1. Introduction

The quantum destructive interference between two different excitation pathways in a three-level atomic system, giving rise to a narrow transparency window within a broad absorption profile, is a well-known phenomenon named as electromagnetically induced transparency (EIT) [1]. Within the transparency window, the dispersion properties is drastically modified, enabling the EIT effect possesses many promising applications in slow light devices, nonlinear optics, ultrafast optical switching, and so on [2–4]. As a kind of similar coherence and interference phenomenon, the analogue of EIT (EIT-like) effect has attracted considerable attentions recently and has been demonstrated in classical systems such as plasmonic structures [5,6], optomechanical systems [7–9], coupled optical resonators [10–13], and metamaterial configurations [14–17]. Especially, the EIT-like effect observed in coupled optical resonators, which is known as coupled-resonator-induced transparency, has been reported based on various types of resonator systems, such as microring resonators [18], whispering-gallery microsphere resonators [12,19], photonic crystal cavities [20–22], and plasmonic resonators [13]. In addition to those resonators, one-dimensional (1D) photonic crystal nanobeam (PCN) cavity [23–25], which has attractive characteristics of ultrasmall mode volume, extremly high quality factor, very compact footprint and can be integrated with complementary-metal-oxide-semiconductor (CMOS) very easily, is a kind of important device and has been demonstrated as a promising candidate for different applications, including high-sensitivity sensors, optical switches, elctro-optic modulator, low-threshold lasers, and so on. Both theoretical and experimental researches on the EIT-like effect and the corresponding slow light effect based on 1D PCN cavities, however, have been so far rarely reported in literature [26].

On the other hand, it is very important for both fundamental and practical applications if the transparency window as well as the dispersion and group velocity of light can be well controlled. This purpose can be partially realized by varying the geometrical parameters of the structures [5,12,13]. This passive modulation method, however, is very inconvenient and relatively impractical since the structures have to be reoptimized or refabricated once their geometrical parameters have been changed. Active modulation methods without reoptimizing or refabricating the structures are then highly desirable. Up to now, a variety of strategies to actively modulate the EIT-like effect and the corresponding slow light effect have been proposed. The most commonly used method is to integrate tunable materials, such as photoactive semiconductors [15,27,28], superconductors [29,30], and graphene [31–34], within or as part of the structures to realize active manipulation of EIT-like effect by utilizing the natural response of these materials to an external stimulus, such as optical pump, temperature, and gate voltage. In particular, among all the tunable materials, graphene, a two-dimensional material with exceptional electronic and photonic properties like ultra-high carrier mobility as well as broadband tunable optical absorption, has attracted considerable attention. The tunability of its surface conductivity and permittivity, achieved by changing the Fermi level with chemical doping or electrostatic gating, has taken a wide interest in active modulation of, for instance, the EIT-like effect and electro-optic effect [31–37].

In this paper, based on a bus waveguide and two 1D PCN cavities, we proposed a design of a coupled PCN cavity-waveguide system. By using the three-dimensional (3D) finite-difference time-domain (FDTD) method, the EIT-like effect of the system is numerically investigated by changing the separation distance between two cavities. A monolayer and three layers of graphene are further integrated into the system independently and a complete on-to-off modulation of the EIT-like transparency window is numerically demonstrated by electrically tuning the graphene’s Fermi level. Theoretical model based on the coupled mode theory (CMT) is put forward to explore the underlying physics behind the generation and modulation of the EIT-like effect and the theoretical results are highly consistent with the numerical results. Finally, the group delay of the system is also numerically investigated. The largest group delay of 5.01 ps within the transparency window corresponding to a time delay of a 1.50 mm distance of free space propagation is achieved and we demonstrated that the group delay can also be actively modulated by changing the Fermi level of graphene, achieving a well-controlled slow light effect.

