# Recent Progress on Graphene-Functionalized Metasurfaces for Tunable Phase and Polarization Control
## 3 Phase Modulation in Graphene Metasurfaces
### Wavefront Shaping in Reflection  
When a metallic mirror is introduced to the active phase shifter with a dielectric spacer, the phase modulation range can be significantly enlarged. The whole structure can be considered as an asymmetric Fabry-Perot resonator, where the large phase range is gained from the multi-path propagation between the top and bottom layers. In other words, the electric current circulating between the top metasurface and the bottom plate forms a magnetic resonance with 360\({}^{\circ}\) phase variation across the resonance frequency. In order to access the 360\({}^{\circ}\) phase range dynamically at a fixed working frequency, a loss tuning mechanism is required in the top layer to tune the structure through the underdamped-critical coupling-overdamped transition. Tuning the resistive loss of the top layer dictates how far the energy goes through the dielectric spacer and how strong the coupling is with the bottom plate, and is therefore an effective way to tune the phase response. The resistive loss changes the reflection intensity and phase, but does not shift the resonance frequency. One can only access half of the 360\({}^{\circ}\) phase range at a fixed frequency with the structure in Figure 4a, where the metallic patches are covered by graphene. The 360\({}^{\circ}\) phase range can be covered by two types of metallic patches with shifted resonance frequencies. The working frequency is chosen in the shaded area in Figure 4b between the two resonances [55]. The THz-TDS measurement has demonstrated a maximum phase modulation range of 243\({}^{\circ}\) at 0.48 THz, which can be further increased if the carrier scattering loss is reduced in graphene. The above design works at sub-THz, where graphene is a lossy surrounding medium, as analyzed in Figure 1a. The necessity of two types of unit cells to access 360\({}^{\circ}\) phase response originates from the inability of graphene to tune the resonance frequency. Another approach to accessing the full phase range is to work at higher frequencies where graphene directly modulates the resonance frequency with different biasing. In 2017, [PERSON]'s group demonstrated more than 230\({}^{\circ}\) phase modulation using the structure in Figure 4c in mid-IR frequencies [56]. Here, graphene is attached to a uniform array of gold antennas with very small gaps. In mid-IR, graphene shows low loss and a tunable dielectric constant with the carrier density, as discussed in Figure 1b. The resonance frequency shifts with the environmental dielectric constant, resulting in a widely-tunable phase range of more than 200\({}^{\circ}\) from 8.5 \(\upmu\)m to 8.7 \(\upmu\)m, as shown in Figure 4d,e. Although the measured reflectivity is quite low due to absorption of the SiN\({}_{x}\) substrate and the low mobility of graphene charge carriers, the proposed design is one of the most promising designs for dynamic beam steering. Despite the similar configurations in Figure 4a,c, graphene plays different roles in the dynamic adjustment depending on the working frequencies. Alternatively, graphene resonators can replace the graphene-metallic resonator structure in the top layer for reflective phase modulation, as summarized in Figure 5. All of them are targeted at mid-IR beam shaping, where the graphene pattern is a plasmonic resonator with tunable resonance frequency. Among various shapes, the graphene ribbons are the most extensively-studied ones, due to their simplicity and ease of biasing at one end of the structure. Specifically, Figure 5a utilizes the spatially-variant graphene ribbons to achieve desired phase profile at the center frequency and to change the Fermi level for dynamic modulation [47; 57]. Figure 5b shows the phase variation with the ribbon width, which serves as the guidance for designing the metasurface array for focusing and steering applications. When the graphene ribbons are separately biased, more degrees of freedom are gained for controlling the functionality temporally and spatially [58; 59].