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Abstract
The performance of an open-loop fiber-optic gyroscope is strongly dependent on the optical characteristics of its polarizer. Here we report the implementation of an in-house fabricated 45° tilted-fiber-grating-based polarizer, for the first time on an ultra-fine diameter polarization-maintaining fiber platform in an open-loop fiber-optic gyroscope. This special in-line polarizer is proven to have the merits of high extinction ratio, broad spectrum, bendability, stretchability, temperature insensitivity, and high reliability, all of which make it a perfect match for practical fiber optic gyros that need to be packaged compactly without affecting performance. Our prototype fiber optic gyroscope has a compact volume of only ϕ35 × 20 mm2, achieving a bias instability of less than 0.1 °/h, full temperature bias stability of less than 1 °/h, and scale factor linearity of better than 200 ppm. This compact and high-performance fiber gyro enabled by TFG polarizer may promise great potential in the field of automation and control.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
A gyroscope is used to measure the rotation angular velocity, and it is the core sensor for the navigation, positioning, and attitude control of moving objects [1–3]. Fiber-optic-based gyroscopes are the latest gyroscope technology that has been continuously developed for several decades. As an all-solid-state sensor with a relatively simple structure, the fiber-optic gyroscope (FOG) is gradually replacing traditional mechanical gyroscopes and becoming the mainstream in the field of inertial navigation [4–6]. FOG has two categories depending on their control methods: closed-loop and open-loop gyroscopes [7]. The closed-loop FOG requires a lithium niobate phase modulator to form closed-loop feedback of the interference phase to compensate for the drift influence and improve the sensing accuracy [8]. This kind of gyroscope usually has a large volume and high cost. On the other hand, the open-loop FOG uses a fiber-wound piezoelectric ceramic modulator, an inline polarizer, and a few other fiber devices [9,10]. It has the advantages of small size, low cost, and moderate accuracy and has received extensive attention in recent years [11]. It enables a wide range of applications in the automation and control fields, such as unmanned aerial vehicles, antenna stabilization, robots, and unmanned autonomous driving [12].
Since the FOG’s sensitivity is proportional to the winding fiber length, the fiber industry has been trying to reduce the fiber diameter to wind more fiber for a given fiber gyroscope spool size [13]. The recently matured ultra-fine 40 µm diameter fiber technology has made the open-loop FOG attractive. However, such a specialty fiber brings many difficulties in manufacturing gyroscope devices, one of which is the fabrication of fiber polarizers, required for the optical path’s polarization reciprocity [14]. Current fiber polarizers for the open-loop FOG are fabricated mainly based on the techniques of birefringent-crystal, surface-plasmon-resonance or micro-optics [15–17]. The former techniques require structurally processing the optical fiber, causing potential mechanical reliability issues. The latter is essentially a free-space solution that can be bulky and high loss. As a result, an alternative fiber polarizer remains to be explored, especially for the ultra-fine diameter fiber. Recently, 45° tilted fiber gratings (TFG) have been shown to have a high extinction ratio, low loss, and high reliability. Moreover, the manufacturing process does not require the macroscopic processing of the fiber [18]. The current research of 45°-TFG concerns primarily common types of optical fibers, and their operation bands are traditional telecommunication bands around 1550 nm or 1310 nm [19–21].
In this work, driven by the demand of highly reliable open-loop FOG, 45°-TFG technology is introduced into the 40 µm ultra-fine diameter polarization-maintaining (PM) fiber platform. The resulting in-house developed in-fiber TFG polarizer exhibits excellent optical characteristics, including high extinction ratio, low insert loss, and high bandwidth. Theoretical research and experimental verification are conducted on the comprehensive characteristics of TFG, such as reliability and mechanical characteristics required by the actual working conditions of compact open-loop FOG. The successful incorporation of this kind of polarizer results in an open-loop FOG with a compact volume of only ϕ35 × 20 mm2 and high performance with bias instability less than 0.1 °/h and full temperature bias stability less than 1 °/h. These characteristics promise that this novel open-loop FOG can find potential applications in many fields, such as automation and control.
