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《Generalized Sagnac Effect》

RuyongWang, Yi Zheng, and Aiping Yao
St. Cloud State University, St. Cloud, Minnesota 56301, USA
(Received 18 March 2004; published 27 September 2004)

Experiments were conducted to study light propagation in a light waveguide loop consisting of linearly and circularly moving segments.We found that any segment of the loop contributes

to the total phase difference between two counterpropagating light beams in the loop. The contribution is proportional to a product of the moving velocity v and the projection of the segment length

l on the moving direction,


4v 
l=c. It is independent of the type of motion and the refractive index of waveguides. The finding includes the Sagnac effect of rotation as a special case and suggests a new fiber optic sensor for measuring linear motion with nanoscale sensitivity.

DOI: 10.1103/PhysRevLett.93.143901 PACS numbers: 42.25.Bs, 03.30+p, 42.81.Pa

The Sagnac effect [1] shows that two counterpropagating
light beams take different time intervals to travel a
closed path on a rotating disk, while the light source and
detector are rotating with the disk.When the disk rotates
clockwise, the beam propagating clockwise takes a longer
time interval than the beam propagating counterclockwise,
while both beams travel the same light path in
opposite directions. The travel-time difference between
them is
t
4A=c2, where A is the area enclosed by the
path and  is the angular velocity of the rotation. Since
the 1970s, the Sagnac effect has found its crucial applications
in navigation as the fundamental design principle
of fiber optic gyroscopes (FOGs) [2,3]. In a FOG, when a
single mode fiber is wound onto a circular coil with N
turns, the Sagnac effect is enhanced by N times so that
the travel-time difference is 2vNl=c2, where v is the
velocity of the rotating fiber, l is the circumference of
the circle, and Nl is the total length of the fiber. The
travel-time difference in a FOG can be expressed by the
phase difference


2
tc=, where  is the free
space wavelength of light. Today, FOGs have become
highly sensitive detectors measuring rotational motion
in navigation [4,5].
It is believed that the Sagnac effect exists only in
circular motion. However, we have discovered that any
moving path contributes to the total phase difference
between two counterpropagating light beams in the
loop. In a previous experiment using a fiber optic conveyor
(FOC), we showed our preliminary result that a segment
of linearly moving glass fiber contributes


4vL=c to the phase difference [6]. Here,

we generalize
our finding for the Sagnac effect with a more complete
study and a series of new experiments. Our experiments
include different types of motion and light paths with a
glass fiber and an air-core fiber [7]. In this study, we found
that in a light waveguide loop consisting of linearly and
circularly moving segments, any segment of the loop
contributes to the phase difference between two counterpropagating
light beams in the loop. The contribution is
proportional to a product of the moving velocity v and the
projection of the segment length
l on the moving direction,



4v 
l=c. It is independent of the type of
motion and the refractive index of waveguides. The total
phase difference of the loop is


4Hl v  dl=c.
This general conclusion includes the Sagnac effect for
rotation as a special case. The finding also suggests a new
fiber optic linear motion sensor having nanoscale sensitivity,
which is much more sensitive than any existing
linear motion detectors. This linear motion sensor may be
applied to accelerometers in navigation and seismology.
Our experiments are different from the Fizeau-type
experiment [8],‘‘drag’’ experiment of a moving medium.
In the Fizeau-type experiment, there is relative motion
between the light path and the medium, water or glass.
The result of the Fizeau-type experiment is dependent on
the refractive index of the medium, and the drag coeffi-
cient is zero when the refractive index is 1. In our experiments,
as in the FOG, there is no relative motion between
the light path and the medium, glass fiber or air-core fiber.
The experimental system in our study consists of a fiber
optic loop, a FOG [9], and a mechanical conveyor, as
shown in Fig. 1(a). A single mode fiber loop with different
configurations was added to the FOG, which was calibrated
for the added length. The loop includes significant
portions of fiber segments that move linearly. The function
of the FOG in this experiment is to transmit and
receive the counterpropagating light beams and to detect
the phase differences between two light beams. Because
the FOG is not rotating in the experiment, the detected
phase differences are caused by the movement of the
added fiber optic loop.
An air-core fiber was used to verify that the phase
difference is independent of the refractive index and
various types of motion. The new photonic band gap fiber
[7] has a hollow air-guiding core for light with a wavelength
of 1310 nm. The fiber was used to construct a twowheel
FOC shown in Fig. 1(a) and a three-wheel FOC, not
shown. Shown in Fig. 1(b) are the detected phase differences
caused by the fiber motion with both configurations.
The length of the fiber loop is 4.1 m and the speeds of the
motion are from 0.001 91 to 0:211 m=s. Each phase difference
is an average of eight measurements. Thirteen
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phase differences range from 0.000 246 to 0.0276 rad. The
maximum conveyor velocity error is 1.7%. The leastsquares
linear regression of the phase differences is
0:0323vL, which closely matches with


