Spatial Characterization Of Two-Photon States - GAP-Optique
Spatial Characterization Of Two-Photon States - GAP-Optique
Spatial Characterization Of Two-Photon States - GAP-Optique
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1. General description of two-photon states<br />
pump<br />
polarization<br />
y<br />
Figure 1.4: The azimuthal angle α between the pump beam polarization and the x<br />
axis defines the position of a single pair of photons on the cone. The x axis is by<br />
definition normal to the plane of emission.<br />
With the origin at the crystal’s center, the z direction parallel to the pump<br />
propagation direction, and the yz plane containing the emitted photons, the<br />
unitary vectors in the coordinate systems of the generated photons transform<br />
as<br />
ˆxs =ˆx<br />
<br />
ˆys =ˆy cos ϕs + ˆz sin ϕs<br />
x<br />
ˆzs = − ˆy sin ϕs + ˆz cos ϕs<br />
ˆxi =ˆx<br />
ˆyi =ˆy cos ϕi − ˆz sin ϕi<br />
ˆzi =ˆy sin ϕi + ˆz cos ϕi. (1.23)<br />
As a single pair of photons defines the yz plane, the coordinate transformations<br />
consider only one of the possible directions of propagation on the cone. The<br />
angle α between the x axis and the pump polarization defines the transverse<br />
position on the cone for one photon pair, as figure 1.4 shows. The pump<br />
polarization is normal to the plane in which the generated photons propagate<br />
when α = 0 ◦ or α = 180 ◦ , and it is contained in the yz plane when α = 90 ◦ or<br />
α = 270 ◦ .<br />
Poynting vector walk-off<br />
This thesis considers an extraordinary polarized pump beam and ordinary polarized<br />
generated photons, an eoo configuration. Therefore, while the refractive<br />
index does not change with the direction for the generated photons, it changes<br />
for the pump beam with the angle between the pump wave vector and the axis<br />
of the crystal. Figure 1.5 shows how the energy flux direction of the pump,<br />
given by the Poynting vector, is rotated from the direction of the wave vector<br />
by an angle ρ0 given by<br />
ρ0 = − 1 ∂ne<br />
. (1.24)<br />
∂θ<br />
ne<br />
where ne is the refractive index for the extraordinary pump beam, and θ is<br />
the angle between the optical axis and the pump’s wave vector. The Poynting<br />
8