1. Introduction

Metamaterials attract significant research interest because of their unusual electromagnetic properties and potential applications [1–6], whose wider utilization requires further improvements in many aspects including broadband and low-loss operation. As one kind of quantum interference effects, electromagnetically induced transparency (EIT) was introduced to metamaterial design to overcome the loss problem. EIT refers to the formation of a narrow transparency window inside original wide absorption band due to destructive interferences between two excitation pathways [7]. Realizing EIT stems from coupled resonances and is potentially useful in many applications, analogous effects using classical systems are widely investigated. Increasing attentions have been paid to classical mimic EIT during the last few years, including EIT-like transmission in metallic nanostructures. So far a variety of mechanisms have been proposed, including RLC circuits [8] optical antennas [9], coupled optical resonators [10–12], and array of metallic nanoparticles [13].

It is well known that surface plasmon resonance (SPR) of metamaterials highly depends on the structure. Among various proposed metallic structures, split-ring resonators (SRRs) have been used for many applications, such as organic sensors [14], left-hand materials [15] and especially in the realization of EIT-like transmission [16–20]. Previously, the electromagnetic responses of SRRs and its coupling with waveguides [21–24] have been investigated. Each SRR has a distributed inductance, L, and a distributed capacitance, C, arising from charge build-up at the notch. If an external incidence is applied perpendicular to the SRR plane, the SRR exhibits a resonant response at a frequency${\omega }_{\text{m}}\text{=(1/}LC{\text{)}}^{1\text{/}2}$.

In this paper, we investigate the EIT-like transmission in side-coupled complementary split-ring resonator (CSRR). Finite difference time domain (FDTD) method is utilized to study the transmission and electromagnetic responses of the structure. Results indicate isolate CSRR can be regarded as mutual coupled U-shaped slot cavities. One cavity can be excited directly by the incidence, called bright mode, while the other cannot and thus a dark mode. When the bright mode experiences interference destruction due to near field interaction with the dark mode, EIT-like transmission can be obtained. Moreover, the destructive interferences between the bright and dark modes depend on suitable frequency detuning and coupling distance between them. Therefore, different shapes of transparency windows can be access.

2. Structure and FDTD model

Figure 1 illustrates the concept diagram of the investigated structure. A bus waveguide and a square ring are etched from the metal plane in side-coupled arrangements. The square ring is split by adding a metallic baffle at the center of the left and right arm, respectively, making a complement structure to the well known SRR. The main structural parameters are the width of the bus waveguide and square ring (w), side length of the ring (L), width of the metallic baffles (g) and coupling distance between the bus waveguide and the CSRR (D). As shown in Fig. 1, the light is incident along the side-coupled CSRR plane with the magnetic field perpendicular to the CSRR plane. The bus waveguide induces the incidence in the insulator region and excite the CSRR through near-field interaction. The insulator and metal in the proposed structure is selected as air and silver, respectively. The frequency-dependent complex permittivity of silver is described by the common used Drude model, ${\epsilon }_{silver}(w)={\epsilon }_{\infty }-{\omega }_P^2/[\omega (\omega +i\gamma )]$ with the parameters set as: ε_∞ = 3.70, γ = 0.018 eV and w_p = 9.1eV for consist with the experimental ones [25]. In the simulations, the structure in the z direction is considered to be infinite (i.e., 2D structure). We note, however, for on-chip optical components, the structures should be made finite. In actual 3D system, the incidence with required polarization can be send through a polarization maintaining fiber, and then evanescent coupled into silica submicrometre- or nanometre-diameter wires (SMNW) through a fiber tape. Since the diameters of the SMNW can be fabricated down to 50 nm [26], we can excite the sample by aligning the center of the SMNW with the bus waveguide directly or inserting a compact coupler [27] to improve the conversion efficiency between them. After go through the sample, the light is coupled back to the conventional fiber by another compact coupler and fiber tape in the same way. The output fiber is connected to with an optical spectrum analyzer to analysis the optical properties of the samples. Due to the limitation of experimental measurements, we just perform 2D FDTD simulations by using Lumerical FDTD Solution Version 7.0 software with perfectly matched layer (PML) absorbing boundary conditions. The grid sizes in x- and y- directions are chosen to be Δx = Δy = 5 nm to ensure a convergence. The structural parameters are set to be L = 250 nm, w = 50 nm, g = 20 nm, and D = 20 nm. To calculate transmittance and analysis the electromagnetic responses, a frequency-domain power or profile monitors monitor is placed at the position Q to record the electric fields E(ω) and the magnetic fields H(ω). Therefore we can simultaneously calculate the complex Poynting vector $\overrightarrow{\text{P}}\text{=}\overrightarrow{\text{E}}(\omega )\times {\overrightarrow{H}}^{*}(\omega )$at a series of user-defined frequencies. Therefore, the time-averaged power flowing across a surface is obtained by$power(\omega )=[{\displaystyle \underset{s}{\int }real(\overrightarrow{P})}•d\overrightarrow{s}]/2$.

