IEEE TRANS. ANTENNAS PROPAGAT., SUBMITTED 1

Multi-Static Response Matrix of a 3-D Inclusion inHalf Space and MUSIC Imaging

Ekaterina Iakovleva, Souhir Gdoura, Dominique Lesselier,Senior Member, IEEE, and Gaele Perrusson

Abstract— Earlier works [1] on the retrieval of 3-D boundeddielectric and/or magnetic inclusions buried in free spaceareextended herein to a half-space burial. Emphasis of the presentcontribution is on the case of a single inclusion, since manyclosed-form mathematical results can be derived in illuminating fashionin that case, yet the proposed approach readily extends to the caseof (an unknown number of) well-separated inclusions. Within theframework of an asymptotic field formulation derived from th eexact contrast-source vector integral formulations satisfied by thetime-harmonic fields and using proper reciprocity relationshipsof the dyadic Green’s functions, the Multi-Static Responsematrix(MSR) of the inclusion is constructed from the leading-order termof the fields —the product of the MSR matrix by its transposeyields the time-reversal operator in matrix form which is proneto so-called DORT analyses, e.g., [2], in electromagnetics. Thesingular value structure of the MSR is then analyzed in detailfor an inclusion which has either dielectric contrast with respectto the embedding half-space, or magnetic contrast, or both,thisbeing done in line with the pioneering study in free space ledin [3]. The work is performed for fixed electric dipole arrays,in the transmit/receive mode at a single frequency. A MUSIC-type algorithm readily follows from that decomposition, yieldinga cost functional the magnitude of which peaks at the inclusioncenter, provided in particular that the number of scattered fielddata at the frequency of operation is larger than the number ofsingular values (up to 5 in the case of simultaneous dielectricand magnetic contrast of a single inclusion). Numerical resultsare presented in order to illustrate the above structure as afunction of the geometric and electromagnetic parameters of theconfiguration. Imaging of a spherical inclusion is then proposedfrom severely noisy synthetic data via a pertinent application ofthe MUSIC algorithm. Images of two such inclusions follow soasto illustrate the potentialities of the present framework beyondthe single-inclusion case.

Index Terms— time-harmonic 3-D electromagnetic scattering- Green’s dyadic function - multi-static response matrix -asymptotic formulation - half-space burial - MUSIC reconstruc-tion

I. I NTRODUCTION

SINGLE-FREQUENCY time-harmonic non-iterative elec-tromagnetic imaging of a collection of small 3-D inclu-

sions in free space, those being characterized by arbitrarycon-trasts of dielectric permittivity and of magnetic permeabilitywith respect to the ones of their embedding medium, has beenconsidered in much detail in a recent contribution [1].

The starting point was an asymptotic expansion of theelectromagnetic field as a function of the (assumed) common

E. Iakovleva is with Centre de Mathematiques Appliquees (CNRS-EcolePolytechnique) 91128 Palaiseau cedex, France.

S. Gdoura, D. Lesselier and G. Perrusson are with Departement deRecherche enElectromagnetisme - Laboratoire des Signaux et Systemes(CNRS-Supelec-UPS 11) 91192 Gif-sur-Yvette, France.

order of magnitude of the size of the inclusions in harmonywith the systematic framework developed in [4], enabling theauthors then to construct the Multi-Static Response (MSR)matrix of the collection of inclusions for a set of distincttransmitters and receivers.

Proper singular value decomposition of the MSR matrixallowed them to carry out —and numerically illustrate— atwo-step analysis which goes as follows:

• first, the calculation of singular values (and the cor-responding eigenvectors), the number of nonzero onesdepending upon the number and the electromagneticnature of the inclusions, and upon the transmitters and/orreceivers’ geometrical arrangement and polarization;

• second, the orthogonal projection of a properly builtvector propagator onto the null space of the MSR matrix,coincidence with an inclusion being associated to a peakof the inverse norm of the projection (yielding an imageof the collection in a prescribed search space by mappingthat inverse norm), within the framework of the so-called MUSIC (MUltiple SIgnal Classification) methodfor non-iterative solution of inverse source and scatteringproblems.

In the present paper, one is extending the above work to thecase of inclusions that are fully buried in a half space. Thistaskis rendered far more complex both at the mathematical leveland at the numerical level due to the specific need to handle thedyadic Green’s functions of the stratified embedding medium,known only through their spectral expansions, yet the sameprocedure: asymptotic field formulation, calculation of theMSR matrix, singular value decomposition, MUSIC-type, non-iterative imaging, applies as it will be shown and illustratedfrom synthetic data thereafter.

However, the main difference with that previous work,beyond in particular the just mentioned subtleties of the Green-based field formulation involved, lies in the fact that one isintending to focus herein onto the singular value structureofthe MSR matrix, analyzed in great (and mostly novel) detail,for only one (typically, ellipsoidal) inclusion.

Indeed, we believe that, as confirmed already by the pioneer-ing study in free space led in [3] and a just submitted sequelby the same authors [5], one needs to soundly understand theproperties of the MSR matrix, before proposing beautiful yetpossibly ill-explained images of multiple inclusions via appli-cation of MUSIC in the half-space, aspect-limited scatteringsituation at hand here.

In order to avoid unnecessary complexity, this is done inthis contribution for planar arrays of electric dipoles in thetransmit/receive mode, but via electromagnetic duality this

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could be easily extended to magnetic dipole arrays, and evento a mix of electric and magnetic sources and receivers.

Evidently, since the end’s user in the field of applicationexpected (e.g., non destructive testing of defects in industrialstructures, identification of small targets in sub-soils orothernatural media, imaging of anomalous biological structures),might still yearn for images, those, from severely noisy data,are provided as well, in the case of the single inclusion (forsimplicity, a sphere) and in the case of two inclusions (twospheres); the latter are well-separated whereby it is meantthat they are far enough from one another to being imagedindependently by the MUSIC procedure.

The work presented here, as just indicated, takes its directinspiration from the monograph [4], and from the researchpaper (involving two of the authors, Iakovleva and Lesselier)[1] —one should also mention, again involving the same twoauthors, [6] which is concerned with to 2-D scalar inversescattering (both in fluid acoustics and TE/TM electromagnet-ics) and half-space burial of cylindrical inclusions. In addition,the thoughtful companion contributions by Chambers andBerryman, [3] and [5], have been very useful to the present-day investigation.

Since it is well-known that the MSR matrix times its conju-gate transpose yields the time-reversal operator in matrixform,which is thus prone to so-called DORT (Decomposition del’Operateur de Retournement Temporel) analyses, one shouldalso refer to a series of works on this subject carried out inelectromagnetics in free space [2] and for targets buried insubsoil [7], [8], those however mostly with emphasis on 2-DTE/TM configurations and always for non-magnetic materials.

Origin of the DORT method itself goes well back to themid 90’s (refer to, e.g., [9]) in fluid acoustics and the recentreview [10] of time-reversal (which is including DORT) isstill a must-read among increasingly many references dealingwith modalities and applications of time-reversal in variousconfigurations of interest which we will not attempt here toreview. However, especially noteworthy in the same line ofreasoning is the work by Devaney [11] —refer also to themany references therein by him and co-workers.

