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sensors Article Backward Secondary-Wave Coherence Errors in Photonic Bandgap Fiber Optic Gyroscopes Xiaobin Xu *, Ningfang Song, Zuchen Zhang and Jing Jin Department of Opto-electronics Engineering, Beihang University, Beijing 100191, China; [email protected] (N.S.); [email protected] (Z.Z.); [email protected] (J.J.) * Correspondence: [email protected]; Tel.: +86-10-8231-6887 Academic Editor: Jörg F. Wagner Received: 25 April 2016; Accepted: 3 June 2016; Published: 8 June 2016 Abstract: Photonic bandgap fiber optic gyroscope (PBFOG) is a novel fiber optic gyroscope (FOG) with excellent environment adaptability performance compared to a conventional FOG. In this work we find and investigate the backward secondary-wave coherence (BSC) error, which is a bias error unique to the PBFOG and caused by the interference between back-reflection-induced and backscatter-induced secondary waves. Our theoretical and experimental results show a maximum BSC error of ~4.7˝ /h for a 300-m PBF coil with a diameter of 10 cm. The BSC error is an important error source contributing to bias instability in the PBFOG and has to be addressed before practical applications of the PBFOG can be implemented. Keywords: photonic bandgap fiber; fiber optic gyroscope; bias error 1. Introduction Current fiber optic gyroscopes (FOGs) have very good performance [1], but inevitably they still have large Shupe effects, Faraday effects and radiation effects due to the fact that conventional silica fiber is very sensitive to these environments. Photonic bandgap fibers (PBFs), a novel kind of fiber in which the light travels not in a conventional silica core but in an air core [2–4], have attracted considerable interest owing to their unique optical properties such as significantly lower nonlinear coefficients and especially, their improved adaptability to temperature, magnetic fields and radiation in comparison with the properties of silica fibers [5,6]. A photonic bandgap fiber optic gyroscope (PBFOG), as illustrated in Figure 1, consists of a broad-spectrum source, a coupler, an integrated optic chip (IOC), and a PBF coil [7,8]. A previous study [9] has reported reductions in the errors induced by the Kerr effect (>170), Shupe effect (~6.5), and Faraday effect (>20) in a PBFOG when compared with the corresponding error values of a conventional FOG. Therefore, the PBFOG represents a substantially improvement of these key properties compared to a FOG, and it has excellent prospects, but as a new kind of FOG, it definitely exhibits some different errors and noises from those which are normally observed in a traditional FOG owing to the use of the PBF. References [7,8] have investigated shot noise, backscattering noise which is defined as the error due to coherence interference between the backscattered waves and primary waves and which also exists in resonant fiber optic gyroscopes using an air-core fiber [10,11], electronic noise, and reflection-induced intensity noise; further, the polarization non-reciprocity error has also been investigated [12]. In this study, for the first time to the best of our knowledge, we find and investigate the backward secondary-wave coherence (BSC) error, which is defined as a bias error caused in the PBFOG by interference between back-reflection-induced and backscatter-induced secondary waves. Sensors 2016, 16, 851; doi:10.3390/s16060851 www.mdpi.com/journal/sensors Sensors 2016, 16, 851 Sensors 2016, 16, 851 2 of 7 2 of 7 Figure 1. Diagram of a photonic bandgap fibe r optic gyroscope . Figure 1. Diagram of a photonic bandgap fiber optic gyroscope. 2. Theoretical Analysis 2. Theoretical Analysis In the PBFOG, the pigtails of the IOC are conventional fibers that have a Ge-doped SiO2 core, In the PBFOG, the pigtails of the IOC are conventional fibers that have a Ge-doped SiO2 core, but but the coil comprises a commercially available 7-cell air-core PBF with hexagonal air holes the coil comprises a commercially available 7-cell air-core PBF with hexagonal air holes triangularly triangularly arranged in the cladding [12]. According to the measurement results in our previous arranged in the cladding [12]. According to the measurement results in our previous study [13], study [13], a strong reflection inevitably occurs at the two fusion splicing points between the PBF a strong reflection inevitably occurs at the two fusion splicing points between the PBF coil and the coil and the IOC pigtails. Both back-reflection-induced secondary waves have very large IOC pigtails. Both back-reflection-induced secondary waves have very large magnitudes (only about magnitudes (only about 3–10 times smaller than that of primary waves for a 300-m