Reduction of reset pulse in resonant frequency servo loop for resonant fiber-optic gyro by an auto-controlled reset technique Huilian Ma, Xiao Lu, and Zhonghe Jin* Micro-Satellite Research Center, Zhejiang University, Hangzhou, China *Corresponding author: [email protected] Received 12 September 2013; revised 20 November 2013; accepted 20 November 2013; posted 26 November 2013 (Doc. ID 197370); published 16 December 2013
Resonant fiber-optic gyro (RFOG) based on the Sagnac effect has the potential to achieve the inertial navigation system requirement with a short sensing coil. A high-accuracy resonant frequency servo loop is indispensable for a high-performance RFOG. A digital proportional-integral (PI) controller is always adopted in the resonant frequency servo loop. The resonant frequency of the optical fiber ring resonator drifts with environmental temperature changes. When the resonant frequency drift is beyond the tracking range of the resonant frequency servo loop, the digital PI controller overflows and outputs a reset signal. A large reset pulse, which is equivalent to a rotation rate error of 26°∕h, has been observed at the output of the RFOG, while a long time is required for returning to the lock-in state simultaneously. To reduce the effect of the overflow resetting in the digital PI controller, an auto-controlled reset technique is proposed and experimentally demonstrated. As a result, the time for returning to the lock-in state is reduced to 5 ms from 8 s. With the integration time of 1 s, the equivalent accuracy of the resonant frequency servo loop is improved to 0.18°∕h. © 2013 Optical Society of America OCIS codes: (060.2800) Gyroscopes; (120.5790) Sagnac effect; (140.4780) Optical resonators. http://dx.doi.org/10.1364/AO.52.008771
1. Introduction
Resonant fiber-optic gyro (RFOG) with no moving parts based on the Sagnac effect [1] is developed as an attractive candidate for inertial navigation systems. Compared with the very successful He–Ne ring laser gyro (RLG), an RFOG based on a passive resonator shows the absence of the gain competition problem [2]. Compared with the interferometric fiber-optic gyro (IFOG) [3,4], an RFOG has potential in realizing IFOG-like performance with a coil length of up to 100 times shorter than those of IFOG in a given performance class [5]. However, its performance achieved to data is still below expectation. In the RFOG, the rotation rate readout is given as the resonant frequency difference between the 1559-128X/13/368771-08$15.00/0 © 2013 Optical Society of America
clockwise (CW) and counterclockwise (CCW) lightwaves propagating in the optical fiber ring resonator (OFRR) through the Sagnac effect. In practice, both the resonant frequency of the OFRR and the central frequency of the laser drift with environment temperature changes, and the resonant frequency difference introduced by the Sagnac effect is very small. Therefore, a high-accuracy resonant frequency servo loop, used to make the central frequency of the laser source track the resonant frequency of the OFRR in one direction, is indispensable for a highperformance RFOG [6]. A low-noise, low-delay proportional-integral (PI) controller implemented on a single field-programmable gated array (FPGA) has been adopted [7–9]. As a result, a bias stability over 1800 s close to the shot-noise limit for the RFOG was achieved [9]. The resonant frequency of the OFRR drifts with environmental temperature changes. The temperature 20 December 2013 / Vol. 52, No. 36 / APPLIED OPTICS
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coefficient of the refractive index of silica fiber in the OFRR is approximately 10−5 ∕°C [10]. A temperature drift of 0.01°C results in a resonant frequency drift of 13.3 MHz with the central laser frequency at 194 THz. The central frequency drift of a commercial fiber laser associated with temperature is at least two orders magnitude less than the resonant frequency drift of the OFRR. Therefore, the thermal-induced resonant frequency drift should be tracked by the laser. The central frequency of the laser source can be tuned by applying a voltage signal to a piezoelectric transducer (PZT) controller, which is included in the laser. The laser frequency tuning range is determined by the maximum voltage supplied by the resonant frequency servo loop and the tuning coefficient of the laser. The temperature fluctuation is 1°C over a few hours in the laboratory without any thermal stabilization, and then the resonant frequency drift of the OFRR is about 200–400 kHz∕s [11]. The resonant frequency drift increases with time. When it is beyond the tracking range of the resonant frequency servo loop, the digital PI controller overflows and outputs a reset signal. A large reset pulse, which is equivalent to a rotation rate error of 26°∕h, has been observed at the output of the RFOG, while a long time is required for returning to the lock-in state simultaneously. In this paper, an auto-controlled reset technique is proposed and successfully applied to the output of the digital PI controller. Experiments show that the lock-in time is reduced to 5 ms from 8 s. With the integration time of 1 s, the equivalent accuracy of the resonant frequency servo loop is improved to 0.18°∕h.pAn angle random walk (ARW) coefficient of 0.0028°∕ h is obtained for the resonant frequency servo loop. To the best of our knowledge, this is the best ARW result reported for a resonant frequency servo loop in an RFOG. 2. Principles and Analysis
Figure 1 shows the experimental setup of the RFOG based on the digital resonant frequency servo loop. The polarization-maintaining fiber (PMF) transmission-type OFRR with twin 90° polarization-axis rotated splices is the key rotation sensing element in the RFOG. To further decrease the polarization error, two in-line polarizers Px and Py are inserted [12,13]. A lightwave from a fiber laser is equally divided by coupler C1 and injected into the OFRR in the CW and the CCW lightwaves. Two phase modulators PM1 and PM2 are driven by sinusoidal waveforms. To attenuate the backscattering noise, different modulation frequencies f 1 and f 2 are used. The amplitudes of the two sinusoidal voltages V 1 and V 2 are carefully optimized to reduce the carrier components [14,15]. The output of photodetector PD2 is demodulated by the digital lock-in amplifier LIA2, which is used as an error signal to the PI controller to lock the laser frequency to the resonant frequency of the CCW lightwave in the OFRR. The demodulated signal of the CW lightwave from LIA1 is used as the open-loop output of the rotation rate. 8772
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Fig. 1. Experimental setup of the RFOG based on the digital resonant frequency servo loop. C1–C3, couplers; PM1 and PM2, phase modulators; PD1 and PD2, photodetectors; LIA1 and LIA2, lock-in amplifiers; OFRR, optical fiber ring resonator; PI, proportionalintegral controller.
