Polarization-Insensitive Graphene Modulator Based on Hybrid Plasmonic Waveguide

: A polarization-insensitive graphene-assisted electro-optic modulator is proposed. The orthogonal T-shaped metal slot hybrid plasmonic waveguide allows the polarization-independent propagation of transverse electric ﬁeld mode and complex mode. By the introduction of dual-layer graphene on the ridge waveguide, the polarization-insensitive modulation depths of the TE mode and complex mode are 0.511 dB/ µ m and 0.502 dB/ µ m, respectively. The 3 dB bandwidth of the modulator we have proposed is about 127 GHz at the waveguide length of 20 µ m. The power consumption of 72 fJ/bit promised potential graphene electro-optic modulator applications for on-chip interconnected information transfer and processing.


Introduction
An electro-optical modulator is an important device in an optical fiber network and photonics integrated circuit [1]. EO modulators based on the bulk or thin-film LiNbO3 technology are mature but are still facing the challenges of relatively large size and high modulation voltage [2]. Though the fabrication of modulators based on silicon is perfectly compatible with the CMOS process, the limited EO coefficient leads to the low modulation efficiency [3]. The III-V semiconductor modulators offer a compact size, but the insertion loss (IL) and the stability are still not enough for practical applications [4]. Due to the intrinsic material characteristics, nonlinear polymer-based modulators have the merits of high EO coefficient, low driving voltage, and low dispersion [5]. However, their favorable performance largely depends on the high performance of poled EO polymer, which is obtained from a complex synthetizing process with low productivity. In addition, the reliability is yet to be proven in practical applications [6].
Graphene is a single-layer two-dimensional honeycomb lattice structure with excellent optoelectronic properties, such as an ultra-wide transmission spectral range, a tuning effect of light absorption, and high electronic conductivity and modulation rates. In addition, the Fermi level of graphene can be tuned by the applied voltage, thereby adjusting the absorption [7]. In 2011, the first graphene electro-optic modulator was experimentally fabricated. Its modulation depth (MD, which we define as the difference between the "on state" and the "off state") was 0.1 dB/µm [8]. After that, a modulator using graphene to form a double-layer flat capacitor was fabricated. The modulation depth was increased to 0.16 dB/µm [9]. Due to the strong mode confinement, the surface plasmon effect has also been introduced into graphene modulators [10,11]. Huang et al. proposed a waveguidecoupled hybrid plasmonic graphene modulator with a wide 3 dB bandwidth that can reach 0.48 THz and a power consumption of about 145 fJ/bit [12], which proves the privilege of a graphene-based hybrid plasmonic waveguide modulator. However, due to the feature of coupled hybrid plasmonic graphene modulator with a wide 3 dB bandwidth that can reach 0.48 THz and a power consumption of about 145 fJ/bit [12], which proves the privilege of a graphene-based hybrid plasmonic waveguide modulator. However, due to the feature of atomic thickness, graphene-based modulators are sensitive to the polarization [13], which implies the independent modulation of the TE mode, transverse magnetic field (TM) mode, or TE-TM complex mode. The simultaneous modulation of both the TE mode and the TM mode is still to be implemented with a new waveguide [14,15].
In this paper, we propose a modulator based on a T-shaped metal slot waveguide and graphene to support the simultaneous modulation of both the TE mode and the TE-TM complex mode. Dual-layer graphene covers the top and one sidewall of the ridge waveguide. The finite element method is used study and optimize the geometric dimensions with respect to the equal modulation of the TE mode and the TE-TM complex mode. The simulation results show that the TE mode and the complex mode can be modulated at the depth of 0.511 dB/µm and 0.502 dB/µm, respectively. The 3 dB bandwidth of this modulator that we have proposed is about 127 GHz at the modulation length of 20 µm.

