# Fundamentals of a
Null Field Method-Surface Equivalence Principle Approach for Scattering by
Dielectric Cylinders

Minas Kouroublakis  Nikolaos L. Tsitsas  George Fikioris

###### Abstract

The null-field method (NFM) and the method of auxiliary sources (MAS) have
been both used extensively for the numerical solution of boundary-value
problems arising in diverse applications involving propagation and scattering
of waves. It has been shown that, under certain conditions, the applicability
of MAS may be restricted by issues concerning the divergence of the auxiliary
currents, manifested by the appearance of exponentially large oscillations. In
this work, we combine the NFM with the surface equivalence principle (SEP) and
investigate analytically the convergence properties of the combined NFM-SEP
with reference to the problem of (internal or external) line-source excitation
of a dielectric cylinder. Our main purpose is to prove that (contrary to the
MAS) the discrete NFM-SEP currents, when properly normalized, always converge
to the corresponding continuous current densities, and thus no divergence and
oscillations phenomena appear. The theoretical analysis of the NFM-SEP is
accompanied by detailed comparisons with the MAS as well as with
representative numerical results illustrating the conclusions.

###### keywords:

Extended Boundary Condition , Null-Field Method , Extended Integral Equation ,
Surface Equivalence Principle , Method of Auxiliary Sources , Scattering ,
Dielectric Cylinder , Convergence , Oscillations

††journal:

[label1]organization=School of Informatics, Aristotle University of
Thessaloniki,city=Thessaloniki, postcode=54124, country=Greece

[label3]organization=School of Electrical and Computer Engineering, National
Technical University of Athens,city=Athens, postcode=15780, country=Greece

##  1 Introduction

The numerical solution of acoustic, electromagnetic and elastic boundary-value
problems involving cylindrical geometries lies at the core of many
applications, including, e.g., the analysis of scattering by periodic
structures [1], the modeling and design of antennas and waveguiding systems
[2], and the investigation of shielding or cloaking devices [3]. In many
cases, the solution of a boundary-value problem is facilitated by the
application of the so-called _surface equivalence principle_ (SEP) [4, 5, 6].
Then, surface current densities are defined on each participating boundary,
which are referred to as _equivalent current densities_. Imposing the boundary
conditions leads to a system of surface integral equations (SIEs), of equal
number to that of the unknown current densities. The numerical method that is
typically used to solve SIEs is the Method of Moments (MoM) [7]. However, MoM
encounters several difficulties, the main one being the evaluation of singular
integrals in order to fill the diagonal elements of the pertinent impedance
matrix. The singularities in the integrals stem from the fact that the
integration domain coincides with the physical boundary.

The above mentioned limitations (of evaluating singular integrals in MoM) led
to the development of alternative methods in which the support of the current
distribution is displaced from the domain on which the boundary conditions are
applied [8, 9, 10]. To this end, two main alternative approaches have been
established:

  1. 1.

Define equivalent current densities on displaced surfaces; enforce boundary
conditions on the physical surfaces.

  2. 2.

Define equivalent current densities on the physical boundaries; enforce
boundary conditions on displaced surfaces. Such boundary conditions are
referred to as _extended boundary conditions_ (EBCs) and the resulting
integral equations as _extended integral equations_ (EIEs).

One of the methods that is based on the first approach is the method of
auxiliary sources (MAS) [11, 12]. The MAS (or close variants of MAS) is also
known as the _source model technique_ [13, 14], the _method of fundamental
solutions_ [15, 16], and the _method of fictitious sources_ [17, 18]. In MAS,
the equivalent surface current densities are defined on fictitious surfaces,
referred to as _auxiliary surfaces_. The pertinent SIEs are then discretized,
with the equivalent current densities replaced by a number of _auxiliary
sources_ (ASs). For the two-dimensional (2-D) geometries dealt with in this
paper, the ASs are infinitely long electric (for transverse magnetic
($\rm{TM}$) polarization) or magnetic (for transverse electric ($\rm{TE}$)
polarization) current filaments, which radiate cylindrical waves described by
zero-order Hankel functions. Despite its interesting features, the
effectiveness and applicability of the MAS are restricted by open issues,
including the placement of the ASs and the convergence of the fields.
Regarding the convergence issues, it was shown in [19, 20, 21, 22, 23] for
perfectly conducting scatterers and in [24] for dielectric scatterers,
that—under certain conditions on the placement of the auxiliary surface, and
for sufficiently large numbers $N$ of ASs—the auxiliary currents diverge and
oscillate while the generated MAS fields converge to the exact ones, provided
that the oscillations are mild. However, for even larger values of $N$, the
oscillations’ amplitudes increase further and numerically corrupted fields are
obtained.

A well-established method that utilizes the second approach is the _Extended
Boundary Condition Method_ (EBCM) [8, 9] also known as _Null Field Method_
(NFM) [25, 26, 27, 28, 29, 30]. The EIEs are discretized, with the equivalent
current densities on the physical boundaries replaced by $N$ discrete sources
(similar to those of MAS). For problems involving perfectly conducting
scatterers, it was shown that a main advantage of NFM over MAS is that in NFM
no divergence and oscillations phenomena occur for the discrete currents [21,
31]. Particularly, for a circular cylinder, in [31] it was shown analytically
that the EIE is solvable, irrespective of the position of the surface where
the EBCs are applied, and that the solution of the discrete version of the
EIE, when properly normalized, converges to the true solution as
$N\rightarrow\infty$. Hence, NFM showcases a very different behavior compared
to MAS despite the close resemblance of the integral equations involved in the
two methods.

In this paper, we consider the problem of (internal or external) line-source
excitation of a dielectric cylinder and combine the concepts of SEP and NFM by
defining two equivalent problems of the original one and determining the
unknown current densities on the physical boundary via a system of two
integral equations stemming from two extended boundary conditions. Our main
goal is to study the convergence properties of the combined NFM-SEP approach
and see how the above discussed conclusions on the solutions of the EIEs for
perfectly conducting cylinders carry over to dielectric cylinders. As in [31],
we bring out stark contrasts to previous findings on the MAS, despite that the
matrices of the NFM-SEP and the MAS are actually the same.

The structure of the paper is as follows. In Section 2.1, we define the
original scattering/excitation problems and apply the SEP to obtain its two
equivalent problems. Next, Section 2.2 clarifies the implementation of the
NFM-SEP to the considered problems. In Section 3, we give the exact solution
for the circular problem and then consider and solve in Section 4 the
continuous version of the involved EIEs to derive analytically this exact
solution. Then, in Section 5, we formulate the discrete version of the NFM-SEP
and obtain the corresponding solutions. There, we prove that (as opposed to
the MAS) no oscillations appear in the discrete NFM-SEP currents, which always
converge to the corresponding continuous current densities. Section 6 contains
a detailed comparative discussion between properties of NFM-SEP and MAS.
Furthermore, in Section 7, we show representative numerical results for
circular and noncircular shapes illustrating some of the conclusions of
Sections 4-6. The paper is closed with conclusions in Section 8.

An $e^{\mathrm{i}\omega t}$ dependence is assumed and suppressed in what
follows, where $\omega$ is the angular frequency, assumed throughout to be
real.

##  2 NFM-SEP

We describe NFM-SEP in the present section, beginning with the two equivalent
problems.

###  2.1 The Original Problem and its Equivalents

Figure 1a shows the cross section of an infinitely-long and parallel to the
$z$-axis dielectric cylinder (Region 2 or $R_{2}$), which is characterized by
dielectric permittivity $\varepsilon_{2}$ and magnetic permeability $\mu_{2}$.
The cylinder is surrounded by a homogeneous dielectric medium, which is
referred to as Region 1 ($R_{1}$) and characterized by respective parameters
$\varepsilon_{1}$ and $\mu_{1}$. The boundary between the two regions is
denoted as $C$ and is of arbitrary shape. We consider two scattering problems:

Figure 1: Scattering of a cylindrical wave by an infinitely-long dielectric
cylinder. (a) Original problem; (b) first equivalent problem for the external
field in $R_{1}$; and (c) second equivalent problem for the internal field in
$R_{2}$.

  1. 1.

A $\rm{TM_{z}}$ or $\rm{TE_{z}}$ incident field propagates in $R_{1}$ and is
scattered by the boundary $C$. The incident wave is either a plane wave
(generated by a source far away from the cylinder), or a cylindrical wave
generated by a $z$-directed electric ($\rm{TM_{z}}$) or magnetic
($\rm{TE_{z}}$) current filament of infinite length within $R_{1}$ and near
$C$. This problem will be referred to as the _external problem_.

  2. 2.

An incident $\rm{TM_{z}}$ or $\rm{TE_{z}}$ field is generated by a
$z$-directed electric or magnetic current filament of infinite length within
$R_{2}$. This problem will be referred to as the _internal problem_.

In both problems, the goal is to find the fields in $R_{1}$ and $R_{2}$. To do
this, we apply the _surface equivalence principle_ (SEP) [4, 5, 6]. According
to the SEP, two equivalent problems of the original one are defined, the first
for the fields in $R_{1}$, and the second for the fields in $R_{2}$.

In the first problem (depicted in Fig. 1(b)), the fields in $R_{1}$ are the
same as in the original problem, the material of $R_{2}$ is replaced by the
material of $R_{1}$, and the fields in $R_{2}$ are set to zero. To satisfy the
boundary conditions on $C$, two surface current densities, an electric
$\mathbf{J}^{s}$ and a magnetic $\mathbf{M}^{s}$, are defined on $C$. These
densities are given by

$\displaystyle\mathbf{J}^{s}$
$\displaystyle=\hat{\mathbf{n}}\times\mathbf{H},$ (1)
$\displaystyle\mathbf{M}^{s}$
$\displaystyle=\mathbf{E}\times\hat{\mathbf{n}},$ (2)

where $\hat{\mathbf{n}}$ is the outward normal unit vector on $C$,
$\mathbf{E}$ and $\mathbf{H}$ is the total electric and magnetic field just
outside $C$. In case of a $\rm{TM_{z}}$ ($\rm{TE_{z}}$) incident wave,
$\mathbf{J}^{s}$ ($\mathbf{M}^{s}$) is $z$-directed, while $\mathbf{M}^{s}$
($\mathbf{J}^{s}$) is tangential to $C$.

In the second equivalent problem (depicted in Fig. 1(c)), the fields in
$R_{2}$ are the same as in the original problem, the material of $R_{1}$ is
replaced by the material of $R_{2}$, and the fields in $R_{1}$ are set to
zero. Now, we need to define the following current densities on $C$

$\displaystyle\tilde{\mathbf{J}}^{s}$
$\displaystyle=-\hat{\mathbf{n}}\times\mathbf{H},$ (3)
$\displaystyle\tilde{\mathbf{M}}^{s}$
$\displaystyle=-\mathbf{E}\times\hat{\mathbf{n}},$ (4)

i.e., the opposites of those defined in the first equivalent problem.

For the external problem, the field in $R_{1}$ is the superposition of the
incident field plus the field generated by the sources defined in (1) and (2),
while the field in $R_{2}$ is that generated by the sources defined in (3) and
(4). For the internal problem, the field in $R_{2}$ is the superposition of
the incident field plus the field generated by the sources defined in (3) and
(4) and the field in $R_{1}$ is that generated by the sources defined in (1)
and (2).

