Quadratic Nonlinearities and Turbulence in a Laser System

  • F. T. Arecchi
  • A. M. Ricca
Conference paper


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 ΣK′K′ VK-K′VK′ arises which is quadratic in the mode amplitudes VK. 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]
$$ - \delta \sum\limits_{{{\text{K}}_1},{{\text{K}}_2}} {{\text{A}}_{{{\text{K}}_1}}^ * {\text{A}}_{{{\text{K}}_2}}^ * {\delta _{{{\text{K}}_1} + {{\text{K}}_2} + {\text{K}},0}}}$$
, to be added in the dynamic equation for the K mode amplitude AK. Notice that hydrodynamics allows for a very wide excitation spectrum (indeed, energy conservation implies ωK′ + ωK″ = ωK) 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.


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    The introduction of a Hamiltonian and the derivation of a dynamic equation as (1.4) from it, is a purely formal device introduced for the sake of clarifying the terminology. The photon rate equations written later are a combination of the semiclassical laser equations, plus second-order terms evaluated by a perturbative approach.Google Scholar
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Copyright information

© Springer Science+Business Media New York 1978

Authors and Affiliations

  • F. T. Arecchi
    • 1
  • A. M. Ricca
    • 2
  1. 1.Istituto Nazionale d’OtticaFirenzeItaly
  2. 2.CISEMilanoItaly

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