COMPARING TWO METHODS TO SOLVE THE LAYERED SPHERE PROBLEM, APPLICATION TO ELECTROMAGNETIC INDUCTION SENSORS.

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1 COMPARING TWO METHODS TO SOLVE THE LAYERED SPHERE PROBLEM, APPLICATION TO ELECTROMAGNETIC INDUCTION SENSORS. Pascal Druyts 1, Chrstphe Craeye 2, ad Marc Achery 1 1 Sgal ad Image Cetre, Ryal Mltary Academy, Av. de la Reassace 30, 1000 Brussels, Belgum, Pascal.Druyts@elec.rma.ac.be. 2 ELTM departmet, UCL, Place du Levat 3, B-1348 Luva-la-Neuve, Belgum. ABSTRACT The layered sphere prblem ca be slved by usg a multple expas each layer ad mpsg the apprprate budary cdts. Ths drect apprach has sme drawbacks because t ca be umercally badly cdted. We shw that a tramss le TL) frmalsm ca be used t slve the prblem ad that ths apprach s better cdted tha the drect apprach. The crrespdg TL s hmgeeus because t has a characterstc mpedace whch vares wth the radus ad whch als depeds the drect. We shw hw the classcal TL expresss ca be geeralzed t take t accut such a hmgeeus TL. Bth frmalsms are appled t a cfgurat represetatve f a metal detectr MD) abve a magetc sl. Fr such a cfgurat, a large umber f the terms s requred the expas ad the drect apprach fals cmputg the hgh rder terms because f umercal saturats. I ctrast, accurate results are btaed wth the TL apprach. Key wrds: layered sphere, multples, trasmss le, characterstc mpedace, metal detectr. 1. INTRODUCTION The slut t the layered sphere prblem s a drect extes f the Me slut fr a sphere [10]. It has bee used t mdel may prblems such as the huma head [7] r a stratfed les [9, 6]. I thse applcat, a drect apprach, whch a lear system f equat s buld ad slved, s used. I ctrast, fr the plaar layered medum, a TL aalgy [5] s fte trduced ad the slut s based a recursve prpagat f the reflect ceffcet frm the frst t the last layer ad the the reverse drect. Ths aalgy allws fr a better uderstadg f the uderlyg physcs ad als leads t mre tractable ad well-kw frmulas. We wll shw that a equvalet TL ca als be defed fr the layered sphere prblem 1. The equvalet TL s hwever hmgeeus because the characterstc mpedace vares wth the radus ad s als fuct f the drect. Ths hmgeety mples that the classcal TL frmulas must be geeralzed. We wll shw hw ths geeralzat ca be perfrmed. I all cases, the classcal frmulas are recvered f a equal mpedace s used fr bth drects. I bth cases, a mdal expas f the felds s used. I the the frst case, the ukw are the ampltudes f the mdes, whereas the secd, they are the reflect ceffcets. There exsts a -lear relat betwee thse ukws. Therefre, bth appraches ca sgfcatly dffer frm a umercal pt f vew. We wll shw that ths deed s the case ad that the TL apprach s better cdted. I [1] a termedate apprach s fllwed the sese that geeralzed reflect ceffcets are cmputed frm the terface ceffcets as the TL frmalsm but the terface ceffcets are drectly cmputed frm the medum electrmagetc EM) prpertes. N characterstc mpedace s defed ad the resultg frmulas are therefre less tractable. Furthermre, the reflect ceffcets are defed as a rat f multple ampltudes stead f a feld r vltage rat as s classcally de fr TL. There therefre exsts a multplcatve factr betwee the ceffcets defed [1] ad thse defed ths paper. Ths mples that the classcal terpretat f the TL trasmss ad reflect ceffcets are lst [1]. Fr example, a reflect ceffcet equal t e des t mply ttal reflect. Appart frm a reduced physcal terpretat, the multplcatve factr ca result umercal prblems. Ideed, the factr s fuct f the radus ad f the medum prpertes ad t ca take extreme values. Ths prblem s recgzed by Chew [1, p. 152] wh suggests a rermalzat. 