1. Introduction

After four decades of development, interferometric fiber optic gyroscope (IFOG) becomes a type of inertial sensing instrument that is capable of detecting rotating motion with high precision and excellent stability [1–4], and has been widely utilized for civilian and military applications [4–10]. It is well known that, the environment variation is one of the most critical factors that degrades the performance of IFOG [4–6, 11]. Among various of environment variations, the temperature variation causes the refractive index and strain of the fiber coil to change, and hence, induces polarization nonreciprocal (PN) phase error that cannot be distinguished from the Sagnac phase shift. As a result, the temperature variation becomes a serious issue that severely deteriorates the bias stability of IFOG [8, 12].

The temperature induced nonreciprocal phase shift was firstly discussed by Shupe [13], which is induced by a time-dependent temperature gradient along the fiber when the counter-propagating light waves travel through the same fiber region at different times. Such thermal effect can be suppressed by adopting sophisticated symmetrical fiber-winding methods and by well designed mechanical packaging which alleviates the temperature gradient [14–19]. It is noteworthy that, temperature variations will also result in PN phase error because of the changing of birefringence and random coupling between two orthogonally polarized light waves. During the changes of temperature, the thermal strain that originates from the difference of thermal expansion coefficients of different parts of the coil will lead to a local photoelastic effect and causes PN phase error [20–23]. In order to suppress the PN phase error in a conventional “minimal scheme” IFOG, a polarizer with high polarization extinction ratio (PER) is indispensable.

Recently, we have developed a “dual polarization” scheme of IFOG to suppress PN phase error in which two orthogonal polarizations are allowed to propagate along the fiber coil simultaneously [24–28]. Since the PN phase error of two orthogonal and balanced polarizations have the same amplitudes but opposite signs, they can be optically compensated and canceled out effectively by summing up the intensities of the two polarizations. Compared with the conventional “polarization-maintaining” scheme that uses only a single-polarization, some recent theoretical and experimental findings indicate several benefits of using two polarizations, such as cost-effective [25] or ultra-simple [26] configuration, utilizing nonreciprocal port [27], and compensation of Faraday effects [30, 31]. We have also preliminarily validated the effectiveness of PN phase error suppression under temperature variations, by using dual-polarization operation in a polarization-maintain (PM) fiber coil [32].

Nevertheless, the PN phase error issue under temperature variation would be more severe for single-mode (SM) fiber coils when compared with PM fiber coils. The thermal effect induces extra strain and enhances a series of intrinsic deformation to the SM fiber, such as twist, bending, transverse pressure, and etc. Since lack of polarization isolation, the SM fiber would experience stronger random coupling between polarizations and unstable birefringence. As a result, it is worthy and important to investigate whether the PN phase error compensation against temperature variations is still valid for SM fiber coils under dual-polarization operation.

In this work, we first present a light propagation model of thermally deformed fiber based on the coupled-wave equations of two orthogonal polarizations, and such model is generally valid for both SM and PM fiber coils. The expression of compensated PN error is derived for a dual-polarization setup of IFOG. Moreover, the effectiveness of optical compensation with respect to PN phase error is theoretically and experimentally investigated by applying a slow and adiabatic temperature change. The results indicate that, the thermal strain induced PN phase error can still be compensated even in SM fiber coils.

The remainder of this paper is organized as follows. In Section 2, we discuss the thermal induced strain in a fiber coil and derive the formulation of light propagation. In Section 3, we present the transfer matrix of the counter-propagating light in two polarizations by dividing the fiber coil into small segments. In Section 4, we present our simulation results and experimental observation of PN phase error suppression under an adiabatic temperature evolution. In Section 5, we conclude with our findings.

2. Propagation model of thermally deformed fiber

We denote the polarizations in x and y directions as e_x, e_y, respectively. For ideal SM fiber with perfect circular core, (e_x, e_y) represent two orthogonal polarized transverse modes which are completely degenerate. On the other hand, strong birefringence in PM fibers makes (e_x, e_y) have different propagating constants. The electric field E of the light propagating in the fiber can be represented as [33]:

A realistic fiber coil is schematically illustrated in Fig. 1(a). The fiber coil is wound by the quadrupole symmetric method and solidified with polymer glue. Besides, a well-designed mechanical shield is adopted to thermally insulate the coil from the environment and provide vibration isolation. Apparently, such a fiber coil is far from an ideal one. A series of imperfections, such as internal core ellipticity, twist, bending, transverse pressure, exist along the fiber, and hence, give a predefined status of birefringence and polarization cross coupling of (e_x, e_y).

