METHODS AND APPLICATIONS OF ANALYSIS. 1999 International PressVol. 6, No. 4, pp. xxx{xxx, De ember 1999 00xLOW FREQUENCY ELECTROMAGNETIC SCATTERING THEORYFOR A MULTI-LAYERED CHIRAL OBSTACLE�CHRISTODOULOS ATHANASIADISy AND IOANNIS G. STRATISyAbstra t. We onsider the problem of s attering of a plane ele tromagneti wave by a multi-layered hiral s atterer with a perfe tly ondu ting ore. We derive integral representations for thetotal exterior ele tri �eld, as well as for the ele tri far �eld pattern of this problem. Using the lowfrequen y approximation, we redu e the s attering problem to an iterative sequen e of problems inpotential theory, and onstru t the leading term approximation of the ele tri far �eld pattern.1. Introdu tion. The study of the s attering of low frequen y ele tromagneti waves by an obsta le was initiated by Lord Rayleigh in 1897. A method of obtainingsu esive approximations in as ending powers of the wave number was des ribed byStevenson [19℄. Signi� ant ontribution to the study of s attering in low frequen ieshas been made by Kleinman and by Jones (we refer to [13℄, [14℄ and [11℄, [12℄, respe -tively). Some important theorems for multiple s attering of ele tromagneti waveswere proved by Twersky [20℄. The present authors [5℄ obtained low frequen y solutionsof the ondu tive problem for Maxwell's equations. Low frequen y ele tromagneti s attering theory has been developed by one of the authors [3℄ for a multi-layereds atterer. In [18℄, Pi ard investigates the limiting pro ess as the angular frequen ytends to zero for the solution of the lassi al Maxwell system. In [1℄, Ammari et al.obtain the limit of the s attered �eld solution of the Drude-Born-Fedorov equations asthe frequen y tends to zero, and show that the asymptoti s depend on the topologi alproperties of the onsidered domain. Low frequen y expansions have also been usedin a ousti s and elasti ity; see, for example, Dassios [8℄, Dassios and Kiriaki [9℄, andone of the authors [2℄.A hiral obje t is a body that annot be brought into ongruen e with its mirrorimage by translation and rotation. Chirality is ommon in a variety of naturallyo urring and man-made obje ts (e.g., DNA in mole ular s ale, heli es). From anoperational point of view, hirality is introdu ed into the lassi al Maxwell equationsby a pair of generalized onstitutive relations in whi h ele tri and magneti �elds are oupled via a new material parameter. The monograph [16℄ by Lakhtakia ontainsa large part of work on s attering problems involving hiral media. Low frequen yexpansions in hiral media have been used by Lakhtakia et al. [17℄, Lakhtakia [15℄,and Jaggard et al. [10℄.The purpose of this work is to explore the low frequen y s attering problem foran ele tromagneti plane wave in ident upon a multi-layered hiral s atterer with aperfe tly ondu ting ore. In Se tion 2, we des ribe the multi-layered hiral s attererand formulate the orresponding s attering problem for the ele tri �eld only. InSe tion 3, we onstru t an integral representation for the total exterior ele