  We investigate on the properties of the recently introduced time-diffracting
(TD) beams in free space. They are shown to be paraxial and quasi-monochromatic
realizations of localized waves, spatiotemporal localized waves travelling
undistorted at arbitrary speeds. The paraxial and quasi-monochromatic regime is
shown to be necessary to observe what can properly be named diffraction in
time. TD beams of finite energy travelling at quasi-luminal velocities are seen
to form substantially longer foci or needles of light than the so-called
abruptly focusing and defocusing needle of light, or limiting TD beam of
infinite speed. Exploring the properties of TD beams under Lorentz
transformations and transformation by paraxial optical systems, we realize that
the relativistically moving nonlinear polarization of material media induced by
a strongly localized fundamental pump wave generates a TD beam at its second
harmonic, whose diffraction-free behavior as a needle of light in free space
can be optimized with a standard $4f$-imager system.
