Super-oscillating beams of light and matter

Super-oscillating functions are band-limited functions that oscillate
locally faster than their higher Fourier component. These functions were
studied in the past for wide ranging applications, including super-directive
antennas, weak measurements of quantum systems, imaging and microscopy. 
Here I will present new applications of super-oscillations for trapping and
manipulation of nano-particles, for generation of electron beams with
sub-diffraction central spot, for realization of nonlinear frequency
converters with arbitrary narrow spectral and thermal bandwidth, and finally
for structured illumination microscopy.