High- T$_{c}$ layered cuprate superconductors are important candidates for negative-refractive- index (NRI) metamaterials (see, e.g., [1, 2]). Indeed, being uniaxial strongly anisotropic materials, they provide different signs of the permittivity tensor components along, ε$_{ab}$ , and transverse, ε$_{c}$ , to the layers in a wide frequency range (see, e.g., [3]), providing a possibility of NRI. These metamaterials are attracting considerable attention because they have the po- tential for subwavelength resolution and aberration-free imaging. Layered superconductors are very promising metamaterials because they are relatively straightforward to fabricate (compared to double negative metamaterials) and do not require negative permeability. 

Experiments for the c -axis conductivity in layered superconductors prove the use of a model in which the superconducting CuO$_{2}$ layers are coupled by the intrinsic Josephson effect through the layers [4]. Thus, a Josephson plasma with anisotropic current capability is produced in layered superconductors. Moreover, the physical mechanisms of the cur- rents along and across the layers are fundamentally different. The current along the layers is similar to the current in bulk superconductors, while the current across the layers is Josephson-type. 

The Josephson current along the c -axis couples with the electromagnetic field inside the insulating dielectric layers, forming Josephson plasma waves (JPWs) (see the review [5] and references therein). Thus, the propagation of electromagnetic waves through the layers is favored by the layered structure. The study of these waves is very important because of their Terahertz frequency range, which is still hardly reachable for electronic and optical devices. 

Like in common plasma waves, JPWs propagate with frequencies above some threshold value (the Josephson plasma frequency ω$_{J}$ ). However, in the frequency range below ω$_{J}$ , the presence of the sample boundary can produce surface Josephson plasma waves (SJPWs) [6], which are analog to the surface plasmon polaritons in metals [7, 8]. Such waves can propagate along the vacuum-layered superconductor interface and damp away from it. 

At frequencies ω higher than the Josephson plasma frequency ω$_{J}$ , the normal-to-the-layers components of both the group velocity and the Poynting vector of propagating JPWs, have signs opposite to the sign of the normal component of the wave-vector k$_{s}$ . This corresponds to a NRI. However, the condition ω > ω$_{J}$ is not sufficient for NRI. This NRI effect can be observed at the vacuum-layered superconductor boundary only in a relatively narrow frequency range , ω$_{J}$ < ω < ω vac 1 = ω$_{J}$ [ ε/ ( ε − 1)] 1 / $^{2}$, where ε is the interlayer dielectric 

