Solitary waves due to χ^(2): χ^(2) cascading

TLDR

The study investigates solitary waves in cascaded χ(2):χ(2) materials and discusses the implications of the robustness hypothesis. The authors derive the governing equations for cascaded χ(2):χ(2) nonlinearities with dispersion or diffraction, reduce them to a nonlinear Schrödinger form under large phase mismatch, and present a comprehensive classification of temporal and spatial solitary‑wave solutions—including arbitrary group‑velocity differences—expressed as combinations of constant, sech, and sech‑tanh terms. The analysis reveals that all solitary‑wave solutions form zero‑parameter or one‑parameter families, encompassing known and novel waveforms, and supports the robustness hypothesis by predicting a two‑parameter family of soliton‑like solutions whose experimental observation requires specific conditions.

Abstract

Solitary waves in materials with a cascaded χ(2):χ(2) nonlinearity are investigated, and the implications of the robustness hypothesis for these solitary waves are discussed. Both temporal and spatial solitary waves are studied. First, the basic equations that describe the χ(2):χ(2) nonlinearity in the presence of dispersion or diffraction are derived in the plane-wave approximation, and we show that these equations reduce to the nonlinear Schrödinger equation in the limit of large phase mismatch and can be considered a Hamiltonian deformation of the nonlinear Schrödinger equation. We then proceed to a comprehensive description of all the solitary-wave solutions of the basic equations that can be expressed as a simple sum of a constant term, a term proportional to a power of the hyperbolic secant, and a term proportional to a power of the hyperbolic secant multiplied by the hyperbolic tangent. This formulation includes all the previously known solitary-wave solutions and some exotic new ones as well. Our solutions are derived in the presence of an arbitrary group-velocity difference between the two harmonics, but a transformation that relates our solutions to zero-velocity solutions is derived. We find that all the solitary-wave solutions are zero-parameter and one-parameter families, as opposed to nonlinear-Schrödinger-equation solitons, which are a two-parameter family of solutions. Finally, we discuss the prediction of the robustness hypothesis that there should be a two-parameter family of solutions with solitonlike behavior, and we discuss the experimental requirements for observation of solitonlike behavior.