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Acta Physica Polonica A

Propagation Effects in Superfluorescence

R. Michalska-Trautman

Published Sep 1, 2016• 1 citation
Publisher

Abstract

The evolution of a superfluorescent pulse is studied in the region of large sample lengths.A solution to the Maxwell-Bloch semiclassical equations is found by using the inverse scattering method.It has the form of a onesoliton solution with varying parameters.Dependence on the pulse width and the retarded time is found without further approximations.These parameters are shown to be independent of the inhomogeneous broadening time.Their values are in good agreement with experiment.Asymmetry of the pulse is also investigated.

DOI
10.12693/aphyspola.130.734

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The evolution of a superfluorescent pulse is studied in the region of large sample lengths.A solution to the Maxwell-Bloch semiclassical equations is found by using the inverse scattering method.It has the form of a onesoliton solution with varying parameters.Dependence on the pulse width and the retarded time is found without further approximations.These parameters are shown to be independent of the inhomogeneous broadening time.Their values are in good agreement with experiment.Asymmetry of the pulse is also investigated.
The inverse scattering method enables a generic nonlinear theory of superfluorescence. In the semiclassical theory used here the process starts from a fluctuating polarization source. The source function is expanded with respect to an appropriate function system, and in this way it can be replaced by a set of normalized random numbers. The Gelfand-Levitan-Marchenko equations are solved approximately. Delay statistics and the pulse shape are studied, and the influence of inhomogeneous broadening is discussed. Results are compared with other theories as well as with experiment.
The fluctuating properties of superradiant and superfluorescent systems are investigated by supplementing the equations of motion for the expectation values of quantum observables with random force terms. In the case of a small atomic system the time evolution of the radiated intensity as well as the intensity fluctuations are computed. We also find the time dependence of the average dipole moment in the presence of chirping. For a pencil-shaped geometry, a generalized system of Maxwell–Bloch equations is analysed. The inhomogeneous broadening is then included. We compare the semiclassical equation for the probability distribution of the atomic variables with the quantum equations based both on normal and antinormal ordering. With the exception of the region in the vicinity of the poles of the Bloch sphere, a close correspondence between the semiclassical and quantum theories is established. The oscillatory superfluorescence is treated similarly to quantum analysis by employing a new method of Lugiato for open systems.
Conservation laws for Maxwell-Bloch equations applied to describe superfluorescence are studied. Modifications due to the initial boundary conditions are discussed. For initial polarization in the form of a constant tipping angle, conservation laws are integrated to give exact dependence of all field moments on the propagation length. Three regions of propagation are distinguished: the linear region, where the field energy is negligibly small, the region of rapid exponential growth, and the saturation region. The characteristic distance, where maximum radiation occurs, is explicitly evaluated; it defines the threshold length for superfluoresence. A small chirping, proportional to the square of the tipping angle, is found. The change of the pulse width with propagation length is estimated.
The coupling of forward and backward propagating waves is studied theoretically in the transition region between superfluorescence (SF) and amplified spontaneous emission (ASE). To accomplish this a computer code has been written that solves Maxwell's equations in time and two spatial dimensions and accounts for counter-propagation, diffraction, and temporal coherence in the Bloch equations. Diffraction is ignored in the present calculations. In the SF regime the coupling is generally weak, as found by others; however, I have found a randomly occurring shot-to-shot coupling that is quite significant in cases where one wave, forward or backward, becomes established from quantum noise first in the amplified rod. In this case the wave that arises second is strongly coupled to the first. In the ASE regime the propagation shows strong forward-backward coupling and the characteristic spatial profile of the inversion that tends to zero at either end and to the small signal value of one in the middle of the amplifier. In the transion region the propagation shows mixed SF and ASE behavior.
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.