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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 23 — Nov. 5, 2012
  • pp: 25421–25431
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Quadrature demodulation with synchronous difference for interferometric fiber-optic gyroscopes

Zinan Wang, Yi Yang, Yongxiao Li, Xiaoqi Yu, Zhenrong Zhang, and Zhengbin Li  »View Author Affiliations


Optics Express, Vol. 20, Issue 23, pp. 25421-25431 (2012)
http://dx.doi.org/10.1364/OE.20.025421


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Abstract

We propose a novel method of quadrature demodulation with synchronous difference for suppressing noise in interferometric fiber-optic gyroscopes (IFOGs). For an IFOG with sine wave phase modulation, an in-phase result and a quadrature result are obtained simultaneously by coherent detection. Eigenfrequency modulation is used and a phase shift of 45° is set between the modulation signal and the reference signal, so that two results have the same expectation of amplitude but with opposite signs. A synchronous difference procedure is carried out for output, in which signals are added up and common noise between two results is eliminated. Theoretical analysis and experimental results show that both short term noise and long term instability of the IFOG are reduced by this method. In experimental comparison with the traditional demodulation method on the same IFOG with a 1982 m fiber coil, this method reduces the bias drift from 0.040°/h to 0.004°/h.

© 2012 OSA

1. Introduction

Optical gyroscopes are rotation sensors detecting the phase shift induced by the Sagnac effect [1

1. E. J. Post, “Sagnac effect,” Rev. Mod. Phys. 39, 475–493 (1967). [CrossRef]

, 2

2. H. J. Arditty and H. C. Lefèvre, “Sagnac effect in fiber gyroscopes,” Opt. Lett. 6, 401–403 (1981). [CrossRef] [PubMed]

]. They are widely used in industrial and military applications such as aircraft, satellites, and remote control devices. With superiority over other types of gyroscopes in precision and cost, interferometric fiber-optic gyroscopes (IFOGs) have been intensively studied over the past few decades, and commercial products are available [3

3. R. A. Bergh, H. C. Lefèvre, and H. J. Shaw, “An overview of fiber-optic gyroscopes,” J. Lightwave Technol. 2, 91–107 (1984). [CrossRef]

, 4

4. H. C. Lefèvre, The Fiber-Optic Gyroscope (Artech House, 1993).

]. In applications, however, the performance of the IFOG is limited by both short term noise and long term instability [5

5. G. B. Malykin, “On the ultimate sensitivity of fiber-optic gyroscopes,” Tech. Phys. 54, 415–418 (2009). [CrossRef]

, 6

6. I. A. Andronova and G. B. Malykin, “Physical problems of fiber gyroscopy based on the Sagnac effect,” Phys. Usp. 45, 793–817 (2002). [CrossRef]

]. Suppressing noise is thus important for IFOG designing.

Among short term noises, the optical intensity noise produced by broadband sources becomes the major component at the output when optical power is relatively high [13

13. R. C. Rabelo, R. T. Carvalho, and J. Blake, “SNR enhancement of intensity noise-limited FOGs,” J. Lightwave Technol. 18, 2146–2150 (2000). [CrossRef]

]. A noise reference is mostly used for intensity noise subtraction. Moeller et al. first applied intensity noise substraction in the IFOG in 1991, by utilizing the unused port of source coupler as a noise reference [12

12. R. P. Moeller and W. K. Burns, “1.06-ptm all-fiber gyroscope with noise subtraction,” Opt. Lett. 16, 1902–1904 (1991). [CrossRef] [PubMed]

]. Similar approaches are studies by Rabelo et al. and Polynkin et al. [13

13. R. C. Rabelo, R. T. Carvalho, and J. Blake, “SNR enhancement of intensity noise-limited FOGs,” J. Lightwave Technol. 18, 2146–2150 (2000). [CrossRef]

,14

14. P. Polynkin, J. Arruda, and J. Blake, “All-optical noise-subtraction scheme for a fiber-optic gyroscope,” Opt. Lett. 25, 147–149 (2000). [CrossRef]

]. From another point of view, Blake et al. showed that intensity noise could be relegated to the quadrature channel by adjusting the modulation depth in sine wave modulated IFOGs, thus noise in the in-phase signal was reduced [9

9. J. Blake and I. S. Kim, “Distribution of relative intensity noise in signal and quadrature channels of a fiber-optic gyroscope,” Opt. Lett. 19, 1648–1650 (1994) [CrossRef] [PubMed]

].

The bias drift is an important parameter representing long term instability of IFOGs. Methods based on polarization controlling were unusually used for bias drift reduction [15

15. R. Ulrich, “Fiber-optic rotation sensing with low drift,” Opt. Lett. 5, 173–175 (1980). [CrossRef] [PubMed]

19

19. O. Çelikel and F. Sametoǧlu, “Assessment of magneto-optic Faraday effect-based drift on interferometric single-mode fiber optic gyroscope (IFOG) as a function of variable degree of polarization (DOP),” Meas. Sci. Technol. 23, 025104 (2012). [CrossRef]

]. In addition, it is also possible for reducing bias drift without changing optical components of the IFOG. A method for bias drift reduction was proposed by Wang et al. in 2011, by subtracting two signals with different polarities at the output [10

10. X. Wang, C. He, and Z. Wang, “Method for suppressing the bias drift of interferometric all-fiber optic gyroscopes,” Opt. Lett. 36, 1191–1193 (2011). [CrossRef] [PubMed]

]. The two signals are obtained in a time division manner by switching the polarity of the modulation function periodically.