4vL=
c
0:0320vL with 
1310 nm. It also matches the
result using a glass fiber [6]. Therefore, it is confirmed that
the contribution to the phase difference in a moving light
waveguide is independent of the refractive index of the
waveguide and independent of the loop configurations.
To study the relationship between the motion of the
fiber and the fiber orientation, we conducted an experiment
in which the fiber zigzags and has an angle  with
respect to the direction of fiber motion. Thus, for a fiber
segment having an actual length of
l, its effective length
is
l cos, which is a projection of the fiber onto the
motion direction. As shown in Fig. 2, our experiment
demonstrates that the effective length contributes the
phase difference, not the actual length; therefore, the
phase difference

is not 4v
l=c, but 4v
l cos=
c
4v 
l=c, which is the dot product between two
vectors.
A further experiment was conducted to study the phase
difference when different segments of the loop moved at
different speeds. Figure 3(a) shows a fiber optic ‘‘parallelogram’’
where the top arm moves with the conveyor
and the bottom arm is stationary.While moving, the two
sidearms, being flexible, are kept the same shape so that
the phase differences in these two sidearms cancel each
other. There is no phase difference in the bottom stationary
arm. Therefore, the detected phase difference is
contributed solely by the motion of the top arm. The
experiment was conducted using a glass fiber and an
air-core fiber. The detected phase differences are shown
in Fig. 3(b). For the glass fiber configuration, the length of
the top arm is 1.455 m and there are 11 turns. The
measured phase differences range from 0.000 482 to
0.121 rad for speeds from 0.000 917 to 0:233 m=s. In
another configuration, the top arm is air-core fiber with
a length
l of 5.23 m. The phase differences range from
0.000 301 to 0.0346 rad for speeds from 0.00186 to
0:207 m=s. The least-squares linear regression of the
phase differences is 0:0317v
l, which agrees with


4v
l=c
0:0320v
l with 
1310

nm.
According to our experiments, we can draw a conclusion
about the generalized Sagnac effect that in a moving
fiber loop or waveguide, a segment
l with a velocity v
contributes


4v 
l=c to the total phase difference
between two counterpropagating beams in the loop.
The contribution

is independent of the refractive
index of the waveguide, and the motion of the segment
FIG. 2 (color online). Phase differences of moving fiber segments
that have the same effective length and different actual
lengths. The effective length is 7.20 m and four actual lengths
are 8.31, 10.2, 14.4, and 27.8 m for 
30, 45, 60, and 75,
respectively. The velocity is 0:091 m=s. Four phase differences
range from 0.0206 to 0.0215 rad, with standard deviations from
0.000 280 to 0.000 743 rad. In this experiment, 4v
l cos=
c
0:032v
l cos
0:0211 rad, which is represented by the
dashed line.
FIG. 1 (color online). Experiment for detecting the phase
difference of two counterpropagating light beams in a moving
fiber loop. (a) Experimental setup. The fiber loop is driven by
the conveyor at a velocity v. The conveyor has a length of 1.5 m
and can move from 0.001 to 0:25 m=s. The diameters of the
wheels are 0.3 m. The FOG consists of a 1310-nm superluminescent
light-emitting diode as the light source and a phase
difference detector that has an output rate of 1162.6 mVper rad.
(b) Phase differences of two counterpropagating beams in a
moving air-core fiber.
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can be either linear or circular. Thus, for the entire loop,
the total phase difference between two counterpropagating
beams in the loop is



