A Broadband Polarization-Insensitive Graphene Modulator Based on Dual Built-in Orthogonal Slots Plasmonic Waveguide

: A broadband polarization-insensitive graphene modulator has been proposed. The dual built-in orthogonal slots waveguide allows polarization independence for the transverse electric (TE) mode and the transverse magnetic (TM) mode. Due to the introduction of metal slots in both the vertical and horizontal directions, the optical ﬁeld as well as the electro-absorption of graphene are enhanced by the plasmonic effect. The proposed electro-optic modulator shows a modulation depth of 0.474 and 0.462 dB/ µ m for two supported modes, respectively. An ultra-low effective index difference of 0.001 can be achieved within the wavelength range from 1100 to 1900 nm. The 3 dB-bandwidth is estimated to be 101 GHz. The power consumption is 271 fJ/bit at a modulation length of 20 µ m. The proposed modulator provides high speed broadband solutions in microwave photonic systems.


Introduction
High performance electro-optic (EO) modulators are highly demanded in wideband optics communication networks [1]. The lithium niobite-based modulators are mature but suffer from a large footprint [2]. Silicon modulators are compatible with CMOS technology, while facing the problem of low EO efficiency [3]. III-V based modulators offer a small footprint; however, the insertion loss and thermal stability are still to be improved [4]. SiC-based waveguides also show application potential [5]. Due to the unsatisfactory EO properties of the above underlying materials, modulators that meet all requires are still elusive. Graphene with excellent EO characteristics has been well studied and widely adopted in optical modulators [6][7][8][9]. In 2011, Liu et al. experimentally presented a graphene modulator with a modulation depth (MD, the difference between "ON" state and "OFF" state) of 0.1 dB/µm [10]. A built-in p-oxide-n-like junction graphene modulator exhibited an MD of 0.16 dB/µm and a power consumption of 1 pJ/bit [11]. The graphene EO modulator with a high 3 dB-bandwidth of 30 GHz has also been demonstrated [12]. However, the performance of above graphene modulators is limited by the weak interaction between the extremely thin graphene flake and the light field that is strongly bounded in the high index dielectric material. If the plasmonic effect is introduced, the light can propagate as a surface plasmon polarizations (SPPs) mode along the waveguide, which implies an enhanced optical field intensity. More optical power will be guided to the dielectric/graphene interface, facilitating the light-graphene interaction [13][14][15]. Huang et al. proposed a waveguide-coupled hybrid plasmonic graphene modulator with a 3 dB-bandwidth of 0.48 THz and an energy consumption of 145 fJ/bit [16]. A hybrid plasmonic graphene modulator with a 3 dB-bandwidth of 0.662 THz and a power consumption of 118.7 fJ/bit promises good potential for hybrid plasmonic graphene EO modulators [17]. Another issue for the graphene modulator is the polarization sensitivity, which originates from the low absorption rate of 2.3% for the electric field polarization perpendicular to the graphene flake surface. Most reported works can only operate in transverse magnetic (TM) or transverse electric (TE) polarization independently [18]. The introduction of polarization control functions will inevitably complexify the device structure [19]. Therefore, a compact and polarization-insensitive graphene modulator with characteristic of broadband operation is desirable.
In this paper, we proposed a graphene modulator based on a dual built-in orthogonal slots waveguide (DBOSW). The geometric parameters of the hybrid plasmonic waveguide are investigated and optimized by finite element method (FEM). The mechanism of polarization insensitivity and broadband modulation is elaborated in detail. Our proposed waveguide structure overcomes the strict polarization dependence of plasmonic-based waveguides. Moreover, the DBOSW graphene modulator realizes broadband modulation with excellent polarization independence, which shows good potential in microwave photonic systems.

Device Structure
As shown in Figure 1, the proposed graphene modulator based on the dual built-in orthogonal slots waveguide consists of a bottom silver layer (Ag, n Ag = 0.14447 + 11.366), a middle silica layer (SiO 2 , n SiO2 = 1.45) and a top silver slot isolated by SiO 2 . Double 0.33 nm-thick graphene flakes are isolated by 15 nm-thick SiO 2 and cover the hybrid waveguide. The metal stacks-graphene contacts form the electrodes [20]. The whole structure is supported by the SiO 2 substrate.
Appl. Sci. 2021, 11, x FOR PEER REVIEW 2 of 16 plasmonic graphene modulator with a 3 dB-bandwidth of 0.662 THz and a power consumption of 118.7 fJ/bit promises good potential for hybrid plasmonic graphene EO modulators [17]. Another issue for the graphene modulator is the polarization sensitivity, which originates from the low absorption rate of 2.3% for the electric field polarization perpendicular to the graphene flake surface. Most reported works can only operate in transverse magnetic (TM) or transverse electric (TE) polarization independently [18]. The introduction of polarization control functions will inevitably complexify the device structure [19]. Therefore, a compact and polarization-insensitive graphene modulator with characteristic of broadband operation is desirable.
In this paper, we proposed a graphene modulator based on a dual built-in orthogonal slots waveguide (DBOSW). The geometric parameters of the hybrid plasmonic waveguide are investigated and optimized by finite element method (FEM). The mechanism of polarization insensitivity and broadband modulation is elaborated in detail. Our proposed waveguide structure overcomes the strict polarization dependence of plasmonic-based waveguides. Moreover, the DBOSW graphene modulator realizes broadband modulation with excellent polarization independence, which shows good potential in microwave photonic systems.

