## 9 Transfer of SAM into OAM of the BSC with \(m\
\begin{array}{c}\mathbf{p}_{|m|}^{ \sigma}\\ sign(m)\mathbf{q}_{|m|}^{\sigma}\end{array} \right),m\text{ are odd},\sigma=TM\] \[\Psi_{inc}^{m,\sigma}=\left( \begin{array}{c}\mathbf{p}_{|m|}^{ \sigma}\\ sign(m)\mathbf{q}_{|m|}^{\sigma}\end{array} \right),m\text{ are even},\sigma=TE\] \[\Psi_{inc}^{m,\sigma}=\left( \begin{array}{c}sign(m)\mathbf{p}_ {|m|}^{\sigma}\\ \mathbf{q}_{|m|}^{\sigma}\end{array} \right),m\text{ are even},\sigma=TM. \tag{52}\]  
and subvectors \(\mathbf{p}^{m}\) and \(\mathbf{q}^{m}\) are given by Equation (13). In particular, for the plane wave incident normally to the array \(\beta=0\) we have \[\mathbf{P}_{|m|}^{TE}=\left( \begin{array}{c}0\\ p_{m,2}^{TE}\\ 0\\ p_{m,A}^{TE}\\ \vdots\end{array} \right),\mathbf{q}_{|m|}^{TE}=\left( \begin{array}{c}q_{m,1}^{ TE}\\ 0\\ q_{m,3}^{TE}\\ 0\\ \vdots\end{array} \right),m\text{ are odd},\] \[\mathbf{P}_{|m|}^{TM}=\left( \begin{array}{c}p_{m,1}^{TM}\\ 0\\ p_{m,3}^{TM}\\ 0\\ \vdots\end{array} \right),\mathbf{q}_{|m|}^{TM}=\left( \begin{array}{c}0\\ q_{m,2}^{TM}\\ 0\\ q_{m,A}^{TM}\\ \vdots\end{array} \right),m\text{ are odd},\] \[\mathbf{P}_{|m|}^{TE}=\left( \begin{array}{c}p_{m,1}^{TE}\\ 0\\ p_{m,3}^{TE}\\ 0\\ \vdots\end{array} \right),\mathbf{q}_{|m|}^{TE}=\left( \begin{array}{c}0\\ q_{m,2}^{TE}\\ 0\\ q_{m,A}^{TE}\\ \vdots\end{array} \right),m\text{ are even},\] \[\mathbf{P}_{|m|}^{TM}=\left( \begin{array}{c}0\\ p_{m,2}^{TM}\\ 0\\ p_{m,M}^{TM}\\ \vdots\end{array} \right),\mathbf{q}_{|m|}^{TM}=\left( \begin{array}{c}q_{m,1 }^{TM}\\ 0\\ q_{m,3}^{TM}\\ 0\\ \vdots\end{array} \right),m\text{ are even}. \tag{53}\]  
By virtue of Equation (45) and \(\mathcal{B}_{ll^{\prime}}^{(m)}=-\mathcal{B}_{ll^{\prime}}^{(-m)}\) the eigenvectors can be decomposed over the polarizations as follows  
\[\mathbf{X}_{c}(\pm m)=\left( \begin{array}{c}\mathbf{x}_{TE}^{m}\\ \pm\mathbf{x}_{TM}^{m}\end{array} \right),\mathbf{Y}_{c}(\pm m)=\left( \begin{array} []{c}\mathbf{y}_{TE}^{m}\\ \pm\mathbf{y}_{TM}^{m}\end{array} \right). \tag{54}\]  
Then it follows from Equation (51)  
\[\Psi_{\sigma}^{m}\approx\left\{ \begin{array}{c}\frac{D_{|m|}^{|m|}}{L_{c}m} [\mathbf{X}_{c}(m)+(-1)^{m}(\mathbf{X}_{c}(-m)],\sigma=TE\\ \frac{D_{|m|}^{|m|}}{L_{c}m}[\mathbf{X}_{c}(m)+(-1)^{m+1}\mathbf{X}_{c}(-m)], \sigma=TM,\end{array} \right., \tag{55}\]  
where  
\[D_{\sigma}^{|m|}=\mathbf{y}_{TE}^{+}\mathbf{p}_{|m|}^{\sigma}+\mathbf{y}_{TM} ^{+}\mathbf{q}_{|m|}^{\sigma}. \tag{56}\]  
Assume that the elliptically polarized plane wave \(\Psi_{inc}^{TE}+\alpha\Psi_{inc}^{TM}\) is incident with small \(\beta\). By taking  
\[\alpha=\frac{D_{TE}^{|m|}}{D_{TM}^{|m|}} \tag{57}\]  
we obtain from Equation (55) that  
\[\Psi_{\sigma}\approx F_{|m|}\mathbf{X}_{c}^{+m},F_{|m|}=\frac{2D_{TE}^{|m|}}{ L_{c}m}. \tag{58}\]  
The scattering function has only a contribution with the positive OAM \(m>0\). Here we introduced the enhancement factor \(F\) which defines to what extent the scattering function is amplified in the near zone. Respectively for \(D_{TE}=-\alpha D_{TM}\) the scattering function has only a contribution with the negative OAM \(m<0\). One can show from Equations (53), (54) and (56) that asymptotically \(D_{TM}^{|m|}\to 0\) for \(\beta\to 0\). From Equation (58) it follows that the enhancement factor for scattering of plane waves in the vicinity of the BSC point is determined by the ratio \(D_{\sigma}/L_{c}\). In what follows we sweep the frequency of the incident wave \(k_{0}\) and the angle of incidence defined by \(\beta\) in the vicinity of the BSCs with OAM \(m=1\) and \(m=2\). Figure 13 illustrates the behavior of the enhancement factor in the plane of the frequency \(k_{0}\) and \(\beta\) calculated with the use of Equation (58). Following the line with \(|\alpha|=1\) we found the maximal enhancement marked by open green circles in Figure 13 for the following parameters.