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particle conductivities σ$_{ab}$ (along the layers) and σ$_{c}$ (across the layers), respectively; ω$_{J}$ = (8 πeDj$_{c}$/ ℏ ε ) 1 / 2 is the Josephson plasma frequency. The latter is determined by the critical Josephson current density j$_{c}$ , the interlayer dielectric constant ε , and the spatial period of the layered structure D = s + d ≈ d . 

Analyzing the relations for ε$_{c}$ (Ω) and ε$_{ab}$ (Ω), we conclude that their real parts have different signs in a wide frequency range: ω$_{J}$ < ω < ω$_{2}$ = ω$_{J}$ γ (or 1 < Ω < Ω$_{2}$ = γ ≫ 1). In this frequency range, the z -components of the group velocity and the Poynting vector of the bulk JPWs are directed opposite to the z -component of the wave-vector k$_{s}$ , and this, at first sight, corresponds to having a NRI. However, a more careful analysis shows that this can only be observed for a much narrower frequency interval. To verify this, one can consider the well-known dispersion relation for the normal component κ s of the JPW wave-vector, 

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that follows directly from Maxwell’s equations. Obviously, the JPWs can propagate only when Re( κ 2 $_{s}$) > 0, where “Re” stands for the real part. Neglecting dissipation, the permit- tivity ε$_{ab}$ is negative for the frequency region ω$_{J}$ < ω < ω$_{2}$ considered here. Consequently, JPWs can propagate if the factor [ k 2 − q $^{2}$/ε$_{c}$ (Ω)] in Eq. (3) is negative. Using Eq. (2) for ε$_{c}$ (Ω), one can conclude that this factor is negative only when 1 < Ω 2 < 1 + q $^{2}$λ2 $_{c}$. For a wave incident, at an angle θ , from the insulator with dielectric constant ε$_{ext}$ onto the layered superconductor, we have q = ( ωε 1 / 2 $_{ext}$/c ) sin θ . Thus, a NRI can be observed for waves with incident angles higher than the critical value θ$_{crit}$ defined by the equation, 

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It is important to note that, due to the negative sign of the permittivity ε$_{ab}$ in Eq. (3), the incident wave penetrates the superconductor for θ > θ$_{crit}$ , and totally reflects from it for θ < θ$_{crit}$ , contrary to the standard case of waves incident onto the interface dividing two usual right-handed media. 

For ε$_{ext}$ < ε , Eqs. (2), (4), and the inequality sin( θ$_{crit}$ ) ≤ 1, provide the conditions, 

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Thus, the NRI for a layered superconductor bounded by an insulator with ε$_{ext}$ < ε can only be observed in the frequency range ω$_{J}$ < ω < ω$_{1}$ = ω$_{J}$ Ω$_{1}$. This frequency window 

