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Physical Review A

Nonlinear propagation of picosecond pulses interacting with a three-level system

J.-Y. Bigot, B. Hönerlage

Published Jul 1, 1987• 9 citations
Publisher

Abstract

We study the evolution in time and in one space coordinate of picosecond laser pulses interacting with a nonlinear medium modeled by three quantum states. This is done by solving numerically the equation of evolution of the density matrix elements together with the Maxwell equation. The nonperturbative treatment of the density matrix allows us to understand the contribution of the different optical transitions and populations to the pulse deformation during its propagation. We apply the model to CuCl where the quantum levels are the fundamental, the excitonic, and the biexcitonic states. Near the two-photon resonance, the pulse is strongly altered and the excitonic and biexcitonic levels show a population inversion. A modulation of the amplitude and a superbroadening of the phase occur during the propagation.

DOI
10.1103/physreva.36.715

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We study the evolution in time and in one space coordinate of picosecond laser pulses interacting with a nonlinear medium modeled by three quantum states. This is done by solving numerically the equation of evolution of the density matrix elements together with the Maxwell equation. The nonperturbative treatment of the density matrix allows us to understand the contribution of the different optical transitions and populations to the pulse deformation during its propagation. We apply the model to CuCl where the quantum levels are the fundamental, the excitonic, and the biexcitonic states. Near the two-photon resonance, the pulse is strongly altered and the excitonic and biexcitonic levels show a population inversion. A modulation of the amplitude and a superbroadening of the phase occur during the propagation.
The theoretical study of the nonlinear dielectric function of three-level systems has shown a memory effect which depends on the inversion of the exciton and biexciton populations created during the excitation and on the relaxation times of these quasiparticles. This effect is studied at low temperatures (2 K) by measuring the time evolution of the transmission of a beam through a CuCl sample. In addition, it has important consequences on the switching properties of bistable devices using CuCl as the active medium.
The coherent propagation of two optical short pulses through a resonant medium consisting of two three-level atoms in the configuration is investigated. A self-consisting analytical solution without steady state or adiabatic approximations is presented. The electric field in the model studied consists of two co-propagating plane waves, each of which is in near resonance with a transition in the absorber. The density matrix of two three level atoms is studied. Also, the reduced density matrices are stated. The present approach for the semiclassical treatment of resonant coherent interactions in three-level atoms represents a generalization of the Maxwell- Bloch equations for a two-level system. The Maxwell-Bloch equations reduced to the non-linear pendulum equation. The solution shows that the two pulses can propagate simultaneously without loss under some conditions about the pulses and the medium. The propagation of the two pulses through the medium without any constrains approves the self -induced transparency (SIT) phenomenon.
The authors develop and apply a novel group-theoretical approach for studying the coherent dynamics of ultrashort pulse propagation in nonlinear optical waveguides and passive semiconductor microresonators. The resonant nonlinearity is modeled by a degenerate three-level system of saturable absorbers in order to allow for a two-dimensional medium polarization. The resulting Maxwell-pseudospin equations are solved in the time domain using the finite-difference time-domain method. Conditions of onset of the self-induced transparency (SIT) regime of propagation are investigated. Numerical evidence of multidimensional solitons localized both in space and in time is given for the planar optical waveguides. Pattern formation and cavity SIT-soliton formation are demonstrated for a passive semiconductor microcavity filled with saturable absorbers.
We consider the problem of the propagation of short-duration light pulses in a two-level medium in the regime of strong interaction, near the resonance, with no restriction on the strength of the coupling between field and medium. We prove that the local variations of the population difference created by the electric field actually induce a nonlinear effect by coupling the fundamental of the field Fourier component to its harmonics. As a result, the multiscale averaging limit of the Maxwell-Bloch system is a new system of coupled nonlinear Schrodinger equations for the slowly varying envelope of the electric field. This system has one integrable limit when the electric field is circularly polarized: it is the scalar nonlinear Schrodinger equation. Moreover, it possesses three constants of the motion and a Hamiltonian formulation. Then we prove that our system is nonintegrable as it does not pass the Painleve test. Finally, a stability analysis shows that it is modulationally unstable and thus it propagates steady pulses of coherent light.
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.
Pulse propagation experiments are performed with coherent picosecond laser pulses in resonance with the donor bound A-exciton (I2) state in CdS. Using bandwidth-limited time-resolved spectroscopy, we investigate the intensity of the transmitted laser pulse as a function of monochromator energy position and time. This method permits the determination of both the amplitude and the phase of the complex electric-field amplitude. Studying the dependence of pulse propagation on the input intensity, for small-area pulses we found at the rear of the sample the spectral hole formation and beating characteristic of a 0π pulse. At large pulse areas close to 2π the pulse propagates nearly unperturbed, showing no oscillatory reshaping or significant absorption losses. This provides clear evidence for self-induced transparency at an excitonic state in solids. Our interpretation is substantiated by an analysis within the framework of the combined Maxwell and optical Bloch equations.