To implement the SEP, it is, thus, first necessary to determine
$\mathbf{J}^{s}$ and $\mathbf{M}^{s}$. This can be accomplished by means of a
system of two linear integral equations resulting from two extended boundary
conditions. In the context of NFM, the first such boundary condition arises
from the first equivalent problem and requires the zeroing of the total
(electric or magnetic) field on an auxiliary surface $C^{(1)}_{\mathrm{aux}}$
inside $R_{2}$. The second extended boundary condition stems from the second
equivalent problem and requires the zeroing of the total field on an auxiliary
surface $C^{(2)}_{\mathrm{aux}}$ inside $R_{1}$.

The above-described “continuous” version of NFM-SEP leads to a system of
linear integral equations. For brevity, we do not write this system down.
Instead, we proceed to describe its discretized version, which is what one
solves in actual practice.

###  2.2 Implementation of NFM-SEP

The discretized versions of the NFM-SEP problems of Fig. 1 are depicted in
Fig. 2. Here, the continuous current densities are approximated by a number of
discrete sources. Specifically, and in case of $\rm{TM_{z}}$ incidence, the
$z$-directed electric current density $\mathbf{J}^{s}$ is replaced by $N$
infinitely-long and parallel to the $z$-axis electric current filaments with
current complex amplitudes $I_{l}$. The tangential magnetic current density
$\mathbf{M}^{s}$ is replaced by $N$ tangential on $C$ magnetic current
filaments, each carrying a voltage $K_{l}$ ($l=0,1,\ldots,N-1$).

Figure 2: Discretization of (a) the external and (b) the internal NFM
equivalent problem for a dielectric cylinder of arbitrary cross section. Black
dots represent the discrete electric and magnetic sources, and black squares
represent the collocation points.

The $z$-directed incident electric field is given by

$E^{\mathrm{inc}}_{j}(\mathbf{r})=-\frac{k_{j}Z_{j}}{4}IH_{0}^{(2)}(k_{j}|\mathbf{r}-\mathbf{r}_{\mathrm{fil}}|),\,\,j=1,2,$
(5)

where $j$ denotes the region in which the filament is positioned, $I$ and
$\mathbf{r}_{\mathrm{fil}}$ are the constant electric current amplitude and
position vector of the filament, $k_{j}=\omega\sqrt{\varepsilon_{j}\mu_{j}}$
and $Z_{j}=\sqrt{\mu_{j}/\varepsilon_{j}}$ are the wavenumber and impedance of
region $j$, and $H_{0}^{(2)}$ is the 0-order and second kind cylindrical
Hankel function.

For external excitation, the total electric field in $R_{1}$ is given by

$\displaystyle E_{z}^{(1)}=$
$\displaystyle-\frac{k_{1}Z_{1}}{4}IH_{0}^{(2)}(k_{1}|\mathbf{r}-\mathbf{r}_{\mathrm{fil}}|)$
$\displaystyle-\frac{k_{1}Z_{1}}{4}\sum_{l=0}^{N-1}I_{l}H_{0}^{(2)}(k_{1}|\mathbf{r}-\mathbf{r}_{l}|)+\frac{k_{1}}{4\mathrm{i}}\sum_{l=0}^{N-1}K_{l}\frac{\partial
H_{0}^{(2)}(k_{1}|\mathbf{r}-\mathbf{r}_{l}|)}{\partial n},$ (6)

while the total electric field in $R_{2}$ has the form

$\displaystyle
E_{z}^{(2)}=-\frac{k_{2}Z_{2}}{4}\sum_{l=0}^{N-1}(-I_{l})H_{0}^{(2)}(k_{2}|\mathbf{r}-\mathbf{r}_{l}|)+\frac{k_{2}}{4\mathrm{i}}\sum_{l=0}^{N-1}(-K_{l})\frac{\partial
H_{0}^{(2)}(k_{2}|\mathbf{r}-\mathbf{r}_{l}|)}{\partial n},$ (7)

where $\mathbf{r}_{l}$ is the position vector of the $l$-th collocated
electric and magnetic discrete source. Imposing the boundary conditions at $N$
collocation points on $C^{(1)}_{\mathrm{aux}}$ and $C^{(2)}_{\mathrm{aux}}$,
we obtain the equations, for $p=0,1,\ldots,N-1$,

$\displaystyle
Z_{1}\sum_{l=0}^{N-1}I_{l}H_{0}^{(2)}(k_{1}|\mathbf{r}^{(1)}_{p,\mathrm{aux}}-\mathbf{r}_{l}|)+\mathrm{i}\sum_{l=0}^{N-1}K_{l}\frac{\partial
H_{0}^{(2)}(k_{1}|\mathbf{r}^{(1)}_{p,\mathrm{aux}}-\mathbf{r}_{l}|)}{\partial
n}=$ $\displaystyle-
Z_{1}IH_{0}^{(2)}(k_{1}|\mathbf{r}^{(1)}_{p,\mathrm{aux}}-\mathbf{r}_{\mathrm{fil}}|),$
(8) $\displaystyle
Z_{2}\sum_{l=0}^{N-1}I_{l}H_{0}^{(2)}(k_{2}|\mathbf{r}^{(2)}_{p,\mathrm{aux}}-\mathbf{r}_{l}|)+\mathrm{i}\sum_{l=0}^{N-1}K_{l}\frac{\partial
H_{0}^{(2)}(k_{2}|\mathbf{r}^{(2)}_{p,\mathrm{aux}}-\mathbf{r}_{l}|)}{\partial
n}=0,$ (9)

where $\mathbf{r}^{(j)}_{p,\mathrm{aux}}$ is the position vector of the $p$-th
collocation point on $C^{(j)}_{\mathrm{aux}}$. Equations (8) and (9)
constitute an $2N\times 2N$ linear system with unknowns the electric $I_{l}$
and magnetic $K_{l}$ current complex amplitudes. Once solved, the fields in
each region are determined from (6) and (7).

##  3 Circular Problem: Exact Solution and its Analytic Continuation

The simplest geometry is that of a circular boundary $C$, as illustrated in
Fig. 3(a): A circular cylinder of radius $\rho_{\mathrm{cyl}}$ is excited by
an external or an internal (electric or magnetic) source lying at
$(\rho_{\mathrm{fil}},0)$ in polar coordinates. For $\rm{TM_{z}}$ incidence,
the sole $z$-component of the incident electric field is

$E_{j}^{\mathrm{inc}}(\rho_{\mathrm{obs}},\phi_{\mathrm{obs}})=-\frac{k_{j}Z_{j}}{4}IH_{0}^{(2)}(k_{j}D_{\mathrm{fil,obs}}),\,\,j=1,2,$
(10)

with $j$ the region in which the filament of amplitude $I$ is located, and
$D_{\mathrm{fil,obs}}$ the distance between the filament position and the
observation point, i.e.,

$D_{\mathrm{fil,obs}}=\sqrt{\rho^{2}_{\mathrm{obs}}+\rho^{2}_{\mathrm{fil}}-2\rho_{\mathrm{obs}}\rho_{\mathrm{fil}}\cos{\phi_{\mathrm{obs}}}}.$
(11)

By the duality principle [5, 6], the incident $\rm TE_{z}$ magnetic field is
given by

$H_{j}^{\mathrm{inc}}(\rho_{\mathrm{obs}},\phi_{\mathrm{obs}})=-\frac{k_{j}}{4Z_{j}}KH_{0}^{(2)}(k_{j}D_{\mathrm{fil,obs}}),\,\,j=1,2,$
(12)

where $K$ is the magnetic current amplitude of the filament.

The analysis that follows concerns $\rm{TM_{z}}$ polarization. The
corresponding results for $\rm{TE_{z}}$ polarization are omitted for brevity,
but can be easily obtained via the duality principle.

Figure 3: Scattering of a cylindrical wave by an infinitely long circular
dielectric cylinder. (a) Original problem; (b) first equivalent problem (for
external field in $R_{1}$); and (c) second equivalent problem (for internal
field in $R_{2}$).

The problem of Fig. 3a is solved analytically by Fourier series [24]. For
external excitation, the total electric fields in $R_{1}$ and $R_{2}$ are
found to be

$\displaystyle
E^{(1)}_{z}(\rho_{\mathrm{obs}},\phi_{\mathrm{obs}})=E_{1}^{\mathrm{inc}}(\rho_{\mathrm{obs}},\phi_{\mathrm{obs}})+\frac{k_{1}Z_{1}I}{4}\sum_{n=-\infty}^{+\infty}H_{n}^{(2)}(k_{1}\rho_{\mathrm{obs}})H_{n}^{(2)}(k_{1}\rho_{\mathrm{fil}})\times$
$\displaystyle\frac{Z_{1}J^{\prime}_{n}(k_{2}\rho_{\mathrm{cyl}})J_{n}(k_{1}\rho_{\mathrm{cyl}})-Z_{2}J_{n}(k_{2}\rho_{\mathrm{cyl}})J^{\prime}_{n}(k_{1}\rho_{\mathrm{cyl}})}{Z_{1}H_{n}^{(2)}(k_{1}\rho_{\mathrm{cyl}})J^{\prime}_{n}(k_{2}\rho_{\mathrm{cyl}})-Z_{2}J_{n}(k_{2}\rho_{\mathrm{cyl}})H^{\prime(2)}_{n}(k_{1}\rho_{\mathrm{cyl}})}e^{\mathrm{i}n\phi_{\mathrm{obs}}},$
(13) $\displaystyle
E^{(2)}_{z}(\rho_{\mathrm{obs}},\phi_{\mathrm{obs}})=-\frac{I}{2\pi\rho_{\mathrm{cyl}}}\sum_{n=-\infty}^{+\infty}J_{n}(k_{2}\rho_{\mathrm{obs}})H_{n}^{(2)}(k_{1}\rho_{\mathrm{fil}})\times$
$\displaystyle\frac{\mathrm{i}Z_{1}Z_{2}}{Z_{1}H_{n}^{(2)}(k_{1}\rho_{\mathrm{cyl}})J^{\prime}_{n}(k_{2}\rho_{\mathrm{cyl}})-Z_{2}J_{n}(k_{2}\rho_{\mathrm{cyl}})H^{\prime(2)}_{n}(k_{1}\rho_{\mathrm{cyl}})}e^{\mathrm{i}n\phi_{\mathrm{obs}}}.$
(14)

