A thorough look at the nonuniqueness of the electromagnetic scattering integral equation solutions as compared to the scalar acoustic ones

Authors


Abstract

[1] It is well-known that solutions of electromagnetic scattering integral equations of the first or second kind (EFIE and MFIE) for perfectly electric or perfectly magnetic conducting scatterers are nonunique for those frequencies which correspond to interior Maxwell resonances of the scatterer; hence, the null spaces of the respective interior problem operators are under concern. In principle, all mathematical facts and proofs regarding this problem and cited in this paper are available from the book by Colton and Kress [1983], yet, these authors mainly concentrate on single and double layer potentials for the scalar acoustic (Dirichlet and Neumann) as well as the magnetic dipole layer ansatz for the perfectly electric conducting (Maxwell) problem and treat the Huygens-type representation, which is more common in the electrical engineering community, not in the same detail. This might be the reason that part of the electrical engineering literature suffers from some confusion regarding the proper null spaces and their physical relevance, in particular, if the electromagnetic problem is considered in 2-D, where it reduces to scalar TM/TE-problems. The present contribution comments on these issues emphasizing that the null spaces of 2-D electromagnetics are the nonphysical null spaces originating from the Huygens-type representation of scalar acoustics.

1. Governing Equations of Acoustics and Electromagnetics

[2] The governing equations for (linear) acoustics

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and electromagnetics (Maxwell's equations)

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essentially differ with regard to the structure of the spatial derivative in both systems of equations: The two Maxwell equations both exhibit the curl and the acoustic governing equations exhibit the gradient (of the pressure p(R, t), R being the vector of position and t time) and the divergence (of the particle velocity v(R, t)); ρ is the mass density and κ is the compressibility of a homogeneous isotropic instantaneously reacting medium [de Hoop, 1995], which replaces vacuum in electromagnetics. The acoustic sources are force densities f(R, t) and dilatation rates h(R, t), and the electromagnetic sources are electric and magnetic current densities Je(R, t) and Jm(R, t), respectively. This difference of the spatial acoustic field behavior gives rise to two mathematically and physically different canonical boundary value problems, the Dirichlet problem (p(R′, t) = nc · t(R′, t) = 0, R′ ∈ Sc, Sc surface of a scatterer of volume Vc with outward normal nc at the point R′, t being surface tractions) and the Neumann problem (nc · v(R′, t) = −g(R′, t) = 0, R′ ∈ Sc, g being surface dilatation rates), whereas the electromagnetic boundary value problem of perfect electric conductivity (nc × E(R′, t) = −Km(R′, t) = 0; Km denotes magnetic surface current density) and perfect magnetic conductivity (nc × H(R′, t) = Ke(R′, t) = 0; Ke denotes electric surface current density) are mathematically dual to each other. Since two-dimensional electromagnetics reduces to two independent scalar TM- and TE-problems, which are formally identical to two-dimensional acoustics, it is obviously of interest to investigate how this splitting affects the splitting of null spaces of the pertinent integral equations.

2. Integral Equations of the First and Second Kind for Scalar Acoustic Scattering

2.1. Representation Theorem

[3] Outside a scatterer with closed surface Sc and outward normal nc a solution of the time harmonic (e−jωt) system of acoustic equations is given in terms of the incident field ϕi(R, ω) (ω is the circular frequency) — we use the letter ϕ instead of the pressure p because in 2-D ϕ can equally stand for electromagnetic field components — and the scattered field ϕs(R, ω):

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this is the Huygens-type Helmholtz representation theorem expressing the physical intuition, that physical secondary sources on Sc radiate the scattered field. Applying (5) to an interior point of Vc, the volume enclosed by Sc, yields ϕs = −ϕi, the so-called extinction theorem.

[4] To obtain the physical secondary sources, integral equations are usually formulated. A proper transition of R from outside Sc to the surface Sc itself yields integral equations of the first and second kind for the respective secondary sources of the exterior Dirichlet- (upper index D(e)) and the exterior Neumann-problem (upper index N(e)):

equation image
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where equation image denotes the identity operator, and the operators equation image, equation image, equation image, equation image are defined as follows:

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G is the scalar three-dimensional free space Green function. Note equation image and equation image are adjoint to each other, whereas equation image and equation image are self-adjoint.

[5] In two spatial dimensions Maxwell equations decouple into scalar TM- and TE-problems, which associate the physical secondary sources in the representation theorem with physical axial and circumferential electric surface current densities if the scatterer is a perfect electric conductor. In that case the TM-problem is a Dirichlet-problem and the TE-problem is a Neumann-problem (Figure 1).

Figure 1.

Integral equations of the first and second kind for the exterior acoustic and electromagnetic scattering problems; rep: representation-type formulation; dlp: double-layer potential; sdl: single-layer potential; mdl: magnetic dipole-layer; unit-vector es = ez × nc with unit-vector ez along the 2-D axis.

