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Sagnac Effect and Fiber Optic Gyroscopes | OMICS International
ISSN: 2090-0902
Journal of Physical Mathematics
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Sagnac Effect and Fiber Optic Gyroscopes

Kecman M*

Independent Researcher, Poland

*Corresponding Author:
Kecman M
Independent Researcher, Poland
Tel: +38765680971
E-mail: [email protected]

Received date: March 31, 2016; Accepted date: July 29, 2016; Published date: August 03, 2016

Citation: Kecman M (2016) Sagnac Effect and Fiber Optic Gyroscopes. J Phys Math 7:189. doi: 10.4172/2090-0902.1000189

Copyright: © 2016 Kecman M. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

Sagnac effect (Sagnac interference) is well described by the author Anderson. There is the original paper translated into English. The generally accepted interpretation is based on different trails (The difference in the length of roads caused by peripheral speed). Interference fringes are evenly lined with dark stripes, move with the changing angular velocity. Theoretical calculations given significant deviations from the experimental results. Experimental data suggest: Sensitivity (accuracy) of the device is proportional to the square of the length of optical fiber. The official interpretations (and others) argue: Sensitivity is proportional to the length of the optical fiber. Exposed is a mathematical analysis of the experimental data.

Keywords

Sagnac; Optical gyroscope; Quadratic regression

Introduction

Sagnac experiment was published 1913 years by the Author which is precisely described device and method of measurement [1,2]. The calculation and experimental results are consistent. Figures 1 and 2 show a model for calculating the phase shift. Sagnac has introduced a surface A due to deviations of the measurement results. On his drawing device is a hatched area [2]. It is better to agree with the measurement of 2π R to model. Sagnac wrote:” After many runs, I have always observed the sense to change as expected (Table 1).

physical-mathematics-measured-shift-interference

Figure 1: Observed and measured shift interference fringe.

physical-mathematics-method-least-squares

Figure 2: The method of least squares.

  EMP-1 EMP-1 EMP-1 EMP-1 EMP-1.2K
FiberLength[m] 205 300 420 700 1200
Bias(Typical)[°/h] 0.5 0.2 0.2 0.03 0.01
ScaleFactor 0.027 0.027 0.012 0.03 0.027
xi(L) 205 300 420 700 1200
yi(n) 1 2,5 5.6 15 50

Table 1: Bias data is reduced to the same peripheral speed, and on the same scale factor.

At the end of the twentieth century made sophisticated optical gyroscope. The device had no moving parts (elements). Optical cable, laser, electronics, and software enable high-precision devices.

Experimental Data

It gives an insight to the technical characteristics. There is no reliable data sensitivity of their devices. Data obtained by experimental methods [3,4].

Linear regressions

y=k1x

Calculate k1give us a sum of squared deviations is minimal.

F=Σ(yi-k1xi)2

From this

dF/dk1=0 get:

k1=(Σxiyi)/(Σ(xi)2)

k1=(Σxiyi)/( Σ(xi)2)Σxiyi=73807Σ(xi)2=2238425F1=Σ(yi-0.033xi)2

k1=0.033 Calculate thesum of the squareddeviations: F1=330

4.2Quadratic regression (QuadReg)

y=k2x2 calculate k2 give us a sum of squared deviations is minimal as shown in Tables 2 and 3.

xi yi xiyi xi2 yi-0.033xi (yi-0.033xi)2
205 1 205 42025 -5.765 33.23
300 2.5 750 90000 -7.4 54.76
420 5.6 2352 176400 -8.26 68.22
700 15 10500 490000 -8.1 65.61
1200 50 60000 1440000 10.4 108.16

Table 2: Sum of squared deviations.

xi [m] yi xi2 xi2yi xi4 yi-k2xi2 (yi-k2xi2)2
205 1 42025 62500 3906 × 106 -1.16 1.35
300 2.5 90000 225000 8100 × 106 -0.61 0.372
420 5.6 176400 9877840 31117 × 106 -0.5 0.25
700 15 490000 8183000 240100 × 106 -0.2 0.04
1200 50 1440000 72000000 2073600 × 106 -0.23 0.053

Table 3: illustration of quadratic regression.

F=Σ(yi-k2xi2)2

From thisdF/dk2=0 get:

k2=(Σxi2yi)/( Σ(xi)4)

k2=(Σxi2yi)/( Σ(xi)4)Σxi2yi=81.46*106Σ(xi)4=2.36*1012F2=Σ(yi-k2xi2)2

k2=0.0000346F2=2.06

Conclusion

F2 << F1 from which it follows that the quadratic regression much better fit with the experimental data.

Predict the sensitivity of optical fiber length L=5000 m.y=0.0000346*50002=856. Bias is 0.00058°/h, and this is one revolution in 70 years. Is necessary revision of theoretical explanations for Sagnac effect? The new theory must contain a square dependence of sensitivity on the length of the fiber. Additional experimental design is also essential. Serious hypothesis will be given.

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