1. Introduction

The interferometric fiber-optic gyroscopes (IFOGs) as one of the most successful fiber sensors have widely been used in inertial navigation systems owing to their features of high sensitivity, small size, and large dynamic range [1,2]. The demand for high-precision IFOGs has increased with the expansion of IFOG applications. However, it is known that the environmental adaptability of the IFOGs, especially the effect of external temperature disturbance has currently become the key and urgent factor that limits their stability and accuracy [3,4]. To solve this, the possible temperature compensation mechanisms in data processing are introduced [5], while this method is inapplicable under severe environmental changes and tends to increase the complexity of the IFOG systems. The fiber coil is the core optical component of the IFOG. The output bias of IFOGs has been proven to be sensitive to varying thermal gradients present across the fiber coils [6]. The phase nonreciprocal error caused by polarization crosstalk in fiber coils has also been found to increase the noise level and degrade the temperature stability of the IFOG output [7,8]. Thus, enhancing the performance of the fiber coil is a measure necessary to improve the precision of IFOGs operating in harsh environments.

Much research has been performed on the temperature characteristics of fiber coils. Adjusting the thermal parameters of the curing adhesive and utilizing potting technology for the coils can reduce the additional stresses under variable temperature conditions [9,10], but with limited effects. Given the improved thermal stability of the coil obtained by the quadrupolar (QAD) winding pattern, the subsequent adoption of an octupole [11], and a 16-polar symmetrical winding pattern are proposed [12]. Results show that a coil with a 16-polar wound exhibits a better suppression for the thermal-induced bias drift of IFOGs as increasing fiber position symmetry. On the other hand, the polarization-maintaining ability of the fiber structure itself for improving the performance of the coil must be emphasized. Currently, the coils in IFOGs are wound by polarization-maintaining fibers (PMFs), including traditional PMFs and photonic crystal fibers (PCFs). Because of their versatile design features, PCFs have attracted great attention for achieving high birefringence [13]. However, a high loss and manufacturing cost make the difficulty of applications in IFOGs. The traditional PMFs with mature fabrication technology have been proven to reduce polarization errors and Farady effects owing to their ideal optical performance for the major applications in optical paths of IFOGs [14], such as the elliptical-core PMF and Panda-type circular-core PMF (C-PMF) [15]. In our previous work [16,17], an elliptical-core Panda-type PMF based on traditional PMFs was proposed to improve the performance of the coil, but a large extinction ratio fluctuation in the coil compromises their advantages in the IFOG. Therefore, further optimizing the PMF designs and a combination of ideal coil winding patterns to enhance polarization performance and thermal stability of fiber coils is extremely meaningful in improving sensing precision over the variable temperature range.

In this paper, we propose two hybrid PMF structures composed of the symmetrical circular stress-applying parts (SAPs) and the elliptical-core with different polarized directions of slow propagation axis, creating a Panda-type horizontal-elliptical core PMF (HE-PMF) by introducing an enhanced geometry effect, and a Panda-type longitudinal-elliptical core PMF (LE-PMF) employed a suppressed geometric birefringence as a comparison to verify the polarization theory and design principles. Both proposed PMFs for IFOGs include analytical models that are systematically given based on the elastic-optic and thermal stress effects. To determine ideal fiber designs, the influence of the structural factors on the modal birefringence of both PMFs is investigated. Other modal performance and the tolerance of fabrication is also comprehensively evaluated via numerical simulations. The Panda-type HE-PMF and LE-PMF are then experimentally fabricated based on optimized geometric parameters and wound into four fiber coils in combination with various configurations such as QAD and 16-polar symmetrical winding patterns. Under static and dynamic temperature conditions, the birefringence, ER, and corresponding assembled IFOG output tests of four fiber coils are performed and compared to those of a conventional Panda-type C-PMF coil. The birefringence of the HE-PMF increases significantly and the ER value is found to be up to 29.96 dB under the static temperature of 25 °C, and wound coils both maintain ultrahigh ER of over 30 dB, corresponding to an almost one-fold improvement over the traditional coil. Furthermore, the dynamic testing results reveal that the HE-PMF coil combined with a 16-polar wound exhibits minimal fluctuations in ER and IFOG output, indicating strongly enhanced polarization performance and thermal stability. The fluctuation in ER of this coil is significantly reduced to 3.0 dB compared to a 6.20 dB fluctuation of the existing PMF coils, and the output bias stability of the assembled IFOG is improved from 0.136 °/h to 0.082 °/h. The results suggest the HE-PMF design with high birefringence combined with a 16-polar symmetry winding pattern contributes to improving the ER property of coils and suppressing the IFOG output drift caused by polarization crosstalk under temperature fluctuations. The LE-PMF, on the other hand, has poor modal properties, as demonstrated by a low birefringence, a small ER, and a larger drift in the related IFOG output. The experimental results reveal great agreement with theoretical analysis and numerical simulations. The advancements of both fibers are critical for boosting the practical availability of PMFs in high-precision IFOGs.

