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A Time-Domain Uniqueness

Theorem for Electromagnetic

Wavefield Modeling in

Dispersive, Anisotropic Media

Adrianus T. de Hoop

Abstract

A uniqueness theorem for the initial-/boundary-value problem arising in the (analytic or compu-tational) modeling of electromagnetic wavefields inarbitrarily dispersive and anisotropic media is pre-sented. It is known that for media where the disper-sion takes place via electrically conductive and/orlinear magnetic hysteresis losses only, a uniquenesstheorem for the initial-/boundary-value problem canbe constructed by using direct time-domain argu-ments in the pertaining energy balance (Poynting’stheorem). The case of arbitrary dispersion in themedium’s electric and magnetic behavior, however,withstands such an approach. Here, as an interme-diate step, the one-to-one correspondence betweenthe causal time-domain field components and mate-rial response functions in the constitutive relationson the one hand and their time Laplace transformsfor (a set of) real, positive values of the transform pa-rameter on the other hand, seems a necessary tool.It is shown that this approach leads to simple, ex-plicit, sufficiency conditions on the relaxation ten-sors describing the medium’s electric and magneticbehavior, in which the property of causality provesto play an essential role.

1. Introduction

One of the issues one is confronted with in the math-ematical modeling – be it with analytical or numeri-cal techniques – of electromagnetic wave phenomenais the question about the uniqueness of the solutionto the problem as it is formulated mathematically.Evidently, such a uniqueness should be expected onaccount of the underlying physics. When investigat-ing wave propagation and scattering problems one

expects the pertaining partial differential equations,constitutive relations, boundary conditions at inter-faces, excitation conditions at exciting sources, ini-tial values at the time window one considers and thecausal relationship that is to exist between the ex-citing sources and the generated wavefield to playa role. For simple media with instantaneous rela-tions between the intensive quantities (that carry thepower flow in the wavefield) and the extensive quan-tities (that carry the momentum of the wavefield),i.e., for lossless media, the time-domain power bal-ance provides a tool to prove uniqueness. This is alsothe case when simple loss mechanisms (such as elec-trically conductive and/or linear magnetic hystere-sis losses) are incorporated. A uniqueness proof forthe case of arbitrary relaxation effects in the mediaseems to withstand such a direct time-domain ap-proach. Since for the class of linear, time-invariant,causally reacting media the constitutive relations areexpressed via time convolutions, it can be expectedthat the time Laplace transformation (under whichtransformation the convolution operation transformsinto a simple product of the constituents) might pro-vide a useful tool. This approach is followed in thepresent paper and applied to the general class of lin-ear, time-invariant, causal, locally reacting, inhomo-geneous, anisotropic media. For this class of media,sufficient conditions for the uniqueness of the elec-tromagnetic wavefield problem are specified for theelectric and magnetic relaxation tensors in the timeLaplace transform domain at real, postitive values ofthe transform parameter. In the procedure, Lerch’stheorem of the one-sided (= causal) Laplace trans-formation plays an essential role.

Prof. Adrianus T. de Hoop isLorentz Chair Emeritus Professor with theLaboratory of Electromagnetic ResearchFaculty of Information Technology and SystemsDelft University of Technology4 Mekelweg2628 CD Delft, the NetherlandsT: 31-15-2785203/31-15-2786620F: 31-15-2786194E: [email protected]

Dedicated to Professor J. Van Bladel

on the occasion of his 80th. birthday.

TheRadio Science Bulletin No 305 (June, 2003) 17

2. Description of the configuration

The configuration for which the uniqueness of theelectromagnetic wavefield problem will be provedconsists of a linear, time-invariant, locally reacting,inhomogeneous, anisotropic medium with arbitraryelectric and magnetic relaxation properties and ofbounded support D ⊂ R3. This part of the con-figuration is embedded in a linear, time-invariant,locally reacting, homogeneous, isotropic, instanta-neously reacting medium with permittivity ε∞ andpermeability µ∞. The unbounded domain occupiedby the embedding is denoted as D∞. The commonboundary of D and D∞ is the bounded closed surface∂D (Figure 1). The constitutive relaxation functionsin D vary piecewise continuously with position withfinite jump discontinuities at a finite number of piece-wise smooth, bounded surfaces (interfaces). Positionin the configuration is specified by the coordinatesx1, x2, x3 with respect to an orthogonal Cartesianreference frame with the origin O and the three mu-tually perpendicular base vectors i1, i2, i3 of unitlength each. In the indicated order, the base vectorsform a right-handed system. The subscript notationfor Cartesian vectors and tensors is used and thesummation convention for repeated subscripts ap-plies. Whenever appropriate, vectors are indicatedby boldface symbols, with x as the position vector.The time coordinate is t. Partial differentiation withrespect to xm will be denoted by ∂m; ∂t is a reservedsymbol indicating partial differentiation with respectto t. Volume source distributions of electric polariza-tion and/or magnetization, with bounded supports,excite a transient electromagnetic field in the config-uration. They start to act at the instant t = 0. Thefield that is causally related to the action of thesesources then vanishes throughout space for t < 0.

3. Formulation of the EM wavefieldproblem

At any point in the configuration where the electro-magnetic field quantities are differentiable they sat-isfy the Maxwell field equations [1, p. 611]

εk,m,p∂mHp − ∂tDk = 0, (1)

εj,n,q∂nEq + ∂tBj = 0, (2)

where

Eq = electric field strength (V/m),Hp = magnetic field strength (A/m),Dk = electric flux density (C/m2),Bj = magnetic flux density (T),

and εk,m,p is the completely antisymmetrical unittensor of rank three: εk,m,p = 1 for k,m, p = evenpermutation of 1, 2, 3, εk,m,p = −1 for k,m, p =odd permutation of 1, 2, 3, εk,m,p = 0 in all othercases.

O

ε∞, µ∞ D∞

∂D

D

εk,q(x, t)µj,p(x, t)

x

νminter-

face

Figure 1: Configuration with inhomogeneous,

anisotropic, dispersive medium (with bounded sup-

port D) embedded in a homogeneous, isotropic,

non-dispersive medium (with unbounded support

D∞).

The constitutive relations are:

Dk(x, t) = Pk(x, t) + εk,q(x, t)(t)∗ Eq(x, t)

for x ∈ D, (3)

Bj(x, t) = Mj(x, t) + µj,p(x, t)(t)∗ Hp(x, t)

for x ∈ D, (4)

where(t)∗ denotes time convolution, and

Dk(x, t) = ε∞Ek(x, t) for x ∈ D∞, (5)

Bj(x, t) = µ∞Hj(x, t) for x ∈ D∞. (6)

In these relations,

Pk = electric source polarization (C/m2),Mj = source magnetization (T),

represent the active parts (source distributions) ofthe medium in D,

εk,q(x, t) = electric relaxation function (F/m·s),µj,p(x, t) = magnetic relaxation function (H/m·s),

of the medium in D, and

ε∞ = electric permittivity (F/m),µ∞ = magnetic permeability (H/m),

of the medium in D∞. In the representation of theconstitutive behavior of the media we have chosento incorporate the action of field exciting source dis-tributions in them through field-independent excita-tion terms as is done in the Kirchhoff theory of activenetworks (Thevenin and Norton representations forthe action of source voltages and source electric cur-rents, respectively). Across any interface Σ of jumpdiscontinuity in constitutive properties the boundaryconditions of the continuity type

εk,m,pνmHp = continuous across Σ, (7)

εj,n,qνnEq = continuous across Σ, (8)

18 TheRadio Science Bulletin No 305 (June, 2003)

hold, where νm is the unit vector along the normalto Σ. This implies that the tangential components ofthe electric and magnetic fields strengths are contin-uous across the interface. The constitutive relaxationfunctions are subject to the causality condition

εk,q(x, t) = 0 for t < 0 and all x ∈ D, (9)

µj,p(x, t) = 0 for t < 0 and all x ∈ D. (10)

Further conditions to be laid upon them with regardto the uniqueness of the electromagnetic wavefieldproblem are investigated further on. On the consti-tutive coefficients of the embedding we impose theconditions ε∞ > 0 and µ∞ > 0.

In the embedding the Green’s tensors (point-source solutions) can be determined analytically [1,Sections 28.8 and 28.12]. From the correspondingHuygens surface source representations over the sur-face ∂D it follows that the outgoing fields in D∞

admit the far-field expansions

Eq, Hp(x, t) =eq, hp(θ, t − |x|/c∞)

4π|x|[

1 + O(|x|−1)]

as |x| → ∞,

(11)

where x is the position vector from the chosen far-field reference center to the point of observation,θ = x/|x| is the unit vector in the direction of obser-vation and c∞ = (ε∞µ∞)−1/2 is the electromagneticwavespeed in D∞. The far-field radiation character-istics are mutually related via

eq = −(µ∞/ε∞)1/2εq,m,pθmhp, (12)

hp = (ε∞/µ∞)1/2εp,n,qθneq. (13)

In the following it will be shown that the problemthus formulated has at most one solution, assumingthat, for each type of excitation at least one solu-tion exists. The proof puts restrictions on the re-laxation functions representing the electric and mag-netic properties of the medium in D. For the mediumin D∞ these simply are ε∞ > 0 and µ∞ > 0 (as in-dicated already).

