Multi-Static Response Matrix of Spherical Scatterers and the Back-Propagation of Singular Fields

Transcription

1 IEEE TRANS. ANTENNAS PROPAGAT., SUBMITTED Multi-Static Response Matrix of Spherical Scatterers and the Back-Propagation of Singular Fields Ekaterina Iakovleva and Dominique Lesselier, Senior Member, IEEE Abstract In view of the development of MUSIC-type, noniterative imaging procedures which are based on the singular value decomposition of the Multi-Static Response matrix (MSR) of a collection of inclusions, the full 3-D electromagnetic case with arbitrary contrasts of permeability and permittivity (including the PEC or PMC limit cases) is investigated. The structure of the MSR matrix of a single inclusion (or a set of well-separated ones) is analyzed first. Emphasis is put on a far-field situation, in which, for simplicity, one considers a fixed electric dipole array operated in the transmit/receive mode at a single frequency, and one uses appropriate far-field forms of the dyadic Green s functions involved. Next, back-propagated electric and magnetic fields that are associated to the singular vectors of the MSR matrix for a single spherical inclusion (or again for a set of well-separated ones) are given in closed form and their spatial behavior is studied in detail, their leading-order values in particular being exhibited. This is carried out for an inclusion which has a dielectric contrast, or a magnetic one, or both contrasts vs. the embedding space. Numerical illustrations (cross-sectional maps of back-propagated fields computed from the singular value decomposition of the MSR matrix), are presented, for one inclusion or two inclusions, in order to illustrate the modeled behavior as a function of the geometric and electromagnetic parameters of the configuration. Index Terms time-harmonic 3-D electromagnetic scattering - Green s dyadic functions - multi-static response matrix - asymptotic formulation - singular value decomposition - backpropagation I. INTRODUCTION THE retrieval of a collection of small volumetric dielectric and/or magnetic inclusions located in some prescribed domain of a homogenous space has been considered earlier [] via a MUSIC-type non-iterative imaging. This approach involves the singular value decomposition of the Multi-Static Response matrix (MSR) of the collection modeled via an asymptotic series expansion of the scattered electromagnetic field. This procedure yields, as is now well-known, sets of singular values and their associated singular vectors and, via a properly defined MUSIC cost functional, provides a map of the search domain exhibiting peaks at the location of the inclusions. Yet in the said reference, though there were some insistence on the behavior expected from the electromagnetic fields that are back-propagated from the measurement devices onto the search domain for a given singular value of the MSR matrix (as a reminder, every such field would be generated by a source E. Iakovleva is with Centre de Mathématiques Appliquées (CNRS-Ecole Polytechnique) 98 Palaiseau cedex, France. D. Lesselier is with Département de Recherche en Électromagnétisme - Laboratoire des Signaux et Systèmes (CNRS-Supélec-UPS ) 99 Gifsur-Yvette cedex, France. distribution constructed from the associated singular vector of the MSR matrix), their precise behavior was not fully analyzed in mathematical terms neither illustrated in explicit numerical fashion. However, in particular since it is generally asserted in the literature that these fields simply focus onto the sought inclusions, with sometimes little elaboration on that statement, mathematically-sound investigation of the back-propagation and its numerical analysis thereof should enable us to better understand how it is to work in practice, and how it might be used for inversion purposes. In particular, most of the scientists involved with so-called DORT analyses as those carried out within the general body of work on time-reversal DORT uses the fact that the product of the MSR matrix by its transpose yields the time-reversal operator in matrix form or with like methods have been exhibiting and analyzing such fields e.g. [], [3], [4], [5], yet in more restrictive frameworks that the one of full 3-D electromagnetics (refer, e.g. to the conclusions of Devaney s aforementioned reference [5], where the need for further work in this full case is strongly emphasized). As for the numerous acoustical analyses, so interesting they may be in many situations (and one will simply quote without claim to exhaustivity [6] and [7] herein), they cannot be extended straightforwardly due to the major difference of the pertinent wave equations in between acoustics and electromagnetics in 3-D configurations. So, the present work is aimed at providing sound foundations to the analysis of the back-propagation of singular electric and magnetic fields resulting from the decomposition of the MSR matrix, in a full 3-D Maxwell setting, and to illustrate them, pioneering works as in [8] being encompassed as well. Also, it intends to delineate the tools of analysis that could be, at a later stage, applied to configurations of sources and receivers different from the one considered here there is specialization for brevity of the paper to an electric dipole array operated in the transmit-receive mode, being noticed that this case is also a challenge per se since it is providing us with strongly aspect-limited data. More complicated geometries of the embedding spaces could be investigated as well refer, e.g., to the half-space burial of 3-D inclusions recently considered in [9]. In accord with the above, the paper is organized as follows. In section II, the model of the scattering problem at hand is proposed and the corresponding Green-based electromagnetic field formulation which one is starting from is given for an arbitrary collection of 3-D bounded inclusions in free space.

