I_\ell ^\sigma  
\propto \left| {A^\sigma  } \right|^2 \left[ {a^2 J_\ell^2 \left( \xi \right) + b^2 J_{\ell + 2\sigma }^2 \left( \xi \right) + 2abJ_{\ell + \sigma }^2 \left( \xi \right)} \right],$$ Below we show that the polarization-dependent intensity distributions (\[eqn:44\]) (see Fig. 2(a)) signify the SOI of light.

The $\ell$- and $\sigma$-dependence of the radial intensity profile (\[eqn:44\]) can be explained via a geometrical-optics (GO) ray picture and the *quantization of caustic* underlying the maximum of the intensity. The rays associated with a Bessel beam are those that form an angle $\theta_0$ with the $z$-axis and touch a cylindrical caustic of radius $\rho  = R_\ell ^\sigma$ [@Berry2008], Fig. 1. The quantization condition for a closed orbit $\Gamma$ is $\oint\limits_\Gamma  {\bf k}  \cdot d{\bf r} = 2\pi \ell$. Using the underlying position (\[eqn:7\]), ${\bf r'}^\sigma   = {\bf r} - \sigma {\bf A}_B$, we observe that the Berry phase changes the effective optical length of a closed orbit on the cylindrical surface, Fig 1(b). For the orbit $\Gamma  = \left\{ {\rho  = R_\ell ^\sigma  ,\varphi  \in \left( {0,2\pi } \right)} \right\}$ it becomes $k_ \bot  \left[ {2\pi \, {\rm sgn}(\ell) R_\ell ^\sigma   - \sigma {\kern 1pt} \Phi _B } \right]$, which yields 