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Dyadic Green’s Functions Inside/Outside a Dielectric
Elliptical Cylinder: Theory and Application
Le-Wei Li, Senior Member, IEEE, Hock-Guan Wee, Student Member, IEEE, and
Mook-Seng Leong, Senior Member, IEEE
Abstract—Dyadic Green’s functions in two regions separated
by an infinitely long elliptical dielectric cylinder are formulated in
this paper. As an application, the plane electromagnetic wave scattering by an isotropic elliptical dielectric cylinder is revisited by
applying these dyadic Green’s functions and the scattering-to-radiation transform. First, the dyadic Green’s functions are formulated and expanded in terms of elliptical vector wave functions. The
general equations are derived from the boundary conditions and
expressed in matrix form. Then the scattering and transmission
coefficients coupled to each other are solved from the matrix equations. To verify the theory developed and its applicability, we revisit
the plane electromagnetic wave scattering (of TE- and TM-polarizations) by an infinitely long elliptical cylinder, and consider it as a
special case of electromagnetic radiation using the dyadic Green’s
function technique. The derived equations and computed numerical results are then compared with published results and a good
agreement in each case is found. Special cases where the elliptical
cylinder degenerates to a circular cylinder and where the material
of the cylinder is isorefractive are also considered, and the same
analytical solutions in both cases are obtained.
Index Terms—Dielectric waveguide, dyadic Green’s functions,
eigenfunction expansions, elliptical cylinder, Mathieu functions,
plane wave scattering.
I. INTRODUCTION
E
LECTROMAGNETIC scattering by normal incident
plane waves has been analyzed by Yeh [1] and Burke [2].
For an oblique incident case, the equations needed to solve for
the scattering and transmission coefficients were derived by
Yeh [3]. The method for solving the problem is to express the
incident, scattered, and transmitted waves expanded in terms
of eigenfunctions. These eigenfunctions are obtained from
the separation of variables method and expressed in terms of
radial and angular Mathieu functions. The continuity boundary
conditions are then used to solve for the coefficients of the
scattered and transmitted waves. Numerical computations were
made in [1], [2], [4], and [5] using this method for the normal
incident plane waves. For the oblique incident plane waves,
Manuscript received October 19, 2001; revised January 26, 2002. This work
was supported by the National University of Singapore under Research Grant
RP981617.
L.-W. Li is with the High Performance Computation for Engineered Systems
(HPCES) Programme, Singapore-MIT Alliance (SMA), and the Department
of Electrical and Computer Engineering, National University of Singapore,
119260 Singapore (e-mail: [email protected]; [email protected]).
H.-G. Wee is with the High Performance Computation for Engineered
Systems (HPCES) Programme, Singapore-MIT Alliance (SMA), 119260,
Singapore.
M.-S. Leong is with the Department of Electrical and Computer Engineering,
National University of Singapore, 119260 Singapore.
Digital Object Identifier 10.1109/TAP.2003.809854
numerical computations were presented by Kim [6] 20 years
later. Instead of scattering problems, this work is motivated by
radiation problems.
It is well known that for radiation problems, the dyadic
Green’s function (DGFs) are very important kernels for integral
equation methods such as method of moments and boundary
element method [7]–[15]. Therefore, our objective here is
to 1) formulate the dyadic Green’s functions in the regions
separated by a dielectric cylinder and their scattering and
transmission coefficients and 2) revisit the plane wave scattering problem but using the dyadic Green’s function technique
and scattering-to-radiation transform approach [16]–[18]. The
former represents an original contribution, while the latter
presents an alternative approach to the classic problem for
verifying the correctness and applicability of the derived dyadic
Green’s functions.
The scattering superposition principle is utilized here for the
formulation of the Green’s functions. The free space dyadic
Green’s function available in [7] in terms of elliptical vector
wave functions is applied first. The scattering dyadic Green’s
functions in both (air and dielectric) regions separated by an elliptical dielectric cylinder are formulated. Boundary conditions
satisfied by the dyadic Green’s functions are matched to derive
a set of general equations governing the coefficients in terms
of wave functions. Then a general matrix equation is obtained
to calculate the coupled coefficients of the scattering dyadic
Green’s functions.
