Spatial Characterization Of Two-Photon States - GAP-Optique

3. Spatial correlations and OAM transfer
Configuration
Weight
1
0
-4 4
0
Spatial
-4 4
distribution OAM content
1
Mode
Figure 3.2: In a collinear configuration a pump beam with oam equal to 1 transfers
its oam into the signal photon when the idler is projected into a Gaussian mode. The
figure shows the pump in blue and the signal and idler in pink. Additionally, it shows
the spatial distribution and the oam content of the signal photon for co-propagating
and counter-propagating configurations.
3.3 OAM transfer in collinear configurations
The aperture of the down-conversion cone is given by the angle of emission
of the photons. In the collinear configuration this angle is zero, and therefore
the cone collapses onto its axis, as shown in figure 1.2 (d). As each photon
propagates in the same direction as all the other photons, equation 3.18 is
valid.
To analyze the transfer of oam from the pump beam to the generated photons,
we will calculate the signal oam content, assuming a lg1 pump beam so
that lp = 1, and an idler photon projected into a Gaussian state li = 0. We
will consider two kinds of collinear configurations: co-propagating and counterpropagating,
both shown in figure 3.2. The collinear counter-propagating generation
of photons requires fulfilling a special phase-matching condition, which
can be achieved by quasi-phase-matching in a periodic structure. Such counterpropagating
generation has been demonstrated in second harmonic generation
[53], parametric fluorescence [54], and spdc in fibers [55]. Chapter 5 explores
another way to generate counter-propagating two-photon states: Raman scattering.
Figure 3.2 shows the calculated signal spatial distribution and oam content
in two cases. The first row shows the collinear co-propagating case. The second
row shows the collinear counterpropagating case.
According to figure 3.2, the collinear case fulfills the selection rule, except
for a minus sign in the counterpropagating case. The phase appears because
the photon’s oam content is evaluated with respect to the pump propagation
direction, as figure 3.3 shows. To take this effect into account equation 3.18
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