Universality and non-differentiability: A new perspective on angular dispersion in optics

TL;DR

This paper introduces the concept of non-differentiable angular dispersion in optics, highlighting its implications for new optical fields and proposing a universal synthesizer to access all AD orders, expanding the scope of optical control.

Contribution

It presents a unified framework for differentiable and non-differentiable angular dispersion, revealing new optical phenomena and outlining the design of a universal AD synthesizer.

Findings

Non-differentiable AD occurs at specific frequencies where the derivative of the propagation angle is undefined.

Non-differentiable AD allows circumventing traditional optical constraints.

A schema for constructing a universal AD synthesizer capable of generating all AD orders.

Abstract

Angular dispersion (AD) is a ubiquitous phenomenon in optics after light traverses a diffractive or dispersive device, whereby each wavelength propagates at a different angle. AD is useful in a variety of applications; for example, modifying the group velocity or group-velocity dispersion of pulsed lasers in free space or optical materials, which are essential ingredients in group-velocity matching and dispersion compensation. Conventional optical components introduce `differentiable' AD, so that the propagation angle can be expanded perturbatively around a fixed frequency, in which only a few low AD-orders are typically relevant. However, this model does not encompass newly emerging classes of propagation-invariant pulsed optical fields, such as `space-time wave packets', which incorporate a new form of AD that we call `non-differentiable AD'. This is a surprising feature: there exists a frequency at which the derivative of the propagation angle with respect to frequency is not defined. Consequently, the propagation angle cannot be expanded perturbatively at this frequency, and a large number of independently controllable AD orders are needed to approximate this condition. Synthesizing these new AD-induced field configurations requires constructing a `universal AD synthesizer' capable of accessing the magnitude and sign of any AD order, a capability missing from any single optical component to date. This Perspective article provides a unified schema for studying differentiable and non-differentiable AD, shows that non-differentiable AD enables circumventing many well-established constraints in optics -- thereby giving rise to new applications, and outlines the requirements for a universal AD synthesizer capable of producing both forms of AD.