Polarization-maintaining fibers (PMFs) have been widely used in fiber-optic sensors and fiber-optic communications, due to their capability to maintain a state of polarization (SOP) along the fiber. The PMF in fiber sensor systems, particularly the fiber-optic hydrophones and fiber-optic gyroscopes, is generally winded into fiber coils to improve sensitivity and reduce size. However, the birefringence, an important fundamental parameter of the PMF, is usually inhomogeneous due to variations in temperature, axial strain, and transverse strain along the fiber. Several methods have been proposed for measuring the birefringence of the PMF over the years. However, these techniques cannot fully determine the birefringence variations induced by temperature, axial strain, and transverse strain. A white light interferometer (WLI) based on a distributed polarization crosstalk analysis technique can provide group-birefringence variation along the fiber under test^[1–3]. In 2022, researchers employed WLIs to verify the strain-induced birefringence distribution model of a single-layer PMF coil in a fiber-optic gyroscope^[4]. In 2006, group birefringence measurement using optical-frequency-domain reflectometry (OFDR) was proposed, based on distributed analysis of the complex Rayleigh-scattering pattern of the fiber under test^[5]. In Ref. [6], a distributed measurement with a spatial resolution of 7.8 cm was demonstrated. However, birefringence measurements based on OFDR can only obtain relative, rather than absolute, axial- and transverse-strain-induced birefringence distributions^[7,8]. Therefore, the existing methods remain limited, as they cannot distinguish the birefringence contributions along the fiber induced by temperature, axial strain, and transverse strain.

We previously reported the use of a Brillouin dynamic grating (BDG) to measure the birefringence distribution of PMFs^[9–11]. The BDG is essentially a moving acoustic wave generated through stimulated Brillouin scattering (SBS), which has enabled many useful applications, such as distributed fiber sensing^[12–15], high-resolution spectrometry^[16], and all-optical delays^[17]. In a PMF, the BDG can be excited by the Brillouin interaction between two counter-propagating pump waves in one polarization state, and can be used to reflect an orthogonally polarized reading wave^[18]. The frequency offset between the two pump waves is set to the Brillouin frequency shift (BFS) of the PMF to excite the strongest BDG. The maximum reflection of the BDG would occur under the phase-matching condition, while the frequency offset between the pump and read waves is called birefringence-induced frequency shift (BireFS). Utilizing the linear relationship between the BireFS and birefringence, a distributed birefringence measurement of the PMF based on BDGs has been demonstrated with a 20-cm spatial resolution^[11]. In Ref. [12], the dependences of BireFS on strain was measured, yielding a strain coefficient of 1.13 MHz/µε, establishing a mechanism for characterizing the birefringence induced by axial strain using BireFS.

In this paper, we extend the birefringence measurement capability of the BDG technique to measure and distinguish all birefringence-related parameters in PMF coils, including those induced by temperature, axial strain, and transverse strain. The differential pulse-width pair Brillouin optical time-domain analysis (DPP-BOTDA) is employed for axial-strain measurement with a 5-cm spatial resolution. For birefringence measurement, an 18.5-ns pump pulse 1 and frequency-upshifted continuous-wave (CW) pump2 are used to generate the BDG, while a 4-ns read pulse is used to detect the BDG, achieving a spatial resolution of 40 cm. Distributed measurements of birefringence, axial strain, axial-strain-induced birefringence, and transverse-strain-induced birefringence were successfully demonstrated over a 1333-m PMF coil in a fiber-optic gyroscope.

In Brillouin-based optical fiber sensing systems, a distributed strain measurement can be realized by measuring the BFS of the fiber. The BFS of the fiber is given by νB=2nVa/λ,(1)where νB is the BFS of the fiber, n is the refractive index, Va is the velocity of the acoustic wave, and λ is the light wavelength in vacuum. The BFS is sensitive to strain and temperature, and it has been widely used in distributed optical fiber sensing systems, such as Brillouin optical time-domain analysis (BOTDA)^[19–23]. For instance, the BFS of the fiber at a stable temperature can be expressed as νB=νBF+CBεε,(2)where νBF is the BFS of the free fiber, CBε is the coefficient of BFS to strain, and ε is the strain applied to the fiber. From Eq. (2), a distributed strain measurement can be realized by a distributed BFS measurement of the fiber and free fiber.

