Quadratic solitons as nonlocal solitons

Abstract

We show that quadratic solitons are equivalent to solitons of a nonlocal Kerr medium. This provides new physical insight into the properties of quadratic solitons, often believed to be equivalent to solitons of an effective saturable Kerr medium. The nonlocal analogy also allows for novel analytical solutions and the prediction of novel bound states of quadratic solitons.

Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media

“…For s=+1, where the χ (2) -system (15) has a family of bright (for d 1 >0) and dark (for d 1 <0) soliton solutions [53], R(k) is positive definite and localized, giving R(x) = (2σ) −1 exp(−|x|/σ). It is possible to show, e.g., that the nonlocal model (16) does not allow collapse in any physical dimension [51], a known property of the χ (2) system (15) not captured by the cascading limit NLS equation. The cascading limit β −1 → 0 is now seen to correspond to the local limit σ → 0, in which the response function becomes a delta function, R(x) → δ(x).…”

“…However, this model wrongly predicts several features that are known not to exist in Eqs. (15) and even for stationary solutions it is inaccurate, since the term ∂ 2 x E 2 is neglected [51]. To obtain a more accurate model we assume a slow variation of the SH field e 2 (x, z) in the propagation direction only (i.e., only ∂ z e 2 is neglected), which leads to the nonlocal equation for the FW…”

“…Following the discussion in the preceeding section about the equivalence between stationary nonlocal and parametric solitons, and the results of Refs. [51,53], the nonlocal NLS equation with such a response function predicts the existence of single fundamental dark soliton solutions with nonmonotonic tails above a certain critical value of σ. As a result bound states involving two or more solitons can be formed.…”

“…One can show that the structure of the nonlocal equation governing the stationary fields resulting from (16) is identical to the conventional nonlocal model for stationary solitons and thus it has the same weakly and strongly nonlocal limits with the same exact bright and dark soliton solutions. We recently showed that the nonlocal model elegantly explains the structural properties of both bright and dark solitons and their bound states and that it provides good approximate quadratic soliton solutions in large regimes of the parameter space [51].…”

“…We will show below that quadratic solitons can be described by nonlocal models. Such models provide simple physical explanations of these properties and many more, building on a simple waveguide analogy [49,50,51]. Consider a fundamental wave (FW) and its second harmonic (SH) propagating along the z-direction in a χ (2) crystal under conditions for type I phase-matching.…”