Low-delay, high-bandwidth frequency-locking loop of resonator integrated optic gyro with triangular phase modulation Yinzhou Zhi,1,2,* Lishuang Feng,1,2 Ming Lei,1,2 and Kunbo Wang1,2 1
Science and Technology on Inertial Laboratory, Beihang University, Beijing 100191, China 2
Precision Opto-mechatronics Technology Key Laboratory of Education Ministry, Beihang University, Beijing 100191, China *Corresponding author: [email protected]
Received 8 August 2013; revised 14 October 2013; accepted 14 October 2013; posted 15 October 2013 (Doc. ID 195430); published 14 November 2013
A frequency-locking loop affects the bandwidth and output of the resonator integrated optic gyro (RIOG). A low-delay, high-bandwidth frequency-locking loop is implemented on a single field-programmable gate array with triangular phase modulation. The signal processing delay is reduced to less than 1 μs. The loop model is set up, and the influences of loop parameters on the bandwidth and unit step response are analyzed; the bandwidth of 10 kHz is obtained with the optimized loop parameters. As a result, the accuracy of the frequency-locking loop is reduced to 1.37 Hz (10 s integrated time). It is equivalent to a rotation rate of 0.005 deg ∕s, which is close to the ultimate sensitivity of the RIOG. Moreover, the bias stability of the RIOG is improved to 0.45 deg ∕s (10 s integrated time) based on the frequency-locking loop. © 2013 Optical Society of America OCIS codes: (130.6010) Sensors; (140.3410) Laser resonators; (140.3370) Laser gyroscopes. http://dx.doi.org/10.1364/AO.52.008024
1. Introduction
The optical passive resonator gyros are inertial rotation sensors based on the Sagnac effect [1]. They are developed after the ring laser gyro (RLG) and the interferometer fiber optic gyro (IFOG) [2]. The optical passive resonator gyros overcome the lockin phenomenon encountered in RLG [3] and have a better performance than IFOG with the same length of sensing coil [4]. The resonator integrated optic gyro (RIOG) and the resonator fiber optic gyro (RFOG) are two different types of optical passive resonator gyros. Compared with RFOG, RIOG employs the waveguide-type ring resonator to substitute for the fiber ring resonator, which is beneficial to the miniaturization of gyro system [5–8]. Moreover, 1559-128X/13/338024-08$15.00/0 © 2013 Optical Society of America 8024
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with the development of the hybrid optoelectronic integrated technology based on silicon, it is possible to realize the RIOG on a single silicon chip [9]. Therefore, RIOG has the potential advantages of high performance, small size, and low cost. It has been widely investigated in recent decades [10–13]. Although the short-term bias stability of the RIOG has been reached, 1.85 × 10−4 rad∕s (0.0106 deg ∕s) for 60 seconds [14], the long-term performance is affected by backscattering noise [15–17], polarization noise [18,19], accuracy of frequency-locking loop [20–22], and so on. Much work is required to improve the performance of the RIOG. To achieve high finesse and high coherent length, a high stability and narrow linewidth laser is employed in RIOG [23,24]. The frequency stability of laser is sensitive to temperature, vibration, and magnetic field [25]. Many methods are proposed to improve the frequency stability of laser, such as
Lamb’s dip [26], Zeeman effect [27], saturated absorption spectroscopy [28], optical cavity [29], and so on. Although the frequencies of lasers are locked to the frequency reference or controlled in a certain range by these methods, they cannot offer real-time frequency regulation with the rotation. In a RIOG system, in order to detect the resonance frequency difference induced by the Sagnac effect, the frequency of the laser is usually locked to the resonance frequency of the clockwise (CW) or counter-clockwise (CCW) lightwave propagating in the resonator [30,31], which varies with the rotation and is also perturbed by environment. Moreover, the dynamic performance of the RIOG is dependent on the bandwidth of the frequency-locking loop [21]. Consequently, a high precision and wide bandwidth frequency-locking loop is indispensable for RIOG. A proportional integrator (PI) has been adopted in the frequency-locking loop based on the commercial data acquisition board (National Instrument DAQ module) and LabVIEW software. Although the loop delay was 682.16 μs, the equivalent bandwidth was 0.16 Hz resulted by 1 s time constant of the low pass filter (LPF), and the frequency-locking process of 0.0052 s was obtained [20]. The bandwidth of frequency-locking loop was limited by the large loop delay contributed by the PI scheme on NI LabVIEW [14,20]. A digital signal processor, as well as a PI module, were all constructed on a single fieldprogrammable gate array (FPGA), and the signal processing delay was reduced to 1.1 μs. However, the bandwidth of the digital lock-in amplifier (LIA) in the frequency-locking loop was 4 Hz [22], which was too narrow to satisfy the high dynamic performance of the RIOG. In this paper, a low-delay, high-bandwidth frequency-locking loop is implemented on a single FPGA with triangular phase modulation. First, the principle of the RIOG based on triangular phase modulation is described, and the gyro output affected by frequency-locking error is analyzed. Second, the signal processing based on FPGA is designed. The signal processing delay is reduced to less than 1 μs to take advantage of accurate timing control and parallel processing capability of FPGA. Third, the frequency-locking model is set up, and the influences of loop parameters on the bandwidth and unit step response are analyzed. The optimized parameters are obtained considering the bandwidth and unit step response comprehensively. The bandwidth of the frequency-locking loop is improved to 10 kHz, which is the best result to our knowledge. Finally, the experimental setup is established, and the tested overshoot and the rise time of the frequencylocking process are 8.5% and 35 μs, respectively, which are consistent with the simulated results. The frequency-locking error is also tested. The accuracy of DFB-FL frequency locked at the CCW lightwave resonance frequency is reduced to 1.37 Hz (10 s integrated time); it is equivalent to a rotation rate of 0.005 deg ∕s, which is close to the ultimate
sensitivity of the RIOG. Moreover, as a main performance of the RIOG, the bias stability of the RIOG is improved to 0.45 deg ∕s (10 s integrated time) based on the frequency-locking loop. 2. System Design
The schematic illustration of the RIOG based on triangular phase modulation is shown in Fig. 1. The lightwave from the distributed feedback fiber laser (DFB-FL) passes through the optical isolator (ISO) and is split into two beams by the Y-branch phase modulator, which is modulated by the antiphase triangular waves. The two beams passed through the couplers C1 and C2 are launched into integrated optical resonator (IOR) through coupler C3, acting as CW and CCW lightwaves. Finally, the CW and CCW lightwaves out of the IOR are detected by the photodetectors PD1 and PD2, respectively. In order to eliminate the backscattering and backreflection noises, the antiphase triangular waves generated by the signal generators SG1 and SG2 are adopted [31,32]. With the triangular phase modulation, the frequency difference between the DFB-FL frequency and the resonance frequency of the CW/CCW lightwave is transferred to the voltage amplitude detected by PD1/PD2 [16]. According to the synchronizing signal Sync2 and PD2, the demodulated voltage amplitude signal from the CCW lightwave is obtained by the frequency-locking loop and is used to lock the DFB-FL frequency to the resonance frequency of CCW lightwave. The demodulated voltage amplitude signal from the CW lightwave, which is obtained by signal detection module according to Sync1 and PD1, provides the gyro output. f
1 dϕt : 2π dt
(1)
Figure 2 gives the principle illustration of triangular phase modulation. When DFB-FL is driven by a linear voltage to realize optical frequency sweep, the resonance curve detected at PD2 is shown in Fig. 2(a). As the phase ϕt and frequency f have
Fig. 1. Schematic illustration of the RIOG based on triangular phase modulation. DFB-FL: distributed feedback fiber laser; ISO: isolator; IOR: integrated optical resonator; SG1, SG2: signal generators; C1, C2, C3: couplers; PD1, PD2: photodetectors. 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
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Fig. 2. Principle illustration of triangular phase modulation. (a) Resonance curve detected at PD2 induced by DFB-FL frequency sweep. (b) Triangular phase modulation corresponds to square wave frequency modulation. (c) Resonance curve and demodulation curve with triangular phase modulation. (d) Output of PD1 and PD2 when the gyro is at rotation state.
