The Application of Helmholtz Decomposition Method to Investigation of Multicore Fibers and Their Application in Next-Generation Communications Systems

Abstract

New optical multicore fibers use their spatial properties in the designs of next-generation systems. To investigate light propagation in such fiber waveguides we use Helmholtz decomposition method.

We consider a waveguide having the constant cross-section S with ideally conducting walls. We assume that the filling of waveguide does not change along its axis and is described by the piecewise continuous functions $\epsilon$ and $\mu$ defined on the waveguide cross section. We show that it is possible to make a substitution, which allows dealing only with continuous functions. Instead of discontinuous cross components of the electromagnetic field $\varvec{E}$ and $\varvec{H}$ we propose to use four potentials $u_e, u_h$ and $v_e, v_h$. Generalizing the Thikhonov-Samarskii theorem, we have proved that any field in the waveguide allows such representation, if we consider the potentials $u_e, u_h$ as elements of the Sobolev space and the potentials $v_e, v_h$ as elements of the Sobolev space ${W}^1_2(S)$.

If $\epsilon$ and $\mu$ are piecewise constant functions, then in terms of four potentials the Maxwell equations reduce to a pair of Helmholtz equations. This fact means that a few dielectric waveguides placed between ideally conducting walls can be described by a scalar boundary problem. This statement offers a new approach to the investigation of spectral properties of waveguides. First, we can prove the completeness of the system of the normal waves in closed waveguides using standard functional spaces. Second, we can propose a new technique for calculating the normal waves using standard finite elements.

A. A. Tiutiunnik—The publication has been prepared with the support of the “RUDN University Program 5-100” and funded by RFBR according to the research projects No. 18-07-00567 and No. 18-51-18005.