# Grating-like Terahertz Metasurface for Large-Deflection-Angle Beam Manipulations
## 2 Principle and Design  
The working principle of the grating-like beam-sweeping metasurface will be discussed in detail. As shown in Figure 1, the metasurface grating is a kind of multilayer structure, among which the key functional layers are the metallic structural layer, which is composed of periodic metasurface elements and slits, the LC layer, and the ground layer. It is noted that the metasurface element is designed with a typical subwavelength structure of a complementary split-ring resonator (CSRR). There is a slit between every two elements whose width is much smaller than the length of the metasurface element, and hence the metasurface exhibits the grating effect. In the geometry of Figure 1, the plane parallel to the metasurface is set as the XOY plane. The period of metasurface grating was described by the grating constant \(D_{x}\). As the plane electromagnetic wave with an \(x\) polarization is incident onto the metasurface grating, the diffraction waves of different orders will be generated, which can be described by the following diffraction equation:  
\[\sin\theta\pm\sin\theta_{0}=m\frac{\lambda}{D_{x}}\ (m=0,\pm 1,\pm 2,\pm 3\ldots) \tag{1}\]  
where \(\theta_{0}\) and \(\theta\) represent the orientation angles of the incident wave and the diffraction wave, respectively, \(\lambda\) is the free-space wavelength, \(D_{x}\) represents the grating constant, and \(m\) denotes the diffraction order, which is an integer. Since the incident wave is perpendicular to the metasurface grating in this scheme, \(\theta_{0}\) is equal to zero, then Equation (1) can be rewritten as [31]  
\[\theta=\sin^{-1}(\frac{m\lambda}{D_{x}})\ (m=0,\pm 1,\pm 2,\pm 3\ldots) \tag{2}\]Equation (2) is the simplified expression of Equation (1) under the case of normal incidence. As the diffraction order \(m\) is equal to zero, we can obtain the 0 th diffraction beam at \(\theta\) = 0\({}^{\circ}\). For \(m\) = \(\pm\)1, the \(\pm\)1 st diffraction beams can be obtained, and the diffraction angle is frequency (or wavelength) dependent, which reflects the dispersion characteristic of the grating. Moreover, for a fixed working frequency, the diffraction beams of different orders (0 th, \(\pm\)1 st, \(\pm\)2 nd, etc.) can also be determined by Equation (2). Additionally, hence, Equation (2) describes the diffraction behavior of a one-dimensional grating under normal incidence. As observed in Figure 1, a series of diffraction waves will be generated at the XOZ plane by the metasurface grating, and the diffraction angle is frequency-dependent. It is interesting to note that the metasurface designed in this work is a multifunctional beam-manipulating device. In our previous work [30], terahertz beam scanning was achieved based on the electrically controlled programmable metasurface. Specifically, the digital coding states of the LC-loaded metasurface elements can be rapidly switched by tuning the bias voltage applied to the LC molecules, and then the coding pattern can be modulated in real-time, which leads to programmable beam manipulation. Nevertheless, herein, it functions as a frequency-scanning device, and no bias voltage is required. According to Equation (2), the diffraction angles for terahertz waves of different frequencies and orders can be calculated. The grating constant is 400 \(\upmu\)m. Due to the weak intensity of high-order diffraction waves, we only consider the first-order diffraction case. Moreover, the optimized working frequency range was determined to be 0.87-1.02 THz according to the subsequent numerical simulation and experimental results. Table 1 presents the incident frequencies and the corresponding diffraction angles calculated based on Equation (2). It can be observed that the diffraction angle can reach up to 59.5\({}^{\circ}\) as the incident frequency is equal to 0.87 THz, and it ranges from 59.5\({}^{\circ}\) to 47.3\({}^{\circ}\) as the frequency increases.