Figure 2 shows the analysis model of the digital resonant frequency servo loop [9]. It consists of three main parts: a laser, a frequency discriminator, and a loop filter. Si f is the linear frequency noise spectrum density, which is added in the loop model as the noise source. Ignoring the initial phase, the output field of the fiber laser is written as Z EFL-out t E0 exp j 2πf 0 t V D Adt ;
(1)
where E0 is the amplitude, f 0 is the central frequency, and A is the frequency turning coefficient of the laser. The laser converts the voltage fluctuations from the servo controller into the frequency shift by the tuning coefficient A. In this paper, A is 36 MHz∕V, provided by the fiber laser manufacturer. The frequency discriminator is composed of PM2, the OFRR, PD2, and the phase sensitive detector (PSD) part of LIA2. It converts the laser frequency fluctuations into the voltage fluctuations by a discriminative slope D (V∕Hz), which is a key parameter in the phase modulation technique [9]. When the phase of the CCW beam is modulated by a sinusoidal wave at rate f 2 before the CCW beam being launched into the OFRR, the field at the entrance of the OFRR is written as Z EPM2-out t E0 exp j2πf 0 t M 2 sin2πf 2 t;
V D Adt (2)
where M 2 is the phase modulation index, which can be expressed as M 2 πV 2 ∕V π2 , in which V π2 is the half-wave voltage for PM2 and V 2 is the amplitude of the sinusoidal wave volt signal. The phase modulation index M 2 is set to 2.405 to suppress the carrier component [14,15]. The OFRR is the core-sensing component in the RFOG. The modulated lightwave output from the
Fig. 2. Model of the digital resonant frequency servo loop.
OFRR is converted into the electric signal by photodetector PD2. Zhang et al. have analyzed the sinusoidal phase modulation technology [16]. It includes a DC component and those harmonic components of f 2. The optical frequency fluctuations both of the laser and the OFRR are reflected on the first harmonic component of f 2. The PSD part of LIA2 is used to demodulate the output of PD2 at the rate of f 2. After being demodulated, the output of the PSD is given by V PSD-out t kLIA xtrt 1 1 kLIA V s V r cos θ V s V r cos4πf 2 tθ ; 2 2 (3) where kLIA is the gain of LIA2, xt V s cos2πf 2 is the first harmonic component output from PD2, in which V s is the amplitude. rt V r cos2πf 2 θ is the reference signal provided by LIA2, in which V r is the amplitude and θ is the phase difference between these two signals. It contains two terms at the output of the PSD. The last term in Eq. (3) can be filtered by the low-pass filter (LPF) part of LIA2 in the loop filter, thus obtaining the amplitude of xt. The loop filter amplifies the voltage amplitude and feedback to the PZT actuator to tune the laser frequency. It includes the LPF part of LIA2 and the PI controller. Therefore, its transfer function is given by Gs kP
1 1 ; 1 τis 1 τLPF s
(4)
where kP is the proportional gain,τi is the integration time, and τLPF is the time constant of the LPF. Based on the mathematic properties of the models mentioned above, the open-loop transfer function for the resonant frequency servo loop is given by 1 1 Hs k 1 e−sτD ; τis 1 τLPF s
(5)
where k is the total gain of the loop contributing from the frequency discriminator, the PI controller, and the laser actuator. τD is the loop delay, mainly induced by LIA2, PI controller, and the laser.
An OFRR temperature change induces a drift of the resonant frequency. Without any thermal stabilization, the resonant frequency drift of the OFRR in the laboratory is about 200–400 kHz∕s [11]. The laser frequency tuning range is determined by the maximum voltage supplied by the resonant frequency servo loop and the tuning coefficient of the laser. The resonant frequency drift increases with time. Limited by a finite output voltage range of the digital-to-analog converter (DAC), the output of the digital PI controller overflows and outputs a reset signal. As a result, a large reset pulse, which is equivalent to a rotation rate error of 26°∕h, was observed at the output of the RFOG, as shown in Fig. 3. The time required for returning to the lock-in state is about 8 s. It is not beneficial in practical applications. In the lock-in state, the central frequency of the laser tracks the resonant frequency of the CCW lightwave in the OFRR. To avoid large reset pulses occurring at the output of the RFOG due to a limited output voltage range of the DAC, an auto-controlled reset technique is applied to the output of the PI controller. The voltage difference ΔV between before and after resetting should be satisfied such that it can tune the laser frequency by exact integer multiples of the free spectral range (FSR) of the OFRR. This auto-controlled reset technique takes advantage of the periodic characteristic of the resonant curve of the OFRR. The output voltage of the PI controller is automatically subtracted ΔV before overflow resetting. Ideally, when the reset voltage difference ΔV is applied to the laser actuator that can tune the laser frequency by exact integer multiples of the FSR, the laser frequency locates the exact same position on the resonant curve as before resetting. A demodulation error appears at the output of the RFOG for nonideal ΔV. Figure 4 shows the simulation results of the effects of the reset voltage difference on the demodulation error in the output of the RFOG with a noninstantaneous resetting time. Four different equivalent frequency deviations of −300 kHz, 0, 300 kHz, and 3.3 MHz are shown in Figs. 4(a)–4(d). To reduce the times of reset, a high ΔV is preferred. However, limited by the output voltage range of the DAC, three multiples of the FSR are set. Figure 4(a) shows that the reset voltage difference produces a laser frequency shift deviating three multiples of the FSR by −300 kHz, which induces a demodulated 20 December 2013 / Vol. 52, No. 36 / APPLIED OPTICS
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Fig. 3. (a) Observation of the large reset pulses occurring on the overflow resetting of the digital PI controller. (b) Closed-loop output with the auto-controlled reset technique applied to the output of the digital PI controller.
