Consequences of non-differentiable angular dispersion in optics: tilted pulse fronts versus space-time wave packets.

Conventional diffractive and dispersive devices introduce angular dispersion (AD) into pulsed optical fields, thus producing so-called 'tilted pulse fronts'. Naturally, it is always assumed that the functional form of the wavelength-dependent propagation angle[s] associated with AD is differentiable with respect to wavelength. Recent developments in the study of space-time wave packets - pulsed beams in which the spatial and temporal degrees of freedom are inextricably intertwined - have pointed to the existence of non-differentiable AD: field configurations in which the propagation angle does not possess a derivative at some wavelength. Here we investigate the consequences of introducing non-differentiable AD into a pulsed field and show that it is the crucial ingredient required to realize group velocities that deviate from c (the speed of light in vacuum) along the propagation axis in free space. In contrast, the on-axis group velocity for conventional pulsed fields in free space is always equal to c. Furthermore, we show that non-differentiable AD is needed for realizing anomalous or normal group-velocity dispersion along the propagation axis, while simultaneously suppressing all higher-order dispersion terms. We experimentally verify these and several other consequences of non-differentiable AD using a pulsed-beam shaper capable of introducing AD with arbitrary spectral profile. Non-differentiable AD is not an exotic phenomenon, but is rather an accessible, robust, and versatile resource for sculpting pulsed optical fields.