As for the so-called MUSIC algorithms, which have beenof large prior use in the signal processing community, onemight point out to the clever comparison of linear samplingand MUSIC made in [12]; linear sampling, e.g., [13], andthe companion factorization method [14], [15] are anothereffective mean to achieve non-iterative imaging. Let us alsorefer to [16] for a demonstration of MUSIC as being —in the electrostatic case (Laplace), yet no limitation of thesaid analysis is expected in electromagnetics— the limit ofthe factorization method when the extended objects which itapplies to are shrinked to small inclusions like ours.

Let us notice the existence of few investigations ofbonafide 3-D buried objects within the full Maxwell’s system ofthe previous linear sampling and similarly minded methods,including [17], [18], whilst to the best of our knowledge thepresent work is the first attempt to analyze in depth the MSRmatrix and correspondingly design a robust MUSIC algorithmfor a small 3-D buried inclusion with arbitrary dielectric andmagnetic contrasts (the material presented readily applying as

well to several inclusions of similar magnitude of size providedthey are well-separated).

The paper is organized as follows. In section 2 the model ofthe scattering problem is given. In section 3 the constructionof the MSR matrix is performed within the asymptotic frame-work. In section 4 its eigenvalue structure is analyzed in fulldetail for either dielectric, or magnetic, or both contrasts. Insection 5 the MUSIC algorithm is described briefly. In section6 illustrative numerical results, such as singular values studiedas a function of the electromagnetic and geometric parametersof the scattering configuration and MUSIC-type images of oneor two inclusions from sparse noisy data, are proposed andcommented upon. A conclusion is given in section 6, mostlyoutlining issues ahead. Necessary results about the calculationof the dyadic Green’s functions in the half-space case, and ofmatrix analysis, are found in the three appendices.

II. T HE MODEL OF THE SCATTERING PROBLEM

Let us consider the following 3-D time-harmonic electro-magnetic scattering problem (the dependencee−iωt is hence-forth implied, the work will be carried out at one givenω.)

A homogeneous volumetric inclusion, which is of the formD = x⋆ + ǫB, with reference (center) pointx⋆, whereB is abounded, smooth (C∞) domain containing the origin, is buriedentirely within the lower half space,R3

− = z < 0, and itlies within an open subsetΩ of R3

−\∂R3− (the planar interface

∂R3− = z = 0 does not cross the inclusion).

In practice one will deal with an inclusion the volume ofwhich is small enough, letting the order of magnitude of itsdiameterǫ ≪ λ−, whereλ−, see next, will be the wavelengthin the lower half-space. In addition, its distance to the interface∂R3

− = z = 0 will be large enough vs.ǫ.Let us notice that whether needed the regularity of the shape

could be considerably weakened, whilst our emphasis here willbe on the simple hypothesis —yet highly versatile in terms ofboth global shape and orientation— of a triaxial ellipsoidalinclusion, and at the numerical level on the case of sphericalshape.

All materials involved in the analysis are assumed to belinear, isotropic, and they are fully characterized, at thesinglefrequency of operation, by their dielectric permittivity andtheir magnetic permeability (both possibly complex-valuedwhenever losses are accounted for, with positive real andimaginary parts). For the two-half-space embedding medium,those read as

(µ, ε)(r) :=

(µ+, ε+) for r ∈ R3

+

(µ−, ε−) for r ∈ R3−

(1)

For the inclusion D, permeability and permittivity, bothconstant-valued, read asµ⋆ and ε⋆. Correspondingly, thewavenumberk is such thatk2(r) = ω2µ(r)ε(r), wherek(r) = k+ for r ∈ R3

+ and k(r) = k− for r ∈ R3− outside

the inclusion, both being chosen with positive imaginary part.This choice of such material characteristics is rather general.

In particular, though one will focus herein onto propagativeelectromagnetic phenomena in the case of electric dipolarilluminations (see next), purely diffusive phenomena likeeddy

IEEE TRANS. ANTENNAS PROPAGAT., SUBMITTED 3

currents disturbed in a conductive embedding half-space illu-minated by electric current loops at quasi-static frequenciescould be considered also (should be in view of the manyapplications in that range of frequency) at a later stage.

The sources of the primary electromagnetic field and thereceivers of the scattered one are placed in the upper halfspace,R3

+ = z > 0, all such elements being at finitedistance from the interface.

We will limit ourselves to the case of a finite numberN ofvertical electric dipoles, each one ideally acting both as trans-mitter and as receiver (we are in the so-called transmit/receivemode), these dipoles being regularly distributed so as to forma unique horizontal squared array of finite size, the step sizeof the array being, as typically, valued to half-a-wavelength inthis upper half space.

Furthermore, we will always consider that the size of thisarray is large with respect to its distance to the inclusion,in tune with the applications which are foreseen as it wasalready mentioned (non-destructive testing, ground probingradar, biological imaging).

Let us emphasize that several mathematical results given inthe present paper will be based upon passing to the limit ofan infinitely large planar source/receiver array, discretesumsover the array aperture being replaced by integral counterpartsas it was already done earlier, e.g., in [1], [3], [5], with goodsuccess. Nevertheless, as it will be seen from the numericalsimulations led for a finite-sized array this does not overlyimpair the conclusions drawn.

As already said, magnetic dipoles, or a mix of dipolesof both electric and magnetic type, other dipole orientationsas well, and distinct source and receiver arrays, could beconsidered also, yielding other MSR matrices. The earlierfree-space analysis [1] that applies to all such arrangements(though at the price of much more notational and mathematicalcomplexity) would again provide us with most useful bricks,and careful application of field duality in particular wouldhelpus to achieve the sought-after results.

From the above geometry of the scattering configuration,even if the illumination/observation aperture is large, one seesthat only aspect-limited data in the reflection mode are madeavailable. In addition, there is no compensation of this lack ofspace diversity provided either by frequency diversity or bypolarization diversity.

Now, in accord with, e.g., the encompassing analysis ofelectromagnetic fields made in [19], the incident electric andmagnetic fields observed at any locationr within the two-half-space medium, in absence of inclusion in the lower halfspace, due to an ideal electric dipole with amplitudeI0 set atarbitrary positionrn in R3 \Ω (it can be located in either halfspace but not inside the inclusion) and directed into arbitraryα′ direction, are

E(n)0 (r) = iωµ(rn)Gee(r, rn) · α′I0

H(n)0 (r) = µ−1(r)µ(rn)Gme(r, rn) · α′I0

The dyadic Green’s functions of the embedding medium foran electric excitationGee and G

me appearing in the abovesatisfy the dyadic differential equations

-

6

h

6

?

e u u u u

u u u u

u u u u

z

xy

µ+, ε+

µ−, ε−

J(n)06

receivers

Fig. 1. Schematic drawing of the configuration under study: two ho-mogeneous dielectric and/or magnetic half spaces, homogeneous inclusionembedded within the lower one, finitely-sized planar array of vertical electricdipole sources and receivers in the upper one placed parallel to the horizontalinterface and operated in the transmit/receive mode (each dipole, here depictedby J

(n)0 , radiates in turn, all dipoles including itself collect theresulting signal,

yielding the MSR matrix).

