## 9 Transfer of SAM into OAM of the BSC with \(m\
eq 0\)  
It is well known that electromagnetic (EM) fields can not only carry energy but also angular momentum. The angular momentum is composed of the spin angular momentum (SAM) and the orbital angular momentum (OAM) describing the polarization and the phase structure distribution of EM fields, respectively. The research on the OAM of EM fields has been in focus of researches since [PERSON] et al. investigated the mechanism of the OAM in laser modes [54; 84]. In contrast to SAM, which has only two possible states of left-handed and right-handed circular polarizations, the states of OAM are in principle unlimited owing to the unique characteristics of spiral flow of propagating EM waves [55]. The OAM has the potential to tremendously increase the spectral efficiency and capacity of communication systems [85]. Among numerous investigations on OAM effects, one of the subjects of intensive recent studies is the link between the near-field chirality and the far-field OAM. For different types of chiral polaritonic lenses, it was shown that the near-field chirality can lead to the tailoring optical OAM in the far-field region [86; 87]. There were many proposals to generate OAM beams by use of chiral plasmonic nanostructures [86], ferrite particles [88], the monolithic integration of spiral phase plates [89], chiral polaritonic lenses [90], and by designer metasurfaces [91], etc. Figure 12: (**a**) Total cross-section for scattering of plane wave with \(\beta=1.3074\) in the vicinity of the Bloch BSC vs. the frequency; (**b**) The plane wave supports giant laminar power current at the point marked in the left panel by open circle. The color bar at the right indicates absolute value of the current. [PERSON] et al. [92] have shown that chiral fields, i.e., electromagnetic fields with nonvanishing optical chirality, can occur next to symmetric nanostructures without geometrical chirality illuminated with linearly polarized light at normal incidence. [PERSON] et al. [93] demonstrated a planar photonic nanostructure with no chirality consisting of a silicon microdisk coupled to two waveguides. The device distinguishes the handedness of an incoming circularly polarized light beam by driving photons with opposite spins toward different waveguides. It was shown theoretically and experimentally that the fundamental resonance of a silicon microdisk resonator can inherit the angular momentum carried by anormally incident light beam and transfer it as linear momentum into one of two output waveguides. Remarkably, the microdisk is not chiral: it responds equally to the left chiral polarization and the right chiral polarization without exhibiting optical activity nor circular dichroism. Instead, it couples light to different waveguides (with opposite linear momenta) depending on the handedness of incoming light and the relative position between the microdisk and the waveguides. The above results have a simple interpretation as an analogue with the spin-orbit interaction [94]. [PERSON] et al. [95] demonstrated circularly polarized dipole results in the unidirectional excitation of guided electromagnetic modes in the near field, with no preferred far-field radiation direction. In the present section we show a similar excitation of the Bloch BSC mode however with OAM in the near field transfer. Other words, we show a transfer of SAM to OAM of bound states in all-dielectric system in the near field of the array. Because of the time-reversal symmetry BSCs with OAM are degenerate with respect to the sign of \(m\). That modifies Equation (47) as follows  
\[\Psi_{\sigma}^{m}\approx\frac{1}{L_{c}}\sum_{\pm}[\mathbf{X}_{c}(\pm m)(Y_{c }(\pm m)^{+}\cdot\Psi_{inc}^{\pm m,\sigma})] \tag{51}\]  
where the incident wave according to Equation (16) is given  
\[\Psi_{inc}^{m,\sigma}=\left( \begin{array}{c}sign(m)\mathbf{p}_{| m|}^{\sigma}\\ \mathbf{q}_{|m|}^{\sigma}\end{array} \right),m\text{ are odd},\sigma=TE\] \[\Psi_{inc}^{m,\sigma}=\left(