Page 72 - Phase Space Optics Fundamentals and Applications

Ambiguity Function in Optical Imaging 53

This is the convolution integral, with respect to the variable a, of the
AF in the object plane

A ob ( f, ) = d
exp (−i2 f
) ob
+ ,
− (2.30)
2 2
with the function

a a

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

f f
= dg exp (i2 ga) ˜ G g + ˜ G ∗ g − (2.31)
2 2
where ˜ G(g) is known as the pupil function of the imaging system. Also,
A G ( f, a) is to be considered equivalently as the AF of the coherent PSF
or as the pupil AF.
Theimageintensityspectrum,whichisaquantityofspecialinterest,
is obtained by setting a equal to 0 in Eq. (2.29):

˜ I im ( f ) = A im f, 0) = d A G ( f, − )A ob ( f, ) (2.32)

Formula (2.31) shows that, in the particular case of free-space prop-
2
agation for which ˜ G( f ) = exp (−i Df ), the pupil AF is a delta
function (a − Lf ). The pupil AF of an ideal (stigmatic) system is
equal to (a).

2.5 The AF of the Image of an
Incoherent Source

2.5.1 Derivation of the Zernike-Van Cittert
Theorem from the
Propagation of the AF
The mutual intensity of a planar incoherent source, in the source plane,
is of the form (x, x ) = S(x) (x − x ), where S(x) is the source density

[note that, for obvious dimensionality considerations, S(x) is not an
intensity distribution in the sense of the present formulation].
Formula (2.2) shows that the AF in the source plane is ˜ S( f ) (a),
where ˜ S( f ) is the spectrum of the source density. According to formula
(2.18), at distance L from the source, the AF is therefore ˜ S( f ) (a− Lf ),
and the mutual intensity can be obtained by formula (2.6) as

a a

x + ,x − = df exp (i2 xf ) ˜ S( f ) (a − Lf )
2 2

a i2 xa
= ˜ S exp ( L) −1 (2.33)
L L