For internal excitation, the respective total fields are

$\displaystyle
E^{(1)}_{z}(\rho_{\mathrm{obs}},\phi_{\mathrm{obs}})=-\frac{I}{2\pi\rho_{\mathrm{cyl}}}\sum_{n=-\infty}^{+\infty}H^{(2)}_{n}(k_{1}\rho_{\mathrm{obs}})J_{n}(k_{2}\rho_{\mathrm{fil}})\times$
$\displaystyle\frac{\mathrm{i}Z_{1}Z_{2}}{Z_{1}H^{(2)}_{n}(k_{1}\rho_{\mathrm{cyl}})J^{\prime}_{n}(k_{2}\rho_{\mathrm{cyl}})-Z_{2}J_{n}(k_{2}\rho_{\mathrm{cyl}})H^{\prime(2)}_{n}(k_{1}\rho_{\mathrm{cyl}})}e^{\mathrm{i}n\phi_{\mathrm{obs}}},$
(15) $\displaystyle
E^{(2)}_{z}(\rho_{\mathrm{obs}},\phi_{\mathrm{obs}})=E_{2}^{\mathrm{inc}}(\rho_{\mathrm{obs}},\phi_{\mathrm{obs}})+\frac{k_{2}Z_{2}I}{4}\sum_{n=-\infty}^{+\infty}J_{n}(k_{2}\rho_{\mathrm{obs}})J_{n}(k_{2}\rho_{\mathrm{fil}})\times$
$\displaystyle\frac{Z_{1}H^{(2)}_{n}(k_{1}\rho_{\mathrm{cyl}})H^{\prime(2)}_{n}(k_{2}\rho_{\mathrm{cyl}})-Z_{2}H^{\prime(2)}_{n}(k_{1}\rho_{\mathrm{cyl}})H^{(2)}_{n}(k_{2}\rho_{\mathrm{cyl}})}{Z_{1}H^{(2)}_{n}(k_{1}\rho_{\mathrm{cyl}})J^{\prime}_{n}(k_{2}\rho_{\mathrm{cyl}})-Z_{2}J_{n}(k_{2}\rho_{\mathrm{cyl}})H^{\prime(2)}_{n}(k_{1}\rho_{\mathrm{cyl}})}e^{\mathrm{i}n\phi_{\mathrm{obs}}}.$
(16)

Note that in each term of all series (13)-(16) the denominator is the same.
This denominator does not vanish for any real frequency value [32, 33], and,
hence, we cannot detect the cutoff frequency of any propagating mode in the
dielectric cylinder. This is in contrast to the case of PEC waveguides excited
by an internal or external source [34], where the denominator of the
expression of the total field within the PEC cavity vanishes for real
frequency values corresponding to the cutoff frequencies. This is because in
dielectric waveguides, the concept of cutoff is related to the inability to
confine a propagating wave near the core [32], whereas in PEC waveguides, to
the inability of a propagating wave to exist. In other words, to detect cutoff
frequencies in a dielectric cylinder, we would need to excite it with sources
that induce some propagating wave along the axis of the cylinder. This cannot
be achieved with an electric or magnetic filament with a constant amplitude as
a source, but rather with a proper combination of both [35], or by exciting
the dielectric cylinder with an oblique plane wave [36].

Needless to say, the four series converge within their region of validity,
i.e., the two series associated with $E_{z}^{(1)}$ ($E_{z}^{(2)}$) converge
for $\rho_{\mathrm{obs}}>\rho_{\mathrm{cyl}}$
($\rho_{\mathrm{obs}}<\rho_{\mathrm{cyl}}$). But, we can show that all series
converge within larger regions; and that each function, thus, defined is in
fact the analytic continuation of the respective field into the extended
region. The four extended regions have circular boundaries and, as seen in
Table 1, are described by the relation between $\rho_{\mathrm{obs}}$ on the
one hand, and either $\rho_{\mathrm{fil}}$ or the “critical distance”
$\rho_{\mathrm{cri}}$ on the other. This distance is defined by

$\rho_{\mathrm{cri}}=\frac{\rho_{\mathrm{cyl}}^{2}}{\rho_{\mathrm{fil}}}.$
(17)

Table 1: Each cell shows the extended region of convergence of a series in (13), (14) or (15), (16). For example, the series in (15) pertains to internal excitation (so that $\rho_{\rm fil}<\rho_{\rm cyl}$) and gives the field in $R_{1}$ (i.e., for observation points $\rho_{\rm obs}>\rho_{\rm cyl}$); but this series converges not only in $\rho_{\rm obs}>\rho_{\rm cyl}$, but in the extended region $\rho_{\rm obs}>\rho_{\rm fil}$. All series diverge in the complementary regions; for example, the series in (15) diverges for $\rho_{\rm obs}<\rho_{\rm fil}$. | $R_{2},\,\rho_{\rm obs}<\rho_{\rm cyl}$ | $R_{1},\,\rho_{\rm obs}>\rho_{\rm cyl}$
---|---|---
external excitation,$\,\rho_{\rm cyl}<\rho_{\rm fil}$ | (14),$\,\rho_{\rm obs}<\rho_{\rm fil}$ | (13),$\,\rho_{\rm obs}>\rho_{\rm cri}$
internal excitation,$\,\rho_{\rm fil}<\rho_{\rm cyl}$ | (16),$\,\rho_{\rm obs}<\rho_{\rm cri}$ | (15),$\,\rho_{\rm obs}>\rho_{\rm fil}$

To derive the results in Table 1, use the large-order asymptotic
approximations (for the Bessel/Hankel functions and their derivatives)
provided in A. For example, when $|n|$ is large, the $n$-th terms of the
series in (13) (scattered field) and (14) are seen to behave like

$\displaystyle\frac{2}{\pi|n|}\frac{\rho_{\mathrm{cri}}^{|n|}}{\rho_{\mathrm{obs}}^{|n|}}e^{\mathrm{i}n\phi_{\mathrm{obs}}},$
(18)
$\displaystyle\frac{\mathrm{i}Z_{1}Z_{2}}{\frac{Z_{1}}{k_{2}}-\frac{Z_{2}}{k_{1}}}\frac{\rho_{\mathrm{obs}}^{|n|}}{\rho_{\mathrm{fil}}^{|n|}}e^{\mathrm{i}n\phi_{\mathrm{obs}}},$
(19)

respectively, where we used the definition (17). By (18), the series
pertaining to $R_{1}$ converges exponentially for
$\rho_{\mathrm{obs}}>\rho_{\mathrm{cri}}$ and diverges exponentially for
$\rho_{\mathrm{obs}}<\rho_{\mathrm{cri}}$, verifying the pertinent entry of
Table 1. Since the derivative of (13) is continuous across $\rho_{\rm
obs}=\rho_{\mathrm{cyl}}$, this extended field is in fact the analytic
continuation of the scattered field (series in (13)) from the original region
$R_{1}$ into part of the complementary region $R_{2}$.

Recall that, in MAS, an “auxiliary surface” is a surface on which MAS
(auxiliary) currents are located. In the case of MAS, it is found in [24] that
placing an auxiliary surface beyond the respective extended region, as defined
in Table 1, results in divergence and unphysical oscillations in the
corresponding auxiliary currents. The conclusions regarding the divergence of
the auxiliary currents are summarized in Table 2.

Table 2: Each cell shows the region of divergence of the MAS auxiliary currents. For example, in case of internal excitation (so that $\rho_{\rm fil}<\rho_{\rm cyl}$), the MAS currents on the internal (external) auxiliary surface $C^{(1)}_{\mathrm{aux}}$ $(C^{(2)}_{\mathrm{aux}})$ diverge when $\,\rho^{(1)}_{\mathrm{aux}}<\rho_{\rm fil}$ ($\,\rho^{(2)}_{\mathrm{aux}}>\rho_{\rm cri}$). | $C^{(1)}_{\mathrm{aux}}$ | $C^{(2)}_{\mathrm{aux}}$
---|---|---
external excitation,$\,\rho_{\rm cyl}<\rho_{\rm fil}$ | $\,\rho^{(1)}_{\mathrm{aux}}<\rho_{\rm cri}$ | $\,\rho^{(2)}_{\mathrm{aux}}>\rho_{\rm fil}$
internal excitation,$\,\rho_{\rm fil}<\rho_{\rm cyl}$ | $\,\rho^{(1)}_{\mathrm{aux}}<\rho_{\rm fil}$ | $\,\rho^{(2)}_{\mathrm{aux}}>\rho_{\rm cri}$

In the next section, we show that the above mentioned findings concerning the
divergence and oscillations in the MAS auxiliary currents have no counterpart
in the case of the EIE.

##  4 Continuous EIE Solution and its Convergence

The key idea is to analytically extract the solution of the circular problem
by applying the NFM. The two equivalent circular problems are depicted in
Figs. 3(b) and (c) respectively. In case of external $\rm{TM_{z}}$ incidence,
according to the first equivalent problem of Fig. 3b, the total electric field
in $R_{1}$ is the following superposition of the incident field and the field
radiated by the current densities $J_{z}^{s}$ and $M_{\phi}^{s}$

$\displaystyle
E_{z}^{(1)}(\rho_{\mathrm{obs}},\phi_{\mathrm{obs}})=E_{1}^{\mathrm{inc}}(\rho_{\mathrm{obs}},\phi_{\mathrm{obs}})-\frac{k_{1}Z_{1}\rho_{\mathrm{cyl}}}{4}\int_{-\pi}^{\pi}J^{s}_{z}(\phi_{\mathrm{cyl}})H_{0}^{(2)}(k_{1}D_{\mathrm{obs,cyl}})\mathrm{d}\phi_{\mathrm{cyl}}$
$\displaystyle-\frac{k_{1}\rho_{\mathrm{cyl}}}{4\mathrm{i}}\int_{-\pi}^{\pi}M^{s}_{\phi}(\phi_{\mathrm{cyl}})\frac{\partial
H_{0}^{(2)}(k_{1}D_{\mathrm{obs,cyl}})}{\partial\rho_{\rm
cyl}}\mathrm{d}\phi_{\mathrm{cyl}},$ (20)

with

$D_{\mathrm{obs,cyl}}=\sqrt{\rho^{2}_{\mathrm{cyl}}+\rho^{2}_{\mathrm{obs}}-2\rho_{\mathrm{cyl}}\rho_{\mathrm{obs}}\cos\mathrm{(}{\phi_{\mathrm{cyl}}}-\phi_{\mathrm{obs}})}$
(21)

the distance between a point on the circular boundary $C$ and the observation
point. According to the second equivalent problem (Fig. 3c), the field in
$R_{2}$ is the following superposition of the fields radiated by $-J_{z}^{s}$
and $-M_{\phi}^{s}$ in an unbounded medium filled with the cylinder’s material

$\displaystyle E_{z}^{(2)}(\rho_{\mathrm{obs}},\phi_{\mathrm{obs}})=$
$\displaystyle-\frac{k_{2}Z_{2}\rho_{\mathrm{cyl}}}{4}\int_{-\pi}^{\pi}(-J^{s}_{z}(\phi_{\mathrm{cyl}}))H_{0}^{(2)}(k_{2}D_{\mathrm{obs,cyl}})\mathrm{d}\phi_{\mathrm{cyl}}$
$\displaystyle-\frac{k_{2}\rho_{\mathrm{cyl}}}{4\mathrm{i}}\int_{-\pi}^{\pi}(-M^{s}_{\phi}(\phi_{\mathrm{cyl}}))\frac{\partial
H_{0}^{(2)}(k_{2}D_{\mathrm{obs,cyl}})}{\partial\rho_{\rm
cyl}}\mathrm{d}\phi_{\mathrm{cyl}}.$ (22)

The unknown current densities $J_{z}^{s}$ and $M_{\phi}^{s}$ in Eqs. (20) and
(22) are determined by enforcing the two boundary conditions discussed in
Section 2.1. The first stems from the first equivalent problem and requires
the zeroing of the total field on the auxiliary surface
$C^{(1)}_{\mathrm{aux}}$, with
$\rho^{(1)}_{\mathrm{aux}}<\rho_{\mathrm{cyl}}$, yielding

