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A method to suppress polarization-fluctuation induced drift in resonator fiber optic gyro (R-FOG) is demonstrated by a polarization-maintaining fiber (PMF) resonator with twin 90° polarization-axis rotated splices. By setting the length difference of the fiber segments between two 90° polarization-axis rotated splicing points to a half of the beat-length of the fiber, a single eigen-state of polarization (ESOP) is excited with incident lightwave linearly polarized along the polarization-axis of the fiber. Compared to the previously reported resonator employing single 90° polarization-axis rotated splice [1], in which two ESOPs are excited, our new scheme avoids the effect from the unwanted ESOP and thus suppresses the polarization-fluctuation induced drift in R-FOG output significantly.
©2010 Optical Society of America
1. Introduction
Resonator fiber optic gyro (R-FOG), detecting rotation signal based on the resonant frequency difference between the clockwise (CW) and the counter clockwise (CCW) propagating lightwaves in a fiber ring resonator according to the Sagnac effect, is expected to satisfy the Inertial Navigation System (INS) requirement (10−7rad/s) with a fiber as short as 5~10 meters [2]. In practice, however, its performance achieved to date is still below expectation due to noises from various effects, such as the Kerr effect caused by the nonlinearity of the fiber [3], the Shupe effect caused by temporally variant temperature distribution along the fiber coil [4], and most importantly, the polarization-fluctuation induced drift caused by the existence of dual eigen-states of polarization (ESOP) in the resonator and by temperature-sensitive birefringence of the fiber [5,6]. Recently, it is proposed to adopt hollow-core photonic bandgap fiber (PBF) to increase the long term stability of interferometer fiber optic gyro (I-FOG) [7] and R-FOG [8], respectively. Because the PBF of air-core has much smaller nonlinearity and temperature dependence than silica core fiber [9], a PBF-made R-FOG suffers expectedly much less from the Kerr effect and Shupe effect. Moreover, the birefringence in PBF is also more stable against temperature, which means using PBF in R-FOG also helps in stabilizing the polarization-fluctuation induced drift. To suppress completely the polarization-fluctuation induced drift, however, requires a scheme to ensure single ESOP excitation in the resonator.
In R-FOG generally made of a polarization-maintaining fiber (PMF), two ESOPs, which are the states of polarization (SOPs) that return to the original ones after one round trip through the resonator [5], are simultaneously excited. The unwanted ESOP appearing as the second peak or dip in the resonant curve gives rise to two kinds of error: intensity-type error and interference-type error [6,10]. To reduce the polarization-fluctuation induced errors, a scheme employing single 90° polarization-axis rotated splice in the resonator was proposed, where two ESOPs are separated each other by half free-spectral-range (FSR) [1]. This method is effective in suppressing the intensity-type error contributed by the unwanted ESOP through off-resonance intensity. The interference-type error, however, cannot be sufficiently reduced by this scheme if the 90° rotated splice is imperfect or polarization dependent loss exists in the resonator, where two ESOPs are not orthogonal to each other [10]. Previously, we proposed, by numerical simulation, a resonator with twin 90° polarization-axis rotated splices to excite single ESOP while the ESOP-to-ESOP phase-separation is π. The bias stability of the R-FOG employing this scheme can achieve the INS requirement [11]. In this preliminary proposal, however, the large increase in the bias drift when ESOP-to-ESOP phase-separation changes from π to 0 has not been explained. Besides, neither experiment demonstration of single ESOP excitation nor comparison of the bias drifts between single and double ESOP excitations has been carried out.
In this letter, after explaining the relationship between the shape of ESOP and the one-turn ESOP-to-ESOP phase-separation in the resonator with twin 90° polarization-axis rotated splices, we experimentally demonstrate, for the first time to the best of our knowledge, the effectiveness of this configuration in bias drift reduction by exciting a single linear ESOP oriented along one polarization-axis of the PMF.
