FIBEROPTIC SENSOR TECHNOLOGY HANDBOOK

The path length AL traveled by light in a time At is
therefore given by:
AL = cAt = (4A/c)fl (5.34)
5.4.5 Sagnac Effect in a Medium
In the case of light propagation in a medium
(see Ref. 5, Subsection 5.4.20) of refractive index n,
(Fig. 5.44) the velocity of propagation must take into
consideration the relativistic addition of the velocity
of light in the medium, i.e., c/n and the tangential
velocity of the medium, i.e., Rfl so that Ccw becomes:
Ccw = (c/n + R.Q)/(l + RQ/nc) =
c/n+RQ(l- l/n2 + ...)
(5.35)
This
given
corresponds to a nonreciprocal phase shift A$,
by:
A$ = 2~Atc/Ao = 2rAt/(A/V)
= (8.AN/aoC)c?
(5.40)
where A = A./n and v = c/n are the wavelength and sc.eed
“.
of light in the medium (n is the refractive index of
the fiber core), respectively. In terms of path length
difference:
AL = A4A012T = (4AN/c)!l (5.41)
For a fiber of length L wound in a coil of diameter D:
A = TD2/4 and N = L/nD so that:
AL = (4~/c)n = (LD/c)n (5.42)
to first
order in VIC.
Similarly Cccw,
is given by:
or:
CCCW = (c/n - IW)/(1 - RQ/nc) =
c/n-RQ(l - l/n2 + ...)
IN A MEDIUM. REFRACTIVE INDEX n
~
c
+RO
c . Cw = ~ +RQ (l– ~)+..
1+% “2
c
Ti- – R()
C&w = —
,–~
=
nc
c
7i-
–Rfl (1-~)+
(5.36)
AI$ = (2TTLD/aoC)Q (5.43)
5.4.6 The Magnitude of the Sagnac Effect
In order to get a feel for the magnitude of
AL, (Fig. 5.45) assume an area A = 100 au2 and a rota-
~~:~orate of 10-3 ~ (i.e., 0.015°/hr or 7 x 10-8 radl
For a single-turn fiber loop enclosing such an
area we get AL . 10-15 cm. This is not a very large
effect considering the diameter of a hydrogen atom is
about IO-8 cm. Clearly, a large number of turns N is
necessary to increase the magnitude of AL.
~Ccw-Cccw=2RQ (1-;)
r
~A, ~2wR[2R &(Ccw-Cccw)l ~ 2rR12R0-2R62(l-p)j
Ccwcccw
C2
“2
THESAME ASIN AVACUUM
‘EE1
Fig. 5.44 A demonstration of the Sagnac relationships
for the rotating single-loop optical fiber.
A=100cm2; Q .CiE * 10-4 rad/sec
~ AL*10-12cm
W. ‘DIAMETER- OF HYDROGEN ATOM- lo-8cm
FoR n = 10-3nE = 10-7 radlsec
Fig. 5.45 Computation of the change in effective optical
length (Sagnac effect) in a rotating
single-loop of optical fiber.
Therefore, At in a medium becomes:
At = tcw - tccw
= 21rR[2Ro - (Ccw - cccw)l/[cc#ccwl
(5.37)
UPon substitution for Ccw - Cccw from above, there is
obtained:
At = 21TR[2RQ - 2RQ(I - l/n2]/[c2/n2]
. 21TR(2RQ/C2) = (4A/c2)Q
(5.38)
which is idential to that in a vacuum. If the medium
is an optical fiber wound in a coil of N turns, then At
becomes:
At = (4AN/c2)fl (5.39)
5.4.7 Methods of Optical Rotation Sensin&
For the sake of completeness, Fig. 5.46 shows
the various schemes for the measurement of AL. On the
extreme right is the multiturn fiber interferometer
method (see Ref. 7, Subsection 5.4.20) mentioned above.
On the extreme left is the ring laser approach (see
Ref. 6, Subsection 5.4.20) and in the middle is the
passive resonator approach (see Ref. 8, Subsection
5.4.20). In both the active and the passive reaonator
approach, a nonreciprocal path length difference AL due
to the Sagnac effect becomes a nonreciprocal change in
the resonance frequency, Af, of the cavity for CW and
ccw propagation where:
Af = (4A/ioP)$l (5.44)
5-16