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Transactions on Engineering and Computing Sciences - Vol. 11, No. 6

Publication Date: December 25, 2023

DOI:10.14738/tecs.116.15919.

Kim, Y. M. (2023). A Study on the Fine Eigenmodes in the Core Region of Coaxial Waveguide Induced by the Dielectric Cladding.

Transactions on Engineering and Computing Sciences, 11(6). 60-66.

Services for Science and Education – United Kingdom

A Study on the Fine Eigenmodes in the Core Region of Coaxial

Waveguide Induced by the Dielectric Cladding

Yeong Min Kim

Kyonggi University, Korea

ABSTRACT

The fine eigenmodes in the core region of the coaxial waveguide induced by the

dielectric cladding are investigated by the finite element method. The relative

permittivity of the core region is significantly higher than that of the cladding area.

By appropriately adjusting the dielectric constant and geometry of the cladding,

electromagnetic waves can be focused on the core region. When the eigenmode

formed by the waveguide resonates with the electromagnetic wave, it can propagate

long distances without seriously losing energy. In this study, eigenmodes localized

in a fine core region are found using the numerical iteration method of the finite

element method. As a result, TEM (Transverse Electro-Magnetic) eigenmodes are

shown in a simple schematic representation. These results will enable the

implementation of more advanced waveguides by utilizing the ideal characteristics

of the cladding layer.

Keyword: dielectric cladding, eigenmode, resonance, finite element method, iteration

method, TEM (Transverse Electro-Magnetic).

INTRODUCTION

Previously, we have studied on the eigenmodes established in the photonic crystals of various

types usingFEM (Finite Element Method) [1][2]. In these studies, air holes or dielectrics were

arranged symmetrically according to the geometry of the photonic crystal waveguide. The main

purpose of this research was to concentrating light onto the core waveguide of the photonic

crystal. At the same time, the leakage of light energy induced into the waveguide line was

minimized. The eigenmodes depicted in the schematic representation illustrate the potential of

these photonic crystal waveguides for microscopic optical circuits. The possibility of achieving

this goal increases as the eigenmodes generated in the waveguide become finer. Although the

usability is different from the photonic crystal, fine eigenmodes can also be obtained from the

core of a waveguide wrapped in a dielectric cladding. Recently, we have studied the fine

eigenmodes in a cylindrical coaxial waveguide core wrapped in a dielectric coating [3]. In this

study, fine eigenmodes of TM and TE have been obtained by using through FEM calculation. By

adjusting the geometrical structure and electrical permittivity of the cladding, it was identified

that it would be possible to establish finer eigenmode generated in the core. There would be a

need to implement this possibility more concretely and progressively. If the thickness of the

cladding is not so thin compared to the core and the dielectric constant is significantly different

from the that region, it may be expected that the finer eigenmodes could be obtained.

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Kim, Y. M. (2023). A Study on the Fine Eigenmodes in the Core Region of Coaxial Waveguide Induced by the Dielectric Cladding. Transactions on

Engineering and Computing Sciences, 11(6). 60-66.

URL: http://dx.doi.org/10.14738/tecs.116.15919

Therefore, this study extends previous work and attempts to obtain finer and more detailed

eigenmodes incylindrical coaxial waveguides. What different from the previous study is that

when performing FEM calculations, the input parameters substituting into the program are

adjusted to obtain the desired eigenmode with reducing try and error. The cylindrical coaxial

waveguide is not much different from previous forms. One difference is that as the program

progresses, the relative thickness and dielectric permittivity relation between the core and the

cladding change little by little to obtain the final result. The cross-sectional area of the

waveguide is decomposed into a mesh structure consisting of triangular elements for FEM

calculation. The calculation of FEM is based on the vector Helmholtz governing equations and is

achieved by obtaining the solution of the eigen matrix equations composed of the edges and

nodes of the triangular elements [4]. The eigen equations are proportional to the number of

triangular elements of the mesh. To increase the resolution of the spectrum, the density of

triangular elements must be increased. Then,the eigen equations become larger and a computer

with high processing capacity is needed. However, because the capacity of personal computers

is limited, it is difficult to handle the inverse matrix of the large size eigen equation. To

overcome this contradiction, FEM uses the Arnoldi algorithm to compress the matrix equation

into the smaller one as mentioned in the previous manuscript. Afterwards, the Krylov-Schur

iteration method is used to find several prominent eigenmodes with the highest reliability [5].

This method is used in this study to find the desired several eigenmodes. Eigenmodes consist of

electric fields and potential pairs. These constitute the column matrix of the similarity

transformation matrix usedin the iterative method. The mathematical derivation process for

FEM has been mentioned in a previous manuscript [1][2]. Therefore, this process is not

mentioned again in this manuscript. In other words, the theory of FEM is omitted in this

manuscript. The description format of this manuscript deviates from the existing order and is

described as follows. This manuscript describes the structure of waveguide for FEM calculation,

the result and discussion of the simulation, and conclusion in order.

THE STRUCTURE OF WAVEGUIDE

Figure 1: Schematic representation of cross section of awaveguide depicted as a mesh of

triangular units Ref. [3].

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Transactions on Engineering and Computing Sciences (TECS) Vol 11, Issue 6, December - 2023

Services for Science and Education – United Kingdom

As mentioned in the previous study, the eigenmode strongly depend on the thickness of the

cladding layerand its dielectric permittivity. Accordingly, it was first obtained the eigenmodes

by varying the thickness of the cladding layer in the coaxial waveguide. The relative dielectric

constant between the core and cladding is fixed to be εi ∶ εo = 11 ∶ 1. This remarkable dielectric

constant ratio is intended to relatively concentrating electromagnetic waves in the core area

according to Snell's law. Figure 1 is a schematic representation of the coaxial waveguide used

in this study. In this figure, the core and cladding areas of the dielectric coaxial waveguide are

separated to facilitate understanding of the structure. By several try and error, it was identified

that among waveguides, the structure of Figure 1 reveal most fine resolution of theeigenmode

spectra. Similar to previous studies, the small and irregular potential distribution in the

cladding region forms a complex spectrum even with a major peak in the core region.

Nevertheless, compared to the spectra of other structures,the eigenmodes obtained from Figure

1 are the best resolution. Therefore, this study establishes a mesh of triangular element based

on Figure 1 and applies FEM on it. The spectrum shown in the next section is the result of

struggle to find the most ideal thickness of the cladding. In this study, the thickness of the

cladding layer is set to be d = Ro − Ri = 0.5 − 0.35(arb. units).

Generally, in FEM calculations, TM (Transverse Magnetic) and TE (Transverse Electric) modes

are distinguished by boundary conditions set on the waveguide surface. As mentioned in the

previous study, the eigenmodes are expressed only as an electric field, because in FEM

calculations, the magnetic field is obtained through the same process as in the electric field.

When obtaining the TE mode, the calculation is performed by canceling the components of the

edges and nodes corresponding to the surface of the waveguide from the eigenequation. Then,

the eigen equations are reduced by their number, allowing only the tangential component ofthe

electric field and excluding the perpendicular component. However, in this study, the

eigenmode is determined by the interface between the core and the cladding rather than the

surface. These TM and TE characteristics are mixed in one eigenmode spectrum to represent

the TEM (Transverse Electro-Magnetic) mode.