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Polarization nonreciprocity suppression of dual-polarization fiber-optic gyroscope under temperature variation Ping Lu, Zinan Wang, Rongya Luo, Dayu Zhao, Chao Peng,* and Zhengbin Li State Key Laboratory of Advanced Optical Communication Systems Networks, Department of Electronics, Peking University, Beijing 100871, China *Corresponding author:
[email protected] Received January 8, 2015; revised March 13, 2015; accepted March 23, 2015; posted March 24, 2015 (Doc. ID 232020); published April 13, 2015 Polarization nonreciprocity (PN) is one of the most critical factors that degrades the performance of interferometric fiber-optic gyroscopes (IFOGs), particularly under varying temperature. We present an experimental investigation of PN error suppression in a dual-polarization IFOG. Both experimental results and theoretical analysis indicate that the PN errors of the two orthogonally polarized light waves always have opposite signs that can be effectively compensated despite the temperature variation. As a result, the long-term stability of the IFOG has been significantly improved. This study is promising for reducing the temperature fragility of IFOGs. © 2015 Optical Society of America OCIS codes: (060.2370) Fiber optics sensors; (060.2800) Gyroscopes. http://dx.doi.org/10.1364/OL.40.001826
Interferometric fiber-optic gyroscopes are rotation sensors detecting the Sagnac phase shift between two counter-propagating light waves. They have been studied over the last two decades and widely utilized for civilian and military applications [1,2]. Polarization nonreciprocity (PN) error is one of the most critical factors that degrades the performance of IFOGs [3,4]. PN errors, first observed in 1976 [5], are caused by the propagating phase differences between the two orthogonal polarized light waves that travel along the same light path. These errors cannot be distinguished from the Sagnac phase shift, and hence, severely deteriorate the bias stability of IFOGs. Conventionally, the PN errors are suppressed by maintaining only one polarization and eliminating the other. For this purpose, a polarization-maintaining (PM) fiber with high birefringence is adopted. Apparently, the temperature variation will cause the refractive index and strain changing of PM fiber [2,3]. The changing of refractive index with temperature will induce ripples in the output results. This phenomenon had been investigated by Kim and Kang [6]. At the same time, temperature variation will also induce random coupling between two orthogonally polarized light waves (we refer to it as “polarization coupling”). As a result, the PM fiber can hardly maintain the single polarization operation, which results in huge PN errors. In order to suppress PN error in conventional “minimal scheme” configuration, a polarizer with high polarization extinction ratio (PER) is indispensable. Recently, the optical compensation scheme has been proposed as an alternative way to suppress PN errors [7–11]. Different from the polarization-maintaining scheme with its single-polarization operation, the optical compensation scheme allows two orthogonal polarizations to propagate simultaneously. Actually, the benefit of using two polarizations has been discussed in some early studies [12–16]. Our recent theoretical and experimental findings further indicate that PN errors of two polarizations possess opposite signs, thus they can be 0146-9592/15/081826-04$15.00/0
optically canceled out when two polarizations are balanced in their intensity. Since the optical compensation is valid regardless of the amplitude of PN errors, one can expect that the performance of this configuration will be constant even when the temperature varies. In this Letter, we will experimentally demonstrate that the optical compensation mechanism still works for varying temperature, and present a theoretical analysis on this phenomenon. The IFOG setup illustrated in Fig. 1 is used for experimentally investigating the optical compensation under varying temperatures. The light waves from a broadband ASE source are polarized separately along two orthogonal directions at polarizer 1 and 2, and the optical delay line ensures that the orthogonal polarizations are incoherent. The light waves travel along the fiber coil in clockwise (CW) and counter-clockwise (CCW) directions, respectively. Two PDs are used for detecting the output signals. As elaborated in our previous work [8], the long-term stability of this IFOG configuration has been remarkably improved due to the optical compensation, when temperature is constant. The setup is placed in a chamber that is thermally insulated from the room environment and vibration
Power Controller
PD1
Polarizer1 PZT pd1
Circulator
Source
Circulator
Coupler1 Delay Lines
PD2
PMCoupler Polarizer2
Fiber Coil
(+) (-) pd2