As an application of the derived dyadic Green’s functions,
we will revisit the plane wave scattering by an elliptical dielectric cylinder, which is considered as a special case of antenna
radiation in the presence of the same elliptical cylinder. The
scattering-to-radiation transform is utilized, where an infinitely
long line source is assumed to be placed at infinity, and this line
source is to generate two types of plane waves of transverse electric (TE) and the transverse magnetic (TM) polarizations illuminating toward the elliptical cylinder. In the scattering-to-radiation transform, the electric and magnetic current densities
and
of the line source that generates the TM and TE modes
are predicted with an unknown constant. Next, the electric and
magnetic fields are derived from the unbounded dyadic Green’s
function. These derived electric and magnetic fields are then
compared to known electric and magnetic fields of TM and TE
plane waves expanded in terms of the Mathieu functions. The
unknown constant can then be found after the comparison is
made. These current densities together with some mathematical properties of the Mathieu functions are then utilized to simplify the equations so as to solve for the scattering and trans-
0018-926X/03$17.00 © 2003 IEEE
LI et al.: DYADIC GREEN’S FUNCTIONS INSIDE/OUTSIDE DIELECTRIC ELLIPTICAL CYLINDER
Fig. 1.
A cross-section view of the elliptical coordinate system.
Fig. 2. Direction of the incident plane wave.
mission coefficients. The coefficients for the special case are
solved for by rewriting the equations into matrices and solving
them numerically.
The derived equations in mathematical form are compared
with published results [1], [3], [6], and a good agreement between the present forms and the existing ones is achieved. To
further prove the validity of the derived equations, we simplified them into those for a special case whereby both the inner
. With
and outer regions are isorefractive, that is,
this property, the orthogonality properties of the Mathieu functions can be used to greatly simplify the equations, and the reflection and transmission coefficients are obtained in analytical
form. The coefficients are solved and compared with the expressions obtained by Uslenghi [19] who formulated the dyadic
Green’s functions and their coefficients for this special case. The
same results as those of Uslenghi [19] are obtained, which is
expected. As for the numerical results, the normalized differential and backscattering cross sections are obtained and compared
with those of Kim [6]. Also, a good agreement is further found
from the comparison.
II. ELLIPTICAL COORDINATES SYSTEM
MATHIEU FUNCTIONS
565
AND
The elliptical coordinate system (Fig. 1) is defined by the
following relations:
functions by the National Bureau of Standards [20] are given
as follows:
(2a)
(2b)
means that when is even, is
where the summation
summed over all even values, and when is odd, is summed
in (2a) and
over all odd values. The coefficients
in (2b) are defined in [20] and [21] and normalized as follows:
and
The values of Mathieu functions are readily available in many
Fortran and C routines or in Mathematica, MatLab, or Maple
in (2a) and
in (2b) can
software packages. So,
be easily obtained, if specifically required, by multiplying (2a)
and
, respectively, and then integrating
and (2b) by
them from 0 to 2 with respect to . We do not need to follow
the complicated procedure or to use the tables in [20] and [21]
to calculate directly these coefficients.
The radial Mathieu functions of the first kind are defined as
(3a)
(1a)
(1b)
(1c)
where
(3b)
The radial Mathieu functions of the second kind are given by
while is the semifocal length of the ellipse. The contour surfaces of constant represent confocal elliptic cylinders, and
those of constant identify confocal hyperbolic cylinders. One
is assumed to coinof the confocal elliptic cylinders at
cide with the boundary of the solid dielectric cylinder. We assume the incident plane wave to propagate in the direction as
shown in Fig. 2.
The mathematical functions used here are the two kinds of
Mathieu functions, the periodic solutions consisting of the even
and
and the nonor odd angular functions
periodic solutions consisting of the even or odd radial funcand
. The definitions of the Mathieu
tions
(4a)
(4b)
(4c)
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 51, NO. 3, MARCH 2003
(4d)
and
while and
stand
where
for the Bessel functions of the first and second kinds, respectively. The radial functions of the third kind representing outtime harmonic system are defined as
going waves in the
(9b)
(5a)
(5b)
The orthogonality properties of the angular Mathieu functions are given as follows:
(9c)
(6a)
(6b)
(6c)
(9d)
where
(7a)
III. FORMULATION OF DYADIC GREEN’S FUNCTIONS
(7b)
The unbounded dyadic Green’s function was obtained by Tai
in terms of elliptical vector wave functions
[7] for
as
and
as
. Note that the
while
orthogonality can only be applied for the same . We can express the angular Mathieu functions of different and given
by McLachlan [21]
(10)
(8a)
(8b)
where represents the position of the source, which is assumed
to be anywhere in general but at infinity in the present applica.
