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Abstract
We experimentally demonstrated that the random walk in a dithered ring laser gyroscope caused by lock-in crossing can be eliminated by lock-in error compensation. Computer simulations demonstrated the effectiveness of this method. The experimental results show that the random walk coefficient can be reduced as predicted. Furthermore, gyroscopes with different random walk performances can achieve nearly the same random walk level after compensation, illustrating that the quantum limit can be approached after removing the lock-in error.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Compared to other kinds of gyroscopes such as fiber optical gyros and hemispherical resonator gyros [1–6], the mechanically dithered ring laser gyroscopes (RLGs) have outstanding stability performance; thus, they are widely used in high-accuracy navigation systems [7,8]. The noise in the RLG output can be divided into bias instability, angle random walk, rate random walk, quantization noise, and rate ramp [9]. In a mechanically dithered RLG, the noise terms of rate random walk, bias instability, and rate ramp are not obvious, and the quantization noise has no accumulation error; therefore, the determining noise term is the angle random walk, and the ultimate performance of the navigation system is limited by this term. A mechanically dithered RLG has two sources of angle random walk: one arising from spontaneous emission of the gain media [10–13], and the other from the random lock-in crossing [3]. The former is the quantum limit of the ring laser gyroscopes.
The RLG has an intrinsic lock-in problem. Because of the coupling of the counter-propagating laser beams when the rotation rate is too low, the beams in the cavity are synchronized with each other, and no beat frequency is generated. The sinusoidal mechanical dither bias method can cause the RLG to work out of the lock-in zone most of the time. To overcome the error accumulation of every lock-in cross moment, noise is injected into the dither driving signal to fully randomize the lock-in error; thus, the angle random walk error is generated [2].
The lock-in error is centered in the lock-in cross moment and exhibits a step character [14]. If the step error can be represented by observable signals, then the compensation can be realized. Many schemes for lock-in error compensation have been proposed [15,16]; however, the experimental realization has never been reported. In this study, using the Fresnel approximation, the lock-in error is expressed in the form of the beat phase and angular acceleration in the lock-in cross moment. By detecting the differential of the two orthogonal beat-frequency signals, the exact time point of lock-in crossing was determined, and the phases and angular acceleration at that point was recorded. Other parameters such as the lock-in threshold can be determined using least-squares fitting. Experimental results show that the random walk coefficient (RWC) can be reduced from $4.6\textrm{e}-4^{\circ}/\sqrt{\textrm{h}}$ to $2.1\textrm{e} - 4^{\circ}/\sqrt{\textrm{h}}$ after compensation. More experiments on other gyroscopes with different random walk performances revealed that they can achieve nearly the same random walk level after compensation. This specified that the residual random walk is cause by quantum limit.
2. Theoretical basis
2.1 Stepping lock-in error
In a two-mode RLG, the beat frequency equation can be described as [17]
where $\varphi$ is the phase of the counter-propagation laser beams; ${\Omega _{in}}$ is the input of the RLG; ${\Omega _L}$ is the lock-in threshold which is determined by the mode coupling due to backscattering; and $K$ is the scale factor. When a mechanical dither bias is adopted, Eq. (1) can be written as where ${\Omega _d}$ is the dither amplitude and ${\omega _d}$ is the dithering frequency. Under this condition, the output error can be expressed asAssuming
Figure 1 shows that the ${E_{error}}$ oscillates most of the time without accumulation error. However, at the moment of the lock-in crossing, a dramatic stepping in the integrated angle error can be observed. Fortunately, we can obtain an approximate analytical expression for the stepping error signal.
At the moment of forward lock-in crossing, the input angular rate $\dot{\varphi }$ becomes zero, and the angular acceleration $\ddot{\varphi }$ reaches its maximum and changes the slowest. Therefore, $\ddot{\varphi }$ can be treated as a constant, and the phase around the lock-in crossing can be expressed as a parabolic curve
where ${\varphi _0}$ is the phase in the lock-in crossing moment, and $t$ is the time that satisfy $t = 0\textrm{s}$ when $\varphi = {\varphi _0}$. Substituting Eq. (5) into Eq. (3), we obtainFigure 2 shows the integration of $\int_{ - r}^r {\sin ({{x^2}} )dx}$ and $\int_{ - r}^r {\cos ({{x^2}} )dx}$ under different values of r. When $r$ is increased to $+ \infty$, the integration approaches to an asymptotic value of $\sqrt{\frac{\pi }{2}}$. In fact, this is the famous Fresnel integration equation [18]
Substituting Eq. (7) into Eq. (6) gives
Similarly, the stepping error in the backward lock-in crossing can be derived as
Considering the sign of $\ddot{\varphi }$ is negative in backward lock-in crossing, the lock-in error can be expressed in a unified complex form as
2.2 Statistical analysis of lock-in error
The lock-in error expressions in Eq. (8) and Eq. (9) can be treated as discrete noise injections change the continuous error accumulation into a discrete form: the stepping error signal is added only at every lock-in crossing moment. If the lock-in phase is fully randomized, stochastic noise would be generated, which has a uniform distribution between [-π, π]. It’s probability density function (PDF) can be expressed as Eq. (11), as shown in Fig. 4.
