  We investigate the phenomenology of nonlinear Compton scattering of polarised
photons by unpolarised electrons in plane-wave backgrounds. The energy and
angular spectra of polarised photons are calculated for linearly- and
circularly-polarised pulses, monochromatic fields and constant crossed field
backgrounds. When the field intensity is in the weakly nonlinear regime,
photons in different polarisation states are predicted to possess very
different energy and angular distributions. We explain this difference by
calculating the spectrum for nonlinear Thomson scattering in a
linearly-polarised monochromatic background and projecting the electron's
trajectory onto the different polarisation directions. Finally, we calculate a
scenario for multi-GeV electrons and by considering the energy and
angular-resolved photon polarisation purity, find one can achieve a GeV-photon
beam with polarisation purities of over $90\%$.