1 strct sesu the equvalet schema s t a TL because the dfferetal equats gverg the vltages ad currets are t thse f a TL but ths has practcal csequeces fr ufrm layers because a aalytc slut exsts each layer ad the dfferetal equat must t be slved. 1

2 The paper s rgazed as fllws. Sect 2 presets the layered sphere prblem ad ts drect slut. The Sect 3 presets the TL apprach ad Sect 4 makes the lk wth the expresss fud [1]. The drect ad TL appraches are cmpared frm a umercal pt f vew Sect 5 fr a cfgurat represetatve f a MD abve a magetc sl. Fally, Sect 6 ccludes ths paper. 2. LAYERED SPHERE The prblem csdered s depcted Fg. 1 fr the 3- layers case. Each layer s assumed hmgeeus ad characterzed by ts electrcal permttvty 2 ɛ ad magetc permeablty µ. Equ. 2) expresses the felds as a weghted sum f mdes characterzed by the type t = T E fr a trasverse electrc feld ad t = T M fr a trasverse magetc feld), the party p = fr a dd feld ad p = e fr a eve feld), m = 0, the rder = 0 ad the drect f prpagat d =, ut). The sphercal multples ca be expressed as fllws: E TE,d, ) HTE,d = m d), j ) η d) 3a) E TM,d, ) HTM,d = d), j ) η md) 3b) where η = µ/ɛ s the characterstc mpedace f the medum ad m d) ad d) are the elemetary sphercal vectr wave fucts: ɛ 3, µ 3 I 2 I 1 R 3 R 2 R 1 α TE,3 R 1 R 2 α TE,3 Fgure 1. Layered sphere gemetry fr 3 layers ad a surces t shw uter layer. Regs are R 1, R 2 ad R 3. Iterfaces are I 1 ad I 2 wth radus R 1 ad R 2. TE multples α TE, are shw. Whe surces are preset a reg R, the ttal felds are frst splt a cdet ad scattered cmpet: E, H ) = E,c, H,c) E,scat, H,scat) 1) where the cdet feld s the feld that wuld be prduced by the surces R f all space was flled by the medum R. Fr the scattered feld, a multple expas ca be used each layer [10, Equ. 12 p. 394 ad Equ. 11,12 p. 416] 3 : E scat, H scat) = d α scat,d E TE,d, ) HTE,d β scat,d E TM,d, ) HTM,d 2) 2 as s classcally de the frequecy dma, fr a cductg medum, the cductvty σ s cmbed wth the permttvty ɛ t defe a equvalet permttvty ɛ σ = ɛ σ/jω ad the subscrpt σ s drpped 3 The tme depedecy assumed ths paper s e jωt whereas Stratt uses e jωt m d) e where u e w e m = N 1 z d) ρ)u e mθ, φ) 4) z d) ρ) m = N 1 w e ρ mθ, φ) r [ ] ρz d) ρ) N 1 v e ρ mθ, φ) 5) d) e m ad v e m s a scalar, all fuct ly f the agular crdates θ, φ): u e mθ, φ) = N 2 v e mθ, φ) = N 2 m are trasverse vectr felds ad { m s θ P m cs θ) s cs mφ P m } cs θ) cs θ s mφ φ { P m cs θ) cs θ s mφ θ m sθ P m s cs θ) cs mφ w e mθ, φ) = N 2 1)P m } φ θ 6) 7) cs θ)csmφ 8) s I the abve equats, R, θ, φ) are the sphercal crdates, r, θ, φ) are the crrespdg ut vectrs, ρ = kr where k = ω ɛµ s the waveumber, [fρ)] meas the dervatve f fρ) wth respect t ρ, P m s the asscated Legedre fuct 4, z ) = j s the sphercal Bessel fuct f frst kd ad z ut) = h s the Hakel fuct f secd kd ad N 2 s a rmalzat factr. We wll dete the multple expas 2) by { α, α, β, β } where α ad β are vectrs as dcated by the lwer bar bult by the ccateat f 4 Tw defts f the asscated Legedre fuct dfferg by a factr 1) m exst. We use the deft [10, Equ. 64 p. 182] 2

