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To improve the thermal stability of a resonator fiber optic gyro (R-FOG), a transmission-type polarizing resonator by inserting two in-line polarizers in a polarization-maintaining fiber resonator with twin 90° polarization-axis rotated splices is proposed and experimentally demonstrated. The in-line polarizers attenuate the unwanted resonance by introducing high loss for the unwanted eigenstates of polarization in the resonator. The desired resonance in the resonator can keep excellent stability in a wide temperature range, thus the temperature-related polarization error in the R-FOG is dramatically suppressed. Both our numerical simulation and experimental verification are carried out, which for the first time to our best knowledge demonstrate that the open-loop output of the R-FOG is insensitive to environmental temperature variations. A bias stability below 2°/h in the temperature range of 36.2°C to 33°C is successfully demonstrated.
©2013 Optical Society of America
1. Introduction
A resonator fiber optic gyro (R-FOG) based on the Sagnac effect [1] has the potential to achieve the inertial navigation system requirement with a short sensing coil, a small volume and light weight. In practice, however, its performance achieved to date is still below expectation. Polarization-fluctuation induced drift, which restricts the application of an R-FOG in a wide temperature range, is one of the most important noises.
In the early research, the single-mode fiber (SMF) was incorporated into the resonator and applied in the R-FOG [2]. An equivalent rotation-rate uncertainty of the R-FOG over a 1 m diameter resonator of 0.5°/h in 100 s was reported. Inside the resonator, a fiber-optic polarization controller was used to compensate for the birefringence in the fiber ring in order to overcome the polarization problem. However, the long-term bias stability was not as good as the short-term one. One of the noise sources is that SMF cannot preserve the polarization effectively due to variations in environmental temperature and pressure.
In order to solve the polarization problem in the SMF, a polarization-maintaining fiber (PMF) resonator has been proposed. The PMF itself preserves the polarization adequately; however, the polarization crosstalk at coupler compromises this condition [3]. A birefringence PMF resonator supports two eigenstates of polarization (ESOPs) that reproduce their polarization states after one roundtrip in the resonator [4]. The unwanted resonance caused by the unwanted ESOP drifts rapidly with respect to the desired resonance due to temperature-dependent birefringence in the PMF. For a typical 10m-length PMF resonator, the two resonances will actually coincide and then cross each other for every 1°C temperature change [5]. Therefore, the polarization-fluctuation induced drift is sensitive to environmental temperature changes and it affects the long-term stability of the R-FOG. To improve the long-term stability, researchers have proposed several structures of the PMF resonators. In a PMF resonator with a single 90° polarization-axis rotated splice [6, 7], the temperature sensitivity is reduced hugely. However, the equal excitation of two circular ESOPs causes interference-type error due to the polarization-dependent loss at coupler. In a PMF resonator with twin 90° polarization-axis rotated splices [8, 9], a single resonance can only be excited at some special temperatures.
To completely suppress the polarization-fluctuation induced drift, however, requires a scheme to ensure a single ESOP excitation in the resonator. A hybrid single-polarization (SP) resonator made with a PMF coupler formed by splicing a section of SP fiber into the resonator has been demonstrated [10]. The addition of the SP fiber into the resonator was used to increase the loss in one of the two ESOPs, thus to eliminate that unwanted resonance. An extinction ratio between the two ESOPs of 40 dB was observed. However, the application of this hybrid resonator to the R-FOG has not been reported. A resonator made of the SP fiber or the fiber polarizer to reduce the polarization-fluctuation induced error in the R-FOG has been evaluated by K. Takiguchi and K. Hotate [11]. It has been shown that the large error induced when using the ordinary PMF can be reduced by 4 orders for a 50 dB loss difference between the two ESOPs in the SP fiber resonator, and a moderate performance gyro (error 10−5-10−4 rad/s) can be achieved. In the evaluation, the effect of the crosstalk in the resonator is neglected. Therefore, the ultimate performance will be deteriorated.
