Experiments to Control Atom Number and Phase-Space Density in ...

The right hand side of equation 7.12 consists of a vector (I−ˆz ′ ˆz ′ )·ˆσ− and a scalar ˆσ ∗ −·ˆǫ.
The first solution is hence
ˆǫ1 = γ(I− ˆz ′ ˆz ′ )·ˆσ− = ˆx ′
ξ1 =
Here γ is the normalization constant.
1
ˆσ
γ
∗ − ·ˆǫ = 1
2
(7.13)
(7.14)
The second solution is easily found as it has to be perpendicular to the propaga-
tion direction (ˆz ′ · ˆǫ2 = 0) and the already known solution (ˆǫ ∗ 2 · ˆǫ1 = 0). The solutions
are thus
ˆǫ2 = ˆy ′
(7.15)
ξ2 = 0. (7.16)
Every polarization can be decomposed into the two eigenpolarizations, one inter-
acting with the atoms (ˆǫ1), and one not (ˆǫ2).
In general the incident field is given by
1
Einc = E0 ˆǫ ′ inc = E0√
(ˆǫ
2 ′ 1 ∓iˆǫ ′ 2). (7.17)
Expanding this field in terms of the eigenpolarization leads to
Einc = E0 (ˆǫinc ·ˆǫ ∗ 1)ˆǫ1 +E0 (ˆǫinc ·ˆǫ ∗ 2)ˆǫ2. (7.18)
The field in the object plane of the imaging system can then be derived by applying the
same ansatz to each term:
Eobj = E0(ˆǫ ′ inc ·ˆǫ ∗ 1)exp
−nξ1σ0
2(1+δ 2
(1+iδ) ˆǫ1 +E0(ˆǫ
)
′ inc ·ˆǫ ∗ 2)ˆǫ2
(7.19)
For the incoming polarization in the experimental setup, ˆǫ ′ inc = 1/ √ 2(ˆx ′ ∓iˆy ′ ), the field
in the object plane simplifies to
Eobj = E0
√2 exp
−nξ1σ0
2(1+δ 2
(1+iδ) ˆǫ1 ∓
)
iE0
√2ˆǫ2, (7.20)
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