Coupled Maxwell-pseudospin equations for investigation of self-induced transparency effects in a degenerate three-level quantum system in two dimensions: Finite-difference time-domain study

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Abstract
We extend to more than one spatial dimension the semiclassical full-wave vector Maxwell-Bloch equations for the purpose of achieving an adequate and rigorous description of ultrashort pulse propagation in optical waveguides containing resonant nonlinearities. Our considerations are based on the generalized pseudospin formalism introduced by Hioe and Eberly [Phys. Rev. Lett. 47, 838 (1981)] for treatment of the resonant coherent interactions of ultrashort light pulses with discrete-multilevel systems. A self-consistent set of coupled curl Maxwell-pseudospin equations in two spatial dimensions and time for the special case of a degenerate three-level system of quantum absorbers is originally derived. Maxwell’s curl equations are considered to be coupled via macroscopic medium polarization to the three-level atom model for the resonant medium. Two distinct sets of pseudospin equations are obtained corresponding to the TE- and TM-polarized optical waves. For the case of TM polarization, the electromagnetic wave is polarized in a general direction in the plane of incidence inducing two dipole transitions in a degenerate three-level system by each E-field component along the propagation axis and in transverse direction. We introduce a dipole-coupling interaction Hamiltonian allowing Rabi flopping of the population difference along and perpendicular to the propagation axis with frequencies depending on the corresponding field components. The relationship between the induced polarization and the state vector components that describe the evolution of the discrete-level system is derived in order to couple the quantum system equations to the Maxwell’s curl equations. The pseudospin equations are phenomenologically extended to include relaxation effects by introducing nonuniform decay times corresponding to the various dipole transitions occurring in a three-level system. The system has been discretized using finite differences on a Yee grid and solved numerically by an iterative predictor-corrector finite-difference time-domain method. Self-induced transparency soliton propagation through a degenerate three-level quantum system of absorbers in two spatial dimensions and time is demonstrated in planar parallel-mirror waveguide geometries.