Biomedical Engineering Reference
In-Depth Information
where
I o
is the input intensity and the phase difference
d
is defined as
d ¼
2
pD=l
ð
17
:
57
Þ
In this equation,
l
is the wavelength, and the net optical path difference,
D
, is defined as
D ¼
2
d
cos
ðyÞþl=
2
¼ð m þ
0
:
5
Þl
ð
17
:
58
Þ
Clearly, from Eq. (17.58) the following equation can be derived
2
d
cos
ðyÞ¼ m l
ð
17
:
59
Þ
in which 2
is the optical path difference or, rather, the difference in the two paths from the
beam splitter, m is the number of fringes, and
d
0 is a normal
or on-axis beam). When a plate, a gas, or a thin minimally absorbing or scattering tissue
slice is assumed with constant index of refraction and is inserted in one of the paths, then
d ¼
y
is the angle of incidence (
y ¼
(
n s -
n air )
L
, in which
L
is the actual length of the substance,
n s is the index of refraction
of the substance, and
n air is the index of refraction of the air.
EXAMPLE PROBLEM 17.8
A thin sheet of clear tissue, such as a section of the cornea of the eye (
1.33), is inserted nor-
mally into one beam of a Michelson interferometer. Using 589 nanometers of light, the fringe pat-
tern is found to shift by 50 fringes. Determine the thickness of the tissue section.
n
Solution
FromEq. (17.59),2
14.72 micrometers, which is the calcu-
lated optical path length. However, the physical length of the tissuemust take into account the index of
refraction of the sample and the air. Thus, as described in the paragraph after Eq. (17.59), the equation
forphysicalpathlengthisL
d
cos(
y
)
¼ m l
, and thus
d ¼
(50
.589)/2
¼
¼
d/(
n s -
n air )
¼
14.72/(1.33 - 1.0)
¼
44.6 micrometers.
All the dual beam interferometry techniques such as the Michelson and Mach-Zehnder
approaches suffer from the limitation that the accuracy depends on the location of the
maxima (or minima) of a sinusoidal variation, as shown in Eq. (17.59). For very accurate
measurements, such as precision spectroscopy, this limitation is severe. Rather than a dual
beam, if the interference of many beams is utilized, the accuracy can be improved con-
siderably. A Fabry-Perot interferometer uses a multiple beam approach, as shown in
Figure 17.10. As can be seen in the figure, the interferometer uses a plane parallel plate to
produce an interference pattern by combining the multiple beams of the transmitted light.
The parallel plate is typically composed of two thick glass or quartz plates that enclose a
plane parallel plate of air between them. The flatness and reflectivity of the inner surfaces
are important and are polished generally better than
/50 and coated with a highly reflec-
tive layer of silver or aluminum. The silver is good for wavelengths above 400 nm in the
visible light region, but aluminum has better reflectivity below 400 nm. These film coatings
must also be thin enough to be partially transmitting (
l
50 nm thickness for silver coatings).
In many instances the outer surfaces of the glass plate are purposely formed at a small
angle relative to the inner faces (several minutes of arc) to eliminate spurious fringe
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