Measurement of Polarization Vectors in Spun Fiber Using Fixed-point Principle

This study establishes a high-precision measurement methodology for determining the polarization vector and the circular-to-linear birefringence ratio of Spun Fiber， addressing the critical need for accurate parameter decoupling in Fiber Optical Current Transformers （FOCT）. By leveraging the fixed-point principle within the framework of quaternion polarization optics， the proposed method identifies the intrinsic rotation axis of the fiber on the Poincaré sphere， thereby circumventing the systematic errors induced by polarization axis misalignment and the complexities of prior axis calibration. The research focuses on the 1 310 nm wavelength band to provide a robust physical basis for optimizing FOCT sensitivity and stability through the precise quantification of residual linear birefringence and inherent circular birefringence. The methodology is centered on the application of quaternion polarization optics， where any polarization-dependent system is characterized by a Jones-Mueller Quaternion （JMQ） whose vector part defines the polarization vector W.Utilizing an experimental configuration comprising an eight-wavelength discrete light source （1 284~1 318 nm）， an electric tunable Polarization Controller （PC） with 0.01 V resolution， and a high-accuracy Polarization Analyzer （PA）， the measurement process exploits the invariance of the output State of Polarization （SOP） when the input SOP is aligned with the fiber's polarization vector. During the direction-finding phase， continuous wavelength scanning is performed while the PC iteratively adjusts the input SOP； the polarization vector's orientation is confirmed when the output SOP trajectory on the Poincaré sphere converges to a stationary “fixed point”， minimizing the rotation radius to near zero. Following the identification of this vector， the magnitude of the birefringence is determined by shifting the fixed point to the Poincaré sphere's pole and moving the SOP to the equator via an output PC. By recording the angular displacement Δϕ of the SOP along the equatorial plane during subsequent wavelength scans， the total beat length Lp is calculated using the variational phase-shift formula Lp=(L⋅Δλ)/(λ0⋅(Δδ/2π))， where L is the fiber length and Δδ is the phase change derived from the trajectory arc. The circular beat length （Lc） and linear beat length （Ll） are then extracted by projecting the synthesized polarization vector onto the Z-axis and the equatorial plane， respectively， enabling a direct physical separation of the two birefringence components. Empirical validation was conducted using Fibercore SHB1250 elliptical polarization-maintaining Spun Fiber across multiple specimens with lengths of 1.20 m， 0.80 m， 0.53 m， and 0.46 m to ensure consistency and repeatability. For the 1.2 m sample， the polarization vector was precisely located at Stokes coordinates.Across the experimental trials， the average circular beat length was measured at 93.347 mm and the average linear beat length at 497.451 mm， yielding a circular-to-linear birefringence ratio of 5.33. These results are highly congruent with the manufacturer's specified circular beat length range of 63~125 mm， verifying the accuracy of the fixed-point detection. Comprehensive error analysis indicates a systematic error of approximately 0.063， primarily stemming from the discrete resolution of the polarization controller and the intrinsic measurement uncertainty of the analyzer. The random error was restricted to 0.002， demonstrating the method's superior immunity to environmental perturbations， such as thermal drift and mechanical vibration， facilitated by the trajectory-fitting approach.The fixed-point principle offers a physically intuitive and mathematically rigorous solution for the characterization of complex optical fibers， providing a significant advantage over Mueller matrix decomposition by eliminating the need for coordinate-dependent calibration. This methodology facilitates the high-precision decoupling of birefringence components， which is essential for suppressing zero-point drift and improving the long-term reliability of FOCT systems in high-voltage sensing applications. The ability to directly output the polarization vector and beat length parameters under varying environmental conditions establishes this approach as a candidate for the standardized testing of complex polarization-maintaining components. Furthermore， the robust performance of the fixed-point identification under broad wavelength scanning demonstrates its potential for broader applications in the standardized detection of non-uniform birefringence in advanced optical waveguides.