Spatial Characterization Of Two-Photon States - GAP-Optique

5. Spatial correlations in Raman transitions
with
F = 4AB + (A + B) w2 g
A + B + w 2 g
G = 4CD + (C + D) w2 g
C + D + w2 . (5.17)
g
As in the previous chapter, this mode function can be written as a superposition
of spherical harmonics
Φs (qs, θs) = (2π)
−1/2
ls
als (qs) exp (ilsθs) . (5.18)
The probability of having a Stokes photon with oam equal to ls is given by the
weight of each spiral mode
F + G
C2ls = (F G)1/2 dqsqs exp − q
4
2
s I 2
G − F
ls q
8
2
s (5.19)
where Ils are the Bessel functions of the second kind. Only even modes appear
in the distribution as a consequence of the symmetry of the Stokes mode
function.
Figure 5.3 shows the weight of the mode ls = 0 as a function of the angle
of emission for different values of the length of the cloud of atoms. For nearly
collinear configurations, the probability of having a Gaussian Stokes photon is
one, therefore, in this case the process satisfies the relation lp+lc = ls +las. For
these kind of configurations, A = D in equation 5.12, and therefore the Stokes
spatial mode function shows cylindrical symmetry in the transverse planes
(xs, ys) and (xas, yas). This is the case in most experimental configurations
[59, 64, 68], and in fact, reference [23] experimentally proves the relationship
lp + lc = ls + las for collinear configurations.
The probability of having a Gaussian Stokes photon decreases as other
modes appear in the distribution in highly noncollinear configurations. The
length of the cloud controls the importance of the other modes and it is possible
to obtain a spatial mode with cylindrical symmetry for any emission angle, by
fulfilling the condition
L = R 1 + 2R2
w2 −1/2
. (5.20)
p
This condition reduces to L = R, when the beam waist is much larger than the
transverse size of the cloud. Any deviation from a spherical volume of interaction
(described by this condition) introduces ellipticity in the mode function.
For this reason, a highly elliptical configuration, as the one described in reference
[60], will not satisfy the oam selection rule.
Figure 5.4 shows the weight of the mode ls = 0 as a function of the length
of the cloud for different values of the emission angle. For any angle, the
probability of having a Gaussian mode is maximum at the length given by
equation 5.20. In a collinear configuration, the Stokes photon always has a
Gaussian distribution, independently of the length of the cloud. As the angle
of emission increases, the length of the cloud becomes more important. The
mode weight is weakly affected by the change of the cloud length when the
length is much longer than the other relevant parameters: ws, wg and R. This
is especially evident for an angle of emission ϕ = 90◦ .
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