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Abstract:
In order to control the group velocity of slow light, a graphene plasmon time crystal slow light waveguide was constructed and used for the waveguide to construct the Zigzag topology interface channel for transmission. When the structure is fixed, the external bias voltage of the graphene nano-disk can be dynamically adjusted to obtain the dispersion curves at different times. The corresponding group velocity is studied. First, the graphene plasmon time crystal is obtained by applying the bias voltages periodically varying with time to different regions of the honeycomb arranged graphene nano-disks. When the time translation symmetry of the crystal is destroyed, the crystal band gap will periodically appear and disappear with time, and exhibit the band topology effect. The Zigzag topology interface is constructed to analyze the topological interface state and its slow light mode existing at different moments. Then the corresponding group velocity is calculated according to the dispersion curve. Finally, a slow light waveguide model is established through numerical simulation, and the field enhancement process is detected at the light energy capture point of the waveguide. Simulation results show that the waveguide designed based on the graphene plasmon time crystal can achieve a good slow light transmission effect, and the group velocity of the light can be dynamically adjusted when the waveguide structure is fixed. Under slow light transmission, the light energy capture point realizes the field enhancement effect. The slow light waveguide with simple structure can be dynamically tuned, and has broad application prospects in slow light modulation devices and optical storage devices.
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Key words:
- slow light /
- graphene /
- plasmonic time crystal /
- dynamical tuning
摘要:为了实现对慢光群速度的控制，构建了石墨烯等离激元时间晶体慢光波导。对波导采用石墨烯等离激元时间晶体来构造用于传输的Zigzag拓扑界面通道，当结构一定时动态调节石墨烯纳米盘的外加偏置电压，即可获得若干不同时刻的色散曲线，并对相应的群速度进行研究。首先，通过对蜂窝状排列的石墨烯纳米盘的不同区域施加随时间周期性变化的偏置电压，来获得石墨烯等离激元时间晶体。当晶体时间平移对称性遭到破坏时，晶体带隙会随着时间周期性出现及消失，进而呈现出能带拓扑效应。接着，构造Zigzag拓扑界面分析不同时刻下存在的拓扑界面态及其慢光模式。然后，根据该色散曲线计算出对应的群速度。最后，通过数值仿真建立慢光波导模型，并在波导光能捕获点检测场增强过程。模拟结果表明：基于石墨烯等离激元时间晶体所设计的波导可以实现很好的慢光传输效果，在波导结构固定时可以动态调节光的群速度。慢光传输下光能捕获点实现了场增强效应。该慢光波导结构简单，可动态调谐，在慢光调制器件和光储存器件中具有广阔的应用前景。
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1. Introduction
The slow light technology has attracted more and more attention due to its quite broad potential application prospect in optical field regulation and optical storage devices[1-3]. Controlling the structure dispersion to realize slow light is one of the commonly used technologies at present. The main method of this technology is to control the light group velocity by designing a slow optical waveguide with a specific structure. Slow light generated in such photonic crystal waveguides has been observed in several studies[4-5]. However, in the previous slow optical waveguide structure, when the group velocity becomes slow, its supported bandwidth is very narrow, and it is also accompanied by a huge group velocity dispersion effect. Scheuer J. et al. used a complex coupled resonant optical waveguide to achieve internal dispersion compensation and solved the problem of distortion[6]. In recent years, Surface Plasmon Polaritons (SPPs) waveguides have been used to realize slow light, which can achieve higher slow light capacity with a fixed structure, but the adjustment of slow light performance is limited[7]. In addition, the large group velocity dispersion effect of the chirped structure can cause seriously distortion of the optical signal, and the device has a limited operating frequency range[8]. Furthermore, the structural design of traditional slow-light devices is complicated and their tunability is poor once the structure is fixed, which limits their practical applications.
Photonic crystals have good topological band effects, and their topological concepts have great potential in the application of photonics, which has attracted extensive attention. Topological photonic structures have subverted some conventional views on wave propagation and manipulation. Applying topological photonic crystals to wave propagation makes it possible to realize new photonic devices with specific funtions, such as sharp bending waves without reflection, robust delay lines, spin polarization switches and non-reciprocal devices[9]. Recently, Chen Xiaodong et al. proposed and proved that the valley states in topological photonic crystals can be used as topological protection to realize light transmission. The design is to place two topological photonic crystals with different topological structures in mirror images to form an interface that can achieve topological protection[10]. Later, Yoshimi et al. proposed a method to realize topological slow optical waveguides in valley photonic crystals[11]. The waveguide structure is based on a semiconductor substrate to realize slow light transmission with a group index greater than 100 in the topological band gap range, but the manufacturing process is complex, and the structure has no tunability and does not involve the change of field strength caused by slow light. With the deepening of research, it is found that the topological band gap structure of valley state can be realized in photonic time crystal materials whose refractive index changes periodically with time[12]. The method is that by breaking the crystal time translation symmetry, photonic time crystals can achieve topologically non-trivial phases[13-15], thereby affecting the propagation of light in the crystal.
