Modulational instability in the nonlocal χ(2)χ(2)-model

We investigate in detail the linear regime of the modulational instability (MI) properties of the plane waves of the nonlocal model for χ(2)χ(2)-media formulated in Nikolov et al. [N.I. Nikolov, D. Neshev, O. Bang, W.Z. Królikowski, Quadratic solitons as nonlocal solitons, Phys. Rev. E 68 (2003) 036614; I.V. Shadrivov, A.A. Zharov, Dynamics of optical spatial solitons near the interface between two quadratically nonlinear media, J. Opt. Soc. Amer. B 19 (2002) 596–602]. It is shown that the MI is of the oscillatory type and of finite bandwidth. Moreover, it is possible to identify regions in the parameter space for which a fundamental gain band exists, and regions for which higher order gain bands and modulational stability exist. We also show that the MI analysis for the nonlocal model is applicable in the finite walk-off case. Finally, we show that the plane waves of the nonlocal χ(2)χ(2)-model are recovered as the asymptotic limit of one of the branches of the plane waves (i.e. the adiabatic branch or the acoustic branch  ) of the full χ(2)χ(2)-model by means of a singular perturbational approach. It is also proven that the stability results for the adiabatic branch continuously approach those of the nonlocal χ(2)χ(2)-model, by using the singular perturbational approach. The other branch of the plane waves (i.e. the nonadiabatic branch or the optical branch) is always modulationally   unstable. We compare the MI results for the adiabatic branch with the predictions obtained from the full χ(2)χ(2)-model in the non-walk-off limit. It is concluded that for most physical relevant parameter regimes it suffices to use the nonlocal model in order to determine the MI properties.