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Optical beams exhibiting a long depth of focus and a minimum spot size can be obtained with the tight focusing of a narrow annulus of radially polarized light, leading to a needle of longitudinally polarized light. Such beams are of increasing interest for their applications, for example in optical data storage, particle acceleration, and biomedical imaging. Hence one needs to characterize the needles of longitudinally polarized light obtained with different focusing optics and incident beams. In this paper, we present analytical expressions for the electric field of such a nearly nondiffracting, subwavelength beam obtained with a parabolic mirror or an aplanatic lens. Based on these results, we give expressions of the transverse and longitudinal full widths at half maximum of the focal lines as a function of the width of the incident annular beam and we compare the performances of the two focusing systems. Then, we propose a practical solution to produce a needle of longitudinally polarized light with a tunable axial extent and a transverse width reaching the theoretical limit of 0.36λ.

In the past few years, there has been a growing interest for direct-field electron acceleration with ultra-intense and ultrafast radially polarized laser beams. This particular acceleration scheme offers the possibility of producing highly collimated mono-energetic relativistic attosecond electron pulses from an initial cloud of free electrons that could be produced by ionizing a nanoparticle. In this paper, we describe how electron energy scales with laser power and we explain how the beam waist size and the pulse duration can be optimized for maximal acceleration. The main conclusion of our work is that an electron can effectively reach the high-intensity optical cycles of this particular beam and be optimally accelerated without the necessity of being released by photoionization near the pulse peak.

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