In a PMF, the BDG can be generated by the SBS interaction of two pump waves. The BireFS of the PMF can be obtained by scanning the frequency offset between the read and pump waves, which is given by^[9]νBire=Δnν/ng,(3)where νBire is the BireFS of the PMF, Δn is the birefringence of the PMF, ν is the frequency of the reading wave, and ng is the group refractive index of the PMF. Compared to a free PMF at a fixed temperature, the birefringence change of the PMF can be expressed as δn=δn(ε)+δn(ξ)+δn(T),(4)where δn, δn(ε), δn(ξ), and δn(T) are the birefringence change, axial-strain-induced birefringence, transverse-strain-induced birefringence, and temperature-induced birefringence of the PMF, respectively. Among them, axial-strain-induced birefringence originates from refractive index anisotropy parallel to the applied axial strain. Transverse-strain-induced birefringence stems from lateral deformation through the Poisson effect. Temperature-induced birefringence emerges from the thermo-optic effect coupled with thermal expansion mismatch. According to Eqs. (2)–(4), for a PMF, the birefringence, strain, axial-strain-induced birefringence, transverse-strain-induced birefringence, and temperature-induced birefringence can be accurately calculated by measuring the BireFS, the BFS, and the temperature of both the PMF and the free fiber.

Figure 1 shows the experimental setup for the distributed BFS and BireFS measurements. A fiber-optic gyroscope coil with a 1333-m PMF based on a quadrupole winding pattern is used as the fiber under test (FUT). The average radius of the fiber-optic gyroscope coil is 80 mm, and the height is 16 mm. Two ends (i.e., 2-m free PMF) of the fiber-optic gyroscope coil are in a loosened condition, which means the strain-induced BFS and BireFS changes can be ignored. For BireFS measurement, a distributed feedback (DFB) laser with a linewidth around 100 kHz and a tunable laser with a wavelength resolution of 0.1 pm were used as light sources. The output from the DFB laser was divided by a 50/50 coupler (OC), and one of the outputs was modulated by an electro-optic modulator (EOM3) driven by a sinusoidal microwave output from a microwave generator (MG). EOM3 generated two first-order sidebands, and the higher-frequency sideband was selected as a CW pump2. The CW pump2 was injected into the fast axis of the FUT with a power of 0.9 mW. The other output was modulated as an 18.5-ns square pulse for pump1 by an electrical pulse output from channel 1 (CH1) of an arbitrary-wave generator (AWG) and EOM1. After being amplified by an erbium-doped amplifier (EDFA1), the pump1 with a peak power of 0.5 W was injected into the fast axis of the FUT through a polarization beam splitter (PBS). The tunable laser output was modulated as a 4-ns square pulse for the read pulse by the output from CH2 of the AWG and EOM2. The peak power of the read pulse was amplified to 0.4 W by EDFA2. The read pulse was injected into the slow axis of the FUT through PBS to read the BDG generated by the Brillouin interaction of two pump waves. A photodetector (PD2) was used to detect the reflection wave of the BDG, before which a fiber Bragg grating (FBG2) was used to filter out the crosstalk of pump2. The BDG reflection spectrum was measured by sweeping the frequency offset of the two lasers.

For BFS measurement, the setup was simplified into a DPP-BOTDA system. The DFB laser was used as a light source, and the tunable laser was turned off. An optimized 8/8.5-ns pulse pair was pre-edited, and the output from CH1 of the AWG had a pulse width difference of 0.5 ns, corresponding to a 5-cm spatial resolution. The BFS was measured by scanning the frequency f of the sinusoidal microwave output from the MG around the BFS νB of the FUT.

The distributed BFS measurements of the fiber-optic gyroscope coil are tested as the temperature is increased from −20°C to 60°C with a step of 20°C, and the results are shown in Fig. 2(a). A 2-m-long relaxed fiber is used as the zero-strain reference fiber. According to Ref. [12], our previous works measured the coefficient of BFS to strain CBε=0.0482MHz/με and the coefficient of BireFS to axial strain CBireε=1.13MHz/με. According to Eq. (2), the fiber-optic gyroscope coil’s distributed axial strain as the temperature increases from −20°C to 60°C is calculated. The linear relationship between the coefficient of BireFS to axial strain CBireε and the axial strain coefficient of birefringence Cnε can be expressed as Cnε=CBireεng/ν,(5)and the calculated axial-strain coefficient of birefringence is Cnε=0.857×10−8/µε. In addition, the experimental fiber-optic gyroscope coils adopted the quadrupole symmetric winding method, and the stacking of each layer of optical fibers generates a periodic stress distribution due to uneven mechanical tension or thermal expansion differences. The interval between layers is fixed, forming periodic peaks in Fig. 2(a).

