We address the existence and properties of hybrid surface waves forming at interfaces between left-handed materials and dielectric birefringent media. The existence conditions of such waves are found to be highly relaxed in comparison to Dyakonov waves existing in right-handed media. We show that left-handed materials cause the coexistence of several surface solutions, which feature an enhanced degree of localization. Remarkably, we find that the hybrid surface modes appear for large areas in the parameter space, a key property in view of their experimental observation.

Peer Reviewed