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Physical Review A

Analytical properties of the initial-value problem for zero-area pulses

Sami A. Shakir

Published Oct 1, 1979• 2 citations
Publisher

Abstract

The inverse scattering technique is used to show that the phase term in the Maxwell-Bloch equations should be set to zero when the slowly varying envelope approximation is assumed for self-induced transparency. A solvable model of two square pulses shifted and inverted with respect to each other is considered, and the threshold for "breather" generation for this profile is shown to correspond to an initial absolute area of $\ensuremath{\surd}2\ensuremath{\pi}$ when the separation between the two pulses is zero, and approaches $\ensuremath{\pi}$ as the separation is increased. When the two pulses are identical, only breathers are found. Upper bounds on the energy of the $0\ensuremath{\pi}$ pulses are derived, along with threshold conditions for soliton generation. For an absolute area less than $\ensuremath{\pi}$ there are no bound solutions.

DOI
10.1103/physreva.20.1579

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The inverse scattering technique is used to show that the phase term in the Maxwell-Bloch equations should be set to zero when the slowly varying envelope approximation is assumed for self-induced transparency. A solvable model of two square pulses shifted and inverted with respect to each other is considered, and the threshold for "breather" generation for this profile is shown to correspond to an initial absolute area of \ensuremath√2\ensuremathπ\ensuremath{\surd}2\ensuremath{\pi}\ensuremath√2\ensuremathπ when the separation between the two pulses is zero, and approaches \ensuremathπ\ensuremath{\pi}\ensuremathπ as the separation is increased. When the two pulses are identical, only breathers are found. Upper bounds on the energy of the 0\ensuremathπ0\ensuremath{\pi}0\ensuremathπ pulses are derived, along with threshold conditions for soliton generation. For an absolute area less than \ensuremathπ\ensuremath{\pi}\ensuremathπ there are no bound solutions.
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.
An analysis is made of the propagation of an ultrashort optical radiation pulse under conditions of inhomogeneous broadening of a resonance line. By dispensing with the approximation of a slowly varying complex pulse envelope, constraints on the duration are removed and any phase modulation is automatically taken into account. It is shown that if the boundary conditions determining the attenuation of the pulse at its leading edges are defined, the appropriate reduced Maxwell–Bloch equations can be integrated using the method of the inverse problem of scattering theory. Higher conservation laws, whose existence has been doubted, are found.
We reveal the existence of an optical self-induced-transparency soliton family in a two-level absorbing system down to the few-cycle limit. Based on the few-cycle envelope approximation, we introduce a systematical method of reducing the Maxwell-Bloch equations beyond the slowly varying envelope approximation and characterize the parameter space to achieve slow and fast lights. Verified by direct numerical simulations with full vectorial Maxwell-Bloch equations, we also give the relationships for a number of optical cycles on area theory and pulse group velocity, which demonstrate possible coherent photon-matter interactions.
A theory based on the method of the inverse scattering problem is presented describing propagation of an ultrashort light pulse under the condition that resonant interaction with atoms of the medium is initially switched on by an additional Stark pulse. A Hamiltonian form of the Maxwell-Bloch equation is found without redefining the spatiotemporal variables. The method of the inverse scattering problem is based on a special spectral problem. Equations of the Gel’fand-Levitan-Marchenko type and a soliton solution are obtained.
The generation of 0π pulses in coherent pulse propagation is studied for a simple initial pulse profile consisting of two separated square pulses with one inverted to give a total pulse area of zero. When the two pulses are close together, it is found that the threshold for lossless propagation occurs at an absolute area of 21/2π. But as the pulses become farther separated, one finds two thresholds. The first one occurs for an absolute area of about π and the second one at about 2π. Although one can have lossless propagation occurring for the absolute area between π and 2π, it is accomplished only by generating very low amplitude 0π pulses. It is only after the second threshold, 2π, is achieved that one can expect to find moderate to large amplitude 0π pulses, and these may be accompanied by several low amplitude 0π pulses.