content_image/372230.jpg content_image/372231.jpg
wavefield h$_{x}$ ( x, y ). Nondiffracting beams with a propagation constant β ∈ ( β$_{c}$, β$_{max}$ ) are wave fields of this kind. 

In Fig. 3 we map different contours of isofrequency β in the k$_{x}$K plane based on the graphical representation of the dispersion equation given in Fig. 2. Thus we consider real values of K again. The first band is represented in Fig. 3(a) providing a surface of maxima at k$_{x}$ = 0 and K = ± π/p reaching β$_{max}$ . Simple closed curves around these points are found for decreasing values of the propagation constant provided β$_{c}$$_{1}$ ≤ β < β$_{max}$ , where β$_{c}$$_{1}$p = 8 . 503 coincides with the upper boundary of the first bandgap. At lower β , spatial frequencies around k$_{x}$ = 0 cannot excite propagating Bloch modes leading to open (isofrequency) curves. This gap is removed in the second sheet of the dispersion equation, which arises for β ≤ β$_{c}$$_{2}$ and it is shown in Fig. 3(b); obviously β$_{c}$$_{2}$p = 8 . 301 corresponds to the upper limit of the second band of allowed spatial frequencies k$_{‖}$ . In this case, the closed curves are centered at the origin of the plane k$_{x}$K (and at multiples of 2 π/p along the K -axis). Similarly to Fig. 3(a), the isofrequency curves are closed when β takes values belonging to the (second) band of allowed frequencies, here β$_{c}$$_{3}$ ≤ β < β$_{c}$$_{2}$ where β$_{c}$$_{3}$p = 7 . 899. Out of this interval, the isofrequency curves are open again. 

## B. Bloch components of the wavefield 

According to Eq. (15), once the amplitudes v$_{0}$ and v$_{1}$ are given for all Bloch wavenumbers K , the periodic function h$_{K}$ and thus the wavefield h$_{x}$ are estimated unambiguously. In fact we have shown that v$_{1}$ = P − 1 1 D − 1 1 D$_{0}$v$_{0}$ , therefore the degrees of freedom are reduced to the amplitudes v$_{0}$ . With the normalization h$_{K}$ (0) = 1 we additionally impose that the summation of the elements of v$_{0}$ is left to unity. As a result, the amplitudes associated with the Bloch modes in the dielectric slab at R$_{0}$ ≡ ( − p + w, 0] yield 

where A$_{0}$ and B$_{0}$ are matrix elements of M$_{0}$ . 

We point out that the denominator in Eq. (20) vanishes if 

<img src='content_image/372230.jpg'>

<img src='content_image/372231.jpg'>

