Page 78 - Phase Space Optics Fundamentals and Applications

Ambiguity Function in Optical Imaging 59

2.8.2 Application to Phase Retrieval
and X-Ray Holotomography
In the case of a phase object, such that T( ) = exp [i
( )], formula
(2.42) can be written in the following form:

2
∗
( f ) − exp (−i Df ) dh exp (−i2 D n hf )˜ (h + f )˜ (h)
˜ I D n
2
= ( f ) + 2 sin( D n f )˜
( f ) (2.44)
,
Neglecting the nonlinear term (the integral term), denoted as NLT D n
in the left-hand side of this equation, we obtain the following estima-
tion of the phase spectrum 35 from the experimentally known ˜ I D n ( f )
by a least-squares fitting as
-
2
n sin( D n f ) ˜ I D n ( f )
˜
( f ) = - (2.45)
2
2
2 sin ( D n f )
n
From this result, it is possible to calculate the nonlinear terms to check
whether they could indeed be neglected. If necessary, it is possible
can be
to take them into account recursively: the calculated NLT D n
( f ) to obtain a new
substracted from the experimentally known ˜ I D n
estimate of the phase spectrum
-
2
n sin( D n f )[ ˜ I D n ( f ) − NLT D n ]
˜
( f ) = - 2 (2.46)
2
2 sin ( D n f )
n
and this process can be continued recursively.
If a single image (N = 1) were used in formula (2.43), the phase
spectrum could not be obtained for the spatial frequencies f such that
2
sin( Df ) 0. Using several images (typically four or five images
36
are used ) allows one to eliminate this defect and to reduce the influ-
ence of the nonlinear terms.
This phase retrieval approach, which has some similarity to the
focus variation method used in electron microscopy, 37 has been im-
plemented in synchrotron X-ray optics (see Fig. 2.3) to provide two-
dimensionalphasemaps,withmicrometerresolution,ofobjectsshow-
ing a nearly uniform absorption but introducing an important phase
modulation. Advantage is taken of the high degree of spatial coher-
ence (due to the small lateral size of the source and the long source-
specimen distance) and the good monochromaticity available on mod-
ern synchrotron beam lines. The phase maps obtained for different
orientations of the object are used as input for a tomographic recon-
struction of the three-dimensional distribution of the electron density
in the sample. This technique named holotomography 35,36,38 is of parti-
cular interest in the case of objects opaque to visible light. It has been