Spatial Characterization Of Two-Photon States - GAP-Optique

3.2. OAM transfer in general SPDC configurations
A special case of the lg modes is the zero-order mode that does not carry oam.
Since the zero-order Laguerre polynomial L 0 0 = 1, the zero-order Laguerre-
Gaussian lg00 = 1 is the Gaussian mode given by
w
1
2
LG00(q) = exp
2π
− w2 q 2
4
. (3.4)
This is not, however, the only mode with oam equal to zero, the same is true
for all other modes lg0p. Spiral harmonics are defined to collect all the modes
with the same oam value, regardless of the value of p, these modes are defined
as
LGl(q) =
LGlp(q) (3.5)
p
=al(q) exp [ilθ].
The phase dependence on l, shown by each spiral harmonic mode, is exploited
to determine the photon’s oam content [52]. Consider, for instance, a photon
with a spatial distribution given by Φ(q), which using the spiral harmonic
modes can be written as
∞
Φ(q) = al(q) exp [ilθ], (3.6)
l=−∞
so that each mode in the decomposition has a well defined oam of l per
photon. Therefore, the probability Cl of having a photon with oam equal to l
is the weight of the corresponding mode in the distribution:
where
Cl =
al(q) = 1
√ 2
∞
0
2π
0
dq|al(q)| 2 q (3.7)
dθΦ(q, θ) exp (−ilθ). (3.8)
If the photon has a well defined oam of l0, the weight of the corresponding
mode Cl0 = 1 and the weights of the other modes Cl=l 0 = 0. If Cl = 0 for
different values of l, the photon state is a superposition of those modes with
different oam values. Compared to the relative simplicity of these calculations,
measuring the oam content is a more complicated task described in appendix
C.
Now that the techniques for the calculation of the oam content are introduced,
the next section will use the spherical harmonics and the oam decomposition
to study the transfer of oam from the pump photon to the signal and
idler.
3.2 OAM transfer in general SPDC configurations
The oam carried by a photon is directly associated to its spatial shape. In
order to simplify the description of the oam transfer mechanisms, this section
considers only the spatial part of the mode function in a spdc configuration
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