Hybrid Plasmonic Slot Waveguide
The proposed graphene modulator with a T-shaped hybrid plasmonic slot waveguide is shown in Figure 1. The device design and simulations are performed at the center wavelength of = 1550 nm. Three silver blocks (Ag, nAg = 0.14447 + 11.366i) and a sandwiched silicon oxide (SiO2, nSiO2 = 1.45) insulator construct the orthogonal horizontal Tshaped slot waveguide. Dual 0.33 nm-thick graphene sheets isolated by hexagonal boron nitride (hBN, nhBN = 1.98) with a thickness of 10 nm cover the top and one sidewall of the plasmonic waveguide. Two metal contacts connect the lower and the upper graphene sheet to the external voltage source, respectively [16]. The whole structure is supported by the SiO2 substrate. The refractive index varies with the wavelength change. Therefore, the real part and imaginary part of the effective refractive index of Ag is confirmed at each wavelength by introducing the interpolation function to offer the precise result in the simulations.  Our proposed dual built-in orthogonal slot waveguide can support two polarizations simultaneously, which overcomes the limitation of the polarization dependence of the common plasmonic waveguides. As shown in Figure 1, the T-shaped metal slot waveguide supports two modes with different polarizations. As shown in Figure 2, Ex and Ey represent the electric field distribution of the TE mode and the TE-TM complex mode, respectively. As shown in Figure 2a, at the optical wavelength of 1550 nm, the TE mode mainly exists in the vertical Ag slot, which is due to the surface plasmon excitation by the B1 and B2 Ag blocks [17]. The complex mode mainly exists in the horizontal slot, partially extending to the vertical slot waveguide, as shown in Figure 2b. Hence, the complex mode can be viewed as a mixed plasmonic mode generated in perpendicular metal slits. Our proposed dual built-in orthogonal slot waveguide can support two polarizations simultaneously, which overcomes the limitation of the polarization dependence of the common plasmonic waveguides. As shown in Figure 1, the T-shaped metal slot waveguide supports two modes with different polarizations. As shown in Figure 2, Ex and Ey represent the electric field distribution of the TE mode and the TE-TM complex mode, respectively. As shown in Figure 2a, at the optical wavelength of 1550 nm, the TE mode mainly exists in the vertical Ag slot, which is due to the surface plasmon excitation by the B1 and B2 Ag blocks [17]. The complex mode mainly exists in the horizontal slot, partially extending to the vertical slot waveguide, as shown in Figure 2b. Hence, the complex mode can be viewed as a mixed plasmonic mode generated in perpendicular metal slits. To better evaluate the performance of the proposed modulator, the parameters of mode power attenuation (MPA), modulation depth (MD), and propagation loss (PL) are introduced. Here, MPA, MD, and PL can be defined as: In the above equation, λ is the incident wavelength, Im (Neff) is the imaginary part of the effective refractive index, the modulation depth is defined as the difference between mode power attenuation ("OFF" state) and mode power attenuation ("ON" state), and propagation loss refers to mode power attenuation ("ON" state).

Graphene Film
Graphene is a novel material with a two-dimensional single-layer honeycomb lattice structure. This unique feature offers graphene controllable optical absorption, considerable carrier mobility, 2.3% uniform optical absorption in a wide band, and other optical properties [18,19]. As shown in Figure 1, the 10 nm-thick hBN is sandwiched by two graphene films, which forms a plate capacitor. The applied bias voltage changes the Fermi level of the graphene and then the light absorption. The intrinsic tuning mechanism is consistent with that of the single-layer graphene modulator. The main difference is the capacitor structure that is constructed by the upper-layer graphene, lower-layer graphene, and the sandwiched hBN insulator.
In simulation, graphene can be defined either as a three-dimensional material with very thin thickness or as a two-dimensional structure without thickness [20]. In 3D model simulations, a fine mesh is demanded to obtain an accurate result, which implies long calculations and low efficiency. Therefore, the 2D model, which could provide similar accuracy and be more efficient, is adopted in this work. In this model, the graphene layer is considered as a surface conductive medium without thickness. In practical simulations, In the above equation, λ is the incident wavelength, Im (N eff ) is the imaginary part of the effective refractive index, the modulation depth is defined as the difference between mode power attenuation ("OFF" state) and mode power attenuation ("ON" state), and propagation loss refers to mode power attenuation ("ON" state).

Graphene Film
Graphene is a novel material with a two-dimensional single-layer honeycomb lattice structure. This unique feature offers graphene controllable optical absorption, considerable carrier mobility, 2.3% uniform optical absorption in a wide band, and other optical properties [18,19]. As shown in Figure 1, the 10 nm-thick hBN is sandwiched by two graphene films, which forms a plate capacitor. The applied bias voltage changes the Fermi level of the graphene and then the light absorption. The intrinsic tuning mechanism is consistent with that of the single-layer graphene modulator. The main difference is the capacitor structure that is constructed by the upper-layer graphene, lower-layer graphene, and the sandwiched hBN insulator.
In simulation, graphene can be defined either as a three-dimensional material with very thin thickness or as a two-dimensional structure without thickness [20]. In 3D model simulations, a fine mesh is demanded to obtain an accurate result, which implies long calculations and low efficiency. Therefore, the 2D model, which could provide similar accuracy and be more efficient, is adopted in this work. In this model, the graphene layer is considered as a surface conductive medium without thickness. In practical simulations, the performance of the modulator is mainly affected by the conductivity, and the conductivity is mainly determined by the Fermi level, µ c , the optical frequency, ω, and the relaxation time, τ, according to the Kubo formula [21,22]. The Fermi level, µ c , as well as the surface conductivity, σ g, are tuned by applying an external voltage. When no voltage or a low voltage is applied, the graphene behaves like metal, with high light absorption, which corresponds to the "OFF" state. However, when a certain voltage is applied, graphene behaves like a dielectric material, which corresponds to the "ON" state. Equation (1) illustrates the relationship between the Fermi level and the applied voltage [23]: where ν F is the Fermi velocity of graphene (ν F = 2.5 × 10 6 m/s) [24] and η is determined by the insulation and thickness of hBN. As shown in Figure 3 below, µ c can be changed from 0.2 to 0.6 eV as the applied bias voltage changes from 0.325 to 2.915 V.
Photonics 2022, 9, x FOR PEER REVIEW 4 of 17 the performance of the modulator is mainly affected by the conductivity, and the conductivity is mainly determined by the Fermi level, µc, the optical frequency, ω, and the relaxation time, τ, according to the Kubo formula [21,22]. The Fermi level, µc, as well as the surface conductivity, σg, are tuned by applying an external voltage. When no voltage or a low voltage is applied, the graphene behaves like metal, with high light absorption, which corresponds to the "OFF" state. However, when a certain voltage is applied, graphene behaves like a dielectric material, which corresponds to the "ON" state. Equation (1) illustrates the relationship between the Fermi level and the applied voltage [23]: where νF is the Fermi velocity of graphene (νF = 2.5 10 6 m/s) [24] and η is determined by the insulation and thickness of hBN. As shown in Figure 3 below, µc can be changed from 0.2 to 0.6 eV as the applied bias voltage changes from 0.325 to 2.915 V.

