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Abstract
A resonant fiber optic gyroscope (RFOG) based on the reciprocal phase modulation-demodulation technique is proposed and demonstrated. The residual amplitude modulation induced error of the phase modulator, and the effect of laser frequency noise are all suppressed thanks to the reciprocity of the proposed signal processing scheme. Compared with the past separate modulation-demodulation RFOG, the angular random walk is improved by a factor of 15 times from 0.08°/√h to 0.0052°/√h, and the bias stability is improved from 0.3°/h to 0.06°/h.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical gyroscopes based on the Sagnac effect are important inertial rotation-rate sensors. The resonant fiber-optic gyroscope (RFOG) uses a recirculating fiber ring resonator (FRR) to increase the Sagnac effect and has a much shorter fiber length to achieve the same shot-noise-limited theoretical sensitivity compared with the well-established interferometric fiber-optic gyroscope (IFOG) [1–3]. More attractively, the RFOG can be realized in a more compact and robust version, that is the resonant micro-optic gyroscope (RMOG) with a centimeter-scale waveguide-type ring resonator or whispering gallery mode resonator [4,5]. However, the reported performance of the RFOG is well below that of the IFOG [6,7]. The major issue is that most parasite effects including Rayleigh backscattering, the polarization fluctuation, the nonlinear optical Kerr effect and the laser frequency noise encountered in the RFOG must be cancelled by additional countermeasures due to lack of a good reciprocal configuration and the usage of a highly-coherent laser source [8–13]. The polarization-fluctuation induced noise can be reduced by optimizing the FRR configuration with adoption of single-polarization (SP) fiber and a 90° polarization-axis rotated splice inside the cavity [14–15]. The Kerr-induced error results from a power difference between the clockwise (CW) and counterclockwise (CCW) waves circulating in the FRR, which can be diminished by implementation of the square-wave intensity modulation technique [10] or adoption of the hollow-core optical fibers [16]. As for the problem brought by Rayleigh backscattering, the most widely used solution is the carrier suppressed separate phase modulation-demodulation technique [8,17]. The CW and CCW input waves to the FRR are phase modulated at different frequencies with two separate electro-optic phase modulators (PMs) which are then synchronously demodulated at the output. The backscattering induced noise which affects the accuracy of the RFOG most seriously is successfully avoided in the rotation-rate detection band. An angle random walk (ARW) of 0.08 °/√h and a bias stability of 0.3 °/h have been reported with a fiber coil length of 18.25 m [14].
The amplitude of a purely phase modulated optical field at a fixed frequency should be constant ideally. However in fact, the pure phase modulation is hardly achieved due to the accompany of an amplitude modulation that is often referred to as a residual amplitude modulation (RAM) [18]. When the CW and CCW waves pass through different PMs and are phase modulated at different frequencies, the parasite RAMs are different so that they cannot be counteracted in the subsequent locking process. What’s more, the RAM increases with the decrease of the linewidth of the laser source. The difference of RAM between the CW and CCW waves finally limits the bias stability of the RFOG. Meanwhile, the intermodulation effect due to the laser frequency noise [11,13] is also nonreciprocal with separate phase modulations applied to the CW and CCW waves, which limits the practical ARW of the RFOG. The phenomenon is especially evident when adopting a miniaturized semiconductor laser as a light source for its white noise is much larger than the other types of the lasers with a same linewidth. Unfortunately, the servo loop with a limited loop bandwidth is only able to suppress the laser frequency noise at low frequency region but has no effect on high-frequency laser noise folded into the rotation-rate detection band through the modulation-demodulation process, which finally causes the deterioration of RFOG ARW.
A common modulation technique is proposed for a three-laser RFOG configuration to deal with problems with Rayleigh backscattering and modulation imperfections noise by Strandjord et al. [7]. The achieved bias stability is about 0.02 °/h with a fiber coil length of 100 m, and the ARW is 0.00293°/√h. Both the bias stability and the ARW are the best results reported to date in an RFOG. However, three lasers are employed and two additional optical phase lock loops are indispensable, which increases the system complexity.
In this paper, a reciprocal modulation-demodulation technique based on a single semiconductor laser is proposed and demonstrated. The effects of the RAM of the phase modulator and the frequency noise of the laser are all effectively reduced. Compared with the past separate modulation-demodulation RFOG [14], the angular random walk is improved by a factor of 15 times from 0.08°/√h to 0.0052°/√h, and the bias stability is improved from 0.3°/h to 0.06°/h.
