Page 24 - Phase Space Optics Fundamentals and Applications

Wigner Distribution in Optics 5

t
described by the real symmetric 2 × 2 matrix H = H .Weuse
this example to introduce the instantaneous frequency of a sig-
nal | f (r)| exp[i2 (r)] as the derivative d /dr =∇ (r) =
t
[∂ /∂x, ∂ /∂y] of the signal’s argument. In the case of a spheri-
cal wave we have d /dr = Hr, and the instantaneous frequency
corresponds to the normal on the spherical wavefront.

Basic example of partially coherent signals include
4. Completely incoherent light with intensity p(r), (r 1 , r 2 ) =
p(r 1 ) (r 1 − r 2 ). Note that p(r) is a nonnegative function.
5. Spatially stationary light, (r 1 , r 2 ) = s(r 1 − r 2 ). We will see later
that the Fourier transform of s(r) is a nonnegative function.

1.3 Wigner Distribution and Ambiguity
Function
In this section we introduce the Wigner distribution and its Fourier
transform, the ambiguity function.

1.3.1 Definitions
We introduce the spatial Fourier transforms of f (r) and (r 1 , r 2 ):

¯ t
f (q) = f (r) exp(−i2 q r) dr (1.10)

t t
¯ (q , q ) = (r 1 , r 2 ) exp − i2 q r 1 − q r 2 dr 1 dr 2 (1.11)
1 2 1 2
Throughout we use the generic form (r 1 , r 2 ), even in the case of
completely coherent light, where we could use the product form
f (r 1 ) f (r 2 ). We thus elaborate on Eq. (1.11) and apply the coordinate
∗
transformation

1
r 1 = r + r r = (r 1 + r 2 )
1
2 2 (1.12)
1
r 2 = r − r r = r 2 − r 1

2
and similarly for q. Note that the jacobian equals 1, so that dr 1 dr 2 =
1

dr dr . The Wigner distribution W(r, q) and ambiguity function 18

A(r , q ) now arise “midway” between the cross-spectral density