2 IEEE TRANS. ANTENNAS PROPAGAT., SUBMITTED Though this is done for a fixed array of electric dipole sources operated in the transmit/receive mode at a single frequency, as just indicated, the model could be easily extended to more general configurations of sources and receivers. In section III, the asymptotic form of the MSR matrix (of a single inclusion or a set of well-separated ones) is introduced, which is involving generalized polarization tensors of inclusions of arbitrary shapes. Then, one puts specific and careful emphasis on the limit case of far fields, which uses the corresponding forms of the dyadic Green s functions involved, and one aims at spherical inclusions mainly notice that MUSIC-type algorithms as employed up to now can retrieve locations of inclusions via mapping of the appropriate functional, but cannot provide their shapes unless some involved work is performed in addition [], so this limitation is felt to be secondary at this stage. Let us emphasize that even if many details in the section III are left to the copious applied mathematical paper [] already mentioned, the material in this section, though described quickly, is self-contained. In section IV, which is where most of the novelty of the present work is found and which provides the key results, the singular vectors of the MSR matrix are exhibited in closed form and the back-propagated electric and magnetic fields that are associated to the singular vectors of the MSR matrix corresponding to its non-zero singular values for a single spherical inclusion (or a set of well-separated ones) are given. Their spatial behavior is then analyzed in depth, with again insistence on far fields. This task is carried out for purely dielectric contrasts of the inclusions vs. the embedding space, for purely magnetic ones, and when both contrasts occur, each such case having its own subtleties (even though the main lines of reasoning remain similar). Numerical illustrations follow in section V. Those are crosssectional maps of back-propagated electric fields which are computed from the singular value decomposition of the MSR matrix (done in numerical fashion) for one inclusion or for two inclusions. This enables us to confirm the modeled behavior as a function of the geometric and electromagnetic parameters of the configuration, and to discuss the suitability of the results for inversion purposes. After a brief outline of pending issues and forthcoming work, two appendices are proposed. The first one gives a descriptive summary of the behavior of the Green dyads involved in the work and provide useful concatenated forms which one is working with in the main text, the second one reminds of results of matrix analysis needed herein as well. II. THE MODEL OF THE SCATTERING PROBLEM Let us consider the following 3-D time-harmonic electromagnetic scattering problem in free space (dependence e iωt is henceforth implied). A finite number m of volumetric inclusions, each being of the form D j = x j + ǫb j, where B j R 3 is a bounded, smooth (C ) domain containing the origin, and ǫ is the order of magnitude of the size of the said inclusions, is considered. It is further assumed that this collection of inclusions m j= D j lies within in open subset Ω of R 3. z receivers J (n) x h y Fig.. Sketch of the configuration under study. (Only one single dipole transmitter in the array and part of the receivers are shown for readability.) All materials (linear, isotropic, at rest) involved are characterized by their dielectric permittivity and magnetic permeability (they are complex-valued if losses are accounted for, with positive real and imaginary parts). For the homogeneous embedding medium they are µ and ε ; those, µ j and ε j, of the jth inclusion D j, also assumed constant, may evidently differ from those of any other inclusion. The free-space wavenumber k is as usual such that k = ω µ ε. Sources of the primary field and receivers of the scattered field are taken as one single set of electric dipoles. Those are of the same vertical polarization, and they are arranged as a horizontal planar array (for simplicity) which is set at a fixed location in R 3 \Ω and which is operated at a single frequency (one does not attempt to compensate lack of space diversity by frequency diversity). In this situation, each dipole successively acts as a transmitter and the vertical electric field at each dipole location (including the one of the transmitter) is assumed to be collected, which is ensuring the availability of the so-called Multi-Static Response (MSR) matrix of the collection of inclusions in a prescribed, fixed configuration of illumination and observation. Let us emphasize that the asymptotic derivation thereafter requires homogeneous and small enough inclusions with size magnitude ǫ λ, where λ is the wavelength (the exact formulation does not require these assumptions). Also, it will be considered that the inclusions are not too close together (one will discuss that hypothesis later again). The incident electric and magnetic fields that are due to a point electrical current density with amplitude I n at position r n in R 3 \ Ω and directed into the ˆα direction are E (n) (r) = iωµ G(r,r n ) ˆα I n () H (n) (r) = G(r,r n) ˆα I n () where G(r,r n ) denotes the well-known dyadic Green s function for an unbounded, homogeneous medium (free space). Appropriate use of the Green theorem straightforwardly yields the Lippman-Schwinger (contrast-source) vector integral formulation for the electric field in the presence of the inclusions (e.g., [3]) as E (n) (r) E (n) (r) = (3)

