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 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]
, 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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References
a) L. Landau and E. Lifshitz, Fluid Mechanics ( Pergamon Press, London, 1959 );
b) R. Graham, Phys. Rev. Lett. 31, 479 (1973);
c) H. Haken, Rev. Mod. Phys. 47, 67 (1975);
d) see e.g. papers by H. Haken and P.C. Martin delivered at the Conference on Statistical Physics, Budapest, 25–29 August, 1975.
W.E. Lamb, Jr., Phys. Rev. 134, A1429 (1964);
see also M. Sargent, III, M.O. Scully and W.E. Lamb, Laser Physics ( Addison-Wesley, Reading, Mass., 1974 ).
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.
J.E. Geusic et al., Appl. Phys. Lett. 12, 306 (1968).
Any slight deviation from a flat gain line is sufficient to break the symmetry, thus giving rise to the privileged mode.
F.T. Arecchi, A.M. Ricca, to be published.
F.T. Arecchi and E. Schulz-Dubois, Laser Handbook ( North-Holland, Amsterdam, 1972 ).
See for instance, A. Yariv, Quantum Electronics, 2nd ed. ( Wiley, New York, 1975 ) p. 430.
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Arecchi, F.T., Ricca, A.M. (1978). Quadratic Nonlinearities and Turbulence in a Laser System. In: Mandel, L., Wolf, E. (eds) Coherence and Quantum Optics IV. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-0665-9_97
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DOI: https://doi.org/10.1007/978-1-4757-0665-9_97
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