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Physics Letters A 312 (2003) 7–10

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Modiﬁed Sagnac experiment for measuring travel-time difference between counter-propagating light beams in a uniformly
moving ﬁber
Ruyong Wang a, Yi Zheng b,∗, Aiping Yao b, Dean Langley c
a St. Cloud State University, St. Cloud, MN 56301, USA b Department of Electrical and Computer Engineering, St. Cloud State University, St. Cloud, MN 56301, USA
c Physics Department, St. John’s University, Collegeville, MN 56321, USA Received 11 March 2003; received in revised form 2 April 2003; accepted 4 April 2003
Communicated by P.R. Holland

Abstract
A ﬁber optic conveyor has been developed for investigating the travel-time difference between two counter-propagating light beams in uniformly moving ﬁber. Our ﬁnding is that there is a travel-time difference t = 2v l/c2 in a ﬁber segment of length
l moving with the source and detector at a speed v, whether the segment is moving uniformly or circularly.  2003 Elsevier Science B.V. All rights reserved.
Keywords: Sagnac effect; Speed of light; Light propagation; Fiber optic gyroscope

1. Introduction
The Sagnac effect [1] shows that two light beams, sent clockwise and counterclockwise around a closed path on a rotating disk, take different time intervals to travel the path. The time difference between them is given generally by t = 4AΩ/c2, where A is the area enclosed by the path, Ω is the angular velocity of the rotation [2]. For a circular path of radius R, the difference can also be represented as t = 2vl/c2, where v = ΩR is the speed of the circular motion and l = 2πR is the circumference of the circle. The
* Corresponding author. E-mail address: zheng@stcloudstate.edu (Y. Zheng).

Sagnac effect is a ﬁrst order effect in v/c and it has been found in many systems ranging in size from the around-the-world Sagnac experiment [3] to small ﬁber optic gyroscopes (FOGs) [4]. In a FOG, when a single mode ﬁber is wound to a coil with N turns, the Sagnac effect is enhanced to t = 4AΩN/c2 or
t = 2vL/c2, where L is the ﬁber length. The traveltime difference in the FOG can also be expressed by the phase shift, φ = 2π tc/λ, where λ is the free space wavelength of light.
The existence and importance of the Sagnac effect have been well acknowledged, although a dozen interpretations can be found [5]. A rotating frame of reference is usually used in explanations, but Sagnac himself interpreted the effect without the use of relativity [1], while general relativity is used by others [6].

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R. Wang et al. / Physics Letters A 312 (2003) 7–10

We designed a modiﬁed Sagnac experiment to examine whether the travel-time difference only appears in rotational motion, or if it also appears in uniform motion. Our experiment is important because it shows that the time-difference effect also occurs for uniform motion.
2. Method
Conceptually, a FOG, shown in Fig. 1(a), could be divided into two semicircular sections with extended ﬁber connecting the end sections as shown in Fig. 1(b). The ﬁber moves when the wheels at the two ends rotate. We call this new device a ﬁber optic conveyor (FOC). The travel-time difference due to uniform motion was investigated by using different lengths for the straight-ﬁber segments with the same semicircular end sections. As a matter of fact, in a FOC, the traveltime difference between two beams through the loop is a summation of all the travel-time differences between two beams in each segment of the loop. Therefore, the comparison of the two travel-time differences of the two FOCs, that have the same semicircular end sections but different lengths for the straight-ﬁber segments as shown in Fig. 1(c), will reveal whether or not the travel-time difference exists in uniformly moving segments of the ﬁber. As in the case of the FOG, the FOC carries multiple turns of ﬁber to increase the travel-time difference.
If the travel-time difference appears in uniform motion, it should be independent of the enclosed area that is a determining factor in the Sagnac effect caused by

rotational motion. Therefore, we designed two additional conveyor conﬁgurations: zero-area (Fig. 1(d)) and ﬁgure-8 (Fig. 1(e)). In the zero-area conﬁguration, the enclosed area of the main part of the conveyor is almost zero except the two ends of the conveyor. The two counter-moving linear segments of the ﬁber are like two shoulder-brushing trains on parallel tracks. In the ﬁgure-8 conﬁguration, from the point of view of rotation, the two enclosed areas have opposite directions so the effective enclosed area of the loop is approximately zero. This means if we conduct the Sagnac experiment by putting the ﬁgure-8 conﬁguration on a rotating disk, the phase shifts of the two counter-wound portions cancel each other.
Our experiments differ from the Fizeau type experiment [7] in which the medium, water or glass, is moving, but the light source and detector are stationary. In our FOC experiment, as in the Sagnac type experiments, the light source and the detector are comoving with the medium.
3. Experiment
The experiment was conducted using a FOC that was constructed by a FOG [8] which was modiﬁed by adding an extra 50 m of single mode ﬁber to the original ﬁber loop (Fig. 2). The extra ﬁber was wrapped onto a polyester ribbon that went around two wheels with diameters of 30 cm to form our FOC. This ﬁber loop was moved by one wheel dragged by a 1.5-m long mechanical conveyor. For the zero-area conﬁguration, additional wheels pushed inward on the loop so the middle portion of the FOC enclosed very

Fig. 1. (a) A FOG including a source and detector rotating with the ﬁber, is transformed into (b) a FOC in a regular conﬁguration. (c) Regular conﬁguration with different lengths for straight-ﬁber segments, (d) zero-area conﬁguration and (e) ﬁgure-8 conﬁguration.

