1. Introduction

The invention of metamaterials has garnered many academic and technological interests due to their extraordinary electromagnetic (EM) properties, which are not easily obtained in the existing photonics and optics world. Plasmonic metamaterials utilize surface plasmons (SPs) which exhibit the most compelling EM properties among the ramifications of the metamaterial [1]. Structures comprising noble metals, such as gold and silver, can overcome the limitations of classical diffraction limit and realize a photonic loop in the sub-wavelength range of light waves; however, significant energy loss limits practical applicability due to the poor binding effect on incoming light and scattering effect [2]. Gr, a two-dimensional material with a hexagonal lattice, has similar properties as noble metals in the THz and infrared regime [3]. Most importantly, Gr can excite SPs with strong localization ability, low transmission loss, high tunability, and high group delay. These properties make Gr metamaterial-based structures advantageous in applications including optical cloaking [4], perfect lens [5], modulators [6], photodetectors [7,8], polarizers [9], bandstop filtering [10], and slow light devices [11].

Bandstop filters are essential components of information processing systems. Recently, plasmonic bandstop filters have received considerable attention from researchers. Conventional metal-based bandstop filters operate well in the visible and near-infrared ranges; however, their applications are limited by the poor SP confinement in the THz frequency range. Contrarily, Gr-based bandstop filters exhibit strong SP confinement providing excellent performance in the THz spectral band. Tunability of resonant frequency without varying structural parameters is another distinguishing attribute for Gr-based structures, whereas tuning characteristics are not easily obtained in metal-based structures where liquid crystals or liquid metals with variable permittivity have been proposed. However, tunning is achievable by simply changing the chemical potential of Gr with an external gate voltage due to Gr’s unique energy band structure. This allows the fine-tuning of resonant frequencies and group velocities in Gr-based bandstop filters. Graphene nanoribbon (GNR)-Gr disk [12], multilayer Gr metamaterial [13], continuously patterned Gr monolayer [14,15], H-shape Gr-metamaterial [16], split-ring resonator [17], and asynchronous crisscross structures [18] were reported; however, extinction ratio and quality factor, two performance parameters of bandstop plasmonic filter, were quite small. Therefore, there is a scope for better Gr metamaterial-based design to improve the performance of the bandstop optical filters.

Gr-based plasmonic structures play a vital role in slow light applications, such as optical buffers and quantum memory, by the tuning properties of SPs originated by the interaction of Gr and incoming light. Previously, we explored multilayered Gr structures to obtain plasmonic modes for developing polarizers and optical switches [6,9]. A slow light device can slow down the speed of light pulses propagating through it. The concept behind slow light is based on the phenomenon of light propagation in a medium with a refractive index that can be manipulated by increasing the group delay of light [19,20]. Group delay, a performance parameter of slow light devices, is a measure of how much the light in a medium is delayed compared to the speed of light in the vacuum. Multilayered Gr and different shapes of GNRs exhibited slow light properties due to light-matter interactions; however, the group delay and delay-bandwidth product of the previous reports were low [13,14,17]. Zhao et al. proposed a ring metamaterial structure comprising noble metals, which provided a high group delay; however, tunability can not be possible without changing structural parameters [21]. Numerous Gr-based plasmonic nanostructures focusing on single and double resonant bandstop modes were reported in the THz frequency regime; however, nanostructures with multiple bandstop modes, such as quadband and pentaband, were rare. Moreover, wide tuning in the THz region was hardly explored. Multiple bandstop modes can provide the flexibility of selecting a mode from multiple bandstop modes without adjusting chemical potential or voltage. Hence, there is scope for significant improvement in slow light device performance with wide tunability for THz interconnects, quantum memory, and optical buffer applications.

In this work, we proposed two multiband THz modulator structures comprising GNR, Gr nanoholes, and polymethyl methacrylate (PMMA). We employed the FDTD simulation method to optimize the structures and evaluated the performance parameters of the devices. These structures exhibited dips in their transmittance spectra that arose from the interaction between the incoming light and Gr-based metamaterials. We utilized four- and five-patterned Gr layers on the PMMA substrate for the quadband and pentaband plasmonic modulators, respectively, where each Gr layer acted as an attenuating medium. Our proposed modulators achieved high ER and group delay at resonant frequencies due to strong light-matter interactions, making them promising for slow light applications. Moreover, the plasmonic modulators demonstrated high tunability by varying the chemical potentials of Gr layers, which acted as a perfect multi-bandstop filter. We analyzed the polarization sensitivity of our proposed modulators when they were used as bandstop filters. Additionally, we provided a comprehensive comparative study of performance parameters among previously reported Gr-based and our proposed THz modulator structures.

