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

RESUMO

A plane-wave optical pulse travels in free space at a group velocity $c$ (the speed of light in vacuum) measured along its propagation axis. It is sometimes thought that spatially structuring the field in free space reduces the group velocity below $c$ because -- intuitively -- the oblique wave vectors undergirding the wave packet increase the average group delay. Here we show that spatiotemporally structuring a pulsed optical beam (or wave packet) can yield -- in principle -- arbitrary group velocities in free space in contradistinction to this commonly held intuition. We devote our attention here to pulses endowed with angular dispersion (AD) where the propagation angle is wavelength dependent. This class of wave packets is particularly pertinent because their group velocity is constant over the transverse field profile and along the propagation axis, thereby yielding an unambiguous group delay. However, significant deviation of the AD-induced group velocity in free space from~$c$ inevitably requires a large numerical aperture that lies deep in the non-paraxial regime, and such wave packets experience AD-induced group-velocity dispersion. The formulation presented here captures a broad range of results, connecting them in a single framework. In Part~II of this tutorial, we describe recently identified `non-differentiable AD' (associated with propagation-invariant space-time wave packets) that helps circumvent the limits associated with conventional (differentiable) AD as outlined here.