voltage error of −382 mV and needs 5.46 ms to get back to the lock-in state. After resetting, the location of the central frequency of the laser on the demodulation curve as point A shows in Fig. 5. Therefore, it induces a negative error and then is locked back to the resonant point. Ideally, when the reset voltage difference tunes the laser frequency by exact
integer multiples of the FSR, no error occurs on the demodulated output as point B shows in Fig. 5. The demodulated voltage error of −99 mV shown in Fig. 4(b) comes from the effect of the limited tuning bandwidth of the laser and thus makes the resetting noninstantaneous. Figure 4(d) shows the simulated results for a large deviation to three multiples of
Fig. 4. Simulation results of the closed-loop output with different reset voltage differences. 8774
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the FSR. As seen from Fig. 4(d), it takes 17.34 ms back to the lock-in state. 3. Experiments
The experimental setup of the RFOG is shown in Fig. 1. A fiber laser source is used, with a central wavelength of 1550 nm, a linewidth of 5 kHz, and apower of 1.6 mW. The coupling coefficient of couplers C2 and C3 are 0.03. The length and the diameter of the OFRR are 14 and 0.12 m, respectively. The measured FSR is 14.7 MHz. The auto-controlled reset technique is also realized on the same FPGA with the digital PI controller. As Fig. 6 shows, two reference voltages, V ref and −V ref , and two set voltages, V set and −V set , are inserted between the maximum positive voltage (V max ) and the minimum negative voltage (−V max ) of the digital PI controller. The voltage difference between the reference and set voltages, V ref and −V set , V set and −V ref , tunes the laser frequency by integer multiples of the FSR. When the central frequency of the laser drifts positively relative to the resonant frequency of the OFRR, the PI output voltage increases progressively. As soon as the PI output voltage reaches the positive reference voltage V ref , it is reset to the negative set voltage −V set . Therefore, the central frequency of the laser can track another resonant frequency quickly. A similar process occurs when the laser frequency drifts negatively relative to the resonant frequency of the OFRR. Considering that the loop needs fine-tuning after being reset, as Fig. 6 shows, two voltage buffer regions are set between the two reference and set voltages to prevent from being reset ceaselessly. The voltage difference between the reference and set voltages is converted to an equivalent frequency deviation through the tuning parameter of the fiber laser. According to the definition of the FSR, both n and L are temperature dependent, thus making the FSR a temperature-dependent parameter. The voltage difference cannot be set ideally to tune the laser
Fig. 5. Locations of the central frequency of the laser on the demodulation curve after PI output resetting with different values.
Fig. 6. Diagram of the proposed auto-controlled reset technique. V max 1.15 V, V ref 0.82 V, and V set 0.41 V.
frequency exactly equivalent to integer multiples of the FSR. Figure 7 shows measurement results of the effects of the reset voltage difference on the demodulated error of the closed-loop output of the RFOG. Corresponding to Fig. 4, four different equivalent frequency deviations of −300 kHz, 0, 300 kHz, and 3.3 MHz are set in Figs. 7(a)–7(d). The resonant frequency of the OFRR drifts with environmental temperature changes in practice, which affects the locking process after reset. Therefore, there are differences between the experimental and simulation results. As seen in Fig. 7(a), as the equivalent frequency deviates three multiples of the FSR by −300 kHz, it induces a demodulated voltage error of −785 mV and needs 6.2 ms to get back to the lock-in state. When the voltage difference is equivalent to three multiples of the FSR, the demodulated voltage error should be theoretically zero. The demodulated error of −590 mV comes from two aspects. One is that the reset voltage used to tune the laser frequency by three multiples of the FSR cannot be exactly set. The other comes from the effect of the limited tuning bandwidth of the laser, which broadens the instant reset. Therefore, it induces a negative error and then is relocked to the resonant point. Figure 7(d) shows a large deviation from three multiples of the FSR. It takes 16 ms to get back to the lock-in state. The aforementioned demodulated error induces a frequency deviation of the lock-in frequency and thus finally induces a rotation rate error. Figure 8 shows the demodulation voltage error and the time required for returning to the lock-in state with different equivalent frequency deviations. Those simulated results are also presented in Fig. 8. As seen in Fig. 8(a), the experimental results agree well with those predicted from simulations. The error changes linearly with the frequency deviation in small frequency deviation, which corresponds to the linear region of the demodulation curve. Limited by the maximum demodulation value, the error tends to be stable in large frequency deviation. Figure 8(b) shows the relationship between the time required for returning to the lock-in state and the equivalent frequency deviation. The bigger frequency deviation is, the longer the lock-in time needs. For the frequency 20 December 2013 / Vol. 52, No. 36 / APPLIED OPTICS
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Fig. 7. Measurement results of the closed-loop output with different reset values.