3 IEEE TRANS. ANTENNAS PROPAGAT., SUBMITTED 3 = m j= D j dr [ iω (µ j µ ) G(r,r ) H (n) (r ) ] +ω µ (ε j ε )G(r,r ) E (n) (r ) where denotes differentiation with respect to the variable r. Via duality and reciprocity a similar form of the scattered magnetic field H (n) (r) H (n) (r) follows whether needed. III. THE ASYMPTOTIC MSR MATRIX For any r away from r n and x j, in accord with the earlier analysis of [], further material being available in [], one arrives at the asymptotic formulation of the electric field as E (n) (r) E (n) (r) = m [ = G(r,x j ) M µ j H(n) (x ) (4) j= ] +(iωµ ) G(r,x j ) M ε j E (n) (x j) + O(ǫ 4 ) Generalized polarization tensors in the above are M µ j = ǫ3 iω (µ j µ )M(µ j /µ ; B j ) M ε j = ǫ3 iω 3 µ (ε j ε )M(ε j /ε ; B j ) where M(q j /q ; B j ) is the polarization tensor associated to the inclusion B j and the contrast q j /q (i.e., µ j /µ or ε j /ε ). It is available in analytically closed form for a triaxial ellipsoid and degenerate shapes (ball), the above formulation holding as well for perfectly conducting or perfectly magnetic inclusions, e.g., []. In the case of a spherical inclusion D j with volume B j, its (diagonal) polarization tensor M(q j /q ; B j ) has the following explicit form M(q j /q ; B j ) = 3q q + q j B j I (5) Let us remark that the rest of the series expansion starts at ǫ 5 for inclusions with a center of symmetry. For fixed r, using to that effect the material given in Appendix I, the following far-field expansions hold as r + : G(r,r ) = h(kr)e ikˆr r (ˆr) + O ( ) r G(r,r ) = ikh(kr)e ikˆr r Λ(ˆr) + O ( ) (6) r where h(kr) = e ikr /4πr, and where the tensors (ˆr) and Λ(ˆr) are given by (4) and (5), respectively, and satisfy in particular = t, Λ = Λ t with upper index t as the mark of transposition. The scalar r is the magnitude of the vector r and ˆr is a unit vector in the direction r. From ()-(4) and (6) the leading-order term in the asymptotic expansion of E (n) (r) E (n) (r) can be written as E (n) (r) E (n) (r) = m [ = e ik(ˆr+ˆrn) xj Λ(ˆr) M µ j Λ(ˆr n ) (7) j= + (ˆr) M ε j (ˆr ] n ) ˆα I n + O(ǫ 4 ) where (ˆr) = h(kr) (ˆr) and Λ(ˆr) = ikh(kr)λ(ˆr). Let us now consider two coincident transmitter and receiver planar arrays {r,...,r N } made of N vertical electric dipoles (with transmitted amplitude I n, n =,...,N) symmetric about the z-axis and centered within the plane z = h in R 3 (Fig. ). For any point x in R 3 \{r,...,r N }, let us introduce the matrices G e (x) and G h (x) C N 3 : [ t G e (x) = e ikˆr x (ˆr ) ẑ,..., e ikˆrn x (ˆrN ) ẑ] (8) G h (x) = [ e ikˆr x Λ(ˆr ) ẑ,...,e ikˆrn x Λ(ˆrN ) ẑ] t (9) Using (4) the Multi-Static Response (MSR) matrix A C N N is formed, and it is decomposed as m A = G(x j )M j G t (x j ) () j= where G(x) = [ G e, G h] (x) and M j = diag(m ε j,m µ j ). As it has been shown already in [], [9], the rank of the matrices G e (x), G h (x) and [G e, G h ](x) does not depend upon the point x in R 3 \ {r,...,r N }. G e has rank 3, rankg h = and rank[g e, G h ] = 5. Let us notice that this result is true for the specific set of sources and receivers considered, and in particular (but this is outside of the scope of the paper) using sets of dipoles directed along two orthogonal directions would yield rankg h = 3 and accordingly rank[g e, G h ] = 6. Also, for m well-resolved inclusions, A would have rank mp, where p = rankg. Now, if the dimension of the signal space, some s, is known or is estimated from the study of the distribution of singular values of A, decomposed as A = UΣV, the MUSIC algorithm applies. In particular one shows that for any vector a R 6, such as G(x) a, and for any x within the search domain, a map of the estimator W(x) as the inverse of the squared Euclidean distance from the Green s vector G(x) a to the signal space, N W(x) = / U i, G(x) a i=s+ peaks (to infinity, in theory) at the centers of the inclusions. Further study on that matter in the present paper will not be attempted, i.e., good examples of MUSIC mapping given by the authors and colleagues are already available, e.g., [], [9], [], which one could refer to, whilst other results are found in the literature in specific settings. Focus, see below, is on introduction of novel, pertinent material on back-propagation. IV. SINGULAR VECTORS OF THE MSR MATRIX AND BACK-PROPAGATED ELECTROMAGNETIC FIELDS Let us investigate the case of a single spherical inclusion positioned at x = x s, + ẑx and characterized by electric permittivity ε and magnetic permeability µ. As is well understood nowadays, e.g., [8], any singular vector of the MSR matrix (see its decomposition above) specifies a current distribution which whenever properly fed to the array elements (here, the electric dipoles) produces a field in the search space denoted as the back-propagated field.