2. Methodology

A unit cell of the Gr-based plasmonic quadband modulator is illustrated in Fig. 1(a). Contrary to the conventional modulator designs, the proposed structure used Gr’s excellent SPP and metal-like properties to enhance the modulator’s performance. The quadband modulator was designed by comprising four Gr monolayers, which were made up of GNR and Gr nanoholes on PMMA substrate as illustrated in Fig. 1(a). Here, the substrate height was selected to be 2 $\mu$m to avoid plasmonic coupling and signal interference. We set the GNR on the modulator’s top Gr layer. Twelve periodic airstrips or eleven GNRs on the Gr monolayer were adopted for our proposed modulators. Resonant modes overlapped each other when we reduced and increased GNRs, as depicted in Fig. S1 of Supplement 1. The chemical vapor deposition (CVD) technique can grow such a Gr monolayer on PMMA [22]. We provided a section of suggested fabrication techniques using CVD and lithography methods, as shown in Section 2 of Supplement 1. Figure 1(b) presents the GNR layer of the proposed structures with length, q of 2.5 $\mu$m, width, w of 10 nm, and pitch, d of 20 nm. The inset shows the zoomed view of the GNR layer. The period, p, of the proposed structures was set to be 3 $\mu$m. The widths of the nanoholes, i = j = k of 150 nm in the rest of the Gr layers, were adopted with length, q of 2.5 $\mu$m. To design a pentaband plasmonic modulator, we utilized an additional Gr layer with a nanohole below, where the width and length of the Gr nanohole were the same as before. Figure 1(c) illustrates the schematic representation of a unit cell of a pentaband plasmonic modulator. The surface current, J, of the Gr layer was modeled by employing the equation of Ohm’s law: J = $\sigma _g$E, where $\sigma _g$ and E are the Gr’s complex surface conductivity and electric field, respectively. We modeled the $\sigma _g$ of Gr by using Kubo’s formulation (see Section 3 of Supplement 1) [23]. This formulation was modelled specifically for the $\sigma _g$ of Gr monolayer; however, it can be conducive to explaining multilayer Gr’s $\sigma _g$ [24]. The interband conductivity term can be neglected due to $E_F \gg k_B T$ in the THz frequency regime where $E_F$, $k_B$, and $T$ denote the Fermi energy level, Boltzmann constant, and temperature, respectively. The $\sigma _g$ of Gr is given by,

 

Fig. 1. (a) Illustration of a quadband plasmonic modulator unit cell. Here, four Gr monolayers were employed, and the structural parameters were: i = j = k = 150 nm, h = 2 $\mu$m. The output transmission signal attenuated and deviated from the source signal due to scattering. (b) Top Gr layer of the proposed structures where the period of the structure, p = 3 $\mu$m, length of GNR, q = 2.5 $\mu$m. The inset shows the magnified view of GNR. Here, the pitch distance of GNR, d is 20 nm, width, w = 10 nm. (c) Schematic view of a pentaband plasmonic modulator unit cell with geometrical parameters: i = j = k = l = 150 nm. $V_{g_n}$ is the applied voltage source of the n-th Gr layer via a metal contact. The $V_{g_n}$ varied the $\mu _{c_n}$ of Gr layers and $\sigma _g$.

