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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 thewavelength-dependent propagation angle associated with AD is differentiablewith respect to wavelength. Recent developments in the study of space-time wavepackets -- pulsed beams in which the spatial and temporal degrees of freedomare inextricably intertwined -- have pointed to the existence ofnon-differentiable AD: field configurations in which the propagation angle doesnot possess a derivative at some wavelength. Here we investigate theconsequences of introducing non-differentiable AD into a pulsed field and showthat it is the crucial ingredient required to realize group velocities thatdeviate from $c$ (the speed of light in vacuum) along the propagation axis infree space. In contrast, the on-axis phase and group velocities are alwaysequal in conventional scenarios. Furthermore, we show that non-differentiableAD is needed for realizing anomalous or normal group-velocity dispersion alongthe propagation axis, while simultaneously suppressing all higher-orderdispersion terms. These and several other consequences of non-differentiable ADare verified experimentally using a pulsed-beam shaper capable of introducingAD with arbitrary spectral profile. Rather than being an exotic phenomenon,non-differentiable AD is an accessible, robust, and versatile resource forsculpting pulsed optical fields.

The consequences of non-differentiable angular dispersion in optics: Tilted pulse fronts versus space-time wave packets