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Abstract
We realize a transmissive single-beam-splitter resonator optic gyro based on a hollow-core photonic-crystal fiber (HCPCF), utilizing a micro-optical coupler formed by pairs of lenses and one filter, which is a new type of resonator fiber optic gyro based on the HCPCF (HC-RFOG). We build a mathematical model of the polarization noise based on the transfer function of this novel transmissive single-beam-splitter resonator. We construct a HC-RFOG and simulate and validate the effects of polarization noise on the gyro system. In addition, we apply an effective method to suppress the polarization noise and prove its efficacy through experiments. The bias stability of the gyro system is successfully improved from 25 °/h to 2 °/h, which indicates a remarkable advance of performance of HC-RFOG.
© 2017 Optical Society of America
1. Introduction
The resonant fiber optic gyro (RFOG) uses a fiber ring resonator to detect angular velocity, as compared with interferometric fiber optic gyros [1], taking advantage of the shorter fiber and good linearity with a wide dynamic range [2]. However, because of the use of highly coherent lasers, the backscattering noise, backreflection noise, Kerr noise, and polarization noise are more important in RFOGs [3–6]. The hollow-core photonic-crystal fiber (HCPCF) that generates light in air exhibits remarkable optical performance. The RFOG based on the HCPCF (HC-RFOG) is a novel gyro, altering the mechanism of the waveguide, which is promising for the reduction of the noise sources mentioned above [7].
As previously reported, all the HC-RFOG systems were tested in laboratories. The coupling between a HCPCF and polarization-maintaining fiber (PMF) is still a difficult issue. Traditional coupling approaches include fused coupling and lapped coupling between different fibers. The coupling loss of these two approaches is relatively larger, and considerable backscattering/backreflection light may be introduced into the system [8]. In this study, the micro-optical coupling is used to realize a novel coupler. Pairs of lenses are used to match the modes of different fibers and all the devices are fixed on one silica floor, reducing the coupling loss and backscattering/backreflection light [9]. However, undesired polarization is easily excited, as the misalignment between optic devices is inevitable [10]. It was validated that in this gyro system, based on the novel HCPCF resonator above, polarization noise is the main source of gyro bias.
Many researchers have analyzed the behaviors of polarization noise in gyros. Ioannidis et al. at University College London examined the transmission characteristics of a polarization-maintaining optical-fiber ring resonator, pointing out that there is large crosstalk of as much as 40% between the fiber modes and splitting of the resonance notch as the phase delay between the birefringent fiber axes approaches 2mπ, where m is an integer, even for very good isolation between the fiber polarization modes [11]. Carrara et al. at Stanford University researched the effect of polarization noise on the gyro output signal drift. Time-averaging techniques were introduced to reduce the drift, by employing birefringence modulation at the common input/output port and at one end of the fiber coil [12]. Takahashi et al. at the Mitsubishi Electric Corporation of Japan studied the effect of polarization coupling on the detection sensitivity of the RFOG and attained a rotation sensitivity of 2 °/h with a time constant of 10 s obtained in about 10 min [13]. Takahashi and Hotate also clarified that the bias caused by polarization noise can be suppressed by setting the fiber polarizers at the lead portions of the resonator, but the requirements for the polarizer parameters are quite severe to achieve the performance required for high-grade aircraft navigation [14].
In this study, the mathematical model and numerical simulations of the corresponding polarization noise are determined, and a reasonable polarization noise suppression method is proposed and validated through experiments. Finally, a bias stability of 2 °/h is successfully demonstrated.
2. Principle and simulation
2.1 Transmissive single-beam-splitter resonator
The transmissive resonator is coupled with pairs of lenses and only a single beam-splitter [15], as illustrated in Fig. 1. All the elements are connected with concentric tubes and are fixed on one silica floor. Only one HCPCF is used to detect the Sagnac effect in the resonator, indicating excellent environment adjustability [16]. The clockwise light and counter-clockwise light are transmitted into the resonator through PMF1 and PMF2, respectively. The two ends of the HCPCF are located at either side of the splitter according to refraction law.
Fig. 1 Transmissive HCPCF ring resonator with single optical beam-splitter. (a) Structure of the novel resonator. (b) Coupling structure and light path of the resonator.
The transfer function E, finesse F, and transmittance ƞ of the resonator can be respectively expressed as
Here, E0 represents the electric field intensity of the incident light, r is the reflection coefficient of the splitter, αc is the coupling loss between different fiber pigtails, αl is the transmission loss of the HCPCF, ω is the light frequency, and τ is the intrinsic time of the resonator. The FSR parameter represents the free spectral range of the resonator, with FWHM denoting the full-width at half-maximum of the resonant curve.