2. Theory

In order to maintain linear-polarization of the orthogonal modes in single-mode fibers while avoiding the destruction of the double-degeneracy caused by the internal defects and external temperature environments during the actual manufacturing, high modal birefringence is introduced to expand the difference between the propagation constants of the polarization modes, further improving the transmission stability of the light wave [18]. The modal birefringence (B) takes into account both geometric birefringence (B_G) and stress birefringence (B_S), which is determined by taking the difference of the effective refractive indices of two orthogonal polarization modes:

It can be concluded from Eq. (6) that the change rate of unguided mode is proportional to the ΔER caused by dynamic temperature loads, while inversely proportional to ER and varies with ER at a faster than exponential rate. The drift of the IFOG output is closely related to the signal power P_f and ΔP_f / P_s of the optical path, with low values of P_f and ΔP_f / P_s both requiring the PMFs with a large ER and a small ΔER, contributing to enhancing polarization-maintaining ability and temperature stability of the fiber coils. Remarkably, achieving high ER values can decrease the impact of thermally induced ΔER on the power variation of the IFOG output signal by up to ten times based on Eq. (6). Thus, under the constrained length of PMFs, improving polarization performance of the PMFs by prioritizing the attainment of a high birefringence and high ER and then in conjunction with a low value of thermally induced fluctuations in ER is extremely meaningful in enhancing temperature stability of IFOGs.

The effective mode area for evaluating the property of PMFs determines the confinement loss and energy concentration of the optical transmission system. It can be calculated as:

The chromatic dispersion (D) can be directly calculated from the effective index of the fundamental mode [21]:

3. Simulation

3.1 Enhancement and suppression effects of geometry in elliptical core

The modal birefringence of PMFs can be obtained by introducing the geometric birefringence that arises from the perturbation in the dielectric constant and magnetic permeability of the material caused by the anisotropy of the core geometry, such as an elliptical-core PMF (as shown in Fig. 1(a)), and by introducing stress birefringence via SAPs with high thermal expansion coefficients to induce an asymmetric stress distribution surrounding a fiber core during high- temperature annealing, such as a Panda-type C-PMF (as shown in Fig. 1(b)). We firstly conducted on two typical PMFs to analyze the polarization theory, geometric and stress effects using the software Comsol Multiphysics based on the full-vector finite-element method (FEM). The solid mechanics and wave optics modules are combined to imitate the propagation of electromagnetic waves and then the effective refractive indices of fundamental modes can be obtained by solving the eigenvalue. The schematic cross-sections and corresponding refractive index profiles of both fibers are shown in Fig. 1, where the cladding radius (W) is 40 µm to achieve optical fiber miniaturization. The radii along the x-axis and y-axis of the elliptical-core in Fig. 1(a) are a and b of fixing at 4 µm and 2.5 µm, respectively, with an ellipticity denoted as e = a/b. For the Panda-type C-PMF shown in Fig. 1(b), the radii of the core (a) and SAPs (R) are 4 µm and 12 µm, respectively, and a gap (d) of 2 µm. The simulation parameters used in modeling have been determined to be utilized in the practical fabrication and relevant for high stress birefringence [22], which are presented in Table 1.

 

Fig. 1. Schematic cross-sections and effective refractive index profiles of the typical (a) elliptical-core PMF, and (b) Panda-type C-PMF.

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Table 1. Simulation parameters for PMFs

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The fundamental electric field distributions for both the x- and y-polarized modes at a wavelength of 1550 nm of the elliptical-core PMF are depicted in Fig. 2(a). The effective refractive index in the x-polarized mode (the major-axis of the elliptical-core) is higher than the y-polarized mode (the minor-axis direction), arising from the geometric shape anisotropy of the core. However, the polarization-maintaining ability of this fiber is limited with a poor birefringence when used alone without constraints from the SAPs. Figure 2(b) shows the electric field of the Panda-type C-PMF. The fundamental electric field of both polarizations is well confined inside the core, while the mode field of x-polarization (the horizontal direction of the SAPs axial connection) is more strongly confined in the core region than that of y-polarization (the vertical direction of the SAPs axial connection), revealing a larger effective refractive index for the x-polarized mode. This comes from that the introduction of high thermal expansion SAPs in the x-direction contributes to the strong confinement in the core region of the x-polarized mode, leading to a high concentration of light [21]. Figures 2(c) and 2(d) present the von Mises stress and stress birefringence distributions in the transverse cross-section of the C-PMF. The greatest stress-induced birefringence value between the core and two SAPs is observed to be around 1.3 × 10⁻³, inducing a modal birefringence optimization of 5.8 × 10⁻⁴, and this is one of the reasons that this fiber is widely used in IFOGs.

 

Fig. 2. Fundamental mode field profiles for x- and y-polarization of the (a) elliptical-core PMF, and (b) Panda-type C-PMF. (c) Von Mises stress, and (d) stress-induced birefringence distributions of the Panda-type C-PMF.

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Based on polarization-maintaining properties of the above PMFs are achieved by introducing geometric and stress birefringence, separately. Figures 3(a) and 3(b) show the schematic cross-sections, structural parameter definitions, and refractive index profiles of the two proposed hybrid PMFs: the Panda-type HE-PMF and LE-PMF. The designed PMFs are both composed of symmetric circular SAPs and an elliptical-core with different polarized directions of the slow propagation axis, creating novel fibers by a collaboration of geometry and stress effects. Specifically, the Panda-type HE-PMF in Fig. 3(a) is characterized by a consistent alignment between the major-axis of the elliptical-core and the SAPs axial connection. This unique configuration strengths lie in that the slow propagation axes of both geometric birefringence (n_x) and stress birefringence (N_x) are superimposed in the same x-polarized direction, contributing to an enhanced modal birefringence by enlarging the difference between the propagation constants of the polarization modes based on Eq. (1). Whereas for the Panda-type LE-PMF in Fig. 3(b), the slow propagation axes of geometric birefringence (n_y) and stress birefringence (N_x) are orthogonal to each other in the x and y-polarized directions. This arrangement results in a reduction of the modal birefringence arising from a suppressed effect of geometric birefringence on stress birefringence. Note in the following numerical simulations, the W of PMFs remains 40 µm, and the mole percentage of the B₂O₃ in SAPs is defined as m. The simulation methodology and parameters discussed above will be utilized in the following analyses.