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Here, $e$, $\hbar$, and $\omega$ are the electron charge, reduced Plunck’s constant, and angular frequency, respectively. $\tau$ is the relaxation time of the Gr layer, which can be expressed by $\tau = \mu E_F / e{V_{F}}^{2}$. The Fermi velocity, $V_{F}$, and mobility, $\mu$ were set to be 10$^6$ m/s, and 0.4$m^2V^{-1}s^{-1}$ respectively [25]. The real and imaginary surface conductivity of Gr as a function of $E_F$ and frequency are shown in Figs. 2(a) and (b). The imaginary part of the Gr conductivity rises while the real part stays constant as $E_F$ increases. Therefore, Gr exhibits more metal-like properties with the increase of $E_F$ producing SPPs. Figures 2(c) and (d) illustrate the real and imaginary surface conductivity of the Gr layer at $E_F$ = 0.6 eV. The refractive index data of PMMA was adapted from Fan et al. [26]. We conducted the study utilizing the three-dimensional (3D) FDTD simulation technique (Ansys Lumerical). The 3D FDTD simulation range was set to be 3 $\mu$m $\times$ 3 $\mu$m $\times$ 35 $\mu$m. We utilized the periodic boundary conditions in the x- and y-directions and 24-steep angle perfectly matched layers (PMLs) in the z-direction to absorb light after the simulation region without back reflections. We employed a transverse magnetic (TM) polarized plane-wave light source on the top of the proposed modulators along the z-direction with a center frequency of 3 THz and a frequency span of 5.7 THz. To consider the room temperature for our study, the temperature of the simulation environment was set to 300 K. FDTD is a time-varying and time-consuming technique: larger mesh sizes shorten the computational time, which results in low accuracy, and smaller mesh sizes provide high accuracy, increasing computational time and memory size. In our work, smaller mesh sizes were utilized in the region of interest to resolve tiny features, whereas bigger mesh sizes were used in other regions. The non-uniform mesh was utilized to complete simulations in a reasonable amount of time while maintaining the memory requirements. We utilized a mesh size of 8 nm in the GNR and Gr nanoholes for better accuracy in our simulations. Near-field monitors were utilized to obtain the spatial electric field profiles. Transmittance and reflectance spectra were recorded by using power monitors.

 

Fig. 2. Evolution of (a) Re ($\sigma _g$) and (b) Im ($\sigma _g$) of Gr layer at different Fermi levels. (c) Re ($\sigma _g$) and (d) Im ($\sigma _g$) parts of the Gr layer at the $E_F$ of 0.6 eV. Gr exhibited more metallic behaviour with increasing Fermi energy levels.

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3. Results and discussion

Interaction between the incident plane wave’s electric field and free electrons or plasmons of the Gr layer resulted in SP resonance on the surface of the Gr layers of the proposed modulators. As a result, the modulators attenuated light by scattering that created dips in transmittance spectra at resonant frequencies. Based on this phenomenon, we utilized each Gr layer as an attenuating medium at the resonant frequency. When the chemical potentials of Gr layers were set to be the same, multiple modes were coupled at the resonance frequency. We varied the chemical potentials of each Gr layer via external sources to obtain distinguishable plasmonic modes primarily originating from each layer (see Section 4 of Supplement 1 for details). Most importantly, this phenomenon will be beneficial for designing tunable plasmonic bandstop filters and slow light devices.

3.1 Bandstop filtering applications

3.1.1 Quadband filter

We utilized the Gr layer’s light-blocking phenomenon to obtain the filtering mechanism by enhancing surface light-matter interactions. This blockage of light was the origin of the bandstop plasmonic filter. Based on this mechanism, we utilized four engineered Gr layers with different chemical potentials, providing four transmittance dips due to strong light-matter interaction at resonant frequencies. Figure 3(a) depicts the transmittance spectra of the proposed quadband plasmonic filter structure. Here, chemical potentials of Gr layers $\mu _{c_1}$ = 0.28, $\mu _{c_2}$ = 0.20, $\mu _{c_3}$ = 0.40, $\mu _{c_4}$ = 0.52 eV were adopted. Chemical potentials were meticulously selected so that the resonant modes did not overlap each other. Four transmittance dips formed at the resonant frequencies of 1.746, 2.487, 4.197, and 5.451 THz, respectively. Figure 3(b) represents the transmitted THz wave in the time domain. The structure experienced a huge deviation from the reference source THz wave because of attenuation by scattering. The transmittance, reflectance, and absorptance spectra are given in Fig. S6(a) of Supplement 1. To get a better understanding, we presented the spatial electric field distributions of our proposed plasmonic quadband modulator in Figs. 3(c)–(f). Each Gr layer primarily localized the electric light field at a resonant frequency. First, second, third, and fourth transmittance dips originated from the second, third, first, and fourth Gr layers. Scattering parameters of the quadband plasmonic modulator exhibited four dips at resonant frequencies, which are provided in Fig. S7(b) of Supplement 1.