3 the expas ceffcets. The superscrpt d= ad d=ut have bee replaced respectvely by a left ad a rght arrw t mprve readablty. Fr the cdet feld, a multple expas ca als be used, but ly utsde the surce vlume: { E, H ) = 0, α, 0, } β, r R 9a) E, H) = { α, 0, β, 0 }, r R 9b) where R ad R are the er sphere ad uter sphercal shell excludg the surce vlume. We assume that all surces are remte frm the terface. Therefre, Equ. 9) ca be used at the terfaces f the surce reg. The ceffcets f the multple expass ca be cmputed frm the surce dstrbut [2]: α = ηk2 E J E TE dv 10) N 1N 1 V s The currets ad vltages are further splt as: V = V V I = I I 14a) 14b) where V, ad I are the α ctrbut ad we have drpped the p, m, ) subscrpt. Oe ca shw that the felds are the related t the vltages ad currets as fllws: E TE = V TE u H TE = j ηρ V TE w r I TE v 15a) 15b) whch shws that the budary cdts traslates t the ctuty f V ad I at the terfaces f the equvalet TL llustrated Fg. 2. where V s s the vlume ctag the surces. The duble arrw dcates that the ceffcet f the utgg multple ca be btaed frm the dt prduct f the curret ad the feld f a cmg multple ad vce Γ 1 Γ 1 V R) V R) V V V 1 V 1 versa. The express fr β ca be btaed by replacg TE wth TM 10). Impsg that the budary cdts are satsfed at each f the s terfaces yelds, fr every t, p, m, ), 2s lear equats the 2s 1) ukws 5 scat, α. Tw addtal cdts ca be derved frm the behavr at the rg ad at fty: α s1 = 0 11) α 1 = 0 12) leavg a square system f lear equats frm whch the ukws ca be determed by smple matrx vers. Ths s wll be called the drect slut t the prblem. 3. TRANSMISSION LINE FORMALISM The TL frmalsm has bee used t slve the plaar layered prblem amgst thers [5]. The aalgy wth a TL s btaed by defg vltages, currets ad mpedaces. The prblem s the slved by prpagatg the reflex ceffcet acrss the terfaces. By aalgy, we defe vltages ad currets as fllws: V TE = α TE N 1 j ρ) α TE N 1 h ρ) 13a) I TE = α TE jn 1 [ρj ρ)] α TE jn 1 [ρh ρ)] η ρ η ρ 13b) 5 fr the TE case R 2 R 1 R 1 Fgure 2. TL mdel. Represetat f vltages ad reflect ceffcets. X ad X represets a quatty cmputed respectvely at the uter ad er terface f reg R R Oe the defe the characterstc mpedaces: η TE,d ±V TE,d I TE,d = ±η j [ ρz d) ρz d) R ρ) ] 16) ρ) A smlar express ca be develped fr the TM case. The plus ad mus sg must be used respectvely fr the utgg ad cmg mde. Whe cmparg t the plaar case, e tes that the characterstc mpedaces are fuct f R ad that they are dfferet fr the cmg ad fr the utgg mde. They hwever satsfy a relat detcal t the e establshed [10, Equ. 43 p. 355] fr cyldrcal waves : η TE,d η TM,d = η 2 17) Oe ca further check that, as expected the far feld ρ 1), bth utgg mpedaces ted tward the ubud medum mpedace: η TM = η TE = η, lm ρ 18) 3

4 We w defe the wave mpedace: Z R) = V R) I R) ad the reflect ceffcet: 19) mdes flwg t the tw-prt V 1, V ) t thse flwg ut V 1, V ) f t. I the abve equat, Γ 1, = η 1 η 1 η η 1 η η 1 26a) Γ R) = V R) V R) 20) T 1, = η η 1 η 1 η 1 η η 1 26b) where the rght arrw dcates that e lks twards the uter regs ad that all surces are lcated sde the radus f terest. Smlar express ca be develped fr the ther drect. We fally defe the prpagatr fr a gve quatty X betwee tw pts at radus R a ad R b as fllws: P [Ra,R b ] X) XR b ) XR a ) 21) The prpagatrs fr V ad Γ ca easly be cmputed frm Equ. 13). We wll use smpler tat whe