Page 34 - Phase Space Optics Fundamentals and Applications

Wigner Distribution in Optics 15

andwhenwecombinethesetworesults,weimmediatelygetEq.(1.29).
It is this property in particular that made the Wigner distribution a
popular tool for the determination of the instantaneous frequency.

1.4.5 Moyal’s Relationship
An important relationship between the Wigner distributions of two
signals and the cross-spectral densities of these signals, which is an
extension to partially coherent light of a relationship formulated by
Moyal 48 for completely coherent light, reads as

∗
W 1 (r, q) W 2 (r, q) dr dq = 1 (r 1 , r 2 ) (r 1 , r 2 ) dr 1 dr 2
2

¯ 1 (q , q ) ¯ (q , q ) dq dq
∗
=
1 2 2 1 2 1 2
(1.30)
This relationship has an application in averaging one Wigner distribu-
tion with another one, which averaging always yields a nonnegative
result.

1.5 One-Dimensional Case and the
Fractional Fourier Transformation
Let us for the moment restrict ourselves to coherent light and to the
one-dimensional case, and let us use normalized coordinates. The sig-
nal is now written as f (x).

1.5.1 Fractional Fourier Transformation
An important transformation with respect to operations in a phase
space is the fractional Fourier transformation, which reads as 49–53

1
2
exp i x + x 2 o cos − 2x o x i
i
f o (x o ) = F (x o ) = √ 2 exp i
i sin sin
( = n ) (1.31)
× f i (x i ) dx i
√ 1
where i sin is defined as | sin | exp[i( ) sgn(sin )]. We mention
4
the special cases F 0 (x) = f (x), F (x) = f (−x), and the common
¯
Fourier transform F /2 (x) = f (x). Two realizations of an optical frac-
tional Fourier transformer have been proposed by Lohmann 50 (see
2 1
Fig. 1.3). For both cases we have sin ( ) = d/2 f ; the normalization
2