4. The electromagnetic field problemin the time Laplace-transform domain

For the general type of dispersive media consideredin the present paper there is, as far as is known,no direct uniqueness proof in the space/time domainbased on energy considerations as is the case for me-dia with simple constitutive behavior [2, Section 9.2].However, because of the causality of both the me-dia’s passive field reponse and the field’s relation toits activating sources, the time Laplace transforma-tion with real, positive transform parameter offers atool to specify certain conditions to be imposed onthe constitutive relaxation functions in order that

the wavefield problem has a unique solution. Therelevant transformation is given by

Eq, Hp(x, s) =

∫ ∞

t=0

exp(−st)Eq, Hp(x, t)dt.

(14)

For the case of physical interest of excitation func-tions and relaxation functions that show at mosta Dirac delta distribution time behavior, the timeLaplace transforms of the field vectors and the re-laxation tensors exist for all s ∈ C; Re(s) > 0, i.e.,for all values of the transform parameter in the righthalf of the complex s-plane. Furthermore, since alltime functions involved are real-valued, their Laplacetransforms take on real values for real values of s. Inrelation to our uniqueness proof we now take s to bea Lerch sequence: s ∈ R; s = s0 + nh, s0 > 0, h >0, n = 0, 1, 2, . . .. Lerch’s theorem [3, p. 63] statesthat if the transformation expressed by Equation(14) is to hold for all s belonging to this sequence,only one (causal) time-domain original correspondsto its related transform. Under the transformation,the time derivative is replaced with a multiplicationby s (if zero-value initial conditions apply, as is thecase) and the time convolution transforms into theproduct of the constituents. Using these properties,Equations (1) - (6) lead, upon time Laplace trans-formation, to

εk,m,p∂mHp − sεk,qEq = sPk for x ∈ D, (15)

εj,n,q∂nEq + sµj,pHp = −sMj for x ∈ D,(16)

and

εk,m,p∂mHp − sε∞Ek = 0 for x ∈ D∞, (17)

εj,n,q∂nEq + sµ∞Hj = 0 for x ∈ D∞. (18)

The interface continuity conditions (7) - (8) are uponLaplace transformation replaced by

εk,m,pνmHp = continuous across Σ, (19)

εj,n,qνnEq = continuous across Σ, (20)

and the far-field expansion (11) by

Eq, Hp(x, s) =eq, hp(θ, s)

4π|x|exp(−s|x|/c∞)

[

1 + O(|x|−1)]

as |x| → ∞.

(21)

Upon contracting Equations (15) and (17) with Ek

and Equations (16) and (18) with Hj and combiningthe results we construct the relations

εm,k,j∂m(EkHj) + sEk εk,qEq + sHj µj,pHp =

−sEkPk − sHjMj for x ∈ D, (22)

and

εm,k,j∂m(EkHj) + sEkε∞Ek + sHjµ∞Hj =

0 for x ∈ D∞. (23)


Integration of Equation (22) over D and applicationof Gauss’ divergence theorem yields

∫

∂D

εm,k,jνmEkHjdA(x) +

∫

D

(sEk εk,qEq + sHj µj,pHp)dV (x) =

−

∫

D

(sEkPk + sHjMj)dV (x), (24)

where νm is the outward unit vector along the nor-mal to ∂D. Next, Equation (23) is integrated overthe domain that is bounded internally by ∂D andexternally by the sphere S∆ of radius ∆ and centerat the far-field reference center, where ∆ is chosen solarge that S∆ completely surrounds ∂D (Figure 2).Subsequent application of Gauss’ divergence theoremleads to

∫

S∆

εm,k,jνmEkHjdA(x) −

∫

∂D


∫

D∞∩D∆

(sEkε∞Ek + sHjµ∞Hj)dV (x)

= 0, (25)

where D∆ is the domain interior to S∆. With the useof the far-field representation (21) in the integrationover S∆, the limit ∆ → ∞ in Equation (25) leadsto (note that the integral over S∆ goes to zero as∆ → ∞)

O

ε∞, µ∞ D∞

∂D

D

∆

S∆

νm

D∆

Figure 2: Configuration used in the derivation

of the time Laplace-transform domain uniqueness

identity. (The limit ∆ → ∞ is taken.)

−

∫

∂D


∫

D∞

(sEkε∞Ek + sHjµ∞Hj)dV (x) = 0. (26)

Addition of Equations (24) and (26) finally yields

∫

D

(sEk εk,qEq + sHj µj,pHp)dV (x) +

∫

D∞

(sEkε∞Ek + sHjµ∞Hj)dV (x) =

−

∫

D

(sEkPk + sHjMj)dV (x), (27)

where the surface integrals over ∂D have canceled inview of the continuity of εm,k,jνmEkHj across ∂D.Equation (27) will be used in the construction of theuniqueness proof.

5. The uniqueness proof

The uniqueness proof starts by assuming that inthe given configuration, for one and the same ex-citation, there exist at least two non-identical fieldsolutions, which we will distinguish by the super-

scripts [1] and [2]. Obviously, P[1]k = P

[2]k = Pk

and M[1]j = M

[2]j = Mj . Consider the differences

in value in the field quantities ∆Eq = E[2]q −E

[1]q and

∆Hp = H[2]p − H

[1]p . Their time Laplace transforms

then satisfy the equations (cf. Equations (15) - (18))

εk,m,p∂m∆Hp − sεk,q∆Eq = 0 for x ∈ D, (28)

εj,n,q∂n∆Eq + sµj,p∆Hp = 0 for x ∈ D, (29)

and

εk,m,p∂m∆Hp − sε∞∆Ek = 0 for x ∈ D∞, (30)

εj,n,q∂n∆Eq + sµ∞∆Hj = 0 for x ∈ D∞. (31)

The same operations that have led to Equation (27)now yield

∫

D

(s∆Ek εk,q∆Eq + s∆Hj µj,p∆Hp)dV (x) +

∫

D∞

(s∆Ekε∞∆Ek + s∆Hjµ∞∆Hj)dV (x) =

0. (32)

Evidently, for real, positive values of s the integrandin the integral over D∞, and hence the integral it-self, is positive for any non identically vanishing ∆Ek

and/or any non identically vanishing ∆Hj through-out D∞. The integral over D shares this propertyif we impose on εk,q and µj,p the condition thatthroughout D they are positive definite tensors ofrank two for all real, positive values of s. Under thiscondition, also the integral over D is positive for anynon identically vanishing ∆Ek and/or any non iden-tically vanishing ∆Hj throughout D. For non iden-

tically vanishing ∆Ek and/or non identically vanish-ing ∆Hj throughout D ∪ D∞ Equation (32) leads,in view of the value zero of the right-hand side to acontradiction. Under the given conditions we there-fore have ∆Ek = 0 and ∆Hj = 0 for x ∈ D ∪D∞,

which implies E[2]k = E

[1]k and H

[2]j = H

[1]j for x ∈

D ∪ D∞. In view of Lerch’s uniqueness theoremof the one-sided Laplace transformation this implies

20 TheRadio Science Bulletin No 305 (June, 2003)

that E[2]k = E

[1]k and H

[2]j = H

[1]j for x ∈ D ∪ D∞

and all t ≥ 0, i.e., there is only one electromagneticfield in the configuration that is causally related tothe action of its exciting sources.

It is noted that the conditions imposed onthe constitutive relaxation functions are specifiedthrough their time Laplace transforms. Strictlyspeaking the pertaining conditions need only holdon a Lerch sequence. In view of the analyticity ofthe transforms in s ∈ C; Re(s) > 0, however, theyhold for all real, positive values of s. The conditionsthus specified are sufficient ones, but at present noweaker conditions seem to be in existence. Also, asimple time-domain counterpart does not seem to ex-ist. This, however, is the same situation as in linear,time-invariant, causal system’s theory.

6. Examples of relaxation functions

Some examples of relaxation functions that arise inthe physics of electric and magnetic materials aregiven below. They all apply to the simple case ofisotropic materials.

Permittivity relaxation function of an isotropic

plasma

For an isotropic plasma the (isotropic, scalar) s-domain permittivity relaxation function as based onthe Lorentz theory of electrons model is [1, pp. 639-640]

ε = ε0

[

1 +1

s

ωp2

s + νc

]

, (33)

where ε0 is the permittivity of vacuum, ωp the elec-tron angular plasma frequency of the plasma andνc the collison frequency. The corresponding time-domain relaxation function is

ε = ε0δ(t) + (ωp2/νc)[1 − exp(−νct)]H(t),

(34)

where δ(t) is the Dirac delta distribution and H(t) isthe Heaviside unit step function. The time-domainmagnetic relaxation function is µ = µ0δ(t), where µ0

is the permeability of vacuum.

Lorentzian absorption line of a dielectric material

For the Lorentzian absorption line of a dielectric ma-terial the (isotropic, scalar) s-domain permittivityrelaxation function is [1, pp. 639-640]

ε = ε0

[

1 +ωp

2

(s + Γ/2)2 + Ω2

]

, (35)

where Γ is a phenomenological damping coefficient,Ω = (ω2

0 − ωp2/3 − Γ2/4)1/2 is the natural angular

frequency of the oscillations of the movable electriccharge, ω0 is the resonant angular frequency of the(Coulomb force) mechanical model of the atom andωp is the angular plasma frequency of the movableelectric charge distribution. The corresponding time-domain relaxation function is

ε = ε0[δ(t) + (ωp2/Ω) exp(−Γt/2) sin(Ωt)H(t)].

