Fiber optic gyroscopes (FOGs) are widely used in motion sensing and inertial navigation technologies to accurately measure angular velocity^[1]. They can be classified into open-loop and closed-loop FOGs^[2]. Open-loop FOGs have slightly lower accuracy than closed-loop ones, but their cost advantage is favored by cost-sensitive industries such as autopilot and robot control^[3,4].

Open-loop FOGs use a lead zirconate titanate (PZT) modulator, which is only 1/100^th the price of the lithium niobate electro-optic modulator used in closed-loop FOGs^[5]. However, piezoelectric modulation via squeezing and stretching optical fiber wound on a ceramic can induce polarization cross talk and modulation birefringence, causing phase errors^[6,7]. The typical co-frequency and in-phase error resulting from the PZT phase modulation has a value approximately ranging from 10−5 to 10−6rad. This error significantly deteriorates the accuracy of rotational phase demodulation, and it is thus a primary error source in open-loop FOGs^[8]. In addition, there are few optical suppression methods available to mitigate modulation parasitic errors, and traditional polarization measurement methods, such as the polarization-extinction ratio (PER), are not even adequate for assessing such minute levels of polarization cross talk. Current research primarily focuses on the system-level analysis and algorithm-based compensation of PZT-modulated FOGs^[9,10], but they all fall short of achieving high precision in FOG development. To address these issues, we developed a polarization-diversity coherent optical frequency domain reflectometry (OFDR) with high-polarization-extinction light launching capability to measure the distributed modulation polarization error. For the higher sensitivity of the FOG, 40 µm ultrafine diameter polarization-maintaining (PM) fiber is employed to increase the length of coiled fiber under a limited volume. However, a corresponding challenge arises due to the unavailability of a reliable fusion splicing approach for this fiber. To overcome this limitation, we proposed and demonstrated two approaches in a spliceless FOG optical configuration, including an inline UV-induced wave plate approach and an inline polarization mode converter (PMC), to suppress the modulation polarization error. By combining these approaches, our experimental results show that the optimized FOG achieved a polarization error suppression of over 15 dB and reduced the full-temperature peak-to-peak gyro error from 25°/h to 0.7°/h, which meets the requirements for a control-level gyroscope with 1°/h bias stability.

The gyroscope used in this work is based on a spliceless all-fiber FOG architecture^[11] with all of its fiber-optical components sequentially fabricated on a 40 µm PM fiber. As shown in Fig. 1, the light source is a superluminescent diode at a center wavelength of 830 nm^[2], in order to gain more sensitivity in a compact form factor. The fused biconical tapered couplers are used for the splitting and combining of light. The polarizer for polarization filtering is an in-line 45° tilted fiber grating (TFG) with a PER of more than 35 dB^[12,13]. The fiber coil, with a diameter of 30 mm, acts as the rotation-sensitive component with a total length of 150 m. The modulator that is used for generating sinusoidal phase modulation is a PZT ring with a diameter of 10 mm. It has a 1 m-long fiber wound at a tension of 0.06 N and its modulation frequency is about 120 kHz.

Using the Jones Matrix method to analyze the optical path of the gyro, we can express the polarization error, Er, resulting from polarization cross talk induced by the phase modulation as^[8]$$\begin{gathered} Er=hlΔϕcos(ωτ/2)cos(2α)sin(2πl/lb)⁢[1+J0(ϕm)−J2(ϕm)]sin(ωt) \\ +Δϕmcos(ωτ/2)cos(2α)sin(2πl/lb)[1+J0(ϕm)+J2(ϕm)]cos(ωt), \end{gathered}$$(1)where h is the polarization cross talk of the modulated fiber, l is the length of fiber wound on the PZT modulator, Δϕ is the birefringent phase difference caused by the modulation, ω is the modulation circular frequency, τ is the transit time of light in the fiber, α is the polarization angle, i.e., α=arctan(Ex/Ey), where Ex and Ey are the amplitudes in x and y polarization axes, lb is the fiber beat length (about 3 mm at room temperature), and Δϕm is the induced phase difference between the two principal axes of PM fiber during the phase modulation process. In this study, Δϕm≈10−4. The demodulated output of the open-loop FOG is I=I0J1(ϕm)sinϕssin(ωt), where ϕs is the phase shift to be measured, which is caused by the rotation of the FOG, and Jn(ϕm) is the nth order Bessel function of the first kind of the modulation depth ϕm^[14].

From (1), the modulation polarization error can be divided by two parts of the quadrature part with the cos(ωt) term and the in-phase part with the sin(ωt) term according to the demodulated signal. The quadrature part can be suppressed to 10−7rad by precise phase locking and can be neglected compared with the in-phase error. Hence, our work mainly focuses on the study of suppressing in-phase error. When the modulation length, frequency, and transit time are determined, the in-phase modulation polarization error is primarily determined by the polarization cross talk, the birefringence phase difference, and the amplitude ratio of the light in the two principal axes. The influence of the modulation-related error of the gyroscope is estimated through theoretical analysis and experiments below.

