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can be expanded if one uses an insulator with large enough permittivity ε$_{ext}$ . Only for insulators with very high permittivity ε$_{ext}$ > ε , the negative refraction can occur in the whole frequency interval ω$_{J}$ < ω < ω$_{2}$ . The frequency range for the existence of the bulk JPWs in superconductors with capacitive interlayer coupling was derived in [10]. If the constant of this coupling tends to zero, the frequency range obtained in [10] coincides with Eq. (5). Below we consider waves in the insulator-layered superconductor system with ε$_{ext}$ < ε for frequencies ω > ω$_{1}$ . 

Surface Josephson plasma waves above ω$_{1}$ .— When ω$_{J}$ < ω < ω$_{2}$ and the factor in Eq. (3) is positive, the z -component κ s of the wave-vector k$_{s}$ becomes imaginary. This means that the wave damps into the layered superconductor. On the other hand, for q > ω √ ε$_{ext}$/c the wave damps also into the insulator above the layered superconductor. These are the characteristic features of the surface waves discussed in this section. 

Consider an interface (the xy -plane) separating an insulator ( z > 0 in Fig. 1) and a layered superconductor ( z ≤ 0). We now consider a linear surface transverse-magnetic monochromatic wave propagating along the x -axis (i.e., proportional to exp[ i ( qx − ωt )]) and decaying into both, the insulator and layered superconductor, away from the interface z = 0. 

Performing the standard procedure for searching surface waves (i.e., solving the Maxwell equations for the insulator and layered superconductor with proper boundary conditions at the interface between them) we obtain the dispersion relation for the surface Josephson plasma waves: 

or, neglecting dissipation, 

Here the dimensionless wave-vector is defined as κ = cq/ω$_{J}$ ε 1 / 2 $_{ext}$. Equation (7) describes two branches of the dispersion curve for the SJPWs (see Fig. 2). The first branch exists in the low frequency range, 0 < ω < ω$_{J}$ , and it was studied before in [6]. The second (predicted here) branch starts from the light line ω = cq/ε 1 / 2 ext (or Ω = κ ) at ω = ω$_{1}$ (point A in Fig. 2), then follows this line, deviates from it at ω ∼ ω$_{2}$ = γω$_{J}$ , and stops at the point where q = γω$_{J}$ ε 1 / $^{2}$/c, ω = ω$_{2}$ (point B in Fig. 2). 

<img src='content_image/349137.jpg'>

<img src='content_image/349136.jpg'>

Thus, there exists a frequency gap, ω$_{J}$ < ω < ω$_{1}$ , in the spectrum of the SJPWs. We emphasize that the NRI should only exist within this gap. When the permittivity ε$_{ext}$ of the 