the relationship according to Eq. (1) [33], the square wave frequency modulation induced by the triangular phase modulation is shown in Fig. 2(b); the rise and fall half-period of the triangular wave correspond to modulation frequency f pm and −f pm , respectively. Figure 2(c) shows the resonance curve and demodulation curve with triangular phase modulation. The amplitude of square wave varying with frequency deviation is observed in the resonance curve. When the gyro is at rotation state, the resonance frequency of CW and CCW lightwave vary in the opposite direction, as shown in Fig. 2(d). When the frequency of DFB-FL is locked to the CCW light resonance frequency, the output of PD2 is a constant offset. However, the output of PD1 is a square wave whose amplitude is proportional to the rotation rate. As the demodulation curve in Fig. 2(c) shows, there is a range of linearity between frequency deviation and demodulation curve adjacent to the zero-offset point, which is the work region of the RIOG, and it limits the maximum sensing angular velocity. In this region, the relationship between frequency deviation Δf and amplitude of the square wave I out can be expressed as I out kΔf ;
(2)
where k is the slope of demodulation curve adjacent to the zero-offset point. In ideal conditions, the frequency of DFB-FL f laser is locked at the resonance frequency of CCW light f ccw . However, the frequency-locking loop is influenced by frequency noise of the DFB-FL, circuit noise, and other optical noises, which produce frequency-locking error f error. Consequently, the output of gyro I gyro can be written as 8026
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I gyro kf cw − f laser ; f laser f ccw f error
3
where f cw is the resonance frequency of CW lightwave. Moreover, the output of the gyro can be driven as I gyro I out − kf error :
(4)
According to Eq. (4), the output of gyro is affected by the frequency-locking error f error, which deteriorates the performance of the gyro. Therefore, the performance of the gyro can be improved by reducing the frequency-locking error. The delay time of the frequency-locking loop, which is one of the significant parameters affecting the frequency-locking error [14,20,21], is composed of optical signal delay, circuit signal delay, and signal processing delay. In an RIOG system, the optical signal delay and the circuit signal delay are at the same order of magnitude 10−9 s, which can be neglected compared with the signal processing delay. Figure 3 gives the signal processing scheme of frequencylocking loop based on FPGA. As can be seen from Fig. 3(a), the sample count, which is cycled from 1 to f s ∕f pm , is created by the sample clock f s of FPGA according to Sync2. The sample control module controls signal processing at different value of the sample count. First, the signal processor samples and accumulates in the positive and negative halfperiod of the square wave, respectively, and then the amplitude of square wave is obtained by subtraction. The demodulated amplitude of square wave is passed through the sequence of LPF and PI module, and finally exported to control the DFBFL. Figure 3(b) shows the corresponding relation
is transformed to the control voltage, which is assigned to the DFB-FL to tune its frequency to the resonance frequency of CCW lightwave. According to Fig. 4, the s-domain model of frequency-locking loop can be derived as Fig. 5, where the total gain of loop is K, τl is the time constant of LPF, T i is the integration time of the PI, and τd is the loop delay time. Therefore, the open-loop transfer function of the frequency-locking loop can be expressed as 1 1 1 e−τd s : Hs K 1 τl s Tis Fig. 3. Schematic illustration of the signal processing scheme. (a) Signal processing scheme of the frequency-locking loop. (b) Signal processing procedure corresponds to the sample count.
between sample count and signal processing procedure. The signal processing delay can be limited in a modulation period by this method, which is contributed to by the accuracy timing control and parallel processing capability of FPGA. When the frequency of modulation triangular wave is 1 MHz, the signal processing delay can be reduced to less than 1 μs. Therefore, the delay time of the frequencylocking loop is reduced as low as possible, which is beneficial for improving the performance of frequency-locking loop. 3. Simulation and Analysis A.
Model Building
The model of the RIOG frequency-locking loop is shown in Fig. 4. It is composed of a frequency sensor, a frequency detector, and a frequency feedback controller. The frequency sensor is IOR, which senses the frequency deviation between frequency of the DFB-FL and resonance frequency of the CCW lightwave. The frequency detector, which is combined by PD, preamplifier (PAM), band pass filter (BPF), and phase-sensitive detector (PSD), converts the frequency deviation to error signal. The frequency feedback controller includes LPF, PI, amplifier (AM), and DFB-FL. Through LPF, PI, and AM, the error signal
(5)
The closed-loop transfer function and error transfer function of the frequency-locking loop can be derived as Eqs. (6) and (7), respectively: Hs 1 Hs K1 T i se−τd s 1 τl sT i s K1 T i se−τd s
Φs
Φe s
(6)
1 1 Hs 1 τl sT i s : 1 τl sT i s K1 T i se−τd s
(7)
According to the final-value theorem [34], when unit step input signal is assigned to the input, the steady state error of the output can be expressed as ess lim sΦe s s→0
lim s→0
1 s
1 τl sT i s 0: 1 τl sT i s AF1 T i se−τd s
(8)
Equation (8) shows the steady state error is zero. Consequently, the frequency of DFB-FL can be precisely locked at the resonance frequency of CCW lightwave, which guarantees the long-term stability of the frequency-locking loop. B. Bandwidth Analysis
Bandwidth is a significant parameter to evaluate the performance of frequency-locking loop. A wide loop bandwidth can not only contribute to quick response
Fig. 4. Schematic model of the frequency-locking loop.
