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We evaluate the effect of half-wave plate (HWP) imperfections inducingintensity leakage to the measurement of Cosmic Microwave Background (CMB)$B$-mode polarization signal with future satellite missions focusing on thetensor-to-scalar ratio $r$. The HWP is modeled with the Mueller formalism, andcoefficients are decomposed for any incident angle into harmonics of the HWProtation frequency due to azimuthal angle dependence. Although we use a generalformalism, band-averaged matrix coefficients are calculated as an example for a9-layer sapphire HWP using EM propagation simulations. We perform simulationsof multi-detector observations in a band centered at 140\,GHz using \LBinstrumental configuration. We show both theoretically and with the simulationsthat most of the artefacts on Stokes parameter maps are produced by the dipoleleakage 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 focalplane scanning cross linking parameters. We develop a maximum likelihood-basedmethod to correct the IP leakage by joint fitting of the Mueller matrixcoefficients as well as the Stokes parameter maps. % by modifying the standardmap-making procedure. We show that the residual leakage after correction leadsto an additional noise limited uncertainty on $r$ of the order of $10^{-7}$,independently of the value of the Mueller matrix coefficients. We discuss theimpact of the monopole signal and the potential coupling with other systematiceffects such as gain variations and detector nonlinearities.

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