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

38 3 (Excited State) Faraday Anomalous Dispersion Optical Filters
the electric field vector E and the resulting frequency dependent Faraday rotation
angle is given by
φ(ω) = ω L
2c [n +(ω) − n − (ω)]. (3.7)
Based on relations 3.6 and 3.7, it is possible to deduce the frequency dependent
transmission of the FADOF device to
∣ T(ω) =
E Out∣∣∣
2
∣
(3.8)
E In
= 1 {
e −k +(ω)L + e −k −(ω)L
4
−2 cos[2φ(ω)]e − 1 2 (k +(ω) + k − (ω)) L } .
In conclusion, the full knowledge of the frequency dependent complex index of
refraction ñ determines the FADOF transmission spectrum. Eq. 3.8 can be decomposed
into the bare atomic absorption
a(ω) = 1 2
{
e −k +(ω)L + e −k −(ω)L } (3.9)
and the cosine–term, which describes, according to Malus–law [92], the projection
of the electric field vector on the basis of the second polarizer A. It is obvious,
that a removal of A transforms Eq. 3.8 into Eq. 3.9. However, the anisotropy
of the index of refraction is based on the applied magnetic field. Thus, a fully
isotropic vapor cell, which is either a consequence of a vanishing magnetic field
or equal circular indices of refraction n ± far from atomic absorption lines, results
in the disappearance of the Faraday rotation angle φ (cf. Eq. 3.7). As a result, the
second polarizer A blocks the light wave, which explains the excellent daylight
rejection of FADOF devices.
3.2.2 Inhomogeneities Along the Propagation
Constant but inhomogeneous complex refractive indices can be easily integrated
in the above formalism. Such inhomogeneities result as a consequence of an inhomogeneous
magnetic field, which affects the atomic transitions along the beam
path. In addition, ESFADOFs are susceptible to develop strong inhomogenieties
along the beam path, due to the necessity of optically pumping the lower
ESFADOF state. The exponential decrease of the pump intensity along the pump
beam trajectory (cf. Fig. 3.1) considerably affects the population of the lower
ESFADOF state and hence the refractive indices. Sec. 3.3 discusses the connection
between both entities and Sec. 3.6 extends the theoretical description to
ESFADOFs.
Let ñ(ω,z) denote the complex refractive indices along e z . Then, according to
Eq. 3.3, any infinitesimal propagation of the electric field vector can be described
as