It is well known that graphene has unique electrical and optical properties, especially it supports surface plasmonic excitation waves, and has relatively low ohmic loss and high tunability[16]. Jin et al. designed monolayer graphene with periodic patterns to achieve topological unidirectional boundary transport by introducing a static magnetic field[17]. Later, Wang Yang et al. realized topological valley plasmonic transport in bilayer graphene metasurfaces for sensing applications[18]. Recently, Guo Xiang et al. designed a graphene SPP equivalent two-dimensional photonic crystal slow light waveguide through photonic crystal line defects, and achieved slow light modulation through the gradual change of chemical potential of graphene in space[19]. Likewise, it is very important to realize the modulation of the slow light transport in the valley state topology of graphene plasmonic crystals in time. In this paper, we propose a novel approach to achieve topological slow light transport in waveguides constructed from graphene plasmonic time crystals. Two-dimensional graphene plasmonic time crystals consist of a set of graphene nanodisks periodically arranged in a honeycomb pattern. By controlling the periodic change of the chemical potential of graphene with time, the time-translational symmetry of graphene plasmonic time crystals is broken. Numerical simulations show that the plasmonic time crystal band gap can open and close periodically with time. Further, we find that the Zigzag edges can support near-zero electromagnetic transport group velocities within the topological bandgap. Numerical simulation results of electromagnetic transmission in slow light waveguide show that the topological waveguide can generate slow light. The advantages of this method are simple structure, good field enhancement effect and dynamic tuning ability, which provides a new way for dynamically realizing optical field regulation.
2. Numerical simulation methods and models
In this paper, a valley state topological slow optical transmission waveguide is implemented based on graphene plasmonic time crystals. As shown in Fig. 1(a) (Color online), the waveguide adopts the Zigzag structure topological interface as the transmission channel, and the top is composed of graphene nanodisks. Graphene is in contact with air, SiO2 is selected as the background material, and Si as the substrate material. It can be seen from the cross-sectional view of the waveguide in the figure that the bias voltage is applied to the graphene nanodisk and the silicon substrate. The silicon pillars are arranged in a triangular lattice, and V1(t), V2(t) and V3(t) are the bias voltages applied to the graphene nanodisks, respectively. This method is based on the photonic crystal realization mechanism of the graphene plasmonic exciton[20]. When the distance between graphene and the substrate is constant, the chemical potential of graphene can be changed by changing the applied voltage.
Figure 1. (a) The three-dimensional schematic of slow light waveguide, with a single layer of graphene nanodisks at the top. The graphene is exposed to air on the top, the background material is SiO2, and the substrate material is Si. Different graphene nanodisks are applied with different bias voltages: V1(t), V2(t) and V3(t). The diagram on the right shows how voltage is applied. (b) The graphene nanodisk’s external bias voltage changes periodically with timeIt is assumed that the relative permittivity of the surrounding environment of graphene is