Device Structure
As shown in Figure 1, the proposed graphene modulator based on the dual built-in orthogonal slots waveguide consists of a bottom silver layer (Ag, nAg = 0.14447+ 11.366), a middle silica layer (SiO2, nSiO2 = 1.45) and a top silver slot isolated by SiO2. Double 0.33 nmthick graphene flakes are isolated by 15 nm-thick SiO2 and cover the hybrid waveguide. The metal stacks-graphene contacts form the electrodes [20]. The whole structure is supported by the SiO2 substrate. Graphene plays a major role in optical modulation of this modulator. As shown in Figure 2a, the transmission spectrum of DBOSW with a modulation length of L = 20 μm covered with graphene-SiO2-graphene stacks but without applied voltage (U = 0 V (μc = 0 eV)) has a sharp drop compared to DBOSW without graphene. This large significant light loss mainly originates from the light absorption effect of graphene, indicating that graphene has an effective light modulation function. Graphene plays a major role in optical modulation of this modulator. As shown in Figure 2a, the transmission spectrum of DBOSW with a modulation length of L = 20 µm covered with graphene-SiO 2 -graphene stacks but without applied voltage (U = 0 V (µ c = 0 eV)) has a sharp drop compared to DBOSW without graphene. This large significant light loss mainly originates from the light absorption effect of graphene, indicating that graphene has an effective light modulation function. Appl. Sci. 2021, 11, x FOR PEER REVIEW 3 of 16 Two modes with different electric field polarizations are supported by the dual builtin orthogonal slots waveguide. The distributions of transversal and longitudinal electric field components of the transverse polarization (TM) mode and the complex-polarization mode (complex mode) are illustrated by |Ex| and |Ey|, respectively. As depicted in Figure  3a, the TM mode is mainly distributed in the middle silica layer, which is attributed to the excitation of surface plasmon polarizations (SPPs) constrained by the top and bottom Ag layers. This Ey-dominated plasmonic mode is similar to the optical mode constrained in the metal slot waveguide [21]. As shown in Figure 3b, the complex mode is mainly distributed in the upper slot and extends into the middle SiO2 layer. It should be noted that the complex mode can be seen as a mix of plasmonic modes supported by the crossed vertical metal slot and horizontal metal slot. In this case, both Ey and Ex components exist. The symmetry of both the horizontal and the vertical metal-insulator-metal (MIM) substructures lead to the spreading of transversal and longitudinal polarizations into the inplane graphene-SiO2-graphene layers, which is remarkably favorable to polarization-insensitive modulation.  Two modes with different electric field polarizations are supported by the dual built-in orthogonal slots waveguide. The distributions of transversal and longitudinal electric field components of the transverse polarization (TM) mode and the complex-polarization mode (complex mode) are illustrated by |E x | and |E y |, respectively. As depicted in Figure 3a, the TM mode is mainly distributed in the middle silica layer, which is attributed to the excitation of surface plasmon polarizations (SPPs) constrained by the top and bottom Ag layers. This E y -dominated plasmonic mode is similar to the optical mode constrained in the metal slot waveguide [21]. As shown in Figure 3b, the complex mode is mainly distributed in the upper slot and extends into the middle SiO 2 layer. It should be noted that the complex mode can be seen as a mix of plasmonic modes supported by the crossed vertical metal slot and horizontal metal slot. In this case, both E y and E x components exist. The symmetry of both the horizontal and the vertical metal-insulator-metal (MIM) substructures lead to the spreading of transversal and longitudinal polarizations into the in-plane graphene-SiO 2 -graphene layers, which is remarkably favorable to polarizationinsensitive modulation.  Two modes with different electric field polarizations are supported by the dual builtin orthogonal slots waveguide. The distributions of transversal and longitudinal electric field components of the transverse polarization (TM) mode and the complex-polarization mode (complex mode) are illustrated by |Ex| and |Ey|, respectively. As depicted in Figure  3a, the TM mode is mainly distributed in the middle silica layer, which is attributed to the excitation of surface plasmon polarizations (SPPs) constrained by the top and bottom Ag layers. This Ey-dominated plasmonic mode is similar to the optical mode constrained in the metal slot waveguide [21]. As shown in Figure 3b, the complex mode is mainly distributed in the upper slot and extends into the middle SiO2 layer. It should be noted that the complex mode can be seen as a mix of plasmonic modes supported by the crossed vertical metal slot and horizontal metal slot. In this case, both Ey and Ex components exist. The symmetry of both the horizontal and the vertical metal-insulator-metal (MIM) substructures lead to the spreading of transversal and longitudinal polarizations into the inplane graphene-SiO2-graphene layers, which is remarkably favorable to polarization-insensitive modulation.  In order to better illustrate the waveguide optimization and characterize the modulation performance of proposed design, the mode power attenuation (MPA), modulation depth (MD) and propagation loss (PL) are defined as follows: where λ is the incident wavelength, Im(N eff ) is the imaginary part of the effective index, MD is defined as the difference between MPA ("ON" state) and MPA ("OFF" state) and PL refers to MPA ("ON" state).

