One- and two-dimensional solitons supported by singular modulation of quadratic nonlinearity

Abstract

We introduce a model of one-and two-dimensional (1D and 2D) optical media with the χ (2) nonlinearity whose local strength is subject to cusp-shaped spatial modulation, χ (2) ∼ r −α , with α > 0, which can be induced by spatially nonuniform poling. Using analytical and numerical methods, we demonstrate that this setting supports 1D and 2D fundamental solitons, at α < 1 and α < 2, respectively. The 1D solitons have a small instability region, while the 2D solitons have a stability region at α < 0.5 and are unstable at α > 0.5. 2D solitary vortices are found too. They are unstable, splitting into a set of fragments, which eventually merge into a single fundamental soliton pinned to the cusp. Spontaneous symmetry breaking of solitons is studied in the 1D system with a symmetric pair of the cusp-modulation peaks.

Spatio-Temporal Mode-locking in Quadratic Nonlinear Media

“…In 1995, optical solitons in a quadratic bulk material were observed by Torruellas et al [23] , and existence of the solitons in χ 2 waveguide were observed experimentally by Schiek et al [24] in 1996. These original results have since been corroborated and extended in many follow-up experiments [25][26][27][28][29][30], demonstrating that quadratic solitons can exist in both the spatial and the temporal domains in waveguides or bulk materials [31][32][33][34]. One of the distinguished properties of quadratic nonlinear media is that it provides stable multidimensional pulse propagation without collapse in any dimension [31,32,35].…”

Multi-soliton states under triangular spatial modulation of the quadratic nonlinearity

“…As mentioned above, a basic result reported in Ref. [27] is that these equations generate fundamental (zerovorticity) localized states pinned to the singular-modulation peak at α < 2, and these states have a stability area at α < 0.5. Under this condition, they tend to be stable and unstable, respectively, at smaller and larger values of the total power, P , at all values of the mismatch, see Eq.…”

“…A discrete version of the localized quadratic nonlinearity was elaborated in the form of a linear lattice with one or two χ (2) -nonlinear sites embedded in it [26].While the delta-like modulation of the local χ (2) coefficient may not be easily realized in the experiment, a more realistic case of the 1D singular modulation, with χ (2) ∼ |x| −α and positive α, was introduced in Ref. [27]. It was found that this modulation format supports quadratic solitons, pinned to the singular peak, for α < 1 (the pinned modes vanish at α = 1), and they are chiefly stable.…”

“…A natural extension of that setting is a symmetric pair of two peaks (similar to the above-mentioned symmetric set of two delta-functions multiplying the nonlinear terms [25]). The consideration of the twin peaks has made it possible to predict effects of the spontaneously symmetry breaking [28] and formation of asymmetric two-soliton states pinned to the two peaks [27]. These results may be used for the design of steering optical beams in the form of spatial solitons.An essential advantage of using the quadratic nonlinearity with this type of the local modulation is that it may be extended to the 2D geometry, by choosing χ (2) (r) ∼ r −α (r is the radial coordinate), while any 2D singular modulation of the self-focusing χ (3) nonlinearity leads to collapse.…”

“…The analysis of the 2D setting, performed in Ref. [27], has revealed that the solitons, pinned to the singular-modulation peak, exist at α < 2, and they have a stability region at α < 0.5. Solitons with embedded vorticity can also be pinned to the singularity peak, but they all are unstable.The availability of stable 2D solitons attached to the singular-modulation peak suggests a possibility to study multi-soliton patterns in multi-peak modulation profiles, subject to natural symmetry conditions.…”