Analog of gravitational force in hyperbolic metamaterials

Subwavelength confinement of light in nonlinear hyperbolic metamaterials due to formation of spatial solitons has attracted much recent attention because of its seemingly counterintuitive behavior. In order to achieve self-focusing in a hyperbolic wire medium, a nonlinear self-defocusing Kerr medium must be used as a dielectric host. Here we demonstrate that this behavior finds a natural explanation in terms of the analog of gravity. A wave equation describing the propagation of extraordinary light inside hyperbolic metamaterials exhibits (2+1)-dimensional Lorentz symmetry. The role of time in the corresponding effective three-dimensional Minkowski space-time is played by the spatial coordinate aligned with the optical axis of the metamaterial. Nonlinear optical Kerr effect ``bends'' this space-time resulting in effective gravitational force between extraordinary photons. In order for the effective gravitational constant to be positive, a negative self-defocusing Kerr medium must be used as a host. If gravitational self-interaction is strong enough, the spatial soliton may collapse into a black hole analog.