 

Fig. 3. (a) Transmittance spectra of the proposed quadband plasmonic filter. Here, $\mu _{c_1}$ = 0.28, $\mu _{c_2}$ = 0.20, $\mu _{c_3}$ = 0.40, and $\mu _{c_4}$ = 0.52 eV were employed. (b) The E-field time-domain waveforms of the proposed quadband plasmonic filter at the normal incidence for TM polarized incident THz light. The transmitted waveform attenuated and deviated from the source THz waveform. The spatial E-field distribution of Gr layers at resonant frequencies of (c) 1.746, (d) 2.487, (e) 4.197, and (f) 5.451 THz for our proposed quadband plasmonic filter. The top, second, third, and bottom Gr layers showed localized E-fields at the resonant frequencies of 4.197, 1.746, 2.487, and 3.385 THz, respectively. The logarithmic scale was utilized to represent the color map of the E-field.

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3.1.2 Pentaband filter

A pentaband plasmonic filter was modeled utilizing multilayered Gr layers with different chemical potentials. Five engineered Gr layers with different chemical potentials were employed to block light by enhancing surface light-matter interactions, as seen in Fig. 1(c). The chemical potentials of five Gr layers were set to be $\mu _{c_1}$ = 0.28, $\mu _{c_2}$ = 0.18, $\mu _{c_3}$ = 0.45, $\mu _{c_4}$ = 0.65, and $\mu _{c_5}$ = 0.50 eV. Five transmittance dips emerged at the resonant frequencies of 1.404, 2.145, 3.171, 4.197, and 5.337 THz, as shown in Fig. 4(a). Figure 4(b) represents the transmitted THz wave in the time domain. Pentaband filter structure experienced a larger attenuation with prolonged time than quadband filter structure due to scattering among Gr layers and localized light field. The transmittance, reflectance, and absorptance spectra are provided in Fig. S6(b) of Supplement 1. We studied the spatial electric field distributions of our proposed structure to elevate our knowledge of the localized electric fields, which were depicted at resonant frequencies in Figs. 4(c)–(g). Scattering parameters of the pentaband plasmonic modulator exhibited five dips at resonant frequencies, as shown in Fig. S7(c) of Supplement 1. The electric field of incoming light was localized at resonant frequencies by considering each Gr layer as an attenuating medium. First, second, third, fourth, and fifth transmittance dips emerged from the second, third, fourth, first, and fifth Gr layers.

 

Fig. 4. (a) Transmittance spectra of the proposed pentaband plasmonic filter. Here, $\mu _{c_1}$ = 0.28, $\mu _{c_2}$ = 0.18, $\mu _{c_3}$ = 0.45, $\mu _{c_4}$ = 0.65, and $\mu _{c_5}$ = 0.50 eV were adopted. (b) The E-field waveforms of the proposed pentaband plasmonic filter in the time domain for TM polarized incident THz light. The spatial E-field distributions of Gr layers at resonant frequencies of (c) 1.404, (d) 2.145, (e) 3.171, (f) 4.197, and (g) 5.337 THz for our proposed pentaband plasmonic filter. The top, second, third, fourth, and bottom Gr layers exhibited localized E-fields at the resonant frequencies of 4.197, 1.404, 2.145, 3.171, and 5.337 THz, respectively. The logarithmic scale was utilized to represent the E-field color map.

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3.1.3 Plasmon tuning

The interaction between the electric field of incident light and the Gr’s plasmons or free electrons resulted in a plasmonic mode on the Gr’s surface. Here, the engineered Gr layer worked as an attenuation medium. When the chemical potentials of Gr layers were set to be the same, multiple modes were coupled, as shown in Figs. S3, S4, and S5 of the Supplement 1. Hence, we differentiated the chemical potentials of the Gr layers via an external source to obtain distinguishable plasmonic modes. Meanwhile, the external voltage source changed the surface conductivity of the Gr layer. Additionally, the resonance frequency of plasmonic modes could be tuned. High tunability without structural modifications for our proposed plasmonic modulators can be achieved by changing surface conductivity via applying $V_{g_n}$. Meanwhile, the surface conductivity varied the $\mu _c$ of Gr layers [6,9,27]. The relationship between $V_{g_n}$ and $\Delta \mu _c$ is given by [28],