tri �eld inwhi h the transmission and boundary onditions, along with the behaviour at in�nity,are in orporated. Moreover, we derive the ele tri far �eld pattern of the onsideredproblem. In Se tion 4, we redu e the s attering problem to a series of problems inpotential theory whi h an be solved su essively in terms of expansions in appropriateharmoni fun tions; we also onstru t the leading term approximation of the ele tri �Re eived Febuary 4, 1998; revised Sept. 15, 1998.yUniversity of Athens, Department of Mathemati s, Panepistimiopolis, GR-157 84 Athens, Gree e(istratis�eudoxos.uoa.gr). 1

2 C. ATHANASIADIS AND I. G. STRATISfar �eld pattern. Finally, Se tion 5 ontains some remarks and omments on otherrelated resear h work.2. Formulation of the problem. The multi-layered hiral s atterer is abounded, onvex, and losed subset of R3 , with a C2-boundary S0. The interior of is divided by means of losed and noninterse ting C2-surfa es into subsets (layers)j , with �j \ �j+1 = Sj , j = 1; 2; : : : ; N . The surfa e Sj�1 surrounds Sj andthere is one normal unit ve tor n(r) at ea h point r of any surfa e Sj pointing atj . Ea h j , j = 1; 2; : : : ; N , is o upied by a homogeneous isotropi hiral mediumwith ele tri permittivity "j , magneti permeability �j , hirality measure �j , andvanishing ondu tivity. The spa e N+1, within whi h lies the origin, is the ore ofthe multi-layered hiral s atterer, and appropriate boundary onditions are satis�edon its surfa e SN , that may be either the perfe t ondu tor ondition, or the diele tri onditions, or an impedan e boundary ondition [7℄.The exterior 0 of the s atterer is an in�nite homogeneous isotropi a hiralmedium with ele tri permittivity "0, magneti permeability �0, and vanishing on-du tivity.We shall onsider the s attering of time-harmoni ele tromagneti plane waves bythe multi-layered hiral s atterer . Let (Ein ;Hin ) and (Es ;Hs ) be the in identand the s attered waves, respe tively. (Es ;Hs ) satis�es the well-known Silver-M�ullerradiation ondition [7℄. The total exterior �eld (E0;H0) in 0 is given byE0 = Es +Ein ;(2.1) H0 =Hs +Hin ;(2.2)and satis�es the Maxwell equationsr�E0 = i!�0H0;(2.3) r�H0 = �i!"0E0(2.4)where ! is the angular frequen y. The total �eld (Ej ;Hj) in ea h hiral layer j ,j = 1; 2; : : : ; N , satis�es the equationsr�Ej = i!�j 2jk2j Hj + �j 2jEj ;(2.5) r�Hj = �i!"j 2jk2j Ej + �j 2jHj(2.6)where 2j = k2j1� k2j�2j(2.7)with k2j = !2"j�j :(2.8)The equations (2.5) and (2.6) are derived by the sour e-free Maxwell url postulatesr�Ej = i!Bj ; r�Hj = �i!Dj(2.9)

LOW FREQUENCY ELECTROMAGNETIC SCATTERING THEORY 3along with the Drude-Born-Fedorov onstitutive relationsDj = "j(Ej + �jr�Ej) ; Bj = �j(Hj + �jr�Hj):(2.10)For the physi al meaning of �j , j , kj and more details about the onstitutive relationsfor hiral media, we refer to [16℄. We note that any solution of (2.3){(2.6) is divergen efree.In order to develop the low-frequen y theory for the s attering by , we onsiderthe de oupling of Ej and Hj , j = 0; 1; : : : ; N in (2.3){(2.6), eliminating the magneti �eld. The equation thus arising is [3℄, [16℄,r�r� Ej = 2jEj + 2�j 2j r�Ej in j ; j = 0; 1; : : : ; N(2.11)where �0 = 0, and 0 = k0 = !