deviation from −1.2 MHz to 1.2 MHz, the measured lock-in time is below 10 ms, which is consistent with the theoretical result. In the practical RFOG, the reset voltage always deviates from the practical one corresponding to integer multiples of the FSR. According to the definition of the FSR, it is given by
FSR
c : nL
(6)
The change of the FSR is mainly caused by n, which changes with temperature. The temperature coefficient of the refractive index of the silica fiber is approximately 10−5 ∕°C and the change rate of the FSR can be expressed as
Fig. 8. Demodulated voltage error and time required for returning to the lock-in state with different frequency deviations. (a) Relationship between the demodulated voltage error and the frequency deviation. (b) Relationship between the time required for returning to the lock-in state and the frequency deviation. 8776
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Fig. 9. Closed-loop output of the residual laser frequency noise. (a) Closed-loop output for the integration time of 1 s. (b) Allan deviation of the closed-loop output.
∂FSR c ∂ n1 c 1 ∂n − 2 −142 Hz∕°C: ∂T L ∂T L n ∂T
(7)
For the application where the temperature changes from −100°C to 100°C, the frequency deviation caused by the change of the FSR is from −14.2 kHz to 14.2 kHz. As seen in Fig. 8(b), the lock-in time for the frequency deviation from −14.2 kHz to 14.2 kHz is below 5 ms. Therefore, the effect of the reset caused by the change of the FSR can be ignored. When the PI controller overflows resetting, large reset pulses occur at the output of the RFOG. Figure 3(b) shows the measurement result of the closed-loop output with the auto-controlled reset technique applied to the output of the digital PI controller. As seen in Fig. 3(b), there is no obvious reset pulse occurring on the resetting points of the digital PI controller. Figure 9 shows the experimental results for long time duration. The calculated residual laser frequency noise is equivalent to a rotation rate of 0.18°∕h (1σ) for the integration time of 1 s. The Allan deviation of the residual laser frequency noise is shown in Fig. 9(b). The Allan deviation is dominated by the white frequency noise at the low integration time and reaches the so-called flicker floor at approximately 500 s and increases again due to the frequency drift. By extrapolating a fitted line with τ−1∕2 (τ is the p integration time), an ARW coefficient of 0.0028°∕ h is obtained. 4. Conclusion
An RFOG based on a digital PI controller with an auto-controlled reset technique is set up. Experiments show that the time required for returning to the lock-in state falls from 8 s to 5 ms. With the integration time of 1 s, the equivalent accuracy of the resonant frequency servo loop is increased to p 0.18°∕h. An ARW coefficient of 0.0028°∕ h is obtained for the resonant frequency servo loop. To the best of our knowledge, this is the best ARW result
reported for a resonant frequency servo loop in an RFOG. The authors would like to acknowledge financial support from the National Natural Science Foundation of China (No. 61377101). References 1. S. Ezekiel and S. K. Balsmo, “Passive ring resonator laser gyroscope,” Appl. Phys. Lett. 30, 478–480 (1977). 2. S. Ezekiel, “Optical gyroscope options: principles and challenges,” in 18th International Optical Fiber Sensors Conference Technical Digest (Optical Society of America, 2006), paper MC1. 3. G. A. Pavlath, “Fiber-optic gyros: the vision realized,” in 18th International Optical Fiber Sensors Conference Technical Digest (Optical Society of America, 2006), paper MA3. 4. A. Ohno, A. Kurokawa, T. Kumagai, S. Nakamura, and K. Hotate, “Applications and technical progress of fiber-optic gyros in Japan,” in 18th International Optical Fiber Sensors Conference Technical Digest (Optical Society of America, 2006), paper MA4. 5. G. A. Sanders, L. K. Strandjord, and T. Qiu, “Hollow core fiberoptic ring resonator for rotation sensing,” in 18th International Optical Fiber Sensors Conference Technical Digest (Optical Society of America, 2006), paper ME6. 6. K. Takiguchi and K. Hotate, “Partially digital-feedback scheme and evaluation of optical Kerr-effect induced bias in optical passive ring-resonator gyro,” IEEE Photon. Technol. Lett. 3, 679–681 (1991). 7. 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). 8. H. Ma, W. Wang, Y. Ren, and Z. Jin, “Low-noise, low-delay digital signal processor for resonant micro optic gyro,” IEEE Photon. Technol. Lett. 25, 198–201 (2013). 9. Z. Jin, X. Yu, and H. Ma, “Resonator fiber-optic gyro employing a semiconductor laser,” Appl. Opt. 51, 2856–2864 (2012). 10. R. Kingslake and B. J. Thompson, Applied Optics and Optical Engineering (Academic, 1965). 11. C. Ciminelli, F. D. Olio, and M. N. Armenise, “High-Q spiral resonator for optical gyroscope applications: numerical and experimental investigation,” IEEE Photon. J. 4, 1844–1854 (2012). 12. H. Ma, X. Yu, and Z. Jin, “Reduction of polarizationfluctuation induced drift in resonator fiber-optic gyro by a resonator integrating in-line polarizers,” Opt. Lett. 37, 3342–33449 (2012). 20 December 2013 / Vol. 52, No. 36 / APPLIED OPTICS
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13. 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). 14. 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).
