Spatial Characterization Of Two-Photon States - GAP-Optique

APPENDIX B
Integrals of
the matrix mode function
Given a n×n symmetric and positive definite matrix A, and two n order vectors
x and b
dx exp − xtAx 2 + ixt
b = (2π)n/2
exp −
det(A) btA−1
b
.
2
(B.1)
To proof this result consider that, since A is positive definite, there exist another
matrix O such that OAO t = D where D is a diagonal matrix. With the
transformation y = Ox, the integral in equation B.1 becomes
dx exp − xtAx 2 + ixt
b = dy exp
− yt Dy
2 + iyt b ′
,
(B.2)
with b ′ = O −1 b.
As the argument of the exponential, in the right side of equation B.1, is
given by the matrix product
− 1
y1
2
y2 . . . yn
⎛
d1
⎜ 0
⎜
⎝ .
0
d2
.
. . .
. . .
. ..
0
0
.
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
0 0 0 dn
y1
y2
.
yn
⎞
⎛
⎟
⎠ + i ⎜
y1 y2 . . . yn ⎜
⎝
the integral can be written in a polynomial form as
dx exp − xtAx 2 + ixt
b
= dy1dy2...dyn exp − y2 1d1
2 + iy1b ′ 1 − y2 2d2
2 + iy2b ′ 2... − y2 ndn
2 + iynb ′
n
(B.3)
(B.4)
69
b ′ 1
b ′ 2
.
b ′ n
⎞
⎟
⎠ ,