Benefited from all these ideas, we see the potential for suppressing simultaneously short term noise and long term instability, by utilizing simple phase modulation and signal processing methods. Specifically, we find the quadrature channel mentioned in Ref. [9

9. J. Blake and I. S. Kim, “Distribution of relative intensity noise in signal and quadrature channels of a fiber-optic gyroscope,” Opt. Lett. 19, 1648–1650 (1994) [CrossRef] [PubMed]

] can also be used as a noise reference. Differently from the noise references used in Ref. [12

12. R. P. Moeller and W. K. Burns, “1.06-ptm all-fiber gyroscope with noise subtraction,” Opt. Lett. 16, 1902–1904 (1991). [CrossRef] [PubMed]

14

14. P. Polynkin, J. Arruda, and J. Blake, “All-optical noise-subtraction scheme for a fiber-optic gyroscope,” Opt. Lett. 25, 147–149 (2000). [CrossRef]

], the quadrature channel contained not only the short term intensity noise from the light source but also the long term instability from the fiber coil. Therefore, the quadrature channel and the in-phase channel can be used together for reducing both short term noise and long term instability, by subtracting two signals in a similar way with Ref. [10

10. X. Wang, C. He, and Z. Wang, “Method for suppressing the bias drift of interferometric all-fiber optic gyroscopes,” Opt. Lett. 36, 1191–1193 (2011). [CrossRef] [PubMed]

].

In this paper we propose a novel method of quadrature demodulation with synchronous difference for suppressing both short term noise and long term instability in IFOGs. Two channels of detected results are simultaneously generated with the same expectation of amplitude but different signs. These two channels are also regarded as each other’s noise references. Synchronous difference between the two results is used as the output, where signals are added up and common noise is eliminated. Theory and experimental results show that both short term noise and long term instability of the IFOG are notably reduced by this method.

The construction of this paper is as follows. In Section 2, we describe the requirement and the process of quadrature demodulation, and present mathematical forms of relevant signals. In section 3, we analyze theoretically how noise is reduced during the synchronous difference procedure. Section 4 reports experimental results on an IFOG for performance comparison between traditional demodulation and the quadrature demodulation with synchronous difference, and Section 5 presents a summary of the work.

2. Theory of quadrature demodulation for the synchronous difference

The polarization maintaining IFOG is one of the typical IFOG structures meeting the requirement of reciprocity [3

3. R. A. Bergh, H. C. Lefèvre, and H. J. Shaw, “An overview of fiber-optic gyroscopes,” J. Lightwave Technol. 2, 91–107 (1984). [CrossRef]

, 20

20. A. Lompado, J. C. Reinhardt, L. C. Heaton, J. L. Williams, and P. B. Ruffin, “Full Stokes polarimeter for characterization of fiber optic gyroscope coils,” Opt. Express 17, 8370–8381 (2009). [CrossRef] [PubMed]

, 21

21. Y. Yang, Z. Wang, and Z. Li, “Optically compensated dual-polarization interferometric fiber-optic gyroscope,” Opt. Lett. 37, 2841–2843 (2012). [CrossRef] [PubMed]

]. Figure 1 shows the open-loop polarization maintaining IFOG used in our experiment, and implementation details of modulation and demodulation.

Fig. 1 Quadrature demodulation with synchronous difference for an open-loop polarization maintaining IFOG. The experiment setup includes a C-Band ASE light source with a spectrum of 40 nm and output power of 13 dBmW, a circulator, a photodetector (PD) with a response of 10 μA/mW and load resistant of 100 KΩ, a Y-junction waveguide with polarizer and phase modulation function, a 1982 m long coil of polarization maintaining fiber (PMF), a dual-channel signal generator, a dual-channel digitizer with flexible resolution of 16–24 bits, and a computer.

In our novel method, the phase modulation function is given by
ϕm(t)=ϕ0[sin(ωmt)cos(ωmt)]=2ϕ0sin(ωmtπ4),
(1)
which has a phase delay of 45° with the reference signal VI(t) = Vf sin(ωmt). When eigenfrequency modulation is used [3

3. R. A. Bergh, H. C. Lefèvre, and H. J. Shaw, “An overview of fiber-optic gyroscopes,” J. Lightwave Technol. 2, 91–107 (1984). [CrossRef]

, 13

13. R. C. Rabelo, R. T. Carvalho, and J. Blake, “SNR enhancement of intensity noise-limited FOGs,” J. Lightwave Technol. 18, 2146–2150 (2000). [CrossRef]

], we have ωm = π/τ, and the modulation introduced phase difference as
Δϕ(t)=ϕm(t)ϕm(tτ)=22ϕ0cos(ωmtπ4),
(2)
This function ensures that the in-phase signal and the quadrature signal have the same expectation of amplitude but with opposite signs. In this case, the 1st harmonic of detected signal is changed from Eq. (26) in Appendix to
I1H(t)=I0ηJ1(ϕb)sin(ϕs)sin(ωmt+3π4)=22I0ηJ1(ϕb)sin(ϕs)sin(ωmt)+22I0ηJ1(ϕb)sin(ϕs)cos(ωmt),
(3)
where I0 is the source intensity coupled to the optical circuit, η is the response of the PD, Jn is the nth Bessel function of the first kind, and ϕb=22ϕ0.

