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Abstract
The measuring accuracy of the fiber optic gyroscope (FOG) for weak signals under very short sampling time is significantly impacted by the quantization error, impeding its application in high-speed measurement and real-time control. In this work, we propose and implement a double-electrode-pair multifunction integrated-optic circuit (MIOC), which contains an additional pair of short electrodes besides the conventional electrode-pair. Taking advantage of the better modulating precision of the additional electrode-pair, the digital feedback is more refined and the quantization error in the FOG output is significantly suppressed. The driving circuits and the control scheme of the proposed MIOC are specially designed for FOGs. Experimental results show that the resolution for extremely small angular rates at short smoothing times is significantly improved. This work provides the potential of the applications in high-speed measuring and controlling systems for high-precision FOGs.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Fiber optic gyroscopes (FOGs) are widely-used rotation-rate sensors based on the Sagnac interferometer, in which the angular rate with respect to inertial space is linearly reflected as the phase difference between the counterpropagating light waves in the fiber loop [1]. Benefit from the intrinsically advantaged configuration and the continuous development in manufacturing, the FOG has exhibited outstanding performances and mass production around the world [2–6].
Similar with other sensors, the FOG output contains some certain types of errors, which behave differently in different applications [7,8]. For the long-time navigation for platforms like submarines, the long-term instability dominates the FOG performance, as the considered smoothing time for FOG output is usually larger than several thousands of seconds and the noise in output is gradually smoothed out [5]. With the suppression of the parasitic nonreciprocities caused by defective devices, manufacturing process and environmental factors, the long-term instability has achieved as low as 10−4 deg/h [9,10], which meets the demands for most navigation equipment. Additionally, some specified techniques are proposed for stabilizing critical parameters, like the loop gain and the mean wavelength, to achieve better performances [1,7]. For the applications, like the positioning and orientating for vehicles, with the smoothing time close to several tens of seconds, the FOG accuracy is mainly limited by the angular random walk (ARW) coefficient, which stands for the white noise in output [1]. Based on the study of the characteristics for each noise component, various suppression methods have been put forward and proved to be effective [11,12].
Besides, there are some significant applications with the requirement of measuring extremely small angular rate at extremely short sampling time (less than several milliseconds). For instance, observation satellites usually employ gyroscopes to sense high-frequency mechanical angular vibration for the rectification of image blurring [13]. The frequency of the vibration to be measured can reach several kilo hertz or even higher. As another example, real-time attitude control asks for the angular-rate sensor to obtain fine resolution at high sampling rate (generally larger than several kilo hertz), which directly determines the control precision and bandwidth [3]. For the evaluation and calibration of inertial test equipment [2], the FOG should be capable of resolving all kinds of weak external interferences acting on the equipment. Unfortunately, under these cases, the quantization error is usually dominant in FOG output [1]. The quantization error ruins the nice features in precision and resolution, which are normally owned by FOGs at long smoothing times, thus seriously limiting the application of FOGs in these important fields.
However, to our best knowledge, the suppressing of quantization error is rarely researched. The quantization error is the unfavorable consequence of digital closed-loop operation in the FOG, where the digital-to-analog converter (DAC) generates the feedback staircase ramp with 1 least-significant-bit (LSB) quantization [14]. Each step in the ramp represents the feedback phase difference to be generated by the multifunction integrated-optic circuit (MIOC) and acts as the output data at the current closed-loop cycle as well [1]. As a result, the quantization in the output corresponds to the equivalent rotation rate for 1LSB of DAC. It comes easily to mind that increasing the feedback-ramp digits by applying a DAC with more data bits naturally solves the problem. But, after increasing the data bits of DAC, the quantized analog voltage for 1LSB can reach several microvolts or even much lower, which is easily lost in electronic noise and transient disturbance, making the suppression of quantization error unobvious. Also, some dynamic properties, like update rate and setting time, for the DACs with more data bits are usually getting worse, which impacts other performances for FOGs.
