Spatial Characterization Of Two-Photon States - GAP-Optique

3. Spatial correlations and OAM transfer
in which the space and the frequency are not correlated. By writing the mode
function in a different coordinate system, it will be possible to show that a
selection rule exists for the oam transfer in spdc.
According to equation 1.6, all the fields have been defined in terms of a
coordinate system related with their propagation direction. To study the oam
transfer, we start by defining the fields in terms of a unique coordinate system,
the one of the pump. This coordinate transformation is possible since there is
a vector Kn such that kn · rn = Kn · rp.
This coordinate transformation is more general than the one of section 1.3,
where the plane that contained all photons was defined as the yz plane. As
a consequence this section considers all possible emission directions over the
cone, while section 1.3 considered only one.
As was done for the vector kn in section 1.3, it is useful to separate the
vector Kn in its longitudinal component Kz n and transversal component Qn. Taking into account the change of coordinates, the spatial part of the mode
function in the semiclassical approximation can be written as
∆kL
Φq(Qs, Qi) ∝ Ep(Qs + Qi)sinc
(3.9)
2
where the delta factor is given by
with
∆k = K z p(Qs + Qi) − K z s (Qi) − K z i (Qi). (3.10)
K z j (Q) =
ω 2 j n2 j
c 2 − |Q|2 . (3.11)
To study the oam transfer, consider the two functions in equation 3.9 separately.
Consider first the sinc function. Due to its dependence on the delta
factor, the sinc function depends only on the magnitudes |Qs +Qi|, Qs = |Qs|,
Qi = |Qi|. And, since
|Qs + Qi| 2 = Q 2 s + Q 2 i + 2QsQi cos(Θs − Θi), (3.12)
the only angular dependence of the function is through the periodical term
cos(Θs − Θi), therefore, using Fourier analysis it can be written in a very
general form as
∆kL
sinc =
2
∞
m=−∞
Fm(Qs, Qi) exp [im(Θs − Θi)], (3.13)
where, importantly, Fm(Qs, Qi) does not depend on the angles Θs and Θi.
On the other hand, if we consider the pump beam as a Laguerre-Gaussian
beam with a oam content of lp per photon, its mode function can be written
as
32
Ep(Qs + Qi) ∝ exp
− w2 p|Qs + Qi| 2
4
(Q x s + Q x i ) + i(Q y s + Q y
i )
l , (3.14)