Spatial Characterization Of Two-Photon States - GAP-Optique

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