(36)

The time-domain magnetic relaxation function isµ = µ0δ(t), where µ0 is the magnetic permeability ofvacuum.

Linear hysteresis in a magnetic material

Linear hysteresis in an isotropic magnetic materialcan be modeled via a Debije type of relaxation func-tion

µ = µ0µr(1 + Γ/s), (37)

where µr is the relative permeability of the materialand Γ is a phenomenological (Landau) damping co-efficient. The corresponding time-domain relaxationfunction is

µ = µ0µr[δ(t) + ΓH(t)]. (38)

The electric properties of the material need furtherspecification.

It is observed that all these relaxation functions sat-isfy the conditions for uniqueness discussed in Sec-tion 5,

7. Conclusion

A time-domain uniqueness theorem for electromag-netic wavefield modeling in arbitrarily dispersive andanisotropic media is presented. Sufficient conditionsfor the uniqueness to be laid upon the electric andmagnetic tensorial relaxation functions are formu-lated in the (causal) time Laplace-transform domainfor real, positive values of the transform parameter.Some simple relaxation functions arising from phys-ical models on an atomic level in plasma and solid-state physics are shown to be in accordance with thecriteria developed.

8. References

1. A. T. de Hoop, Handbook of Radiation and Scat-

tering of Waves, London, Academic Press, 1995.2. J. A. Stratton, Electromagnetic Theory, New

York, McGraw-Hill, 1941.3. D. V. Widder, The Laplace Transform, Princeton,

Princeton University Press, 1946.


Advanced Applications of theField Equivalence Principle

in Numerical ElectromagneticModelling Techniques

Hendrik Rogier

Abstract

The field equivalence principle plays a very impor-tant role in a large variety of full-wave electro-magnetic field solvers, since it permits partition-ing the simulation domain into a number of subre-gions that can have separate electromagnetic descrip-tions. These formulations can consist, for example,of boundary integral equations with different Green’sfunction kernels, but also finite elements and finite-difference time-domain techniques can be used forcertain subdomains, resulting in hybrid formalisms.In this contribution, we first review the field equiv-alence theorem in a very general form. We thenpresent some advanced applications of this princi-ple in some of our recent work. Both pure boundaryintegral equation approaches, as well as hybrid tech-niques that combine the boundary integral equationmethod with finite elements and finite-differences intime-domain are discussed.

1. Introduction

In this contribution, we review the classical Huygensprinciple [1,2], stating that each point on a wavefrontcan be regarded as a new source of waves. This prin-ciple, introduced by Huygens in 1690 in connectionwith optical effects, plays a fundamental role in anypart of physics that can be treated by means of fieldtheory. Although the principle is of fundamental im-portance in electromagnetics, only a limited numberof papers have dealt with the subject. Existing liter-ature on the Huygens principle is closely related tothe reaction concept [3] and to surface integral rep-resentations [4–6]. We show that the more generalform of the Huygens principle, i.e. the principle offield equivalence, can be applied to a wide range ofstate-of-the-art numerical modelling problems. Ourpaper is structured as follows. We start by review-ing the Huygens principle in Section 2. FollowingLindell, a mathematical foundation can be given to

the field equivalence theorem [7]. We show in Sec-tion 2 that applying field equivalence allows one topartition a complex simulation domain into severalsubregions that can be treated by different mod-elling techniques, provided that unknown virtual cur-rents are introduced at the subdomains’ boundaries.In this contribution we focus on rigorous full-wavetechniques such as the finite element (FE) method,the boundary integral equation (BIE) technique andthe finite-difference time-domain (FDTD) approach,but, within the same framework, asymptotic tech-niques can also be incorporated, such as geometricalor physical optics and diffraction theories [8]. Thebasic application of this subdomain technique for ahybrid FE-BIE method (as, for example, describedin [9,10]) is discussed in Section 3. For the bound-ary integral equation subdomains, Love’s field equiv-alence principle is used whereas for the finite elementsubregions, Schelkunoff’s field equivalence theoremis applied. In Sections 4 and 5, we present moreadvanced applications in recent formalisms that wehave developed. First, the boundary integral ap-proach is applied to model shielding enclosures inan efficient way [11]. Finally, the application of fieldequivalence is illustrated for the FDTD part of a newhybrid FDTD-BIE formalism [12–14].

2. The Huygens principle –Field equivalence

Let us now review the Huygens principle applied toa very simple configuration, as shown in Fig. 1. Weconsider a simple source configuration radiating infree space. A closed surface S is placed around thesource, partitioning the simulation domain into aninterior and an exterior subdomain. Following Huy-gens principle, we can replace the original sourceconfiguration by virtual current sources Jvirtual andKvirtual on S. These current soures are called equiva-lent sources for the exterior field problem, since they

Hendrik Rogier is with theDepartment of Information TechnologyGhent University St. Pietersnieuwstraat 41B-9000 Ghent, Belgium.T: ++32/92643343F: ++32/92643593.E: [email protected].



22 TheRadio Science Bulletin No. 305 (June 2003)

produce the same field in Ve as the original source.Furthermore, they lead to a zero field inside S.

J, K

E, H

S

J, K

E, H

VVe i

Jvirtual Kvirtual

n

E, H

J, K

E=0, H=0

partitioning intotwo subdomains

replacing original sourcesby virtual sources

Figure 1: The Huygens principle.

Although the Huygens principle has been knownfor a long time, not much attention has been devotedto a rigorous proof of the field equivalence principle.Rumsey [3] used the reaction principle to derive theHuygens principle, whereas Tai [4] relied on Green’sfunction theory. Quite recently, Lindell derived a di-rect rigorous mathematical formulation to state theprinciple of field equivalence [7]. He starts by defin-ing a truncation function, which is zero inside S andone in the exterior subdomain.

PVe(r) = 1, r ∈ Ve; PVe

(r) = 0, r /∈ Ve

The left- and right-hand sides of Maxwell’s curl lawsare then multiplied with this function, yielding

(∇× E) PVe= −jωµ · H PVe

− K PVe

(∇× H) PVe= jωε · E PVe

+ J PVe

The equations can then be rewritten in terms of trun-cated fields (PVe

E, PVeH) and sources (PVe

J, PVeK)

as follows:

∇× (PVeE) = − jωµ · PVe

H − PVeK

− [E × (−n)] δS

∇× (PVeH) =jωε · PVe

E + PVeJ

+ [(−n) × H] δS

In the right-hand sides, two new sources appear.They represent an electric current layer Jvirtual =(−n) × H and a magnetic current layer Kvirtual =E × (−n) on S. Because of the definition of thetruncation function, the truncated fields and sourcesare zero in the interior subregion. A dual formalismcan be applied for the inner subregion Vi. Now, anelectric current layer Jvirtual = n×H and a magneticcurrent layer Kvirtual = E × n are introduced on S.Because of these currents, the fields and sources arenow set to zero in the exterior subregion.

The introduction of the equivalent currents on Scancels fields inside Vi. This means that the origi-nal materials and sources inside Vi may be changedwithout affecting the fields in the exterior subregion.Placing equivalent currents on S thus allows us tochange the original configuration into a new prob-lem which is easier to model. Unfortunately, theseequivalent current sources are in fact unknown, sincethey depend upon the tangential fields on S. The ex-act value of the virtual current sources thus requiresthe solution of the global EM problem. We can, how-ever, consider the equivalent current sources as un-knowns of the problem to solve. Now, an EM fielddescription can be found for each subregion definedin the global simulation space. This description doesnot depend on the materials and sources present inthe other subdomains. The global EM problem isfinally obtained by expressing the continuity of thetangential electric and magnetic fields at all the in-terfaces between the different subregions. This re-sults in a global matrix system, the solution set ofwhich comprises the equivalent current sources at allthe interfaces. It may seem suprising that the effectof the entire field distribution inside Vi can be re-placed by electric are magnetic current sources onlylocated at its boundary S. The theorem of field unic-ity, however, shows that knowledge of the tangentialelectric and magnetic boundary fields on S is indeedsufficient to reconstruct the unique field distributioninside Vi. At frequencies that do not correspond to aresonance frequency of the cavity Vi, even the knowl-edge of only the tangential electric field or the tan-gential magnetic field leads to a unique field distri-bution in Vi. Similar conclusions can be formulatedfor the exterior subregion Ve, provided the radiationcondition is enforced at infinity [15].

TheRadio Science Bulletin No. 305 (June 2003) 23

3. Application to hybrid FE-BIEtechniques

The concepts of field equivalence and partitioningof the simulation domain can be applied to modelcomplex electromagnetic (EM) configurations moreefficiently. These structures, which typically com-bine inhomogeneous features with an open three-dimensional simulation space, can be described withhybrid techniques that combine a rigorous boundaryintegral equation (BIE) for the open domain with thefinite element method (FE) or the finite difference-time domain (FDTD) technique for the inhomoge-neous part of the problem. Let us first focus on a hy-brid technique, combining the finite element methodwith the boundary integral approach.