Here, $\epsilon _0$, e, $\epsilon _{\mbox {PMMA}}$, $\Delta E_F$, and $V_{Dirac}$ represent the permittivity of vacuum, the charge of an electron, PMMA layer’s permittivity, change of Fermi level, and the Dirac voltage of Gr layer, respectively. Voltage source varied the change of the Gr layer’s chemical potential, as can be seen in Fig. S8 of Supplement 1. Here, one assumption was made that the impedance of metal contacts is negligible; therefore, it is not considered in our study. A color map of transmittance for our proposed plasmonic modulators displays the structure’s wide tunability evolution at different chemical potentials, shown in Fig. 5. The third and the fourth resonant modes of our proposed quadband and pentaband modulators, respectively, were highly tunable, which were consistent with previous report [6]. These plasmonic modulators can be operated in the wide range from 1.25 to 5.85 THz by varying the $\mu _c$ of Gr layers from 0.12 to 0.80 eV. It can be inferred from Fig. 5 that the dips in transmittance are blue-shifted when chemical potentials rise. Initially, we adopted the incident light’s frequency span from 0.25 to 5.85 THz in our study. When we increased the scanning parameters, the frequency span of incident light needed to increase, as can be seen in Fig. S10 of Supplement 1. However, this will not contribute any new information to our paper. Moreover, we obtained the highest performance for lower chemical potential cases of pentaband and quadband Gr plasmonic modulators. Thus, the evaluations of quadband and pentaband plasmonic modes are not presented beyond the frequency of 5.85 THz here. However, we simulated three more cases for our proposed quadband and pentaband Gr modulator structures beyond the previously reported cases (see Fig. S10 of the Supplement 1).

 

Fig. 5. Evolution of plasmonic (a) quadband, and (b) pentaband transmittance dips at different chemical potentials of Gr layers. We applied a TM-polarized normal incidence plane wave THz light to our proposed modulators. Transmittance dips and peaks were blue-shifted by increasing the chemical potentials of Gr layers. White dashed lines represent the different case studies of our proposed modulators.

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3.1.4 Impact of polarization angles

We analyzed the effect of the polarization angles of the incident light on the transmittance spectra of our proposed modulators, as illustrated in Fig. 6. Here, $\theta$ represents the polarization angle of incoming light. The electric field of TM polarized incident light ($\theta$ = 0$^{\circ }$) was strongly coupled with the proposed modulators that exhibited transmittance minima at resonant frequencies. Transmittance gradually elevated as $\theta$ increased and reached a maximum at $\theta$ = 90$^\circ$ shown in Fig. 6. The symmetric nature of the proposed modulators provided the same transmittance spectra at $\theta$ = 0$^\circ$ = 180$^\circ$ and $\theta$ = 90$^\circ$ = 270$^\circ$. Lower-frequency resonances were less sensitive to polarization angles compared to high-frequency resonances. It can be observed that high-frequency resonances were confined to the polarization angle range from 45$^\circ$ to 315$^\circ$ and 135$^\circ$ to 225$^\circ$. Two high-frequency resonances existed around 4.197, 5.451 THz, and 4.197, 5.337 THz for our proposed quadband and pentaband plasmonic modulators.

 

Fig. 6. Color plot of transmittance spectra at different $\theta$s of incoming light for our proposed (a) quadband and (b) pentaband plasmonic modulators. $\theta$ = 0$^{\circ }$ and $\theta$ = 90$^{\circ }$ represented the TM and TE polarized incident light, respectively.

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3.1.5 Performance analysis

We enumerated the quality factor, Q, for our proposed quad and penta bandstop plasmonic structures. Q provides the relation between full-width half-maximum (FWHM) and resonant frequency, $f_c$ that is given by,

Calculated Q of 4.38, 4.04, 12.27, and 7.97 were obtained at resonant frequencies of 1.746, 2.487, 4.197, and 5.451 THz for the proposed quadband plasmonic modulator. Moreover, we achieved Q of 3.7, 5.38, 5.06, 9.2, and 7.8 at resonant frequencies of 1.404, 2.145, 3.171, 4.197, and 5.337 THz for the proposed pentaband plasmonic modulator. The top GNR layer provided the highest Q compared to Gr nanoholes. Q gives the idea of the filter’s selectivity. As Q increases, our filter becomes more selective with decreasing FWHM. Therefore, it can be inferred that the resonance modes at the frequencies of 4.197 and 4.197 THz exhibited high selectiveness for our proposed quadband and pentaband plasmonic filters, respectively. Previously, several plasmonic nanostructures with a low Q-factor of less than 10 properly functioned as a filter [29,30]. Thus, our proposed structures have the capability to operate as a filter in the THz frequency regime.

