Spatial Characterization Of Two-Photon States - GAP-Optique

where
5.2. Orbital angular momentum correlations
Under these conditions, the mode function Φq (qs, qas) can be written as
Φq (qs, qas) = (ABCD)1/4
π
× exp − A
4 (qx s + q x as) 2 − B
4 (qx s − q x as) 2
× exp − C
4 (qy s + q y as) 2 − D
4 (qy s − q y as) 2
A = w2 pR2 2R2 + w2 +
p
w2 s
2
B = w2 s
2
C = w2 s
2
D = w2 pR 2 cos 2 ϕs
2R 2 + w 2 p
(5.12)
+ L 2 sin 2 ϕs + w2 s
. (5.13)
2
The specific characteristics of the state are determined by the size of the atomic
cloud in the longitudinal and transverse planes, L and R; by the beam waist of
the pump and control beams, wp,c; by the waist of the Stokes and anti-Stokes
modes ws; and by the angle of emission ϕs. The next section describes the
effect off all these parameters on the oam transfer from the pump and the
control beams to the pair Stokes and anti-Stokes.
5.2 Orbital angular momentum correlations
Since the pump and control beams are Gaussian beams, lp = lc = 0, the
analysis of the oam transfer reduces to the study of the oam content of the
Stokes and anti-Stokes pair. The Stokes oam content becomes the only free
variable, after projecting the anti-Stokes into a Gaussian mode
u(qas) = Nas exp
− w2 g
4 (qx2 as + q y2
as)
(5.14)
with beam width wg at the center of the cloud. The Stokes mode function
defined as
Φs (qs) = dqasΦ (qs, qas) u(qas) (5.15)
becomes
(F G)(1/4)
Φs (qs) = exp −
1/2
(2π) F
4 (qx s ) 2 − G
4 (qy s ) 2
, (5.16)
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