Present theoretical work investigates the dominance of polarization modes i.e.Extraordinary ( \(E-mode\) ) and Ordinary ( \(O-mode\) ) effects caused by an externally applied magnetic field on the self-focusing of higher-order elegant Hermite-cosh-Gaussian (EHChG) laser beams in a magnetized plasma medium. In the present investigation author have taken transverse electromagnetic (TEM) modes with mode indices (m, n) of EHChG lasers having different values (0, 0), (0, 2), and (0, 4) and explored. In addition, the most generally utilized Wentzel, Kramers, Brillouin (WKB) and paraxial ray approximations (PRA) through parabolic wave equation approach are employed to establish the general form of differential equations for beam-width parameters in both transverse dimensions of the EHChG laser beam. The behavior of beam-width parameters with dimensionless distance of propagation is thoroughly examined for various magnetic field and polarization modes with considered beams (EHChG). The dominance of magnetic field and polarization modes emphasizes the complexity in beam width parameters during propagation.