According to the PDF lock-in phase ${\varphi _0}$, the PDF of every lock-in error $\Delta E$ can be further obtained as Eq. (12), which is shown in Fig. 5(a).
Figure 5(b) shows the Monte Carlo computer simulation of the lock-in error distribution with 100000 sample points in total. The simulation results agreed with the theoretical distribution.
Using the parameters in Eq. (4), the lock-in error can be numerically calculated to be in the range of $[{ - 0.0019^{\prime\prime},0.0019^{\prime\prime}} ].$ The variance of $\Delta E$ is
A dither cycle has two lock-in crossings; therefore the variance in the unit time is
Using the parameters in Eq. (4), we can obtain that $RWC_L = 6.2e-4^{\circ}/\sqrt{\textrm{h}}$. The total random walk in the output of RLG is the superposition of both the quantum noise and the random lock-in error. Assuming that the random walk induced by quantum noise is $RWC_Q = 2e-4^{\circ}/\sqrt{\textrm{h}}$, the total random walk is
Figure 6 shows the Monte Carlo simulation of the angle random walk noise for a duration of 10 h. The red solid line shows the angle random walk generated from the lock-in error, and red dashed lines are the boundaries of ${\pm} 6.2e - 4^{\circ}/\sqrt{\textrm{h}}$. the angle random walk error was mostly inside the upper and lower boundaries. This is similar for quantum-induced angle random walk (green lines) and the total random walk (blue lines).
3. Experiment and discussion
3.1 Experiments
The output beat frequency of RLG is often sensed using two photodiodes placed in the heterodyne plane of the combining prism, and their phases are in quadrature. Assuming that the phase of one photodiode ${\varphi _{H1}}$ has an offset of $\varepsilon$ from the phase of $\varphi$ in Eq. (2), their relationship can be expressed as
The stepping errors at the forward and backward lock-in crossings in Eq. (8) and Eq. (9) can be expressed as
Letting
Both ${\Omega _L}$ and $\varepsilon$ can be treated as constants in relatively stable environments. Therefore, a and $b$ can also be treated as constants. Although the ${\Omega _L}$ and $\varepsilon$ are unknown, they can be obtained through curve fitting. Eq. (18) can be written as
Based on Eq. (20), a mixed signal processing circuit board was developed to realize the lock-in error compensation. A dual-channel high-speed analog-to-digital converter (ADC) with a sample frequency 50 MHz was used to sense the two beat-frequency signals simultaneously. After that, the signals were sent to a field programmable gate array (FPGA) chip. The software processing procedure is shown in Fig. 7. The FPGA dealt with the real-time data stream to determine the exact time point of the lock-in crossing. Once the crossing was detected, variables related to lock-in error are calculated and recorded in memory, such as $\ddot{\varphi }$, $\sin {\varphi _{H1}}$, $\sin {\varphi _{H2}}$.
3.2 Result and discussion
Figure 8 shows the output of one RLG for a duration of 10 h. Figure 8(a) is the time-domain data before lock-in error compensation, and Fig. 8(b) shows the data after compensation. By comparing Fig. 8(a) and Fig. 8(b), random fluctuations were found to decrease after compensation.
Allan variance analyses of the data shown in Fig. 8 were carried out. The following equation was used to fit the Allan variance curve [19]
Figure 9 shows the Allan variance curves at different time intervals before and after compensation. the Allan variance was found decreased after compensation. Using least-square fitting of Eq. (21), the RWCs before and after compensation can be calculated as $4.55e - 4^{\circ}/\sqrt{\textrm{h}}$ and $2.05e - 4^{\circ}/\sqrt{\textrm{h}}$ respectively. The results state that the random walk source from the lock-in can be removed with the compensation, and residual random walk is from the quantum noise. The quantum noise is generated by the spontaneous radiation of upper energy level, and cannot be removed.