Fig. 5. S-domain model of the frequency-locking loop. 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
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Fig. 6. Bandwidth influenced by the loop parameters. (a)–(d) are amplitude-frequency responses of the closed-loop transfer function when K, T i , τl , and τd take different values, respectively.
but also reduce the time of eliminating steady state error. Moreover, a wide bandwidth of frequency-locking loop is beneficial to improve the dynamic performance of the RIOG [33]. The bandwidth of the frequency-locking loop is determined by the parameters K, T i , τl , and τd . Figure 6 gives the relationships between the bandwidth and these four parameters. As can be seen from Fig. 6(a), the bandwidth becomes wider along with a larger K value, but when it is larger than 0.2,
the overshoot is observed on amplitude-frequency responses of closed-loop transfer function, which indicates the loop is not stable and begins to oscillate. In contrast, the effect of T i is shown in Fig. 6(b), and its contribution is greater than K. Especially when T i is smaller than 2 × 10−4 s, the bandwidth of loop is deteriorated sharply. A smaller τl will not only narrow the bandwidth but also induces the overshoot, which will lead to the oscillation of the loop, as can be seen from Fig. 6(c). The bandwidth varies
Fig. 7. Unit step response influenced by the loop parameters. (a)–(d) are unit step responses of the closed-loop transfer function when K, T i , τl , and τd take different values, respectively. 8028
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Fig. 8. Amplitude-frequency response and unit step response with the optimized parameters. (a) Amplitude-frequency response of the closed-loop transfer function. (b) Unit step response of the closed-loop transfer function.
slightly with different values of τd. However, τd mainly affects the stability of loop; as shown in Fig. 6(d), a larger τd will produce a much serious overshoot. Therefore, a smaller τd is needed for the stability of loop. C.
Unit Step Response Analysis
Unit step response, which reflects the dynamic property of closed-loop system, is the output response when input is unit step signal. Overshoot and rise time are two significant indexes of unit step response, where overshoot is used to evaluate the system damping ratio and rise time is used to estimate the system response speed, respectively. In order to obtain the ideal performance of loop, the overshoot δ% is usually selected in the range of 5%–15%, and the shorter rise time tr is considered [34]. Figure 7 gives unit step response of the closed-loop transfer functions when the parameters K, T i , τl and τd take different values. As can be seen from Fig. 7(a), when K varies from 0.04 to 5, the rise time is kept shorter, and the loop is transformed from underdamping state to over-damping state. However, the overshoot becomes bigger, which influences the stability of loop. Figure 7(b) shows the unit step response when T i varies from 5 × 10−4 to 4 × 10−6 , the loop experiences the process from under-damping state to over-damping state as well, but the rise time is dramatically reduced; these results are consistent with its influence on bandwidth, as shown in Fig. 6(b). The influences of τl and τd on unit step response are similar, as shown in Fig. 7(c) and 7(d). The overshoot and rise time are both reduced along with the two parameters decrease. As a result, a smaller τl and τd are theoretically beneficial to improve the performance of loop. But a smaller τl will bring about more noises [21], the value selected should be comprehensively considered. Based on the above analysis, the optimized parameters are obtained when the bandwidth and unit step response of the closed-loop transfer function are comprehensively considered. The parameters K, T i , τl and τd are set to 0.12, 2.5 × 10−6 , 1.6 × 10−5 , and 1 × 10−6 , respectively. The amplitude-frequency response and unit step response of the closed-loop transfer function are shown in Fig. 8. It can be seen
that the 3 dB bandwidth is about 10 kHz, the overshoot δ% is 8.8%, and the rise time tr is 34 μs. 4. Experiments and Discussion
The experimental setup of RIOG is established according to Fig. 1. The IOR is fabricated by siliconbased silica waveguide, and the refractive index n; the length L, the area A, the finesse F, the full width at half-maximum, and the Q-factor of the IOR are 1.46, 12.8 cm, 11.34 cm2 , 59, 27.5 MHz, and 7.1 × 106 , respectively. The center wavelength of DFB-FL is 1550 nm, and the responsivities of the PD1 and PD2 are both 0.92 A∕W. When the input intensity of IOR is 73.4 μW, the shot noise equivalent rotation of the RIOG is 15.5 deg ∕h (0.0043 deg ∕s) for an integration time of 10 s [35]. The frequency of modulation triangular wave is 1 MHz. The sample clock f s of FPGA is 100 MHz. According to the signal processing scheme analyzed above, the signal processing time of frequency-locking loop is reduced to less than 1 μs. Because the control voltage of DFB-FL is proportional to the input frequency of IOR, it can be used to test the unit step response of frequency-locking loop. The optimized parameters of the frequencylocking loop are applied in the RIOG system, and the tested frequency-locking process is shown in Fig. 9. As can be seen from Fig. 9, the overshoot δ% and the rise time tr of laser control voltage are 8.5% and 35 μs, respectively, which is almost consistent with the simulated results as shown in Fig. 8.
Fig. 9. Frequency-locking process of the RIOG. 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
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Fig. 10. Experiment results of the frequency-locking error and gyro output. (a) Equivalent frequency-locking error. (b) Output of the RIOG.
After the frequency-locking, the output of PD2 is a constant voltage, which demonstrates that the frequency of DFB-FL is locked to the resonance frequency of CCW lightwave. According to Fig. 4, the frequency-locking error can be detected at PSD output of the frequency-locking loop. The tested results of frequency-locking error are shown in Fig. 10(a). The accuracy of DFB-FL frequency locked at the CCW lightwave resonance frequency is reduced to 1.37 Hz (10 s integrated time). It is equivalent to a rotation rate of 0.005 deg ∕s, which is close to the ultimate sensitivity of the RIOG. Moreover, the output of RIOG is tested based on the optimized frequency-locking loop. Figure 10(b) shows that with the output of the RIOG for an hour, a zero bias stability of 0.45 deg ∕s (10 s integrated time) is obtained. Obviously, the zero bias stability is far from the accuracy of frequency-locking loop, which is mainly induced by the nonreciprocal noises existing in the RIOG system. Backscattering, backreflection, polarization, Kerr effect, and temperature drift are still to be suppressed to improve the performance of RIOG. However, the noise caused by frequencylocking error has been reduced close to the ultimate sensitivity, and it can be neglected compared with the nonreciprocal noises at present. 5. Conclusion
