Non-differentiable angular dispersion as an optical resource

TL;DR

This paper explores the concept of non-differentiable angular dispersion in optical fields, demonstrating its potential as a resource for novel optical properties and its importance near the non-differentiable frequency for practical applications.

Contribution

It introduces the idea of non-differentiable angular dispersion as a new optical resource and analyzes its benefits and robustness near the non-differentiable frequency.

Findings

Operating near the non-differentiable frequency enhances the robustness of the optical field.

Non-differentiable angular dispersion can be quantified by a Schmidt number.

Proximity to the non-differentiable frequency reduces errors and spectral uncertainties.

Abstract

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.