Spatial Characterization Of Two-Photon States - GAP-Optique

3.2. OAM transfer in general SPDC configurations
where the identity Q exp[iθ] = Q x + iQ y was used. Using equation 3.12, the
last equation can be written as
Ep(Qs + Qi) ∝ exp − w2 p(Q2 s + Q2 i + 2QsQi
cos(Θs − Θi))
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l × Qs exp (iΘs) + Qi exp (iΘi)
and by using the binomial theorem it becomes
(3.15)
lp
lp
Ep(Qs + Qi) ∝ Q
l
l=0
l sQ lp−l
i exp [ilΘs + ilpΘi − ilΘi]
× exp − w2 p(Q2 s + Q2 i + 2QsQi
cos(Θs − Θi))
. (3.16)
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By replacing equations 3.16 and 3.13 into equation 3.9, the two-photon spatial
mode function becomes
Φ(Qs, Qi) ∝ exp − w2 p(Q2 s + Q2 i + 2QsQi
cos(Θs − Θi))
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∞
× Gn(Qs, Qi) exp [inΘs + i(lp − n)Θi], (3.17)
n=−∞
where the value of Gn(Qs, Qi) does not depend on Θs and Θi. The index
(lp − n) associated to Θi is fixed for each value of lp and ls, according to the
phase dependence of the last equation. This relation can be written as the
selection rule
lp = ls + li. (3.18)
Therefore, the angular momentum carried by the pump is completely transferred
to the signal and idler photons.
The geometry of the spdc configuration restricts the validity of the selection
rule since it was deduced from equation 3.9, and that equation describes
only the configuration in which all possible emission directions are taken into
account. Collinear configurations achieve this condition by definition, but in
the type-I noncollinear configurations, all the experiments up to now have measured
only a certain portion of the whole cone. Under this condition the transfer
of oam from the pump to the signal and idler is not governed necessarily by
equation 3.18, in agreement with previous reports [32, 30, 31].
Using numerical methods to calculate the oam content of the downconverted
photons, the next section explores the oam transfer mechanisms in
collinear configurations. This configurations are special cases where the generated
photons propagate in the same direction and therefore, the detection
system detects all photons from the emission cone.
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