 

Fig. 1 (a) The schematic of a fiber coil and a cross section of it. (b) The temperature distribution on a cross section of the fiber coil. The temperatures are represented by colors (red, orange, and blue), from high to low; the yellow part shows the the polymer glue between the fibers. (c) The thermal induced stress applied on a single fiber because of thermal expansion under temperature gradient; the dash ellipse presents the deformation of the fiber core schematically.

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Under temperature variations, the deformation of fiber causes extra strain and stress. As a result, the birefringence and polarization cross coupling vary with the temperature. Birefringence leads to different propagation constants β_x and β_y of (e_x, e_y). For given propagation length L, a phase difference Δϕ = ΔβL = (β_y − β_x)L is accumulated between the orthogonal polarizations. Such phase is not constant under temperature variations and will induce ripples on the output light intensity [35]. On the other hand, the thermally induced strain also alters the elliptically deformed core of a twisted fiber, which gives a varying of the polarization cross coupling coefficient κ under temperature gradient.

The propagation constants and polarization cross coupling coefficients in deformed SM fiber can be depicted by an effective model [33]:

Eqs. (4) and (5) indicate that the real parts of β_x,y and imaginary parts of κ_ij are linearly proportional to the core ellipticity e. The thermal induced strain leads to large deformation of the fiber core, resulting in stronger birefringence and polarization cross coupling. On the other hand, the torsion rate ϕ_t contributes an intrinsic polarization cross coupling strength ϕ_t F₂, and also projects part of the deformation term eF₁(ν) onto the polarization cross coupling coefficients. While it is reported that, no extra twisting would appear under external temperature variations and ϕ_t would remain basically the same [35]. This fact is not counter-intuitive since in a glued and solidified fiber coil, the thermal stress is not likely to introduce extra torsion.

Specifically, the deformation of fiber originates from the temperature gradient in cross section of the fiber coil [20, 23]. As illustrated in Fig. 1(b), the differences in the thermal expansion coefficients of different parts of the fiber (such as fiber core α₁ and cladding α₂, shown in Fig. 1(c)) causes the stress concentrated on the core-cladding interface and lead to thermal strain on the core region. Under a temperature difference ΔT, the thermal induced strain can be expressed as Δε = (α_T1 − α_T2)ΔT, and the stress is Δσ = ΔεE correspondingly, where E is the Young’s modulus of the fiber core. The photoelastic coefficient C, is used to transform the thermal stress into a change of core ellipticity. It is reported that, in the case of the original core ellipticity is small (such as SM fiber), the thermal strain induced ellipticity is much more significant than its geometrical anisotropy [36]. The relation between the thermal induced ellipticity Δe and temperature gradient ΔT becomes:

Eq. (6) gives that the thermal strain directly affects the core ellipticity, and hence, induces extra birefringence and polarization cross coupling. The light propagation in realistic fiber coil is indeed quite temperature dependent.

We here focus on the spatially temperature distribution upon the cross section of the fiber coil. Even for a well-design and shielded fiber coil, such temperature distribution always exists because any ambient temperature changes would make the thermal equilibrium difficult to reach. Above discussions are also valid for a slow and adiabatic temperature changing. Therefore, the temporal temperature variation would induce PN phase error that degrades the long term stability performance of IFOGs.

It is noteworthy that, the Shupe effect [13] concerns the time-dependent temperature gradient along the fiber. The refractive index at the same region changes as the temperature changes, and the difference of refractive index that the counter-propagating light waves experience, leads to a nonreciprocal phase. By using sophisticated symmetric winding methods, the Shupe effect can be effective suppressed because two symmetric locations z and L − z along the fiber are physically at the same area of fiber coil. While differently, the temperature gradient in cross section of the fiber coil deforms the fiber, and hence, introduces PN phase error.

3. Polarization nonreciprocity suppression in fiber coil

The propagation of light waves in the fiber coil can be regarded as a thermally induced perturbation with respect to the predefined state of internal strain. In an IFOG, the fiber coil is completely solidified and packaged by polymer glue. A series of elastic deformations, such as core ellipticity, bending, transverse pressure and twist, are formed during the drawing process of the fiber as well as the winding process of the coil, which gives a predefine state of birefringence and polarization cross coupling.

