## 7 Emergence of the BSC in Scattering
### Scattering of Plane Waves in the Vicinity of the Quasi-BSCs with OAM  
Next, consider the effect of the BSCs with \(m=2\) given by Equation (28) on the cross-section. We begin with the TE plane waves incident onto the array normally (\(k_{z}=0\)). Then we have from Equations (13)-(30) that \(p_{l2}^{TE}=0,q_{l2}^{TE}\
eq 0\) for odd \(l\), and \(p_{l2}^{TE}\
eq 0,q_{l2}^{TE}=0\) for even \(l\). Therefore as Equation (16) shows there are only Type II solutions for scattered waves with the amplitudes \((a_{2k+1},b_{2k})\). Table 1 shows that they belong to the same type of BSCs with \(m=2\). Therefore in the vicinity of \(R_{BSC}=0.471\) this BSC is coupled with the TE continuum and gives the resonant contribution in the cross-section \(\sigma_{TE,TE}\) that is demonstrated in Figure 11a,b. As for the scattering of the TM plane waves there are no resonant features as shown in Figure 10b by dash line. One can see in Figure 11c bright features of the differential cross-sections near the eigenfrequency of the BSC caused by the resonant contribution of the amplitude \(A_{2}\) at the azimuthal angles \(\phi=0,\pm 90^{\circ},180^{\circ}\):  
\[\frac{d\sigma}{d\phi}=\sigma_{0}|\sum_{m}A_{m}cos(m\phi)|^{2}. \tag{49}\]  
Figure 10: Total cross-section for scattering of plane wave incident by the angle \(\phi\) onto the array. (**a**) Scattering of TE plane wave is strongly affected by the presence of the symmetry protected Type I BSC (24) with the eigenfrequency \(k_{0}=4.24\) for \(R=0.3\), \(\epsilon=12\); (**b**) Scattering of TM plane wave is strongly affected by the presence of the symmetry protected Type II BSC (25) with the eigenfrequency \(k_{0}=4.7504\) for \(R=0.3\), \(\epsilon=15\).  
It is clear that for the sphere radius close to \(R_{BSC}=0.471\) the BSC solution dominates in the near field zone. The solution can be presented as  
\[\Psi=\alpha\Psi_{BSC}+\Phi \tag{50}\]  
where \(\alpha\) has a resonant behavior over frequency \(k_{0}\) with the resonant width \(\gamma\sim|R-R_{BSC}|\). Analytical expression for the resonant width can be derived following Refs. [77; 78]. Thus we have slowly decaying quasi BSC modes above the light cone similar to those considered in Ref. [82]. That effect is important for concentration of light by touching spheres [52; 83] notified as the harvesting capability of the system. Figure 11d illustrates the harvesting capability of the array of spheres in the vicinity of the BSC (28). Solid blue line shows the contribution of the background \(\phi=||\Phi||\) where \(||\cdots||\) is the norm of vector \(\Phi\). We do not present here the scattering of plane incident normally to the array at the vicinity of the BSC with \(m=\pm 1\). The results are very similar to those shown in Figure 11 except that differential cross-section has two maxima around the azimuthal angle equal \(\phi=0\) and \(\phi=\pi\) while scattering near the BSC with \(m=\pm 2\) shows maxima at \(\phi=0,\pi/2,\pi,3\pi/4\).  
Figure 11: The effect of the BSC (28) with \(m=2,k_{0}=3.086,R=0.471\) in: (**a**) differential cross-section vs. frequency and the azimuthal angle; (**b**) total cross-sections for different radii of the spheres close to the BSC radius (27) for plane wave illuminating the array normally; (**c**) Frequency behavior of the amplitudes \(A_{m}\) in the expansion (49); (**d**) Harvesting capability of the quasi BSC at \(R=0.473\). Dash red line shows the contribution of the BSC into the scattering function, blue solid line shows background \(\phi\).