# Polarization-Insensitive Phase Modulators Based on an Embedded Silicon-Graphene-Silicon Waveguide
## 2 Polarization-Insensitive Waveguide Structure  
As is known, graphene is an attractive material for electro-optic modulators due to its strong interaction with light and electrically-controllable optical conductivity [14]. It has been reported that the structure of a sandwiched graphene-silicon waveguide will enhance the interaction between graphene and light [21]. However, sandwiched graphene-silicon waveguides are generally polarization-sensitive, since the TE and TM modes experience an unbalanced refractive index and loss on the horizontal and vertical axes. To achieve polarization insensitivity, the structure and size of the graphene waveguide should be specifically designed, since they have much influence on the effective refractive index (_n_eff) for TE and TM modes. Here, we design a novel embedded structure in which an inner waveguide is embedded into an outer one. As is shown in Figure 1, an inner silicon waveguide with 400 nm width and 460 nm height is surrounded by two 0.7 nm graphene layers. The two graphene layers, which are separated by a 10 nm thick Si\({}_{3}\)N\({}_{4}\) spacer, are also surrounded by an outer waveguide with the width and height of 800 nm and 800 nm. For fabrication, it is better to replacing the rectangular silicon-graphene waveguide with a slanted silicon-graphene waveguide for the inner waveguide [22]. Metal electrodes are in contact with the graphene layers, so that the two graphene layers can form capacitor structures.  
For a polarization-insensitive waveguide, the real-part difference of an effective refractive index between TE and TM modes (\(\Delta Re=Re(n_{\textit{eff\_TE}})-Re(n_{\textit{eff\_TM}})\)) should be less than 5 \(\times\) 10\({}^{-4}\)[23], and the insertion loss difference between TE and TM modes (\(\Delta\alpha=\alpha(TE)-\alpha(TM)\)), which results from the imaginary-part difference of the refractive effective refractive index between TE and TM modes, should be less than 4 \(\times\) 10\({}^{-3}\) dB/\(\mu\)m [11; 24; 25; 26], such as the polarization sensitive \(<\) 1 dB with the waveguide length of 250 \(\mu\)m (i.e., 4 \(\times\) 10\({}^{-3}\) dB/\(\mu\)m) [26]. The insertion loss can be calculated with \(\alpha=-40\pi\textit{lge\cdot Im}(n_{\textit{eff\_TM}})/\lambda\). The outer waveguide around the small one enhances light-graphene interaction which is the key design element for achieving effective electro-optic modulation. In the horizontal  
Figure 1: The cross-sectional view of the polarization-insensitive waveguide.  
and vertical directions, the light in the waveguide is influenced by the graphene, to make sure that the waveguide is polarization-insensitive to the TM and TE modes. With the finite element method, \(\Delta Re\) and \(\Delta\alpha\) of the proposed waveguide are calculated in the case of different widths and heights of the inner waveguide, as shown in Figure 2. In the simulation, there is no voltage on the graphene and the incident light is set as \(\lambda\) = 1550 nm. From Figure 2a, it can be observed that \(\Delta Re\) and \(\Delta\alpha\) are kept to be less than 1.5 \(\times\) 10\({}^{-3}\) and 1.4 \(\times\) 10\({}^{-2}\) dB/\(\upmu\)m, respectively, with the inner waveguide width ranging from 300 nm to 500 nm. From Figure 2b, it can be observed that \(\Delta Re\) and \(\Delta\alpha\) are kept to be less than 1.5 \(\times\) 10\({}^{-3}\) and 4 \(\times\) 10\({}^{-2}\) dB/\(\upmu\)m, respectively, with the inner waveguide height ranging from 410 nm to 530 nm. The waveguide width and height can be optimized to make sure there is a small value of \(\Delta Re\) and \(\Delta\alpha\) for a polarization-insensitive waveguide.