Quaternion Approach to the Measurement of the Local Birefringence Distribution in Optical Fibers

A method to structure the Jones quaternion utilizing Pauli matrices is proposed, improving the quaternion theory of polarization optics. It is proved that the Stokes quaternion is the double product of the Jones quaternion and its Hermitian transpose and that the rotation axis of the output three-adjacent-point Stokes quaternions is a mapping from the rotation axis of the three-adjacent-reflection-point Stokes quaternions in the optical fiber and that the rotation angle among the output three adjacent points is the doubleness of the rotation angle among the three adjacent reflection points in the optical fiber. The three-point quaternion method to measure birefringence distribution is proposed and implemented, the experimental result shows that the average birefringence is 0.263/m in the fiber under test, and the corresponding average beat length is 26 m. We also demonstrated the paraxial approximate quaternion Baker-Campbell-Hausdorff formula of exponential quaternions multiplying for cascaded optical components; then, a quaternion interpolation method to estimate the error of birefringence measurement is proposed, and the relative error calculated was less than 5%. The previously cited works indicate that the quaternion algorithm is very effective in the research of the polarization in optical fibers.