  We investigate the third-harmonic generation (THG) of s-wave superconductors
under microwave pulse irradiation. We consider the effect of paramagnetic
impurities on the THG intensity of dirty superconductors. The nonlinear
response function is calculated using the method of the quasiclassical Green
function. It is shown that the amplitude mode is included as the vertex
correction and makes a predominant contribution to the THG intensity. When the
effect of paramagnetic impurities is weak, the THG intensity shows a peak at
the temperature at which the superconducting gap is about the same as the
frequency of the incident pulse, similarly to in experiments. As the effect of
paramagnetic impurities is strengthened, the peak of the THG intensity
disappears. This indicates that time-reversal symmetry breaking due to
paramagnetic impurities eliminates the well-defined amplitude mode. The result
of our calculation shows that the existence of the amplitude mode can be
confirmed through the THG intensity. The result of a semiquantitative
calculation is in good agreement with the experimental result, and it also
shows that the diamagnetic term is negligible.
