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Abstract
Fiber optic gyroscopes (FOGs) suffer from the scale-factor inaccuracy induced by the wavelength instability of the broadband source, which remains a bottleneck both in theory and in practical application. In this work, we propose a simple but effective technique for reducing the wavelength dependence of the scale factor by employing the size of the digital-ramp register as the actuator in the closed-loop scheme for nulling the ramp-reset-induced errors, instead of the conventionally-used feedback-chain gain. Experiments show that, for the tactical-grade FOG equipped with the super-luminescent diode (SLD) operating under temperatures from −40 °C to +60 °C, the proposed technique reduces the compensated scale-factor inaccuracy to 282 ppm, with respect to 2065ppm in the conventional case. This technique relaxes the stringent requirements on the wavelength stability of SLDs, which contributes to the large-scale production and application of tactical-grade FOGs.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The continuous advancements in gyroscope technologies have significantly promoted the developments of high-precision inertial navigation systems (INSs), among which fiber optic gyroscopes (FOGs) [1], ring laser gyroscopes (RLGs) [2,3], microelectromechanical (MEMS) gyroscopes [4], and hemispherical resonator gyroscopes (HRGs) [5] are technically mature and commercially available for various applications. Recently, benefit from photonic innovative solutions, optical gyroscopes, including FOGs and RLGs, are progressing towards chip-based design and implementation to achieve miniaturization, lower power consumption, and lower cost [6–8].
In various fields across the world, the FOG has exhibited outstanding performances, in which the Sagnac effect is magnified and measured using the counterpropagating interfering light waves in the passive optical-fiber loop and the assembly of discrete optical components with fiber provides more flexibility in design and manufacture. The intrinsically advantaged configurations, together with a number of brilliant techniques [9–14], contribute to the performances of very low noise and very high bias stability for the FOG, which have supported tactical, navigational, and strategic applications [15–20]. However, the scale-factor accuracy of the FOG is not satisfactory, especially when comparing with the RLG, which has become an acknowledged shortcoming in the FOG applications [21]. The reason lies in that, different from the RLG that employs a narrowband laser [22,23], the FOG relies on the broadband light source driving the Sagnac interferometer [9], which plays an important role in avoiding the correlation-related errors generated in the interferometer and significantly promotes the long-term accuracy and stability of the FOG [24,25]. However, the broadband source brings some inevitable defects, and the best-known one is the intrinsic wavelength instability leading to the scale-factor inaccuracy of the FOG [26,27].
Focusing on the scale factor of the FOG, its impacting factors mainly contains the wavelength of the interfering light waves, the geometry of the interference loop, and the operating biasing point determined by the signal processing technique [28,29]. Benefit from the closed-loop operations [30] and the reciprocal biasing modulation-demodulation [28], the scale-factor accuracy is nearly unaffected by the signal processing [1]. Besides, the geometrical parameters of the interference loop can be effectively extracted and then be compensated in the FOG output in real time [31–33]. As a result, it is the optical-wavelength fluctuations with respect to time, temperature and optical power that significantly impacts the scale-factor accuracy, which has been proved and widely accepted [12,34,35].
In order to improve the accuracy of the scale factor, a number of methods have been put forward to eliminate the wavelength instability in the FOG [9,29,36–40], which are demonstrated to be effective both in theory and in practice. The broadband source has been redesigned and optimized to achieve better wavelength stability [38,39]. Besides, some specially-designed filters are added to the exit of the source or to the entrance of the photodetector to make sure that the detected optical spectrum remains stable after the broadband light being filtered [40]. Also, the wavelength distribution at photodetection can be measured out and then is utilized to cancel out the scale-factor fluctuation [9,29]. What is more, the narrowband laser has been introduced as the source for the FOG to achieve better scale-factor performances [22]. However, these proposed techniques usually lead to the deterioration of other FOG performances, like the angular random walk and the long-term bias stability, and often require a number of optical devices with large volume or the assembly with complicated configurations, which are not applicable in small-sized tactical-grade FOGs.