2. Operation principle and polarization error analysis of an open-loop fiber gyroscope
The open-loop FOG schematic is shown in Figure 1(a). The light from a low-temporal coherence broadband light source passes through a coupler (coupler I) and enters a polarizer to ensure the purity of the coherent polarization mode, followed by a higher-order transverse mode spatial filter, which is made of a tight fiber coil. The filtered light is divided into two counterpropagating beams at the second coupler (coupler II). The returning beams are combined at coupler II and filtered again by the spatial mode filter followed by the polarizer. Finally, the light passes through coupler I and enters the detector. Having the polarizer (or polarization mode filter) and spatial mode filter in the FOG is crucial to guarantee its mode reciprocity [9] Due to the Sagnac effect, the optical path difference Φs of the two counterpropagating beams is proportional to the rotational angular velocity Ω,
where, L is the fiber length, D is the fiber loop diameter, λ is the center wavelength of the light source, and c is the light speed in vacuum. The returned and recombined coherent light is filtered again by the polarizer and spatial filter and then enters into the photodetector. The path difference can be obtained by demodulating the interference light intensity, and hence the rotational angular velocity of the carrier can be obtained by Eq. (1). According to Eq. (1), the gyroscope’s sensitivity benefits from a short wavelength. A preferable operating wavelength of the open-loop FOG is around 850 nm instead of the standard telecom band 1550 nm or 1310 nm. Furthermore, a longer fiber gives better sensitivity, which is realized by winding a smaller-diameter optical fiber. This work uses a newly developed ultra-fine diameter fiber (3 µm core diameter and 40 µm cladding diameter). It has a tiger-type stress zone to facilitate the polarization-maintaining requirement. The ultra-fine diameter also enables a tight bending radius, further increasing the winding fiber length on the FOG. The fiber’s numerical aperture (NA) ∼ 0.22 is much larger than a standard fiber, giving much less bending loss. As a result, our FOG’s 200 m fiber loop’s smallest bend is only 10 mm with negligible bend loss. This fact suggests a miniaturized open-loop FOG with the structure shown in Figure 1(b) can be realized while maintaining a good performance.A PM fiber is used to suppress polarization-related error in the FOG. However, there is still a certain degree of polarization crosstalk in the PM fiber, and the fusion splice points between devices could introduce mutual conversion of polarization states. Due to apparent polarization mode dispersion in PM fiber, parasitic interference errors are produced between the clockwise and counter-clockwise beams in the Sagnac fiber loop. This error source is one of the most crucial ones for the FOG, which cannot be separated from the Sagnac effect by modulation and demodulation. This error often manifests as a quasi-periodical fluctuation of the zero bias against the environment temperature, which is often unregulated in practice in open-loop FOG. Hence, controlling the polarization error becomes the key to improving the gyroscope’s bias stability. It usually involves placing a polarizer on the FOG’s common port between the coupler I and II to filter out accumulated polarization crosstalk generated by multiple components such as the coupler, fiber coil, modulator, etc.