Design and Optimization
To confirm the influence of the waveguide dimensions on the mode distribution, simulations based on the FDTD method were carried out. Based on Maxwell's equations in wave optics, the two-dimensional wave optics module in the software (Lumerical Ltd., Vancouver, BC V6E 2M6, CA) was chosen. The mode in the waveguide was first studied to obtain different polarizations. Then, parametric scanning and global calculation were performed to determine the optimal waveguide size by monitoring the mode field status and modulation performance. In this process, the wavelength was set to be = 1550 nm. The frequency, , was determined by the light speed and the wavelength. The dielectric constant, ε, of hBN is 8.85424 10 / . Here, the modulation (MD) is defined as the difference between the mode power attenuation (MPA) values at µc = 0 eV and µc = 1 eV. The propagation loss (PL) is the power attenuation (MPA) at µc = 0 eV.

w1
The TE mode and the complex mode in this hybrid plasmonic waveguide can be regarded as orthogonal polarizations supported by the Ag-SiO2-Ag structure. The Re(Neff)

Design and Optimization
To confirm the influence of the waveguide dimensions on the mode distribution, simulations based on the FDTD method were carried out. Based on Maxwell's equations in wave optics, the two-dimensional wave optics module in the software (Lumerical Ltd., Vancouver, BC V6E 2M6, CA) was chosen. The mode in the waveguide was first studied to obtain different polarizations. Then, parametric scanning and global calculation were performed to determine the optimal waveguide size by monitoring the mode field status and modulation performance. In this process, the wavelength was set to be λ = 1550 nm.
The frequency, f 0 , was determined by the light speed and the wavelength. The dielectric constant, ε, of hBN is 8.85424 × 10 −12 F/m. Here, the modulation (MD) is defined as the difference between the mode power attenuation (MPA) values at µ c = 0 eV and µ c = 1 eV. The propagation loss (PL) is the power attenuation (MPA) at µ c = 0 eV.

w 1
The TE mode and the complex mode in this hybrid plasmonic waveguide can be regarded as orthogonal polarizations supported by the Ag-SiO 2 -Ag structure. The Re(N eff ) of the TE mode as a changing relationship of the Ag block width, w 1 , is shown in Figure 4a. An unremarkable Re(N eff ) variation is observed, which can be attributed to the lateral TE mode confinement, which is insensitive to the width change of Ag block B2 and Ag block B3. Conversely, the complex mode distribution is constrained by the T-shaped metal slots. Therefore, the Re(N eff ) of the complex mode decreases with the increment in w 1 , which is due to the enhanced light absorption occurring in the gap between Ag blocks B2 and B3, as shown in Figure 4b. of the TE mode as a changing relationship of the Ag block width, w1, is shown in Figu 4a. An unremarkable Re(Neff) variation is observed, which can be attributed to the later TE mode confinement, which is insensitive to the width change of Ag block B2 and A block B3. Conversely, the complex mode distribution is constrained by the T-shaped met slots. Therefore, the Re(Neff) of the complex mode decreases with the increment in w which is due to the enhanced light absorption occurring in the gap between Ag blocks B and B3, as shown in Figure 4b.
As Figure 5 shows, for the TE mode, no significant MD and PL changes are observe However, the MD and PL of the complex mode decrease with increases in w1. To obta the same MD, w1 was chosen to be 200 nm, at which the real part of the effective inde difference Re(∆Neff) of the TE and that of the complex modes are both around 0.05.

h1
The height, h1, of the horizontal slot largely affects the complex mode field distribu tion. Therefore, the Re(Neff) of the TE mode, as a changing relationship of the Ag gap, h is investigated. As Figure 6a shows, the limited Re(Neff) change of the TE mode at differen µc values is observed when h1 varies from 35 to 45 nm. Conversely, the Re(Neff) of th complex mode decreases rapidly with the increment in h1, which is due to the fast redu tion in plasmonic mode confinement with increases in the metal gap distance. As Figure 5 shows, for the TE mode, no significant MD and PL changes are observed. However, the MD and PL of the complex mode decrease with increases in w 1 . To obtain the same MD, w 1 was chosen to be 200 nm, at which the real part of the effective index difference Re(∆N eff ) of the TE and that of the complex modes are both around 0.05. of the TE mode as a changing relationship of the Ag block width, w1, is shown in Figure  4a. An unremarkable Re(Neff) variation is observed, which can be attributed to the lateral TE mode confinement, which is insensitive to the width change of Ag block B2 and Ag block B3. Conversely, the complex mode distribution is constrained by the T-shaped metal slots. Therefore, the Re(Neff) of the complex mode decreases with the increment in w1, which is due to the enhanced light absorption occurring in the gap between Ag blocks B2 and B3, as shown in Figure 4b. As Figure 5 shows, for the TE mode, no significant MD and PL changes are observed. However, the MD and PL of the complex mode decrease with increases in w1. To obtain the same MD, w1 was chosen to be 200 nm, at which the real part of the effective index difference Re(∆Neff) of the TE and that of the complex modes are both around 0.05.

