Assessment of a Rubidium ESFADOF Edge-Filter as ... - tuprints

3.6 Extension to Excited State FADOFs: ESFADOFs 55
deliver more insight. Approximating these processes by the effective transfer rate
A 20 is accurate enough for the current disccussion. Hence A 20 is a function of the
transition rates of all possible decay pathways 2 → η ′ → ··· → η → 0, where the
2 → 1 → 0 pathway is explicitly excluded. This yields:
1
A 20 =
1
A
+ ··· + 1 + ··· (3.58)
2η
′ A η0
By inserting data from the Kurucz atomic line database [144] A 20 = 2 MHz results.
Moreover, after inserting the expression for the energy–pooling transfer rate
(Eq. 3.53) into the system of rate equations 3.54–3.57, the nonlinear character
of this process becomes visible. Mixed and quadratic terms of the number densities
appear. These terms suspend the linear behavior of the rate equations. In
addittion, the complexity considerably increases, when extending the description
to the complete Zeeman–splitted case. Hence, estimations can only be computed
by numerical solutions. However, restricting the model to the fine structure only,
allows to estimate the effect of the energy–pooling quenching. Fig. 3.6 has been
obtained by numerically solving the system of the rate equations 3.54–3.57 based
on Rb atoms, while neglecting the spatial dependency. A typical cell temperature
of T Cell = 170 ◦ C has been assumed. The saturation of the vapor is clearly visible
and the energy–pooling quenching of the excited atoms binds 6% of the total
vapor density to the higher excited level 2. This result is of particular importance
for Sec. 5.3.2.
3.6.3 Radiation Trapping
Radiation trapping is always an issue when optically pumping atomic vapor. The
pump radiation, which is scattered off the pump beam by the atomic vapor, does
not leave the atomic vapor directly. In fact, there always exists a non–vanishing
probability that the scattered photons are reabsorbed by the atomic vapor prior to
leaving the vapor. These photons are re–emitted afterwards and can undergo a cycle
of absorptions and re–emissions, before they eventually leave the vapor. Thus,
radiation trapping increases the number density of excited state atoms, as there
is also an important probability that these photons re–enter the primary pumped
regions. Theoretical descriptions of this phenomenon go back to the pioneering
works of Holstein, who published in 1947 a description based on an integro–
differential equation, known as the Holstein–equation [138]:
d
dt
∫V
n(r,t) = −An(r,t) + A n(r ′ ,t)G(r,r ′ )dr ′ . (3.59)
A is the Einstein coefficient, n(r) is the number density of the excited state and
∫
V G(r,r′ )dr ′ is the radiation trapping operator.
Since 1947, radiation trapping has been the subject to extensive research
throughout the years and it is still a fascinating research–field today [139]. A comprehensive
introduction to this subject together with an exhausting survey of this