$\sigma$-dependent OAM, the non-paraxial fields exhibit $\sigma$-dependent intensity distributions related to the modified position operator. The real-space field of the circularly-polarized Bessel beam, calculated via the Fourier transformation (\[eqn:A6\]) of Eq. (\[eqn:41\]), is $$\mathbf{E}_\ell ^\sigma   \propto A^\sigma  \left( {\begin{array}{*{20}c}
   {\frac{{1 + \sigma }}
{2}J_\ell  \left( \xi  \right) - \sigma b\,e^{i\left( {\sigma  - 1} \right)\varphi } J_{\ell  + \sigma  - 1} \left( \xi  \right)}  \\
   {\frac{{1 - \sigma }}
{2}J_\ell  \left( \xi  \right) + \sigma b\,e^{i\left( {\sigma  + 1} \right)\varphi } J_{\ell  + \sigma  + 1} \left( \xi  \right)}  \\
   { - i\sigma \sqrt {2ab} e^{i\sigma \varphi } J_{\ell  + \sigma } \left( \xi  \right)}  \\
 \end{array} } \right)e^{ik_{\parallel}  z + i\ell \varphi },$$ where $\left( {\rho ,\varphi ,z} \right)$ are the cylindrical coordinates in real space, $a = \cos ^2 \left( {\theta _0 /2} \right)$, $b = \sin ^2 \left( {\theta _0 /2} \right)$, $\xi=k_\bot\rho$, $k_ \bot = k\sin \theta _0$, $k_{\parallel} = k\cos \theta _0$, $J_n (\xi)$ are the Bessel functions of the first kind, and the field components are written in the basis $\left(\frac{{\bf e}_x+i{\bf e}_y}{\sqrt{2}},\frac{{\bf e}_x-i{\bf e}_y}{\sqrt{2}},{\bf e}_z\right)$. This field has a cylindrically symmetric intensity distribution, $I_\ell ^\sigma   = \left| {\mathbf{E}_\ell ^\sigma  } \right|^2$, given by $$\label{eqn:44}
