Spatial Characterization Of Two-Photon States - GAP-Optique

1. General description of two-photon states
Interaction volume approximation
In the experimental cases studied here, the crystals are cuboids with two square
faces. While typical crystal faces have areas of about 1cm 2 , the pump beam
waist is only some hundreds of micrometers. By assuming that the transversal
dimensions of the crystal are much larger than the pump, the integral over the
volume becomes
V
dV →
∞
−∞
∞ L/2
dx dy dz, (1.26)
−∞ −L/2
where L is the length of the crystal. After integrating over x and y, the mode
function reduces to
where
Φ(qs, Ωs, qi, Ωi) ∝ exp − w2 p
4 ∆20 − w2 p
4 ∆21 − T0
4 (Ωs + Ωi) 2
L/2
× dz exp [i∆kz] (1.27)
−L/2
∆0 =q x s + q x i
∆1 =q y s cos ϕs + q y
i cos ϕi − k z s sin ϕs + k z i sin ϕi
∆k =k z p − k z s cos ϕs − k z i cos ϕi − q y s sin ϕs + q y
i sin ϕi
+ ∆0 tan ρ0 cos α + ∆1 tan ρ0 sin α. (1.28)
Wave vector and group velocity
The use of a Taylor series expansion of kz s,i , simplifies the delta factors defined
in the previous paragraph. Because the wave vector magnitude is a function
of the frequency deviation Ω, the expansion around the origin reads
k z n = k z0 ∂k
n + Ωn
z n
+ Ω 2 ∂
n
2kz n
∂2Ωn ∂Ωn
+ . . . , (1.29)
where k z0
n is the magnitude of the wave vector at the central frequency ω 0 n, and
the first partial derivative of the wave vector magnitude with respect to the
frequency is the group velocity Nn.
Taking only the first order terms of the Taylor expansion, and considering
momentum conservation, the delta factors become
∆0 =q x s + q x i
∆1 =q y s cos ϕs + q y
i cos ϕi − NsΩs sin ϕs + NiΩi sin ϕi
∆k =Np(Ωs + Ωi) − NsΩs cos ϕs − NiΩi cos ϕi − q y s sin ϕs + q y
i sin ϕi
+ ∆0 tan ρ0 cos α + ∆1 tan ρ0 sin α. (1.30)
Exponential approximation of the sinc function
The mode function in equation 1.27, depends on the integral of an exponential
function of z. By integrating the exponential over the length of the crystal, the
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