  Terahertz nonlinear optics is a viable method to interrogate collective
phenomena in quantum materials spanning ferroelectrics, charge-density waves,
and superconductivity. In superconductors this includes the Higgs amplitude and
Josephson phase modes. We have investigated the nonlinear c-axis response of
optimally doped La$_{1.85}$Sr$_{0.15}$CuO$_4$ using high-field THz time domain
spectroscopy (THz-TDS) at field strengths up to $\sim$80 kV/cm. With increasing
field, we observe a distinct red-shift of the Josephson plasma edge and
enhanced reflectivity (above the plasma edge) arising from third harmonic
generation. The non-monotonic temperature dependent response is consistent with
nonlinear drive of the Josephson Plasma Mode (JPM) as verified with comparison
to theoretical expectations. Our results add to the understanding that, using
THz light, the JPM (in addition to the Higgs mode) provides a route to
interrogate and control superconducting properties.
