  Nonlinear optics of semiconductors is an important field of fundamental and
applied research, but surprisingly the role of excitons in the coherent
processes leading to harmonics generation has remained essentially unexplored.
Here we report results of a comprehensive experimental and theoretical study of
the three-photon process of optical second harmonic generation (SHG) involving
the exciton resonances of the noncentrosymmetric hexagonal wide-band-gap
semiconductor ZnO in the photon energy range of 3.2-3.5 eV. Resonant
crystallographic SHG is observed for the 1s(A,B), 2s(A,B), 2p(A,B), and 1s(C)
excitons. We show that strong SHG signals at these exciton resonances are
induced by the application of a magnetic field when the incident and the SHG
light wave vectors are along the crystal z-axis where the crystallographic SHG
response vanishes. A microscopic theory of SHG generation through excitons is
developed, which shows that the nonlinear interaction of coherent light with
excitons has to be considered beyond the electric-dipole approximation.
Depending on the particular symmetry of the exciton states SHG can originate
from the electric- and magnetic-field-induced perturbations of the excitons due
to the Stark effect, the spin as well as orbital Zeeman effects, or the
magneto-Stark effect. The importance of each mechanism is analyzed and
discussed by confronting experimental data and theoretical results for the
dependencies of the SHG signals on photon energy, magnetic field, electric
field, crystal temperature, and light polarization. Good agreement is obtained
between experiment and theory proving the validity of our approach to the
complex problem of nonlinear interaction of light with ZnO excitons. This
general approach can be applied also to other semiconductors.
