Time-diffracting beams: On their nature, diffraction-free propagation as needles of light, and nonlinear generation

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.