# Wide-Band Wide-Beam Circularly-Polarized Slot-Coupled Antenna for Wide-Angle Beam Scanning Arrays
## 3 Design Methodology
Moreover, \(S_{11}^{th}\) is the desired threshold and \(f_{q}\) [\(f_{q}\triangleq f_{\min}+(q-1)\times\frac{(f_{\max}-f_{\min})}{(Q-1)}\)] is the \(q\)-th (\(q=1,\ldots,Q\)) frequency sample, \(Q\) being the number of spectral components analyzed with full-wave (\(FW\)) simulations. Finally, \(\mathcal{H}\{\{\,.\}\}\) is the Heaviside's function, equal to \(\mathcal{H}\{\xi\}=1\) if \(\zeta>0\) and \(\mathcal{H}\{\xi\}=0\), otherwise. As for the wide-beam features, the _HPBW_ cost term (\(\gamma=HPBW\)) is given by  
\[ \begin{array}{rl}\Phi_{HPBW}(\underline{\Omega})=\frac{1}{Q\times M}\sum_{q= 1}^{Q}\sum_{m=1}^{M}\\ \frac{\left\{HPBW^{th}-HPBW\left(f_{q},\varphi_{m}\big{|}\underline{\Omega} \right)\right\}}{HPBW^{th}}\times\\ \times\mathcal{H}\Bigl{\{}HPBW^{th}-HPBW\big{(}f_{q},\,\varphi_{m}\big{|} \underline{\Omega}\big{)}\Bigr{\}}\end{array}  \tag{6}\]  
where \(HPBW^{th}\) is the user-defined requirement, while \(\varphi_{m}\left(\varphi_{m}=(m-1)\times\frac{360}{M}\left[\text{deg}\right]\right)\) is the \(m\)-th (\(m=1,\ldots,M\)) azimuth angular sample [Figure 1a], \(M\) being the number of elevation planes considered for the numerical evaluation of the _HPBW_. The last two cost function terms in (3) (i.e., \(\gamma=AR\) and \(\gamma=XPD\)) are related to the _CP_ and they are defined as follows  
\[ \begin{array}{l}\Phi_{AR}(\underline{\Omega})=\frac{1}{Q\times V\times M}\sum _{q=1}^{Q}\sum_{v=1}^{V}\sum_{m=1}^{M}\\ \frac{\left\{AR^{(dB)}\left(f_{q},\theta_{v},\varphi_{m}\left|\underline{ \Omega}\right)-AR^{th}\right\}}{AR^{th}}\times\\ \times\mathcal{H}\Big{\{}AR^{(dB)}\left(f_{q},\,\theta_{v},\,\varphi_{m}\left| \underline{\Omega}\right)-AR^{th}\right\}\end{array}  \tag{7}\]  
and  
\[ \begin{array}{l}\Phi_{XPD}(\underline{\Omega})=\frac{1}{Q\times V\times M} \sum_{q=1}^{Q}\sum_{v=1}^{V}\sum_{m=1}^{M}\\ \frac{\left\{XPD^{th}-XPD^{(dB)}\left(f_{q},\theta_{v},\varphi_{m}\left| \underline{\Omega}\right)\right\}}{XP^{\underline{\Omega}\underline{\Omega} \underline{\Omega}}}\times\\ \times\mathcal{H}\Big{\{}XPD^{th}-XPD^{(dB)}\left(f_{q},\,\theta_{v},\,\varphi _{m}\left|\underline{\Omega}\right)\right\}.\end{array}  \tag{8}\]  
In the previous expressions, \(\theta_{v}=(v-1)\times\frac{90}{(V-1)}\) [deg] is the \(v\)-th (\(v=1,\ldots,V\)) elevation angle [Figure 1a], \(AR^{th}\) is the maximum _AR_ given by [6]  
\[ \begin{array}{l}AR^{(dB)}\big{(}f_{q},\,\theta_{v},\,\varphi_{m}\left| \underline{\Omega}\right)=\\ =20\times\log_{10}\left(\frac{\left|E_{C}\left(f_{q},\,\theta_{v},\,\varphi_{ m}\left|\underline{\Omega}\right)\right|+\left|E_{X}\left(f_{q},\,\theta_{v},\, \varphi_{m}\left|\underline{\Omega}\right)\right|\right)}{\left|E_{C}\left(f_{ q},\,\theta_{v},\,\varphi_{m}\left|\underline{\Omega}\right)\right|-\left|E_{X} \left(f_{q},\,\theta_{v},\,\varphi_{m}\left|\underline{\Omega}\right)\right| \right)}\right)\end{array}  \tag{9}\]  
where the subscripts \"\(\mathcal{C}\)\" and \"\(\mathcal{X}\)\" denote the co-polar and the cross-polar field components, respectively (i.e., \(\mathcal{C}\gets LHCP\) and \(\mathcal{X}\gets RHCP\) if a _LHCP_ antenna is designed, and vice-versa for _RHCP_ operation), equal to  
\[\left\{ \begin{array}{l}E_{C}\big{(}f_{q},\,\theta_{v},\,\varphi_{m}\left| \underline{\Omega}\right)=\mathbf{E}\big{(}f_{q},\,\theta_{v},\,\varphi_{m} \left|\underline{\Omega}\right)\cdot\left(\widehat{\rho}_{\mathcal{C}}\right) ^{*}\\ E_{X}\big{(}f_{q},\,\theta_{v},\,\varphi_{m}\left|\underline{\Omega}\right)= \mathbf{E}\big{(}f_{q},\,\theta_{v},\,\varphi_{m}\left|\underline{\Omega} \right)\cdot\left(\widehat{\rho}_{\mathcal{X}}\right)^{*},\end{array} \right. \tag{10}\]  
\(\mathbf{E}\big{(}f_{q},\,\theta_{v},\,\varphi_{m}\left|\underline{\Omega} \right)=E_{\theta}\big{(}f_{q},\,\theta_{v},\,\varphi_{m}\left|\underline{ \Omega}\right)\widehat{\theta}+E_{\varphi}\big{(}f_{q},\,\theta_{v},\,\varphi _{m}\left|\underline{\Omega}\right)\widehat{\varphi}\) being the far-field electric field, \((\cdot)\) is the dot product, and \((\cdot)^{*}\) stands for complex conjugate.