Moreover, we calculated the ER, a performance parameter of the filter, which was solved by [31],

Here, $T_{TM}$ and $T_{TE}$ are the transmittances of TM and TE polarized incident light, respectively. We obtained a high difference between the transmittance under TM and TE polarized light for our proposed quadband and pentaband plasmonic modulators shown in Fig. S9 of Supplement 1. ERs of 4.6, 12.8, 14.4, and 22.2 dB were achieved at resonant frequencies for the quadband plasmonic filter. Similarly, we found ERs of -0.4, 10.0, 16.1, 15.2, and 22.3 dB at resonant frequencies for the pentaband plasmonic filter.

Furthermore, we calculated the attenuation of the bandstop filters, a performance parameter of the plasmonic filter, by solving $-10log_{10}(T_{TM})$. Attenuations of 14.8, 19.1, 16.6, and 22.5 dB were obtained at resonant frequencies for the quadband plasmonic filter. Similarly, we achieved attenuations of 14.8, 22.1, 24.6, 20.3, and 24.5 dB at resonant frequencies for our proposed pentaband plasmonic filter.

3.2 Slow light applications

We investigated the emergence of slow-light properties, such as group delay and phase, for our proposed THz modulator structures. Figure 5 demonstrates the evolution of plasmonic resonances with various cases of chemical potentials at different Gr layers. White dashed lines of Fig. 5 indicated the various cases of quadband and pentaband plasmonic modulators. The chemical potentials of different Gr layers for these cases are provided in Tables S1 and S2 of Supplement 1. The transmittance peaks had a strong destructive interference effect, generating a strong dispersion with a positive group delay. Meanwhile, the phase of our proposed modulators experienced a sharp change while the frequency approached transmission peaks. This phenomenon is depicted in Figs. 7 and 8. The group delay, $\tau _g$ was calculated by,

Here, $\omega$ and $\theta _p$ are the angular frequency of light and the phase of the transmittance coefficient, respectively. $\theta _p$ was enumerated by solving $\theta _p = arg(t)$ where $t$ denotes the modulator’s transmittance coefficient. Fig. 7 illustrates the impact of changing the Gr’s chemical potential on slow light properties for our proposed quadband modulator. We can change the $\tau _g$ and $\theta _p$ of our proposed quadband plasmonic modulator by varying the $\mu _{c_n}$ of Gr layers. $\tau _g$ and $\theta _p$ were blue-shifted by increasing $\mu _{c_n}$ of Gr layers, as shown in Fig. 7. We utilized the $\mu _{c_n}$ of Gr layers from 0.12 to 0.57 eV. The chemical potentials of case 1 and case 4 were set to be $\mu _{c_1}$ = 0.16, $\mu _{c_2}$ = 0.12, $\mu _{c_3}$ = 0.35, $\mu _{c_4}$ = 0.37 and $\mu _{c_1}$ = 0.32, $\mu _{c_2}$ = 0.28, $\mu _{c_3}$ = 0.55, $\mu _{c_4}$ = 0.57 eV, respectively. The chemical potentials of the other two cases varied between the chemical potentials of case 1 and case 4. We obtained a maximum group delay of 1.02 ps for the proposed quadband plasmonic modulator.

 

Fig. 7. Group delay and phase spectra of the quadband plasmonic modulator at (a) case 1, (b) case 2, (c) case 3, and (d) case 4. White dashed lines of Fig. 5(a) indicate the case studies of our proposed modulator.

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Fig. 8. Group delay and phase spectra of the pentaband plasmonic modulator at (a) case 1, (b) case 2, (c) case 3, (d) case 4, and (e) case 5. White dashed lines of Fig. 5(b) indicate the case studies of our proposed modulator.

