Page 68 - Phase Space Optics Fundamentals and Applications

Ambiguity Function in Optical Imaging 49

2.2.3 Contrast Transfer Functions
Considering T(x) = exp [−B(x) +i
(x)], where exp [−B(x)] and
(x)
are the absorption and the phase modulations, respectively, of the
object, we can introduce in formula (2.11) the approximation

Df Df Df Df
T ∗ x + T x − 1 − B x + − B x −
2 2 2 2

Df Df
− i
x + −
x − (2.14)
2 2
which should be valid under the conditions 0 < B(x) 1 (weak
absorption) and |
(x) −
(x − Df )| 1. This last condition is the
6
slowly varying phase condition which is less restrictive and more
precise than the weak phase condition |
(x)| 1. Under such condi-
tions, the intensity spectrum takes a simple linear form
2
2
˜ I D ( f ) = ( f ) − 2 cos ( Df ) ˜ B( f ) + 2 sin ( Df )˜
( f ) (2.15)
where the factors of ˜ B( f ) and ˜
( f ) are called the absorption-transfer
function (ATF) and the phase-transfer function (PTF) respectively.
Formula (2.14) can be generalized to the case of an imaging system
with aberrations other than defocusing. In electron microscopy, for
which primary spherical aberration characterized by the coefficient
C S is unavoidable, the following formula is to be used. 8, 9
˜ I D ( f ) = ( f ) − 2 cos [ ( f )] ˜ B( f ) + 2 sin [ ( f )]˜
( f ) (2.16)

with
3 4
C S f
2
( f ) = Df +
2

2.3 Propagation through a Paraxial Optical
System in Terms of AF
2.3.1 Propagation in Free Space
Let us consider the propagation in free space, with mean direction
along the z axis, of a partially coherent beam. The mutual intensity in
the z = D plane is given in terms of the mutual intensity in the z = 0
plane as

a a 1 (x + a/2 − )
2
D x + ,x − = d exp i d
2 2 D D
2
(x − a/2 − )
× exp −i 0 ( , ) (2.17)
D