Resolving topological obstructions to vectorial structured field control

TL;DR

This paper introduces a mathematical framework to analyze and overcome topological limitations in the control of vectorial optical fields, enabling more effective design of layered optical devices.

Contribution

It provides a novel method to determine the minimal number of layers needed to achieve continuous transformations in vectorial field control.

Findings

Established a framework for analyzing topological constraints

Determined minimal depth for layered transformations

Improved design strategies for optical polarization fields

Abstract

The use of structured matter, such as optical retarders, for vectorial control is a well-established and widely employed technique in modern optics, and has driven continued advances in the manipulation of complex, spatially varying vectorial fields. However, achieving arbitrary field conversion typically requires the use of cascaded elements, as intrinsic physical and fabrication constraints fundamentally limit individual devices to a restricted subset of transformations. This results in an overall continuous transformation potentially failing to be continuous at the level of the parameters of the cascade, leading to detrimental engineering consequences such as the introduction of complex, discontinuous aberrations that disrupt important topological properties of the underlying matter field. In this work, we establish a novel mathematical framework for analyzing the topological difficulties that emerge in the decomposition of an overall transformation into individual layers, and for determining the minimal depth required to overcome them. The strategy introduced provides a general pathway for optimizing designs for vectorial field control and matter field generation, with particular significance for the manipulation of topological phases in optical polarization fields, such as Stokes skyrmions, where continuity is of vital importance.