In this study, we propose a simplified model for estimating the cross-polarization modulation in a bulk semiconductor optical amplifier (SOA), assuming that the induced birefringence is the main contribution to this nonlinear phenomenon. Using a semi-classical model by sections and the effective index method, we determine the longitudinal distribution of the induced birefringence along the optical waveguide and the total phase shifts experienced by the orthogonal components of the electric field of a data beam. The proposed model is numerically and analytically solved. These results are compared with experimental data, providing good agreement. Particularly, we analyze the angular displacements in the PoincarÉ representation of the polarization state of the data beam propagating simultaneously with a control beam, whose input power was varied from $-$30 to $+$2 dBm. The results show a maximum relative error of 13.9% and 13.3% in, respectively, the numerical and analytical solutions.