The role of losses in determining hyperbolic material figures of merit

Uniaxial materials have achieved new prominence in photonics because they can have hyperbolic spectral regions with metallic (ε<0) and dielectric (ε>0) permittivities along different crystal axes. In the lossless case, this results in an open hyperboloid dispersion relation, allowing materials to support highly confined modes with extremely large wavevectors. However, even small losses change the character of the hyperbolic dispersion from open hyperboloids to closed surfaces with finite maximum k, significantly limiting the extent to which highly-confined modes can be achieved. Here, we derive a simple analytic formula for the dispersion relation in the presence of loss and show that for some typical materials the maximum wavevector in hyperbolic materials is roughly ten times the free-space. The scaling of the maximum wavevector is derived, and it is shown that there is a universal scaling relation between the propagation length and the wavelength, which implies that the shortest wavelengths in any hyperbolic material are strongly attenuated.