## 2 Continuous Beam Steering MS Reflector Concept  
The schematic structure of the designed MS reflector and its equivalent electric circuit model are shown in Figure 1.  
Figure 1a shows the basic MS reflector structure composed of a printed metal shape on a dielectric substrate. The printed metal shape on the front side can be a patch, a slot, a ring [1; 36], or more complex geometries, such as a K-shape [14], connected via a metal surface used as a ground plane at the backside. Figure 1b presents the equivalent circuit of lumped elements with inductance, capacitance, and resistance components for the destined unit cell [11]. The plane wave electric field component interacts with the patches' edges, arousing surface charges (marked by \(+/-\) signs in Figure 1a); this is modeled as a capacitance \(C\). The magnetic flux developing from the current (marked by dashed arrows in Figure 1a) between the patch and the ground plane through the via is modeled as an inductance \(L\). The dielectric and ohmic losses are modeled as a resistance \(R\), which is the dissipation resistive part of the unit cell. When the unit cells are arranged in a 2-D periodic configuration, the surface is characterized as an effective impedance surface, which was defined by [PERSON]'s work [11] with the surface impedance parameter \(Z_{S}\), which is shown in Equation (1):  
\[Z_{S}=\frac{1}{1/R+j(\omega C-1/\omega L)} \tag{1}\]  
where around the angular frequency values of \(\omega=1/\sqrt{LC}\), the imaginary part of the \(Z_{S}\) becomes dominant, and is shown in Equation (2):  
\[Z_{S}=\frac{j\omega L}{1-\omega^{2}L\zeta} \tag{2}\]  
Figure 1: MS reflector structure and basic unit cell model: (**a**) side view of the MS structure interacting with a plane wave and (**b**) MS unit cell equivalent circuit model. The values of \(C\) and \(L\) are determined by the unit cell's dimensions, geometry, materials, and PCB properties. The parallel resonance frequency \(f_{res}\) of the circuit model around which the radiation is manipulated is shown in Equation (3):  
\[f_{res}=\frac{1}{2\pi\sqrt{LC}} \tag{3}\]  
The bandwidth of the resonance frequency [37] is shown in Equation (4):  
\[BW=\frac{1}{(RC\omega_{0})}=\frac{(L\omega_{0})}{R}\propto\frac{L}{C} \tag{4}\]  
The Q-factor is shown in Equation (5):  
\[Q=\frac{1}{BW}\propto\frac{C}{L} \tag{5}\]  
According to the phased array theory, each unit cell location on the MS is defined at its center [38]. A 2-D surface in the \(XY\) plane with a spatial array arrangement of a fixed distance and a 90\({}^{\circ}\) angle between the unit cells is defined as S (\(x_{j},y_{i}\)), \(j=1\), 2,..., \(N\) and \(i=1\), 2,..., \(M\), where \(N\) and \(M\) are integer numbers that lead to an array of \(N\times M\) unit cells. This unit cell array with the desired phase distribution throughout the surface allows for focusing radiation to the desired point [9] or radiation steering to the desired direction [8]. A scheme of a beam-steering side cross-section view that is representative of an MS reflector is shown in Figure 2.  
Figure 2 shows a scheme of radiation steering using an MS reflector, where \(L_{1}\), \(L_{2}\), and \(L_{N}\) are incident rays toward the surface marked in red arrows. The optical path difference (OPD) between the reflected rays from unit cells \(X_{j}\) to \(X_{j+1}\) is defined as \(\Delta L\) and is described in Equation (6):  
\[\Delta L=\Delta X\sin\theta \tag{6}\]  
where \(\Delta X\) is the array constant. For beam steering, \(\Delta L\) is constant for all different paths from adjacent cells \(X_{j}\) and \(X_{j+1}\) throughout the surface. The conversion of \(\Delta L\) into a phase difference is shown in Equation (7):  
\[\Delta\varphi_{x}=\frac{360\cdot\Delta L}{\lambda} \tag{7}\]  
\[Z_{S}=\frac{1}{1/R+j(\omega C-1/\omega L)} \tag{1}\]0 MS reflector side cross-section view scheme of the beam-steering principle. where \(\lambda\) is the wavelength at the working frequency.