2. Principle
To study the relationship between the shape of ESOP and the ESOP-to-ESOP one-turn phase-separation ∆φ, we build a mathematical model as shown in Fig. 1 . Differing from the original suggestion in [11] where polarizers were placed at each input and output ports, polarizers are placed at the input ports only, which we proved gives less drift by avoiding the interference between the wanted and unwanted ESOPs [12]. Two couplers are adopted with intensity loss coefficients ri,j and coupling constants κ i,j, where i = x, y and j = 1,2, respectively. Four fiber segments with length of L u (u = 1~4) are connected with angles of θ v (v = 1, 2) between polarization axes. Using the Jones transfer matrix of each optical component in the resonator, we can calculate the one-turn transfer matrix of the resonator [1,5,11,12]. With the ESOP determined as the eigen-vector of the matrix, the phase of ESOP and the transmission loss after one-turn propagation are specified by the argument and the amplitude of the eigen-value, respectively [1].
Fig. 1 Mathematical model of R-FOG with a resonator adopting twin 90° polarization-axis rotated splices. E j,i: input electric field, E j,o: output electric field (j = cw, ccw); C1,2: coupler; P1,2: polarizer; L 1,2,3,4: fiber length; θ1,2: rotated angle in splicing; PD1,2: photodetector; Ω: rotation rate.
Figures 2(a) ~(i) depict the calculated shapes of ESOP as a function of ESOP-to-ESOP one-turn phase-separation from 0 to 2π with π/4 increment, respectively. In the ideal case θ 1 = θ 2 = 90° and no coupler crosstalk, two ESOPs denoted by horizontal and vertical solid black lines in each square coincide with the PMF polarization-axes. In an exaggerated case of θ 1 = 91°, θ 2 = 80° and crosstalk of 20 dB, two ESOPs shown in blue and red dotted lines (colored online) change from circular to linear and then to circular again as ∆φ varies from zero to π and then to 2π. With a state-of-the-art splicing machine, the misalignment angle at the splicing point can be made within ± 0.5°, so the ESOPs can be much closer to the polarization-axes.
Fig. 2 Calculated shapes of ESOP as a function of ESOP-to-ESOP phase-separation after one-turn transmission through the resonator. (a) to (i) correspond to the case of 0, π/4, π/2, 3π/4, π, 5π/4, 3π/2, 7π/4, 2π, respectively. Solid black lines represent the ESOPs when θ1 = θ2 = 90° and without coupler crosstalk, and blue and red (colored online) dotted lines stand for the ESOPs when θ1 = 91°, θ2 = 80° and coupler crosstalk of 20 dB.
Notably, in the special case as shown in Fig. 2(e), when
i.e.,where ∆β stands for the birefringence of the PMF, B the beat length of PMF, and Δl the length difference between (L 1 + L 4) and (L 2 + L 3), two ESOPs are linearly shaped and of almost the same orientations as the PMF polarization-axes. Therefore, by setting Δl = B/2, we can realize the condition shown in Fig. 2(e), and thus can excite a single ESOP selectively in the resonator with a linearly polarized input lightwave appropriately oriented with respect to thepolarization axes of the PMF. The condition of single ESOP excitation removes the polarization-fluctuation induced bias drift due to both the intensity-type error and the interference-type error from the unwanted ESOP. In addition, the large increase in bias drift as the ESOP-to-ESOP phase-separation changes from π to 0 described in Ref [11]. can be explained now by our calculated results: ∆φ = 0 gives the equal excitation of two circular ESOPs (Fig. 2(a)) resulting in large interference-type and intensity-type errors, and ∆φ = π corresponds to the single excitation of one of the polarization-axis oriented linear ESOPs (Fig. 2(e)), respectively.3. Experimental details
Figure 3 shows the conceptual drawing for measuring the polarization-fluctuation induced drift in R-FOG. Lightwave from a distributed feedback (DFB) fiber laser (FL) (linewidth less than 10 kHz) is equally divided and injected into the resonator in CW and CCW directions. Polarizers are placed only at the input ports of the resonator. To track the resonant frequency, the input lightwave is modulated with the so-called digital serrodyne modulation scheme [12–14], in which the phase of the lightwave is modulated by a stepwise sawtooth waveform with 2π-amplitude and a certain repeating frequency. Several other functions, including a countermeasure to Rayleigh backscattering noise, a countermeasure to Kerr-effect induced noise, and an auto-correction of the imperfect serrodyne phase modulation waveform amplitude, can also be incorporated into the digital serrodyne modulation scheme by modifying the modulation waveform [12–14].
