Page 40 - Phase Space Optics Fundamentals and Applications

Wigner Distribution in Optics 21

expressed in the form of Eq. (1.48), this would mean y =− x . Note
that the matrices U r ( ), U g ( ), and U f ( , − ), corresponding to a ro-
tator R( ), a gyrator G( ), and an antisymmetric fractional Fourier
transformer F( , − ), respectively, are quaternions, and that every
separable fractional Fourier transformer F( x , y ) can be decomposed
as F(ϑ, ϑ) F( , − ).
We easily verify—for instance, by expressing the unitary matrix U
in the form of Eq. (1.48)—that the input-output relation for a phase-
space rotator can be expressed in the form

r o − iq = U(r i − iq ) (1.50)
o i
which is an easy alternative for Eq. (1.39). Phase-space rotators are
considered in greater detail in Chap. 3 by Tatiana Alieva.

1.6.3 More General Systems—Ray-Spread
Function
First-order optical systems are a perfect match for the Wigner distribu-
tion, since their point-spread function is a quadratic-phase function.
Nevertheless, an input-output relationship can always be formulated
for the Wigner distribution. In the most general case, based on the
relationships (1.5) and (1.9), we write

W o (r o , q ) = K(r o , q , r i , q )W i (r i , q ) dr i dq (1.51)
o o i i i
with

1
1
1
1
∗
K(r o , q , r i , q ) = h r o + r , r i + r h r o − r , r i − r
o i 2 o 2 i 2 o 2 i

× exp − i2 q r − q r dr dr (1.52)
t

t
o o i i o i
Relation (1.52) can be considered the definition of a double Wigner
distribution; hence, the function K has all the properties of a Wigner
distribution, for instance, the property of realness.
Let us think about the physical meaning of the function K.Ina
formal way, the function K is the response of the system in the space-
frequency domain when the input signal is described by a product of
two Dirac functions W i (r, q) = (r−r i ) (q−q ); only in a formal way,
i
since an actual input signal yielding such a Wigner distribution does
not exist. Nevertheless, such an input signal could be considered as
a single ray entering the system at the position r i with direction q .
i
Hence, the function K might be called the ray-spread function of the
system.