to a numerical aperture of NA=0.8.

![(a-c) Cross section plot of the electric field amplitude, $|\bf E|$, for a sub-wavelength aperture (diameter=200nm) in a thin layer of silver (thickness=200nm, refractive index $n=0.12-3.7i$ at a wavelength $\lambda=600$nm) illuminated from below. The yellow lines represent the boundary of the structure. (a) Intensity OEi ensuring the largest transmission (transmission enhancement factor 2.1 with respect to the tightest Bessel beam and 1.55 with respect to the Airy disk illumination). (b) Incident intensity OEi without the structure. (c) Tightest Bessel beam illumination. (d-f) Electric field amplitude, $|\bf E|$, in a cross section for a high refractive index ($n=1.8$) microparticle (diameter=800nm) illuminated from below with a wavelength ($\lambda=504$nm). (d) Momentum OEi ensuring the largest momentum transfer (enhancement factor 49.3 with respect to the plane wave and 1.33 with respect the Airy disk illumination). (e) Incident momentum OEi without the structure. (f) Plane wave illumination. []{data-label="fig:oei"}](noei.pdf){width="10.3cm"}

In the first example (Fig. \[fig:oei\] a-c), we use the intensity operator associated to the measure $m^{(0)}$ to determine the largest transmission through a sub-wavelength aperture (diameter=200nm) in a thin layer of silver (thickness=200nm). The incident light field considered is linearly polarised and the transmission is determined across the output surface 