  In this paper we investigate the impact of using a polarization modulator in
the Short Wavelenght Instrument for the Polarization Explorer (SWIPE) of the
Large Scale Polarization Explorer (LSPE). The experiment is optimized to
measure the linear polarization of the Cosmic Microwave Background at large
angular scales during a circumpolar long-duration stratospheric balloon
mission, and uses multi-mode bolometers cooled at 0.3 K. The 330 detectors
cover 3 bands at 140 GHz, 220 GHz and 240 GHz. Polarimetry is achieved by means
of a large rotating half-wave plate (HWP) and a single wire-grid polarizer in
front of the arrays. The polarization modulator is the first
polarization-active component of the optical chain, reducing significantly the
effect of instrumental polarization. A trade-off study comparing stepped vs
spinning HWPs drives the choice towards the second. Modulating the CMB
polarization signal at 4 times the spin frequency moves it away from $1/f$
noise from the detectors and the residual atmosphere. The HWP is cooled at 1.6
K to reduce the background on the detectors. Furthermore its polarized emission
combined with the emission of the polarizer produces spurious signals modulated
at $2f$ and $4f$. The $4f$ component is synchronous with the signal of interest
and has to characterized to be removed from cosmological data.
