Page 49 - Phase Space Optics Fundamentals and Applications

30 Chapter One

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spatial-frequency variable q, hence k(r, q, r , q ) = K(r , q ), in which
case the resulting space-frequency distribution is shift-covariant (see
Sec. 1.4.2).
1.8.1 Multicomponent Signals—Auto-Terms
and Cross-Terms
The Wigner distribution, like the mutual coherence function and the
cross-spectral density, is a bilinear signal representation. In the case
of completely coherent light, however, we usually deal with a linear
signal representation. Using a bilinear representation to describe co-
herent light thus yields cross-terms if the signal consists of multiple
components. The two-component signal f (r) = f 1 (r) + f 2 (r) yields
the Wigner distribution

(r, q)
W f (r, q) = W f 1 (r, q) + W f 2

1
+ 2Re f 1 r + r f ∗ r − r exp(−i2 q r ) dr (1.73)
1
t
2 2 2
(r, q) and
and we notice a cross-term in addition to the auto-terms W f 1
(r, q). In the case of two point sources (r − r 1 ) and (r − r 2 ), for
W f 2
instance, the cross-term reads
1 t
2 r − (r 1 + r 2 ) cos[2 (r 1 − r 2 ) q)]
2
1
It appears at the position (r 1 + r 2 ), i.e., in the middle between the
2
(r, q) = (r − r 2 ), and is
two auto-terms W f 1 (r, q) = (r − r 1 ) and W f 2
modulated in the q direction. We can get rid of this cross-term when
we average the Wigner distribution with a kernel that is narrow in the
r direction and broad in the q direction. We thus remove the cross-term
without seriously disturbing the auto-terms.
The occurrence of cross-terms is also visible from the general
condition 45,88

1
1
1
1

W f r + r , q + q W f r − r , q − q
2 2 2 2

1 1
1

1
= W f r + r , q + q W f r − r , q − q
2 2 2 2

t
t
× exp[−i2 (q r − q r )] dr dq (1.74)

which, for r = q = 0, reduces to

2 1 1 1 1
W (r, q) = W f r + r , q + q W f r − r , q − q dr dq
f
2
2
2
2
(1.75)