  We report on a comprehensive experimental and theoretical study of optical
third harmonic generation (THG) on the exciton-polariton resonances in the
zinc-blende semiconductors GaAs, CdTe, and ZnSe subject to an external magnetic
field, representing a topic that had remained unexplored so far. In these
crystals, crystallographic THG is allowed in the electric-dipole approximation,
so that no strong magnetic-field-induced changes of the THG are expected.
Therefore, it comes as a total surprise that we observe a drastic enhancement
of the THG intensity by a factor of fifty for the $1s$-exciton-polariton in
GaAs in magnetic fields up to 10 T. In contrast, the corresponding enhancement
is moderate for CdTe and almost neglectful for ZnSe. In order to explain this
strong variation, we develop a microscopic theory accounting for the optical
harmonics generation on exciton-polaritons and analyze the THG mechanisms
induced by the magnetic field. The calculations show that the increase of THG
intensity is dominated by the magnetic field enhancement of the exciton
oscillator strength which is particularly strong for GaAs in the studied range
of field strengths. The much weaker increase of THG intensity in CdTe and ZnSe
is explained by the considerably larger exciton binding energies, leading to a
weaker modification of their oscillator strengths by the magnetic field.