In this work, for the purpose of reducing quantization error, we shift the focus to MIOC, which is driven by the DAC and acts as the modulator for generating quantized feedback phase difference in the closed loop [15]. We propose and implement a double-electrode-pair MIOC different from the conventional MIOC typically with one pair of electrodes, by which the number of feedback-ramp digits is increased and the quantization error is significantly attenuated. The corresponding driving circuits and control scheme for the proposed MIOC are specially designed. Experiments have been performed to validate the suppression of quantization and the improvement of the resolution for rapidly changing weak signals with the double-electrode-pair MIOC applied in the FOG.
2. Quantization error in FOGs
In order to achieve low bias drift and high scale-factor accuracy, FOGs typically operate in closed-loop mode using digital feedback phase-ramp technique [14]. The staircase ramp, in which the amplitude of each step represents the real-time feedback phase difference, is applied to the MIOC to null out the rotation-induced Sagnac phase difference. For a N-bit digital staircase ramp, automatic overflow happens as the digital number exceeds (2N−1). The DAC transforms a series of digital numbers in the staircase ramp into the corresponding analog voltages in a dynamic range between zero and (2N−1)VLSB, where VLSB is the output voltage for the LSB in the ramp (normally also the LSB of DAC). To avoid any detection error during the automatic overflow, the gain of the feedback path is adjusted to satisfy 2NVLSB = 2Vπ, where Vπ is the required voltage to produce a π-rad phase shift for the MIOC (typically around 3 V). Therefore, 1LSB in the digital phase ramp stands for a phase difference of ΔφQ = 2π/2N and the analog voltage of VLSB = 2Vπ/2N (Table 1), which is a relatively small value with respect to the noise and disturbance in electronics.
Table 1. Correspondences among digital number, phase difference and analog voltage
Meanwhile, the amplitude of each step in the digital staircase ramp acts as the output for the corresponding closed-loop cycle. As a result, the digital output of the FOG has a significant quantization error of 1LSB for the step-duration time, which is typically set as the transition time τ through the fiber coil.
Referring to the relationship between the rotation rate and the Sagnac phase difference [1], the quantized rotation rate ΩQ for 1LSB in the digital staircase ramp is represented as
where c is the speed of light in vacuum, L and D represent the fiber length and the coil diameter respectively, and λ is the mean wavelength. For example, a high-precision FOG is configured with L = 5500 m, D = 250 mm, λ = 1560 nm, and the staircase-ramp digits N = 15. ΩQ is calculated as 2.14 deg/h, four orders of magnitude worse than the long-term stability of the FOG, which is approximately 1 × 10−4 deg/h. The quantization error significantly affects the resolution of FOGs in the measurements with high transmission rate and for the detections of rapidly varying signals.3. Implementation of a FOG with double-electrode-pair MIOC
3.1 Design of MIOC with two pairs of electrodes
In order to suppress the quantization error, we have designed and implemented a new-type MIOC containing an additional pair of electrodes to the right of the primary ones beside the branches of the Y-junction, as illustrated in Fig. 1(b). Like conventional MIOCs, the double-electrode-pair MIOC is fabricated by annealed proton exchange (APE) method using lithium niobate (LiNbO3) crystal as substrate [1,16,17]. As a x-cut y-propagation MIOC, the applied electric field is along the optical axis (z axis) of LiNbO3 crystal and the electrodes are placed on both sides of the waveguide in pull-push connection in order to obtain high modulation efficiency [1,18].
Fig. 1 Designs of (a) the conventional MIOC and (b) the double-electrode-pair MIOC, where le represents the length of the primary electrodes and k denotes the ratio between the lengths of the primary and additional electrodes.
Here, we only consider the phase variations of the transmitted light (extraordinary light for x-cut y-propagation MIOC). Referring to electro-optic Pockels effect, with the modulation of electric field ε applied to the electrode length of le, the index of refraction seen by the light wave is modified and the induced phase shift Δφ can be expressed as [19]
where γ33 denotes the electro-optic coefficient along z-axis of LiNbO3 crystal, ne is the refractive index of the transmitted light, Г represents the electro-optic overlap integral factor, and λ0 is the wavelength in vacuum. Therefore, Vπ, the required voltage to generate a π-rad phase shift, is deduced aswhere Ge = Vπ/ε, denoting the gap between the voltage-controlled electrodes.Seen from Eq. (3), Vπ is inversely proportional to le. Illustrated as the double-electrode-pair MIOC in Fig. 1(b), the length of the additional electrodes is 1/k of that of the primary ones. Consequently, Vπ corresponding to the additional electrodes, Vπ-a, is k times that of the primary electrodes Vπ-p. The value of Vπ-p, like Vπ for conventional MIOCs, is in the range of a few volts. By choosing k, Vπ-a can be as large as several tens of volts, providing the potential to obtain smaller phase variation with larger modulating voltages. The total phase variation seen by the transmitted light wave is the sum of the voltage-induced phase shifts for the two electrode-pairs. Figure 2 shows the fabricated double-electrode-pair MIOC, in which the ratio k between the two lengths is chosen to be 8.