2. Structure and design

The proposed coupled PCN cavity-waveguide system is schematically visualized in Fig. 1(a), which consists of a silicon bus waveguide and two 1D PCN cavities, $C_1$ and $C_2$. The bus waveguide and two cavities are placed on a silica substrate with refractive index $n_{\textrm{SiO}_{2}} = 1.47$. The refractive index of silicon is 3.4 and the bus waveguide and two cavities are 500 nm wide and 220 nm thick. Considered our numerical simulations are conducted in a very small wavelength range, the material dispersions of silicon and silica are negligible in such a small range and hence are not considered in our following calculations. Cavity $C_2$ is covered with graphene sheets on top of it and the Fermi level of graphene can be efficiently tuned through the electrical gating. Graphene is incorporated into the simulation by using the surface conductivity material model. According to the Kubo formula [38], the surface conductivity ($\sigma _\textrm{g}$) of graphene, which consists of the intraband and interband terms, is given by

 

Fig. 1. Schematic perspective view (a), top view (b) and left view (c) of the coupled PCN cavity-waveguide system. Notice that cavity $C_2$ is covered with graphene. PCN cavities are symmetric with respect to their centers. (d) Theoretical model of two resonant cavities coupled to a bus waveguide.

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The top view of the proposed system is illustrated in Fig. 1(b), where the coupling distance between $C_1$ and the bus waveguide is denoted as $g$ and the coupling distance between two cavities is denoted as $d$. $g$ is fixed at 50 nm in this work to guarantee the coupling between $C_1$ and the bus waveguide is near the critical coupling so that a fundamental resonant mode with highly confined field can be well excited in $C_1$.

The design of $C_1$ and $C_2$ follows the principle of Bloch mode index matching and the mode-gap effect in order to minimize the scattering losses and thus to effectively confine the electromagnetic fields in the middle of the cavities [39,40]. The cavities are symmetric with respect to their centers and each side of them is designed to have a reflector with nine uniform holes and a tapered section with six gradually varied holes. The lattice constant $a$ (i.e., the center-to-center spacing between two nearest neighboring holes) and the hole radius $r$ of the reflector are set to be $a = 407.25$ nm and $r = 0.278a$, respectively. From both ends of the tapered section toward the cavity center, the lattice constant $a_i$ is tapered from $0.98a$ to $0.76a$ and the hole radius $r_i$ is tapered from $0.272a$ to $0.211a$, where $i = 1, 2, \ldots , 6$ is the hole index. Besides, it is well known that in order to form a distinct EIT-like effect, the resonance frequency detuning between two cavities should satisfy the critical condition, $\delta \omega = {\omega _1} - {\omega _2} \ll {{({\omega _1} + {\omega _2})} \mathord {\left / {\vphantom {{({\omega _1} + {\omega _2})} 2}} \right.} 2}$ [5,41], where $\omega _1$ and $\omega _2$ are the resonant frequencies of $C_1$ and $C_2$, respectively. Therefore, a small difference between the structural parameters of $C_1$ and $C_2$ should be introduced to make sure the resonance frequency detuning $\delta \omega$ between $C_1$ and $C_2$ is as slight as possible to match this coherent condition. And thus the radius of the air hole in the center of $C_2$, $r_6$, is intentionally set to be 45.5 nm, which is different from $r_6$ of $C_1$. Figure 1(c) is a sectional view of the coupled PCN cavity-waveguide system. One can see that a controlled circuit is applied to the graphene to change the Fermi level of graphene. Both the wavelength and the loss of the resonant mode of $C_2$ can thus be changed by electrically tuning of the Fermi level of graphene, which can interfere with the resonant mode of $C_1$ excited directly by the travelling wave incident in the bus waveguide.

To numerically investigate the performances of the proposed coupled PCN cavity-waveguide system, the 3D FDTD method is used in this work. The perfectly matched layer absorbing boundary conditions are applied in all directions to avoid any unwanted reflection from the boundaries. The moderate mesh grid is adopted to obtain a well convergence of the calculated results during the numerical calculations. The input wave with transverse-electric (TE, i.e., electric field in the $x$-$y$ plane as shown in Fig. 1(a)) polarization is incident in the waveguide from the left side of it. A receiver plane is placed at the right end of the waveguide to record the transmission power and thus the transmission is obtained correspondingly.

