1. Introduction
The theory of characteristic modes (TCM) based on the electric field integral equation (EFIE) for a perfectly electric conductor (PEC) was proposed by Harrington and Mautz in 1971 [
1]. The resulting characteristic modes (CMs) are not dependent on the excitation source but on the structure and material of the object. Due to this superior property, it has been employed as a great potential analysis tool for antenna design [
2], such as in the reduction of the cross-sections of radar antenna [
3], the improvement of multiple-input-multiple-output antenna [
4], etc.
Subsequently, TCM was generalized to material objects by employing a volume integral equation (VIE) [
5] and a surface integral equation (SIE) [
6]. Unfortunately, the SIE-based TCM was reported to generate spurious modes [
7]. In order to remove these spurious modes, there are some valuable formulations [
8,
9,
10,
11,
12,
13], including building a dependent relationship between equivalent electric and magnetic currents [
8,
9,
10,
11,
12,
13,
14] and making the right-hand side of the generalized eigenvalue equation (GEE) related to only radiated power [
11,
12,
13]. Thus far, the spurious modes have almost been suppressed.
However, few researchers have discussed why the spurious modes appear when using SIE-based TCM, especially the traditional TCM based on the symmetric PMCHWT by Chang and Harrington [
6] (CH-sPMCHWT-TCM). Miers and Lau [
15] think that internal resonance may partly explain this phenomenon because sPMCHWT could not remove internal resonance. However, the spurious modes still exist when sPMCHWT is changed to asymmetric PMCHWT (CH-PMCHWT-TCM). Thus, the internal resonance should not be responsible for the spurious modes [
16]. As a remedy, some researchers [
8,
10,
16] found that the spurious (non-physical) modes do not obey the dependent relationship of equivalent electric and magnetic currents, while the physical modes do. Namely, the dependent relationship can be taken as a post-process technology to separate the physical and non-physical modes. Bernabeu-Jiménez et al. [
17] pointed out that the spurious modes can be excited by an internal source for the infinite lossless dielectric cylinder, which means that the spurious modes may exist in reality and are renamed as non-radiation modes. However, the spurious modes of lossless material objects are found to be coincident with the CMs of the reverse problem, where the material parameters of the object and the background are swapped [
18].
The CMs for material objects are independent of the excitation source, so they are expected to be applicable to analyses of both scattering and radiation problems. It is noted that existing SIE-based formulations are essentially established in the scattering system framework. Therefore, the formulations are suitable for scattering problems indisputably. It is uncertain whether the resulting CMs include those of radiation problems, especially when the sources are inside the objects. Recently, our previous work [
19] revealed that EFIE-based TCM [
1] is not appropriate for wave-port-fed transmitting PEC structures.
There are some existing works that apply TCM to deal with real antenna systems;, mainly by using a mixed potential integral equation with the spatial domain Green’s functions of multilayer mediums [
20,
21] or volume and surface integral equations [
22,
23], other than the forementioned SIE-based TCM. It is worth mentioning that the radiation problem we will consider is that the source is inside the material object.
In this paper, we will discuss four physical models with reverse media and with the excitation source in either domain in order to demonstrate the reason why CH-PMCHWT-TCM is prone to generate spurious modes. We find that the impedance operator of the PMCHWT equation mixes the two different media without specified excitation and is also the same whether the excitation is inside or outside. For this reason, we define the CMs of the four cases by using specified boundary conditions.
2. Traditional Theory for Four Physical Problems
It is known that the PMCHWT equation solves a mixed problem with two different media and gives a unique solution for a specified excitation. However, for a characteristic mode problem that is independent of the excitation, if improperly defined, the generalized eigenvalue equation (GEE) may not be able to distinguish (i) the two domains if their media are reversed and (ii) the excitation from inside or outside. Let us start with the four cases, as shown in
Figure 1. Cases (A) and (B) are scattering problems to be excited from the outside, but the media are reversed. Cases (C) and (D) are radiation problems to be excited from the inside.
For the scattering problem (A), in terms of the extinction theorem, the surface equivalent currents
would generate
in
V1 if
V1 were filled with the same medium as in
V2, while
would generate zero fields in
V2 if
V2 were filled with the same medium as in
V1, i.e.,
where
, and the operators are defined as
When the observation points approach the surface
S from the inside and outside, respectively, we would recognize that
and
, with
being the principal-value part of
. After the discretization of (1)–(4) by the Galerkin method with the RWG basis functions
and singularity treatment [
24], we can obtain
where the matrix elements are
By denoting
we may rewrite (7)–(8) compactly as
with
Adding (15) to (14), one obtains the classical PMCHWT equation
where
. Make
, with
in which the superscript “
H” denotes the Hermitian transpose. As one of the traditional ways, the GEE is defined by
, i.e.,
which is the so-called CH-PMCHWT-CM [
6,
16]. Unfortunately, this formulation would generate many spurious modes, as reported in [
7,
8,
9,
10,
11,
12].