2.2. Single- and Double-Layer Potentials

[6] For the Dirichlet boundary condition the representation theorem (5) can be thought of as a single-layer potential representation [Kellogg, 1929], whereas the Neumann boundary condition yields a double-layer potential representation. Alternatively, because each term in the Helmholtz-integral is by itself a solution of the homogeneous acoustic equations, a double layer potential ansatz

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as well as a single-layer potential ansatz

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can be postulated for the Dirichlet- and the Neumann-problem. As a consequence, the exterior scattering problem yields integral equations of the second kind

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which are adjoint to (8), (9). This will be of considerable importance regarding the proper physical interpretation of respective null spaces of the operators.

[7] For 2-D electromagnetics the above (TM-)Dirichlet double-layer as well as the (TE-)Neumann single-layer secondary sources can be interpreted as magnetic surface current densities if the scatterer is perfectly electric conducting. Note: Mathematically, magnetic current densities can be equivalent to a perfect electric conductor once they satisfy proper integral equations. Of course, this does not comply with physical intuition, hence, we refer to these secondary sources as mathematical secondary sources.

2.3. Nonuniqueness of Integral Equation Solutions

2.3.1. Null Spaces of the Integral Equations for the Interior Resonance Problem

[8] All the above inhomogeneous integral equations do have nonunique solutions if the corresponding homogeneous equations have nontrivial solutions. But homogeneous equations account for the interior resonance problem, hence, the corresponding integral equations for the interior problem and their pertinent null spaces should be related to the integral equations for the exterior scattering problem.

[9] From the proper formulation of the representation theorem for the interior Dirichlet and Neumann resonance problems (a single-layer potential for the Dirichlet- and a double-layer potential for the Neumann-problem), the integral equations of the first kind

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can be deduced for the physically equivalent sources (upper index D(i), N(i)). It is obvious that the operators agree with the respective exterior operators, hence, if there exist nontrivial null spaces N(equation image) = V, N(equation image) = U for the interior problems, they exist for the exterior problems as well. As a matter of fact, V is nontrivial for all ω = ωjD, j = 1, 2, 3, …, for which the homogeneous Helmholtz-equation has nonzero solutions for RVc satisfying a Dirichlet boundary condition on Sc; in 2-D electromagnetic terminology: V contains axial electric surface current densities on the perfectly electric conducting surface at the pertinent resonant frequencies. Interesting enough: These current densities do not contribute to the exterior Helmholtz-representation of the scattered field, hence, they do not affect the unique solvability of the exterior Dirichlet-problem (this is not recognized by Cho [1990]). As for N(equation image) = U, this null space is nontrivial for the Neumann resonance frequencies ω = ωjN, j = 1, 2, 3, …, and, in 2-D electromagnetics, its elements correspond to circumferential electric current densities on the perfectly electric conducting surface Sc. These current densities to neither radiate into the exterior of the scatterer.

[10] So far with the integral equations of the first kind. To investigate the nonuniqueness of the solutions of the integral equations of the second kind, we must once more formulate them for the (homogeneous) interior resonance problem. Based on the representation theorem we find

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and based on the potential layers we find

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for the Dirichlet double-layer and the Neumann single-layer secondary sources.

[11] Now we use physical intuition instead of a mathematical proof (which, as already mentioned, can be found in [Colton and Kress, 1983]) to postulate null spaces

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for the representation theorem integral equations (20), (21): Interior resonances and their respective secondary sources are deduced from the homogeneous Helmholtz-equation, which has the physical Huygens-type Helmholtz-integral (5) as a solution defining physically relevant secondary sources for the respective boundary conditions. These can either be found from the integral equations of the first kind (18), (19) or from the integral equations of the second kind (20), (21), whence nontrivial null spaces (22), (23) for ω = ωjD,N must exist for the integral equations of the second kind.

2.3.2. Null Spaces of the Integral Equations for the Exterior Scattering Problem

[12] We have already mentioned that solutions of the integral equations of the first kind for the exterior scattering problem may be contaminated by elements of the null spaces V and U of the interior resonance problem. If integral equations of the second kind are considered, these null spaces appear as well for the interior resonance problem, leaving the question of null spaces of the integral equations (8), (9) for the exterior scattering problem involving the operators equation image and equation image still open. At that point we have to recognize that the operator equation image for the exterior Dirichlet scattering problem is adjoint to the operator equation image for the interior Neumann resonance problem, and a similar “cross-wise” adjoint relationship is observed for the exterior Neumann scattering and the interior Dirichlet resonance problem. Consulting Fredholm's alternative we must conclude, that null spaces U′ and V′ exist for the representation-type second kind integral equations for the exterior Dirichlet- and Neumann-problem

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which, for ω = ωjN,D, have the same dimension as the null spaces U and V for the respective interior problem second kind integral equations, i.e., U and U′ have the same dimension for ω = ωjN, and V and V′ have the same dimension for ω = ωjD. For all other frequencies all null spaces are trivial.

