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

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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.

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Authors and Affiliations

  1. Instituto de Física, Universidade Federal de Goiás, 74.690-900, Goiânia, Goiás, Brazil

    Wesley B. Cardoso

  2. Dipartimento di Fisica e Astronomia “Galileo Galilei” and CNISM, Università di Padova, via Marzolo 8, 35131, Padova, Italy

    Luca Salasnich

  3. Istituto Nazionale di Ottica (INO) del Consiglio Nazionale delle Ricerche (CNR), Sezione di Sesto Fiorentino, via Nello Carrara, 1 – , 50019, Sesto Fiorentino, Italy

    Luca Salasnich

  4. Department of Interdisciplinary Studies, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv, 69978, Israel

    Boris A. Malomed

  5. Laboratory of Nonlinear-Optical Informatics, ITMO University, St. Petersburg, 197101, Russia

    Boris A. Malomed

Authors
  1. Wesley B. Cardoso
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  2. Luca Salasnich
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  3. Boris A. Malomed
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Correspondence to Wesley B. Cardoso.

Additional information

Contribution to the Topical Issue: “Theory and Applications of the Lugiato-Lefever Equation”, edited by Yanne K. Chembo, Damia Gomila, Mustapha Tlidi, Curtis R. Menyuk.

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Cardoso, W.B., Salasnich, L. & Malomed, B.A. Zero-dimensional limit of the two-dimensional Lugiato-Lefever equation. Eur. Phys. J. D 71, 112 (2017). https://doi.org/10.1140/epjd/e2017-80060-7

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  • DOI: https://doi.org/10.1140/epjd/e2017-80060-7

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