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Ambiguity Function in Optical Imaging 47

The AF and the WDF are related to each other by double Fourier
transformation over the position and frequency variables:

A(f, a) = dx dg exp [i2 (a · g − f · x)]W(x, g)

W(x, g) = da df exp [i2 (f · x − a · g)]A(f, a) (2.7)

The property (x , x) = (x, x ) of the mutual intensity shows that the
∗
WDF is real and that the AF, which is in general complex, satisfies the
∗
relation A(−f, −a) = [A(f, a)] .
In the exit plane of an object of transmittance T(x), under uni-
form coherent illumination, the mutual intensity (x + a/2, x − a/2)
is equal to the product T(x + a/2)T (x − a/2) which is, according
∗
to Chap. 5, the product-space representation of the signal T(x). The
corresponding AF and WDF, which are referred to as the AF and
the WDF associated to T(x), were introduced in optics by Papoulis 2
3
and Bastiaans, respectively. They are redundant representations
of T(x).
PSO representations are useful tools to characterize the perfor-
mances of optical systems. They provide elegant approaches to the
description and processing of optical signals or images. It has been
4
shown recently by Nugent (see also Ref. 5) that the concept of AF can
be used to unify the various noninterferometric approaches to X-ray
phase retrieval.
To simplify the formulation of the following sections, we most often
consider one-dimensional fields, in which case the two-dimensional
vectors are replaced by scalars, without a real loss of generality, be-
cause the extension to the general case is usually straigthforward.

2.2 Intensity Spectrum of a Fresnel
Diffraction Pattern Under
Coherent Illumination

2.2.1 General Formulation
For simplicity, let us consider a plane wave, of wavelength , incident
along the z direction on a thin object of transmittance T(x) in the plane
z = 0. In the conditions of Fresnel diffraction, the wave function in a
plane z = D is

2
−1/2 (x − )
D (x) =| D| exp −i d exp i T( ) (2.8)
4 D