# Zero-dimensional limit of the two-dimensional Lugiato-Lefever equation

## TL;DR

This paper investigates the zero-dimensional limit of the 2D Lugiato-Lefever equation, revealing stable localized optical modes, including vortex solitons, through variational, numerical, and analytical methods.

## Contribution

It introduces a detailed analysis of tightly confined optical modes in the 2D LL model, including vortex states, using variational, numerical, and Thomas-Fermi approximations.

## Key findings

- Stable pixel-like localized modes identified.
- Vortex solitons with embedded vorticity are stable over broad parameter ranges.
- Multi-ring vortex states with spiral phase are also stable.

## Abstract

We study effects of tight harmonic-oscillator confinement on the electromagnetic field in a laser cavity by solving the two-dimensional Lugiato-Lefever (2D LL) equation, taking into account self- focusing or defocusing nonlinearity, losses, pump, and the trapping potential. Tightly confined (quasi-zero-dimensional) optical modes (pixels), produced by this model, are analyzed by means of the variational approximation, which provides a qualitative picture of the ensuing phenomena. This is followed by systematic simulations of the time-dependent 2D LL equation, which reveal the shape, stability, and dynamical behavior of the resulting localized patterns. In this way, we produce stability diagrams for the expected pixels. Then, we consider the LL model with the vortical pump, showing that it can produce stable pixels with embedded vorticity (vortex solitons) in remarkably broad sta- bility areas. Alongside confined vortices with the simple single-ring structure, in the latter case the LL model gives rise to stable multi-ring states, with a spiral phase field. In addition to the numeri- cal results, a qualitatively correct description of the vortex solitons is provided by the Thomas-Fermi approximation.