("0�0)1=2 is the free-spa e wave number in 0, and inview of (2.8), we get k2j = "j�j"0�0 k20 ; j = 0; 1; : : : ; N:(2.12)The s attered ele tri �eld Es satis�es the Silver-M�uller radiation onditionlimr!1 �r�r�Es (r) + ik0rEs (r)� = 0(2.13)uniformly in all dire tions.On the surfa es Sj , j = 0; 1; : : : ; N � 1, we have the transmission onditionsn�Ej = n�Ej+1;(2.14) n�r�Ej = "j+1"j 2j 2j+1 n�r�Ej+1 + 2j ��j � "j+1"j �j+1� n�Ej+1 :(2.15)The relation (2.15) arises from the standard diele tri ondition n�Hj = n�Hj+1on Sj by substituting the magneti �eld from (2.5).We onsider that the ore is a perfe t ondu tor, that isn�EN = 0; on SN :(2.16)The proof that the above transmission problem has a unique solution an be foundin [6℄.The orresponding problem, arising by eliminating the ele tri �eld, onsists ofthe equations in j , j = 1; 2; : : : ; Nr�r�Hj = 2j Hj + 2�j 2j r�Hj ;(2.17)the transmission onditions on Sj , j = 1; 2; : : : ; N � 1n�Hj = n�Hj+1;(2.18) n�r�Hj = �j+1�j 2j 2j+1 n�r�Hj+1 + 2j��j � �j+1�j �j+1�n�Hj+1;(2.19)the boundary ondition on SNn�r�HN � �N 2N n�HN = 0;(2.20)

4 C. ATHANASIADIS AND I. G. STRATISand the radiation onditionlimr!1�r�r�Hs (r) + ik0rHs (r)� = 0; uniformly in all dire tions;(2.21)with �0 = 0 and 0 = k0. This problem an be treated by an entirely analogouspro edure to that of the following se tions.Remark. Instead of being a perfe t ondu tor, the ore may be onsidered tobe hiral and penetrable. In this ase, the ele tri �eld EN+1, in the interior of the ore N+1, will satisfy (2.11) with j = N +1, and the transmission onditions (2.14),(2.15) for j = N , on SN . This problem an be studied essentially as in Se tions 3 and4. 3. The exterior �eld. In this se tion, we onstru t an integral representationof the total exterior ele tri �eld in whi h all the transmission and boundary on-ditions are in orporated. In order to do this, we begin with the following integralrepresentation for the ele tri �eld, [20℄, [3℄:(3.1) Es (r) = ZS0h(r�Es (r0)) � (n� ~�(r; r0))� (n�Es (r0)) � (rr0 � ~�(r; r0))i ds(r0); r 2 0where ~�(r; r0) is the free-spa e dyadi Green's fun tion~�(r; r0) = ��~I + k�20 rr� eikojr�r0j4�jr� r0j ;(3.2)with ~I = e1 e1 + e2 e2 + e3 e3 the identity dyadi (ej , j = 1; 2; 3, are the artesian unit ve tors). Inserting (2.1) in (3.1) and taking into a ount that Ein is asolution of the equation (2.11) for j = 0, we obtain(3.3) E0(r) = Ein (r) + ZS0h(r�E0(r0)) � (n� ~�(r; r0))� (n�E0(r0)) � (rr0 � ~�(r; r0))i ds(r0) :Using the transmission onditions on S0, the relation (3.3) be omes(3.4) E0(r) = Ein (r)� "1 20"0 21 ZS0 n � h(r�E1(r0))� ~�(r; r0)i ds(r0)+ 20 "1"0�1 ZS0 n�hE1(r0)� ~�(r; r0)i ds(r0)�ZS0 n�hE1(r0)� (rr0 � ~�(r; r0))i ds(r0) :Now, applying su essively the dyadi form of the divergen e theorem and taking intoa ount that Ej and ~� are solutions of (2.11) in j , j = 1; 2; : : : ; N , and 0 respe -tively, and introdu ing the transmission onditions (2.14), (2.15) and the boundary ondition (2.16), we obtain thatE0(r) = Ein (r) +Es ( )(r) +Es (a)(r)(3.5)where Es ( ) and Es (a) denote, respe tively, the hiral and the a hiral parts of theele tri s attered �eld Es , and are given by4�Es ( )(r)� "N"0 �2Nk20 ZSN (r�EN (r0)) � (n� ~�(r; r0)) ds(r0)

