content_image/358421.jpg content_image/358420.jpg content_image/358418.jpg content_image/358419.jpg
The two-dimensional Helmholtz equation 

may be of help in order to find any localized solution of the field h$_{x}$ . However it is preferable an alternative approach. Based on the Bloch theorem [18], we may describe the propagating wavefield h$_{x}$ as a superposition of Bloch modes having the form 

<img src='content_image/358421.jpg'>

<img src='content_image/358420.jpg'>

where k$_{x}$ is the (real) component of a wave vector along the x -axis, and K is the so-called Block wave number, which generally is multivalued for a given k$_{x}$ . Moreover, h$_{K}$ ( y + p ) = h$_{K}$ ( y ) is a periodic function, which is normalized as h$_{K}$ (0) = 1 for convenience, and a$_{K}$ is simply a field amplitude. 

Assuming that sources are sufficiently far from the focal region of the localized beam, the role of evanescent waves [19] is negligible in our analysis and, therefore, they are disregarded setting a$_{K}$ = 0 if Im( K ) ̸ = 0. In this sense we also neglect material losses imposing that the relative dielectric constant of the medium is a real parameter of the problem [Im( ϵ ) = 0]. We point out that this is not strictly true in a realistic problem; plasma-like media with ϵ < 0 are necessarily dispersive since energy density considerations lead to the condition d ( ωϵ ) /dω > 0, and the presence of dispersion in general signifies dissipation of energy [20]. However, it is possible to neglect the absorption in a transparency window where the imaginary part of ϵ is very small in comparison with its real part. 

In the case of the stratified medium shown in Fig. 1, the periodic function is conveniently written as the addition of two independent terms, h$_{K}$ = h + K + h − $_{K}$. Using the column vector 

we may write [21] 

<img src='content_image/358418.jpg'>

<img src='content_image/358419.jpg'>

where α is an integer that refers to a unique slab. The domains R$_{α}$ ≡ ( y$_{α}$$_{−}$$_{1}$, y$_{α}$ ] where y$_{0}$ = 0, y$_{1}$ = w and y$_{α}$$_{+2}$ = y$_{α}$ + p ; for instance, R$_{α}$ is associated with a region where the 