$\displaystyle-\frac{k_{1}Z_{1}\rho_{\mathrm{cyl}}}{4}\int_{-\pi}^{\pi}J^{s}_{z}(\phi_{\mathrm{cyl}})H_{0}^{(2)}(k_{1}D^{(1)}_{\mathrm{cyl,aux}})\mathrm{d}\phi_{\mathrm{cyl}}$
$\displaystyle+\frac{k_{1}\rho_{\mathrm{cyl}}}{4\mathrm{i}}\int_{-\pi}^{\pi}M^{s}_{\phi}(\phi_{\mathrm{cyl}})\frac{\rho_{\mathrm{cyl}}-\rho^{(1)}_{\mathrm{aux}}\cos(\phi_{\mathrm{cyl}}-\phi_{\mathrm{aux}})}{D^{(1)}_{\mathrm{cyl,aux}}}H_{1}^{(2)}(k_{1}D^{(1)}_{\mathrm{cyl,aux}})\mathrm{d}\phi_{\mathrm{cyl}}=$
$\displaystyle\frac{k_{1}Z_{1}}{4}IH_{0}^{(2)}(k_{1}D^{(1)}_{\mathrm{fil,aux}}),$
(23)

where $D^{(1)}_{\mathrm{fil,aux}}$ is the distance between the filament $I$
and a point on $C^{(1)}_{\mathrm{aux}}$ with coordinates
$(\rho^{(1)}_{\mathrm{aux}},\phi_{\mathrm{aux}})$, i.e.,

$D^{(1)}_{\mathrm{fil,aux}}=\sqrt{\rho^{2}_{\mathrm{fil}}+(\rho^{(1)}_{\mathrm{aux}})^{2}-2\rho_{\mathrm{fil}}\rho^{(1)}_{\mathrm{aux}}\cos\phi^{(1)}_{\mathrm{aux}}},$
(24)

and $D^{(1)}_{\mathrm{cyl,aux}}$ is the distance from a point on $C$ to a
point on $C^{(1)}_{\mathrm{aux}}$

$D^{(1)}_{\mathrm{cyl,aux}}=\sqrt{\rho^{2}_{\mathrm{cyl}}+(\rho^{(1)}_{\mathrm{aux}})^{2}-2\rho_{\mathrm{cyl}}\rho^{(1)}_{\mathrm{aux}}\cos(\mathrm{\phi_{\mathrm{cyl}}}-\phi^{(1)}_{\mathrm{aux}})}.$
(25)

From the second equivalent problem, and for
$\rho^{(2)}_{\mathrm{aux}}>\rho_{\mathrm{cyl}}$, we obtain

$\displaystyle-\frac{k_{2}Z_{2}\rho_{\mathrm{cyl}}}{4}\int_{-\pi}^{\pi}J^{s}_{z}(\phi_{\mathrm{cyl}})H_{0}^{(2)}(k_{2}D^{(2)}_{\mathrm{cyl,aux}})\mathrm{d}\phi_{\mathrm{cyl}}$
$\displaystyle+\frac{k_{2}\rho_{\mathrm{cyl}}}{4\mathrm{i}}\int_{-\pi}^{\pi}M^{s}_{\phi}(\phi_{\mathrm{cyl}})\frac{\rho_{\mathrm{cyl}}-\rho^{(2)}_{\mathrm{aux}}\cos(\phi_{\mathrm{cyl}}-\phi^{(2)}_{\mathrm{aux}})}{D^{(2)}_{\mathrm{cyl,aux}}}H_{1}^{(2)}(k_{2}D^{(2)}_{\mathrm{cyl,aux}})\mathrm{d}\phi_{\mathrm{cyl}}=0,$
(26)

where

$D^{(2)}_{\mathrm{cyl,aux}}=\sqrt{\rho^{2}_{\mathrm{cyl}}+(\rho^{(2)}_{\mathrm{aux}})^{2}-2\rho_{\mathrm{cyl}}\rho^{(2)}_{\mathrm{aux}}\cos(\mathrm{\phi_{\mathrm{cyl}}}-\phi^{(2)}_{\mathrm{aux}})}.$
(27)

Together, (23) and (26) consist a $2\times 2$ system of Fredholm linear
integral equations, in which the involved kernels are even and
$2\pi$-periodic. Such systems can be solved explicitly by using Fourier series
(as described, e.g., in Appendix A of [19]). Let

$\displaystyle J^{s}_{z}(\phi_{\mathrm{cyl}})$
$\displaystyle=\sum_{n=-\infty}^{+\infty}I_{n}^{s}e^{\mathrm{i}n\phi_{\mathrm{cyl}}},$
(28) $\displaystyle M^{s}_{\phi}(\phi_{\mathrm{cyl}})$
$\displaystyle=\sum_{n=-\infty}^{+\infty}M_{n}^{s}e^{\mathrm{i}n\phi_{\mathrm{cyl}}},$
(29)

where $I_{n}^{s}$ and $M_{n}^{s}$ are the respective Fourier coefficients.
Then, we multiply Eqs. (23) and (26) with
$e^{\mathrm{i}n\phi^{(j)}_{\mathrm{aux}}}$, and integrate with respect to
$\phi^{(j)}_{\mathrm{aux}}$, with $j=1,2$. Next, we interchange the orders of
integration and invoke the addition theorems (77)-(79) to introduce the
Fourier coefficients

$\displaystyle
H_{n}^{(2)}(k_{1}\rho_{\mathrm{cyl}})J_{n}(k_{1}\rho^{(1)}_{\mathrm{aux}})e^{-\mathrm{i}n\phi_{\mathrm{cyl}}}=\frac{1}{2\pi}\int_{-\pi}^{\pi}H_{0}^{(2)}(k_{1}D^{(1)}_{\mathrm{cyl,aux}})e^{-\mathrm{i}n\phi^{(1)}_{\mathrm{aux}}}\mathrm{d}\phi^{(1)}_{\mathrm{aux}}$
(30) $\displaystyle
H_{n}^{(2)}(k_{2}\rho^{(2)}_{\mathrm{aux}})J_{n}(k_{2}\rho_{\mathrm{cyl}})e^{-\mathrm{i}n\phi_{\mathrm{cyl}}}=\frac{1}{2\pi}\int_{-\pi}^{\pi}H_{0}^{(2)}(k_{2}D^{(2)}_{\mathrm{cyl,aux}})e^{-\mathrm{i}n\phi^{(2)}_{\mathrm{aux}}}\mathrm{d}\phi^{(2)}_{\mathrm{aux}}$
(31) $\displaystyle
H_{n}^{{}^{\prime}(2)}(k_{1}\rho_{\mathrm{cyl}})J_{n}(k_{1}\rho^{(1)}_{\mathrm{aux}})e^{-\mathrm{i}n\phi_{\mathrm{cyl}}}=$
(32)
$\displaystyle-\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{\rho_{\mathrm{cyl}}-\rho^{(1)}_{\mathrm{aux}}\cos(\phi_{\mathrm{cyl}}-\phi^{(1)}_{\mathrm{aux}})}{D^{(1)}_{\mathrm{cyl,aux}}}H_{1}^{(2)}(k_{1}D^{(1)}_{\mathrm{cyl,aux}})e^{-\mathrm{i}n\phi^{(1)}_{\mathrm{aux}}}\mathrm{d}\phi^{(1)}_{\mathrm{aux}}$
(33) $\displaystyle
H_{n}^{(2)}(k_{2}\rho^{(2)}_{\mathrm{aux}})J^{\prime}_{n}(k_{2}\rho_{\mathrm{cyl}})e^{-\mathrm{i}n\phi_{\mathrm{cyl}}}=$
(34)
$\displaystyle-\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{\rho_{\mathrm{cyl}}-\rho^{(2)}_{\mathrm{aux}}\cos(\phi_{\mathrm{cyl}}-\phi^{(2)}_{\mathrm{aux}})}{D^{(2)}_{\mathrm{cyl,aux}}}H_{1}^{(2)}(k_{2}D^{(2)}_{\mathrm{cyl,aux}})e^{-\mathrm{i}n\phi^{(2)}_{\mathrm{aux}}}\mathrm{d}\phi^{(2)}_{\mathrm{aux}}.$
(35)

In this way, we obtain the algebraic system with respect to $I^{s}_{n}$ and
$M^{s}_{n}$

$\displaystyle
I^{s}_{n}H_{n}^{(2)}(k_{1}\rho_{\mathrm{cyl}})+\frac{1}{\mathrm{i}Z_{1}}M^{s}_{n}H_{n}^{{}^{\prime}(2)}(k_{1}\rho_{\mathrm{cyl}})=-\frac{I}{2\pi\rho_{\mathrm{cyl}}}H_{n}^{(2)}(k_{1}\rho_{\mathrm{fil}})$
(36) $\displaystyle
I^{s}_{n}J_{n}(k_{2}\rho_{\mathrm{cyl}})+\frac{1}{\mathrm{i}Z_{2}}M^{s}_{n}J^{\prime}_{n}(k_{2}\rho_{\mathrm{cyl}})=0$
(37)

Importantly, the radii of the auxiliary surfaces (i.e.,
$\rho_{\mathrm{aux}}^{(1)}$ and $\rho_{\mathrm{aux}}^{(2)}$) have cancelled
out and do not appear. Solving the system (36) and (37), yields, via (28) and
(29), the explicit expressions of the current densities

$\displaystyle
J^{s}_{z}(\phi_{\mathrm{cyl}})=-\frac{I}{2\pi\rho_{\mathrm{cyl}}}\times$
$\displaystyle\sum_{n=-\infty}^{+\infty}\frac{Z_{1}H_{n}^{(2)}(k_{1}\rho_{\mathrm{fil}})J^{\prime}_{n}(k_{2}\rho_{\mathrm{cyl}})}{Z_{1}H_{n}^{(2)}(k_{1}\rho_{\mathrm{cyl}})J^{\prime}_{n}(k_{2}\rho_{\mathrm{cyl}})-Z_{2}H_{n}^{{}^{\prime}(2)}(k_{1}\rho_{\mathrm{cyl}})J_{n}(k_{2}\rho_{\mathrm{cyl}})}e^{\mathrm{i}n\phi_{\mathrm{cyl}}}$
(38) $\displaystyle
M^{s}_{\phi}(\phi_{\mathrm{cyl}})=\frac{I}{2\pi\rho_{\mathrm{cyl}}}\times$
$\displaystyle\sum_{n=-\infty}^{+\infty}\frac{\mathrm{i}Z_{1}Z_{2}H_{n}^{(2)}(k_{1}\rho_{\mathrm{fil}})J_{n}(k_{2}\rho_{\mathrm{cyl}})}{Z_{1}H_{n}^{(2)}(k_{1}\rho_{\mathrm{cyl}})J^{\prime}_{n}(k_{2}\rho_{\mathrm{cyl}})-Z_{2}H_{n}^{{}^{\prime}(2)}(k_{1}\rho_{\mathrm{cyl}})J_{n}(k_{2}\rho_{\mathrm{cyl}})}e^{\mathrm{i}n\phi_{\mathrm{cyl}}}$
(39)

By the large-$n$ formulas of A, the $n$-th terms of the series in Eqs. (38)
and (39) behave, respectively, like

$\displaystyle\frac{Z_{1}}{Z_{1}+Z_{2}\frac{k_{2}}{k_{1}}}\frac{\rho_{\mathrm{cyl}}^{|n|}}{\rho_{\mathrm{fil}}^{|n|}}e^{\mathrm{i}n\phi_{\mathrm{cyl}}},$
(40)
$\displaystyle-\mathrm{i}\frac{Z_{1}Z_{2}}{\frac{Z_{1}}{k_{2}}-\frac{Z_{2}}{k_{1}}}\frac{\rho^{|n|+1}_{\mathrm{cyl}}}{|n|\rho^{|n|}_{\mathrm{fil}}}e^{\mathrm{i}n\phi_{\mathrm{cyl}}}.$
(41)

Thus, both series are convergent; this is true even if the auxiliary surfaces
are placed beyond the extended regions mentioned in the previous section. In
other words, the NFM integral equations—as opposed to those of MAS [24]—are
always solvable.