A low-delay, high-bandwidth frequency-locking loop is implemented on FPGA with triangular phase modulation. The signal processing delay is reduced to less than 1 μs to take advantage of the accurate timing control and parallel processing capability of FPGA. The frequency-locking model is set up, and the bandwidth of the loop is 10 kHz based on the optimized parameters. The experimental setup is established, and the simulated results are demonstrated. The experiments show that the accuracy of frequency-locking loop is reduced to 1.37 Hz (10 s integrated time); it is equivalent to a rotation rate of 0.005 deg ∕s, which is close to the ultimate sensitivity of the RIOG. Moreover, the bias stability of the RIOG is improved to 0.45 deg ∕s (10 s integrated time) based on the frequency-locking loop. Although the frequency-locking error has been reduced close to the ultimate sensitivity, the nonreciprocal noises, 8030
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such as backscattering, backreflection, polarization, Kerr effect, and temperature drift, still greatly affect the zero bias stability. The countermeasures suppressing the nonreciprocal noises are still required to improve the performance of RIOG. The authors would like to acknowledge the financial support from National Natural Science Foundation of China (No. 61171004) and Institute of opto-electronic Technique in Beihang University. References 1. H. J. Arditty and H. C. Lefovre, “Sagnac effect in fiber gyroscopes,” Opt. Lett. 6, 401–403 (1981). 2. S. Ezekiel and S. R. Balsamo, “Passive ring resonator gyroscope,” Appl. Phys. Lett. 30, 478–480 (1977). 3. D. M. Shupe, “Fiber resonator gyroscope: sensitivity and thermal nonreciprocity,” Appl. Opt. 20, 286–289 (1981). 4. L. Hong, C. Zhang, and L. Feng, “Effect of phase modulation nonlinearity in resonator micro-optic gyro,” Opt. Eng. 50, 094404 (2011). 5. S. Donati, Electro-optical Instrumentation Sensing and Measuring with Lasers (Prentice-Hall, 2004), pp. 187–233. 6. C. Ciminelli, F. Dell’Olio, C. E. Campanella, and M. N. Armenise, “Numerical and experimental investigation of an optical high-Q spiral resonator gyroscope,” in ICTON (IEEE Photonics Society, 2012), paper Th.A4.5. 7. C. Ciminelli, F. Dell’Olio, M. N. Armenise, F. M. Soares, and W. Passenberg, “High performance InP ring resonator for new generation monolithically integrated optical gyroscopes,” Opt. Express 21, 556–564 (2013). 8. F. Dell’Olio, C. Ciminelli, and M. N. Armenise, “Theoretical investigation of InP buried ring resonators for new angular velocity sensors,” Opt. Eng. 52, 024601 (2013). 9. N. M. Barbour, “Inertial navigation sensors,” RTO-EN-SET116 (2011). 10. K. Iwatsuki, M. Saruwatari, M. Kawachi, and H. Yamazaki, “Waveguide-type optical passive ring-resonator gyro using time division detection scheme,” Electron. Lett. 25, 688–689 (1989). 11. C. Vannahme, H. Suche, S. Reza, R. Ricken, V. Quiring, and W. Sohler, “Integrated optical Ti:LiNbO3 ring resonator for zero bias stability,” ECIO, Copenhagen, Denmark (2007). 12. C. Ciminelli, F. Dell’Olio, and M. N. Armenise, “High-Q spiral resonator for optical gyroscope applications: numerical and experimental investigation,” IEEE Photon. J. 4, 1844–1854 (2012). 13. C. Ciminelli, C. E. Campanella, F. Dell’Olio, C. Campanella, and M. N. Armenise, “Theoretical investigation on the scale factor of a triple ring cavity to be used in frequency sensitive resonant gyroscopes,” J. Eur. Opt. Soc. 8, 13050 (2013). 14. H. Mao, H. Ma, and Z. Jin, “Polarization maintaining silica waveguide resonator optic gyro using double phase modulation technique,” Opt. Express 19, 4632–4643 (2011).
15. K. Iwatsuki, K. Hotate, and M. Higashiguchi, “Effect of Rayleigh backscattering in an optical passive ring-resonator gyro,” Appl. Opt. 23, 3916–3924 (1984). 16. Z. Jin, Z. Yang, H. Ma, and D. Ying, “Open-loop experiments in a resonator fiber-optic gyro using digital triangle wave phase modulation,” IEEE Photon. Technol. Lett. 19, 1685–1687 (2007). 17. H. Ma, Z. He, and K. Hotate, “Reduction of backscattering induced noise by carrier suppression in waveguide-type optical ring resonator gyro,” J. Lightwave Technol. 29, 85–90 (2011). 18. K. Takiguchi and K. Hotate, “Evaluation of the output error in an optical passive ring-resonator gyro with a 90 polarizationaxis rotation in the polarization-maintaining fiber resonator,” IEEE Photon. Technol. Lett. 3, 88–90 (1991). 19. X. Yu, H. Ma, and Z. Jin, “Improving thermal stability of a resonator fiber optic gyro employing a polarizing resonator,” Opt. Express 21, 358–369 (2013). 20. Y. Ren, Z. Jin, Y. Chen, and H. Ma, “Optimization of the resonance frequency servo loop technique in the resonator micro optic gyro,” J. Zhejiang Univ. Sci. C. 12, 942–950 (2011). 21. H. Ma, X. Lu, L. Yao, X. Yu, and Z. Jin, “Full investigation of the resonant frequency servo loop for resonator fiber-optic gyro,” Appl. Opt. 51, 5178–5185 (2012). 22. H. Ma, W. Wang, Y. Ren, and Z. Jin, “Low-noise low-delay digital signal processor for resonant micro optic gyro,” IEEE Photon. Tech. L. 25, 198–201 (2013). 23. K. Suzuki, K. Takiguchi, and K. Hotate, “Monolithically integrated resonator micro-optic gyro on silica planar lightwave circuit,” J. Lightwave Technol. 18, 66–72 (2000). 24. F. Zarinetchi and S. Ezekiel, “Observation of lock-in behavior in a passive resonator gyroscope,” Opt. Lett. 11, 401–403 (1986).
25. G. Galzerano and P. Laporta, “Single-frequency diode-pumped Yb:KYF4 laser around 1030 nm,” Opt. Express 15, 3257–3264 (2007). 26. E. I. Moses and C. L. Tang, “High-sensitivity laser wavelength-regulation spectroscopy,” Opt. Lett. 1, 115–117 (1977). 27. G. A. Massey, M. K. Oshman, and R. Targ, “Generation of single-frequency light using the FM laser,” Appl. Phys. Lett. 6, 10–11 (1965). 28. Y. Ohtaa, S. Maehar, K. Hasebe, Y. Kurosaki, and M. Ohkawa, “Frequency stabilization of a semiconductor laser using the Rb saturated absorption spectroscopy,” Proc. SPIE 6115, 1–10 (2006). 29. F. Rogister, D. W. Sukow, A. Gavrielides, P. Mégret, O. Deparis, and M. Blondel, “Experimental demonstration of suppression of low-frequency fluctuations and stabilization of an externalcavity laser diode,” Opt. Lett. 25, 808–810 (2000). 30. L. K. Strandiord and G. A. Sanders, “Resonator optic gyro employing a polarization rotating resonator,” Proc. SPIE 1585, 163–172 (1991). 31. K. Iwatsuki, K. Hotate, and M. Higashiguchi, “Backscattering in an optical passive ring-resonator gyro: experiment,” Appl. Opt. 25, 4448–4451 (1986). 32. L. Feng, M. Lei, H. Liu, Y. Zhi, and J. Wang, “Suppression of backreflection noise in a resonator integrated optic gyro by hybrid phase-modulation technology,” Appl. Opt. 52, 1668–1675 (2013). 33. C. H. Lefevre, The Fiber-Optic Gyroscope (Artech, 1993), pp. 157–167. 34. K. Ogata, Modern Control Engineering, 5th ed. (Prentice-Hall, 2010), pp. 320–485. 35. M. Lei, L. Feng, Y. Zhi, and H. Liu, “Effect of intensity variation of laser in resonator, integrated optic gyro,” Appl. Opt. 52, 1–7 (2013).