 

Fig. 3. Schematic cross-sections, parameter definitions, and effective refractive index profiles of the designed Panda-type (a) HE-PMF, and (b) LE-PMF.

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3.2 Fiber designs and parameters selection

Numerical simulations are carried out to determine the target structure dimensions of proposed PMFs. The parameters of the core (including a, b, and e) and SAPs (including R, d, and m) are swept to calculate modal birefringence at 1550 nm. We firstly focus on the target of high birefringence by exploring the core parameters a and b (from 2 μm to 6 μm with 0.2 μm spacing) with SAPs parameters R fixed at 12 µm, d fixed at 2 µm, and m fixed at 30%. The selection of the parameters R, d, and m will be discussed later. A colormap of the calculated modal birefringence as functions of a and b is displayed in Fig. 4. It is noteworthy that the Panda-type HE-PMF with a > b and the Panda-type LE-PMF with a < b are distinguished based on the values of the major and minor-axes of the elliptical-core. The elliptical-core becomes a circular-core when a = b, corresponding to the traditional Panda-type C-PMF. From Fig. 4, the modal birefringence of the HE-PMF is significantly higher than that of the C-PMF and LE-PMF. This is because the HE-PMF structure makes the most of the superposition effects of both the geometric and stress birefringence. On the other hand, the LE-PMF exhibits lower modal birefringence values, as the geometric birefringence exerts a suppressed effect due to their orthogonal feature of propagation axes. Furthermore, the modal birefringence decreases gradually with increasing b of the LE-PMF when a is fixed, attributed to stronger suppression of geometric birefringence in the y-polarized direction to modal birefringence. For the HE-PMF, with increasing a in the major-axis direction from 2 µm to 6 µm, the modal birefringence would increase from 5.0 × 10⁻⁴ to 8.1 × 10⁻⁴ given the contribution in a superposition of the core geometric shape anisotropy formed enhanced geometric birefringence to high modal birefringence. Considering the existing fiber manufacturing facility restriction, the excessive ratio of the major-axis to the minor-axis in an elliptical-core induces transmission and splicing loss. Thus, the points in the black circles of a = 4.1 µm and b = 2.7 µm for the HE-PMF (corresponding to e = 1.5), and a = 2.7 µm and b = 4.1 µm for the LE-PMF (corresponding to e = 1/1.5) are chosen as the final design sizes, with modal birefringence equal to 7.15 × 10⁻⁴ and 5.31 × 10⁻⁴, respectively. Additionally, within the region of 4 µm ≤ a ≤ 6 µm and 2 µm ≤ b < 4 µm, the HE-PMF maintains high modal birefringence with larger than 6.20 × 10⁻⁴, while within the region of 2 µm ≤ a < 4 µm and 4 µm ≤ b ≤ 6 µm, the LE-PMF has a modal birefringence with smaller than 5.80 × 10⁻⁴ and larger than 3.8 × 10⁻⁴. It is believed that the modal properties of both designs are quite tolerant to fabrication errors while meeting the polarization ability of fiber sensors.

 

Fig. 4. A colormap of the modal birefringence as functions of a and b at 1550 nm for R = 12 µm, d = 2 µm, and m = 30%. In the black circles are the final design points.

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We further investigate the influence of SAPs on modal performance by sweeping the other three parameters R, d, and m, with the elliptical-core fixed at a = 4.1 µm, b = 2.7 µm in the HE-PMF, and a = 2.7 µm, b = 4.1 µm in the LE-PMF. Figures 5(a) and 5(b) present colormaps on modal birefringence values of all combinations of R (from 8 μm to 16 μm with 0.5 μm spacing) and d (from 1 μm to 4 μm with 0.2 μm spacing) in the HE-PMF and the LE-PMF, respectively. The modal birefringence increases significantly with decreasing d and increasing R. The main reasons are that SAPs are closer to the core for a smaller d, and increasing R means an expansion in the area of SAPs, both of which are the direct factors causing higher stress birefringence. We choose the point R = 12.5 µm and d = 2 µm in the black circle as the final designs for both PMFs given the fabrication constraints. Within the whole swept region of 1 µm ≤ d ≤ 4 µm, and 11.5 µm ≤ R < 16 µm, the HE-PMF exhibits a high birefringence larger than 6.20 × 10⁻⁴. Additionally, the modal birefringence of the Panda-type C-PMF, HE-PMF, and LE-PMF as functions of the parameter m is displayed in Fig. 5(c). In contrast to the traditional C-PMF with keeping smaller than 5.80 × 10⁻⁴ of birefringence in the whole range of m, the HE-PMF characterized by enhanced geometric effect shows a superior birefringence property, while the LE-PMF with suppressed geometric birefringence has a lower one. Within the region of 18% ≤ m ≤ 30%, the HE-PMF achieves a high modal birefringence larger than 6.25 × 10⁻⁴, while the LE-PMF maintains a birefringence smaller than 5.40 × 10⁻⁴. Eventually, to obtain larger birefringence with reasonable parameter values, we choose the point m = 30% with the modal birefringence equal to 7.34 × 10⁻⁴ for the HE-PMF and 5.40 × 10⁻⁴ for the LE-PMF as the target designs.