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Additionally, we calculated delay-bandwidth product (DBP) by solving DBP = $\tau _g \times \Delta f$. Here, $\Delta f$ is the bandwidth of slow light delay spectra. We require slow light devices as a buffer in integrated systems where DBP is an important performance parameter for buffer components. Slow light performance analysis of our proposed quadband plasmonic modulator is enlisted in Table 1. We adopted cases from Fig. 5(a) where white dashed lines indicated the four cases of our proposed quadband plasmonic modulator. DBP can be adjusted without changing structural parameters by varying $\mu _c$ via an external voltage source. A maximum DBP of 1.268 was obtained for our proposed quadband plasmonic modulator. The impact of varying the chemical potential of Gr layers on slow light properties for our proposed pentaband plasmonic modulator was depicted in Fig. 8. Here, we varied the $\mu _{c_n}$ of Gr layers from 0.12 to 0.80 eV in our study. Increasing $\mu _c$ of Gr layers provided blue-shifted $\tau _g$ and $\theta _p$, as can be seen in Fig. 8. The chemical potentials of case 1 and case 5 were considered to be $\mu _{c_1}$ = 0.24, $\mu _{c_2}$ = 0.12, $\mu _{c_3}$ = 0.40, $\mu _{c_4}$ = 0.60, $\mu _{c_5}$ = 0.45 and $\mu _{c_1}$ = 0.36, $\mu _{c_2}$ = 0.20, $\mu _{c_3}$ = 0.60, $\mu _{c_4}$ = 0.80, $\mu _{c_5}$ = 0.57 eV, respectively. The $\mu _{c_n}$ of the other three cases were adopted between the $\mu _{c_n}$ of these two cases. We achieved a maximum group delay of 1.41 ps for the proposed pentaband plasmonic modulator, which was the Gr modulator’s highest reported group delay. The group delay and phase were blue-shifted by increasing the Gr layers’ $\mu _{c_n}$. Thus, we can alter the slow-light properties of our proposed modulators by varying the $\mu _{c_n}$ of Gr layers, which allows us to create external voltage-controlled tunable slow-light devices. Table 2 provides the slow light performance analysis of our proposed pentaband plasmonic modulator. We adopted cases from Fig. 5(b) where white dashed lines indicated the five cases of our proposed pentaband plasmonic modulator. Moreover, the group delay of our proposed Gr modulators could be increased with increasing Gr monolayer stages, and we deduced a relation between the Gr stage numbers and the maximum group delay of Gr-based modulator structures (see Section 10 of Supplement 1 for details). A calculated maximum DBP of 1.725 was achieved for our proposed pentaband plasmonic modulator.

Table 1. Calculated $\tau _g$ and DBP for various case studies of quadband plasmonic modulator

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Table 2. Calculated $\tau _g$ and DBP for various case studies of pentaband plasmonic modulator

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4. Comparative analysis

Table 3 shows a detailed comparative analysis of the performance parameters among the previously reported studies and our proposed modulators. Our obtained $\tau _g$ outperformed previously reported works. Xiong et al. reported a continuous staircase-shaped structure on the Gr monolayer [15]. Their proposed structure provided a high ER of 16.56 dB with a $\tau _g$ of 0.488 ps. However, a staircase shape in the Gr layer makes the fabrication procedure complex. Gao et al. proposed a modulator comprising four GNRs on silicon substrate [14]. Low ER was the limiting factor of their modulator structure. Liu et al. demonstrated a similar kind of GNR modulator, which had similar disadvantages as previous work including low ER and small operating range [32]. Hu et al. reported an asymmetrical Gr-based structure; however, the operating range of the structure covered a small region of the THz spectrum. In comparison to existing Gr-based devices, our suggested quadband and pentaband plasmonic modulator structures offer higher ER with a $\tau _g$ of 1.41 ps, which is quite large.

Table 3. Comparative performance analysis of Gr structures

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5. Conclusions

In this paper, we proposed two multilayered Gr-based quadband and pentaband plasmonic modulators and utilized the FDTD technique to conduct the performance analysis of these modulators. The Gr layers strongly interacted with the incident light’s electric field, resulting in SP resonance at resonant frequencies due to scattering. Varying applied voltage changed the Gr layer’s chemical potential, allowing the modulators to tune the resonant frequencies without modifying structural parameters. Meanwhile, the change in chemical potentials shifted the Gr’s surface conductivity. The proposed modulators exhibited high tunability over a wide operating range of THz spectrum. We obtained a noteworthy improvement in performance for the pentaband plasmonic modulator compared to the quadband plasmonic modulator. Our designed pentaband plasmonic modulator had more than a quarter group delay compared to the quadband plasmonic modulator; however, the quadband plasmonic modulator provided a 25% higher Q-factor. We achieved a maximum optical group delay of 1.41 ps and ER of 22.3 dB for our proposed pentaband plasmonic modulator, which is significantly higher than previously reported works. Therefore, the proposed high-performance modulators will benefit optical filtering and slow light applications, specifically in optical buffers.