Biomedical Engineering Reference
In-Depth Information
17.7.2 Change of Optical Path Through the Observed Object
Let us now consider the quasi-monochromatic coherent wave emitted by a nano-sized
prismatic object. The actual formation of the image is similar to a typical lens hologram of
a phase object illuminated by a phase grating
[19]
. The FT of the image of the nano-object
extended to the complex plane is an analytical function. If the FT is known in a region,
then, by analytic continuation,
F
(
sp
) can be extended to the entire domain. The resolution
obtained in this process is determined by the frequency
ω
sp
captured in the image. The
image can be reconstructed by a combination of phase retrieval and suitable algorithms.
The image can be reconstructed from an
F
(
ω
ω
sp
) such that
ω
,
ω
sp,max
, where
ω
sp,max
is
sp
determined by the wavefronts captured by the sensor.
The fringes generated by different diffraction orders that were analyzed in
Sections 17.4
and 17.5
experience phase changes through the passage of the nano-object that provide
depth information. This type of optical setup to observe phase objects was used in phase
hologram interferometry as a variant of the original setup proposed by Burch and Gates
[20]
and Spencer and Anthony
[21]
. When the index of refraction in the medium is
constant, the rays going through the object are straight lines. If a prismatic object is
illuminated with a beam normal to its surface, the optical path
s
op
through the object is
given by the integral:
ð
n
i
ð
s
op
ð
x
;
y
Þ
5
x
;
y
;
z
Þ
d
z
(17.21)
where the direction of propagation of the illuminating beam is the
z
coordinate and the
analyzed plane wavefront is the plane
X
Y
;
n
i
(
x
,
y
,
z
) is the index of refraction of the
object through which light propagates.
The change experienced by the optical path is:
ð
t
0
½
Δ
s
op
ð
x
;
y
Þ
5
n
i
ð
x
;
y
;
z
Þ
2
n
o
d
z
(17.22)
where
t
is the thickness of the medium. By assuming that
n
i
(
x
,
y
,
z
)
5
n
c
where
n
c
is the
index of refraction of the observed objects,
Eq. (17.22)
then becomes:
Δ
s
op
ð
x
;
y
Þ
5
ð
n
c
2
n
o
Þ
t
(17.23)
By transforming
Eq. (17.23)
into phase differences and making
n
o
5
n
s
, where
n
s
is the
index of refraction of the saline solution containing the observed objects, one can write:
2
p
ð
Δφ
5
n
c
2
n
s
Þ
t
(17.24)
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