∇×µ−1(r)∇×Gee(r, r′) − ω2ε(r)Gee(r, r′) =

= µ−1(r)Iδ(r − r′)

Gme(r, r′) = ∇×G

ee(r, r′)

(2)

I unit dyad, plus proper radiation conditions at infinity —continuity of the transverse components of both dyads andjumps of their normal components at any surface of disconti-nuity of the electromagnetic parameters would proceed fromthe above.

A detailed expression of the solutions of (2) is given inAppendix I, another, somewhat less symmetric, form beingfound in [20], [21], whilst one could use also the alternativerepresentations in multiply layered media proposed in [22],among many other worthwhile forms.

The corresponding magnetic-magnetic and electric-magnetic Green’s dyadsGmm and G

em in the embeddingmedium associated to a magnetic excitation follow by dualityin straightforward fashion. As for the reciprocity relationshipssatisfied by the four Green dyads, either by direct applicationof the reciprocity theorem, or more tediously by starting fromthe dedicated forms given in Appendix I, one is able to arriveat

µ(r′)Gee(r, r′) = µ(r) [Gee(r′, r)]t

ε(r′)Gmm(r, r′) = ε(r) [Gmm(r′, r)]t (3)

k2(r′)Gme(r, r′) = k2(r) [Gem(r′, r)]t

with upper indext as the mark of transposition.Let us notice that if the first two relationships are well-

known, the third one, which is linking the magnetic-electricand electric-magnetic dyads, is not so readily found to ourknowledge, in the standard textbooks at least, see [19] and[23]. Also, one is introducing the primed Green’s functionG

em′

(r, r′) = −Gem(r, r′) = ∇′×G

mm(r, r′).

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Then, appropriate use of a vector-dyadic Green’s theoremyields the Lippman-Schwinger (contrast-source) integralfor-mulation for the electric field in the presence of the inclusionas

E(n)(r) − E

(n)0 (r) =

=

∫

D

dr′[−

iωε−ε(r)

(µ⋆ − µ−)Gem′

(r, r′) ·H(n)(r′)

+ω2µ− (ε⋆ − ε−)Gee(r, r′) · E(n)(r′)

]

Via duality and reciprocity a similar form of the scatteredmagnetic fieldH(n)(r) − H

(n)0 (r) follows.

Notice that,de facto, we are placing ourselves in a so-callednear-field configuration, the transmitter/receiver array and thescatterer being placed at finite, commensurable distances fromthez = 0 interface, and all fields and dyadic Green’s functionsabove being calculated with no hypothesis on ranges.

However, extension to far-field setups could be carried outfrom a proper far-field asymptotics of the dyadic Green’sfunctions in this two-half-space case. This would be in lineto[1] (free-space model in 3-D electromagnetics) and [6] (fullmathematical and numerical analysis in 2-D for inclusionsburied in a half-space), again keeping in mind that the apertureof illumination/observation has to remain large enough.

As for generalization to multiple inclusions likewise in freespace [1], it would be straightforward, by simply carryingout the integration over all volumes of the inclusions. Fur-thermore, for a layering, e.g., two or more planar interfacesencountered, it would be sufficient to replace term-to-termtheactual dyadic Green’s functions by those pertinent, the latterbeing constructed via generalized transmission and reflectioncoefficients, e.g., [19].

III. T HE CONSTRUCTION OF THE ASYMPTOTIC

MULTI -STATIC RESPONSE MATRIX

Let us again emphasize at this stage of the analysis thatthe asymptotic derivation the result of which is describednext obliges us to deal with a small enough inclusion, byletting ǫ ≪ λ−, where λ− is the probing wavelength asseen in the lower half-space, whilst their distance to theinterface z = 0 should be large enough vs.ǫ. The latterrestriction could however be lifted though at the price offurther complexity, refer to [1] and remarks below. As forthe exact field representation, size of inclusion and distance tointerface are of no worry.

In accord with the earlier analysis of [1] since one is treatingthe same type of Lippman-Schwinger integral formulationsand dealing with similarly behaving dyadic Green’s functions,and referring the reader for any further material of generalscope to [4], [24], the main result from which the MSR matrixis built up reads as follows:

For any observation pointr away from source pointrn

(again both might be in the same half space or in differentones) and from the inclusion centerx⋆, the following asymp-totic expansion of the electric field (again by duality one will

get the magnetic field) as a function of the size of the inclusionholds:

E(n)(r) − E

(n)0 (r) =

= −ω2ε(rn)µ− k2− G

em′

(r,x⋆) · Mµ · H

(n)0 (x⋆)

+µ−

iωG

ee(r,x⋆) ·Mε ·E

(n)0 (x⋆) + O(ǫ4)

(4)

being stressed that the remainder of the series expansion startsat ǫ5 for an inclusion with a center of symmetry and no moreǫ4, which then adds to the expected accuracy of the leadingterm.

In the above, the so-called Generalized Polarization Tensors(GPT) play an essential role. Those (proportional to thevolumeǫ3) read as

Mµ = ǫ3

iωε−ε(r)

µ(rn)

k2−k2(rn)µ−

(µ⋆ − µ−)M (µ⋆/µ−; B)

Mε = ǫ3 iω3 (ε⋆ − ε−)M (ε⋆/ε−; B)

where M(q⋆/q0; B) is the polarization tensor associated toinclusionB with contrastq⋆/q0 (i.e., µ⋆/µ− or ε⋆/ε−). Thelatter is available in analytically closed form for a triaxial el-lipsoid and degenerate shapes (ball), refer to [1] and referencestherein, e.g., [25].

The above formulation is valid as well for a perfectlyelectric conductor (PEC) by simply letting the permittivity andthe magnetic permeability of the inclusion pass properly to∞and0, respectively, the case of a perfectly magnetic conductor(PMC) following by direct application of duality.

In the case of a spherical inclusionD its polarization tensorM(q⋆/q0; B) has the following explicit form

M(q⋆/q0; B) =3q0

2q0 + q⋆|B| I (5)

If one were to consider an inclusion close to the interfacez = 0 (e.g., magnitude of the inclusion sizeǫ of the order ofits distance to this interface) the above formulation wouldstillbe correct upon introduction of a proper GPT accounting forinteraction with the interface.1 Let us point out also that at firstorder (and we are focusing onto this first order) any regularenough volumetric inclusion can be represented in canonicalfashion by an ellipsoid with the same polarization tensor [4].

So far, the electromagnetic formulation has been set upfor an arbitrarily located and orientated dipole source andanarbitrary observation point (the only limitation is that they liesomewhat far from the inclusion).

Let us now only consider for brevity2 two coincident trans-mitter and receiver arrays made ofN vertical electric dipoles(with fixed transmitted amplitudesI0) centered within the

1Only the case of a spherical inclusion should be amenable, bysolving aboundary value problem in a bi-spherical coordinate system, to a closed formexpression; other cases should require brute-force numerical calculation from,e.g., a boundary integral formulation.

2The general case, if of interest to an end’s user, could be consideredlikewise, but it would require us to keep intricate notations like in [1] andthereupon go through tedious calculations, at the price of comprehensibilityto the reader, so we have opted for this specific situation here.

