  In multiband superconductors, multiple collective modes exist associated with
the multiple order parameters. Oscillations of the amplitude and the relative
phase of the order parameters are called Higgs and Leggett modes, respectively.
Recently, it has been suggested that nonmagnetic impurity scattering would
enhance nonlinear coupling between the Higgs mode and an electromagnetic wave
with a frequency located in the superconducting gap region, while its effect on
the Leggett mode is still unresolved. Here, we theoretically investigated the
nonlinear optical response of multiband Bardeen-Cooper-Schrieffer-type
superconductors in the presence of nonmagnetic impurities with a density matrix
approach extending the Mattis-Bardeen model of linear response. We found that
the drastic enhancement of nonlinear optical response due to the nonmagnetic
impurity scattering occurs only for the Higgs modes and not for the Leggett
mode. As a result, both the light-induced dynamics of the superconducting gaps
and the resulting third-harmonic generation are dominated by the Higgs modes.
We also examined the role of quasiparticle excitations to find that they give
the subdominant contribution to the third-harmonic generation.
