Page 73 - Phase Space Optics Fundamentals and Applications

54 Chapter Two

This result, which is the expression of the Van Cittert-Zernike
theorem, 14 can also be obtained from the WDF which is equal to
S(x − Lf ), since the WDF is easily found to be equal to S(x) in the
plane of the incoherent source.

2.5.2 Partial Coherence Properties in the
Image of an Incoherent Source 11
The image of an incoherent source (or equivalently in the image of
an object under incoherent illumination) by a nonideal optical system
shows some degree of coherence because the light from each point
of the primary source is spread over a finite area in the image; it is
convenient to introduce the image AF, which characterizes this im-
age completely, including its coherence properties, and has a simple
expression. 11

A im ( f, a) = d A G ( f, a − ) ˜ S( f ) ( ) = A G ( f, a) ˜ S( f ) (2.34)

2.5.3 The Pupil AF as a Generalization
of the OTF
In the case a = 0, formula (2.32) gives the image intensity spectrum as

˜ I im ( f ) = A im ( f, 0) = A G ( f, 0) ˜ S( f ) (2.35)

This formula shows that A G ( f, 0) is identical to the well-known
optical transfer function (OTF). Since formula (2.34) is obviously a gen-
eralization of formula (2.35), the pupil AF is to be considered as a
generalization of the OTF.
According to Eq. (2.18), if the image is recorded at a distance D
from its normal position (defocusing), a is to be replaced by a − Df in
11
Eq. (2.34). This means that the defocused pupil AF is A G ( f, a − Df ).
The defocused OTF is consequently A G ( f, − Df ). Therefore, the
pupil AF contains the values of the OTF for any value of the defo-
cusing distance. More precisely, as first pointed out in Ref. 15, the
pupil AF can be seen as a polar display of the OTF for variable defo-
cusing distance: the OTF is displayed, as represented schematically in
Fig. 2.1, along lines going through the origin of coordinates in the
( f , a) representation (see Chapt. 5).
This connection between the OTF and the pupil AF has been
used 16−24 for designing pupil phase masks (phase apodizers) which
increase the depth of focus without losing lateral resolution and light-
gathering power. Furthermore, various effects such as the behavior
of the Strehl ratio and the sensitivity to spherical aberration, 25 or the