Page 333 - Phase Space Optics Fundamentals and Applications

314 Chapter Ten

10.2.2 The Linear Canonical Transform
and the WDF
A property of central importance in this chapter is the relationship of
the WDF to the LCT. When an optical signal is input to a QPS, the
LCT describes the relationship between the signal at the output and
input to the system. The parameters of the LCT depend on the type of
system. QPSs are systems made up of any number of sequential thin
lensesandfreespaceaswellasmanyotherlosslessopticalelementsfor
which the paraxial approximation is valid. The LCT also has meaning
in quantum mechanics. The LCT is mathematically defined as

exp(− j /4)

u M (x ) = L M {u(x)}(x ) = √
B
∞

A 2 2 D 2

× u(x) exp j x − xx + x dx (10.7)
B B B
−∞
where L M {u(x)}(x ) is the operator notation for the LCT and M is a

matrix that contains the parameters of the LCT

x = M x AB x (10.8)
k k = CD k
where AD − BC = 1. This is simply the ray transfer matrix that is
commonly applied in geometrical optics. It maps the position and
angle of an input ray to those of the output. Collins 26 first pointed
out the relationship between the ray transfer matrix and the LCT.
Remarkably, this relationship can be extended to include the WDF as
defined in Eq. (10.9).

{u M (x )}(x ,k ) = {u(x)}(Ax + Bk, Cx + Dk) (10.9)
Therefore, if an LCT is applied to a signal, the WDF of the signal under-
goes a simple coordinate transformation. This operation is affine, 29–31
meaning that a given area on the WDF plane is conserved under this
coordinate shift. A noticeable and very useful property of the LCT-
WDF matrix relationship is that the combined matrix of several opti-
cal systems placed in series, each with its own matrix, can be found by
multiplyingtheindividualsystemmatrices.Thusratherthancalculate
a series of LCTs to determine the output of the constituent subsystems,
a single LCT can be determined that approximates the entire system.

10.2.3 The Phase-Space Diagram
The PSD is an illustrative plan-view outline of the WDF of a sig-
nal. This diagrammatic approximation can be a very useful source of