 

Fig. 1 Concept diagram of the side-coupled CSRR in plasmonic waveguide system.

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3. EIT in isolate CSRR

Since the width of metallic baffles g is typically no more than 50 nm, isolate CSRR shown in Fig. 1 can be taken as two symmetric U-shaped slot cavities coupling via baffles. Due to the side-coupled arrangements, the bottom U-shaped cavity couples strongly to the bus waveguide, so called the bright mode. However, the upper U-shaped cavity cannot directly couple to the bus waveguide, supporting the dark mode. When there is no interfere between them (i.e., g > 100 nm), the dark mode cannot be excited and the CSRR is equivalent to an isolate bright mode (inset in Fig. 2(a) ). According to simulations in Fig. 2(a) and Fig. 2(b), the isolate bright mode exhibits a resonant dip at 1102 nm, with strong magnetic field trapped into it (i.e., opaque to the bus waveguide). Squeezing the width of the metallic baffles to 20 nm (inset in Fig. 2(c)), the dark mode is excited through tunneling coupling with the bright mode due to the greatly enhanced near-field interferences between them. As a result, EIT-like transmission appears at original stop band, with two new resonant dips generating at 1046 nm and 1166 nm, respectively, as shown in Fig. 2(c). The magnetic field of the CSRR system at original dip is shown in Fig. 2(d). We can see the interference between the bright and dark modes gives rise to the suppression of field in the bright mode. Consequently, the incidences originally trapped in the bright mode become transparent to the bus waveguide. More specifically, both the transmittance and the magnetic field intensity at 1102 nm are greatly enhanced in the bus waveguide.

 

Fig. 2 (a) Transmission spectra of side-coupled U-shaped slot cavity and (b) magnetic field at 1102 nm. (c) Transmission spectra of the CSRR system and (d) magnetic field at 1102 nm. Insets: corresponding diagram of the model.

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In order to analyze the electromagnetic response of the side-coupled CSRR, the magnetic and electric field of the CSRR at the new resonant dips 1046 nm and 1166 nm are plotted in Figs. 3(a) -3(d), respectively. We can see both the bright and dark modes are strongly excited with no field suppression between them at these frequencies. Note that, the metallic baffles (i.e., as inductive elements) are surrounded by stronger magnetic field while the notch cavity (i.e., as captive elements) are surrounded by stronger electric field. By varying the asymmetry degree and width of metallic baffles, the effective inductance L, capacitance C and coupling strength of CSRR will change, which modify the resonant frequencies and transmitted intensities, respectively. Consequently, a wider range of designs can be accessed in CSRR.

 

Fig. 3 (a) Magnetic field and (b) electric field of the CSRR at 1046 nm. (c) Magnetic field and (d) electric field of the CSRR at 1166 nm.

