Biomedical Engineering Reference
In-Depth Information
experiments have their central wavelengths at 500
650 nm with a spectral FWHM
(full width at half maximum) of 10
15 nm.
In these derivations, paraxial approximation has also been assumed to simplify the results
since
z
a
and
z
s
for our hologram recording geometry are typically much longer than the
extent of each cell hologram. However, such an approximation was not used in the
numerical reconstruction process of the cell images, which will be further discussed later in
this chapter.
D
0
in
Eq. (8.1)
can be further expanded into:
ððððð
W
ðΔ
x
; Δ
y
; η; νÞ
exp
2
j
π
λ
x
0
1
2
x
D
Þ
2
y
0
1
2
y
D
Þ
2
D
0
5
z
s
ð
1
ð
2
ðλ
z
s
Þ
d
x
0
1
d
y
0
1
d
x
0
2
d
y
0
2
d
exp j
π
λ
2
2
x
0
2
2
x
D
Þ
y
0
2
2
y
D
Þ
z
s
1
ð
1
ð
ν
which physically represents the background illumination reaching the sensor plane and it
carries no spatial information regarding the cells' structure or distribution. For typical
illumination configurations,
D
0
constitutes a uniform or slowly varying background, and
hence can be digitally subtracted out without an issue.
Equations (8.1)
(8.4)
are rather important to explain the key parameters in our partially
coherent lensless on-chip holography scheme utilizing an incoherent light source emanating
through a large aperture.
Equation (8.1)
includes the background illumination (
D
0
term) and
the self-interference of the scattered waves (the terms that are proportional to
2
and
j
c
1
j
2
), both of which represent the classical diffraction that occurs between the object and
the sensor planes under the paraxial approximation. Note also that these self-interference
terms in
Eq. (8.1)
are scaled with
P
(0,0) as the physical extent of the spatial coherence at
the cell plane is not a determining factor for self-interference.
j
c
2
j
Equation (8.2)
, on the other hand, represents the interference between these two scatterers
located at the object plane. Just like self-interference, the cross-interference term,
I
C
(
x
D
,
y
D
),
is also not useful for holographic reconstruction of object images. Since this cross-
interference term is proportional to the amplitude of
P
two scatterers that are
far from each other can still interfere effectively if a small aperture size is used (hence wide
P
). The
ðð
2
a
=λ
0
z
a
Þ;
0
Þ;
P
0
z
a
/
D
a
) (where
D
a
is the aperture width),
the cross-interference
I
C
(
x
D
,
y
D
) from these two scattered fields will generate strong but
undesired cross-interference patterns at the sensor plane. This conclusion is also supported by
the fact that the coherence diameter at the object plane is in the order of
B
(
λ
0
z
a
/
D
a
), as
estimated by van Cittert-Zernike theorem. Therefore, as another advantage of using a large
aperture that is illuminated by an incoherent light source, this cross-interference noise term,
I
C
(
x
C
,
y
D
), will only contain the contributions of a limited number of cells within the imaging
ðð
2
a
=λ
0
z
a
Þ;
Þ
term predicts that, if (2
a
,
λ
0
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