a sphere of directions in ${\bf k}$-space. The operators ${\bf \hat L}$ and ${\bf \hat S}$ do not keep this subspace invariant, i.e., their action on a transverse mode results in a non-zero longitudinal component [@AB; @EN]. However, this subspace is invariant for the total AM operator ${\bf \hat J}$, and one can divide it into two parts consistent with the transversality condition: $$\label{eqn:2}
{\bf \hat J} = {\bf \hat L'} + {\bf \hat S'},~{\bf \hat L'} = {\bf \hat L} - {\bm \kappa } \times \left( {{\bm \kappa } \times {\bf \hat S}} \right),~{\bf \hat S'} = {\bm \kappa }\left( {{\bm \kappa } \cdot {\bf \hat S}} \right),$$ where ${\bm \kappa } = {\bf k}/k$ and the modified OAM and SAM operators ${\bf \hat L'}$ and ${\bf \hat S'}$ can be regarded as projections of the operators ${\bf \hat L}$ and ${\bf \hat S}$ onto the transversality subspace [@EN].

The modified SAM operator ${\bf \hat S'}$ is proportional to the helicity operator $\hat \sigma  = {\bm \kappa } \cdot {\bf \hat S}$, whereas the OAM operator can be written as ${\bf \hat L'} = {\bf \hat r'} \times {\bf k}$ with $$\label{eqn:3}
{\bf \hat r'} = {\bf \hat r} 