coherent population oscillation (CPO) [4], stimulated Raman scattering (SRS) [5], stimulated Brillouin scattering (SBS) [6], Bragg fiber [7], photonic crystal waveguides or cavities [8,9], surface plasmon waveguide [10], etc. . Among these schemes for achieving slow light, electromagnetically induced transparency is one of the most important approaches. Electromagnetically induced transparency, a spectrally narrow transparency window accompanied with extreme dispersion, is a coherent process, resulting from the quantum inference of pump and probe laser beams tuned at different transitions [11–15]. This transparency window accompanied with an extreme dispersion results in a substantial reduction of the group velocity of light. However, the experiment of quantum EIT requires a special setup with extreme constraints [16]. Recent discoveries of analogues of EIT in many classical systems, including photonic crystal [17], plasma [18–22], drop-filter cavity-waveguide system [23], metamaterial [24–29] and coupled-resonator [30,31], have opened new ways to overcome these issues. However, up to now, most EIT-like metamaterials reported in the literature have been designed to exhibit only the single-band slow light effect. Instead, designing a complex media that possesses the multiple transparency windows with slow light properties has many important applications in multi-band filters [32] and multi-band slow light devices (e.g. delay line).

In Ref [33], a bull’s-eye-shaped structure is presented to achieve the multi-peak of EIT metamaterial. Inspired by the literature [33], this paper proposes a more straightforward multi-band slow light metamaterial in microwave frequency by employing cut wires of different sizes and parallel to each other. Contrasting with some previous schemes, our proposed structure is more convenient for fabrication. Moreover, the electrical field distributions and slow light properties of the metamaterial are also investigated by theory and simulation in detail. Simulation results show our proposed structure can achieve the multi-band slow light effects and possess multi-band EIT-like properties. In addition, the impacts of structure parameters on transparency windows are also investigated. Simulation results show the transparency windows can be tuned by adjusting the structural parameters of metamaterial. Finally, the equivalent RLC circuit model and the synthesis method of the multi-band slow light metamaterial are presented. To verify the validity of the synthesis method, a specific example is given. The simulation results are in a good agreement with the proposed goal. Therefore, our research opens a door to a quick and accurate construction the multi-band slow light metamaterial.

2. Realization of the multi-band slow light metamaterial

The unit cell of the proposed multi-band slow light metamaterial is shown in Fig. 1 . It is composed of three cut wires of different sizes and parallel to each other. Three cut wires are placed in the same plane. Their lengths are shorter in order while other parameters are identical. Due to the different sizes of three cut wires, their resonance frequency is different. The incident light is perpendicularly to the plane of the multi-band metamaterial with electric field along the cut wire. Under excitation by incident light, metal strips serve as dipole antennas that induce strongly localized electric and magnetic fields. Because the lengths of three cut wires are not identical, the strip I will experience the stronger coupling to the radiation field. The strip II undergoes the weaker coupling to the radiation field compared with the strip I. When these constituents are brought into the two wires structure, this asymmetry configuration induces the anti-parallel currents on each metal strip which cause the destructive interference of scattering fields and result in a pronounced transparency window [34, 35]. Similarly, the strip II experiences the stronger coupling to the radiation field comparing with the strip III. The interference between resonances of the strip II and the strip III induces a new transparency window which does not affect the original transparency window. Therefore, three cut wires result in two EIT windows, which thereby achieves the multi-band slow light metamaterial. From the above, the more cut wires included in asymmetric double wires EIT structure, as shown in Fig. 1, will support the more EIT window in different frequency.

 

Fig. 1 Configuration of the multi-band slow light metamaterial.

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The proposed metamaterial structure is constructed on the teflon substrate with dielectric constant of ε=2.2, and the cut wires are resumed copper with a electric conductivity 5.96 × 10⁷ S/m. The thickness of copper is 0.018 mm. One period of the multi-band slow light metamaterial is shown in Fig. 1. The structure parameters of the simulated multi-band metamaterial are l₁ = 14 mm, l₂ = 12 mm, l₃ = 10 mm. E-fields polarized along x axis and H-fields polarized along y axis are employed. To verify the validity of our proposed the multi-band slow light metamaterial, the transmission spectra of four different cut wires configurations are shown in Fig. 2 . It is seen from Fig. 2, the dependent strip I shows strong resonance at 9.1 GHz. The dependent strip II shows strong resonance at 10.3 GHz as well. When we put these two constituents into double wires, due to breaking the length symmetry of double wires, the destructive inference between double wires induces a pronounced transparency window at 8.9 GHz. During this process, on the one hand, the radiative element transfers the exciting electromagnetically energy to the subradiative element through the coupling of near field. On the other hand, the strip II also shows strong resonance under the polarization by electromagnetic fields. Because of these two parts of effects, the resultant transmission spectrum emerges an EIT window. When we bring the third wire into the double wires, an additional EIT-like window occurs at 10.6 GHz and does not affect the original EIT window. These characters effectively verify that the multi-band transparency windows are achieved by simple cut wire configuration.