[13] Obviously, it is of interest now to investigate the physical relevance of the null spaces V′ and U′, and, in particular to answer that question, a look at the potential-layer-type integral equations of the second kind is mandatory: We find that the respective interior resonance operator equation image for the Dirichlet-problem is adjoint to the interior resonance representation-type operator equation image for the Dirichlet-problem, and a similar finding holds for the Neumann-problems. So, if V is nontrivial, V′ is nontrivial having the same dimension, and, if U is nontrivial, U′ is nontrivial having the same dimension. The elements of V′ therefore comprise mathematical secondary sources on a Dirichlet boundary related to a double-layer potential, and the elements of U′ comprise mathematical secondary sources on a Neumann boundary related to a single-layer potential, and, as we already pointed out for 2-D electromagnetics, these are “mathematically adjoint secondary sources” with no physical relevance [Cho, 1990]. Hence, the solutions of the representation-type second kind integral equations for the exterior Dirichlet and Neumann scattering problems may be contaminated by elements of the nonphysical null spaces U′ and V′ (and these contaminate the Helmholtz-representation of the scattered field as well, because they produce a nonzero field in the exterior when inserted into this representation).

[14] Since the exterior potential-layer-type operators of the second kind are adjoint to the exterior representation-type operators of the second kind — Dirichlet adjoint to Dirichlet, Neumann adjoint to Neumann — we find, on behalf of Fredholm's alternative, U and V as their pertinent null spaces, i.e., the elements of neither U nor V do contribute to the exterior scattered field, making the field representation unique. But: Also due to Fredholm's alternative, we have to recognize a solvability condition for the corresponding integral equations: The incident field must be orthogonal to the mathematical secondary sources of the interior resonance problem, i.e., the interior Neumann mathematical secondary sources as elements of U′ for the exterior Dirichlet problem, and the interior Dirichlet mathematical secondary sources as elements of V′ for the exterior Neumann problem. This seems to contradict the unconstrained unique solvability of the exterior scattering problems, but a reformulation of integral equations expressing a balanced contribution of physical and mathematical secondary sources fixes this contradiction. It should be mentioned that solvability conditions of the above type for the exterior representation-type integral equations of the second kind can be shown to hold for any incident field, and this is possibly the reason, why they are not explicitly considered in the electric engineering literature [Cho, 1990].

3. Integral Equations of the First and Second Kind for Electromagnetic Scattering

3.1. Representation Theorem

[15] The electromagnetic counterpart to the scalar representation theorem (5) turns out to be (we use the Franz-Larmor-representation in terms of Hertz- and Fitzgerald-vectors):

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The electromagnetic Huygens-type physical secondary sources appear as electric and magnetic surface current densities. Again, when applied to interior points of the scattering volume, (28) and (29) realize the extinction theorem. By the way, (29) results from (28) through application of the duality transform.

[16] Defining the operators

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with nc · Ψ = 0 yields the following integral equations of the first and second kind (EFIE and MFIE) for the exterior scattering problem for the case of perfect electric conductivity of Sc:

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where equation image is again the identity operator.

[17] The corresponding integral equations for the case of perfect magnetic conductivity of Sc

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are dual to (32), (33), hence, they do not define a new mathematical problem as it is true for acoustics. As mentioned already, the reason for that is the appearance of the curl equally in Faraday's as well as in Ampère-Maxwell's law.

[18] Interesting enough, (33) can be alternatively written in terms of the (tangential) magnetic field strength H in nc × H = Ke, whence the operator equation image as the adjoint operator to equation image appears:

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I is the unit tensor of rank two, and ncnc is a dyadic product.

3.2. Magnetic Dipole-Layer Ansatz

[19] Alternatively to (28), (29) the magnetic dipole-layer ansatz

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with nc · ΨD = 0 also solves the Maxwell exterior scattering problem for perfectly electric conducting scatterers, resulting in the integral equation of the second kind

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for mathematical secondary sources ΨD, namely magnetic current densities (magnetic dipole layers) on a perfectly electric conducting object.

[20] Note: The integral equation of the first kind resulting from an electric dipole-layer ansatz for the perfect electric conductor is identical to (28), and, for reasons mentioned above (duality), the electric dipole-layer ansatz for the case of perfect magnetic conductivity will not be considered.