As the reflectivity of the HCPCF at the end is only about 40 dB or even lower, the HCPCF is sliced for simplification. Figure 2(a) shows that a lower αc leads to a higher F and a larger ƞ. The mode mismatch loss is reduced by an optimizing lens, and αc ~0.3 dB can be reached in the simulations. However, the coupling loss of the real structure is about 1.2 dB. The deterioration of the coupling efficiency is mostly ascribed to two reasons. First, there is an additional scattering loss at the end of the HCPCF. As it is spliced radially by the machine, the microstructure of the fiber is destroyed at the end surface. Secondly, the finite collimating precision of the light path introduces undesired losses.
Fig. 2 Simulations of characteristic parameters of resonator. (a) Resonant curves, (b) fineness F, (c) transmittance ƞ, and (d) key parameter F√ƞ of the resonator.
As this resonator is designed for the RFOG, the parameters of the resonator need to be optimized to improve the gyro sensitivity. A simulation of the relationship between F and r at different αc is given in Fig. 2(b). It can be seen that a higher F can be derived with a smaller r. Besides, A simulation of the relationship between ƞ and r at different αc is given in Fig. 2(c). It is clear that a bigger r introduces a larger ƞ. Therefore, F and ƞ have the opposite trends as to r. From the mathematical model of the RFOG, it can be observed that the F√ƞ parameter forms the key parameter of interest [17]; larger F√ƞ values correspond to higher sensitivities. For any αc value, there is an optimized r value to maximize F√ƞ, as can be observed from Fig. 2(d). With the actual coupling loss αc between the HCPCF pigtails obtained as 1.2 dB and an HCPCF length of 50 m, a section of the simulation is highlighted in Fig. 2(d) as a red virtual plane and the corresponding projection is plotted as a red “real” line. When r is set to 0.1, a maximum F√ƞ value of about 3.8 can be achieved with F = 12 and ƞ = 10%.
2.2 Modeling and simulation of polarization noise
There are two fundamental light polarization eigenstates that are excited in the HCPCF, and they are orthogonal. Ideally, those two eigenstates are independent as light transfers along the fiber. However, there is a non-ignorable crosstalk between those two eigenstates for the effects of temperature, stress, and twist along the axial direction, introducing a significant aberrance in the resonant curve and leading to a gyro output drift.
In the designed HCPCF resonator, the initial power ratio of the two polarization eigenstates is determined by the angle error (φCW and φCCW) between the light polarization direction and fiber polarization axis. Thus, the initial power ratios of the incident light of the clockwise and counter-clockwise paths can be respectively written as
σCW and σCCW are the phase delay, between the two fundamental light polarization eigenstates, in the PMFs of the clockwise and counter-clockwise paths, respectively.
The evaluation process of the polarization noise is presented clearly below, with simplified transfer functions of the novel resonator. In this condition, the characteristic functions of the resonator can be expressed as
Here, Rr1_CW is the reflecting coupling function from PMF1 to the left pigtail of HCPCF, Rr1_CCW is that from the left pigtail of HCPCF to PMF1, Rr2_CW is that from the right pigtail of HCPCF to PMF2, and Rr2_CCW is that from PMF2 to the right pigtail of HCPCF. Rt_CW and Rt_CCW are the transmissive coupling functions between the two HCPCF pigtails in opposite directions. Tl_CW and Tl_CCW are the transmission matrices of the HCPCF in opposite directions. ωτx and ωτy are the phase delay of the two polarization states in the HCPCF, separately. θS is the phase difference introduced by the Sagnac effect between the clockwise and counter-clockwise paths of the resonator as there is an angle–velocity input along the sensitive axis of the gyro. C is the polarization crosstalk matrix between different elements of the coupling structure:
Here, Δθ is the angle alignment error between different elements of the coupling structure. In this case, the amplitude transfer functions ECW and ECCW and the intensity transfer functions ICW and ICCW can be expressed as
By substituting Eqs. (2), (3), and (4) into Eq. (5), we can obtain the final form of ICW and ICCW, considering the undesired polarization, as follows:
Here, Φ = ωτx is the phase delay of the major polarization state, introduced by the Sagnac effect, in the HCPCF. When the light frequency ω increases by a FSR, Φ increases by 2π. pi (i = 0, 1, 2) is the parameter related to σCW and ω(τx – τy), and ui is the parameter related to σCCW and ω(τx – τy). In this simplified model, qj = vj (j = 0, 1, 2, 3, 4) is the parameter related to ω(τx – τy). In real applications, the laser is locked to the resonant frequency of the resonator. Thus, Φ is equal to 2kπ + ϕ, where k is an integer and ϕ approaches zero; θS also approaches zero. We can obtain a first-order approximate solution of Eq. (6) through the Taylor expansion:
The gyro output is proportional to the phase difference between the maximum points of ICW and ICCW. Thus, the derived functions of ICW and ICCW can be expressed as
The gyro output Ωpl, considering the undesired polarization, can be derived by computing the zeroes of the numerator polynomials of Eq. (8). Thus, the Ωpl and the gyro bias ΔΩ caused by the polarization noise can be expressed as
Although the simplified analysis described above allows an understanding of the physics behind our approach, it is not precise enough to be used for quantitative analysis. In the following, we make some corrections to the transfer functions. It should be noted that the reflection coefficients of the two polarization eigenstates are different and the polarization crosstalk matrices between the elements are different. Thus, the transfer functions of the clockwise and counter-clockwise paths are modified, as shown in the left and right columns of Eq. (10), respectively.
