# Recent Progress on Graphene-Functionalized Metasurfaces for Tunable Phase and Polarization Control
## 4 Polarization Modulation in Graphene Metasurfaces
### Graphene-Based Half-Wave Plate
Although all the designs show very high PCR above 90%, this does not account for the absorption. The peak efficiency listed in Table 1 is the maximum value of \(|R_{cross}|^{2}\), which indicates the net power into the cross-polarization. Due to the different relaxation times and different intensities of the field localization, the net efficiency varies from 50 to 90%. Still, these HWPs are efficient due to the blocking of the transmission channel. \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline
**Ref. ** & **Structure** & **Freq. ** & **Static Bandwidth** & \(\mathbf{r}\) & **Peak Eff. ** & **Lattice Size** & **Tickness** & **Dynamic Tunability** \\ \hline
[85] & L-shaped G pattern + back mirror & 38.9 THz & \(\sim 3\)\% & \(\sim 0.9\) s & \(\sim 7\)\% & 0.02\(\lambda\) & 0.23\(\lambda\) & \(\sim 13\)\% dynamic bandwidth with & E\({}_{f}\) from 0.7 to 0.9 eV \\ \hline
[86] & L-shaped G slot + back mirror & 31.4 THz & \(\sim 5\)\% & 1 p\(\sim 3\)\% & 1 p\(\sim 80\)\% & 0.016\(\lambda\) & 0.28\(\lambda\) & \(\sim 30\)\% and \(\sim 20\)\% dynamic bandwidth & with E\({}_{f}\) from 0.7 to 1 eV \\ \hline
[87] & L-shaped G pattern + back mirror & 6.65 THz & \(\sim 50\)\% & 1 p\(\sim 80\)\% & 0.17\(\lambda\) & 0.33\(\lambda\) & \(\sim 69\)\% dynamic bandwidth with & E\({}_{f}\) from 0.7 to 1 eV \\ \hline
[92] & rectangle G holes + back mirror & 46.8 THz & \(\sim 5\)\% & 1 p\(\sim 4\)\% & 0.03\(\lambda\) & 0.25\(\lambda\) & \(\sim 32\)\% dynamic bandwidth with & E\({}_{f}\) from 0.6 to 1 eV \\ \hline
[93] & elliptical G pattern + back mirror & 22.5 THz & \(\sim 9\)\% & 1 p\(\sim 80\)\% & \(\sim 80\)\% & 0.015\(\lambda\) & 0.16\(\lambda\) & \(\sim 28\)\% dynamic bandwidth with & E\({}_{f}\) from 0.6 to 0.9 eV \\ \hline
[91] & I-shaped metallic resonator + G ribbons + back mirror & 0.67 THz & \(\sim 96\)\% & 1 p\(\sim 9\)\% & 0.27\(\lambda\) & 0.25\(\lambda\) & E\({}_{f}\) from 0 to 0.6 eV, polarization varies & \\  & & & & & & & & from cross-LP to ellipse to CP \\ \hline
[90] & H-shaped G holes + back mirror & 35.7 THz & 11\% & 1 v\(\sim 70\)\% & \(\sim 70\)\% & 0.02\(\lambda\) & 0.26\(\lambda\) & \(\sim 37\)\% dynamic bandwidth with & E\({}_{f}\) from 0.6 to 1 eV \\ \hline
[94] & Cross G pattern + back mirror & 7.7 THz & 25.8\% & 1 p\(\sim 80\)\% & 0.05\(\lambda\) & 0.25\(\lambda\) & \(\sim 77\)\% dynamic bandwidth with & \\  & & & & & & & & & \\  & & & & & & & & & \\  & & & & & & & & & \\  & & & & & &The lattice size is apparently one or two orders of magnitude smaller than the vacuum wavelength taking advantage of the strong localization of graphene plasmons. As the localization increases with frequency, the lattice size is smaller in higher frequency designs. Due to such a small feature size, all the proposed designs are very robust to the variation of the incidence angle up to 40\({}^{\circ}\) or more [88; 90; 93; 94]. This is the merit of the graphene HWP compared to the metallic counterpart besides electrical tunability. Interestingly, the thickness of the dielectric spacer layer separating the graphene pattern and the back mirror is always around 0.25\(\lambda\) if the thickness means the single optical path taking the refractive index of the dielectric into account. The round trip through the dielectric with the 180\({}^{\circ}\) phase delay upon back reflection enables the enhancement of interference in the reflection side. Finally, the dynamic tunability of the graphene-based HWP through the electrical control of the Fermi level is discussed in the last column of Table 1. Generally, an increase of E\({}_{F}\) leads to reduced loss and stronger plasmonic resonance with blue-shifted resonance frequency and thus helps increase the PCR. Accordingly, a decrease of E\({}_{F}\) causes lower PCR and narrower static bandwidth. The lower limit of the dynamic bandwidth is set by PCR \(>\) 0.5. In addition, E\({}_{F}\) above 1 eV is avoided considering the breakdown of the capacitor and the extraordinarily high biasing voltage, which sets the upper limit of the dynamic bandwidth.