IEEE TRANS. ANTENNAS PROPAGAT., SUBMITTED 5

planez = h in R3+ (Fig. 1). For any space pointx in R3

−\∂R3−,

let us introduce the matricesGe(x) andGh(x) ∈ CN×3 as

Ge(x) = µ+

[G

ee,T (x, r1) · z, . . . ,Gee,T (x, rN ) · z]t

Gh(x) = k2+

[G

me,T (x, r1) · z, . . . ,Gme,T (x, rN ) · z]t

whererp, p = 1, . . . , N denotes the position of thepth arrayelement andT denotes the transmitted part (since source andobservation points are separated by the interface) of the dyadicGreen’s functions. Let us note that reciprocity here means that

µ+ Gee,T (x, r) = µ−

[G

ee,T (r,x)]t

k2+ G

me,T (x, r) = −k2−

[G

em′,T (r,x)]t

Using equation (4), the Multi-Static Response matrixA ∈CN×N is readily formed, and is decomposed as

A = G(x⋆)

[M

ε 00 M

µ

]Gt(x⋆) (6)

whereG(x) =[Ge, Gh

](x).

IV. A NALYSIS OF THE EIGENVALUE STRUCTURE OF THE

MSR MATRIX

In the Cartesian coordinate system, see Fig. 1, the positionof the jth dipole can be written asrj = xrj,x + yrj,y + zh =rj,s + zh, whereh > 0 and the position of the center of theinclusionD asx⋆ = xs,⋆ + zx⋆, x⋆ < 0. We will study thefollowing three cases:

1) Dielectric inclusions:In this case magnetic permeabilityµ⋆ = µ−. By letting G(x) = Ge(x) in equation (6) the MSRmatrix A can be rewritten asA = G(x⋆)M

ε Gt(x⋆). Then,let us define the symmetric positive semidefinite matrixQ =(G∗G

)(x⋆), with upper index∗ as the mark of Hermitian

adjoint, as

Q = µ+µ−

n∑

j=1

Gee,T (x⋆, rj) · zz · Gee,T (rj ,x⋆)

From the closed-form spectral expression of the dyadicGreen’s function as is provided by equation (15) in AppendixI,we have the rigorous integral relationship:

µ−z ·Gee,T (r,x⋆) =

=

+∞∫

−∞

dks µ−z · Gee,T

(ks, h, x⋆) eiks·(rs−xs,⋆)

=

+∞∫

−∞

dks b(ks) f−(ks, h, x⋆) eiks·(rs−xs,⋆)

where spectral scalar and tensor coefficientsf± and b areintroduced as

f±(ks, h, x⋆) = e∓i(kz∓h−kz±x⋆) (7)

b(ks) =i

ω2a0

[−kxkz− ,−kykz− , k2

s

]

letting ks = xkx + yky, ks = |ks|, a0 = ε+kz− + ε−kz+and

kz± =√

k2± − k2

s .

The upper sign in equation (7) is chosen whenh < 0 andx⋆ > 0; the lower sign is chosen whenh > 0 andx⋆ < 0. Toensure satisfaction of the radiation condition, we requirethatℜe(kz±) > 0 and ℑm(kz±) > 0 over all valueskx and ky

in the integration. Thus,|ksf±(ks, h, x⋆)| → 0 as |ks| → ∞becausef± is exponentially small wheneverℑm(kz±) → ∞.The above Fourier representation ofµ−z · Gee,T (r,x⋆) thusis an uniformly convergent integral.

From the duality principle we can write the Fourier repre-sentation ofµ+G

ee,T (x⋆, r) · z as

µ+Gee,T (x⋆, r) · z =

=

+∞∫

−∞

dks bt(−ks) f+(−ks, x⋆, h) e−iks·(rs−xs,⋆)

=

+∞∫

−∞

dk′s b

t(k′s) f−(k′

s, h, x⋆) eik′s·(rs−xs,⋆)

Always assuming that the aperture array is large with respectto its physical distance to the inclusion, in the limit thatthe number of array elements becomes large and that thespacing between elements becomes small, the inner productQ = G∗G can be replaced with an integral performedover the aperture of the array. By lettingN(ks, rs,xs,⋆) =b(ks) f−(ks, h, x⋆) eiks·(rs−xs,⋆) we obtain

Q(h, x⋆; k+, k−) =

=

+∞∫

−∞

drs dk′s dks N

∗(k′s, rs,xs,⋆)N(ks, rs,xs,⋆)

=

+∞∫

−∞

dks

(b∗b)(ks)

(f−f−

)(ks, h, x⋆)

(8)

where the dyadic function(b∗b)(ks) is given by

(b∗b)(ks) =

1

|a0|2ω4×

×

k2x|k

2z−

| kxky|k2z−

| −kxkz−k2s

kxky|k2z−

| k2y|k

2z−

| −kykz−k2s

−kxkz−k2s −kykz−k2

s k4s

The above Fourier representation (8) is a convergent inte-gral. We observe that

(b∗b)(0) = 0; all diagonal elements of

b∗b are even functions whereas all non diagonal elements of

b∗b are odd functions; the function(f−f−)(ks) is even.From the above we easily conclude that the3-by-3 matrixQ

is real-valued, and diagonal with nonzero diagonal elements,which are the square norms of the vector columns ofG(x⋆).Thus, the matrixG(x⋆) has rank3. From the material re-minded in Appendix II, it follows that rankA = 3 and thevector columns ofG(x⋆) belong to the range ofA as isdenoted byR(A). Therefore, for any vectora 6= 0 ∈ C3,the vectorG · a belongs toR(A).

Moreover, in this supposedly infinite array case, the vectorcolumns of the matrixG are orthogonal; so, if the inclusionD is a ball (which means that the polarization tensorM

ε

is proportional to the identity matrix, refer to equation (5)),then the nonzero singular values of the matrixA are simply

IEEE TRANS. ANTENNAS PROPAGAT., SUBMITTED 6

proportional to the diagonal elements ofMε, and the corre-

sponding left singular vectors are normalized vector columnsof G = Ge(x⋆).

In this case (the ball), in view of the structure of the dyadicfunction

(b∗b)(ks), the matrix AA∗ admits two distinct

eigenvalues, one eigenvalue being of multiplicity2. The eigen-vector ofAA∗ (or left singular vector ofA) corresponding tothe eigenvalue ofAA∗ of multiplicity 1 (or singular value ofA) is the third normalized vector column ofG(x⋆). Notice thatthis eigenvalue is the largest eigenvalue whenever|k+| ≥ |k−|and the smallest eigenvalue whenever|k−| ≫ |k+|), as isillustrated by the numerical examples proposed thereafter.

2) Permeable inclusions:The analysis can be carried out insimilar fashion as previously, yet results significantly differ andwe feel that one should still consider every step to comprehendthem properly. So the details are provided for.

The starting point is now that dielectric permittivityε⋆ =ε−. Then, by lettingG(x) = Gh(x) in equation (6), the MSRmatrix A can be rewritten asA = G(x⋆)M

µ Gt(x⋆). Letus now calculate the symmetric positive semidefinite matrixQ =

(G∗G

)(x⋆) by

Q = −k2+k2

−

n∑

j=1

Gme,T (x⋆, rj) · zz ·Gem′,T (rj ,x⋆)

From equation (18) in Appendix I, using the duality principle,we have that

−k2−z · Gem′,T (r,x⋆) =

=

+∞∫

−∞

dks k2−z · G

em,T(ks, h, x⋆) eiks·(rs−xs,⋆)

=

+∞∫

−∞

dks c(ks) f−(ks, h, x⋆) eiks·(rs−xs,⋆)

wheref−(ks, h, x⋆) is defined by (7) and where tensor coef-ficient

c(ks) =k2−ε+

a0

[−ky, kx, 0

]

The above Fourier representation of−k2−z ·Gem′,T (r,x⋆) is

an uniformly convergent integral. We see also that the thirdvector column ofG(x) = Gh(x) is zero.