Graphene Film
Graphene is an emerging material with single layer carbon atoms in the form of hexagons. This unique structure offers graphene optical properties of controllable light absorption, considerable carrier mobility, uniform light absorption rate of 2.3% in a broadband, etc. [10,[22][23][24][25][26][27][28][29][30]. These favorable properties imply the graphene has good potential in the application of high-speed, low-footprint and low-power consumption devices.
As shown in Figure 2b, a plate capacitor is formed by applying bias voltage to graphene layers. Light absorption is then controlled by the applied external electric field via the electrostatic doping effect. Since the interband transition in graphene is affected by Fermi level with no consideration of sign, both graphene layers can simultaneously be "transparent" at the high driving voltage, which refers to the "ON" state. An efficient numerical calculation method is important [31]. As has been reported [32], the graphene layer can be modeled as a 3D bulk material with a constant thickness or a 2D surface structure. To obtain accurate results, a smaller grid size is required in 3D simulations, which implies longer calculation time and larger memories requirement. On the basis of accuracy, it is desirable to model graphene as a 2D layer for higher efficiency. In this work, the graphene layer is considered a surface conductivity dielectric with a thickness of zero. Benefit from the reduced number of meshes, the calculation time reduces with accuracy guaranteed. The surface conductivity σ g of the graphene may be adjusted by its chemical potential µ c . This process can be well described by the Kubo formula [33], in which τ corresponds to the relaxation time, the intraband transition τ 1 is 1.2 ps and the interband transition τ 2 is 15 fs. Graphene layers exhibit special physical properties at the epsilon-near-zero (ENZ) point, µ c (ENZ) ≈ 0.5 eV [34]. When µ c is larger than µ c (ENZ), graphene shows the characteristics of the dielectric medium. Conversely, graphene behaves like a metallic layer and strong light absorption happens when µ c is smaller than µ c (ENZ). Equation (4) illustrates the relationship between chemical potential and bias voltage [30]: where v F is the Fermi velocity of graphene (v F = 2.5 × 10 6 m/s) [35]] and η = ε r ε 0 /de is derived from the parallel capacitor model (ε r and d are the permittivity and thickness of SiO 2 isolation layer, respectively). As shown in Figure 4, µ c can shift from 0.2 to 0.6 eV as the applied voltage changes from 0.608 to 5.442 V. Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 16

Design Optimization
To confirm the effect of waveguide dimension on the mode field distribution, the impact of slot width change on the field confinement and metal absorption is investigated, as shown in the Figure 5a-d (g = h = 50) and Figure 5e-h (g = h = 200 nm), respectively. When g and h increase, the field confinement profiles become slack, which handicap the light-graphene interaction. When a smaller slot width (e.g., 50 nm) is adopted, a higher metal absorption can be expected, since the mode power extends more into the metal layers. When a relatively larger slot width (e.g., 200 nm) is adopted, modes are mainly distributed in SiO2, which implies a relatively lower optical loss. Therefore, systematic dimension analysis is necessary to balance field confinement and metal absorption. In the following geometric discussions, MD is temporarily defined as the difference between MPA (μc = 1 eV) and MPA (μc = 0 eV), and PL refers to MPA (μc = 1 eV) to stress the comparison. The effective index of DBOSW is also calculated at μc = 0 and 1 eV. The polarization dependence of proposed design is studied by comparison of Neff at different μc and geometric parameters.

hb
As mentioned above, the TM mode and complex mode in this hybrid waveguide can be regarded as compound polarizations supported by the vertical, as well as the horizon-

Design Optimization
To confirm the effect of waveguide dimension on the mode field distribution, the impact of slot width change on the field confinement and metal absorption is investigated, as shown in the Figure 5a-d (g = h = 50) and Figure 5e-h (g = h = 200 nm), respectively. When g and h increase, the field confinement profiles become slack, which handicap the light-graphene interaction. When a smaller slot width (e.g., 50 nm) is adopted, a higher metal absorption can be expected, since the mode power extends more into the metal layers. When a relatively larger slot width (e.g., 200 nm) is adopted, modes are mainly distributed in SiO 2 , which implies a relatively lower optical loss. Therefore, systematic dimension analysis is necessary to balance field confinement and metal absorption. In the following geometric discussions, MD is temporarily defined as the difference between MPA (µ c = 1 eV) and MPA (µ c = 0 eV), and PL refers to MPA (µ c = 1 eV) to stress the comparison. The effective index of DBOSW is also calculated at µ c = 0 and 1 eV. The polarization dependence of proposed design is studied by comparison of N eff at different µ c and geometric parameters.

Design Optimization
To confirm the effect of waveguide dimension on the mode field distribution, the impact of slot width change on the field confinement and metal absorption is investigated, as shown in the Figure 5a-d (g = h = 50) and Figure 5e-h (g = h = 200 nm), respectively. When g and h increase, the field confinement profiles become slack, which handicap the light-graphene interaction. When a smaller slot width (e.g., 50 nm) is adopted, a higher metal absorption can be expected, since the mode power extends more into the metal layers. When a relatively larger slot width (e.g., 200 nm) is adopted, modes are mainly distributed in SiO2, which implies a relatively lower optical loss. Therefore, systematic dimension analysis is necessary to balance field confinement and metal absorption. In the following geometric discussions, MD is temporarily defined as the difference between MPA (μc = 1 eV) and MPA (μc = 0 eV), and PL refers to MPA (μc = 1 eV) to stress the comparison. The effective index of DBOSW is also calculated at μc = 0 and 1 eV. The polarization dependence of proposed design is studied by comparison of Neff at different μc and geometric parameters.

hb
As mentioned above, the TM mode and complex mode in this hybrid waveguide can be regarded as compound polarizations supported by the vertical, as well as the horizon-