Unfortunately, in tactical-grade FOGs, the problem in achieving scale-factor accuracy is more significant. Tactical-grade FOGs have the bias stability of 1°/h to 0.01°/h and are usually applicated in small-sized vehicles, like the tactical missiles [41]. With the requirements of miniature size, large-scale production, high level of consistency and production efficiency, and low cost for the tactical-grade FOGs, the miniaturized super-luminescent diode (SLD) is typically utilized as the broadband source [42]. However, its wavelength instability is more significant than that for the Er-doped fiber source, that are often employed in navigation-grade and strategic-grade FOGs, especially under stringent thermal conditions or in long-term operations [43]. Furthermore, the wavelength-variation tendency for the SLD is usually different from one to another making the compensation for the scale factor difficult to implement, which is a disaster in the FOG production and application in large scale.
In this paper, rather than painfully eliminating the wavelength fluctuations, we propose a scheme to reduce the wavelength dependence of the scale factor without adding any extra device, fundamentally decreasing the scale-factor error induced by the wavelength instability. In this case, the scale factor relies on the feedback-chain gain of the FOG instead, which has better stability and compensability than the wavelength of the SLD. With the wavelength-dependence-reduction scheme applied, the scale factor shows better accuracy than that with the conventional scheme, under the conditions of manually-modified wavelengths and stringent thermal environments, which has been verified experimentally. The reduction of the wavelength dependence for the scale factor is meaningful for relaxing the stringent requirements on SLDs, which contributes to the production-efficiency promotion and the cost reduction for tactical-grade FOGs. Also, this proposed technique is referable for realizing the high scale-factor accuracy in navigation-grade and strategic-grade FOGs.
2. Wavelength dependence of the scale factor in a conventional FOG
2.1 Scale factor of the FOG when excluding the effect of ramp-reset
To facilitate the following derivations for the scale factor, the optical frequency v is employed instead of the frequently-used wavelength λ, where v = c/λ with c representing the light speed in vacuum. The Sagnac effect is expressed as the rotation-induced phase difference Δφs at the frequency of v and shown as
In conventional FOGs, in order to improve the linearity of scale factor, the closed-loop scheme for nulling the rotation-induced phase difference is implemented by the feedback technique using the digital ramp [1,30], which is illustrated as the solid arrows and the connected blocks in Fig. 1. The multifunction integrated-optic circuit (MIOC) is usually employed as the electro-optic phase modulator, which is based on the Pockels effect. When excluding the practical ramp-reset effect, the accumulation in the generation of the ramp and the first-order difference in making the feedback difference through the MIOC are balanced each other, which are shown as the blue blocks in Fig. 1. Consequently, each step ΔQf in the digital ramp results in the feedback phase difference of Δφf nulling the rotation-induced phase difference, which is calculated as [1,44,45]
Fig. 1. The schematic of the conventional FOG, in which the closed-loop scheme for the rotation rate is implemented by the feedback phase difference generated by the ramp and the closed-loop scheme for the reset error is implemented by adjusting the feedback-chain gain. As the closed-loop cycle for the rotation rate is the same as the transition time τ, setting z−1 = e-jwτ, 1/(1-z−1) and (1-z−1) stand for the transfer functions of the ramp generation and the first-order difference, respectively, using the Z-transform.
Similarly, for the purpose of improving the sensitivity and the signal-to-noise ratio, an extra biasing phase modulation is also applied to the MIOC, leading to the phase difference between the two interfering light waves as Δφm = 2πΔtmv, where Δtm is the temporal delay generated by the biasing phase modulation. Although, nonlinearity and dispersion exist in Δtm in a similar manner as Eq. (2), they have minor effect in the following derivations. As the biasing modulation typically varies in a square-wave-like form with the cycle of 2τ, where τ is the transition time for the light propagating in the fiber coil, Δtm is simplified as the temporally alternating constants of ± tm as shown in Fig. 1.
