Single-Photon Atomic Cooling - Raizen Lab - The University of ...

produces a quadrupole magnetic field.
The form of the magnetic field produced in this configuration can be
arrived at by first considering the form of the magnetic field produced by a
single circular loop carrying a current I in a region of permittivity µ which is
given by [56]
where
Bz = µI
2π
1
K(k
(R + ρ) 2 + (z) 2
2 ) + R2 − ρ2 − z2 Bρ = µI z
2πρ (R + ρ) 2 + (z) 2
k 2 =
(R − ρ) 2 + z2E(k2
)
−K(k 2 ) + R2 + ρ2 − z2 (R − ρ) 2 + z2E(k2 )
4Rρ
(R + ρ) 2 + z 2
(2.36a)
(2.36b)
(2.37)
and K(k 2 ) and E(k 2 ) are the complete elliptic integrals of the first and second
kind, respectively [57]. The terms in this expression are clarified in Fig. 2.4.
Figure 2.4: Geometry of Eq. 2.36. The red loop of current produces a magnetic
field at the point (ρ,z). The magnetic field is decomposed into components
parallel to ˆz and to ˆρ.
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