SOLUTION OF THE HOLSTEIN EQUATION OF RADIATION TRAPPING IN ONE-DIMENSIONAL GEOMETRIES BY THE GEOMETRIC-QUANTIZATION TECHNIQUE

We solve the Holstein equation of radiation trapping in an atomic vapo r cell by the geometric quantization technique (GQT). In the GQT, the rate equation for the excited-state density is transformed into an ''e quivalent'' Schrodinger equation for an associated quasiparticle. The problem of finding the complete set of radiation escape factors is thu s reduced to searching quantized energy values for the quasiparticle l ocked into the vapor cell. We combine already known solutions for the trapping factors at high opacities with new results for the phase jump at the vapor cell boundary to arrive, within the framework of the GQT technique, at solutions that are valid at all opacities. The phase fa ctors are independent of geometry, and we derive an explicit represent ation for them by the Wiener-Hopf technique in the most simple geometr y, a half-space. Our approach enables an analytical computation of the trapping factors in all practically occurring line shapes (including Voigt lines and hyperfine-split lines), all opacities, and all modes i n one-dimensional (1D), 2D, and 3D geometries allowing for variable se paration. We present results obtained in 1D geometries at all opacitie s that show discrepancies within 5% for the lowest-order trapping fact or and even less (at the level of approximately 0.1%) for higher-order modes, in agreement with the predictions of the GQT theory that has b een developed.