LOW FREQUENCY ELECTROMAGNETIC SCATTERING THEORY 5� k20 NXj=1 "j"0�2j Zj (r�Ej(r0)) � (rr0 � ~�(r; r0)) dv(r0)� k20 NXj=1 "j"0�j Zj (r�Ej(r0)) � ~�(r; r0) dv(r0)� k20 NXj=1 "j"0�j Zj Ej(r0) � (rr0 � ~�(r; r0)) dv(r0); r 2 0(3.6)and4�Es (a)(r) = �0�N ZSN (r�EN (r0)) � (n� ~�(r; r0)) ds(r0)+ k20 NXj=1�1� "j"0� Zj Ej(r0) � ~�(r; r0) dv(r0)+ NXj=1��0�j � 1�Zj (r�Ej(r0)) � (rr0 � ~�(r; r0)) dv(r0); r 2 0 :(3.7)The surfa e integrals in (3.6), (3.7) express the e�e t of the ore, while the volumeintegrals express the ontribution of ea h layer to the exterior �eld.The behavior of the s attered ele tri wave in the region of radiation is des ribedby the ele tri far �eld pattern E1(r), whi h is de�ned [7℄ by the relationEs (r) = E1(r)h(k0r) +O� 1r2� ; r !1 ;(3.8)uniformly in all dire tions where h(x) = eix=(ix) is the zeroth-order spheri al Hankelfun tion of the �rst kind. Substituting the asymptoti forms, [3℄,~�(r; r0) = � ik04� (~I � r r)h(k0r) e�ik0 r�r0 + O� 1r2� ; r !1(3.9) rr0 � ~�(r; r0) = � k204� (~I � r)h(k0r) e�ik0 r�r0 +O� 1r2� ; r !1(3.10)in (3.6), (3.7) and, in view of (3.8), we obtainE1(r) = E( )1 (r) +E(a)1 (r)(3.11)where4�E( )1 (r) = ik30 "N"0 �2N ZSN (r�EN (r0)) � �n� (~I � r r)� e�ik0 r�r0ds(r0)+ k40 NXj=1 "j"0�2j Zj (r�Ej(r0)) � (~I � r) e�ik0 r�r0dv(r0)+ ik30 NXj=1 "j"0�j Zj (r�Ej(r0)) � (~I � r r) e�ik0 r�r0dv(r0)+ k40 NXj=1 "j"0�j Zj Ej(r0) � (~I � r) e�ik0 r�r0dv(r0)(3.12)

6 C. ATHANASIADIS AND I. G. STRATISand 4�E(a)1 (r) = �ik0 �0�N ZSN (r�EN (r0)) � �n� (~I � r r)� e�ik0 r�r0ds(r0)� ik30 NXj=1�1� "j"0�Zj Ej(r0) � (~I � r r) e�ik0 r�r0dv(r0)� k20 NXj=1��0�j � 1�Zj (r�Ej(r0)) � (~I � r) e�ik0 r�r0dv(r0) :(3.13)4. The low frequen y approximation. We onsider an in ident plane waveEin (r) = peik0d�r where the unit ve tor d des ribes the dire tion of propagation, the onstant ve tor p 2 R3 gives the polarization, and are su h that p � d = 0. Ein isanalyti at k0 = 0 and it an be expanded into the onvergent power seriesEin (r) = p 1Xm=0 (ik0)mm! (d � r)m :(4.1)We assume the standard low frequen y hypothesis k0a � 1, where a is the radius ofthe smallest ir ums ribable sphere around the s atterer, [3℄, [8℄, [9℄, [13℄, [14℄. We onsider the following expansion of the �elds Ej into onvergent power series of k0:Ej(r) = 1Xm=0 (ik0)mm! �(j)m (r)(4.2)where �(j)m (r) are independent of k0. For the justi� ation of this expansion, see theAppendix. Inserting the expansion (4.2) into (2.11) and equating equal powers of k0,we obtain for j = 0; 1; : : : ; N , m = 0; 1; 2; : : : , that(4.3) r�r��(j)m (r)= m(m� 1) "j�j"0�0 h�j r�r��(j)m�2(r) + 2�j r��(j)m�2(r) + �(j)m�2(r)i :Let us note that r�E = 0 renders r��(j)m = 0. In view of this relation, (4.3) be omesthe Lapla e equation for m = 0; 1, and the Poisson equation