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Ambiguity Function in Optical Imaging 57

imaging system with pupil AF A G ( f, )is

˜ I im ( f, ) = d A G ( f, − )A ob ( f, ) exp (i2 ) (2.40)

which is a Fourier transform. Consequently,

A ob ( f, )A G ( f, − ) = d exp (−i2 ) ˜ I im ( f, ) (2.41)

Supposing A G ( f, − ) to be known and ˜ I im ( f, ) to be measured as
a function of , we see this last formula provides the possibility to
obtain the object AF.
This approach was suggested 34 in the context of X-ray analyzer-
based imaging which is a Schlieren-type technique (See Fig. 2.2) based
on Bragg diffraction by a perfect crystal (analyzer) acting as a filter
in Fourier space. 34 The image spectrum is equal to ˜ T( f ) ˜ G( f ), where
˜ T( f ) is the object spectrum and ˜ G( f ) is the complex reflectivity of
the crystal for a plane wave of offset angular position = f with
respect to the exact Bragg angular position. Therefore ˜ G( f ) is the
pupil function of the system.

Object Image

Crystal analyzer

FIGURE 2.2 Principle of the X-ray analyzer-based imaging system. A
quasi-parallel and quasi-monochromatic beam is diffracted, after
transmission through the object, by a plate of perfect silicon crystal. The
arrangement is such that Bragg diffraction occurs, corresponding to
reflecting planes parallel to the crystal surface. The Bragg diffraction process
is highly sensitive to the direction of the rays. Images are recorded across the
diffracted beam for different angular settings of the crystal.