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

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