We apply coherent detection to get amplitudes and signs of the in-phase signal and the quadrature signal. The reference sine wave VI(t) = Vf sin(ωmt) is used as the local oscillator (LO). The quadrature LO is a cosine wave VQ(t) = Vf cos(ωmt), which has a π/2 phase difference with VI(t). The detected signal is multiplied by two LOs respectively, with amplitude scaling as
1VfI(t)VI(t)=22I0ηJ1(ϕb)sin(ϕs)+h.c.,
(4)
1VfI(t)VQ(t)=22I0ηJ1(ϕb)sin(ϕs)+h.c..
(5)
Here h.c. stands for high-frequency components, which do not affect final results. After low-pass filtering (LPF), we get useful DC components
II=22I0ηJ1(ϕb)sin(ϕs),
(6)
IQ=22I0ηJ1(ϕb)sin(ϕs).
(7)

Differently with the traditional demodulation in Appendix, the rotation induced phase shift is simultaneously detected by two signals as
ϕI=arctan2IIJ2(ϕb)I2HJ1(ϕb)=ϕs,
(8)
ϕQ=arctan2IQJ2(ϕb)I2HJ1(ϕb)=ϕs,
(9)
where the 2nd harmonic I2H and ϕb are derived in a traditional way (see Appendix). By substituting ϕI and ϕQ for ϕs in Eq. (21) in Appendix respectively, we get two detection results of the rotation rate with different signs as ΩI = −Ω, and ΩQ = Ω. As shown by Eq. (8) and Eq. (9), the factor 2 and the arc-tangent function ensure that scale factors of the two final outputs are linear and have theoretical values of −1 and 1, respectively.

3. Analysis of noise reduction in the synchronous difference

We have described the signal processing in demodulation from Eq. (3) to Eq. (9), where signals are in their ideal forms. In practice, detected results have error terms induced by noise, which are changing over time [10

10. X. Wang, C. He, and Z. Wang, “Method for suppressing the bias drift of interferometric all-fiber optic gyroscopes,” Opt. Lett. 36, 1191–1193 (2011). [CrossRef] [PubMed]

, 21

21. Y. Yang, Z. Wang, and Z. Li, “Optically compensated dual-polarization interferometric fiber-optic gyroscope,” Opt. Lett. 37, 2841–2843 (2012). [CrossRef] [PubMed]

]
ΩI(t)=Ω+N+(t)N(t)+N1(t),
(10)
ΩQ(t)=Ω+N+(t)+N(t)+N2(t),
(11)
where N+(t) is common noise between ΩI and ΩQ, and N(t) is differential noise between them. N1(t) and N2(t) are independent random noise parts. It should be noticed here that common noise and differential noise are defined between two outputs ΩI and ΩQ, in spite of different signs with Ω. As ΩQ(t) and ΩI(t) are obtained simultaneously, synchronous difference is conveniently carried out in the novel method to eliminate common noise as
Ωout(t)=ΩQ(t)ΩI(t)2=Ω+N(t)+N2(t)N1(t)2.
(12)

Equation (10) and Eq. (11) show that ΩI(t) and ΩQ(t) are orthogonal detection results of the same target Ω. Two signals modulated on the light wave travel through the same circuit simultaneously, thus both short term noise and long term instability in two results have a considerable common part in N+(t).

First, we study short term noise reduction with the elimination of N+(t). Short term noise mainly affecting the IFOG performance includes thermal noise, shot noise, and light intensity noise as [11

11. W. K. Burns, R. P. Moeller, and A. Dandridge, “Excess noise in fiber gyroscope sources,” IEEE Photonic. Tech. Lett. 2, 606–608 (1990). [CrossRef]

, 13

13. R. C. Rabelo, R. T. Carvalho, and J. Blake, “SNR enhancement of intensity noise-limited FOGs,” J. Lightwave Technol. 18, 2146–2150 (2000). [CrossRef]

]
<iN2>=<iT2>+<iS2>+<iI2>=(4kTRL+2e<i>+<i>2Δν)B,
(13)
where subscripts T, S, and I stands for thermal, shot, and intensity, respectively. k is the Boltzmann constant, T is absolute temperature, e is electron charge, Δν is the source bandwidth, RL is detection load resistance, <i> is mean electric current at the PD, and B is the detection bandwidth. As optical intensity noise is proportional to the square of the light intensity, it becomes the major component of short term noise at the output when the optical power reaching the detector is more than a few tens of microwatts [13

13. R. C. Rabelo, R. T. Carvalho, and J. Blake, “SNR enhancement of intensity noise-limited FOGs,” J. Lightwave Technol. 18, 2146–2150 (2000). [CrossRef]

].

