Spatial Characterization Of Two-Photon States - GAP-Optique

4. OAM transfer in noncollinear configurations
4.1 Ellipticity in noncollinear configurations
In the experimental implementation of spdc, the detection system selects the
photons that are emitted in a certain spatial direction. In a collinear configuration,
the selection is not a problem since all the photons are emitted in the
same direction. But, in a noncollinear configuration, the photons are emitted in
different azimuthal directions described by the angle α, therefore the detection
system selects only a section of the full cone. As the symmetry is broken when
only a portion of the cone is considered, the oam transfer mechanism can not
be described by the selection rule in 3.18. This section studies this mechanism
in a simple spdc configuration by calculating the spatial distribution of the
signal photon after fixing the oam content of the pump and idler photons.
As a first example, consider a Gaussian pump beam lp = 0 and an idler
photon projected into a Gaussian mode li = 0, given by
u(qi) = Ni exp − w2 i
4 (qx2 i + q y2
i )
, (4.1)
the spatial distribution of the signal photon is given by the normalized mode
function
Φs(qs) = Ns dqiΦq(qs, qi)u(qi). (4.2)
This integral has an analytical solution for simple spdc configurations. Consider
a degenerate spdc process with negligible Poynting vector walk-off. After
suppressing the correlations between space and frequency the spatial part of
the two-photon state is given by
with
Φq(qs, qi) = exp
− w2 p
4 ∆2 0 − w2 p
4 ∆2 1
∆0 =q x s + q x i
∆1 =q y s cos ϕs + q y
i cos ϕi
∆k = − q y s sin ϕs + q y
i
sin ϕi.
L
sinc
2 ∆k
(4.3)
(4.4)
Therefore, using the sinc to exponential approximation, the signal mode function
defined by equation 4.2 is given by
Φs(qs) =Ns exp
× exp
−
w2 pw2 i
4(w2 p + w2 qx2 s
i )2
− 4w2 pγ 2 L 2 cos ϕs 2 sin ϕs 2 + (w 2 p cos ϕs 2 + γ 2 L 2 sin ϕs)w 2 i
4(w 2 p cos ϕs 2 + γ 2 L 2 sin ϕs 2 + w 2 i )
q y2
s
(4.5)
When the coefficients of the variables q x s and q y s are equal, the signal mode
function reduces to a Gaussian and the selection rule is fulfilled. These coefficients
are equal in collinear configurations, or in noncollinear configurations in
38
.