Fig. 1. IFOG setup in experiment: an amplified spontaneousemission (ASE) source (central wavelength: 1550 nm, bandwidth: 70 nm), a single-mode coupler (Coupler 1, 50:50), two circulators, two photo-detectors (PDs), two polarizers (polarizationextinction, ratio: 30 dB), a biaxial polarization-maintaining coupler (PM-coupler), and a quadrupole polarization-mode fiber (PMF) coil (length: 2 km, diameter: 14 cm, ΔnT 0 5 × 10−4 ). © 2015 Optical Society of America
April 15, 2015 / Vol. 40, No. 8 / OPTICS LETTERS
proofed. To avoid the vibration from the air compressor, the chamber has first been heated to about 40°C, and then the temperature is allowed to drop naturally. This process is adiabatic and happens slowly, which ensures that the whole chamber reaches thermal equilibrium. 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 experimental results are presented in Figs. 2(a) and 2(b). As the experimental results illustrate, when temperature varies, the rotation velocities obtained from both PD1 and PD2 exhibit considerably large fluctuation. Moreover, several ripples also exist on the curves that are quite similar with the birefringence ripples reported by Kim and Kang [6]. Obviously, the period of the ripples depend on the rate of temperature dropping. At the start of the experiment, the temperature was dropping more rapidly, and hence, the ripples’ periods are shorter in time. Nevertheless, it is noticed that the rotation velocities from PD1 and PD2 always have opposite signs. When summing them up, the fluctuation and ripples cancel out due to optical compensation, and the overall result is much more stable than that of either single PD. These results experimentally indicate that the optical compensation mechanism still works even when the temperature changes dramatically. Allan variance analysis is used for a quantitative and comprehensive evaluation of the IFOG’s performance, as presented in Fig. 2(c). The detailed noise indices
40
PD1
20
Sum
Temperature (°C)
Ω (deg/h)
60
0
PD2
−20 −40 0
3
(a)
6
9
34 32 30 28 26 24 22 0
12
time (h)
36
4
(b)
8
12
time (h)
σ(τ) (deg/h)
1
10
0
PD1
PD2
sum
10
Δn ΔnT 0
dΔn T − T 0 . dT
(1)
Equation (1) indicates the two polarizations traveling in different velocities along the fiber coil. We denote their propagation constants as βx and βy , respectively. Apparently, they vary with temperature. At the same time, polarization couplings exist in the PM fiber coil that represent the limited PER of the non-ideal PM fiber. Polarization coupling is described by a parameter κ, which also randomly fluctuates according to the temperature variations. In the proposed IFOG configuration, two orthogonal polarizations (denoted as X, Y) propagate simultaneously and mutually couple with each other. This behavior can be readily modeled by the CMT equations [Eq. (A1)], and then the non-reciprocal phase shifts at PD1 and PD2 (i.e., PN error) can be obtained as Eq. (2):
−1
−2
0
1
10
2
10 τ (s)
(c) PD1
20
Sum
Temperature (°C)
40
0
PD2
−20 −40 0
3
6
time (h)
9
3
10
60
Ω (deg/h)
are given in Table 1, including rate ramp (RR), rate random walk (RRW), bias instability (BI), angle random walk (ARW), and quantization noise (QN). The compensated results are much lower than those of the two individual PDs in RR, RRW, and BI, which indicates that the long-term stability of the IFOG has been significantly improved by optical compensation, even in the environment with varying temperature. To understand the experimental results, we applied a coupled-mode theory (CMT) analysis on the propagating and coupling of two polarized light waves [17]. Although the Jones matrix method is a conventional choice for this problem, it assumes a lumped model of the fiber coil that makes it inconvenient for studying distributed effect, particularly for the varying temperature and random polarization couplings. The detailed derivations are presented in Appendix A. As mentioned, the temperature variation will cause the refractive index and strain changing of PM fiber, and hence, the birefringence also depends on temperature. Noting the effective refractive index of the x and y polarization as nx and ny , the birefringence can be written as Δn nx − ny . The birefringence can be expended as Taylor series around a given temperature T 0 as follows [6]:
10
10 −1 10
(d)
1827
12
(e)
10
ϕpd2
36 34 32
Γw2 α3 sin ϕ ; w1 α1 Γw2 α3 cos ϕ Γw1 α3 sin ϕ : − arctan w2 α2 Γw1 α3 cos ϕ
ϕpd1 arctan
(2)
The CMT model indicates that ϕpd1 and ϕpd2 depend on the birefringence ΔnT and polarization coupling κT; both are varying with temperature. Nevertheless, ϕpd1
30 28 26 24 22 0
4
8
Table 1. Allan Variance Indices
12
time (h)
Fig. 2. Experiment and simulation results in time domain: (a) demodulated angular velocities in experiment results, (b) temperature outputs of experiment results, (c) Allan variance of experiment results, (d) demodulated angular velocities in simulation results, and (e) temperature outputs of simulation results.
