The hyperfine Paschen-Back Faraday effect

We investigate experimentally and theoretically the Faraday effect in an atomic medium in the hyperfine Paschen-Back regime, where the Zeeman interaction is larger than the hyperfine splitting. We use a small permanent magnet and a micro-fabricated vapour cell, giving magnetic fields of the order of a Tesla. We show that for low absorption and small rotation angles, the refractive index is well approximated by the Faraday rotation signal, giving a simple way to measure the atomic refractive index. Fitting to the atomic spectra, we achieve magnetic field sensitivity at the $10^{-4}$ level. Finally we note that the Faraday signal shows zero crossings which can be used as temperature insensitive error signals for laser frequency stabilisation at large detuning. The theoretical sensitivity for $^{87}$Rb is found to be $\sim 40$ kHz/$^\circ$C.