# Novel Beam Scan Method of Fabry-Perot Cavity (FPC) Antennas
## 2 Materials and Methods
### Operation Principle of the Proposed Antenna
to be considered as the bias lines significantly affect the fields in the cavity, which ultimately harms the antenna's radiation. Even if PIN-diodes were applied for a discrete beam-steering, bias lines would still be located in the radiation area, which would also adversely affect the antenna performance. However, when using the active AMC instead of the active PRS, all bias lines can be placed beneath the ground plane (see Figure 3b), which hardly affects the radiation performance. Therefore, in these regards, we can conclude that the AMC ground plane is far more favorable than the PRS in respect to numerous future phase-control applications. So far, we have described the reason why the AMC ground plane is introduced. From now on, we are going to explain how to define a proper reflection phase of each AMC cell to obtain a desired scanned beam. For a scanned beam, each phase of fields transmitted through the PRS should be equal on the tilted target wavefront. These phases, \(\angle R_{-n}\) to \(\angle R_{n}\) in Figure 2b, can be derived as  
\[\angle R_{\pm n}=-\beta\cdot\left(\sqrt{p^{2}+h^{2}}+2 nl\right)+\sum_{i=1}^{n} \left(\angle\Gamma_{i}+\phi_{\pm xi}\right)+\angle T_{n+1}\left(n\ \geq\ 1\right) \tag{1}\]  
where \(\beta\) is the propagation constant in free space, \(h\) is the height of the PRS, \(p\) is the period of each PRS and AMC cell, \(l\) is the length of a ray path between the AMC and PRS, \(\phi_{\pm xn}\) is the reflection phase of each \(\pm n^{th}\) AMC cell, and \(\angle\Gamma_{n}\) and \(\angle\Gamma_{n}\) are the reflection and transmission phases of the \(\pm n^{th}\) PRS cell, respectively. The cell number \(n\) is represented in Figure 2.  
Equation (1) means that we can control all the transmitting rays' phases by adequately changing each AMC cell's reflection phase. However, there are three phase-uncontrollable rays directly passing through the PRS without any reflection by the AMC, which are depicted as red short-dashed lines in Figure 2b. Among the phase-uncontrollable rays, the blue-dashed one (see Figure 2b) must be the most influential on beam-scanning as they are normally bouncing between the PRS and the ground plane, which yields the strongest radiation power. Therefore, we set the strongest ray's phase as the reference \(\angle R_{n\!rf}\) when designing a target wavefront, which is obtainable through a simulation where there is a superstrate of three PRS cells that the phase-uncontrollable fields pass through. According to the simulation, \(\angle R_{n\!rf}\) is about \(-130^{\circ}\) above 1 mm from the center of the PRS at the design frequency. Now, the relationship among the phases \(\angle R_{\pm n}\) on the inclined target wavefront (see Figure 2b) can be derived with regard to \(\angle R_{n\!rf}\) as follows:  
\[\angle R_{\pm n}=\angle R_{n\!rf}\mp\beta\cdot\left(n+1\right)p\sin\theta_{t }\ \ \ \ \left(n\geq 1\right) \tag{2}\]  
Finally, by combining the Equations (1) and (2), the required reflection phase of each AMC cell can be derived as  
\[\phi_{\pm x1\ \text{or}\ \pm y1}=\beta\cdot\left(\sqrt{p^{2}+h^{2}}+2l\mp 2p \sin\theta_{t}\right)-\angle\Gamma_{1}^{Pol.}-\angle\Gamma_{2}^{Pol.}+\angle R _{n\!rf}\ \ \ \ \left(n=1\right) \tag{3}\]  
Figure 3: Overview of FTC antennas consisting of **(a)** an active PRS and **(b)** an active AMC. It is desirable for the bias lines to be placed beneath the ground plane as they hardly affect the FP resonance in the cavity. \[\phi_{\pm xn\text{ or }\pm yn}=\beta\cdot(2l\mp p\sin\theta_{t})-\angle T_{n}^{Pol.}+ \angle T_{n}^{Pol.}-\angle T_{n+1}^{Pol.}\quad(n\geq 2) \tag{4}\]  
where \(\phi_{\pm xn}\) and \(\phi_{\pm yn}\) are the reflection phases of the \(\pm n^{th}\) AMC cell along the \(x\)- and \(y\)-axis, respectively, and the superscript \(Pol.\) means the polarization mode of fields incident on the PRS.