tion, while and satisfy the relation of
The expansion of a combination of even and odd modes means
(8c)
(8d)
The vector wave functions are expressed in terms of Mathieu
functions as follows:
means that when is even, or is summed over
where
is odd, or is summed over all
even numbers, and when
,
,
, and
odd numbers. By using (2a) and (2b),
are defined as
(11a)
(11b)
(9a)
and
LI et al.: DYADIC GREEN’S FUNCTIONS INSIDE/OUTSIDE DIELECTRIC ELLIPTICAL CYLINDER
567
Substituting (10), (13a), and (13b) into the boundary equations,
we can further rewrite (15a) as
(12a)
(12b)
.
where
Subsequently, we consider two regions, namely, Region 1
outside of the dielectric cylinder (with permittivity and permeability ) and Region 2 inside of the dielectric cylinder (with
permittivity and permeability ). We can now assume that
the scattering dyadic Green’s functions due to an electric source
in Region 1 are defined as
(16a)
and (15b) as
(13a)
(13b)
,
,
, and
denote the scattering cowhere
,
,
, and
represent the transefficients while
mission coefficients; both sets to be determined. Also, in the
and
with
above expressions,
and
. The first superscript in
and
corresponds to the region where the field point is located, while the second superscript corresponds to the region
where the source is located. Region 1 is the surrounding space,
i.e., air; and Region 2 is inside the dielectric cylinder.
Using the principle of scattering superposition, we have
(14a)
(14b)
The boundary conditions at
tions are written as follows:
for the dyadic Green’s func(15a)
(15b)
(16b)
These two equations will be used later to solve for the unknown
coefficients.
IV. THEORETICAL APPLICATIONS: EM SCATTERING
To demonstrate how these scattering coefficients are
obtained, we will apply the afore-derived dyadic Green’s
functions to two specific cases of plane wave scattering by
a dielectric cylinder: one for the TM polarization incidence
and the other for the TE polarization incidence. To revisit this
classic problem, we do not follow what exists in the literature
for the scattering. Instead, we will make use of the concept
of scattering-to-radiation transform and the dyadic Green’s
functions derived previously to formulate the problem.
A. Scattering by TM-Polarized Plane Wave
We are now to find an electric current source located at
infinity that will generate a TM plane wave. In the scattering-to-radiation transform, the source distributions at infinity
are important and necessary because they are considered to
generate incident plane waves. Once this source is formulated,
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 51, NO. 3, MARCH 2003
then the scattered fields can be obtained directly from radiation
theory. The source can be found using the procedure given by
Li et al. [16]. Assume that the electric current source has the
form
and (16b) as
(17)
is the amplitude of the electric field at
, i.e.,
where
on the surface of the cylinder; and is to be determined.
The electric field can be found from
(18)
Meanwhile, the electric field of a TM incident plane wave can
be expressed as
(21b)
To solve for the scattering coefficients, we express the vector
wave functions in terms of Mathieu functions in (21a) and (21b)
and separate them in terms of the - and -components. There
will be four equations, with two coming from (21a) and the
other two from (21b). These four equations can then be further
separated into eight independent equations after the orthogonal
properties of the angular Mathieu functions in Section II are employed. The eight equations obtained are given as follows:
(19)
Equating (18) and (19) and using the following asymptotic relations of the radial Mathieu functions of the first kind:
we can obtain
(22a)
as
The asymptotic expressions of the outgoing radial Mathieu
functions are given as follows:
(22b)
(20a)
(22c)
(20b)
where denotes the cylindrical radial coordinate. By using the
electric current source, it can be shown that
(22d)
and
Multiplying (16a) and (16b) with (17) and using the above
volume integral relations, we have (16a) as
(21a)
(23a)
LI et al.: DYADIC GREEN’S FUNCTIONS INSIDE/OUTSIDE DIELECTRIC ELLIPTICAL CYLINDER
for
(23a)–(23d) can be expressed in a matrix form by varying ,
i.e.,
;
where
(23b)
for
;
for
; and
(23c)
(23d)
for
569
.
Equations (22a)–(22d) show that the scattering coefficients
are coupled to all the transmission coefficients. Equations
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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 51, NO. 3, MARCH 2003
These four coefficients are the same as those given in [19],
which shows the correctness of our formulation. It should be
,
,
, and
are zero due to the
pointed out that
normal incidence of the plane wave.
B. Scattering by TE-Polarized Plane Wave
Now, we focus on the scattering of a TE plane wave by the
cylindrical structure. The analysis is very much similar to that
in the previous section, but now we will use a magnetic current
source at infinity to generate the TE incident wave. Using the
duality theorem, we can obtain the magnetic current source from
the electric current source in the previous section
(24)
where the intermediate quantity is defined as
..
.
..
.