The polarization error in the fiber optic gyroscope can be quantified as [22]
where, ε is the polarizer’s amplitude extinction ratio, p is the degree of polarization of light source, H is the polarization crosstalk of fiber, and N = L/Ld (L and Ld correspond to the length of fiber loop and the polarization decoherence length, respectively). Low-polarization light sources with p about 0.1 are primarily used for FOG. The fiber length is about 200 m, and the coherence length is about 0.1 m, giving the value of N about 2000. For the 40 µm ultra-fine diameter polarization-maintaining fiber used in our FOG, H is about 20 dB/km, which is very close to the performance of commercially available panda-type polarization-maintaining fibers. For medium-precision fiber optic gyroscopes, the polarizer’s extinction ratio should be better than 25 dB to achieve better than 10−6 rads phase error according to Eq. (2). Besides, since open-loop FOG’s broadband light source has a typical wavelength bandwidth of about 30-40 nm, the polarizer’s operating bandwidth must match that bandwidth. In addition, the polarizer’s insertion loss dramatically impacts the open-loop FOG’s signal-to-noise ratio, i.e., the random walk coefficient (RWC) [23]. These facts all point to an open-loop FOG polarizer that should have a high extinction ratio, broad bandwidth (greater than 50 nm), low insertion loss (less than 3 dB), and high-reliability long-term usage.3. In-fiber polarizer fabrication, characterizations, and system construction
45°-TFG technology is developed in-house using a scanning phase mask UV exposure configuration for fabricating the required in-fiber polarizer. As shown in Figure 2(a), the TFG polarizer has a series of index modulation planes periodically arrayed and tilted at the corresponding Brewster angle, i.e., 45ο in a weakly modulated fiber. When two orthogonal fiber polarization modes incident onto the fiber grating, only s-polarization (TE) light experiences Fresnel reflection and thus radiation loss. When this process is phase-matched with an appropriate grating period, it will be resonantly enhanced, leading to an intense polarization-dependent loss. This polarizer exhibits exceptionally high reliability because the manufacturing process is contactless and thus barely changes its structural strength. Besides, it has been shown elsewhere this type of polarizer can be broadband, low loss, high extinction ratio, and well-integrated with fiber optic system as inline fiber device [18–21,24]. It is therefore highly desired for an all-fiber open-loop FOG.
Fig. 2. (a) TFG structure and polarizing principle, (b) photo of the re-coated in-fiber polarizer on a 40 µm fiber, (c) polarization extinction ratio and insert loss with different lengths, inset: PER as a function of wavelength.
In order to ensure the mode field matching and the reliability of fiber splicing, the 40 µm PM fiber for the fiber coil is used to fabricate the in-fiber polarizer. A high concentration of Ge element is added in the fiber core preform manufacturing (the flow ratio SiCl4: GeCl4 = 73:27) to enhance its UV photosensitivity. The fabricated fiber is hydrogen-loaded at 85 °C and under 100 atmospheres for 48 hours [25] to enhance UV photosensitivity. The 45°-TFG is fabricated on an in-house scanning phase-mask UV inscription setup [26]. The 248 nm KrF excimer laser’s pulse energy used in writing is 15 mJ, and its repetition rate is 100 Hz. The phase mask has a nominal period of 974 nm and is tilted by 33.8° [19] to write 45°-TFG inside the fiber. An example of the inscribed and re-coated polarizer is shown in Figure 2(b). It is first annealed at 85 °C for 48 h to stabilize the TFG before subsequent optical characterizations. A series of TFGs are fabricated using the same writing receipt except for their grating lengths. Their measured extinction ratio increases approximately linearly with the writing length, as shown in Figure 2(c). With a grating length of 40 mm, a polarization extinction ratio (PER) of more than 30 dB and an insertion loss of about 0.9 dB around the center wavelength can be achieved. In addition, the polarizer shows PER beyond 25 dB over a bandwidth of 86 nm, as shown in Figure 2(c), which are adequate for the open-loop FOG.
A comprehensive study of this polarizer’s properties required by a practical open-loop FOG is also studied. First of all, the TFG must be coiled within a small space to miniaturize the FOG, as shown in Figure 1(b), and it thus is inevitably subjected to bending stress. There could be two effects on the TFG: period change and index modulation. They can lead to the shift in center wavelength and PER. However, our experimental results shown in Figure 3 suggest that even the tightest millimeter-scale bending has little effect on the PER, and 0.1 N equivalent tensile force barely changes the center wavelength of the broadband polarizer (less than 1 nm wavelength shift). Numerical simulations are also conducted to study the impact of bending and tension on the TFG’s PER using the volume current method [27]. Several factors are taken in account, including perturbations of the effective index and modal field distribution of the guided mode in the fiber caused by bending and tension and perturbation of the grating period due to the tensile strain. As shown in Figures 3(a) and 3(b), the numerical results are well agreement with the experimental results. Therefore, the fabricated TFGs can be safely coiled in a compact package with negligible performance degradation. Practical fiber optic gyroscopes must operate in a wide temperature range, e.g., from -40 °C to 60 °C. The measured and theoretically simulated PER of the annealed polarizer at different temperatures are shown in Figure 3(c). It suggests that the in-fiber polarizer offers excellent temperature stability with PER varying within only 0.5 dB over the whole temperature range.
