Page 71 - Phase Space Optics Fundamentals and Applications

52 Chapter Two

ray to the corresponding outgoing ray, in the paraxial approximation,

/
with each ray being represented by a column vector , where
x
x and denote the position and the direction of the ray, respectively.
It turns out that the ray-transfer matrix is equal to the inverse of
matrix M of Eq. (2.26).
Therefore, the correspondance between geometrical optics and the
PSO formulation pointed out in Ref. 3 for the WDF appears to be also
valid for the AF.
This PSO method is thus a convenient and elegant tool to describe
the propagation of a coherent or partially coherent beam in any system
comprising coaxial lenses, and it is possible to use the WDF instead of
the AF. The PSO method is much simpler than the method based on
the propagation of mutual intensity which would involve convolution
integrals or Fourier transformations.

2.4 The AF in Isoplanatic (Space-Invariant)
Imaging

The mutual intensity im (x, x ) in the image plane is given in terms
of the mutual intensity ob (x, x ) in the object plane (for convenience,

the magnification is set equal to 1) as

∗
im (x, x ) = d d G(x − )G (x − ) ob ( , ) (2.27)

where G(x) is the coherent point-spread function (PSF) of the imaging
system. The image AF is therefore

a
A im ( f, a) = dx exp (−i2 xf ) d G x + −
2

a

× d G ∗ x − − ob ( , ) (2.28)
2

By introducing new variables
= ( + )/2, = − , and t = x −
in this integral expression, we obtain directly 11−13

A im ( f, a) = dt d
exp [−i2 f (t +
)]

a − a −

× d G t + G ∗ t − ob
+ ,
−
2 2 2 2

= d A G ( f, a − )A ob ( f, ) (2.29)