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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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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.
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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
THE RESULTS AND DISCUSSION
Figure 2: Schematic representation of TEM mode-1.
Figure 2 schematically represents some of the TEM modes obtained through FEM calculations.
Small and complex potential peaks in the cladding area are present, but are not of a concern
level. In this figure, the electric potential is represented as a three-dimensional solid
representation and as an equipotential contour in a cross section of the waveguide. As can be
seen from the expression of the potential, there are small and complex components in the
spectrum, making it difficult to determine their eigenmodes. These components appear
concentrated in the cladding and interface. It is guessed that these components are due to the
heterogeneous material properties of the core and cladding. Equipotential contours are used as
complementary to determine the eigenmodes for each spectrum along with the electric field.
The spectrum for the electric potential as a whole describes the eigenmodes well along with the
electric field. As expected, the spatial distribution of the electric field is concentrated in the core
region. The dielectric cladding with a low dielectric constant suppressed the expansion of the
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eigenmode beyond the interface. These spectra have such a clear resolution of the electric field
that the eigenmodes can be distinguished even in a small area. These eigenmodes form
concentric circles from the center and extend to the interface between the core and the
cladding. They show that the eigenmodes of tangential components are more developed than
that of in the radial direction. Reflecting these characteristics of each spectrum, the eigenmodes
are determined as shown in Figure 2. In particular, in the case of low-order spectra (a) and (b),
an eigenmode is formed concentrated at the center ofthe core. In the application field, attention
will be focused on these eigenmodes so that they can be applied to fields with high utility.
Figure 3: Schematic representation of TEM mode-2.
Figure 3 schematically represents another TEM modes obtained through FEM calculations.
These spectra are TEM eigenmodes similar to TM modes. They form in small areas of the core
and exhibit a clear spatial distribution of electric fields. Here too, small and complex potential
peaks between the cladding and the interface are included, and these cannot be excluded from
all spectra. All potentials nevertheless contribute, together with the overall electric field, to
impose eigenmodes on their respective spectra. These spectra are a fine reproduction of those
seen in reference [6] from a small region of the core. Compared to them in reference [6], the
eigenmodes for the spectrum shown in Figure 3 can be easily imposed. The eigenmodes
corresponding to each spectrum are inserted in Figure 3.
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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
Figure 4: Schematic representation TEM mode-3.
Another type of spectrum appearing in the core region is shown in Figure 4. The characteristics
that appear at the center of the core extend to the interface and determine the eigenmode
properties of the spectrum. The higher the order of the spectrum, the more complex the
spectrum becomes. But in this study, even if the spectrum exists in a small area of the core, the
resolution is not so bed. No matter how complex the eigenmode is formed inside the core, it is
(a) TEM
11
(b) TEM
21
(c) TEM
22
(d) TEM 22
(e) TEM 23
(f) TEM 32
(g) TEM 43
(h) TEM 43
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Transactions on Engineering and Computing Sciences (TECS) Vol 11, Issue 6, December - 2023
Services for Science and Education – United Kingdom
difficult to extend beyond the interface, and its component exists only weakly in the cladding
area. Electromagnetic waves limited by the interface are concentrated in the core and form an
eigenmode that reflects the characteristics of the structure. The spectra shown in Figure4 reveal
TEM eigenmodes similar to the TE mode. In the spectra, it is identified that the eigenmodes develop
sequentially as the order increases. Figure 4(a) shows a spectrum similar to that in Figure 2(b).
However, when comparing the spectra in Figure 2(b) and Figure 4(1) in detail, it is reasonable
to impose each eigenmode as TEM 12 and TEM 11, respectively. If this eigenmode resonates
with electromagnetic waves injected from the outside, its energy can be propagated over long
distances. It can be asserted that the eigenmodes organized in each figure exhibit similar
functions. The above results can be said to meet the main purpose of focusing the eigenmodes
only on the core and forming the finer characteristics in detail by wrapping the core with the
dielectric cladding.
CONCLUSION
The main purpose of this study was to identify finer and detailed eigenmodes formed in the core of
a coaxial waveguide. As a result of FEM calculations, TEM modes concentrated in the core region
were obtained, and eigenmodes were assigned to them and shown in the schematic
representation. From these results, it wasfound that the eigenmodes generated in the core of
the coaxial waveguide are greatly dependent on the dielectric cladding. Furthermore, it was
discovered that the desired eigenmode can be obtained by controlling the dielectric cladding
and that this may be practically utilized in application fields.
Reference
[1] Yeong Min Kim, A STUDY ON THE EIGENMODES OF PCF VARYING THE POSITION OF THE DIELECTRIC
HOLE SBYFEM, Global Journal of Engineering Science and Researches, July 2016, 3(7).
[2] Yeong Min Kim, A Study on the Contribution of a Buffer Coated with a Perfect Conductor to Constructing
Eigenmodes in Square HAPCF, Journal of Electrical and Electronic Engineering, March 2019, 7(1), pp 36-
41.
[3] Yeong Min Kim, A Study on the Eigen Properties of the Coaxial Waveguide of Cylindrical Form,
Transactions on Engineering and Computing Sciences, Oct., 25, 2023, 11(5).
[4] C. J. Reddy, Manohar D. Deshpande, C. R. Cockrell, and Fred B. Beck, NASA Technical Paper, 1994, 3485.
[5] V. Hernandez, J. E. Roman, A. Tomas, V. Vidal, Krylov-Schur Methods in SLEPc, June, 2007, Available at
Available at http://www.grycap.upv.es/slepc.
[6] Yeong Min Kim, Jong Soo Lim, A Study on the Eigen-properties on Varied Structural 2-Dim. Waveguidesby
Krylov-Schur Iteration Method, Journal of the IEIE, February 2014, 51(2).