 

Fig. 2 The transmission coefficients of different metamaterials.

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However, From Fig. 2, it can been seen that transparency windows exhibit asymmetry in frequency, so that it can be explained by the fact that the three metal strips of the multi-band EIT metamaterial have different length, and thus they have different resonance frequencies. Usually, transparency windows of EIT metamaterials appear symmetric only when their resonance elements have the same resonance frequencies, and the symmetry can be easily broken by the difference between the resonance frequencies whether the individual spectra of the resonance elements are symmetric or not. Therefore, the proposed multi-band metamaterial exhibits the asymmetric spectra. Worth noting is that the asymmetric EIT windows usually have sharper spectral profile, which is very useful for achieving stronger dispersion (slower light) and higher sensitivity [18, 28].

We also investigate the multi-band slow light properties of metamaterial which can be illustrated by employing propagating electromagnetic pulses [13]. We consider a Gaussian-shaped pulse centered at 8.9 GHz, normally propagating through the metamaterial with a thickness of 1 cell in the z direction. The simulation results are shown in Fig. 3(a) . From Fig. 3(a), we can see that the peak of incident Gaussian pulse with the center frequency at 8.9 GHz appears at 35.193 ns and the peak of transmitted pulse appears at 37.118 ns. Hence, the pulse is delayed ~1.925 ns, while the delay time of the pulse across free space with the same thickness is only 0.01 ns. To demonstrate the delay properties of metamaterial in another transparency window, we also simulate a Gaussian-shaped pulse centered at 10.6 GHz, normally propagating through the metamaterial with a thickness of 1 cell in the z direction. From the simulation result revealed in Fig. 3(b), the transmitted pulse in the second EIT window is delayed by ~0.858 ns, which is about 85.8 times longer than the delay time of a pulse propagating through free space with the same thickness. Interestingly, the delay properties of pulse at two EIT-like windows do not disturb each other. This property can realize double channel slow light equipment for various potential applications.

 

Fig. 3 (a) The simulated time pulse signal with Gaussian-shaped pulse centered at 8.9 GHz (b) The simulated time pulse signal with Gaussian-shaped pulse centered at 10.6 GHz.

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To understand the underlying electromagnetic physics of the multi-band slow light metamaterial, we investigate the electric field distributions of the multi-band structure in detail. Figure 4 shows the simulation results of the E-fields at frequencies (a)-(e) of Fig. 2 which correspond to 8.6, 8.9, 10.1, 10.6, 12.5 GHz, respectively. The color represents the intensity of E-field. In Fig. 4, we can clearly observe the formation process of multi-band transparency windows. In Fig. 4(a), before the onset EIT, we can see that the electrical fields of strip I are much stronger than strip II, especially near the ends of the strip I. As the frequency is increased, EIT window occurs at the overlap of absorption band of the independent metal wire, which is further verified from Fig. 4(b). In Fig. 4(b), we can see the electrical fields of strip II are much stronger than strip I, i.e., the electrical fields of strip I is suppressed due to the existence of strip II. The destructive inference between two metallic wires results in a pronounced transparency window at 8.9 GHz. When frequency is further increased, the prominent electrical fields appear on strip II and strip III, and the EIT window occurring at higher frequency repeats the former process as shown in Fig. 4(c)-(e).

 

Fig. 4 The electrical field distributions of the multi-band slow light metamaterial at frequencies (a) 8.6 GHz, (b) 8.9 GHz, (c) 10.1 GHz, (d) 10.6 GHz, (e) 12.5 GHz, respectively. Red (green) corresponds to a large (small) electric field.

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Finally, we investigate the influences of structure parameters on the transparency windows. For comparison, we plot the transmission coefficients of the multi-band slow light metamaterial for a reference structure with the same parameters as Fig. 1 (red solid curves) and for two structures with l₂ = 11 mm (blue curves) and l₂ = 13 mm (magenta curves) while keeping other parameters constant (see Fig. 5(a) ). As shown in Fig. 5(a), the strip II affects the spectral response the most. To the first transparency window as an example, when the length of strip II is decreased, the first transparency window widens and its strength enhances while the second window is just opposite. This is because the induced transparency window will become narrow and sharp when the difference between lengths of two metal strips become small [35,36]. In other words, at a low degree of asymmetry, the transparency window is extremely sharp and is accompanied by steep phase change [35]. Therefore, by adjusting the asymmetric degree of metamaterial, we can control the phase change of resonance, and thus control the width of resonance and the sharp degree of transparency window [34]. Figure 5(b) shows the changes in transmission spectra induced by the different widths w of metal strip. The blue, the red and the magenta solid lines present the transmission spectra for various widths w of metal strip from 0.5 mm to 1.5 mm with an increment of 0.5 mm. We can see when the widths of strips are increased, the transparency windows become narrow and the strengths decrease at the same time.