As stated in section 2, the quantization error is directly related to 1LSB in the feedback ramp. For the same quantized analog voltage, the induced quantized phase difference is reduced by k times with the application of the additional electrodes. In other words, the equivalent number of binary digits for the modulating digital ramp is increased by n = log2k. In this case, there is no need to exhaustively refine the modulation voltage by increasing the data bits in DAC, and the smaller quantization error is achieved in optical phase variation. The double-electrode-pair MIOC behaves like a vernier caliper in generating phase difference and modulating the transmitted light wave, where the primary electrode-pair undertakes most modulating signal, providing the range of modulation, and the additional electrode-pair determines the modulating accuracy and quantization.
3.2 Driving circuits for MIOC
With the conventional MIOC, suppose the binary digits in the digital ramp register is N as large as the number of data bits in DAC, shown in Fig. 3(a). As for the double-electrode-pair MIOC, we can expand n more digits for the ramp register. To take advantage of high update rate and short setting time normally possessed by DACs with less digits, the (N + n)-bit digital ramp is divided into two parts and transferred to two DACs respectively, as illustrated in Fig. 3(b).
Fig. 3 Configurations of data bits in DACs (a) for the conventional MIOC and (b) for the double-electrode-pair MIOC.
The first several binary-digits in the ramp register from (N + n-1) to m are configured to the high data bits in DAC1 from (N-1) to (m-n). The rest digits in the ramp register (m-1) to 0 are connected to the low data bits (m-1) to 0 of DAC2. As the required voltage to obtain the same phase modulation for the additional electrodes is k times higher than that for the primary electrodes and k = 2n, the highest configured digit in DAC2 (m-1) is n bits more significant than the digit (m-n-1) following the lowest configured digit (m-n) in DAC1, which results in (N + n) data bits for the total DAC inputs. The choice of m is not fixed, as long as the outputs for both DACs are in good precision. The other unconfigured bits in the two DACs are left blank in the process so as not to influence generation of the feedback ramp.
Schematically shown in Fig. 4, the phases generated by two modulation chains containing two parts of the ramp register are added together as the light wave propagates through the two electrode-pairs, implementing the modulation by (N + n)-bit phase ramp. In another word, we construct an equivalent (N + n)-bit DAC for the FOG using two N-bit DACs. We can obtain nice dynamic characteristics provided by N-bit DACs and n more input data bits at the same time.
The driving chain for each electrode-pair contains a part of the digital ramp register, a DAC and a buffer amplifier. The serial DAC provides controllable reference voltages for the two DACs, adjusting the gains for the two modulation chains to track Vπ-a and Vπ-p independently.
3.3 Vπ control scheme
The thermal dependence of Vπ for the LiNbO3 MIOC is up to −800 ppm/°C [20]. In conventional FOGs, the amplitude of the staircase ramp applied to the MIOC is classically controlled by four-state modulation technique to realize real-time and accurate tracking of Vπ [1].
