Definition of phase velocity for non-differentiable angular dispersion

Determine a physically appropriate definition of phase velocity for optical wave packets exhibiting non-differentiable angular dispersion, particularly for applications involving phase matching in nonlinear optics.

Background

For non-differentiable angular dispersion, the propagation angle changes rapidly near the non-differentiable frequency, making the usual phase-velocity definition difficult to apply. The paper adopts the value obtained at the non-differentiable frequency for consistency with conventional angular-dispersion wave packets, but identifies the lack of a settled definition as a problem relevant to nonlinear-optical phase matching.

References

Nevertheless, more work is needed to determine a definition of phase velocity in presence of non-differentiable AD, which is important in applications involving phase matching in nonlinear optics.

— Limits on the free-space group velocity of optical wave packets incorporating angular dispersion. Part~II, non-differentiable angular dispersion: tutorial  (2609.19580 - Hall et al., 17 Sep 2026) in Section 2.2, “Non-differentiable AD for on-axis propagation”