## 2 Configuration and Structure  
### 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, \(n_{\text{Ag}}\) = 0.14447 + 11.366), a middle silica layer (SiO\({}_{2}\), \(n_{\text{SiO2}}\) = 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.  
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 \(\upmu\)m covered with graphene-SiO\({}_{2}\)-graphene stacks but without applied voltage (U = 0 V (\(\mu_{\text{c}}\) = 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.  
Figure 1: (**a**) Two-dimensional (2D) cross section; (**b**) three-dimensional (3D) view of the graphene modulator with dual built-in orthogonal slots waveguide.  
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 \(\mid\!E_{\mathrm{x}}\!\mid\) and \(\mid\!E_{\mathrm{y}}\!\mid\), 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 \(E_{\mathrm{y}}\)-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 \(E_{\mathrm{y}}\) and \(E_{\mathrm{x}}\) 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 polarization-insensitive modulation.  
Figure 3: The electric field distributions of (**a**) transverse magnetic (TM) polarized mode and (**b**) complex-polarization mode. The color bars and arrows represent the intensity and direction of the electric field, respectively.  
Figure 2: (**a**) Transmission spectrum of the dual built-in orthogonal slots waveguide (DBOSW) with and without graphene; (**b**) optical absorption in the graphene capacitor structure.  
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:  
\[\text{MPA}=40\pi(\log_{10}\text{e})\text{Im}(N_{\text{eff}})/\lambda \tag{1}\]  
\[\text{MD}=\text{MPA}(\text{OFF})-\text{MPA}(\text{ON}) \tag{2}\]  
\[\text{PL}=\text{MPA}(\text{ON}) \tag{3}\]  
where \(\lambda\) is the incident wavelength, \(\text{Im}(N_{\text{eff}})\) 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).