$$\label{eqn:45}
k_ \bot  R_\ell ^\sigma   = \left| \ell  + \sigma {\kern 1pt} \frac{\Phi _B }{2\pi } \right|~.$$ Similar Berry-phase effects appear in quantum quantization problems [@Quantization], e.g., the half-integer Hall effect in graphene [@Graphene]. Note also the exact correspondence between the GO caustic (\[eqn:45\]) and the wave OAM (\[eqn:42\]), $|L_z|=k_{\bot}R_\ell ^\sigma$, which reflects the OAM interpretation as ${\bf r}\times{\bf k}$ for the rays. Figure 2 shows $\ell$- and $\sigma$-dependent intensity distributions (\[eqn:44\]) of the Bessel beams vs. the GO caustics (\[eqn:45\]). Spin-dependent splitting of caustics and intensity maxima are the optical analogues of the fine spin-orbit splitting of levels in quantum systems. The $\sigma$-dependence in radial distributions of non-paraxial vortex fields can be observed experimentally by tightly focusing paraxial light with different polarizations, cf. [@Gorod].

![(Color online) Transverse intensity distributions at $z=0$ of the asymmetric Bessel beams with $\delta=\pi/2$ and $\theta_0 =3\pi/8$ marked by quantum numbers $(\ell,\sigma)$. Dashed lines indicate the $\ell$- and $\sigma$-dependent transverse shifts of the centers of gravity, Eq. (\[eqn:46\]), i.e., orbital and spin Hall effects of light in free space. We have verified that the centers of gravity calculated numerically from the intensity distributions and theoretically from Eq. (\[eqn:46\]) coincide.[]{data-label="fig3"}](Fig3_2jpeg.eps){width="8.6cm"}

Finally, we consider the Hall effects 