Diffraction-free optical beams propagate freely without change in shape and scale. Monochromatic beams that avoid diffractive spreading require two-dimensional transverse profiles, and there are no corresponding solutions for profiles restricted to one transverse dimension. Here, we demonstrate that the temporal degree of freedom can be exploited to efficiently synthesize onedimensional pulsed optical sheets that propagate self-similarly in free space. By introducing programmable conical (hyperbolic, parabolic, or elliptical) spectral correlations between the beam's spatio-temporal degrees of freedom, a continuum of families of axially invariant pulsed localized beams is generated. The spectral loci of such beams are the reduced-dimensionality trajectories at the intersection of the light-cone with spatio-temporal spectral planes. Far from being exceptional, self-similar axial propagation is a generic feature of fields whose spatial and temporal degrees of freedom are tightly correlated. These one-dimensional 'space-time' beams can be useful in optical sheet microscopy, nonlinear spectroscopy, and non-contact measurements.Diffractive spreading is a fundamental feature of freely propagating optical beams that is readily observed in everyday life. Diffraction sets limits on the optical resolution in microscopy, lithography, and photography; on the maximum distance for free-space optical communications and standoff detection; and on the precision of spectral analysis 1,2 . As a result, there has been a long-standing fascination with socalled 'diffraction-free' beams whose change in shape and scale during propagation is curbed when compared to other beams of comparable transverse size 3 . Monochromatic diffraction-free beams have sculpted two-dimensional (2D) transverse spatial profiles that confirm to Bessel 4 , Mathieu 5 , or Weber 6 functions, among other examples (see Refs. [7,8] for recent taxonomies). The situation is altogether different for monochromatic beams with one transverse dimension -or optical sheets -where there are only two possible diffraction-free solutions: the cosine wave that lacks spatial localization and the Airy beam that maintains a localized intensity profile but whose center-of-mass undergoes a transverse shift with propagation 9,10 . Indeed, a conclusive argument by Michael Berry 11 identified the Airy beam as the only such monochromatic one-dimensional (1D) profile. Optical nonlinearities can be exploited to thwart diffractive spreading 12,13 , and in some cases chromatic dispersion is required in the medium to restrain the diffraction of pulsed beams [14][15][16] . However, most applications require free-space diffraction-free beams.Here, we exploit the temporal degree of freedom (DoF) in conjunction with the spatial DoF to realize a variety of diffraction-free pulsed solutions having arbitrary 1D transverse profiles. By establishing a correlation between the spatial and temporal DoFs, diffractive spreading is reined-in and the time-averaged spatial profile propagates self-similarly. ...

Controlling the group velocity of an optical pulse typically requires traversing a material or structure whose dispersion is judiciously crafted. Alternatively, the group velocity can be modified in free space by spatially structuring the beam profile, but the realizable deviation from the speed of light in vacuum is small. Here we demonstrate precise and versatile control over the group velocity of a propagation-invariant optical wave packet in free space through sculpting its spatio-temporal spectrum. By jointly modulating the spatial and temporal degrees of freedom, arbitrary group velocities are unambiguously observed in free space above or below the speed of light in vacuum, whether in the forward direction propagating away from the source or even traveling backwards towards it.

Introducing correlations between the spatial and temporal degrees of freedom of a pulsed optical beam (or wave packet) can profoundly alter its propagation in free space. Indeed, appropriate spatio-temporal spectral correlations can render the wave packet propagation-invariant: the spatial and temporal profiles remain unchanged along the propagation axis. The spatio-temporal spectral locus of any such wave packet lies at the intersection of the light-cone with tilted spectral hyperplanes. We investigate (2+1)D propagation-invariant 'space-time' light sheets, and identify 10 classes categorized according to the magnitude and sign of their group velocity and the nature of their spatial spectrum -whether the low spatial frequencies are physically allowed or forbidden according to their compatibility with causal excitation and propagation. We experimentally synthesize and characterize all 10 classes using an experimental strategy capable of synthesizing space-time wave packets that incorporate arbitrary spatio-temporal spectral correlations.

Diffraction places a fundamental limitation on the distance an optical beam propagates before its size increases and spatial details blur. We show here that imposing a judicious correlation between spatial and spectral degrees of freedom of a pulsed beam can render its transverse spatial profile independent of location along the propagation axis, thereby arresting the spread of the time-averaged beam. Such correlation introduced into a beam with arbitrary spatial profile enables spatio-temporal dispersion to compensate for purely spatial dispersion that underlies diffraction. As a result, the spatio-temporal profile in the local time-frame of the pulsed beam remains invariant at all positions along the propagation axis. One-dimensional diffraction-free space-time beams are described - including non-accelerating Airy beams, despite the well-known fact that cosine waves and accelerating Airy beams are the only one-dimensional diffraction-free solutions to the monochromatic Helmholtz equation.