Biomedical Engineering Reference
In-Depth Information
One of the key steps for recording high-quality in-line holograms with a spatially
incoherent source emanating from a large aperture is to bring the cell plane close to
the detector array by ensuring
z
s
{
z
a
, where
z
a
defines the distance between the
incoherently illuminated aperture plane and the object/cell plane, and
z
s
defines the distance
between the object/cell plane and the sensor array (see
Figure 8.1
). While the total
aperture-to-detector distance (
z
s
1
z
a
) and the overall device length remain almost
unchanged, conventional lensless in-line holography approaches typically choose to utilize
z
a
{
z
s
. Therefore, in addition to an incoherent source used with a large aperture, our choice
of
z
s
{
z
a
is also quite different from the mainstream lensless in-line holographic imaging
approaches.
To better quantify the impact of these differences on the detected holograms and their
reconstructions, we assume two point scatterers (separated by 2
a
) located at the object/cell
plane (
z
5
z
a
) with a field transmission of
t
(
x
,
y
)
5
1
1
c
1
(
x
1
a
,
y
), where
the intensities of
c
1
and
c
2
denote the strength of the scattering process for these two point
sources and
δ
(
x
,
y
) is a Dirac-delta function in space. Subcellular elements that make up a
cell can be represented by such point scatterers. Let us further assume that there is a large
aperture (of arbitrary shape) that is positioned at
z
5
0 with a transmission function of
p
(
x
,
y
)
and that the digital recording device (e.g., a CMOS (complementary-symmetry metal-oxide-
semiconductor) or CCD (charge-coupled device) sensor array) is positioned at
z
5
z
a
1
z
s
,
where typically
z
a
is 3
δ
(
x
2
a
,
y
)
1
c
2
δ
10 cm and
z
s
is 0.5
2 mm.
Assuming that a spatially incoherent light source uniformly illuminates the aperture
p
(
x
,
y
),
the cross-spectral density at the aperture plane can be written as:
W
ð
x
1
;
y
1
;
x
2
;
y
2
; νÞ
5
S
ðνÞ
p
ð
x
1
;
y
1
Þδð
x
1
2
x
2
Þδð
y
1
2
y
2
Þ
where (
x
1
,
y
1
) and (
x
2
,
y
2
) denotes two arbitrary points on the aperture plane and
S
(
v
)
represents the power spectrum of the incoherent source having its center frequency at
v
0
(corresponding to the center wavelength of
0
). After propagating over a distance of
z
a
in
free space, one can write the cross-spectral density at the object plane (just before
interacting with the cells) as
[63]
:
λ
ðð
pðx; yÞ
exp j
2
d
x
d
y
S
ðνÞ
2
exp
2
j
2
πν
q
π
WðΔx; Δy; η; νÞ
5
z
a
ðxΔx
1
yΔyÞ
cz
a
λ
ðλ
z
a
Þ
),
c
is the
speed of light, and (
x
1
,
y
1
)(
x
2
,
y
2
) denotes two arbitrary points on the object plane. After
interacting with the specimen, that is,
t
(
x
,
y
), the cross-spectral density right after the object
plane can be written as:
Δ
x
5
x
0
1
2
x
0
2
; Δ
y
5
y
0
1
2
y
0
2
; η
5
ð
x
0
1
1
x
0
2
=
ÞΔ
x
1
ð
y
0
1
1
y
0
2
=
ÞΔ
y
; λ
5
(
c
/
ν
where
2
2
t
x
0
1
;
y
0
1
ÞU
x
0
2
;
y
0
2
Þ
W
ðΔ
x
; Δ
y
; η; νÞU
ð
t
ð
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