## 2 Numerical Analysis of Beam-Steering Performance Based on 1-Bit Quantization  
This study presented a design based on 1-bit control to simplify the configuration and implement a multi-function transmitarray, which is different from the previously suggested reconfigurable transmitarray design. Equation (1) shows the calculation of the phase delay characteristics of each unit cell's position (\(\overrightarrow{r}_{mn}\)) deduced from the relationship between the phase \(\psi_{mn}^{inc}\), the source phase \(\psi_{s}\), and the desired beam steering direction (\(\hat{r}_{o}\)), and developed the beam steering characteristics by reconfiguring the phase distribution through electrical control when an EM wave incident from the outside reaches the transmitarray. Here, \(R_{mn}\) and \(k\) represent each distance from the propagation constant and the source. \[\psi_{mn}^{tr}=k\Big{(}R_{mn}-\overrightarrow{r}_{mn}\cdot\hat{r}_{o}\Big{)}+ \psi_{mn}^{inc}+\psi_{s} \tag{1}\]  
The phase control ability of the unit cell results in quantization loss and therefore affects the efficiency of the transmitarray and the beam steering performance. The beam steering angle range, control margin of error, and efficiency were analyzed to estimate the beam steering performance of the transmitarray created using only a PIN diode having ON/OFF switching characteristics. Before analyzing the beam steering margin of error, an 8 \(\times\) 8 transmitarray in which unit cells are arranged at a half-wavelength period of the operating frequency was assumed to configure a focal distance (F/D, ratio of the focal distance F to the aperture size D) carrying the optimal feeding efficiency, and total efficiency was calculated according to the quantization level, the results of which are presented in Table 1. The spillover (\(\eta_{s}\)), illumination (\(\eta_{i}\)), and total efficiency (\(\eta_{t}=\eta_{s}+\eta_{i}\)) can be calculated by the approximation method for aperture efficiency as follows [23] (Equations (2)-(4)). To calculate the radiation pattern of transmitarray, the aperture field can be expressed by (2), where the cos\({}^{qf}\) is feed power pattern, \(q_{f}\) is the feed pattern power factor, \(\theta_{f}\) is the angle between source and (\(m\),\(n\))th element, \(\overrightarrow{r}_{f}\) is the position vector of the feed, \(|T_{mn}|\) is the transmission magnitude of the (\(m\),\(n\))th element, \(A_{p}\) is aperture area, and \(T\) is the total number of elements. \[I_{i}\approx\frac{\cos^{q_{f}}\Big{(}\theta_{f}(m,n)\Big{)}}{\left| \overrightarrow{r}_{mn}-\overrightarrow{r}_{f}\right|}\cdot e^{-|k| \overrightarrow{r}_{mn}-\overrightarrow{r}_{f}|}\cdot|T_{mn}|e^{i\phi_{mn} ^{tr}} \tag{2}\]  
\[\eta_{s}=\frac{2q_{f}+1}{2\pi}\sum_{m=1}^{M}\sum_{n=1}^{N}\frac{F\cos^{2q_{f} }\Big{(}\theta_{f}\Big{)}}{\left|\overrightarrow{r}\right|^{3}}\Delta x\Delta y \tag{3}\]  
\[\eta_{i}=\frac{1}{A_{p}}\frac{\left[\int_{S}I_{i}dS\right]^{2}}{\left[\int_{S }|I_{i}dS\right]^{2}}=\frac{1}{A_{p}}\frac{\left|\sum_{i=1}^{T}I_{i}\Delta x \Delta y\right|^{2}}{\sum_{i=1}^{T}\left|I_{i}\right|^{2}\Delta x\Delta y} \tag{4}\]Additionally, the beam steering margin of error was analyzed according to the transmitarray's quantization level under the condition of the optimal focal distance, and the results are shown in Figure 2. The calculation confirms that the maximum efficiency decreases as the quantization level decreases, and a maximum of 1.5 degrees of beam steering error is observed within the desired steering range of 0 to 45 degrees and 2.9 dB of quantization loss compared to an optimal condition when a transmitarray unit cell carries a phase control characteristic of 1-bit. The design complexity and cost of the transmitarray were lowered using a low level of quantization to be used in high-security areas and facilities such as border security and a reconnaissance such as land vehicles or UAVs.