Spatial Characterization Of Two-Photon States - GAP-Optique

4. OAM transfer in noncollinear configurations
100µ m
1mm
pump crystal
-4 4
-4 4
1
0
1
0
after 1mm
-4 4
-4 4
0.1
0
1
0
output mode
Figure 4.6: Due to the crystal birefringence, the oam content of a beam changes as
the beam passes trough the material. The number of new oam modes introduced
by the birefringence increases for more focused beams. The figure shows the mode
content of a Gaussian beam after traveling 1 mm in the crystal, and at the output of
the 5- mm-long crystal.
to the value of the width in y, and therefore the ellipticity disappears.
4.3 Effect of the Poynting vector walk-off on the OAM
transfer
Up to now, this chapter only considered spdc configurations where the Poynting
vector walk-off was not relevant. However, since the selective detection of
one section of the cone affects the oam transfer, the distinguishability introduced
by the walk-off should affect it as well.
To understand the effect of the walk-off on the oam transfer, consider that
the spatial shape of the pump is modified as it passes trough the crystal due to
the birefringence. A Gaussian photon thus acquires more modes when traveling
in a crystal. This section explains this effect using theoretical calculations and
experimental results.
4.3.1 Theoretical calculations
The displacement introduced by the walk-off, explained in section 1.3, changes
the pump beam spatial distribution and therefore its oam content. The transverse
profile of the pump, at each position z inside the nonlinear crystal, can
be written as
Ep (qp,z)=E0 exp
−q 2 p
w 2 p
4
z
+ i
2k0
∞
Jn (zqp tan ρ0) exp {inθp}
p n=−∞
(4.7)
where Jn are Bessel functions of the first kind. Based on this expression, figure
4.6 shows the oam content of a Gaussian pump beam after traveling through
a 5 mm crystal. New modes appear and become important as the pump beam
gets narrower.
The oam content of the pump with respect to z is different at different
positions inside the crystal. Pairs of photons produced at the beginning or at
the end of the crystal are effectively generated by a pump beam with different
spatial properties, as figure 4.6 shows. This change in the pump beam is
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