FIBEROPTIC SENSOR TECHNOLOGY HANDBOOK

Fig. 5.35
SIGNAL
ELECTRICAL
REBALANCE
UPPER
~1 lppoRT
IER
MASS
DIODE
LASER
3dB COUPLER
iii=
CASE
DIAPHRAGMS
LOWER SUPPORT
lkFIBER
)1‘t’ /u
m1
s
3dB COUPLER
PHOTODIODES
A two-fiber phase-change interferometric
fiberoptic accelerometer.
where A is the cross sectional area of the fiber, AT is
the magnitude of the change of the tensile stress in
each fiber, and the 2 is due to the presence of two fibers.
The resulting strain AS = .4L/L is given by:
AS = AT/Y = ma/2YA (5.20)
attached to a spring with a spring constant k, the resonant
frequency is given by:
combining Eqs. (5.26) and (5.27) there is obtain-
Thus,
ed:
fr = (1/2n)(k/m)112. (5.27)
fr = (1/2n)(2YA/Lm)l/2 (5.28)
As above, A = m(d/2)2. To further emphasize the dependence
of the resonant frequency, fr, on the fiber parameters,
Eq. (5.28) may be written as:
fr = [Yd2/8mLm]1/2 (5.29)
Comparing Eqs. (5.25) and (5.29) we see that the optical
fiber physical parameters appear in the form Yd2/
Lm. Therefore, if the expression d2/Lm in Eq. (5.25)
is decreased in order to decrease ~in, the value
of fr given by Eq. (5.28) is also decreased. The minimum
detectable acceleration (a~n) and longitudinal
resonant frequency are shown in Figs. 5.36 and 5.37 as
functions of m and d. In each case the length of fibe
L is taken to be one cm and the value of A is 10 -5
radian. Alternately, if a mass of one gram % ~ chosen,
the mass in grams on the abscissa in both Figs. 5.36
and 5.37 can be replaced by the length in centimeters.
3.0 t.
where Y is Young’s modulus for the fiber.
Consider next an optical beam propagating in
one of*the fibers. Its phase shift, $ , in traveling
the length L, as given in Subsection 4.2.1 and Eqs.
(4.7) and (4.8), is:
4 = zn~~l~o (5.21)
where h. is the optical wavelength in vacuum and n is
the fiber core’s refractive index. The’ quantity Aoln
is the wavelength of the light in the fiber core. In
general, the change in @ per fiber (twice this for two
fibers) may be written as it was in Eq. (5.1), namely:
A+ = 2m(nAL + LAn)/Ao (5.22)
with the wave number k = 2iI/lo. For the case of a tensile
strain, however, the AL term dominates and one may
write:
A+ = 2nnAL/~o = 2nnLAS/lo (5.23)
Fig. 5.36
10.0
10 2.0 3.0
MASS (grams)
The variation of sensitivity in micrograms
as a function of the mass in grams in a
fiberoptic accelerometer for different
sizes of fibers.
Substituting into Eq. (5.23) from Eq. (5.20) and because
A - n(d/2)2:
A~ = 4nLma/Y~d2 (5.24)
where d is the fiber diameter.
- h
-$
d
m -1-
Solving Eq. (5.24) for ~n in terms of A$~n yields:
amin = AoYd2A$tin/4nLm (5.25)
Referring to Fig. 5.35, the effective spring
force F, required to displace the mass m a distance z,
along the axis of the fiber, is given by:
F = -2YAz/L = -kz (5.26)
from which it follows that 2YA/L = k, where k is the
effective spring constant. However, when a mass m is
.--— ------.-----
I 1 1 1
10 2.0 3.0
MASS(grams)
Fig. 5.37 The variation of resonant frequency as a
function of the mass in a fiberoptic accelerometer
for different sizes of fibers.
5-13