$ {\varepsilon _{{\rm{r}}1}} $ and$ {\varepsilon _{{\rm{r}}2}} $ , respectively. The relative permittivity of the substrate SiO2 is$ {\varepsilon _{{\rm{r}}1}} $ =3.9, and the relative permittivity of the air on the upper surface of the graphene is$ {\varepsilon _{\rm{{r}}2}} $ =1. In our numerical simulation, the TM mode is considered and the electric field form of TM mode is assumed to be[21]:$$ \left\{ {\begin{array}{*{20}{c}} {{E_x} = A{{\rm{e}}^{{\rm{i}}\beta x - {Q_1}\textit{z}}},}\;{{E_y} = 0,}{{E_\textit{z}} = B{{\rm{e}}^{{\rm{i}}qx - {Q_1}x}},}\;{(\textit{z} > 0)} \\ {{E_x} = A{{\rm{e}}^{{\rm{i}}\beta x + {Q_2}\textit{z}}},}\;{{E_y} = 0,}{{E_\textit{z}} = B{{\rm{e}}^{{\rm{i}}qx + {Q_2}x}},}\;{(\textit{z} < 0)} \end{array}} \right.\quad. $$ (1) Substituting equation (1) into Maxwell's equations, adding the surface conductivity of graphene
$ \left( {{\sigma _{\rm{g}}}} \right) $ to participate in the formula transformation, and matching the corresponding boundary conditions, the dispersion relation of the TM mode is obtained as[22]:$$ \frac{{{\varepsilon _{{\rm{r}}1}}}}{{\sqrt {{\beta ^2} - \dfrac{{{\varepsilon _{{\rm{r}}1}}{\omega ^2}}}{{{c^2}}}} }} + \frac{{{\varepsilon _{{\rm{r}}2}}}}{{\sqrt {{\beta ^2} - \dfrac{{{\varepsilon _{{\rm{r}}2}}{\omega ^2}}}{{{c^2}}}} }} = - \frac{{{\sigma _{\rm{g}}}i}}{{\omega {\varepsilon _0}}}\quad, $$ (2) where
$ {\varepsilon _0} $ is the vacuum permittivity in free space,$ \omega $ is the angular frequency of the plasmon, and$ c $ is the propagation speed of light in vacuum. In the whole calculation process, we only consider the case of the propagation constant$\beta \ll \dfrac{\omega}{c}$ , so equation (2) can be simplified to the following form[23]:$$ \beta = {\varepsilon _0}\left(\frac{{{\varepsilon _{{\rm{r}}1}} + {\varepsilon _{{\rm{r}}2}}}}{2}\right)\frac{{2\omega i}}{{{\sigma _{\rm{g}}}}}\quad, $$ (3) where
$ \beta $ is the propagation constant based on graphene SPP, the surface conductivity of graphene$ {\sigma _{\rm{g}}} $ can be adjusted with temperature T, chemical potential$ {\mu _{\rm{c}}} $ , scattering rate$ \tau $ and angular frequency$ \omega $ , which consists of two parts: intra-band electron scattering${\sigma _{{{\rm{intra}}} }}$ and inter-band electron transition${\sigma _{{{\rm{inter}} }}}$ , according to Kubo Formula[23]:$$ {\sigma _{\rm{g}}} = {\sigma _{{{\rm{intra}}} }} + {\sigma _{{{\rm{inter}}} }}\quad, $$ (4) $$ {\sigma _{{{\rm{intra}}} }} = - {\rm{j}}\frac{{{{\rm{e}}^2}{{\rm{k_B}}}T}}{{{\text{π}} {{{\hbar}} ^2}(\omega - {\rm{j}}{\tau ^{ - 1}})}}\left[ {\frac{{{\mu _{\rm{c}}}}}{{{{\rm{k_B}}}T}} + 2\ln ({{\rm{e}}^{{{ - {\mu _{\rm{c}}}} / {{{\rm{k_B}}}T}}}} + 1)} \right] \quad,$$ (5) $$ {\sigma _{{{\rm{inter}}} }} = \frac{{ - {\rm{j{e}}^2}}}{{4{\text{π}} {{\hbar}} }}\ln \left( {\frac{{2\left| {{\mu _{\rm{c}}}} \right| - \left( {\omega - {\rm{j}}{\tau ^{ - 1}}} \right){{\hbar}} }}{{2\left| {{\mu _{\rm{c}}}} \right| + \left( {\omega - {\rm{j}}{\tau ^{ - 1}}} \right){{\hbar}} }}} \right)\quad, $$ (6) where
$ {{\rm{k_B}}} $ is the Boltzmann constant,$ {\rm{e}} $ is the charge of the electron, and${{\hbar}} $ is the reduced Planck constant. Specifically, the chemical potential of graphene$ {\mu _{\rm{c}}} $ can be effectively tuned by an externally applied voltage[20-21, 24-25].$$ {\mu _c}\left( t \right) = {{\hbar}} {v_{\rm{F}}}\sqrt {\frac{{{\text{π}} {C_{{\rm{ox}}}}{V_{\rm{g}}}\left( t \right)}}{{\rm{e}}}} \quad,$$ (7) where
${v_{\rm{F}}}$ is the Fermi velocity,$ {C_{{\rm{ox}}}} $ is the gate capacitance[26], and$ {V_{\rm{g}}}\left( t \right) $ is the applied voltage that changes periodically with time. In this paper, the curve of the applied bias voltage of graphene nanodisk changing periodically with time is shown in Fig. 1(b) (Color online). By changing the bias voltages V1(t), V2(t) and V3(t) applied to the graphene nanodisks, the graphene chemical potentials$ {\mu _{{\rm{c}}1}}(t) $ ,$ {\mu _{{\rm{c}}2}}(t) $ and$ {\mu _{{\rm{c}}3}}(t) $ are changed.3. Numerical simulation results and discussion
3.1 Graphene plasmonic time crystal model and properties
