  The twisted light modes used in modern atomic physics experiments can be
contaminated by small admixtures of plane wave radiation. Although these
admixtures hardly reveal themselves in the beam intensity profile, they may
seriously affect the outcome of high precision spectroscopy measurements. In
the present study we propose a method for diagnosing such a plane wave
contamination, which is based on the analysis of the magnetic sublevel
population of atoms or ions interacting with the "twisted + plane wave"
radiation. In order to theoretically investigate the sublevel populations, we
solve the Liouville-von Neumann equation for the time evolution of atomic
density matrix. The proposed method is illustrated for the electric dipole $5s
\, {}^{2}\mathrm{S}_{1/2} \, - \, 5p \, {}^{2}\mathrm{P}_{3/2}$ transition in
Rb induced by (linearly, radially, or azimuthally polarized) vortex light with
just a small contamination. We find that even tiny admixtures of plane wave
radiation can lead to remarkable variations in the populations of the
ground-state magnetic sublevels. This opens up new opportunities for
diagnostics of twisted light in atomic spectroscopy experiments.
