Page 56 - Phase Space Optics Fundamentals and Applications

Wigner Distribution in Optics 37

A definition directly in terms of the signal f (x) reads

1 1
P (x, u; w, z) = S f x + t sin ,u + t cos ; W − z(t)
f
2
2
× exp(−i2 ut sin )
1 1
×S ∗ f x − t sin ,u − t cos ; W − dt (1.90)
2
2
where the fractional Fourier transform W − (x) of the window w(x)
arises and where we have used the relationship
(x 2 ,u 2 ; W ) = exp[i (u 2 x 2 − u 1 x 1 )] S f (x 1 ,u 1 ; w)
S F

(1.91)
x 2 cos sin x 1
with =
u 2 − sin cos u 1

The -rotated smoothed interferogram P (x, u; w, z) is related to the
f
Wigner distribution W f (x, u) with the kernels 89,92
K(x, u) = W w (−x, −u)¯z(−x cos + u sin ) (1.92a)

¯
K(x ,u ) = A w (−x + t sin , −u + t cos )z(t) dt (1.92b)

Note that for = 0, the distribution P (x, u; w, z) reduces to the one
f
originally introduced, which was based on a combination of win-
1
dowed Fourier transforms in the u direction, while for = it
2
reduces to the version that combines these windowed Fourier trans-
forms in the x direction. 90
The rotated version of the smoothed interferogram is a versatile
method to remove cross-terms. To illustrate this, we show two nu-
merical examples. Consider first the signal

8
3x
f (x) = exp − {exp[i 1 (x)] + exp[i 2 (x)]}
x o
2
1 (x) = h 1 x + a 1 cos(2 u 1 x)
with
2
2 (x) = h 2 x + a 2 cos(2 u 2 x)
consisting of two components with instantaneous frequency h 1 x −
a 1 u 1 sin(2 u 1 x) and h 2 x − a 2 u 2 sin(2 u 2 x), respectively; note that the
instantaneous frequencies cross at x = 0. In the numerical simulation,
the variables take the values x o = 128, h 1 = 192, h 2 = 64, u 1 = u 2 = 2,
2
−a 1 = a 2 = 8/ . A Hann(ing) window w(x) = cos ( x/X) rect(x/X)
with width X = 128 is used for the calculation of the windowed
Fourier transform S f (x, u; w). The values of the normalized second-
/4
/2
order central moments are 0 xx = 1, xx = 1.38, and xx = 0.07.