Page 134 - Phase Space Optics Fundamentals and Applications

The Radon-Wigner Transform 115

By applying the definition in Eq. (4.12), it is straightforward to obtain
⎧
+∞
x
⎪ W g x, x dx for = 0,
⎪ −
⎪ sin tan 2
⎪
⎪−∞
⎪
⎪
⎨ +∞

RW g (x , ) = W g (x , ) d for = 0
⎪
⎪−∞
⎪
⎪
⎪ +∞
⎪
⎪
⎩ W g (x, x ) dx for =
2
−∞
⎧ ⎫
+∞
⎪ x ⎪
⎪ W f ax + b x ,cx ⎪
⎪ sin − tan ⎪
⎪ ⎪
⎪ ⎪
⎪−∞ ⎪
⎪ ⎪
⎪ x ⎪
⎪ x ⎪
⎪ +d − dx for = 0, ⎪
⎪ sin tan 2 ⎪
⎪ ⎪
⎨ ⎬
= +∞

⎪ W f (ax + b ,cx + d ) d for = 0 ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪−∞ ⎪
⎪ ⎪
⎪ ⎪
⎪ ⎪
⎪ +∞ ⎪
⎪ ⎪
⎪ ⎪
⎪ W f (ax + bx ,cx + dx ) dx for = ⎪
⎩ 2 ⎭
−∞
∝ RW f (x , ) (4.27)

where the mapped coordinates for the original RWT are given by
a tan − b x

tan =− , x = sin (4.28)

c tan − d a sin − b cos
Let us consider in the following examples a spatially coherent light
distribution f (x), with wavelength , that travels along a system that
imposes a transformation in the input characterized by an abcd trans-
form. Special attention is usually paid to the cases = 0, /2 since,
according to Eqs. (4.6) and (4.7), the modulus squared of the abcd
transform in Eq. (4.24) and its FT are then obtained, respectively.
1. Coherent propagation through a (cylindrical) thin lens. In this case
the associated M matrix for the transformation of the light field
is given by

1 0
M L = 1 (4.29)
f 1
with f being the focal length of the lens. Thus, the RWT for the
transformed amplitude light distribution is given in this case by
tan

RW g (x , ) ∝ RW f x , , tan =− f ,

tan − f
x
x = sin (4.30)

sin