# Recent Progress on Graphene-Functionalized Metasurfaces for Tunable Phase and Polarization Control
## 3 Phase Modulation in Graphene Metasurfaces
### In-Plane Plasmonic Wavefront Shaping and Coupling with Out-of-Plane Propagation  
Apart from wavefront shaping in free space, control of the SP waves propagating along the graphene surface holds significant promises towards minimization of the photonic integrated circuits. Compared to the noble metal counterparts, graphene plasmonics show low loss and strong confinement together with the tunability via electrical or chemical doping. As will be shown in the following literature, in-plane and out-of-plane beam shaping focus on applications in the THz to mid-IR frequencies taking advantage of the spatially- and temporally-tunable \(n_{sp}\) of the SP wave with the Fermi level following the discussion in Figure 1b. Figure 5: Graphene nanoribbons for reflective wavefront shaping. (**a**) Graphene ribbon element on top of the dielectric/metal substrate. (**b**) Variation of reflectivity and phase with the ribbon width. Reproduced with permission from [47], Copyright The authors, 2015. (**c**) Smooth 360\({}^{\circ}\) phase variation is covered by tuning the Fermi level of graphene, and it is shifted with a well-maintained shape from 4 THz to 6 THz. Adapted from [58], Copyright IOP Publishing Ltd., 2017. (**d**) Far away from the optimum frequency, the phase variation is either too sharp or too weak to use. Reproduced with permission from [59], Copyright IEEE, 2018. Wavefront shaping is possible by properly selecting the ribbon width at the fixed Fermi level from (**b**), or by individual gate control over an uniform graphene nanoribbon array following (**c,d**). Fundamentally, the plasmonic wavefront is manipulated via a nonuniform conductivity pattern along graphene. Several ways are proposed to achieve such a conductivity pattern [65]. Figure 6a uses the uneven ground plane to create different distances between the flat graphene layer and the highly-doped silicon ground plane [66]. When a fixed biasing voltage is applied, the electric fields experienced by the graphene inside and outside of the double-convex region are different, which in turn produces an inhomogeneous conductivity pattern for spatial Fourier transformation in Figure 6b. With a proper design of the uneven ground plane, the focal length is even variable with the voltage [67]. Chemical doping, though without dynamic control, is another popular way to change the optical properties of graphene. Figure 6c shows that the adsorption of proper molecules on graphene leads to chemical doping through charge transfer. By patterning two types of organic molecules on graphene, a plasmonic metasurface can be realized with any gradient effective refractive index profile to manipulate SP beams as desired [43]. A multiscale theoretical approach combining the first-principles electronic structure calculations and finite-difference time-domain simulations is developed to reconcile the band structure modification by the molecules and the mesoscopic effect on the SP wave propagation. The designed plasmonic Luneburg lens and SELFOC lens are in a notably subwavelength size of around one tenth of the free space wavelength. By creating vacancies with the designed shape in the graphene layer (Figure 6d), this enables excitation of the SP wave from free-space illumination and manipulation of the SP wavefront [68]. Figure 6e,f shows the plasmonic superfocusing and plasmonic vortex beam generation with a well-designed vacancy geometry. The size of the focusing hotspot is far below the diffraction limit considering the strong plasmonic localization in graphene. Figure 6: In-plane plasmonic wavefront engineering. (**a**) Monolayer graphene on top of the uneven ground plane made of highly-doped silicon. Different distances between the graphene layer and the ground plane lead to two conductivities and two refractive indices of the SP wave in and out of the lentil area. (**b**) Simulated electric field of the SP wave through the graphene Fourier lens with point source excitation. Reproduce with permission from [66], Copyright Aptara Inc., 2012.