3.1. The BIE subdomain– Love’sField Equivalence Theorem [16]

n

Jvirtual = −n x H

Kvirtual = E x −n

r r(ε( ), µ( ))

J, K

E=0, H=0

(ε, µ)

n

Jvirtual = −n x H

Kvirtual = E x −n

(ε, µ)

(ε, µ)

Figure 2: Love’s field equivalence theorem.

Let us first discuss the use of field equivalencewhen applying the boundary integral equation tech-nique to the exterior subdomain. Let the exteriorsubregion Ve be homogeneous with parameters (ε, µ).The unknowns of the integral equation are the equiv-alent sources introduced on S. Applying the Huy-gens principle then allows us to replace the arbri-tary materials of the interior subregion Vi with thesame homogeneous dielectric as in the exterior sub-domain, as shown in Fig. 2. In the resulting ho-mogeneous free-space problem, one easily obtains aboundary integral equation description with the free-space Green’s function as the kernel.

3.2. The FE part– Schelkunoff’s FieldEquivalence Theorem [17]

n

Jvirtual = n x H

Kvirtual = E x n

J, K

E=0, H=0

r r(ε( ), µ( ))

r r(ε( ), µ( ))

nS

J, K

E, H

E=0, H=0

Kvirtual = E x n

Figure 3: Schelkunoff’s field equivalence theorem.

The interior subdomain can be described by a fi-nite element method after invoking field equivalencefollowing Schelkunoff. Because of the introductionof virtual sources on S, fields are zero in the exte-rior domain. Thus, we are free to place a perfectlyconducting metallic box in this domain, completelyenclosing the interior subregion, as shown in Fig. 3.This box can then be placed arbitrary close to thesurface S. For the closed configuration, one easily ob-tains an FE description. This description is coupledto the boundary integral equation for the exteriorsubproblem through the unknown virtual magneticcurrents Kvirtual on S.

3.3. Extension to periodic structures

n-1C nC n+1C

n-1c nc n+1c... ...S S

S

S

d dd

ε , µ

ε , µ

ε , µε , µ

ε , µ

1

1 1

S

2 22

1 1

222

22 2

1

Figure 4: A simple periodic configuration.

By invoking of the Floquet-Bloch theorem, whichlimits the analysis to a single unit cell, the hybridFE-BIE approach can be easily extended to peri-odic structures. The analysis of these structures is ofimportance in practical configurations, such as grat-ings in optical applications and absorbers in anechoic


chambers [18–20]. Application of the field equiva-lence principle provides several mechanisms to ex-press periodicity of the field within the unit cell. Forillustration purposes, we consider the simple periodicconfiguration shown in Fig. 4. The unit cell of theperiodic structure consists of a homogeneous and un-bounded subregion S1 and a bounded but potentiallyinhomogeneous subregion S2. For the nth unit cell,subdomain S1 is bounded by contour Cn, whereas S2

is bounded by contour cn. Both contours conincide,but their orientation is different.

d dd

n-1C(1)

nC(1)

n+1C(1)

n-1(1)

c n+1(1)

cn(1)

c... ...

S

S

ε , µS

ε , µ

1ε , µ

1

1 1

2

n+1nn-1 C(2)

C(2)

C(2)

1

22

1

Figure 5: Boundary integral representation with

periodic Green’s function.

A rigorous treatment of subregion S1 is obtained bymeans of a boundary integral representation, appliedover all contours Cn. Since this formulation is ob-tained by means of field equivalence, the virtual elec-tric and magnetic currents along Cn appear as un-knowns. By means of the Floquet-Bloch theory, theanalysis can be limited to the contour Cn of a singleunit cell, provided a periodic Green’s function [21]is used as the integral kernel in the boundary in-tegration. Assuming for now that subdomain S2 ishomogeneous as well, one can give a similar bound-ary integral description for S2, now integrating overthe contour cn. Since the contributions of contoursCn and cn along the boundaries of the unit cell can-cel out in neighbouring cells, the final description isschematically represented in Fig. 5. The boundaryintegral representation with periodic Green’s func-

tion kernel proceeds along contours C(1)n and C

(2)n

for the unbounded domain S1 and along contours

c(1)n and c

(2)n for the bounded domain S2.

yx

z

e - j k dy

S

S

d

Cε , µ

ε , µ

(2)

1(1)

(1)1

ε , µ1 1

1(2)

SC

2

2

1

2cn

n

n

Figure 6: Boundary integral representation with

free space Green’s function and periodic boundary

condition.

An alternative formulation can be derived forsubregion S2 as follows: In the previous analysis westarted from an integral representation for the fieldinside S2. The integral was taken over all contours cn

present in the periodic configuration. Now focus on

region S2 in the nth unit cell. From the field equiv-alence principle, it is knowns that only the virtualelectric and magnetic currents on cn contribute tothe field inside S2 of the nth unit cell, whereas thecontributions of the virtual currents on all other con-tours give rise to a zero field. This proves that an in-tegral representation with a much simpler free spaceGreen’s function kernel leads to equivalent results.However, this formulation results in a higher num-ber of unknowns, since the unknowns on the bound-ary do not cancel out in the present formulation. In-stead, the Floquet-Bloch condition should be appliedfor these virtual currents on the unit cell boundary,giving rise there to a periodic boundary condition,as shown in Fig. 6. This line of reasoning also holdswhen S2 is inhomogeneous and an FE descriptionis applied. The FE formulation proceeds as in freespace. Only the periodic boundary condition mustbe imposed along the boundaries of the unit cell.

Which of the two mechanisms described above ismore efficient depends on the simulation geometryat hand. A more detailed discussion together withsome examples can be found in [20].

4. An Advanced application: A BIEtechnique for modelling enclosures

The problem of a metallic structure with a slot rep-resents one of the oldest practical configurations forwhich the application of field equivalence is relevantas a means for solving the EM problem (see, for ex-ample, [22,23]). Here we consider a metallic enclo-sure Sencl with apertures Saper. These structuresplay a very important role in electromagnetic com-patibility issues. We are interested in the efficiencyof the shielding when there is a current inside theenclosure. A straightforward approach proceeds byreplacing all metallic parts of the enclosure with un-known electric currents and by invoking Love’s fieldequivalence theorem as in Section 3.1, both insideand outside the enclosure [24]. Quite recently, wehave developed a more efficient technique for suchconfigurations [11]. The domains inside and outsidethe enclosure are modelled with the BIE approach.To restrict the number of unknowns, we do not use aconventional boundary integral equation, since thisrequires discretizing the complete enclosure. Mag-netic virtual current sources K are placed over theaperture only. We then invoke Schelkunoff’s fieldequivalence theorem.

4.1. Field equivalence applied to theenclosure problem

To solve for the unknown aperture currents, we ap-peal to the field equivalence and first partition theconfiguration into two subproblems. A boundary in-tegral equation formulation is then derived for bothsubdomains with specific kernel functions. For theinterior problem we use the Green’s function of the


closed metallic enclosure. This kernel function canbe obtained in the form of a three-dimensional seriesby modal expansion. The convergence of this seriesis accelerated by means of the Ewald transform tech-nique [25].

(a)

or

(b)

(c)

or

Sencl

Sapern

Kvirtual = E x −nJ, K

J, K

E, H

E=0, H=0

Kvirtual = E x n

n Saper

J, K

E=0, H=0

E, H

Kvirtual = E x n

n Saper

+

Saper

J, K

E=0, H=0

E, H

Kvirtual = E x n

n

Jvirtual = −n x H

+

E, H

E=0, H=0

J, K

n

Kvirtual = E x −nSaper

+

Figure 7: Field equivalence principle for mod-

elling enclosures.

The exterior problem can be treated in a similarfashion, by considering the magnetic currents K over

the apertures as unknowns and by using the Green’sfunction of a PEC metallic box in free space as thekernel of the integral equation, as shown in Fig. 7a.Contrary to the interior problem, however, no ana-lytic solutions can be found for an elementary dipoleradiating in free space in the presence of a PEC box.A numerical technique proceeds by defining electriccurrents K on S to replace the box and by applyingLove’s field equivalence principle, as in Section 3.1and illustrated in Fig. 7b, to find a BIE descriptionfor the Green’s function. This approach, however,drastically increases the number of unknowns. Theintroduction of additional unknowns on the PEC boxcan be avoided by approximating the exterior regionby a half-space (Fig. 7c), for which a simple analyt-ical Green’s function can be used. This approxima-tion gives satisfactory results for apertures of moder-ate size that are not too close to the edges of the boxand are sufficiently far away from the enclosure.

4.2. Example: Enclosure withvertical aperture

50 cm

50 cm

50 cm

3 m

5 cm x 20 cm

Figure 8: Enclosure with vertical aperture.

Let us now illustrate the technique by means of anexample. We consider a metallic enclosure with avertical aperture, as shown in Fig. 8. We are inter-ested in the shielding effiency at 3m with a dipolesource placed in the middle of the enclosure.

