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

Abstract

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.

Distributed Fiber Birefringence Measurement Using Pulse-Compression Φ-OTDR

“…The experimental result showed that the average local birefringence of the fiber was 0.220 rad/m in the 17 km single-mode fiber under test, corresponding to a beat length of 28.5 meters. In 2015, Liu et al [16] proposed and implemented the three-point quaternion method to measure birefringence distribution for single-mode fiber. The experimental result showed that the average birefringence was 0.236 rad/m in the 1 km single-mode fiber under test, and the corresponding average beat length was 26.6 meters.…”

“…However, such methods have strict requirements for the polarization alignment-the angles between the polarization of the input linearly polarized light and the principal axis of the Wollaston prism should be precisely adjusted to 45°. The scattering-based birefringence measurement scheme includes two typical types, which are Brillouin-scattering (BS) based [11,12] birefringence measurement systems and Rayleighscattering (RS) based [13][14][15][16][17] birefringence measurement systems. As for the BS based systems, Lu et al [12] proposed a system with beat period detection of homodyne Brillouin optical time-domain reflectometry (BOTDR), but this system's signal-to-noise ratio (SNR) highly depends on the input pulses' state of polarization (SOP) and it takes 80 seconds to complete once measurement.…”

“…However, this kind of systems strictly needs two input lightwaves along the principal axes of the fiber and multiple input pulses with different frequencies. After that, Liu et al [16] adopted the quaternion approach with direct detection to avoid multiple pulses, but the detection module involves free space optical components, which is too complicated. For example, when measuring the state of polarization, if detection module involves free space optical components, these components need to be adjusted during the measurement to ensure that no additional errors are introduced, such as the polarization alignment of the optical components.…”

“…Compared with previous methods [6][7][8][9][10][11][12][13][14][15][16][17], the proposed method only needs a single pulse with arbitrary SOP to retrieve the distribution of birefringence. Besides, the detection module only uses a 90° optical hybrid and two balance photo-detectors, which does not involve any polarization alignment.…”

Quaternion algebra for Stokes–Mueller formalism

“…. From the formula (10) it immediately follows that a very similar transformation (rotation) applies to the Stokes quaternion (see also Liu et al [8]):…”