3. Results and discussions

3.1 Generation of EIT-like effect

At first, the influence of the coupling strength between $C_1$ and $C_2$ on the EIT-like effect is numerically investigated based on the proposed coupled PCN cavity-waveguide system when there is no graphene covered on top of cavity $C_2$. By gradually altering the separation distance between $C_1$ and $C_2$, the strength of near-field coupling between $C_1$ and $C_2$ can thus be changed correspondingly, leading to an alteration of the transmission spectrum. Figure 2(a) shows the normalized transmission spectra of the system for different coupling distances $d$ between $C_1$ and $C_2$. We can observe from Fig. 2(a) that when $d$ is sufficiently large, $d = 850$ nm for instance (blue curve), so that the effect of $C_2$ could be neglected, a transmission dip appears at the resonant frequency of $\omega _\textrm{A} = 201.04$ THz, implying that the incident light is almost completely coupled into $C_1$ and a resonant mode with frequency of $\omega _1=\omega _\textrm{A}$ is formed in $C_1$ in this case. This behavior can be verified by inspecting the magnetic field $H_z$ profile of the system at the transmission dip (point A in Fig. 2(a)), as shown in Fig. 2(b). One can clearly see from Fig. 2(b) that a resonant mode excited by the incident wave from the bus waveguide is highly confined within the middle region of $C_1$. When $d$ decreases to 550 nm (cyan curve in Fig. 2(a)), a narrow and small transparency window appears at the resonant frequency, indicating $C_2$ is weakly coupled to $C_1$ by the evanescent field of $C_1$ in this case. With the further decreasing of $d$, the coupling strength between two cavities increases, resulting in an increase of the value of the transmission peak. The bandwidth of the transparency window becomes wider at the same time. When $d$ is small enough, $d = 350$ nm for instance (red curve in Fig. 2(a)), the value of the transmission peak at the resonant frequency of $\omega _\textrm{C} = 201.05$ THz is as high as $\sim 87\%$, implying the light is almost entirely transmitted through the waveguide in this case. Moreover, two symmetric transmission dips are formed and the frequency splitting is larger than the linewidth of the transparency window. This is a typical EIT-like spectral response, which can be confirmed by inspecting the magnetic field $H_z$ profile of the system at the transmission peak (point C in Fig. 2(a)), as shown in Fig. 2(d). The localized field in $C_1$ is highly suppressed and the light propagates smoothly through the waveguide. At the same time, a resonant mode with frequency of $\omega _2 \approx \omega _\textrm{C}$ is formed in $C_2$ in this case.

 

Fig. 2. (a) Normalized transmission spectra of the system without graphene for different coupling distances $d$. (b) Magnetic field $H_z$ profile at point A denoted in Fig. 2(a) when $d = 850$ nm. (c)-(e) Magnetic field $H_z$ profiles at points B, C and D denoted in Fig. 2(a) when $d = 350$ nm.

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When the EIT-like effect occurs, the resonant mode of the coupled PCN cavity-waveguide system is split into two resonant modes, anti-symmetric mode with wavelength is blue shifted and symmetric mode with wavelength is red shifted, which can be clearly seen in the transmission spectra in Fig. 2(a). Figures 2(c) and 2(e) represent the magnetic field $H_z$ profiles of the system at two transmission dips (point B and point D in Fig. 2(a)) when $d = 350$ nm, respectively. Figure 2(c) shows the symmetric mode, where the resonant frequency $\omega _\textrm{B}$ is 200.91 THz, which is red shifted compared to $\omega _\textrm{A} = 201.04$ THz. It can be observed that the magnetic field phases of two cavities are consistent. Figure 2(e) displays the anti-symmetric mode, where resonant frequency $\omega _\textrm{D}$ is 201.18 THz which is blue shifted compared to $\omega _\textrm{A} = 201.04$ THz. It can be observed that the magnetic field phase difference between two cavities is $\pi$. In both modes, the light propagating in the waveguide is truncated, and about $\sim 90\%$ of the energy is confined in two cavities to form two dips in the transmission spectra.