Previously, we proposed an improved scheme for decreasing the polarization-fluctuation induced drift by inserting two in-line polarizers in the PMF transmission-type resonator with twin 90° polarization-axis rotated splices [12]. Inside this novel resonator, the unwanted resonance is suppressed by the in-line polarizers and the relative drift between the two ESOPs is slowed by twin 90° polarization-axis rotated splices. Thus, the desired resonance can keep excellent stability at different temperatures and this novel resonator can operate in a wide temperature range. The initial experimental result has been demonstrated. However, its excellent temperature characteristics have not been evaluated by experiments in a wide temperature range. For the purpose to clarify the physical mechanism of the polarizing resonator, detailed principle and simulation of this polarizing resonator is firstly demonstrated in this paper. A simple formula estimating the bias drift due to the temperature-related polarization-fluctuation is derived. The excellent temperature characteristics are measured in a wide temperature range. Experiment results show that the open-loop output of the R-FOG is insensitive to environmental temperature variations. A bias stability below 2°/h in the temperature range of 36.2°C to 33°C is successfully demonstrated. To the best of our knowledge, this high temperature-stability is the best ever demonstrated in an R-FOG.
2. Principle and Simulation
2.1 Model of the fiber resonator and eigenstates of polarization
A fiber resonator is the core-sensing element in an R-FOG and fabricated by two fused PMF tap couplers C1 and C2, as shown in Fig. 1 . l1 and l4, l2 and l3 are the pigtails of C1 and C2, respectively. Two in-line polarizers PX and PY are spliced to the pigtails l4 and l2 to block the fast-axis and the slow-axis lightwaves, respectively. The pigtails l1 and l4 of C1 are 90° polarization-axis rotated spliced to the pigtails l2 and l3 of C2, respectively.
To analyze the polarization properties of the fiber resonator, Jones matrices are used to describe optical components and polarization cross couplings in the resonator. Take the clockwise (CW) lightwave as an example; the transmission matrix from port 4 of C1 to port 2 of C2 is given by
where εf2 is the polarization extinction ratio (PER) of the polarizer PY. β and Δβ are the average propagation constant and the birefringence of the PMF, respectively. The propagation constants of the slow axis (x) and fast axis (y) can be described byThe transmission matrix from ports 2 to 4 of coupler Ci (i = 1, 2) is given by
where θti (i = 1, 2) represents the equivalent angular misalignment to describe the crosstalk in the coupler for the through port. kx and ky are the coupling coefficients for the slow axis and fast axis, respectively. The coupling matrix from ports 1 to 4 of the coupler C1 or from ports 2 to 3 of the coupler C2, is expressed aswhere θki (i = 1, 2) represents the effective angular misalignment to describe the crosstalk in the coupler for the cross port.The transmission matrix from port 4 of the coupler C2 to port 2 of the coupler C1 is given by
where εs2 is the PER of the polarizer PX.Based on above descriptions, the one roundtrip matrix for the CW lightwave in the resonator is written as
where αt is the total one roundtrip loss, including the propagation loss of the PMF, excess losses in polarizers and couplers. In Eq. (6), the polarization-dependent loss in the coupler is not considered, thus, kx = ky = k. The other parameters are list in Eq. (7). p11,cw, p12,cw, p21,cw, p22,cw are the components of the matrix, respectively. l, Δl1 and Δl2 are the length and the length difference of the resonator, respectively.The eigenvalue λm,cw (m = 1, 2) and eigenvector νm,cw (m = 1, 2) of the matrix Scw are the two key parameters given by the relation [5]
The eigenvectors, referred to the ESOPs in the resonator, represent those states of polarization at port 4 of the coupler C1 that reproduce themselves after one roundtrip in the resonator. The complex eigenvalues represent the corresponding roundtrip transmission coefficients of the two ESOPs.According to Eq. (8), Figs. 2(a) -2(j) depict the calculated shapes of the two ESOPs as a function of one roundtrip phase separation between the two ESOPs. Figures 2(a), 2(c), 2(e), 2(g) and 2(i) are for the case that no polarizers are added. The two ESOPs marked by black solid lines and red dotted lines change from circular to linear and then to circular again as the phase separation changes from 0 to 2π. Thus, the two ESOPs vary dramatically with temperature induced phase separation when no polarizers are added. Figures 2(b), 2(d), 2(f), 2(h) and 2(j) are for the case that two in-line polarizers are added. The two ESOPs always keep in linear as the phase difference changes. Thus, the resonator by inserting two in-line polarizers is expected to achieve stable ESOPs.