quattes are csdered at the terfaces. P [,j] X) s the prpagatr betwee terfaces ad j. X ad X stads fr X cmputed respectvely at the er ad uter terface f reg R. Oe ca easly shw that: ad Z = η 1 Γ Γ = η 22) Γ 1 η η η Z η Z η 23) Thse relats betwee Z ad Γ reduces t the classcal results [8, Sec ] whe η = η whch s vald fr a plaar layered medum. Usg the abve expresss ad tg that the wave mpedace s ctuus acrss a terface, e fds the fllwg relat betwee the reflect ceffcets at bth sdes f a terface: where Γ 1 = Γ 1, D 1, Γ 1 Γ, 1 Γ 24) D 1, = Γ, 1 Γ 1, T, 1 T 1, η = 1 η η 1 η 25) η 1 η η 1 η s the determat f the scatterg matrx f the tw-prt descrbg the terface. The scatterg matrx relates the are the elemetary reflect ad trasmss ceffcet f the terface whe lkg frm reg R 1 t R. The ceffcets Γ, 1 ad T, 1 crrespd t the ther drect ad ca be cmputed by terchagg wth 1 ad the frward ad backward mpedaces 26). Oe ca easly check that T 1, = 1 Γ 1,. Fr symmetrc mpedaces, the abve expresss are detcal t the classcal Fresel ceffcets. I that case, D 1, = 1 ad Γ 1, = Γ, 1 as requred fr Fresel ceffcets [8, Sec ]. The terpretat f the express relatg Γ bth sdes f a terface by tme-dma multple reflexs s well-kw fr symmetrc les [8, p. 102]. Ths terpretat als hlds fr the hmgeeus les csdered here as ca be see by trducg the Taylr develpmet 1/1 x) = 1 x x ). Ntg that Γ s ull the uter reg, usg P [,] X) t prpagate Γ frm the uter t the er terface f each reg ad Equ. 24) t crss the terfaces, we ca cmpute Γ recursvely frm the uter t the er reg. Γ ca smlarly be prpagated frm the er t the uter reg. Oe ca the prpagate the vltages frm the surce reg t the uter ad t the er regs. Fr ths, we frst have t cmpute the scattered vltage at the terfaces f the surce reg. Oe ca shw that the fllwg express ca be used: Γ V scat, = V c, Γ P [,1] V ) V c, 1 Γ Γ 27) were V c, ad V c, are the vltages crrespdg t the mdal cdet felds respectvely the uter ad er terface f the surce reg. A smlar expresss ca be establshed fr V scat, 4. COMPARISON WITH EXPRESSIONS IN CHEW I [1], trasmss ad reflect ceffcets have als bee defed. Fr the TE case, the geeralzed trasmss ad reflect ceffcet used [1] are defed as ) 4

5 R,1 = α / Chew α ad T,1 = α 1 / α respectvely. I ctrast, ur trasmss ad reflect ceffcet are defed by Γ = V / V ad T,1 = V 1/ V. Frm 13), e fds that: Γ = R j k R ),1 h k R ) Chew h k T,1 = T 1 R ),1 h k R ) ad smlar relats fr the TM case. 28a) 28b) The same relat lks the elemetary reflect ad trasmss ceffcets appearg [1] R,1 ad T,1 ) ad thse appearg ths paper Γ,1 ad T,1 ). Takg ths trasfrmat t accut, e ca recver the express [1] frm thse preseted ths paper. Ths yelds a crss-check fr ur expresss. Oe tes that the factr relatg bth defts ca vary large prprts. Ths may lead t umercal prblems. Ths was recgzed by Chew wh recmmeds a rermalzat. I ur apprach, the ceffcets are aturally rmalzed. Furthermre, ur ceffcets ca be terprated as ther classcal TL cuterpart. Fr example, a reflect f e meas a ttal reflect. Such prpertes whch ease the terpretat f the results are lst [1]. 