The setup of a dual-polarization IFOG is schematically illustrated in Fig. 2(a). For better understanding of the details of the PN phase error compensation, we adopted an equivalent configuration to our previous setup as shown in Fig. 2(b). To be specific, we depart a Lyot depolarizer (DP3) into several discrete components, including two polarization beam combiner/splitters (PBC/PBS), two photo-detectors (PDs), a tunable optical attenuator, and an optical delay line of 5m in length. As a result, the intensities of two orthogonal polarizations can be picked up by PD1 and PD2, respectively, rather than optically summed up in a Lyot depolarizer and picked up by one PD. The setup of Fig. 2(a) is consistently adopted for the formulation, simulation, and experiment in this work.

 

Fig. 2 (a) The schematic of dual polarization IFOG setup adopted in this work; (b) The Lyot depolarizer setup in our previous work. setup.(a) is equivalent to setup.(b) but consists of several discrete components. (c) The schematic of fiber twists among segments.

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We use the Jones matrix method to depict the propagating behavior of the fiber coil, in which the birefringence and polarization cross coupling is addressed. As shown in Fig. 2(c), the fiber coil is divided into M fiber segments and the propagating properties of (e_x, e_y) are assumed to be constant in each segment. The transfer matrices of the fiber coil in the clockwise (CW) and counterclockwise (CCW) directions can be written as:

In the above expression, the matrix T(θ_n) represents the intrinsic torsion of the twist fiber, depicted by a rotation angle θ_n of (e_x, e_y) between the adjacent fiber segments. The matrix K(κ_n) models the birefringence and polarization cross coupling of (e_x, e_y). According to Eqs. (3), they can be explicitly expressed as:

As stated above, the matrix elements ${\kappa }_n^{ii}$ and ${\kappa }_n^{ij}$ represent the birefringence and polarization cross coupling, respectively, which vary under temperature variations due to thermally induced fiber deformations. The matrix in the CCW direction can be similarly written as $\mathbf{K}({\kappa }_n^{*})$. Owing to energy conservation, we have ${\mathbf{M}}_{\text{cw}}^{*}={\mathbf{M}}_{\text{ccw}}$.

The PN error can be compensated in the dual-polarization IFOG scheme. We can separately obtain the intensities of the two orthogonal polarizations from the PD1 and PD2 as shown in Fig. 2(a). The PN phase error in either one of the two polarizations can be derived as follows, they are remarkably and visibly large under temperature variation:

As shown in Eqs. (12) and (13), the PN phase error of the two orthogonal polarizations have similar amplitudes (both proportional to $\left|C_2^r{C}_3^r\right|\mathrm{\Gamma }(z_{23}^r)\sin ({\varphi }_{23}^r)$), but opposite polarities with respect to d, namely, −(1 − d)k₁ for ϕ₁ and (1 + d)k₂ for ϕ₂, respectively. Therefore, the overall PN phase error can be compensated by summing up the light intensity of PD1 and PD2, as:

The formulation stated above is generally valid for both SM and PM fiber coils, but their different properties of birefringence and polarization cross coupling would result in different patterns of PN phase error. The overall birefringence B_T is determined by the intrinsic anisotropy birefringence B_int and the extra birefringence B_ext induced by thermal perturbations together, given by [37]:

4. Results and discussions

In the setup shown in Fig. 2(a), we use a low DOP, broad band, amplified spontaneous emission (ASE) light source with central wavelength of 1550 nm and spectrum width of 40 nm. The Lyot depolarizers (DP1 and DP2) on the two arms are made by PM fibers with lengths of 2m/4m and 8m/16m, respectively, and fused by 45° rotation of the fiber axis. The beat length of the PM fiber used in the depolarizers is approximately 1.96mm. The fiber coils consist of 2km SM/PM fiber that are wound by the quadrupole symmetric method. The piezoelectric transducer (PZT) modulates a 52kHz sinusoidal signal on the propagating light. The output signals are detected by PD1 and PD2, respectively, and acquired by using NI-5922 with with output rate of 10 Hz with a sampling length of 0.2 million.

We place the whole IFOG setup into a thermally insulated chamber on an independent foundation, in order to minimize the impact of uncontrolled room environment and vibration. To avoid the vibration from the air compressor, the chamber has first been heated to about 40°C, and then the temperature drops slowly and naturally. This process is assumed to be adiabatic and slow. We carried out a 12-hour long-term stability test, targeting the earths local equivalent rotation velocity, 9.666°/h at our lab location (39.99°N).

