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Physical Review A

Monte Carlo study of population and alignment relaxation by trapped line radiation

Akiyoshi Hishikawa, Takashi Fujimoto, Peter Erman

Published Jul 1, 1995• 16 citations
Publisher

Abstract

Population and alignment relaxation for the Doppler broadening case in the infinite plane parallel geometry has been studied in the range of the optical thickness at the line center between ${10}^{\mathrm{\ensuremath{-}}3}$ and ${10}^{2}$. The obtained population and alignment relaxation rates are compared with analytical estimates. The validity of two approximations used in the estimates, i.e., the complete frequency redistribution of absorbed and successively emitted photons and the pure-parabolic distribution of excited atoms, is discussed. Also studied is the mixing of the population and the alignment, which is due to the spatial anisotropy of the geometry. This mixing leads to the self-alignment effect, which develops at a sufficiently late time after excitation. A separate simulation with the alignment effect omitted has revealed that, at a large optical thickness, the population relaxation rate agrees with the result with the alignment effect included. It is therefore concluded that the previous estimate on the influence of the self-alignment to the population relaxation rate is too large. The velocity distribution of the excited atoms in the direction perpendicular to the planes shows an effectively lower (higher) temperature at the center (boundary) than the base temperature (\ensuremath{\approxeq}5%) under the self-alignment condition.

DOI
10.1103/physreva.52.189

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Self-trapping of resonance radiation in gadolinium chloride is experimentally and theoretically investigated at 77\ifmmode^\circ\else\textdegree\fi{}K. The dependence of the self-reversed line shapes on crystal thickness is used to distinguish between different elementary line profiles. By comparing the observed dependence of the peak separation on crystal thickness with theory, it is shown that the lines have Lorentzian profile. This indicates that the elementary profile of the lines is determined by lifetime broadening due to relaxation processes. The mean free path and width of the lines are also obtained by this method. A direct observation of the line profile seems to be difficult in the present case. The Boltzmann equation for transport of photons in planar geometry is solved numerically for Lorentzian and Gaussian line profiles. The dependence of line shape on crystal thickness and on direction of observation is calculated. Results of calculations of the amplification of nonresonant lines accompanying the resonance emission, and the lengthening of the period of the fundamental decay mode of the resonance radiation in the crystal, are also presented.

