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Physical Review A

Superfluorescence pumped by a short pulse of arbitrary shape

C. T. Lee

Published Jan 1, 1984• 1 citation
Publisher

Abstract

Recognizing that the complete initial inversion required in superfluorescence (SF) can only be achieved in a finite time, we use two different master equations for two different stages of SF: one describing the initiation of SF during pumping and another one governing the free evolution of SF after pumping. The pumping process is treated as a Poisson process with varying rate to reflect a pump pulse of arbitrary shape. The exact solution to the first master equation is obtained and used as the initial condition for the second master equation. The statistical properties of the tipping angle and the delay time are then calculated explicity. It is shown that the effects of pumping process on these two quantities can be reduced to a parameter called effective head-start time (EHT). Two different expressions for this EHT have been derived: one with saturation effects and another one without.

DOI
10.1103/physreva.29.217

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Recognizing that the complete initial inversion required in superfluorescence (SF) can only be achieved in a finite time, we use two different master equations for two different stages of SF: one describing the initiation of SF during pumping and another one governing the free evolution of SF after pumping. The pumping process is treated as a Poisson process with varying rate to reflect a pump pulse of arbitrary shape. The exact solution to the first master equation is obtained and used as the initial condition for the second master equation. The statistical properties of the tipping angle and the delay time are then calculated explicity. It is shown that the effects of pumping process on these two quantities can be reduced to a parameter called effective head-start time (EHT). Two different expressions for this EHT have been derived: one with saturation effects and another one without.
Recent work is reviewed where dynamic diffraction coupling is examined in Superfluorescence SF with use of a one-way semiclassical model in which diffraction and transverse density variations are rigorously included. The Cs data are correctly simulated for the first time in conjunction with both an average tipping angle and quantum fluctuation (either longitudinal or transverse) at the initiating stage. This comprehensive review also encompasses rigorously the effects of pump dynamics on the SF evolution. The terminology is unified. Specification of certain pump beam initial conditions at a given frequency results in specific SF characteristics at another frequency as recently observed in CH3F and Ba. Pump dynamics are studied in both the average value of the tipping angle and quantum fluctuation at the initiation stage.
For a pencil-shaped extended medium with Fresnel number equal to 1, we have quantum mechanically derived a description of the initiation of superfluorescence in terms of Maxwell-Bloch equations with a fluctuating source due to the zero-point fluctuations of the vacuum field. By the introduction of classical behavior, these equations are extended to include nonlinear behavior due to decreasing atomic inversion. The principal assumption in the derivation is that the main features of superfluorescence are governed by the interaction of atoms with field modes inside two small solid angles around the pencil axis. The delay ${\ensuremath{\tau}}_{D}$, defined as the time at which the mean-squared tipping angle of the collective Bloch vector attains the value 1, turns out to be given by ${\ensuremath{\tau}}_{D}=(\frac{{\ensuremath{\tau}}_{R}}{4}){[\mathrm{ln}{(2\ensuremath{\pi}N)}^{\frac{1}{2}}]}^{2}$, where ${\ensuremath{\tau}}_{R}$ is the radiation time for collective decay and NNN is the number of atoms. The corresponding effective initial tipping angle roughly equals 2(N)12\frac{2}{{(N)}^{\frac{1}{2}}}(N)21​2​. A Fokker-Planck equation is derived to describe the statistics of the initial development of the tipping angle. The variance $\ensuremath{\Delta}{\ensuremath{\tau}}_{D}$ of the delay of the superfluorescence pulse satisfies approximately $\ensuremath{\Delta}{\ensuremath{\tau}}_{D}=\frac{2.3}{\mathrm{ln}N}$. A brief comparison with previous treatments of superfluorescence is given.
A model is presented for the dynamical evolution of superfluorescence from an optically pumped three-level system. The full propagation, transverse, and diffraction effects are taken into account. With the use of a previously developed algorithm, a computational simulation was constructed from this model and results are presented and discussed. In particular, it is shown that the injected coherent pump-pulse initial characteristics, such as on-axis area, temporal and radial width and shape, can have significant deterministic effects on the superfluorescent pulse delay time, peak intensity, temporal width, and shape. Thus, by specifying certain initial properties of the injected pump pulse, the superfluorescent pulse can be shaped and altered. The results predict the conditions under which an injected light pulse of a given frequency can be used to generate, shape, and control a second light pulse of a different frequency via a nonlinear medium, thus demonstrating a new aspect of the phenomenon of light control by light.
Numerical simulations of an extended sample of three-level atoms with slow dipole dephasing exhibit a superfluorescent pulse shortly after incoherent pumping begins. The operator Maxwell-Bloch equations are simulated by classical Langevin equations. The counterpropagating fields are solved for in the (one-dimensional) limit of the Fresnel number near unity, with open-end boundary conditions. An initial, ``quiet'' buildup of the inversion is followed by superfluorescence. Oscillation of the populations, coherences, and fields is observed. The superfluorescence delay time is compared with analytical predictions made in the linear regime. A transition to amplified spontaneous emission is found as the pump rate is decreased.