corresponding GO rays tangent to the caustic. Scalar phases are color-coded for $\ell=-1$, $\theta_0=\pi/4$. Points $A$ and $A'$ on the caustic are connected by two paths: the straight line and the Poynting-flow helix. The phase matching yields the phase difference $2\pi\ell$ between the paths and quantization of the caustic radius. For circularly-polarized waves, the helical path brings about an additional Berry phase $\sigma\Phi_B$, Eq. (\[eqn:45\]).[]{data-label="fig1"}](Fig1_3jpeg.eps){width="8.6cm"}

Application to Bessel beams
===========================

Importantly, our theory has a number of directly observable consequences. As the simplest example we take non-paraxial vector Bessel-beam solutions which are eigenmodes of $\hat J_z$ constructed from plane waves with well-defined helicity $\sigma$ (cf. [@EN; @Li; @Hawton; @Bessel]). The angular spectrum of such beams is $$\label{eqn:41}
{\bf \tilde E}_\ell ^\sigma  
%\left( {\theta ,\phi } \right) 
= {\bf e}^\sigma \left( {\theta ,\phi } \right) \tilde E_\ell ^\sigma  \left( {\theta ,\phi } \right),~{\tilde E}_\ell ^\sigma  = A^\sigma  \delta \left( {\theta  - \theta _0 } \right)e^{i\ell \phi },$$ where $A^\sigma$ is a constant amplitude, $\theta_0$ is the polar angle of conical distribution of the ${\bf k}$-vectors, Fig. 1(a), and no summation over $\sigma$ is implied here.

For the $z$-components of OAM and SAM, Eqs. (\[eqn:10\]) and (\[eqn:11\]), or (\[eqn:38\]) and (\[eqn:39\]), of 