for m � 2; in the latter ase, the right-hand side of (4.3) is dependent on the (m�2)-th order oeÆ ient whi his known from previous steps.The boundary onditions (2.14) and (2.15) are transformed inton��(j)m (r) = n��(j+1)m (r)(4.4)n�r��(j)m (r) = �j�j+1 n�r��(j+1)m (r) + m(m� 1)"0�0 ��2j "j�j n�r��(j)m�2(r)� �2j+1"j+1�j+1 n�r��(j+1)m�2 (r) + "j�j ��j � "j+1"j � n��(j+1)m�2 (r)�;(4.5)on Sj , j = 0; 1; : : : ; N � 1, and the boundary ondition (2.16) be omesn��(N)m (r) = 0 on SN :(4.6)

LOW FREQUENCY ELECTROMAGNETIC SCATTERING THEORY 7It is known, [3℄, that the fundamental dyadi ~�(r; r0) and the dyadi rr0� ~�(r; r0)have the following expansions:~�(r; r0) = 1Xm=0 (ik0)mm! ~�m(r; r0);(4.7) rr0 � ~�(r; r0) = 1Xm=0 (ik0)mm! ~Æm(r; r0)(4.8)where ~�m(r; r0) = �jr� r0jm�14�(m+ 2) �(m+ 1)~I � m� 1jr� r0j2 (r� r0) (r� r0)� ;(4.9) ~Æm(r; r0) = 14� (m� 1)jr� r0jm�3(r� r0)� ~I:(4.10)Substituting (4.1), (4.2), (4.7), and (4.8) into (3.5){(3.7) and equating equal powers ofk0, the following integral relation among the oeÆ ients �j0;�j1; : : : ;�jm is obtained:�(0)m (r) = p(d � r)m +F( )m (r) +F(a)m (r)(4.11)where4�F( )m (r) = �2N "N"0 mX�=0�m���(�� 1) ZSN (r��(N)��2(r0)) � (n� ~�m��(r; r0)) ds(r0)+ 1"0 mX�=0 NXj=1�m���(�� 1)�2j "j Zj (r��(j)��2(r0)) � ~Æm��(r; r0) dv(r0)+ mX�=0 NXj=1�m���(�� 1)"j"0�j Zj (r��(j)��2(r0)) � ~�m��(r; r0) dv(r0)+ mX�=0 NXj=1�m���(�� 1)"j"0�j Zj �(j)��2(r0) � ~Æm��(r; r0) dv(r0)(4.12)and4�F(a)m (r) = �0�N mX�=0�m��ZSN (r� �(N)� (r0)) � (n� ~�m��(r; r0)) ds(r0)� mX�=0 NXj=1�m���1� "j"0� �(�� 1) Zj �(j)��2(r0) � ~�m��(r; r0) dv(r0)+ mX�=0 NXj=1�m����0�j � 1�Zj (r��(j)� (r0)) � ~Æm��(r; r0) dv(r0):(4.13)In order to derive the low-frequen y expansion for the ele tri far �eld patternE1(r), the series (4.2) ande�ik0r�r0 = 1Xm=0(�1)m (ik0)mm! (r � r0)m(4.14)

8 C. ATHANASIADIS AND I. G. STRATISare substituted into (3.8) to giveE1(r) = E( )1 (r) +E(a)1 (r)(4.15)where now4�E( )1 (r) = �"N"0 �2N 1Xm=0 (ik0)m+3m! mX�=0�m��(�1)� ZSN (r��(N)m��(r0))� (n� (~I � r r))(r � r0)�ds(r0) + 1Xm=0 (ik0)m+4m! mX�=0�m��(�1)�� NXj=1 "j"0�j Zj (r� �(j)m��(r0)) � (~I � r)(r � r0)� dv(r0)+ 1Xm=0 (ik0)m+4m! mX�=0�m��(�1)� NXj=1 "j"0�j Zj �(j)m��(r0)� (~I � r)(r � r0)� dv(r0)� 1Xm=0 (ik0)m+3m! mX�=0�m��(�1)� NXj=1 "j"0�j� Zj (r� �(j)m��(r0)) � (~I � r r)(r � r0)�dv(r0)(4.16)and4�E(a)1 (r) = � �0�N 1Xm=0 (ik0)m+1m! mX�=0�m��(�1)� ZSN (r��(N)m��(r0))� (n� (~I � r r))(r � r0)� ds(r0) + 1Xm=0 (ik0)m+3m! mX�=0�m��(�1)�� NXj=1�1� "j"0�Zj �(j)m��(r0)) � (~I � r r)(r � r0)�dv(r0)+ 1Xm=0 (ik0)m+2m! mX�=0�m��(�1)� NXj=1��0�j � 1�� Zj (r� �(j)m��(r0)) � (~I � r)(r � r0)� dv(r0):(4.17)In order to onstru t the leading-term approximation as k0 ! 0, we �rst notethat the low frequen y oeÆ ients �(j)0 orrespond to k0 = 0 and, in view of (2.7),(2.12), to j = 0. Consequently, �(j)0 is a solution of the ele trostati problem and,hen e, �(j)0 is the gradient of a s alar fun tion; therefore, sin e the oeÆ ient of ik0 ontains the term r��(N)0 , it is equal to zero. Moreover, the oeÆ ient of (ik0)2 is(4.18) � 14� �0�N ZSN �r� �(N)1 (r0)� � �n� (~I � r r)� ds(r0)+ 14� �0�N ZSN �r��(N)0 (r0)� � �n� (~I � r r)� (r � r0) ds(r0)