The phase change of the modulator δϕ is realized by periodically stretching and squeezing the optical fiber fixed on the ceramic outer circle, which can be estimated as δϕ≈0.78βlε, where β is the propagation constant of the fiber polarization mode and ε is the applied strain on the fiber^[16]. The modulation depth of open-loop FOG with sinusoidal modulation ϕm is calculated by ϕm=2δϕsin(ωτ/2). When ϕm=1.84rad, the gyroscope operates at its highest sensitivity. In this scenario, the maximum strain of the fiber is approximately 4.5×10−7. The optical fiber is wound on the outer circle of the piezoelectric ceramic under significant tension, and the angle between the principal axis of the optical fiber and the deformation direction of the piezoelectric ceramic is θ. As shown in Fig. 2(b), fiber bending, together with stretching and squeezing in the modulation process, generates a parasitic birefringence inevitably at the same frequency as the modulation. The propagation constant difference due to birefringence, Δβ, can be estimated by the following formula^[9]: Δβ=πn3(P12−P11)(1+ν)(2−3ν1−ν)aRε/λ(cosθ−sinθ),(2)where n is the refractive index of the fiber, a is the cladding radius, R is the radius of PZT, λ is the operating wavelength, p11 and p12 are the elastic-optic coefficients of the fiber, and ν is Poisson’s ratio. During the drawing process, the PM fiber’s polarization principal axis rotates slowly, causing Δβ to change gradually along its propagation axis. With the parameters used, the birefringent phase difference caused by modulation is approximately 7×10−3rad. In addition, the large initial tension used in winding the piezoelectric ceramic modulator and the transverse force exerted on the fiber by the piezoelectric deformation will increase the polarization cross talk significantly. This cross talk, h, can be estimated by the following equations^[15]: h=(1/16lb)sin2(2χ)[1−cos(2πlbK/λ)]2,(3)wherein χ=arctan[ε(0.04fν+4εYνπa2)(p11−p12)sin2θ/πlbΔn−ε(0.04fν+4εYνπa2)(p11−p12)cos2θ/πlb],(4)K=[Δn2+ε2(p11−p12)2(0.04fν/πa+4ενYa)2−2ε(p11−p12)Δn(0.04fν/πa+4ενYa)cos(2θ)]1/2,(5)where Δn is the birefringence of the fiber, Y is the Young’s modulus, and f is the winding tension of the PZT modulator. Based on Eqs. (3) to (5), when the ϕm is 1.84 rad, the average value of polarization cross talk is about −30dB/m. Accurate measurement of polarization cross talk and validating theoretical calculations serve as the foundation for further research on error suppression methods. Traditional optical time-domain distributed measurement methods often have spatial resolutions exceeding 0.2 m. However, the length of the coiled fiber on the PZT modulator is only about 1 m. To accurately measure the distributed polarization cross talk when the modulator is in action, an OFDR system is developed to measure Rayleigh scattering from two polarization principal axes^[16–18]. The in-house OFDR system achieves a remarkable sensitivity of up to −140dB and a spatial resolution of 8 µm, comparable to commercial systems’ performance in the telecom bands. Its schematic diagram is shown in Fig. 2(a).

The output of a mode-hop-free wavelength-swept laser with a wavelength range of 815 to 845 nm is linearly polarized using an in-house developed 45° TFG with a PER exceeding 50 dB. It is then divided into two parts through a coupler. Fifty percent of the light is kept as the local oscillator light, while the other 50% is sent, via an optical circulator, to the measurement path that contains the fiber wound around the piezoelectric ceramic. The polarization state of the light launched into the PM fiber on the PZT modulator is controlled by a polarization controller (PC). In the experiment, it is aligned with the fast axis of the PM fiber. At the far end of the fiber wound on the PZT modulator, the measured PER exceeds 40 dB. The OFDR measurement is employed to confirm that the intensity difference of the Rayleigh backscattered light between the S-wave and P-wave states also surpasses 40 dB, which is expected because of the superior polarization cross talk performance (−45dB/m) of the PM fiber used for the FOG. The Rayleigh scattered light on the optical fiber returns to the main path through the circulator, which is then split into two principal polarization states (S-wave and P-wave) via a polarization beam splitter (PBS). Subsequently, they are mixed with two copies of the local oscillator reference light, obtained via a PBS (the upper one in Fig. 2(a), to form a frequency domain interference signal). The cross talk of the polarization state of the fiber on the modulator can then be obtained via inverse Fourier transform of the record interference signal. This OFDR has a spatial resolution of 8 µm and a sensitivity of −140dB. When the modulation depth is 1.84 rad, the obtained experimental results are shown in Fig. 2(c). The Rayleigh scattering feature is clearly shown on the OFDR trace for the S-wave and P-wave. Initially, the scattered S-wave is much larger than the P-wave, which agrees with the fact that the light is launched into the fast axis. After the fiber-fixing point on the PZT ceramic, the light starts to couple into the slow axis, and the coupled power increases monotonically with the distance. When the light exits the modulator (the fiber fixing point at the distance end), the cross talk reaches about −32dB, which moderately agrees with our theoretical analysis above.

