  Nonlinear magneto-optics is a very sensitive fingerprint of the electronic,
magnetic, and atomic structure of surfaces, interfaces, and thin ferromagnetic
films. Analyzing theoretically the nonlinear magneto-optical Kerr effect for
thin films of Fe(001) and at Fe surfaces we demonstrate exemplarily how various
electronic material properties of ferromagnets , such as the $d$-band width,
the magnetization, the substrate lattice constant, and the film-thickness
dependence can be extracted from the calculated nonlinear Kerr spectra.
Furthermore, we show how the substrate $d$ electrons (Cu(001)) affect the
nonlinear Kerr spectra even without being themselves spin-polarized and without
film-substrate hybridization. We show that the Kerr rotation angle in second
harmonic generation is enhanced by one to two orders of magnitude compared to
the linear Kerr angle and how symmetry can be used to obtain the direction of
magnetization in thin films and at buried interfaces from nonlinear
magneto-optics.