4 IEEE TRANS. ANTENNAS PROPAGAT., SUBMITTED 4 Much attention is devoted to that field since it is expected (or surmised) to focus onto the scatterers, and as a matter of fact, if properly computed, to provide an image of these scatterers (say, peaks of high magnitude at or near the scatterer locations), each singular vector being hopefully associated to one scatterer only in the collection. This procedure as mentioned already in the introduction has been investigated in a number of studies in electromagnetics, e.g. [], [3], [4], [8], [5], to mention but a few main references, and far many more exist in simpler situations of acoustics. Here, as already said before, one is attempting to enlarge previous work [] in terms of the mathematical findings and to provide an useful interpretation of the back-propagation as well, in a original full Maxwell setting. So, let us focus on the calculation of the back-propagated field produced at point r s +ẑx by the N ideal vertical electric dipoles Three situations of interest emerge. ) Dielectric inclusions: In this case, the magnetic permeability µ = µ, and from (5) and () the MSR matrix A can be rewritten as A = m e Ge (x )[G e (x )] t, where the depolarization factor of the sphere is given by m e = ǫ 3 iω 3 µ 3ε (ε ε ) B () ε + ε and the matrix G e (x ) is given by (8). In accord with what has been shown earlier in [], the vector columns of G e (x ) are orthogonal, and it immediately follows that the jth nonzero singular value of A is given by σ j = m e G e j with corresponding left singular vector U j = G e j / Ge j, j =,, 3, where G e j denotes the jth vector column of Ge (x ). For arbitrary points r = r s + ẑr z and x = x s + ẑx z, let us define a function f(r,x) by f(r,x) = N g(r,r n )g(x,r n ) () n= where g(r,r n ) = g(r n,r) = h(kr n )e ikˆrn r. Upon noticing that g(r,r n ) = k ˆr nˆr n g(r,r n ), from (8) the left singular vectors which correspond to the three nonzero singular values of A can be written as U = x z g(x, r) k G e, U = y z g(x, r) k G e U 3 = (k + z )g(x, r) k G e 3 where g(, r) = [g(,r ),..., g(,r N )] t, and where the square norms of the vector columns of the matrix G e can be written as G e = x z f(x,x )/k 4, G e = y z f(x,x )/k 4, and G e 3 = (k + z) f(x,x )/k 4. From () and () the back-propagated electric and magnetic fields which are produced by one single electric dipole positioned at r n and directed into the vertical ẑ direction are given by E (n) (r) = iωµ h(kr n )e ikˆrn r (r n ) ẑ I n H (n) (r) = ik h(kr n)e ikˆrn r Λ(r n ) ẑ I n or E (n) (r) = iωµ (ẑ + k ) z g(r,rn )I n (3) H (n) (r) = s ẑ g(r,r n )I n where = s + ẑ z. By letting I n = U,n = x z g(x,r n )/(k G e ) and substituting this into the above equation for the electric field and summing (3) over the array index n, the total back-propagated electric field denoted by E e (r) becomes E e (r) = iωµ k 4 G e (k ẑ + z ) x z f(r,x ) (4) In like manner, we can derive the back-propagated electric fields as E e (r) = iωµ k 4 G e (k ẑ + z ) y z f(r,x ) (5) E e 3 (r) = iωµ k 4 G e 3 (k ẑ + z )(k + z )f(r,x ) (6) The back-propagated magnetic fields correspondingly read H e (r) = k G e s ẑ x z f(r,x ) H e (r) = k G e s ẑ y z f(r,x ) H e 3 (r) = k G e 3 s ẑ (k + z )f(r,x ) Noticing that the spherical Bessel function of zero order is given by the angular integral j (kr) = sin(kr) kr = dˆk eik r 4π it follows from () that for any point r = r s + ẑx the function f(r,x ) behaves like j (k r x ). Thus, in the plane z = x, the back-propagated electric fields E e (r), Ee (r) and E e 3(r) behave like E e (r) ( s + ẑ) x j (k r x ) (7) E e (r) ( s + ẑ) y j (k r x ) (8) E e 3(r) ( s + ẑ)j (k r x ) (9) and the back-propagated magnetic fields H e (r), He (r) and (r) as H e 3 H e (r) s ẑ x j (k r x ) H e (r) s ẑ y j (k r x ) H e 3 (r) s ẑ j (k r x )

5 IEEE TRANS. ANTENNAS PROPAGAT., SUBMITTED 5 ) Permeable inclusion (ε = ε ): In this case, from (5) and () the MSR matrix A can be rewritten as A = m h G h (x )[G h (x )] t, where the depolarization factor now reads as m h = ǫ3 iω 3µ (µ µ ) µ + µ B () and the matrix G h (x ) is given by (9). Since the vector columns of G h (x ) are orthogonal (again, refer to []), it follows immediately that the jth nonzero singular value of A is given by σ j = m h Gh j with corresponding left singular vector U j = G h j / Gh j, j =,, where Gh j denotes the jth vector column of G h (x ). The singular vectors which correspond to the two nonzero singular values of A can be written as U = yg(x, r) G h, U = xg(x, r) G h where the square norms of the vector columns of G h are given by G h = y f(x,x ), G h = x f(x,x ). By letting I n = y g(x,r n )/ G e and substituting this into the above equation for the electric field and summing (3) over the array index n, the backpropagated electric fields E h (r) and E h (r) become E h (r) = iωµ k G h (k ẑ + z ) y f(r,x ) E h (r) = iωµ k G h (k ẑ + z ) x f(r,x ) Analogously, we can derive the magnetic fields H h (r) = G h s ẑ y f(r,x ) H h (r) = G h s ẑ x f(r,x ) In the plane z = x the back-propagated electric fields E h (r),e h (r) behave as E h (r) ( s + ẑ) y j (k r x ) () E h (r) ( s + ẑ) x j (k r x ) () and the back-propagated magnetic fields H h (r),hh (r) as H h (r) s ẑ y j (k r x ) H h (r) s ẑ x j (k r x ) Let us emphasize that, in this case, E h 3(r) and H h 3(r) since the third vector column of the matrix G h (x ) =. 