 

Fig. 5 The transmission spectra (a) with variable l₂ (b) with variable w.

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3. Synthesis of the multi-band slow light metamaterial

As to the authors’ best knowledge, there are no papers reporting the synthesis design method for such a multi-band structure with three cut wires. In order to obtain some rules of thumb for design the multi-band slow light metamaterial, we propose a simple model which clearly describes the EM behaviors of the multi-band metamaterial. We model the coupled metallic wires as three RLC circuits coupled by two capacities, as shown in Fig. 6 . The leftmost subcircuit corresponds to the longest wire (strip I) and contains a voltage source representing the action by external electric field. The medial subcircuit responds to quasi-dark state [12] circuit—the metal strip II, and also contains a voltage source since the independent metal strip II can act as a bright component. The rightmost subcircuit represents the strip III.

 

Fig. 6 The equivalent circuits of the multi-band slow light metamaterial.

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To the sake of simplification, we negative the mutual inductances between wires, the mutual capacities between non adjacent metal wires, and the loss of substrate in RLC circuit model. In order to obtain the calculating formula of inductances and capacitances of the proposed multi-band metamaterial, we assume that the width w of cut wires, the separation s between adjacent cut wires and the length l of cut wires.

The inductance L produced by cut wires can be estimated by the following formula [37]:

The detailed derivation process of inductance L can be seen from Ref [37].The expression of capacity is also estimated by the following formula:

The resistance R_{i (i = 1, 2, 3)} of equivalent circuit shown in Fig. 6 is composed of two parts: R_rad and R_c. Because the three metal strips serve as dipole antennas, the radiation resistance R_rad of metal strip is estimated by the radiation resistance of dipole antenna:

In the case of lossy conductors, we add a series resistance R_c to take into account the losses in the conductor. Because the metal wire is very thin, it is difficult to decide. Therefore, the total series resistance can be estimated by the form of R_c = R₀l, where R₀ = ρ/ (wt) is the per-unit-length resistance, withρbeing the electrical resistivity of the metal. Hence, the final expression of the conductor loss is

At last, we analyze the coupled capacity C_c. Because of the electromagnetic brink effects, the coupled capacity is difficult to be decided. Hence, the coupled capacitance C_c (C₁₂ or C₂₃ in Fig. 6) can be estimated by the following formula:

Using the equivalent circuit depicted in Fig. 6 and Eqs. (1) – (7), we can obtain the analytical formula for the transparency window of metamaterial at center frequency:

Because of involving to the complete elliptic integral of the first kind with unknown parameters, the explicit solutions of physical parameters of cut wires cannot be given. However, by employing the Eq. (8), we can get the numerical solution of physical parameters of cut wires. To testify the validity of the proposed synthesis method, we give a specific example. To obtain the multi-band transparency window (multi-band slow light effect) with the center frequency of 7.9 GHz and 10 GHz, we assume the length 0.526${\lambda }_{01}$ and width 0.263${\lambda }_{01}$of the unit cell, the length 0.421${\lambda }_{01}$ and width 0.026${\lambda }_{01}$of the longest cut wire, the width 0.026${\lambda }_{01}$other two cut wires, and the separations 0.026${\lambda }_{01}$between cut wires (${\lambda }_{01}$corresponds to the wavelength of 7.9 GHz). We can get the lengths of other two cut wires are 13.8 mm and 10.2 mm by Eq. (8). Three cut wires with above parameters are simulated by CST Microwave Studio. Simulation results are showed in Fig. 7 . It is seen from Fig. 7, the center frequencies of the multi-band transparency windows are 7.9 GHz and 10.2 GHz, respectively, which are in a good agreement with the design goal.

 

Fig. 7 The numerical simulation results of three cut wires

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4. Summary

In this paper, we have demonstrated a metamaterial exhibiting the properties of multi-band slow light. The metamaterial is consisted of three cut wires of different sizes and parallel to each other. The resonance frequencies are different because the lengths of three metal wires are different. Under the polarization of electrical field, because three cut wires experience strong resonance, two-two overlaps of absorption bands of three cut wires induce two transparency windows. The calculated transmission spectra also reveal that the additional wire added in the single band double wires structure induces a new transparency window while not affecting the original window. Moreover, the slow light properties and electrical field distributions of the metamaterial are also investigated in detail, which further confirms the multi-band slow light effects and the nature of EIT. In addition, the impacts of structure parameters on transparency windows are also investigated. Simulation results show the transparency windows can be tuned by adjusting the structural parameters of metamaterial. Finally, we propose the equivalent RLC circuit model of multi-band slow light metamaterial and the synthesis design method. Simulation results show that the synthesis method accurately predicts the center frequencies of transparency windows, which opens a door to a quick and accurate construction for the multi-band slow light metamaterial. Our proposed metamaterial structure with all above features confirms the versatility of metamaterials toward various potential applications.