  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.