h1
The height, h1, of the horizontal slot largely affects the complex mode field distribution. Therefore, the Re(Neff) of the TE mode, as a changing relationship of the Ag gap, h1, is investigated. As Figure 6a shows, the limited Re(Neff) change of the TE mode at different µc values is observed when h1 varies from 35 to 45 nm. Conversely, the Re(Neff) of the complex mode decreases rapidly with the increment in h1, which is due to the fast reduction in plasmonic mode confinement with increases in the metal gap distance.

h 1
The height, h 1 , of the horizontal slot largely affects the complex mode field distribution. Therefore, the Re(N eff ) of the TE mode, as a changing relationship of the Ag gap, h 1 , is investigated. As Figure 6a shows, the limited Re(N eff ) change of the TE mode at different µ c values is observed when h 1 varies from 35 to 45 nm. Conversely, the Re(N eff ) of the complex Photonics 2022, 9, 609 6 of 16 mode decreases rapidly with the increment in h 1 , which is due to the fast reduction in plasmonic mode confinement with increases in the metal gap distance. tion depth of the TE mode rises with increases in h1, while the MD of the complex mod drops down as h1 increases from 35 to 45 nm. The electric field distributions of the T mode and the complex mode (h1 = 35 nm and 40 nm) are shown in the insets of Figure 7 Due to the asymmetric waveguide structure, the enlargement of h1 leads to the asymmetri distortion of the two modes. Moreover, the unbalanced placement of the graphene layer results in the inverse modulation when the metal gap, h1, is increasing. However, the sam MD can be obtained at h1 = 40 nm. A similar scenario can be observed in the change in PL and the same PL of the TE mode and complex mode is realized at h1 = 41 nm. Therefore h1 was compromisingly chosen to be 40 nm.    Figure 7. Due to the asymmetric waveguide structure, the enlargement of h 1 leads to the asymmetric distortion of the two modes. Moreover, the unbalanced placement of the graphene layers results in the inverse modulation when the metal gap, h 1 , is increasing. However, the same MD can be obtained at h 1 = 40 nm. A similar scenario can be observed in the change in PL, and the same PL of the TE mode and complex mode is realized at h 1 = 41 nm. Therefore, h 1 was compromisingly chosen to be 40 nm.  Figure 7. Due to the asymmetric waveguide structure, the enlargement of h1 leads to the asymmetric distortion of the two modes. Moreover, the unbalanced placement of the graphene layers results in the inverse modulation when the metal gap, h1, is increasing. However, the same MD can be obtained at h1 = 40 nm. A similar scenario can be observed in the change in PL, and the same PL of the TE mode and complex mode is realized at h1 = 41 nm. Therefore, h1 was compromisingly chosen to be 40 nm.

w 2
In this design, w 2 which is crucial for the restraint of the TE mode, refers to the gap size between Ag blocks B1 and B2 as well as B1 and B3. Similarly, the Re(N eff ) of the TE mode and the complex mode, as a changing relationship between the width of this slot, w 2 , is investigated. As shown in Figure 8a,b, predictably, the Re(N eff ) decreases as w 2 varies from 35 to 45 nm, which is attributed to the weakening of the mode confinement. Consequently, the MD of the TE mode reduces with the increasing of w 2 , as shown in Figure 9. For the complex mode, a stronger electric field can be observed at w 2 = 45 nm compared to that at w 2 = 35 nm. As a result, the MD of the complex mode increases with the increment in w 2 . However, both modes have the same MD of about 0.53 dB/um at w 2 = 40 nm.

w2
In this design, w2 which is crucial for the restraint of the TE mode, refers to the gap size between Ag blocks B1 and B2 as well as B1 and B3. Similarly, the Re(Neff) of the TE mode and the complex mode, as a changing relationship between the width of this slot, w2, is investigated. As shown in Figure 8a,b, predictably, the Re(Neff) decreases as w2 varies from 35 to 45 nm, which is attributed to the weakening of the mode confinement. Consequently, the MD of the TE mode reduces with the increasing of w2, as shown in Figure 9. For the complex mode, a stronger electric field can be observed at w2 = 45 nm compared to that at w2 = 35 nm. As a result, the MD of the complex mode increases with the increment in w2. However, both modes have the same MD of about 0.53 dB/um at w2 = 40 nm.
The PLs of both modes decrease when w2 shifts from 35 to 45 nm, which is due to reduced mode penetration into the Ag block of B1. At w2 = 40 nm, the PL difference between the two modes is about 0.01 dB/µm. For the purpose of polarization independence, w2 was chosen to be 40 nm.