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15. 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). 16. X. Zhang, H. Ma, Z. Jin, and C. Ding, “Open-loop operation experiments in a resonator fiber-optic gyro using the phase modulation spectroscopy technique,” Appl. Opt. 45, 7961– 7965 (2006).
Resonant fiber-optic gyro (RFOG) based on the Sagnac effect has the potential to achieve the inertial navigation system requirement with a short sensing coil. A high-accuracy resonant frequency servo loop is indispensable for a high-performance RFOG. A digital proportional-integral (PI) controller is always adopted in the resonant frequency servo loop. The resonant frequency of the optical fiber ring resonator drifts with environmental temperature changes. When the resonant frequency drift is beyond the tracking range of the resonant frequency servo loop, the digital PI controller overflows and outputs a reset signal. A large reset pulse, which is equivalent to a rotation rate error of 26°∕h, has been observed at the output of the RFOG, while a long time is required for returning to the lock-in state simultaneously. To reduce the effect of the overflow resetting in the digital PI controller, an auto-controlled reset technique is proposed and experimentally demonstrated. As a result, the time for returning to the lock-in state is reduced to 5 ms from 8 s. With the integration time of 1 s, the equivalent accuracy of the resonant frequency servo loop is improved to 0.18°∕h. © 2013 Optical Society of America OCIS codes: (060.2800) Gyroscopes; (120.5790) Sagnac effect; (140.4780) Optical resonators. http://dx.doi.org/10.1364/AO.52.008771
1. Introduction
Resonant fiber-optic gyro (RFOG) with no moving parts based on the Sagnac effect [1] is developed as an attractive candidate for inertial navigation systems. Compared with the very successful He–Ne ring laser gyro (RLG), an RFOG based on a passive resonator shows the absence of the gain competition problem [2]. Compared with the interferometric fiber-optic gyro (IFOG) [3,4], an RFOG has potential in realizing IFOG-like performance with a coil length of up to 100 times shorter than those of IFOG in a given performance class [5]. However, its performance achieved to data is still below expectation. In the RFOG, the rotation rate readout is given as the resonant frequency difference between the 1559-128X/13/368771-08$15.00/0 © 2013 Optical Society of America
clockwise (CW) and counterclockwise (CCW) lightwaves propagating in the optical fiber ring resonator (OFRR) through the Sagnac effect. In practice, both the resonant frequency of the OFRR and the central frequency of the laser drift with environment temperature changes, and the resonant frequency difference introduced by the Sagnac effect is very small. Therefore, a high-accuracy resonant frequency servo loop, used to make the central frequency of the laser source track the resonant frequency of the OFRR in one direction, is indispensable for a highperformance RFOG [6]. A low-noise, low-delay proportional-integral (PI) controller implemented on a single field-programmable gated array (FPGA) has been adopted [7–9]. As a result, a bias stability over 1800 s close to the shot-noise limit for the RFOG was achieved [9]. The resonant frequency of the OFRR drifts with environmental temperature changes. The temperature 20 December 2013 / Vol. 52, No. 36 / APPLIED OPTICS
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coefficient of the refractive index of silica fiber in the OFRR is approximately 10−5 ∕°C [10]. A temperature drift of 0.01°C results in a resonant frequency drift of 13.3 MHz with the central laser frequency at 194 THz. The central frequency drift of a commercial fiber laser associated with temperature is at least two orders magnitude less than the resonant frequency drift of the OFRR. Therefore, the thermal-induced resonant frequency drift should be tracked by the laser. The central frequency of the laser source can be tuned by applying a voltage signal to a piezoelectric transducer (PZT) controller, which is included in the laser. The laser frequency tuning range is determined by the maximum voltage supplied by the resonant frequency servo loop and the tuning coefficient of the laser. The temperature fluctuation is 1°C over a few hours in the laboratory without any thermal stabilization, and then the resonant frequency drift of the OFRR is about 200–400 kHz∕s [11]. The resonant frequency drift increases with time. When it is beyond the tracking range of the resonant frequency servo loop, the digital PI controller overflows and outputs a reset signal. A large reset pulse, which is equivalent to a rotation rate error of 26°∕h, has been observed at the output of the RFOG, while a long time is required for returning to the lock-in state simultaneously. In this paper, an auto-controlled reset technique is proposed and successfully applied to the output of the digital PI controller. Experiments show that the lock-in time is reduced to 5 ms from 8 s. With the integration time of 1 s, the equivalent accuracy of the resonant frequency servo loop is improved to 0.18°∕h.pAn angle random walk (ARW) coefficient of 0.0028°∕ h is obtained for the resonant frequency servo loop. To the best of our knowledge, this is the best ARW result reported for a resonant frequency servo loop in an RFOG. 2. Principles and Analysis
Figure 1 shows the experimental setup of the RFOG based on the digital resonant frequency servo loop. The polarization-maintaining fiber (PMF) transmission-type OFRR with twin 90° polarization-axis rotated splices is the key rotation sensing element in the RFOG. To further decrease the polarization error, two in-line polarizers Px and Py are inserted [12,13]. A lightwave from a fiber laser is equally divided by coupler C1 and injected into the OFRR in the CW and the CCW lightwaves. Two phase modulators PM1 and PM2 are driven by sinusoidal waveforms. To attenuate the backscattering noise, different modulation frequencies f 1 and f 2 are used. The amplitudes of the two sinusoidal voltages V 1 and V 2 are carefully optimized to reduce the carrier components [14,15]. The output of photodetector PD2 is demodulated by the digital lock-in amplifier LIA2, which is used as an error signal to the PI controller to lock the laser frequency to the resonant frequency of the CCW lightwave in the OFRR. The demodulated signal of the CW lightwave from LIA1 is used as the open-loop output of the rotation rate. 8772
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Fig. 1. Experimental setup of the RFOG based on the digital resonant frequency servo loop. C1–C3, couplers; PM1 and PM2, phase modulators; PD1 and PD2, photodetectors; LIA1 and LIA2, lock-in amplifiers; OFRR, optical fiber ring resonator; PI, proportionalintegral controller.
