# Recent Progress on Graphene-Functionalized Metasurfaces for Tunable Phase and Polarization Control
## 4 Polarization Modulation in Graphene Metasurfaces
### Graphene-Based Quarter-Wave Plate  
The quarter-wave plate (QWP) converts a linear polarization into a circular one, by engineering the two orthogonal eigenmodes with equal amplitude and 90\({}^{\circ}\) phase delay. Figure 8a is an  
Figure 7: Leaky waves from graphene plasmonic structures with a dynamically-steerable direction. **(a)** A graphene sheet on the back-metalized substrate with the isolated poly-silicon gating pads for space-dependent DC biasing. The surface reactance is sinusoidally modulated with adequate biasing voltage to each pad. The periodic modulation offers effective momentum to transfer the surface wave into free-space radiation. **(b)** The radiation direction of the leaky wave is dynamically shifted when different numbers of pads are contained in a modulation period. Reproduced with permission from [72], Copyright IEEE, 2014. **(c)** The silica-graphene grating with the silver substrate and a slit for THz beam scanning. All the graphene ribbons are biased with the same voltage. The leaky beam direction is determined and tuned by the effective refractive index of the SP wave in graphene. **(d)** The near-field plot of the radiation with different biasing voltages applied. Biasing voltage is 256.5 mV (left) and 53.2 mV (right). The refractive index of the SP wave is 1.27 and 2.60, resulting in the radiation towards 13\({}^{\circ}\) and \(-\)18\({}^{\circ}\), respectively. Reproduced with permission from [73], Copyright Elsevier B.V., 2015. **(e)** Graphene leaky wave antenna for two-dimensional beam scanning with the simplified two groups of gating pads. One group on the left offering monotonic impedance variation along the y direction mainly determines the radiation in the azimuthal plane, and the other provides sinusoidal impedance modulation along the x direction to decide the elevation angle of the radiation. **(f)** Radiation pattern in different directions via simulation by simply changing the two groups of biasing voltages. Reproduced with permission from [74], Copyright Optical Society of America, 2016.  
asymmetric graphene nanocross with distinct plasmon resonances along the two arms [80]. The 45\({}^{\circ}\) linearly-polarized (LP) beam is changed into a circularly-polarized (CP) beam at 7.92 \(\upmu\)m when the graphene Fermi level is 0.75 eV. The working wavelength is blue-shifted with the further increase of the graphene Fermi level. However, the conversion efficiency is low, and the quarter-wave plate is strictly satisfied at a single frequency. Thus, the bandwidth is very narrow (estimated as 1% from [80]) once a certain biasing voltage is applied. Figure 8b enlarges the bandwidth to 40% due to the interaction of the top graphene grating and the bottom gold grating with opposite phase retardation dispersion [81]. Variation of the carrier density changes the graphene from transparent to conductive, leading to a polarization switch from co-polarized LP to CP with \(E_{F}=0\) and 0.5 eV. The form birefringence of the graphene grating is further enhanced in Figure 8c by introducing a periodic gradient instead of the binary pattern [82]. The increased birefringence shifts the working frequency to a lower value. The bandwidth of the quarter-wave plate relative to the center frequency is therefore increased as compared to the binary design. In order to dynamically move the working frequency in a wide bandwidth, liquid crystal (LC) is integrated into the graphene grating with additional electrical modulation of the birefringence. The theoretical study indicates by electrically controlling the liquid crystal director angle that the dynamic bandwidth of such hybrid quarter-wave plate achieves 78% centered at 1.15 THz. The design in Figure 8d has shown similar dynamic bandwidth by independently biasing the top graphene pattern and the bottom graphene film [83]. The bottom layer composed of seven layers of graphene acts as a reflector with a tunable reflection phase, which is used to compensate the difference of phase delay due to the frequency shift.