Solution of the Holstein equation of radiation trapping by the geometrical quantization technique. III. Partial frequency redistribution with Doppler broadening

Abstract

We introduce an analytical method to investigate radiation trapping problems with Doppler frequency redistribution. The problem is formulated within the framework of the Holstein-Biberman-Payne equation. We interpret the basic integro-differential trapping equation as a generalized wave equation for a four-dimensional (4D) classical system (an associated quasiparticle). We then construct its analytical solution by a semiclassical approach, called the geometrical quantization technique (GQT). Within the GQT, it is shown that the spatial and frequency variables can be separated and that the frequency part of the excited atom distribution function obeys a stationary Schr\"odinger equation for a perturbed oscillator. We demonstrate that there is a noticeable deviation of the actual spectral emission profile from the Doppler line in the region of small opacities. The problem of calculating the spatial mode structure and the effective radiation trapping factors is reduced to the evaluation of wave functions and quantized energy values of the quasiparticle confined in the vapor cell. We formulate the quantization rules and derive the phase factors, which allow us to obtain analytically the complete spectrum of the trapping factors in 1D geometries (layer, cylinder, sphere) and other (2D and 3D) geometries when the separation of space variables is possible. Finally, we outline a possible extension of our method to treat radiation trapping effects for more general experimental situations including, for instance, a system of cold atoms.