A Constitutive Description of Nonlinear Metamaterials Through Electric, Magnetic, and Magnetoelectric Nonlinearities

Nonlinear metamaterials provide a host of interesting phenomena which, like for their linear counterpart, can be described using homogenized, effective properties. Following the convention used in nonlinear optics, the response of nonlinear metamaterials can be expressed as a power series of the incident fields. However, contrarily to most materials used in nonlinear optics that only possess an electric nonlinear response, nonlinear metamaterials often show electric, magnetic, and magnetoelectric nonlinear responses within a single unit cell. In this chapter, we present two complementary approaches to determine all the effective nonlinear susceptibilities of nonlinear metamaterials. First we present a coupled-mode theory that provides insight into the origin of the various nonlinear susceptibilities that arise in nonlinear metamaterials according to the symmetry of the unit cell. This approach also leads to a description of the effect of the finite size of the unit cells, often called spatial dispersion. Second, we present a retrieval approach based on transfer matrices that can be used to determine the effective nonlinear susceptibilities from either simulated or experimental results. We finally demonstrate how to use this approach by applying it to the case of dual-gap varactor-loaded split ring resonators.