(°/h2 )
RR RRW (°/h3∕2 ) BI (°/h) ARW (°/h1∕2 ) QN (rad)
PD1
PD2
Sum
16.0810 1.7381 0.0620 0.0025 8.6 × 10−8
13.4939 1.6951 0.0685 0.0025 8.6 × 10−8
0.4479 0.3459 0.0261 0.0024 8.6 × 10−8
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OPTICS LETTERS / Vol. 40, No. 8 / April 15, 2015
and ϕpd2 always have opposite signs. Moreover, the PN error after summing up is given as Eq. (3): ϕsum arctan
Γw2 − w1 α3 sin ϕ : (3) w1 α1 w2 α2 Γw1 w2 α3 cos ϕ
It is readily noticed that, once the intensities of two polarizations are balanced (w1 w2 ), the overall PN errors can be eliminated regardless of the temperature varying. According to the CMT, we present a numerical simulation to reproduce the experimental phenomena, and the results are illustrated in Figs. 2(d) and 2(e). The simulation uses the same parameters as those in experiment (Fig. 1). As expected, the simulation behaves similarly to the experiment presented above [Figs. 2(a) and 2(b)]. Since the polarization-coupling coefficient κ is randomly varying with temperature, the rotation velocities detected at PD1 and PD2 fluctuate over a large time scale, but the compensated output remains stable. When temperature drops, the birefringence of the PM fiber changes thus producing periodical phase difference between the two polarizations. As a result, ripples appear with periods proportional to the rate of change of the temperature. Fitting with the experimental data gives dΔn∕dT −1.5 × 10−9 ∕K, which is approximately the same order as in [6]. As demonstrated experimentally and theoretically, the optical compensation effectively suppresses PN errors under varying temperatures, and thus improves the IFOG’s performance in long-term stability. This characteristic is very promising since the temperature fragility is one of the top challenges of the IFOGs. The optical compensation scheme requires the intensity of two polarizations balanced as much as possible, as w1 w2 indicated in Eq. (3). In the practical case, the polarization-dependent loss of those optical components will lead to imperfect balance that introduces residual PN errors. In our experiment, a power controller is placed in one arm of IFOG to balance the two polarizations carefully. It should be interesting and important to discuss the Shupe effect [17] in the proposed scheme. First reported in 1980, a time-dependent temperature gradient along the fiber will induce nonreciprocal phase shift, which is indistinguishable from the phase shift caused by rotation. This thermally induced nonreciprocity arises when the counter-propagating light waves travel through the same fiber region at different times. This error is not PN error thus cannot be readily compensated by two polarizations. In our experiment and theoretical analysis, the thermodynamic process is assumed as quasi-static that happens infinitely slowly. As a result, the temperature difference of a fiber region in single round-trip time has been ignored. The Shupe effect will also induce residual PN errors. To minimize the Shupe effect, the fiber coil is quadrupole wound and well thermally insulated from the room environment by the temperature champer in the experiment. It is noteworthy that the optical compensation mechanism allows spatial temperature gradient along the fiber. For convenience and simplicity, the CMT analysis assumes all the fiber coil possesses the same birefringence
Δn and polarization coupling coefficient κ, and an analytical solution has been obtained. For spatialdependent parameters, the equation can still be solved numerically, which indicates that PN errors can be compensated in the same manner. Similar discussions are elaborated in our previous works [7–11], which indicate that PN errors of two polarization always have opposite signs that originate from the intrinsic symmetry of the two orthogonal polarizations. In conclusion, we have presented an investigation of PN error suppression of IFOGs under varying temperature by utilizing optical compensation. The experiment indicates that the long-term stability of the IFOG has been improved considerably, especially under the conditions of varying temperature. Due to the temperatureinduced changes of birefringence and polarization coupling, the PN errors at individual PDs exhibit considerably large fluctuation and ripples, but the compensated output remains stable. This phenomena has been observed in experiment, and well explained by CMT equations. The PN errors of the two orthogonal polarized light always have opposite signs that can be compensated regardless of varying temperature. This should be a promising feature to overcome the temperature fragility of IFOGs. Appendix A