3) Dielectric and permeable (ε ε and µ µ ) or perfectly conducting (σ = and µ = ) inclusion: In this case the MSR matrix A is given by A = G(x )M G t (x ), G(x ) = [G e, G h ](x ), and M = diag(m e I, mh I) with me and m h given by () and (), respectively. First, let us consider the Hermitian positive semidefinite matrix Q C 6 6 given by Q = G t (x )G(x ) which is of the form of (6), where b i = G e i, d i = G h i c = G h, Ge = x zf(x,x ) c = G h, Ge = y zf(x,x ) ( x,y = y x herein denotes the inner product of the vectors x and y C n ). The eigenvalues λ j, j =,...,6 of Q are given by (7) and are associated to the eigenvectors V j by (8). It is easy to show that the nonzero singular values of A are the nonzero diagonal elements of the matrix M Λ and the corresponding left singular vectors of the A are the first five normalized vector columns of the matrix U = G(x )V. Therefore, the singular vectors of A can be written as U,5 = U,4 = U 3 = k λ,5 ( α,5 x z + k β,5 x ) g(x, r) k λ,4 ( α,4 y z k β,4 y ) g(x, r) k G e 3 (k + z )g(x, r) where α j and β j, j =, 5 and j =, 4 are given by (9) and (3), respectively. Here we note that λ 3 = G e 3. As in the two previous cases, a similar analysis yields the back-propagated electric fields E e,h,5 (r) = = iωµ k 4 λ,5 (k ẑ + z ) x (α,5 z + k β,5 )f(r,x ) E e,h,4 (r) = = iωµ k 4 λ,4 (k ẑ + z ) y (α,4 z k β,4 )f(r,x ) E e,h 3 (r) = iωµ k 4 G e 3 (k ẑ + z )(k + z )f(r,x ) and the back-propagated magnetic fields H e,h,5 (r) = = k s ẑ x (α,5 z + k β,5 )f(r,x ) λ,5 H e,h,4 (r) = = k s ẑ y (α,4 z k β,4 )f(r,x ) λ,4 H e,h 3 (r) = k G e 3 s ẑ (k + z )f(r,x ) Note that E e,h 3 = E e 3 and H e,h 3 = H e 3. From the material above, we can observe that the backpropagated electric (magnetic) fields E e,h,5 (r) (He,h,5 (r)) behave as the linear combination of the electric fields E e (r) (H e (r)) and E h (r) (H h (r)), respectively; the backpropagated electric (magnetic) fields E e,h,4 (r) (He,h,4 (r)) behave as the linear combination of the electric fields E e (r) (H e (r)) and Eh (r) (Hh (r)), respectively.

6 IEEE TRANS. ANTENNAS PROPAGAT., SUBMITTED 6 Fig.. One dielectric inclusion: maps of the amplitudes of the backpropagated electric fields F e, Fe, Fe 3, in the plane z =, associated to the three nonzero singular vectors provided by the SVD algorithm as applied to the calculated MSR matrix. Fig. 3. One dielectric inclusion: same display of back-propagated fields as in Fig., in the plane x =. To summarize, beyond the sometimes involved mathematical details, the study of the three contrast cases along similar lines of reasoning yet with specific attention to their peculiarities, enable us to conclude the following: The back-propagated field behaves as the Bessel function j or as its spatial partial derivatives up to third order in the plane z = x. From the definition of the function j (k r x ), the resolution is confirmed to be λ/ in the plane z = x. If one is considering well-separated inclusions, i.e., from the definition of the MSR matrix A (equation ()) in the case of m inclusions, whenever [G e (x i )] G e (x j ) = and/or [G h (x i )] G h (x j ) = for i, j =,...,m (in practice whenever the magnitude of such quantities is small enough with respect to the one of the other elements), then each inclusion can be imaged independently from the others. V. NUMERICAL ILLUSTRATIONS Let us refer to the configuration under study as is sketched in Fig.. All geometrical dimensions henceforth are in meters. Permittivities and permeabilities of the background medium, ε and µ, are the values in air. The frequency of operation is f = 5 MHz. The corresponding wavelength λ is.6. The planar transmitter/receiver array, symmetric about the axis z, is consisting of 7 7 vertical electric dipoles distributed at the nodes of a regular mesh with a half-a-wavelength step size, and it is placed at h = 5λ. The MSR matrix in the examples below is computed by using the leading order of the electric fields, the singular value decomposition (SVD) being performed thereupon in pure numerical fashion by standard application of a SVD Matlab subroutine; the computation of the back-propagated Fig. 4. One dielectric inclusion: same display of back-propagated fields as in Fig., in the plane y =. fields from the dipole array into the search space then follows via application of the formulas given in the previous section, e.g., (4), (5), (6) for the pure dielectric case. In the first numerical example we consider the case of one spherical inclusion which is centered at x = with diameter ǫ =., permittivity ε = 5ε, and permeability µ = µ. The magnitudes of the back-propagated electric fields in the planes z =, x = and y =, associated to every one of the three non-zero singular values, are shown in Figs., 3 and 4, respectively. Following, e.g., [9], the matrix AA admits two distinct eigenvalues in the case of one dielectric inclusion, the smallest

