# Liquid Crystal-Tuned Planar Optics in Terahertz Range
## 4 Tunable Dielectric Metasurfaces Based on LCs  
For metallic metamaterials, a phase change from 0 to \(2\pi\) cannot be achieved by changing the resonant frequency. Moreover, the low forward-scattering discounts the efficiency of wavefront modulation. Therefore, researchers introduced the concept of a metasurface into the THz band and designed a THz dielectric metasurface. Similar to glass in visible light, high-impedance silicon is transparent in the THz band. Additionally, silicon is perfectly compatible with existing nanofabrication technology, making it an ideal candidate for THz planar optics. By coding the phase of the THz wavefront, functions such as focusing, deflection and vortex beam generation can be realized. By appropriately designing the unit structure of dielectric metasurfaces, a subwavelength phase gradient can be realized in the two-dimensional plane. The phase introduced by dielectric metasurfaces is generally based on two different principles. One is the resonance phase related to the structural parameters of the unit. Dielectric meta-atoms interact with incident THz waves to excite Mie resonance and add a certain phase to the incident wave [96; 97]. The resonant phase is expressed as [98]:  
\[\varphi_{\text{R}}=\frac{2\pi}{\lambda}n_{\text{eff}}d \tag{1}\]  
\(\lambda\) is the operating wavelength, \(n_{\text{eff}}\) is the effective refractive index of the dielectric silicon unit which is determined by the lengths and widths of meta-atoms, and \(d\) is the height of the dielectric silicon unit. Due to the coupling of spin angular momentum and orbital angular momentum, the metasurface also produces geometric phase effects on incident electromagnetic waves. The geometric phase, also known as the Pancharatham-Berry phase (PB phase), is related to the evolution of the polarization state in space propagation and only depends on the spatial distribution of the optical axis of the structural unit. Assuming that the azimuthal angle of the optical axis is \(\alpha\), its Jones matrix is [99]:  
\[T(r,\varphi)=\cos\left(\frac{\delta}{2}\right) \begin{pmatrix}1&0\\ 0&1\end{pmatrix} +\text{i}\,\sin\left(\frac{\delta}{2}\right) \begin{pmatrix} \cos 2\alpha&\sin 2\alpha\\ \sin 2\alpha&-\cos 2\alpha\end{pmatrix}  \tag{2}\]  
Figure 6: Schematic diagrams of (**a**) a THz programmable metasurface [89] and (**b**) an LC spatial phase modulator [90]. (**c**) Transmissive programmable metasurface composed of two layers of Fano metamaterial sandwiched by LCs [91]. (**d**) Resonant switch-based multibit coding [92]. Schematic diagrams of (**e**) transflective spatial THz wave modulator [94] and (**f**) dual-color spatial THz wave modulators for single-pixel imaging [95]. where \(\delta\) is the retardation phase of the structure. When the left/right circularly polarized light incident \(E_{in}(r,\varphi)=E_{0}(r,\varphi)\cdot(1,i\sigma)^{T},\sigma=+1\) for left and \(\sigma=-1\) for right, the output beam can be expressed as:  
\[E_{out}(r,\varphi)=T(r,\varphi)E_{in}(r,\varphi)=E_{0}\cos\left(\frac{\delta} {2}\right)\binom{1}{i\sigma}+iE_{0}\sin\left(\frac{\delta}{2}\right)\exp(i2 \alpha\sigma)\binom{1}{-i\sigma} \tag{3}\]  
This indicates that the chirality of the incident light will be reversed and carries a phase delay of \(\pm 2\alpha\) accordingly. In the THz band, silicon-based metasurfaces are often fabricated by photolithography and deep reactive ion etching (DRIE). The manufacture is matured and available for mass production. As the function of the dielectric metasurface is strictly bound to the structure, it is totally fixed once the structure is fabricated. Methods for the realization of a tunable THz dielectric metasurface have become an important research direction. To introduce tunability to the THz dielectric metasurface, LCs are integrated. The tunability is divided into two species. One uses LCs as a birefringent medium to provide a tunable anisotropic refractive index. It allows LCs to be used as a tunable wave plate or a special environmental medium. In 2018, [PERSON] et al.