  High index dielectric nanoantennas excited at Mie-type resonances have
exhibited enormous enhancement of optical nonlinearity. Such nanostructures
have been actively studied by researchers in the last years. The present work
provides the first numerical analysis study of the optical Kerr effect of
nanocomposites consisting of high refractive index (GaP) spheres at the
wavelength of 532~nm. This is done by means of three-dimensional
finite-difference time-domain (FDTD) simulations. The effective nonlinear
refractive index of $0.8$~$\mu$m thick nanocomposites and metasurfaces is
evaluated. It is shown that the optical Kerr nonlinearity of the nanocomposites
rises by orders in proximity to Mie resonances and may exceed the second-order
refractive index of the bulk material. It is revealed that the sign of the
effective optical Kerr coefficient is inverted near the Mie resonances. This
effect may be of interest in developing nonlinear optical metadevices.
