Constraining magnetic fields through plasma lensing: application to the Black Widow pulsar

In regions with strongly varying electron density, radio emission can be magnified significantly by plasma lensing. In the presence of magnetic fields, magnification in time and frequency will be different for two circular polarizations. We show how these effects can be used to measure or constrain the magnetic field parallel to the line of sight, $B_\parallel$, as well as its spatial structure, $\sigma_{B_\parallel}$, in the lensing region. In addition, we discuss how generalized Faraday rotation can constrain the strength of the perpendicular field, $B_\perp$. We attempt to make such measurements for the Black Widow pulsar, PSR~B1957+20, in which plasma lensing was recently discovered. For this system, pressure equilibrium suggests $B\gtrsim 20\,$G at the interface between the pulsar and companion winds, where the radio eclipse starts and ends, and where most lensing occurs. We find no evidence for large-scale magnetic fields, with, on average, $B_\parallel=0.02\pm0.09\,$G over the egress lensing region. From individual lensing events, we strongly constrain small scale magnetic structure to $\sigma_B<10\,$mG, thus excluding scenarios with a strong but rapidly varying field. Finally, from the lack of reduction of average circular polarization in the same region, we rule out a strong, quasi-transverse field. We cannot identify any plausible scenario in which a large magnetic field in this system is concealed, leaving the nature of the interface between the pulsar and companion winds an enigma. Our method can be applied to other sources showing plasma lensing, including other eclipsing pulsars and fast radio bursts, to study the local properties of the magnetic field.