 

Fig. 5. Colormaps of the modal birefringence as functions of R and d at 1550 nm in the Panda-type (a) HE-PMF for a = 4.1 µm, b = 2.7 µm, and (b) LE-PMF for a = 2.7 µm, b = 4.1 µm. (c) Modal birefringence contrast of the C-PMF, HE-PMF, and LE-PMF versus m for R = 12.5 µm, d = 2 µm. In the black circles are the final design points.

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3.3 Modal properties analysis

The stress-optical coupling effects and other modal properties, including effective mode area (A_eff), nonlinear coefficient (γ), and chromatic dispersion (D) for both target PMF structures are comprehensively analyzed to evaluate the overall optical performance and fabrication feasibility of designs. Figures 6(a) and 6(b) present the stress-induced birefringence (N_x-N_y) distributions in transverse cross-sections at 1550 nm of the Panda-type HE-PMF and LE-PMF, respectively. One can see that the greatest birefringence between the outer ellipse and two SAPs in the HE-PMF is around 1.40 × 10⁻³, and the stress-induced birefringence of the elliptical-core region is 7.04 × 10⁻⁴. Similarly, for the LE-PMF, the highest stress-induced birefringence is 1.25 × 10⁻³, and the value of the elliptical-core region is 5.97 × 10⁻⁴.

 

Fig. 6. Stress-induced birefringence (N_x-N_y) distributions for the (a) HE-PMF, and (b) LE-PMF.

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We further discuss the A_eff, γ, and D for the x and y-polarized modes of both designed PMFs over the whole C + L band ranging from 1530 nm to 1625 nm. The results are displayed in Fig. 7, where it is observed from Figs. 7(a) and 7(b) that the A_eff and γ values gradually increase and decrease, respectively, with increasing wavelength accordingly over the entire band. Table 2 summarizes the calculated A_eff and γ for the designed PMFs and the fused common C-PMF at 1550 nm. The A_eff values are found to be within the range of (24.63, 25.13) µm² for the HE-PMF, and (24.56, 25.17) µm² for the LE-PMF, while the γ values are within (7.225, 7.853) km⁻¹W⁻¹ for the HE-PMF and (7.213, 7.876) km⁻¹W⁻¹ for the LE-PMF. Especially, at a typical wavelength of 1550 nm, the A_eff values of both PMF structures are 24.73 μm² and 24.66 μm², respectively, making them excellent fusion matches with common PMFs such as output branches of Y-waveguide in IFOGs and ideal selections for practical fabrication with low loss. The γ values at 1550 nm are 7.72 km⁻¹W⁻¹ and 7.74 km⁻¹W⁻¹, respectively, indicating a high concentration of light. Also, both structures maintain single-mode characteristics, and the fundamental electric field is well limited in the core region over the whole wavelength range. It is revealed by Figs. 7(c) and 7(d) that the difference between D values for the two designs is small, within (4.07, 5.47) ps/nm/km and (4.11, 5.49) ps/nm/km over the whole wavelength range, respectively, and of 5.25 ps/nm/km and 5.26 ps/nm/km at 1550 nm. The proposed PMF structures show ideal modal properties for further manufacturing and are suitable for applications in the optical paths for IFOGs.

 

Fig. 7. Calculated A_eff and γ versus wavelength in the (a) HE-PMF, and (b) LE-PMF. Calculated D in the (c) HE-PMF, and (d) LE-PMF.

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Table 2. The performance of fused PMFs in IFOGs at 1550 nm.

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Considering the potential performance degradation resulting from fluctuations in fiber structure parameters during the fabrication process. The sensitivity of modal properties of both PMFs over the whole C + L band is thoroughly assessed. Figure 8 illustrates the effect of a ± 5% variation around the designed values in core parameters on the modal performance. It is found that a ± 5% variation in the major-axis (a) causes at most ± 0.62% change in birefringence, ± 4.06% in the A_eff, ± 4.24% in the γ, and ± 4.74% in the D of the HE-PMF, while in the major-axis (b) causes ± 2.68% change in birefringence, ± 4.10% in the A_eff, ± 4.27% in the γ, and ± 5.21% in the D of the LE-PMF. Further investigations numerically suggest that a ± 5% fluctuation in the minor-axis (b) leads to ± 2.12% change in birefringence, ± 4.46% in the A_eff, ± 4.66% in the γ, and ± 17.69% in the D of the HE-PMF, while in the minor-axis (a) leads to ± 2.44% change in birefringence, ± 4.45% in the A_eff, ± 4.66% in the γ, and ± 16.11% in the D of the LE-PMF. Although the dispersion deviations caused by the minor-axis of elliptical cores are larger in both designs, the D values after experiencing fluctuations are within (3.35, 5.67) ps/nm/km for the HE-PMF, and within (3.44, 5.65) ps/nm/km for the LE-PMF over the whole band. Remarkably, the calculated deviations of the birefringence, A_eff, γ, and D at 1550 nm also demonstrate excellent tolerance for fabrication errors, revealing the robustness of both designs.

 

Fig. 8. Modal birefringence in y-polarization versus wavelength for a variation of ± 5% in the (a) a, and (b) b for the HE-PMF, (c) a, and (d) b for the LE-PMF; A_eff as functions of wavelength in the (e) a, and (f) b for the HE-PMF, (g) a, and (h) b for the LE-PMF; γ as functions of wavelength in the (i) a, and (j) b for the HE-PMF, (k) a, and (l) b for the LE-PMF; D as functions of wavelength in the (m) a, and (n) b for the HE-PMF, (o) a, and (p) b for the LE-PMF.

