In this paper, we relate the evolution equation of the electric field and magnetic field vectors of the polarized light ray traveling along with a coiled optical fiber on the unit 2-sphere S² into the nonlinear Schrödinger's equation by proposing new kinds of binormal motions and new kinds of Hasimoto functions in addition to commonly known formula of the binormal motion and Hasimoto function. All these operations have been conducted by using the orthonormal frame of spherical equations that is defined along with the coiled optical fiber lying on the unit 2-sphere S². We also propose perturbed solutions of the nonlinear Schrödinger's evolution equation that governs the propagation of solitons through electric field (E) and magnetic field (M) vectors. Finally, we provide some numerical simulations to supplement the analytical outcomes.

We review the geometric evolution of a linearly polarized light wave coupling into an optical fiber and the rotation of the polarization plane in the three dimensional semi-Riemannian manifold. The optical fiber is assumed to be a one-dimensional object imbedded in a 3D semi-Riemannian manifold along the paper. Thus, in the 3D semi-Riemannian manifold, we demonstrate that the evolution of a linearly polarized light wave is associated with the Berry phase or more commonly known as the geometric phase. The ordinary condition for parallel transportation is defined by the Fermi-Walker parallelism law. We define other Fermi-Walker parallel transportation laws and connect them with the famous Rytov parallel transportation law for an electric field $\mathcal{E}$, which is considered as the direction of the state of the linearly polarized light wave in the optical fiber in the 3D semi-Riemannian manifold. Later, we define a special class of magnetic curves called by Bishop electromagnetic curves ( $\mathcal{L}\mathcal{B}\mathcal{E}\mathcal{M}$–curves), which are generated by the electric field $\mathcal{E}$ along the linearly polarized monochromatic light wave propagating in the optical fiber. In this way, not only we define a special class of linearly polarized point-particles corresponding to $\mathcal{L}\mathcal{B}\mathcal{E}\mathcal{M}$–curves of the electromagnetic field along with the optical fiber in the 3D semi-Riemannian manifold, but we also calculate, both numerically and analytically, the electromagnetic force, Poynting vector, energy-exchanges rate, optical angular and linear momentum, and optical magnetic-torque experienced by the linearly polarizable point-particles along with the optical fiber in the 3D semi-Riemannian manifold.