The recored temperature changing curves T(t) are presented in Fig. 3(a)(b), for the SM and PM fiber coil experiments, respectively. Accordingly, Fig. 3(c)(d) illustrate the simulation results which are calculated by assuming a distribution of thermal expansion coefficients, Δα_T(L_n, t) over the fiber segment and time, for the realistic temperature evolution. The parameters applied in simulation model, including the core ellipticity, progation constants, polarization cross coupling coeffients, as well as their sensitivity with respect to temperature, are listed in Table 1, which are derived from the structural and material parameter of the fiber coils (see Appendix for details). It is noticed that the SM and PM fiber coils exhibit different patterns with respect to temperature variations. The PN phase error of the two orthonormal polarizations can be effectively compensated as predicted by the theory, both for the SM and PM fiber coils. However, the results of two uncompensated single polarizations in the SM and PM fiber behave differently. For SM fiber coils, the PN phase error shows some large ripples (the amplitudes ranging from ~ −400°/h to ~ 800°/h on PD1 curve), with relatively long periodicity over time (roughly 40 mins). Differently, PM fiber coils exhibit much smaller ripples (the amplitudes ranging from ~ 38°/h to ~ 42°/h on PD1 curve) but shorter periodicity (roughly 30 mins) in the time scale. For both types of fiber coils, the ripples’ periods are shorter in time when the temperature drops more rapidly.

 

Fig. 3 Simulation and experiment results for comparison between the SM fiber coil and the PM fiber coil. (a)(b) are temperature data acquired by sensors. (c)(d) are simulation results. (e)(f) are experiment results.

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Table 1. The derived parameters applied in the wave-propagation model.

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Such phenomena can be explained by the intrinsic different properties of birefringence and polarization cross coupling for SM and PM fiber. According to the propagation model presented in Sections 2 and 3, the amplitude of the ripples is determined by the strength of polarization cross coupling κ, while the periodicity over time is related to the phase Δϕ = δβL accumulated through the birefringence. As we stated above, because of the absence of polarization isolation which exists in the PM fiber, the polarization cross coupling in the SM fiber is remarkably strong when the fiber is thermally deformed. As a result, the amplitude of the ripples is much larger in the SM fiber than the PM fiber. On the other hand, the overall birefringence in SM fiber is much weak that in PM fiber, even under the thermal perturbation, and hence, it explains why the ripples in the SM fiber have much longer periodicity comparing with those in the PM fiber. Moreover, the birefringence is proportional to the temperature gradient. At the beginning of experiments, the temperature drops more rapidly, and hence, a larger temperature gradient makes the ripples’ periods shorter in time.

The experiment results of the dual-polarization IFOG setup of SM and PM fiber coils (Fig. 2(a)) are presented in Fig. 3(e)(f), respectively. Because of the limited precision of the tunable attenuator, the power of the two orthonormal polarizations are slightly different. We adopted a weighted averaging for the compensated output, as Ω_comp = (Ω₁ + Ω₂/w)/2, in which the weight is the maximum amplitudes difference of the output of PD1 and PD2, w = ΔΩ₁/ΔΩ₂. It is noticed that the experiment results agree well with the simulation prediction. In Fig. 3(e), the outputs from the two separate polarizations of the SM fiber coil drift intensely during the temperature variations, and some large ripples (ranging from ~ −400° h to − / to ~ 700°/h PD1 curve) are readily observed. In comparison, the results of the PM fiber are shown in Fig. 3(f) illustrate a series of much smaller (ranging from ~30°/h to ~40°/h on PD1 curve) but more rapid ripples. Such observations are in agreement with our simulations. Conventionally, the holding length (h parameter) is used to depict the maximum length that a linear polarized light can propagate in a PM fiber without significant polarization fading, which is in the order of 1 × 10⁻⁵/m. Applying the similar concept to a typical SM fiber, h is roughly 6.3 × 10⁻³/m [38]. The difference in the ripples’ amplitudes of the SM and PM fiber agrees with the estimations from h parameter.