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Science and Technology on Inertial Laboratory, Beihang University, Beijing 100191, China 2
Precision Opto-mechatronics Technology Key Laboratory of Education Ministry, Beihang University, Beijing 100191, China *Corresponding author: [email protected]
Received 8 August 2013; revised 14 October 2013; accepted 14 October 2013; posted 15 October 2013 (Doc. ID 195430); published 14 November 2013
A frequency-locking loop affects the bandwidth and output of the resonator integrated optic gyro (RIOG). A low-delay, high-bandwidth frequency-locking loop is implemented on a single field-programmable gate array with triangular phase modulation. The signal processing delay is reduced to less than 1 μs. The loop model is set up, and the influences of loop parameters on the bandwidth and unit step response are analyzed; the bandwidth of 10 kHz is obtained with the optimized loop parameters. As a result, the accuracy of the frequency-locking loop is reduced to 1.37 Hz (10 s integrated time). It is equivalent to a rotation rate of 0.005 deg ∕s, which is close to the ultimate sensitivity of the RIOG. Moreover, the bias stability of the RIOG is improved to 0.45 deg ∕s (10 s integrated time) based on the frequency-locking loop. © 2013 Optical Society of America OCIS codes: (130.6010) Sensors; (140.3410) Laser resonators; (140.3370) Laser gyroscopes. http://dx.doi.org/10.1364/AO.52.008024
1. Introduction
The optical passive resonator gyros are inertial rotation sensors based on the Sagnac effect [1]. They are developed after the ring laser gyro (RLG) and the interferometer fiber optic gyro (IFOG) [2]. The optical passive resonator gyros overcome the lockin phenomenon encountered in RLG [3] and have a better performance than IFOG with the same length of sensing coil [4]. The resonator integrated optic gyro (RIOG) and the resonator fiber optic gyro (RFOG) are two different types of optical passive resonator gyros. Compared with RFOG, RIOG employs the waveguide-type ring resonator to substitute for the fiber ring resonator, which is beneficial to the miniaturization of gyro system [5–8]. Moreover, 1559-128X/13/338024-08$15.00/0 © 2013 Optical Society of America 8024
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with the development of the hybrid optoelectronic integrated technology based on silicon, it is possible to realize the RIOG on a single silicon chip [9]. Therefore, RIOG has the potential advantages of high performance, small size, and low cost. It has been widely investigated in recent decades [10–13]. Although the short-term bias stability of the RIOG has been reached, 1.85 × 10−4 rad∕s (0.0106 deg ∕s) for 60 seconds [14], the long-term performance is affected by backscattering noise [15–17], polarization noise [18,19], accuracy of frequency-locking loop [20–22], and so on. Much work is required to improve the performance of the RIOG. To achieve high finesse and high coherent length, a high stability and narrow linewidth laser is employed in RIOG [23,24]. The frequency stability of laser is sensitive to temperature, vibration, and magnetic field [25]. Many methods are proposed to improve the frequency stability of laser, such as
Lamb’s dip [26], Zeeman effect [27], saturated absorption spectroscopy [28], optical cavity [29], and so on. Although the frequencies of lasers are locked to the frequency reference or controlled in a certain range by these methods, they cannot offer real-time frequency regulation with the rotation. In a RIOG system, in order to detect the resonance frequency difference induced by the Sagnac effect, the frequency of the laser is usually locked to the resonance frequency of the clockwise (CW) or counter-clockwise (CCW) lightwave propagating in the resonator [30,31], which varies with the rotation and is also perturbed by environment. Moreover, the dynamic performance of the RIOG is dependent on the bandwidth of the frequency-locking loop [21]. Consequently, a high precision and wide bandwidth frequency-locking loop is indispensable for RIOG. A proportional integrator (PI) has been adopted in the frequency-locking loop based on the commercial data acquisition board (National Instrument DAQ module) and LabVIEW software. Although the loop delay was 682.16 μs, the equivalent bandwidth was 0.16 Hz resulted by 1 s time constant of the low pass filter (LPF), and the frequency-locking process of 0.0052 s was obtained [20]. The bandwidth of frequency-locking loop was limited by the large loop delay contributed by the PI scheme on NI LabVIEW [14,20]. A digital signal processor, as well as a PI module, were all constructed on a single fieldprogrammable gate array (FPGA), and the signal processing delay was reduced to 1.1 μs. However, the bandwidth of the digital lock-in amplifier (LIA) in the frequency-locking loop was 4 Hz [22], which was too narrow to satisfy the high dynamic performance of the RIOG. In this paper, a low-delay, high-bandwidth frequency-locking loop is implemented on a single FPGA with triangular phase modulation. First, the principle of the RIOG based on triangular phase modulation is described, and the gyro output affected by frequency-locking error is analyzed. Second, the signal processing based on FPGA is designed. The signal processing delay is reduced to less than 1 μs to take advantage of accurate timing control and parallel processing capability of FPGA. Third, the frequency-locking model is set up, and the influences of loop parameters on the bandwidth and unit step response are analyzed. The optimized parameters are obtained considering the bandwidth and unit step response comprehensively. The bandwidth of the frequency-locking loop is improved to 10 kHz, which is the best result to our knowledge. Finally, the experimental setup is established, and the tested overshoot and the rise time of the frequencylocking process are 8.5% and 35 μs, respectively, which are consistent with the simulated results. The frequency-locking error is also tested. The accuracy of DFB-FL frequency locked at the CCW lightwave resonance frequency is reduced to 1.37 Hz (10 s integrated time); it is equivalent to a rotation rate of 0.005 deg ∕s, which is close to the ultimate
sensitivity of the RIOG. Moreover, as a main performance of the RIOG, the bias stability of the RIOG is improved to 0.45 deg ∕s (10 s integrated time) based on the frequency-locking loop. 2. System Design
The schematic illustration of the RIOG based on triangular phase modulation is shown in Fig. 1. The lightwave from the distributed feedback fiber laser (DFB-FL) passes through the optical isolator (ISO) and is split into two beams by the Y-branch phase modulator, which is modulated by the antiphase triangular waves. The two beams passed through the couplers C1 and C2 are launched into integrated optical resonator (IOR) through coupler C3, acting as CW and CCW lightwaves. Finally, the CW and CCW lightwaves out of the IOR are detected by the photodetectors PD1 and PD2, respectively. In order to eliminate the backscattering and backreflection noises, the antiphase triangular waves generated by the signal generators SG1 and SG2 are adopted [31,32]. With the triangular phase modulation, the frequency difference between the DFB-FL frequency and the resonance frequency of the CW/CCW lightwave is transferred to the voltage amplitude detected by PD1/PD2 [16]. According to the synchronizing signal Sync2 and PD2, the demodulated voltage amplitude signal from the CCW lightwave is obtained by the frequency-locking loop and is used to lock the DFB-FL frequency to the resonance frequency of CCW lightwave. The demodulated voltage amplitude signal from the CW lightwave, which is obtained by signal detection module according to Sync1 and PD1, provides the gyro output. f
1 dϕt : 2π dt
(1)
Figure 2 gives the principle illustration of triangular phase modulation. When DFB-FL is driven by a linear voltage to realize optical frequency sweep, the resonance curve detected at PD2 is shown in Fig. 2(a). As the phase ϕt and frequency f have
Fig. 1. Schematic illustration of the RIOG based on triangular phase modulation. DFB-FL: distributed feedback fiber laser; ISO: isolator; IOR: integrated optical resonator; SG1, SG2: signal generators; C1, C2, C3: couplers; PD1, PD2: photodetectors. 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
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Fig. 2. Principle illustration of triangular phase modulation. (a) Resonance curve detected at PD2 induced by DFB-FL frequency sweep. (b) Triangular phase modulation corresponds to square wave frequency modulation. (c) Resonance curve and demodulation curve with triangular phase modulation. (d) Output of PD1 and PD2 when the gyro is at rotation state.
