Page 48 - Phase Space Optics Fundamentals and Applications

Wigner Distribution in Optics 29

moments is symplectic (up to a positive constant) as described in
Eq. (1.68), its propagation through a first-order optical system is com-
pletely described by the invariance of this positive constant and the
ABCD law (1.69).

1.7.4 Measurement of Moments
Several optical schemes to determine all 10 second-order moments
have been described. 72,82–87 We mention in particular Ref. 87, which is
based on a general scheme that also can be used for the determination
of arbitrary higher-order moments pqrs with

p q r s
pqrs E = W(x, y, u, v) x u y v dx dy du dv ( p, q, r, s ≥ 0)
(1.70)
Note that for q = s = 0 we have intensity moments

p r
p0r0 E = W(x, y, u, v)x y dx dy du dv

r
p
= x y (x, x; y, y) dx dy ( p, r ≥ 0) (1.71)
which can easily be measured. The 10 second-order moments can be
determined from the knowledge of the output intensities of four first-
order optical systems, where one of them has to be anamorphic. For
the determination of the 20 third-order moments, for instance, we thus
find the need of using a total of six-first-order optical systems: four
isotropic systems and two anamorphic systems. For the details of how
to construct appropriate measuring schemes, we refer to Ref. 87.

1.8 Coherent Signals and the Cohen Class
The Wigner distribution belongs to a broad class of space-frequency
functions known as the Cohen class. 30 Any function of this class is
described by the general formula

1
1

C f (r, q) = f r o + r f ∗ r o − r k(r, q, r , q )
2 2
t
t
t

× exp − i2 q r − r q + r q dr o dr dq (1.72)
o
and the choice of the kernel k(r, q, r , q ) selects one particular function

of the Cohen class. The Wigner distribution, for instance, arises for
k(r, q, r , q ) = 1, whereas k(r, q, r , q ) = (r − r ) (q − q ) yields the

ambiguity function. In this chapter we restrict ourselves to the case

that k(r, q, r , q ) does not depend on the space variable r and the