  We extend the theory of astrophysical maser propagation through a medium with
a Zeeman-split molecular response to the case of a non-uniform magnetic field,
and allow a component of the electric field of the radiation in the direction
of propagation: a characteristic of radiation with orbital angular momentum. A
classical reduction of the governing equations leads to a set of nine
differential equations for the evolution of intensity-like parameters for each
Fourier component of the radiation. Four of these parameters correspond to the
standard Stokes parameters, whilst the other five represent the $z$-component
of the electric field, and its coupling to the conventional components in the
$x-y$-plane. A restricted analytical solution of the governing equations
demonstrates a non-trivial coupling of the Stokes parameters to those
representing orbital angular momentum: the $z$-component of the electric field
can grow from a background in which only Stokes-$I$ is non-zero. A numerical
solution of the governing equations reveals radiation patterns with a radial
and angular structure for the case of an ideal quadrupole magnetic field
perpendicular to the propagation direction. In this ideal case generation of
radiation orbital angular momentum, like polarization, can approach 100 per
cent.
