# CubeSat-Based Observations of Lunar Ice Water Using a 183 GHz Horn Antenna: Design and Optimization
## 2 Design Methodology
In the initial step of our study, we must calculate the parameters of the corrugated antenna see Figure 1. Considering the center frequency of 183 GHz, which corresponds to a wavelength of 1.6655 mm. Follows we calculate Corrugated horn antenna parameters with variable depth:  
Input radius equals  
\[a_{i}=\frac{3x\lambda_{c}}{2\pi}. \tag{1}\]  
Outer radius \(a_{i}\). with \(-15\) dB edge-taper gain and \(20^{\circ}\) half-cone angle [8] equals:  
\[a_{0}=1.95\lambda_{c}. \tag{2}\]  
Figure 1: Corrugated horn antenna geometrical characteristics [26]. Slot pitch \(P\) can be chosen in the range of  
\[\frac{\lambda_{c}}{10}\leq p\leq\frac{\lambda_{c}}{5}. \tag{3}\]  
Slot pitch-to-width ratio \(\delta\) in the range  
\[0.7\leq\delta\leq 0.9. \tag{4}\]  
Length of horn antenna  
\[L=5\lambda_{c}\leq L\leq 10\lambda_{c}. \tag{5}\]  
Number of slots  
\[N=\frac{L}{P}. \tag{6}\]  
Number of variable depth-slots  
\[N_{\text{MC}}=5\leq N_{\text{MC}}\leq 7. \tag{7}\]  
Width of the slot teeth  
\[W=\delta P. \tag{8}\]  
Figure 1 illustrates the general geometry of a corrugated horn antenna. By applying Formulas (1)-(8), we can simply calculate the required variables. When constructing the teeth of the antenna from the input radius to the aperture, the hyperbolic corrugated surface profile \(a_{j}\) and slot-depth \(d_{j}\) (\(1<j<N-1\)) will vary based on the pitch number. Therefore, these variables must be computed as an array of numbers. Following in Table 1, we can see the primary geometry of corrugated horn antenna. To be capable of computing number of teeth and their geometry, we apply Table 1 Information to Formulas (9)-(14). To compute array antenna \(a_{j}\) and \(d_{j}\), we proceed via following formulas:  
\[a_{i}=\frac{3x\lambda_{c}}{2\pi}. \tag{1}\]0  
\[a_{i}=\frac{3x\lambda_{c}}{2\pi}. \tag{1}\]1  
\[a_{i}=\frac{3x\lambda_{c}}{2\pi}. \tag{1}\]2  
\[a_{i}=\frac{3x\lambda_{c}}{2\pi}. \tag{1}\]3 \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Optimized Values (mm)** \\ \hline \(a_{i}\) & 0.4777 \\ \(L\) & 6\(\lambda_{c}\) = 9.99 \\ \(P\) & \(\frac{\lambda_{c}}{5}\) = 0.3331 \\ \(\delta\) & 0.7 \\ \(W\) & 0.23317 \\ \(a_{0}\) & 3.2477 \\ \(N_{\text{MC}}\) & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Corrugated horn antenna geometric data. To compute \(d_{j}\), we must take into account \(1\leq j\leq N_{MC}+1\) then slot depth of the \(j\)th slot is  
\[d_{j}\ =\left\{\delta-\frac{j-1}{N_{MC}}\Bigg{[}\delta-\frac{1}{4}\text{exp} \left(\frac{1}{2.114\big{(}K_{c}a_{j}\big{)}^{1.134}}\right]\Bigg{\}}\lambda_{c}.\right. \tag{13}\]  
For \(N_{MC}+2\leq j\leq N\) then the slot depth of \(j\)th slot is  
\[d_{j}=\frac{\lambda_{c}}{4}\text{exp}\left[\frac{1}{2.114\big{(}K_{c}a_{j} \big{)}^{1.134}}\right]-\left(\frac{j-N_{MC}-1}{N-N_{MC}-1}\right)\Bigg{\{} \frac{\lambda_{c}}{4}exp\left[\frac{1}{2.114\big{(}K_{c}a_{0}\big{)}^{1.134}} \right]\Bigg{\}}-\frac{\lambda_{0}}{4}exp\left[\frac{1}{2.114\big{(}K_{c}a_{0 }\big{)}^{1.134}}\right]. \tag{14}\]  
In Figure 2, we show the subsequence of \(a_{j}\) and \(d_{j}\) corrugated antenna from waveguide to aperture. Next to those calculations, we proceed to plot antenna 2D sketch. From Formulas (12)-(14), we realize the coordinates of corrugated antenna teeth depend on \(j\)th of each slot. This property assists us to figure antenna shape more accurately. From \(N_{MC}\) to \(N\), the scale of slots approximately stays unchanged, only \(a_{j}\) gradually rises up. See Figure 2 for more clearances. In this section, we present the computed data that forms the foundation for designing the corrugated horn antenna. In the next section, we conduct simulations of the corrugated horn antenna and apply the smooth-walled algorithm to transform it into a smooth-walled design. Through optimization, we further refine the smooth-walled antenna to achieve enhanced performance characteristics. Finally, we provide the necessary data required for the fabrication of the smooth-walled antenna, facilitating its practical realization.