For the traditional PM open-loop FOG, the linearly polarized light filtered by the polarizer is aligned with the principal axis of the fiber, and the polarization is maintained along the whole optical path, i.e., α=0. Substituting the analyzed and measured polarization cross talk, birefringent propagation constant, and the gyroscope’s design parameters into Eq. (1), we can obtain a modulation polarization error of approximately 8×10−6rad. The corresponding gyroscope bias error reaches 15°/h^[11], which is larger than the requirements of <1°/h for the widely used gyroscope in motion gesture control.

From Eq. (1), as the polarization angle α approaches 45°, i.e., the light amplitudes of x and y polarization axes are equal, the polarization error reaches a minimum value. It can be obtained by inserting a λ/4 wave plate following the polarizer with the wave plate axis aligned 45° to that of the polarizer^[19]. Because there is no reliable splicing method for our 40 µm ultrafine PM fiber, both the polarizer and the wave plate are required to be fabricated in the form of fiber in-line. Fortunately, the fiber-optic industry has already developed versatile ultraviolet (UV) inscription technology for in-line fiber-optic devices. The employed polarizer is based on a 45° TFG^[20,21]. The wave plate can be fabricated using the UV inscription method as shown in Fig. 3(a).

A 248 nm KrF excimer laser with a single-pulse energy density of 20mJ/cm2 is used for wave plate inscription. A cylindrical lens is used to focus the light into the fiber core. The cylindrical lens and reflecting mirror are mounted on a linear scanning stage, enabling the scanning of the UV beam along the fiber. The scanning speed is 50 µm/s, and the required length of the wave plate is precisely controlled by adjusting the scanning time. The fiber is hydrogen loaded with 100 atmospheres at 85°C for 48 h for sensitization treatment^[22]. The optical fiber is rotated under the microscope to align its principal axis at 45° to the incident direction of UV light. The refractive index of the optical fiber increases with the absorption of UV light^[23–25]. Since the front part of the optical fiber facing the incoming UV beam absorbs more light, the corresponding UV-induced refractive index change is higher than that of the back part of the fiber, as shown in Fig. 3(b), which in turn causes birefringence, as shown in Fig. 3(c)^[23,24]. The irradiation fluence and the wave plate length control the phase retardation of this UV-induced fiber in-line birefringence wave plate, which is monitored in situ as the wave plate is being fabricated. As shown in Fig. 3(d), without the need to cut the fiber for fabricating individual devices, a broadband light source is first coupled into the ultrafine fiber, and then it is filtered by a 45° TFG polarizer fabricated before the wave plate, resulting in highly linearly polarized output light with a PER of at least 35 dB. Since the principal axis of the wave plate is 45° to the principal axis of the fiber, as its phase delay grows, the polarization state will change from linear polarization to elliptical polarization and to circular polarization when the phase retardation is π/2. This kind of polarization change can be monitored using the PER value of the output light; the result is shown in Fig. 3(e) for a 20 mm long wave plate. With the increasing UV fluence and thus phase delay, the initial high PER drops gradually to almost zero, indicating the conversion of linearly polarized light to nearly perfect circularly polarized light.

The inscribed λ/4 wave plate needs to be annealed at 100°C for 24 h for long-term stability^[26]. Such a practice usually results in approximately 10% birefringence degradation. Therefore, in a practical inscription, the wave plate needs to be overinscribed by about 10% after the PER reaches a minimum value near zero. However, the degradation mentioned above is related to various factors such as fiber characteristics and hydrogen-loading sensitization. Considering the large dispersiveness of process control and the axis alignment error, the PER after annealing could be controlled within 1.8 dB in our work, as shown in Fig. 3(f) for 10 samples, indicating the technical soundness of our in situ monitoring method. Using these samples’ average PER of 1.34 dB, α is calculated to be 4.4° away from the optimal value of 45°, and in this case, the modulation polarization error could be suppressed by 8 dB. It is worth noting that, from Eq. (1), the in-line inscription of the λ/4 wave plate could also suppress the quadrature error by the same proportion. However, the modulation polarization error should be suppressed to more than 12 dB, and the gyro accuracy can reach 1°/h. It is therefore necessary to combine other technical solutions to suppress the error.

