  Several prominent forthcoming Cosmic Microwave Background polarisation
experiments will employ a Continuously Rotating Half-Wave Plate (CRHWP), the
primary purpose of which is to mitigate instrumental systematic effects on
relatively large angular scales, where the $B$-mode polarisation signal
generated by primordial gravitational waves is expected to peak. The use of a
CRHWP necessitates demodulating the time-ordered data during the early stages
of data processing. The standard approach is to ``lock in'' on the polarisation
signal using the known polarisation modulation frequency and then use Fourier
techniques to filter out the remaining unwanted components in the data.
However, an alternative less well-studied option is to incorporate the
demodulation directly into the map-making step. Using simulations, we compare
the performance of these two approaches to determine which is most effective
for $B$-mode signal recovery. Testing the two techniques in multiple
experimental scenarios, we find that the lock-in technique performs best over
the full multipole range explored. However, for the recovery of the largest
angular scales ($\ell < 100$) we find essentially no difference in the recovery
of the signal between the lock-in and map-making approaches, suggesting that a
parallel analysis based on the latter approach could represent a powerful
consistency check for primordial $B$-mode experiments employing a CRHWP. We
also investigate the impact of a detector-differencing step, implemented prior
to demodulation, finding that in most scenarios it makes no difference whether
differencing is used or not. However, analysing detectors individually allows
the point at which information from multiple detectors is combined to be moved
to later stages in the analysis pipeline. This presents alternative options for
dealing with instrumental systematic effects that are not mitigated by the
CRHWP.