3.3. Nonuniqueness of Integral Equation Solutions

3.3.1. Null Spaces of the Integral Equations for the Maxwell Interior Resonance Problem

[21] Again, to investigate the nonuniqueness of the integral equation solutions for the exterior Maxwell scattering problem we have to formulate the (homogeneous) integral equations for the pertinent interior Maxwell resonance problem. From the Franz Larmor representation (28), (29) we find

equation image
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as integral equations of the first kind and

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as integral equations of the second kind. As anticipated, neither the first kind nor the second kind integral equations differ for the perfectly electric and the perfectly magnetic conducting surface. Hence, in either case, the pertinent null spaces equation image contain the physical Maxwell resonance surface current densities for the Maxwell resonance frequencies ω = ωjM, j = 1, 2, 3, …. Note: Unlike acoustic scattering, electromagnetic scattering does not have to distinguish between two (unprimed) null spaces! As a consequence of the existence of the null space M, the exterior representation-type integral equations of the first kind are not uniquely solvable for ω = ωjM, yet the Franz Larmor field representations are unique because the (physical) resonant currents do not radiate into the exterior.

[22] As an alternative to (42), from the magnetic dipole-layer ansatz (37), (38) for the perfectly electric conducting surface we find the homogeneous integral equation of the first kind:

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Introducing ΨtD = nc × ΨD this integral equation can be reformulated in terms of the adjoint operator equation image:

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just as it was the case for (36).

3.3.2. Null Spaces of the Integral Equations for the Maxwell Exterior Scattering Problem

[23] The same words as in section 2.3.2 apply: We have already mentioned that solutions of the integral equations of the first kind for the exterior Maxwell scattering problem may be contaminated by elements of the null space M of the interior Maxwell resonance problem. If integral equations of the second kind are considered, this null space appears as well for the interior Maxwell resonance problem, leaving the question of the null space of the integral equations (33), (35) for the exterior Maxwell scattering problem involving the operator equation image still open. Obviously, this operator equation image is not adjoint to equation image, but, fortunately, we can rely on the version (36) of the exterior representation-type integral equation of the second kind, which is adjoint to the interior equation. Hence, Fredholm's alternative assures us that equation image has a null space M′, which is trivial for ω ≠ ωjM, and which has the same dimension as M for ω = ωjM. Again, as for all primed null spaces, M′ contains mathematical secondary sources: magnetic surface current densities (magnetic dipoles) on a perfectly electric conducting surface Sc; this is obvious from the integral equation (45) arising from the magnetic dipole-layer ansatz for the interior resonance problem. As a consequence, the Franz Larmor representation of the exterior scattered field is not unique for ω = ωjM, because, as in the acoustic case, the “primed mathematical secondary sources” radiate into the exterior. To make things complete, the integral equation of the second kind for the magnetic dipole-layer exterior scattering problem has once more the (physical) null space M, making the magnetic dipole-layer exterior field ansatz unique because the elements of M do not radiate when inserted into the Km-integral of (28) (the exterior electromagnetic dual problem has the same Franz Larmor field representation); unfortunately, for ω = ωjM, a solvability condition must hold, but as for acoustics it can be overcome by a “balanced play” of electric and magnetic surface current densities on a perfectly electric conducting surface.

[24] The basic facts regarding integral equations of the first and second kind for exterior acoustic as well as electromagnetic scattering are summarized in Figure 1.

4. Conclusion

[25] As already mentioned, in two spatial dimensions electromagnetic scattering reduces to two scalar problems, a TM-Dirichlet-problem and a TE-Neumann-problem for perfectly electric conducting surfaces, and, via duality, a TM-Neumann-problem and a TE-Dirichlet-problem for perfectly magnetic conducting surfaces (Figure 1). Since we find the primed null space M′ for the electromagnetic exterior problem second kind integral equations as derived from the representation theorem, we anticipate to find primed null spaces for the resulting 2-D integral equations as well, and, by comparison with the 2-D/3-D acoustic case, this is indeed confirmed (Figure 1). Therefore, as a conclusion, we state the existence of mathematical (nonphysical) secondary sources for the 2-D electromagnetic exterior scattering problems, which contaminate the solutions of integral equations of the second kind at the pertinent interior resonance frequencies; we do not find the physical secondary sources as elements of the unprimed null spaces [Morita et al., 1990]. Of course, for the case of perfectly magnetic conducting surfaces, the representation of the field is in terms of magnetic current densities on magnetic conducting surfaces, but the mathematical secondary sources which make the solutions of the integral equations of the second kind for both the electric and magnetic conducting surfaces nonunique, represent magnetic current densities on perfectly electric conducting surfaces. It seems that these conclusions become only obvious if the representation theorem as well as the dipole-layer ansatz is exploited to represent scattered fields [Colton and Kress, 1983].

Acknowledgments

[26] I am greatly indebted to Rainer Kress who was kind enough to proofread my lecture notes on the above subject; numerous discussions with him finally clarified all open “engineering” questions, which came along while reading his book co-authored by David Colton. Dedicated to Thomas Weiland on the occasion of his 50th birthday.

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