5. NUMERICAL RESULTS R Fgure 3. Magetc ad electrc feld fr sphercal cl free space surruded by tw cductg shells. Real part f H r ) z-axs ad real part f H θ. ) ad E φ ) x-axs a ampltude varyg as s θ werehas the magetc feld has tw cmpets, alg θ ad r respectvely wth a s θ ad cs θ depedecy. There s vsble dfferece betwee the felds cmputed by the tw appraches. Ths ad the fact that the result s agreemet wth ur physcal uderstadg 7 yelds a gd valdat f ur mplemetat. We have appled the drect ad TL appraches t a threelayered test case. The terfaces have a radus f e ad tw meters. The cductvty f the er ad uter layer s σ = 10 5 S/m. All ther EM prpertes are equal t thse f free-space. h TX ẑz θ TX z TX R TX The surce s a sphercal cl [4, Sec. 8.5] f radus R s = 1.5m, lcated the mddle f the termedate layer. A sphercal cl s desged t geerate a multple feld characterzed by t = T E, = 1, m = 0, p = e. A cmg 6 ad a utgg multple are geerated, respectvely fr R < R s ad R > R s. Ar Sl R sl r The rat betwee the cmg ad utgg multple ca be fud by efrcg the ctuty f the tagetal electrc feld the surce sphere. I ctrast, due t the presece f a surface curret, the magetc feld wll be dsctuus the surce sphere. Frm ths dsctuty, e ca cmpute the curret dstrbut ad check that the curret s alg φ ad has a s θ dstrbut as expected fr a sphercal cl. I Fg. 3, we shw H r the x-axs ad H θ ad E φ the z-axs because, frm 3), t s apparet that fr the csdered multples, the electrcal feld s alg φ ad has 6 fr lw frequeces the magetc feld sde the surce sphere s hmgeeus ad drected alg bz. Ths requremet usually mtvates the use f a sphercal cl Fgure 4. MD abve sl. The cl prject s see as a hrztal le at z = z TX. Cl radus s a TX ad sphercal crdates are R = R TX, θ = θ TX ad φ = 0 2π The secd example, descrbed Fg. 4, ams at mdelg a MD abve a magetc sl. We have shw [3] that mdelg the sl by a sphere ca be advatageus t aalyze the effect f the sl the sgature f a bured me. Hwever, t be realstc, the radus f the sl sphere must be chse large whe cmpared t 7 ctuty at the terfaces, fast decay f the felds sde the cductrs, ctuty f the electrcal feld the surce sphere, etc. 5

6 1.4 x 10 7 the ampltude f the cdet multples are 8 : B z z Fgure 5. Magetc duct B z the z-axs fr 10cm crcular cl 50cm abve 10m sl sphere wth µ r = 5. Cmpars betwee half-space slut ) ad multple expas f rder 50 ), 200. ) ad 500 dstgushable frm half-space slut). the detectr ftprt. Ideed, the multple expas f the cdet feld cverges fast at sme dstace frm the surce sphere the sphere whch the cl les but the cvergece becmes very bad whe e appraches the surce sphere. Keepg the heght f the cl abve the grud cstat whle creasg the sl sphere radus, the ar-sl terface peetrates deeper t the slwcvergece reg ad t keep a gd apprxmat f the cdet felds at the terface, a large umber f multples s requred. Fg. 5) shws the magetc feld the z-axs fr a crcular cl abve a magetc sl sphere characterzed by µ r = 5. The parameters are a TX = 10cm, R sl = 10m ad h TX = 50cm. O the z axs, the magetc feld s vertcal ad fr large sl sphere radus, t shuld cverge t the feld a half-space cfgurat. The half-space slut s cmpared t a multple expas t rder 50, 200 ad 500. The maxmum rder fr whch the drect apprach ca be used s 54. Abve, umercal prblems ccur. The TL apprach des t have ths lmtat ad develpmets f rder 1000 ad mre have bee cmputed. Fr rder 200, the feld s well apprxmated belw the terface. Fr rder 500 ad abve, there s vsble dfferece betwee the half-space slut ad the multple expas. The dsctuty vsble fr lw-rder apprxmats s due t the fact that the exact cdet feld s used sde the surce reg stead f ts multple expas. The dsctuty the dcates that