Small Beam Nonparaxiality Arrests Self-Focusing of Optical Beams
Physical Review Letters
The behaviour of grey solitons incident on Kerr defocusing interfaces is studied within the framework of Helmholtz theory. We report that the refraction of grey solitons is governed by a generalized Snell's law whose validity extends to arbitrary angles. Unlike their bright or black counterparts, grey solitons may exhibit either external or internal refraction depending solely on the soliton greyness parameter. References Generalized Snell's law for bright, black and grey solitons Spatial
... itons Spatial soliton refraction at interfaces separating two nonlinear media has been traditionally studied in terms of the paraxial Nonlinear Schrödinger (NLS) equation, which limits the validity of results to vanishingly small angles of incidence . This restriction is removed in a Helmholtz nonparaxial framework [2,3], in which a Nonlinear Helmholtz (NLH) equation  describes the evolution of a broad beam (when compared to the wavelength) propagating at arbitrary angles.  A.B. Aceves, J.V. Moloney and A.C. Newell, "Theory of light-beam propagation at nonlinear interfaces. I. Equivalent-particle theory for a single interface", Phys. Rev.  G. Fibich, "Small beam nonparaxiality arrests self-focusing of optical beams", Phys. Rev. Lett. 76, 4356-4359 (1996)  J. Sánchez-Curto, P. Chamorro-Posada and G.S. McDonald, "Helmholtz solitons at nonlinear interfaces", Opt. Lett. 32, 1126-1128 (2007)  J. Sánchez-Curto, P. Chamorro-Posada and G.S. McDonald, "Dark solitons at nonlinear interfaces", Opt. Lett. 35, 1347-1349 (2010)  P. Chamorro-Posada and G.S. McDonald, "Helmholtz dark solitons", Opt. Lett. 28, 825-828 (2003). Grey soliton refraction is predicted to be strongly dependent on the greyness parameter F. This effect is verified in the snapshots below, which show numerical results of grey solitons impinging on a nonlinear interface (n 02 =1.0124n 01 and α 2 =4α 1 ) at 30 o . Soliton greyness parameters are increasing from F=0.2 to F=0.5, in intervals of 0.1. The set of figures reveals that the net angle of refraction θ nt decreases as F grows, resulting in a transition of grey soliton refraction from external to internal regimes. This behaviour is completely governed by the generalized Snell's law. Larger values of F increase the angular component associated with the intrinsic grey excitation of the refracted soliton θ 0t , thus reducing the net angle of refraction θ nt . Bright solitons 1. θ ni and θ nt are the net angles of incidence and refraction of a soliton, respectively. They account for the total angle between the propagation direction of the soliton hump (bright) or dip (dark) and the nonlinear interface. 2. θ 0i =θ i -θ ni and θ 0t =θ t -θ nt represent the intrinsic angles of the incident and refracted grey soliton relative to the propagation direction of the background wave (white arrow) supporting the corresponding dark soliton, respectively. For bright and black solitons, θ 0i =θ 0t =0. 3. γ ± is a nonlinear correction where ± corresponds to either Kerr focusing (+) or defocusing (-) media.