Data processing is illustrated in
Figure 2 . Empirically, it was found that the following steps, incorporated into
a C program, provided satisfactory analysis. Step 1: Recorded spectra
from the eye and from the reference lens were read from disc, and, for
array-mode, the data from the 64 elements at each wavelength were
added. Wavelength was converted to its reciprocal: wave number,χ
.
Figure 2A shows typical recorded spectra from the eye,
V(χ), and from the reference lens,
V L(χ). Response units are
10
6 electrons per wavelength sample. Step 2:
Spectra were corrected for small nonlinearities of the photodetector at
each wavelength, giving corrected values
V′(χ) and
V L′(χ). Step 3:
Figure 2B illustrates the derivation of the reflectance spectrum of the eye,
\[r({\chi}){=}r_{\mathrm{L}}({\chi})\ V{^\prime}({\chi})/V_{\mathrm{L}}{^\prime}({\chi}),\]
where
r L(χ) is the reflectance
of the reference lens, which was derived from Fresnel’s
equation.
12 Step 4: The sloping baseline of the
reflectance spectrum (due to the lipid layer
9 15 ) was fit
by the Marquardt
19 method with the function
\[p({\chi}){=}{[}A{+}B{\chi}{+}C{\chi}^{2}{+}D{\chi}^{3}{]}{[}1{+}Ev({\chi}){]}{[}1{+}Fh({\chi}){]},\]
where
A, B, C, D, E, and
F are adjusted to
provide a least squares fit.
v(χ),
h(χ) are
derived from changes in the reflection spectra from the reference lens,
with vertical and lateral displacement, respectively; their inclusion
in
Equation 3 provides a modest improvement to the accuracy of fit. In
the fitting program, more weight was given to wavelengths with greater
photodetector response
(Fig. 2A) . The resulting fit,
p(χ),
is shown by the dashed line in
Figure 2B . Step 5: The fractional
deviation of the reflectance from this fit, that is,[
r(χ) −
p(χ)]/
p(χ) is
shown by the black curve in
Figure 2C . Step 6: This deviation was then
reexpressed as a function of α = 2
nχ, where
n is the refractive index of the layer of interest. The
refractive indices at 588 nm for tears
20 and
cornea
21 were taken as 1.337 and 1.376, respectively, and
their dispersion constant
12 was assumed to be the same as
for water. A Fourier analysis of this spectrum as a function of α was
then performed and is shown in
Figure 2E . (Subtraction of
p(χ) in step 5 improves the Fourier transform by reducing
artifacts due to steps at the ends of the spectrum). Interference
between reflections from two surfaces gives rise to a peak in the
Fourier transform whose frequency corresponds to the thickness,
t, of a layer (or more precisely, the depth of the second
surface behind the air surface). This is shown by rewriting
Equation 1 for normal incidence, that is,
\[m{=}2n{\chi}t{=}{\alpha}t.\]
Thus, the frequency of the corresponding peak (number of cycles
per unit of α) is given by
Unless otherwise noted, a Hann window was used to reduce the side
lobes associated with large peaks, such as that near 3μ
m.
19 Step 7: The depth of the largest peak from step 6
was used as a starting value (for
T in
Eq. 5 below) for a
second Marquardt fit to
r(χ) with the function
\[q({\chi}){=}s({\chi})\ p{^\prime}({\chi}),\]
where
\[s({\chi}){=}{[}1{+}G\ \mathrm{cos}\ (2{\pi}T{\alpha}{+}H)\ \mathrm{exp}(\mathrm{-}J{\chi}){]},\]
and
p′(χ) is given by
Equation 3 (but with new
constants
A through
F).
A, B, C, D, E, F,
G, T, H, and
J are adjusted to provide a least squares
fit.
p′(χ) is found to be similar but not identical with
p(χ). The cos(2π
Tα +
H) term
gives the “spectral oscillations,” whereas the decay term,
exp(−
Jχ), is empiric, providing a better fit to the
spectrum.
17 The function,
s(χ), is shown as
the thick gray curve in
Figure 2C .
T is a new estimate of
thickness, which generally has better repeatability than estimates from
Fourier analysis. Values of thickness,
T, corresponding to
different peaks, can be obtained by limiting the Fourier analysis in
step 6 to different thickness ranges; for example, for a range 20 to
100 μm in
Figure 2E , the peak near 55 μm would be studied. Step 8:
The fractional deviation of reflectance from
p′, that is,[
r(χ) −
p′(χ)]/
p′(χ) is
calculated and a second Fourier analysis is performed resulting in
Figure 2F . Typically this Fourier transform is found to have somewhat
smaller artifacts than the original transform
(Fig. 2E) . Step 9: The
fractional deviation of reflectance from the overall fit,
q,
that is, [
r(χ) −
q(χ)]/
q(χ) is calculated and is shown in
Figure 2D . Another Fourier analysis is then performed, resulting in
Figure 2G . This eliminates or reduces the peak in the Fourier transform
near
T, so that nearby weaker peaks may be studied better.
In summary, the least squares fit of step 7 gives the best estimate of
thickness, the Fourier transform of step 8 indicates the presence of
interference from surfaces at different depths behind the air surface,
whereas the transform of step 9 allows study of interference peaks near
a major peak.