w2
In this design, w2 which is crucial for the restraint of the TE mode, refers to the gap size between Ag blocks B1 and B2 as well as B1 and B3. Similarly, the Re(Neff) of the TE mode and the complex mode, as a changing relationship between the width of this slot, w2, is investigated. As shown in Figure 8a,b, predictably, the Re(Neff) decreases as w2 varies from 35 to 45 nm, which is attributed to the weakening of the mode confinement. Consequently, the MD of the TE mode reduces with the increasing of w2, as shown in Figure 9. For the complex mode, a stronger electric field can be observed at w2 = 45 nm compared to that at w2 = 35 nm. As a result, the MD of the complex mode increases with the increment in w2. However, both modes have the same MD of about 0.53 dB/um at w2 = 40 nm.
The PLs of both modes decrease when w2 shifts from 35 to 45 nm, which is due to reduced mode penetration into the Ag block of B1. At w2 = 40 nm, the PL difference between the two modes is about 0.01 dB/µm. For the purpose of polarization independence, w2 was chosen to be 40 nm.   The PLs of both modes decrease when w 2 shifts from 35 to 45 nm, which is due to reduced mode penetration into the Ag block of B1. At w 2 = 40 nm, the PL difference between the two modes is about 0.01 dB/µm. For the purpose of polarization independence, w 2 was chosen to be 40 nm.

h 2
In Figure 1, h 2 represents the height of the proposed plasmonic waveguide. It is necessary to confirm the impact of h 2 on the characteristics of hybrid plasmonic waveguide at µ c = 0 and µ c = 1 eV. As shown in Figure 10a, the Re(N eff ) of the TE mode gradually increases with increases in waveguide height, while the Re(N eff ) of the complex mode gradually declines with increases in h 2 , as shown in Figure 10b. at µc = 0 and µc = 1 eV. As shown in Figure 10a, the Re(Neff) of the TE mode gradually increases with increases in waveguide height, while the Re(Neff) of the complex mod gradually declines with increases in h2, as shown in Figure 10b.
As Figure 11 shows, as h2 varies from 250 to 300 nm, the MD difference between th two modes first shrinks then gradually expands. The same MD for the two modes can b obtained at h2 = 285 nm. In addition, the increment in h2 is accompanied by the gradua decrement of PL in both modes. The insets show the electric field distributions of the two modes at h2 = 250, 285, and 300 nm. At h2 = 285 nm, the PL difference between two mode is about 0.005 dB/µm. Therefore, the hybrid waveguide height, h2, was chosen to be 285 nm.

wB1
In this work, wB1 represents the width of the Ag block B1, as shown in Figure 1. Sim ilarly, the Re(Neff) of the TE mode and the complex mode all decrease when wB1 varie from 50 nm to 150 nm. As shown in Figure 12a,b, no obvious Re(Neff) change happen when wB1 is larger than 100 nm. As Figure 11 shows, as h 2 varies from 250 to 300 nm, the MD difference between the two modes first shrinks then gradually expands. The same MD for the two modes can be obtained at h 2 = 285 nm. In addition, the increment in h 2 is accompanied by the gradual decrement of PL in both modes. The insets show the electric field distributions of the two modes at h 2 = 250, 285, and 300 nm. At h 2 = 285 nm, the PL difference between two modes is about 0.005 dB/µm. Therefore, the hybrid waveguide height, h 2 , was chosen to be 285 nm.

h2
In Figure 1, h2 represents the height of the proposed plasmonic waveguide. It is necessary to confirm the impact of h2 on the characteristics of hybrid plasmonic waveguide at µc = 0 and µc = 1 eV. As shown in Figure 10a, the Re(Neff) of the TE mode gradually increases with increases in waveguide height, while the Re(Neff) of the complex mode gradually declines with increases in h2, as shown in Figure 10b.
As Figure 11 shows, as h2 varies from 250 to 300 nm, the MD difference between the two modes first shrinks then gradually expands. The same MD for the two modes can be obtained at h2 = 285 nm. In addition, the increment in h2 is accompanied by the gradual decrement of PL in both modes. The insets show the electric field distributions of the two modes at h2 = 250, 285, and 300 nm. At h2 = 285 nm, the PL difference between two modes is about 0.005 dB/µm. Therefore, the hybrid waveguide height, h2, was chosen to be 285 nm.

wB1
In this work, wB1 represents the width of the Ag block B1, as shown in Figure 1. Similarly, the Re(Neff) of the TE mode and the complex mode all decrease when wB1 varies from 50 nm to 150 nm. As shown in Figure 12a,b, no obvious Re(Neff) change happens when wB1 is larger than 100 nm.