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4. Experiments

4.1 Static test

In the following, the practical fabrication of the proposed Panda-type HE-PMF and LE-PMF based on simulation optimization is achieved by employing the modified chemical vapor deposition method. In order to ensure reliable comparisons between experimental and simulated evolutions for modal properties, the parameters of the PMFs here should be kept as consistent as possible. The detailed geometric parameters, cutoff wavelength (C-λ), and mode field diameter (MFD) for each PMF in experiments are listed in Table 3. The whole disks and cross-sections scanned by the electron microscope of both designed 1-km PMFs are shown in Fig. 9. The operation wavelength is 1550 nm. The modal birefringence values of both PMF structures are numerically and experimentally reported via comparison with a conventional Panda-type C-PMF under a static temperature of 25 °C. The results presented in Table 3 exhibit a good agreement between numerical and experimental data, thus confirming that the analytical models based on the coupled wave optics theory and the solid mechanics theory provide reliable simulations of polarization-related phenomena in proposed PMFs. Moreover, the modal birefringence of the HE-PMF with optimized structural parameters is measured to be 7.63 × 10⁻⁴, corresponding to a significant improvement of 2.01 × 10⁻⁴ compared to the C-PMF arising from a strong enhanced geometric anisotropy of a superimposed effect, verifying that the proposed HE-PMF has a higher polarization-maintaining ability. On the other hand, the LE-PMF shows a slight reduction in birefringence by 0.25 × 10⁻⁴, which is attributed to a suppressed geometric birefringence resulting from a minor ellipticity in the longitudinal-elliptical core. Anyway, the results reveal a consistent trend with the designed principles in Section 3.2, confirming the validity of the polarization theory and designed models. Hence, the geometric effect plays a critical role in determining the polarization performance of PMFs.

 

Fig. 9. Scanning electron microscopy of the cross-sections of the (a) HE-PMF, and (b) LE-PMF. (c) The whole disks of designed 1-km fibers.

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Table 3. Modal birefringence comparison for fiber designs at 1550 nm under 25 °C.

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The ER property under static temperature is measured to evaluate the polarization-maintaining ability of PMFs. The whole disks of both designed PMFs and a C-PMF with 1-km length are separately fused in the ER test system, as depicted in Fig. 10. The light from the superluminescent diode (SLD) entered ER meter (ERM) to acquire the measured ER values after passing through a semi-closed loop (SCL) optical path that spliced one output branch of Y-waveguide with one end of the PMF. The prepared PMFs are fused into the SCL path with 0° to ensure the same-axis alignment coupling between PMF pigtails (A is the 0° fused point), thereby minimizing the angular error and achieving stable transmission of the polarization state. The experimental results of the ER tests under a static temperature are presented in Table 4. The better polarization-maintaining property of the HE-PMF with optimized parameters is proven by the ultrahigh ER value of 29.96 dB, corresponding to a great increase of 11.92 dB compared to a C-PMF. This superior performance is attributed to the significantly improved values of the modal birefringence resulting from a contribution of the enhanced geometric birefringence in this fiber design. Whereas the LE-PMF exhibits a decrease in the ER value of 0.25 dB relative to the conventional C-PMF, which can be understood for the same reason as that discussed in the previous analysis of birefringence performance. Furthermore, considering that the practical application of geometric PMFs is mainly confined to the transmission and fusion loss of the aspect ratio in an elliptical-core, and thus the losses (l_s) in both designs are calculated by using a power meter (PM) to measure the power at point A of the input port (P_A) and output port B (P_B) in Fig. 10(a). The results from Table 4 show that the tested PMFs are welded into the optical path with ideal loss values. Meanwhile, the SCL path of the test system is similar to that of the IFOG, which makes the fabricated PMFs feasible for the next sensing applications.

 

Fig. 10. (a) Schematic diagram, and (b) corresponding experimental equipment setup of the ER test under the static temperature. SLD: superluminescent diode; ERM: extinction ratio meter.

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Table 4. The ER (dB) and loss (dB/km) testing results of fiber designs at 1550 nm under 25 °C

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Then, the polarization performance of the designed PMF structures including the HE-PMF and LE-PMF are further evaluated by winding them into four coils with various winding configurations, including the quadrupole (QAD) and 16-polar winding patterns. Figure 11 presents the structural diagrams of the QAD and 16-polar winding ones. The two sides of the fiber midpoint are divided into equal section 1 (white cladding) and section 2 (gray cladding), and the arrows represent the direction of the fiber from its starting point to its ending point. The 16-polar method is wound by sixteen layers as one cycle based on the quadrupole (QAD) one. The first four layers are wound in the same way as the QAD, the fifth to eighth layers are in the opposite row of arranging the fibers to the QAD, and the last nine to sixteen layers are wound oppositely to the first eight layers. It can be obtained that the fiber using the 16-polar pattern at the symmetrical layers on both sides of the midpoint is closer in the radial and axial directions than using the QAD one. Also, the times of crossing multiple layers in a 16-polar one are significantly reduced, both of which are the key factors suppressing the thermally induced error of the IFOG caused by temperature perturbations [5]. The frameless coils using QAD winding patterns are designated HE/LE-coil-4, and employing sixteen-polar symmetrical wound patterns are named HE/LE-coil-16, as shown in Fig. 12.

 

Fig. 11. Winding diagram of the (a) QAD pattern, and (b) 16-polar pattern.

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Fig. 12. The coils wound by HE-PMF with (a) QAD pattern, (b) 16-polar pattern; by LE-PMF with (c) QAD pattern, (d) 16-polar pattern. (e) Measured ER result of the HE-coil-16.

