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 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
! 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
! 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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