First, we explored the energy band topology of 2D graphene plasmonic crystals composed of graphene nanodisk arrays arranged in a triangular lattice. The time crystal structure of graphene plasmon is shown in Figure 2 (a). The dotted line area is the crystal cell, and the solid line part is the Brillouin region of the crystal, where
$\varGamma-{{M}}-{K}-\varGamma$ is the reduced wedge of Brillouin region, a is the crystal lattice constant, a=40 mm, and the radius of graphene nanodisk is expressed as r and r=0.21a.$ {\mu _{{\rm{c}}1}} $ ,$ {\mu _{{\rm{c}}2}} $ and$ {\mu _{{\rm{c}}3}} $ are the chemical potentials of different graphene nanodiscs, respectively. From equation (7), it can be seen that the chemical potentials of these graphene nanodiscs can also change periodically with time, which can be written as$ {\mu _{{\rm{c}}1}}(t) $ ,$ {\mu _{{\rm{c}}2}}(t) $ and$ {\mu _{{\rm{c}}3}}(t) $ [27-29]. The chemical potential of graphene nanodisks can be flexibly controlled in time by applying periodic changed bias voltage of V1(t), V2(t) and V3(t). Here, we calculate the energy bands of graphene plasmonic time crystals at several moments in one cycle, as shown in Fig. 2(b) (Color online), where the black line depicts that, when t=0 s (i.e.$\Delta {\mu _{\rm{c}}} $ =$ {\mu _{{\rm{c}}1}} - {\mu _{{\rm{c}}2}} $ =0 eV), the chemical potentials of the graphene nanodisk are${\mu _{{\rm{c}}1}} $ =$ {\mu _{{\rm{c}}2}}$ =0.3 eV and${\mu _{{\rm{c}}3}}$ =0.6 eV, respectively, the two energy bands degenerate at the K point and intersect at the Dirac point, and there is obviously no band gap. During the periodic change of chemical potential with time, the time-translation symmetry of the graphene plasmonic time crystal is broken, the Dirac cone dispersion will be split, and the band gap will be opened[30]. The green, blue and red curves in Fig. 2(b) depict the band structures of the following three time nodes:${t_1} $ =1 e−12s (i.e.$\Delta {\mu _{\rm{c}}} $ =0.1 eV),${t_2} $ =1.5 e−12s (i.e.$\Delta {\mu _{\rm{c}}} $ =0.12 eV) and${t_3} $ =2 e−12s (i.e.$\Delta {\mu _{\rm{c}}} $ =0.14 eV), respectively. It is clear that the crystal band gap undergoes a process from closing to opening over time. It is worth noting here that this graphene plasmonic time crystal energy band change over time can be performed without changing the geometry.
Figure 2. (a) Schematic diagram of graphene plasmon time crystal structure. (b) Energy band diagrams of graphene plasmon time crystals at four different moments in a cycle of external bias voltage changeIn order to verify the properties of graphene plasmonic time crystals, a numerical simulation study is designed in this work. As shown in Fig. 3(d), in a region composed of graphene plasmonic time crystals, an excitation source is placed at point P, and the relationship between the chemical potentials of graphene nanodisks satisfies the variation law shown in Fig. 3 (a). Here, electromagnetic waves with a frequency of 46.50 THz are excited from the excitation source. We further explored the change trend of crystal energy band gap with the change of graphene chemical potential. By analyzing the opening and closing of the band gap, the propagation phenomenon occurs when the frequency of the excitation wave is in the conduction band; however, when the frequency is in the band gap, the wave will not propagate in this time interval[31-33]. The simulation results are shown in Figure 3 (Color online), in which screenshots are taken of the electric fields at four time nodes during the transmission process. With the periodic change of
$ \Delta {\mu _{\rm{c}}} $ in time, the transmission and inhibition alternate phenomenon occurs in the timing of${t_{\rm{b}}} $ =3.20 e−12s,${t_{\rm{c}}} $ =4.16 e−12s,${t_{\rm{d}}} $ = 5.82 e−12s and${t_{\rm{f}}} $ =8.24 e−12s, respectively. This result can well illustrate that the graphene plasmonic time crystal can realize the periodic opening and closing of the energy band gap with time by periodically adjusting the chemical potential.