7 IEEE TRANS. ANTENNAS PROPAGAT., SUBMITTED 7 eigenvalue being of multiplicity. The eigenvector of AA (or the left singular vector of A) corresponding to the largest eigenvalue of AA (or singular value of A) is the third normalized vector column of the matrix G e (x ). Hence, the back-propagated field F e as is displayed in Figs., 3 and 4 corresponds to the field E e 3 defined by (6). This F e is in effect associated to the first singular column vector which is computed from the MSR matrix by the SVD routine, then corresponding with the first singular value of that matrix yielded by this routine. F e corresponds to the one, Ee, defined by (4), and now is associated to the second singular column vector (and the second singular value) being computed; and F e 3 corresponds to the one, Ee, defined by (5), and likewise is associated to the third singular column vector and singular value. In the plane z =, which is parallel to the transmit/receive planar array, one reaches as expected an excellent (transverse) resolution, at least if one is looking at the z-component of F e, whose monopole-like visual aspect reflects the fact that its asymptotic value is given by the Bessel function (see (9)). The other two components of F e exhibit a dipole-like behavior, in tune with the action of a first-order partial derivative vs. x or vs. y (for the x- and y-component, respectively) upon the Bessel function. As for the behaviors of F e and Fe 3 in the same z = plane, dipole-like aspects (for their z-components) associated to first-order partial derivatives, and multipole-like aspects (for the others) associated to second-order partial derivatives, as it could be deducted from (7) and (8), are observed. However, due to the occurrence of identical singular values, F e and Fe 3 cannot be separated from one another, which is leading, for any given component, to a mix of their respective values. For example, one does not see a pure x-derivative or a pure y-derivative for the z-components, but a sum of the two, a pure xx-derivative or a pure xy-derivative for the x- components, but a sum of the two, etc. In the planes x = and y =, the above aspects are observed, and a mostly similar discussion could be led. Yet the main fact of interest is now that the resolution in those two planes has considerably decayed, since one is exhibiting spots which are strongly spread out in a direction parallel to the z-axis. Indeed, the fixed source/receive antenna array as is assumed in this study cannot be expected to provide a point-spread function (which is what the backpropagation brings out) with good longitudinal resolution. Similar phenomena have been depicted earlier, see, e.g., in - D settings, [3], [4], [] one should move the array around the search space to collect additional data if one were willing to improve the resolution in other directions. In the second numerical example we consider two dielectric inclusions centered at points x = (.6,.6, ) and x = (.6,.6, ) (the separation distance is of the order of.7 wavelengths) with same diameter ǫ =.. The back-propagated electric field (the six vector fields, each one associated to one of the six nonzero singular values) in the plane z = for two inclusions with different dielectric contrasts (ε = 5ε, ε = ε ) is displayed in Fig. 5 and the one for two identical dielectric inclusions (ε = ε = ε ) is displayed in Fig. 6. In this case, the inclusions are far enough from another not to physically interact (multiple electromagnetic scattering if any is very weak, and the leading order term of the scattered field is just the sum of the leading orders of the scattered fields due to two isolated inclusions). As a consequence, they can be imaged independently through mapping of a MUSIC functional in the search space, see [], [], and references therein. But the two cases markedly differ in terms of the back-propagation of fields. If the two inclusions are of different contrasts, the two sets of nonzero singular values correspondingly differ (being noticed that one still has for each inclusion a value of multiplicity one and a value of multiplicity two). The resulting back-propagated fields then appear as just a superimposition of the back-propagated fields calculated for each inclusion in independent fashion as is easily seen by comparison of the results displayed in Fig. 5 for the two inclusions with those obtained for one inclusion displayed in Fig.. Let us notice however that the images are still not so squarely interpreted, since the ranging (which as already said is coming from the SVD routine applied to the MSR matrix) F e, Fe,... Fe 6 of the back-propagated fields sees a fair amount of interlacing: F e is associated to the sphere of high contrast, F e to the one of low contrast, Fe 3 and Fe 4 (which are themselves combinations as discussed earlier) to the one of high contrast, and the last two fields F e 5 and F e 6 to the one of low contrast. If the two inclusions are identical, this is affecting the multiplicity of the singular values, and as a consequence the expected focusing properties of the back-propagated fields essentially disappear. Maybe better stated, from the aspect of the maps retrieved in this case, there is no more successive focusing on either one or the other (each singular vector is not associated to one given inclusion as before) but superimposition of patterns (still monopole-, dipole-, multipole-like, as a