Fig. 3 Conceptual drawing of the setup for basic experiments to measure the polarization-fluctuation induced drift in the R-FOG with twin 90° polarization-axis rotated splices. Digital feedback schemes are incorporated to control the laser central frequency and the amplitude of the phase modulation waveform. Counter-measure for the Rayleigh backscattering noise is also adopted. FL: fiber laser; ISO: isolator; C: coupler; PM: phase modulator; POL: polarizer; PD: photo diode; LIA: lock-in amplifier; Atten: attenuator; FG: function generator.
At first, resonant curves under different ESOP excitation conditions were observed by modifying ∆φ. In this basic experiment, the first and the forth PMF segments (denoted as L 1 and L 4 in Fig. 3) were heated to adjust ∆φ. The measured resonant curves by scanning the central frequency of the input lightwave are shown in Fig. 4 . In Fig. 4(a), only a single linear ESOP was excited while ∆φ was adjusted to π. In Fig. 4(b), two elliptical ESOPs were closely located in phase but with different excitation ratio while ∆φ was π/2. In Fig. 4(c), two nearly-circular ESOPs almost coincided and were equally excited while ∆φ is set to π/5. In Fig. 4(d), equal excitation of two linear ESOPs was achieved by inserting a λ/2 waveplate in front of the input port of the resonator (C2 in Fig. 3) and rotating the polarization of the input lightwave by 45°, while ∆φ was once again kept in π. The experimental results shown in Fig. 4 demonstrate our theoretical analyses given in Fig. 2. Particularly, when setting Δl = B/2 or ∆φ = π, a single ESOP is excited in the resonator with a lightwave linearly polarized along one polarization-axis of the PMF.
Fig. 4 Resonant characteristics obtained in the clockwise output by scanning the central frequency of the input lightwave, (a) under the ideal condition that the ESOP-to-ESOP phase-separation is equal to π, and the input lightwave is linearly polarized along one of the polarization axis of the PMF; (b) under the condition that the ESOP-to-ESOP phase-separation is equal to π/2; (c) under the condition that the ESOP-to-ESOP phase-separation is equal to π/5; and (d) under the same ESOP-to-ESOP phase-separation condition of (a), but the input lightwave is linearly polarized along with 45° tilted from the polarization axis of the PMF.
Next, R-FOG bias stability was recorded under different ESOP-excitation situations. A digital control scheme using FPGA shown in Fig. 3 was applied to lock the central frequency of FL to the CCW resonant frequency and to determine the exact 2π amplitude of the phase modulator as discussed in detail in [15]. Besides, the digital serrodyne phase modulation waveform was designed to reduce the Rayleigh backscattering induced noise [13]. The time constant of the lock-in amplifier (LIA) was set 1 s. Bias drift corresponding to the excitation situations shown in Fig. 4(a)~(d) are illustrated in Fig. 5 , giving the bias drifts of 4.1 × 10−4 rad/s, 1.2 × 10−2 rad/s, 1.8 × 10−2 rad/s, and 2.2 × 10−3 rad/s, respectively, all over 150 seconds. It can be seen clearly that the bias drift is reduced significantly by exciting a single ESOP in the resonator while keeping Δl = B/2.
4. Conclusion
In summary, our analyses and experiments demonstrated the effectiveness of our proposed scheme of R-FOG with a resonator adopting twin 90° polarization-axis rotated splices in suppressing the polarization-fluctuation induced drift in R-FOG. It is shown that, by adjusting the lengths of the fiber segments between the twin 90° polarization-axis rotated splices to a half of the beat length of PMF, we can excite a single linear ESOP oriented along one polarization-axis of the PMF. It is also demonstrated that the single ESOP excitation in the resonator reduces the bias drift effectively compared with exciting dual ESOPs.
In this basic experiment, the condition Δl = B/2 is maintained by heating a part of the fiber segment. In practice, it is possible to maintain it mechanically, for example, by using a piezo fiber stretcher to adjust Δl. Especially, in an R-FOG using polarization-maintaining PBF resonator, environmental temperature-change induced birefringence variation is much smaller than that in a silica core fiber. Combining with quadrupole winding technique [16,17] developed as a countermeasure for the Shupe effect, environmental change induced variation in Δl can be very small, and thus there is no fundamental difficulty in maintaining the condition. Therefore, we can expect our proposed scheme is feasible for building high grade R-FOGs.
References and links
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