Non-linear theory of superfluorescence

The superfluorescence phenomenon-the generation of coherent electromagnetic radiation in an open laser without mirrors-was predicted by Dicke' in 1954 and first observed experimentally by Skribanowitz eta/. ' in 1973. A large number of theoretical and experimental papers has up to the present been devoted to the superfluorescence effect (see, e.g., Refs. 3 to 5 and the reviews 6 to 8). In Refs. 3 ,4 a consistent quantum-mechanical theory of the initial stage of superfluorescence (linear theory of superfluorescence) was developed and it was shown that one can in fact in all stages of the process use the semiclassical Maxwell-Bloch equations taking into account the quantum nature of the electromagnetic field by specifying at the initial time a small normally distributed polarization of the medium. In a medium with an inverted population this polarization starts the development of an instability as a result of which radiation pulses are formed at the ends of the laser. The linear theory enables one to describe the initial stage of the development of these pulses-to construct a statistics of "delay times. " To describe the pulse shape generated for a given initial polarization of the medium it is necessary to go beyond the framework of the linear theory and this is the object of the present paper. We make somewhat idealized assumptions, the most important of which are homogeneity of the problem (neglect of diffraction) and neglect of the homogeneous line broadening. In addition, we consider only waves traveling in one direction. This, at first sight very restrictive, assumption, can, as we show, easily be justified by the nature of the answer found. In the present paper we also neglect inhomogeneous broadening. Strictly speaking, we shall assume that the time for inhomogeneous broadening appreciably exceeds the length of the first pulse. In principle it is possible to take inhomogeneous broadening into account but this requires the development of a more detailed theory using the inverse scattering method. Here we present an elementary theory variant which is based upon joining up the solution of the linearized problem with the well known self-similar solution of the Maxwell-Bloch equations. We show that under reasonable restrictions on the length of the laser the superfluorescence pulse shape turns out to be universal, depending self-similarly on the length of the sample, and is determined by a single undetermined parameter-the pulse delay time which is uniquely connected with the value of the polarization fluctuations at the opposite end of the laser. This conclusion is well confirmed by comparison with laboratory experiments on the observation of superfluorescence in cesium vapor5 and with numerical solutions of the Maxwell-Bloch equation. After that we study the self-similar solution itself which plays an important role not only in superfluorescence problems but also in amplification problem^. ^-'^ Although the self-similar solution cannot be expressed in terms of elementary functions, in the limiting case of interest for the theory of superfluorescence it is possible to obtain a simple explicit expression which well approximates the first pulse.