  Introducing angular dispersion into a pulsed field associates each frequency
with a particular angle with respect to the propagation axis. A perennial yet
implicit assumption is that the propagation angle is differentiable with
respect to the frequency. Recent work has shown that the existence of a
frequency at which the derivative of the propagation angle does not exist --
which we refer to as non-differentiable angular dispersion -- allows for the
optical field to exhibit unique and useful characteristics that are
unattainable by endowing optical fields with conventional angular dispersion.
Because these novel features are retained in principle even when the specific
non-differentiable frequency is not part of the selected spectrum, the question
arises as to the impact of the proximity of the spectrum to this frequency. We
show here that operating in the vicinity of the non-differentiable frequency is
imperative to reduce the deleterious impact of (1) errors in implementing the
angular-dispersion profile, and (2) the spectral uncertainty intrinsic to
finite-energy wave packets in any realistic system. Non-differential angular
dispersion can then be viewed as a resource -- quantified by a Schmidt number
-- that is maximized in the vicinity of the non-differentiable frequency. These
results will be useful in designing novel phase-matching of nonlinear
interactions in dispersive media.
