Loss-compensated and active hyperbolic metamaterials.

We have studied the dispersion relations of multilayers of silver and a dye-doped dielectric using four methods: standard effective-medium theory (EMT), nonlocal-effect-corrected EMT, nonlinear equations based on the eigenmode method, and a spatial harmonic analysis method. We compare the validity of these methods and show that metallic losses can be greatly compensated by saturated gain. Two realizable applications are also proposed. Loss-compensated metal-dielectric multilayers that have hyperbolic dispersion relationships are beneficial for numerous applications such as subwavelength imaging and quantum optics.