Spatial Characterization Of Two-Photon States - GAP-Optique
Spatial Characterization Of Two-Photon States - GAP-Optique
Spatial Characterization Of Two-Photon States - GAP-Optique
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3.2. OAM transfer in general SPDC configurations<br />
A special case of the lg modes is the zero-order mode that does not carry oam.<br />
Since the zero-order Laguerre polynomial L 0 0 = 1, the zero-order Laguerre-<br />
Gaussian lg00 = 1 is the Gaussian mode given by<br />
<br />
w<br />
1<br />
2<br />
LG00(q) = exp<br />
2π<br />
<br />
− w2 q 2<br />
4<br />
<br />
. (3.4)<br />
This is not, however, the only mode with oam equal to zero, the same is true<br />
for all other modes lg0p. Spiral harmonics are defined to collect all the modes<br />
with the same oam value, regardless of the value of p, these modes are defined<br />
as<br />
LGl(q) = <br />
LGlp(q) (3.5)<br />
p<br />
=al(q) exp [ilθ].<br />
The phase dependence on l, shown by each spiral harmonic mode, is exploited<br />
to determine the photon’s oam content [52]. Consider, for instance, a photon<br />
with a spatial distribution given by Φ(q), which using the spiral harmonic<br />
modes can be written as<br />
∞<br />
Φ(q) = al(q) exp [ilθ], (3.6)<br />
l=−∞<br />
so that each mode in the decomposition has a well defined oam of l per<br />
photon. Therefore, the probability Cl of having a photon with oam equal to l<br />
is the weight of the corresponding mode in the distribution:<br />
where<br />
Cl =<br />
al(q) = 1<br />
√ 2<br />
∞<br />
0<br />
2π<br />
0<br />
dq|al(q)| 2 q (3.7)<br />
dθΦ(q, θ) exp (−ilθ). (3.8)<br />
If the photon has a well defined oam of l0, the weight of the corresponding<br />
mode Cl0 = 1 and the weights of the other modes Cl=l 0 = 0. If Cl = 0 for<br />
different values of l, the photon state is a superposition of those modes with<br />
different oam values. Compared to the relative simplicity of these calculations,<br />
measuring the oam content is a more complicated task described in appendix<br />
C.<br />
Now that the techniques for the calculation of the oam content are introduced,<br />
the next section will use the spherical harmonics and the oam decomposition<br />
to study the transfer of oam from the pump photon to the signal and<br />
idler.<br />
3.2 OAM transfer in general SPDC configurations<br />
The oam carried by a photon is directly associated to its spatial shape. In<br />
order to simplify the description of the oam transfer mechanisms, this section<br />
considers only the spatial part of the mode function in a spdc configuration<br />
31