Modulational instability and beam splitting in the nonlinear light propagation in sodium vapour

A linearly polarized laser beam propagating through sodium vapour is known to break up into its circular polarization components ('beam splitting'). We clarify the underlying mechanism by showing that spatially periodic perturbations of the polarization state of a plane wave will be amplified exponentially during propagation due to a modulational instability. For a Gaussian beam in one transverse spatial dimension we find even as well as odd active eigenmodes for polarization perturbations. Their two-dimensional generalizations elucidate the reason for the radial and lateral splitting that is observed in a number of experiments. In the limiting case of a cubic nonlinearity the splitting can also be expected from a variational approach.