yellow showing super-oscillating regions.[]{data-label="fig6"}](MaziluFig6.pdf){width="12cm"}

On a final note, we remark that squeezing light below its diffraction limit may be associated with the effect of super-oscillations [@Berry:2006p2578]. This refers specifically to the ability to have a local $k$-vector (gradient of the phase) larger than the spectral bandwidth of the original field. To visualize this effect, in the case of OEi spot size optimized beams, we have calculated the spectral density of the radial wave-vector for the smallest planar spot [@Dennis:12p9892]. As shown in Fig. \[fig6\], this spectral density clearly identifies a spectral bandwidth (white background in Fig. \[fig6\]). Regions of the beam which exhibit locally larger wave-vectors than the ones supported by this spectral band width correspond to super-oscillating regions. The local wave vector is defined as $\partial_r \arg(u(r))$ where $\arg(u)$ defines the phase of the analytical signal $u$. In this particular case, we observe that super-oscillations occur in the dark region of the beam. Additionally, when the ROI is large compared to the Gaussian beam waist $w_0$, there are no super-oscillating regions. These only appear when the beam starts to be squeezed.

Smallest focal spot using Bessel beams {#s:sec34}
--------------------------------------

The paraxial approximation employed above in the case of LG 