Figure 3. (a) When μc3=0.6 eV, the relationship between ∆μc and t. (d) A region composed of 5×10 graphene plasmon time crystals. P is the position of the excitation source. (b), (c), (e) and (f) Screenshots of four moments in the propagation process at the time nodes of${t_{\rm{b}}} $ =3.20 e−12s,${t_{\rm{c}}} $ =4.16 e−12s,${t_{\rm{d}}} $ =5.82 e−12s and${t_{\rm{f}}} $ =8.24 e−12s, respectively3.2 Zigzag topological boundary structure theory and model
When the time-translational symmetry of the graphene plasmonic time crystal is broken, the Dirac cone dispersion is not preserved. From the point of view of group theory, when the inversion symmetry is broken, the group symmetry of the point (K or K′) will be reduced from C3v to C3. In the energy band, the Dirac cone is destroyed, and the two energy bands originally degenerated to the point K will be opened[34-35]. By analyzing the change of the orbital angular momentum at the valley of the energy band after the energy band is opened, we obtain the phase distribution at point K at time of
${t_1} $ =1 e−12s (i.e.$\Delta {\mu _{\rm{c}}} $ =0.1 eV),${t_2} $ =1.5 e−12s (i.e.$\Delta {\mu _{\rm{c}}} $ =0.12 eV) and${t_3} $ =2 e−12s (i.e.$\Delta {\mu _{\rm{c}}} $ =0.14 eV). As shown in Figure 4 (Color online), it can be seen that, at different times, the two energy bands appear the phase distribution of Left-handed Circular Polarization (LCP) and Right-handed Circular Polarization (RCP) at the degenerate point K, respectively, that is, in the process of$ \Delta {\mu _{\rm{c}}} $ changing with time, the energy valley at any moment has the circularly polarized orbital angular momentum in the opposite direction after the time crystal energy band is opened. Therefore, we construct electromagnetic transport at the topological edge through electromagnetic modes that can be loaded with different orbital angular momentums, which provides a theoretical basis for the use of topological boundary to construct slow light waveguides.
Figure 4. The phase distributions of Left-handed Circularly Polarized (LCP) and Right-handed Circularly Polarized (RCP) of the time crystal appear at point K, which are expressed as the component of the electric field in the direction Z and the in-plane Poynting vector (Px, Py). (a) and (d) the phase distribution diagram of point K at the time of$ \Delta {\mu _{\rm{c}}} $ =0.1 eV; (b) and (e) the phase distribution diagram of point K at the time of$ \Delta {\mu _{\rm{c}}} $ =0.12 eV; (c) and (f) the phase distribution diagram of point K at the time$ \Delta {\mu _{\rm{c}}} $ =0.14 eVIn this study, Zigzag topological boundaries are used to construct slow light waveguides. This topology is constructed from graphene plasmonic time crystals, which generate “temporal topological boundary states”, they are temporal analogs of topological edge states[13]. Breaking the time-translational symmetry of graphene plasmonic time crystals will lead to the opening of Dirac points at band degeneracy, thus forming a full band gap in which topologically protected boundary modes exist. The red dotted line in Figure 5(a) (Color online) is the Zigzag topology interface. By sampling the area within the black dotted line, a finite period (N=19) supercell model is established in the commercial simulation software COMSOL Multiphysics. Part of the simulation results are shown at the bottom of Fig. 5(a) (Color online), and it can be seen that the electric field distribution is concentrated at the boundary. We also calculated the dispersion relation of this boundary mode at different times (Fig. 5(b)), and the gray area represents the projected body energy band diagram. Fig. 5(b) also depicts the projected energy bands of three boundary states at the time
${t_1} $ =1 e−12s,${t_2} $ =1.5 e−12s and${t_3} $ =2 e−12s[36]. Obviously, when$ \Delta {\mu _{\rm{c}}} $ changes with time, there is always a corresponding boundary state at each moment, which means that the time simulation based on the Zigzag topology boundary state of the graphene plasmon time crystal is realized.