-60

-40

-20

0

20

40

60

80

0 200 400 600 800 1000

Shie

ldin

g E

ffic

ienc

y (d

B)

Frequency (MHz)

(a)(b)(c)

Figure 9: Shielding efficiency.

For the 50cm enclosure, we calculate the shieldingefficiency up to 1 GHz by means of three different


techniques. In Fig. 9, good agreement is seen be-tween the hybrid BIE technique (curve (a)) discussedin the previous subsection, a low frequency approxi-mation (curve (b)) and a boundary integral equationapproach in which the complete enclosure is discre-tised (curve (c)).

5. Another advanced application:The FDTD-BIE formalism

KvirtualJ virtual or

S

V i

V e

Figure 10: Modelling a subdomain with the

FDTD-BIE technique.

Another part of our recent research activities fo-cussed on combining the boundary integral equationformulation with the finite-difference time-domaintechnique. Again consider a simulation domain sub-divided by a closed surface S into an interior andexterior subregion, as shown in Fig. 10. We start byapplying field equivalence with equivalent currentson S. We then discretize the currents on S by usingsubdomain expansion functions. Therefore, S is di-vided into rectangular cells and the electric and mag-netic currents on its surface are expanded in rooftopbasis functions. First, a BIE description under theform of a system matrix is derived for the exteriordomain Ve. The Love field equivalence principle isinvoked as discussed in Section 3.1. We then pro-ceed by modelling the interior subdomain by meansof FDTD. This is done by bringing each rooftop ba-sis function for expanding the electric or magneticcurrents into an FDTD simulation. This leads to asystem matrix which describes the EM field for Vi.

5.1. Field equivalence applied to theFDTD subdomain

To find such a system matrix, the number of FDTDsimulations to describe Vi can be quite large. In-deed, the number of required FDTD runs equals thenumber of basis functions to represent the magneticand electric currents on S. Therefore, special mea-sures are taken to limit the CPU-time for each FDTDsimulation. This is done by reducing transients bylowering the quality factor of the closed simulationdomain. We invoke Schelkunoff’s field equivalence

theorem, but now, the metallic boundary does notcoincide with S. In the region formed by the bound-ary of the simulation domain and the surface S, fieldsare zero. In this region, absorbing material can beinserted. This reduces transients without affectingthe global EM field behavior.

n

Jvirtual = n x H

Kvirtual = E x n

J, K

E=0, H=0

n

Kvirtual = E x n

Jvirtual = n x H

S

J, K

E, H

E=0, H=0

(Berenger) absorber or ABC

Figure 11: Field equivalence principle for mod-

elling an FDTD subdomain.

5.2. Example: An FE-FDTD-BIEtechnique

0.9 cm 1.8 cm

θ

xy

z

ε = 2.33r

ε = 12r

0.9 cm0.9 cm

0.14 cm0.4 cm

0.9 cm

1.8 cm

0.2 cm

ε = 12r

0.75 cm

Figure 12: A dipole radiating above a substrate

in the presence of an FSS.

The techniques of combining the boundary inte-gral equation with the FDTD technique can be ex-tended with a finite element part. This example il-lustrates our formalism that combines the three full-wave techniques, based on the theory presented inSections 3.1, 3.2 and 5.1. Let us consider a Hertzian


dipole located on the central axis above a multi-layered substrate, as shown in Fig. 12. The sub-strate has dimensions 0.9cm × 0.9cm × 0.4cm andis composed of four dielectric layers with alternatingrelative permittivity of 2.33 and 12. The elementarydipole is located 0.14cm above the substrate. Thesubtrate together with the dipole is placed into onesubregion modelled with finite elements. Above thesubstrate and dipole, a frequency selective surface(FSS) is placed, consisting of a 4 by 4 array of dielec-tric patches. Each patch has dimensions 0.9cm by0.9cm by 0.2cm and consists of a dielectric with per-mittivity 12. Each patch is embedded in an FDTD-subregion. Finally all the subregions are coupled byapplying the BIE technique.

3

2.5

2

1.5

1

0.5

0

0.5

1

1.5

2

2.5

3FE-FDTD-BIE

FE-BIE

3

2.5

2

1.5

1

0.5

0

0.5

1

1.5

2

2.5

3FE-FDTD-BIE

FE-BIE

Figure 13: Radiation pattern in the xz-plane

(left) and in the yz-plane (right).

Now consider the configuration with the dipoleoscillating at 8 GHz. In Fig. 13, the radiation pat-terns in the xz-plane and yz-plane are shown at afrequency of 8GHz. Good agreement is observed inthe results obtained with FE-FDTD-BIE and FE-BIE code.

6. Conclusions

In this contribution, we have reviewed the Huygensprinciple under the form of the field equivalence the-orem. We have proposed some basic and advancedapplications of field equivalence to electromagneticmodelling techniques. These formalisms include theboundary integral equation method, the finite ele-ment method, a new modelling technique for enclo-sures, and a new hybrid FDTD-boundary integralequation method.

7. Acknowledgement

The research of H. Rogier is supported by a grantof the DWTC/SSTC, MOTION project.

The author would like to thank Prof. Daniel DeZutter and Prof. Chalmers Butler for carefully proof-reading the manuscript.

8. References

1. R. E. Collin, Field Theory of Guided Waves, IEEEPress, New-York, 1991.

2. R. F. Harrington, Time-Harmonic Electromag-

netic Fields, Electrical and Electronic EngineeringSeries. McGraw-Hill, New York, 1961.

3. V. H. Rumsey, “Some new forms of huygen’s prin-ciple,” IEEE Trans. Antennas Propag., vol. 7, no.5, pp. 103–116, Dec. 1959.

4. C. T. Tai, “A concise formulation of huygen’s prin-ciple for the electromagnetic field,” IEEE Trans.

Antennas Propag., vol. 8, no. 6, pp. 634–634,Nov. 1960.

5. J. C. Monzon, “On surface integral equations: Va-lidity of huygen’s principle and the equivalenceprinciple in inhomogeneous bianisotropic media,”field,” IEEE Trans. Microwave Theory Tech.,vol. 41, no. 11, pp. 1995–2001, Nov. 1993.

6. C. C. Lu and W; C. Chew, “The use of huygen’sequivalence principle for solving 3-D volume inte-gral equations of scattering,” IEEE Trans. An-

tennas Propag., vol. 43, no. 5, pp. 500–507, May.1995.

7. I. V. Lindell, Methods for Electromagnetic Field

Analysis, Oxford Engineering Science Series.Clarendon Press, Oxford, 1992.

8. M. Alaydrus, V. Hansen, and T. F. Eibert,“Hybrid2: combining the three-dimensional hy-brid finite element-boundary integral techniquefor planar multilayered media with the uniformgeometrical theory of diffraction,” IEEE Trans.

Antennas Propag., vol. 50, no. 1, pp. 67–74, Jan2002.

9. H. Rogier, F. Olyslager, and D. De Zutter, “Ahybrid FE/IE approach for the eigenmode analy-sis of complex anisotropic dielectric waveguides,”Radio Science, Spec. Issue on Comp. EM, vol.31, no. 4, pp. 999–1010, Jul.-Aug. 1996.

10. T. F. Eibert and V. Hansen, “3-D FEM/BEM-hybrid approach based on a general formulationof Huygens’ principle for planar layered media,”IEEE Trans. Microwave Theory Tech., vol. 45,no. 7, pp. 1105–1112, July 1997.

11. W. Wallyn, D. De Zutter, and H. Ro-gier,“Prediction of the shielding and resonant be-haviour of multisection enclosures based on mag-netic current modelling,” IEEE Trans. Electro-

magnetic Compatibility, vol. 44, no. 1, pp. 130–138, Invited Paper, Special issue in memory of M.Kanda, Feb. 2002.

12. H. Rogier, F. Olyslager, and D. De Zutter, “Anew hybrid FDTD-BIE approach to model electro-magnetic scattering problems,” IEEE Microwave

Guided Wave Lett., vol. 8, no. 3, pp. 138–140,Mar 1998.


13. H. Rogier, D. De Zutter, and F. Olyslager, “Rig-orous analysis of frequency selective surfaces of fi-nite extent using a hybrid finite difference timedomain-boundary integral equation technique,”Radio Science, vol. 35, no. 2, pp. 483–494, Mar-Apr 2000.

14. H. Rogier, D. De Zutter, and F. Olyslager, “A newhybrid FDTD-BIE technique with BiCG solverfor the analysis of EM scattering problems,” Int.

Journal of Electr. and Comm., vol. 54, no. 4, pp.227–232, Jul-Aug 2000.

15. C. Muller, Foundations of the Mathematical The-

ory of Electromagnetic Waves, Die Grundlehrender mathematischen Wissenschaften. ClarendonPress, Berlin, 1969.

16. A. E. H. Love, “The integration of the equations ofpropagation of electric waves,” Phil. Trans. Roy.

Soc. London, Ser. A, vol. 197, pp. 1–45, 1901.

17. S. A. Schelkunoff, “Some equivalence theorems ofelectromagnetics and their application to radia-tion problems,” Bell Syst. Tech. J., vol. 15, pp.92–112, 1936.