The optical intensity noise at the IFOG output can be expressed by [9

9. J. Blake and I. S. Kim, “Distribution of relative intensity noise in signal and quadrature channels of a fiber-optic gyroscope,” Opt. Lett. 19, 1648–1650 (1994) [CrossRef] [PubMed]

, 13

13. R. C. Rabelo, R. T. Carvalho, and J. Blake, “SNR enhancement of intensity noise-limited FOGs,” J. Lightwave Technol. 18, 2146–2150 (2000). [CrossRef]

]
nPD(t)=ηnS(t)gQ(t),
(14)
where ns(t) is the inherent intensity noise of the light source, and gQ(t) = g[t − (π/4ω)] is the IFOG transfer function when Eq. (1) is used as phase modulation. Here g(t) is the IFOG transfer function when traditional sine wave modulation is used (see Appendix). The light intensity received by the PD is given by
IPD=ID+nPD(t)=η(I0+nS(t))gQ(t),
(15)
where ID is the detection signal without intensity noise (see Appendix). The noise part around the 1st harmonic of sin(ωt) in the beat signal nS(t)gQ(t) should be considered, which finally affects detection results. If the 45° phase shift is not set between the modulation signal and reference signal, the distribution of this part of noise in the in-phase signal and the quadrature signal will not be equal in most cases [9

9. J. Blake and I. S. Kim, “Distribution of relative intensity noise in signal and quadrature channels of a fiber-optic gyroscope,” Opt. Lett. 19, 1648–1650 (1994) [CrossRef] [PubMed]

]. In our modulation, a phase shift of 45° is set between the modulation signal and the reference signal, and the eigenfrequency modulation is used, so that two channels have the same magnitude of signal theoretically. According to the symmetry of two channels in this case, the power of intensity noise should be commonly distributed in the in-phase signal and the quadrature signal as
<[nI(t)]2><[nQ(t)]2>12<[n1H(t)]2>,
(16)
where, nI(t), nQ(t) are effective intensity noise in II(t) and IQ(t), and n1H (t) is the noise part around the 1st harmonic of sin(ωt) in nPD(t). From this point of view, two channels have a considerable common part of short term noise in N+(t), although signals have opposite signs.

Numerical simulation for IFOG outputs with intensity noise is carried out according to Eq. (15). White Gaussian noise is added to the source intensity for modeling the influence of intensity noise. Simulation for a source intensity SNR of 35dB is shown by Fig. 2. ΩI(t), ΩQ(t), Ωout (t), and ΩT (t) have mean values of −9.287°/h, 9.161°/h, 9.224°/h, and 9.358°/h, respectively. Noise amplitudes quantified by standard deviation are 1.667°/h, 1.726°/h, 0.604°/h, and 0.623°/h for ΩI(t), ΩQ(t), Ωout (t), and ΩT (t), respectively. Here ΩI(t) is negative in accordance with Eq. (10). The noise in Ωout (t) is obviously reduced by the difference procedure, as its deviation is much lower than single channel results ΩI(t) and ΩQ(t). The noise amplitude in ΩT (t) is also lower than ΩI(t) and ΩQ(t), for the difference noise part between II2 and IQ2 is reduced by the procedure given by Eq. (28) in Appendix. The noise in Ωout (t) is lower than ΩT (t), indicating that intensity noise reduction by the linear difference procedure is more effective.

Fig. 2 Numerical simulation for IFOG outputs with optical intensity noise. The signal-to-noise ratio (SNR) of the source intensity is 35 dB. Theoretical value of the rotation rate is 9.667°/h in accordance with the following experiment. 5000 samples are given, where the sampling time is 0.15 s. Single channel results ΩI and ΩQ, differential result Ωout, and the traditional demodulation result ΩT are put forward for comparison.

For further comparison of four outputs in intensity noise limited IFOGs, simulation of noise amplitudes at different SNRs of source intensity are given in Fig. 3(a). For each output at each SNR, the noise amplitude is represented by the standard deviation of 5000 samples, where simulation parameters are the same as in Fig. 2. It can be seen that noise amplitudes in ΩI(t) and ΩQ(t) are close to each other in accordance with Eq. (16). Ωout (t) has the lowest noise amplitude among the four outputs in all considered SNRs, and its curve is much lower than single channel results ΩI(t) and ΩQ(t). This is a clear evidence for the reduction of noise due to the synchronous difference procedure. Furthermore, the elimination of N+(t) in Eq. (12) is proved more effective than the difference noise reduction in Eq. (28), as the curve for Ωout (t) is lower than the curve for ΩT (t). When the SNR is higher than 40 dB, the intensity noise reduction is not so obvious as low SNRs, which in practice may be submerged by other kind of noise such as phase noise induced by vibration of light circuit. In other words, performance of different demodulation methods is similar to each other when noise amplitude is very low.

Fig. 3 (a) Numerical simulation for output noise amplitude versus SNR of source intensity. Noise amplitudes are quantified as standard deviations for ΩI, ΩQ, Ωout, and ΩT. (b) Numerical simulation for output bias versus SNR of source intensity. Biases are calculated by mean values of ΩI, ΩQ, Ωout, and ΩT.

As a conclusion of short term noise analysis, the quadrature demodulation with synchronous difference is better than either the single channel demodulation or the traditional demodulation in intensity noise limited IFOGs. Ωout (t) has the lowest noise amplitude, and its bias is robust against short term noise.