It can also be shown in a similar fashion that the eight equations
are
..
.
..
.
The scattering coefficients can then be found using (22a)–(22d).
These eight equations satisfied by the scattering coefficients are
found to have the same form as that given by Yeh [3] and Kim
[6].
Next, we would like to constrain our equations to a special
.
case where both regions are isorefractive, i.e.,
The wavenumbers in both regions are the same in this case,
. The analysis of this problem
i.e.,
was conducted by Uslenghi [19] for the TM normal incidence
) using the dyadic Green’s function. With
plane wave (
the same wavenumber, the orthogonal properties of the angular
Mathieu can be used, and this leads to a one-to-one matching
between the field modes on both sides of the interface. From
(22a) to (23d), applying the conditions in [19], the scattering
and transmitting coefficients are given as
(25a)
(25b)
(25c)
(25d)
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571
and
(25h)
.
for
V. NUMERICAL APPLICATIONS
(25e)
for
After we implemented the theoretical formulas into our
codes, we then computed them numerically and plotted some
results graphically shown in this section. To make the calculations, we first simplify the formulas and define some physical
quantities.
A. Far Field Expressions
For TM plane wave illumination, the scattered electric field
can be found easily using the formula given by Tai [7]
(26)
(25f)
for
;
The scattered magnetic field can be found using the Maxwell’s
equations.
In the far zone, we can approximate EM fields in the elliptical
coordinate system to those in the circular cylindrical coordinate
,
, and
). With the asymptotic
system using (
expressions for the radial Mathieu functions, the - and -components of the electric and magnetic fields can then be written
as
(27a)
(27b)
(27c)
(25g)
(27d)
for
; and
(27e)
(27f)
(27g)
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Under TE plane wave illumination, the scattered magnetic field
is obtained as
The backscattering cross section (also known as radar cross section) for both TE and TM incident plane waves are defined as
(34)
(28)
The expressions of far-field - and -components are
The backscattering cross section is a measure of the scattered
power in the backward direction. The cross-polarized components are defined for the TM case below
CP
(29a)
where the TM polarized components are obtained for a TM incident plane wave; and
CP
(29b)
(29c)
(35a)
(35b)
where the TE polarized components are obtained for a TM incident plane wave. For the TE incident wave, the expressions
are the same as those of the TM incident wave except that the
scattering coefficients for the TE case are used.
From the above analysis, it is seen that the present method
of scattering-to-radiation transform together with the dyadic
Green’s functions provides exactly the same results as in the
conventional boundary point matching technique.
C. Numerical Results
(29d)
B. Differential and Backscattering Cross Section
The differential cross section represents the density of the
power scattered by the cylinder in various directions. It depends
on the cylinder material property, size and shape of the cylinder,
and polarization state and frequency of the incident wave beam.
The differential cross section is defined as
(30)
For an incident plane wave, we have
(31)
where is the intrinsic impedance, i.e.,
field expressions, we have
. From the far(32)
By substituting these expressions, the differential cross sections
for the TE and TM incident plane waves take the same form and
are expressed as
(33)
The numerical results of the normalized differential and
backscattering cross sections for scattering of TM and TE
polarized incident plane waves by an elliptical dielectric
cylinder are computed. The normalized differential cross
) as a function of for various cases
sections (
are obtained. Due to the symmetry of the problem, only the
at
and of
angular regions of
at
are necessary. For backscattering
cross sections, we will show how the backscattered power
varies if the cross-sectional area of the ellipchanges as
tical cylinder constant remains unchanged. Various dielectric
medium parameters are considered in the case studies. In all
the numerical computations, the relative permeability of the
). Also,
dielectric cylinder is assumed to be unity (i.e.,
and used here represent the physical major and minor axes
of the elliptical cylinder, respectively. The convergence of
the series expansion is also considered and tested. From the
test, it is realized that the convergence is very fast when the
backscattering cross sections as given by Kim are computed.
Only six terms or elements at most are needed to achieve very
accurate results of relative error less than 0.1%, for example,
is calculated.
when
After the development of the codes under Mathematica software, we are able to readily calculate radiated fields due to any
antenna in the presence of the elliptical dielectric cylinder. However, we focus in this paper on only the scattering problem. Also,
we do not intend to show many results, and instead we will show
one set of results. Table I presents the backscattering cross secfor various
values. A comparison
tions of
is made between the results obtained herein and the results by
Kim. It is seen that these two sets of values are almost identical,
within the allowed computing accuracy.