Taking the transpose of−k2−z·Gem′,T we obtain the Fourier

representation ofk2+G

me,T (x⋆, r) · z:

k2+G

me,T (x⋆, r) · z =

=

+∞∫

−∞

dk′s c

t(k′s) f−(k′

s, h, x⋆) eik′s·(rs−xs,⋆)

Analogously to the previous purely dielectric contrast case,the inner productQ can be written as an integral:

Q(h, x⋆; k+, k−) =

=

+∞∫

−∞

dks

(c∗c)(ks)

(f−f−

)(ks, h, x⋆)

(9)

wherein

(c∗c)(ks) =

∣∣∣∣k2−ε+

a0

∣∣∣∣2

k2y −kxky 0

−kxky k2x 0

0 0 0

Here, we see that(c∗c)(0) = 0 and all diagonal elements

of the nonzero2-by-2 submatrix ofc∗c are even functionswhile non diagonal elements ofc∗c are odd ones.

From the convergence of the Fourier representation (9) weconclude that the3-by-3 real matrixQ is diagonal, its two firstnonzero diagonal elements being the square norms of two firstnonzero vector columns ofG(x⋆). Thus, the matrixG(x⋆) hasrank 2. From Appendix II it follows that rankA = 2 and therange ofA is spanned by two first nonzero vector columns ofG.3 Therefore, for any vectora = as + zaz ∈ C

3, such thatas 6= 0, the vectorG · a belongs toR(A).

As previously, in this supposedly infinite array case, thevector columns of the matrixG are orthogonal; and if the in-clusionD is a ball (the polarization tensorMµ is proportionalto the identity matrix), then the nonzero singular values ofthematrix A are proportional to the two first diagonal elementsof the polarization tensor and the corresponding left singularvectors are two first nonzero normalized vector columns ofG = Gh(x⋆).

In view of the dyadic function(c∗c)(ks), it follows that

the singular values ofA might be identical (this is true in thecase of a spherical inclusion, of course), i.e., the matrixAA∗

might have one nonzero eigenvector of multiplicity2.3) Dielectric and permeable inclusions, or PEC ones:In

this case one has bothµ⋆ 6= µ− and ε⋆ 6= ε− with specificvalues in the PEC case. Then, from equations (6), (8) and (9)the inner productQ =

(G∗G

)(x⋆), whereG = [Ge, Gh], is

given by

Q(h, x⋆; k+, k−) =

=

+∞∫

−∞

dks d(ks)(f−f−

)(ks, h, x⋆)

(10)

wheref−(ks, h, x⋆) is defined by (7), the tensord(ks) by

d(ks) =

[b∗b b

∗c

c∗b c

∗c

](ks)

and where dyadic coefficientc∗b is given by

(c∗b)(ks) =

ik2−ε+

ω2|a0|2

kxkykz− k2

ykz− −kyk2s

−k2xkz− −kxkykz− kxk2

s

0 0 0

Combining the results of the two previous cases, in viewof the structure of the dyadc∗b, we are able to conclude thatthe 6-by-6 matrix Q is Hermitian of the form of the matrixgiven by equation (21) in Appendix III, and has rank5. ThusA has rank5 as well and, since the nonzero vector columns ofG(x⋆) belong to the range ofA (refer to Appendix II), then

3In this case, the singular values with corresponding singular vectors ofAcan be found analogously to Appendix II, whereG ∈ CN×3 is partitioned asG = [G2|0], G2 ∈ CN×2 with rankG2 = 2. Then the matrixA becomesA = G2M2Gt

2 with nonsingular2-by-2 matrix M2.

IEEE TRANS. ANTENNAS PROPAGAT., SUBMITTED 7

for any vectora ∈ C6 such thatG · a 6= 0 the vectorG · abelongs toR(A).4

From Appendix III it follows that the vector columns ofGe

and the nonzero vector columns ofGh are linearly independentbut not orthogonal; now, if the inclusionD is a ball the matrixM = diag(Mε,Mµ) is diagonal. LetΛ = diag(λ1, . . . , λ6)be the matrix of the eigenvalues ofQ given by (22), and areassociated to the eigenvectors denoted byU = [u1, . . . , u6].It is easy to show that the nonzero singular values ofAare the nonzero diagonal elements of the matrixMΛ andthe corresponding left singular vectors of theA are first fivenormalized vector columns of the matrixG(x⋆)U .

To conclude this section, let us emphasize that therank of the matrixQ(h, x⋆, k+, k−), defined by equations(8), (9), (10), i.e., the rank ofA, does not depend upon thegeometrical parametersh, x⋆ and the propagation constantsk+, k− since the diagonal elements ofQ are the square normsof the vector columns of the matrixG. Therefore, the rankof A does not depend upon the position of inclusionx⋆.However, these parameters are expected to have influence onthe magnitudes of the elements ofQ, i.e., on the respectivemagnitudes of the nonzero singular values of the matrixA, asis shown indeed in section VI.

V. THE MUSIC ALGORITHM

If the dimension of the signal space,s, is known or isestimated from the singular value decomposition ofA, definedby A = UΣV ∗, then the MUSIC (standing for MultipleSignal Classification) algorithm applies, examples of thatbeing found, e.g., in [6], [1], [11], [14]. In particular, for theconfiguration depicted in Fig. 1, it is shown thats = 3 in thecase of a single dielectric inclusion;s = 2 in the case of asingle permeable inclusion ands = 5 in the case of a single,both dielectric and permeable (or PEC) inclusion.

Furthermore, for any vectora ∈ Cp, where the dimensionp of the vectora has been chosen accordingly to the threecases considered above, such asG(x) · a 6= 0, and for anyspace pointx within the search domain, a map of the estimatorW (x) defined as the inverse of the squared Euclidean distancefrom the Green’s vectorG(x) · a to the signal space,

W (x) = 1/

N∑

i=s+1

|〈Ui, G(x) · a〉|2

peaks (to infinity, in theory) at the center of the inclusionx⋆. Let us emphasize the fact that this algorithm implies thatN > s.

Let us point out here, in view of equation (20), thatthe functionW (x) does not contain any information aboutthe shape and the orientation of the inclusion. Yet, if theposition of the inclusions is found (approximately at least) viaobservation of the map ofW , then one could attempt, using thedecomposition (6), to retrieve the polarization tensors (whichare of orderǫ3), and to infer the contrastsε⋆/ε0 or µ⋆/µ0

themselves. Also, in the specific case of a general ellipsoid,

4We impose the conditionG · a 6= 0 because the matrixG has one zerovector column. See also the case of a permeable inclusion.

the recently proposed analysis [5] which yields the orientationof this ellipsoid from the behavior of the singular values forseveral illuminations might be applied here.