To identify the reason, we may check the solutions with the boundary condition (15) and find that these spurious modes do not observe it, as reported in [
16]. This means that these modes are not physical solutions to scattering problem (A). Then, what boundary condition do these spurious modes satisfy? There exist problems (B), (C) and (D), as displayed in
Figure 1, all of which can obtain a GEE similar to (19) if following the conventional way, as discussed below.
For the scattering problem (B) in
Figure 1, the corresponding equations to (14)–(15) are
Adding these two equations, one obtains the same PMCHWT equation as (17), and then the same GEE as (19), but the boundary condition (BC) needing to meet is (21), not (15).
For the radiation problem (C) in
Figure 1, by using the same procedure as described for problem (A), we can obtain the corresponding equations to (14)–(15) as
Again, the addition of these two equations gives the same PMCHWT equation as (17) and the same GEE as (19), but the BC needing to meet is (23).
Finally, for radiation problem (D), the corresponding equations to (14)–(15) are
It is apparent that the PMCHWT and GEE equations for this problem are also (17) and (19), respectively; however, the BC needing to be met becomes (25).
In summary, the PMCHWT Equation (17) and the GEE (19) cannot distinguish the four cases in
Figure 1. One must use one of the boundary conditions (BCs) of (15), (21), (23), or (25) to select suitable modes.
Referring to (8), the solution of (15) is
with
Similarly, solutions of (21), (23), and (25) can also been obtained, respectively, as follows:
There are two schemes to judge which one of the four problems the modes
solved from GEE (19) satisfy. One way is to submit the found characteristic modes
into (15), (21), (23), or (25) and then see whether the corresponding error of matrix-vector multiplication
is sufficiently small. The error is defined as
Another way is to calculate the correlation coefficient of the known and new characteristic electric and magnetic currents as in [
16], where the known characteristic modal electric and magnetic currents
are constructed by
, and the new modal electric
are obtained by submitting the known characteristic magnetic currents
into (26), (28), (29) or (30), i.e.,
. Analogously, the new magnetic currents
(
) are obtained from the known
. For brevity, the resulting new modal currents computed from (26), (28)–(30) are denoted as
with symbol ‘K’ = ’A’, ’B’, ’C’ or ’D’, respectively. For instance, if the
n-th mode is the solution of problem (A),
should be the same as
. A similar scheme applies to other problems. In order to judge the equality, we can define the correlation coefficients as
One mode is identified to satisfy one of the four problems if both of its correlation coefficients and are close to 1 for ‘K’ = ’A’, ’B’, ’C’ and ’D’.
One example is given to observe which one of the four cases the modes solved from GEE (19) satisfy. A cubic dielectric resonator (DR) (a = b = c = 25.4 mm) in [
15] is considered since it has been frequently applied as a verified example [
11,
12,
13,
14]. Its constitutive parameters are
and
, and the background is the air, i.e.,
and
. The cubic DR is meshed as 528 triangular patches.
The real part of characteristic eigenvalues solved from GEE (19) is displayed in
Figure 2, with a frequency range from 2.0 GHz to 4.5 GHz. Good agreements can be observed with the results in [
18]. The solid red lines represent the so-called physical characteristic modes of scattering problem (A), and the black lines denote the spurious modes of scattering problem A.
In order to understand the underlying physical meaning of these spurious modes, we apply the BCs of the four problems (A), (B), (C), and (D). The errors and correlation coefficients of the first 70 lower-order modes at a frequency point of 2.0 GHz are demonstrated in
Figure 3. It is noted that the evaluated standard is less than 1 × 10
−6 for the error scheme, while the standard is bigger than 0.95 for the correlation coefficient scheme. It is obviously observed that some modes of the first 70 modes satisfy the BC of scattering problem (A), but some modes satisfy the BC of scattering problem (B). For instance, the correlation coefficients
of modes 1–11 are close to zero, and its error is more than 1 × 10
−6, which means these modes truly do not satisfy the BC of problem (A). Thus, these modes are not the solutions to problem (A) and are indeed spurious modes. However, their correlation coefficients
are bigger than 0.95, and the corresponding error is less than 1 × 10
−6. This implies that these spurious modes may be solutions to problem (B). On the contrary, modes 12–14 are solutions to problem (A) due to
,
and
. The further comparison of visualized electric and magnetic currents can lead to the same conclusion as discussed below.
