Influence of the Vector Order Parameter on the Dynamics of 3D Ultrashort Pulses in Carbon Nanotubes

In this paper, we propose a model for taking into account the vector order parameter for studying the evolution of an ultrashort optical pulse in a medium with carbon nanotubes. We consider the electric field of the pulse in the three-dimensional geometry. To describe the dynamics of the considered system we use the phenomenological theory developed by A.Z. Patashinskii and V.L. Pokrovskii. Given the specificity of the problem at hand, it is assumed that the vector order parameter is related to the electric field directed along the nanotube axis. For the clarity, we consider a ferroelectric medium with the polarization, which has three nonzero spatial components. Based on the Ginzburg-Landau theory, we obtain the governing equation of motion for the vector order parameter. Based on Maxwell’s equations, we obtain the effective nonlinear wave equation in a cylindrical coordinate system. We use the approximation when the accumulation of charge can be neglected. Therefore, the cylindrical symmetry in the field distribution is preserved. The system of these two equations allows us to analyze the dependence of the shape of three-dimensional ultrashort optical pulses on the distance from the critical point of phase transition. That important observation can in practice be used to identify experimentally the critical point. We show that ultrashort optical pulse propagates stably without secondary wave radiation. The energy of the electromagnetic pulse is preserved in a localized region. Also a comparative analysis of two models of the order parameter (scalar order parameter and vector order parameter) is performed. The influence of the order parameter model on the dynamics of the pulse is determined. The proposed technique allows us to carry out the spectroscopy by changing the order parameter when probing the medium with ultrashort pulses.