Biomedical Engineering Reference
In-Depth Information
splitting the horizontal and vertical polarization components to create two spatially
separated and phase-shifted interferograms on the camera.
This setup introduces a phase shift between the two parallel interferograms, which can be
explained as follows. In the reference arm, a quarter (
/4) waveplate is oriented along the
horizontal axis, so that the light transmitted through it has a 90
phase difference between
the horizontal and vertical components. On the other hand, in the sample arm, the light
remains linearly polarized at 45
, so there is no phase difference between the horizontal and
vertical components. When the beams are combined by BS
2
, there is a
λ
90
shift
between the interference patterns formed by the horizontal and the vertical components.
The Wollaston prism outputs two perpendicularly polarized beams, separating the
horizontal or vertical polarization components for each of the interferograms, yielding a
phase shift of
α5
α
between the interferograms.
The two interferograms,
I
1
and
I
2
, acquired by the digital camera, can be mathematically
written as follows:
2
p
I
1
5
I
R
1
I
S
1
I
S
I
R
½
cos
ðφ
OBJ
1φ
C
Þ
2
p
(14.1)
I
2
5
I
R
1
I
S
1
I
S
I
R
½
cos
ðφ
OBJ
1φ
C
1αÞ
where
I
R
and
I
S
are the reference and sample intensity distributions, respectively;
φ
OBJ
is the
spatially varying phase associated with the object; and
φ
C
is the spatially varying phase of the
φ
α
interferometer without the object present. Note that
can be digitally measured, by
fitting the background interference signal (interference pattern without sample) in each
interferogram to a sine wave. The wrapped object phase
φ
OBJ
is computed as
[6]
:
C
and
(14.2)
exp
ð2
j
φ
C
Þ
Im
F
Re
F
F
5
½
I
1
2
I
2
1
jHT I
1
2
f
I
2
g
;
φ
OBJ
5
arctan
1
2
exp
ð j
αÞ
where
HT
denotes a Hilbert transform. The final object phase is then obtained using an
unwrapping algorithm. Note that this process removes most common noise and background
elements thus producing highly stable phase measurements. In addition, since the
interferograms
I
1
and
I
2
are acquired in a single exposure, without the need for raster
scanning, fast phenomena can be visualized. Using
Eq. (14.1)
and recalling that
HT
{cos(
φ
)}
5
sin(
φ
), cos(
φ
)
5
0.5[exp(
jφ
)
2
exp(
2jφ
)], and sin(
φ
)
5
0.5
j
[exp(
2jφ
)
2
exp(
jφ
)], one
2
I
S
I
p
exp
ð jφ
OBJ
Þ;
can easily show that
F 5
and thus
Eq. (14.2)
produces the desired
signal.
To demonstrate the utility of the DQPM system (
Figure 14.1
), we imaged live unstained
MDA-MB-468 human breast cancer cells in standard growth medium which contained
phenol red growth medium.
Figure 14.2A
shows a typical light microscopic image of the
sample through the optical system, demonstrating the very low visibility achieved with
simple brightfield imaging of this sample. The specific setup incorporated a HeNe laser
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