2. Problem description
2.1 RAM of the phase modulator
The RAM of the PM yields a power variation when the light is phase modulated. Figure 1 shows the measured RAM of the LiNbO3 PM used in our RFOG system. The red solid line is the voltage signal applied to the PM and the blue dashed line is the normalized optical intensity at the output of the PM. The half-wave voltage of the PM is 2.35 V. As shown in Fig. 1, the peak-to-peak amplitude of the sawtooth wave is 9.4 V corresponding to a phase change of 4π. In the ideal case of pure phase modulation, the power of the output light should be constant. However, obvious power variation with a period consistent with the sawtooth voltage signal can be observed. The sinusoidal intensity variation in one period may come from the interference effect resulted from multiple reflections and scatterings in the LiNbO3 waveguides. A variety of effects can give rise to the RAM [18]. Here we will not investigate the detailed source mechanisms of the RAM but focus on the RAM noise problem in the phase modulation-demodulation technique.
The sinusoidal phase modulation-demodulation scheme is widely used in the RFOG. Figure 2 shows the ideal phase modulation-demodulation process with a purely phase- modulated signal. Figure 2(a) shows the RFOG at rest which corresponds to zero rotation rate and the laser frequency (fL) is kept on the resonant frequency of the FRR (fR). The detected signal of the FRR is an ideal second harmonic with the frequency of 2fm and fm is the modulation frequency. After synchronously demodulation at fm and the filter action with the low-pass filter (LPF), the demodulated output signal is zero. Figure 2(b) shows the RFOG in rotation with rate of Ω. There is a difference ΔfR between the laser frequency and the resonant frequency of the FRR due to the Sagnac effect. The detected signal contains the first harmonic with the frequency of fm. Then the demodulated output signal from the LPF is proportional to the rotation rate Ω.
Fig. 2. Ideal sinusoidal phase modulation-demodulation process in an RFOG. (a) At rest. (b) In rotation.
However, in the actual situation, the first harmonic related to the RAM of the PM, exists in the detected signal and will be demodulated as a bias error as shown in Fig. 3. The RAM-induced rotation-rate error is non-reciprocal to the CW and CCW waves when PMs of different characteristic are applied. The difference of RAM between the CW and CCW waves finally limits the bias stability of the RFOG.
2.2 Laser frequency noise
It is well known that an RFOG requires a highly coherent laser source. The effect of the laser frequency noise can be divided into two components, one of which is the frequency noise in the rotation band which is called the low-frequency laser noise, and the other is the high-frequency laser noise well above the rotation-rate detection band [12,13]. Generally, the central frequency of the laser source is locked to the resonant frequency of the FRR via a high performance laser-frequency servo loop so as to reduce the effect of the low-frequency laser noise. However, the high-frequency noise which cannot be reduced by the laser-frequency servo loop due to the limited loop bandwidth is still a remaining issue, since the laser frequency noise at even harmonics of the modulation frequency is down converted to the low-frequency gyro output through modulation and demodulation process, as known as the intermodulation effect [11,13]. The effect of the high-frequency laser noise was also appreciated by Qiu et al. [12].
Figure 4 shows the effect of the laser frequency noise at the second harmonic with the frequency of 2fm in the phase modulation-demodulation process. fm is the modulation frequency. The RFOG is at rest and the central frequency of the laser (fL) is locked to the resonant frequency of the FRR (fR). A phase modulated signal with the frequency of fm is input to the FRR. As shown in Fig. 4, the first harmonic with the frequency of fm arisen from the laser frequency noise at 2fm exists in the detected signal which will be synchronously demodulated as bias error. In the past separate phase-modulation RFOG, different modulation frequencies are applied to the CW and CCW waves in order to reduce the backscattering induced noise [8,17]. The difference of laser frequency noise between the two counter-propagating waves yields the demodulation error at the gyro output and finally limits the gyro performance, which is much more observable while a semiconductor laser is adopted as a result of the laser frequency white noise at high frequency range.
3. Reciprocal modulation-demodulation RFOG
The system configuration of the reciprocal modulation-demodulation RFOG is shown in Fig. 5. Three PMs are used, a primary PM and two secondary PMs. The primary phase modulator, PM0, is used for gyro signal modulation and demodulation. Two secondary phase modulators, PM1 and PM2, are driven by two different sinusoidal voltage waves at different frequencies f2 and f3, which enhance the suppression of the nonreciprocal Rayleigh backscattering induced noise [19]. The output light from the semiconductor laser is phased modulated by PM0 at frequency f1 and then divided into two beams by a 3 dB coupler. The two beams are further phase modulated by PM1 and PM2 and then coupled to the FRR in CW and CCW directions. Two lengths of 5-meter-long single-polarization fibers are spliced into the FRR to suppress the nonreciprocal polarization-fluctuation induced noise. The laser beam in CW direction is detected by PD2 and fed back to the semiconductor laser by the proportional-integral (PI) controller to lock the laser central frequency to the resonant frequency of the FRR. The laser beam in CCW direction is detected by PD1 and then demodulated and filtered to give out the rotation signal.
Compared with the past separate modulation-demodulation technique where the CW and CCW beams are phase modulated and then synchronously demodulated at different frequencies, the reciprocal technique performs the modulation before light being splitting which makes most of the RAM and laser frequency noise reciprocal to the CW and CCW waves. Detailed explanation is as follows.