With the temporal delays in the fiber loop and the MIOC at each individual frequency, the optical spectrum, which represents the power spectral density of the electric field vector, at the output of the interferometer, is expressed as [47]
At photodetection, when approximately treating the responsivity for the optical spectrum as a constant k0 within the optical spectral range of vrange, the output voltages Vc1 and Vc2 corresponding to each square-wave modulation state (Δtm = +tm and Δtm = −tm) are shown as [30]
In the closed-loop scheme, the tracking error for the rotation rate should converge to zero, which is demodulated as [30]
As the ramp-step ΔQf acts as the rotation-rate output of the FOG at the same time, referring to Eqs. (1), (2) and (6), the scale factor, which is the ratio between the FOG output and the input rotation rate is calculated as
Equation (7) shows that, in this case, the scale factor mainly depends on the actual feedback-chain gain and is, to first order, frequency-independent, considering that the dispersion effect is fairly small. Here, we highlight that Eq. (7) holds true when the effect of ramp-reset is gated out [1]. Practically, as the uprising ramp will impossibly exceed the rails of electronics, the reset of ramp inevitably occurs [30,48]. When the effect of ramp-reset is taken into account, the properties of the scale factor will become completely different.
2.2 Scale factor of the FOG when including the effect of ramp-reset
In the conventional all-digital closed-loop scheme, the reset of the ramp happens at the automatic overflow of the digital ramp register. For a N-bit ramp register, the reset process generates the digital amplitude of ΔQr = 2N applied to the MIOC. Similar to Eq. (2), the phase difference Δφr generated by ΔQr via the feedback chain is [44,48]
In order to avoid any reset-induced error, Eq. (10) should converge to zero, which is implemented by the closed-loop scheme for the reset error demonstrated as the dashed arrows in Fig. 1. The solution to Eq. (10) is [49]
where $\bar{v}$ is the mean frequency determined by the closed-loop scheme for the reset error and is completely different from $\tilde{v}$ in Eq. (6) obtained from the closed-loop scheme for the rotation rate. Referring to Eq. (8), the feedback-chain gain becomesPlugging Eq. (12) into Eq. (7), the scale factor is shown as
2.3 Discussion on the accuracy of the scale factor
With the expressions of the scale factors in Eqs. (7) and (13), the dominant factors, that contribute to the scale factors with and without the ramp-reset effect, are proved to be the unconstrained feedback-chain gain and the mean frequency, respectively.
The unconstrained feedback-chain gain is, to first order, wavelength-independent, but it varies with temperature in the order of −800 ppm/°C (ppm, i.e. parts per million) [1]. Luckily, the thermal dependence is approximately regular and relatively stable, which is easily compensated to several ppm/°C.
The stability for the mean frequency is with more uncertainties, which depends on the applied broadband source and the application environments. For the SLD massively employed in tactical-grade FOGs, it has a typical mean-frequency drift of nearly 400 ppm/°C with temperature and of nearly 40 ppm/mA with driving current. The aging effect and the feedback effect of the returning light make the drift even more serious [1]. Additionally, the mean-frequency variation for the SLD is usually ruleless, and changes from one element to another and with different application conditions, especially for the SLD without cooling inside. As for the Er-doped fiber source, which is typically used in navigation-grade and strategic-grade FOGs, its mean-frequency drift is significantly smaller than that of the SLD, as the energy levels of rare-earth are much more stable than those of the semiconductors.
Usually, when using the Er-doped fiber source, the stability of the scale factor in linear relationship with the mean frequency can be guaranteed. However, for the SLD with severe and ruleless variation of the mean-frequency, if the scale factor gets rid of the mean frequency and depends on the unconstrained feedback-chain gain instead, the scale factor will see more regular thermal dependence and better accuracy under different conditions. In that case, the stringent requirements on SLDs will be relaxed in the production of tactical-grade FOGs with large scale, which is meaningful in the production-efficiency promotion and in the cost reduction. The technique for the reduction of the wavelength dependence for the scale factor is demonstrated in Section 3.
