Standard and embedded solitons in nematic optical fibers

Abstract

A model for a non-Kerr cylindrical nematic fiber is presented. We use the multiple scales method to show the possibility of constructing different kinds of wavepackets of transverse magnetic (T M ) modes propagating through the fiber. This procedure allows us to generate different hierarchies of nonlinear partial differential equations (P DEs) which describe the propagation of optical pulses along the fiber. We go beyond the usual weakly nonlinear limit of a Kerr medium and derive a complex modified Korteweg-de Vries equation (cmKdV) which governs the dynamics for the amplitude of the wavepacket. In this derivation the dispersion, self-focussing and diffraction in the nematic are taken * Fellow of SNI, Mexico † Correspondence author. E-mail: zepeda@fisica.unam.mx ‡ Fellow of SNI, Mexico 1 into account. It is shown that this cmKdV equation has two-parameter families of bright and dark complex solitons. We show analytically that under certain conditions, the bright solitons are actually double embedded solitons. We explain why these solitons do not radiate at all, even though their wavenumbers are contained in the linear spectrum of the system. We study (numerically and analytically) the stability of these solitons. Our results show that these embedded solitons are stable solutions, which is an interesting property since in most systems the embedded solitons are weakly unstable solutions. Finally, we close the paper by making comments on the advantages as well as the limitations of our approach, and on further generalizations of the model and method presented.

Moving embedded lattice solitons

“…At first, ES were found in nonlinear-optical models including quadratic (χ (2) ) nonlinearities [1]- [6], and later they were discovered in hydrodynamic [7] and liquidcrystal dynamical models [8]. Quite recently, ES were found in discrete systems too, viz., in a finite-difference version of a higher-order NLS (nonlinear Schrödinger) equation [9], and in a model of an array of linearlycoupled waveguides with the χ (2) and χ (3) (cubic) nonlinearities [10].…”

“…It is worth observing that the coefficient in front of the nonlinear term in the cmKdV equation may be positive or negative, depending on the physical application. For example, in the description of light propagation in liquidcrystal waveguides, this coefficient is positive [8]. On the other hand, the derivation of the cmKdV equation for strongly dispersive waves in a weakly nonlinear medium by means of a multiple-scale expansion, presented in Ref.…”

“…If ρ − 2δ = 6 Eq. (2) is precisely the cmKdV equation studied in [8]. It is worth observing that the coefficient in front of the nonlinear term in the cmKdV equation may be positive or negative, depending on the physical application.…”

“…The ES may be stable against small perturbations in the linear approximation, but general arguments and direct simulations demonstrate that they are usually unstable at the second order, which is why they are often called "semi-stable" solutions. Nevertheless, examples of completely stable ES have been also found [8,13,15], and their stability has been demonstrated in Refs. [13,15].…”

“…Motivated by the recent discoveries of continuous families of ES in the complex modified Korteweg-de Vries (cmKdV) equation [8] and in a generalized third-order nonlinear Schrödinger equation [13,15], in the present paper we will consider a differential-difference equation of the following form:…”