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tokens: ‘s’ in space–time, because for isotropic geodesic motion we have d s = 0 but dΣ ≠ 0. This method has been successfully used to study various non-linear processes occurring in magnetic dipole fields of pulsars and magnetars (Abishev et al. 2016 ), as well as in an electric dipole field (Vasili’ev et al. 2017 ). Calculations have shown that in this case the rays of electromagnetic waves are bent, and the electromagnetic wave itself splits at η 1 ≠ η 2 into two normal modes with mutually perpendicular polarization. These modes propagate along different rays at different speeds. At the present time, another polarization effect of non-linear electrodynamics of vacuum is observed (Taverna et al. 2014 ; Caniulef et al. 2016 ; Mignani et al. 2017 ): polarized thermal emission from X-ray-dim isolated neutron stars (the case of RX J1856.5−3754). Recent papers (Pétri 2013 , 2015 ) assume that neutron stars can also have a quadrupole magnetic field, the intensity of which can reach the intensity of dipole fields. To date, there has not been any study of the non-linear electrodynamic effect of the magnetic quadrupole field on the propagation of pulses of electromagnetic radiation. Therefore, the purpose of this article is to calculate the non-linear effect of the magnetic dipole and quadrupole fields on the propagation of electromagnetic waves. 2 FORMULATION OF THE PROBLEM We place the origin of the Cartesian coordinate system at the point where the pulsar or magnetar is located. Then, the vector of magnetic induction will have the form where B 1 is the vector of the dipole magnetic induction and B 2 is the vector of the quadrupole magnetic induction. In the Maxwellian approximation, we have where m is the magnetic dipole moment of the pulsar or magnetar. The components of the quadrupole magnetic induction vector B 2 in a spherical coordinate system r , θ, φ, as shown in Pétri ( 2015 ), will have the form where R s is the neutron star radius, B is a constant having a dimension of magnetic induction, and χ 1 ∈ [0, ] and χ 2 ∈ [0, 2] are two angles specifying the particular geometry of the quadrupole magnetic field. However, for further calculations it is more convenient for us to use a rectangular Cartesian coordinate system. Redesignating the constants B , χ 1 , and χ 2 in accordance with relations from the expression (8) we obtain where the notations were introduced f 1 = cos ξ, f 2 = sin ξcos χ 2 , f 3 = sin ξsin χ 2 , at that The B 0 constant can be expressed in terms of the magnetic induction of a quadrupole B N at a point θ = /2, φ = /4 on the surface of a neutron star: Suppose that an electromagnetic pulse is emitted from a certain point r = r s = { x s , y s , z s } at time t = t s . We assume that at the point r = r d = { x d , y d , z d } there is an electromagnetic radiation detector. We orient the axes of the Cartesian coordinate system so that the source and the electromagnetic radiation detector lie in the XOZ plane, and the Z - axis is