Biomedical Engineering Reference
In-Depth Information
Here,
k
0
5
2
π
/
λ
0
is the wave number in the free space with
λ
0
the wavelength in the free
!
space, and
n
ð
Þ
is the complex refractive index. If the field is decomposed into the incident
!
!
field
U
ð
I
Þ
ð
and scattered field
U
ð
S
Þ
ð
Þ
Þ;
!
!
!
Þ
5
U
ð
I
Þ
ð
Þ
1
U
ð
S
Þ
ð
U
ð
Þ
(12.2)
then, the wave equation becomes
!
!
!
2
1
k
0
n
2
m
Þ
U
ð
S
Þ
ð
ðr
Þ
5
F
ð
Þ
U
ð
Þ
(12.3)
!
!
2
2
with
F
ð
Þ
2
ð
2
π
n
m
=λ
0
Þ
ðð
n
ð
Þ=
n
m
Þ
2
1
Þ;
and
n
m
is the refractive index of the medium.
!
The
F
is known as the object function. Based on Green's theorem, the formal solution
to
Eq. (12.3)
can be written as:
ð
Þ
ð
G
!
!
2
!
0
jÞ
!
0
Þ
!
0
Þ
d
3
!
0
U
ð
S
Þ
ð
Þ
52
ðj
F
ð
U
ð
(12.4)
with
G
(
r
)
5
exp(
in
m
k
0
r
)/(4
π
r
) the Green's function. Since the integrand contains the
!
unknown variable,
U
ð
Þ;
we employ an approximation to obtain a closed form solution for
!
U
ð
S
Þ
ð
The first Born approximation is the simplest we can introduce when the scattered
field is much weaker than the incident field (
U
(S)
Þ:
{
U
(I)
), in which case the scattered field is
given by the following equation:
ð
G
!
!
2
!
0
jÞ
!
0
Þ
!
0
Þ
d
3
!
0
U
ð
S
Þ
ð
U
ð
I
Þ
ð
Þ
2
ðj
F
ð
(12.5)
This approximation provides a linear relation between the object function and the scattered
field
U
ð
S
Þ
ð
!
By taking the Fourier transform of both sides of
Eq. (12.5)
, we obtain the
following relation, known as the Fourier diffraction theorem
[1]
:
Þ:
ik
z
π
U
ð
S
Þ
F
z
1
5
0
ð
K
x
;
K
y
;
K
z
Þ
5
ð
k
x
;
k
y
;
Þ
(12.6)
Here,
F
and
U
ð
S
Þ
are the 3D and 2D Fourier transform of
F
and
U
(S)
, respectively;
k
x
and
k
y
are the spatial frequencies corresponding to the spatial coordinate
x
and
y
in the transverse
image plane, respectively;
z
1
5
0 is the axial coordinate of the detector plane, which is the
plane of objective focus in the experiment. (
K
x
,
K
y
,
K
z
), the spatial frequencies in the object
frame, define the spatial frequency vector of (
k
x
,
k
y
,
k
z
) relative to the spatial frequency
vector of the incident beam (
k
x
0
,
k
y
0
,
k
z
0
)
, and
k
z
is determined by the relation
k
z
5
q
ð
2
n
m
k
0
Þ
2
k
x
2
k
y
:
For each illumination angle, the incident wave vector changes,
and so does (
K
x
,
K
y
,
K
z
)
.
As a result, we can map different regions of the 3D frequency
spectrum of the object function with various 2D angular complex E-field images.
After completing the mapping, we can take the inverse Fourier transform of
F
to get the 3D
distribution of the complex refractive index.
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