[Paper Review] Optical Nonlinear Dark X-Waves

Fabio Baronio, S. Wabnitz+4 more|arXiv (Cornell University)|Aug 31, 2016

Nonlinear Waves and Solitons1 references3 citations

TL;DR

This paper proposes analytical and numerical solutions for spatiotemporal optical dark X-waves in (2+1)D hyperbolic nonlinear Schrödinger equations (NLSE), derived via a transformation from the KP-II equation. The key result is the demonstration of long-distance propagation (tens of nonlinear lengths) of dark X solitary waves before breakup due to modulation instability, enabling new pathways for excitation and control in nonlinear optics and related fields.

ABSTRACT

We introduce spatiotemporal optical dark X solitary waves of the (2+1)D {hyperbolic} nonlinear Schrödinger equation (NLSE), that rules wave propagation in a self-focusing and normally dispersive medium. These analytical solutions are derived by exploiting the connection between such NLSE and a well known equation of hydrodynamics, namely the type II Kadomtsev-Petviashvili (KP-II) equation. As a result, families of shallow water X soliton solutions of the KP-II equation are mapped into optical dark X solitary wave solutions of the NLSE. Numerical simulations show that optical dark X solitary waves may propagate for long distances (tens of nonlinear lengths) before they eventually break up, owing to the modulation instability of the continuous wave background. This finding opens a novel path for the excitation and control of X solitary waves in nonlinear optics.

Motivation & Objective

• To overcome the lack of analytical solutions for nonlinear X-waves in free-propagation media, particularly dark X-waves over finite backgrounds.
• To extend the known family of bright nonlinear X-waves to include dark counterparts in self-focusing, normally dispersive media.
• To establish a theoretical and numerical framework for the existence and stability of dark X-waves in the (2+1)D hyperbolic NLSE.
• To map hydrodynamic shallow water X-soliton solutions of the KP-II equation onto optical dark X-waves via a nonlinear transformation.
• To assess the stability of these dark X-waves under modulation instability and transverse perturbations in realistic experimental conditions.

Proposed method

• Utilize a nonlinear transformation linking the (2+1)D hyperbolic NLSE to the KP-II equation, enabling the transfer of known KP-II soliton solutions to optical dark X-waves.
• Apply the transformation $ u(t,y,z) riangleq ig[ ho_0 + ho_0 ilde{ ho}( au, u, ho)ig]^{1/2} imes ext{exp}ig[iig( ho_0 ilde{ ho} z - ( ho_0/c_0)ig)ig] $, where $ ilde{ ho} $ satisfies the KP-II equation.
• Leverage two families of KP-II solitons: the two-soliton X-shaped solution and the Toda-type solution, to generate distinct dark X-wave profiles.
• Perform numerical integration of the hyperbolic NLSE to validate the analytical predictions and assess long-term dynamics and stability.
• Analyze the role of modulation instability (MI) of the continuous wave background as the dominant destabilizing mechanism, using parameter regimes relevant to experiments in Kerr media and parametric converters.
• Compare analytical X-wave profiles with numerical results at $ z = 10 $ to confirm the accuracy of the analytical approximation and observe fission dynamics at the X-node.

Experimental results

Research questions

• RQ1 Can analytical dark X-waves be constructed in the (2+1)D hyperbolic NLSE, which describes self-focusing and normally dispersive media?
• RQ2 Do families of KP-II shallow water solitons map into physically observable optical dark X-waves via a nonlinear transformation?
• RQ3 How stable are these dark X-waves during propagation, and what is the dominant instability mechanism?
• RQ4 Can the Toda-type KP-II soliton solution generate a distinct dark X-wave profile with resonant notch formation at the intersection point?
• RQ5 To what extent do numerical simulations of the NLSE reproduce the analytical predictions, especially regarding waveform distortion and long-distance propagation?

Key findings

• Analytical dark X-waves are derived for the first time in the (2+1)D hyperbolic NLSE, using the KP-II equation as a bridge to known soliton solutions.
• The two-soliton X-shaped solution of KP-II maps into a dark X-wave with two asymptotic dark line solitons, each with intensity depth $ rac{1}{2}( ho_1 + ho_2)^2 $ and angles $ an^{-1}( ho_1 - ho_2) $ from the transverse axis.
• The Toda-type KP-II solution generates a dark X-wave that undergoes fission at the intersection point, forming a large-amplitude solitary notch due to resonant dispersive excitation.
• Numerical simulations confirm that the analytical solution provides an excellent approximation of the NLSE dynamics at $ z = 10 $, validating the transformation method.
• Modulation instability of the continuous wave background is identified as the primary cause of long-term instability, with observable effects only beyond $ z = 10-20 $, corresponding to $ Z > 30-60 L_{nl} $ in real-world distances.
• Transverse instabilities of the constituent line solitons do not manifest in simulations due to their long-range nature and shallow depth, indicating that MI of the background is the dominant destabilizing factor.