The classical three-level resonant system in quantum EIT [4] can be applied to explain the underlying physical mechanism of the EIT-like effect appears in our proposed coupled PCN cavity-waveguide system. The incident wave transmitted in the waveguide can be regarded as the ground state $\left | 0 \right \rangle$. Since cavity $C_1$ has strong ability to confine light within the middle region of it, when the light that corresponds to the resonant frequency of $C_1$ is incident from the left side of the bus waveguide, it can be easily coupled into $C_1$. This kind of resonant mode of $C_1$ which can be directly excited by the incident wave can serve as a bright mode (denoted as excited state $\left | 1 \right \rangle$), which is highly radiative and exhibits a relatively low quality factor. On the other hand, due to the existence of cavity $C_2$, when the separation distance between $C_1$ and $C_2$ is close enough, a resonant mode will be excited in $C_2$ via near-field coupling with $C_1$. This kind of resonant mode of $C_2$ can serve as a dark mode (denoted as excited state $\left | 2 \right \rangle$), which is characterized by a relatively high quality factor and could not be excited by the incident wave directly. The dark mode of $C_2$ can interact with the bright mode of $C_1$ through near-field coupling, resulting in a distinct transparency window at the resonant frequency, which is an EIT-like effect. This kind of EIT-like effect is derived from a special coherent effect: the destructive interference between two optical pathways, namely, $\left | 0 \right \rangle \to \left | 1 \right \rangle$ and $\left | 0 \right \rangle \to \left | 1 \right \rangle \to \left | 2 \right \rangle \to \left | 1 \right \rangle$, resulting in a strong suppression of electromagnetic field in $C_1$, as shown in Fig. 2(d).

3.2 Tunability of EIT-like effect based on graphene

Next, to achieve a dynamically modulation of the EIT-like effect, graphene is introduced into the original structure with $d = 350$ nm where a pronounced EIT-like transparency window appears and is placed on top of the dark mode cavity $C_2$, as shown in Fig. 1(a). When $C_2$ is covered with graphene sheets on top of it, the graphene sheets serve as a variable-index optical cladding upon the top surface of $C_2$. The optical properties of graphene are greatly affected by the Fermi level and the number of layers of graphene and thus both the real and imaginary parts of the effective index of $C_2$ varied correspondingly. Therefore, by changing the Fermi level of graphene, which can be obtained by tuning the external gate voltages, the EIT-like effect can thus be correspondingly modulated.

Figure 3(a) shows the normalized transmission spectra of the coupled PCN cavity-waveguide system when $C_2$ is covered by a monolayer graphene with different Fermi levels (blue balls). We can see from Fig. 3(a) that when the Fermi level of graphene $E_F$ is 0.55 eV, the influence of graphene on the transparency window is negligible and the EIT-like effect remains nearly unchanged. The transmission at the transparency peak is $\sim 86\%$ in this case. As $E_F$ decreases from 0.55 eV to 0.40 eV, the transmission at the transparency peak is gradually reduced and the transparency frequency is slightly red shifted. The transmission at the transparency peak reduces to $\sim 49\%$ when $E_F$ decreases to 0.40 eV, resulting in a modulation depth of $\sim 2.4$ dB. The transparency window can be actively controlled by tuning the Fermi level of a monolayer graphene, the modulation strength, however, is relatively small since the transparency peak still exists when $E_F$ decreases to 0.40 eV. In order to achieve a complete on-to-off modulation of the EIT-like transparency peak, three layers of graphene are deposited on top of cavity $C_2$ and the normalized transmission spectra of the system for different Fermi levels are displayed in Fig. 3(b) (blue balls). It is clearly seen that the transparency window undergoes a significantly enhanced modulation compared with the situation when only a monolayer graphene exists. The transmission at the transparency peak declines quickly from $\sim 86\%$ to $\sim 45\%$ when $E_F$ decreases from 0.55 eV to 0.40 eV, leading to a modulation depth of $\sim 2.8$ dB. At the same time, the transparency frequency has an obvious red shift. When $E_F$ arrives at 0.4 eV, the transparency window merges into a broad absorption profile and only a single transmission dip remains, thus completing an on-to-off modulation of the EIT-like transparency peak. In addition, from Fig. 3 one can see that in both cases of a monolayer and three layers of graphene, the frequency differences between two transmission dips remain unchanged when $E_F$ decreases. This situation is very different from that of the coupling distance-modulated EIT-like effect, where the bandwidth between two transmission dips widens gradually as the coupling distance between $C_1$ and $C_2$ decreases.