The nonlinear parts in the scale-factor expressions of Eqs. (7) and (13) lead to the variations of the scale factor with the amplitude and the sign (plus or minus) of the outputs for FOGs, which are often referred to as the scale-factor nonlinearity and the scale-factor asymmetry in the analysis and measurement of the scale factor. In this work, we focus on the wavelength dependence of the scale factor and the nonlinear parts can be temporarily omitted. The quantitative study for the nonlinear errors in the MIOC is the future work.
3. Wavelength-dependence reduction scheme for the scale factor
In the conventional scheme shown in Fig. 1, the feedback-chain gain is the one to be adjusted to null the reset-induced error, which inevitably results in the feedback-chain gain and then the scale factor being forced to vary with the mean frequency. In order to break up the linear frequency-dependence of the scale factor, a challenge has to be overcame, which is to make the feedback-chain gain independent of the mean frequency while avoiding the reset-induced error at the same time.
Here, the focus is shifted to the size of the digital-ramp register ΔQr, which corresponds to the reset-induced phase difference. As is stated in Section 2.2, we used to take ΔQr being constant for granted. However, referring to Eq. (12), if ΔQr is treated as a variant to compensate for the change in the mean frequency, the feedback-chain gain becomes unconstrained again. The closed-loop scheme for the reset error remains unchanged except that the actuator is transferred to the size of ΔQr instead of the feedback-chain gain, which nearly decouples the feedback-chain gain from the mean frequency. It is worth mentioning that the number of digits for the digital-ramp register needs to be extended properly in adjusting the size of ΔQr to avoid the following quantization error.
The actions on the ramp by controlling the feedback-chain gain ${\bar{K}_\textrm{f}}$ and the size of the digital-ramp register ΔQr are compared as Fig. 2. When adjusting ${\bar{K}_\textrm{f}}$, the amplitude of the ramp is altered integratedly (Fig. 2(a)), where the step (ΔQf) and the peak-to-peak amplitude (ΔQr) of the ramp are simultaneously amplified by the same factor (a). As a result, when the variation of the mean frequency leads to the change in the feedback-chain gain, the ramp-step, which also acts as the FOG output, is altered, thus impacting the scale factor. However, when the variation of the mean frequency just influences the peak-to-peak amplitude (ΔQr) of the ramp (Fig. 2(b)), the ramp-step remains unaffected and the scale factor then becomes nearly unrelated to the mean frequency. For both the two cases, the variations in the peak-to-peak amplitude of the ramp are identical, so as to avoid the reset-induced error [48]. It is the different consequences acted on the ramp step that brings about the distinctions on the scale factor.
Fig. 2. The actions on the ramp to null the reset-induced error (a) by adjusting the feedback-chain gain, which results in the simultaneous modifications of the ramp step and the peak-to-peak amplitude of the ramp, leading to the scale factor (the ratio between the ramp step and the input rotation rate) dependent on the mean frequency, and (b) by adjusting the size of the digital-ramp register, which results in the modification of the peak-to-peak amplitude of the ramp only, leading to the ramp step and then the scale factor nearly independent of the mean frequency.
Therefore, we can build up two working modes in the FOG, that adjust the feedback-chain gain and the size of the digital-ramp register in the closed-loop operation for the reset error corresponding to Fig. 2(a) and (b). The two modes can be switched upon working conditions or upon external directives. The proposed mode, in which the scale factor is nearly unrelated to the mean frequency, is especially meaningful for the tactical-grade FOGs under stringent environments, in which the mean frequency of the SLD varies severely and irregularly. What is more, this mode can be treated as an alternative for high-precision FOGs under long-term or inaccessible applications like the satellites or the submarines. With the wavelength (frequency)-dependence reduction of the scale factor, even if the light source is malfunctioning with a scrambling of the optical spectrum, the scale-factor accuracy is guaranteed at the level of the stability of the compensated feedback-chain gain.