The details of the compensated results for the SM and PM fiber coils are presented in Fig. 4(a)(b), in which the residual PN errors are illustrated. The long term stability of the IFOG is quantitative evaluated by adopting the Allan variance analysis, as presented in Fig. 4(c)(d), and the statistics are shown in Table 2. Comparing the compensated and uncompensated output, we find that there are quite obvious correlations between them. The ideal compensation condition d = 0 can not be rigorously fulfilled, because the tunable attenuator adopted to balance the power of the two polarizations is limited in precision. According to Eq. (14), the residual PN phase error is also proportional to $d\cdot \left|C_2^r{C}_3^r\right|$, and hence, it shows similar pattern over time compared to the uncompensated output. It also explains the fact noticed from Fig. 4 that, the optical compensation is more effective in the PM fiber coil than the SM one: when d ≠ 0, the residual error is proportional to the polarization cross coupling strength, which is much stronger in the SM fiber than the PM one.

 

Fig. 4 The compensated outputs and Allan variance analysis results for comparison between the SM fiber coil and the PM fiber coil. (a)(b) are compensated outputs in experiments. (c)(d) are experiment curves analyzed by Allan variance.

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Table 2. Comparison of angle random walk (ARW) and bias instability (BI).

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The theoretical and experimental results presented above consistently show that, the thermal induced strain introduces some extra birefringence and polarization cross coupling into the fiber coil, and hence, causes PN phase error. Nevertheless, the PN phase error can still be effectively compensated in a dual-polarization IFOG, even under varying temperature. For a given d, the amplitudes of the PN phase error is determined by the polarization cross coupling strength, therefore, small-diameter optical fiber may be beneficial to suppress thermal induced PN phase error owing to its robustness with respect to thermal induced deformation. On the other hand, the birefringence induces some temporal ripples to the output. For a strong birefringence in the PM fiber, the ripples have short periodicity in time scale that may be harmful to the short term stability of IFOG. Differently, the weak birefringence in the SM fiber could make the ripples spread on a scale of hours or even days, which would degrade the long term stability of IFOG.

It is noteworthy that, there always exist some extra stress and strain after the fiber coil has been wound, glued, solidified and packaged. Such predefined strain also introduce birefringence and polarization cross coupling, and may slowly be released in a long time. It is expected that the process of strain releasing is quite similar to an adiabatic temperature dropping, and the IFOG outputs would behave similarly. A sufficient strain releasing can be quite beneficial to improve the stability performance of IFOG.

5. Conclusion

In this work, we have presented an analysis of the thermal induced PN phase error for the SM and PM fiber coils in the dual-polarization IFOG, respectively. By adopting coupled mode equations, we first derive a propagation model of two orthogonal polarizations when strain is thermally induced. Moreover, the expressions of the PN phase error associated with two orthogonal polarizations are obtained by Jones matrix method in which the fiber coil is divided into a series of small and thermally constant segments. It is theoretically proved that, the PN phase error can still be effectively compensated, even under temperature variations.

In order to verify the proposed model, we demonstrated a dual-polarization IFOG setup which is equivalent to our previous studies, and applied an adiabatic and slowly temperature dropping on it. The experimental observations confirm that the PN phase error can still be compensated. Some particularly temporal ripples are exhibited during temperature changing process while the SM and PM fiber coils show distinct patterns. Such results are consistently explained by the theory and simulation results, by assuming thermally induced birefringence and polarization cross coupling in the deformed fibers. The amplitudes of the ripples are proportional to the polarization cross coupling strength while the ripples’ periodicity is related to the birefringence. Owing to the intrinsic different properties of the SM and PM fiber, they behave different.

The birefringence and polarization cross coupling caused by strain may remarkably influence the stability of IFOG under environment variations. We proved that the optical compensation mechanism in dual-polarization scheme works for both the SM and PM coils, even under temperature variations. This should be a promising feature to overcome the temperature fragility of IFOG.

Appendix

The parameters of the SM and PM fiber coils applied in the simulation are listed in Table 3, in which the optical, thermal and structural parameters are included.

Table 3. Structural and material parameters applied in the simulation.

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Funding

National Natural Science Foundation of China (NSFC) (91736207, 61575002).

