  We explore a plasma based analogue of a helical undulator capable of
providing circularly and elliptically polarised betatron radiation. We focus on
ionisation injection configurations and in the conditions where the laser pulse
driver can force collective betatron oscillations over the whole trapped
electron bunch. With an analytical model and by employing three dimensional
simulations and radiation calculations, we find that circularly or elliptically
polarised laser drivers can force helical betatron oscillations, which produce
circularly/elliptically polarised betatron x-rays. We assess the level of
polarisation numerically and analytically, and find that the number of
circularly polarised photons can be controlled by tuning the laser pulse driver
polarisation. We show the production of betatron radiation that is circularly
polarised up to < 40% close to regions of maximum photon flux. The total flux
of circularly polarised betatron radiation drops for elliptically polarised
drivers, and is negligible when using linear polarisation. Our results can be
tested today in current experimental facilities.