According to Fig. 5, the demodulation signal of the CW wave is used as an error signal to keep the laser frequency on the resonant frequency of the FRR in the CW direction. When the laser frequency is close to the resonant frequency, the demodulation output from LPF2 is approximately proportional to the difference between the laser frequency and the resonant frequency given by [20]
where fL is the laser frequency, fR_CW is the resonant frequency of the FRR in the CW direction, and KCW represents the slope of the demodulation curve. Considering the effect of the RAM and the laser frequency noise, Eq. (2) is modified asIn a practical RFOG system, the laser frequency is always locked to the resonant frequency of the FRR in the CW direction, therefore, the demodulation output signal of the CW wave should be zero, that is
Then the demodulation output signal of the CCW wave becomes
The difference between the resonant frequencies in the CW and CCW directions represents the rotation-rate induced frequency shift which is proportional to the rotation rate. Therefore, the demodulation signal of the CCW wave is proportional to the rotation rate which is not affected by both the RAM-induced error and the laser frequency noise thanks to the reciprocity of the signal modulation technique.
4. Experiment results
A reciprocal phase-modulation RFOG depicted in Fig. 5 is setup and tested. The semiconductor laser works at 1550 nm with a linewidth of 3 kHz. The sensing element is a hybrid single-polarization FRR with a fiber coil length of 29 m, a diameter of 12 cm and a finesse of 14.7. The primary modulation frequency f1 is 240 kHz and the secondary modulation frequencies f2 and f3 are 134 Hz and 150 Hz, respectively. The theoretical sensitivity limited by the shot noise of the photodetector is given by [21]
The power spectral density (PSD) of a 5-minute test of the gyro output is shown in Fig. 6. The PSD of the random noise SΩ is 0.198(°/h)2/Hz in the bandwidth of 2 Hz to 5 Hz.
According to the relationship of the PSD of the measured rotation rate and the ARW expressed in Eq. (8),
where hr and Hz are both units, SΩ is the PSD of the gyro output. An calculated ARW from Eq. (8) is 0.0052°/√h. Compared with the ARW of 0.08°/√h obtained in the past separate modulation-demodulation technique [14], the reciprocal scheme improves the ARW by more than 15 times.The reciprocal phase-modulation RFOG system is then mounted on a high-accuracy rotation table. Figure 7 shows the measured results and the Allan deviation analysis. For the comparison, results from the separate phase-modulation system are also added. Figure 7(a) shows the gyro output of the reciprocal phase-modulation RFOG. A peak-to-peak bias fluctuation is 2.6 °/h, a standard deviation of the bias stability is 0.37 °/h, and the bias is 18.8 °/h. Figure 7(b) is the measured result of the past separate phase-modulation system [14]. A peak-to-peak bias fluctuation is about 35 °/h, a standard deviation of the bias stability is 4.6°/h, and the bias is 93.8 °/h. Figures 7(c) and 7(d) show the Allan deviation analysis of the two systems. The bias stability of the reciprocal phase-modulation RFOG is 0.06°/h at an integration time of 370 s which is improved by a factor of 5 times compared with that of 0.3°/h obtained in the past separate phase-modulation system.
Fig. 7. Measurement results of the RFOG at rest. (a) Gyro output and (c) Allan deviation analysis of the reciprocal phase-modulation RFOG. (b) Gyro output and (d) Allan deviation analysis of the separate phase-modulation RFOG.
The sinusoidal rotation measurement results of the reciprocal phase-modulation RFOG with an integration time of 1s is shown in Fig. 8. The rotation table is set to work under sinusoidal swing mode. The gyro output data are sampled when the rotation amplitudes of the sinusoidal swing are set to ±0.1°/s, ±0.005°/s, ±0.001°/s, ±0.0005°/s, ±0.0002°/s, and ±0.0001°/s, respectively. According to Fig. 6, the random noise is 1.24 × 10−4°/s in the bandwidth of 1 Hz which agrees well with the measured minimum rotation rate of 0.0001°/s.
5. Conclusions
In conclusion, an RFOG based on the reciprocal modulation-demodulation technique is proposed, and the ARW is improved by a factor of 15 times from 0.08°/√h to 0.0052°/√h. In a test time span of 1800 s, the gyro output keeps stable and the bias stability achieves 0.06 °/h in an integration time of 370 s. At the integration time of 1 s the minimum sinusoidal rotation rate of 0.0001°/s is successfully measured. The reciprocal RFOG not only effectively suppresses the RAM-induced error of the phase modulator but also reduces the effect of the laser frequency noise of the semiconductor laser, which has great value in the integration and miniaturization of the RFOG.
Funding
National Natural Science Foundation of China (61675181, 61377101).
Acknowledgments
The authors would like to acknowledge the financial support from the National Natural Science Foundation of China.
Disclosures
The authors declare no conflicts of interest.
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