VI. N UMERICAL ILLUSTRATIONS (DIELECTRIC CASE)

A. The general set-up

Let us refer to the configuration under study as sketched inFig. 1. All dimensions henceforth are in meters. Permittivitiesof the upper and lower half spaces areε+ = ε0, ε− = 4ε0, thepermeabilities being all valued atµ0, with ε0 andµ0 the valuesin air. The temporal frequency of operation isf = 300 MHz.The corresponding wavelengthλ+ in the upper half space thusis 1. The planar transmitter/receiver array, symmetric aboutthe axis z, is consisting of8 × 8 vertical electric dipolesdistributed at the nodes of a regular mesh with a half-a-wavelength step size, typically located half-a-wavelength awayfrom the interface (h = 0.5).

The dyadic Green’s functions needed are calculated fromthe expressions given in Appendix I once the double spectraldomain integrals involved are transformed into single Fourier-Bessel ones using classical means [20], [21].

The integration itself (noticing one is interested only intotransmitted parts of the dyadic Green’s functions) is performedvia a carefully validated yet standard MATLAB subroutine. Inparticular one does not attempt to fasten the computations asit was investigated in those references, our only goal at thepresent stage being to provide values of the MSR matrix asinitial data of the problem, to illustrate the singular structure ofthat MSR matrix, and to exemplify how the MUSIC algorithm(which involves the calculation ofG(x) throughout the searchdomain), might then work.

B. Distribution of singular values ofA

We consider the case of one spherical inclusion with diame-ter ǫ = 0.1, permittivity ε⋆ = 5ε−, and permeabilityµ⋆ = µ0.Results are in Figs. 2-4.

Letting the inclusion be centered atx⋆ = (0, 0,−1), thedistribution of the three first nonzero singular values ofAas a function of the array heighth is shown in Fig. 2, forh ∈ [0.25, 0.5, 0.75, . . . , 2.75]. (All other singular values arealmost zero, see Fig. 4 as an illustration.)

Here we are in the case of|k−| ≫ |k+|; thus the matrixA admits two distinct eigenvalues with the largest eigenvaluebeing of multiplicity 2; the eigenvector ofAA∗ (or singularvector ofA) corresponding to the smallest eigenvalue ofAA∗

(or singular value ofA) is the third normalized vector columnof G(x⋆).

For large values ofh in the Fourier representation ofQ,the functionf−(ks, h, x⋆) is decaying rapidly as|ks| → ∞.Therefore, the magnitudes of the nonzero singular valuesdecrease whenh is increased. The same conclusion holdswhen h is fixed and |x⋆| → ∞. So, it can be expectedthat going farther and farther away of the interface (far-fieldhypothesis) and adding noise should degrade the performanceof the MUSIC imaging algorithm.

Now, let us keeph = 0.5 and the same inclusion center asbefore. The distribution of three first nonzero singular values

IEEE TRANS. ANTENNAS PROPAGAT., SUBMITTED 8

0 0.5 1 1.5 2 2.5 310

−1

100

101

102

Distance from the interface, h

log 10

(σj)

σ1

σ2

σ3

Fig. 2. Distribution of the three largest singular values (the first two areconfounded) ofA as a function of the distance between the transmitter/receiverarray and the interfacez = 0.

of A as a function of the contrast|ε−/ε+| = |ε−/ε0| is ex-hibited in Fig. 3. Hereε− ∈ ε0[1/4, 1/3, 1/2, 1, 2, 2.3, 3, 4, 5].

As it is shown by this numerical simulation, the matrixAA∗

has an eigenvalue of multiplicity3 for ε− ≈ 2.3ε0 (for theparametersh, x⋆ andλ± chosen above).

We observe that for|ε−| < 2.3|ε0| the matrixAA∗ admitsone eigenvalue of multiplicity1 and one (smaller) one ofmultiplicity 2; and for |ε−| > 2.3|ε0| the matrixAA∗ admitsone eigenvalue of multiplicity1 and one (larger) one ofmultiplicity 2.

Also, if |k+| < |k−|, the magnitudes of the singular valuesof A are larger than those of the singular values ofA when|k+| > |k−|. This in particular means that the case|k+| < |k−|(rather realistic if the probing array is in air and the inclusionin some subsoil, artificial or biological material) should bebetter tailored to MUSIC imaging sinceσ1 = σ2 do not tendto zero.

Let us still keeph = 0.5, ε− = 4ε0, and ε⋆ = 5ε−, butmove the inclusion at given fixed depth along they axis, withcenter atx⋆ = (0, y,−1). The distribution of the singularvalues ofA as a function of the position of the inclusion isshown in Fig. 4, wherey = [−1.5,−1,−0.5, 0, 0.5, 1, 1.5]. Asone main observation, one has that the rank of the matrixAdoes not depend upon the position of the inclusionx⋆.

C. Application of the MUSIC imaging algorithm

The parameters of the media involved are the same as before(ε− = 4ε0, andε⋆ = 5ε−), andh = 0.5, permeabilities beingall valued atµ0.

The case of one spherical inclusion (diameterǫ = 0.1) isconsidered first. It is centered atx⋆ = (0.15, 0.23,−1). Thedistribution of singular values of the MSR matrix (noisy data,with 30 dB signal-to-noise ratio) is exhibited in Fig. 5(a),illustrating that as expected 3 of them (or better said, 1 ofmultiplicity 2, 1 of multiplicity 1) characteristic of the signalsubspace are strongly emerging from those (61 of them, sincethe MSR matrix is 64 x 64) of the noise subspace.

0 1 2 3 4 510

−1

100

101

102

| ε−,j

/ ε+|

log 10

(σj)

σ1

σ2

σ3

σ1

σ1, σ

2

σ3

σ2, σ

3

Fig. 3. Same distribution of singular values as in Fig. 2 in function of thepermittivity contrast|ε−/ε+| = |ε−/ε0|.

−1.5 −1 −0.5 0 0.5 1 1.5

10−15

10−10

10−5

100

σn, n>3

y

log 10

(σj)

σ1

σ2

σ3

Fig. 4. Same distribution of singular values as in Fig. 2 in function of theposition of the inclusionx⋆ = (0, y,−1). Notice that all values of orderhigher than 3 are in the numerical noise.

Maps of the estimatorW (x) in cross-sectional planesx =0.15, y = 0.23 and z = −1 are displayed in Fig. 6(a), 6(b)and 6(c), respectively. The corresponding 3D plot is displayedin Fig. 5(b), the surface shown being associated to values ofhalf the peak magnitude.

Let us notice that the resolution in a direction parallel to thetransmitter/receiver array looks like a fraction of the onein adirection perpendicular to it, as already observed [1], [6], [7],[8]. Typical spheroidal-like shapes of iso-magnitude surfacesare observed as well; again, this phenomenon is independentof the shape of the inclusion itself, and is only linked tothe specific aspect-limited arrangement of illumination andobservation chosen.

Let us now consider two identical spherical inclusions (withdiameterǫ = 0.1) centered atx1 = (0.4,−0.6,−0.75) andx2 = (−0.4, 0.6,−1.5). The distribution of singular valuesof the MSR matrix (noisy data again with 30 dB signal-to-noise ratio) is exhibited in Fig. 7(a), illustrating that 6of them —the 3 expected per inclusion, yet with no more

IEEE TRANS. ANTENNAS PROPAGAT., SUBMITTED 9

1 5010

−3

10−2

10−1

100

101

102

Singular Values of A (64×64), λ=1

Singular Value Number, σj

log 10

(σj)

(a) (b)

Fig. 5. (Dielectric contrasts only): distribution of the singular values ofA for 8 × 8 singly-polarized dipolar transmitters and receivers in the case of noisydata with 30dB signal-to-noise ratio (a); corresponding 3Dplots of W (x) (b).

