  We report comparative experimental and theoretical studies of second and
third harmonic generation from a 20nm-thick indium tin oxide layer in proximity
of the epsilon-near-zero condition. Using a tunable OPA laser we record both
spectral and angular dependence of the generated harmonic signals close to this
particular point. In addition to the enhancement of the second harmonic
efficiency close to the epsilon-near-zero wavelength, at oblique incidence
third harmonic generation displays unusual behavior, predicted but not observed
before. We implement a comprehensive, first-principles hydrodynamic approach
able to simulate our experimental conditions. The model is unique, flexible,
and able to capture all major physical mechanisms that drive the electrodynamic
behavior of conductive oxide layers: nonlocal effects, which blueshift the
epsilon-near-zero resonance by tens of nanometers; plasma frequency redshift
due to variations of the effective mass of hot carriers; charge density
distribution inside the layer, which determines nonlinear surface and magnetic
interactions; and the nonlinearity of the background medium triggered by bound
electrons. We show that by taking these contributions into account our
theoretical predictions are in very good qualitative and quantitative agreement
with our experimental results. We show that by taking these contributions into
account our theoretical predictions are in very good qualitative and
quantitative agreement with our experimental results. We expect that our
results can be extended to other geometries where ENZ nonlinearity plays an
important role.