coil composed of 3–10 times smaller than that of primary waves for a 300-m coil composed of a PBF with a loss of a PBF with a loss of 20 dB/km). Obviously, those back-reflection-induced secondary waves produce 20 dB/km). Obviously, those back-reflection-induced secondary waves produce an offset at the detector an offset at the detector and add intensity-type noise because the coherence length of the and add intensity-type noise because the coherence length of the broad-spectrum source is always broad-spectrum source is always considerably less than the path mismatch length between the two considerably less than the path mismatch length between the two IOC pigtails, and thus, there is no IOC pigtails, and thus, there is no interference between these two sets of secondary w aves [7,9]. interference between these two sets of secondary waves [7,9]. However, we found that the strong However, we found that the strong back-reflection-induced secondary waves at one fusion splicing back-reflection-induced secondary waves at one fusion splicing point inevitably interfere with the point inevitably interfere with the backscatter-induced secondary waves within the other pigtail, backscatter-induced secondary waves within the other pigtail, thereby causing a BSC error. This thereby causing a BSC error. This BSC error is a phase-type error that does not exist in a BSC error is a phase-type error that does not exist in a conventional FOG, and it seriously affects the conventional FOG, and it seriously affects the PBFOG performance. PBFOG performance. As illustrated in Figure 2a, before the primary-wave (black arrows) output from the IOC enters As illustrated in Figure 2a, before the primary-wave (black arrows) output from the IOC enters the PBF coil, secondary waves (W A and W B) are generated owing to the strong back-reflection at the PBF coil, secondary waves (WA and WB ) are generated owing to the strong back-reflection at fusion splicing points A and B. W A has an optical path of L(W A) = nIOC |OO1| + n SiO2 |O1 A| from the fusion splicing points A and B. WA has an optical path of L(WA ) = nIOC |OO1 | + nSiO2 |O1 A| from common input/output end O. nIOC and nSiO2 is, respectively, the refractive index of the IOC and the common input/output end O. nIOC and nSiO2 is, respectively, the refractive index of the IOC and conventional fiber. It is well known that the backscattering is randomly distributed along the fiber, conventional fiber. It is well known that the backscattering is randomly distributed along the fiber, and each backscattering point causes a secondary wave, but not all of those backscattering-induced and each backscattering point causes a secondary wave, but not all of those backscattering-induced secondary waves are able to interfere with W A, because it is the broad-spectrum light in the optical secondary waves are able to interfere with WA , because it is the broad-spectrum light in the optical circuit and the interference intensity is maximum when the optical path difference (OPD) is 0, but circuit and the interference intensity is maximum when the optical path difference (OPD) is 0, but decreases dramatically and rapidly as the OPD increases [14]. Therefore, i f a secondary wave W A1 decreases dramatically and rapidly as the OPD increases [14]. Therefore, if a secondary wave WA1 induced by the backscattering at point A1 has an optical path of L(W A1)= nIOC|OO2 | + nSiO2|O2B| + induced by the backscattering at point A1 has an optical path of L(WA1 )= nIOC |OO2 | + nSiO2 |O2 B| + nair |BA1 |from the common input/output end O and L(W A1) = L(W A), then the interference occurs nair |BA1 |from the common input/output end O and L(WA1 ) = L(WA ), then the interference occurs between W A and W A1 and the intensity is maximum that is given by the first term in: between WA and WA1 and the intensity is maximum that is given by the first term in: 2 I in splicing cos(m` ) 2 I in pR mm`2ϕ)2 q RBB RRBB11 q1{2cos( BSC AR A1 cospΦ m “I2I pR ARRA1 q1{2 ϕ cospΦ 1 q1 ` 2Iininαin in αin αinsplicing 1/ 2 IBSC 1/ 2 1{2 2 3 22 `2Iin α α2coilpR R A RqA1{2 cospΦ ϕ3 q3 )`2I α cospΦ ϕ q in2α I in2splicing in 2splicing cos( 2 Iininαinin α 3splicing m m` m` BBR B1 q cos( RpR coil coilA RA1 1 m splicing coil BR 1 4) 4 1/ 2 1/ 2 (1) (1) which equation describes the summed intensity (as detected by the detector) of all the interference which equation describes the summed intensity (as detected by the detector) of all the interference phenomena involved (that is, IBSC ). Here nair is the refractive index of the PBF. In Equation (1), Iin phenomena involved (that is, IBSC). Here nair is the refractive index of the PBF. In Equation (1), Iin denotes the light intensity output from the IOC. Further, αin , αsplicing , and αcoil represent the losses at denotes the light intensity output from the IOC. Further, α in, α splicing, and α coil represent the losses at the input channel (from the IOC pigtail to detector), fusion splicing at points A and B, and PBF coil, the input channel (from the IOC pigtail to detector), fusion splicing at points A and B, and PBF coil, respectively. The typical values of αin , αsplicing , and αcoil are ~8 dB, ~1.5 dB and ~6.6 dB for a 300-m respectively. The typical values of α in, α splicing, and α coil are~8 dB, ~1.5 dB and ~6.6 dB for a 300-m PBF PBF coil, respectively. Parameters RA and RB represent the back-reflection coefficients at points A and coil, respectively. Parameters RA and RB represent the back-reflection coefficients at points A and B, B, respectively. Similarly, RA1 and RB1 represent the backscattering coefficients at points A1 and B1 , respectively. Similarly, RA1 and RB1 represent the backscattering coefficients at points A1 and B1, respectively. Parameters ϕ1 , ϕ2 , ϕ3 , and ϕ4 denote the random phases between the two secondary respectively. Parameters φ1, φ2, φ3 , and φ4 denote the random phases between the two secondary waves involved in the interference. Φm is the modulation phase. waves involved in the interference. Фm is the modulation phase. Sensors 2016, 16, 851 Sensors 2016, 16, 851 3 of 7 3 of 7 (a) (b) Figure 2. Se condary waves induce d by back-re flection and backscatte r of primary waves . (a) Be fore Figure 2. Secondary waves induced by back-reflection and backscatter of primary waves. (a) Before the primary waves e nter the PBF coil; (b) Afte r the primary waves have propagate d through 1 loop the primary waves enter the PBF coil; (b) After the primary waves have propagated through 1 loop of of the PBF coil. the PBF coil. Interference between W B and W B1 can similarly be examined, and the resulting interference Interference between WB and WB1 similarly bethe examined, and resulting interference intensity is given by the second term in can Equation (1 ). On other hand, asthe illustrated in Figure 2b, when the primary-wave outputterm fromin the PBF coil after propagating coil once, in backward intensity is given by the second Equation (1). On the other through hand, asthe illustrated Figure 2b, secondary waves (W′ A,output W′ B, W′from A1, W′the B1) arise, and after interference can also occur between W′ A and W′ A1, when the primary-wave PBF coil propagating through the coil once, backward 1 1 1 1 1 and W′ B and W′ B1 with the intensities given by the last two terms in Equation (1). In fact, there also secondary waves (W A , W B , W A1 , W B1 ) arise, and interference can also occur between Wexist A 1 1 1 other backward secondary waves that propagate over multiple loops; however, their contribution W A1 , W B and W B1 with the intensities given by the last two terms in Equation (1). In fact, there to also the BSC backward error is very small, and the relevant terms have been neglected in Equation exist other secondary waves that propagate over multiple loops; however,(1). their contribution In the standard operation, a square terms wave with τ and amplitude of ±π/8, to the BSC error is veryFOG small, and the relevant have eigenfrequency been neglectedfin Equation (1). and a sawtooth wave with amplitude π are added and applied to the IOC to modulate primary In the standard FOG operation, a square wave with eigenfrequency f τ and amplitude of ˘π/8, waves for coherence detection and closing-loop operation [14]. The demodulation value of the and a sawtooth wave with amplitude π are added and applied to the IOC to modulate primary interference intensity between the primary waves indicates the angle velocity of the FOG. The waves for coherence detection and closing-loop operation [14]. The demodulation value of the back-reflection-induced and backscatter-induced secondary waves also propagate through the IOC, interference intensity between the primary waves indicates the angle velocity of the FOG. The and therefore, these waves are also modulated by the square and sawtooth waves; thus, the back-reflection-induced backscatter-induced secondary waves also propagate through the IOC, modulation phase Фm and in Equation (1) includes two parts: the square-wave-induced phase (Ф m_SQ) andand therefore, these waves are also modulated by the square and sawtooth waves; thus, the modulation the sawtooth-wave-induced phase (Фm_SA). Because the secondary waves propagate twice phase Φm in (1) of includes parts: the square-wave-induced phase of (Φfm_SQ )Ф and through theEquation same branch the IOC,two Фm_SQ has an amplitude of ±π/2 and frequency τ, and m_ SAthe sawtooth-wave-induced has an amplitude of 4π.phase (Φm_SA ). Because the secondary waves propagate twice through the same branch of thetoIOC, an amplitude of ˘π/2 and frequency , and Φ According the Φ theory of coherence detection, because Фm_SQ has of thef τsame modulation m_SQ has m_SA has an amplitude of (f4π. frequency τ) as that of the primary waves, the demodulation value of IBSC cannot be separated from the angle velocity the FOG which is the demodulation ofhas thethe primary interference, According to theof theory of coherence detection, becauseresult Φm_SQ same waves’ modulation frequency leading a bias error (namely, the BSC error) [14].ofOn the other be hand, Фm_SA leads (f τ )thus as that of thetoprimary waves, the demodulation value IBSC cannot separated from to theBSC angle error’s variation with the sawtooth wave, and there are two complete periods when the sawtooth velocity of the FOG which is the demodulation result of the primary waves’ interference, thus leading changes from 0 the (rad) to π (rad).[14]. Under conditions, the maximum BSCerror’s error isvariation given to awave bias error (namely, BSC error) Onextreme the other hand, Φm_SA leads to BSC by: with the sawtooth wave, and there are two complete periods when the sawtooth wave changes from 0 2 maximum 2 error is given by: (rad) toBSC π (rad). Under extreme the BSC Errormax splicing RA RA1 conditions, RB RB1 2splicing coil RA RA1 3splicing coil RB RB1 2LD / c 2splicing coil a a a (2) a 2 3 BSCError “ pαLsplicing R A R A1 ` diameter R B R B1 ` α α2coil length R A R A1of `α α2coil R B R B1 q{p2πLD{λc α2splicing αcoil q (2) splicing splicing wheremax D and represent the and fiber the coil, respectively, λ the¨wavelength, and c the velocity of light. where D and L represent the diameter and fiber length of the coil, respectively, λ the wavelength, and c the velocity of light. Sensors 2016, 16, 851 Sensors 2016, 16, 851 4 of 7 4 of 7 3. Experimental Results 3. Experimental To verify theResults existence of the BSC error and measure its magnitude in a practical FOG, we promoted a method and established experimental as illustrated in To verify the existence of the BSC the errorcorresponding and measure its magnitude setup, in a practical FOG, we Figure 3, where the light launched from an amplified spontaneous emission (ASE) source enters the promoted a method and established the corresponding experimental setup, as illustrated in Figure 3, IOC through single-mode fiber coupler, and it travels backward the enters detector to the where the light alaunched from an amplified spontaneous emission (ASE) to source theowing IOC through back-reflection and backscattering. The ASE source has a power of ~5 mW and the IOC a single-mode fiber coupler, and it travels backward to the detector owing to the back-reflection is anda proton-exchanged 3 circuit with the half-wave voltage of IOC ~5 V.isIn fact, the setup mainly aims backscattering. TheLiNbO ASE source has a power of ~5 mW and the a proton-exchanged LiNbO 3 at determining the first two terms in Equations (1) or (2), which are the dominant BSC error terms. circuit with the half-wave voltage of ~5 V. In fact, the setup mainly aims at determining the first two Moreover, our setup a configuration that is nearlyBSC identical to the Moreover, FOG configuration; terms in Equations (1)has or (2), which are the dominant error terms. our setupsquare has a and sawtooththat waves are applied IOC, configuration; and a lock-insquare amplifier is used towaves demodulate the configuration is nearly identicalto to the the FOG and sawtooth are applied interference intensity (I BSC ) at the detector so as to better simulate the actual FOG operation. The one to the IOC, and a lock-in amplifier is used to demodulate the interference intensity (IBSC ) at the detector difference in our case isthe that two lengths of PBFs (and not adifference PBF coil)in are connected to the IOC so as to better simulate actual FOG operation. The one our case is that twotwo lengths pigtails to eliminate interference between the primary waves whose demodulation value cannot be of PBFs (and not a PBF coil) are connected to the two IOC pigtails to eliminate interference between separated from the measurement results. From the aspect of BSC error determined by the first two the primary waves whose demodulation value cannot be separated from the measurement results. termsthe in Equation (1 ), this does notfirst matter, because this part (1), of the error does From aspect of BSC errordifference determined by the two terms in Equation thisBSC difference does not not depend on the subsequent PBF coil. Considering the IOC’s half-wave voltage and the signal matter, because this part of the BSC error does not depend on the subsequent PBF coil. Considering generator’s