−1−0.5

00.5

1

−1

−0.5

0

0.5

1

−2

−1.5

−1

−0.5

0

X axisY axis

Z a

xis

2

4

6

8

10

12

14x 10

7

(a)

−1−0.5

00.5

1

−1

−0.5

0

0.5

1

−2

−1.5

−1

−0.5

0

X axisY axis

Z a

xis

2

4

6

8

10

12

14x 10

7

(b)

−1−0.5

00.5

1

−1

−0.5

0

0.5

1

−2

−1.5

−1

−0.5

0

Z a

xis

X axisY axis

2

4

6

8

10

12

14x 10

7

(c)

Fig. 6. (Dielectric contrasts only): color maps ofW (x) in cross-sectional planesx = 0.15 (a), y = 0.23 (b) andz = −1 (c) in the case of noisy data with30dB signal-to-noise ratio (refer to 5(a) and 5(b)).

specific multiplicity— are emerging from those (58 of them)of the noise subspace. (Here, let us refer to [1] for the caseof multiple inclusions.)

Maps of the estimatorW (x) in cross-sectional planesx = 0.4,−0.4, y = −0.6, 0.6 and z = −0.75,−1.5are displayed in Fig. 8(a), 8(b) and 8(c), respectively. Thecorresponding 3D plot is displayed in Fig. 7(b), the surfacesshown being associated to values of half the peak magnitude.

Similar observations as before about transverse vs. longi-tudinal resolution and aspect of the maps hold. Let us noticein addition that the images produced are less sharp than inthe case of one single inclusion, while it is obvious thatif the two inclusions were about one atop the other, theywould be very difficult to differentiate. Let us remind thatno assumption on the number of inclusions is made in thereconstruction procedure whereas the specific configuration ofstudy (transmit/receive mode, with fixed location of the array)cannot yield an isotropic resolution.

VII. C ONCLUSION

To conclude, let us first emphasize that only a small numberof numerical examples is shown in the present paper, in ratheracademic situations, with no claim to covering all possiblesituations of interest.

If the results still tend to substantiate the mathematicalanalysis before, numerical work should certainly be pursuedespecially if one has a given application in mind, beyond theproof-of-concept which one has focused onto in the presentpaper.

Also, in view of the size of the inclusion(s) it is expectedthat the asymptotic MSR matrix used as data in the examplesis quite accurate (and as good as those data that might beprovided by a brute-force finite-element or moment method,being said the smallness of the inclusion would be a challengeper seto those methods) whilst adding severe noise as it wasdone when applying the MUSIC algorithm is already a firststep towards the handling of real data.

IEEE TRANS. ANTENNAS PROPAGAT., SUBMITTED 10

1 5010

−3

10−2

10−1

100

101

102

Singular Values of A (64×64), λ=1

Singular Value Number, σj

log 10

(σj)

(a) (b)

Fig. 7. (Dielectric contrasts only): distribution of the singular values ofA for 8 × 8 singly-polarized dipolar transmitters and receivers in the case of noisydata with 30dB signal-to-noise ratio (a); corresponding 3Dplots of W (x) (b).

−1−0.5

00.5

1

−1

−0.5

0

0.5

1

−2

−1.5

−1

−0.5

0

X axisY axis

Z a

xis

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5x 10

7

(a)

−1−0.5

00.5

1

−1

−0.5

0

0.5

1

−2

−1.5

−1

−0.5

0

X axisY axis

Z a

xis

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5x 10

7

(b)

−1−0.5

00.5

1

−1

−0.5

0

0.5

1

−2

−1.5

−1

−0.5

0

Z a

xis

X axisY axis

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5x 10

7

(c)

Fig. 8. (Dielectric contrasts only): color maps ofW (x) in cross-sectional planesx = 0.4,−0.4 (a), y = −0.6, 0.6 (b) andz = −0.75,−1.5 (c)in the case of noisy data with 30dB signal-to-noise ratio (refer to 7(a) and 7(b)).

Evidently larger inclusions might/should lead to less accu-rate MSR matrices, with no possibility to mimic the discrep-ancies via addition of noise to asymptotic data, and it is oneof the topics of current interest to the authors.

As for the quite worthwhile topic of uncertain or stochasti-cally varying embedding media, it seems that a lot remain tobe done about it once said that an averaging process shouldprovide us with suitable dyadic Green’s functions associatedto the embedding medium.

If much lies ahead in terms of the mathematical analysisitself (for a magnetic transmitter/receiver array already, and/orfor separate transmitter and receiver arrays, with or withoutsimilar nature and polarization), one believes that most toolsneeded are already made available from the present investiga-tion, the case of transmitters/receivers set in the far-field beingslightly more demanding at the level of the treatment of thedyadic Green’s functions at least.

A possibly numerous collection of inclusions would call

for generalization of the work that has been previously ledin free space [1]. Again most tools appear already available,whereas it seems difficult to get highly meaningful resultsbeyond the case of well-separated inclusions for which oneis de factotreating a collection of independent scatterers —yet, if two inclusions get too close together they appear as asingle equivalent inclusion.

More pressing might then be to find or to confirm effectivemeans to retrieve the shape and orientation parameters of ageneral triaxial ellipsoid, borrowing, e.g., some of the solutionfrom the present investigation and some from [5], being saidthat a thorough numerical study will also be needed.

Other situations certainly remain challenging. The diffusivecase, as hinted to already, must be addressed not only becauseof specific mathematical setting involved but also if oneintends to apply the present framework to the demanding casesof eddy-current non-destructive testing and low-frequencyEarth probing, for example.

IEEE TRANS. ANTENNAS PROPAGAT., SUBMITTED 11

As for inclusions close to an interface (say, at a distanceof a few ǫ), and coupled to it, this is a generic configurationpossibly found in many practical fields of interest; the lackof an easy model of this interaction is expected to hinder theinvestigation yet a combination of closed-form solutions andnumerical machinery might at least enable us to attack thecase of a shallowly buried sphere.

APPENDIX ITHE DYADIC GREEN’ S FUNCTIONS FOR A HALF SPACE

We consider the electric-electric dyadic Green’s functionasthe solution to (2), when the source pointr

′ lies either in theupper half space,R3

+, or in the lower half space,R3−.

In the Cartesian coordinate system, referring again to Fig.1,the primary (P ) electric-electric dyadic Green’s function (theone occurring in a homogeneous infinite space) is given by

Gee,P =

[I +

∇∇

k2±

]eik±|r−r

′|

4π|r − r′|(11)

where the upper sign, ”+”, is chosen whenr, r′ ∈ R3+ and the

lower sign, ”–”, whenr, r′ ∈ R3−.