Now, it is true that substitution of the expressions (38) and (39) for the
surface current densities into the integrals (20) for the fields do yield the
exact solutions in (13). As this is not immediately apparent (due to the
complexity of the involved expressions), we provide a direct verification in
B.

By following the same steps for internal excitation
($\rho_{\mathrm{fil}}<\rho_{\mathrm{cyl}}$), we find

$\displaystyle
J_{z}^{s}(\phi_{\mathrm{cyl}})=-\frac{I}{2\pi\rho_{\mathrm{cyl}}}\times$
$\displaystyle\sum_{n=-\infty}^{+\infty}\frac{Z_{2}J_{n}(k_{2}\rho_{\mathrm{fil}})H^{{}^{\prime}(2)}_{n}(k_{1}\rho_{\mathrm{cyl}})}{Z_{2}J_{n}(k_{2}\rho_{\mathrm{cyl}})H_{n}^{{}^{\prime}(2)}(k_{1}\rho_{\mathrm{cyl}})-Z_{1}J^{\prime}_{n}(k_{2}\rho_{\mathrm{cyl}})H^{(2)}_{n}(k_{1}\rho_{\mathrm{cyl}})}e^{\mathrm{i}n\phi_{\mathrm{cyl}}}$
(42) $\displaystyle
M_{\phi}^{s}(\phi_{\mathrm{cyl}})=\frac{I}{2\pi\rho_{\mathrm{cyl}}}\times$
$\displaystyle\sum_{n=-\infty}^{+\infty}\frac{\mathrm{i}Z_{1}Z_{2}J_{n}(k_{2}\rho_{\mathrm{fil}})H^{(2)}_{n}(k_{1}\rho_{\mathrm{cyl}})}{Z_{2}J_{n}(k_{2}\rho_{\mathrm{cyl}})H^{{}^{\prime}(2)}_{n}(k_{1}\rho_{\mathrm{cyl}})-Z_{1}J^{\prime}_{n}(k_{2}\rho_{\mathrm{cyl}})H^{(2)}_{n}(k_{1}\rho_{\mathrm{cyl}})}e^{\mathrm{i}n\phi_{\mathrm{cyl}}}.$
(43)

The conclusions are similar. The series are always (irrespective of the
positions of the auxiliary surfaces) convergent; and the current densities
represented by these series always lead to the correct fields given in Section
3.

##  5 Discrete Version of NFM-SEP

###  5.1 The Linear System and its Explicit Solution

In the previous section, we considered the continuous version of NFM-SEP. In
actual practice, we apply the discretized version described in Section 2.2.
For the circular problem, we now formulate this system and obtain its closed-
form solution.

Figure 4: Discretization of (a) the external and (b) the internal NFM
equivalent problem for the circular dielectric cylinder.

The continuous surface current densities of the previous section are replaced
by discrete sources, as shown in Fig. 4. Specifically, for external
$\rm{TM}_{z}$ incidence, the electric current density
$J_{z}^{s}(\phi_{\mathrm{cyl}})$ is replaced by $N$ $z$-directed electric
current filaments with amplitudes $I_{l}$, and the magnetic current density
$M_{\phi}^{s}(\phi_{\mathrm{cyl}})$ by $N$ $\phi$-directed magnetic current
filaments with amplitudes $K_{l}$ ($l=0,1,\ldots,N-1$). The amplitudes $I_{l}$
and $K_{l}$ are determined by satisfying the boundary conditions at $N$
collocation points on the polar angles $2p\pi/N$ of $C^{(1)}_{\mathrm{aux}}$
and $C^{(2)}_{\mathrm{aux}}$. For $p=0,1,\ldots,N-1$, this procedure leads to

$\displaystyle
Z_{1}\sum_{l=0}^{N-1}I_{l}H_{0}^{(2)}(k_{1}b^{(1)}_{p-l})+\mathrm{i}\sum_{l=0}^{N-1}K_{l}\frac{\rho_{\mathrm{cyl}}-\rho^{(1)}_{\mathrm{aux}}\cos(\frac{2\pi(p-l)}{N})}{b^{(1)}_{p-l}}H_{1}^{(2)}(k_{1}b^{(1)}_{p-l})$
$\displaystyle=-IZ_{1}H_{0}^{(2)}(k_{1}d^{(1)}_{p}),$ (44) $\displaystyle
Z_{2}\sum_{l=0}^{N-1}I_{l}H_{0}^{(2)}(k_{2}b^{(2)}_{p-l})+\mathrm{i}\sum_{l=0}^{N-1}K_{l}\frac{\rho_{\mathrm{cyl}}-\rho^{(2)}_{\mathrm{aux}}\cos(\frac{2\pi(p-l)}{N})}{b^{(2)}_{p-l}}H_{1}^{(2)}(k_{2}b^{(2)}_{p-l})=0,$
(45)

with

$b^{(j)}_{p-l}=\sqrt{(\rho_{\mathrm{cyl}})^{2}+(\rho^{(j)}_{\mathrm{aux}})^{2}-2\rho_{\mathrm{cyl}}\rho^{(j)}_{\mathrm{aux}}\cos\left(\frac{2\pi(p-l)}{N}\right)},\quad
j=1,2$ (46)

the distance between the $p$-th collocation point on $C^{(j)}_{\mathrm{aux}}$
and the $l$-th source point on $C$, and

$d^{(1)}_{p}=\sqrt{(\rho^{(1)}_{\mathrm{aux}})^{2}+(\rho_{\mathrm{fil}})^{2}-2\rho^{(1)}_{\mathrm{aux}}\rho_{\mathrm{fil}}\cos\left(\frac{2\pi
p}{N}\right)}$ (47)

the distance from the excitation source to the $p$-th collocation point on
$C^{(1)}_{\mathrm{aux}}$.

Equations (44) and (45)—which are discretized versions of Eqs. (23) and
(26)—are an $2N\times 2N$ linear system of equations with unknowns the complex
amplitudes $I_{l}$ and $K_{l}$. In block form, the system is

$\begin{bmatrix}\mathbf{Z}_{11}&\mathbf{Z}_{12}\\\
\mathbf{Z}_{21}&\mathbf{Z}_{22}\end{bmatrix}\begin{bmatrix}\mathbf{I}\\\
\mathbf{K}\end{bmatrix}=\begin{bmatrix}\mathbf{V}\\\ \mathbf{0}\end{bmatrix},$
(48)

where $\mathbf{I}$ and $\mathbf{K}$ are $N$-dimensional vectors with elements
$I_{l}$ and $K_{l}$, respectively, while the $N$-dimensional vector
$\mathbf{V}$ incorporates the contributions from the excitation source
according to the right hand side of Eq. (44). Importantly, each block
$\mathbf{Z}_{11}$, $\mathbf{Z}_{12}$, $\mathbf{Z}_{21}$ and $\mathbf{Z}_{22}$
is a circulant matrix, namely, a square matrix in which all rows are composed
of the same elements and each row is rotated one element to the right relative
to the preceding row [37, 38]. Hence the matrix in (48) is a $2\times 2$ block
matrix composed of circulant blocks of order $N$. Block matrices with
circulant blocks were analyzed in [39], where efficient inversion algorithms
were established.

We solve (48) by utilizing the discrete Fourier transform (DFT) (similar to
the procedure in [24]). We consider the following DFT pairs of $I_{l}$ and
$K_{l}$,

$I_{l}=\sum_{m=0}^{N-1}I^{(m)}e^{\mathrm{i}2\pi
lm/N},\,\,K_{l}=\sum_{m=0}^{N-1}K^{(m)}e^{\mathrm{i}2\pi
lm/N},~{}l=0,1,\ldots,N-1$ (49)
$I^{(m)}=\frac{1}{N}\sum_{p=0}^{N-1}I_{p}e^{-\mathrm{i}2\pi
mp/N},\,\,K^{(m)}=\frac{1}{N}\sum_{p=0}^{N-1}K_{p}e^{-\mathrm{i}2\pi
mp/N},~{}m=0,1,\ldots,N-1.$ (50)

The DFT of any convolution sum $\sum_{l=0}^{N-1}\alpha_{p-l}\beta_{l}$ with
$\alpha_{l}=\alpha_{-l}=\alpha_{N-l}$ equals $N\alpha^{(m)}\beta^{(m)}$. Thus,
application of the DFT to (44) and (45) gives the following $2\times 2$ system

$\displaystyle
B^{(m)}_{1}I^{(m)}+\frac{\mathrm{i}}{Z_{1}}B_{2}^{(m)}K^{(m)}=\frac{I}{N}D^{(m)}$
(51) $\displaystyle
B_{3}^{(m)}I^{(m)}+\frac{\mathrm{i}}{Z_{2}}B_{4}^{(m)}K^{(m)}=0,$ (52)

for all $m=0,1,\ldots,N-1$, with

$\displaystyle B_{1}^{(m)}$
$\displaystyle=\frac{1}{N}\sum_{p=0}^{N-1}H_{0}^{(2)}(k_{1}b^{(1)}_{p})e^{-\mathrm{i}2\pi
mp/N},$ (53) $\displaystyle B_{2}^{(m)}$
$\displaystyle=\frac{1}{N}\sum_{p=0}^{N-1}\frac{\rho_{\mathrm{cyl}}-\rho^{(1)}_{\mathrm{aux}}\cos(\frac{2\pi
p}{N})}{b^{(1)}_{p}}H_{1}^{(2)}(k_{1}b^{(1)}_{p})e^{-\mathrm{i}2\pi mp/N},$
(54) $\displaystyle B_{3}^{(m)}$
$\displaystyle=\frac{1}{N}\sum_{p=0}^{N-1}H_{0}^{(2)}(k_{2}b^{(2)}_{p})e^{-\mathrm{i}2\pi
mp/N},$ (55) $\displaystyle B_{4}^{(m)}$
$\displaystyle=\frac{1}{N}\sum_{p=0}^{N-1}\frac{\rho_{\mathrm{cyl}}-\rho^{(2)}_{\mathrm{aux}}\cos(\frac{2\pi
p}{N})}{{b^{(2)}_{p}}}H_{1}^{(2)}(k_{2}b^{(2)}_{p})e^{-\mathrm{i}2\pi mp/N},$
(56) $\displaystyle D^{(m)}$
$\displaystyle=-\frac{1}{N}\sum_{p=0}^{N-1}H_{0}^{(2)}(k_{1}d_{p}^{(1)})e^{-\mathrm{i}2\pi
mp/N}.$ (57)

Then, by solving (51) and (52), we obtain

$\displaystyle I^{(m)}$
$\displaystyle=\frac{I}{N}\frac{Z_{1}D^{(m)}B_{4}^{(m)}}{Z_{1}B_{1}^{(m)}B_{4}^{(m)}-Z_{2}B_{2}^{(m)}B_{3}^{(m)}}$
(58) $\displaystyle K^{(m)}$
$\displaystyle=\frac{I}{N}\frac{\mathrm{i}Z_{2}Z_{1}D^{(m)}B_{3}^{(m)}}{Z_{1}B_{1}^{(m)}B_{4}^{(m)}-Z_{2}B_{2}^{(m)}B_{3}^{(m)}},$
(59)

which, when combined with (49), provide the explicit, closed-form solution of
the system (44) and (45), for any finite $N$. We now study the large-$N$
behavior of this solution.

