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.
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 (BG) and stress birefringence (BS), 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 Pf and ΔPf / Ps of the optical path, with low values of Pf and ΔPf / Ps 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]:
where neff is the effective refractive index, and c is the velocity of light in vacuum.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.
Table 1. Simulation parameters for PMFs
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.
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 (nx) and stress birefringence (Nx) 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 (ny) and stress birefringence (Nx) 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.
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.
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.
The stress-optical coupling effects and other modal properties, including effective mode area (Aeff), 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 (Nx-Ny) 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.
We further discuss the Aeff, 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 Aeff and γ values gradually increase and decrease, respectively, with increasing wavelength accordingly over the entire band. Table 2 summarizes the calculated Aeff and γ for the designed PMFs and the fused common C-PMF at 1550 nm. The Aeff 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 Aeff 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 Aeff and γ versus wavelength in the (a) HE-PMF, and (b) LE-PMF. Calculated D in the (c) HE-PMF, and (d) LE-PMF.
Table 2. The performance of fused PMFs in IFOGs at 1550 nm.
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 Aeff, ± 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 Aeff, ± 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 Aeff, ± 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 Aeff, ± 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, Aeff, γ, and D at 1550 nm also demonstrate excellent tolerance for fabrication errors, revealing the robustness of both designs.
