Derivation of a modified Korteweg–de Vries model for few-optical-cycles soliton propagation from a general Hamiltonian

Abstract

Propagation of few-cycles optical pulses in a centrosymmetric nonlinear optical Kerr (cubic) type material described by a general Hamiltonian of multilevel atoms is considered. Assuming that all transition frequencies of the nonlinear medium are well above the typical wave frequency, we use a long-wave approximation to derive an approximate evolution model of modified Korteweg-de Vries type. The model derived by rigorous application of the reductive perturbation formalism allows one the adequate description of propagation of ultrashort (few-cycles long) solitons.

Models for supercontinuum generation beyond the slowly-varying-envelope approximation

“…In such simple systems, all high-order terms, including dispersion ones, are self-contained into a compact field propagation model, which goes beyond the envelope approximation and thus becomes pertinent in few-cycle pulse propagation. We recall that there is a rigorous mathematical procedure to derive the sG and mKdV models from the Maxwell-Bloch equations for a two-level system [26], which can be extended to a general Hamiltonian formulation in the latter case [43]. In the present paper we show that both mKdV and sG models are able to describe the process of multiple-octavespan supercontinuum generation, with an initial high-intensity ultrashort input pulse.…”

“…We assume that the resonance lines of the medium can be separated in two sets: those with frequency well above the central frequency of the pulse and those with frequency well below it. For the former set, the optical pulse can be considered formally as a low-frequency wave; then using the so-called long-wave approximation, the one-dimensional propagation is shown to obey a mKdV-type model [26,43]. For the latter set, it is a shortwave approximation that leads to a sG-type model [26,44].…”

Few-cycle solitons in supercontinuum generation dynamics

“…In the case of the UV transitions, it has been shown that the mKdV model as given by Eq. (1) generalizes without modification to a general Hamiltonian, with an arbitrary number of atomic levels [18]. In the case of the IR transitions, the generalization is not so simple.…”

Derivation of a generalized double-sine-Gordon equation describing ultrashort-soliton propagation in optical media composed of multilevel atoms

“…Most of the theoretical investigations concerned only FCPs propagating in nonlinear optical media described by two-level Hamiltonians. However, in two recent works [50,51] we extended the existing studies to a more general physical situation involving N -level Hamiltonians in the framework of the reductive perturbation method (multiscale analysis) [52]. First, in the long-wave approximation regime we gave in Ref.…”

“…First, in the long-wave approximation regime we gave in Ref. [50] a detailed mathematical derivation of the mKdV equation for a general N -level Hamiltonian. We assumed that the absorption spectrum of the nonlinear medium does not extend below some cutoff frequency, and that the typical frequency of the FCP is much less than the latter; therefore, a Kerr medium which has no transition in the infrared was actually described in Ref.…”