content_image/355151.jpg
Let us consider that β and γ may be tuned at convenience. Thus β ≤ k$_{‖}$ < √ β 2 + γ 2 so that we might apply any spectral band (i.e. ordinates in Fig. 2) arbitrarily. In our model it is reasonable to think of √ β 2 + γ 2 denoting the cut-off frequency ¯ β$_{c}$ = 2 $_{π}$√$_{¯}$ ϵ$_{d}$/λ$_{0}$ associated with the uniform external medium of dielectric constant ¯ ϵ$_{d}$ where the source field is driven. In a great number of examples given below ¯ β$_{c}$ = β$_{max}$ yielding ¯ ϵ$_{d}$ = 2 . 89; in this medium the wavelength ¯ λ$_{d}$ = λ$_{0}$$_{/}$√$_{¯}$ ϵ$_{d}$ = 324 nm and the minimum spot size of the zero-order Bessel beam is (Δ$_{x}$)$_{min}$ = 2 . 253 /γ = 116 nm where the maximum value of γ = ¯ β$_{c}$ at β = 0. In other hand, our model neglects filtering (apodizing) effects and aberrations on the wave fields induced at the boundaries of the system [10]. However, conventional techniques for its compensation might be employed in order to find a good agreement with our results. Otherwise the theory remains valid leaving the appropriate estimation of the spectrum a$_{K}$ . 

## IV. SUBWAVELENGTH TRANSVERSE PATTERNS: NUMERICAL RESULTS 

To illustrate the focus generation along the z -axis, we perform a numerical simulation in the periodic layered medium of p = 450 nm (and ϵ$_{d}$ = 2 . 25) with metallic films of w = 50 nm and ϵ$_{m}$ = − 15 . 0. In Fig. 5 we show the field intensity | h$_{x}$ | 2 for a nondiffracting beam of propagation constant β = 18 . 90 µ m − 1 ( β = β$_{c}$$_{1}$ ), which spatial spectrum is given by Eq. (26) with γ = 4 . 462 µ m − 1 ( γp = 2 . 008). Thus the maximum value of k$_{‖}$ reaches a value coinciding with β$_{max}$ , i.e. β$_{c}$$_{1}$ ≤ k$_{‖}$ < β$_{max}$ corresponding to the first band of allowed frequencies shown in Fig. 2. The isofrequency curve K = K ( k$_{x}$ ) is also depicted in the inset of the figure, where excited frequencies k$_{x}$ are shaded in blue. The intensity is maximum at the origin but the pattern exhibits no radial symmetry. The field distribution along the abscissa, 

<img src='content_image/355151.jpg'>

resembles that of the source field that would propagate in the homogeneous dielectric medium of ¯ ϵ$_{d}$ = 2 . 89. We point out that some discrepancies arise in Eq. (27) if some spatial fre- quencies k$_{x}$ are excited but, because of the presence of a gap, they do not contribute to the resulting propagating wavefield; this case will be treated ahead. From Eq. (27) we derive that the FWHM of the intensity peak along this direction is inversely proportional to γ following Δ$_{x}$ = 2 . 253 /γ . In our case Δ$_{x}$ = 505 nm, which is above the wavelength in the dielectric host λ$_{d}$ = λ$_{0}$/ √ ϵ$_{d}$ (= 367 nm). However, the behavior along the ordinate is signif- 