The above analysis suggests that the polarization error cannot be reduced to a value less than 1°/h using the wave plate alone. An in-line polarization mode converter is proposed to be combined with the wave plate to further suppress the error. Modulation polarization error is proportional to the induced birefringence phase difference. If the polarization modes are exchanged at the midpoint of the optical fiber, and the birefringence generated by modulation is equal at two locations with the same distance from the midpoint, then Δϕ=∫0l/2Δβ(z)dz−∫l/2lΔβ(z)dz=0,(6)where z is the location of the fiber. This means the phase retardation due to birefringence effectively cancels out. However, traditional PZT ceramic modulators use a unipolar spiral fiber winding form^[11], and the midpoint is in the center of the piezoelectric ceramic. It is impossible to put a PMC there. As shown in Fig. 4(a), we resort to a bipolar symmetrical winding form, of which the fiber midpoint is placed away outside of the ceramic, and the lead fiber of the PMC located at the midpoint is wound in the vicinity of the piezoelectric ceramic to ensure that the modulation birefringence at the symmetrical position is the same.

A traditional method of making a PMC is to cleave an optical fiber to obtain two cleaved fibers and rotate these two fibers’ axis so that one fiber’s fast axis is aligned with the other one’s slow axis and then splice them together. However, our fiber cannot be spliced, so we developed a modified taper technology to fabricate such a polarization mode converter. As shown in Figs. 4(b) and 4(c), a 2 mm-long segment of optical fiber is subjected to a hydrogen–oxygen flame fusion process and slowly stretched to a length of 8 mm. This process results in the formation of an adiabatic double taper structure with a parabolic shape. The diameter of the tapered region is approximately 6 µm. A 90° rotation is located at the narrowest zone of the taper, effectively achieving the desired polarization mode conversion. The tapered PMC is packaged and protected in a quartz tube^[27].

By utilizing the fabrication methods described above, a total of 20 PMCs are produced, and the polarization angle variation during fabrication is assessed using a PER meter, as shown in Fig. 4(d). Statistical analysis reveals that the polarization angle variation is 86°±4°, and the deviation from 90° could be attributed to inadvertent rotation during the cooling process of the optical fiber. It is estimated that the modulation polarization error can be suppressed by an average of 11 dB and a minimum of 6 dB.

The enhanced FOG configuration, incorporating both a wave plate and a PMC, is depicted in Fig. 5(a). These devices will be added sequentially in the experiment for comparative testing. The modulation polarization error is proportional to sin(2πl/lb), where l represents the fiber length and lb represents the beat length. For stress birefringence fibers, the beat length is proportional to temperature^[28]. Consequently, the modulation polarization error exhibits quasi-periodic sinusoidal fluctuation characteristics with temperature. Suppression of modulation polarization error is evaluated by analyzing the amplitude variation of bias with temperature fluctuation, with results presented in Fig. 5(b).

The test results reveal that the traditional open-loop FOG suffers from a significant modulation polarization error. The peak-to-peak temperature-dependent fluctuation is approximately 25°/h. However, by incorporating an in-line λ/4 wave plate, the modulation polarization error suppression is about 7.1 dB, which is consistent with our analysis above. Subsequently, both an in-line λ/4 wave plate and an in-line PMC are introduced simultaneously in the open-loop FOG. The bias caused by the modulation polarization error with temperature fluctuation is buried in the gyroscope’s noise. The peak-peak value is only 0.7°/h, and the additional suppression is better than 8.4 dB. The combined suppression from the wave plate and the PMC is better than 15 dB, which successfully fulfills the accuracy requirements with a bias error of less than 1°/h. Allan variance and gyro dynamic characteristics have also been tested, as shown in Fig. 5(c). The result shows that the gyro bias instability, which can be calculated by dividing the bottom of the curve by 0.664, reaches 0.1°/h, which meets the requirements of control-level applications.

Polarization cross talk in the fiber wound on the PZT ceramic modulator is carefully characterized both theoretically and experimentally with a self-developed OFDR. A new scheme of open-loop FOG is proposed to overcome the polarization modulation error due to the polarization cross talk in the modulator. The corresponding critical components under the new scheme, in-line λ/4 wave plate, bipolar symmetric wound PZT modulator, and PMC, are introduced and discussed. Our experimental investigations suggest that under the proposed scheme with practical components fabricated in-house, the polarization modulation error can be significantly reduced by more than 15 dB. The open-loop FOG can achieve a bias error of less than 1°/h. The findings may open the door to cost-effective FOG, with a wide application scope, including but not limited to motion sensing, gesture control, and inertial navigation.