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The coils are fused at 0° in the SCL optical path to measure the ER values using the same test method as fibers in Fig. 10. The parameters of the fiber coils used in the experiments, including the coil length (L), height (H), inner diameter (D_i), outer diameter (D_o), number of layers (L_s), number of turns in each layer (T_s), winding pattern (W), and the tested loss (l_s) and ER results of each coil are all listed in Table 5. As illustrated by comparing Table 4 and Table 5, the difference in the loss and ER results between fibers and the corresponding wound fiber coils is found to be small for the same fiber length of 1-km. Although the polarization-maintaining performance of the fiber coil is affected by the additional stress resulting from the layer and turn change during the fiber winding into a coil, the ER of the coil remains stable after the release of residual stress during the coil aging. Moreover, it is obtained from Table 5 a significant advantage in two coils wound by designed HE-PMF compared to the C-PMF coil via one test. This can be demonstrated by consistently keeping the ultrahigh ER of the HE-PMF coils in different winding configurations, such as ER values of 31.93 dB for the HE-coil-16 (corresponding to an almost one-fold increase, as shown in Fig. 12(e)) and 30.56 dB for the HE-coil-4 (an increase of 12.54 dB), suggesting that the geometric superposition in this fiber contributes to the high polarization-maintaining ability of coils. While both LE-PMF coils have a decrease in ER values compared to a C-PMF coil, arising from a low birefringence formed by the suppressed geometric effect. The experimental results further confirm the reliability of the proposed theoretical models and design principles.

Table 5. Size parameters and measured loss and ER of different fiber coils at 1550 nm under static test

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The HE-PMF and LE-PMF coils close-packed with QAD and 16-polar winding patterns are employed to typical sensor IFOGs to verify the static performance of designed PMFs for IFOG applications at room temperature conditions. In detail, the four coils equipped with heat shields are individually fused at 0° in the full-closed loop (FCL) optical path of the IFOG system for four separate static tests. The whole optical path is connected by five fused points of A-E, as presented in Fig. 13(a). The light from the SLD returns to the coupler pigtail after through the FCL path with different PMF coils, and the optical power received at point B for each experiment is over 40 μW. The light is then directed to the detector to complete the final splicing, and the electrical signal is detected at 2 V, both of which strengthen to prove the high-precision splicing of the optical path and are applicable for the operation of a high-performance IFOG. The rotation rate output can be directly acquired by demodulation of the detector signal sent to the digital closed-loop control signal process circuit. A closed-loop IFOG is kept at rest to perform a static output test with the sampling frequency of 1 Hz at a temperature of 25 °C, the experiment setup process is shown in Fig. 13(b). The IFOG output tests of the different fiber coils hold 4 hours at static condition while the rest devices of the IFOG keep the same.

 

Fig. 13. (a) Schematic diagram, and (b) experimental equipment setup of an IFOG test system.

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The testing results of IFOGs with different coils under static conditions are illustrated in Fig. 14. The static output data are smoothed to eliminate noise. One can see that the amount of output drift in IFOGs with different coils differs under static temperature maintained at 25 °C. The IFOG output results of two coils wound by HE-PMF in Fig. 14(a) are both smoother and quickly converge to initial values after experiencing smaller fluctuations. The calculated output bias stability (OBS) of the IFOGs with HE-coil-4 and HE-coil-16 is found to be 0.002421 °/h and 0.001931 °/h, respectively, which are greatly better than the conventional C-PMF coil IFOG of 0.003006 °/h. Also, the angle random walk performance of the IFOG with the HE-coil-16 is reduced to 0.000624 °/√h. This improvement comes from that excellent polarization-maintaining performance with high birefringence and ER induced by a superposition effect of an enhanced geometric birefringence in obtained HE-PMF coils contributes to improving the static stability of the IFOG in the entire testing process. However, there are relatively larger fluctuations for IFOGs with LE-PMF coils characterized by violent vibrations in many periods from Fig. 14(b), and the OBS properties are about 0.003705 °/h and 0.003154 °/h for the LE-coil-4 and LE-coil-16, respectively, which mainly results from a poor polarization performance for the LE-PMF. The measured results of the IFOG static output initially confirm that the HE-PMF design is a promising candidate for achieving high-performance optical fiber sensors.

 

Fig. 14. The static smoothed output bias of IFOG with the (a) HE-PMF, and (b) LE-PMF coils.

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4.2 Dynamic test

Next, we experimentally report the polarization ability and thermal properties of both designed PMFs and the corresponding wound fiber coils in dynamic variable temperature environments. The dynamic temperature is performed between -40 °C and 70 °C in the following tests, which is the actual operating range of the IFOG. The comparison of modal birefringence between C-PMF and both designed PMFs at 1550 nm for varying temperatures is firstly discussed, as shown in Fig. 15. The results of such a study reveal that the birefringence property presents linear evolutions directly related to the temperature, and the birefringence with significant maximum values is remained for the HE-PMF over the entire temperature range, indicating the superior polarization performance in dynamic conditions. Conversely, the birefringence values of the LE-PMF are smaller than the conventional C-PMF. The tests under variable temperature validate the correctness of our polarization theory and simulation models once again, while also proving the accuracy of experimental results by presenting consistent birefringence values at a constant temperature point of 20 °C with those obtained at static temperature in Section 4.1.

 

Fig. 15. Comparison of birefringence test results for varying temperatures.