LOW FREQUENCY ELECTROMAGNETIC SCATTERING THEORY 9+ 14� NXj=1��0�j � 1�Zj �r��(j)0 (r0)� � (~I � r) dv(r0):If we onsider the low frequen y expansions for the magneti �eldsHj(r) = 1Xm=0 (ik0)mm! (j)m (r); r 2 j ; j = 0; 1; : : : ; N(4.19)where (j)m are independent of k0, and substitute (4.19), (4.2) into (2.5), we obtainthe relation(4.20) r��(j)m (r)�m(m� 1) "j�j"0�0 �2j r��(j)m�2(r)= m�j("0�0)�1=2m�1(r) +m(m� 1) "j�j"0�0 �2j r��(j)m�2(r):>From the relation (4.20), we have that the �rst integral of (4.18) is a multiple of asurfa e integral of the form ZSN N0 (r0) � n ds(r0);whi h is equal to zero, [3℄. The other integrals of (4.18) vanish sin e they ontainr � �(j)0 (r). Therefore, the oeÆ ient of (ik0)2 vanishes and the ele tri far �eldpattern is of order O(k30) as k0 ! 0. In parti ular, the leading term approximation,as k0 ! 0, isE1(r) = (ik0)34� �� �02�N ZSN �r� �(N)2 (r0)� � �n� (~I � r r)� ds(r0)+ �0�N ZSN �r��(N)1 (r0)� � �n� (~I � r r)� (r � r0) ds(r0)+ NXj=1�1� "j"0�Zj �(j)0 (r0) � �~I � r r� dv(r0)+ NXj=1��0�j � 1�Zj �r��(j)1 (r0)� � (~I � r) dv(r0)�+O(k40); k0 ! 0:(4.21)Hen e, the leading term approximation as k0 ! 0 of the ele tri far �eld pattern oforder k30 in the low frequen y ase does not depend on the hirality measures �j andhas the same form as for the a hiral multi-layered s atterer [3℄.On the other hand, the k40 oeÆ ient of E( )1 (r) is given by(4.22) NXj=1 "j"0�j Zj �(j)0 (r0) � �~I � r� dv(r0)� "N"0 �2N ZSN �r��(N)1 (r0)� � �n� (~I � r r)� ds(r0)whi h is learly nonzero. Hen e, as far as the appearan e of hirality is the leadingterm approximation as k0 ! 0 of the ele tri far �eld pattern is on erned, the\ hirally signi� ant" exponent of k0 is 4.

10 C. ATHANASIADIS AND I. G. STRATISRemark. The most ommon approa h for using the low-frequen y method inele tromagneti s onsists in working with the ele tri (or the magneti ) �eld only: [3℄,[5℄, [12℄, [13℄. However, one ould work with both the ele tri and magneti �elds; inour ase, the de omposition of E and H into suitable Beltrami �elds, [4℄, [14℄, shouldthen be taken into a ount.5. Spe ial ases.A hiral multi-layered s atterer. When the hirality measures �j, j = 1; 2; : : : ;N are equal to zero (i.e., when the s atterer is a hiral multi-layered), then all theterms involving hirality disappear, and the orresponding results of [3℄ for an a hiralmulti{layered s atterer are re overed.Homogeneous (non-layered) hiral s atterer. When �j = �, "j = ", �j = �,j = 1; 2; : : : ; N , and there is no ore, then the obsta le is a non-layered homogeneouspenetrable hiral s atterer. The existen e and uniqueness of solutions to this problemhas been established in [4℄. In this ase, the analysis of the pre eding se tions givesthe following results:(a) The total exterior ele tri �eld is given by E0 = Ein +Es ( ) +Es (a) where(5.1) 4�Es ( )(r) = � ""0 k20�� � Z(r�E(r0)) � (rr0 � ~�(r; r0)) dv(r0)+ Z(r�E(r0) � ~�(r; r0) dv(r0) + Z E(r0) � (rr0 � ~�(r; r0)) dv(r0)�;(5.2) 4�Es (a)(r) = k20 �1� ""0�ZE(r0) � ~�(r; r0) dv(r0)+��0� � 1�Z(r�E(r0)) � (rr0 � ~�(r; r0)) dv(r0);and E(r0) is the ele tri �eld in .