In this work, the non-linear effect of the magnetic dipole and quadrupole fields on the propagation of electromagnetic waves in the eikonal approximation of the parametrized post-Maxwell electrodynamics of the vacuum is calculated. Equations of motion for electromagnetic pulses transmitted in a strong magnetic field of a pulsar by two normal modes with mutually orthogonal polarization are constructed. The difference Δt in propagation times of normal waves from the common source of electromagnetic radiation to the receiver is calculated. It is shown that the forward part and the 'tail' by length cΔt of any hard radiation pulse due to the non-linear electromagnetic influence of the magnetic dipole and quadrupole fields turn out to be linearly polarized in mutually perpendicular planes, and the remaining part of the pulse must have elliptical polarization. © 2018 The Author(s). Published by Oxford University Press on behalf of the Royal Astronomical Society.