  We developed the theory which describes the Floquet engineering of surface
electronic modes in bulk mercury telluride (HgTe) by a circularly polarized
electromagnetic field. The analysis shows that the field results in appearance
of the surface states which arise from the mixing of conduction and valence
bands of HgTe. Their branches lie near the center of the Brillouin zone and
have the Dirac dispersion characteristic for topological states. Besides them,
the irradiation induces the gap between the conduction and valence bands of
HgTe. Thus, the irradiation can turn mercury telluride into topological
insulator from gapless semiconductor. It is demonstrated that the optically
induced states differ substantially from the non-topological surface states
existing in HgTe without irradiation. The structure of the found states is
studied both analytically and numerically in the broad range of their
parameters.
