${\cal PT}$-symmetric coupler with $\chi^{(2)}$ nonlinearity

TL;DR

This paper introduces a ${ m PT}$-symmetric dimer with quadratic nonlinearity, revealing unique bifurcation behaviors and stability properties distinct from the standard cubic case, with implications for optical waveguide design.

Contribution

It proposes a ${ m PT}$-symmetric model with $ ext{chi}^{(2)}$ nonlinearity, analyzing its bifurcations and stability, which is novel compared to existing cubic nonlinear models.

Findings

Bifurcations depend on whether the first or second harmonic is vanishing.

Multiple bifurcation types, including saddle-center and pitchfork, are identified.

Numerical simulations confirm stability and evolution of solutions.

Abstract

We introduce the notion of a ${\cal PT}$-symmetric dimer with a $\chi^{(2)}$ nonlinearity. Similarly to the Kerr case, we argue that such a nonlinearity should be accessible in a pair of optical waveguides with quadratic nonlinearity and gain and loss, respectively. An interesting feature of the problem is that because of the two harmonics, there exist in general two distinct gain/loss parameters, different values of which are considered herein. We find a number of traits that appear to be absent in the more standard cubic case. For instance, bifurcations of nonlinear modes from the linear solutions occur in two different ways depending on whether the first or the second harmonic amplitude is vanishing in the underlying linear eigenvector. Moreover, a host of interesting bifurcation phenomena appear to occur including saddle-center and pitchfork bifurcations which our parametric variations elucidate. The existence and stability analysis of the stationary solutions is corroborated by numerical time-evolution simulations exploring the evolution of the different configurations, when unstable.