(b) The ele tri far �eld pattern is given by E1 = E( )1 +E(a)1 where(5.3) 4�E( )1 (r) = ""0 k30�� k0� Z(r�E(r0)) � (~I � r)e�ik0 r�r0dv(r0)+ i Z(r�E(r0)) � (~I � r r)e�ik0 r�r0dv(r0) + k0 Z E(r0) � (~I � r)e�ik0 r�r0dv(r0)�;(5.4) 4�E(a)1 (r) = �k20�ik0�1� ""0�Z E(r0) � (~I � r r)e�ik0 r�r0dv(r0)+��0� � 1�Z(r�E(r0)) � (~I � r)e�ik0 r�r0dv(r0)�:This expression is ompletely analogous to the one given in [17℄ after elimination ofthe terms ontaining the magneti �eld there.( ) The low-frequen y oeÆ ients are given by (4.11) where

LOW FREQUENCY ELECTROMAGNETIC SCATTERING THEORY 11(5.5) 4�F( )m (r) = ""0� mX�=0�m���(�� 1)�� Z(r����2(r0)) � ~Æm��(r; r0) dv(r0)+ Z(r����2(r0)) � ~�m��(r; r0) dv(r0) + Z���2(r0) � ~Æm��(r; r0) dv(r0)�;(5.6) 4�F(a)m (r) = � mX�=0�m����1� ""0��(�� 1) Z���2(r0) � ~�m��(r; r0) dv(r0)���0� � 1�Z(r���(r0)) � ~Æm��(r; r0) dv(r0)�:(d) In the low frequen y expansion for the ele tri far �eld pattern, we have4�E( )1 (r) = ""0�2 1Xm=0 (ik0)m+4m! mX�=0�m��(�1)� Z(r��m��(r0))� (~I � r)(r � r0)� dv(r0) + ""0� 1Xm=0 (ik0)m+4m! mX�=0�m��(�1)�� Z�m��(r0) � (~I � r)(r � r0)� dv(r0)� ""0� 1Xm=0 (ik0)m+3m!� mX�=0�m��(�1)� Z(r��m��(r0)) � (~I � r r)(r � r0)� dv(r0)(5.7)4�E(a)1 (r) = �1� ""0� 1Xm=0 (ik0)m+3m! mX�=0�m��(�1)� Z�m��(r0)� (~I � r r)(r � r0)� dv(r0) +��0� � 1� 1Xm=0 (ik0)m+2m! mX�=0�m��(�1)�� Z(r��m��(r0)) � (~I � r)(r � r0)� dv(r0):(5.8)6. Appendix A. In this Appendix, we establish that the ele tri �elds maybe expanded into onvergent power series of k0. For simpli ity, and without lossof generality, we onsider that is a non-layered homogeneous hiral s atterer (ofele tri permittivity ", magneti permeability �, and hirality measure �) immersedin free spa e 0.Let � be the intrinsi impedan e (� = p�=") of , and let k2 = !2"�. Let usnote that by the de�nition of k0: k2 = "�"0�0 k20 :(6.1)Re all the well-known Bohren transform [16℄E = QL � i�QR ; H = 1i� QL +QR ;(6.2)

12 C. ATHANASIADIS AND I. G. STRATISof the ele tri and magneti �elds E and H in into left-handed and right-handedBeltrami �elds QL and QR, respe tively, satisfyingr�QL = LQL ; ;r�QR = � RQR(6.3)where the wave numbers orresponding to QL and QR are given by L = k(1� k�)�1; R = k(1 + k�)�1:(6.4)Sin e E and H are divergen e free, an immediate onsequen e of (A.3) is�QL + 2LQL = 0; �QR + 2RQR = 0; in :(6.5)It is known, see e.g., [21℄, that QL depends analyti ally on L, for L 6= 0 (and QRon R). Sin e, by (6.4), L and R are rational fun tions of k, QL and QR dependanalyti ally on k; hen e, by (6.2), E and H are both analyti in k, and in view of(6.1) in k0. Thus, we have established the desired property.A knowledgment. The authors thank Dr. W.S. Weiglhofer (University of Glas-gow, U.K.) for onstru tive dis ussions. In addition, they thank the referees for sub-stantial omments that led to the present form of the paper. They also a knowledgepartial �nan ial support from the General Se retariat of Resear h and Te hnology ofthe Ministry of Development (Gree e), within the PENED 1994 programme.REFERENCES[1℄ H. Ammari, M. Laouadi, and J.