# Dual-Polarized Multi-Beam Fixed-Frequency Beam Scanning Leaky-Wave Antenna
## 2 Methods  
Figure 1 shows the proposed dual-polarized multi-beam fixed-frequency beam scanning LWA, which is composed of six SPPs transmission lines with impedance modulation and a DC control circuit. The DC control circuit of the varactor diode is placed on the right side of the antenna array. Each parallel routing DC bias line is loaded with an inductor to separate the DC path from the RF path and avoid mutual interference. The end of the bias line is connected to the ground plane through metal vias. The entire LWA array contains three groups of dual-polarized antennas. The first group of dual-polarized antennas contains six modulation units per modulation period, the second group contains eight modulation units, and the third group contains ten modulation units. They are labeled as antennas 1 st, 2 nd, 3 rd, 4 th, 5 th, and 6 th from top to bottom. Antennas 1 and 5 employ the varactor diode SMV2202-040 LF, with a capacitance range of 0.31 pF to 3.14 pF. Antenna 3 utilizes the varactor diode SMV2203-040 LF, with a capacitance range of 0.44 pF to 4.71 pF. Their bias voltage varies from 0 to 20 V. Antennas 2, 4, and 6 utilize the varactor diode MAVR-000120-1141, with a capacitance range of 0.14 pF to 1.1 pF within a voltage variation range of 0 to 12 V. Ports 1, 3, 5, 7, 9, and 11 are feeding ports, and ports 2, 4, 6, 8, 10, and 12 are connected to matched loads to absorb the remaining electromagnetic energy. The LWA array is arranged in an X-Y-X-Y-X-Y polarization to maximize space utilization and reduce crosstalk between antennas. The antenna is designed on an F4B substrate with a thickness of 3 mm, \(\varepsilon_{r}=3.5\), and \(tan\delta=0.001\). The dimensions of each part are indicated in Table 1. Assuming that the leaky wave direction is along the transmission line, the surface impedance \(Z_{s}\) can be described as  
\[Z_{s}(x)=jX_{s}[1+Mcos\bigg{(}\frac{2\pi x}{p}\bigg{)}] \tag{1}\]  
where \(X_{s}\) represents the average reactance of the surface, \(M\) represents the modulation factor, and \(p\) represents the modulation period. In this design, the SPPs transmission line structure is used as the modulation unit, and the surface impedance is modulated by loading the varactor diode device and varying the notch depth to obtain the beam steering at a fixed frequency. The radiation schematic of the periodic LWA is shown in Figure 2. Unlike resonant antenna, LWA is a kind of traveling wave antenna. The electromagnetic energy will continuously leak into the free space, and the unradiated energy is absorbed by the matching load. It has demonstrated that sinusoidal impedance surface modulation can efficiently convert non-radiative traveling waves into radiative leaky waves [17]. When \(p\) is determined, \(M\) is also determined. The fundamental mode of the SPPs antenna is slow wave mode. For \(-1\)th harmonic radiation, the main beam angle can be approximately calculated as  
\[\theta_{-1}=arcsin(\sqrt{1+X^{\prime 2}}-\frac{2\pi}{k_{0}p}) \tag{2}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Values (mm)** & **Parameter** & **Values (mm)** \\ \hline \(L_{1}\) & 480 & \(d_{1}\) & 7.3 \\ \(L_{2}\) & 472 & \(d_{2}\) & 7.3 \\ \(L_{3}\) & 480 & \(d_{3}\) & 6.1 \\ \(L_{4}\) & 450 & \(d_{4}\) & 6.1 \\ \(L_{5}\) & 460 & \(d_{5}\) & 5.2 \\ \(L_{6}\) & 465 & \(d_{6 In which \(X^{\prime}=X_{s}/\eta_{0}\) is the average surface reactance, \(\eta_{0}\) is the free-space wave impedance, and \(k_{0}\) is the free-space wave number. By changing \(p\), the LWA can obtain beams with different radiation angles. The wave number \(k_{x}\) in the direction of propagation along the SPPs transmission line can be expressed as  
\[k_{x}=k_{0}\sqrt{1+[\frac{jX_{s}(1+Mcos\left(\frac{2\pi x}{p}\right))}{\eta_{0 }}]^{2}} \tag{3}\]  
Similarly, the leaky-wave antenna radiation angle \(\theta\) can be described as  
\[\theta=arcsin\left(\frac{k_{x}}{k_{0}}\right) \tag{4}\]  
To implement dual polarization, two different impedance surfaces are built on a single-sided comb structure.