###  5.2 Discrete Currents and Fields they Generate: Large-$N$ behavior

We specifically ask whether the two limits

$J^{s}_{\rm{limit}}(\phi_{\mathrm{cyl}})=\lim_{N\to\infty}\frac{NI_{l}}{2\pi\rho_{\mathrm{cyl}}}\,\,\,\mbox{and}\,\,\,M^{s}_{\rm{limit}}(\phi_{\mathrm{cyl}})=\lim_{N\to\infty}\frac{NK_{l}}{2\pi\rho_{\mathrm{cyl}}}$
(60)

exist; and, if they do exist, whether they coincide with the surface current
densities (38) and (39). The question thus posed is a natural one because [24]
shows that, in the case of MAS, the analogous limits do not always exist.

We proceed as follows: Replace the Hankel functions in (53)–(57) by the
expressions given in (77)-(79); interchange the order of summation in the
resulting expressions; and then use the identity

$\sum_{p=0}^{N-1}e^{\mathrm{i}2\pi
p(n-m)/N}=\begin{cases}N,~{}~{}\rm{if~{}n-m=multiple~{}of~{}N}\\\
0,~{}~{}\rm{otherwise}\end{cases}$ (61)

This procedure results in the exact expressions (for $m=0,\ldots,N-1$)

$\displaystyle D^{(m)}$
$\displaystyle=-\sum_{q=-\infty}^{+\infty}J_{qN+m}(k_{1}\rho^{(1)}_{\mathrm{aux}})H^{(2)}_{qN+m}(k_{1}\rho_{\mathrm{fil}}),$
(62) $\displaystyle B^{(m)}_{1}$
$\displaystyle=\sum_{q=-\infty}^{+\infty}J_{qN+m}(k_{1}\rho^{(1)}_{\mathrm{aux}})H^{(2)}_{qN+m}(k_{1}\rho_{\mathrm{cyl}}),$
(63) $\displaystyle B^{(m)}_{2}$
$\displaystyle=-\sum_{q=-\infty}^{+\infty}J_{qN+m}(k_{1}\rho^{(1)}_{\mathrm{aux}})H^{\prime(2)}_{qN+m}(k_{1}\rho_{\mathrm{cyl}}),$
(64) $\displaystyle B^{(m)}_{3}$
$\displaystyle=\sum_{q=-\infty}^{+\infty}J_{qN+m}(k_{2}\rho_{\mathrm{cyl}})H^{(2)}_{qN+m}(k_{2}\rho^{(2)}_{\mathrm{aux}}),$
(65) $\displaystyle B^{(m)}_{4}$
$\displaystyle=-\sum_{q=-\infty}^{+\infty}J^{\prime}_{qN+m}(k_{2}\rho_{\mathrm{cyl}})H^{(2)}_{qN+m}(k_{2}\rho^{(2)}_{\mathrm{aux}}).$
(66)

Now, assume for simplicity that $N$ is odd, so that (49) is written as

$\displaystyle I_{l}$
$\displaystyle=I^{(0)}+\sum_{m=1}^{(N-1)/2}I^{(m)}\cos\left(\frac{2\pi
lm}{N}\right),~{}l=0,\ldots,N-1,$ (67) $\displaystyle K_{l}$
$\displaystyle=K^{(0)}+\sum_{m=1}^{(N-1)/2}K^{(m)}\cos\left(\frac{2\pi
lm}{N}\right),~{}l=0,\ldots,N-1.$ (68)

Then, for large $N$ and $0\leq m\leq(N-1)/2$, by (73)-(76), it is seen that
the $q=0$ terms in (62)-(66) are dominant. Thus, as $N\to\infty$, we get

$\displaystyle
I^{(m)}\sim-\frac{I}{N}\frac{Z_{1}H_{m}^{(2)}(k_{1}\rho_{\mathrm{fil}})J^{\prime}_{m}(k_{2}\rho_{\mathrm{cyl}})}{Z_{1}H_{m}^{(2)}(k_{1}\rho_{\mathrm{cyl}})J^{\prime}_{m}(k_{2}\rho_{\mathrm{cyl}})-Z_{2}H_{m}^{{}^{\prime}(2)}(k_{1}\rho_{\mathrm{cyl}})J_{m}(k_{2}\rho_{\mathrm{cyl}})},$
(69) $\displaystyle
K^{(m)}\sim\frac{I}{N}\frac{\mathrm{i}Z_{1}Z_{2}H_{m}^{(2)}(k_{1}\rho_{\mathrm{fil}})J_{m}(k_{2}\rho_{\mathrm{cyl}})}{Z_{1}H_{m}^{(2)}(k_{1}\rho_{\mathrm{cyl}})J^{\prime}_{m}(k_{2}\rho_{\mathrm{cyl}})-Z_{2}H_{m}^{{}^{\prime}(2)}(k_{1}\rho_{\mathrm{cyl}})J_{m}(k_{2}\rho_{\mathrm{cyl}})},$
(70)

in which $\rho_{\mathrm{aux}}^{(1)}$ and $\rho_{\mathrm{aux}}^{(2)}$ have
cancelled out. Upon substituting these expressions into (67) and (68) and
setting $2\pi l/N=\phi_{\mathrm{cyl}}$, we see that the limits defined in (60)
are finite and given by the series

$\displaystyle
J^{s}_{\rm{limit}}(\phi_{\mathrm{cyl}})=-\frac{I}{2\pi\rho_{\mathrm{cyl}}}\times$
$\displaystyle\biggl{[}\frac{Z_{1}H_{0}^{(2)}(k_{1}\rho_{\mathrm{fil}})J^{\prime}_{0}(k_{2}\rho_{\mathrm{cyl}})}{Z_{1}H_{0}^{(2)}(k_{1}\rho_{\mathrm{cyl}})J^{\prime}_{0}(k_{2}\rho_{\mathrm{cyl}})-Z_{2}H_{0}^{{}^{\prime}(2)}(k_{1}\rho_{\mathrm{cyl}})J_{0}(k_{2}\rho_{\mathrm{cyl}})}+$
$\displaystyle
2\sum_{m=1}^{+\infty}\frac{Z_{1}H_{m}^{(2)}(k_{1}\rho_{\mathrm{fil}})J^{\prime}_{m}(k_{2}\rho_{\mathrm{cyl}})}{Z_{1}H_{m}^{(2)}(k_{1}\rho_{\mathrm{cyl}})J^{\prime}_{m}(k_{2}\rho_{\mathrm{cyl}})-Z_{2}H_{m}^{{}^{\prime}(2)}(k_{1}\rho_{\mathrm{cyl}})J_{m}(k_{2}\rho_{\mathrm{cyl}})}\cos(m\phi_{\mathrm{cyl}})\biggr{]},$
(71) $\displaystyle
M^{s}_{\rm{limit}}(\phi_{\mathrm{cyl}})=\frac{I}{2\pi\rho_{\mathrm{cyl}}}\times$
$\displaystyle\biggl{[}\frac{\mathrm{i}Z_{1}Z_{2}H_{0}^{(2)}(k_{1}\rho_{\mathrm{fil}})J_{0}(k_{2}\rho_{\mathrm{cyl}})}{Z_{1}H_{0}^{(2)}(k_{1}\rho_{\mathrm{cyl}})J^{\prime}_{0}(k_{2}\rho_{\mathrm{cyl}})-Z_{2}H_{0}^{{}^{\prime}(2)}(k_{1}\rho_{\mathrm{cyl}})J_{0}(k_{2}\rho_{\mathrm{cyl}})}+$
$\displaystyle
2\sum_{m=1}^{+\infty}\frac{\mathrm{i}Z_{1}Z_{2}H_{m}^{(2)}(k_{1}\rho_{\mathrm{fil}})J_{m}(k_{2}\rho_{\mathrm{cyl}})}{Z_{1}H_{m}^{(2)}(k_{1}\rho_{\mathrm{cyl}})J^{\prime}_{m}(k_{2}\rho_{\mathrm{cyl}})-Z_{2}H_{m}^{\prime(2)}(k_{1}\rho_{\mathrm{cyl}})J_{m}(k_{2}\rho_{\mathrm{cyl}})}\cos(m\phi_{\mathrm{cyl}})\biggr{]},$
(72)

which are Fourier series whose coefficients are even in $m$. Therefore,
$J^{s}_{\rm{limit}}(\phi_{\mathrm{cyl}})$ and
$M^{s}_{\rm{limit}}(\phi_{\mathrm{cyl}})$ are equal to the
$J_{z}^{s}(\phi_{\mathrm{cyl}})$ and $M_{\phi}^{s}(\phi_{\mathrm{cyl}})$ given
by (38) and (39), respectively. Hence, it follows that the large-$N$ limits of
the fields generated by $I_{l}$ and $K_{l}$ exist and equal the fields
generated by $J_{z}^{s}(\phi_{\mathrm{cyl}})$ and
$M_{\phi}^{s}(\phi_{\mathrm{cyl}})$. Recall that these were shown in B to be
the true fields.

Repeating the calculations for internal $\rm{TM_{z}}$ excitation, we reached
the same conclusions.

##  6 NFM-SEP vs. MAS

We summarize our hitherto results by comparing to corresponding ones which can
be found in [24] and which pertain to MAS. Numerical methods are often
compared between themselves, but the comparison we now perform (between NFM-
SEP and MAS) is more appropriate than usual; this is so because the matrices
of the two methods are, apart from trivial differences 111The trivial
differences follow by comparing (44) and (45) to (11) of [24]: We obtain the
MAS matrix as follows. Multiply the top two blocks (i.e., the first $N$ rows)
of our NFM-SEP matrix by $-k_{1}/4$; multiply the bottom two blocks by
$k_{2}/4$; block-transpose. The multiplications of the blocks of the NFM-SEP
matrix by the above coefficients correspond to respective multiplications on
both sides of (44) and (45)., the same. Only the right-hand-side vectors
present nontrivial differences. Importantly, we note that this coincidence of
the NFM-SEP and MAS matrices continues to hold also for non-circular shapes.

  * 1.

Continuous integral equations: The NFM-SEP continuous integral equations are
always solvable; but those of MAS are nonsolvable if the interior surface has
radius smaller than a certain threshold value, or if the exterior surface has
radius larger than an other threshold value; all thresholds depend on the type
of excitation (external or internal) and are given in Table 2.

  * 2.