18. N. Marly, D. De Zutter, and H. Pues, “A surfaceintegral equation approach to the scattering andabsorption of doubly periodic lossy structures,”IEEE Trans. Antennas Propag., vol. 36, no. 1,pp. 14–22, Feb. 1994.

19. B. Baekelandt, F. Olyslager, and D. De Zut-ter, “An analysis of uniaxial bianisotropic two-dimensional periodic and non-periodic structuresusing a boundary integral equation method,” Ra-

dio Sci., vol. 32, no. 4, pp. 1361–1374, Jul. 1997.

20. H. Rogier, B. Baekelandt, F. Olyslager, and D. DeZutter, “The FE-BIE technique applied to some 2-D problems relevant to electromagnetic compati-bility: Optimal choice of mechanisms to take intoaccount periodicity,” IEEE Trans. Electromag-

netic Compatibility, vol. 42, no. 3, pp. 246–256,Aug. 2000.

21. B. Baekelandt, D. De Zutter, and F. Olyslager,“Arbitrary order asymptotic approximation of aGreen’s function series,” Int. Journal of Electr.

and Comm., , no. 4, pp. 224–230, July 1997.

22. J. D. Shumpert and C. M. Butler, “Penetrationthrough slots in conducting cylinders - part 1: TEcase,” IEEE Trans. Antennas Propag., vol. 46,no. 11, pp. 1612–1621, Nov. 1998.

23. J. D. Shumpert and C. M. Butler, “Penetrationthrough slots in conducting cylinders - part 1: TMcase,” IEEE Trans. Antennas Propag., vol. 46,no. 11, pp. 1622–1628, Nov. 1998.

24. F. Olyslager, E. Laermans, D. De Zutter, S. Criel,R. De Smedt, N. Lietaert, and A. De Clercq, “Nu-merical and experimental study of the shielding ef-fectiveness of a metallic enclosure,” IEEE Trans.

Electromagnetic Compatibility, vol. 41, pp. 311–316, Aug 1999.

25. T. F. Eibert, J. L. Volakis, D. R. Wilton, andD. R. Jackson, “Hybrid FE/BI modeling of 3-D doubly periodic structures utilizing triangularprismatic elements and an MPIE formulation ac-celerated by the Ewald transformation,” IEEE

Trans. Antennas Propag., vol. 47, no. 5, pp.843–850, May 1999.


Closed Form Solutions of

Maxwell’s Equations in the

Computer Age Frank OlyslagerIsmo V. Lindell

Abstract

We consider four closed form solutions of Maxwell’sequations that have been derived or were revisitedduring the past decade. First we focus on twoproblems from electrostatics: the electrostatic im-age for the dielectric sphere and a generalisation ofthe Thompson and Lampard capacitor. Then wegive two solutions of the full Maxwell equations inthe time-harmonic regime: the Green dyadics fora homogeneous bianisotropic medium with similarmedium dyadics and the scattering of a plane wave ata diaphanous wedge. Although we have studied theseproblems with some rigour in the past we present, ineach case, a new, shorter and more direct solution.

1. Introduction

In this paper we present closed form solutions of anumber of electromagnetic field problems for whichclosed form solutions were constructed by the au-thors during the past ten years. Professor Van Bladelhimself, has devoted a considerable amount of his sci-entific studies to closed form solutions of Maxwell’sequations. Another motivation to bring this subjectin honour of the eigthieth birthday of Professor VanBladel is the fact that Professor Van Bladel broughtthe present authors together in 1992. This has ledto a fruitful cooperation resulting in numerous pub-lications on theoretical electromagnetics.

In an age where powerful simulation techniquesrun on powerful computers allowing the accurate so-lution of exeedingly complex field problems one couldwonder about the usefulness of closed form solutionsof Maxwell’s equations. This argument can be coun-tered by the following motivations:

• Closed form solutions provide insight in elec-tromagnetic field behaviour and immediatelyreveal the influence of parameters to the fields.

• Closed form solutions are forever. Once a prob-lem is solved, it is solved. This in contrary toa new numerical solution technique which ismost often open for improvements. Once animprovement is found the old technique loosesits usefulness.

• Closed form solutions are good benchmarkcases for numerical simulation techniques or forthe calibration of measurements.

• Constructing closed form solutions gives a cer-tain satisfaction, which was perhaps our mostimportant motivation.

In this paper we present four recent solutions ofelectromagnetic problems that allow closed form so-lutions:

• The first problem is a new approach to the solu-tion of the electrostatic image for the dielectricsphere.

• The second problem is a two-dimensional elec-trostatic problem. In the 1950s an interest-ing capacitance relation was found betweenfour almost touching conductors, the so-calledThompson and Lampard capacitor. This rela-tion was generalised by the authors to symmet-ric configurations with M conductors.

• The third problem is a closed form Greendyadic for a certain class of homogeneous bi-anisotropic media.

• The fourth problem is the scattering at a di-aphanous wedge. The field singularity at thetip of wedges has been one of the key problemsthat were tackled by Professor Van Bladel.

Frank Olyslager is with theDepartment of Information TechnologyGhent University, St. Pietersnieuwstraat 41B-9000 Ghent, Belgium.T: ++32/92643344F: ++32/92643593E: [email protected]

Ismo V. Lindell is with the

Electromagnetics Laboratory

Helsinki University of Technology, PO BOX 3000

FIN-02015 HUT, Espoo, Finland

T: ++358/94512266

F: ++358/94512267

E: [email protected]




2. Electrostatic image for the sphererevisited

2.1. Historic preamble

The basic problem of an electric point charge out-side a PEC sphere in terms of an image point chargewas solved by Thomson (Kelvin) in 1845 [1] by notic-ing that two point charges of opposite sign producea potential function whose zero potential surface isa sphere and, thus, one of the point charges couldbe replaced by a PEC sphere at zero potential. Thesame principle was independently found in 1861 byCarl Neumann [2] who also subsequently solved themagnetostatic image problem. Because the perfectmagnetic conductor was not known at the time, hissolution was actually a generalization of the Kelvinimage because it assumed finite permeability. Theimage solution consisting of a point source and aline source was published in 1883 in an appendix ofa book on hydrodynamics [3].

Because Neumann’s solution was not widelyknown for a century, it was reinvented by many au-thors. The first of these known to the present au-thors was Yossel in 1971 [4] who used a method sim-ilar to that of Neumann. Others followed withoutreference to any of the previous ones, Efremov andProsvirnii in 1985 [5,6], Poladian in 1988 [7] and Lin-dell in 1992 [8]. Finally, reviewing papers on im-age theories [9], the authors found Neumann’s orig-inal work through a reference in an old review arti-cle [10], p.321. Still, even after this, other indepen-dent work on this topic appeared [11]. That Neu-mann’s image principle was really not known, wasreflected by a sentence in the much cited monographby Smythe [12], p.127:

We find that there is not, as in the anal-ogous two-dimensional case, a simple im-age solution for the dielectric sphere andthe point charge. We have to use spheri-cal harmonics to solve this problem.

Other similar statements can be found in more recenttexts [13,14].

2.2. Operational solution

The solution for a point charge Q defined by thecharge density

%(r) = Qδ(r − uzb) (1)

in air in front of a dielectric sphere centered at theorigin with permittivity εrεo and radius a (Figure1), can be derived in many ways. Expanding the po-tential function in spherical harmonics we can eas-ily solve the potential reflected (scattered) from thesphere as [15]

φr(r) = −Qεo

∞∑

n=1

(εr − 1)n

(εr + 1)n+ 1

a2n+1

bn+1

Pn(cos θ)

4πrn+1, (2)

for r > a. The image of the point source is basi-cally the source of this potential confined inside theregion r < a in air. Its expression can be identi-fied from (2) by manipulating the sum and compar-ing the result with the potential from a line sourceor by deriving an integral equation for the unknownsource. Another method is to expand in the θ and ϕcoordinates only and solve the remaining differentialequation in the r variable as that of a transmissionline [16]. Let us use yet another method applyingHeaviside operator calculus [17]. This technique alsois close to Engheta’s representation [18] but does notrequire any fractional differentiations.

Let us invoke the differentiation rule fromMaxwell’s Treatise [19], p.209

∂nz

1

r= (−1)nn!