Physical Review•Mar 1, 1965•12 citations
Population and alignment relaxation for the Doppler broadening case in the infinite plane parallel geometry has been studied in the range of the optical thickness at the line center between 10\ensuremath−3{10}^{\mathrm{\ensuremath{-}}3}10\ensuremath−3 and 102{10}^{2}102. The obtained population and alignment relaxation rates are compared with analytical estimates. The validity of two approximations used in the estimates, i.e., the complete frequency redistribution of absorbed and successively emitted photons and the pure-parabolic distribution of excited atoms, is discussed. Also studied is the mixing of the population and the alignment, which is due to the spatial anisotropy of the geometry. This mixing leads to the self-alignment effect, which develops at a sufficiently late time after excitation. A separate simulation with the alignment effect omitted has revealed that, at a large optical thickness, the population relaxation rate agrees with the result with the alignment effect included. It is therefore concluded that the previous estimate on the influence of the self-alignment to the population relaxation rate is too large. The velocity distribution of the excited atoms in the direction perpendicular to the planes shows an effectively lower (higher) temperature at the center (boundary) than the base temperature (\ensuremath{\approxeq}5%) under the self-alignment condition.
Abstract The problem of transfer of line radiation in sodium vapor under photoexcitation by laser radiation of a resonance line with wavelength λ = 589.6 nm is numerically solved. The influence of Doppler frequency shifts on photon emission coherent in the atomic reference system is taken into account in the formation of an emission line profile using the model of partial frequency redistribution. In the case of a small optical thickness of a medium τ0 ≤ 1, the Doppler frequency redistribution in the laboratory reference system yields higher intensity values in the emission line profile core in comparison with the model of complete frequency redistribution. For an optically dense medium, thermal motion of atoms leads to an increase of reabsorption in the emission line core, and intensity increases in the line wings as compared to the complete redistribution model. The dependence of the Holstein trapping factor on optical thickness is closer to the analytic dependence than the complete redistribution model. This is explained by the fact that the frequencies of emitted photons in the laboratory reference system fall into the spectral profile core, which increases the effects of resonance radiation trapping in a dense medium.
Previously derived equations describing recoil effects in saturated absorption by a gas of two-level atoms or molecules are solved numerically for the case of standing plane-wave excitation and large Doppler broadening. The numerical techniques and accuracy estimates are discussed briefly. The lineshapes obtained are well approximated by the superposition of two Lorentzian functions whose width, heights, and separation parametrise the lineshape. The dependence of these parameters on saturation, recoil splitting, and relaxation rates is described and compared with corresponding results in the rate equation approximation and in the weak-probe-beam approximation. Saturation broadening depends strongly on these latter approximations, but very little on recoil splitting. When multiphoton processes are retained in the equations, increasing saturation causes a slight decrease in the fitted separation of the peaks. The ratio of the heights of the two Lorentzians is closely related to the ratio of relaxation rates for the level populations.
We have measured the atomic lifetime of the argon 2 p 4 level by the laser-induced-fluorescence spectroscopy to be 30.5 ns. This contradicts the recent measurement using the high-frequency plused electron-beam excitation. We conjecture that, in their experiment, the radiation trapping not only affects the decay of the upper-level population but also relaxes the alignment initially produced by the electron-collision excitation in the upper-level population. Thus, the directly observed decay of the emission-line intensity is different from the decay of the upper-level population. We have observed the radiative relaxation of alignment on the neon 2 p 2 level which is a complementary effect to coherence narrowing.
We have studied trapping of resonance radiation under conditions of moderate to low line-center optical depths in sodium-argon mixtures. We report measured effective radiative decay rates, which are compared with predictions of the Post [Phys. Rev. A 33, 2003 (1986)] and the Milne [J. Lond. Math. Soc. 1, 40 (1926)] theories of radiation trapping. These theories are expected to be valid for spectral lines with low to moderate line-center optical depths, where the commonly used Holstein [Phys. Rev. 72, 1212 (1947); 83, 1159 (1951)] theory of radiation trapping is expected to break down. The experiment was performed under conditions in which the trapping was dominated by either the Doppler-broadened Gaussian or by the impact-broadened Lorentzian line-shape function. The measured effective radiative decay rates agree well with those predicted by the Milne and Post theories over the range of low to moderate optical depth.
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 2^{2}2S1/2{\mathit{S}}_{1/2}S1/2​ state to either the 6 2^{2}2P3/2{\mathit{P}}_{3/2}P3/2​ (metastable state) at 535 nm or 6 2^{2}2P1/2{\mathit{P}}_{1/2}P1/2​ (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 ground- and 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 ground- and metastable-state densities than would be practical for efficient filter operation. This is done to illustrate several aspects of the radiation trapping processes.
Self-trapping of resonance radiation in gadolinium chloride is experimentally and theoretically investigated at 77\ifmmode^\circ\else\textdegree\fi{}K. The dependence of the self-reversed line shapes on crystal thickness is used to distinguish between different elementary line profiles. By comparing the observed dependence of the peak separation on crystal thickness with theory, it is shown that the lines have Lorentzian profile. This indicates that the elementary profile of the lines is determined by lifetime broadening due to relaxation processes. The mean free path and width of the lines are also obtained by this method. A direct observation of the line profile seems to be difficult in the present case. The Boltzmann equation for transport of photons in planar geometry is solved numerically for Lorentzian and Gaussian line profiles. The dependence of line shape on crystal thickness and on direction of observation is calculated. Results of calculations of the amplification of nonresonant lines accompanying the resonance emission, and the lengthening of the period of the fundamental decay mode of the resonance radiation in the crystal, are also presented.