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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x<br />
Wave fronts<br />
1.3. Approximations and other considerations<br />
Poynting vector<br />
0<br />
Wave vector<br />
Figure 1.5: As a consequence of the birefringence, the Poynting vector is no longer<br />
parallel to the wave vector. The energy flows in an angle ρ0 with respect to propagation<br />
direction.<br />
x<br />
<br />
y z<br />
Figure 1.6: The Poynting vector moves away from the wave vector in the direction of<br />
the pump beam polarization. The angles α and ρ0 characterize the displacement. A<br />
nonradial effect such as the Poynting vector walk-off inevitably brakes the azimuthal<br />
symmetry of the properties of the photons on the cone.<br />
vector walk-off displaces the effective transversal shape of the pump beam<br />
inside the crystal in the pump polarization direction, as shown in figure 1.6.<br />
The vector p = z tan ρ0 cos αˆx + z tan ρ0 sin αˆy, describes the magnitude and<br />
direction of this displacement.<br />
By including the Poynting vector walk-off, and by using the same coordinate<br />
system for all the fields, the mode function becomes<br />
<br />
Φ(qs, Ωs, qi, Ωi) ∝ dV dqp exp −<br />
V<br />
w2 p<br />
4 q2 p − T0<br />
4 (Ωs + Ωi) 2<br />
<br />
× exp i(q x p − q x s − q x i )x <br />
<br />
0<br />
× exp [i(q y p + k z s sin ϕs − k z i sin ϕi − q y s cos ϕs − q y<br />
i cos ϕi)y]<br />
× exp [i(k z p − k z s cos ϕs − k z i cos ϕi − q y s sin ϕs + q y<br />
i sin ϕi)z]<br />
× exp [i(q x p tan ρ0 cos α + q y p tan ρ0 sin α)z]. (1.25)<br />
p<br />
z<br />
9