4Il
v  dl=c:
This general conclusion includes the Sagnac effect of
rotation as a special case. In fact, the Sagnac result can
be derived from this general equation by assuming a
circular motion of the loop and using Stokes’s theorem,



4Il
v  dl=c
4ZZAr  v  dA=c

4ZZA
2k  dA=c
8A=c:
This generalization provides a design principle for a
new fiber optic linear motion sensor (FOLMS), which has
a high sensitivity and a high stability. The basic structure
of this sensor can be similar to that shown in Fig. 3(a).
The linear motion of the top arm of the sensor is detected
with a phase difference


4vN
l=c. Because two
beams share the same optical path, the sensor is optically
stable. Just as a FOG detects the rotational motion of an
object, a FOLMS can detect the relative linear motion
between two objects fixed on the top and bottom arms of
the parallelogram. The optical technologies developed in
recent decades for the FOG can be utilized for the
FOLMS; therefore, the sensitivities and stability of the
FOLMS are comparable to that of the FOG. The sensitivity
of a FOG can be 107 rad of the phase difference
[5].With N
l
500 m and 
106 m, a FOLMS can
detect a linear velocity of
v


c=4N
l
4:8 nm=s
which is a nanoscale velocity.
This nanoscale sensitivity linear motion sensor can
detect the very small relative motion appearing in an
accelerometer, which is important in navigation and seismology.
The common design of an accelerometer is a
spring-mass system. Improving the sensitivity of an accelerometer
requires improving the sensitivity of detecting
the linear movement of the mass relative to the base.
Utilizing a FOLMS to detect the relative motion between
the mass and the base will greatly increase the sensitivity
of the accelerometer. It can be foreseen that an accelerometer
using the FOLMS combined with the fiber optic
gyroscope may be beneficial for navigation because both
use the same technology and both are very stable
optically.
We thank Dean Langley for useful discussions and
help. We thank Robert Moeller of the Naval Research
Laboratory for technical assistance and NRL for the
loan of the FOG. The air-core fiber was purchased from
Crystal Fiber, Denmark.
[1] G. Sagnac, C. R. Acad. Sci. Paris 157, 708 (1913).
[2] V. Vali and R. Shorthill, Appl. Opt. 16, 290 (1977).
[3] W. Leeb, G. Schiffner, and E. Scheiterer, Appl. Opt. 18,
1293 (1979).
[4] H. Lefevre, The Fiber-Optic Gyroscope (Artech House,
Boston, 1993).
[5] W. K. Burns, Optical Fiber Rotation Sensing (Academic
Press, Boston, 1994).
[6] R.Wang, Y. Zheng, A. Yao, and D. Langley, Phys. Lett. A
312, 7 (2003).
[7] R. F. Cregan et al., Science 228, 1537 (1999).
[8] H. L. Fizeau, C. R. Acad. Sci. Paris 33, 349 (1851).
[9] R. P. Moeller et al., in Fiber Optical and Laser Sensors
XI, edited by R. P. DePaula, SPIE ProceedingsVol. 2070
(SPIE–The International Society for Optical
Engineering, Boston, MA, 1993), p. 255.
FIG. 3 (color online). Experiment for studying the phase
difference when different segments of the loop move at different
speeds. (a) Experimental setup. The light from a source is
split into two beams that counterpropagate in the fiber which is
wound onto a parallelogram. The bottom arm is fixed while the
top arm is moving. The phase difference can be enhanced by
multiple turns of the fiber on the parallelogram. The coupler,
source, and detector are replaced by the FOG in the experiment.
(b) Phase differences caused by the linear motion of the top
arm of the fiber optic loop.
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