Spatial Characterization Of Two-Photon States - GAP-Optique

4. OAM transfer in noncollinear configurations
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Figure 4.1: As the pump beam waist increases, the Gaussian mode in the signal oam
distribution becomes more important. The left part of the figure, where wp = 100 µm,
shows several non-Gaussian modes in blue, while the right part, where wp = 1000 µm,
shows only a Gaussian mode in gray. The other parameters used to generate this
figure are listed in table 4.1.
Table 4.1: The parameters used in the theoretical calculations in section 4.2.
Parameter Value
Crystal ppktp
L 10 mm
ρ0
0◦ Laser cw diode
wp 100 − 1000 µm
λp
405 nm
λ0 s
ϕs
810 nm
1◦ Figure 4.2 shows the probability of finding a non-Gaussian mode as a function
of the pump beam waist. The presence of non-Gaussian modes implies a
violation of the selection rule in the configuration considered here. The figure
shows the probability Cls = 0 for emission angles 1, 5, 10 ◦ . As the collinear
angle ϕ becomes larger, the probability to detect signal photons with ls = 0
increases. In a highly noncollinear configuration a large pump waist does not
guarantee the generation of a Gaussian mode, that is, it does not imply the
validity of the selection rule.
Reference [30] introduced the noncollinear length defined as Lnc = wp/ sin ϕ.
Lnc quantifies the strength of the violation of the selection rule due to the angle
of emission and the pump beam waist. The authors defined this quantity
based on the fact that when Lq y sin ϕ is small the sinc function (that introduces
ellipticity) tends to one. As q y is of the order of 1/wp the condition can be
written as L sin ϕ/wp ≪ 1, or L ≪ Lnc. In this regime the ellipticity of the
mode function is small, and thus the selection rule lp = ls + li is fulfilled. This
is the case for nearly all the experiments using the oam of photons generated
in spdc [7, 12, 28, 56, 58, 25, 26, 29]. Instead, if the crystal length is larger
than or equal to the noncollinear length L ≥ Lnc, a strong violation of the
selection rule is expected. The experiments reported in references [31, 32, 57]
are in this regime since the authors used highly focused pump beams or long
crystals.
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