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We study the enhancement of the magnetic dipole induced excitationprobability of the hyperfine ground state of Doppler-broadened muonic hydrogen($p \mu^{-}$) by a nanosecond laser pulse in the mid-infrared range withGaussian temporal shape such that the pulse bandwidth is broader than theDoppler width at 10~K. The enhancement is achieved by shrinking thecross-section of the laser pulse and placing the muonic hydrogen medium in amultipass cavity, while preserving the total irradiated target volume. Wenumerically solve a set of Maxwell-Schr$\ddot{\rm o}$dinger equations to obtainthe excitation probability and the total efficiency for various densities ofthe muonic hydrogen atomic medium and at various positions in the multipasscavity. For the typical range of densities of muonic hydrogen atoms at majorproton accelerator facilities such as the J-PARC (density $\sim$$10^5$cm$^{-3}$), the laser propagation effect is insignificant. For suchcases, the total efficiency increases by an order of two for 100 reflectionswith a uniform polarization. If the density exceeds the value of$10^{17}$cm$^{-3}$ as might be in the future advances, the laser propagationeffect has to be taken into account, and the total efficiency decreases withthe number of reflections giving rise to a pulsed polarization of the beam. Ourstudy can serve as a guideline for the development of a polarized muonic beamfor a precise measurement of the ground state hyperfine splitting of muonichydrogen, or for $\mu$SR experiments.

Enhancement of the laser-induced excitation probability of the hyperfine ground state of muonic hydrogen by a multipass cavity setup