Biomedical Engineering Reference
In-Depth Information
On the other hand, if the object is a mostly transparent cell on the reflective substrate (the
light propagates through it, reflects from the substrate and propagates back), the physical
thickness is
hðx; yÞ
5
λ
ϕðx; yÞ
ðn
2
n
0
r
(7.5)
2
π
Þ
where (
n
2
n
0
) is the refractive index difference between the cell and the air. Here,
λ
r
5
λ=
2 is half of the wavelength of light, because the light travels through the sample
twice.
As mentioned before, the phase
Δϕ
in these equations can only vary from 0 to 2
π
, which
corresponds to optical thickness variation of 0 to
λ
. If an object is thicker, then this results
in 2
π
discontinuities in the phase image that need to be unwrapped.
7.2.3 Curvature Correction
The angular spectrum method described earlier is based on the assumption that the
reference and object waves are both ideal plane waves. However, in the real setup, each
wave has its wave front curvature, resulting in a curvature mismatch
[3]
.
Consider the complex field captured by a CCD array (
Figure 7.3
).
R
is the wave's radius
o
f
curvature centered at point
C
, which can be determined experimentally for a given set
u
p,
r
is the vector from the center of the CCD matrix (point
O
) to an arbitrary point
A
, and
r
0
is
the vector from the center
o
f the CCD matrix to the projection of the center of curvature on
t
h
e CCD matrix
P
. Then
p
x
2
jrj
5
1
y
2
, where
x
and
y
are the coordinates of
A
and
p
x
0
1
y
0
jr
0
j
5
, where
x
0
and
y
0
are the coordinates of
P
.
The phase mismatch can be compensated numerically, by multiplying the original “flat”
field
E
0
(
x
,
y
;
z
5
0) by the phase factor exp[i
ϕ
], where
ϕ
5
kd
is the phase difference
between
A
and
O
and
d
is the optical path difference:
p
CP
2
p
CP
2
d
5
CA
2
CO
5
1
PA
2
1
PO
2
2
(7.6)
A
r
R
O
r
0
C
R
P
Figure 7.3
Curvature compensation.
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