Wave-turbulence approach of supercontinuum generation: Influence of self-steepening and higher-order dispersion

We analyze the influence of self-steepening and higher-order dispersion on the process of optical wave thermalization. This study is aimed at developing a thermodynamic formulation of supercontinuum generation in photonic crystal fibers. In the highly nonlinear regime of supercontinuum generation, the optical field exhibits a turbulent dynamics that may be described by the kinetic wave theory. In this respect, the phenomenon of spectral broadening inherent to supercontinuum generation may be interpreted as a natural process of thermalization, which is characterized by an irreversible evolution of the optical field toward a thermodynamic equilibrium state. The numerical simulations show that, under certain conditions, the intensity of the field spontaneously concentrates around two specific frequencies. The theory reveals that these particular frequencies are selected in such a way that the corresponding wave packets propagate with the same group velocity, which is shown to also match the average group velocity of the optical field. This “velocity-locking” effect and the underlying process of spectral fission are interpreted in the light of a fundamental property of statistical equilibrium thermodynamics. It is shown that a velocity locking is required, in the sense that it prevents “a macroscopic internal motion in the wave system.” Furthermore, the kinetic wave theory sheds light on the role of self-steepening in the incoherent regime of supercontinuum generation. A family of equilibrium spectra is derived in the presence of self-steepening. They are found in quantitative agreement with the numerical simulations of the generalized nonlinear Schrödinger equation, without adjustable parameters.