-C. N�ed�ele , Low frequen y behavior of solutions to ele -tromagneti s attering problems in hiral media, SIAM J. Appl. Math., 58 (1998), pp.1022{1042.[2℄ C. Athanasiadis, The multi-layered ellipsoid with a soft ore in the presen e of a low-frequen ya ousti wave, Quart. J. Me h. Appl. Math., 47 (1994), pp. 441{459.[3℄ , Low-frequen y ele tromagneti s attering theory for a multi-layered s atterer, Quart. J.Me h. Appl. Math., 44 (1991), pp. 55{67.[4℄ C. Athanasiadis and I. G. Stratis, Ele tromagneti s attering by a hiral obje t, IMA J.Appl. Math., 58 (1997), pp. 83{91.[5℄ , The ondu tive problem for Maxwell's equations at low frequen ies, Appl. Math. Lett.,10 (1997), pp. 101{105.[6℄ , On the s attering of ele tromagneti waves by a pie ewise homogeneous hiral obsta le,in Mathemati al Methods in S attering Theory and Biomedi al Te hnology, G. Dassios, etal., ed., Addison-Wesley-Longman, London, 1998, pp. 173{185.[7℄ D. Colton and R. Kress, Integral Equation Methods in S attering Theory, Wiley, New York,1983.[8℄ G. Dassios, Low frequen y s attering theory for a penetrable body with an impenetrable ore,SIAM J. Appl. Math., 42 (1982), pp. 272{280.[9℄ G. Dassios and K. Kiriaki, The low-frequen y theory of elasti wave s attering, Quart. Appl.Math., 42 (1984), pp. 225{248.[10℄ D. L. Jaggard, X. Sun, and J. C. Lin, On the hiral Ri ati equation, Mi rowave Opt. Te hn.Lett., 5 (1992), pp. 107{112.[11℄ D. S. Jones, Low frequen y ele tromagneti radiation, J. Inst. Math. Appl., 23 (1979), pp.421{447.[12℄ , S attering by inhomogeneous diele tri parti les, Quart. J. Me h. Appl. Math., 38(1985), pp. 135{155.[13℄ R. E. Kleinman, Low frequen y solution of ele tromagneti s attering problems, in Delft Sym-posium, ???ed., Pergamon, Oxford, 1967, pp. 891{905.[14℄ R. E. Kleinman and B. Vainberg, Full low-frequen y asymptoti expansion for se ond-orderellipti equations in two dimensions, Math. Meth. Appl. S i., 17 (1994), pp. 989{1004.[15℄ A. Lakhtakia, Rayleigh s attering by a bianisotropi ellipsoid in a biisotropi medium, Int. J.Ele troni s, 6 (1991), pp. 1057{1062.

LOW FREQUENCY ELECTROMAGNETIC SCATTERING THEORY 13[16℄ , Beltrami Fields in Chiral Media, World S ienti� , Singapore, 1994.[17℄ A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Dilute random distribution of small hiral spheres, Appl. Opti s, 29 (1990), pp. 3627{3632.[18℄ R. Pi ard, On the low frequen y asymptoti s in ele tromagneti theory, J. Reine Angew.Math., 354 (1984), pp. 50{73.[19℄ A. F. Stevenson, Solution of ele tromagneti s attering problems as power series in the ratio(dimension of s atterer)/wavelength, J. Appl. Phys., 24 (1953), pp. 1134{1142.[20℄ V. Twersky, Multiple s attering of ele tromagneti waves by arbitrary on�gurations, J. Math.Phys., 8 (1967), pp. 589{610.[21℄ P. Werner, On the exterior boundary value problem of perfe t refra tion for stationary ele -tromagneti wave �elds, J. Math. Anal. Appl., 7 (1963), pp. 348{396.