Figure 5. (a) Schematic diagram of the Zigzag interface based on graphene plasmon time crystals, in which the bottom is the calculation model of the finite period super cell unit and the simulation electric field distribution results. (b) The dispersion curves of the Zigzag interface mode at different times3.3 Slow light phenomenon and field enhancement effects
By further analyzing the dispersion relation of the above boundary modes, we can deduce a phenomenon that the group velocity is zero (slow light) at the extreme point of the dispersion curve. In order to better study the group velocity of guided modes existing in this boundary state, the partial dispersion curves at three different times are plotted in Fig. 6(a) (Color online). It is obvious that the dispersion is accompanied by severe bending, which leads to the existence of slow light mode in the gap. Based on the dispersion curves of these boundary modes, the change curve of the group velocity with angular frequency w can be calculated with the relationship
${v_{\rm{g}}} = {{{\rm{d}}\omega } /{{\rm{d}}k}}$ [37-38]. As shown in Figure 6(b) (Color online), the red curve is the group velocity at${t_1} $ =1 e−12s and$\Delta {\mu _{\rm{c}}} $ =0.1 eV, the group velocity tends to zero, and the frequency to 50.2370 THz. At${t_2} $ =1.5 e−12s and$\Delta {\mu _{\rm{c}}} $ =0.12 eV, the electromagnetic wave group velocity is close to zero at the frequency of 50.1149 THz, which is depicted in Fig. 6(b) by the blue curve. At${t_3} $ =2 e−12s and$\Delta {\mu _{\rm{c}}} $ =0.14 eV, the group velocity is close to zero at a frequency of 49.9292 THz, which is marked by the green curve. Numerical calculation results show that the group velocity of electromagnetic waves with different frequencies reach zero at different moments, that is, the topological boundary can realize slow light that its certain frequency electromagnetic wave group velocity near zero at different moments.
Figure 6. (a) The boundary mode dispersion curve supported by the topological boundary under different$ \Delta {\mu _{\rm{c}}} $ . (b) The relationship between group velocity and frequency under different$ \Delta {\mu _{\rm{c}}} $ Finally, with the slow light waveguide model given in Fig. 1(a), a two-dimensional modeling is carried out in the simulation software COMSOL, the Zigzag boundary structure is used as the transmission channel, and the excitation source is located at P (as shown in Fig. 1(a)). The chemical potential of graphene nanodisks can be flexibly controlled in time by applying a bias voltage that varies periodically with V1(t), V2(t), and V3(t). The graphene SPP wave with a certain frequency is emitted from point P, and the whole process of its transmission in the waveguide over time can be obtained. Figure 7 (Color online) shows the electric field intensity distribution results of plane waves with frequencies of 50.2370 THz, 50.1147 THz and 49.9292 THz propagating in the waveguide respectively. In this simulation, the plane wave with a frequency of 50.2370 THz is continuously excited from point P at
$ {t_0}$ =0 s , and reaches a state where the group velocity is close to zero after traveling along the waveguide for time$ 1\;{{\rm{e}}^{ - 12}}{\rm{s}} $ , and the SPP wave transmission stagnates at the position of X=307.998 nm. Figure 7(a) shows the electric field screenshot of a plane wave with a frequency of 50.2370 THz at time$ {t_1} $ =1 e−12s during the transmission process. After the plane wave group velocity is close to zero, the excitation in the slow light mode is continued, and it is found that the electric field intensity at the position of X=307.998 nm (the green dot in Fig. 7(a)) reaches the maximum value at t=1.4 e−12s, and the electric field intensity at this time is much greater than that at the excitation source. Plotting the electric field intensity in the waveguide at t=1.4 e−12s in Fig. 7 (b), it can be found that the field intensity B at the light energy capture point is greater than the field intensity A at the excitation source. By plotting the change process of the electric field intensity at the green dot position in the time period from$ {t_0} $ =0 s to t=1.4 e−12s is depicted in Fig. 7(c), we can see that the waveguide achieves a continuous superposition of the electric field strength at the light energy trapping point (green dot) and is higher than the enhancement effect of the excitation source due to the field enhancement effect in slow light transmission[4]. Figure 7(d) shows the electric field distribution of the SPP wave with a frequency of 50.1147 THz from point P, which is continuously excited from time$ {t_0} $ =0 s, and propagates along the waveguide for time$1.5\;{{\rm{e}}^{ - 12}}{\rm{s}}$ . When SPPs wave is transmitted