Herein, we can select any one physical mode and spurious mode from
Figure 2 and
Figure 3 to demonstrate their current distributions. For convenience, modes 1 and 13 are considered, where mode 1 is the first spurious mode in
Figure 2 and mode 13 is the first physical mode in
Figure 2 for scattering problem (A). The corresponding characteristic electric and magnetic currents are displayed in
Figure 4 and
Figure 5, compared with the new ones from the boundary conditions (BCs) of the four cases. Symbols J and M represent known electric and magnetic currents obtained from Equation (19), while symbols J’(M) and M’(J) represent new currents by submitting M and J into one BC of the four problems. The prefixes A, B, C and D represent the BCs of problems (A), (B), (C), and (D). As previously mentioned, original currents should be matched with the resulting new currents if this mode satisfies a relevant BC. In
Figure 4, it is apparently observed that the original currents (J, M) of mode 1 agree well with the new currents (B-J’(M), B-M’(J) from the BC of problem (B), which demonstrates mode 1 obeys the BC of problem (B). Similarly, mode 13 obeys the BC of problem (A), as displayed in
Figure 5.
Furthermore, the modal powers of the first 70 lower-order modes are demonstrated in the bottom sub-figure of
Figure 3. Here,
means the radiated power of the
n-th mode for problem A or dissipated power for problem B, and
indicates the dissipated power of the
n-th mode of problem (A) or radiated power of problem (B). We can find that the physical modes are modes with a unitary radiated power of problem (A), and the spurious modes are modes with zero radiated power of problem (A). Thus, these spurious modes are non-radiation modes for problem (A). However, the normalized radiated power of these modes are equal to one for problem (B).
The discussions above explain from the perspective of boundary conditions why the spurious modes are indeed solutions to problem (B), as found in [
18], for lossless materials. However, the interesting phenomenon is that none of these modes satisfies the BCs of radiation problems (C) or (D). The characteristic modes are often considered to be source-free since GEE (19) does not contain an excitation source. Does this mean that the characteristic modes are appropriate for both scattering problems and radiation problems? Obviously, problems (A) and (C) are, respectively, the scattering and radiation problems. However, no one mode is found to satisfy the BC of problem (C). The characteristic modes of scattering problem (A) are not suitable for radiation problem (C). In other words, scattering systems and radiation systems are distinct. In a word, CH-PMCHWT-TCM indeed does not identify the boundary or object media for an external source scattering problem and also does not extract characteristic modes of an internal source radiation problem. The proper definitions are needed for the four cases.
3. Definition of Radiation Characteristic Modes
It is clear that the problems (A), (B), (C), and (D) are different with specific boundary conditions (15), (21), (23), and (25), although with the same PMCHWT Equation (17). Therefore, we understand that (i) the scattering problem (A) is to solve (17) subject to (15), (ii) the scattering problem (B) is to solve (17) subject to (21), (iii) the radiation problem (C) is to solve (17) subject to (23), and (iv) the radiation problem (D) is to solve (17) subject to (25).
In addition, our previous work [
11,
14] has addressed scattering problem (A) and proposed adequate formulations. The boundary condition is truly one of the key points. Another key point is the proper definition of the GEE, i.e., the right-hand side of the GEE should be related to radiation power [
11,
12,
13]. These two points are taken into account in the following. Actually, we are chiefly concerned about scattering problem (A) and radiation problem (C) in engineering applications. Thus, we first derive the characteristic mode formulation of radiation problem (C) and then extend it to other problems.
For radiation problem (C), we substitute the solution of (17) to (23). Applying (29) to (17) and multiplying both sides by
, we obtain
in which
and
is an identity matrix. It should be noted that (34) is the total power of the system.
In this paper, we adopt the GEE form as in [
11,
12,
13], where the right-hand side of the equations should be related to radiated power in order to obtain the maximal radiation characteristic. Therefore, the GEE is defined by
with radiated power operators
which are proved in
Appendix A.
Following the same process, GEEs for case (D) in
Figure 1 can be given as
with the following operators:
Alternatively, due to by (23), which would give and , we may replace with for problem (C). In the same way, we may replace with for problem (D).
Similarly, GEEs for scattering problems (A) and (B) are shown as follows:
in which
,
,
and
can be obtained by replacing
in
,
,
and
with
. Similarly,
,
,
and
can be obtained by replacing
in
,
,
and
with
. Due to
, which would give
and
, we may replace
with
for problem (A). Likewise, we may replace
with
for problem (B). Specifically, we may replace
with
for problem (A) and with
for problem (B) when the material is lossless.