  We investigate, theoretically and numerically, the dependence of a material's
nonlinear-optical response on the linear relative electric permittivity
$\epsilon$ and magnetic permeability $\mu$. The conversion efficiency of
low-order harmonic-generation processes, as well as the increase rate of
Kerr-effect nonlinear phase shift and nonlinear losses from two-photon
absorption (TPA), are seen to increase with decreasing $\epsilon$ and/or
increasing $\mu$. We also discuss the rationale and physical insights behind
this nonlinear response, particularly its enhancement in $\epsilon$-near-zero
(ENZ) media. This behavior is consistent with the experimental observation of
intriguingly high effective nonlinear refractive index in degenerate
semiconductors such as indium tin oxide [\textit{Alam et al., Science 352
(795), 2016}] (where the nonlinearity is attributed to a modification of the
energy distribution of conduction-band electrons due to laser-induced electron
heating) and aluminum zinc oxide [\textit{Caspani et al., Phys. Rev. Lett. 116
(233901), 2016}] at frequencies with vanishing real part of the linear
permittivity. Such strong nonlinear response can pave the way for a new
paradigm in nonlinear optics with much higher conversion efficiencies and
therefore better miniaturization capabilities and power requirements for
next-generation integrated nanophotonics.