Fig. 2 Calculated shapes of the two ESOPs as a function of one roundtrip phase separation between the two ESOPs in the resonator. Resonator without in-line polarizer for (a), (c), (e), (g) and (i); resonator with in-line polarizers for (b), (d), (f), (h) and (j). The simulation parameters: k = 0.03, θt1 = θt2 = 6°, l1 = 3.8m, l2 = 3.2m, l3 = 3.9m, l4 = 3.35m, εf2 = εs2 = 27dB.
2.2 Resonance
The incident lightwave at port 1 of the coupler C1 is given by
where the first number and the second one in the subscript denote different couplers and the related port, relatively.The light field E14 at port 4 of the coupler C1, coupling from the light field E11 at port 1, can be projected onto the two ESOPs, written as
where V = (ν1,cw ν2,cw), a and b are the field amplitudes of the two ESOPs.Considering the lightwave circulating in the resonator many turns, the total light fields of the two ESOPs at port 4 of the coupler C1 are given by
The light fields of the two ESOPs via the pigtails l1, l2 and the polarizer PY, at last exit at port 3 of the coupler C2, described as follows
where αC2 and αPY are the excess losses of the coupler C2 and the polarizer PY, respectively. According to Eq. (12), the resonant curves as a function of one roundtrip phase separation between the two ESOPs is shown in Fig. 3 . The resonators with polarizers added or not are marked by red dotted line and black solid line, respectively. The resonator without in-line polarizer is discussed first. When the phase separation is 0, the two ESOPs are circular as shown in Fig. 2(a). The input linear-polarization light excites two equal-amplitude ESOPs. Their resonances are shown in Fig. 3(a). The desired one is distorted seriously, whose ESOP and the related lineshape are sensitive to the birefringence induced phase fluctuation at this time, and cause the largest polarization-fluctuation induced drift. As the phase separation increases to 0.5π, the effect of the unwanted resonance is weakened, as shown in Fig. 3(b). When the phase separation increases to π, the two ESOPs are linearly shaped and of almost the same orientations as the polarization axes of the PMF, as shown in Fig. 2(e). The unwanted resonance induced by ESOP2 is the smallest, as shown in Fig. 3(c); the desired resonance becomes strongest and results in the smallest polarization-fluctuation induced drift. If the phase separation continually increases to 1.5π, as shown in Fig. 3(d), the desired resonance will be influenced by the approaching unwanted resonance. Setting the phase separation equivalent to π is an ideal working condition for the R-FOG. Consequently, this scheme only operates at some special temperatures and cannot operate over a wide temperature range.Fig. 3 Simulated resonant curves as a function of one roundtrip phase separation between the two ESOPs in the resonator. (a) to (d) correspond to the case of 0, 0.5π, π, 1.5π, respectively. The simulation parameters: k = 0.03, θt1 = θt2 = 6°, θk1 = θk2 = 8°, l1 = 3.8 m, l2 = 3.2m, l3 = 3.9 m, l4 = 3.35 m, εf2 = εs2 = 27dB.
When the polarizers are inserted into the resonator, the unwanted resonance is almost suppressed. The resonant curve marked by red dotted line shown in Fig. 3(a) is enlarged in Fig. 4 for clearer observation. Compared to the suppressed resonance (ESOP2) by the in-line polarizers, the desired resonance (ESOP1) is 4 × 107 times larger.
Fig. 4 Resonant curves of the resonator with in-line polarizers when the one roundtrip phase separation between the two ESOPs is 0.
Moreover, according to Eq. (7), the temperature sensitivity of the resonator could be reduced hugely by decreasing the length difference Δl1, between twin 90° polarization-axis rotated splices. In an ideal condition that Δl1 is 0, the birefringence-induced one roundtrip phase separation between the two ESOPs, ΔβΔl1 is independent on the temperature. Conclusively, inside this novel resonator, the unwanted resonance is suppressed by the in-line polarizers and the relative drift between the two ESOPs is slowed by twin 90° polarization-axis rotated splices. Thus, the desired resonance can keep excellent stability at different temperatures and this novel resonator can be expected to operate in a wide temperature range.
