Self-Organization, Coherence and Turbulence in Laser Optics

Abstract

In the last decades, rapid progress in modern nonlinear science was marked by the development of the concept of dissipative soliton (DS). This concept is highly useful in many different fields of science ranging from field theory, optics, and condensed matter physics to biology, medicine, and even sociology. This chapter aims to present a DS appearance from random fluctuations, development, and growth, the formation of the nontrivial internal structure of mature DS and its breakup, in other words, a full life cycle of DS as a self-organized object. Our extensive numerical simulations of the generalized cubicquintic nonlinear Ginzburg-Landau equation, which models, in particular, dynamics of mode-locked fiber lasers, demonstrate a close analogy between the properties of DS and the general properties of turbulent and chaotic systems. In particular, we show a disintegration of DS into a noncoherent (or partially coherent) multisoliton complex. Thus, a DS can be interpreted as a complex of nonlinearly coupled coherent "internal modes" that allows developing the kinetic and thermodynamic theory of the nonequilibrious dissipative phenomena. Also, we demonstrate an improvement of DS integrity and, as a result, its disintegration suppression due to noninstantaneous nonlinearity caused by the stimulated Raman scattering. This effect leads to an appearance of a new coherent structure, namely, a dissipative Raman soliton.

A Phase-Space Approach to Non-stationary Nonlinear Systems

“…The "condensation" means a flow of energy to zero wavenumbers 0 p → that is the increase of long-range correlations and the suppression of fluctuations in direct analogy with minimization of dispersion of a quantum system transiting to a classical state (Fig. 2) [11]. Simultaneously, that results in the minimization of the Hamiltonian for the well-known (1+1)-dimensional cubic nonlinear Schrödinger equation which describes an evolution of slowly varying wave in a nonlinear medium [9,12]:…”

“…x  outside the condensate [8,13]. As a result, the condensate evolves toward the Rayleigh-Jeans equilibrium distribution [11]:…”

“…The condensation in the vicinity of 0 p = is illustrated by shading. Thus, a bridge to the statistical mechanics is in the offing, and such invention is relevant to the description not only coherent solitons but also to the study of the dissipative solitons (DS) and the turbulent phenomena [11,14].…”

“…A soliton (i.e., a coherent "condensate") has the minimal entropy so that the rest of entropy concentrates in the small-scale fluctuations with large 2 x  outside the condensate[8,13]. As a result, the condensate evolves toward the Rayleigh-Jeans equilibrium distribution[11]: correlation scales: a long-range one defined by a negative "chemical potential"  , and a short-range one defined by a momentum cut-off at cut p which is caused by the nonlinear and dissipative effects (  is the Heaviside function, seeFig. 3).…”