Page 75 - Phase Space Optics Fundamentals and Applications

56 Chapter Two

The WDF tomographic reconstruction is based on the formula

I D (x) = df W(x − Df, f ) (2.38)

which shows that I D (x) is the projection of the WDF in the (x, f )
space along a direction that can be varied by the position z = D of the
recording plane. The operation that allows the WDF reconstruction is
an inverse Radon transform. 27, 28 The feasibility of the tomographic
WDF reconstruction has been discussed in Refs. 27 to 31.
The AF reconstruction is considered 28−31 to be simpler, because
there is no need of inverse Radon transform (the term phase-space to-
mography is therefore not really appropriate in this case); we only need
to perform the Fourier transformation of the measured I D (x). The re-
lation ˜ I D ( f ) = A( f, − Df ) shows that the intensity spectra represent
the variations of the AF along the radial lines a =− Df in the ( f, a)
space, as depicted in Fig. 2.1. To sample the AF over the complete
( f, a) space, it is necessary to use negative as well as positive values
of the distance D; this is not possible presently in X-ray optics because
appropriate lenses are not available.
The process of AF reconstruction was first considered in the
case of one-dimensional structures. 28, 29 A two-dimensional struc-
ture can be considered as an ensemble of one-dimensional y struc-
tures T(x 0 ,y); the corresponding AFs are A(x 0 ; f y ,a y ), from which
T(x 0 ,y) can be derived as a function of y according to Eq. (2.35). For
this purpose, a convenient setup, actually a one-dimensional prop-
agator system, has been proposed by Liu and Brenner 32 (see also
Ref. 33): A cylindrical lens of focal length F produces an exact image
(corresponding to no effective propagation) in the x direction, while
there is propagation over the object image distance in the y direction;
this distance can be varied by using several cylindrical lenses.

2.7 Another Possible Approach to AF
Reconstruction
The AF in the exit plane of an object illuminated by a tilted plane wave
of tilting angle is

a a a
! "
dx exp (−i2 fx)T x + exp i2 x + T ∗ x −
2 2 2
! a " a
× exp −i2 x − = A ob ( f, a) exp i2 (2.39)
2
where A ob ( f, a) is the AF associated to the transmittance T(x). By
setting = / , the intensity spectrum of the image delivered by an