Biomedical Engineering Reference
In-Depth Information
image of the sample. For the case that the object is not imaged sharply in
the hologram plane HP during the hologram recording process, e.g., due to
mechanical instability of the experimental setup or thermal effects, in a second
evaluation step a further propagation of the object wave to the image plane can
be carried out for subsequent focus correction. The propagation of O(
x, y, z
0
)
to the image plane
z
IP
that is located at
z
IP
=
z
0
+∆
z
in the distance ∆
z
to HP can be carried out by a Fresnel transformation [26, 31, 35] or by a
convolution algorithm [34, 36, 37]:
O(
x, y, z
IP
=
z
0
+∆
z
)=
F
−
1
F {
exp
i
πλ
∆
z
ν
2
+
µ
2
.
(9.9)
O(
x, y, z
0
)
}
In (9.9)
λ
is the applied laser light wave length,
ν, µ
are the coordinates in
frequency domain and
F
denotes a Fourier transformation. During the propa-
gation process, the parameter ∆
z
is chosen in such a way that the holographic
amplitude image appears sharply, in correspondence to a microscopic image
under white light illumination. A further criterion for a sharp image of the
sample is that diffraction effects due to the coherent illumination appear min-
imized in the reconstructed data. As a consequence of the applied algorithms
and the parameter model for the phase difference model ∆
ϕ
HP
in (9.8), the re-
sulting reconstructed holographic images do not contain the disturbing terms
“twin image” and “zero order.” In addition, the method allows in compari-
son to propagation by Fresnel transformation, as e.g., in [31, 35], a sharply
focused image of the sample in the hologram plane. The propagation of O by
(9.9) enables in this way the evaluation of image plane holograms containing
a sharply focused image of the sample and effects no change of the image scale
during subsequent refocusing. In the special case that the image of the sam-
ple is sharply focused in the hologram plane with ∆
z
=0andthus
z
IP
=
z
0
,
the reconstruction process can be accelerated because no propagation of O by
(9.9) is required.
From O(
x, y, z
IP
), in addition to the absolute amplitude
that
represents the image of the sample, the phase information ∆
ϕ
S
(
x, y, z
0
)ofthe
sample is reconstructed simultaneously:
|
O(
x, y, z
IP
)
|
∆
ϕ
S
(
x, y, z
0
)=
φ
O
(
x, y, z
0
)
− φ
O
0
(
x, y, z
0
)
= arctan
{
O(
x, y, z
IP
)
}
(mod2
π
)
.
(9.10)
{
O(
x, y, z
IP
)
}
After removal of the 2
π
ambiguity by a phase unwrapping process [1], the
data obtained by (9.10) can be applied for quantitative phase contrast mi-
croscopy, which is the main topic of interest for the described experiments.
For an incident light geometry as depicted in the left panel of Fig. 9.13, the
topography
z
s
can be calculated on the phase distribution ∆
ϕ
S
(
x, y, z
0
):
z
s
(
x, y, z
0
)=
λ
∆
ϕ
S
(
x, y, z
0
)
2
.
2
π
λ
4
π
=
∆
ϕ
s
(
x, y, z
0
)
.
(9.11)
