Page 135 - Phase Space Optics Fundamentals and Applications

116 Chapter Four

A careful calculation for the case of = 0 leads to
2
RW g (x 0 , 0) = |g(x 0 )| ∝ RW f (x 0 , 0) (4.31)
while for the value = /2 the following result is obtained

RW g x /2 , ∝ RW f x /2 sin , , tan =− f

2
(4.32)
Note that the effect of this propagation through a thin lens of
focal length f is also physically equivalent to the illumination of
the incident light distribution by a spherical wavefront whose
focus is located at a distance = f from the input plane. Thus,
thesameresultsdiscussedherecanbeappliedstraightforwardly
to that case.
2. Free-space (Fresnel) propagation. If we consider now the Fresnel
approximation for the propagation of a transverse coherent light
distribution f (x) by a distance z, namely,

+∞

i 2

g (x) = f (x ) exp (x − x) dx (4.33)
z
−∞
the transformation matrix M is given by

1 − z
M F = (4.34)
0 1
and, therefore, the transformed RWT can be calculated through
the expression

RW g (x , ) ∝ RW f (x , ), tan = tan − z,

x
x = sin (4.35)

sin + z cos
For the projection with = 0, one obtains
2
RW g (x 0 , 0) = |g(x 0 )| ∝ RW f (x , ), tan =− z,

x 0
x = sin (4.36)

z
and for the orthogonal projection = /2 the following result
is achieved

RW g x /2 , ∝ RW f x /2 , (4.37)
2 2