Physical Review Letters 93 (2004) 143901

Generalized Sagnac Effect

Ruyong Wang, 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,     4 v    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.

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     4 vL=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 direc-

tion,     4 v    l=c . It is independent of the type of

motion and the refractive index of waveguides. The total

phase

difference

of

the

loop

is

  



4 

H
l

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 sensi-

tivity, 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 experi-

ments, 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 cali-

brated for the added length. The loop includes signiﬁcant

portions of ﬁber segments that move linearly. The func-

tion 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 wave-

length of 1310 nm. The ﬁber was used to construct a two-

wheel FOC shown in Fig. 1(a) and a three-wheel FOC, not

shown. Shown in Fig. 1(b) are the detected phase differ-

ences 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 dif-

ference is an average of eight measurements. Thirteen

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Physical Review Letters 93 (2004) 143901

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     4 vL=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 4 v l=c , but

4 v l cos =c   4 v    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     4 v 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     4 v    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. 1

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 superlumi-

nescent light-emitting diode as the light source and a phase

difference detector that has an output rate of 1162.6 mV per rad.

(b) Phase differences of two counterpropagating beams in a

moving air-core ﬁber.

FIG. 2

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, 4 v l cos =

c   0:032v l cos   0:0211 rad, which is represented by the

dashed line.

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Physical Review Letters 93 (2004) 143901

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     4 vN 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 10ÿ7 rad of the phase difference [5]. With N l  500 m and    10ÿ6 m, a FOLMS can detect a linear velocity of

v    c =4 N l  4:8 nm=s

FIG. 3

Experiment for studying the phase

difference when different segments of the loop move at differ-

ent 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.

can be either linear or circular. Thus, for the entire loop, the total phase difference between two counterpropagating beams in the loop is

I     4  v   dl=c :
l

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,

I

ZZ

    4  v   dl=c   4  r   v   dA=c 

l

A

ZZ  4  2
k   dA=c   8 
A=c :

A

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

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 Proceedings Vol. 2070 (SPIE –The International Society for Optical Engineering, Boston, MA, 1993), p. 255.

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