  The diffraction limited resolution of light focused by a lens was derived in
1873 by Ernst Abbe. Later in 1952, a method to reach sub-diffraction light
spots was proposed by modulating the wavefront of the focused beam. In a
related development, super-oscillating functions, i.e. band limited functions
that locally oscillate faster than their highest Fourier component, were
introduced and experimentally applied for super-resolution microscopy. Up till
now, only simple Gaussian-like sub-diffraction spots were used. Here we show
that the amplitude and phase profile of these sub-diffraction spots can be
arbitrarily controlled. In particular we utilize Hermite-Gauss, Laguerre-Gauss
and Airy functions to structure super-oscillating beams with sub-diffraction
lobes. These structured beams are then used for high resolution trapping and
manipulation of nanometer-sized particles. The trapping potential provides
unprecedented localization accuracy and stiffness, significantly exceeding
those provided by standard diffraction limited beams.
