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