4. Experiments
4.1 Verification of the wavelength-dependence of the scale factor
For the purpose of verifying the wavelength dependence of the scale factor under the conventional and the newly-proposed schemes, a tactical-grade FOG is built up, which is configured with a fiber coil with the fiber length of approximately 400 m.
First, as a brief and direct verification, we inject two Er-doped fiber sources to the built-up FOG, of which the power outputs are identical but the central wavelengths are approximately 1531 nm and 1560 nm, respectively. Other parts of the FOG are unmodified. The scale factors of the FOG are read out under the rotation inputs from the turntable with constant temperature [50]. The measured variations of the scale factors under different schemes and under different central wavelengths are summarized as Table 1.
Table 1. Measured scale factors and the corresponding variations under different schemes and under different central wavelengths
Under the conventional scheme, the variation of the measured scale factors is nearly consistent with that of the central wavelength, which conforms to Eq. (13). Comparatively, with the newly-proposed scheme, the scale factor sees much smaller variation than the central wavelength, which primarily demonstrates the reduction of the wavelength dependence of the scale factor.
Then, to obtain the scale factors at continuously varying wavelengths, an SLD centered at the wavelength of near 1550 nm is applied, as shown in Fig. 3.
Fig. 3. (a) The built-up tactical-grade FOG and (b) its schematic for measuring the relationship between the scale factor and the optical frequency (wavelength), in which the optical spectrum analyzer (OSA) is used to obtain the central wavelengths of the super-luminescent diode (SLD) under different equilibrium temperature inside determined by the thermal control circuit. (OSA: YOKOGAWA AQ6370, programmable power supply: ROHDE & SCHWARZ HMP4040, optical fiber in the fiber coil and the pigtails of components: polarization maintaining with outer diameter of 135µm).
The central wavelength of the SLD is manually modified by adjusting the equilibrium temperature in the closed-loop temperature control for the SLD. In the meanwhile, the drive current for the SLD is adjusted to make sure that the output power remains unchanged. The fiber coil is placed under constant room temperature and is unaffected by the thermal control inside the SLD, in order to avoid the thermally-induced variation of the coil geometry (L and D in Eqs. (7) and (13)). Under different equilibrium temperature inside the SLD, the central wavelength λc of the SLD is read out by the optical spectrum analyzer (OSA) and the central frequency vc is obtained by vc =c/λc. With different central frequencies, the scale factors of the FOG are calculated out by the FOG outputs under different input rotation rates generated by the turntable [50].
It is worth mentioning that the central frequency vc read out from the OSA is completely different from the mean frequency $\bar{v}$ defined in Section 2.2, which is determined by the closed-loop operation for the reset error under the constrains of the FOG parameters. In practice, when the SLD sees apparent optical-spectrum modifications from manual manipulations or under stringent thermal environments, the measured central frequency and the mean frequency we concern about share similar variations. So, the central frequency is utilized here instead of the mean frequency, which is hard to quantitively evaluate, to demonstrate the relationships between the scale factor and the optical frequency under the conventional and the newly-proposed schemes, which are shown as the curves in Fig. 4, respectively.
Fig. 4. The measured scale factors at different central frequencies of the SLD, which are normalized to the ones with the equilibrium temperature of 25 °C inside the SLD. For the central frequency varying from 191.38 THz to 193.59 THz (11481 ppm), the variations of the scale factor (Δ) under the conventional and the proposed schemes are 11797ppm and 1068 ppm, respectively, which testifies the effectiveness of the wavelength-dependence-reduction technique. At each central frequency, the measurement of the scale factor is conducted based on the method in Ref. [50] with the applied rotation rates of 0 °/s, ±0.1 °/s, ±0.2 °/s, ±0.5 °/s, ±1 °/s, ±2 °/s, ±5 °/s, ±10 °/s, ±20 °/s, ±50 °/s and ±100 °/s. The FOG output sampling rate is 1 Hz and the sampling time is 40 s under each rate.
