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
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
5
z r
2
z d , and d
5
z s
2
tot
a
ω
),
1
2
dn ðωÞ
ω 0
ω2ω 0
c 0
1
2 !
d 2 n ðωÞ
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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