The unstable biases in ΩI(t) and ΩQ(t) can be written in terms of common part B+(t), differential part B(t), and independent parts B1(t) and B2(t). Considering only long term instability, detection results can be written as
ΩI(t)=Ω+BI(t)=Ω+B+(t)B(t)+B1(t),
(17)
ΩQ(t)=Ω+BQ(t)=Ω+B+(t)+B(t)+B2(t),
(18)
Ωout(t)=Ω+B(t)+B2(t)B1(t)2.
(19)
Here the eliminated common part B+(t) also stands for the bias component independent of the polarity of the rotation rate. In many cases of practical IFOGs, B+(t) is the main part of total bias [10

10. X. Wang, C. He, and Z. Wang, “Method for suppressing the bias drift of interferometric all-fiber optic gyroscopes,” Opt. Lett. 36, 1191–1193 (2011). [CrossRef] [PubMed]

]. The elimination of B+(t) is thus effective for bias drift reduction in these IFOGs, where the main part of bias drift is intrinsically independent of the polarity of the rotation rate.

Both short term noise and long term instability are predicted to be reduced in Ωout, due to the elimination of the common noise term N+(t). In this way, gyroscope precision is enhanced. These are proved by the following experimental results.

4. Experimental results

The open-loop IFOG shown in Fig. 1 was horizontally placed on a stable optical bench, sensing only the Earth’s rotation rate. The theoretical value is 9.666°/h, as the laboratory latitude is 39.99° north and the rotation rate of the earth is 15.041°/h [23

23. Y. Zhao, Y. Zheng, Y. Lin, and B. Li, “Step by step improvement of measurement methods for earth’s rotary rate using fiber optic gyro,” Measurement 44, 1177–1182 (2011). [CrossRef]

]. The sampling rate and the resolution of the digitizer were 2 MHz and 22 bits respectively, and the theoretical resolution of the IFOG was 0.007°/h accordingly. The source intensity was 13 dBmW, so that intensity noise was a big part of short term noise at the output. A phase modulation frequency of 52.35 KHz was used, which was an accurately measured value of the coil eigenfrequency. The traditional sinusoidal demodulation and the quadrature demodulation were both carried out for comparison. The test was carried out in a laboratory on the 4th floor of a building, where human activities might introduce additional noise.

IFOG output data comparison is shown in Fig. 4. We can see short term noise in Ωout is reduced notably by the synchronous difference in comparison with ΩI and ΩQ, as the output data is more concentrated. In the time domain results, we should also notice that there are some high spikes due to environmental acoustic vibrations. These spikes are mainly caused by human activities near our laboratory and appear only occasionally, thus they do not contribute much to long term statistical properties. Besides the occasional spikes, there are also contentious acoustic vibrations which have similar influence with other short term noise such as intensity noise and thermal noise. All these kinds of short term noise can be treated as a whole from a mathematical perspective, and be discussed together in the comparison.

Fig. 4 Experimental results for IFOG outputs. The test was during a period of 50 min, where the sample time was 0.148 s. ΩI, ΩQ, Ωout, and ΩT are put forward for comparison. The test was carried out in a laboratory on the 4th floor of a building, where human activities might introduce additional noise due to acoustic vibration of light paths. This additional noise did not affect our comparison.

Distribution of ΩI and ΩQ is illustrated in Fig. 5 as a scatter diagram, where solid curves represent fitting curves of their PDFs with a Gaussian distribution model. The scatter center is on the 135° line in accordance with Eq. (3). To quantify the short term noise reduction, we use the standard deviation of each output data to calculate the SNR improvement as [13

13. R. C. Rabelo, R. T. Carvalho, and J. Blake, “SNR enhancement of intensity noise-limited FOGs,” J. Lightwave Technol. 18, 2146–2150 (2000). [CrossRef]

]
RdB=20log[min(σI,σQ)σout],
(20)
where σI = 1.048°/h, σQ = 1.045°/h, and σout = 0.732°/h are standard deviation values for ΩI, ΩQ, and Ωout, respectively. The enhancement has a value of at least RdB=3.09 dB from a single channel result to the final output, as an evidence of short term reduction in the synchronous difference procedure.

Fig. 5 Distribution of ΩI and ΩQ detected in experiment. The solid curves represent probability density functions (PDFs) of ΩI, ΩQ, and Ωout. ΩI has a mean value of −6.80°/h and standard deviation of 1.048°/h. ΩQ has a mean value of 6.43°/h and standard deviation of 1.045°/h. Ωout has a mean value of 6.61°/h and standard deviation of 0.732°/h. They all have negative biases, as their absolute mean values are lower than the theoretical value of 9.666°/h. The stable bias is not included in IFOG noise for it can be calibrated.

The short term noise performance of Ωout is similar to ΩT, which has a standard deviation of σT = 0.720°/hσout. This result is consistent with theoretical analysis at high SNRs of source intensity in Fig. 3, where intensity noise difference between Ωout and ΩT is small and easily emerged by other kinds of noise. To further evaluate the performance difference between Ωout and ΩT, Allan variance method is used [24

24. F. L. Walls and D. W. Allan, “Measurements of frequency stability,” Proc. IEEE 74, 162–168 (1986). [CrossRef]

]. As shown in Fig. 6, it is the square root of the Allan variance σ(τ) versus cluster time τ. A significant phenomenon is that long term noise is notably suppressed, for the curve goes down rapidly when τ becomes larger. The numerical value of bias drift is obtained by reading the bottom of the curve. The bias drift of the traditional demodulated result ΩT is 0.040°/h, and the value is reduced to 0.004°/h by the quadrature demodulation method for the same IFOG.

