Biomedical Engineering Reference
In-Depth Information
Converging
Lens
L
P
θ t
θ t
S
air
FIGURE 17.10 Fabry-Perot multiple beam interferometer.
patterns that can arise from the glass itself acting as the parallel plate interferometer. When
L
is fixed, the instrument is referred to as a Fabry-Perot etalon. The nature of the superpo-
sition at point
P
is determined by the path difference between successive parallel beams, so
D ¼
2
n f L
cos
y t ¼ m l ðminimumÞ ¼
2
L
cos
y t
ð
17
:
60
Þ
using
1 for air.
Other beams from different points of the source but in the same plane and making the
same angle
n f ¼
y t with the axis satisfy the same path difference and also arrive at
P
. With
L
fixed, Eq. (17.60) for
D
is satisfied for certain angles
y t and the fringe system is the familiar
concentric rings due to the focusing of
fringes of equal
inclination, as shown in
Figures 17.11a and b. If the thickness of
varies with time, a detector
will record an
L
D
interferogram as a function of time, as shown in Figure 17.11c.
Variation in the fringe pattern irradiance of the Fabry-Perot as a function of phase or
path difference is known as the fringe profile. The sharpness of the fringes is important
to the ultimate resolving power of the instrument. Using the trig identity cos
d ¼
1-
2 sin 2 (
d
/2), the transmittance can be written as
2
2
2
sin 2
T ¼ I T = I i ¼
1
1
þ½
4
r
1
r
Þ
ðd=
2
ÞÞ
ð
Airy function
Þ
ð
17
:
61
Þ
where r is the reflectivity, and the term in the square brackets in the denominator is known
as the coefficient of finesse,
, and thus,
F
sin 2
T ¼
1
1
þ F
ðd=
2
ÞÞ
ð
17
:
62
Þ
It should be noted that as 0
< r <
1, then 0
< F < 1
. The coefficient of finesse, F, also
represents a certain measure of fringe contrast:
F ¼ðð I T Þ max ð I T Þ min Þ= I T min
¼ð T
T
Þ= T
ð
17
:
63
Þ
max
min
min
It should be noted here that
T max ¼
1 when sin(
d
/2)
¼
0, and
T min ¼
1 /(1
þ F
) when
sin(
d
/2)
¼
1. Given
r
, the fringe profile can be plotted as shown in Figure 17.12. As seen
in the figure,
T ¼ T max ¼
1at
d ¼
m2
p
and
T ¼ T min ¼
1/(1
þ F
)at
d ¼
(m
þ
½)2
p
. Further,
T max ¼
T min is never zero but approaches zero as r approaches one.
Also, the transmittance peaks sharply at higher values of
1 regardless of
r
, and
, remaining near zero for most
of the region between the fringes. It should be noted that the Michelson interferometer
r
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