Andrei I. Maimistov

Present state of self-induced transparency theory

Abstract Recently it has been shown that the traditional McCall-Hahn model of self-induced transparency may be extended to new physical situations taking into account polarization effects. Some of these models are completely integrable and may be solved exactly by the inverse scattering transform method. In this survey we summarize the results which have been obtained in this way. The main part of the review is devoted to the systematic application of the inverse scattering transform technique to self-induced transparency models with degenerate energy levels (two- and three-level configurations), models with nonresonant self-interaction, and models of two-photon resonance. We discuss briefly some theoretical results for the self-induced transparency of polaritons and numerical results for the initial stages of formation of optical solitons.

Solitary electromagnetic wave propagation in the asymmetric oppositely directed coupler

We consider electromagnetic waves propagating in the system of coupled waveguides. One of the system components is a standard waveguide fabricated from nonlinear material having positive refraction index and another component is a waveguide produced from an artificial material having negative refraction index. In this waveguide a wave propagates in the direction opposite to direction of energy flux. We suppose that constituting metamaterial has linear properties. It is found that the coupled nonlinear solitary waves propagating both in the same direction are exist in this oppositely directed coupler due to linear coupling between nonlinear positive refractive waveguide and linear negative refractive waveguide. The corresponding analytical solution is found and it is used for numerical simulation to illustrate that the results of the solitary wave collisions are sensible to the relative velocity of the colliding solitary waves. The effect of linear losses in negative refractive waveguide is considered numerically.

Coherent Pulse Propagation

Propagation of an ultrashort optical pulse is often associated with the self-induced transparency phenomenon. But not always the optical pulse propagation can be described in the frames of the models based on the systems of completely integrable equations. By assuming, however, that the duration of an optical pulse is less than the characteristic relaxation times, we should admit the synchronous interaction of the resonance atoms with the electromagnetic field. Then they say that the system possesses a phase memory. A number of effects were considered in Chapter 2, where the evolution of an ultrashort pulse in the course of its propagation through the medium was not taken into account. These are the coherent transient processes. The phenomena arise when an ultrashort pulse spreads in a system with the phase memory are called the transient processes. The self-induced transparency, examined in the previous chapter, is a typical representative of the coherent transient processes. In the current chapter we shall consider several examples of the similar phenomena, which theory is based on the sets of non-completely integrable equations. Strictly speaking, there are no true solitons in such systems, but steady pulses have soliton-like properties.

Quasi-flat bands in waveguide arrays

Coupled electromagnetic waves propagating in a waveguide array are discussed. This waveguide array can be considered as a one dimensional photon crystal composed from the unit cell containing three different waveguides, namely, two positive refraction index waveguides and one waveguide with negative refraction index. Spectrum of the linear modes of electromagnetic waves in this device contains the quasi-flat band.

Inelastic interaction of ultrashort pulses of polarized light with an ensemble of isolated quantum dots

A system of equations is formulated describing the evolution of a slowly varying envelope of an arbitrarily polarized ultrashort pulse of electromagnetic radiation in a medium with its resonant properties determined by an ensemble of isolated quantum dots. It is assumed that the concentration of quantum dots is small and that the whole system is equivalent to a gas of resonant four-level atoms. Particular solutions are found that correspond to the propagation of a stationary optical pulse. It is shown by numerical solution of the generalized truncated Maxwell-Bloch equations that steady-state propagation is possible only for circularly polarized light pulses, whereas the pulses of arbitrary polarization either decay and experience the dispersion-related broadening or are converted into circularly polarized solitary waves.

Coherent effects in a thin film of metamaterial

The refraction is theoretically considered of ultimately short pulses at interface of two dielectrics that contains a thin film of nonlinear metamaterial. For the model of metamaterial composed of nanoparticles and magnetic nanocircuits (splitring resonators) the equations are obtained suitable for describing the coherent responses of such film. The numerical simulation demonstrates the emergence of oscillatory echo in inhomogeneous system of meta-atoms. It is supposed that the reported methods are applicable for investigation of thin metamaterial films.

Vector π pulse soliton in coherent optical amplifiers

We found a novel type of vector soliton pulse in a medium with linear loss and nonlinear gain from the coherent resonant interaction of light with two-level atoms exhibiting a degenerate upper state.

Propagation of videopulse through a thin layer of two-level atoms possessing permanent dipole moments

The excitation of a thin layer of two-level permanent dipole moment atoms by ultimately short (less than field oscillation period) electromagnetic pulses (videopulse) is observed. The numerical analysis of matter equations free of rotating wave approximation and relaxation reveals a strong affect of local field and Stark effect on temporal behavior oftransmitted field. Specifically it is demonstrated that a dense film irradiated by videopulse emits a short response with a delay much longer even than the characteristic cooperative time of atom ensemble. It is supposed that the local field in the thin layer of permanent dipole atoms is able to re-pump the atomic subsystem. The close analogy to nonlinear pendulum motion is discussed.

Resonant parametric interaction of electromagnetic waves in quadratic nonlinear medium

We derive nonlinear evolution equations describing parametric up- and down-conversion under the conditions when either of the two interacting waves is in resonance with a material transition, e.g., with plasmonic oscillations. Using perturbation theory in the limit of the large wave-number mismatch we drive analytical expressions for two families of quasi-solitonic solutions. If material transition is in resonance with the second-harmonic then the quasi-solitons are of the Nonlinear-Schrodinger type. If the fundamental frequency is in resonance then the reduced system is the paraxial wave equation coupled to the nonlinear classical oscillator. The latter system is integrated analytically and localized solutions are presented.

Inverse Scattering Transform Method

Let us consider the Fourier transform method for solving the linear partial differential equation. It is known that the Fourier transformation converts this equation into a linear ordinary differential equation, which can be easily integrated. Thus we obtain the evolution of Fourier components of the initial data in the problem under consideration. Inverse Fourier transformation yields the solution of initial equation in the integral form. It would be more correct to say that the Fourier transforms method permits to solve the Cauchy problem of a linear partial differential equation. This is the main reason why this method is attractive to apply in mathematical physics.