Spatial Characterization Of Two-Photon States - GAP-Optique

APPENDIX A
The matrix form
of the mode function
According to section 1.3, the normalized two-photon mode function, after some
approximations, reads
Φ(qs, Ωs, qi, Ωi) ∝ exp − w2 p
4 ∆2 0 − w2 p
4 ∆21
× exp − (γL)2
4 ∆2k − T 2 0
4 (Ωs + Ωi) 2
× exp − w2 s
2 |qs| 2 − w2 i
2 |qi| 2 − 1
2B2 Ω
s
2 s − 1
2B2 Ω
i
2
i . (A.1)
The argument of the exponential function is a second order polynomial. Each
term is the product of at most two variables (qx s , qy s , qx i , qy i , Ωs, Ωi) and a coefficient
f = a
4 qx2 s + b
4 qy2 s + c
4 qx2 i + d
4 qy2
h
i + . . . +
2 qx s q y s + . . . + z
2 ΩsΩi. (A.2)
Such a polynomial can be written as the product of a matrix A with the
coefficients as elements, and a vector x of the variables
f = 1
x qs 4
qy s qx i q y
i Ωs Ωs
⎛
⎜
⎜
⎝
a
h
i
j
k
h
b
m
n
p
i
m
c
s
t
j
n
s
d
v
k
p
t
v
f
l
r
u
w
z
⎞ ⎛
q
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
l r u w z g
x s
qy s
qx i
q y
i
Ωs
⎞
⎟ ,
⎟
⎠
Ωi
(A.3)
therefore, the mode function can be written using matrix notation as
Φ(qs, Ωs, qi, Ωi) ∝ exp − 1
2 xt
Ax . (A.4)
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