# Control of spatially rotating structures in diffractive Kerr cavities

## TL;DR

This paper demonstrates how to control the rotation of Turing patterns in nonlinear optical cavities using beams with orbital angular momentum, enabling precise manipulation of pattern dynamics for various applications.

## Contribution

It introduces a method to control the angular velocity of Turing patterns in Kerr cavities through tailored optical pumping with specific OAM properties.

## Key findings

- Pattern rotation velocity depends on OAM and ring radius.
- Full control over pattern rotation achieved with cylindrical vector beams.
- Multiple concentric rings can rotate at different speeds, enabling complex pattern dynamics.

## Abstract

Turing patterns in self-focussing nonlinear optical cavities pumped by beams carrying orbital angular momentum (OAM) $m$ are shown to rotate with an angular velocity $\omega = 2m/R^2$ on rings of radii $R$. We verify this prediction in 1D models on a ring and for 2D Laguerre-Gaussian and top-hat pumps with OAM. Full control over the angular velocity of the pattern in the range $- 2m/R^2 \le \omega \le 2m/R^2$ is obtained by using cylindrical vector beam pumps that consist of orthogonally polarized eigenmodes with equal and opposite OAM. Using Poincar\'e beams that consist of orthogonally polarized eigenmodes with different magnitudes of OAM, the resultant angular velocity is $\omega = (m_L + m_R)/R^2$, where $m_L, m_R$ are the OAMs of the eigenmodes, assuming good overlap between the eigenmodes. If there is no, or very little, overlap between the modes then concentric Turing pattern rings, each with angular velocity $\omega = 2m_{L,R}/R^2$ will result. This can lead to, for example, concentric, counter-rotating Turing patterns creating an 'optical peppermill'-type structure. Full control over the speeds of multiple rings has potential applications in particle manipulation and stretching, atom trapping, and circular transport of cold atoms and BEC wavepackets.