Radiation trapping

Radiation trapping, imprisonment of resonance radiation, radiative transfer of spectral lines, line transfer or radiation diffusion is a phenomenon in physics whereby radiation may be "trapped" in a system as it is emitted by one atom and absorbed by another.
Classical Description.
Classically, one can think of radiation trapping as a multiple scattering phenomena, where a photon is scattered off of multiple atoms in a cloud. This motivates treatment as a diffusion problem. As such, one can primarily consider the mean free path of light, defined as the reciprocal of the density of scatterers and the scattering cross section.
formula_1
One can assume for simplicity that the scattering diagram is isotropic, which ends up being a good approximation for atoms with equally populated sublevels of total angular momentum. In the classical limit, we can think of the electromagnetic energy density as what is being diffused. So, we consider the diffusion constant in three dimensions
formula_2
where formula_3 is the transport time. The transport time accounts for both the group delay between scattering events and Wigner's delay time, which is associated with an elastic scattering process. It is written as
formula_4
where formula_5 is the group velocity. When the photons are near resonance, the lifetime of an excited state in the atomic vapor is equal to the transport time, formula_6, independent of the detuning. This comes in handy since the average number of scattering events is the ratio of the time spent in the system to the lifetime of the excited state (or equivalently, the scattering time). Since in a 3D diffusion process the electromagnetic energy density spreads as formula_7, we can find the average number of scattering events for a photon before it escapes.
formula_8
Finally, the number of scattering events can be related to the optical depth formula_9 as follows. Since formula_10, the number of scattering events scales with the square of the optical depth.
Derivation of the Holstein Equation.
In 1947, Theodore Holstein attacked the problem of imprisonment of resonance radiation in a novel way. Foregoing the classical method presented in the prior section, Holstein asserted that there could not exist a mean free path for the photons. His treatment begins with the introduction of a probability function formula_11, which describes the probability that a photon emitted at formula_12 is absorbed within the volume element formula_13 about the point formula_12. Additionally, one can enforce atom number conservation to write
formula_15
where formula_16 represent the number increase and decrease in population of excited atoms and formula_17 is the number density of excited atoms. If the reciprocal lifetime of an excited atom is given by formula_18, then formula_19 is given by
formula_20
Then formula_21 is then obtained by considering all other volume elements, which is where the introduction of formula_22 becomes useful. The contribution of an outside volume formula_23 to the number of excited atoms is given by the number of photons emitted by that outside volume formula_23 multiplied by the probability that those photons are absorbed within the volume formula_13. Integration over all outside volume elements yields
formula_26
Substituting formula_21 and formula_19 into the particle conservation law, we arrive at an integral equation for the density of excited atoms - the Holstein equation.
formula_29
Finding the Escape Probability of Photons from the Holstein Equation.
Now to find the escape probability of the photons, we consider solutions by ansatz of the form
formula_30
Observing the Holstein equation, one can note that these solutions are subject to the constraint
formula_31
Aided by the exchange symmetry of formula_32, namely that formula_33, one can use variational methods to assert that from formula_34,
formula_35
Completing the square and introducing the escape probability formula_36, whose definition follows from that all particles must either be absorbed or escape with a summed probability of 1, an equation in terms of the escape probability is derived.
formula_37
Numerical Methods for Solving the Holstein Equation.
Many contemporary studies in atomic physics utilize numerical solutions to Holstein's equation to both show the presence of radiation trapping in their experimental system and to discuss its effects on the atomic spectra. Radiation trapping has been observed in a variety of experiments, including in the trapping of cesium atoms in a magneto-optical trap (MOT), in the spectroscopic characterization of dense Rydberg gases of strontium atoms, and in lifetime analyses of doped ytterbium(III) oxide for laser improvement.
To solve or simulate the Holstein equation, the Monte Carlo method is commonly employed. An absorption coefficient is calculated for an experiment with a certain opacity, atomic species, Doppler-broadened lineshape, etc. and then a test is made to see if the photon escapes after formula_38 flights through the atomic vapor (see Figure 1 in the reference).
Other methods include transforming the Holstein equation into a linear generalized eigenvalue problem, which is more computationally expensive and requires the usage of several simplifying assumptions - including but not limited to that the lowest eigenmode of the Holstein equation is parabolic in shape, the atomic vapor is spherical, the atomic vapor has reached a steady state after the near-resonant laser has been shut off, etc.