an apparent partial conversion from SAM to OAM in non-paraxial light with the total AM being constant [@BA; @Li], akin to the spin-to-orbit AM conversion upon focusing of polarized light [@Li; @Beksh; @AMC1; @AMC2; @Oscar]. Indeed, in the Richards-Wolf approximation [@RW], the focusing represents a geometric conical redirection of partial plane waves with their helicity being conserved. It is described exactly by the same transformation operator $\hat U (\theta,\phi)$ that describes transition to the helicity basis [@Oscar].

![(Color online) (a) Intensity distributions, Eq. (\[eqn:45\]), marked by quantum numbers $(\ell,\sigma)$ for Bessel beams with $\theta_0=3\pi /8$. The spin-dependent profiles are shown for $\ell=4$ with $\sigma=-1$, 0 (scalar case), and 1. Dashed circles indicate the GO caustics (\[eqn:45\]). (b) Radial intensity profiles of the scalar ($\sigma=0$) or paraxial ($\theta_0\rightarrow0$) Bessel beams $I_{|\ell|}=J_{\ell}^2(\xi)$. (c) The SOI splitting of the profile of the polarized non-paraxial Bessel beam $I_{\pm4}^{\sigma}(\xi)$, Eq. (\[eqn:44\]), at different values of $\theta_0$; vertical lines indicate GO caustics (\[eqn:45\]), cf. (a). (d) The GO caustics (\[eqn:45\]) marked by ${\rm sgn}(\ell)(\ell,\sigma)$ as functions of $\theta_0$. (e) The SOI splitting of the maxima of intensity (\[eqn:44\]) \[cf. (c)\] as dependent on $\theta_0$, approaching the GO limit (\[eqn:45\]) at $|\ell| \gg 1$.[]{data-label="fig2"}](Fig2_6jpeg.eps){width="8.6cm"}

Simultaneously with a 