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Periodicity in the medium renders P$_{α}$$_{+2}$ = P$_{α}$ and D$_{α}$$_{+2}$ = D$_{α}$ which yield further sim- plifications in the determination of M$_{α}$ . We leave the reader the long but straightforward evaluation of the elements of the matrix M$_{α}$ . In general M$_{α}$$_{+1}$ ̸ = M$_{α}$ ; however M$_{α}$$_{+2}$ = M$_{α}$ and, more importantly, the matrix M$_{α}$ is unimodular [det( M$_{α}$ ) = 1] and its trace is inde- pendent of α . 

In agreement with Eq. (16), v$_{α}$ is an eigenvector of the matrix M$_{α}$ providing the eigen- value exp ( iKp ). Consequently the determinant of the matrix M$_{α}$ − exp ( iKp ) I vanishes, where I is the identity matrix. The resultant equation may be cast simply as cos ( Kp ) = ( A$_{α}$ + D$_{α}$ ) / 2, where the unimodularity property of the matrix M$_{α}$ , A$_{α}$D$_{α}$ − B$_{α}$C$_{α}$ = 1, should be employed. Thus we have derived the dispersion equation, which explicitly reads 

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As expected, the dispersion equation (18) is independent of the integer α . Our interest is focused in real values of K , for which the right-hand side of Eq. (18) should be comprised between − 1 and 1. 

In Fig. 2 we plot the real-valued K solutions of equation (18) for a multilayer medium composed of thin metallic films of width w = 50 nm, separated a distance p = 450 nm, and embedded in a dielectric medium of ϵ$_{d}$ = 2 . 25. At a frequency ω = 3 . 427 fs − 1 (wavelength λ$_{0}$ = 550 nm in the vacuum) our plasma-like medium has a relative dielectric constant ϵ$_{m}$ = − 15 . 0. A large bandgap in the interval k$_{‖}$p ∈ (0 , 4 . 571) is followed by some two others in (5 . 126 , 7 . 899) and (8 . 301 , 8 . 503), together with the evanescent-wave regime at k$_{‖}$p > 8 . 737. As a consequence, real values of β are limited by β$_{max}$ = 8 . 737 /p in our example ( β$_{max}$ = 19 . 42 µ m − $^{1}$). This boundary value is reached at K = ± π/p rather than at a zero value [22]. Let us point out that, if the superposition shown in Eq. (5) is such that k$_{‖}$ is higher than the cut-off frequency 

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occurring if k$_{‖}$p > 7 . 711, the wavefields are all of evanescent nature in the metal and in the dielectric; here β$_{c}$ = 17 . 14 µ m − $^{1}$. The cut-off frequency is plotted with a horizontal red line in Fig. 2. Interestingly, these evanescent waves may arise at values of k$_{‖}$p such that Eq. (18) gives a Bloch wave number with Im( K ) = 0. Thus, the evanescent waves are resonantly coupled leading to propagating Bloch-type constituents of the diffraction-free 