w B1
In this work, w B1 represents the width of the Ag block B1, as shown in Figure 1. Similarly, the Re(N eff ) of the TE mode and the complex mode all decrease when w B1 varies from 50 nm to 150 nm. As shown in Figure 12a,b, no obvious Re(N eff ) change happens when w B1 is larger than 100 nm. from 50 nm to 150 nm. The differences between the two modes first decrease then enlarg gradually. The same MD can be obtained at wB1 = 100 nm. The increment in wB1 is accom panied by the gradual decrement of PL for both modes. The insets in Figure 13 show th electric field distribution of the two modes at wB1 = 50, 100, and 150 nm. At wB1 = 100 nm the PL difference between two modes is only about 0.005 dB/µm. With comprehensiv consideration of MD, the hybrid waveguide height, wB1, was chosen to be 100 nm.

hB2 and hB3
The height of Ag blocks B2 and B3 may have an influence on the performance of th modulator. Because of the complete height of the plasmonic slot waveguide, h2 is stable When hB2 is increasing, the height of block B3 decreases accordingly. Therefore, the con firmation of hB2 would give the answer to hB3. As shown in Figure 14, Re(Neff) is invest gated as a changing relationship of hB2 for the TE mode and the complex mode. When h varies from 70 nm to 170 nm, the fluctuation in the Re(Neff) of the TE mode is only aroun 0.002 at µc = 0 eV. At µc = 1 eV, the Re(Neff) of the TE mode gradually rises from 2.084 t The MD and PL, as a changing relationship of w B1 for the TE mode and the complex mode, are shown in the Figure 13. The MD of the TE mode decreases as w B1 varies from 50 nm to 150 nm. Conversely, the MD of the complex mode increases when w B1 varies from 50 nm to 150 nm. The differences between the two modes first decrease then enlarge gradually. The same MD can be obtained at w B1 = 100 nm. The increment in w B1 is accompanied by the gradual decrement of PL for both modes. The insets in Figure 13 show the electric field distribution of the two modes at w B1 = 50, 100, and 150 nm. At w B1 = 100 nm, the PL difference between two modes is only about 0.005 dB/µm. With comprehensive consideration of MD, the hybrid waveguide height, w B1 , was chosen to be 100 nm. The MD and PL, as a changing relationship of wB1 for the TE mode and the complex mode, are shown in the Figure 13. The MD of the TE mode decreases as wB1 varies from 50 nm to 150 nm. Conversely, the MD of the complex mode increases when wB1 varies from 50 nm to 150 nm. The differences between the two modes first decrease then enlarge gradually. The same MD can be obtained at wB1 = 100 nm. The increment in wB1 is accompanied by the gradual decrement of PL for both modes. The insets in Figure 13 show the electric field distribution of the two modes at wB1 = 50, 100, and 150 nm. At wB1 = 100 nm, the PL difference between two modes is only about 0.005 dB/µm. With comprehensive consideration of MD, the hybrid waveguide height, wB1, was chosen to be 100 nm.

hB2 and hB3
The height of Ag blocks B2 and B3 may have an influence on the performance of the modulator. Because of the complete height of the plasmonic slot waveguide, h2 is stable. When hB2 is increasing, the height of block B3 decreases accordingly. Therefore, the confirmation of hB2 would give the answer to hB3. As shown in Figure 14, Re(Neff) is investigated as a changing relationship of hB2 for the TE mode and the complex mode. When hB2 varies from 70 nm to 170 nm, the fluctuation in the Re(Neff) of the TE mode is only around 0.002 at µc = 0 eV. At µc = 1 eV, the Re(Neff) of the TE mode gradually rises from 2.084 to

h B2 and h B3
The height of Ag blocks B2 and B3 may have an influence on the performance of the modulator. Because of the complete height of the plasmonic slot waveguide, h 2 is stable. When h B2 is increasing, the height of block B3 decreases accordingly. Therefore, the confirmation of h B2 would give the answer to h B3 . As shown in Figure 14, Re(N eff ) is investigated as a changing relationship of h B2 for the TE mode and the complex mode. When h B2 varies from 70 nm to 170 nm, the fluctuation in the Re(N eff ) of the TE mode is only around 0.002 at µ c = 0 eV. At µ c = 1 eV, the Re(N eff ) of the TE mode gradually rises from 2.084 to 2.088. In comparison, the Re(N eff ) of the complex mode primarily drops when h B2 is lower than 105 nm then rises up again.
2.088. In comparison, the Re(Neff) of the complex mode primarily drops when hB2 is lowe than 105 nm then rises up again.
As shown in Figure 15, the MDs of the TE mode and the complex mode graduall decrease. The same MD value for two modes can be obtained at hB2 = 120 nm. As hB2 in creases, the PL of the TE mode monotonously decreases. The smallest PL difference be tween the two modes is 0.005 dB/µm at hB2 = 100 nm. The PL difference is about 0.00 dB/µm at hB2 = 120 nm. The insets show the electric field distributions of the TE mode an the complex mode at hB2 = 100, 120, and 160 nm. To obtain better polarization independ ence, hB2 was set to be 120 nm. Since h2 is 285 nm, h1 is 40 nm, and hB2 is 120 nm, hB3 was se to be 125 nm.   As shown in Figure 15, the MDs of the TE mode and the complex mode gradually decrease. The same MD value for two modes can be obtained at h B2 = 120 nm. As h B2 increases, the PL of the TE mode monotonously decreases. The smallest PL difference between the two modes is 0.005 dB/µm at h B2 = 100 nm. The PL difference is about 0.007 dB/µm at h B2 = 120 nm. The insets show the electric field distributions of the TE mode and the complex mode at h B2 = 100, 120, and 160 nm. To obtain better polarization independence, h B2 was set to be 120 nm. Since h 2 is 285 nm, h 1 is 40 nm, and h B2 is 120 nm, h B3 was set to be 125 nm.
Photonics 2022, 9, x FOR PEER REVIEW 10 of 17 2.088. In comparison, the Re(Neff) of the complex mode primarily drops when hB2 is lower than 105 nm then rises up again. As shown in Figure 15, the MDs of the TE mode and the complex mode gradually decrease. The same MD value for two modes can be obtained at hB2 = 120 nm. As hB2 increases, the PL of the TE mode monotonously decreases. The smallest PL difference between the two modes is 0.005 dB/µm at hB2 = 100 nm. The PL difference is about 0.007 dB/µm at hB2 = 120 nm. The insets show the electric field distributions of the TE mode and the complex mode at hB2 = 100, 120, and 160 nm. To obtain better polarization independence, hB2 was set to be 120 nm. Since h2 is 285 nm, h1 is 40 nm, and hB2 is 120 nm, hB3 was set to be 125 nm.