function of the derivatives involved) associated to each inclusion. Let us emphasize, at this stage, that if the two identical inclusions were much closer to one another than as of now, whilst remaining fairly uncoupled from a electromagnetic point-of-view, most of the components of the back-propagated fields would not be easily attributed anymore to one specific inclusion save the monopole-like map of the z-component of F e and Fe Finally, we consider a single, simultaneously dielectric and permeable inclusion centered at the center of coordinates (ε = 5ε and µ = µ ). The back-propagated electric field in the plane z = is shown in Fig. 7. Here, the interpretation of the maps is easier, the first three fields F e,h F e,h, Fe,h 3,, being assigned to the dielectric part, the last two to the magnetic part this ranging at the frequency of operation remains the same for realistic values of permeability contrast, but the influence of frequency matters, and much lower frequencies, for example, might lead to rearrangement in favor of the magnetic part (this investigation is out of the scope of the paper). With the dielectric part, again notice that the displayed back-

8 IEEE TRANS. ANTENNAS PROPAGAT., SUBMITTED 8 Fig. 5. Two dielectric inclusions with different contrast: maps of the amplitudes of the back-propagated electric fields F e, Fe,, Fe 6, in the plane z =, associated to the six nonzero singular vectors provided by the SVD algorithm as applied to the calculated MSR matrix. Fig. 6. Same display as in Fig. 5 for two dielectric inclusions with identical contrast. propagated field F e,h corresponds to the third and largest singular value, i.e., to the field E e,h 3 whose asymptotic behavior both from electric and magnetic contributions is only involving the Bessel function or its first partial derivatives. F e,h (resp. F e,h 3 ) combine electric and magnetic contributions of same asymptotic behavior, which again does not disturb the observed patterns. As for the magnetic part, the multiplicity of the singular values has a strong impact: since, for a single permeable inclusion, refer to [9], the matrix AA has one nonzero eigenvector of multiplicity, F e,h 5 and F e,h 6 mix together here. VI. CONCLUSION The above investigation, both in terms of mathematics and numerics, has aimed at a better understanding of the behavior of back-propagation in the framework of MUSIC-type noniterative imaging procedures. To do so, and in order to consider a more realistic situation than in most previous studies, one

9 IEEE TRANS. ANTENNAS PROPAGAT., SUBMITTED 9 Fig. 7. One simultaneously dielectric and permeable inclusion: maps of the magnitudes of the back-propagated electric fields E e,h,,ee,h 5, in the plane z =, associated to the five nonzero singular vectors provided by the SVD algorithm as applied to the calculated MSR matrix. has attacked the full 3-D electromagnetic case, and one has attempted to provided a comprehensive set of novel results (to the best of our knowledge) which would be useful in that respect. For certain, further numerical experimentation is still needed at this stage (the examples above have no pretense to exhaustivity). Nevertheless, one can conclude that back-propagation is an useful tool, with some limitations (see below), being emphasized that the MUSIC-type imaging procedure itself already yields a robust localization of the inclusions as is exemplified earlier [], [], [9] and in effect is well-known from the increasingly abundant literature on the subject. In particular, data which back-propagation can bring to an end-user might indeed enable her/him to acquire supplementary information on the imaged structures, if the appropriate field components are looked at carefully. However, the behavior of several field components is generally not very simple; complex, often combined patterns appear, and in particular it is confirmed herein that one cannot attach one field pattern to one singular value as straighforwardly as one might have hoped for. Also, spreading along directions orthogonal to the planar array remains a strong limitation in any potential separation of scatterers, but this is usual with antenna arrays and is not at all specific to the present treatment. As for proximity of scatterers from one another, even though they could be safely considered as signing independently into the data, this might still result in apparently strong coupling if one proceeds as one does here via the singular value decomposition of the MSR matrix and consequent mapping of back-propagated fields. Forthcoming works in electromagnetics should in particular deal with extensions of the modeling to ellipsoidal inclusions since orientation and aspect ratio of inclusions matter in any imaging experiment; ellipsoids are rather versatile objects in such terms yet they are amenable to exact calculations since their polarization tensors are known in closed form. Though the numerical work is slightly more cumbersome with them than with the spheres, a Matlab code is readily applicable in that situation already; but the needed underlying mathematical analysis of the back-propagated field behavior itself is still in progress at the present time (noticing that the diagonal form