Angular structure of finite-field vacuum birefringence in magnetars
Vacuum birefringence in a strong magnetic field is often modelled by combining the finite-field splitting for perpendicular propagation with the familiar $\sin^2θ$ angular factor. That separation is exact only at leading response order. Continuing our previous study of the field-strength dependence of the birefringence, we examine the angular dependence of the full one-loop refractive indices in the local Heisenberg--Euler theory. We show analytically that, on the positive-birefringence branch, the usual factorized form systematically overestimates the exact splitting away from perpendicular propagation. The relative correction is largest close to the magnetic-field direction, where the absolute birefringence itself vanishes. Compact analytic expansions expose the sign, scaling, and angular location of the correction throughout the magnetar-relevant range $B/B_{\rm cr}\le100$, while an independent proper-time calculation verifies the underlying one-loop coefficients. In centred-dipole applications the correction is sub-percent for moderate magnetar fields but reaches several percent for the strongest sources; surface averaging and radial phase accumulation reduce its physical weight as the field decreases away from the stellar surface. The result separates finite-field amplitude and angular corrections and provides controlled analytic benchmarks for precision polarization transport in strongly magnetized environments.