Fig. 3. Influences of (a) bending and (b) tensile force (c) temperature on the TFG, and (d) estimation of the grating lifetime.
Reliability is another important consideration for polarizers used in FOG. Arrhenius model is used to study the TFG’s long-term reliability. It is formulated as follows [28,29]
The above study shows that the fabricated TFGs exhibit excellent characteristics favored by the open-loop FOG. A prototype is assembled based on the open-loop FOG schematic, as shown in Figure 1. The 200 m optical fiber is wound on a ring skeleton with a diameter of only 30 mm using a quadrupole symmetrical winding method. An in-house packaged miniaturized SLD light source is adopted, which has a center wavelength of ∼830 nm, a 3 dB spectral width of ∼30 nm, and a polarization degree of ∼0.1. It is mounted on a heat sink that has thermal contact with the mechanical structure of our open-loop FOG, and no active cooling is applied. The output power of the light source is more than 500 µW under the driving current of 60 mA. Considering the butt coupling efficiency between the light source and a cleaved 40 µm polarization-maintaining fiber and the loop-back optical path loss in the FOG, the optical power reaching the detector is about 8 µw. The two couplers are in-house developed fused taper-type couplers using the 40 µm polarization-maintaining fiber mentioned above. The beam splitting ratio error is less than 5%, and the polarization-maintaining capability is greater than 20 dB. The optical phase modulator is a sinusoidally-driven piezoelectric ceramic modulator with wound fiber. It can fulfill our system requirements of compactness and low cost. However, its bandwidth is limited to about 100 kHz due to its macroscopic size and capacitance. Note that this bandwidth is much lower than the proper/eigen frequency of about 510 kHz for our 200-m fiber ring. Hence, the Sagnac phase is demodulated via a low-frequency multi-harmonic digital demodulation algorithm [30]. These polarization-maintaining components are spliced together using a customized setup with less than 0.5 dB splicing loss and better than 1° axis alignment accuracy. All the optical components, including the TFG polarizer, are compactly installed, combined with the control circuit to construct the miniaturized open-loop fiber-optic gyroscope with a volume of only ϕ35 × 20 mm2. Notice that because the tilted grating is not sensitive to bending, the size of FOG may be further reduced in the future.
4. Determinations of gyroscope performances
According to the previous analysis, the polarization error in the open-loop FOG polarizer that needs to be suppressed is mainly manifested as the quasi-periodic fluctuation of zero bias with temperature [31]. Therefore, in order to test the applicability in the open-loop FOG, TFGs of different lengths, i.e., different PERs, are fabricated into the same gyroscope one at a time to measure the full-temperature zero bias of the gyroscope. The measured results are plotted in Figure 4(a). The test results show that the gyroscope’s zero-bias assembled with a low PER TFG polarizer fluctuates periodically with the temperature. With the increase of the grating length as well as the PER, the amplitude of bias fluctuation with the temperature dramatically decreases. Meanwhile, the average bias value tends to approach the celestial component of the earth's rotation speed, which is -8.4 ο/h at the test location. The FOG’s polarization-related errors can be effectively suppressed when the grating length exceeds 30 mm with the corresponding PER close to 25 dB. Using a 35 mm long TFG polarizer, the full temperature bias stability, quantified as the standard deviation of the measured data as shown in Figure 4(a), has a value of 0.75 °/h with a corresponding value of -8.9 ο/h. As this result suggests, the optimization of the TFG polarizer’s PER can significantly improve the temperature performance of our open-loop FOG. With a sufficient PER value, the FOG’s average bias approaches the theoretical earth’s rotation speed. In turn, this finding strongly indicates that the polarization error is the main error of this kind of FOG, and the polarizer is a key component for improving its performance. The performance of this open-loop FOG prototype with the best polarizer installed, i.e., a 40 mm TFG polarizer as shown in Figure 2, is further characterized using IEEE 952-2008 protocol. A synchronous digital demodulation algorithm directly demodulates the Sagnac phase of this FOG, and the resulting output is very linear with the angular rotation of the test platform, where the FOG is mounted. This kind of digital demodulation technique has been investigated and reported elsewhere [30]. The scale factor error, shown in Figure 4(b), is less than 200 ppm, strongly indicating that our FOG using TFG has good and stable coherent contrast. The suppressed polarization-related error because of the adoption of TFG has no more deleterious effect on the gyro’s rotation output.