µ c
After the confirmation of the geometric dimensions, the relationship between the modulation performance and the Femi level, µ c , was investigated. As Figure 16 shows, when µ c rises to 0.4 eV, the Re(N eff ) values of the TE mode and the complex mode reach maximums of 2.1742 and 2.1142, respectively. As µ c shifts to 0.6 eV, the Re(N eff ) values of the TE mode and the complex mode gradually reduce to 2.1601 and 2.1012, respectively. At µ c = 0.6 eV, Re(∆N eff ) is only 0.06.

µc
After the confirmation of the geometric dimensions, the relationship between the modulation performance and the Femi level, µc, was investigated. As Figure 16 shows, when µc rises to 0.4 eV, the Re(Neff) values of the TE mode and the complex mode reach maximums of 2.1742 and 2.1142, respectively. As µc shifts to 0.6 eV, the Re(Neff) values of the TE mode and the complex mode gradually reduce to 2.1601 and 2.1012, respectively. At µc = 0.6 eV, Re(∆Neff) is only 0.06. As shown in Figure 17, a sharp drop in PL can be observed with the µc increasing from 0.2 eV to 0.6 eV. Therefore, we define the "ON" state at µc= 0.6 and the "OFF" state at µc = 0.2 eV. In this case, the MDs of the TE mode and the complex mode are 0.502 dB/µm and 0.511 dB/µm, respectively. The difference between MDTE-mode and MDcomplex mode is only 0.007 dB/µm at λ = 1550 nm. As shown in Figure 17, a sharp drop in PL can be observed with the µc increasing from 0.2 eV to 0.6 eV. Therefore, we define the "ON" state at µ c = 0.6 and the "OFF" state at µ c = 0.2 eV. In this case, the MDs of the TE mode and the complex mode are 0.502 dB/µm and 0.511 dB/µm, respectively. The difference between MD TE-mode and MD complex mode is only 0.007 dB/µm at λ = 1550 nm.

Optical Bandwidth
The optical bandwidth, as a changing relationship of the optical wavelength, was investigated with the Fermi level, µc, increasing from 0.2 eV to 0.6 eV. As Figure 18  At µc = 0.2 eV, the Im(Neff) of the TE mode varies from 0.01613 to 0.02869 when the wavelength varies from 1.1 to 1.9 µm, as shown in Figure 19. Meanwhile, the Im(Neff) of the complex mode increases from 0.01655 to 0.02897, which mainly results from the en-

Optical Bandwidth
The optical bandwidth, as a changing relationship of the optical wavelength, was investigated with the Fermi level, µc, increasing from 0.2 eV to 0.6 eV. As Figure 18

Frequency Response and Power Consumption
To calculate the frequency response and power dissipation, we made the modulator model equivalent to the circuit model, which is shown in Figure 20, and the dynamic response and power consumption of the proposed modulator were investigated. Here, Cair, which is composed of two electrodes and the air is around 12 fF [25]. The capacitor, C, consists of dielectric capacitance, CD, and the quantum capacitance, CQ. In simulation calculations, we consider graphene to be undoped, and we set the quantum capacitance to CQ = 0. Therefore, the capacitor, C, can be represented by: At µ c = 0.2 eV, the Im(N eff ) of the TE mode varies from 0.01613 to 0.02869 when the wavelength varies from 1.1 to 1.9 µm, as shown in Figure 19. Meanwhile, the Im(N eff ) of the complex mode increases from 0.01655 to 0.02897, which mainly results from the enhancement of the graphene-light interaction due to the increasing dielectric constant with increases in wavelength. The MPA change of both modes is very limited within the wavelength range. The MDs of the TE mode are 0.43145 dB/µm and 0.48145 dB/µm at λ = 1.1 µm and 1.9 µm, respectively. Similarly, the MDs of the complex mode are 0.47636 dB/µm and 0.49797 dB/µm at λ = 1.1 µm and 1.9 µm, respectively. The proposed device exhibits better polarization independence at longer wavelengths.