Fig. 6 Allan variance analysis of the IFOG. The bias drift of the traditional demodulated result is 0.040°/h, and the value is reduced to 0.004°/h by the quadrature demodulation with synchronous difference.

Experiments are carried out on an open-loop configuration of the IFOG. With adjusted design, this method is also promising to be applied in closed-loop IFOGs [25

25. O. Çelikel and S. E. San, “Design details and characterization of all digital closed-loop interferometric fiber optic gyroscope with superluminescent light emitting diode,” Opt. Rev. 16, 35–43 (2009). [CrossRef]

].

5. Conclusions

In conclusion, we have shown a novel method of quadrature demodulation for IFOGs, where the synchronous difference of two detected results is used as a low-noise output. Both short term noise and long term instability are reduced by the synchronous difference procedure. Experimental results on an open-loop IFOG are consistent with theory analysis. In comparison with the traditional way of demodulation, the bias drift of the same IFOG is reduced by one order of magnitude using this method.

The most important advantage of the method is that it need not change the optical structure of the IFOG, so that it is possible to be applied in most sine wave modulated IFOGs for noise reduction without increasing their cost or complexity. In addition, it is a method that simultaneously suppresses short term noise and long term instability, which are both significant limitations of IFOGs. Therefore, this quadrature demodulation method is very profitable for designing high precision IFOGs.

Appendix: Traditional sinusoidal phase modulation and demodulation for IFOGs

In the IFOG, the Sagnac phase shift is given by [4

4. H. C. Lefèvre, The Fiber-Optic Gyroscope (Artech House, 1993).

]
ϕs=2πLDλcΩ,
(21)
where L and D are the length and the diameter of the fiber coil, c is the speed of light in vacuum, λ is the effective wave length of the light source, and Ω is the rotation rate. The interference signal received by the PD can be written as
ID=ηI0g(t)=12ηI0{1+cos[ϕs+Δϕ(t)]}
(22)
where I0 is the source intensity, η is the response of the PD, and g(t) is the is the transfer function of the IFOG. For sinusoidal modulation ϕm(t) = ϕ0 sin(ωmt), we have
Δϕ(t)=ϕm(t)ϕm(tτ)=2ϕ0sin(ωmτ/2)cos(ωmt),
(23)
In this case, g(t) can be expanded by Bessel functions as
g(t)=12{1+J0(ϕb)cos(ϕs)+2n=1(1)nJ2n(ϕb)cos[2nωm(tτ2)]cos(ϕs)+2n=1(1)nJ2n1(ϕb)cos[(2n1)ωm(tτ2)]sin(ϕs)}
(24)
where Jn is the nth Bessel function of the first kind, and ϕb = 2ϕ0 sin(ωmτ/2). The eigenfrequency ωm = π/τ is suggested for optimal detection, so that ϕb = 2ϕ0 [3

3. R. A. Bergh, H. C. Lefèvre, and H. J. Shaw, “An overview of fiber-optic gyroscopes,” J. Lightwave Technol. 2, 91–107 (1984). [CrossRef]

,13

13. R. C. Rabelo, R. T. Carvalho, and J. Blake, “SNR enhancement of intensity noise-limited FOGs,” J. Lightwave Technol. 18, 2146–2150 (2000). [CrossRef]

]. In demodulation, ϕb is determined by J4(ϕb)/J2(ϕb) = I4H/I2H, and ϕs is obtained by
ϕs=arctanI1HJ2(ϕb)I2HJ1(ϕb).
(25)
Here InH is the amplitude of the nth harmonic. It is usually assumed that the light intensity and the modulation depth are stable, i.e., I0 and ϕb are both constants. Then only the 1st harmonic is used for detection as
I1H(t)=I0ηJ1(ϕb)sin(ϕs)cos(ωmtπ2)=I0ηJ1(ϕb)sin(ϕs)sin(ωmt+π),
(26)
whose amplitude is denoted by I1H, and the Sagnac phase shift is determined as
ϕs=arcsin[I1HI0ηJ1(ϕb)].
(27)

In traditional demodulation, the amplitude of the first harmonic is determined by a single channel result, or obtained as
I1H=II2+IQ2,
(28)
disregarding the phase of I1H (t). Here II is the amplitude of the in-phase signal of the 1st harmonic, and IQ is the amplitude of the quadrature signal of the 1st harmonic. Then the detection result is derived by Eq. (25) or Eq. (27).

References and links

1.

E. J. Post, “Sagnac effect,” Rev. Mod. Phys. 39, 475–493 (1967). [CrossRef]

2.

H. J. Arditty and H. C. Lefèvre, “Sagnac effect in fiber gyroscopes,” Opt. Lett. 6, 401–403 (1981). [CrossRef] [PubMed]

3.

R. A. Bergh, H. C. Lefèvre, and H. J. Shaw, “An overview of fiber-optic gyroscopes,” J. Lightwave Technol. 2, 91–107 (1984). [CrossRef]

4.

