Spatial Characterization Of Two-Photon States - GAP-Optique

v =γ 2 L 2 Np sin ϕi − γ 2 L 2 Ns cos ϕs sin ϕi − w 2 pNs sin ϕs cos ϕi
+ γ 2 L 2 Np cos ϕi tan ρ0 sin α − γ 2 L 2 Ns cos ϕs cos ϕi tan ρ0 sin α
− γ 2 L 2 Ns sin ϕs sin ϕi tan ρ0 sin α − γ 2 L 2 Ns cos ϕi sin ϕs tan ρ0 2 sin α 2
w =γ 2 L 2 Np sin ϕi − γ 2 L 2 Ni sin ϕi cos ϕi + w 2 pNi sin ϕi cos ϕi
+ γ 2 L 2 Np cos ϕi tan ρ0 sin α − γ 2 L 2 Ni cos ϕi 2 tan ρ0 sin α
+ γ 2 L 2 Ni sin ϕi 2 tan ρ0 sin α + γ 2 L 2 Ni sin ϕi cos ϕi tan ρ0 2 sin α 2
z =γ 2 L 2 N 2 p − γ 2 L 2 NpNi cos ϕi − γ 2 L 2 NpNs cos ϕs + γ 2 L 2 NsNi cos ϕs cos ϕi
+ T 2 0 − w 2 pNsNi sin ϕs sin ϕi − γ 2 L 2 NsNi cos ϕs sin ϕi tan ρ0 sin α
− γ 2 L 2 NpNs sin ϕs tan ρ0 sin α + γ 2 L 2 NsNi cos ϕi sin ϕs tan ρ0 sin α
− γ 2 L 2 NsNi sin ϕs sin ϕi tan ρ0 2 sin α 2 + γ 2 L 2 NpNi sin ϕi tan ρ0 sin α.
(A.5)
This set of expressions is far more useful than compact. The matrix terms
become simpler in some particular spdc configurations that are treated in
chapters 2 and 4. For instance, when the pump polarization is parallel to the
x axis, the matrix terms become
a =w 2 p + 2w 2 s + γ 2 L 2 tan ρ 2 0
b =w 2 p cos ϕs 2 + 2w 2 s + γ 2 L 2 sin ϕs 2
c =w 2 p + 2w 2 i + γ 2 L 2 tan ρ 2 0
d =w 2 p cos ϕi 2 + 2w 2 i + γ 2 L 2 sin ϕi 2
f = 2
B2 s
g = 2
B2 i
+ T 2 0 + γ 2 L 2 (Np − Ns cos ϕs) 2 + w 2 pN 2 s sin ϕs 2
+ T 2 0 + γ 2 L 2 (Np − Ni cos ϕi) 2 + w 2 pN 2 i sin ϕi 2
h =m = −γ 2 L 2 sin ϕs tan ρ0
i =w 2 p + γ 2 L 2 tan ρ0 2
j =s = γ 2 L 2 sin ϕi tan ρ0
k =t = γ 2 L 2 tan ρ0(Np − Ns cos ϕs)
l =u = γ 2 L 2 tan ρ0(Np − Ni cos ϕi)
n = − γ 2 L 2 sin ϕs sin ϕi + w 2 p cos ϕs cos ϕi
p = − γ 2 L 2 sin ϕs(Np − Ns cos ϕs) − w 2 pNs cos ϕs sin ϕs
r = − γ 2 L 2 sin ϕs(Np − Ni cos ϕi) + w 2 pNi cos ϕs sin ϕi
v =γ 2 L 2 sin ϕi(Np − Ns cos ϕs) − w 2 pNs cos ϕi sin ϕs
w =γ 2 L 2 sin ϕi(Np − Ni cos ϕi) + w 2 pNi cos ϕi sin ϕi
z =γ 2 L 2 N 2 p − γ 2 L 2 Np(Ni cos ϕi + Ns cos ϕs) + γ 2 L 2 NsNi cos ϕs cos ϕi
+ T 2 0 − w 2 pNsNi sin ϕs sin ϕi.
(A.6)
The matrix notation extends to functions of the mode function. For instance,
the purity of the spatial part of the two-photon state, given by equation 2.11,
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