We introduce the 2D Fourier transform as

Gee(r, r′) =

1

(2π)2

+∞∫

−∞

dks Gee

eiks·(rs−r′s) (12)

where Gee

= Gee

(ks, z, z′) denotes the spectral dyadicGreen’s function and whereks = xkx + yky and r =rs + zz. The reflected (R) and transmitted (T ) electric-electricdyadic Green’s functions are now accordingly dealt with inthe spectral domain. The reflected part reads as

Gee,R

± =

(I(1) +

1

k2±

∇∇′

)gTM

R,± (13)

+1

k2s

(∇s × z)(∇′s × z)

(gTM

R,± + gTER,±

)

where the upper sign is chosen whenz, z′ > 0 and the lowersign is chosen whenz, z′ < 0. The transmitted part reads as

Gee,T

± =µ∓

µ±

[(I(2) +

1

k2∓

∇∇

)gTM

T,± (14)

+1

k2s

(∇s × z)(∇s × z)

(gTM

T,± −µ±

µ∓

kz∓

kz±

gTET,±

) ]

or in the matrix form,

Gee,T

± =1

k2s

k2y −kxky0

−kxky k2x 0

0 0 0

gTE

T,± +µ∓

µ±k2∓k2

s

(15)

×

k2xkz±kz∓ kxkykz±kz∓±kxkz∓k2

s

kxkykz±kz∓ k2ykz±kz∓ ±kykz∓k2

s

±kxkz±k2s ±kykz±k2

s k4s

gTM

T,±

where the upper sign is chosen whenz′ > 0, z < 0 and thelower sign whenz′ < 0, z > 0.

Here, one has set

gTM/TER,± (ks, z, z′) =

i

2kz±

RTM/TE± e±ikz± (z+z′) (16)

gTM/TET,± (ks, z, z′) =

i

2kz±

TTM/TE± e∓i(kz∓z−kz±z′) (17)

whereRTM/TE± (resp.T TM/TE

± ) are standard reflection (resp.transmission) coefficients ofTM/TE planar waves, fromR3

+

to R3− (those with the upper sign ”+”) and fromR3

− to R3+

(those with the lower sign ”–”) which are given by

RTM± = ±

ε−kz+− ε+kz−

ε+kz− + ε−kz+

T TM± =

2ε∓kz±

ε+kz− + ε−kz+

RTE± = ±

µ−kz+− µ+kz−

µ+kz− + µ−kz+

T TE± =

2µ∓kz±

µ+kz− + µ−kz+

In the above expressions,kz± =√

k2± − k2

s , with positive

imaginary part; k2s = k2

x + k2y; k+ and k− are the wave

numbers inR3+ (R3

−), respectively. Other quantities involvedare ∇s = x∂x + y∂y, ∇ = ∇s + z∂z, ∇′

s = −∇s, ∇ =

∇± =kz±

kz∓∇s + +z∂z; I

(1) = diag(−1,−1, 1) and I(2) =

I(2)± = diag

(kz±

kz∓,

kz±

kz∓, 1

).

The magnetic-electric dyadic Green’s function producedwith source point either atr′ ∈ R3

+ or at R3− can be

derived from the second equation of (2) and by using the setof equations (11), (13), (14). Here, for brevity, we provideonly the matrix form of the spectral dyadic Green’s function

Gme,T

± (ks, z, z′), where the upper sign is chosen whenz′ >0, z < 0 and the lower sign whenz′ < 0, z > 0:

Gme,T

± =1

k2s

∓ikxkykz∓ ±ik2

xkz∓ 0∓ik2

ykz∓ ±ikxkykz∓0−ikyk2

s ikxk2s 0

gTET,± + (18)

µ∓

µ±

1

k2s

±ikxkykz± ±ik2

ykz± ikyk2s

∓ik2xkz± ∓ikxkykz±−ikxk2

s

0 0 0

gTMT,±

APPENDIX IITHE MAIN RESULTS OF MATRIX ANALYSIS INVOLVED

Let us define a matrixG ∈ Cm×n, n ≤ m and introduceM ∈ Cn×n as a symmetric nonsingular matrix. If the rankof G is n, then the symmetric matrixA ∈ Cm×m defined byA = GMGt has rankn.

Now, let G = UQ be a polar decomposition ofG withdefinite positive matrixQ = (G∗G)1/2, the matrixU havingorthonormal columns (i.e.,U∗U = I, whereI is then-by-nidentity matrix).

Since rankG = n, thenU is uniquely determined. So, thematrix A can be rewritten as

A =(UQ

)M

(UQ

)t

SinceQMQt is symmetric, using the Takagi’s factorization(or singular value decomposition of symmetric matrices, see[26]), the matrixQMQt can be rewritten as

QMQt = WΣW t (19)

IEEE TRANS. ANTENNAS PROPAGAT., SUBMITTED 12

where matrix W ∈ Cn×n is unitary and whereΣ =diag(σ1, . . . , σn), σ1 ≥ . . . ≥ σn > 0. Therefore,

A =(UW

)Σ

(UW

)t

with(UW

)∗(UW

)= I.The nonzero singular values ofA are

the diagonal elements ofΣ. The first n left singular vectorsof A are the columns of the matrixUW ∈ Cm×n form anorthonormal basis for the space ofA, then the range of thematrix A, R(A), is spanned by the vector columns ofUW .

The orthogonal projectionP ∈ Cm×m onto R(A)⊥ =N (A∗) is given by

P = I − (UW )(UW )∗ = I − UU∗ (20)

Now, we observe thatP does not depend upon the matrixM .Moreover, from the polar decomposition ofG, it follows thatfor any vectora 6= 0 ∈ Cn the vectorG · a is in the rangeof A, or in other words, any linear combination of the vectorcolumns of the matrixG belongs toR(A).

APPENDIX IIITHE EIGENVALUES AND EIGENVECTORS OF THE MATRIXA

IN THE GENERAL CONTRAST CASE[3]

Let Q ∈ C6×6 be a matrix of the form:

Q =

b1 0 0 0 h1 00 b2 0 h2 0 00 0 b3 0 0 0

0 h2 0 d1 0 0

h1 0 0 0 d2 00 0 0 0 0 0

(21)

whereb1d2 − h1h1 6= 0 andb2d1 − h2h2 6= 0.The eigenvalues ofQ are given by

λ1,5 =1

2

(b1 + d2 ±

√(b1 − d2)2 + 4|h1|2

)

λ2,4 =1

2

(b2 + d1 ±

√(b2 − d1)2 + 4|h2|2

)(22)

λ3 = b3, λ6 = 0

and are associated to the eigenvectors

u1,5 =1√

|h1|2 + |λ1,5 − b1|2(h1e1 + (λ1,5 − b1)e5)

u2,4 =1√

|h2|2 + |λ2,4 − b2|2(h2e2 + (λ2,4 − b2)e4)

u3 = e3, u6 = e6

whereej , j = 1, . . . , 6 is an orthogonal basis inR6.

ACKNOWLEDGMENT

E. Iakovleva would like to thank W. C. Chew for usefuldiscussions on the formulation of the dyadic Green’s functionsand means to achieve it in most optimal fashion. Interactionwith D. H. Chambers has led all authors to a better com-prehension of the behavior of the MSR matrix, whilst theclose and lasting cooperation with H. Ammari has been muchvaluable to their present work. The Feb. 2004-Jan. 2005 post-doctoral support of E. Iakovleva by ACI Jeune Chercheur 9041is acknowledged.

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