the relationship according to Eq. (1) [33], the square wave frequency modulation induced by the triangular phase modulation is shown in Fig. 2(b); the rise and fall half-period of the triangular wave correspond to modulation frequency f pm and −f pm , respectively. Figure 2(c) shows the resonance curve and demodulation curve with triangular phase modulation. The amplitude of square wave varying with frequency deviation is observed in the resonance curve. When the gyro is at rotation state, the resonance frequency of CW and CCW lightwave vary in the opposite direction, as shown in Fig. 2(d). When the frequency of DFB-FL is locked to the CCW light resonance frequency, the output of PD2 is a constant offset. However, the output of PD1 is a square wave whose amplitude is proportional to the rotation rate. As the demodulation curve in Fig. 2(c) shows, there is a range of linearity between frequency deviation and demodulation curve adjacent to the zero-offset point, which is the work region of the RIOG, and it limits the maximum sensing angular velocity. In this region, the relationship between frequency deviation Δf and amplitude of the square wave I out can be expressed as I out kΔf ;
(2)
where k is the slope of demodulation curve adjacent to the zero-offset point. In ideal conditions, the frequency of DFB-FL f laser is locked at the resonance frequency of CCW light f ccw . However, the frequency-locking loop is influenced by frequency noise of the DFB-FL, circuit noise, and other optical noises, which produce frequency-locking error f error. Consequently, the output of gyro I gyro can be written as 8026
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I gyro kf cw − f laser ; f laser f ccw f error
3
where f cw is the resonance frequency of CW lightwave. Moreover, the output of the gyro can be driven as I gyro I out − kf error :
(4)
According to Eq. (4), the output of gyro is affected by the frequency-locking error f error, which deteriorates the performance of the gyro. Therefore, the performance of the gyro can be improved by reducing the frequency-locking error. The delay time of the frequency-locking loop, which is one of the significant parameters affecting the frequency-locking error [14,20,21], is composed of optical signal delay, circuit signal delay, and signal processing delay. In an RIOG system, the optical signal delay and the circuit signal delay are at the same order of magnitude 10−9 s, which can be neglected compared with the signal processing delay. Figure 3 gives the signal processing scheme of frequencylocking loop based on FPGA. As can be seen from Fig. 3(a), the sample count, which is cycled from 1 to f s ∕f pm , is created by the sample clock f s of FPGA according to Sync2. The sample control module controls signal processing at different value of the sample count. First, the signal processor samples and accumulates in the positive and negative halfperiod of the square wave, respectively, and then the amplitude of square wave is obtained by subtraction. The demodulated amplitude of square wave is passed through the sequence of LPF and PI module, and finally exported to control the DFBFL. Figure 3(b) shows the corresponding relation
is transformed to the control voltage, which is assigned to the DFB-FL to tune its frequency to the resonance frequency of CCW lightwave. According to Fig. 4, the s-domain model of frequency-locking loop can be derived as Fig. 5, where the total gain of loop is K, τl is the time constant of LPF, T i is the integration time of the PI, and τd is the loop delay time. Therefore, the open-loop transfer function of the frequency-locking loop can be expressed as 1 1 1 e−τd s : Hs K 1 τl s Tis Fig. 3. Schematic illustration of the signal processing scheme. (a) Signal processing scheme of the frequency-locking loop. (b) Signal processing procedure corresponds to the sample count.
between sample count and signal processing procedure. The signal processing delay can be limited in a modulation period by this method, which is contributed to by the accuracy timing control and parallel processing capability of FPGA. When the frequency of modulation triangular wave is 1 MHz, the signal processing delay can be reduced to less than 1 μs. Therefore, the delay time of the frequencylocking loop is reduced as low as possible, which is beneficial for improving the performance of frequency-locking loop. 3. Simulation and Analysis A.
Model Building
The model of the RIOG frequency-locking loop is shown in Fig. 4. It is composed of a frequency sensor, a frequency detector, and a frequency feedback controller. The frequency sensor is IOR, which senses the frequency deviation between frequency of the DFB-FL and resonance frequency of the CCW lightwave. The frequency detector, which is combined by PD, preamplifier (PAM), band pass filter (BPF), and phase-sensitive detector (PSD), converts the frequency deviation to error signal. The frequency feedback controller includes LPF, PI, amplifier (AM), and DFB-FL. Through LPF, PI, and AM, the error signal
(5)
The closed-loop transfer function and error transfer function of the frequency-locking loop can be derived as Eqs. (6) and (7), respectively: Hs 1 Hs K1 T i se−τd s 1 τl sT i s K1 T i se−τd s
Φs
Φe s
(6)
1 1 Hs 1 τl sT i s : 1 τl sT i s K1 T i se−τd s
(7)
According to the final-value theorem [34], when unit step input signal is assigned to the input, the steady state error of the output can be expressed as ess lim sΦe s s→0
lim s→0
1 s
1 τl sT i s 0: 1 τl sT i s AF1 T i se−τd s
(8)
Equation (8) shows the steady state error is zero. Consequently, the frequency of DFB-FL can be precisely locked at the resonance frequency of CCW lightwave, which guarantees the long-term stability of the frequency-locking loop. B. Bandwidth Analysis
Bandwidth is a significant parameter to evaluate the performance of frequency-locking loop. A wide loop bandwidth can not only contribute to quick response
Fig. 4. Schematic model of the frequency-locking loop.
