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Ambiguity Function in Optical Imaging 55

a

O f

a = –λDf

FIGURE 2.1 Schematic PSO representation. The AF along the line a =− Df
corresponds to the intensity spectra ˜ I D ( f ) at distance D from the reference
plane. The integration of formula (2.37) is to be performed along lines
parallel to the f axis.

26
focal shift, have been studied by considerations based on the behav-
ior of the pupil AF. These applications are detailed in Chapt. 5.

2.6 Phase-Space Tomography
The idea of phase-space tomography 4, 5, 27−33 is to reconstruct the AF
(or the WDF) in the plane z = 0 from a set of intensity measurements
I D (x) in planes z = D n . The mutual intensity can then be derived
from the reconstructed AF (or WDF) by using formula (2.6). This is of
interest for the characterization of the optical field in the plane z = 0.
If an object of transmittance T(x) is placed in this plane, we obtain

a a a a
inc x + ,x − T x + T ∗ x −
2 2 2 2

= df exp (i2 fx)A( f, a) (2.36)
where inc is the mutual intensity of the incident beam. With x = a/2,
this is reduced to

∗
inc (a, 0)T(a)T (0) = df exp (i fa)A( f, a) (2.37)
As inc (a, 0) and the modulus of T(0) can be measured independently,
this last formula allows the determination (up to a constant phase
factor) of the complex function T(a) from the AF.