Fig. 2. Experimental setup. The mechanical conveyor is used to move the ﬁber optic loop and carry the light source and detector in the FOG at the same speed.

R. Wang et al. / Physics Letters A 312 (2003) 7–10

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little area. For the ﬁgure-8 conﬁguration the ribbon was twisted between the two ends. The FOG traveled with the mechanical conveyor and its uniform motion did not cause any phase shift because the FOG is only sensitive to the rotational movement. The light source for the FOG was a 1310-nm superluminescent LED. The FOG with the extended ﬁber was calibrated and its output rate is 1162.6 mV per radian of detected phase shift. The phase shift of the FOG is linearly proportional to vL at a constant rate of 0.03200 radians per m2/s.
The experiment was repeated with 24 different arrangements of conveyor speeds, ﬁber lengths, and the three different FOC conﬁgurations shown in Fig. 1. The conveyor speeds were between 3 and 9 cm/s. The loops had perimeters of 2.5, 4.0, 8.0, and 16.0 m; in each case there were three turns of the ﬁber wound on the loop. The remainder of the 50 m of ﬁber was wound on a coil of 9 cm in diameter which moved uniformly with the FOG so as not to produce any phase shift. When the conveyor ran at various stable speeds, 500 to 1 500 samples were recorded for each measurement. Measurements were repeated about eight times for each of the 24 arrangements of the FOC. Means and standard deviations of phase shifts were calculated for each measurement, and these were averaged for each of the 24 arrangements. Those means of the phase shifts are shown in Fig. 3 and are linearly pro-

portional to the speed of moving ﬁber in the laboratory and ﬁber length. Least-square linear regression values of the FOC output rates are 0.03200 ± 0.00528, 0.03300 ± 0.00274, and 0.03171 ± 0.00282 radians per m2/s for zero-area, ﬁgure-8, and regular conﬁgurations, respectively, and 0.03229 ± 0.00365 radians per m2/s for the data as a whole; these are all in agreement with the FOG output rate due to pure rotation:
φ = 4πvL/cλ = 0.03200 vL.
4. Discussion
As shown in Fig. 3, the phase shift or the traveltime difference between two counter-propagating light beams in the moving optic ﬁber was clearly observed in all different conﬁgurations of FOCs. The phase shift
φ, and therefore, the travel-time difference t are proportional to both the total length and the speed of the moving ﬁber whether the motion is circular or uniform. Other tests using smaller end wheels for the FOC and ﬁber loops with additional curves also conﬁrmed the same ﬁnding. To test the linearity of the phase shift to v and L independently, the phase shifts were normalized by the speeds and were found to be linearly proportional to the moving ﬁber length in Fig. 4.
It is interesting to notice that when the straight-ﬁber segment is made shorter and shorter, a FOC with the

Fig. 3. FOC phase shifts versus products of ﬁber speeds and lengths. Symbol + is for the zero-area conﬁguration, × for the ﬁgure-8 conﬁguration, and ◦ for the regular conﬁguration. The straight line was found by a least square linear regression of all measurements.

Fig. 4. Speed-normalized phase shifts versus ﬁber lengths. The symbols used are the same as in Fig. 3. The straight line was found by a least square linear regression of all data.

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R. Wang et al. / Physics Letters A 312 (2003) 7–10

regular conﬁguration reduces to a FOG, and a FOC with the ﬁgure-8 conﬁguration reduces to two counterwound, counter-rotating FOGs, of which the phase shifts augment each other. While the phase shift in the Sagnac effect is often expressed using the enclosed area as a factor, our results indicate that the length and speed of the moving ﬁber are the fundamental factors, rather than the enclosed area.
Finally, we found the phase shift due to uniform motion of a ﬁber segment was linearly proportional to its length, by subtracting the φ of the shortest loop from each φ of longer loops for a given speed. We conclude that a segment of uniformly moving ﬁber with a speed of v and a length of
l contributes φ = 4πv l/cλ or t = 2v l/c2, like a segment of circularly moving ﬁber does. This travel-time difference observed in our experiment is another ﬁrst order effect in v/c, like the Sagnac effect.
The travel-time difference t = 2v l/c2 in our experiment is independent of the refractive index n, as in the Sagnac type experiment. It implies that the same should occur in vacuum (refractive index n = 1), which could be tested by using a FOC with a hollowcore single-mode ﬁber in which light is guided in vacuum or in air [9]. The analysis of the implication of travel-time difference in a uniformly moving vacuum light-guide to the propagation of light in vacuum will be very interesting.

5. Conclusion
The travel-time difference of two counter-propagating light beams in moving ﬁber is proportional to both the total length and the speed of the ﬁber, regardless of whether the motion is circular or uniform. In a segment of uniformly moving ﬁber with a speed of v and a length of l, the travel-time difference is 2v l/c2.
Acknowledgements
We thank Robert Moeller of the Naval Research Laboratory for technical assistance and NRL for the loan of the FOG. We thank Xiaonan Shen for assistance.
References
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Boston, 1994. [5] F. Hasselbach, M. Nicklaus, Phys. Rev. A 48 (1993) 143. [6] L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields,
Pergamon, Reading, 1962. [7] H.L. Fizeau, C. R. Acad. Sci. Paris 33 (1851) 349. [8] R.P. Moeller, et al., in: Fiber Optic and Laser Sensors XI, in:
SPIE, Vol. 2070, 1993, p. 255. [9] R.F. Cregan, et al., Science 228 (1999) 1537.