Fig. 5. S-domain model of the frequency-locking loop. 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
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Fig. 6. Bandwidth influenced by the loop parameters. (a)–(d) are amplitude-frequency responses of the closed-loop transfer function when K, T i , τl , and τd take different values, respectively.
but also reduce the time of eliminating steady state error. Moreover, a wide bandwidth of frequency-locking loop is beneficial to improve the dynamic performance of the RIOG [33]. The bandwidth of the frequency-locking loop is determined by the parameters K, T i , τl , and τd . Figure 6 gives the relationships between the bandwidth and these four parameters. As can be seen from Fig. 6(a), the bandwidth becomes wider along with a larger K value, but when it is larger than 0.2,
the overshoot is observed on amplitude-frequency responses of closed-loop transfer function, which indicates the loop is not stable and begins to oscillate. In contrast, the effect of T i is shown in Fig. 6(b), and its contribution is greater than K. Especially when T i is smaller than 2 × 10−4 s, the bandwidth of loop is deteriorated sharply. A smaller τl will not only narrow the bandwidth but also induces the overshoot, which will lead to the oscillation of the loop, as can be seen from Fig. 6(c). The bandwidth varies
Fig. 7. Unit step response influenced by the loop parameters. (a)–(d) are unit step responses of the closed-loop transfer function when K, T i , τl , and τd take different values, respectively. 8028
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Fig. 8. Amplitude-frequency response and unit step response with the optimized parameters. (a) Amplitude-frequency response of the closed-loop transfer function. (b) Unit step response of the closed-loop transfer function.
slightly with different values of τd. However, τd mainly affects the stability of loop; as shown in Fig. 6(d), a larger τd will produce a much serious overshoot. Therefore, a smaller τd is needed for the stability of loop. C.
Unit Step Response Analysis
Unit step response, which reflects the dynamic property of closed-loop system, is the output response when input is unit step signal. Overshoot and rise time are two significant indexes of unit step response, where overshoot is used to evaluate the system damping ratio and rise time is used to estimate the system response speed, respectively. In order to obtain the ideal performance of loop, the overshoot δ% is usually selected in the range of 5%–15%, and the shorter rise time tr is considered [34]. Figure 7 gives unit step response of the closed-loop transfer functions when the parameters K, T i , τl and τd take different values. As can be seen from Fig. 7(a), when K varies from 0.04 to 5, the rise time is kept shorter, and the loop is transformed from underdamping state to over-damping state. However, the overshoot becomes bigger, which influences the stability of loop. Figure 7(b) shows the unit step response when T i varies from 5 × 10−4 to 4 × 10−6 , the loop experiences the process from under-damping state to over-damping state as well, but the rise time is dramatically reduced; these results are consistent with its influence on bandwidth, as shown in Fig. 6(b). The influences of τl and τd on unit step response are similar, as shown in Fig. 7(c) and 7(d). The overshoot and rise time are both reduced along with the two parameters decrease. As a result, a smaller τl and τd are theoretically beneficial to improve the performance of loop. But a smaller τl will bring about more noises [21], the value selected should be comprehensively considered. Based on the above analysis, the optimized parameters are obtained when the bandwidth and unit step response of the closed-loop transfer function are comprehensively considered. The parameters K, T i , τl and τd are set to 0.12, 2.5 × 10−6 , 1.6 × 10−5 , and 1 × 10−6 , respectively. The amplitude-frequency response and unit step response of the closed-loop transfer function are shown in Fig. 8. It can be seen
that the 3 dB bandwidth is about 10 kHz, the overshoot δ% is 8.8%, and the rise time tr is 34 μs. 4. Experiments and Discussion
The experimental setup of RIOG is established according to Fig. 1. The IOR is fabricated by siliconbased silica waveguide, and the refractive index n; the length L, the area A, the finesse F, the full width at half-maximum, and the Q-factor of the IOR are 1.46, 12.8 cm, 11.34 cm2 , 59, 27.5 MHz, and 7.1 × 106 , respectively. The center wavelength of DFB-FL is 1550 nm, and the responsivities of the PD1 and PD2 are both 0.92 A∕W. When the input intensity of IOR is 73.4 μW, the shot noise equivalent rotation of the RIOG is 15.5 deg ∕h (0.0043 deg ∕s) for an integration time of 10 s [35]. The frequency of modulation triangular wave is 1 MHz. The sample clock f s of FPGA is 100 MHz. According to the signal processing scheme analyzed above, the signal processing time of frequency-locking loop is reduced to less than 1 μs. Because the control voltage of DFB-FL is proportional to the input frequency of IOR, it can be used to test the unit step response of frequency-locking loop. The optimized parameters of the frequencylocking loop are applied in the RIOG system, and the tested frequency-locking process is shown in Fig. 9. As can be seen from Fig. 9, the overshoot δ% and the rise time tr of laser control voltage are 8.5% and 35 μs, respectively, which is almost consistent with the simulated results as shown in Fig. 8.
Fig. 9. Frequency-locking process of the RIOG. 20 November 2013 / Vol. 52, No. 33 / APPLIED OPTICS
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Fig. 10. Experiment results of the frequency-locking error and gyro output. (a) Equivalent frequency-locking error. (b) Output of the RIOG.
After the frequency-locking, the output of PD2 is a constant voltage, which demonstrates that the frequency of DFB-FL is locked to the resonance frequency of CCW lightwave. According to Fig. 4, the frequency-locking error can be detected at PSD output of the frequency-locking loop. The tested results of frequency-locking error are shown in Fig. 10(a). The accuracy of DFB-FL frequency locked at the CCW lightwave resonance frequency is reduced to 1.37 Hz (10 s integrated time). It is equivalent to a rotation rate of 0.005 deg ∕s, which is close to the ultimate sensitivity of the RIOG. Moreover, the output of RIOG is tested based on the optimized frequency-locking loop. Figure 10(b) shows that with the output of the RIOG for an hour, a zero bias stability of 0.45 deg ∕s (10 s integrated time) is obtained. Obviously, the zero bias stability is far from the accuracy of frequency-locking loop, which is mainly induced by the nonreciprocal noises existing in the RIOG system. Backscattering, backreflection, polarization, Kerr effect, and temperature drift are still to be suppressed to improve the performance of RIOG. However, the noise caused by frequencylocking error has been reduced close to the ultimate sensitivity, and it can be neglected compared with the nonreciprocal noises at present. 5. Conclusion
