Spatial Characterization Of Two-Photon States - GAP-Optique
Spatial Characterization Of Two-Photon States - GAP-Optique
Spatial Characterization Of Two-Photon States - GAP-Optique
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
4.2. Effect of the pump beam waist on the OAM transfer<br />
which the spdc parameters satisfy the relationship<br />
γ 2 L 2 w<br />
=<br />
2 pw4 i<br />
w4 i + 4w2 p(w2 i + w2 p) cos ϕ2 s<br />
. (4.6)<br />
In any other case, the coefficients of the variables q x s and q y s are different, and<br />
the signal mode function becomes elliptical. The ellipticity of the spatial profile<br />
implies the presence of non-Gaussian modes, and therefore it can be used as a<br />
qualitative probe that the selection rule is not fulfilled (lp = ls + li).<br />
As the ellipticity is an effect of the partial detection, it can be controlled<br />
by changing the total shape of the cone, or by changing the detector angular<br />
acceptance. The phase matching conditions define the shape of the cone as a<br />
function of the emission angle ϕs, the pump beam waist wp, and the length of<br />
the crystal L; the detector angular acceptance is a function of the waist of the<br />
spatial modes ws, wi. The next two sections use both the mode decomposition<br />
and the ellipticity to describe the effect of all these parameters on the signal<br />
oam content in a more general scenario than considered in this section.<br />
4.2 Effect of the pump beam waist on the OAM transfer<br />
<strong>Of</strong> all the spdc parameters that affect the signal ellipticity, the easiest to<br />
control is the pump beam waist. To change the angle of emission or the crystal<br />
length implies changing of the geometrical configuration. While changing the<br />
pump beam waist just requires adding lenses in the beam path. In the first<br />
part of this section, numerical calculations show the role of the pump beam<br />
waist on the signal oam content. The second part describes the experimental<br />
corroboration of this effect.<br />
4.2.1 Theoretical calculations<br />
Consider a spdc configuration as the one in equation 1.31. With a Gaussian<br />
pump beam and an idler photon projected into a Gaussian mode, the oam content<br />
of the signal photon can be used to describe the oam transfer mechanism<br />
in spdc. The selection rule is fullfilled only if ls = 0. Equivalently, one could<br />
choose to use the idler photon to study the oam transfer after projecting the<br />
signal into a Gaussian mode.<br />
In a degenerate type-i spdc process characterized by the parameters in<br />
table 4.1, a Gaussian pump beam, with wavelength λ 0 p = 405 nm illuminates<br />
a 10 mm ppktp crystal. The crystal emits signal and idler photons with a<br />
wavelength λ 0 s = λ 0 i = 810 nm. Both photons propagate at ϕs,i = 1 ◦ , after<br />
the crystal they traverse a 2f system, and finally the idler photon is projected<br />
into a Gaussian mode with wi → ∞, so that only idler photons with qi = 0<br />
are considered.<br />
Figure 4.1 shows the signal oam content for two values of the pump beam<br />
waist: wp = 100 µm in the left and wp = 1000 µm in the right. In the<br />
distributions, each bar represents a mode ls, with a weight in the distribution<br />
Cls given by the height of the bar. In the left part of the figure, where the<br />
pump beam waist is smaller, the distribution shows several modes. For larger<br />
waists, the Gaussian mode becomes the only important mode, as seen in the<br />
right part of the figure.<br />
39