Limits on the free-space group velocity of optical wave packets incorporating angular dispersion. Part~II, non-differentiable angular dispersion: tutorial

Abstract: In Part~I of this tutorial, we showed that introducing angular dispersion (AD) into a collimated optical pulse allows for tuning the wave packet group velocity in free space. In principle, the group velocity can take on arbitrary values, whether superluminal, subluminal, or even negative. However, any significant deviation of the group velocity from $c$ (the speed of light in vacuum) necessitates propagation at large angles with respect to the optical axis in the non-paraxial regime. In Part~II of this tutorial, we describe the recent discovery of a new form of AD we refer to as `non-differentiable AD', which circumvents the limits imposed by conventional (differentiable) AD. Non-differentiable AD does \textit{not} refer to an AD profile that is discontinuous, contains a kink, or features a singularity. Rather, a non-differentiable AD profile is continuous, smooth, and is differentiable everywhere except for one wavelength at which the derivative is not defined. We show that salutary features follow from this non-differentiability with respect to tuning the group velocity of a pulsed beam in free space; namely, significant deviations in the group velocity away from $c$ are achievable in the paraxial regime while remaining free of group-velocity dispersion and all higher-order dispersive effects. This tutorial presents an outline of these recent results for tuning the group velocity of a wave packet via non-differentiable AD, connecting them with the corresponding results associated with conventional AD -- as outlined in Part~I. Moreover, we show that non-differentiable AD undergirds the unique characteristics of propagation-invariant space-time wave packets, thus unifying a large swathe of results in a single conceptual framework.