Page 70 - Phase Space Optics Fundamentals and Applications

Ambiguity Function in Optical Imaging 51

where

a a

A T ( f, a) = dx exp (−i2 xf )T x + T ∗ x −
2 2
The transmission by a thin lens of focal length F is of special
2
2
interest. This lens behaves as an object of transmittance exp (−i x /
F). With inc (x + a/2,x − a/2) being the mutual intensity in the lens
entrance surface, the AF in the lens exit surface is calculated as
2 2
(x + a/2) − (x − a/2)
A( f, a) = dx exp −i2 xf − i
F

a a
× inc x + ,x −
2 2

! a " a a
= dx exp −i2 x f + inc x + ,x −
F 2 2

a
= A inc f + ,a (2.23)
F
The corresponding formula for the WDF is easily found as
x

W(x, g) = W inc x, g + (2.24)
F
2.3.3 Propagation in a Paraxial Optical
System
Consider the process of propagation from an input plane to a thin lens
over a distance D 1 , then transmission by this lens of focal length F,
and finally propagation to the output plane over a distance D 2 . Per-
forming the corresponding transformations successively, according to
Eqs. (2.20) and (2.22), it is easy to obtain the output AF in terms of the
input AF:

D 1 a D 2
A out ( f, a) = A in f − f + ,a − a
F F F

D 1 D 2
− f D 1 + D 2 − (2.25)
F
This linear transformation of variables can be represented by the
matrix
⎛ ⎞
D 1 1
1 −
F F
⎜ ⎟
M = ⎝ ⎠ (2.26)
D 1 D 2 − D 1 − D 2 1 − D 2
F F
which can also be obtained by multiplication of the matrices corre-
sponding to the elementary successive transformations. This matrix
shows a remarkable similarity with the ray-transfer matrix of geomet-
rical optics (see, for instance, Ref. 10) which transforms an ingoing