single-end output, wethe setsignal the square wave frequency 500 kHz an square amplitude of the IOC’s half-wave voltage and generator’s single-endtooutput, wewith set the wave ±1.25 V, and the kHz sawtooth wave is set toof vary from 0 V the to 10 V. Based on the analysis frequency to 500 with an amplitude ˘1.25 V, and sawtooth wave is set to varymentioned from 0 V above, the demodulation result of I BSC should have two sinusoidal periods when the sawtooth to 10 V. Based on the analysis mentioned above, the demodulation result of IBSC should havewave two changes from 0 V when to 10 V, its peak-to-peak value can0 V betoeasily which value directly sinusoidal periods theand sawtooth wave changes from 10 V, resolved, and its peak-to-peak value can 1/2 or I in (R BR B1)1/2 or their sum according to which 1/2 1/2 reflects I in α splicing (R A R A1 ) splicing point exists in the be easily resolved, which value directly reflects Iin αsplicing (RA RA1 ) or Iin (RB RB1 ) or their sum system. according to which splicing point exists in the system. Figure 3. Expe rime ntal se tup for me asuring the BSC e rror. Figure 3. Experimental setup for measuring the BSC error. First, a length of PBF is normally connected to the IOC pigtail at point B through fusion First, a length of PBF is normally connected to the IOC pigtail at point B through fusion splicing. splicing. The micrograph of the splicing between the PBF and IOC pigtail is shown in the inset in The micrograph of the splicing between the PBF and IOC pigtail is shown in the inset in Figure 4a. Figure 4a. Since it is 0° splicing, the secondary wave W B is gen erated and its interference with W B1 Since it is 0˝ splicing, the secondary wave WB is generated and its interference with WB1 occurs occurs simultaneously. The lock-in amplifier’s output after demodulation of the interference simultaneously. The lock-in amplifier’s output after demodulation of the interference intensity IBSC intensity IBSC is described by the dashed curve in Figure 4a where the fixed bias induced by such is described by the dashed curve in Figure 4a where the fixed bias induced by such phenomena as phenomena as Earth rotation has been taken out. Obviously, the output varies with the voltage of Earth rotation has been taken out. Obviously, the output varies with the voltage of the sawtooth wave, the sawtooth wave, and there are two periods when the voltage changes from 0 V to 10 V, a result and there are two periods when the voltage changes from 0 V to 10 V, a result that agrees well with that agrees well with the abovementioned analysis. The peak-to-peak value of the output is ~130 μV, the abovementioned analysis. The peak-to-peak value of the output is ~130 µV, which implies that which implies that (RBRB1)1/2 ~3.5 × 10 −7, and therefore, the maximum BSC error in this case is (RB RB1 )1/2 ~3.5 ˆ 10´ 7 , and therefore, the maximum BSC error in this case is ~1.36˝ /h according ~1.36°/h according to Equation (2) for a 300-m PBF coil with a diameter of 10 cm. Next, a second PBF to Equation (2) for a 300-m PBF coil with a diameter of 10 cm. Next, a second PBF is also similarly is also similarly connected to the other IOC pigtail at point A by fusion splicing. Th e resulting connected to the other IOC pigtail at point A by fusion splicing. The resulting output is described by output is described by the solid line in Figure 4a. The peak-to-peak value of the output is ~450 μV, the solid line in Figure 4a. The peak-to-peak value of the output is ~450 µV, and consequently, the and consequently, the total maximum BSC error is ~4.7°/h according to Equation (2). The total maximum BSC error is ~4.7˝ /h according to Equation (2). The experimental results indicate that experimental results indicate that interference between W A and W A1 contributes more to the BSC interference between WA and WA1 contributes more to the BSC error than that between WB and WB1 , error than that between W B and W B1, which can be explained by the fact that the backscatter which can be explained by the fact that the backscatter coefficient in the PBF is higher than that in coefficient in the PBF is higher than that in conventional fiber s [15]. conventional fibers [15]. Sensors 2016, 16, 851 Sensors 2016, 16, 851 5 of 7 5 of 7 (a) (b) Figure 4. Test results of (a) the BSC e rror and (b) bias stability in the PBFOG whe n the PBFs are Figure 4. Test results of (a) the BSC error and (b) bias stability in the PBFOG when the PBFs are connecte d to to the the IOC IOC pigtails pigtails through throughnormal normalfusion fusionsplicing. splicing.Inset: Inset: micrograph micrograph showing showing the the fusion fusion connected splicing be twe e n PBF and conve ntional fibe r at point A or