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Transactions on Engineering and Computing Sciences - Vol. 11, No. 5

Publication Date: October 25, 2023

DOI:10.14738/tecs.115.15656.

Kim, Y. M. (2023). A Study on the Eigen Properties of the Coaxial Waveguide of Cylindrical Form. Transactions on Engineering and

Computing Sciences, 11(5). 69-77.

Services for Science and Education – United Kingdom

A Study on the Eigen Properties of the Coaxial Waveguide of

Cylindrical Form

Yeong Min Kim

Kyonggi University, Korea

ABSTRACT

In this study, the eigen properties of the coaxial waveguide of cylindrical form is

investigated by using finite element method. The eigenmatrix equation constructed

from the Helmholtz vector equation is too large to derivethe results using a personal

computer. Therefore, the eigen equations are compressed using the Arnoldi

algorithmand after that the results are derived using the Krylov-Schur iteration

method. The similarity transformation matrixused during this process contains the

desired eigenmode pair.The eigenmodes are simultaneously included inthecolumn

matrix components of the transform matrix. These are represented with the pairs

of the electric field and electric potential. The eigenmodes have been divided into

two classes: transverse magnetic modes and transverseelectric modes. As results,

in order to more clearly reveal the characteristics of the eigenmodes, these results

are shown in the figure.

Keyword: coaxial waveguide, similarity transformation, eigenmode, transverse magnetic

mode, transverse electric mode, iteration method.

INTRODUCTION

In the previous paper, it was mentioned that the eigenmodes of a waveguide or the cavity are

related to the properties of the constituent materials and geometric structure [1]. In this paper,

the eigenmode formed in a cavity whose horizontal length of the rectangular is 1.5 times larger

than that in reference [2] has been revealed. The eigenmodes were promoted similar to them

of reference [2] as the order increased. In particular, it was confirmed that as the order of the

eigenmode increases, complex characteristics appear in the feature of electric field. From these

results, it was identified that various types of eigenmodes can be expected simply by changing

the geometric structure. A rectangular cavity is obtained by a cross-section of infinitely

extended waveguide. Therefore, the cavity exhibits the transverse cross-sectional

characteristics of the waveguide. Meanwhile, there are results showing that when the

composition of the waveguide changes, the characteristics of the eigenmode take on various

forms. Photonic crystal waveguides are the most representative example of this result [3][4].

In these studies, various dielectrics are symmetrically inserted into the waveguide, and the

eigenmodes of square or cylindrical photonic crystals are determined. The results obtained

from these studies showed that the eigenmodes varied depending on the relative position and

permittivity of the inserted dielectrics. Based on such studies, various waveguide

characteristics can be expected to be able to obtained through changes in the geometrical

structure and diversity of constituent materials.

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Transactions on Engineering and Computing Sciences (TECS) Vol 11, Issue 5, October - 2023

Services for Science and Education – United Kingdom

In this study, the eigen properties of the coaxial waveguide is investigated by FEM (Finite

Element Method). So far, many experiments and studies have been reported on coaxial

waveguides. Among those, we have published the studies and results on the eigen properties of

coaxial waveguides at reference [2][3]. In reference [2], the waveguide has a structure similar

to that of this study, but the core region is an empty space without material. The waveguide in

reference [3] is similar to that in this study, but the core and the cladding region wrapping it

have different material properties. In this manuscript, the results of a study on those

waveguides that similar to previous studies in terms of composite materials and geometry is

revealed. This study is intended to reveal the unique characteristics of coaxial waveguides, such

as in these references, and is therefore extending those studies with a review and supplement

from them.

For the calculation of FEM, the space of waveguide cross section is divided into the triangular

mash. Using the edge and node of the mesh element, the vector Helmholtz equation is

reconstructed into the eigen matrix form. The size of the eigen matrix equation is proportional

to the number of meshes made up with edges and nodes.

However, the accuracy of calculation increases as the number of triangular elements making up

the mesh increases. To overcome this contradiction, FEM uses the Arnoldi algorithm to

compress the matrix equations. Afterwards, the Krylov-Schur circulation method is used to find

several eigenmodes with the highest reliability. This method is used in this study to find the

desired eigenmodes and these results are revealed with the schematic representation. As stated

in previous studies, this study omits the theory and formula derivation process for FEM.

Therefore, 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 COAXIAL WAVEGUIDE AND FEM

Figure 1: The coaxial waveguide and the mesh for FEM calculation

Figure 1 is the schematic representation of the transverse cross section of coaxial waveguide.

A waveguide exhibits a homogeneous dielectric constant in the longitudinal direction, and its

value changes in a stepwise manner in the transverse direction. The relative dielectric

constants are assumed to be εin = 10 and εout=1 for the dielectric core and cladding

respectively. Therefore, the generated eigenmodes are expected to be concentrated in the core

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Kim, Y. M. (2023). A Study on the Eigen Properties of the Coaxial Waveguide of Cylindrical Form. Transactions on Engineering and Computing Sciences,

11(5). 69-77.

URL: http://dx.doi.org/10.14738/tecs.115.15656

region with high dielectric constant according to Snell's law. The mesh is as first set as two- dimensional body centered square unit cell and then refined into a circularly symmetric

structure. Each unit cell is composed of four triangular elements. In this study, cross sectional

space is divided into triangular elements of 15 × 15 × 4 = 900. Here, the number 15 can be

artificially converted to a another set externally to perform FEM calculations. Among the

triangular elements of the mesh, the edges and nodes form the electric field and electric

potential respectively.

In particular, the edges of the triangular elements that make up the mesh must form a consistent

direction with their neighbors, as shown in the figure 2.

Figure 2: The mesh, edges and nodes of triangular elements.

Figure 2(a) is the relationship between edges used previously when performing FEM, and (b)

is the correlation between edges used in the current study. In the case of Figure 2(b), the

rotational direction is 1 → 2 → 3 depending on the color of each edge. The electric field vectors

at each edge obtained in these directions are combined according to reference [1] and marked

at the barycentric point of the triangular element. In Figure 2(b), the barycentric point of each

triangular element is indicated by a black dot. Since figure (b) decomposes the space in more

detail compared to (a), the accuracy of calculations is expected to be further improved. The

edge and the node numbers are in this study is 1204 and 421 respectively. The node points of

the triangular element describe the electric potential which is simultaneously obtained with

the electric field. These values are also used when drawing equipotential contours by

interpolation as shown in (c). The electric potential at a point inside the triangular element can

be obtained by combining the shape function as reference [1].

The numbers of edges and nodes increase as the mesh number increased. The matrix eigen

equation is composited with edges and nodes of the triangular mesh. Therefore, the size of the

matrix eigen equation increase proportional to the number of the mesh. The size of the matrix

eigen equation constructed by adding the number of edges and nodes is 1625 × 1625. In order

to handle the matrix equation of this size, sophisticated algorithm must be used. In this study,

the Arnoldi algorithm is applied to compress this equation so that it can be easily handled. As a

result, the compressed matrix equation has the size of 20 × 20 square matrix. Afterwards, this

square matrix is transformed into a Schur form using the Krylov-Schur iteration method [5]. In

iteration process, the Schur matrix is established by similarity transformation matrices of 20 ×

1625. The eigenmodes of the coaxial waveguide are described by column matrices of the

similarity transforming matrix. Those column matrices contain the contribution from the edges