# Recent Progress on Graphene-Functionalized Metasurfaces for Tunable Phase and Polarization Control
## 4 Polarization Modulation in Graphene Metasurfaces
### Graphene-Based Half-Wave Plate  
Half-wave plates (HWPs) as the cross-polarization converter require 180\({}^{\circ}\) phase retardation and a comparable intensity of the two eigenmodes. Different from the QWP, the HWP is a real challenge through a single layer of anisotropic graphene pattern. As mentioned in Section 3.1, each eigenmode is an electric dipole with 180\({}^{\circ}\) or less phase shift across the resonance. Spectral shifting of two dipole resonances from the orthogonal directions of the anisotropic element leads to a phase difference of  
Figure 8: Graphene metasurfaces as active quarter-wave plates. (**a**) Graphene asymmetric nanocross for LP to CP conversion. The 45\({}^{\circ}\) LP beam is converted into a CP beam at 7.92 \(\upmu\)m when the graphene Fermi level is 0.75 eV. The operation frequency blue shifts with the increase of the Fermi level. Reproduced with permission from [80], Copyright Optical Society of America, 2013. (**b**) Hybrid metasurface composed of graphene sandwich grating and gold grating separated by a polymer spacer. The form birefringence in graphene grating and gold grating adds up to offer a constant phase delay of 90\({}^{\circ}\) between two eigenmodes over a wide bandwidth. Reproduced with permission from [81], Copyright The authors, 2015. (**c**) Graphene grating sandwich with well-designed distance and in-plane gradient on top of the LC layer. The spatial gradient of the graphene grating increases the form birefringence, and the electrical control of the LC molecule direction leads the QWP to have a dynamic bandwidth of 78%. Reproduced with permission from [82], Copyright The authors, 2018. (**d**) A graphene sheet patterned with butterfly holes and backed by seven layers of graphene with separate biasing for wideband LP to CP conversion. The bottom seven layers of graphene show a tunable reflection phase in order to compensate the difference of the phase delay due to the frequency shift over a wide bandwidth. Reproduced with permission from [83], Copyright The authors, 2017.  
less than 180\({}^{\circ}\). Therefore, the monolayer design working in the transmission mode only rotates the LP wave by a few degrees [84], and almost all the graphene HWPs work in the reflection mode with the anisotropic pattern backed with a metallic mirror at a proper distance. Table 1 summarizes the key features of these HWPs working in THz and mid-IR (all theoretical designs), from which we can generalize the rule of designing graphene HWP to meet the needs of different applications. The first three designs [85; 86; 87] in Table 1 share the same L-shaped graphene pattern (G pattern) or the complementary slot. We found that by properly selecting the arm length and width, the performance of the HWP can be narrowband [85], dual-frequency [86], or broadband [87], depending on the spectral separation of the two eigenmodes and their quality factors. Besides the L-shaped design, the graphene layer is patterned into various anisotropic shapes, from which the sinusoidal [88] and \(\phi\)-shaped designs [89] also gain very wide bandwidth. Here, the bandwidth is specific to the static bandwidth when a fixed Fermi level is given to graphene. It is estimated with the polarization conversion ratio (PCR) above 0.5, where PCR is defined as \(|R_{cross}|^{2}/(|R_{cross}|^{2}+|R_{co}|^{2})\). \(R_{cross}\) and \(R_{co}\) are the reflection coefficients of the cross- and co-polarized beams. The maximum PCR in all cases is close to one, indicating a complete polarization conversion. The carrier relaxation time \(\tau\) used in the modelings varies from 0.5 ps to 1 ps, and it should be at least larger than 0.02 ps to maintain the polarization conversion effect [90]. The Fermi level is relatively high, close to 1 eV in most cases. An exception is [91], where the graphene Fermi level is 0 eV. The birefringence in this study originates from two layers: the I-shaped metallic resonator and the graphene ribbon with orthogonal orientations, leading to a broad operation bandwidth of around 96%.