In our redesigned MIOC, Vπ is different for the two electrode-pairs. Although the ratio of electrode-lengths provides a corresponding Vπ scaling, Vπ for the two kinds of electrodes may vary disproportionately and asynchronously in practice. It is necessary to separately control the gains of the modulation chains connecting to the two electrode-pairs. However, Vπ for the additional electrodes Vπ-a is as high as several tens of volts, which is not compatible with the FOG circuits typically driven by low voltages. Here, we proposed an improved four-state modulation technique based on waveform decomposition and time division. As shown in Fig. 5, the conventional modulation-induced phase difference Δφm(t) is sequentially configured as four states S1: -φ1, S2: -φ2, S3: φ1, and S4: φ2 in one period of 2τ with each state lasting τ/2. φ1 and φ2 generally satisfy φ1 + φ2 = 2π and φ1<π, such as φ1 = 3π/4, φ2 = 5π/4. In the new modulation scheme, Δφm(t) is decomposed into Δφm1(t) and Δφm2(t). In modulation mode 2, Δφm1(t) occupies the square-wave part of Δφm(t) and the extra part is given to Δφm2(t). During mode 1, Δφm2(t) is empty and Δφm1(t) equals Δφm(t).
Δφm1(t) and Δφm2(t) are generated by the modulation chains connecting to the primary and additional electrode-pairs, respectively. In this way, the small portion Δφm2(t) in four-state phase difference connecting to the additional electrodes can easily be realized by low-voltage driving electronics.
With Δφm1(t) and Δφm2(t) applied to the counterpropagating light waves by two electrode-pairs, the Vπ tracking errors for the primary and additional electrodes are reflected in the interference light intensity.
Seen from mode 1 in Fig. 6, as the four modulation-states are entirely included in Δφm1(t), the tracking error for Vπ-p can be demodulated and then eliminated. The gain error in Vπ-p control loop can be represented by a coefficient A multiplying φ1 and φ2. The interference intensity at each modulation state (PS1-1, PS2-1, PS3-1 and PS4-1) and the demodulation result for Vπ-p-tracking error (Dem1) can be calculated as
where P0 is the interference intensity without phase difference, Δφs and Δφf denote the Sagnac phase difference and the feedback phase difference respectively, and Δφs + Δφf ≈0 in closed-loop operation. When A = 1, the gain of modulation chain for the primary electrodes correctly matches Vπ-p. The sign of the demodulation results in Eq. (4) indicates that the gain of the modulation chain needs to be decreased (A>1) or increased (A<1), as the signs of the slopes of the raised-cosine function near φ1 and φ2 are opposite.Similarly, in mode 2, since φ1 in Δφm1(t) has been calibrated by mode 1, the gain in the modulation chain for the additional electrodes is ultimately represented by Bφ2, where B≠1 denotes the existing of gain error for Vπ-a. The interference intensities (PS1-2, PS2-2, PS3-2 and PS4-2) and the demodulation result for Vπ-a-tracking error (Dem2) in modulation mode 2 are listed as
Mode 1 and mode 2 are alternatively in operation at different times. The operation period for each modulation mode determines both the accuracy and the speed of the Vπ controlling, which are to be balanced especially when Vπ changes rapidly. In practical applications, the period of milliseconds, which generally involves several hundreds of cycles for Vπ controlling, is a good choice for high-precision FOGs. For the FOGs working under stable environment, in which Vπ varies very slowly, the modulation mode can be set to a longer period. Identical to the conventional four-state modulation, the Vπ-errors are demodulated at the frequency independent of the modulation-demodulation for the rotation rate. The Vπ-errors for the two electrode-pairs are eliminated with two separated closed-loops by adjusting the outputs of the serial DACs in the corresponding modulation chains.
4. Experiments
Experiments are conducted in a high-precision FOG with the fiber length of nearly 5500 m, the coil diameter of 250 mm, and the mean wavelength of 1560 nm. With the conventional MIOC, the number of phase-ramp digits N is set to 15. Then, the conventional MIOC is replaced by the double-electrode-pair MIOC manufactured with the additional-electrode length nearly 1/8 of the primary-electrode length (k = 8). In the two cases, the loop gain in the FOG is modified to be consistent in order to achieve the same closed-loop bandwidth. To obtain better performances for measuring dynamic signals, the modulation-demodulation for the rotation rate is operating at the 3rd harmonic of the proper frequency, which has no influence on the MIOC and the Vπ-tracking. The output cycle of the FOG is set to the transition time, which is about 27.6 μs.
The FOG is place on a stable platform to measure the component of the earth rotation rate, which is about 9.6 deg/h in the experiments theoretically. With the conventional and the double-electrode-pair MIOCs, the measured rotation rates in sequential closed-loop cycles are shown in Fig. 7, respectively.
