We study linear electromagnetic waves guided by an interface between a dielectric medium and liquid metacrystal (LMC) as well as by a slab of LMC. We derive the dispersion equation for such waves and find their transverse structure for the arbitrarily oriented anisotropy axis of LMC controlled by an external constant electric field. A high degree of anisotropy can be reached in LMC near the resonance of the inclusions (meta-atoms), and it allows propagation of known Dyakonov surface waves below the resonance frequency and also specific surface modes above the resonance frequency (in the hyperbolic dispersion regime) along the LMC–dielectric interface. Both types of surface waves are characterized by sensitivity to the parameters (radiation frequency, orientation of the optical axis, etc.) and have hybrid (mixed ordinary and extraordinary) polarization. In the LMC slab there can exist two families of eigenmodes: low-k and high-k modes, with the latter arising only in the hyperbolic dispersion regime. In the case when the normal to the slab interface is along the resonance cone generatrix in the LMC, a very high density of eigenmodes can take place that results from the inherent property of the hyperbolic medium—infinite density of photonic states.
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