A low-delay, high-bandwidth frequency-locking loop is implemented on FPGA with triangular phase modulation. The signal processing delay is reduced to less than 1 μs to take advantage of the accurate timing control and parallel processing capability of FPGA. The frequency-locking model is set up, and the bandwidth of the loop is 10 kHz based on the optimized parameters. The experimental setup is established, and the simulated results are demonstrated. The experiments show that the accuracy of frequency-locking loop is reduced to 1.37 Hz (10 s integrated time); it is equivalent to a rotation rate of 0.005 deg ∕s, which is close to the ultimate sensitivity of the RIOG. Moreover, the bias stability of the RIOG is improved to 0.45 deg ∕s (10 s integrated time) based on the frequency-locking loop. Although the frequency-locking error has been reduced close to the ultimate sensitivity, the nonreciprocal noises, 8030
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such as backscattering, backreflection, polarization, Kerr effect, and temperature drift, still greatly affect the zero bias stability. The countermeasures suppressing the nonreciprocal noises are still required to improve the performance of RIOG. The authors would like to acknowledge the financial support from National Natural Science Foundation of China (No. 61171004) and Institute of opto-electronic Technique in Beihang University. References 1. H. J. Arditty and H. C. Lefovre, “Sagnac effect in fiber gyroscopes,” Opt. Lett. 6, 401–403 (1981). 2. S. Ezekiel and S. R. Balsamo, “Passive ring resonator gyroscope,” Appl. Phys. Lett. 30, 478–480 (1977). 3. D. M. Shupe, “Fiber resonator gyroscope: sensitivity and thermal nonreciprocity,” Appl. Opt. 20, 286–289 (1981). 4. L. Hong, C. Zhang, and L. Feng, “Effect of phase modulation nonlinearity in resonator micro-optic gyro,” Opt. Eng. 50, 094404 (2011). 5. S. Donati, Electro-optical Instrumentation Sensing and Measuring with Lasers (Prentice-Hall, 2004), pp. 187–233. 6. C. Ciminelli, F. Dell’Olio, C. E. Campanella, and M. N. Armenise, “Numerical and experimental investigation of an optical high-Q spiral resonator gyroscope,” in ICTON (IEEE Photonics Society, 2012), paper Th.A4.5. 7. C. Ciminelli, F. Dell’Olio, M. N. Armenise, F. M. Soares, and W. Passenberg, “High performance InP ring resonator for new generation monolithically integrated optical gyroscopes,” Opt. Express 21, 556–564 (2013). 8. F. Dell’Olio, C. Ciminelli, and M. N. Armenise, “Theoretical investigation of InP buried ring resonators for new angular velocity sensors,” Opt. Eng. 52, 024601 (2013). 9. N. M. Barbour, “Inertial navigation sensors,” RTO-EN-SET116 (2011). 10. K. Iwatsuki, M. Saruwatari, M. Kawachi, and H. Yamazaki, “Waveguide-type optical passive ring-resonator gyro using time division detection scheme,” Electron. Lett. 25, 688–689 (1989). 11. C. Vannahme, H. Suche, S. Reza, R. Ricken, V. Quiring, and W. Sohler, “Integrated optical Ti:LiNbO3 ring resonator for zero bias stability,” ECIO, Copenhagen, Denmark (2007). 12. C. Ciminelli, F. Dell’Olio, and M. N. Armenise, “High-Q spiral resonator for optical gyroscope applications: numerical and experimental investigation,” IEEE Photon. J. 4, 1844–1854 (2012). 13. C. Ciminelli, C. E. Campanella, F. Dell’Olio, C. Campanella, and M. N. Armenise, “Theoretical investigation on the scale factor of a triple ring cavity to be used in frequency sensitive resonant gyroscopes,” J. Eur. Opt. Soc. 8, 13050 (2013). 14. H. Mao, H. Ma, and Z. Jin, “Polarization maintaining silica waveguide resonator optic gyro using double phase modulation technique,” Opt. Express 19, 4632–4643 (2011).
15. K. Iwatsuki, K. Hotate, and M. Higashiguchi, “Effect of Rayleigh backscattering in an optical passive ring-resonator gyro,” Appl. Opt. 23, 3916–3924 (1984). 16. Z. Jin, Z. Yang, H. Ma, and D. Ying, “Open-loop experiments in a resonator fiber-optic gyro using digital triangle wave phase modulation,” IEEE Photon. Technol. Lett. 19, 1685–1687 (2007). 17. H. Ma, Z. He, and K. Hotate, “Reduction of backscattering induced noise by carrier suppression in waveguide-type optical ring resonator gyro,” J. Lightwave Technol. 29, 85–90 (2011). 18. K. Takiguchi and K. Hotate, “Evaluation of the output error in an optical passive ring-resonator gyro with a 90 polarizationaxis rotation in the polarization-maintaining fiber resonator,” IEEE Photon. Technol. Lett. 3, 88–90 (1991). 19. X. Yu, H. Ma, and Z. Jin, “Improving thermal stability of a resonator fiber optic gyro employing a polarizing resonator,” Opt. Express 21, 358–369 (2013). 20. Y. Ren, Z. Jin, Y. Chen, and H. Ma, “Optimization of the resonance frequency servo loop technique in the resonator micro optic gyro,” J. Zhejiang Univ. Sci. C. 12, 942–950 (2011). 21. H. Ma, X. Lu, L. Yao, X. Yu, and Z. Jin, “Full investigation of the resonant frequency servo loop for resonator fiber-optic gyro,” Appl. Opt. 51, 5178–5185 (2012). 22. H. Ma, W. Wang, Y. Ren, and Z. Jin, “Low-noise low-delay digital signal processor for resonant micro optic gyro,” IEEE Photon. Tech. L. 25, 198–201 (2013). 23. K. Suzuki, K. Takiguchi, and K. Hotate, “Monolithically integrated resonator micro-optic gyro on silica planar lightwave circuit,” J. Lightwave Technol. 18, 66–72 (2000). 24. F. Zarinetchi and S. Ezekiel, “Observation of lock-in behavior in a passive resonator gyroscope,” Opt. Lett. 11, 401–403 (1986).
25. G. Galzerano and P. Laporta, “Single-frequency diode-pumped Yb:KYF4 laser around 1030 nm,” Opt. Express 15, 3257–3264 (2007). 26. E. I. Moses and C. L. Tang, “High-sensitivity laser wavelength-regulation spectroscopy,” Opt. Lett. 1, 115–117 (1977). 27. G. A. Massey, M. K. Oshman, and R. Targ, “Generation of single-frequency light using the FM laser,” Appl. Phys. Lett. 6, 10–11 (1965). 28. Y. Ohtaa, S. Maehar, K. Hasebe, Y. Kurosaki, and M. Ohkawa, “Frequency stabilization of a semiconductor laser using the Rb saturated absorption spectroscopy,” Proc. SPIE 6115, 1–10 (2006). 29. F. Rogister, D. W. Sukow, A. Gavrielides, P. Mégret, O. Deparis, and M. Blondel, “Experimental demonstration of suppression of low-frequency fluctuations and stabilization of an externalcavity laser diode,” Opt. Lett. 25, 808–810 (2000). 30. L. K. Strandiord and G. A. Sanders, “Resonator optic gyro employing a polarization rotating resonator,” Proc. SPIE 1585, 163–172 (1991). 31. K. Iwatsuki, K. Hotate, and M. Higashiguchi, “Backscattering in an optical passive ring-resonator gyro: experiment,” Appl. Opt. 25, 4448–4451 (1986). 32. L. Feng, M. Lei, H. Liu, Y. Zhi, and J. Wang, “Suppression of backreflection noise in a resonator integrated optic gyro by hybrid phase-modulation technology,” Appl. Opt. 52, 1668–1675 (2013). 33. C. H. Lefevre, The Fiber-Optic Gyroscope (Artech, 1993), pp. 157–167. 34. K. Ogata, Modern Control Engineering, 5th ed. (Prentice-Hall, 2010), pp. 320–485. 35. M. Lei, L. Feng, Y. Zhi, and H. Liu, “Effect of intensity variation of laser in resonator, integrated optic gyro,” Appl. Opt. 52, 1–7 (2013).
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