Effect of Instrumental Polarization with a Half-Wave Plate on the $B$-Mode Signal: Prediction and Correction

We evaluate the effect of half-wave plate (HWP) imperfections inducing intensity leakage to the measurement of Cosmic Microwave Background (CMB) $B$-mode polarization signal with future satellite missions focusing on the tensor-to-scalar ratio $r$. The HWP is modeled with the Mueller formalism, and coefficients are decomposed for any incident angle into harmonics of the HWP rotation frequency due to azimuthal angle dependence. Although we use a general formalism, band-averaged matrix coefficients are calculated as an example for a 9-layer sapphire HWP using EM propagation simulations. We perform simulations of multi-detector observations in a band centered at 140\,GHz using \LB instrumental configuration. We show both theoretically and with the simulations that most of the artefacts on Stokes parameter maps are produced by the dipole leakage on $B$-modes induced by the fourth harmonics $M^{(4f)}_{QI}$ and $M^{(4f)}_{UI}$. The resulting effect is strongly linked to the spin-2 focal plane scanning cross linking parameters. We develop a maximum likelihood-based method to correct the IP leakage by joint fitting of the Mueller matrix coefficients as well as the Stokes parameter maps. % by modifying the standard map-making procedure. We show that the residual leakage after correction leads to an additional noise limited uncertainty on $r$ of the order of $10^{-7}$, independently of the value of the Mueller matrix coefficients. We discuss the impact of the monopole signal and the potential coupling with other systematic effects such as gain variations and detector nonlinearities.