Precision spectroscopy of the $A^2\Pi$ $\leftarrow$ $X^2\Sigma^+$ transition in BaF

High-resolution spectroscopy on the $A^2\Pi$ - $X^2\Sigma^+$ electronic system of $^{138}$Ba$^{19}$F is performed using a cold molecular beam produced by a buffer gas source. The hyperfine structure in both $X^2\Sigma^+$ ground and $A^2\Pi$ excited states is fully resolved and absolute transition frequencies of individual components are measured at the sub-MHz level making use of frequency-comb laser calibration. Sets of molecular constants for the $X^2\Sigma^+$($v=0,1$) and $A^2\Pi$($v=0,1$) levels are determined, with improved accuracy for the $T_{v',v''}$ band origins and spin-orbit interaction constants for the $A^2\Pi$ excited states, that represent the presently measured highly accurate transitions for low-$J$ states as well as previously determined transition frequencies in Fourier-transform emission studies for rotational levels as high as $J \geq 100$. The extracted molecular constants reproduce the measured transition frequencies at the experimental absolute accuracy of 1 MHz. The work is of relevance for future laser cooling schemes, and is performed in the context of a measurement of the electron dipole moment for which BaF is a target system.