Lebesgue-integral-inspired matrix decomposition of optical fibers with arbitrary spinning rates

Optical chiral materials have become a heated topic of research in recent decades, continually injecting new vitality into both the fundamental theory of optics and novel applications. An optical fiber with linear birefringence can be considered a chiral medium when it is twisted along the axial direction. However, when the spinning rate is non-uniform and varies arbitrarily along the fiber axis, obtaining the emerging polarization state of the light wave is challenging. In this work, inspired by the Lebesgue integral, we innovatively propose a method to decompose optical fibers with arbitrary spinning rates by using elliptical birefringence matrix. Mathematical derivation is provided, and the correctness of the scheme is also verified by comparing it with the results obtained from traditional coupled-mode theory and linear birefringence matrix method. The scheme proposed in this work provides great convenience for obtaining the polarization matrix of chiral fibers and is also applicable to other chiral materials.