Dependence of the Nonlinear-Optical Response of Materials on their Linear $\epsilon$ and $\mu$

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.