  Novel materials, with enhanced light-matter interaction capabilities, play an
essential role in achieving the lofty goals of nonlinear optics. Recently,
Epsilon-Near-Zero (ENZ) media have emerged as a promising candidate to enable
the enhancement of several nonlinear processes including refractive index
modulation and harmonic generation. Here, we analyze the optical nonlinearity
of ENZ media to clarify the commonalities with other nonlinear media and its
unique properties. We focus on transparent conducting oxides (TCOs) as the
family of ENZ media with near zero permittivity in the near-infrared (telecom)
band. We investigate the instantaneous and delayed nonlinearities. By
identifying their common origin from the band nonparabolicity, we show that
their relative strength is entirely determined by a ratio of the energy and
momentum relaxation (or dephasing) times. Using this framework, we compare ENZ
materials against the many promising nonlinear media that have been
investigated in literature and show that while ENZ materials do not radically
outpace the strength of traditional materials in either the fast or slow
nonlinearity, they pack key advantages such as an ideal response time,
intrinsic slow light enhancement, and broadband nature in a compact platform
making them a valuable tool for ultrafast photonics applications for decades to
come.
