# Polarization-Insensitive Phase Modulators Based on an Embedded Silicon-Graphene-Silicon Waveguide
## 3 Graphene-Based Polarization-Insensitive Phase Modulator  
The graphene's complex optical conductance \(\sigma\) = \(\sigma_{\text{intra}}\) + \(\sigma_{\text{inter}}\) can be deduced from Kubo formalisms [27]:  
\[\sigma_{\text{intra}}=\sigma_{0}\frac{4\mu_{c}}{\pi}\frac{1}{\hbar(\Gamma_{1} -i\omega)} \tag{1}\]  
\[\sigma_{\text{inter}}=\sigma_{0}\left[1+\frac{1}{\pi}\text{arctan}\frac{\hbar \omega-2\mu_{c}}{\hbar\Gamma_{2}}-\frac{1}{\pi}\text{arctan}\frac{\hbar\omega +2\mu_{c}}{\hbar\Gamma_{2}}-\frac{i}{2\pi}\ln\left(\frac{\left(\hbar\omega+2 \mu_{c}\right)^{2}+\left(\hbar\Gamma_{2}\right)^{2}}{\left(\hbar\omega-2\mu_{ c}\right)^{2}+\left(\hbar\Gamma_{2}\right)^{2}}\right)\right] \tag{2}\]  
where \(\sigma_{\text{intra}}\) and \(\sigma_{\text{inter}}\) are the intraband conductivity and interband conductivity of the graphene, \(\sigma_{0}\) is the universal optical conductance, \(\hbar\) is reduced Planck constant, and \(\omega\) = 2\(\pi c\)/\(\lambda\) is the angular frequency of incident light. \(\Gamma_{1}\) = 1/\(\tau_{1}\) and \(\Gamma_{2}\) = 1/\(\tau_{2}\), where \(\tau_{1}\) and \(\tau_{2}\) are intraband relaxation time and interband relaxation time of the graphene, respectively. The chemical potential \(\mu_{c}\) of the graphene is related to the applied voltage \(V_{g}\) by \(\mu_{c}\) = \(\hbar\omega\) = \({}_{F}\) (\(\pi\,\)1 \(\eta(V_{g}+V_{0})\,\)1)\({}^{1/2}\)[28], with the Fermi velocity \(v_{F}\) \(\approx\) 9.5 \(\times\) 10\({}^{5}\) m/s and the voltage offset \(V_{0}\) caused by the natural doping. \(H\) = (\(\varepsilon_{0}\varepsilon\))/(ed) is a constant, \(\varepsilon\) and d represent the permittivity and distance of the dielectric between the two graphene layers, and \(e\) is the charge of an electron. The permittivity of graphene \(\varepsilon_{g}\) is related to the optical conductance \(\sigma\) by \(\varepsilon_{g}\) = 1 + \(i\cdot\sigma\)/(\(\omega\varepsilon_{0}\Delta\)). Figure 3 shows the permittivity of graphene as a function of the chemical potential \(\mu_{c}\), in which \(\Delta\) = 0.7 nm stands for the single layer graphene's effective thickness. As shown in Figure 3, graphene's permittivity gets close to zero when the chemical potential (\(\mu_{c}\)) is around 0.5 eV, and the point is called the epsilon near zero (ENZ) point. In our simulation, the typical parameters are \(\lambda\) = 1550 nm, \(T\) = 300 K, \(\tau_{1}\) = 1.5 \(\times\) 10\({}^{-14}\) s and \(\tau_{2}\) = 1.2 \(\times\) 10\({}^{-12}\) s [29]. Figure 2: Difference of the real- and imaginary-part of effective refractive index between TE and TM modes in the case of different waveguide widths (**a**) and heights (**b**). As is known, the electrically-controllable optical conductivity of graphene can effectively affect a waveguide mode's effective refractive index. We have designed a polarization-insensitive phase modulator, based on the proposed waveguide shown in Figure 1. The inner silicon waveguide with the width and height of 400 nm and 460 nm, respectively, 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 the outer waveguide, with a width and height of 800 nm and 800 nm, respectively. Figure 4a,b show the electric fields of the TE and TM mode in the case of \(\mu_{c}\) = 0.5 eV. Figure 4c,d show the effective refractive index of the TE and TM mode in the case of different chemical potentials of graphene. Graphene's chemical potential can be changed by applying different voltages on the electrodes of the proposed waveguide. The insets in Figure 4c,d show \(\Delta Re\) and \(\Delta\alpha\) in the case of different chemical potentials of graphene, respectively, the maximum \(\Delta Re\) of 1.2 \(\times\) 10\({}^{-4}\), and \(\Delta\alpha\) of 2.9 \(\times\) 10\({}^{-3}\) dB/\(\upmu\)m verifies the polarization-insensitive modulation of the proposed waveguide structure.