h b
As mentioned above, the TM mode and complex mode in this hybrid waveguide can be regarded as compound polarizations supported by the vertical, as well as the horizontal Ag-SiO 2 -Ag structure. The MD, PL and Re(N eff ) of the proposed modulator as a function of bottom Ag slab thickness h b are shown in the Figure 6. The analysis of h b is implemented by a similar way adopted in the MIM waveguide model analysis, because it is also with thin metal layers [21]. Compared to complex mode, the TM mode is almost dominated by the longitudinal polarization, therefore the dimension change in vertical direction has a larger impact on the TM mode. Figure 6a shows that Re(N eff ) of the TM mode is more sensitive to h b than that of complex mode. The Re(N eff ) of theTM mode keeps decreasing with the rising of h b , which decreases from 1.6993 to 1.6464 at µ c = 1 eV and 1.718 to 1.6654 at µ c = 0 eV. Meanwhile, Re(N eff ) of complex mode slowly decreases from h b = 50 nm, and reaches to a stable value at h b = 150 nm. In addition, as shown in Figure 6c, the change in h b has a significant impact on the PL of the TM mode that decreases with the increasing of h b and reaches a stable value at h b = 150 nm. In contrast, PL of complex mode remains at around 0.15 dB/µm that is caused by the fraction of the TM mode distributed in the metal layer. This leads to a larger optical loss. The MD change of two modes versus h b is very small, which is related to the limited influence of h b on the light field distribution. The Re(∆N eff ) of two modes versus h b is shown in Figure 6b. Here, Re(∆N eff ) is determined by tal Ag-SiO2-Ag structure. The MD, PL and Re(Neff) of the proposed modulator as a function of bottom Ag slab thickness hb are shown in the Figure 6. The analysis of hb is implemented by a similar way adopted in the MIM waveguide model analysis, because it is also with thin metal layers [21]. Compared to complex mode, the TM mode is almost dominated by the longitudinal polarization, therefore the dimension change in vertical direction has a larger impact on the TM mode. Figure 6a shows that Re(Neff) of the TM mode is more sensitive to hb than that of complex mode. The Re(Neff) of theTM mode keeps decreasing with the rising of hb, which decreases from 1.6993 to 1.6464 at μc = 1 eV and 1.718 to 1.6654 at μc = 0 eV. Meanwhile, Re(Neff) of complex mode slowly decreases from hb = 50 nm, and reaches to a stable value at hb = 150 nm. In addition, as shown in Figure 6c, the change in hb has a significant impact on the PL of the TM mode that decreases with the increasing of hb and reaches a stable value at hb = 150 nm. In contrast, PL of complex mode remains at around 0.15 dB/μm that is caused by the fraction of the TM mode distributed in the metal layer. This leads to a larger optical loss. The MD change of two modes versus hb is very small, which is related to the limited influence of hb on the light field distribution. The Re(ΔNeff) of two modes versus hb is shown in Figure 6b. Here, Re(ΔNeff) is determined by Since a low Re(ΔNeff) is more desirable, hb is chosen to be 180 nm.  Figure 7 shows the MD, PL, Re(Neff) and Re(ΔNeff) of two modes in the DBOSW modulator versus the middle silica layer thickness h. As shown Figure 7a, Re(Neff) of both modes reduce consistently with the increasing of h. Furthermore, when h = 100 nm, the Re(Neff) of the TM mode and complex mode show a very close minimum, which can be confirmed in Figure 7b. Though Re(ΔNeff) can already be restrained at a low level (< 0.115) when h is 50 or 200 nm, the thickness of silica layer apparently has a more significant impact on the polarization dependence. As shown in Figure 7c, MD and PL of two modes decrease simultaneously with the increasing of h. These trends are attributed to less confinement of the light field in the middle silica layer as well as the weakening of the interaction with graphene. This is depicted in the illustration in Figure 7c. As h decreases, the electric field intensity of the TM mode and the y-axis electric field component of complex mode increases in the middle silica layer. The increment of mode power in the metal layer Since a low Re(∆N eff ) is more desirable, h b is chosen to be 180 nm. Figure 7 shows the MD, PL, Re(N eff ) and Re(∆N eff ) of two modes in the DBOSW modulator versus the middle silica layer thickness h. As shown Figure 7a, Re(N eff ) of both modes reduce consistently with the increasing of h. Furthermore, when h = 100 nm, the Re(N eff ) of the TM mode and complex mode show a very close minimum, which can be confirmed in Figure 7b. Though Re(∆N eff ) can already be restrained at a low level (<0.115) when h is 50 or 200 nm, the thickness of silica layer apparently has a more significant impact on the polarization dependence. As shown in Figure 7c, MD and PL of two modes decrease simultaneously with the increasing of h. These trends are attributed to less confinement of the light field in the middle silica layer as well as the weakening of the interaction with graphene. This is depicted in the illustration in Figure 7c. As h decreases, the electric field intensity of the TM mode and the y-axis electric field component of complex mode increases in the middle silica layer. The increment of mode power in the metal layer invites higher optical loss. To be noted, the MD difference between the TM mode and complex mode remains at a low level when h varies from 50 to 200 nm. However, the PL of the TM mode becomes lower than that of complex mode when h is higher than 95 nm. To constrain the performance difference between the TM mode and complex mode, h is chosen to be 100 nm. invites higher optical loss. To be noted, the MD difference between the TM mode and complex mode remains at a low level when h varies from 50 to 200 nm. However, the PL of the TM mode becomes lower than that of complex mode when h is higher than 95 nm. To constrain the performance difference between the TM mode and complex mode, h is chosen to be 100 nm.

ht
When hb = 180 nm, h = 100 nm, Re(Neff) and Re(ΔNeff), the MD and PL of two modes in the DBOSW modulator versus the top layer thickness ht are investigated. As shown in Figure 8a, Re(Neff) of the TM mode decreases with the increment of ht at μc = 0 and 1 eV. When ht is 110 nm, Re(Neff) of the complex mode exhibits a minimum of 1.6722 and 1.6469 at μc = 0 and 1 eV, respectively. It is noteworthy that both modes share the same Re(Neff) when ht is 200 nm and μc is 1 eV. This scenario also can be found when ht is 210 nm and μc is 0 eV. In Figure 8 (b), a maximum Re(ΔNeff) is first presented with the increase of ht as μc = 0 and 1 eV. Then, a minimum Re(ΔNeff) of 6 × 10 −4 (μc = 1 eV) appears when ht is 200 nm. A minimum Re(ΔNeff) of 1 × 10 −4 (μc = 0 eV) can be observed when ht is 210 nm. The different behaviors of Re(Neff) and Re(ΔNeff) in Figure 8a and Figure 8b can be explained by the change of fringing field distribution as well as the variation in the slot region. This phenomenon is similar to what happens to the MIM waveguide [20]. As shown in Figure 8c, the PLs of both the TM mode and the complex mode decrease till ht rises up to 200 nm. As ht increases from 50 nm to 200 nm, the MD of the complex mode decreases by 57.1%, while the MD of the TM mode only reduces by 7.0%. Obviously, the change of ht has a more remarkable impact on the complex mode than that on the TM mode. This can be explained by the field distribution difference that can be observed from the illustration in Figure 8c

