Biomedical Engineering Reference
In-Depth Information
Before reaching the detector plan, this cross-spectral density function will also propagate
for another distance of
z
s
. Therefore, the cross-spectral density at the detector plane can be
written as:
W
D
ð
x
D1
;
y
D1
;
x
D2
;
y
D2
;νÞ
ðððð
WðΔx;Δy;η;νÞt
ðx
0
1
;y
0
1
Þtðx
0
2
;y
0
2
Þh
C
ðx
0
1
;x
D1
;y
0
1
;y
D1
;νÞh
C
ðx
0
2
;x
D2
;y
0
2
;y
D2
;νÞ
d
x
0
1
d
y
0
1
d
x
0
2
d
y
0
2
5
where (
x
D1
,
y
D1
) and (
x
D2
,
y
D2
) refer to two arbitrary points on the detector plane
(within the hologram region of each cell) and
h
C
ðx
0
; x
D
; y
0
; y
D
; νÞ
5
exp
1
z
s
exp j
2
πz
s
j
λ
λ
2
2
j
λz
s
x
0
2
x
D
Þ
y
0
2
y
D
Þ
ð
1
ð
:
At the sensor plane (
x
D
,
y
D
), one can then write the optical intensity
I
(
x
D
,
y
D
) as:
ð
W
D
ð
I
ð
x
D
;
y
D
Þ
5
x
D
;
y
D
;
x
D
;
y
D
; νÞ
d
ν
Assuming
t
(
x
,
y
)
5
1
1
c
1
δ
(
x
2
a
,
y
)
1
c
2
δ
(
x
1
a
,
y
), the optical intensity
I
(
x
D
,
y
D
) can be
further expanded into four terms, each with a different physical meaning, i.e.:
Iðx
D
; y
D
Þ
5
I
s
ðx
D
; y
D
Þ
1
I
C
ðx
D
; y
D
Þ
1
H
1
ðx
D
; y
D
Þ
1
H
2
ðx
D
; y
D
Þ
where:
2
S
0
2
S
0
y
D
Þ
5
D
0
1
j
c
1
j
Þ
1
j
c
2
j
P
P
I
s
ð
x
D
;
ð
0
;
0
ð
0
;
0
Þ
(8.1)
2
2
ðλ
0
z
a
z
s
Þ
ðλ
0
z
a
z
s
Þ
exp j
4
c
2
c
1
S
0
ðλ
0
z
a
z
s
Þ
2
a
λ
0
z
a
;
πax
D
λ
0
z
s
P
I
C
ð
x
D
;
y
D
Þ
5
0
1
c
:
c
:
(8.2)
2
1
c
S
0
ðλ
0
z
a
Þ
pð
2
x
D
UM
1
aUMUF;
2
y
D
UMÞ
h
C
ðx
D
; y
D
Þ
H
1
ðx
D
; y
D
Þ
5
2
c
1
U
:
c
:
(8.3)
1
c
S
0
Þ
h
C
ð
H
2
ð
x
D
;
y
D
Þ
5
c
2
U
p
ð
2
x
D
U
M
2
a
U
M
U
F
;
2
y
D
U
M
x
D
;
y
D
Þ
:
c
:
(8.4)
2
ðλ
0
z
a
Þ
where “c.c.” and “
” denote the complex conjugate and convolution operations,
respectively,
M
5
(
z
a
/z
s
),
F
5
(
z
a
1
z
s
/
z
a
), and
P
is the 2D spatial Fourier transform of the
arbitrary aperture function
p
(
x
,
y
). It should be emphasized that (
x
D
,
y
D
) in these equations
is restricted to the extent of the cell/object hologram, rather than extended to the entire FOV
of the detector array. Furthermore,
h
C
ð
x
D
2
1
y
D
2
x
D
;
y
D
Þ
5
ð
1
=
j
λ
0
U
F
U
z
s
Þ
exp
ð
j
ðπ=λ
0
U
F
U
z
s
Þð
ÞÞ
and it represents the 2D coherent impulse response of free space over a distance of
Δ
z
s
. For the incoherent illumination source, we have assumed that the spectral
bandwidth is much smaller than its center frequency
V
0
,i.e.,
S
(
v
)
D
S
0
z
5
F
(
v
2
v
0
). This is an
appropriate assumption since the light sources (LEDs) that we have typically used in our
δ
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