of the polarization tensor of an ellipsoid allows a mostly similar approach). Proper material and illustrations thereof will thus be presented at a later stage. As for studying the case of half-space embedding media [9], provided that introduction of far-field forms enables us to get meaningful interpretations of the back-propagated field patterns, it should be worthwhile as well. Other issues of interest would be also about MUSIC imaging and back-propagation for multiple spheres with larger size than those assumed herein, say, with radii up to halfa-wavelength or so (refer to [4] for a clever first analysis in that context). Finally, the present work has been led in the propagation regime, but the diffusive one especially in view of nondestructive evaluation of conductive damaged parts via eddy currents deserves also a full investigation, being stated that the field asymptotics as introduced here and specialized in [9] to half-space burial is a valid starting point. APPENDIX I FAR-FIELD PATTERNS OF THE DYADIC GREEN S FUNCTIONS IN FREE SPACE Let g(r,r ) denotes the scalar Green s function in free space given by g(r,r ) = e ik r r /4π r r. It is well known that

10 IEEE TRANS. ANTENNAS PROPAGAT., SUBMITTED for fixed r the following far-field expansions hold: ( ) g(r,r ) = h(kr)e ikˆr r + O r g(r,r ) = ikh(kr)e ikˆr r ˆr ( ) + O r where h(kr) = e ikr /4πr. The scalar r is the magnitude of the vector r and ˆr = ˆxˆr x + ŷˆr y + ẑˆr z is a unit vector in the direction r. Then, using the vector identity (ψa) = ψ A + ψ A, one easily arrives at the following expansions, which hold uniformly in ˆr for any vector a and any fixed r : G(r,r ) a = h(kr)e ikˆr r ˆr ( ) (ˆr a) + O r G(r,r ) a = g(r,r ) a = ikh(kr)e ikˆr r ˆr ( ) (3) a + O r as r +. At this stage it is useful to introduce the tensors (which is nothing else than a projection operator) and Λ such that (ˆr) a = ˆr (ˆr a) Λ(ˆr) a = ˆr a or, in more detail, which are given by and (ˆr) = I ˆrˆr (4) Λ(ˆr) = ˆr z ˆr y ˆr z ˆr x (5) ˆr y ˆr x The following properties are satisfied as it can be easily checked: (ˆr) = t (ˆr), (ˆr) = (ˆr) and Λ(ˆr) = Λ t (ˆr), Λ(ˆr) Λ t (ˆr) = (ˆr). So, the far-field expansions introduced in (3) become in concatenated form G(r,r ) = h(kr)e ikˆr r (ˆr) + O ( r G(r,r ) = ikh(kr)e ikˆr r Λ(ˆr) + O ) ( ) r APPENDIX II THE EIGENVALUES AND EIGENVECTORS OF THE MATRIX A IN THE GENERAL CONTRAST CASE [8], [9] Let Q C 6 6 be a matrix of the form b c b c Q = b 3 c d c d where b d c c and b d c c. The eigenvalues of Q are given by λ,5 = (b + d ± ) (b d ) + 4 c λ,4 = (b + d ± ) (b d ) + 4 c λ 3 = b 3, λ 6 = (6) (7) and are associated to the eigenvectors V,5 = α,5 ê + β,5 ê 5, V,4 = α,4 ê + β,4 ê 4 V 3 = ê 3, V 6 = ê 6 (8) where ê j, j =,...,6 is an orthogonal basis in R 6 and the coefficients α j and β j are given by α j = c c + λ j b, β λ j b j = c + λ j b (9) for j = and 5, α j = c c + λ j b, β λ j b j = c + λ j b (3) for j = and 4. REFERENCES [] H. Ammari, E. Iakovleva, D. Lesselier, and G. Perrusson, MUSICtype electromagnetic imaging of a collection of small three-dimensional bounded inclusions, SIAM J. Scientific Computing, to appear, 6. [] H. Tortel, G. Micolau, and M. Saillard, Decomposition of the time reversal operator for electromagnetic scattering, J. Electromagn. Waves Applic., vol. 3, pp , 999. [3] G. Micolau and M. Saillard, DORT method as applied to electromagnetic subsurface sensing, Radio Science, vol. 38, pp , 3. [4] G. Micolau, M. Saillard, and P. Borderies, DORT method as applied to ultrawideband signals for detection of buried objects, IEEE Trans. Geosci. Remote Sensing, vol. 4, pp. 83-8, 3. [5] A. J. Devaney, Time reversal imaging of obscured targets from multistatic data, IEEE Trans. Antennas Propagat., vol. 53, pp. 6 6, 5. [6] C. Prada, S. Manneville, D. Spoliansky, and M. Fink, Decomposition of the time reversal operator. Detection and selective focusing on two scatterers, J. Acoust. Soc. Am., vol. 99, pp , 996. [7] M. Fink and C. Prada, Acoustical time-reversal mirrors, Inverse Problems, vol. 7, R R38,. [8] D. H. Chambers and J. G. Berryman, Analysis of the time-reversal operator for a small spherical scatterer in an electromagnetic field, IEEE Trans. Antennas Propagat., vol. 5, pp , 4. [9] E. Iakovleva, S. Gdoura, D. Lesselier and G. Perrusson, Multi-static response matrix of a 3-D inclusion in half space and MUSIC imaging, IEEE Trans. Antennas Propagat., submitted, Sept. 6 (available as report LS/6/5 from [] D. H. Chambers and J. G. Berryman, Target characterization using decomposition of the time-reversal operator: electromagnetic scattering from small ellipsoids, Inverse Problems, submitted. [] H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, Lecture Notes in Mathematics, vol. 846, Springer-Verlag, Berlin, 4. [] H. Ammari, E. Iakovleva, and D. Lesselier, A MUSIC algorithm for locating small inclusions buried in a half-space from the scattering amplitude at a fixed frequency, Multiscale Modeling Simulation, vol. 3, pp , 5. [3] W. C. Chew, Waves and Fields in Inhomogeneous Media, nd ed., IEEE Press, New-York 995. [4] D. H. Chambers and J. G. Berryman, Radar imaging of spheres in 3D using MUSIC, Report UCRL-ID-5577, Lawrence Livermore National Laboratory, Jan. 3.