Bulk materials possessing a relative electric permittivity $varepsilon$ close to zero exhibit giant Kerr nonlinearities. However, harnessing this response in guided-wave geometries is not straightforward, due to the extreme and counter-intuitive properties of epsilon-near-zero materials. Here we investigate, through rigorous calculations of the Kerr nonlinear coefficient, how the remarkable nonlinear properties of such materials can be exploited in several different types of structures, including bulk films, plasmonic nanowires, and metal nanoapertures. We find the largest Kerr nonlinear response when both the modal area and the group velocity are simultaneously minimized, corresponding to omnidirectional field enhancement. The physical insights developed will be key for understanding and engineering nonlinear nanophotonic systems with extreme nonlinearities and point to new design paradigms.
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