Pn(cos θ)

rn+1. (3)

This inserted into (2) gives us

φr(r) = − εr − 1

εr + 1

Qa

εob

∞∑

n=1

(−c∂z)n

(n− 1)!(n+ α)

1

4πr, (4)

where we denote for brevity c = a2/b, α = 1/(εr +1).Now, the source of the reflected potential is the re-flection image %r(r) which can be obtained from thePoisson equation as

%r(r) = −εo∇2φr(r)

=εr − 1

εr + 1

Qa

b

∞∑

n=1

(−c∂z)n

(n− 1)!(n+ α)∇2 1

4πr. (5)

Inserting

∇2 1

4πr= −δ(r) = −δ(z)δ(ρ), (6)

we can write (5) in the form

%r(r) = − εr − 1

εr + 1

Qa

b

∞∑

n=1

(−c∂z)n

(n− 1)!(n+ α)δ(z)δ(ρ)

= − εr − 1

εr + 1

Qa

b

(

− c

α+ 1∂z +

c2

α+ 2∂2

z

− c3

2!(α+ 3)∂3

z + · · ·)

δ(z)δ(ρ). (7)

(7) represents a solution as a multipole expansion atthe origin. It is known how extended sources can bereplaced by multipoles [20,21]. Here we wish to dothe opposite: replace a multipole expansion by anextended source to get a simpler expression for theimage source. The actual polarization in the dielec-tric sphere is dipolar which is shown by the fact thatthere is no monopole term in (7). Thus, it appearsnatural to express the image charge in terms of thedipole moment density function pr(z) as

%r(r) = ∂zpr(z)δ(ρ). (8)


In operational form the moment density can be iden-tified from (7) as

pr(z) =εr − 1

εr + 1

Qa

b

∞∑

n=0

cn+1(−∂z)n

n!(n+ α+ 1)δ(z). (9)

Proceeding as

∂c[cαpr(z)] =

εr − 1

εr + 1

Qa

b∂c

∞∑

n=0

cn+α+1(−∂z)n

n!(n+ α+ 1)δ(z)

=εr − 1

εr + 1

Qa

b

∞∑

n=0

cn+α(−∂z)n

n!δ(z)

=εr − 1

εr + 1

Qa

bcαe−c∂zδ(z) =

εr − 1

εr + 1

Qa

bzαδ(z − c),

(10)we have a differential equation for the moment den-sity. Requiring as an additional condition vanishingof pr(z) for z > a, we write the solution as

pr(z) =εr − 1

εr + 1

Qa

b(z/c)αU(c− z)U(z), (11)

where U(x) denotes the Heaviside unit step function.In terms of original parameters we have

pr(z) =εr − 1

εr + 1

Qa

b(zb/a2)1/(εr+1)U(c− z)U(z),

(12)which finally gives Neumann’s image charge densityfunction as

%r(r) =εr − 1

(εr + 1)2Qa

b(z/c)−εr/(εr+1)U(c− z)U(z)

−εr − 1

εr + 1

Qa

bδ(c− z). (13)

Thus, the image of a point charge in a dielectricsphere consists of a point charge at the Kelvin imagepoint and a line charge between the center and theKelvin image point as shown in Figure 1. The lineimage has an integrable singularity at the center ofthe sphere.

z0 a ba /b2

er

Figure 1: Point charge in front of adielectric sphere showing the Kelvin image

and image line charge.

3. A generalisation of the Thompsonand Lampard capacitor

3.1. An introduction

12

34

P12

P41

P34

P23

c

Figure 2: Four almost touching

conducting shells.

In 1956 another remarkable problem in two-dimensional electrostratics was solved [22,23]. Itwas shown that a special relation holds between theelements in the capacitance matrix of four almosttouching conducting shells of arbitrary shape (Fig-ure 2). The elements C13 and C24, due to the fieldsinside the shell of conductors, always satisfy

exp(−π|C13|ε0

) + exp(−π|C24|ε0

) = 1. (14)

If from symmetry it follows that C13 = C24 thenthese capacitances are fixed, |C13| = |C24| =(ε0/π) ln 2. These fixed values for the capacitanceelements are used to build capacitance standardsconsisting of a number of metal cylinders embed-ded in a metal box and isolated from each otherby thin pieces of mica. The result for four conduc-tors derived [22,23] was generalised to five and sixalmost touching conductors by Fiebiger and Fleis-chhauer [24] and Elnekave [25]. The present authorshave recently revisted the Thompson and Lampardtheorem and generalised it to an arbitrary numberof conductors [26]. A summary of these results arepresented here. First we give a short proof of (14)and then consider symmetric configurations of an ar-bitrary number of conductors. Recently also Jack-son [27] revisited this interesting problem.

3.2. The Schwarz-Christoffelconformal mapping

The Schwarz-Christoffel conformal mapping [28] al-lows the mapping of the interior of an arbitrary poly-gon, and by extension every closed curve, to a half-plane. This means that we can map the configu-ration of Figure 2 to the upper-half of the complexz = ζ + jη-plane as indicated on Figure 3. The free-dom in the transformation is used to map the pointP41 on z41 = ±∞ and the point P23 on z23 = 0. Themapping of the points P12 and P34 is then fixed andgiven by z12 = −a and z34 = b where a and b dependon the particular geometry of the curve c in Figure2.


P12 P34P23

z-plane

z

jh

1 2 3 4

Figure 3: Complex z-plane with theconductors of Figure 2 mapped

to the real axis.

Let us now determine C13 and C24 for the config-uration in Figure 3 or quivalently Figure 2. For C13

we place conductor 1 at a potential 1V and the otherconductors at zero potential. The complex potentialw(z) is then

w(z) = φ(z) + jψ(z) =j

πln

√1 + a2

z + a. (15)

The capacitance C13 is now readily found to be

C13 = ε0[ψ(z = b) − ψ(z = 0)] =ε0π

lna

a+ b. (16)

The capacitance C24 is found by interchanging therole of a and b

C24 =ε0π

lnb

a+ b. (17)

Combining (16) and (17) immediately yields the re-lation (14).

3.3. Symmetric configuration ofM-conductors

C1

C2

Figure 4: Symmetrical configuration

with M conductors.

Let us consider a symmetric configuration ofM >3 conductors such as shown in Figure 4. We lookonly at the fields and capacitances for the interiorproblem. The external capacitances can be short cir-cuited by embedding the whole structure in a metalcasing as illustrated in Figure 4. All the conductors,inclusive the casing, are isolated from each other byinfinitesimal gaps. If no metal casing is added all thecapacitances derived below have to be multiplied bya factor of two. Remark that the capacitance matrix

for such a symmetric configuration is a circulant ma-trix. Such capacitance matrices have special proper-ties as has been studied by Nitsch et. al. [29]. LetC1 denote the capacitance between a conductor andits second neighbour, C2 the capacitance between aconductor and its third neighbour, etc.. When M isan even number there are N = (M − 2)/2 differentsuch capacitances Ci (i = 1, 2, ..., N) and when M isan odd number there are N = (M−3)/2 different ca-pacitances. We now present equations to determinethese values Ci. Let us first introduce new quantitiesxi

xi = exp(−πCi

ε0), (18)

i = 1, 2, ..., N . From (14) one can show that

xi +

N∏

j=1

xτij

j = 1, (19)

with τij = 2min(i, j) except when M is even andj = N in which case τij = τiN = i. Using the re-lation (19) one finds it possible to express xi−1 as afunction of xj with j ≤ i alone,

xi−1 = 1 − 1 − xi

xσN

N−1∏

j=i

1

x2j

, (20)

with σ = 1 when M is even and σ = 2 when M isodd. Applying this expression recursively allows usto express each xi (i = 1, 2, ..., N) as a rational func-tion of xN only. A polynomial expression for xN isthen found by inserting all the rational expressionsfor the xi in (19) for i = 2. When M is even it ispossible to determine xN , and by iterative use of (20)all xi, in closed form. When M = 2L with L evenone can show that

xN =1 +

√α

2, (21)

with α = xL/2−1 for the configuration with L con-ductors. When M = 2L, with L odd, one can showthat

xN =α+ 3

4, (22)

with α = x(L−1)/2−1 for the configuration with Lconductors. When M is odd it is in general not pos-sible to solve for the values of xi in closed form. InTable 1 the values of xN are listed for M up to 20.The equation (20) then allows the determination ofthe other values xi with i < N .


M xN

4 12

5√

5−12

6 34

7 − 23 + 2

√7

3 cos π−arctan(3√

3)3

8 2+√

24

9 −1 + 2 cos π9

10 5+√

58

11 0.91898594722899477978

12 2+√

34

13 0.94188363485210405210

14 712 + 2

√7

12 cos π−arctan(3√

3)3

15 0.95629520146761127586

16 2+√

2+√

24

17 0.96594619936780355656

181+cos π

9

219 0.97272260680544474721

20 4+√

2√

5+√

58

Table 1: Values of xN for a symmetric

configuration of M conductors.

4. Closed form Green dyadics forbianisotropic media

4.1. The quest

During the past twenty years the electromagneticfield in bianisotropic media [30] has been the sub-ject of considerable analyses. The present authorshave been mainly interested in closed form field so-lutions in homogeneous bianisotropic media. It turnsout that only specific classes of bianisotropic mediaallow some progress toward closed form solutions ofMaxwell’s equations. An overview of our findingscan be found in [31].

The basic field problem is the Green functionproblem. Indeed it presents the field due to an ele-mentary Dirac source, i.e., it presents the impulseresponse of the medium. The Green function foran homogeneous isotropic medium was perhaps firstpublished in [32]. As is explained in [31] this hasbeen generalized in several steps to several classes ofanisotropic and bianisotropic media. To give a flavorof our work on this matter we consider a special bian-isotropic medium where all the medium parametersare proportional to the same dyadic [33]. We optedfor this medium because it is a good illustration ofmany other cases we have studied without requiringexcessively complicated manipulations. The consitu-tive relations of the medium are given by

D = ε · E + ξ · H, (23)

B = ζ · E + µ · H, (24)

with ε = ετ , µ = µτ , ζ = ζτ and ξ = ξτ . In [33] theGreen dyadic for this medium was derived only inclosed form for the special case where ζ+ξ = ±2

√εµ.

