Anisotropic nonlinear response of periodic square or cubic composites

It is well known that the linear electrical response of periodic composites with a microstructure that has sufficiently high symmetry is isotropic. I.e., the macroscopic conductivity or permittivity of such a system, in the absence of an external magnetic field, is a scalar quantity. This is the case for cubic microstructures in three dimensional composites, and for square as well as for triangular and hexagonal or honeycomb microstructures in columnar or thin-film composites. By contrast, it turns out that the macroscopic nonlinear response, in general, is non-isotropic even in such highly symmetric composite structures. We discuss a simple square network model and show that a nonlinear conductivity can be quite anisotropic despite the high (square) symmetry of the system. We also show that a similar model of a metal/dielectric composite with square symmetry can exhibit intrinsic optical bistability that is also strongly anisotropic.