 

Fig. 3. Normalized transmission spectra of the system when $C_2$ is covered by (a) a monolayer and (b) three layers of graphene for different Fermi levels at $d = 350$ nm. The blue balls denote the numerical results calculated by the FDTD method while the solid red curves show the analytical fitting results obtained by the CMT.

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Furthermore, due to the real part of the effective index of $C_2$ increases with a decreasing $E_F$, the corresponding resonant frequency of $C_2$, $\omega _2$, is slightly red-shifted, leading to an increasing resonance frequency detuning $\delta \omega$ between $C_1$ and $C_2$. The increasing $\delta \omega$ will result in the appearing of two asymmetrical transmission dips in the transmission spectrum. That’s why in Fig. 3 that the transmission dip at low frequency widens and its absolute transmission value increases while the transmission dip at high frequency remains almost unchanged in terms of its absolute transmission value.

3.3 Theoretical analysis for EIT-like effect

To further explore the physical origin of the active modulation on the EIT-like effect in our proposed coupled PCN cavity-waveguide system, we employ the CMT [42,43] to analyze the EIT-like effect theoretically. The theoretical model of two resonant cavities, $C_1$ and $C_2$, coupled to the bus waveguide is depicted in Fig. 1(d). It should be noted that only the internal losses of the cavities and waveguide coupling loss are considered in this model. According to the CMT, the amplitudes of the resonant modes of two cavities are represented as $a_1$ and $a_2$ and can be expressed as

 

Fig. 4. (a) Normalized transmission spectrum of the system without graphene at $d = 350$ nm. The blue balls denote the numerical results calculated by the FDTD method while the solid red curves show the analytical fitting results obtained by the CMT. (b) Fitting parameters $\omega _{d1}$, $\omega _{d2}$ and $\kappa _c$ as a function of $d$ when there is no graphene covered on top of cavity $C_2$. (c)-(d) Fitting parameters $\omega _{d1}$, $\omega _{d2}$ and $\kappa _2$ as a function of Fermi level $E_F$ when $C_2$ is covered by (c) a monolayer and (d) three layers of graphene.

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When $C_2$ is covered with a monolayer and three layers of graphene, the corresponding transmission spectra are also analytical fitted according to the CMT and shown in Figs. 3(a) and 3(b) with red solid curves, respectively. The coupling distance $d$ between two cavities is fixed at 350 nm during the fitting process. As shown in Fig. 3, the fitting results are in good agreement with the numerical results. The corresponding fitting parameters, $\omega _{d1}$, $\omega _{d2}$ and $\kappa _2$, as a function of Fermi level $E_F$ are plotted in Figs. 4(c) and 4(d). We can see from Figs. 4(c) and 4(d) that in both cases of a monolayer and three layers of graphene, the decay rate $\kappa _2$ of $C_2$ increases significantly as $E_F$ decreases from 0.55 eV to 0.40 eV, indicating that the decrease of Fermi level of graphene will increase the intrinsic loss of $C_2$. Due to the intrinsic loss of $C_2$ increases, the transmission at the transparency peak is thus considerably reduced, leading to a modulation depth of $\sim 2.4$ dB for a monolayer graphene and of $\sim 2.8$ dB for three layers of graphene. It can thus be concluded that the active control of the EIT-like effect by changing the Fermi level of graphene results from the change of the decay rate of the dark mode cavity. From the fact that the value of decay rate $\kappa _2$ in the case of three layers of graphene is much bigger than that in the case of a monolayer graphene, which means the intrinsic loss of $C_2$ increases with the increasing number of layers of graphene, now it can be well understood that why the modulation strength in the case of three layers of graphene is much stronger than that in the case of a monolayer graphene. In addition, the frequencies of two transmission dips, $\omega _{d1}$ and $\omega _{d2}$, are red-shifted slightly while the difference between them keeps unchanged in both cases when $E_F$ decreases. The reason for this phenomenon is that due to the frequency splitting and the linewidth of transparency window are determined by the coupling strength between $C_1$ and $C_2$, while this coupling strength remains nearly unchanged since $d$ is fixed at 350 nm as $E_F$ varies, so that the difference between $\omega _{d1}$ and $\omega _{d2}$ is almost constant and so is the linewidth of transparency window.