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CSRRs with different asymmetry are shown in the insets of Fig. 4 . Here, asymmetry is obtained by shifting the baffles apart from the arm center with an offset distance s. This results in separation of two resonant frequencies, i.e. blue-shifting of dark-mode resonances and red-shifting of bright-mode resonances, which can be explained from the LC resonance model. We know that the effective capacitance is built up within the U-shaped notch cavity and thus increasing the asymmetry degree of the CSRR results in changes in the effective capacitance C. Consequently, the LC resonance frequencies of the bright and dark modes become different. Moreover, the frequency detuning between them increases as the asymmetry degree increasing, resulting in a broader transparency window as shown in Figs. 4(a)-4(c). As an example, the electromagnetic responses of the CSRR with asymmetry degree s = 10 is plotted in Fig. 5(a) -5(d). We can see the bright mode concentrates the longer resonant wavelength due to owing a larger capacitance, which is constant with the theory analysis.

 

Fig. 4 Transmission spectra of CSRR with different asymmetry degree s: (a) s = 0 nm; (b) s = 5 nm; (c) s = 10 nm.

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Fig. 5 (a) Magnetic field and (b) electric field of CSRR with asymmetry degree (i.e. s = 10 nm) at 1023 nm. (c) Magnetic field and (d) electric field of CSRR with asymmetry degree (i.e. s = 10 nm) and 1188 nm.

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We also investigate the influence of width of metallic baffles g on the transparency window of CSRR. According to the transmission spectra in Figs. 6(a) -6(c), we can see the baffle length g is another important parameter controlling transmitted intensity and resonant frequency. Increased coupling length gives rise to greater coupling loss and weaker near field interference. The former is the reason of lower transmittance while the latter is the reason of a narrower transparency window. Here, the shift of the resonant frequencies also results from the change of the effective capacitance L and effective capacitance C in the CSRR model.

 

Fig. 6 Evolution of the transmission spectra of the CSRR with different baffle width g: (a) g = 20 nm; (b) g = 30 nm; (c) g = 40 nm.

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4. EIT in cascaded CSRRs

In this section, we investigate the characteristics of cascaded CSRRs with a coupling distance d = 20 nm. As a cell of the cascaded CSRRs, CSRR with single baffle (inset in Fig. 7(a) ) is simulated and shown in Fig. 7(a) and Fig. 7(b) to compare with the characteristics of cascaded CSRRs (inset in Fig. 7(c)) in Fig. 7(c) and Fig. 7(d). Compare the transmission spectra in Fig. 7(a) and Fig. 7(c), we can see EIT-like transmission at visible (768 nm) and near-infrared (1055 nm) range, simultaneously. Moreover, the magnetic fields in Fig. 7(b) and Fig. 7(d) demonstrate destructive interferences do exist in the bottom CSRR. The influences of the coupling distance d on the transmission characteristics are also studied by increasing d from 20 nm to 50 nm with a step of 10 nm. As shown in Figs. 8(a) -8(d), the transparency window at 768 nm become distorted and disappeared gradually as d increases due to the weak near-field interaction and great coupling loss. Therefore, appropriate coupling strength (i.e. coupling distance) is a necessary condition to obtain EIT-like transmission.

 

Fig. 7 (a) Transmission spectra of CSRR with single baffle and (b) magnetic field at 1055 nm. (c) Transmission spectra of cascaded CSRRs and (d) magnetic field at 1055 nm. Insets: corresponding diagram of the structure.

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Fig. 8 Transmission spectra of the coupled CSRRs with different coupling distances d: (a) 20 nm; (b) 30 nm; (c) 40 nm; (d) 50 nm.

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5. Conclusion

In summary, by using the FDTD method, we have investigated the EIT-like transmission in side-coupled CSRR. Results illustrate side-coupled CSRRs behave different electromagnetic responses with metallic SRRs. Further investigating the coupling between two individual CSRR, we have obtained a cascaded structure that supports EIT-like transmission at visible and near-infrared region, simultaneously. By adjusting the width (i.e., coupling distance) and the position of the metallic baffles (i.e., asymmetry degree) in CSRR, we are able to control the transmitted intensity and the resonance frequency of the CSRR, resulting in different shapes of transparency windows.