to${t_2} $ =1.5 e−12s, the group velocity at X=667.879 nm decreases to zero. In addition, we also continuously observed the electric field at the light energy capture point X=667.879 nm (blue dot in Fig. 7 (d)), and obtained the cross-section of the electric field distribution at the boundary at t=2.98 e−12s, as shown in Fig. 7 (e). This result shows that at time t=2.98 e−12s, the electric field intensity at the light energy capture point reaches the maximum D and is greater than the electric field intensity C at point P. Figure 7(f) depicts the superposition process of the electric field strength at the blue dot from time$ {t_0} $ =0 s to t=2.98 e−12s. Similarly, as shown in Fig. 7(g), the SPP wave with a frequency of 49.9292 THz is continuously excited from point P at time$ {t_0} $ =0 s, and reaches a near-zero group velocity after traveling along the waveguide for$ 2\;{{\rm{e}}^{ - 12}}{\rm{s}} $ . The light energy trapping is located at X=1029.122 nm (red dot in Fig. 7(g)). Continuous excitation is carried out in the slow light mode, and it is observed that the electric field at the light energy capture place reaches the maximum intensity F at time t=3.22 e−12s, which is much greater than the electric field intensity E at the excitation source, as shown in Figure 7(j). During the time period from$ {t_0} $ =0 s to t=3.22 e−12s, the change process of the electric field intensity at the position of the red dot is depicted in Fig. 7(j), which also experienced a process of superposition of the electric field intensity. The above simulation process takes SPP waves with three different frequencies, and modulates the light group velocity close to zero by changing the applied bias voltage on the graphene nanodisk respectively. It can be seen that under the influence of the field enhancement effect, the waveguide realizes the continuous superposition of electric field intensity at the light energy capture point, which is higher than that at the excitation source. These results show that the slow light waveguide can make the SPP waves with different frequencies stop at different times and positions, and the field enhancement effect occurs at the corresponding optical energy capture points. In this work, graphene plasmonic time crystals are used to construct transmission waveguides, realizing the modulation of slow light group velocity simply by tuning the chemical potential of graphene. Compared with traditional waveguides such as photonic crystal line defect waveguides[19], its structure is simpler. Even if the structure is fixed, the waveguide performance can be flexibly modulated by changing the applied voltage, which greatly improves the electrical tunability of slow light waveguides. In addition, the SPP wave propagates at the topological boundary as an evanescent wave whose electric field amplitude is continuously attenuated, and the longer the transmission distance, the stronger the attenuation. By applying continuous excitation at the excitation source and recording the field strength superposition process at the optical energy capture point of the waveguide, we intuitively show the electric field change caused by the field strength superposition effect in time, which is not found in the previous slow light research work[11]. Since the modes in the waveguide are not discrete, degeneracy of the forward and backward modes will occur, resulting in energy loss. Therefore, when the optical group velocity is zero in the time sequence, the phenomenon of completely stopping the optical group velocity, the so-called "stationary rainbow", does not actually occur.
Figure 7. (a, d, g) Screenshots of the electric field intensity distribution at t1, t2 and t3. (b, e, h) Cross-sectional electric field diagrams at the Zigzag boundary at different time t. (c, f, j) The changing process of the electric field at the capture point of light energy4. Conclusion
In this paper, the time crystal band gap is periodically opened and closed by dynamically controlling the periodic change of the external voltage of the graphene nanodisk with time. Slow light transmission with the group velocity near zero is realized by the slow light waveguide made of graphene plasmon time crystal. The simulation results show that when the light propagates in the waveguide, a group velocity close to zero can be obtained, which is accompanied by the appearance of the field enhancement effect at the corresponding optical energy trapping point of the waveguide. In the case of adjusting the external bias voltage, the operating frequency of the slow optical waveguide can be effectively adjusted. Our research will be applied in the future to devices in the fields of on-chip light buffering and enhanced light-matter interactions.
