# Polarization-Insensitive Phase Modulators Based on an Embedded Silicon-Graphene-Silicon Waveguide
## 3 Graphene-Based Polarization-Insensitive Phase Modulator
As is shown in Figure 4c, the real-part of the effective refractive index \(Re(n_{\textit{eff}})\) for the waveguide changes sharply when 0.4 eV \(<\)\(\mu_{c}\) \(<\) 0.7 eV. The phase change can be simply achieved in this range of \(\mu_{c}\). Figure 4d shows that the image-part of the effective refractive index, \(Im(n_{\textit{eff}})\), experiences a large change in the range of 0.45 eV \(<\)\(\mu_{c}\) \(<\) 0.55 eV, and the \(Im(n_{\textit{eff}})\) is also large for the loss in this range. To compromise a relative large phase shift and low absorption \(\alpha\) within the range of \(\mu_{c}\), we set the operating range of 0.56 eV \(<\)\(\mu_{c}\) \(<\) 0.66 eV, corresponding to the operating voltage of 4.36 V \(<\)\(V_{g}\) \(<\) 6.1 V. The maximum variation of the real-part of the effective refractive index, \(\Delta n_{\textit{eff}}\) = 0.013, which seems to be much larger than conventional electro-optic modulator, is realized by LiNbO\({}_{3}\) (maximum value \(\Delta n_{\textit{eff}}\) = 0.0016) [30]. The phase shift is related to \(\Delta n_{\textit{eff}}\) by \(\Delta\varphi\) = (2\(\pi/\lambda\)) \(\times\)\(\Delta n_{\textit{eff}}\)\(\times\)\(L\) and the length of \(\pi\) phase shift \(L_{\pi}\) = 59.6 \(\upmu\)m. The approximate linear relationship between phase and chemical potential in the range of 0.56 eV \(<\)\(\mu_{c}\) \(<\) 0.66 eV for a 60 \(\upmu\)m long waveguides is shown in Figure 5a with a red line. To evaluate the loss in the selected range, propagation length \(Lp\) is studied and related to \(Im(n_{\textit{eff}})\) by \(Lp=1/(2 Im(n_{\textit{eff}})2\pi/\lambda)\)[31], which is the length when output power decays to 1/e of its input power, as shown in Figure 5a with a blue line. When \(\mu_{c}\) = 0.56 eV, the loss index is 0.063 dB/\(\upmu\)m and the propagation length \(Lp\) = 69.1 \(\upmu\)m, while for \(\mu_{c}\) = 0.66 eV, the propagation length is 343 \(\upmu\)m. The insertion loss is less than 2.12 dB at \(\mu_{c}\) = 0.6 eV for a 60 \(\upmu\)m long waveguide. In Figure 5b, the proposed PM operates with polarization insensitivity when the incident light wavelength is from 1300 nm to 1800 nm. The equivalent electrical circuit model of the proposed polarization-insensitive PM is shown in Figure 6a. The \(R_{\text{total}}\) is the equivalent resistance of each side of graphene and composed of the metal-graphene contact resistance \(R_{\text{c}}\) and the resistance of graphene \(R_{\text{g}}\)[32], and the \(R_{\text{total}}\) is almost offered by the metal-graphene contact resistance, so here we set \(R_{\text{total}}\)\(\approx\)\(R_{\text{c}}\) = 400 \(\Omega\)-\(\upmu\)m [33]. In Figure 6b, the total capacitance consists of three parts, C = C\({}_{1}\) + C\({}_{2}\) + C\({}_{3}\). Each part can be calculated by \(C_{i}\) = \(\varepsilon_{0e}\)\(\varepsilon_{i}\)/\(d_{i}\) (\(i\) = 1,2,3) and \(\varepsilon_{r}\) is the relative permittivity between the two graphene layers. \(S_{i}\) is stands for the area of the plate capacitor, and d\({}_{i}\) is the distance of the plate capacitor. With the function  
Figure 3: Permitivity of graphene under different chemical potentials. of \(f_{\rm 3 dB}=1/(2\pi\cdot 2R_{\rm total}C)\), the 3-dB modulation bandwidth of the proposed PM is estimated to be \(f_{\rm 3 dB}=135.6\) GHz. Figure 4: (**a**,**b**) are the electric fields of the TE and TM modes, respectively; (**c**,**d**) are the real- and imaginary-part of effective refractive index of TE and TM modes in the cases of different chemical potentials of graphene, respectively. Figure 5: (**a**) Phase change and propagation length \(Lp\) in the range of 0.56 eV \(<\mu_{c}<\) 0.66 eV, with the incident optical wavelength of 1550 nm; (**b**) Difference of the real- and imaginary-part of the effective refractive index between TE and TM modes in the case of different wavelengths.