The use of geometrical constraints opens many new perspectives in photonics and in fundamental studies of nonlinear waves. By implementing surface structures in vertical cavity surface emitting lasers as manifolds for curved space, we experimentally study the impacts of geometrical constraints on nonlinear wave localization. We observe localized waves pinned to the maximal curvature in an elliptical-ring, and confirm the reduction in the localization length of waves by measuring near and far field patterns, as well as the corresponding dispersion relation. Theoretically, analyses based on a dissipative model with a parabola curve give good agreement remarkably to experimental measurement on the transition from delocalized to localized waves. The introduction of curved geometry allows to control and design lasing modes in the nonlinear regime.Studies on the effect of geometry on wave propagation can be tracked back to Lord Rayleigh in the early theory of sound [1], and extend to recent investigation as, for example, analogs of gravity in Bose-Einstein condensates [2], optical event horizon in fiber solitons [3,4], Anderson localization [5], random lasing [6] and celestial mechanics in metamaterials with transformation optics [7][8][9][10]. In addition to linear optics in curved space [11], geometrical constraints affect shape preserving wave packets, including localized solitary waves [12], extended Airy beams [13], and shock waves [14,15]. Curvature also triggers localization by trapping the wave in extremely deformed regions [16].The effect of geometry on nonlinear phenomena suggests that a confining structure may alter the conditions for observing spatial or temporal solitary waves, which are due to the nonlinear compensation of linear diffraction, or dispersion effects [17]. In general, self-localized nonlinear waves require an extra formation power to bifurcate from the linear modes [18], but geometrical constraints may reduce or cancel this threshold. In this work, we use the vertical cavity surface emitting lasers (VC-SELs) as a platform and fabricate a series of curved surfaces, including circular-ring and elliptical-ring cavities, as well as a reference "cold" device with no trapping potential. By introducing symmetry-breaking in geometry, i.e., passing from a circular-ring to an elliptical one, we realize experimentally lasing on nonlinear localized waves when the ellipticity of the ring is larger than a critical value [19]. With different curved geometries, we measure the corresponding lasing characteristics both in the near and far fields, as well as the dispersion relation, to verify the localized modes pinned at the maximal curvature. With state-of-the-art semiconductor technologies, microcavities in VCSELs have allowed small mode volumes and ultrahigh qualify factors, for applications from sensors, optical interconnects, printers, optical storages, FIG. 1. (a) Schematic diagram of our electrically driven GaAs-based vertical cavity surface emitting laser (VCSEL) with a curved potential on the emissio...