Fig. 7 Quantization in FOG output with the conventional MIOC and the double-electrode-pair MIOC, respectively.
Configured with the conventional MIOC, the quantization in the output is around 2.14 deg/h. As for the double-electrode-pair case, the quantization is reduced to about 0.30 deg/h. As expressed in section 3.1, with k = 8 for the additional electrodes, the number of phase-ramp digits is theoretically increased by 3 (n = 3). The reduced quantization is consistent with the result calculated by Eq. (1) with N + n = 18, which is 0.27 deg/h theoretically.
From Fig. 7, we find that the reduced quantization is much less than the amplitude of the output noise, which is mainly generated in optics and electronics. In this case, the Allan deviation result at short smoothing time for the FOG output is not obviously modified, as it is dominated by the noise rather than the quantization. However, the function of the reduced quantization lies in improving the resolution for the signals sensed by the FOG, which are extremely weak and rapidly changing. For example, by employing a high-precision FOG to evaluate the test precision and reliability of another testing platform, the measuring results with the conventional and double-electrode-pair MIOCs are shown in Fig. 8, which reveal the external interference acting on the FOG.
Fig. 8 The results are the external interference acting on a testing platform measured by a high-precision FOG with (a) the conventional MIOC and (b) the double-electrode-pair MIOC, respectively.
Using the double-electrode-pair MIOC, the weak signals hidden in the large quantization induced by the conventional MIOC are clearly recovered. With the quantization significantly reduced, we can see much smaller signals without the need of smoothing the original data. As for the smoothing process, it is possible to obtain the slowly varying signals. But, some high-frequency components, which may be valuable in data analysis, are lost at the same time. The improvement in the resolution provided by the double-electrode-pair MIOC enables us to characterize the weak signals more accurately.
As shown in Fig. 8(b), with better resolution, the root-mean-square (RMS) value of the white noise in the output is estimated to be around 0.80 deg/h. For the case in Fig. 8(a), the RMS value of the quantized output is calculated to be 1.51 deg/h and we cannot directly observe the white-noise component in the output. The FOG configured with the double-electrode-pair MIOC provides the potential to perform in-depth research on the noise inside the FOG, promote the FOG accuracy close to theoretical limit, measure mechanical and electrical interference, and detect weaker seismic signals. As a sensor with higher resolution, the FOG is applicable in the control systems with higher bandwidth and accuracy.
5. Conclusion
In this work, for the purpose of suppressing the quantization error in the FOG, a new design of MIOC with double electrode-pairs is proposed and its method of application is systematically researched. The feedback ramp is redistributed to the modulation chains based on the relationship between the Vπ of the two electrode-pairs. The customized driving circuits and the Vπ control scheme based on four-state-waveform decomposition and time division are provided. Experiments show that, with the double-electrode-pair MIOC, the resolution for weak signals at short smoothing time is significantly improved.
The proposed double-electrode-pair MIOC has the same modulation principle as the conventional one, expect for an injected additional pair of electrodes, which is fabricated in the same material and method. So, the new modulator is inherently stable and reliable. Besides, the proposed modulation technique realizes the accurate tracking of Vπ for both electrode-pairs, providing robustness in practical applications.
However, in the design of the new device, as the applied voltages for the two electrode-pairs are usually different, there exists electric field along the transmitting direction of light between the primary and additional electrodes, possibly resulting in some extra modulation. It may cause some additional noise in the light and may make the ARW of the FOG a little bit worse. This type of error is under research now and the design of the electrodes needs to be optimized in the future work. Additionally, as a new device, its manufacturing process is still to be improved, especially in maintaining the properties for both electrode-pairs, which is important in maintaining the performances of the FOG in practical applications.
Besides the suppression of quantization, the double-electrode-pair MIOC provides the potential for improving the scale-factor accuracy and the DAC linearity under small rotation rate. Furthermore, the MIOC can be configured with more than two electrode-pairs to obtain more better performances for FOGs.
In conclusion, with the double-electrode-pair MIOC, the quantization under short smoothing time is greatly reduced, enlarging the application of FOGs to high-speed measuring and controlling systems.
Funding
National Key Scientific Instrument and Equipment Development Projects of China (2013YQ040877).
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