h t
When h b = 180 nm, h = 100 nm, Re(N eff ) and Re(∆N eff ), the MD and PL of two modes in the DBOSW modulator versus the top layer thickness h t are investigated. As shown in Figure 8a, Re(N eff ) of the TM mode decreases with the increment of h t at µ c = 0 and 1 eV. When h t is 110 nm, Re(N eff ) of the complex mode exhibits a minimum of 1.6722 and 1.6469 at µ c = 0 and 1 eV, respectively. It is noteworthy that both modes share the same Re(N eff ) when h t is 200 nm and µ c is 1 eV. This scenario also can be found when h t is 210 nm and µ c is 0 eV. In Figure 8b, a maximum Re(∆N eff ) is first presented with the increase of h t as µ c = 0 and 1 eV. Then, a minimum Re(∆N eff ) of 6 × 10 −4 (µ c = 1 eV) appears when h t is 200 nm. A minimum Re(∆N eff ) of 1 × 10 −4 (µ c = 0 eV) can be observed when h t is 210 nm. The different behaviors of Re(N eff ) and Re(∆N eff ) in Figures 8a and 8b can be explained by the change of fringing field distribution as well as the variation in the slot region. This phenomenon is similar to what happens to the MIM waveguide [20]. As shown in Figure 8c, the PLs of both the TM mode and the complex mode decrease till h t rises up to 200 nm. As h t increases from 50 nm to 200 nm, the MD of the complex mode decreases by 57.1%, while the MD of the TM mode only reduces by 7.0%. Obviously, the change of h t has a more remarkable impact on the complex mode than that on the TM mode. This can be explained by the field distribution difference that can be observed from the illustration in Figure 8c. With comprehensive consideration, a balanced h t of 200 nm is chosen.

w
For the waveguide structure in Figure 1, the width of the Ag strips w will definitely affect the mode field distribution. Therefore, Re(N eff ), Re(∆N eff ), MD and PL of the TM mode and complex mode in the DBOSW modulator versus the Ag strips w are investigated, when h b = 180 nm, h = 100 nm, and h t = 200 nm. As shown in Figure 9a, Re(N eff ) of the TM mode gradually increases from 1.631 to 1.675 at µ c = 0 eV, when w varies from 50 to 150 nm. Re(N eff ) of the TM mode remains at 1.675, without changing with the increment in w. Inversely, Re(N eff ) of complex mode decreases from 1.712 to 1.673 at µ c = 0 eV, when w varies from 50 to 150 nm. A negligible change of Re(N eff ) of the complex mode is observed when w is over 150 nm. When µ c = 1 eV, Re(N eff ) of the TM mode and complex mode exhibit similar inverse variation with the increment in w, which leads to a turning point of Re(∆N eff ) at w = 150 nm, as shown in Figure 9b.

w
For the waveguide structure in Figure 1, the width of the Ag strips w will definitely affect the mode field distribution. Therefore, Re(Neff), Re(ΔNeff), MD and PL of the TM mode and complex mode in the DBOSW modulator versus the Ag strips w are investigated, when hb = 180 nm, h = 100 nm, and ht = 200 nm. As shown in Figure 9(a), Re(Neff) of the TM mode gradually increases from 1.631 to 1.675 at μc = 0 eV, when w varies from 50 to 150 nm. Re(Neff) of the TM mode remains at 1.675, without changing with the increment in w. Inversely, Re(Neff) of complex mode decreases from 1.712 to 1.673 at μc = 0 eV, when w varies from 50 to 150 nm. A negligible change of Re(Neff) of the complex mode is observed when w is over 150 nm. When μc = 1 eV, Re(Neff) of the TM mode and complex mode exhibit similar inverse variation with the increment in w, which leads to a turning point of Re(ΔNeff) at w = 150 nm, as shown in Figure 9b.
As w changes from 50 nm to 200 nm, the MDs of the TM mode and complex mode decrease by 55.1 and 30.5%, respectively. Both modes have the same MD at w = 160 nm. When w ranges from 50 to 150 nm, the PL of the TM mode decreases from 0.16 to 0.14 dB/μm, while PL of complex mode from 0.21 to 0.15 dB/μm. Both PLs become stable when w is larger than 150 nm, as shown in Figure 9c. The decrement of the MD and PL can be attributed to the extension of the mode field region with the increment in w, which weakens the graphene-light interaction strength, as shown in the illustration in Figure 9c. Since the TM mode and complex mode are distributed differently in the middle slot and the top metal slot, the increment in w has a greater impact on the MD of the TM mode. Moreover, the enlargement of the mode field area will weaken the field intensity, and then the metal absorption, leading to a lower PL. From the above, Re(ΔNeff) shows a minimum, while the MDs and PLs of the two modes are very close at w = 150 nm. Therefore, we choose w to be 150 nm in following work.

