# Wide-Band Wide-Beam Circularly-Polarized Slot-Coupled Antenna for Wide-Angle Beam Scanning Arrays
## 3 Design Methodology
Moreover, \(\widehat{\rho}_{\mathcal{C}}\) and \(\widehat{\rho}_{\mathcal{X}}\) are the polarization unit vectors for the two _CP_s [6]  
\[\left\{ \begin{array}{l}\widehat{\rho}_{\mathcal{C}}=\frac{1}{\sqrt{2}} \left(\widehat{\theta}+j\widehat{\varphi}\right)\\ \widehat{\rho}_{\mathcal{X}}=\frac{1}{\sqrt{2}}\left(\widehat{\theta}-j \widehat{\varphi}\right)\end{array} \right.. \tag{11}\]  
Finally, \(XPD^{th}\) is the minimum _XPD_ being  
\[ \begin{array}{l}XPD^{(dB)}\big{(}f_{q},\,\theta_{v},\,\varphi_{m}\left| \underline{\Omega}\right)\\ =10\times\log_{10}\left(\frac{G_{C}\big{(}f_{q},\,\theta_{v},\,\varphi_{m} \left|\underline{\Omega}\right)}{G_{X}\big{(}f_{q},\,\theta_{v},\,\varphi_{m} \left|\underline{\Omega}\right)}\right)\end{array}  \tag{12}\]  
where  
\[ \begin{array}{l}G_{\mathcal{C}/\mathcal{X}}\big{(}f_{q},\,\theta_{v},\, \varphi_{m}\left|\underline{\Omega}\right)=\frac{2\pi}{\eta_{0}P_{\mathit{acc }}\left(f_{q}|\underline{\Omega}\right)}\\ \times\big{|}E_{\mathcal{C}/\mathcal{X}}\big{(}f_{q},\,\theta_{v},\,\varphi_{m} \left|\underline{\Omega}\right)\big{|}^{2}\end{array}  \tag{13}\]  
is the gain related to the \(\mathcal{C}/\mathcal{X}\)-th field component, respectively, \(\eta_{0}\) is the free-space impedance, while \(\mathcal{P}_{\mathit{acc}}\left(f_{q}\left|\underline{\Omega}\right)\) is the accepted power at the antenna terminals for a given incident power \(\mathcal{P}_{\mathit{inc}}\)  
\[\mathcal{P}_{\mathit{acc}}\big{(}f_{q}\left|\underline{\Omega}\right)=\left(1 -10^{\left|\frac{5^{(dB)}}{11}\left(f_{q}\left|\Omega\right)\right|}/10 \right)\times\mathcal{P}_{\mathit{inc}}.\]  
The overall _SbD_-driven design work-flow consists of the following procedural steps:  
1. _Input phase._ Define the bounds of the target's operating band, \(f_{\min}\) and \(f_{\max}\), the required _CP_ (i.e., _LHCP_ or _RHCP_), and the threshold value for each key performance indicator, \(\underline{\Gamma}^{th}=\left\{S_{11}^{th},\,HPBW^{th},\,AR^{th},\,XPD^{th}\right\}\). Perform the following operationsSet \(L_{a}=\frac{\lambda_{0}}{2}\), \(\lambda_{0}\) being the free-space impedance at the central frequency \(f_{0}=\frac{(f_{\min}+f_{\max})}{2}\); * Select from an off-the-shelf data-sheet the material/thickness of the \(l\)-th (\(l=1,\ldots,\mathcal{L}\)) layer \(\ell_{l}\); * Compute the width of the feeding line \(W_{f}\) to yield the desired characteristic impedance \(Z_{0}\) (p. 148, Equation 3.197 [8]); * Derive an analytic guess, \(\widetilde{R}_{p}\), for the radius of the primary element of the radiator as detailed in [33] (p. 846, Equation 14.69), then set its optimization range \(\Omega_{1}^{\left(\min\right)}\) and \(\Omega_{1}^{\left(\max\right)}\) as a percentage of \(\widetilde{R}_{p}\), being \(\Omega_{1}=R_{p}\); * Define the optimization bounds of the remaining \((K-1)\)\(DoF_{8}\left[\Omega_{k}^{\left(\min\right)}\), \(\Omega_{k}^{\left(\max\right)}\right]\), \(k=2,\ldots,K\);
2. _Surrogate model (SM) Building._ Use the Latin hypercube sampling (_LHS_) strategy [32] to generate a training set \(\mathbb{T}_{0}=\left\{\left[\underline{\Omega}^{\left(s\right)},\,\Phi\!\left( \underline{\Omega}^{\left(s\right)}\right)\right]\); \(s=1,...,S_{0}\right\}\) of \(S_{0}\) training samples, \(\Phi\!\left(\underline{\Omega}^{\left(s\right)}\right)\) being the cost function associated with the \(s\)-th (\(s=1,...,S_{0}\)) input sample \(\underline{\Omega}^{\left(s\right)}\in\left[\underline{\Omega}^{\left(\min \right)},\,\underline{\Omega}^{\left(\max\right)}\right]\), computed with a \(FW\) simulation. Starting from \(\mathbb{T}_{0}\), build a computationally-fast \(SM\) of (3) based on the ordinary Kriging (\(OK\)) learning-by-examples (_LBE_) technique [32; 33; 34; 35; 36];
3. _Design Initialization_ (\(i=0\))--Define an initial swarm of \(P\) particles, \(\mathbb{P}_{0}=\left\{\underline{\Omega}_{0}^{\left(p\right)};\,p=1,\ldots,P\right\}\), with random velocities \(\mathbb{V}_{0}=\left\{\underline{v}_{0}^{\left(p\right)};\,p=1,\ldots,P\right\}\);
4.