Solution to $2N\times 2N$ system: In the case of NFM-SEP, the aforesaid
solution (discrete currents), when appropriately normalized, always converge
as $N\to\infty$. By contrast, the interior (exterior) MAS currents diverge if
the interior (exterior) surface is too small (too large). Since the $2N\times
2N$ system can be viewed as a discretization of the system of continuous
integral equations, the divergence of the normalized solution to the former
system is a simple corollary of the nonsolvability of the latter system.

  * 3.

Oscillations: In MAS, the aforementioned divergence of the normalized currents
manifests itself as rapid, unnatural, exponentially large oscillations in the
corresponding MAS currents [24]. On the other hand, NFM-SEP does not give rise
to such oscillations.

  * 4.

Field convergence: In both NFM-SEP and MAS, the fields generated by the
discrete currents always converge to the true field (regardless of the
positions of the auxiliary surfaces and of the occurrence/nonoccurrence of
oscillations).

  * 5.

Numerical issues: Field convergence is a theoretical result, which might not
hold true when the hardware and/or software are imperfect, which is always the
case in practice. This is why the lack of oscillations in NFM-SEP is an
advantage of NFM-SEP (when compared to MAS), in accordance with the principle
that algorithms involving large intermediate results are generally unstable
[23, 40]

  * 6.

Singularities of analytic continuation: The extended convergence regions we
showed in Table 1 are in fact regions into which the physical fields can be
analytically continued. As discussed in [24], the singularities of these
analytic continuations play an important role to the convergence/divergence of
MAS. As shown conclusively in the present paper, the aforementioned
singularities are not relevant to the convergence/divergence of NFM-SEP. This
finding is a generalization of a similar finding for perfectly conducting
(PEC) scatterers [31]. In fact, for the PEC case the issue had caused some
controversy: see [31, 21, 23, 41] and the relevant references therein.

##  7 Numerical Results

In this section, we show representative numerical results illustrating the
main conclusions of the paper, both for circular as well as for elliptical
cylinders. Throughout, the relative permittivity of the $R_{2}$ medium is
taken as $\varepsilon_{r2}=4.2$. As in the theoretical part, we consider only
$\rm{TM_{z}}$ excitation (both internal and external); for $\rm{TE_{z}}$
excitation the conclusions were similar.

###  7.1 Circular Geometry

Figures 5 and 6 concern the circular problem with $N=40$,
$k_{1}\rho_{\mathrm{cyl}}=2$, $k_{1}\rho_{\mathrm{fil}}=4$ for external
excitation, and $k_{1}\rho_{\mathrm{fil}}=1$ for internal excitation. Thus,
$k_{1}\rho_{\rm cri}=1$ and $k_{1}\rho_{\rm cri}=4$ for external and internal
excitation, respectively. Figures 5(a)-(d) compare the continuous surface
current densities $J^{s}_{z}(\phi_{\mathrm{cyl}})$ and
$M^{s}_{\phi}(\phi_{\mathrm{cyl}})$ of Section 4 with the normalized currents
$NI_{l}/(2\pi\rho_{\rm cyl})$ and $NK_{l}/(2\pi\rho_{\rm cyl})$ of Section 5,
for $k_{1}\rho^{(1)}_{\mathrm{aux}}=1.5$ and
$k_{1}\rho^{(2)}_{\mathrm{aux}}=2.5$. Perfect coincidence is observed. These
plots remain unaltered, when we change the radii of the auxiliary surfaces to
$k_{1}\rho^{(1)}_{\mathrm{aux}}=0.5$ and $k_{1}\rho^{(2)}_{\mathrm{aux}}=10$.
This verifies numerically that the NFM-SEP currents always converge to the
expected values. In stark contrast, the real parts of the corresponding MAS
currents on $C^{(1)}_{\mathrm{aux}}$ and $C^{(2)}_{\mathrm{aux}}$ (denoted,
respectively, as $NI^{(1)}_{l}/2\pi\rho_{\mathrm{cyl}}$ and
$NI^{(2)}_{l}/2\pi\rho_{\mathrm{cyl}}$) oscillate rapidly, as shown in Figs. 5
(e)-(f). It is worth stressing that these oscillations are hardware/software
_independent_ [19, 23]. The resulting NFM-SEP fields are compared to the exact
fields in Figs. 6(a)-(d). Once again the results are coincident; they remain
the same if we change the positions of $C^{(1)}_{\mathrm{aux}}$ and
$C^{(2)}_{\mathrm{aux}}$ to $k_{1}\rho^{(1)}_{\mathrm{aux}}=0.5$ and
$k_{1}\rho^{(2)}_{\mathrm{aux}}=10$. The MAS fields, despite the oscillations
shown in Figs. 5(e)-(f), also coincide for $N=40$ with the exact fields; this
phenomenon is explained in detail in [24].

Figure 5: Real (red solid lines) and imaginary parts (blue dashed lines) of
(a) continuous electric current density $J_{z}^{s}(\phi_{\mathrm{cyl}})$ and
(b) continuous magnetic current density $M_{\phi}^{s}(\phi_{\mathrm{cyl}})$
together with real and imaginary parts (circles) of normalized NFM currents
$NI_{l}/2\pi\rho_{\mathrm{cyl}}$ and $NK_{l}/2\pi\rho_{\mathrm{cyl}}$ for
external $\rm{TM_{z}}$ excitation. (c) and (d) are as (a) and (b) but for
internal $\rm{TM_{z}}$ excitation. For (a)-(d):
$k_{1}\rho^{(1)}_{\mathrm{aux}}=1.5$, $k_{1}\rho^{(2)}_{\mathrm{aux}}=2.5$ (or
$k_{1}\rho^{(1)}_{\mathrm{aux}}=0.5$, $k_{1}\rho^{(2)}_{\mathrm{aux}}=10$) and
$N=40$. (e) and (f) show the oscillations of the MAS currents of
$C^{(1)}_{\mathrm{aux}}$ for internal and $C^{(2)}_{\mathrm{aux}}$ for
external excitation with $k_{1}\rho^{(1)}_{\mathrm{aux}}=0.5$,
$k_{1}\rho^{(2)}_{\mathrm{aux}}=10$ and $N=40$.

Figure 6: Real (red solid lines) and imaginary parts (blue dashed lines) of
the exact electric fields (a) $E_{z}^{(1)}(\phi_{\mathrm{obs}})$ and (b)
$E_{z}^{(2)}(\phi_{\mathrm{obs}})$ together with real and imaginary parts of
(a) $E_{z,\mathrm{NFM}}^{(1)}(\phi_{\mathrm{obs}})$ and (b)
$E_{z,\mathrm{NFM}}^{(2)}(\phi_{\mathrm{obs}})$ (circles) for external
$\rm{TM_{z}}$ excitation with $N=40$, $k_{1}\rho^{(1)}_{\mathrm{aux}}=0.5$,
$k_{1}\rho^{(2)}_{\mathrm{aux}}=10$, $k_{1}\rho^{(1)}_{\mathrm{obs}}=10$, and
$k_{1}\rho^{(2)}_{\mathrm{obs}}=1$. (c) and (d) are as (a) and (b) but for
internal $\rm{TM_{z}}$ excitation. (e) and (f) illustrate the negative effect
of oscillations (for $N=46$) of the MAS currents of $C^{(1)}_{\mathrm{aux}}$
on the computations of the real and imaginary parts of $E_{z}^{(1)}$ for
external excitation.

If $N$ is increased to $N=46$, the NFM-SEP fields remain correct. For the MAS
solution, on the other hand, this small increase in $N$ causes a large
increase in the amplitude of the oscillating currents $I^{(2)}_{l}$, resulting
in corrupted field calculations; see Figs. 6(e)-(f). We point out that such
corrupted results are hardware/software _dependent_. This illustrates the
numerical issues discussed in Section 6, and showcases the advantage of NFM-
SEP over MAS.

###  7.2 Elliptical Geometry

The circular geometry is simple enough to draw analytical and explicit
conclusions, which we illustrated numerically in the previous section. To
demonstrate that those conclusions carry over to noncircular problems, we
consider an elliptical boundary (Fig. 7) with large semi axis $k_{1}a=2$ and
small semi axis $k_{1}b=1.6$ (i.e., eccentricity $e=0.6$). The excitation
source lies on the positive semi-major axis at $k_{1}d=4$ for external
excitation, and $k_{1}d=1$ for internal excitation. The auxiliary surfaces
$C^{(1)}_{\mathrm{aux}}$ and $C^{(2)}_{\mathrm{aux}}$ are scaled versions of
$C$, with scaling factors $\sigma^{(1)}_{\mathrm{aux}}$ and
$\sigma^{(2)}_{\mathrm{aux}}$, respectively. Results are shown in Figs. 8 and
9 (Fig. 9 now compares to FEM simulations rather than to exact results). It is
apparent (compare to Figs. 5 and 6 for the circle) that all results concerning
convergence/divergence and existence/nonexistence of oscillations remain
unchanged. Contrast, in particular, the correct NFM-SEP fields of Figs. 9
(e)-(f) to the corrupted MAS fields in the same figure; this illustrates the
negative effect of MAS oscillations (even if these oscillations differ in
appearance from those of the circle).

Figure 7: Discretization of the (a) external and (b) internal NFM equivalent
problem for the elliptical dielectric cylinder. Black dots represent the
collocated electric and magnetic discrete sources, and black squares represent
the collocation points.

Figure 8: Real (solid lines) and imaginary parts (dashed lines) of the NFM (a)
electric current amplitudes $I_{l}$ and (b) magnetic current amplitudes
$K_{l}$ for external $\rm{TM_{z}}$ excitation of the elliptical cylinder for
$N=40$, $\sigma^{(1)}_{\mathrm{aux}}=0.33$ and
$\sigma^{(2)}_{\mathrm{aux}}=5$. (c) and (d) are as (a) and (b) but for
internal $\rm{TM_{z}}$ excitation. (e) and (f) illustrate, respectively, the
oscillations of the MAS currents of $C^{(1)}_{\mathrm{aux}}$ for external and
$C^{(2)}_{\mathrm{aux}}$ for internal excitation.

Figure 9: Real (red solid lines) and imaginary parts (blue dashed lines) of
(a) $E_{z,\mathrm{NFM}}^{(1)}(\phi_{\mathrm{obs}})$ and (b)
$E_{z,\mathrm{NFM}}^{(2)}(\phi_{\mathrm{obs}})$ for $N=40$,
$\sigma^{(1)}_{\mathrm{aux}}=0.33$, $\sigma^{(2)}_{\mathrm{aux}}=5$,
$\sigma^{(1)}_{\mathrm{obs}}=5$, and $\sigma^{(2)}_{\mathrm{obs}}=0.5$ along
with the corresponding FEM simulations (noted with circles) for external
$\rm{TM_{z}}$ excitation. (c) and (d) are as (a) and (b) but for internal
excitation. (e) and (f) illustrate the negative effect of oscillations of the
$C^{(1)}_{\mathrm{aux}}$ MAS currents on the computations of the $E^{(1)}_{z}$
electric field for $N=44$ and external excitation.

