Solitons, modes of nonlinear waveguides, and saturable nonlinear couplers

Abstract

This thesis is concerned with solitons, modes of nonlinear waveguides and nonlinear dynamics in saturable nonlinear couplers. Firstly, dark solitons in two and three dimensions are considered. In the case of two dimensional dark solitons it is found that two-photon absorption (intensity dependent loss) leads to broadening and attenuation of dark solitons, whereas gain is shown to lead to their narrowing and amplification. The relations describing the adiabatic evolution of dark solitons in weakly perturbed media are presented and shown to be in excellent agreement with numerical results. In comparison with fundamental bright solitons, dark solitons are shown to be less sensitive to the perturbations. An exact analytical stationary solution representing a dark soliton in the presence of both two-photon absorption and Raman gain is also obtained. In the case of three-dimensional dark solitons (vortex solitons) a Hamiltonian approach is used to find an approximate stationary solution. It is found that in the presence of loss, gain or two-photon absorption the soliton evolves adiabatically. Also, stationary solutions (vortex solitons) in the presence of gain and two-photon absorption are found numerically. Secondly, the stability of vortex-like stationary solutions of (2+1 )-dimensional nonlinear Shrödinger equation in a self-focusing saturating nonlinearity is investigated analytically and numerically. These solutions represent three dimensional self-trapped beams with a dark spot surrounded by bright rings of varying intensity. Using an operator-theoretic approach, It is shown that the fundamental bound state of the family is stable to a symmetric perturbation but unstable to an asym m etric perturbation (that breaks the azimuthal symmetry of the beam, i.e. transverse instabilities). The higher-order states are also found to display transverse (m odulation) instabilities. The development of the instabilities is shown to lead to emission of filaments which spiral away as the beam propagates. The number of these filaments are predicted using numerical stability analysis the results of which are in complete agreement with the numerical beam propagation. Thirdly, Modal characteristics of light waves trapped in a thin self-defocusing film bounded by an infinite self-defocusing cladding, whose nonlinear coefficient is different from that of the film, are investigated. It is shown that both gray and dark solitary waves can be trapped in the film when the linear refractive index of the film, nf, is smaller than that of the cladding, n₀, and the effective index, nₑ, is nf < nₑ <n₀. On the other hand, a series of trapped dark oscillatory solitary waves results when nₑ < min{nf, n₀}. Also, stability of the fundamental asymmetric mode trapped in a thin linear film bounded by self-focusing media is investigated numerically. The discrepancy between numerical and analytical methods, which has been reported in the literature, is resolved. The existence of a class of quasi-periodic solutions resulting from perturbation of the asymmetric modes is demonstrated. In addition, the effect of loss on the propagation characteristics of the asymmetric modes is investigated. Finally, nonlinear couplers composed of two different saturable nonlinear cores are examined. It is found that the detrimental effect of nonlinear saturation in switching can be greatly reduced when the cores of the coupler are nonlinearly mismatched. As a result of the presence of both linear and nonlinear mismatch novel bifurcation diagrams emerge which are utilized to design new devices. In addition, the switching behavior of a saturable nonlinear coupler with gain is investigated. It is found that nonlinear mismatch can improve the switching characteristics of active saturable nonlinear couplers. Show more