We use coupled-model theory (CMT) to analyze the propagating and coupling of the orthogonal polarized light waves [18]. Two orthogonal polarization (denoted as x and y) mutually couple to each other in transmission. The CMT equations along z direction are given as Eq. (A1), where E x and E y are the electric fields of two polarizations, βx and βy are their propagation constants, and k12 and k21 are the coupling coefficients: dE x −jβx zE x κ12 zE y ; dz dE y −jβy zE y κ21 zE x : dz
(A1)
Then the wave equations along z direction are obtained as Eq. (A2). The equation along −z direction can be derived similarly: d2 E dE x x − βx βy κ12 κ21 E jβ β x 0; x y dz dz2 d2 E dE y y − βx βy κ12 κ21 E jβ β y 0: x y dz dz2
(A2)
βz and κz are spatial-dependent in general. For simplicity, we assume they are constant along the fiber when quasi-thermal equilibrium is reached. The analytical solution of Eq. (A2) along z direction is Eq. (A3): E x z
1 2
E xy z j
Δβ −jβ1 z Δβ −jβ2 z 1 e e 1− ; B B
κ 21 −jβ1 z e − e−jβ2 z ; 2B
(A3)
April 15, 2015 / Vol. 40, No. 8 / OPTICS LETTERS
where B, Δβ, β, β1 , and β2 are given as follows: B β
q Δβ2 − κ 12 κ 21 ; β x βy ; 2
Δβ
βx − βy ; 2
β1;2 β B:
(A4)
E ij z denotes the complex electric fields after roundtrip of the fiber coil, in which z and −z presents the propagating directions, and i, j represent two polarizations. The solution along −z direction can be obtained similarly. When exiting the coil, the electric fields are constituted by two CW and CCW components simultaneously, as jϕs Eij z E E −ij z, in which ϕs contains the ij ze Sagnac phase shift and the modulated phase shift. Therefore, the signals at two PDs are proportional to the corresponding intensity of light, and the related PN errors are derived as follows: I pd1 hw1 E x E x w2 E yx E yx i; I pd2 hw2 E y E y w1 E xy E xy i; Γw2 α3 sin ϕ ; w1 α1 Γw2 α3 cos ϕ Γw1 α3 sin ϕ : − arctan w2 α2 Γw1 α3 cos ϕ
ϕpd1 arctan ϕpd2
(A5)
Here, w1 and w2 are weight of power of two polarization x and y, respectively. α1 , α2 and α4 are parameters that related to the birefringence Δn and polarization coupling coefficient κ: α1 u1 u2 α2 u1 − u2 α3
1 Γ cos2BL jκ 21 κ 12 j; jBj2
(A6)
where Δβ2 Δβ2 Γ 1 − cos2BL jBj2 jBj2 1 1 − sin2BL. u2 jΓΔβ B B u1 1
(A7)
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ϕ argκ12 κ 21 is the coupling phase. Since the matrix is Hermitian that the coupling coefficient is conjugate, the PN errors of two polarizations have opposite signs. Finally, the PN error of the optically compensated result is Eq. (A8): ϕsum arctan
Γw2 − w1 α3 sin ϕ : w1 α1 w2 α2 Γw1 w2 α3 cos ϕ (A8)
This work was supported by 973 Program of China (No. 2013CB329205) and the National Natural Science Foundation of China (No. 61307089). References 1. E. J. Post, Rev. Mod. Phys. 39, 475 (1967). 2. H. C. Lefèvre, The Fiber-Optic Gyroscope (Artech House, 1993). 3. G. Zhang, The Principles and Technologies of Fiber-Optic Gyroscope (National Defense Industry, 2008). 4. I. A. Andronova and G. B. Malykin, Phys. Usp. 45, 793 (2002). 5. V. Vali and R. W. Shorthill, Appl. Opt. 15, 1099 (1976). 6. D. Kim and J. U. Kang, Opt. Express 12, 4490 (2004). 7. Y. Yang, Z. Wang, and Z. Li, Opt. Lett. 37, 2841 (2012). 8. P. Lu, Z. Wang, Y. Yang, D. Zhao, S. Xiong, Y. Li, C. Peng, and Z. Li, IEEE Photon. J. 6, 7200608 (2014). 9. Z. Wang, Y. Yang, P. Lu, Y. Li, D. Zhao, C. Peng, Z. Zhang, and Z. Li, IEEE Photon. J. 6, 7100208 (2014). 10. Z. Wang, Y. Yang, P. Lu, C. Liu, D. Zhao, C. Peng, Z. Zhang, and Z. Li, Opt. Express 22, 4908 (2014). 11. Z. Wang, Y. Yang, P. Lu, R. Luo, Y. Li, D. Zhao, C. Peng, and Z. Li, Opt. Lett. 39, 2463 (2014). 12. G. B. Malykin, Opt. Spectrosc. 76, 484 (1994). 13. I. A. Andronova, V. M. Gelikonov, and G. V. Gelikonov, Radiophys. Quantum Electron. 41, 980 (1998). 14. I. A. Andronova, G. V. Gelikonov, and G. B. Malykin, Quantum Electron. 29, 271 (1999). 15. I. A. Andronova, G. V. Gelikonov, and G. B. Malykin, Proc. SPIE 3736, 423 (1999). 16. V. M. Gelikonov, G. V. Gelikonov, and I. A. Andronova, Radiophys. Quantum Electron. 51, 296 (2008). 17. D. M. Shupe, Appl. Opt. 19, 654 (1980). 18. K. Zhang and D. Li, Electromagnetic Theory for Microwaves and Optoelectronics (Publishing House of Electronics Industry, 1993).