The distributed axial strain and axial-strain-induced birefringence are shown in Fig. 2(b). The fluctuation of the axial strain over the fiber can be distinguished, which indicates uneven axial strain distribution in the fiber-optic gyroscope coil. We can see the symmetric distribution of the axial strain, which is induced by the quadrupole winding pattern. From −20°C to 60°C, the whole axial strain variation distribution decreases with increasing temperature, which could be attributed to using high-temperature adhesive during the fiber-optic gyroscope coil winding process. At −20°C, the maximum axial strain (axial-strain-induced birefringence) is 852 µε (7.3×10−6) at the position of 697.5 m, and the minimum axial strain (axial-strain-induced birefringence) is −748με (−6.4×10−6) located at the position of 1324.5 m, resulting in a fluctuation of 1600 µε (1.37×10−5) over the entire fiber. The results imply that different disturbance factors during the fiber-optic gyroscope coil winding process and environmental changes can impact the axial strain (axial-strain-induced birefringence) of the fiber-optic gyroscope coil.

Similarly, the average birefringence of the free fiber with a length of 2 m is considered the initial state birefringence. The distributed BireFS measurements of the fiber-optic gyroscope coil are tested as the temperature is increased from −20°C to 60°C with a step of 20°C. According to Eq. (3), the birefringence distributions of the fiber-optic gyroscope coil are calculated, and the results are shown in Fig. 3(a).

It can be seen that the birefringence of the fiber-optic gyroscope coil decreases as the temperature is increased. For the free fiber such as the beginning region A (inside the dashed rectangle), the birefringence decreases from 5.0×10−4 to 4.62×10−4 as temperature increases from −20°C to 60°C. As shown in Fig. 3(b), the distributed birefringence of the fiber-optic gyroscope coil induced by axial strain and transverse strain as the temperature ranges from −20°C to 60°C is calculated by subtracting the birefringence of the 2-m free fiber, and it can be expressed as δn(ε+ξ)T=ΔnT−ΔnFT,(6)where δn(ε+ξ)T is the birefringence induced by axial strain and transverse strain at T°C; ΔnT and ΔnFT are the birefringence of the fiber-optic gyroscope coil and free fiber at T°C, respectively. The temperature coefficient of birefringence is CnT=−4.7×10−7/°C achieved by linear fitting the birefringence of free fiber from −20°C to 60°C, as shown in Fig. 4.

From Fig. 3(b), we can see that the strain-induced birefringence variation decreases as temperature increases from −20°C to 60°C. Similar to the distribution of axial-strain-induced birefringence, strain-induced birefringence has a symmetric distribution along the fiber. At −20°C, the maximum strain-induced birefringence is 7.06×10−6 at the position of 698.5 m, and the minimum strain-induced birefringence is −4.68×10−6 at 1283 m, resulting in a fluctuation of 1.17×10−5 over the entire fiber. The change trend is consistent with the strain change trend.

As shown in Fig. 5, the distributed transverse-strain-induced birefringence of the fiber-optic gyroscope coil from −20°C to 60°C is calculated by the strain-induced birefringence subtracting the axial-strain-induced birefringence at T °C, and it can be expressed as δn(ξ)T=Δn(ε+ξ)T−Δn(ε)T,(7)where δn(ε)T and δn(ξ)T are the birefringence induced by axial strain and transverse strain at T°C, respectively.

From −20°C to 60°C, the whole transverse-strain-induced birefringence variations of each layer fiber are less than 1.0×10−6, while the crossover regions exhibit transverse-strain-induced birefringence-variation peaks, as shown in the inset and there exists a period of 31.3 m. At −20°C, the birefringence changes at the peak induced by axial strain and transverse strain are shown in Fig. 6, indicating that the peak is located at the fiber layer changing region. The existence of the transverse-strain-induced birefringence variation peaks could be attributed to several factors. The first factor is the measurement accuracy difference between the axial-strain-induced birefringence and strain-induced birefringence measurements. It is well known that, for the crossover regions with very short fibers, the axial-strain-induced birefringence measured with a 5-cm spatial resolution is more precise than strain-induced birefringence measured with a 40-cm spatial resolution. The second factor is that the crossover-region fiber experiences considerable transverse strain as temperature is increased from −20°C to 60°C.

This work comprehensively characterizes the birefringence parameters of PMFs employing the enhanced BDG technique. The birefringence-induced frequency shift is systematically analyzed with respect to axial strain, transverse strain, and temperature. The measured birefringence coefficients for axial strain and temperature are 0.857×10−8/με and −4.7×10−7/°C, respectively. Within a single layer of the fiber-optic gyroscope coil, the transverse-strain-induced birefringence variations are smaller than those induced by axial strain or temperature. However, significant birefringence changes are observed at interlayer crossing transitions due to transverse strain.

This study is based on a frameless fiber-optic gyroscope coil wound by a quadrupole symmetry process and encapsulated with high-temperature glue. While different winding processes, materials, and structure designs may lead to varying effects, the characterization method presented here remains applicable. We believe this work may offer valuable technical guidance for the design and fabrication of high-precision fiber-optic gyroscopes.