Localized solutions of Lugiato-Lefever equations with focused pump

TL;DR

This paper derives and analyzes localized solutions of the Lugiato-Lefever equations with focused pumping in 1D and 2D, revealing stable confined optical modes with potential applications in photonic pixels.

Contribution

It introduces exact analytical solutions for localized modes with focused pump in LL equations and demonstrates their stability through numerical simulations.

Findings

Exact solutions for 1D confined modes with focused pump

Stable localized modes in 2D via variational and numerical methods

Analytical and numerical evidence of mode stability

Abstract

Lugiato-Lefever (LL) equations in one and two dimensions (1D and 2D) accurately describe the dynamics of optical fields in pumped lossy cavities with the intrinsic Kerr nonlinearity. The external pump is usually assumed to be uniform, but it can be made tightly focused too -- in particular, for building small pixels. We obtain solutions of the LL equations, with both the focusing and defocusing intrinsic nonlinearity, for 1D and 2D confined modes supported by the localized pump. In the 1D setting, we first develop a simple perturbation theory, based in the sech ansatz, in the case of weak pump and loss. Then, a family of exact analytical solutions for spatially confined modes is produced for the pump focused in the form of a delta-function, with a nonlinear loss (two-photon absorption) added to the LL model. Numerical findings demonstrate that these exact solutions are stable, both dynamically and structurally (the latter means that stable numerical solutions close to the exact ones are found when a specific condition, necessary for the existence of the analytical solution, does not hold). In 2D, vast families of stable confined modes are produced by means of a variational approximation and full numerical simulations.