  Three-dimensional topological insulator films in contact with magnetic layers
exhibit intriguing magneto-optical and magnetoelectric phenomena, but little is
known beyond the linear response regime. We demonstrate that the presence of
two electric fields parallel to the surface of the film induces a perpendicular
AC polarization, whereas a static polarization can be induced by applying a
single circularly polarized electromagnetic wave. These phenomena are
determined by the second-order susceptibility, which also controls the manner
in which the anomalous Hall current is modulated by a perpendicular AC electric
field. If the modulation frequency is low, the Hall effect is topologically
protected and the nonlinear response vanishes.