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The following ER experiments conducted on wound fiber coils allow us to further evaluate the performance of obtained PMFs under dynamic variable temperature conditions. Based on the experimental setup for the ER test at a static temperature, the different coils are placed in the thermostat for the ER tests under variable conditions. Figure 16 exhibits the ER results of the HE-PMF and LE-PMF coils via comparison with a conventional C-PMF coil using a QAD winding pattern. One can see that a significantly improved polarization-maintaining performance and thermal stability of the two coils wound by the HE-PMF are proven by consistently maintaining the higher ER values and a great decrease in the fluctuation range of ER peaks for both HE-PMF coils across full temperature. Specifically, not only have the ER values at 20 °C been greatly enhanced (from 16.17 dB to 30.41 dB in HE-coil-4 and 31.90 in HE-coil-16), but also the higher ER performance has been kept at extremely high and low-temperature conditions, corresponding to more than one-fold increase relative to a C-PMF coil. Remarkably, the maximum fluctuation between the ER peaks is improved to 3.53 dB in the HE-coil-4 and 3.01 dB in the HE-coil-16, compared to a 6.20 dB variation of the existing PMF coils, indicating excellent temperature stability of ER property in HE-PMF coils. The results demonstrate that the coil combined HE-PMF design with high birefringence and a 16-polar symmetry winding pattern can effectively suppress polarization crosstalk under temperature fluctuations, which means that the proposed HE-coil-16 is capable of promoting the application of high-performance IFOG in a wider temperature range and of great significance for realizing high-accuracy of fiber sensors according to the theoretical analysis of the PMFs for IFOGs in Section 2. In contrast, it can be obtained from the results that the LE-PMF coils exhibit smaller values of the ER, while there is a larger fluctuation range over the whole various temperature loads compared to other coils. The maximum fluctuation of ER caused by temperature is 7.07 dB in the LE-coil-4 and 6.93 dB in the LE-coil-16, which is higher than the C-PMF coil, indicating poor thermal stability. These both come from the fact that a limited polarization property characterized by low birefringence poses a hindrance to the enhancement of ER values. The other reason is explained that a stronger thermal stress disturbance arising from SAPs material under changing temperature loads due to fabrication technology directly affects the dynamic stability of the output signal in the coils. Moreover, it has been seen that compared to a coil with the QAD winding method, better ER stability in the coils with a 16-polar symmetrical winding pattern can be achieved by smaller temperature drift owing to a remarkably small error of residual stress caused by an excellent spatial symmetry, following the law of coil-winding theory [12]. Also, the ER values of different coils at points of 20 ℃ are basically the same as the static temperature in Section 4.1, confirming the validity of the experimental results.

 

Fig. 16. The ER test results of different fiber coils for varying temperatures.

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To further assess the dynamic output accuracy of the designed PMFs in the applications of IFOGs, the tests are conducted on the IFOGs assembled with different coils under dynamic temperatures based on the IFOG static experimental setup. A closed-loop IFOG system is known to minimize the impact of device drift on property evaluations of fiber coils. However, a precise characterization of the stability of coils under dynamic conditions requires shielding temperature fluctuations experienced by devices other than the coils. This is the primary reason that the experimental IFOG system is divided into two parts, as illustrated in Fig. 13(b). The tested coil is assembled onto a separate IFOG-1 and placed in a thermostat to evaluate the temperature stability of the coil under dynamic conditions, while the rest of the devices remain the same and are placed on the other frame called IFOG-2, kept at a static room temperature. This experimental setup allows us to isolate the tested coil and ensure that it is subjected to the specific temperature loads of interest. The testing process begins by allowing the thermostat temperature to stabilize for 0.5 h at 20 °C before initiating the dynamic temperature program for data collection. The entire dynamic temperature loads applied to the IFOG-1 are selected as a broad range, from -40 °C to 70 °C, including both a low-temperature process (20 °C–40 °C-20 °C), and a high-temperature process (20 °C-70 °C-20 °C). The temperature changes occur uniformly at a rate of 1 °C/min, and each integer temperature condition is held for 10 minutes to implement a stepped change, with the goal of the full effect of the dynamic loads to be experienced at a steadily rapid rate on fiber coils. The experiment process and the temperature loads applied to the coils for the IFOGs under dynamic tests are depicted in Fig. 17.

 

Fig. 17. Dynamic experimental process and temperature loads for IFOG testing.