C1 is the polarization crosstalk matrix between PMF1 and the splitter, C2 is that between the splitter and the HCPCF, C3 is that between the HCPCF and the splitter, and C4 is that between the splitter and PMF2.
Δθi is the angle error between different elements of the splitter. By adding those corrected transfer functions to the evaluation process (from Eqs. (5) to (9)), we can derive an accurate expression for the gyro bias caused by polarization noise. As the formula derivation process is too complex and just involves iterations, it is not included here, to avoid repetition. The detailed simulation results of the polarization noise are given below. Here, the length of the HCPCF is 50 m, r is 0.1, αc is 1.2 dB, and αl is 20 dB/km.
A simulation of the intensity transfer function I of a single path of the resonator, considering the undesired polarization, is presented in Fig. 3. Φ is in range of ± 0.005π and ω(τx – τy) is in range of ± π. The I of every ω(τx – τy) in Fig. 3 is normalized. The black line on the top face of the cube in Fig. 3 represents the maximum points of I of every ω(τx – τy). It can be seen that the bias of the maximum point changes with ω(τx – τy).
The top inset of Fig. 4 shows the simulated I with different φ, as ω(τx – τy) is set to 0.1 × 2π. The effect of the undesired polarization, with different φ, on the maximum point of I can be clearly derived from the Φ–I plane, as shown in the left bottom inset. In the magnified portion, the right bottom inset, it is clear that the I curve moves rightward as φ increases from 0° to 30°. Thus, the maximum points of I are different with different φ. In real conditions, the φCW and φCCW cannot be identical. Thus, there is a difference between the maximum points of ICW and ICCW, introducing a gyro bias.
In order to analyze the characteristics of the gyro bias caused by undesired polarization, considering the actual technical merit, φCW is set to zero and φCCW is set to 5.7°. In this case, the phase difference Δσ (between σCCW and σCCW) is equal to σCCW. The simulation of gyro bias, flowing with Δσ and ω(τx – τy), is presented in Fig. 5 (a) as a colored surface. Three planes are inserted in Fig. 5 (a) at ω(τx – τy) = –0.3 × 2π, 0, and 0.3 × 2π. The intercepted curves are presented in Fig. 5 (b). These curves are sinusoidal with Δσ and their averages change with ω(τx – τy).
Fig. 5 Simulation of gyro bias considering undesired polarization. (a) Gyro bias flowing with Δσ and ω(τx – τy). (b) Intercepted curves of gyro bias at ω(τx – τy) = –0.3 × 2π, 0, and 0.3 × 2π.
As the environment (such as temperature or stress) of the PMFs changes, the Δσ, mainly determined by the length and birefringence of the PMFs, changes. This introduces a gyro bias drift, leading to a deterioration of system precision.
This bias drift mentioned above can be suppressed by using polarization-correlated phase modulation (PCPM) technology. As Δσ is linearly modulated in the range of 2π, the spectral distribution of the bias drift moves to a higher frequency band. When the center frequency of the bias drift is much higher than the bandwidth of the gyro system, it can be filtered out from the gyro output. With this countermeasure against the polarization noise, the gyro bias at different ω(τx – τy) converges to the red line on the front face of the cube in Fig. 5 (a).
In contrast, ω(τx – τy) also changes with temperature, as shown in Eq. (12), though the HCPCF in the resonator has a much smaller birefringence than that of the PMF.
Here, Δn is the birefringence of the HCPCF, is the linear expansion coefficient, and is the birefringence temperature coefficient. The parameters on the right of Eq. (12) are all constant, implying that ω(τx – τy) linearly changes with temperature.
This induces a gyro bias flow along the red line on the front face of the cube in Fig. 5 (a), introducing a bias drift. To reduce this drift, a polarizing resonator structure, which can remove the effect of undesired polarization on the gyro from the mechanism, should be realized. We have been focused on this novel project and have derived some outstanding results, and a research article of the polarizing resonator based on HCPCF is in preparation.
