Page 77 - Phase Space Optics Fundamentals and Applications

58 Chapter Two

If the crystal is rotated by an angle from its peak position in
the incident beam, this pupil function is changed into ˜ G( f − / ).
It is easy to show from formula (2.31) that the pupil AF A G ( f, a)is
then changed to A G ( f, a) exp (i2 a / ); we then obtain for the image
intensity spectrum the same formula as Eq. (2.41) with = / .
This shows that, as expected, the rotation of the crystal analyzer is
equivalent to a change in the direction of the incident beam.
This technique is sensitive to the object structure in one dimension.
To overcome this limitation, it should be just necessary, for each an-
gular position of the crystal analyzer, to perform a 90 rotation of the
◦
object in its plane, to obtain finally two-dimensional information.

2.8 Propagation-Based Holographic Phase
Retrieval from Several Images

2.8.1 Fresnel Diffraction Images as In-Line
Holograms
The holographic features of Fresnel diffraction images are clearly
shown by writing formula (2.11), with T( ) = 1 + ( ), as
2
2
∗
exp (i Df ) ˜ I D ( f ) = ( f ) + ˜ ( f ) + exp (2i Df )˜ (− f )

∗
+ dh exp (−i2 Dh f )˜ (h + f )˜ (h) (2.42)
The sum of the two first terms corresponds to the reconstructed object;
the next term corresponds to the out-of-focus image (at distance 2D)of
∗
the conjugate object (x); the integral term is negligible if | (x)|= 1
(weak object). The importance of these perturbation terms can be
strongly reduced by performing the following summation based on
N images recorded at different distances D n :

1 , N 2 ˜ (− f )
∗
( f ) exp i D n f = ( f ) + ˜ ( f ) +
˜ I D n
N n=1 N
, N
2 −1
× exp 2i D n f + N
n=1

∗
× dh ˜ (h + f )˜ (h)
, N
× exp (−i2 D n hf ) (2.43)
n=1
The quantity on the left-hand side may be calculated from digitally
recorded images; this allows a good reconstruction of the object if the
perturbation terms are nearly canceled by this summation procedure.