Wide range linear magnetometer based on a sub-microsized K vapor cell

$^{39}$K atoms have the smallest ground state ($^2S_{1/2}$) hyperfine splitting of all the most naturally abundent alkali isotopes and, consequently, the smallest characteristic magnetic field value $B_0 = A_{^2S_{1/2}}/\mu_B \approx 170$ G, where $A_{^2S_{1/2}}$ is the ground state's magnetic dipole interaction constant. In the hyperfine Paschen-Back regime ($B \gg B_0$, where $B$ is the magnitude of the external magnetic field applied on the atoms), only 8 Zeeman transitions are visible in the absorption spectrum of the $D_1$ line of $^{39}$K, while the probabilities of the remaining 16 Zeeman transitions tend to zero. In the case of $^{39}$K, this behavior is reached already at relatively low magnetic field $B > B_0$. For each circular polarization ($\sigma^-,\sigma^+$), 4 spectrally resolved atomic transitions having a sub-Doppler width are recorded using a sub-microsized vapor cell of thickness $L = 120 - 390$ nm. We present a method that allows to measure the magnetic field in the range $0.1 - 10$ kG with micrometer spatial resolution, which is relevant in particular for the determination of magnetic fields with a large gradient (up to 3 G$/\mu$m). The theoretical model describes well the experimental results.