Spatial Characterization Of Two-Photon States - GAP-Optique

4.2. Effect of the pump beam waist on the OAM transfer
which the spdc parameters satisfy the relationship
γ 2 L 2 w
=
2 pw4 i
w4 i + 4w2 p(w2 i + w2 p) cos ϕ2 s
. (4.6)
In any other case, the coefficients of the variables q x s and q y s are different, and
the signal mode function becomes elliptical. The ellipticity of the spatial profile
implies the presence of non-Gaussian modes, and therefore it can be used as a
qualitative probe that the selection rule is not fulfilled (lp = ls + li).
As the ellipticity is an effect of the partial detection, it can be controlled
by changing the total shape of the cone, or by changing the detector angular
acceptance. The phase matching conditions define the shape of the cone as a
function of the emission angle ϕs, the pump beam waist wp, and the length of
the crystal L; the detector angular acceptance is a function of the waist of the
spatial modes ws, wi. The next two sections use both the mode decomposition
and the ellipticity to describe the effect of all these parameters on the signal
oam content in a more general scenario than considered in this section.
4.2 Effect of the pump beam waist on the OAM transfer
Of all the spdc parameters that affect the signal ellipticity, the easiest to
control is the pump beam waist. To change the angle of emission or the crystal
length implies changing of the geometrical configuration. While changing the
pump beam waist just requires adding lenses in the beam path. In the first
part of this section, numerical calculations show the role of the pump beam
waist on the signal oam content. The second part describes the experimental
corroboration of this effect.
4.2.1 Theoretical calculations
Consider a spdc configuration as the one in equation 1.31. With a Gaussian
pump beam and an idler photon projected into a Gaussian mode, the oam content
of the signal photon can be used to describe the oam transfer mechanism
in spdc. The selection rule is fullfilled only if ls = 0. Equivalently, one could
choose to use the idler photon to study the oam transfer after projecting the
signal into a Gaussian mode.
In a degenerate type-i spdc process characterized by the parameters in
table 4.1, a Gaussian pump beam, with wavelength λ 0 p = 405 nm illuminates
a 10 mm ppktp crystal. The crystal emits signal and idler photons with a
wavelength λ 0 s = λ 0 i = 810 nm. Both photons propagate at ϕs,i = 1 ◦ , after
the crystal they traverse a 2f system, and finally the idler photon is projected
into a Gaussian mode with wi → ∞, so that only idler photons with qi = 0
are considered.
Figure 4.1 shows the signal oam content for two values of the pump beam
waist: wp = 100 µm in the left and wp = 1000 µm in the right. In the
distributions, each bar represents a mode ls, with a weight in the distribution
Cls given by the height of the bar. In the left part of the figure, where the
pump beam waist is smaller, the distribution shows several modes. For larger
waists, the Gaussian mode becomes the only important mode, as seen in the
right part of the figure.
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