English

Аннотация

A review is presented for theoretical results demonstrating the creation of stable spatially confined 0D (zero-dimensional), 1D, and 2D modes in the framework of the Lugiato-Lefever (LL) equations, which are fundamental models of externally driven nonlinear passive optical cavities. The confinement is imposed, in the 2D setting, by the tight harmonic-oscillator (HO) potential, or, in the framework of the 1D and 2D LL equations, by the tightly focused 1D or 2D pump term. The 2D modes, which are strongly confined by the HO potential, and driven by the zero-vorticity or vortical pump, realize effectively 0D pixels, with the respective vorticity. These solutions are obtained by means of the perturbation theory (in the 1D case), variational approximation (VA) and Thomas-Fermi approximation, as well as in the numerical form. The 1D LL equation with the tightly focused pump, which is approximated by the delta-function, gives rise to an exact solution of the codimension-one (non-generic) type, provided that the equation includes a cubic loss term, along with the linear one. In addition to the codimension-one analytical solution, generic ones are obtained in the numerical form, featuring shapes which are close to those of the analytical solution. These 1D modes are completely stable. The 2D LL equation including the focused pump with vorticity S = 0, 1, 2, ..., produces pump-pinned modes, that are found by means of VA and numerically. Stability regions are identified for these modes in the system’s parameter space. Under the action of the self-focusing cubic nonlinearity, those vortex modes which are unstable spontaneously transform into necklace-shaped states. On the other hand, the defocusing nonlinearity maintains stability of the vortex modes, at least, up to S = 5.