Page 53 - Phase Space Optics Fundamentals and Applications

34 Chapter One

and we need averaging in the x direction. We may thus benefit from a
rotation of the kernel, or let the original kernel operate on the Wigner
distribution of the fractional Fourier transform of the signal,

C f (x, u) = K(x cos + u sin , −x sin + u cos ) ∗ ∗W f (x, u)
x u
(1.80a)
(x, u) (1.80b)
C F (x, u) = K(x, u) ∗ ∗W F
x u
To find the optimal rotation angle o , we may proceed as follows.

Let m and m be the first- and second-order moments of the intensity
x xx
2
|F (x)| of the fractional Fourier transform F (x),

1 1
2
m = xW F (x, u) dx du = x |F (x)| dx (1.81a)
x
E E

1 2 1 2 2

m xx = x W F (x, u) dx du = x |F (x)| dx (1.81b)
E E
and let m be the mixed moment
xu

1

m xu = xu W F (x, u) dx du (1.81c)
E
The propagation laws for the first- and second-order moments
through a rotator read

m cos sin m x
x = (1.82a)
m u − sin cos m u

m xx m xu cos sin m xx m xu cos − sin
m m = − sin cos sin cos
xu uu m xu m uu
(1.82b)
/2 /2 /4 1 /2
Notethatm u = m x ,m uu = m xx ,andm xu = m xx − (m xx +m xx ),and
2
that all second-order moments follow directly from the measurement
of the intensity profiles of only three fractional Fourier transforms:
¯
F 0 (x) = f (x), F /2 (x) = f (x), and F /4 (x). While the second-order
moment m can be expressed as
xx
2
2
m xx = m xx cos + m uu sin + m xu sin 2 (1.83)
2
the second-order central moment = m −(m ) can be expressed
xx xx x
as
2
2
xx = xx cos + uu sin + (m xu − m x m u ) sin 2 (1.84)