Frequency Response and Power Consumption
To calculate the frequency response and power dissipation, we made the modulator model equivalent to the circuit model, which is shown in Figure 20, and the dynamic response and power consumption of the proposed modulator were investigated. Here, Cair, which is composed of two electrodes and the air is around 12 fF [25]. The capacitor, C, Figure 19. Im(N eff ) and MPA of the TE mode and the complex mode vary with wavelength.

Frequency Response and Power Consumption
To calculate the frequency response and power dissipation, we made the modulator model equivalent to the circuit model, which is shown in Figure 20, and the dynamic response and power consumption of the proposed modulator were investigated. Here, C air , which is composed of two electrodes and the air is around 12 fF [25]. The capacitor, C, consists of dielectric capacitance, C D , and the quantum capacitance, C Q . In simulation calculations, we consider graphene to be undoped, and we set the quantum capacitance to C Q = 0. Therefore, the capacitor, C, can be represented by: resistance, RS, of graphene and the resistance, Rc, which is composed by metal-graphene contact. Here, Rs and RC can be set as 100 Ω/□ [26] and 100 Ω-µm [16], respectively. The total resistance, Rtotal, can be calculated by [27]: 2 where wg1 is 1120 nm and wg2 is 1400 nm. Then, Rtotal is 28.9 Ω. Theoretically, the 3 dB bandwidth (f3-dB) of the proposed modulator is determined by the delay effect of the resistors and capacitors. Hence, f3-dB can be evaluated by [28]: With above optimized parameters, the bandwidth is around 127 GHz. When the modulation length, L, is 20 µm and the applied bias voltage varies from 0.325 V to 2.915 V (∆U = 2.59 V), the power consumption can be estimated by: Then, we have an Ebit of 0.072 pJ/bit.

Discussion
To clarify the merits of proposed design, the theoretical performances of the polarization-insensitive graphene modulators that have been reported are comprehensively compared with that of this work. As shown in Table 1, the device we proposed has a higher bandwidth and f3-dB as well as the smallest Ebit compared to those proposed in other studies. Moderate MD and ΔMD values can also be obtained. The characteristics of Re(ΔNeff) are also better than in most reported works. These favorable characteristics mainly originate from the hybrid waveguide structure, which provides a balanced mode In Equation (5), L is the modulation length of this modulator, while w ol = 620 nm represents the overlap width of dual-layer graphene. ε r and d are the permittivity and thickness of the hBN isolation layer. The total resistance, R total , is composed of the sheet resistance, R S, of graphene and the resistance, R c , which is composed by metal-graphene contact. Here, R s and RC can be set as 100 Ω/ [26] and 100 Ω-µm [16], respectively. The total resistance, R total , can be calculated by [27]: where w g1 is 1120 nm and w g2 is 1400 nm. Then, R total is 28.9 Ω. Theoretically, the 3 dB bandwidth (f 3-dB ) of the proposed modulator is determined by the delay effect of the resistors and capacitors. Hence, f 3-dB can be evaluated by [28]: With above optimized parameters, the bandwidth is around 127 GHz. When the modulation length, L, is 20 µm and the applied bias voltage varies from 0.325 V to 2.915 V (∆U = 2.59 V), the power consumption can be estimated by: Then, we have an E bit of 0.072 pJ/bit.

Discussion
To clarify the merits of proposed design, the theoretical performances of the polarizationinsensitive graphene modulators that have been reported are comprehensively compared with that of this work. As shown in Table 1, the device we proposed has a higher bandwidth and f 3-dB as well as the smallest E bit compared to those proposed in other studies. Moderate MD and ∆MD values can also be obtained. The characteristics of Re(∆N eff ) are also better than in most reported works. These favorable characteristics mainly originate from the hybrid waveguide structure, which provides a balanced mode field. The metal slot that supports the plasmonic mode offers the opportunity for wideband operation. It exhibits the favorable features of a small footprint, high speed, and low power consumption.

Conclusions
A polarization-insensitive graphene-assisted electro-optic modulator is proposed. The orthogonal T-shaped metal slot structure offers polarization insensitivity. The proposed dual-layer graphene structure could enhance the light interaction and reduce the effective width of graphene, which is favorable for increasing bandwidth and reducing power consumption. Simulations with finite element algorithms show that the modulation depths of the TE mode and the complex mode supported by the proposed modulator are 0.502 dB/µm and 0.511 dB/µm, respectively. The 3 dB bandwidth is about 127 GHz when the modulation length is 20 µm. The power consumption may be restrained to 72 fJ/bit. The proposed modulator with a favorable bandwidth and relatively low power consumption has potential applications in high-speed on-chip interconnected information transfer and processing.

Conflicts of Interest:
The authors declare no conflict of interest.