Long-wave instability in optical parametric oscillators

We show that the full dynamical equations for optical parametric oscillators in a large aspect ratio cavity for the case of long-wave instability is reduced to a Ginzburg–Landau equation near the instability threshold. This equation enables us to introduce the concept of optical vortices in χ(2) medium. A criterion of supercritical and subcritical instability is given as well as a condition of the Benjamin–Feir instability.