References and links

1. V. Vali and R. W. Shorthill, “Fiber ring interferometer,” Appl. Opt. 15(5), 1099–1100 (1976). [CrossRef]   [PubMed]  

2. Y. Paturel, J. Honthaas, H. Lefèvre, and F. Napolitano, “One nautical mile per month fog-based strapdown inertial navigation system: A dream already within reach?” Gyroscopy Navig. 5(1), 1–8 (2014). [CrossRef]  

3. H. C. Lefèvre, The Fiber-Optic Gyroscope, 2nd. ed. (Artech House, 2014).

4. G. A. Sanders, S. J. Sanders, L. K. Strandjord, T. Qiu, J. Wu, M. Smiciklas, D. Mead, A. Arrizon, W. Ho, and M. Salit, “Fiber optic gyroscope development at Honeywell,” Proc. SPIE 9852, 985207 (2016). [CrossRef]  

5. J. Napoli, “20 years of KVH fiber optic gyro technology,” Proc. SPIE 9852, 98520A (2016).

6. G. A. Pavlath, “Fiber optic gyros from research to production,” Proc. SPIE 9852, 985205 (2016). [CrossRef]  

7. S. Mitani, T. Mizutani, and S. Sakai, “Current status of fiber optic gyro efforts for space applications in Japan,” Proc. SPIE 9852, 985208 (2016). [CrossRef]  

8. S. Minakuchi, T. Sanada, N. Takeda, S. Mitani, T. Mizutani, Y. Sasaki, and K. Shinozaki, “Thermal Strain in Lightweight Composite Fiber-Optic Gyroscope for Space Application,” J. Lightwave Technol. 33(12), 2658–2662 (2015). [CrossRef]  

9. Y. N. Korkishko, V. A. Fedorov, V. E. Prilutskiy, V. G. Ponomarev, I. V. Morev, D. V. Obuhovich, S. M. Kostritskii, A. I. Zuev, V. K. Varnakov, A. V. Belashenko, E. N. Yakimov, G. V. Titov, A. V. Ovchinnikov, I. B. Abdul’minov, and S. V. Latyntsev, “Fiber optic gyro for space applications. Results of R&D and flight tests,” in Proceedings of IEEE International Symposium on Inertial Sensors and Systems (IEEE, 2016), pp. 37–41.

10. Z. Tan, C. Yang, Y. Li, Y. Yan, C. He, X. Wang, and Z. Wang, “A low-complexity sensor fusion algorithm based on a fiber-optic gyroscope aided camera pose estimation system,” Sci. China Inf. Sci. 59, 042412 (2016). [CrossRef]  

11. H. C. Lefèvre, “Potpourri of comments about the fiber optic gyro for its 40th anniversary, and how fascinating it was and it still is!” Proc. SPIE 9852, 985203 (2016). [CrossRef]  

12. H. C. Lefèvre, “The fiber-optic gyroscope, a century after Sagnac’s experiment: The ultimate rotation-sensing technology?” Comptes Rendus Physique 15(10), 851–858 (2014). [CrossRef]  

13. D. M. Shupe, “Thermally induced nonreciprocity in the fiber-optic interferometer,” Appl. Opt. 19(5), 654–655 (1980). [CrossRef]   [PubMed]  

14. F. Mohr and P. Kiesel, “Thermal sensitivity Of sensing coils for fibre gyroscopes,” Proc. SPIE 0514, 305–308 (1984). [CrossRef]  

15. N. J. Frigo, “Compensation of linear sources of non-reciprocity in Sagnac interferometers,” Proc. SPIE 0412, 268–271 (1983). [CrossRef]  

16. M. Chomát, “Efficient suppression of thermally induced nonreciprocity in fiber-optic Sagnac interferometers with novel double-layer winding,” Appl. Opt. 32(13), 2289–2291 (1993). [CrossRef]   [PubMed]  

17. P. B. Ruffin, C. M. Lofts, C. C. Sung, and J. L. Page, “Reduction of nonreciprocity noise in wound fiber optic interferometers,” Opt. Eng 33(8), 2675–2679 (1994). [CrossRef]  

18. C. M. Lofts, P. B. Ruffin, M. D. Parker, and C. C. Sung, “Investigation of the effects of temporal thermal gradients in fiber optic gyroscope sensing coils,” Opt. Eng 34(10), 2856–2863 (1995). [CrossRef]  

19. F. Mohr, “Thermooptically induced bias drift in fiber optical Sagnac interferometers,” J. Lightwave Technol. 14(1), 27–41 (1996). [CrossRef]  

20. N. Lagakos, J. A. Bucaro, and J. Jarzynski, “Temperature-induced optical phase shifts in fibers,” Appl. Opt. 20(13), 2305–2308 (1981). [CrossRef]   [PubMed]  