Fig. 4. The measured results for the open-loop fiber optic gyroscope. (a) bias as a function of the temperature with different TFG PERs ;(b) test results for response rate; (c) static test results for Allan variance; (d) test results for Attitude performance
Allan Variance is used to evaluate the static characteristics of the gyro, i.e., RWC for white noise and bias instability (BI) [32]. As shown in Figure 4(c), on the Allan variance curve, RWC can be estimated, by dividing the stability at 1 s by 60, to be 0.012 ο/$\surd h$, and BI can be calculated, by dividing the bottom of the curve by 0.664, to be 0.07 ο/h. Notice that the figure shows that the quantization noise part (the left part) presents an unusual flat segment, which is mainly due to a weak disturbance in sampling signal caused by insufficient electric isolation in the control circuit of our compact FOG. The shot-noise limited sensitivity of this FOG is about 0.25 µrad/√Hz [22], corresponding to 0.32 ο/h at a bandwidth of 1 Hz. This theoretical value is smaller than we observed in the experiment, which suggests that the RWC may further be improved by optimizing the control circuit and its algorithm. The north-seeking algorithm is used to verify the gyro’s attitude determination performance. As shown in Figure 4(d), the gyro’s sensitive axis is perpendicular to the rotation axis of the turntable, and it rotates once to measure the angular velocity of the earth’s rotation at different angles. The test was conducted in Xi'an, China, with 34.4° north latitude. When the gyro’s sensitive axis coincides with the north direction, the earth’s rotational speed component is the largest in the negative direction, which is 360/24×cos(33.4°)≈12.52 ο/h. From the fitted sinusoidal curve, the true north direction is measured to be 238.3° while the calibrated north direction is 238°, both clockwise to the turntable’s start position.
The reliability of this FOG prototype can be estimated using the parts stress analysis method [33]. Our preliminary analysis shows that the mean time between failures (MTBF) of the FOG prototype can exceed 106 hours. In addition, the highly accelerated life testing (HALT) is also employed to verify its reliability [34]. The study indicates that it can simultaneously withstand temperature cycles from -60 °C to 90 °C at 15 °C/min and random vibration of 30 g, proving that this gyroscope has excellent reliability.
5. Conclusion
We have studied the critical parameters of an in-fiber polarizer for compact open-loop FOG, which consists of a 45° tilted fiber grating on an ultra-fine polarization-maintaining gyroscope fiber. The experiment results indicate that this kind of polarizer has a high polarization extinction ratio, low insertion loss, and broadband operation bandwidth. Its performance is barely affected by coiling into a very tight package and is very stable over time. All these characteristics are preferred by the control-level open-loop fiber optic gyroscope. The relationship between the performance of the in-line polarizer and the gyroscope performance was verified for the first time by accessing different lengths of TFG and thus different PER values. The open-loop FOG prototype with a 40-mm polarizer installed is shown to have bias instability of less than 0.1 °/h, full-temperature bias stability of less than 1 °/h, north-seeking accuracy within 1°, and a compact volume of ϕ35 × 20 mm2. The gyroscope passes the strict HALT test, suggesting its superior reliability. This high-performance, miniaturized, and highly reliable fiber optic gyroscope can hold great promise in the next generation of autonomous vehicles and the robotic industry.
Funding
National Natural Science Foundation of China (61975166).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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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