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The coils wound by different fibers and winding techniques are assembled in IFOG setups for separate dynamic experiments to investigate the output stability of the IFOGs, and then the resultant IFOG output data are subjected to a 100 s smoothing process for optimal analysis. Figures 18 and 19 show separate curves that provide a comparison of the output bias of IFOGs employing designed PMF coils and those of an IFOG with the conventional C-PMF coil, and the close-up view of low- and high-temperature conditions is shown in the two insets. Comprehensive performance evaluations of the different coils implemented in the IFOGs under dynamic conditions are summarised in Table 6. One can see that the output bias of IFOGs equipped with different coils exhibits a slight disparity in drift during the initial temperature point, which can be attributed to the acquisition of all starting data for the IFOGs after the complete attainment of temperature stabilization at 20 °C. The test results from Fig. 18 reveal that compared to the C-PMF coil IFOG, the IFOGs with HE-PMF coils show optimal temperature stability, with smaller drift and smoother distributions across the full dynamic temperature loading. Remarkably, the coil for an IFOG with the HE-PMF of structural optimization in conjunction with a 16-polar winding technique maintains minimal output fluctuations even in the strong thermal stress disturbance environment of temperature extremes, with a remarkably improved output global drift (OGD) of only 0.42 °/h and OBS of 0.0815 °/h at full temperature, significantly enhancing the accuracy of the IFOGs compared to the OGD of 0.58 °/h and OBS of 0.136 °/h for the C-PMF coil IFOG. This improvement directly results from excellent polarization properties of high birefringence and ER values at extreme temperature points and the excellent ER stability in HE-coil-16 under dynamic conditions, which are in line with the theoretical analysis and numerical simulations of the PMFs for IFOGs. Specifically, during the low-temperature process, both HE-PMF coils start to show notable advantages with less drift as the temperature cools down to -10 °C of 3123 s onwards. When the temperature gradually reduces to around -40 °C from 6000 s to 8400 s, the IFOGs utilizing HE-coil-4 and HE-coil-16 exhibit output fluctuations of 0.20 °/h and 0.19 °/h, respectively, significantly better than the C-PMF coil IFOG of 0.26 °/h. The HE-coil-16 IFOG consistently has the best output stability during the temperature rise from -40 °C to the starting temperature point of 20 °C over a period of 7800-14400 s. For the high-temperature process, a sudden drift emerges in the C-PMF coil IFOG at the start of the temperature rise of 15000 s, while the output results of IFOGs with HE-PMF coils are smoother. As the temperature increases up to 70 °C for 19800-20400 s, the maximum drift values observed are only 0.256 °/h for the HE-coil-4 IFOG and 0.225 °/h for the HE-coil-16 IFOG, which are greatly lower than the 0.32 °/h for the C-PMF coil IFOG. In addition, when the temperature load is reduced to 20 °C during the last 1200 s of the tests, IFOGs employing both HE-PMF coils converge quickly to initial values after experiencing fluctuations, with the HE-coil-16 IFOG demonstrating exceptional performance, exhibiting only 0.019 °/h deviation from the initial value. This is greatly enhanced compared to the 0.068 °/h off the origin of C-PMF coil IFOG, again indicating the remarkable temperature stability of the IFOG using the coil combined with the designed HE-PMF and a 16-polar winding pattern under dynamic loading.

 

Fig. 18. Comparison test of smoothed output bias of IFOG with C-PMF and HE-PMF coils.

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Fig. 19. Comparison test of smoothed output bias of IFOG with C-PMF and LE-PMF coils.

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Table 6. The performance evaluations of different coils in IFOGs under dynamic temperature.

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From the comparison results in Fig. 19, IFOGs equipped with LE-PMF coils reveal poor thermal stability with larger drift and multiple sudden vibrations across the full temperature range. The fluctuations are particularly pronounced at the extreme temperature points of -40 °C and 70 °C by strong thermal stress perturbations, resulting from a weak polarization performance of the LE-PMF coils with lower birefringence and larger ER fluctuations under dynamic conditions. Among the three types of coils, the LE-coil-4 IFOG exhibits a maximum OGD of 0.6 °/h and OBS of 0.151 °/h over the entire process, indicating limited temperature stability of the LE-PMF coils. Specifically, the IFOGs with LE-PMF coils show several violent fluctuations during the initial 7200 s of cooling from 20 °C to -40 °C, as well as during the 14400-20400 s of warming from 20 °C to 70 °C. The LE-coil-4 IFOG displays the drift of up to 0.27 °/h and 0.33 °/h at extreme temperature points of -40 °C and 70 °C, respectively. The LE-coil-16 IFOG demonstrates a large drift of 0.24 °/h and 0.31 °/h at extremely low and high-temperature, respectively. Furthermore, the IFOGs with LE-PMF coils fail to stabilize back to initial values when the temperature decreases to 20 °C. The drift values from the origin of the LE-coil-4 and LE-coil-16 are measured to be 0.072 °/h and 0.1 °/h, respectively, which are larger than the conventional IFOG. These justify the design principles again. The advances in both fiber designs contribute to greatly facilitating the development and production of fiber sensing coils and serve as a valuable guide for the engineering applications of fiber coils for IFOGs and improving the accuracy of IFOGs at full temperature.

5. Conclusion

We have presented two types of hybrid PMFs by introducing enhanced and suppressed core geometry effects, including Panda-type HE-PMF and Panda-type LE-PMF. The theoretical analysis and numerical models of both obtained PMFs for the IFOGs are developed. Comprehensive simulations of performance are investigated by scanning the structural parameters of fibers to obtain target designs. Experimental evaluations of modal performance and temperature stability conducted from aspects of optimal fiber structures, fiber coils, and welded IFOGs are then carried out under static and dynamic conditions, demonstrating a good agreement with the theory and simulations. The proposed HE-PMF with high polarization-maintaining ability by superimposing geometric birefringence shows an ER of up to 29.96 dB in a static environment, and the high ER has remained over 30 dB in fiber coils with different winding patterns. In case of a wide range of dynamic temperature conditions, the ER fluctuation of the HE-PMF coil equipped with a 16-polar wound is as much as one time lower than that of a conventional PMF coil, and the output bias stability of this composed IFOG is improved from 0.136 °/h to 0.082 °/h, significantly enhancing the temperature stability and accuracy of the IFOG. This result confirms that the HE-PMF structure with excellent polarization-maintaining performance combined with a 16-polar symmetry winding pattern can effectively suppress the polarization crosstalk and drift error of the IFOG under dynamic temperature fluctuations, which greatly improves the prospects of the IFOG for inertial navigation. We further measured the properties of the LE-PMF and compared the results to traditional PMF and HE-PMF. The LE-PMF exhibits poor polarization properties with lower birefringence and ER, and the corresponding wound coils also show larger fluctuations in ER under dynamic environments, which is the key cause of increasing the output drift of the IFOGs. The advances of both fibers are extremely meaningful for the selection and production of PMFs for the manufacture of IFOGs and contribute to overcoming temperature fragility.