Figure 2 shows the analysis model of the digital resonant frequency servo loop [9]. It consists of three main parts: a laser, a frequency discriminator, and a loop filter. Si f is the linear frequency noise spectrum density, which is added in the loop model as the noise source. Ignoring the initial phase, the output field of the fiber laser is written as Z EFL-out t E0 exp j 2πf 0 t V D Adt ;
(1)
where E0 is the amplitude, f 0 is the central frequency, and A is the frequency turning coefficient of the laser. The laser converts the voltage fluctuations from the servo controller into the frequency shift by the tuning coefficient A. In this paper, A is 36 MHz∕V, provided by the fiber laser manufacturer. The frequency discriminator is composed of PM2, the OFRR, PD2, and the phase sensitive detector (PSD) part of LIA2. It converts the laser frequency fluctuations into the voltage fluctuations by a discriminative slope D (V∕Hz), which is a key parameter in the phase modulation technique [9]. When the phase of the CCW beam is modulated by a sinusoidal wave at rate f 2 before the CCW beam being launched into the OFRR, the field at the entrance of the OFRR is written as Z EPM2-out t E0 exp j2πf 0 t M 2 sin2πf 2 t;
V D Adt (2)
where M 2 is the phase modulation index, which can be expressed as M 2 πV 2 ∕V π2 , in which V π2 is the half-wave voltage for PM2 and V 2 is the amplitude of the sinusoidal wave volt signal. The phase modulation index M 2 is set to 2.405 to suppress the carrier component [14,15]. The OFRR is the core-sensing component in the RFOG. The modulated lightwave output from the
Fig. 2. Model of the digital resonant frequency servo loop.
OFRR is converted into the electric signal by photodetector PD2. Zhang et al. have analyzed the sinusoidal phase modulation technology [16]. It includes a DC component and those harmonic components of f 2. The optical frequency fluctuations both of the laser and the OFRR are reflected on the first harmonic component of f 2. The PSD part of LIA2 is used to demodulate the output of PD2 at the rate of f 2. After being demodulated, the output of the PSD is given by V PSD-out t kLIA xtrt 1 1 kLIA V s V r cos θ V s V r cos4πf 2 tθ ; 2 2 (3) where kLIA is the gain of LIA2, xt V s cos2πf 2 is the first harmonic component output from PD2, in which V s is the amplitude. rt V r cos2πf 2 θ is the reference signal provided by LIA2, in which V r is the amplitude and θ is the phase difference between these two signals. It contains two terms at the output of the PSD. The last term in Eq. (3) can be filtered by the low-pass filter (LPF) part of LIA2 in the loop filter, thus obtaining the amplitude of xt. The loop filter amplifies the voltage amplitude and feedback to the PZT actuator to tune the laser frequency. It includes the LPF part of LIA2 and the PI controller. Therefore, its transfer function is given by Gs kP
1 1 ; 1 τis 1 τLPF s
(4)
where kP is the proportional gain,τi is the integration time, and τLPF is the time constant of the LPF. Based on the mathematic properties of the models mentioned above, the open-loop transfer function for the resonant frequency servo loop is given by 1 1 Hs k 1 e−sτD ; τis 1 τLPF s
(5)
where k is the total gain of the loop contributing from the frequency discriminator, the PI controller, and the laser actuator. τD is the loop delay, mainly induced by LIA2, PI controller, and the laser.
An OFRR temperature change induces a drift of the resonant frequency. Without any thermal stabilization, the resonant frequency drift of the OFRR in the laboratory is about 200–400 kHz∕s [11]. The laser frequency tuning range is determined by the maximum voltage supplied by the resonant frequency servo loop and the tuning coefficient of the laser. The resonant frequency drift increases with time. Limited by a finite output voltage range of the digital-to-analog converter (DAC), the output of the digital PI controller overflows and outputs a reset signal. As a result, a large reset pulse, which is equivalent to a rotation rate error of 26°∕h, was observed at the output of the RFOG, as shown in Fig. 3. The time required for returning to the lock-in state is about 8 s. It is not beneficial in practical applications. In the lock-in state, the central frequency of the laser tracks the resonant frequency of the CCW lightwave in the OFRR. To avoid large reset pulses occurring at the output of the RFOG due to a limited output voltage range of the DAC, an auto-controlled reset technique is applied to the output of the PI controller. The voltage difference ΔV between before and after resetting should be satisfied such that it can tune the laser frequency by exact integer multiples of the free spectral range (FSR) of the OFRR. This auto-controlled reset technique takes advantage of the periodic characteristic of the resonant curve of the OFRR. The output voltage of the PI controller is automatically subtracted ΔV before overflow resetting. Ideally, when the reset voltage difference ΔV is applied to the laser actuator that can tune the laser frequency by exact integer multiples of the FSR, the laser frequency locates the exact same position on the resonant curve as before resetting. A demodulation error appears at the output of the RFOG for nonideal ΔV. Figure 4 shows the simulation results of the effects of the reset voltage difference on the demodulation error in the output of the RFOG with a noninstantaneous resetting time. Four different equivalent frequency deviations of −300 kHz, 0, 300 kHz, and 3.3 MHz are shown in Figs. 4(a)–4(d). To reduce the times of reset, a high ΔV is preferred. However, limited by the output voltage range of the DAC, three multiples of the FSR are set. Figure 4(a) shows that the reset voltage difference produces a laser frequency shift deviating three multiples of the FSR by −300 kHz, which induces a demodulated 20 December 2013 / Vol. 52, No. 36 / APPLIED OPTICS
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Fig. 3. (a) Observation of the large reset pulses occurring on the overflow resetting of the digital PI controller. (b) Closed-loop output with the auto-controlled reset technique applied to the output of the digital PI controller.