2.3 Polarization-fluctuation induced drift
For simplicity, a Lorentz function [13] is used to approximate the resonant curves described by Eq. (11). The light fields of the two ESOPs exiting at port 3 of the coupler C2 are expressed as
where a1 and b1 are the peak values of the optical amplitudes for the two ESOPs described in Eq. (12), fd is the resonant frequency separation between the two ESOPs, Γ1 and Γ2 are the full-width at half-maximum (FWHM) of the resonant curves for the two ESOPs, given byThe new eigenvectors are given byThe detected intensity at the photodetector is written aswhere the first component I1 and the second component I2 are the intensities of ESOP1 and ESOP2, respectively. The third component is the interference between ESOP1 and ESOP2.Assuming ESOP1 is the desired resonance and it is used for signal detection, however, ESOP2 is considered as the unwanted one. The resonant frequency shift Δfpol,cw of ESOP1 due to the influence of ESOP2 can be solved from
The solution isThe polarization analysis for the counterclockwise (CCW) lightwave is similar to that for the CW one. Thus, the total resonant frequency shift Δfpol between the CW and CCW lightwaves is given by
As seen from Eq. (19), the polarization-fluctuation induced frequency error is composed of two components. The first one is the intensity itself of ESOP2, and the second one is the interference between the two ESOPs. When the separation between the two ESOPs is large enough, Eq. (19) can be simplified asAs seen from Eq. (20), the relationship between the polarization-fluctuation induced frequency error and the resonant parameters is given clearly. The error is proportional to the amplitude ratio of b1 to a1, the FWHM Γ1 and Γ2 of the two ESOPs, and inversely proportional to the separation fd between the two ESOPs. Thus, an unwanted resonance with a small and sharp lineshape and far away from the desired one is preferred to suppress the polarization-fluctuation induced drift.According to Eq. (19), the calculated polarization-fluctuation induced error is shown in Fig. 5 . Without polarizers applied, the drift fluctuates largely with the phase separation between the two ESOPs. The largest drift, as high as 35662 °/h, appears at the phase separation equivalent to 0. The two ESOPs’ resonances coincide and the desired one is distorted seriously. When two in-line polarizers with PERs of 27 dB are added, the maximum polarization-fluctuation induced drift are reduced down to 24.4°/h. Compared to the resonator without in-line polarizers, the polarization error is reduced at least by 1462 benefiting from the large suppression of the unwanted resonance. Unfortunately, compared to the calculated shot noise limit 0.3°/h [2], a higher PER is required to make the R-FOG operate in a wide temperature range. As seen in Fig. 5, the polarization error is reduced to 16 °/h for a PER of 30 dB, 0.45°/h for a PER of 60 dB, and 0.01°/h for a PER of 90 dB, respectively. The near shot noise limit sensitivity could be achieved by improving the PER of each polarizer to 60dB.
Fig. 5 Calculated polarization-fluctuation induced error. The simulation parameter: k = 0.03, θt1 = θt2 = 6°, θk1 = θk2 = 8°, l1 = 3.8m, l2 = 3.2m, l3 = 3.9m, l4 = 3.35m. εs2 and εf2 are the PERs of the polarizer PX and PY, respectively.
3. Experiment
A PMF resonator is fabricated by two fused PMF tap couplers C1 and C2 with coupling ratios of 97/3 (3%), as shown in Fig. 1. The excess losses of the two couplers are 0.1dB. The diameter D of the resonator is about 0.12 m. l1, l2, l3 and l4 are 3.8m, 3.2m, 3.9m and 3.35m, respectively. The PERs for both of the two in-line polarizers are 27dB. In order to measure the thermal stability of the PMF resonator, the resonator is mounted on a thermoelectric module. The temperature of one face of the thermoelectric module can be changed by adding a voltage to it. The setting temperature is stabilized by a temperature controller (LFI-3751 from Wavelength Electronics Inc.). The laser central frequency is scanned respect to time. The resonances are measured with temperature changing from 27°C to 40°C, as shown in Fig. 6 . Only one resonance is observed. The resonances at different temperatures keep excellent stability and do not exchange powers between the two ESOPs. The measured result is agreement with the simulation shown in Fig. 3. The total loss of the resonator is 0.8 dB mainly attributed to two in-line polarizers. The measured finesse is about 21.
