Bleaching Wave Phenomena

Abstract

The propagation of a coherent beam in an absorbing medium is described as an energy transport process in which all the transporting particles are moving with the same direction and energy. In many problems involving a stationary medium, the coherent beam is the only significant energy flux. In this case the temperature is governed by a secondorder partial differential equation called the beam equation. If the medium is bleachable (the transport proceeds more freely as the temperature increases), then the temperature profiles exhibit a wavelike propagation. In general, the beam equation can be integrated once leaving a first-order equation. Three known explicit solutions to the firstorder equation are presented: laser heating of a stationary plasma; laser irradiation of a two level substance; and the laser induced breakdown wave. A CONSEQUENCE of the second law of thermodynamics is that any natural process tends to smooth macroscopic nommiformities. The leveling of the nonuniformity may occur by either macroscopic or microscopic motion, or both. If macroscopic motion dominates, then a dynamical process ensues. If microscopic motion dominates, then a transport process ensues. In recent years the development of powerful energy sources such as lasers and electron beams has made it possible to create media having very high temperatures and large temperature gradients. If the energy deposition is rapid, the early stages may be dominated by a transport process (such as thermal conduction) in a stationary medium. In such cases, hydrodynamic motion becomes important only at later times and in fluid media. A transport process may exhibit somewhat different forms involving separate mathematical descriptions. If the mean free path of the transporting particles is small compared to the characteristic distance in which temperature changes significantly, then one sees diffusive transport which is governed by the diffusion equation. However, if the mean free path is comparable to or large compared to the characteristic distance, then one has nondiffusive transport and the mathematical description is not at all obvious. In the extreme case however, the mathematics is very simply described by what may be called the beam equation. This is the limit in which all the transporting particles have the same energy and direction, i.e., a coherent beam. It is the mathematics of coherent transport which is examined in detail in this paper. Of special interest is the case where energy transport proceeds more freely as the temperature increases, leading to bleaching wave phenomena. This kind of phenomenon can appear for both diffusive and coherent transport.