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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A. The matrix form of the mode function<br />
Each of the terms of the matrix is defined by comparing the product of the<br />
polynomial f with the argument of the exponential in equation 1.34. The<br />
elements of matrix A are<br />
a =w 2 p + 2w 2 s + γ 2 L 2 tan ρ 2 0 cos α 2<br />
b =w 2 p cos ϕs 2 + 2w 2 s + γ 2 L 2 sin ϕs 2 − 2γ 2 L 2 sin ϕs cos ϕs sin α tan ρ0<br />
+ γ 2 L 2 cos ϕs 2 tan ρ0 2 sin α 2<br />
c =w 2 p + 2w 2 i + γ 2 L 2 tan ρ 2 0 cos α 2<br />
d =w 2 p cos ϕi 2 + 2w 2 i + γ 2 L 2 sin ϕi 2 + 2γ 2 L 2 sin ϕi cos ϕi tan ρ0 sin α<br />
+ γ 2 L 2 cos ϕi 2 tan ρ0 2 sin α 2<br />
f =2B −2<br />
s + T 2 0 + γ 2 L 2 N 2 p + γ 2 L 2 N 2 s cos ϕs 2 + w 2 pN 2 s sin ϕs 2 − 2γ 2 L 2 NpNs cos ϕs<br />
− 2γ 2 L 2 NpNs sin ϕs tan ρ0 sin α + 2γ 2 L 2 N 2 s cos ϕs sin ϕs sin α tan ρ0<br />
+ γ 2 L 2 N 2 s sin ϕs 2 tan ρ0 2 sin α 2<br />
g =2B −2<br />
i<br />
+ T 2 0 + γ 2 L 2 N 2 p + γ 2 L 2 N 2 i cos ϕi 2 + w 2 pN 2 i sin ϕi 2 − 2γ 2 L 2 NpNi cos ϕi<br />
+ 2γ 2 L 2 NpNi sin ϕi tan ρ0 sin α − 2γ 2 L 2 N 2 i cos ϕi sin ϕi tan ρ0 sin α<br />
+ γ 2 L 2 N 2 i sin ϕi 2 tan ρ0 2 sin α 2<br />
h = − γ 2 L 2 sin ϕs tan ρ0 cos α + γ 2 L 2 cos ϕs tan ρ0 2 sin α cos α 2<br />
i =w 2 p + γ 2 L 2 tan ρ0 2 cos α<br />
j =γ 2 L 2 sin ϕi tan ρ0 cos α + γ 2 L 2 cos ϕi tan ρ0 2 sin α cos α<br />
k =γ 2 L 2 Np tan ρ0 cos α − γ 2 L 2 Ns cos ϕs tan ρ0 cos α<br />
− γ 2 L 2 Ns sin ϕs tan ρ0 2 sin α cos α<br />
l =γ 2 L 2 Np tan ρ0 cos α − γ 2 L 2 Ni cos ϕi tan ρ0 cos α<br />
+ γ 2 L 2 Ni sin ϕi tan ρ0 2 sin α cos α<br />
m = − γ 2 L 2 sin ϕs tan ρ0 cos α + γ 2 L 2 cos ϕs tan ρ0 2 sin α cos α<br />
n = − γ 2 L 2 sin ϕs sin ϕi + w 2 p cos ϕs cos ϕi + γ 2 L 2 cos ϕs sin ϕi sin α tan ρ0<br />
− γ 2 L 2 sin ϕs cos ϕi sin α tan ρ0 + γ 2 L 2 cos ϕs cos ϕi tan ρ0 2 sin α 2<br />
p = − γ 2 L 2 Np sin ϕs + γ 2 L 2 Ns cos ϕs sin ϕs − w 2 pNs cos ϕs sin ϕs<br />
+ γ 2 L 2 Np cos ϕs tan ρ0 sin α − γ 2 L 2 Ns cos ϕs 2 tan ρ0 sin α<br />
+ γ 2 L 2 Ns sin ϕs 2 tan ρ0 sin α − γ 2 L 2 Ns sin ϕs cos ϕs tan ρ0 2 sin α 2<br />
r = − γ 2 L 2 Np sin ϕs + γ 2 L 2 Ni sin ϕs cos ϕi + w 2 pNi cos ϕs sin ϕi<br />
+ γ 2 L 2 Np cos ϕs tan ρ0 sin α − γ 2 L 2 Ni cos ϕs cos ϕi tan ρ0 sin α<br />
− γ 2 L 2 Ni sin ϕs sin ϕi tan ρ0 sin α + γ 2 L 2 Ni cos ϕs sin ϕi tan ρ0 2 sin α 2<br />
s =γ 2 L 2 sin ϕi tan ρ0 cos α + γ 2 L 2 cos ϕi tan ρ0 2 sin α cos α<br />
t =γ 2 L 2 Np tan ρ0 cos α − γ 2 L 2 Ns cos ϕs tan ρ0 cos α − γ 2 L 2 Ns sin ϕs tan ρ0 2 sin α cos α<br />
u =γ 2 L 2 Np tan ρ0 cos α − γ 2 L 2 Ni cos ϕi tan ρ0 cos α + γ 2 L 2 Ni sin ϕi tan ρ0 2 sin α cos α<br />
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