Spectroscopy of the ${\mathbf{X}}^{\mathbf{2}}{\mathbf{\Sigma }}^{+}(\mathbf{v}=\mathbf{2})\to {\mathbf{A}}^{\mathbf{2}}{\mathbf{\Pi }}_{\mathbf{1}\mathbf{/}\mathbf{2}}(\mathbf{v}=\mathbf{1})$ Transition in MgF: Hyperfine Structures and Spectroscopic Constants

We report spectroscopic results of the $X^2{\mathrm{\Sigma }}^{+}(v=2)\to A^2{\mathrm{\Pi }}_{1\mathrm{/}2}(v=1)$ transition in magnesium monofluoride (MgF). Using Doppler-free Laser-Induced Fluorescence (LIF) spectroscopy on the $X^2{\mathrm{\Sigma }}^{+}(v=2)\to A^2{\mathrm{\Pi }}_{1\mathrm{/}2}(v=1)$ transition, we resolved 47 hyperfine components distributed over 11 transition lines in X and A states. An effective Hamiltonian – comprising contributions from vibrational, rotational, $\mathrm{\Lambda }$-doubling, and hyperfine interactions – was presented to model the energy structure of the $A^2{\mathrm{\Pi }}_{1\mathrm{/}2}(v=1)$ state. The spectroscopic parameters, including the rotational constant, the $\mathrm{\Lambda }$-doubling parameter, and the hyperfine interaction constants, were extracted using a least-square fitting and Markov Chain Monte Carlo (MCMC) procedure. Our study reveals that the spectroscopic constants show subtle changes compared to the $A^2{\mathrm{\Pi }}_{1\mathrm{/}2}(v=0)$ state. These results provide critical spectroscopic benchmarks for optimizing optical cycling schemes in MgF, thereby advancing optical cycling efficiency in the magneto-optical trapping of MgF.