Exact solution of the pi-pulse problem

John A. Armstrong, E. Courtens

The exact solution describing steady-state resonant pulse propagation in a two-level inverted medium is given. This steady-state pulse is often referred to as a 180° or π-pulse. No approximation of a slowly varying envelope is made, and hence the solution satisfies the second-order wave equation as well as the equations of motion of the density matrix of the system. As a result the theory is able to describe very short pulses, whose width can be of the order of the period of the optical carrier. The model for the system includes a nonresonant loss and a finite value of the transverse relaxation time T2, but neglects dispersion in the host medium. The steady-state pulse is found to be frequency modulated and asymmetric in shape.

Joshua E. Rothenberg, D. Grischkowsky, Anne C. Balant

R. Michalska-Trautman

The generation of an optical breather in coherent, resonant pulse propagation is studied. The initial field envelope is assumed to be in the form of a hyperbolic secant, with one half inverted so that the total area of the pulse is equal to zero. The Maxwell–Bloch equations for an inhomogeneously broadened atomic line are solved by the inverse-scattering method for all values of the independent variables x and t. The scattering amplitudes are found for any value θ of the absolute area of the initial pulse. For θ = 2πn the reflection coefficients are rational functions of the spectral parameter, and the inverse problem can be solved exactly. It is shown that the x-dependent poles of the reflection coefficients are responsible for the transient pulse shape. Such poles appear only in the presence of inhomogeneous broadening. The threshold for transparency is found at θ = 2π. At this value of the area, a breatherlike decaying solution is obtained. At θ = 4π a breather appears. Its energy amounts to approximately 70% of the initial pulse energy. For θ = 2π and 0 = 4π the solutions are analyzed in detail.

K. T. Hecht

Piotr Garbaczewski, Robert Olkiewicz

Boris I. Mantsyzov

The gap 2\ensuremath{\pi} pulse or gap solitary wave of self-induced transparency at an inhomogeneously broadened line of a resonant Bragg periodical structure has been found theoretically as an exact solution of the coupled-mode Maxwell-Bloch equations. In the case of a slight violation of the Bragg condition at exact two-level resonance, the analytical solution for the gap 2\ensuremath{\pi} pulse with phase modulation has been obtained. Furthermore, the existence of an oscillating pulse that periodically changes in amplitude and in the sign of its velocity has been numerically shown.