On the other hand, when the separation distance $d$ between C1 and $C_2$ changes, the coupling strength $\kappa _c$ changes correspondingly, as shown in Fig. 4(b). But $C_1$’s intrinsic loss $\kappa _1$ and the coupling strength $\kappa _w$ remain almost unchanged since the influence of the changing of $d$ on the intrinsic loss of $C_1$ and the coupling strength between $C_1$ and the bus waveguide is negligible. When $C_2$ is covered with graphene, the changing of Fermi level of graphene mainly affect the intrinsic loss of $C_2$, resulting in a varying $\kappa _2$, as shown in Figs. 4(c) and 4(d). The intrinsic loss of $C_1$ and the coupling strength between $C_1$ and the bus waveguide, however, are still nearly unaffected. Therefore, in both cases of altering the separation distance and tuning the Fermi level of graphene, $\kappa _1$ and $\kappa _w$ are almost constants. From the curve fittings, we obtained that $\kappa _1$ is about 0.03 THz and $\kappa _w$ is about 0.09 THz.

3.4 Tunability of Group delay

It is well known that when the EIT-like effect occurs, the dramatic change of the transmission phase within the transparency window will result in a significantly decrease in group velocity of incident light, leading to a slow light phenomenon. Group delay $\tau$ is always introduced to describe the slow light capability, which is expressed as $\tau = \frac {{d\varphi }}{{2\pi d\omega }}$, where $\varphi$ is the transmission phase shift. When there is no graphene covered on top of cavity $C_2$, the group delay $\tau$ as a function of frequency for different coupling distances $d$ between two cavities is shown in Fig. 5(a). The positive and negative group delay can be observed in the vicinity of the transparency window, which correspond to slow and fast light effects respectively. It is clear a remarkable positive group delay peak $\tau _0$ exists at the transparency frequency, indicating a significant slow light effect arises at this frequency.

 

Fig. 5. (a) Group delay $\tau$ of the system without graphene for different coupling distances $d$. (b)-(c) Group delay $\tau$ of the system when $C_2$ is covered by (b) a monolayer and (c) three layers of graphene for different Fermi levels $E_F$ at $d = 575$ nm.

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The dependence of the group delay $\tau _0$ at the transparency frequency on coupling distance $d$ between two cavities is illustrated in Fig. 6(a). An increase of $\tau _0$ is observed for increased $d$. This trend persists up to $d = 575$ nm, where a maximum $\tau _0$ of 5.01 ps is achieved, which corresponds to a time delay of a 1.50 mm distance of free space propagation. With further increasing of $d$, $\tau _0$ decreases quickly and at last disappears for big enough $d$ (for instance $d = 850$ nm in Fig. 6(a)). The appearing of a maximum group delay can be explained by viewing the whole system as an effective medium. Then the susceptibility ($\chi$) of such an effective medium can be written as [5]

 

Fig. 6. The dependence of the group delay $\tau _0$ at the transparency frequency on (a) coupling distance $d$ when there is no graphene covered on top of cavity $C_2$ and on (b) Fermi level $E_F$ when $C_2$ is covered by a monolayer (blue circles) and three layers of graphene (cyan squares) at $d = 575$ nm.

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The group delays of the system under different Fermi levels $E_F$ when $C_2$ is covered by a monolayer and three layers of graphene are shown in Figs. 5(b) and 5(c), respectively. The corresponding $\tau _0$ as a function of $E_F$ is also plotted in Fig. 6(b). All of these group delays are calculated under the condition of $d = 575$ nm, where the biggest group delay is obtained. One can observe from these figures that when the system is integrated with graphene, it gradually loses the slow light capability with the decreasing of Fermi level, suggesting that the slow light effect can be actively tuned by changing the Fermi level of graphene. We can see from Fig. 6(b) that the achievable maximum group delay is 4.71 ps for a monolayer graphene and 3.13 ps for three layers of graphene, suggesting that the modulation range of group delay in the case of a monolayer graphene is wider than that in the case of three layers of graphene. This phenomenon can be understood from the fact that any leakage from the dark mode cavity $C_2$ will result in a significant reduction of group delay. As we stated above, when $C_2$ is covered by a monolayer graphene, its intrinsic loss is much smaller than that when it is covered by three layers of graphene, it therefore possesses a larger modulation range of group delay. The evolution of $\tau _0$ in dependence on $E_F$ in the case of a monolayer graphene, however, is similar to that in the case of three layers of graphene, as shown in Figs. 5(b) and 5(c). Considered our proposed system has the capability of switching the group delay and controlling the amount of the group delay by tuning the Fermi level of graphene, it is very suitable for applications in designing compact slow light devices with ultrafast response.

