78-02 Research exposition (monographs, survey articles)
We consider the phenomenon of an optical soliton controlled (e.g. amplified) by a much weaker second pulse which is efficiently scattered at the soliton. An important problem in this context is to quantify the small range of parameters at which the interaction takes place. This has been achieved by using adiabatic ODEs for the soliton characteristics, which is much faster than an empirical scan of the full propagation equations for all parameters in question.
We consider scattering of small-amplitude dispersive waves at an intense optical soliton which constitutes a nonlinear perturbation of the refractive index. Specifically, we consider a single-mode optical fiber and a group velocity matched pair: an optical soliton and a nearly perfectly reflected dispersive wave, a fiber-optical analogue of the event horizon. By combining (i) an adiabatic approach that is used in soliton perturbation theory and (ii) scattering theory from Quantum Mechanics, we give a quantitative account for the evolution of all soliton parameters. In particular, we quantify the increase in the soliton peak power that may result in spontaneous appearance of an extremely large, so-called champion soliton. The presented adiabatic theory agrees well with the numerical solutions of the pulse propagation equation. Moreover, for the first time we predict the full frequency band of the scattered dispersive waves and explain an emerging caustic structure in the space-time domain.
Physics of short optical pulses is an important and active research area in nonlinear optics. In what follows we theoretically consider the most extreme representatives of short pulses that contain only several oscillations of electromagnetic field. Description of such pulses is traditionally based on envelope equations and slowly varying envelope approximation, despite the fact that the envelope is not “slow” and, moreover, there is no clear definition of such a “fast” envelope. This happens due to another paradoxical feature: the standard (envelope) generalized nonlinear Schrödinger equation yields very good correspondence to numerical solutions of full Maxwell equations even for few-cycle pulses, a thing that should not be. In what follows we address ultrashort optical pulses using Hamiltonian framework for nonlinear waves. As it appears, the standard optical envelope equation is just a reformulation of general Hamiltonian equations. In a sense, no approximations are required, this is why the generalized nonlinear Schrödinger equation is so effective. Moreover, the Hamiltonian framework greatly contributes to our understanding of “fast” envelope, ultrashort solitons, stability and radiation of optical pulses. Even the inclusion of dissipative terms is possible making the Hamiltonian approach an universal theoretical tool also in extreme nonlinear optics.