###  7.3 NFM-SEP vs. MAS: Speed of convergence

In view of the comparisons in Sections 6, 7.1, 7.2, we can ask whether MAS
presents any fundamental advantages over NFM-SEP. Since the matrix condition
numbers of the two methods are identical, the obvious quantity to compare is
the speed of convergence. Fig. 10 shows results from both methods, for $N=10$,
together with the exact solution. In this case, NFM-SEP appears slightly
better (increasing $N$ to 11 gives MAS results roughly as good as the shown
NFM-SEP results). More generally (and as far as the speed of convergence is
concerned) any differences we observed between the two methods were slight
(and in some cases, MAS was slightly better than NFM-SEP). A more systematic
study of the convergence speed requires finer measures of comparisons and is
beyond the scope of this paper.

Figure 10: Real (a) and imaginary (b) parts of $E^{(2)}_{z}$ for
$\mathrm{TM}_{z}$ external excitation, as computed from the exact solution
(14), NFM for $N=10$, and MAS for $N=10$, with
$k_{1}\rho^{(1)}_{\mathrm{aux}}=1$, and $k_{1}\rho^{(2)}_{\mathrm{aux}}=4$.

##  8 Conclusions

A numerical approach based on the combination of the NFM with the SEP was
investigated analytically. The combined NFM-SEP was applied for the case of a
cylindrical dielectric scatterer excited by an external or internal line
source. It was shown that the normalized discrete NFM-SEP currents always
converge to the respective continuous surface current densities. This comes in
stark contrast to previous findings on the MAS, in which the involved discrete
currents diverge and oscillate under certain conditions. To this end, we
provided a comparative detailed discussion of NFM-SEP with MAS to highlight
many different characteristics of the two methods despite the fact that both
methods have the same matrix. The analysis was supplemented by representative
numerical results for circular and noncircular shapes verifying the
theoretical predictions. An interesting future work direction is to use the
NFM-SEP for computing the eigenvalues of dielectric waveguides.

## CRediT authorship contribution statement

Minas Kouroublakis: Conceptualization, Methodology, Software, Validation,
Formal Analysis, Investigation, Visualization, Writing - Original Draft,
Writing - Review & Editing.
Nikolaos L. Tsitsas: Conceptualization, Methodology, Investigation, Writing -
Review & Editing.
George Fikioris: Conceptualization, Methodology, Investigation, Writing -
Review & Editing.

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Data availability

No data was used for the research described in the article.

##  Appendix A Useful Bessel-function formulas

This appendix lists certain well-known formulas involving the cylindrical
Hankel and Bessel functions [42]. Firstly, for
$\nu\to+\infty,\,z\in\mathbb{C},\,z\neq 0$, we have the following large-order
asymptotic approximations:

$J_{\nu}(z)\sim\frac{1}{\sqrt{2\pi\nu}}\left(\frac{ez}{2\nu}\right)^{\nu},$
(73)
$H^{(2)}_{\nu}(z)\sim+\mathrm{i}\sqrt{\frac{2}{\pi\nu}}\left(\frac{ez}{2\nu}\right)^{-\nu},$
(74)
$J^{\prime}_{\nu}(z)\sim\sqrt{\frac{\nu}{2\pi}}\left(\frac{ez}{2\nu}\right)^{\nu}\frac{1}{z},$
(75)
$H^{\prime(2)}_{\nu}(z)\sim-\mathrm{i}\sqrt{\frac{2\nu}{\pi}}\left(\frac{ez}{2\nu}\right)^{-\nu}\frac{1}{z}.$
(76)

Secondly, the addition theorem for the Hankel function $H_{0}$ is,

$H_{0}^{(2)}(\sqrt{x_{1}^{2}+x_{2}^{2}-2x_{1}x_{2}\cos\theta})=\sum_{n=-\infty}^{+\infty}J_{n}(\min\left\\{x_{1},x_{2}\right\\})H_{n}^{(2)}(\max\left\\{x_{1},x_{2}\right\\})e^{\mathrm{i}n\theta}$
(77)

where $x_{1}>0$, $x_{2}>0$, and $\theta$ is real. For $x_{2}>x_{1}$, we obtain
the following expressions for the derivatives of (77) with respect to $x_{1}$
and $x_{2}$, respectively,

$\displaystyle\frac{x_{1}-x_{2}\cos\theta}{\sqrt{x_{1}^{2}+x_{2}^{2}-2x_{1}x_{2}\cos\theta}}H_{1}^{(2)}(\sqrt{x_{1}^{2}+x_{2}^{2}-2x_{1}x_{2}\cos\theta})=$
$\displaystyle-\sum_{n=-\infty}^{+\infty}J^{\prime}_{n}(x_{1})H_{n}^{(2)}(x_{2})e^{\mathrm{i}n\theta},$
(78)
$\displaystyle\frac{x_{2}-x_{1}\cos\theta}{\sqrt{x_{1}^{2}+x_{2}^{2}-2x_{1}x_{2}\cos\theta}}H_{1}^{(2)}(\sqrt{x_{1}^{2}+x_{2}^{2}-2x_{1}x_{2}\cos\theta})=$
$\displaystyle-\sum_{n=-\infty}^{+\infty}J_{n}(x_{1})H^{\prime(2)}_{n}(x_{2})e^{\mathrm{i}n\theta}.$
(79)

Finally, we note the following Wronskian identity,

$J_{n}(z)H_{n}^{{}^{\prime}(2)}(z)-J^{\prime}_{n}(z)H_{n}^{(2)}(z)=\frac{2}{\mathrm{i}\pi
z}.$ (80)

##  Appendix B Verification of (38) and (39)

In this appendix, we verify that the surface current densities in (38) and
(39) yield the correct fields in (13) and (14). For the field in $R_{1}$, use
the addition theorems (77)-(79) to rewrite (20) as follows:

$E_{z}^{(1)}(\rho_{\mathrm{obs}},\phi_{\mathrm{obs}})=-\frac{k_{1}Z_{1}}{4}IH_{0}^{(2)}(k_{1}D_{\mathrm{fil,obs}})\\\
-\frac{k_{1}Z_{1}\rho_{\mathrm{cyl}}}{4}\sum_{n=-\infty}^{+\infty}J_{n}(k_{1}\rho_{\mathrm{cyl}})H_{n}^{(2)}(k_{1}\rho_{\mathrm{obs}})e^{\mathrm{i}n\phi_{\mathrm{obs}}}\int_{-\pi}^{\pi}J^{s}_{z}(\phi_{\mathrm{cyl}})e^{-\mathrm{i}n\phi_{\mathrm{cyl}}}\mathrm{d}\phi_{\mathrm{cyl}}\\\
-\frac{k_{1}\rho_{\mathrm{cyl}}}{4\mathrm{i}}\sum_{n=-\infty}^{+\infty}J^{\prime}_{n}(k_{1}\rho_{\mathrm{cyl}})H_{n}^{(2)}(k_{1}\rho_{\mathrm{obs}})e^{\mathrm{i}n\phi_{\mathrm{obs}}}\int_{-\pi}^{\pi}M^{s}_{\phi}(\phi_{\mathrm{cyl}})e^{-\mathrm{i}n\phi_{\mathrm{cyl}}}\mathrm{d}\phi_{\mathrm{cyl}}$
(81)

The integrals are $2\pi$ times the $n$-th Fourier series coefficient
$I^{s}_{n}$ (of $J_{z}^{s}$) and $M^{s}_{n}$ (of $M_{\phi}^{s}$). Thus, by
(38) and (39), we have

$E_{z}^{(1)}(\rho_{\mathrm{obs}},\phi_{\mathrm{obs}})=-\frac{k_{1}Z_{1}}{4}IH_{0}^{(2)}(k_{1}D_{\mathrm{fil,obs}})\\\
-\frac{k_{1}Z_{1}I}{4}\sum_{n=-\infty}^{+\infty}J_{n}(k_{1}\rho_{\mathrm{cyl}})H_{n}^{(2)}(k_{1}\rho_{\mathrm{obs}})\times\\\
\frac{Z_{1}H_{n}^{(2)}(k_{1}\rho_{\mathrm{fil}})J^{\prime}_{n}(k_{2}\rho_{\mathrm{cyl}})}{Z_{1}H_{n}^{(2)}(k_{1}\rho_{\mathrm{cyl}})J^{\prime}_{n}(k_{2}\rho_{\mathrm{cyl}})-Z_{2}H_{n}^{{}^{\prime}(2)}(k_{1}\rho_{\mathrm{cyl}})J_{n}(k_{2}\rho_{\mathrm{cyl}})}e^{\mathrm{i}n\phi_{\mathrm{obs}}}\\\
-\frac{k_{1}I}{4\mathrm{i}}\sum_{n=-\infty}^{+\infty}J^{\prime}_{n}(k_{1}\rho_{\mathrm{cyl}})H_{n}^{(2)}(k_{1}\rho_{\mathrm{obs}})\times\\\
\frac{\mathrm{i}Z_{1}Z_{2}H_{n}^{(2)}(k_{1}\rho_{\mathrm{fil}})J_{n}(k_{2}\rho_{\mathrm{cyl}})}{Z_{1}H_{n}^{(2)}(k_{1}\rho_{\mathrm{cyl}})J^{\prime}_{n}(k_{2}\rho_{\mathrm{cyl}})-Z_{2}H_{n}^{{}^{\prime}(2)}(k_{1}\rho_{\mathrm{cyl}})J_{n}(k_{2}\rho_{\mathrm{cyl}})}e^{\mathrm{i}n\phi_{\mathrm{obs}}},$
(82)

which after some simple manipulations yields the exact solution (13).
Application of the same procedure to (22) yields

$E_{z}^{(2)}(\rho_{\mathrm{obs}},\phi_{\mathrm{obs}})=-\frac{k_{2}Z_{2}I}{4}\sum_{n=-\infty}^{+\infty}J_{n}(k_{2}\rho_{\mathrm{obs}})H_{n}^{(2)}(k_{2}\rho_{\mathrm{cyl}})\times\\\
\frac{Z_{1}H_{n}^{(2)}(k_{1}\rho_{\mathrm{fil}})J^{\prime}_{n}(k_{2}\rho_{\mathrm{cyl}})}{Z_{1}H_{n}^{(2)}(k_{1}\rho_{\mathrm{cyl}})J^{\prime}_{n}(k_{2}\rho_{\mathrm{cyl}})-Z_{2}H_{n}^{{}^{\prime}(2)}(k_{1}\rho_{\mathrm{cyl}})J_{n}(k_{2}\rho_{\mathrm{cyl}})}e^{\mathrm{i}n\phi_{\mathrm{obs}}}\\\
-\frac{k_{2}I}{4\mathrm{i}}\sum_{n=-\infty}^{+\infty}J_{n}(k_{2}\rho_{\mathrm{obs}})H_{n}^{{}^{\prime}(2)}(k_{2}\rho_{\mathrm{cyl}})\times\\\
\frac{\mathrm{i}Z_{1}Z_{2}H_{n}^{(2)}(k_{1}\rho_{\mathrm{fil}})J_{n}(k_{2}\rho_{\mathrm{cyl}})}{Z_{1}H_{n}^{(2)}(k_{1}\rho_{\mathrm{cyl}})J^{\prime}_{n}(k_{2}\rho_{\mathrm{cyl}})-Z_{2}H_{n}^{{}^{\prime}(2)}(k_{1}\rho_{\mathrm{cyl}})J_{n}(k_{2}\rho_{\mathrm{cyl}})}e^{\mathrm{i}n\phi_{\mathrm{obs}}},$
(83)

which after some manipulations and use of the identity (80), yields (14).

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