B. splicing between PBF and conventional fiber at point A or B. In a real closed-loop PBFOG, the BSC error can seriously affect the bias stability. Bias stability In a real closed-loop PBFOG, the BSC error can seriously affect the bias stability. Bias stability is is defined as the random variation in bias as computed over specified finite sample time and defined as the random variation in bias as computed over specified finite sample time and averaging averaging time intervals according to IEEE standard [16]. The secondary waves W A and W A1, W B time intervals according to IEEE standard [16]. The secondary waves WA and W (see , WFigure are B and W B1 The and W B1 are not reciprocal, because they propagate along different optical pathsA1 2a). not reciprocal, because they propagate along different optical paths (see Figure 2a). The environment environment (temperature, vibration, and so on) definitely has different influences on different (temperature, vibration, and so on) definitely has different influences on different optical paths, as a optical paths, as a result, their phase differences (φ1, φ2 ) are random environment-dependent result, their phase differences (ϕ , ϕ2 ) are random environment-dependent variables. When those variables. When those secondary1waves interfere, their interference intensities and the induced BSC secondary waves interfere, their interference intensities and the induced BSC error would also vary error would also vary randomly, so the PBFOG bias becomes unstable if it includes this error. In randomly, so the PBFOG bias becomes unstable if it includes this error. In order to verify the BSC error’s order to verify the BSC error’s influence on PBFOG bias stability, a real PBF coil, instead of the two influence on PBFOG bias stability, a real PBF coil, instead of the two lengths of PBFs, is connected to lengths of PBFs, is connected to the IOC pigtails with normal fusion splicing on the basis of the the IOC pigtails with normal fusion splicing on the basis of the experimental setup in Figure 3, and experimental setup in Figure 3, and forms a complete PBFOG. A signal-processing electronic circuit forms a complete PBFOG. A signal-processing electronic circuit substitutes the lock-in amplifier and substitutes the lock-in amplifier and signal generator to implement the digital closing-loop signal generator to implement the digital closing-loop operation in the PBFOG [14]. The test result of operation in the PBFOG [14]. The test result of the PBFOG bias is shown in Figure 4b, where the the PBFOG bias is shown in Figure 4b, where the fixed bias induced by such as earth rotation has also fixed bias induced by such as earth rotation has also been taken out. The test result indicates that been taken out. The test result indicates that the bias remarkably fluctuates and the corresponding bias the bias remarkably fluctuates and the corresponding bias stability is ~0.75°/h (standard deviation, stability is ~0.75˝ /h (standard deviation, 10 s integration time). 10 s integration time). Although backscatter within the fiber cannot be eliminated, back-reflection can be reduced Although backscatter within the fiber cannot be eliminated, back-reflection can be reduced through an angled fiber end face in fiber termination in order to suppress the BSC error. A tilted angle through an angled fiber end face in fiber termination in order to suppress the BSC error . A tilted ˝ of 8 is a commonly used value for conventional fiber and it is often applied in fiber connectors to angle of 8° is a commonly used value for conventional fiber and it is often applied in fiber suppress back-reflection [17]. Although a larger cleavage angle can further reduce the back reflection, connectors to suppress back-reflection [17]. Although a larger cleavage angle can further reduce the this will come at the cost of increased splicing loss, so in the experimental FOG, both the pigtails back reflection, this will come at the cost of increased splicing loss, so in the experimental FOG, both of the IOC and PBF coil are cleaved at a tilted angle of ~8˝ to guarantee both the suppression of the pigtails of the IOC and PBF coil are cleaved at a tilted angle of ~8° to guarantee both the back-reflection at the interface and the proper fusion splicing loss, as shown in Figure 5a [7,9–11,18]. suppression of back-reflection at the interface and the proper fusion splicing loss, as shown in After fusion splicing, according to the test results, the reflectance at the interface (see Figure 5b) is Figure 5a [7,9–11,18]. After fusion splicing, according to the test results, the reflectance at the reduced from greater than ´20 dB to ~´54 dB, so the intensity of both WA and WB decreases to be interface (see Figure 5b) is reduced from greater than −20 dB to ~−54 dB, so the intensity of both W A negligible compared to the primary waves. Under this condition, the BSC error is also negligible and W B decreases to be negligible compared to the primary waves. Under this condition, the BSC according to the experimental result, as illustrated in Figure 5c that gives the test result of the BSC error is also negligible according to the experimental result, as illustrated in Figure 5c that gives the error; the corresponding PBFOG performance is shown in Figure 5d. Obviously, when the BSC error is test result of the BSC error; the corresponding PBFOG performance is shown in Figure 5d. suppressed, the bias stability is dramatically improved to ~0.4˝ /h (standard deviation, 10 s integration Obviously, when the BSC error is suppressed, the bias stability is dramatically improved to ~0.4 °/h time). Therefore, we can conclude that the BSC error is an important bias error source that seriously (standard deviation, 10 s integration time). Therefore, we can conclude that the BSC error is an affects the PBFOG performance, and it has to be addressed before any practical application of the important bias error source that seriously affects the PBFOG performance, and it has to be addressed PBFOG is considered. before any practical application of the PBFOG is considered. Sensors 2016, 16, 851 Sensors 2016, 16, 851 6 of 7 6 of 7 (a) (b) (c) (d) Figure 5. Micrographs Micrographsshowing showing(a) (a)the thecleaved cleaveangles d angles of pigtails of PBF the PBF coil IOC and and IOC(b) and (b) Figure 5. of pigtails of the coil and cross cross section of the corresponding point after fusion splicing; (c) Test results of the BSC error and (d) section of the corresponding point after fusion splicing; (c) Test results of the BSC error and (d) bias bias stability in the PBFOG whe n the PBF coil is connecte d to the IOC pigtails through ~8° fusion ˝ stability in the PBFOG when the PBF coil is connected to the IOC pigtails through ~8 fusion splicing. splicing. 4. Conclusions 4. Conclusions In summary, we found and investigated a bias error (namely, the BSC error) in the PBFOG that is In we found andback-reflection-induced investigated a bias errorand (namely, the BSC error) in the PBFOG that caused summary, by interference between backscatter-induced secondary waves. is caused by interference between back-reflection-induced and backscatter-induced secondary The BSC error was theoretically analyzed and experimentally verified. Our experimental results waves. The BSC error was theoretically and aexperimentally verified. Our the experimental ˝ /h showed a BSC error of ~4.7 for a 300-m analyzed PBF coil with diameter of 10 cm. Therefore, BSC error results show ed a BSC error of ~4.7°/h for a 300-m PBF coil with a diameter of 10 cm. Therefore, the is an important error source in the PBFOG, and it seriously affects the long-term stability of PBFOG. BSC error is an important error source in the PBFOG, and it seriously affects the long-term stability This error has to be addressed before the practical application of the PBFOG is considered, and our of PBFOG. Thisfocus error to be addressed before the practical application of the PBFOGthe is future work will on has further studying possible measures to suppress the BSC error to improve considered, ourPBFOG. future work will focus on further studying possible measures to suppress the bias stabilityand of the BSC error to improve the bias stability of the PBFOG. Acknowledgments: This work was supported by National Natural Science Foundation of China (NSFC) under Grants 61575012 andThis 61575013. Ackno wledgments: work was supporte d by National Natural Scie nce Foundation of China (NSFC) unde r Grants 61575012 and 61575013. Author Contributions: Xiaobin Xu conceived and theoretically analyzed the SIM error and wrote the paper; Ningfang Song and Xiaobin Xu promoted the experimental methods and setup; Zuchen Zhang performed the Author Contributions: Xiaobin Xu conce ive d and theore tically analyze d the SIM e rror and wrote the pape r; experiments; Jing Jin involved in the analysis of the data. Ningfang Song and Xiaobin Xu promote d the e xpe rime ntal me thods and se tup; Zuchen Zhang pe rforme d the Conflicts The authors declare no conflict of interest. e xpe rimeof nts;Interest: Jing Jin involve d in the analysis of the data. Conflicts of Interest: The authors de clare no conflict of inte re st. References 1. Lefevre, H.C. The fiber-optic gyroscope, a century after Sagnac’s experiment: The ultimate rotation-sensing References technology? C. R. Phys. 2014, 15, 851–858. [CrossRef] 1. Le fe vreC.M.; , H.C. 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