voltage error of −382 mV and needs 5.46 ms to get back to the lock-in state. After resetting, the location of the central frequency of the laser on the demodulation curve as point A shows in Fig. 5. Therefore, it induces a negative error and then is locked back to the resonant point. Ideally, when the reset voltage difference tunes the laser frequency by exact
integer multiples of the FSR, no error occurs on the demodulated output as point B shows in Fig. 5. The demodulated voltage error of −99 mV shown in Fig. 4(b) comes from the effect of the limited tuning bandwidth of the laser and thus makes the resetting noninstantaneous. Figure 4(d) shows the simulated results for a large deviation to three multiples of
Fig. 4. Simulation results of the closed-loop output with different reset voltage differences. 8774
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the FSR. As seen from Fig. 4(d), it takes 17.34 ms back to the lock-in state. 3. Experiments
The experimental setup of the RFOG is shown in Fig. 1. A fiber laser source is used, with a central wavelength of 1550 nm, a linewidth of 5 kHz, and apower of 1.6 mW. The coupling coefficient of couplers C2 and C3 are 0.03. The length and the diameter of the OFRR are 14 and 0.12 m, respectively. The measured FSR is 14.7 MHz. The auto-controlled reset technique is also realized on the same FPGA with the digital PI controller. As Fig. 6 shows, two reference voltages, V ref and −V ref , and two set voltages, V set and −V set , are inserted between the maximum positive voltage (V max ) and the minimum negative voltage (−V max ) of the digital PI controller. The voltage difference between the reference and set voltages, V ref and −V set , V set and −V ref , tunes the laser frequency by integer multiples of the FSR. When the central frequency of the laser drifts positively relative to the resonant frequency of the OFRR, the PI output voltage increases progressively. As soon as the PI output voltage reaches the positive reference voltage V ref , it is reset to the negative set voltage −V set . Therefore, the central frequency of the laser can track another resonant frequency quickly. A similar process occurs when the laser frequency drifts negatively relative to the resonant frequency of the OFRR. Considering that the loop needs fine-tuning after being reset, as Fig. 6 shows, two voltage buffer regions are set between the two reference and set voltages to prevent from being reset ceaselessly. The voltage difference between the reference and set voltages is converted to an equivalent frequency deviation through the tuning parameter of the fiber laser. According to the definition of the FSR, both n and L are temperature dependent, thus making the FSR a temperature-dependent parameter. The voltage difference cannot be set ideally to tune the laser
Fig. 5. Locations of the central frequency of the laser on the demodulation curve after PI output resetting with different values.
Fig. 6. Diagram of the proposed auto-controlled reset technique. V max 1.15 V, V ref 0.82 V, and V set 0.41 V.
frequency exactly equivalent to integer multiples of the FSR. Figure 7 shows measurement results of the effects of the reset voltage difference on the demodulated error of the closed-loop output of the RFOG. Corresponding to Fig. 4, four different equivalent frequency deviations of −300 kHz, 0, 300 kHz, and 3.3 MHz are set in Figs. 7(a)–7(d). The resonant frequency of the OFRR drifts with environmental temperature changes in practice, which affects the locking process after reset. Therefore, there are differences between the experimental and simulation results. As seen in Fig. 7(a), as the equivalent frequency deviates three multiples of the FSR by −300 kHz, it induces a demodulated voltage error of −785 mV and needs 6.2 ms to get back to the lock-in state. When the voltage difference is equivalent to three multiples of the FSR, the demodulated voltage error should be theoretically zero. The demodulated error of −590 mV comes from two aspects. One is that the reset voltage used to tune the laser frequency by three multiples of the FSR cannot be exactly set. The other comes from the effect of the limited tuning bandwidth of the laser, which broadens the instant reset. Therefore, it induces a negative error and then is relocked to the resonant point. Figure 7(d) shows a large deviation from three multiples of the FSR. It takes 16 ms to get back to the lock-in state. The aforementioned demodulated error induces a frequency deviation of the lock-in frequency and thus finally induces a rotation rate error. Figure 8 shows the demodulation voltage error and the time required for returning to the lock-in state with different equivalent frequency deviations. Those simulated results are also presented in Fig. 8. As seen in Fig. 8(a), the experimental results agree well with those predicted from simulations. The error changes linearly with the frequency deviation in small frequency deviation, which corresponds to the linear region of the demodulation curve. Limited by the maximum demodulation value, the error tends to be stable in large frequency deviation. Figure 8(b) shows the relationship between the time required for returning to the lock-in state and the equivalent frequency deviation. The bigger frequency deviation is, the longer the lock-in time needs. For the frequency 20 December 2013 / Vol. 52, No. 36 / APPLIED OPTICS
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Fig. 7. Measurement results of the closed-loop output with different reset values.
