## 9 Transfer of SAM into OAM of the BSC with \(m\
(i) For the case of the BSC with \(m=1\) the optimal parameters are \(k_{0}=k_{0c}+0.0025\), \(\beta=0.0052\), \(\alpha=0.63+0.77i\) for \(R=R_{c}-0.0005\); (ii) For the case of the BSC with \(m=2:k_{0}=k_{0c}+0.02\), \(\beta=0.031\), \(\alpha=0.31+0.94i\) for \(R=R_{c}-0.003\). Fixing these parameters except \(\beta\) we plot the lowest eigenvalue of the matrix \(\widehat{L}\) in Figure 14 and the values of \(|D_{\sigma}|\) Equation (56) in Figure 15 versus \(\beta\). From these Figures one can see that, first, the enhancement is determined by the lowest eigenvalue \(L_{cm}\) while \(D_{TE}\) is almost constant. Second, the value \(D_{TM}\) grows from zero. Therefore, to achieve enhancement one has to inject a plane wave with elliptic polarization. In what follows we take for simplicity the circular polarization \(|\alpha|=1\) of the incident wave. Figure 14: The lowest eigenvalue \(|L_{c}|\) of matrix (45) in the vicinity of the BSCs with OAM. Figure 13: Enhancement factor \(|F_{m}|\) vs. \(k_{0}\) and \(\beta\). White line corresponds to polarization (57) \(|\alpha|=1\). Open circles mark of maximal enhancement. Because of the smallness of the eigenvalue \(L_{cm}\) in Equation (55) EM fields given by the scattering function can reach extremely high values near the spheres. Clearly this is an effect of the BSCs with infinitely high quality factor that presents a possibility to enormously enhance the incident light [40; 52; 78]. In Figure 16 we demonstrate that the enhancement is very sensitive to the choice of the sphere radius in the vicinity of \(R_{c}\) when other parameters are tuned to the BSC point. Thanks to carrying OAM the BSC with \(m\
eq 0\) supports vortical power currents [96] as demonstrated in Figure 17. Owing to the enhancement of the scattered field in the near zone the spinning currents can reach giant values with respect to the incident power currents as demonstrated in Figure 18. All currents are measured in terms of the incident power with \(\beta=0.00517\) for the case \(m=1\) and \(\beta=0.0307\) for the case \(m=2\). The value of the current is extremely high inside the spheres but rapidly drops outside the spheres as shown in Figure 19. As soon as the polarization is linear, for example \(\alpha=0\), vortical currents around the array vanish as demonstrated in Figure 20.  
Figure 16: Values of the maximal enhancement factor \(|F|\) vs. \(k_{0}\) and \(\beta\) as dependent on radius of spheres for \(|\alpha|=1\). \[\alpha=\frac{D_{TE}^{|m|}}{D_{TM}^{|m|}} \tag{57}\]0 The values \(D_{\sigma}\) given by Equation (56) in the vicinity of the BSCs with OAM. \[\alpha=\frac{D_{TE}^{|m|}}{D_{TM}^{|m|}} \tag{57}\]1 Pointing current circulates around the spheres when circularly polarized light is injected. Currents around other spheres are repeating periodically. \[\alpha=\frac{D_{TE}^{|m|}}{D_{TM}^{|m|}} \tag{57}\]2 Iso surfaces of constant angular component of the power current. \[\alpha=\frac{D_{TE}^{|m|}}{D_{TM}^{|m|}} \tag{57}\]3 Value of angular component of the power current around the spheres at distance \(r\) from the center of sphere and \(z=0\). Figures 21 and 22 demonstrate that the orbital angular momentum of the BSCs affects the scattering of plane waves with linear polarization. The effect is a conversion of the incident polarizations \(TE\to TM\) and _visa versa_. For the normally incident waves \(\beta=0\) there is no polarization conversion and no resonant peaks in the total cross-sections \(TM\to TM\). Once the angle of incidence deviates from zero \(\beta\
eq 0\) all three total cross-sections acquire resonant response as shown in \[\alpha=\frac{D_{TE}^{|m|}}{D_{TM}^{|m|}} \tag{57}\]4 Note that there is polarization conversion when the frequency is far from the BSC frequencies. The absence of polarization conversion is clearly seen in the differential cross-section \(TE\to TM\) as shown in \[\alpha=\frac{D_{TE}^{|m|}}{D_{TM}^{|m|}} \tag{57}\]5 It is also remarkable that this cross-section distinctively reflects the value of the OAM \(m\).