21. F. Mohr and F. Schadt, “Bias error in fiber optic gyroscopes due to elasto-optic interactions in the sensor fiber,” Proc. SPIE 5502, 410–413 (2004). [CrossRef]  

22. X. Li, W. Ling, Y. Wei, and Z. Xu, “Three-dimensional model of thermal-induced optical phase shifts in rotation sensing,” Chin. Opt. Lett. 13(9), 090603 (2015). [CrossRef]  

23. S. Ogut, B. Osunluk, and E. Ozbay, “Modeling of thermal sensitivity of a fiber optic gyroscope coil with practical quadrupole winding,” Proc. SPIE 10208, 1020806 (2017). [CrossRef]  

24. Yi Yang, Zinan Wang, and Zhengbin Li, “Optically compensated dual-polarization interferometric fiber-optic gyroscope,” Opt. Lett. 37(14), 2841–2843 (2012). [CrossRef]   [PubMed]  

25. Z. Wang, Y. Yang, P. Lu, Y. Li, D. Zhao, C. Peng, Z. Zhang, and Z. Li, “All-depolarized interferometric fiber-optic gyroscope based on optical compensation,” IEEE Photon. J. 6(1), 7100208 (2014). [CrossRef]  

26. Z. Wang, Y. Yang, P. Lu, R. Luo, Y. Li, D. Zhao, C. Peng, and Z. Li, “Dual-polarization interferometric fiber-optic gyroscope with an ultra-simple configuration,” Opt. Lett. 39(8), 2463–2466 (2014). [CrossRef]   [PubMed]  

27. Z. Wang, Y. Yang, P. Lu, C. Liu, D. Zhao, C. Peng, Z. Zhang, and Z. Li, “Optically compensated polarization reciprocity in interferometric fiber-optic gyroscopes,” Opt. Express 22(5), 4908–4919 (2014). [CrossRef]   [PubMed]  

28. Z. Wang, Dual-Polarization Two-Port Fiber-Optic Gyroscope(Springer, 2017). [CrossRef]  

29. B. Szafraniec and G. A. Sanders, “Theory of polarization evolution in interferometric fiber-optic depolarized gyros,” J. Lightwave Technol. 17(4), 579–590 (1999). [CrossRef]  

30. P. Liu, X. Li, X. Guang, Z. Xu, W. Ling, and H. Yang, “Drift suppression in a dual-polarization fiber optic gyroscope caused by the Faraday effect,” Opt. Commun. 394, 122–128 (2017). [CrossRef]  

31. P. Liu, X. Li, X. Guang, G. Li, and L. Guan, “Bias error caused by the Faraday effect in fiber optical gyroscope With double sensitivity,” IEEE Photonics Technol. Lett. 29(15), 1273–1276 (2017). [CrossRef]  

32. P. Lu, Z. Wang, R. Luo, D. Zhao, C. Peng, and Z. Li, “Polarization nonreciprocity suppression of dual-polarization fiber-optic gyroscope under temperature variation,” Opt. Lett. 40(80), 1826–1829 (2015). [CrossRef]   [PubMed]  

33. J. Sakai and T. Kimura, “Birefringence and polarization characteristics of single-mode optical fibers under elastic deformations,” IEEE J. Quantum Electron. 17(11), 1041–1051 (1981). [CrossRef]  

34. D. H. Kim and J. U. Kang, “Sagnac loop interferometer based on polarization maintaining photonic crystal fiber with reduced temperature sensitivity,” Opt. Express 12(19), 4490–4495 (2004). [CrossRef]   [PubMed]  

35. J. Sakai and T. Kimura, “Birefringence caused by thermal stress in elliptically deformed core optical fibers,” IEEE J. Quantum Electron. 18(11), 1899–1909 (1982). [CrossRef]  

36. W. P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11(3), 963–983 (1994). [CrossRef]  

37. S. L. A. Carrara, B. Y. Kim, and H. J. Shaw, “Elasto-optic alignment of birefringent axes in polarization-holding optical fiber,” Opt. Lett. 11(7), 470–472 (1986). [CrossRef]   [PubMed]  

38. M. Isubokawa and Y. Sasaki, “Limitation of transmission distance and capacity due to polarisation dispersion in a lightwave system,” Electron. Lett. 24(6), 350–352 (1988). [CrossRef]