For a slow light device, only talking about the group delay is not enough to describe the slow-light effect thoroughly. The bandwidth should be considered at the same time. The delay-bandwidth product (DBP) is a critical parameter indicating the buffering capacity of a slow light device. DBP is defined as $\rm {DBP} = {\tau _0}\Delta \omega$, where $\Delta \omega$ is the full width at half maximum bandwidth [44]. When there is no graphene covered on $C_2$, the DBP of our system changes with different $d$ and the obtained maximum DBP is about 0.16. When $C_2$ is covered by a monolayer and three layers of graphene, the DBPs change with different $E_F$, and the corresponding maximum DBPs are about 0.14 and 0.13, respectively.

4. Conclusions and Discussion

In conclusion, we proposed a design of a coupled PCN cavity-waveguide system consisting of a silicon bus waveguide and two 1D PCN cavities to investigate the EIT-like effect. One of the cavities serves as the bright mode cavity and the other one serves as the dark mode cavity. By changing the near-field coupling strength between two cavities, a pronounced transparency window is formed in the transmission spectrum of the system, resulting in a distinct EIT-like phenomenon which is caused by the destructive interference resulting from the near-field coupling of the bright and dark modes. By further integrating with graphene on top of the dark mode cavity, the generated EIT-like transparency window can be actively tuned and a complete on-to-off modulation of the EIT-like effect is realized by electrically tuning the graphene’s Fermi level without reoptimizing or refabricating the structure. Theoretical analysis based on the CMT indicated that the active modulation of the EIT-like effect is attributed to the change of the decay rate of the dark mode cavity. In addition, the largest group delay of 5.01 ps within the transparency window corresponding to a time delay of a 1.50 mm distance of free space propagation is achieved and an active modulation of the group delay is also demonstrated by changing the Fermi level of graphene, achieving a well-controlled slow light effect.

We noted that the EIT-like phenomena have been intensively studied in the literature. On one hand, in Ref. [26] for example, Shi et al. designed several structures consist of a waveguide side-coupled with PCN cavities and experimentally observed the all-optical EIT-like phenomenon. The line shape of the transparency resonance is tuned and the very narrow linewidth of the EIT-like peak is achieved. In order to tune the EIT-like peak, the nanoelectromechanical systems-based comb drive actuator is used, which is, however, fairly complicated and inconvenient. On the other hand, a variety of graphene-based devices have been proposed to realize active control of the EIT-like effect. For example, in Ref. [31] Cheng et al. proposed a device composed of periodically patterned graphene nanostrips, and in Ref. [32] Wang et al. proposed a structure which consists of a graphene waveguide and two side-coupled graphene ribbons. By varying the Fermi level of graphene, the dynamically wavelength tunable EIT-like effects are achieved. The transmissions at the transparency frequencies in their structures, however, remain almost unchanged during the modulation process, while a complete on-to-off modulation of the EIT-like effect is realized in our system. In Ref. [33], Zhao et al. proposed a grating-coupled double-layer graphene hybrid system to investigate the EIT-like effect at far-infrared frequencies. The excellent tunability of transparency window is verified, the capacity of switching the group delay, however, is not been demonstrated in their structures. Overall, compared with other kinds of devices, our proposed coupled PCN cavity-waveguide system has excellent features like compact footprint, CMOS compatibility and excellent modulation performances. With these advantages and the impressive EIT-like resonance characteristics, our proposed structure may provide a new platform for applications in chip-integrated slow light devices, tunable switches, optical modulators and high-sensitive sensors.