g
Another parameter to be confirmed in the proposed design is the gap width g between two upper metal parts. When hb = 180 nm, h = 100 nm, w = 150 nm and ht = 200 nm, Re(Neff), Re(ΔNeff), MD and PL of the two modes in the DBOSW modulator versus metal slot width g are investigated. As shown in Figure 10, similar variations of Re(Neff) and Re(ΔNeff) to that in Figure 9 can be observed.
The difference is that the change of g has a greater impact on Re(ΔNeff) than that on Re(Neff). As g increases, Re(ΔNeff) decreases firstly and then gradually increases. At g = 50 nm, Re(ΔNeff) shows the maximum of 0.2707 and 0.2691 for μc = 0 and 1 eV, respectively. The minimum Re(ΔNeff) of 6 × 10 −4 and 3 × 10 −4 can be found at g = 125 nm for μc = 0 and 1 eV, respectively.
As shown in Figure 10c, the MD of the complex mode is less sensitive to the variation of g than the TM mode, which is due to the enhanced interaction strength between the x- As w changes from 50 nm to 200 nm, the MDs of the TM mode and complex mode decrease by 55.1 and 30.5%, respectively. Both modes have the same MD at w = 160 nm. When w ranges from 50 to 150 nm, the PL of the TM mode decreases from 0.16 to 0.14 dB/µm, while PL of complex mode from 0.21 to 0.15 dB/µm. Both PLs become stable when w is larger than 150 nm, as shown in Figure 9c. The decrement of the MD and PL can be attributed to the extension of the mode field region with the increment in w, which weakens the graphene-light interaction strength, as shown in the illustration in Figure 9c. Since the TM mode and complex mode are distributed differently in the middle slot and the top metal slot, the increment in w has a greater impact on the MD of the TM mode. Moreover, the enlargement of the mode field area will weaken the field intensity, and then the metal absorption, leading to a lower PL. From the above, Re(∆N eff ) shows a minimum, while the MDs and PLs of the two modes are very close at w = 150 nm. Therefore, we choose w to be 150 nm in following work.

g
Another parameter to be confirmed in the proposed design is the gap width g between two upper metal parts. When h b = 180 nm, h = 100 nm, w = 150 nm and h t = 200 nm, Re(N eff ), Re(∆N eff ), MD and PL of the two modes in the DBOSW modulator versus metal slot width g are investigated. As shown in Figure 10, similar variations of Re(N eff ) and Re(∆N eff ) to that in Figure 9 can be observed.

μ. c Response
The relationships between Re(Neff) (Re(ΔNeff)), PL, Im(Neff) and the chemical potential μc also have been investigated. Figure 11a shows that Re(Neff) of the TM mode first increases from 1.6699 to the peak value of 1.6772 as μc increases from 0 to 0.4 eV. Subsequently, it decreases to 1.6511 as μc continues to increase to 1 eV. Similar trends for Re(Neff) of the complex mode can also be observed. Furthermore, the Re(ΔNeff) remains at a low level with an average of 3.6 × 10 −4 in the range of μc = 0 to 1 eV. Figure 11b shows that the PL decreases rapidly as μc decreases from 0.2 to 0.6 eV. Therefore, the "ON" and the "OFF" state of DBOSW modulator is defined at μc = 0.6 and 0.2 eV, respectively. In this case, the MD of the TM mode up to 0.474 dB/μm can be obtained, while the MD of the complex mode is 0.462 dB/μm. The calculated ΔMD (|MDTM mode − MDComplex mode|) is only 0.012 dB/μm at λ = 1550 nm. The difference is that the change of g has a greater impact on Re(∆N eff ) than that on Re(N eff ). As g increases, Re(∆N eff ) decreases firstly and then gradually increases. At g = 50 nm, Re(∆N eff ) shows the maximum of 0.2707 and 0.2691 for µ c = 0 and 1 eV, respectively. The minimum Re(∆N eff ) of 6 × 10 −4 and 3 × 10 −4 can be found at g = 125 nm for µ c = 0 and 1 eV, respectively.
As shown in Figure 10c, the MD of the complex mode is less sensitive to the variation of g than the TM mode, which is due to the enhanced interaction strength between the x-axis electric field component of complex mode caused by the smaller g and the lateral metal slot. This leads to a higher loss and the resulting lower MD. In contrast, for the TM mode, the variation of g only changes its mode area, which is similar to the impact of w on the TM mode, as shown in the illustration in Figure 10c. The almost unchanged PL and linearly decreasing MD of the TM mode depicted in Figure 10c is consistent with the EA graphene modulator based on the MIM waveguide [36]. Since the difference of MD and PL between the two modes remains at a low level, when g is larger than 125 nm, g is chosen to be 125 nm in this design.

µ c Response
The relationships between Re(N eff ) (Re(∆N eff )), PL, Im(N eff ) and the chemical potential µ c also have been investigated. Figure 11a shows that Re(N eff ) of the TM mode first increases from 1.6699 to the peak value of 1.6772 as µ c increases from 0 to 0.4 eV. Subsequently, it decreases to 1.6511 as µ c continues to increase to 1 eV. Similar trends for Re(N eff ) of the complex mode can also be observed. Furthermore, the Re(∆N eff ) remains at a low level with an average of 3.6 × 10 −4 in the range of µ c = 0 to 1 eV. Figure 11b shows that the PL decreases rapidly as µ c decreases from 0.2 to 0.6 eV. Therefore, the "ON" and the "OFF" state of DBOSW modulator is defined at µ c = 0.6 and 0.2 eV, respectively. In this case, the MD of the TM mode up to 0.474 dB/µm can be obtained, while the MD of the complex mode is 0.462 dB/µm. The calculated ∆MD (|MD TM mode − MD Complex mode |) is only 0.012 dB/µm at λ = 1550 nm.
in the DBOSW modulator versus g, respectively; the illustration in (c) shows the variation of electric field distribution for both modes versus g.

