Eur. Phys. J. D (2017) 71: 112
https://doi.org/10.1140/epjd/e2017-80060-7

Regular Article

Zero-dimensional limit of the two-dimensional Lugiato-Lefever equation^*

¹ Instituto de Física, Universidade Federal de Goiás, 74.690-900 Goiânia, Goiás, Brazil
² Dipartimento di Fisica e Astronomia “Galileo Galilei” and CNISM, Università di Padova, via Marzolo 8, 35131 Padova, Italy
³ Istituto Nazionale di Ottica (INO) del Consiglio Nazionale delle Ricerche (CNR), Sezione di Sesto Fiorentino, via Nello Carrara, 1 – 50019 Sesto Fiorentino, Italy
⁴ Department of Interdisciplinary Studies, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
⁵ Laboratory of Nonlinear-Optical Informatics, ITMO University, St. Petersburg 197101, Russia

^a e-mail: wesleybcardoso@gmail.com

Received: 28 January 2017
Received in final form: 9 March 2017
Published online: 18 May 2017

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 stability 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 numerical results, a qualitatively correct description of the vortex solitons is provided by the Thomas-Fermi approximation.