deviation from −1.2 MHz to 1.2 MHz, the measured lock-in time is below 10 ms, which is consistent with the theoretical result. In the practical RFOG, the reset voltage always deviates from the practical one corresponding to integer multiples of the FSR. According to the definition of the FSR, it is given by
FSR
c : nL
(6)
The change of the FSR is mainly caused by n, which changes with temperature. The temperature coefficient of the refractive index of the silica fiber is approximately 10−5 ∕°C and the change rate of the FSR can be expressed as
Fig. 8. Demodulated voltage error and time required for returning to the lock-in state with different frequency deviations. (a) Relationship between the demodulated voltage error and the frequency deviation. (b) Relationship between the time required for returning to the lock-in state and the frequency deviation. 8776
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Fig. 9. Closed-loop output of the residual laser frequency noise. (a) Closed-loop output for the integration time of 1 s. (b) Allan deviation of the closed-loop output.
∂FSR c ∂ n1 c 1 ∂n − 2 −142 Hz∕°C: ∂T L ∂T L n ∂T
(7)
For the application where the temperature changes from −100°C to 100°C, the frequency deviation caused by the change of the FSR is from −14.2 kHz to 14.2 kHz. As seen in Fig. 8(b), the lock-in time for the frequency deviation from −14.2 kHz to 14.2 kHz is below 5 ms. Therefore, the effect of the reset caused by the change of the FSR can be ignored. When the PI controller overflows resetting, large reset pulses occur at the output of the RFOG. Figure 3(b) shows the measurement result of the closed-loop output with the auto-controlled reset technique applied to the output of the digital PI controller. As seen in Fig. 3(b), there is no obvious reset pulse occurring on the resetting points of the digital PI controller. Figure 9 shows the experimental results for long time duration. The calculated residual laser frequency noise is equivalent to a rotation rate of 0.18°∕h (1σ) for the integration time of 1 s. The Allan deviation of the residual laser frequency noise is shown in Fig. 9(b). The Allan deviation is dominated by the white frequency noise at the low integration time and reaches the so-called flicker floor at approximately 500 s and increases again due to the frequency drift. By extrapolating a fitted line with τ−1∕2 (τ is the p integration time), an ARW coefficient of 0.0028°∕ h is obtained. 4. Conclusion
An RFOG based on a digital PI controller with an auto-controlled reset technique is set up. Experiments show that the time required for returning to the lock-in state falls from 8 s to 5 ms. With the integration time of 1 s, the equivalent accuracy of the resonant frequency servo loop is increased to p 0.18°∕h. An ARW coefficient of 0.0028°∕ h is obtained for the resonant frequency servo loop. To the best of our knowledge, this is the best ARW result
reported for a resonant frequency servo loop in an RFOG. The authors would like to acknowledge financial support from the National Natural Science Foundation of China (No. 61377101). References 1. S. Ezekiel and S. K. Balsmo, “Passive ring resonator laser gyroscope,” Appl. Phys. Lett. 30, 478–480 (1977). 2. S. Ezekiel, “Optical gyroscope options: principles and challenges,” in 18th International Optical Fiber Sensors Conference Technical Digest (Optical Society of America, 2006), paper MC1. 3. G. A. Pavlath, “Fiber-optic gyros: the vision realized,” in 18th International Optical Fiber Sensors Conference Technical Digest (Optical Society of America, 2006), paper MA3. 4. A. Ohno, A. Kurokawa, T. Kumagai, S. Nakamura, and K. Hotate, “Applications and technical progress of fiber-optic gyros in Japan,” in 18th International Optical Fiber Sensors Conference Technical Digest (Optical Society of America, 2006), paper MA4. 5. G. A. Sanders, L. K. Strandjord, and T. Qiu, “Hollow core fiberoptic ring resonator for rotation sensing,” in 18th International Optical Fiber Sensors Conference Technical Digest (Optical Society of America, 2006), paper ME6. 6. K. Takiguchi and K. Hotate, “Partially digital-feedback scheme and evaluation of optical Kerr-effect induced bias in optical passive ring-resonator gyro,” IEEE Photon. Technol. Lett. 3, 679–681 (1991). 7. 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). 8. H. Ma, W. Wang, Y. Ren, and Z. Jin, “Low-noise, low-delay digital signal processor for resonant micro optic gyro,” IEEE Photon. Technol. Lett. 25, 198–201 (2013). 9. Z. Jin, X. Yu, and H. Ma, “Resonator fiber-optic gyro employing a semiconductor laser,” Appl. Opt. 51, 2856–2864 (2012). 10. R. Kingslake and B. J. Thompson, Applied Optics and Optical Engineering (Academic, 1965). 11. C. Ciminelli, F. D. Olio, and M. N. Armenise, “High-Q spiral resonator for optical gyroscope applications: numerical and experimental investigation,” IEEE Photon. J. 4, 1844–1854 (2012). 12. H. Ma, X. Yu, and Z. Jin, “Reduction of polarizationfluctuation induced drift in resonator fiber-optic gyro by a resonator integrating in-line polarizers,” Opt. Lett. 37, 3342–33449 (2012). 20 December 2013 / Vol. 52, No. 36 / APPLIED OPTICS
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13. 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). 14. 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).
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15. 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). 16. X. Zhang, H. Ma, Z. Jin, and C. Ding, “Open-loop operation experiments in a resonator fiber-optic gyro using the phase modulation spectroscopy technique,” Appl. Opt. 45, 7961– 7965 (2006).
