Page 27 - Phase Space Optics Fundamentals and Applications

8 Chapter One

We remark the clear physical interpretations of the Wigner distri-
butions.

1. The Wigner distribution of a point source f (r) = (r − r o ) reads
W(r, q) = (r − r o ), and we observe that all the light originates
from one point r = r o and propagates uniformly in all direc-
tions q.
t
2. Its dual, a plane wave f (r) = exp(i2 q r), also expressible in
o
¯
the frequency domain as f (q) = (q − q ), has as its Wigner
o
distribution W(r, q) = (q − q ), and we observe that for all
o
positions r the light propagates in only one direction q .
o
3. The Wigner distribution of the spherical wave f (r) =
t
exp(i r Hr) takes the simple form W(r, q) = (q − Hr), and
we conclude that at any point r only one frequency q = Hr,
the instantaneous frequency, manifests itself. This corresponds
exactly to the ray picture of a spherical wave.
4. Incoherent light, (r+ r , r− r ) = p(r) (r ), yields the Wigner

1
1
2 2
distribution W(r, q) = p(r). Note that it is a function of the space
variableronly,andthatitdoesnotdependonthefrequencyvari-
able q: the light radiates equally in all directions, with intensity
profile p(r) ≥ 0.
5. Spatially stationary light, (r + r , r − r ) = s(r ), is dual to
1

1
2 2
incoherent light: its frequency behavior is similar to the space
behavior of incoherent light and vice versa, and ¯(q), its intensity
s
function in the frequency domain, is nonnegative. The duality
between incoherent light and spatially stationary light is, in fact,
the Van Cittert-Zernike theorem.
The Wigner distribution of spatially stationary light reads as
s
W(r, q) = ¯(q); note that it is a function of the frequency variable
q only, and that it does not depend on the space variable r.It
thus has the same form as the Wigner distribution of incoherent
light, except that it is rotated through 90 in the space-frequency
◦
domain. The same observation can be made for the point source
and the plane wave; see examples (1) and (2), which are also
each other’s duals.
We illustrate the Wigner distribution of the one-dimensional spher-
2
ical wave f (x) = exp (i hx ), (see example (3) above), by a numerical
simulation. To calculate W(x, u) practically, we have to restrict the in-
tegration interval for x . We model this by using a window function

w( x ), so that the Wigner distribution takes the form
1
2

1
1
x w
1
P(x, u; w) = f x + x w 1 ∗ − x f ∗ x − x
2 2 2 2

× exp (−i2 ux ) dx (1.17)