Spatial Characterization Of Two-Photon States - GAP-Optique

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