H. C. Lefèvre, The Fiber-Optic Gyroscope (Artech House, 1993).

5.

G. B. Malykin, “On the ultimate sensitivity of fiber-optic gyroscopes,” Tech. Phys. 54, 415–418 (2009). [CrossRef]

6.

I. A. Andronova and G. B. Malykin, “Physical problems of fiber gyroscopy based on the Sagnac effect,” Phys. Usp. 45, 793–817 (2002). [CrossRef]

7.

P. Y. Chien and C. L. Pan, “Triangular phase-modulation approach to an open-loop fiber-optic gyroscope,” Opt. Lett. 16, 1701–1703 (1991). [CrossRef] [PubMed]

8.

D. A. Jackson, A. D. Kersey, and A. C. Lewin, “Fibre gyroscope with passive quadrature detection,” Electron. Lett. 20, 399–401 (1984). [CrossRef]

9.

J. Blake and I. S. Kim, “Distribution of relative intensity noise in signal and quadrature channels of a fiber-optic gyroscope,” Opt. Lett. 19, 1648–1650 (1994) [CrossRef] [PubMed]

10.

X. Wang, C. He, and Z. Wang, “Method for suppressing the bias drift of interferometric all-fiber optic gyroscopes,” Opt. Lett. 36, 1191–1193 (2011). [CrossRef] [PubMed]

11.

W. K. Burns, R. P. Moeller, and A. Dandridge, “Excess noise in fiber gyroscope sources,” IEEE Photonic. Tech. Lett. 2, 606–608 (1990). [CrossRef]

12.

R. P. Moeller and W. K. Burns, “1.06-ptm all-fiber gyroscope with noise subtraction,” Opt. Lett. 16, 1902–1904 (1991). [CrossRef] [PubMed]

13.

R. C. Rabelo, R. T. Carvalho, and J. Blake, “SNR enhancement of intensity noise-limited FOGs,” J. Lightwave Technol. 18, 2146–2150 (2000). [CrossRef]

14.

P. Polynkin, J. Arruda, and J. Blake, “All-optical noise-subtraction scheme for a fiber-optic gyroscope,” Opt. Lett. 25, 147–149 (2000). [CrossRef]

15.

R. Ulrich, “Fiber-optic rotation sensing with low drift,” Opt. Lett. 5, 173–175 (1980). [CrossRef] [PubMed]

16.

K. Bohm, P. Marten, K. Petermann, E. Weidel, and R. Ulrich, “Low-drift fiber gyro using a superluminescent diode,” Electron. Lett. 17, 352–353 (1981). [CrossRef]

17.

E. Jones and J. W. Parker, “Bias reduction by polarisation dispersion in the fibre-optic gyroscope,” Electron. Lett. 22, 54–56 (1986). [CrossRef]

18.

S. L. A. Carrara, B. Y. Kim, and H. J. Shaw, “Bias drift reduction in polarization-maintaining fiber gyroscope,” Opt. Lett. 12, 214–216 (1987). [CrossRef] [PubMed]

19.

O. Çelikel and F. Sametoǧlu, “Assessment of magneto-optic Faraday effect-based drift on interferometric single-mode fiber optic gyroscope (IFOG) as a function of variable degree of polarization (DOP),” Meas. Sci. Technol. 23, 025104 (2012). [CrossRef]

20.

A. Lompado, J. C. Reinhardt, L. C. Heaton, J. L. Williams, and P. B. Ruffin, “Full Stokes polarimeter for characterization of fiber optic gyroscope coils,” Opt. Express 17, 8370–8381 (2009). [CrossRef] [PubMed]

21.

Y. Yang, Z. Wang, and Z. Li, “Optically compensated dual-polarization interferometric fiber-optic gyroscope,” Opt. Lett. 37, 2841–2843 (2012). [CrossRef] [PubMed]

22.

D. Kim and J. Kang, “Sagnac loop interferometer based on polarization maintaining photonic crystal fiber with reduced temperature sensitivity,” Opt. Express 12, 4490–4495 (2004). [CrossRef] [PubMed]

23.

Y. Zhao, Y. Zheng, Y. Lin, and B. Li, “Step by step improvement of measurement methods for earth’s rotary rate using fiber optic gyro,” Measurement 44, 1177–1182 (2011). [CrossRef]

24.

F. L. Walls and D. W. Allan, “Measurements of frequency stability,” Proc. IEEE 74, 162–168 (1986). [CrossRef]

25.