However, in [34] a more general medium was stud-ied that allowed a closed form without any furtherrestrictions, seemingly a contradiction with the re-sult of [33]. In [34] a very different approach wasused to solve the Green dyadic problem. Here we re-solve this contradiction by showing that it is indeedpossible to derive the Green dyadic for the mediumin [33] without any restrictions. We do this using anew approach.

4.2. Solution

The Green dyadic G(r) relates electric current den-sities to the electric field through the relation

e(r) = −jω∫

V

G(r − r′) · J(r′)dV ′, (25)

and satifies the equation

L ·G(r) = −δ(r)I, (26)

with

L = −(∇×I− jωξ) ·µ−1 · (∇×I+ jωζ)+ω2ε. (27)

For the medium under consideration we can writethis as

L =1

µτ ·(τ−1 ·∇×I−jωA+I)·(τ−1 ·∇×I−jωA−I),

(28)with

A± =ξ − ζ ±

√

(ξ + ζ)2 − 4εµ

2. (29)

The Green dyadic, i.e., the solution of (26), is givenformally by

G(r) = −(τ−1 · ∇ × I − jωA−I)

−1

·(τ−1 · ∇ × I − jωA+I)−1µτ

−1δ(r). (30)

Using a partial fraction expansion for matrices, wecan write this in an alternative form as

G(r) =A+ −A−

jω[G−(r) −G+(r)], (31)

withG±(r) = (∇× I − jωA±τ)

−1δ(r). (32)

We assume that A+ 6= A−. The contrary case(A+ = A−) is exactly the case for which a closedform solution was obtained in [33]. Let us now de-

compose −jωA±τ into a symmetric part S± and an

anti-symmetric part a± × I. This allows us to write(32) as

G±(r) = [(∇+ a±)× I +S±]−1δ(r) = e−a±·rH±(r),(33)

with H±(r) given by

H±(r) = (∇× I + S±)−1δ(r). (34)


Using dyadic analysis [21] one can write (34) as

H±(r) = [∇∇−(∇·S±)×I+S−1

± detS±]h±(r), (35)

where h±(r) is a scalar Green function that is thesolution of

(S± : ∇∇ + detS±)h±(r) = δ(r), (36)

given by

h±(r) =e−jD±

4πD±, (37)

with D± =

√

detS±

√

S−1

± : rr. Substitution of (37)in (35), (33) and (31) results in the requested closed

form expression for G(r).

5. The diaphanous wedge

5.1. The Meixner series

Van Bladel has devoted a considerable amount of hisscientific energy to the study of singular fields in elec-tromagnetics [20]. Singular fields not only occur inthe source region of Green dyadics but also at thetips of wedges and cones. The study of wedges turnsout to be a very difficult problem and full closed formsolutions for the scattering of plane waves at wedgesonly exist for impenetrable wedges, e.g., perfectlyconducting wedges. There is one exception [35] of apenetrable wedge that allows a closed form solutionfor the plane wave scattering problem. This wedgeis the so-called diaphanous wedge. A scatterer iscalled diaphanous [36] if the phase velocity of wavesinside the scatterer is the same as the phase velocityof waves outside the scatterer. Diaphanous scatter-ers are also called isorefractive [37]. The importanceof this closed form result for a penetrable wedge isenormous. Indeed, one of the discussion points inthe literature has been the question of whether ornot the Meixner series correctly predicts the singu-lar behaviour of the fields at the edge of a wedge andwhether or not this series converges. Meixner [38,39]assumed that the fields near the wedge could be ex-panded radially as a Taylor series multiplied by asingular factor. From the analysis in [35] it followsthat the Meixner technique indeed correctly predictsthe singular behaviour of the fields but that the seriesis not correct.

In this paper we do not introduce the whole ma-chinery of [35] but rather we consider a more directapproach. We limit the discussion to the perpen-dicular incidence of a TM plane wave on a wedge.This problem is solved using a Kantorovich-Lebedevtransform.

5.2. Solution

x

f=w

f=-w

e m- -

e m+ +

r

f

Figure 5: A diaphanous wedge with

opening angle 2w.

Consider the geometry of Figure 5 consisting of awedge oriented parallel to the z-axis with openingangle 2w. Inside the wedge −w < φ < w the mediumparameters are ε− and µ−. Outside the wedgew < |φ| < π the medium parameters are ε+ and µ+.For a diaphanous wedge, k = ω

√ε−µ− = ω

√ε+µ+.

A TM plane wave eiz(r, φ) = ejkr cos(φ) is incident on

the wedge. The fields inside and outside the wedgesatisfy the Helmholtz equation

∇2ez(r, φ) + k2ez(r, φ) = 0. (38)

At the wedge interface, i.e., at φ = ±w, ez(r, φ) and1µ

∂∂φez(r, φ) are continuous.

Let us now introduce the Kantorovich-Lebedevtransform [36] and its inverse

F (ν) =

+∞∫

0

f(r)

rH(2)

ν (kr)dr, (39)

f(r) = −1

2

j∞∫

−j∞

νJν(kr)F (ν)dν, (40)

in which capital letters imply Kantorovich-Lebedevdomain quantities. After transformation the equa-tion (38) simply becomes

d2

dφ2Ez(φ) + ν2Ez(φ) = 0. (41)

The Kantorovich-Lebedev transform of the incidentfield is [40]

Eiz(φ) = −2j

ν

e−j π2

ν

sin(πν)cos(νφ) = α cos(νφ), (42)

where the second identity defines the parameter α.Due to the symmetry of the problem the fields in-side and outside the wedge are symmetric about thex-axis. This means that we can represent the totalfields inside the wedge as

Ez(φ) = A cos(νφ), (43)


and the scattered fields outside the wedge as

Esz(φ) = B cos[ν(φ− π)], (44)

for w < φ < π. We now determine A and B. Thecontinuity relations at φ = w yield

B cos[ν(w − π)] + α cos(νw) = A cos(νw), (45)

− B

µ+sin[ν(w − π)] − α

µ+sin(νw) = − A

µ−sin(νw).

(46)

Solving for A yields

A = −2j(Γ + 1)

ν

e−j π2

ν

g(ν), (47)

withg(ν) = sin(νπ) + Γ sin[ν(π − 2w)], (48)

and

Γ =µ− − µ+

µ− + µ+=ε+ − ε−ε+ + ε−

. (49)

From (47), (43) and (40) the fields inside the wedgeare then found to be

ez(r, φ) = −j(Γ + 1)

j∞∫

−j∞

e−j π2

ν

g(ν)Jν(kr) cos(νφ)dν

=

+∞∑

n=0

π(Γ + 1)τne−j π

2νnJνn

(kr) cos(νnφ)

π cos(νnπ) + Γ(π − 2w) cos[νn(π − 2w)],

(50)

with τn = 2 when n > 0 and τ0 = 1 and where νn

are the solutions of g(ν) = 0. To evaluate the inte-gral we have applied Cauchy’s theorem. The polesνn are the so-called Greenberg static modes [41,42].The fields outside the wedge are found in a similarmanner. From (50) it is seen that a single Taylor se-ries multiplied by one singular factor is not sufficientto represent the fields as was proposed by Meixner.

6. Conclusions

It has been shown that closed form solutions ofMaxwell equations are not relics from the nineteenthor first half of the twentieth century. No, we haveshown that this is an active and facinating domainof research. We hope that the reader has graspedsome of the beauty of Maxwell’s equations and theirsolutions, a beauty which is and has been admiredby Professor Van Bladel!

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Editor-in-Chief:Imrich ChlamtacDistinguished Chair inTelecommunicationsProfessor of Electrical EngineeringThe University of Texas at DallasP.O. Box 830688, MS EC33Richardson, TX 75083-0688email: [email protected]

Aims & Scope:The wireless communication revolution is bringing fundamental changes todata networking, telecommunication, and is making integrated networks areality. By freeing the user from the cord, personal communicationsnetworks, wireless LAN's, mobile radio networks and cellular systems,harbor the promise of fully distributed mobile computing andcommunications, any time, anywhere. Numerous wireless services are alsomaturing and are poised to change the way and scope of communication.WINET focuses on the networking and user aspects of this field. It providesa single common and global forum for archival value contributionsdocumenting these fast growing areas of interest. The journal publishesrefereed articles dealing with research, experience and management issuesof wireless networks. Its aim is to allow the reader to benefit fromexperience, problems and solutions described. Regularly addressed issuesinclude: Network architectures for Personal Communications Systems,wireless LAN's, radio , tactical and other wireless networks, design andanalysis of protocols, network management and network performance,network services and service integration, nomadic computing,internetworking with cable and other wireless networks, standardization andregulatory issues, specific system descriptions, applications and userinterface, and enabling technologies for wireless networks.

WirelessNetworks

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The journal of mobile communication, computation and information

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Euro 62 / US$ 65(including mailing and handling)

Wireless Networks ISSN 1022-0038Contact: Mrs. Inge HeleuFax +32 9 264 42 88 E-mail [email protected] members/Institutions: contact Baltzer Science Publishers

P R E S Smca mca

Wireless Networks is a jointpublication of the ACM andBaltzer Science Publishers.Officially sponsored by URSI

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