Biomedical Engineering Reference
In-Depth Information
Here A./ is the electric field amplitude per unit wavenumber. When this field enters
the interferometer, it splits into two, and half of each beam goes to the detector. As
mentioned above, the other half is sent back to the light source. Therefore, assuming
that there are no losses at all through the optical path as a result of the reflections
and transmissions, the electric field at the detector is
Z
A./ cos.2.x
C
x/
!t/d
(4.8)
where x is the OPD created between the two splitted beams. The intensity
measured by the detector is the square of the electric field. In addition, the measured
intensity is the time average of the intensity over a time interval that is long enough
relative to the wave time period, which is in the range of 10
15
s, although it can be
very short, say few nanoseconds. Therefore, it gives
Z
1
2
E
D
A./ cos.2 x
!t/d
C
*
1
4
Z
D
2
E
I.x/
D
j
E
j
D
A./cos .2x
!t/ d
A./cos .2 .x
C
x/
!t/d
2
+
Z
C
(4.9)
Here
the
triangular
brackets
represent
averaging
over
time.
Opening
the
squared parenthesis will
give four terms. Two of the terms will
be of the
form
˝'
A
2
./ cos.2x
!t/ cos.2
0
x
!
0
t/d d
0
˛
. Because of the time
dependence of each cosine, the time average will vanish for
0
,orthese
¤
terms will give
˝R
A
2
./ cos
2
.2x
!t/d
˛
D
R
A
2
./
˝
cos
2
.2x
!t/
˛
d
where we inserted the time averaging to the only term that depends on time. The
time average of cos
2
is 1=2, and therefore, each such term gives .1=2/
R
A
2
./d .
This integral simply gives the total intensity of the light source. The two other
terms are equal to
˝'
A
2
./ cos.2.x
C
x/
!t/ cos.2
0
x
!
0
t/d d
0
˛
.
Using trigonometric identities, and the fact that the wavenumbers must be identical
in order that the time average will not vanish, each of these terms is equal to
.1=2/
R
A
2
./ cos.2x/d . Substituting these to Eq.
4.9
will give
Z
Z
A
2
./ cos.2x/d
:
1
2
A
2
./d
C
I.x/
D
(4.10)
The first term is a constant that is equal to the total intensity emitted from each
pixel region of the image. The second term is more important, as it depends on
the OPD, x. Actually, the second term can be identified as the real part of the
Fourier transformation of A
2
./. Recall that Eq.
4.10
is the intensity measured by
the detector when performing the measurement for many different OPDs x.
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