Modulational instability induced by quadratic nonlinearities

The large nonlinear phase shifts obtained with cascaded nonlinearities [1] has attracted renewed interest in the parametric interaction of a fundamental frequency (FF) beam with its doubled frequency (DF) [2]. A pure phase-shift, not accompanied by any energy transfer, may be achieved at any phase mismatch, by exploiting the nonlinear eigenmodes of the interaction [3,4]. These are symbiotic pair of phase-locked FF and DF cw’s which propagate without mutually exchanging energy but independently experiencing a nonlinear phase-shifts. Here we predict that when dispersion is taken into account this cw eigenmodes are subject to modulational instability (MI) [5]. This entails that two sideband pairs, equally detuned by the most unstable perturbation frequency from the FF and DF, may spontaneously grow from noise. MI is commonly believed to occur only in cubic media through four-photon interactions involving the decay of two pump photons into down and up-shifted photon twins. Conversely we anticipate that MI may occur with cascaded nonlinearity via combined three-photon processes. A photon at frequency 2ω0 produces a photon pair at frequencies (ω0 − Ω, ω0 + Ω), whereas photons at the sideband frequencies 2ω0 ± Ω are created by annhilating two photons at frequencies ω0 and ω0 ± Ω, respectively.