## 2 Transmitarray Antenna Design
### Array of MS  
The whole circular TA shown in Figure 6a is established by 660 elements with a diameter of 160 mm (\(D=160\) mm), and from the plot, we can figure out the approximate distribution of Unit Cell A and B. This is because that Unit Cell A covers the phase shift from 0\({}^{\circ}\) to 156\({}^{\circ}\) and Unit Cell B covers from 156\({}^{\circ}\) to 360\({}^{\circ}\), and the related phase at the center of TA is 0\({}^{\circ}\), so all elements are near the center based on Unit Cell B and away from the center of array; the elements based on Unit Cell A and those based on Unit Cell B will appear one after another. After confirming whether Unit Cell A or Unit Cell B is to be used for every element on the array, the corresponding relationship in Figure 5a is used to determine the length of slot for each element, so the final TA will be presented as the structure in Figure 6a. In this process, the required phase \(\varphi_{xy}\) at the position of \(xy^{th}\) element on the array is based on (1) [37]:  
\[\varphi_{xy}=k(R_{xy}-\overset{\wedge}{r_{0}}\cdot\overset{\rightarrow}{r_{ xy}})+\varphi_{0} \tag{1}\]  
where \(k\) is the wave number, \(R_{xy}\) is the distance between the feed position and the center position of \(xy\)th element on the array, \(\overset{\rightarrow}{r_{0}}\) is the main beam direction, \(\overset{\rightarrow}{r_{xy}}\) is the position vector of \(xy\)th element, and \(\varphi_{0}\) represents the constant phase at the central point of the TA. Equation (1) is worked to calculate  
Figure 5: Transmission coefficients: (**a**) transmission phase and (**b**) transmission magnitude. all the detailed phase values on the array as showed in Figure 6b. With the approximate functions of the element's excitation, the radiation pattern can be simplified into the scalar form [24]:  
\[RP(\theta,\varphi)=\sum_{x=1}^{X}\sum_{y=1}^{Y}\frac{\cos^{q}(\theta_{f}(x,y))}{ \left|\overrightarrow{r_{xy}-\overrightarrow{r_{f}}}\right|}\cdot e^{j\theta \cdot\varphi}\cdot e^{j\varphi\cdot\varphi} \tag{2}\]  
\[RP_{n}(\theta,\varphi)=20\times\log_{10}(\frac{\left|RP(\theta,\varphi) \right|}{\max(\left|RP(\theta,\varphi)\right|)}) \tag{3}\]  
where \(\theta\) is in the range of \(-90^{\circ}\) to \(90^{\circ}\) and \(\varphi\) is in the range of \(0\)-\(360^{\circ}\), \(\theta_{f}(x,y)\) is the spherical angle in the feed's coordinate system, \(q\) is the feed pattern factor, which is used in the design of horn antenna with the value of \(6.5\), \(\overrightarrow{r_{f}}\) is the position vector of the feed, and \(\hat{\mu}\) is the observation direction. Based on (2), the normalized radiation pattern \(RP_{n}(\theta,\varphi)\) can be achieved and the main beam has a beam width from \(-10^{\circ}\) to \(10^{\circ}\) with the SLL of \(-19\) dB. The element at the center of TA is selected carefully. It is true that \(\varphi_{0}\) can be an theoretically arbitrary value because the focusing effect of the TA is based on the related phase distribution, which means that the related phase at the center being \(0^{\circ}\) will make the calculation of phase distribution easier so the phase at other positions will change based on the related \(0^{\circ}\) to imitate the function of optical lens. However, in this process, the transmission efficiency is regarded as 1, which means no loss is taken into consideration. However, in the actual situation, the design of the TA has to face the problem of transmission loss inevitably and this is always a research direction for the TA. What's more, in the design of TA, there are mainly three parts of approximations needed to be paid attention. First, the transmitted phase and amplitude values obtained from the cell analysis in Figure 5 are in the case of normal incident plane wave, but in fact, elements in the array except the central one are illuminated by different degrees of oblique incident, so there is some deviation. Second, full \(360^{\circ}\) phase shifting is needed to achieve the focusing effect of the TA, as ideally, the theory requests continuous varying of phase.