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A new calculus for the treatment of Rytovs law in the optical fiber
یک حساب جدید برای قانون درمان Rytov در فیبر نوری-2020 In the present paper, we investigate the geometric properties of the linearly polarized light wave
(LPLW) and the homothetic motion of the polarization plane traveling in optical fiber in threedimensional
Riemannian manifold. We examine the behavior of the polarized plane for the
conditions that the electric field makes a constant angle with the Frenet vectors {e1, e2, e3} of the
curve related to the optical fiber that can be considered as a space curve in Riemannian 3-space.
Moreover, we give the relation between the Fermi–Walker parallel transportation laws and the
homothetic motion of the polarization plane in Riemannian 3-space. The key technique here we
use for examining this approach is to use quaternion algebra. We give the parametric equations of
the Rytov curves that are traced curves of the polarization vector via quaternion product and a
matrix that is similar to a Hamilton operator. By means of this matrix a new motion is defined
and this motion is proven to be homothetic. For this one-parameter homothetic motion, we prove
some theorems about the motion of the polarization plane traveling in optical fiber in threedimensional
Riemannian manifold. Then, we obtain the characterization of the electric field and
generate the electromagnetic trajectories ( M-trajectories) along the (LPLW) in the optical fiber
using the variational approach. Finally, we give various examples with Maple codes to confirms
the theoretical results. Keywords: Applications to physics | Vector fields | Magnetic flows | Ordinary differential equations | Quaternion Algebras |
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