  Polarization of the cosmic microwave background (CMB) can probe new
parity-violating physics such as cosmic birefringence (CB), which requires
exquisite control over instrumental systematics. The non-idealities of the
half-wave plate (HWP) represent a source of systematics when used as a
polarization modulator. We study their impact on the CMB angular power spectra,
which is partially degenerate with CB and miscalibration of the polarization
angle. We use full-sky beam convolution simulations including HWP to generate
mock noiseless time-ordered data, process them through a bin averaging
map-maker, and calculate the power spectra including $TB$ and $EB$
correlations. We also derive analytical formulae which accurately model the
observed spectra. For our choice of HWP parameters, the HWP-induced angle
amounts to a few degrees, which could be misinterpreted as CB. Accurate
knowledge of the HWP is required to mitigate this. Our simulation and
analytical formulae will be useful for deriving requirements for the accuracy
of HWP calibration.
