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

40 3 (Excited State) Faraday Anomalous Dispersion Optical Filters
describes the atomic polarizability of one single atom in the vicinity of its atomic
transitions ( |ω γ,γ ′ −ω| ≪ ω γ,γ ′). The sum over γ ′ includes all allowed transitions,
which emerge from the lower level γ. h = ¯h2π denotes the Planck constant, A JJ’
the Einstein coefficients and Sr
γγ′ their line strengths. Eq. 3.16 is the complex
representation of the Lorentz absorption profile, while the real part describes the
dispersion and the imaginary part the absorption. They are linked through the
Kramers–Kronig relation [133].
The probability P(γ,γ ′ ) of exciting the transition γ → γ ′ , when irradiating the
atom with light of the frequency ω = ω γγ ′, is linked via the oscillator strength
f(γ,γ ′ ) to the line strength Sr
γγ′ , as
P(γ,γ ′ ) = 2ω2 e 2
m e c 3 ∣ ∣f(γ,γ ′ ) ∣ ∣ and (3.17)
f(γ,γ ′ ) = 2m e
3¯he 2 ω γ,γ ′
g γ
S γ,γ ′. (3.18)
e denotes the elementary charge, m e the electron mass and g γ the degeneracy of
the lower level. Eq. 3.17 holds as long as the underlying dipole approximation is
valid, i.e. as long as the wavelength of the exciting light wave exceeds the atomic
dimensions. This assumpton is well satisfied, as optical frequencies exceed atomic
dimensions by several orders of magnitude; e.g. the wavelengths of the atomic
transitions of interest (780 nm and 543 nm; cf. Fig. 3.1) satisfy this assumption
well, as the Rubidium atom does not exceed 5 Å in diameter.
be-
By taking the degeneracies of the atomic transitions into account Sr
γγ′
comes
S γγ′
r
= ∑
MM ′ ∣ ∣∣〈γM|̂dr
∣ ∣ γ ′ M ′〉∣ ∣ ∣
2
, (3.19)
where M and M’ index the degeneracies and ̂d r denotes the dipole operator, which
links the atomic wavefunctions |γM〉 and |γ ′ M ′ 〉. Thus, 〈γM|̂d r |γ ′ M ′ 〉 is the transition
matrix and the sum together with the factor of 1/g γ in Eq. 3.18 results in
an average over all γ sublevels. However, as the applied magnetic field lifts the
degeneracies of the atomic levels, all sublevels have to be taken into account individually
and the line strength Sr
γγ′ becomes
Sr γγ′ ∣ ∣
= ∣〈γ|̂d r γ ′〉∣ ∣2
(3.20)
and g γ = 1. Hence, the total atomic polarizability α tot
r (ω) is achieved by summing
over all ground state levels
α tot
r
(ω) = ∑N(γ)αr(ω). γ (3.21)
γ
The number densities N(γ) account for the population of the level γ and the sum
over γ ′ in Eq. 3.16 refer to all upper state levels.