Effects of nonuniform reflective boundaries and line competition on radiation trapping

Abstract

We develop a radiation diffusion equation for an infinite slab geometry where the slab may have different reflectivities on either surface, and for conditions where complete frequency redistribution is valid. Additionally we consider the effect of two competing trapped transitions. Specifically we consider the emission from the thallium 7 S&&2 state to either the 6 P3/2 (metastable state) at 535 nm or 6 Pl~2 (ground state) at 378 nm. High densities of the long-lived metastable level can be produced in a number of ways. The escape factors (which are simply related to the decay rates) and excited-atom spatial profiles are calculated for a variety of conditions for the first several natural modes of the system. The spatial modes for this system are asymmetric. The asymmetry can vary dramatically with only small changes in either of the two lower-state densities. The correct description of the spatial modes is critical in order to determine the overall excited-atom profile accurately. For the simpler systems (i.e. , a single transition and nonreflective boundaries) usually considered the mode profiles do not change appreciably as the optical depth is changed (given conditions of high optical depth and a single-line broadening mechanism). The fundamental mode decay rate is shown for a fixed metastable-state density as a function of the ground-state density. The dependence of the fundamental mode escape factor for the first resonance transition on the optical depth (or ground-state density) is in general more complicated than for the case of a single trapped transition. For conditions considered here a simple power rule cannot be used to scale the decay rates as a function of line-center optical depth. Also shown is the fundamental mode escape factor as a function of buffer gas pressure for fixed groundand metastable-state densities.The calculations presented here allow us to model the narrow-bandwidth thallium fluorescence filter. This filter will be discussed in more detail in a separate paper. A thorough review of atomic resonance filters was done by Gelbwachs [IEEE J. Quantum Electron. 24, 1266(1988]. We have considered a much broader range of groundand metastable-state densities than would be practical for efficient filter operation, This is done to illustrate several aspects of the radiation trapping processes.PACS number(s): 32.50. +d, 32.90.+a

Influence of specularly reflecting boundaries on radiation trapping in a plane-parallel slab

“…At very high opacities, the many absorption-reemission processes difFuse the radiation, regardless of the kind of reflection at the wall. At very low opacities, the trapping factors cannot change by much anyway [10]. Figure 5 shows the difFerence in go as a function of R (=R+ =R ) for low, intermediate, and high opacities for a Doppler line shape.…”

“…( shape. These errors are usually tolerable, so that the often-used method of approximating specularly reflecting walls by diffusely reflecting walls [10,15] can now be justified. One could even use specularly reflecting walls in the rare cases when the walls are actually diffuse, since we can now compute specularly reflecting walls faster.…”

“…These assumptions are discussed in Refs. [5] and [10] 1994 The American Physical Society ping factors, and aj-the expansion coefficients of the initial distribution into the eigenmodes. Equation (2) expresses the fact that an initial distribution of excited atoms n (z, 0) can be expanded into eigenmodes that independently decay with individual exponential time constants that are each eigenmode's trapping factor times the natural lifetime of the excited state.…”

“…However, the linewidth of a resonance line usually is only several GHz, and the optical properties of reflecting materials will rarely change within such a small frequency range. If more than one resonant transition is present, the excited-state distributions can be computed by the method of either [10] or [13]; in both cases we combine results for the single lines (either the Green's functions or the trapping factors and eigenmodes) and can assume R to be frequency independent within each resonant line. More important, the reflection coefFicient may depend on the angle of incidence.…”