μ. c Response
The relationships between Re(Neff) (Re(ΔNeff)), PL, Im(Neff) and the chemical potential μc also have been investigated. Figure 11a shows that Re(Neff) of the TM mode first increases from 1.6699 to the peak value of 1.6772 as μc increases from 0 to 0.4 eV. Subsequently, it decreases to 1.6511 as μc continues to increase to 1 eV. Similar trends for Re(Neff) of the complex mode can also be observed. Furthermore, the Re(ΔNeff) remains at a low level with an average of 3.6 × 10 −4 in the range of μc = 0 to 1 eV. Figure 11b shows that the PL decreases rapidly as μc decreases from 0.2 to 0.6 eV. Therefore, the "ON" and the "OFF" state of DBOSW modulator is defined at μc = 0.6 and 0.2 eV, respectively. In this case, the MD of the TM mode up to 0.474 dB/μm can be obtained, while the MD of the complex mode is 0.462 dB/μm. The calculated ΔMD (|MDTM mode − MDComplex mode|) is only 0.012 dB/μm at λ = 1550 nm.

Optical Bandwidth
To confirm the optical bandwidth of the DBOSW modulator with the above geometric parameters, the relationships between Re(Neff) and Re(ΔNeff) of the two modes as a

Optical Bandwidth
To confirm the optical bandwidth of the DBOSW modulator with the above geometric parameters, the relationships between Re(N eff ) and Re(∆N eff ) of the two modes as a function of the incident wavelength are investigated when µ c ranges from 0.2 to 0.6 eV. As shown in Figure 12a, Re(N eff ) decreases with the increment in wavelength. When λ = 1.1 µm and µ c = 0.6 eV, the maximum Re(N eff ) of the TM mode and the complex mode is 1.7135 and 1.7163, respectively. When λ = 1.1 µm and µ c = 0.2 eV, Re(N eff ) of the TM mode and complex mode is 1.7096 and 1.7127, respectively. When λ = 1.9 µmand µ c = 0.6 eV, Re(N eff ) of the TM mode and complex mode is 1.6438 and 1.6427, respectively. When λ = 1.9 µm and µ c = 0.2 eV, Re(N eff ) of the TM mode and complex mode is 1.6586 and 1.6574, respectively. It can be observed that a very low Re(∆N eff ) of around 0.001 can be obtained within a wavelength range from 1.1 to 1.9 µm.
In Figure 12b, when the wavelength varies from 1.1 to 1.9 µm, Im(N eff ) of the TM mode quickly increases from 0.01207 to 0.02164 at µ c = 0.2 eV; meanwhile, Im(N eff ) of the complex mode increases from 0.01160 to 0.02177 at µ c = 0.2 eV, which mainly results from the enhancement of the graphene-light interaction due to the increasing dielectric constant with the increment in optical wavelength. To be mentioned, the MPA change of both two modes is very limited within the wavelength range. The MDs of the TM mode are 0.40918 dB/µm at λ = 1.1 µm and 0.46931 dB/µm at λ = 1.9 µm, respectively. While, the MDs of the complex mode are 0.38416 dB/µm at λ = 1.1 µm and 0.46415 dB/µm at λ = 1.9 µm, respectively. Therefore, the proposed modulator shows good polarization-independence characteristics over a broad optical bandwidth. the enhancement of the graphene-light interaction due to the increasing dielectric constant with the increment in optical wavelength. To be mentioned, the MPA change of both two modes is very limited within the wavelength range. The MDs of the TM mode are 0.40918 dB/μm at λ = 1.1 μm and 0.46931 dB/μm at λ = 1.9 μm, respectively. While, the MDs of the complex mode are 0.38416 dB/μm at λ = 1.1 μm and 0.46415 dB/μm at λ = 1.9 μm, respectively. Therefore, the proposed modulator shows good polarization-independence characteristics over a broad optical bandwidth.

Frequency Response
Here, the 3 dB-bandwidth (f3dB) and power consumption of the proposed DBOSW modulator with optimized geometric parameters are also confirmed. An equivalent circuit model is adopted to estimate the dynamic response of the DBOSW modulator, which is shown in Figure 13.

Frequency Response
Here, the 3 dB-bandwidth (f 3dB ) and power consumption of the proposed DBOSW modulator with optimized geometric parameters are also confirmed. An equivalent circuit model is adopted to estimate the dynamic response of the DBOSW modulator, which is shown in Figure 13. complex mode increases from 0.01160 to 0.02177 at μc = 0.2 eV, which mainly results from the enhancement of the graphene-light interaction due to the increasing dielectric constant with the increment in optical wavelength. To be mentioned, the MPA change of both two modes is very limited within the wavelength range. The MDs of the TM mode are 0.40918 dB/μm at λ = 1.1 μm and 0.46931 dB/μm at λ = 1.9 μm, respectively. While, the MDs of the complex mode are 0.38416 dB/μm at λ = 1.1 μm and 0.46415 dB/μm at λ = 1.9 μm, respectively. Therefore, the proposed modulator shows good polarization-independence characteristics over a broad optical bandwidth.

