A BRIEF DISCUSSION ON THE
DIFFERENT FORMULATIONS OF THE
COUPLED MODE METHOD IN CHIRAL
MEDIA: APPLICATION TO THE
PARALLEL-PLATE CHIROWAVEGUIDE
A. Go ´ mez, A. Vegas, and M. A. Solano
University of Cantabria
Dpto. De Ingenierı ´a de Comunicaciones
Avda. de los Castros s/n
39005 Santander, Spain
Received 10 January 2004
ABSTRACT: The application of the coupled-mode method (CMM) for
calculating the propagation constants and the electromagnetic-field distribu-
tion in chirowaveguides is well known. In this paper, two different formula-
tions of the method are revised and compared, and are then extended to
partially filled parallel-plate waveguides. © 2004 Wiley Periodicals, Inc.
Microwave Opt Technol Lett 42: 181–185, 2004; Published online in
Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.
20245
Key words: chiral media; parallel-plate waveguides; coupled-mode
method
1. INTRODUCTION
The application of the coupled-mode method (CMM) to the study
of electromagnetic-field propagation inside a waveguide that is
fully or partially loaded with isotropic, anisotropic, and bi-isotro-
pic materials is not new [1, 2]. This method consists in expanding
a certain set of the electromagnetic-field components in terms of
basis functions that correspond to the empty waveguide-field com-
ponents.
Two different formulations of the method have been proposed
for chirowaveguides. In the first one, the components of the
electric field E
and the magnetic field H
are expanded as a lineal
combination of the electric and magnetic fields of the modes of the
same waveguide, but without the chiral media, that is, the empty
waveguide. We will call this EH formulation. In this formulation,
it is essential to express the longitudinal amplitudes of the elec-
tromagnetic field as a function of the amplitudes of the transverse
components. Two different strategies can be used to do this. In the
first one, which is normally used in the literature for bi-isotropic
media, it is not necessary to solve any systems of equations;
instead, the amplitudes are obtained directly (direct EH formula-
tion) [3]. In the other strategy, E
z
and H
z
are not expressed directly
as functions of the transverse components. This implies that it is
necessary to solve a set of linear equations and, therefore, a matrix
inversion must be performed (indirect EH formulation) [4]. In both
EH formulations the normal components of H
vanish at the perfect
electric walls because they are expanded in terms of functions
( H
-field for an empty waveguide), which are zero at these walls.
However, if a chiral media is in contact with a perfect electric
conductor wall, the normal components of H
are not zero on the
wall. Instead, the normal components of the magnetic induction B
are zero. This drawback of the EH formulation can be remedied if
the components of the magnetic induction B
are expanded [5],
instead of the components of H
. This is called the EB formulation.
A good discussion of the EB and EH formulations can be found
in [6].
The goal of this paper is to extend the EB formulation to
parallel-plate waveguides partially filled with chiral media, which
has only been developed for totally filled waveguides. Also, if
discontinuities between these kinds of structures need to be char-
acterized, it is necessary to find the magnetic field. Then, we also
propose a way of finding H
(which is not directly obtained as in the
EH formulation) from the EB formulation. Finally, we discuss the
advantages and disadvantages and point out a common error found
in the literature with regard to EH formulations.
2. THEORY
2.1. The EH-Formulations
Let us consider the problem (illustrated in Fig. 1) where the planes
y = 0 and y = b are perfect electric walls, thereby constituting a
parallel-plate waveguide. Inside this waveguide there is a chiral
material characterized by the following constitutive relations [7]:
D
=E
- j
0
0
H
, (1a)
B
= H
+ j
0
0
E
, (1b)
where is the dielectric permittivity, is the magnetic permeabil-
ity, and is the Pasteur parameter. The way to obtain the algebraic
system of equations for EH formulations, which is based on the
different form of relating E
z
, D
z
, H
z
, and B
z
by means of Eq. (1),
can be found in [2, 3] (the direct strategy) and in [4] (the indirect
strategy). It must be noted that in [3] the development for the H
z
component [Eq. (1)] is not correct, because a term independent of
the transverse coordinates is not included. For magnetic media, it
has been proved [8] that the absence of this term produces incor-
rect results, both in the propagation constants and in the electro-
magnetic field. The same occurs for bi-isotropic media [4]. On the
other hand, the direct strategy produces, in general, poorer con-
vergence results than the indirect strategy for the propagation
constants (that is, for the same number of basis functions, the value
of the propagation constant of any mode is more accurate using
indirect rather than direct strategy) and also poorer results in the
profile of the electromagnetic field. This apparent and surprising
behavior (because one might think that better results would be
obtained with direct strategy, where no matrix inversion must be
performed than with indirect strategy, where matrix inversion must
be performed) is due to another remarkable situation based on
indirect strategy, which ensures a much better numerical treatment
of the discontinuous nature of the longitudinal components of the
electromagnetic field [4, 7]. The complete expressions for these
two formulations can be seen in detail in [4].
2.2. The EB Formulation
In the EB formulation, the magnetic field H
is not developed
directly using T-functions. In this case, H
is substituted by the
magnetic induction B
. The expansions are given by
Figure 1 Parallel-plate waveguide partially filled with a slab of chiral
medium (PEC: perfect electric conductor)
MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 42, No. 3, August 5 2004 181