Biomedical Engineering Reference
In-Depth Information
To understand how the dispersive properties (i.e., real parts of the RI) may be obtained,
consider a Michelson interferometer with a reference field,
E
r
(
ω
), and a sample field,
E
s
(
ω
),
described as:
E
r
ðωÞ 5
sðωÞ
U
exp
ð
i
ðω=
c
0
Þ
2
z
r
Þ
(14.5a)
p
I
s
ðωÞ
E
s
ðωÞ 5
U
exp
ð
i
ðω=
c
0
Þ
2
z
d
Þ
exp
ð
i
ðω=
c
0
ÞnðωÞ
2
ðz
s
2
z
d
ÞÞ
(14.5b)
where
z
r
,
z
d
, and
z
s
are the distances from the beamsplitter to the reference mirror,
dispersive medium, and scatterer, respectively;
S
(
ω
) is the spectrum of the source field,
I
s
(
) is the real
part of the RI of the sample
[61]
. The sample can be described by bulk absorption and
scattering coefficients
ω
) is the sample field intensity,
c
0
is the speed of light in vacuum, and
n
(
ω
μ
a
and
μ
s
, respectively, such that the sample field intensity may be
2
written as
I
s
(
z
d
)). Thus, the interferometric signal, after
elimination of DC background terms, may be expressed as:
ω
)
5 jS
(
ω
)
j
exp(
2
(
μ
1μ
s
)2(
z
s
2
a
p
I
s
ðωÞI
r
ðωÞ
U
exp
ð
i
ðω=c
0
Þ
2
ðz
0
I
ðωÞ 5
2
2dnðωÞÞÞ
(14.6)
c
0
Þ
2
ðz
0
5
2
I
r
ðωÞ
exp
ð2μ
tot
ðωÞdÞ
U
exp
ð
i
ðω=
2
dnðωÞÞÞ
2
s
,
z
0
where
I
r
z
d
is the sample thickness. To
analyze the information contained in the dispersion of the signal, we employ a Taylor series
expansion of
n
(
5 jSj
,
μ
5μ
1μ
5
z
r
2
z
d
, and
d
5
z
s
2
tot
a
ω
),
1
2
dn
ðωÞ
dω
ω
0
ω2ω
0
c
0
1
2
!
d
2
n
ðωÞ
dω
ω2ω
0
c
0
nðωÞ 5
nðω
0
Þ 1
1
?
5
nðω
0
Þ 1Δ
nðωÞ
2
ω
0
(14.7)
where
n
(
ω
0
) may be evaluated at an arbitrary frequency or wavelength, and
Δn
incorporates
all the high-order
ω
-dependent terms from the Taylor series expansion. Thus,
Eq. 14.6
may
be rewritten as follows:
c
0
Þ
U
2
ðz
0
I
ðωÞ 5
2
I
r
ðωÞ
exp
ð2μ
tot
ðωÞdÞ
U
exp
ð
i
ðω=
2
dnðω
ÞÞÞ
U
exp
ð2
i
ðω=
c
0
Þ
U
2
d
Δ
nðωÞÞ
(14.8)
0
Equation 14.8
clearly shows that the measured signal contains three distinct parts. The first
part attenuates the signal intensity, providing access to spectroscopic measurement of
μ
tot
.
This term may be analyzed using SOCT techniques such as the dual-window method
[62]
,
or in the case of a thin sample, the complex signal
I
(
ω
) may simply be demodulated. The
second term of
I
(
) describes the linear phase and can be obtained using SDPM. Let us
consider that the sample thickness,
d
, may be expanded as multiples of the source's
coherence length,
l
c
, and a subcoherence length deviation,
ω
1δl
c
with
m
being an integer. Thus, a fast Fourier transform (FFT) of
Eq. 14.8
, ignoring the third term
for now, reveals a peak located at 2
n
0
ml
c
, corresponding to the sample thickness.
Additionally, the phase at the same point gives
δ
, given as
d5ml
c
φ5
(
2
π
/
λ
0
)
2n
0
δ
l
c
, with
λ
0
being the center
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