Page 37 - Phase Space Optics Fundamentals and Applications

18 Chapter One

leading to
2
∂|F (x)|

= uW f ( ,u) ( − x) d du
∂
=0

d
=− uW f (x, u) du
dx

2
andthensubstituting,fromEq.(1.29), uW f (x, u) du =| f (x)| d (x)/
2
(x)| and
dx. By measuring two intensity profiles around = 0, |F o
2 2 2
|F − o (x)| for instance, approximating ∂|F (x)| /∂ by [|F o (x)| −
2
2
(x)| ]/2 o , and integrating the result, we get | f (x)| d (x)/dx.
|F − o
2
2
After dividing this by the intensity | f (x)| =|F 0 (x)| , which can be
2 2
approximated by [|F o (x)| +|F − o (x)| ]/2, we find an approximation
for the phase derivative d (x)/dx, which after a second integration
yields the phase (x). Together with the modulus | f (x)|, the signal
f (x) can thus be reconstructed. This procedure can be extended to
other members of the class of Luneburg’s first-order optical systems,
to be considered next, in particular by using a section of free space
instead of a fractional Fourier transformer. 57
1.6 Propagation of the Wigner Distribution
In this section, we study how the Wigner distribution propagates
through linear optical systems. We therefore consider an optical sys-
tem as a black box, with an input plane and an output plane, and
focus on the important class of first-order optical systems. A contin-
uous medium, in which the signal must satisfy a certain differential
equation, is considered in Sec. 1.6.5, but without going into much
detail.

1.6.1 First-Order Optical Systems—Ray
Transformation Matrix
An important class of optical systems is the class of Luneburg’s first-
58
order optical systems. This class consists of a section of free space (in
the Fresnel approximation), a thins lens, and all possible combinations
of these. A first-order optical system can most easily be described in
terms of its (normalized) ray transformation matrix 59

−1
r o W 0 A B W 0 r i
= (1.38)
q 0 W −1 C D 0 W q
o i
which relates the position r i and direction q of an incoming ray to
i
the position r o and direction q of the outgoing ray. In normalized
o