phase singularity (Dirac string) along the positive $z$-axis in Eq. (\[eqn:8\]) [@Hawton], allowing a smooth transition to the paraxial case, $\theta  \to 0$.

It is worth noticing that the transformation to the helicity basis is associated with the transition to the local coordinate frame with the $z$-axis attached to the current ${\bf k}$-vector, which induces pure gauge Coriolis-type potential ${\bf \hat A}= - i\hat U^\dag  \partial _{\bf k} \hat U$, i.e., $( {{\bf \hat A} } )_{ij}  =  - i{\bf e}_i^*  \cdot \left( {\partial _{\bf k} } \right){\bf e}_j $, where $\mathbf{e}_{1,2,3}  \equiv \left( {\mathbf{e}^ +  ,\mathbf{e}^ -  ,\bm{\kappa }} \right)$ [@Photon; @Coriolis]. At the same time, non-canonical operators and commutation relations (\[eqn:2\])–(\[eqn:5\]) essentially owe their origin to the projection onto the transversality subspace, which is equivalent to the diagonalization of the potential ${\bf \hat A}$ [@Photon; @Coriolis]: ${\bf \hat A}_B={\rm dg}{\bf \hat A}$, i.e., $$\label{eqn:9}
( {{\bf \hat A}_B } )_{ij}  =  - i{\bf e}_i^*  \cdot \left( {\partial _{\bf k} } \right){\bf e}_j \delta _{ij}~.$$ While such diagonalization (which uncouples the two helicity components) is an adiabatic approximation for a nearly-transverse paraxial wave beam propagating in an inhomogeneous medium [@Coriolis; @Berry1987], it is exact for transverse plane waves in 