# Design of a Flat-Panel Metasurface Reflectarray C-Band Antenna
## 2 Operation Principle of the Reflectarray Antenna  
The operation principle of a flat-panel metasurface reflectarray is shown in Figure 1.  
Figure 1: Operation principle of a flat-panel metasurface reflectarray antenna. The incident electromagnetic waves radiated from a source antenna are reflected by the flat-panel metasurface. The metasurface consists of an array of unit cells, each of which shifts the phase of the incident waves in order to propagate them in the desired direction. The array factor AF(\(\theta\), \(\phi\)) of a flat-panel metasurface reflector array with \(M\times N\) unit-cells can be written as (1) and (2) when the metasurface reflector is located on the _xy_-plane [15]:  
\[\mathrm{AF}(\theta,\phi)=\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}A_{nm}e^{i\beta_{0}(md _{x}u+md_{y}v)} \tag{1}\]  
\[\left\{ \begin{matrix}u=\sin\theta\cos\phi+\beta_{x}\\ v=\sin\theta\sin\phi+\beta_{y}\end{matrix} \right. \tag{2}\]  
where \(A_{nm}\) is the complex excitation of the (\(m\), \(n\))-th unit cell, \(C(m\), \(n\)) and \(k_{0}\) is the free space wavenumber. \(\beta_{x}\) and \(\beta_{y}\) are the progressive phase shift between the adjacent unit-cells in the \(x\) and \(y\) directions, respectively. The progressive phase shifts can be expressed as (3) in order to create reflected waves focused in the (\(\theta_{r}\), \(\phi_{r}\)) direction. \(d_{x}\) and \(d_{y}\) are the period of the unit cell in the \(x\) and \(y\) directions, respectively. \[\left\{ \begin{matrix}\beta_{x}=-\sin\theta_{0}\cos\phi_{0}\\ \beta_{y}=-\sin\theta_{0}\sin\phi_{0}\end{matrix} \right. \tag{3}\]  
The phase shift (\(\Delta\phi_{nm}\)) on the \(nm\)-th unit cell, \(C(m\), \(n\)), can be calculated by using the period of unit cell, \(\beta_{x}\), and \(\beta_{y}\). It is critical to compensate the phase of the incident waves using the unit-cell structure to control the reflection angle. The phase of the reflected wave at \(C(m\), \(n)\) can be calculated by adding the phase of the incident wave to the relative phase shift due to the unit cell. The required phase shift distribution \(\Delta\phi\) to obtain a desired reflection angle can be analytically calculated by (4):  
\[\Delta\phi_{nm}=2N\pi+k_{0}\left(\left|\stackrel{{\rightharpoonup} }{{R}}_{mn}\right|-\stackrel{{\rightharpoonup}}{{d}}_{mn}\cdot \theta_{0}\right)\,(N=0,1,2,\cdots) \tag{4}\]  
where \(\stackrel{{\rightharpoonup}}{{R}}_{mn}\) is the position vector from the source antenna to \(C(m\), \(n)\), \(\stackrel{{\rightharpoonup}}{{d}}_{mn}\) is the position vector from the center of the metasurface reflector to \(C(m\), \(n)\), \(k_{0}\) is the wavenumber and \(\stackrel{{\rightharpoonup}}{{r}}_{0}\) is the unit vector in the direction of the reflected waves. The unit cell of the flat-panel reflectarray, which operates at 5.8 GHz (\(C\)-band), is composed of dual square rings, as shown in Figure 2. The dual square rings are printed on a widely used commercially available 0.8-mm-thick (Ts) FR4 substrate (\(\varepsilon_{r}=4.4\) and tan \(\delta=0.04\)). The unit cell is placed above a solid metal GND plane, as shown in Figure 2. The width and length (Wring) of the outer ring vary from 6 to 16 mm, whereas the unit cell size (D\({}_{\mathrm{cell}}\)) is fixed at 25 mm. The widths of the inner and outer rings (T\({}_{\mathrm{r1}}\), T\({}_{\mathrm{r2}}\)) and the gap between the rings (G\({}_{\mathrm{1}}\)) are also fixed to 1 mm for design simplicity. The metal thickness and the air gap (G\({}_{\mathrm{2}}\)) between the GND plane and the dual square ring are 18 \(\upmu\)m and 2.76 mm, respectively. All the detailed unit cell dimensions are summarized in Table 1.  
The phase of the reflected wave changes as the size of the double square ring changes because the impedance of the unit cell is a function of the length and width of the square ring [16; 17]. To analyze the phase response of the unit cell according to the size of the dual square rings, an infinite array condition was assumed, and calculation was done by a commercial 3D full-wave simulator, Ansys HFSS.