A Conjugate Linearly Polarized Light Wave Along an Optical Fiber with the Berry Phase Model and Its Magnetic Trajectories According to the Conjugate Frame

In this article, we study how a linear polarized wave that is going along an optical fiber works, which is known not only as a curve on a Lie group but also as a rotation of the polarization plane. What we are trying to show in this article is that linear polarized light waves ( (Formula presented.) s) are related to the Berry phase. Moreover, we give magnetic curves created by (Formula presented.) traveling in the electromagnetic trajectories and the optical fiber generated by the electric field (Formula presented.) of the (Formula presented.) moving through the optical fiber. With this described method, we present a mathematical model to conveniently generate the relationships between an optical fiber and the optical angular momentum in a three-dimensional Lie group. The conjugate frame we used in this article removes unnecessary bending around the tangent and enables a more dynamic characterization, which can still be applied even when the second derivative of the curve is zero.