Propagation of ultra-short optical pulses in cubic nonlinear media

Canonical and Singular Propagation of Ultrashort Pulses in a Nonlinear Medium

“…If we include a plasma oscillation to account for low-frequency behavior, this is exactly the equation used by BLMS [1] and derived in other contexts by Kuznetsov and collaborators in the nineteen eighties. It is also connected with the work of both Alterman and Rauch [4] and Schaffer and Wayne [5] who realized that, to leading order, the propagation characteristics of short pulses are captured by looking at the deformation of the right going Riemann invariant E 0 . Depending on which effect is dominant, we will choose z 0 accordingly.…”

“…Many of the ideas and some of the equation derivations, albeit in different formats, have appeared already in the literature; for example, in the recent works of Balakin et al (BLMS; [1]) which in turn were informed by results going back to the eighties [2,3]; the papers of Alterman and Rauch [4] and Schaffer and Wayne [5], and the results of Kolesik and Moloney [6]. Indeed, in many cases the authors appeared to be unaware of each other's results.…”

Ultrashort pulses and short-pulse equations in 2+1 dimensions

“…These equations may find applications in various physical contexts where the study of ultrashort electromagnetic pulses is important; such contexts include nonlinear metamaterials, nonlinear optical waveguide structures, nonlinear dielectric media, and others. Since both SPE-I and SPE-II actually generalize the (1 + 1)-dimensional SPE [12], they can be used for the study of transverse (diffraction-induced) dynamics of ultrashort pulses in such settings. Suitable assumptions on the nature of the electric and magnetic field and the form of the permittivity and permeability under which the equations can be derived were provided.…”

“…Equation (5) is the (2 + 1)-dimensional generalization of the 1D Klein-Gordon type model derived in the context of nonlinear fiber optics [12,23] (in this case, μ = const) and nonlinear metamaterials [13,[24][25][26] (in this case, ∂ ωμ = 0). Below, we will analyze the latter (more general) case, and assume that both permittivity and permeability are frequency dependent.…”

“…From the physical point of view, the interest in the SPE model arises from the fact that its few-cycle pulse solutions have been shown to compare more favorably to the ones of the original Maxwell's equations, as compared to pertinent solutions of the more traditional nonlinear Schrödinger (NLS) model [12,14]. Furthermore, this model is also interesting from a mathematical point of view, due to the existence of an infinite hierarchy of conserved quantities [15], its connection to the sG model and, thus, to its complete integrability [16].…”