In Fig. 4, the scale factor at each central frequency is normalized to the one with the equilibrium temperature of 25 °C inside the SLD. Under the central frequency ranging from 191.38 THz to 193.59 THz, of which the variation is 11481 ppm, the scale factor suffers from the variation of 11797ppm and shows an approximately linear relation with respect to the central frequency as predicted by Eq. (13), when operating in the conventional scheme. With the newly-proposed scheme applied, the wavelength dependence of the scale factor is significantly reduced, which is nearly an order of magnitude smaller than the conventional case (from 11797ppm to 1068 ppm). The residual scale-factor variation may be due to the dispersion effect of the MIOC and the fluctuation of the unconstrained feedback-chain gain induced by the self-heating of electronics. In this condition, the thermally-induced variation of the scale factor usually shows good compensability, and we can obtain satisfactory scale-factor accuracy under stringent thermal environments with appropriate compensation, which is demonstrated in the following section.
The applied rotation rates in the evaluation of scale-factor accuracy are up to 100 °/s, for the purpose of avoiding the nonlinearity and asymmetry of the scale factor induced by the imperfection of the MIOC, which may be dominant when the applied rotation rate is relatively small (experimentally shown to be less than several degrees per second), so as to focus on the wavelength dependence of the scale factor. Theoretically, there is no limit in the rotation rates for the operation of the proposed scheme.
When operating under the proposed scheme, the nonlinearity and the asymmetry of the scale factor will become a little bit higher than those with the conventional scheme. The reason lies in that the nonlinear part in the expression for the scale factor under the conventional scheme (NL2 in Eq. (13)) includes the nonlinearity caused by the reset, which has some compensation effect to the nonlinearity generated by the feedback ramp-step (NL1 in Eq. (7)). What is more, the noise in the FOG outputs will be increased a little bit under the proposed scheme, as the unconstrained feedback-chain gain suffers from some fluctuations even after being temperature compensated.
Although the reduction of the wavelength dependence for the scale factor using the proposed scheme has been verified with both the Er-doped fiber source and the SLD, the amount of reduction is not in the same level for the two kinds of sources. The reasons may be the distinctions in the shape and the width of the optical spectrums, which affect the mean frequency through Eq. (10), and are needed to be further analyzed.
4.2 Scale-factor accuracy under stringent thermal environments
Under stringent thermal conditions, the fiber-coil geometry, that is the product of the coil diameter and the fiber length, in the scale-factor expressions is approximately variated linearly with respect to the temperature due to the intrinsic thermal expansion, of which the influence can be easily removed using a linear model in temperature compensation. Besides, the scale-factor accuracy under thermal variations depends on the mean frequency and the unconstrained feedback-chain gain with the conventional and the newly proposed schemes, respectively.
The experimental FOG is placed on a turntable equipped with a temperature chamber to measure the scale factors at several temperature between -40 °C and +60 °C. Figure 5 shows that the variation of the scale factors is calculated to be 4828 ppm under the conventional operating scheme (Fig. 5(a)) and 41635 ppm under the proposed scheme (Fig. 5(b)). The result is as expected, as the unconstrained feedback-chain gain containing the modulator efficiency and the gain of the driving electronics sees the variation of several hundreds of ppm/°C. However, the key point lies in that the thermally-induced variation of the unconstrained feedback-chain gain usually has better compensability than that of the mean frequency. Using the common temperature compensation based on polynomial fitting, the scale-factor inaccuracy is easily suppressed to several hundreds of ppm (282 ppm in this experiment) with the proposed scheme, while the compensated scale factor still suffers from the inaccuracy of several thousands of ppm (2065ppm in this experiment) with the conventional scheme applied.
Fig. 5. The scale factors at different environmental temperature from −40 °C to +60 °C before and after the compensation based on the polynomial fitting using the temperature, which are normalized to the ones at the room temperature. (a) Under the conventional scheme, the scale-factor variation (Δ) can be compensated to 2065ppm from 4828 ppm. (b) Under the proposed wavelength-dependence-reduction scheme, the scale-factor variation (Δ) is compensated to 282 ppm from 41635 ppm, which demonstrates better compensability and better scale-factor accuracy after compensation.