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图 1 (a) 慢光波导的三维示意图，顶部是单层石墨烯纳米盘。石墨烯与空气接触，背景材料是 SiO2，衬底材料是Si。对不同的石墨烯纳米圆盘分别加载不同的偏置电压：V1(t)，V2(t)和V3(t)，右上图为电压的施加方式。(b) 石墨烯纳米盘外加偏置电压随时间周期性变化的曲线
Figure 1. (a) The three-dimensional schematic of slow light waveguide, with a single layer of graphene nanodisks at the top. The graphene is exposed to air on the top, the background material is SiO2, and the substrate material is Si. Different graphene nanodisks are applied with different bias voltages: V1(t), V2(t) and V3(t). The diagram on the right shows how voltage is applied. (b) The graphene nanodisk’s external bias voltage changes periodically with time
图 2 (a) 石墨烯等离激元时间晶体结构示意图。(b)在一个外加偏置电压变化周期内，石墨烯等离激元时间晶体在4个不同时刻的能带图
Figure 2. (a) Schematic diagram of graphene plasmon time crystal structure. (b) Energy band diagrams of graphene plasmon time crystals at four different moments in a cycle of external bias voltage change
图 3 (a) 当
$ {\mu _{{\rm{c}}3}} $ =0.6 eV时，$ \Delta {\mu _{\rm{c}}} $ 与t的关系。(d) 5×10石墨烯等离激元时间晶体组成的一个区域，P是激发源的位置。(b)，(c)，(e)和(f)是传播过程中4个时刻的截图，时间节点分别是${t_{\rm{b}}} $ =3.20 e−12s,${t_{\rm{c}}} $ =4.16 e−12s,${t_{\rm{d}}} $ =5.82 e−12s和${t_{\rm{f}}} $ =8.24 e−12sFigure 3. (a) When μc3=0.6 eV, the relationship between ∆μc and t. (d) A region composed of 5×10 graphene plasmon time crystals. P is the position of the excitation source. (b), (c), (e) and (f) Screenshots of four moments in the propagation process at the time nodes of
${t_{\rm{b}}} $ =3.20 e−12s,${t_{\rm{c}}} $ =4.16 e−12s,${t_{\rm{d}}} $ =5.82 e−12s and${t_{\rm{f}}} $ =8.24 e−12s, respectively
图 4 时间晶体在K点分别出现了左旋圆极化(LCP)和右旋圆极化(RCP)的相位分布，表示为电场在Z方向上的分量和面内坡印亭矢量(Px, Py)。(a)和(d)是在
$ \Delta {\mu _{\rm{c}}} $ =0.1 eV时刻，K点的相位分布图；(b)和(e)是在$ \Delta {\mu _{\rm{c}}} $ =0.12 eV时刻，K点的相位分布图；(c)和(f)在$ \Delta {\mu _{\rm{c}}} $ =0.14 eV时刻，K点的相位分布图Figure 4. The phase distributions of Left-handed Circularly Polarized (LCP) and Right-handed Circularly Polarized (RCP) of the time crystal appear at point K, which are expressed as the component of the electric field in the direction Z and the in-plane Poynting vector (Px, Py). (a) and (d) the phase distribution diagram of point K at the time of
$ \Delta {\mu _{\rm{c}}} $ =0.1 eV; (b) and (e) the phase distribution diagram of point K at the time of$ \Delta {\mu _{\rm{c}}} $ =0.12 eV; (c) and (f) the phase distribution diagram of point K at the time$ \Delta {\mu _{\rm{c}}} $ =0.14 eV
图 5 (a) 基于石墨烯等离激元时间晶体构成的Zigzag边界示意图。其中，底部分别是有限周期超胞单元的计算模型和仿真电场分布结果。(b)不同时刻下Zigzag边界模的色散曲线
Figure 5. (a) Schematic diagram of the Zigzag interface based on graphene plasmon time crystals, in which the bottom is the calculation model of the finite period super cell unit and the simulation electric field distribution results. (b) The dispersion curves of the Zigzag interface mode at different times
图 6 (a) 不同
$ \Delta {\mu _{\rm{c}}} $ 下拓扑边界所支持的边界模色散曲线。(b)不同$ \Delta {\mu _{\rm{c}}} $ 下群速度随频率的变化关系图Figure 6. (a) The boundary mode dispersion curve supported by the topological boundary under different
$ \Delta {\mu _{\rm{c}}} $ . (b) The relationship between group velocity and frequency under different$ \Delta {\mu _{\rm{c}}} $ 
图 7 (a, d, g) t1，t2和t3时刻电场强度分布截图。(b, e, h) 不同t时刻Zigzag边界处截面电场图。(c, f, j) 光能捕获点处电场变化过程
Figure 7. (a, d, g) Screenshots of the electric field intensity distribution at t1, t2 and t3. (b, e, h) Cross-sectional electric field diagrams at the Zigzag boundary at different time t. (c, f, j) The changing process of the electric field at the capture point of light energy
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