Quadratic Nonlinearities and Turbulence in a Laser System

Abstract

As is well known, the cubic nonlinearity in a many mode laser system is responsible for mode competition and for the concentration of the emitted power into one or a few excited modes. On the other hand, wide classes of nonlinear, nonequilibrium systems are known where the existence of nonlinearities with different symmetry (e.g., the Navier-Stokes nonlinearity in a driven fluid) gives rise to the spread of the energy initially fed into one mode over a wide spectrum of excited modes. These phenomena go under the general name of turbulence. When they occur in a driven fluid, the nonlinearity is the Navier-Stokes nonlinearity V·∇V where V is the velocity field. By Fourier-transforming this nonlinearity V·∇V within a finite volume, a mode-mode coupling as Σ_K′K′ V_K-K′V_K′ arises which is quadratic in the mode amplitudes V_K. An example of amplitude equations with a cubic as well as a quadratic non-linearity is given in the Fourier expansion of the Benard instability. Here, the quadratic interaction takes the form[1]

$$ - \delta \sum\limits_{{{\text{K}}_1},{{\text{K}}_2}} {{\text{A}}_{{{\text{K}}_1}}^ * {\text{A}}_{{{\text{K}}_2}}^ * {\delta _{{{\text{K}}_1} + {{\text{K}}_2} + {\text{K}},0}}}$$

(1.1)

, to be added in the dynamic equation for the K mode amplitude A_K. Notice that hydrodynamics allows for a very wide excitation spectrum (indeed, energy conservation implies ω_K′ + ω_K″ = ω_K) as well as for three-dimensional structures, so that the momentum closure condition implied by the δ-function in Eq.(1.1) gives rise to the well-known hexagonal structures of the Benard instability.