O. Çelikel and S. E. San, “Design details and characterization of all digital closed-loop interferometric fiber optic gyroscope with superluminescent light emitting diode,” Opt. Rev. 16, 35–43 (2009). [CrossRef]

OCIS Codes
(060.2370) Fiber optics and optical communications : Fiber optics sensors
(060.2800) Fiber optics and optical communications : Gyroscopes

ToC Category:
Sensors

History
Original Manuscript: July 13, 2012
Revised Manuscript: September 14, 2012
Manuscript Accepted: October 9, 2012
Published: October 25, 2012

Citation
Zinan Wang, Yi Yang, Yongxiao Li, Xiaoqi Yu, Zhenrong Zhang, and Zhengbin Li, "Quadrature demodulation with synchronous difference for interferometric fiber-optic gyroscopes," Opt. Express 20, 25421-25431 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-23-25421


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References

  1. E. J. Post, “Sagnac effect,” Rev. Mod. Phys.39, 475–493 (1967). [CrossRef]
  2. H. J. Arditty and H. C. Lefèvre, “Sagnac effect in fiber gyroscopes,” Opt. Lett.6, 401–403 (1981). [CrossRef] [PubMed]
  3. R. A. Bergh, H. C. Lefèvre, and H. J. Shaw, “An overview of fiber-optic gyroscopes,” J. Lightwave Technol.2, 91–107 (1984). [CrossRef]
  4. H. C. Lefèvre, The Fiber-Optic Gyroscope (Artech House, 1993).
  5. G. B. Malykin, “On the ultimate sensitivity of fiber-optic gyroscopes,” Tech. Phys.54, 415–418 (2009). [CrossRef]
  6. I. A. Andronova and G. B. Malykin, “Physical problems of fiber gyroscopy based on the Sagnac effect,” Phys. Usp.45, 793–817 (2002). [CrossRef]
  7. P. Y. Chien and C. L. Pan, “Triangular phase-modulation approach to an open-loop fiber-optic gyroscope,” Opt. Lett.16, 1701–1703 (1991). [CrossRef] [PubMed]
  8. D. A. Jackson, A. D. Kersey, and A. C. Lewin, “Fibre gyroscope with passive quadrature detection,” Electron. Lett.20, 399–401 (1984). [CrossRef]
  9. J. Blake and I. S. Kim, “Distribution of relative intensity noise in signal and quadrature channels of a fiber-optic gyroscope,” Opt. Lett.19, 1648–1650 (1994) [CrossRef] [PubMed]
  10. X. Wang, C. He, and Z. Wang, “Method for suppressing the bias drift of interferometric all-fiber optic gyroscopes,” Opt. Lett.36, 1191–1193 (2011). [CrossRef] [PubMed]
  11. W. K. Burns, R. P. Moeller, and A. Dandridge, “Excess noise in fiber gyroscope sources,” IEEE Photonic. Tech. Lett.2, 606–608 (1990). [CrossRef]
  12. R. P. Moeller and W. K. Burns, “1.06-ptm all-fiber gyroscope with noise subtraction,” Opt. Lett.16, 1902–1904 (1991). [CrossRef] [PubMed]
  13. R. C. Rabelo, R. T. Carvalho, and J. Blake, “SNR enhancement of intensity noise-limited FOGs,” J. Lightwave Technol.18, 2146–2150 (2000). [CrossRef]
  14. P. Polynkin, J. Arruda, and J. Blake, “All-optical noise-subtraction scheme for a fiber-optic gyroscope,” Opt. Lett.25, 147–149 (2000). [CrossRef]
  15. R. Ulrich, “Fiber-optic rotation sensing with low drift,” Opt. Lett.5, 173–175 (1980). [CrossRef] [PubMed]
  16. K. Bohm, P. Marten, K. Petermann, E. Weidel, and R. Ulrich, “Low-drift fiber gyro using a superluminescent diode,” Electron. Lett.17, 352–353 (1981). [CrossRef]
  17. E. Jones and J. W. Parker, “Bias reduction by polarisation dispersion in the fibre-optic gyroscope,” Electron. Lett.22, 54–56 (1986). [CrossRef]
  18. S. L. A. Carrara, B. Y. Kim, and H. J. Shaw, “Bias drift reduction in polarization-maintaining fiber gyroscope,” Opt. Lett.12, 214–216 (1987). [CrossRef] [PubMed]
  19. O. Çelikel and F. Sametoǧlu, “Assessment of magneto-optic Faraday effect-based drift on interferometric single-mode fiber optic gyroscope (IFOG) as a function of variable degree of polarization (DOP),” Meas. Sci. Technol.23, 025104 (2012). [CrossRef]
  20. A. Lompado, J. C. Reinhardt, L. C. Heaton, J. L. Williams, and P. B. Ruffin, “Full Stokes polarimeter for characterization of fiber optic gyroscope coils,” Opt. Express17, 8370–8381 (2009). [CrossRef] [PubMed]
  21. Y. Yang, Z. Wang, and Z. Li, “Optically compensated dual-polarization interferometric fiber-optic gyroscope,” Opt. Lett.37, 2841–2843 (2012). [CrossRef] [PubMed]
  22. D. Kim and J. Kang, “Sagnac loop interferometer based on polarization maintaining photonic crystal fiber with reduced temperature sensitivity,” Opt. Express12, 4490–4495 (2004). [CrossRef] [PubMed]
  23. Y. Zhao, Y. Zheng, Y. Lin, and B. Li, “Step by step improvement of measurement methods for earth’s rotary rate using fiber optic gyro,” Measurement44, 1177–1182 (2011). [CrossRef]
  24. F. L. Walls and D. W. Allan, “Measurements of frequency stability,” Proc. IEEE74, 162–168 (1986). [CrossRef]
  25. O. Çelikel and S. E. San, “Design details and characterization of all digital closed-loop interferometric fiber optic gyroscope with superluminescent light emitting diode,” Opt. Rev.16, 35–43 (2009). [CrossRef]

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