Table 1 lists the lock-in error compensation results of four RLGs. We can see that although the RWCs of the four samples differ greatly before compensation, the RWCs can nearly reach the same value after compensation, which is about $2.1e - 4^{\circ}/\sqrt{\textrm{h}}$. It specified that for RLGs with fixed optical path structures, the quantum limits are nearly the same although they may have different lock-in thresholds.
Table 1. Random walk reduction of different RLGs
4. Conclusion
In a mechanically dithered RLG, the lock-in error is mainly centered at the lock-in crossing moment and takes on a stepping characteristic. Using the Fresnel integration property, an approximate analytical expression of each lock-in error was obtained. A dedicated signal processing circuit board based on FPGA and high-speed ADC was developed to realize the lock-in error compensation. The experimental results show that the random walk error is reduced after the lock-in compensation as the simulations predicted. Experiments on more RLGs verified that gyros with different performance can reach the same level of RWC after lock-in error compensation. The residual random walk is caused by the spontaneous radiation. These results also indicate that the quantum limit is nearly fixed for gyroscopes with the same design parameters and manufacturing processes.
Funding
National Defense Science and Technology Innovation Zone (22-TQ05-00-TS-01-043); Science Foundation for Indigenous Innovation of National University of Defense Technology (22-ZZCX-063); National Natural Science Foundation of China (62375285).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. E. J. Post, “Sagnac Effect,” Rev. Mod. Phys. 39(2), 475–493 (1967). [CrossRef]
2. W. W. Chow, “The ring laser gyro,” Rev. Mod. Phys. 57(1), 61–104 (1985). [CrossRef]
3. J. R. Wilkinson, “Ring lasers,” Prog. Quant. Electron. 11(1), 1–103 (1987). [CrossRef]
4. M. Faucheux, D. Fayoux, and J. J. Roland, “The ring laser gyro,” J. Opt. 19(3), 101–115 (1988). [CrossRef]
5. A. D. King, “Inertial navigation-forty years of evolution,” Geo. Rev. 13(3), 140–149 (1998).
6. W. W. Chow, “Multioscillator laser gyros,” IEEE J. Quantum Electron. 16(9), 918–936 (1980). [CrossRef]
7. Z. Fan, H. Luo, and S. Hu, “Instantaneous phase method for readout signal processing of body dithered ring laser gyro,” Appl. Opt. 50(20), 3455–3460 (2011). [CrossRef]
8. A. Bambini and S. Stenholm, “Theory of a dithered ring laser gyroscope: a floquet-theory treatment,” Phys. Rev. A 31(1), 329–337 (1985). [CrossRef]
9. IEEE Aerospace and Electronic Systems Society, “IEEE standard specification format guide and test procedure for single-axis laser gyros,” in IEEE Std 647-2006 (2006).
10. J. D. Cresser, W. H. Louisell, P. Meystre, W. Schleich, and M. O. Scully, “Quantum noise in ring-laser gyros. I. Theoretical formulation of problem,” Phys. Rev. A 25(4), 2214–2225 (1982). [CrossRef]
11. J. D. Cresser, D. Hammonds, W. H. Louisell, P. Meystre, and H. Risken, “Quantum noise in ring-laser gyros. II. Numerical results,” Phys. Rev. A 25(4), 2226–2234 (1982). [CrossRef]
12. J. D. Cresser, “Quantum noise in ring-laser gyros. III. Approximate analytic results in unlocked region,” Phys. Rev. A 26(1), 398–409 (1982). [CrossRef]
13. W. Schleich, C. S. Cha, and J. D. Cresser, “Quantum noise in a dithered-ring-laser gyroscope,” Phys. Rev. A 29(1), 230–238 (1984). [CrossRef]
14. Z. Fan, H. Luo, G. Lu, and S. Hu, “Theoretical research on lock-in error compensation for mechanically dithered ring laser gyro,” Act. Opt. Sin. 31(11), 1112006 (2011). [CrossRef]
15. S. Song, J. Lee, S. Hong, and D. Chwa, “New random walk reduction algorithm in ring laser gyroscopes,” J. Opt. 12(11), 115501 (2010). [CrossRef]
16. Y. K. Gao and Z. L. Deng, “A new method for eliminating the lock-in error of mechanically dithered ring laser gyro,” Chinese Jou. Las. 34(3), 354–358 (2007).
17. Y. N. Jiang, Ring Laser Gyro (Tsinghua University Press, 1985).
18. A. E. Siegman, Lasers (University Science Books, 1986).
19. M. Wei, M. Yang, and X. Qu, “Allan variance analysis of random error in ring laser gyro,” J. Proj. Roc. Mis. & Guid. 25, 495–498 (2005).
