Page 28 - Phase Space Optics Fundamentals and Applications

Wigner Distribution in Optics 9

P(x,u;w) P(x,u;w)

x x

u u
(a) (b)

FIGURE 1.2 Numerical simulation of the (pseudo) Wigner distribution
2
P(x, u; w) W(x, u) = (u − hx) of the spherical wave f (x) = exp(i hx ),
1

for the case that w( x )is (a) a rectangular window and (b) a Hann(ing)
2
window.
The function P(x, u; w) is called the pseudo-Wigner distribution. It
is common to choose an even window function w( x ) = w(− x ),
1
1
2 2
1 2
so that we have w( x )w (− x ) =|w( x )| . Figure 1.2 shows the
1
∗
1
2 2 2
2
(pseudo) Wigner distribution of the signal f (x) = exp(i hx ), which
reads as

2
2
x

x
w 1 exp[−i2 (u−hx)] dx = F w 1 (u−hx) (u−hx)

2 2
where we have chosen a rectangular window of width X in Fig. 1.2a

x
x
w 1 = rect
2
X
and a Hann(ing) window of width X in Fig. 1.2b

2 x x
w 1 = cos rect
x
2
X X

1 2
Note the effect of F |w( x )| , which results in a sinc-type behav-
2
ior in the case of the rectangular window, P(x, u; w) = sin[ (u −
hx)X]/ (u − hx), and in a nonnegative but smoother version in the
case of the Hann(ing) window.
1.3.3 Gaussian Light
Gaussian light is an example that we will treat in greater detail. The
cross-spectral density of the most general partially coherent Gaussian