Frequency Response
Here, the 3 dB-bandwidth (f3dB) and power consumption of the proposed DBOSW modulator with optimized geometric parameters are also confirmed. An equivalent circuit model is adopted to estimate the dynamic response of the DBOSW modulator, which is shown in Figure 13. Figure 13. The equivalent circuit model. Figure 13. The equivalent circuit model.
In this model, the capacitance C air exists between two electrodes and the air, which is 12 fF [37]. The capacitance C that consists of the two graphene layers and SiO 2 isolation layer includes both the dielectric capacitance C D and the quantum capacitor C Q . Assuming the graphene layer is undoped (n s = 0), then we have C Q = 0. Hence, the capacitance C can be described by the following capacitor model: where L = 20 µm and w ol = 1385 nm are the graphene modulation length and the overlap width of two graphene layers, respectively. ε r and d are the permittivity and the thickness of SiO 2 isolation layer, respectively. R total is the total resistance, which includes the graphene sheet resistance R s of 100 Ω/ [38], and the metal electrode contact resistance to graphene layer R c of 150 Ω-µm [20]. According to Equation (6) below, R total is calculated to be about 33.85 Ω [24,30], where the effective graphene width w g is set to be 1.885 µm.
Based on Equation (8), the 3 dB-bandwidth is dominantly limited by RC delay. According to Equations (5)-(7), the DBOSW modulator shows a high f 3dB of 101 GHz. For the proposed device structure, the silica as an isolation layer that is with a relatively low permittivity and a small thickness of 15 nm is favorable to compensate the disadvantages of the large effective width and overlapping area of graphene layers. In addition, the sheet resistance Rs and contact resistance R c are values adopted in related Ref. [17].
The power consumption (E bit = C total (∆U) 2 /4) is estimated to be 0.271 pJ/bit at L = 20 µm, where the applied voltage change ∆U is 4.834 V. This is in accordance with the change of chemical potential from 0.2 to 0.6 eV.
To clarify the merits of proposed design, the theoretical performance of the reported polarization-insensitive graphene modulators is comprehensively compared with that of this work. As shown in Table 1, the proposed modulator has the largest optical bandwidth and the smallest Re(∆N eff ). Moderate MD and ∆MD also can be obtained. The characteristics of f 3dB and Ebit are also better than most reported works. The proposed DBOSW modulator can be implemented through the following fabrication process. The fabrication process starts with a commercial silica wafer. The E-beam evaporation deposits a silver layer with a thickness of 180 nm (h b ) onto the silica surface, serving as a bottom layer. Subsequently, the plasma-enhanced chemical vapor deposition (PECVD) can be used to deposit the 100 nm-thick SiO 2 (h) middle layer. Next, the thermal evaporation is used to deposit a 200 nm-thick (h t ) silver layer followed by electron-beam lithography (EBL) and inductively coupled plasma etching. The 125 nm-wide metal slot substructure (g) is then filled with SiO 2 by PECVD. The subsequent surface chemical mechanical polishing as well as the etching to three layers above can form a hybrid ridge with a width of 425 nm (w × 2 + g). After that, the bottom graphene sheet grown can be wet-transferred to the hybrid ridge waveguide. The graphene-covered region can be defined by EBL processing to remove the excess graphene. Whereas, the graphene on the other side extends 500 nm to form an electrode contact. A 15 nm-thick SiO 2 isolation layer is deposited on the bottom graphene flake. The second graphene flake is transferred onto the SiO 2 isolation layer and handled with the same process applied on the bottom graphene layer, constructing a complete hybrid waveguide structure.
Considering the structural complexity of the proposed DBOSW, the influence of manufacturing errors on the modulator is also evaluated. Firstly, the influence of inequality of the top Ag strip δ (the asymmetry of DBOSW) on the modulator performance is investigated, as shown in Figure 14. The geometric parameters used here are the optimal values discussed in above sections. It can be seen that the DBOSW modulator exhibits stable polarization insensitivity in the fluctuation of one side w from 135 to 165 nm (−15 nm > δ > 15 nm). The maximum Re(∆N eff ) at δ = 15 nm is 4 × 10 −3 , which is still lower than the optimal value in Ref. [40]. ∆MD remains at a low level, which is still competitive compared with reported works. The undulation of ∆MD relates to the difference between the mutative MD of the TM mode and the relatively constant MD of the complex mode, as shown in Figure 14c. PLs exhibit good stability. As discussed in Section 3, though the proposed design has a relatively large fabrication tolerance, g needs to be treated carefully. A significant negative effect on Re(N eff ) may damage the modulation performance at small g.

Conclusions
In summary, to achieve the polarization-insensitive modulation and enhance the modulation capability of the graphene layer, a broadband polarization-insensitive graphene modulator based on plasmonic effect with DBOSW is proposed and comprehensively investigated. By introducing the two slot-like structure in the waveguide, the DBOSW can support two modes simultaneously, which enables the modulation of both the longitudinal and the transverse polarization electric fields in a broadband bandwidth. When the external bias voltage shifts from 0.608 to 5.442 V, the MD of 0.474 and 0.462 dB/μm for the TM mode and complex mode can be obtained, respectively. A low effective index difference with an average of 0.001 is achieved within a wide optical band range from 1.1 to 1.9 μm. The 3 dB-bandwidth of 101 GHz and a power consumption of 0.271 pJ/bit at a modulation length of 20 μm can be obtained. The proposed graphene modulator has potential in microwave photonic systems.
Author Contributions: W. C. proposed the idea, carried out the simulations and wrote the manuscript. Y. X. and Y. G. plotted the figures and provided suggestions. X. W. supervised the simulations. X. S. and D. Z. supervised the whole work, interpreted the results and improved the manuscript. All authors reviewed the manuscript.

Conclusions
In summary, to achieve the polarization-insensitive modulation and enhance the modulation capability of the graphene layer, a broadband polarization-insensitive graphene modulator based on plasmonic effect with DBOSW is proposed and comprehensively investigated. By introducing the two slot-like structure in the waveguide, the DBOSW can support two modes simultaneously, which enables the modulation of both the longitudinal and the transverse polarization electric fields in a broadband bandwidth. When the external bias voltage shifts from 0.608 to 5.442 V, the MD of 0.474 and 0.462 dB/µm for the TM mode and complex mode can be obtained, respectively. A low effective index difference with an average of 0.001 is achieved within a wide optical band range from 1.1 to 1.9 µm. The 3 dB-bandwidth of 101 GHz and a power consumption of 0.271 pJ/bit at a modulation length of 20 µm can be obtained. The proposed graphene modulator has potential in microwave photonic systems.
Author Contributions: W.C. proposed the idea, carried out the simulations and wrote the manuscript. Y.X. and Y.G. plotted the figures and provided suggestions. X.W. supervised the simulations. X.S. and D.Z. supervised the whole work, interpreted the results and improved the manuscript. All authors reviewed the manuscript. All authors have read and agreed to the published version of the manuscript.