Combined Lensing in the Vicinity of Neutron Stars with Dipole–Quadrupole Magnetospheres: Nonlinear Electrodynamics with General-Relativistic Ray-Tracing and Observational Implications

We consider a model problem of polarization-dependent light bending and time delays in the vicinity of neutron stars endowed with magnetar-strength magnetic fields (B∼1015G), combining an effective-metric formulation of Heisenberg–Euler nonlinear electrodynamics with general-relativistic ray tracing. The spacetime geometry is analyzed using both the Kerr metric and a quadrupole-deformed q-metric, characterized by a quadrupole parameter varying in the range q∈[10−3,0.5]. In addition, the impact of complex magnetic-field topologies is examined by introducing a magnetic quadrupole component alongside the dipole configuration. The simulations performed in this study demonstrate that the inclusion of the quadrupole deformation parameter significantly modifies photon trajectory deflections compared to the standard Kerr solution. We further quantify the geometric dilution of the photon beam, finding a cross-section expansion ratio of approximately 4.7×1013 for rays reaching Earth. This strong dilution imposes stringent constraints on the detectability of polarization-dependent signatures and time-delay echoes. Finally, characteristic illustrations are presented for trajectory distortions, bending-angle distributions, and intensity valleys produced by the combined gravitational and magnetic lensing effects.