In this simple analysis we have also assumed that a$_{K}$ has a well behavior and it does not present discontinuities. In the far field zone | h$_{x}$ | 2 decreases inversely proportional to the radial coordinate r , excepting those directions θ where stationary points satisfy K ′′ s = 0. In these cases the dispersion curve is flat leading to a significantly slow attenuation of the radiated power, at least much slower than r − $^{1}$. In Fig. 6(a) we plot the solutions k$_{xs}$ of the equation K ′′ s = 0 at different values of β . Considering that the dispersion equation (18) may be written as cos( Kp ) = f ( k$_{x}$ ), the solutions of the equation K ′′ s = 0 are also the frequencies k$_{xs}$ satisfying Let us analyze the intensity patterns of diffraction-free beams under the presence of bandgaps [Im( K ) = 0] within the source spectral window | k$_{x}$ | < γ . In Fig. 7 we plot | h$_{x}$ | 2 of wavefields with propagation constant β = β$_{c}$$_{3}$ corresponding to the lower limit of the second band. In Fig. 7(a) γp = 2 . 552 that yields ( k$_{‖}$ ) max = β$_{c}$$_{2}$ . This means that the source excites entirely the second band of allowed spatial frequencies with no gaps; therefore Eq. (27) is valid and Δ$_{x}$ = 2 . 253 /γ (= 397 nm). Along the y -axis, the narrow peak around the origin has a FWHM Δ$_{y}$ = 62 . 52 nm and again is accompanied by a high sidelobe at the other side of the metallic film. The formation of this secondary focus is again the constructive 13 As shown in Fig. 3(a), the isofrequency curves for β$_{c}$$_{1}$ < β < β$_{max}$ mimic ellipses that lack of flat sections so that K ′′ s = 0 has no real solutions. At β = β$_{c}$$_{1}$ , the isofrequency curve represented also in the inset of Fig. 5 is flat at the origin of the plane k$_{x}$K taking a shape in X. In this case ( k$_{xs}$, K$_{s}$ ) = (0 , 0) for which K ′ s = ± 0 . 604 and φ$_{s}$ = ± 31 deg. See Fig. 6(b) showing the estimated (positive) values of φ$_{s}$ also for other propagation constants, β . Our analysis conclude that the self-guiding phenomenon is expected at angles θ = ± 59 deg (and θ = ± 121 deg), in agreement with the transverse pattern shown in Fig. 5. When β$_{c}$$_{3}$ < β < β$_{c}$$_{1}$ , solving Eq. (30) leads to increasing values of | k$_{xs}$ | and therefore increasing values of | φ$_{s}$ | , as shown in Figs. 6(a) and (b) respectively. Interestingly, k$_{xs}$ = 0 becomes again a solution if β = β$_{c}$$_{3}$ (here giving K$_{s}$ = ± π/p ) for which φ$_{s}$ = ± 42 deg; also k$_{xs}$ = ± 3 . 32 /p gives K ′′ s = 0 where φ$_{s}$ = ± 73 deg. Moreover, a minimum of four values of k$_{xs}$ may be found for β < β$_{c}$$_{3}$ in which the zero-diffraction condition K ′′ s = 0 holds. We may conclude the general rule that a larger number of solutions comes out when (positive) β decreases reaching new bandgaps for the frequency k$_{‖}$ as shown in Fig. 2. f ′′ s ( 1 − f 2 s ) + f ′ 2 s f$_{s}$ = 0 . (30)