Page 25 - Phase Space Optics Fundamentals and Applications

6 Chapter One

(r 1 , r 2 ) and its Fourier transform ¯ (q , q ),
1
2

1
1
1

1
¯ q + q , q − q = r + r , r − r
2 2 2 2
t
× exp[−i2 (q r + r q )] dr dr
t

t
= W(r, q) exp(−i2 r q ) dr

t

= A(r , q ) exp(−i2 q r ) dr (1.13)
and their definitions follow as

t
1
1
W(r, q) = r + r , r − r exp(−i2 q r ) dr
2 2

1 1

t
= ¯ q + q , q − q exp(i2 r q ) dq (1.14)
2 2

1 1 t
A(r , q ) = r + r , r − r exp(−i2 r q ) dr
2 2

1 1

t

= ¯ q + q , q − q exp(i2 q r ) dq (1.15)
2 2
We immediately notice the realness of the Wigner distribution, and
the Fourier transform relationship between the Wigner distribution
and the ambiguity function:

t t
A(r , q ) = W(r, q) exp[−i2 (r q − q r )] dr dq

= F[W(r, q)](r , q ) (1.16)
This Fourier transform relationship implies that properties for the
Wigner distribution have their counterparts for the ambiguity func-
tion and vice versa: moments for the Wigner distribution become
derivatives for the ambiguity function, convolutions in the Wigner
domain become products in the ambiguity domain, etc.
We like to present the cross-spectral density , its spatial Fourier
transform ¯ , the Wigner distribution W, and the ambiguity function A
at the corners of a rectangle (see Fig. 1.1). Along the sides of this rect-
angle we have Fourier transformations r → q and r → q and their

inverses, while along the diagonals we have double Fourier transfor-
mations (r, r ) → (q , q) and (r, q) → (q , r ).

A distribution according to the definitions in (1.14) was introduced
2
in optics by Dolin and Walther 3,4 in the field of radiometry; Walther
called it the generalized radiance. A few years later it was reintroduced,
mainly in the field of Fourier optics. 5–11 The ambiguity function was