Biomedical Engineering Reference
In-Depth Information
I
HP
(
x, y, z
0
)=O(
x, y, z
0
)O
∗
(
x, y, z
0
)+ R(
x, y, z
0
)R
∗
(
x, y, z
0
)
+O(
x, y, z
0
)R
∗
(
x, y, z
0
)+R(
x, y, z
0
)O
∗
(
x, y, z
0
)
=
I
O
(
x, y, z
0
)+
I
R
(
x, y, z
0
)
+2
I
O
(
x, y, z
0
)
I
R
(
x, y, z
0
) cos ∆
ϕ
HP
(
x, y, z
0
)
,
(9.7)
with
I
O
=OO
∗
=
2
(* denotes the conjugate com-
plex term). The parameter ∆
ϕ
HP
(
x, y, z
0
)=
φ
R
(
x, y, z
0
)
2
and
I
O
=RR
∗
=
|
O
|
|
R
|
− φ
O
(
x, y, z
0
)isthe
phase difference between O and R at
z
=
z
0
. In the presence of a sample in the
optical path of O, the phase distribution represents the sum
φ
O
(
x, y, z
0
)=
φ
O
0
(
x, y, z
0
)+∆
ϕ
S
(
x, y, z
0
), where
φ
O
0
(
x, y, z
0
) denotes the pure object wave
phase and ∆
ϕ
S
(
x, y, z
0
) represents the optical path length change that is ef-
fected by the sample. For areas without a sample, ∆
ϕ
HP
(
x, y, z
0
) is estimated
by a mathematical model [31, 35]:
∆
ϕ
HP
(
x, y, z
0
)=
φ
R
(
x, y, z
0
)
− φ
O
0
(
x, y, z
0
)
=2
π
K
x
x
2
+
K
y
y
2
+
L
x
x
+
L
y
y
.
(9.8)
The parameters
K
x
,K
y
in (9.8) describe the divergence of the object wave and
the properties of the applied microscopy lens. The constants
L
x
,L
y
denote the
linear phase difference between O and R due to the off-axis geometry of the
experimental setup. For quantitative phase measurement from
I
HP
(
x, y, z
0
)
in a first step, the complex object wave O(
x, y, z
=
z
0
) in the hologram
plane is determined pixel wise by solving a set of equations that is obtained
from insertion of (9.8) in (9.7). For that purpose, neighboring intensity val-
ues within a square area of 5
5 pixels around a given hologram pixel are
considered by application of a spatial phase shifting algorithm (for details
see [9] and [23]). The utilized algorithm is based on the assumption that
only ∆
ϕ
HP
(
x, y, z
0
)=
φ
R
(
x, y, z
0
)
×
− φ
O
0
(
x, y, z
0
) between the object wave
O(
x, y, z
0
) and the reference wave R(
x, y, z
0
) varies rapidly spatially in the
hologram plane. In addition, because of the spatial phase shifting algorithm,
the object wave's intensity has to be assumed constant within an area of about
5
5 pixels around a given point of interest of the hologram. These require-
ments can be fulfilled by an adequate relation between the magnification of
the microscope lens and the image recording device. Therefore, the magnifi-
cation of the microscope lens is chosen in such a way that the smallest imaged
structures of the sample that are restricted by the resolution of the optical
imaging system due to the Abbe criterion are over sampled by the CCD sen-
sor. In this way the lateral resolution of the reconstructed holographic phase
contrast images is not decreased by the spatial phase shifting algorithm [31].
The parameters
K
x
,K
y
,L
x
,L
y
in (9.8) cannot be obtained directly from
the geometry of the experimental setup with an adequate accuracy and for
this reason are adapted once before the measurements by an iterative fitting
process in an area of the hologram without sample [31, 34].
The evaluation of digital holographic phase contrast images requires, in
correspondence to microscopy with white light illumination, a sharply focused
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