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}$${\mathit{S}}_{1/2}$ state to either the 6 $^{2}$${\mathit{P}}_{3/2}$ (metastable state) at 535 nm or 6 $^{2}$${\mathit{P}}_{1/2}$ (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.
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}$${\mathit{S}}_{1/2} state to either the 6 ^{2}$${\mathit{P}}_{3/2} (metastable state) at 535 nm or 6 ^{2}$${\mathit{P}}_{1/2} (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.
The decay rate of the Hg 61P1 level was measured as a function of cold spot temperature (Hg density) and buffer gas pressure in cylindrical, sealed fused silica cells. The decay rates were obtained using a time-resolved laser-induced 185 nm fluorescence experiment with multi-step excitation. Cold spot temperatures from 25 to 100 °C were studied. The Hg densities for this temperature range and with no buffer gas yield the lowest possible decay rates due to radiation trapping with partial frequency redistribution. Decay rates with argon buffer gas pressures of 3 and 30 Torr were also studied. The results are in agreement with published data from a discharge afterglow experiment. Monte Carlo simulations of radiation transport in the cells, including the effects of hyperfine and isotope structure, the effects of foreign gas broadening, and partial frequency redistribution are compared to the experimental data. Reasonably good agreement is obtained, however there is evidence of quenching of Hg 61P1 atoms in collisions with ground state Hg and Ar atoms. An analytic formula for the fundamental mode trapped decay rate of the 61P1 level, which is applicable over a substantial region of parameter space, was devised from the Monte Carlo results.
Several calculations have been carried out using the transition operator technique. The radiative decay in a two-level quantum system with excited state coupled by an external perturbation has been investigated and an expression obtained for P(t), the probability of finding the atom in its excited state at time t. This is seen to possess a steady value for yt >> 1 and a third term which decays at nearly half the rate of its second. Next radiative decay of an atom with two close-lying excited states is considered and |bq| 2 , the spectral density for spontaneous emission from the uppermost level, calculated. This shows that the proximity of the upper two levels makes the line-shape non-Lorentxian. Heisenberg equations of motion were then derived for the transitions operators of an ℓ-level atom undergoing radiative decay, the possibility of overlapping pairs of levels being ignored. These master equations were then used to obtain the spectral profiles of certain atomic lines, employing the so-far avoided Markoff approximation. First the spectral density of the scattered radiation was found for transitions between levels 5P3/2 and 4S (ground state), and 6S and 5P 3/2, in the potassium atom when driving fields of arbitrary strength coupled levels 6S and 4P3/2, and 4P 3/2 and 4S. Triple-peaked profiles were obtained. Spectral profiles for emission were then derived for transitions between levels 3P and (ground state), and 3P and in the hydrogen atom when a driving field of arbitrary strength coupled levels The former profile was double the latter quadruple-peaked. Lastly master equations were derived for an atom, with two excited levels undergoing radiative decay, when allowance was made for thepossibility of their overlapping. A driving field of arbitrary magnitude coupled the uppermost and ground states. The spectrum for the scattered light resulting from transitions between the uppermost and ground state was found and graphs computed for various field strengths and separations of the excited states.
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 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.
The ground electronic state of even such simple atoms as those of alkali elements has the set of sublevels. These sublevels are hyperfine (HF) components and magnetic sublevels. The collisions with particles of nonmagnetic buffer gas or even with walls (when walls have proper coating) do not mix practically the populations of ground state HF sublevels, if the orbital moment of electrons in the ground state is equal to zero. Due to this reason these sublevels may be extremely long lived. Because of this, for instance, the so-called effects of optical orientation and optical pumping are apparent in atoms of alkali metals. The radiation can pump practically all the atoms (with which there is effective interaction) to a single HF component. In a situation in which optical pumping is a minor effect, the monochromatic absorption line consists of separate lines corresponding to intra-atomic transitions. The optical pumping is not significant, for example, if the radiation intensity is so low that now pumping between HF sublevels occurs in the time it takes the atom to traverse the light beam. Under the conditions of optical pumping and when other nonlinear effects are present, the absorption line profile can undergo considerable alteration. Many theoretical and experimental investigations have been done in the laser spectroscopy of resonant transitions of atoms (atoms of alkali metals) under the conditions of optical pumping within ground state HF structure. A characteristic feature of all such investigations is the velocity-selective excitation of absorbing atoms. Also, as a rule, the value of HF splitting is not very big in comparison to Doppler broadening in these experiments. Under such conditions, the absorption spectra consists of Doppler- broadened lines with nonlinear resonances, which emerge when two fields of different frequencies or different propagation directions are employed. When only one monochromatic traveling wave is employed, optical pumping leads to formation of a single smooth absorption line occupying a position between the resonant transitions frequencies associated with HF components. As HF splitting grows, the line profile remains smooth, its width is approximately equal to the size of the HF splitting. The strength of the optical pumping from one ground state HF component to another is greater for greater HF splitting in comparison to the Doppler width. In this regard, Cs and Rb are the most typical of the alkali atoms (the HF splitting exceeds the Doppler width by a factor close to 20). This is the qualitative picture of the absorption line profile (particularly for a monochromatic radiation) under the conditions of optical pumping between the HF sublevels of the electronic ground state, this qualitative picture had formed by now.
View Video Presentation: https://doi.org/10.2514/6.2021-3107.vid This work is a portion of a combined theoretical and experimental effort to develop a controllable excited state barium vapor source to be used as a notch filter. The energy pooling properties of the low-lying levels of barium, which make it ideal for this application, have been the subject of several past kinetics studies. This work was utilized to develop a 6-level kinetics model, incorporating collisional and radiative decay and, most critically, diffusional quenching. This model demonstrates efficient pumping of the metastable manifold in the 6s5d 3DJ levels, particularly the 6s5d 3D2 level which will be utilized by the proposed filter. A vacuum vapor cell and its implementation into a counter-propagating, optically pumped absorption spectroscopy arrangement are also described. Measurements of the optically pumping transition (6s2 1S0-- 6s6p 3P01, 791 nm) and the proposed filtering transition (6s5d 3D2 -- 6s5f 3F02, 355 nm) have been performed with this system. To our knowledge, these are the first measurements of this 355 nm filtering transition. These measurements demonstrate a non-Maxwellian lineshape, suggesting a non-thermalized metastable 6s5d 3D2 level. The measured ground state depletion and 3D2 population agree with the model trends but show quantitative discrepancies which require further study. A detailed theoretical analysis of the effects on filter performance of incorporating an argon buffer gas into the vapor source is also presented. This work shows promise for creating a narrowband absorption feature with increased optical depth.
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.