LI et al.: DYADIC GREEN’S FUNCTIONS INSIDE/OUTSIDE DIELECTRIC ELLIPTICAL CYLINDER
TABLE I
NORMALIZED BACKSCATTERING CROSS SECTIONS log[ (; )] OF A
LOSSLESS DIELECTRIC CYLINDER FOR TE- AND TM-POLARIZED INCIDENT
PLANE WAVES, WHERE ka VARIES, a=b = 5, = 180 , AND = 90
573
is made and numerical results are obtained and found in very
good agreement with existing results. Further exploration
of applying the DGFs is expected in the future analysis of
various practical antenna systems where an elliptical dielectric
cylindrical waveguide is involved.
REFERENCES
VI. CONCLUSION
Two objectives have been achieved in this paper: 1) the
formulation of the dyadic Green’s functions inside and outside
of an elliptical dielectric cylindrical waveguide and 2) the
theoretical and numerical reconsideration of plane wave scattering by an isotropic dielectric elliptical cylinder but using the
DGFs and the scattering-to-radiation transform. The scattering
problem has been solved by directly using the scattering dyadic
Green’s function. To obtain the scattering coefficients of dyadic
Green’s functions, the boundary conditions on the interfaces
are matched. The equations satisfied by the scattering and
transmission coefficients have been derived and found to be the
same as those given by Yeh [3] and Kim [6]. To further confirm
the correctness of the method and to explore the accuracy and
convergence of the algorithm developed, numerical analysis
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Le-Wei Li (S’91–M’92–SM’96) received the B.Sc.
degree in physics from Xuzhou Normal University,
Xuzhou, China, in 1984, the M.Eng.Sc. degree,
China Research Institute of Radiowave Propagation
(CRIRP), Xinxiang, China, in 1987, and the Ph.D.
degree in electrical engineering from Monash
University, Melbourne, Australia, in 1992.
In 1992, he worked at La Trobe University (jointly
with Monash University), Melbourne, as a Research
Fellow. Since 1992, he has been with the Department
of Electrical and Computer Engineering, National
University of Singapore, where he is currently an Associate Professor. Since
1999, he has also been with the High Performance Computations on Engineered
Systems (HPCES) Programme of the Singapore-MIT Alliance (SMA), where
he is a SMA Fellow. His current research interests include electromagnetic
theory, radiowave propagation and scattering in various media, microwave
propagation and scattering in tropical environment, and analysis and design
of various antennas. In these areas, he coauthored Spheroidal Wave Functions
in Electromagnetic Theory (New York: Wiley, 2001), 30 book chapters, more
than 160 international refereed journal papers, 25 regional refereed journal
papers, and more than 170 international conference papers. He is a member
of the Editorial Board member of Journal of Electromagnetic Waves and
Applications.
Dr. Li is a Member of the MIT-based Electromagnetic Academy. He received
the Best Paper Award from the Chinese Institute of Communications for his
paper published in Journal of China Institute of Communications in 1990, and
the Prize Paper Award from the Chinese Institute of Electronics for his paper
published in Chinese Journal of Radio Science in 1991. He received a Ministerial Science & Technology Advancement Award from the Ministry of Electronic
Industries, China, in 1995 and the National Science & Technology Advancement Award with medal from the National Science & Technology Committee,
China, in 1996.
Hock-Guan Wee (S’98) received the B.Eng. degree in electrical engineering
(with first class honors) from the National University of Singapore (NUS) in
1999 and the S.M. degree in high performance computations of engineered systems from Singapore-MIT Alliance, NUS, in 2000. Since 2000, he has been
pursuing the Ph.D. degree at the Department of Electrical and Computer Engineering, NUS.
He is a Research Scholar at NUS. His current interest is on propagation and
scattering of electromagnetic waves, numerical techniques, and fast algorithms
in electromanetics and microwave applications.
Mook-Seng Leong (M’81–SM’98) received the
B.Sc. degree in electrical engineering (with first
class honors) and the Ph.D. degree in microwave
engineering from the University of London, U.K., in
1968 and 1971, respectively.
He is currently a Professor of electrical engineering at the National University of Singapore. His
main research interests include antenna and waveguide boundary-value problems. He is a coauthor
of Spheroidal Wave Functions in Electromagnetic
Theory (New York: Wiley, 2001). He is a member
of the Editorial Board for Microwave and Optical Technology Letters and
Wireless Mobile Communications.
Dr. Leong is a member of the MIT-based Electromagnetic Academy and a
Fellow of the Institution of Electrical Engineers, London. He received the 1996
Defense Science Organization (DSO) R&D Award from DSO National Laboratories, Singapore, in 1996.