3. Experimental research
The HCPCF resonator was realized according to the analysis and design above, as shown in Fig. 6 (a). The fiber ring was compactly wound in order to reduce the inner/outer diameter difference so that the effect of temperature gradients on the gyro bias is suppressed. The obtained resonant curve of this novel resonator is presented in Fig. 6 (b) (red line), with F ~12 and ƞ = 5%. The blue line indicates the sweep voltage used to change the laser frequency, and we note that the parameter F exhibits a good consistency with the estimated value. However, ƞ decreases in practice because of the larger coupling loss between the HCPCF and PMF. The P-P1 and P-P2 in Fig. 6 (b) represent the intensity of the primary polarization eigenstate and the intensity of the undesired polarization eigenstate, respectively. The polarization extinction ratio (PER) of the resonator can be described as −20 × lg(P-P2/P-P1). The PER is about 15 dB, as shown in Fig. 6 (b), which renders necessary an analysis of the polarization noise of the HC-RFOG based on this novel resonator.
The HC-RFOG system was constructed and is illustrated in Fig. 7. This gyro system consists of an optical part and electrical part. The optical part includes the HCPCF resonator, PCPM part (constructed by TEC), optical circulators C1/C2, and integrated optical modulator (IOM), among others. The electrical part includes a narrow-linewidth laser, photodetectors PD1/PD2, and FPGA (including modulation/demodulation module DM1/DM2, signal processing module, and PID controlling module), among others.
A single modulation signal with frequency f1 is used to modulate the incident light and demodulate the gyro output. Two carrier suppression signals with different frequencies f2/f3 are added to either path of the IOM to suppress the backreflection noise and backscattering noise, respectively [18, 19]. In the PCPM module, pieces of PMFs ahead of the resonator are winded on the TEC and are covered by a cooling fin. A square signal is used to drive the TEC to modulate the Δσ. The differential-mode signal of the two paths of the gyro, splittering the common-mode noise, is used to lock the laser frequency to the resonant frequency of the resonator, and the common-mode signal is treated as the gyro output [20].
The bias test in a whole temperature circle (equivalent to ω(τx – τy) in the range of 2π) was realized by placing the resonator in a temperature-controlled cabinet. The temperature in the cabinet was changed linearly, and the bias–temperature (bias–ω(τx – τy)) curve was derived, as shown in Fig. 8 with a black line. The gray area in Fig. 8 is the projection of the simulation of the gyro bias surface in Fig. 5 (a), and the red line is the average result of the gyro bias of every ω(τx – τy). The test result remains in the gray area and fluctuates around the simulation result. This validates the polarization noise model.
Fig. 8 Comparison between simulation and test result of gyro bias in a whole temperature circle (equivalent to ω(τx – τy) in the range of 2π).
A test verifying the effectiveness of the PCPM is given in the Fig. 9. The blue curve in Fig. 9 is the PSD of a traditional test of the gyro without PCPM, as well as the red curve is the PSD of an improved test of the gyro with PCPM. With this countermeasure against the polarization noise, the fluctuation of the gyro bias is observed to be reduced about 20dB in the sampling bandwidth of 1Hz.
The noise suppression method mentioned above was introduced into the HC-RFOG system, and series of gyro tests were carried out. The HC-RFOG system was fixed on a static platform. The static test with PCPM and static test without PCPM were both performed. The Allan variance was used to analyze the gyro data, as shown in Fig. 10. The gyro bias stability is improved from 25 °/h to 2 °/h, which indicates a remarkable advance of performance of HC-RFOG.
Fig. 10 Static tests of the gyro system analyzed by Allan variance method. The bias stability is improved from 25 °/h to 2 °/h.
4. Conclusions
A transmissive single-beam-splitter resonator based on a HCPCF is realized with a micro-optical coupler. A HC-RFOG, considered to be the future development trend of RFOGs, was constructed based on this novel resonator. The transfer functions considering the undesired polarization were deduced. The effects of the undesired polarization on gyro bias were theoretically analyzed and experimentally validated. With the application of the PCPM technique, the polarization noise was effectively suppressed and a bias stability of 2 °/h was successfully demonstrated in this HC-RFOG.
Whereas, the PCPM technique cannot remove the polarization noise entirely, because of an inevitable undesired polarization eigenstate excitated in the resonator. To further improve the performance of the HC-RFOG, we are now studying a polarizing transmissive single-beam-splitter resonator, and the achievements of the research of this promising project will be discussed in detail in future articles.
Funding
National Natural Science Foundation of China (NSFC) (61473022).
Acknowledgments
The authors would like to acknowledge the financial support from the NSFC, thank the colleagues for supporting our experiments, and thank the reviewers from OE for giving meaningful comments.
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