Here, it is worth mentioning that, in the process of temperature compensation using polynomial fitting, the order of the polynomial should not be very high, which is important for on-chip implementation. Figure 6 shows that, with the polynomial order higher than two, the scale-factor accuracy under the wavelength-dependence-reduction scheme shows advantages over that under the conventional case. Second-order polynomial is enough to obtain good scale-factor accuracy with the error of several hundreds of ppm, when the wavelength dependence is reduced.
Fig. 6. The compensated scale-factor inaccuracy with different polynomial orders applied in temperature compensation. With the polynomial order as low as two, the proposed wavelength-dependence-reduction scheme provides much smaller residual scale-factor inaccuracy than that in the conventional case, which demonstrates the practicability of the newly-proposed scheme.
We have to say that, restricted by the thermal dependence of the unconstrained feedback-chain gain, the dispersion effect, and the nonlinearity in the MIOC, the scale-factor inaccuracy can only achieve the scale of several hundreds of ppm with the reduction of the wavelength dependence, which is no better than the conventional case with the Er-doped fiber source or with the SLD having very good wavelength stability. However, the proposed scheme with the reduction of the wavelength dependence provides advantageous and stable scale-factor performances regardless of the individual difference for the source elements even under scrambled optical spectrums, and regardless of different application environments.
For the FOG in practice, we need to mention that the passive optical components apart from the source, such as the optical fiber in the fiber coil, behave like an integrated optical-spectrum filter (with the transfer function of H(v)) and are likely to influence the wavelength distribution at photodetection. Theoretically, the optical spectrum at photodetection in Eqs. (5) and (10) will be determined by the product of the optical spectrum of p(v) at the source output and the transfer function of H(v), instead of p(v) only [47,51]. As a result, the wavelength variations with respect to the internal or environmental temperature will be more complex, which makes the reduction of wavelength dependence more distinctive and meaningful. Using the optical fiber and other passive components with the passband range much wider than the optical spectrum or with the specially-engineered ones, like the air-core photonic bandgap fiber [52], the wavelength drifts induced by the optical components and temperature are likely to be reduced.
5. Conclusion
The wavelength dependence of the scale factor in the FOG is systematically studied in this work. When excluding the effect of the ramp reset, the scale factor depends on the feedback-chain gain itself and nearly independent of the optical wavelength. It is the feedback-chain gain constrained by the closed-loop scheme for nulling the ramp-reset error that builds up the relationship between the scale factor and the wavelength. By shifting the actuator of the closed-loop scheme for the reset error from the feedback-chain gain to the size of the digital-ramp register, the scale factor becomes nearly unrelated to the wavelength and depends on the unconstrained feedback-chain gain itself again. Experiments show that, under manually-modified wavelengths and under actual stringent thermal environments, the scale-factor accuracy is significantly improved with the proposed wavelength-dependence-reduction scheme, compared with the case under the conventional scheme.
In the deductions of the relationship between the scale factor and the wavelength, the closed-loop schemes for the rotation rate and for the reset error are considered to be independent, which is actually related to each other. Also, in order to obtain more accurate expressions for the scale factor, the dispersions and the nonlinearity in the MIOC are necessarily to be specially evaluated, which is the future work.
The wavelength-dependence reduction of the scale factor is now just employed in tactical-grade FOGs. We believe that our proposed technique is also referable for the navigation-grade and strategic-grade FOGs. As improving the wavelength stability of the broadband source remains a bottleneck, with the wavelength-dependence reduction of the scale factor, the research may be transferred to improve the dispersion of the MIOC and the fluctuation of the feedback-chain gain, especially for the applications in thermostatically controlled environments, which may open up a potential route in achieving better scale-factor accuracy (less than 1 ppm) for high-precision FOGs.
Funding
Chinese Aeronautical Establishment (20170851009).
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.
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