Fig. 6 Measured resonances for the resonator integrating in-line polarizers from temperature 27°C to 40°C.
Figure 7 shows a basic configuration of the R-FOG based on the sinusoidal phase modulation. All the fibers in the system are polarization maintaining. A lightwave from a narrow-linewidth laser (linewidth less than 5 kHz) is divided into two equivalent beams by coupler C3. The LiNbO3 phase modulators PM1 and PM2 are driven by sinusoidal waves with modulation frequencies f1 and f2, respectively. The amplitudes of the two sinusoidal voltages V1 and V2 are carefully calibrated to suppress the carrier [14]. The CW and the CCW lightwaves from the resonator are detected by the InGaAs PIN photodetectors, PD1 and PD2 (Model 2053 from New Focus Inc.), respectively. The output of PD2 is fed back through the lock-in amplifier LIA2 to the PI servo controller to reduce the reciprocal noises in the R-FOG [9]. The demodulated signal of the CW lightwave from LIA1 is used as the open-loop readout of the rotation rate.
Fig. 7 Basic configuration of the R-FOG based on the PMF resonator integrating two in-line polarizers
When the temperature of the resonator decreases from 36.2°C down to 33°C, a stable open-loop output of the R-FOG is observed, as shown in Fig. 8(a) . The integration time is 1 s. The resonance frequency drift is approximately 1.33 GHz/°C [15]. The temperature change of 3.2 °C over 800s corresponds to the frequency drift 5.32 MHz/s. As seen in Fig. 8 (a), this large frequency drift has not affected the open-loop output of the R-FOG. Figure 8(b) further illustrates the nature of the gyro stability where the uncertainty in the rotation rate is plotted versus integration time. As seen in Fig. 8(b), the rate uncertainty is approximately inversely proportional to the square root of the integration time with the random walk coefficient of 0.2°/√h and the bias stability is below 2 °/h for an integration time of 100 s. This is encouraging, because it shows that the gyro performance is limited by random noise rather than by temperature-related bias instabilities or drift. The temperature related polarization-fluctuation noise is dramatically suppressed by the PMF resonator combining twin 90° polarization-axis rotated splices with in-line polarizers. The residual error can contribute to Kerr effect and the instability of the optical source.
Fig. 8 Open-loop output thermal stability. (a) Open-loop output vs. temperature. (b) Allan deviation of open-loop output.
The relative resonance distance between the two ESOPs varies with temperature, which has been demonstrated in Fig. 3. Besides, the resonance itself also drifts rapidly with the temperature [15], which is tracked by the PI servo controller in the CCW direction. However, the primary feedback loop could not track the resonant frequency drift in a large temperature range due to a limit output voltage range of the servo controller. Thus, the open-loop output thermal stability is only measured in a temperature range of 3.2°C in this paper. But the ability of R-FOG operating in a wide temperature range is believable, because the wanted resonance keeps excellent stability measured from temperature 14 °C to 40°C, which is shown in Fig. 6 and in [12].
When the resonator is placed in the room environment and free from the temperature controller, the open-loop output of the RFOG is shown in Fig. 9(a) . The drift is not apparent. The Allan deviation of the output shown in Fig. 9(b) is similar to that in Fig. 8(b). The gyro performance is limited by random noise rather than by drift.
Fig. 9 Open-loop output under room temperature, (a) measured for an hour. (b) The Allan deviation of open-loop output.
4. Conclusion
The excellent temperature characteristic of an R-FOG employing a PMF resonator combining twin 90° polarization-axis rotated splices with in-line polarizers is first demonstrated in this paper. The polarization model of the resonator integrating two in-line polarizers is introduced firstly. A simple formula estimating the bias drift due to the temperature-related polarization fluctuation is derived. The desired resonance keeps stable in a wide temperature range through suppression that of the unwanted one. Experiment results show that the open-loop output of the R-FOG is insensitive to environmental temperature changes. This work is of great importance in making an R-FOG operating in a wide temperature range.
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