Page 35 - Phase Space Optics Fundamentals and Applications

16 Chapter One

d f d f d f

input output input output
(a) (b)

FIGURE 1.3 Two optical realizations of the fractional Fourier transformer.

width w is related to the distance d and the focal length of the lens f
1
2
2
by w tan( ) = o d for Fig. 1.3a and by w sin = o d for Fig. 1.3b.
2
1.5.2 Rotation in Phase Space
In terms of the ray transformation matrix, which is introduced and
treated in greater detail in Sec. 1.6, the fractional Fourier transformer
is represented by

−1
x o w 0 cos sin w 0 x i
= (1.32)
u o 0 w −1 − sin cos 0 w u i
and after normalization, w −1 x =: x and wu =: u, we have the form

x o cos sin x i
= (1.33)
u o − sin cos u i
The input-output relation of a fractional Fourier transformer in terms
of the Wigner distribution is remarkably simple; if W f denotes the
denotes that of F (x), we have
Wigner distribution of f (x) and W F
(x, u) = W f (x cos − u sin ,x sin + u cos ) (1.34)
W F
and we conclude that a fractional Fourier transformation corresponds
to a rotation in phase space.

1.5.3 Generalized Marginals—Radon
Transform
2
On the analogy of the two special cases | f (x)| = W f (x, u) du and
¯ 2
| f (u)| = W f (x, u) dx, which correspond to projections along the u