the cdet feld s t well apprxmated by the multple expas. Ths s further hghlghted whe cmpared t the half-space slut. T uderstad the rg f the umercal prblems the drect apprach, we frst te that accrdg t 10) [2], α TE e0 = I TX N 1 2πa TX ηk 2 h kr TX )P 1 θ TX ) 29) where I TX s the cl curret ad the ther parameters are shw Fg. 4. At lw frequeces, the small-argumet apprxmat f the Hakel fuct ca be used yeldg ρ 1 whch creases qute fast wth. Fr large rders, a umercal uderflw wll ccur. I the TL apprach, the cdet mdal electrc feld s used stead f the multple ampltude. The feld at the terface s prprtal t α TE e0j kr sl ). Usg aga the small-argumet apprxmat, the feld at the terface s prprtal t R sl /R TX ) whch decreases slwly wth because the radus rat s clse t e. Therefre, the exctat requred fr the TL apprach ca be cmputed up t hgh rder wthut umercal uderflws. T mtgate the prblem the drect apprach, e culd rermalzes the multples, chsg N 1 a way that the electrc feld at the sl terface s equal t e fr all rders. Hwever, the fast decay f the cdet multple ampltude s t the ly umercal prblem f the drect apprach. Fr reass smlar t thse dscussed abve, the ampltude f the ceffcet the matrx that must be verted spa several rders f magtude, leadg t a ll-cdted system. Addtal cdtg techques were trduced the cde, but eve s, the drect apprach fals much faster tha the TL apprach. The expresss that must be cmputed the TL apprach als clude Bessel fucts whch ca take extreme values. Numercal saturat ca hwever easly be avded by usg small r large argumet develpmet f the Bessel fucts. Takg the TE utgg mpedace fr example, Equ. 16) shws that fr small ρ, ths mpedace s prprtal t ρ. We cclude ths sect by tg that umercal prblems ca als ccur fr lw rder develpmets. A typcal example s a gd cductr a hmgeeus feld. I the cductr, the waveumber k becmes cmplex ad very large fr hgh cductvty. The Bessel fuct h kr cductr ) wll the verflw. The prblem ca aga be slved the TL apprach usg bg argumet apprxmat f the Bessel fucts. Fr the drect apprach a slut s t csder the crrespdg layer as perfect electrc cductr PEC) ad t mdfy the budary cdt accrdgly. Ths hwever cmplcates the cde as bth PEC ad -PEC budary cdts must be mplemeted ad requres t defe a cductvty threshld. 8 by symmetry, ly t = T E, m = 0, p = e multples have t be csdered 6

7 6. CONCLUSION A TL frmalsm has bee develped t slve the layered sphere prblem. I ths frmalsm, the reflect ceffcets are prpagated frm the er t the uter layer ad frm the uter t the er layer. The vltages, frm whch the felds ca be cmputed, are the prpagated frm the surce layer bth twards the er ad uter sphere. The expresss are geeral ad allw fr ay surce dstrbut ay layer. Ths s ctrast wth may expresss fud the lterature whch are specfc t a gve surce such as a dple r a plae wave. The express btaed have bee cmpared wth thse preseted [1] where reflect ad trasmss ceffcets are als defed but the TL aalgy s t trduced. Usg the TL frmalsm, the elemetary reflect ad trasmss ceffcets receve a mre cmpact express by the use f characterstc mpedaces. We have shw that the ceffcets used [1] ad thse used ths paper are equal up t a multplcatve factr. Our ceffcets ca be terprated as ther classcal TL cuterpart. Fr example, a reflect f e meas a ttal reflect. Thse prpertes whch ease the terpretat f the results are lst [1]. Furthermre, the multplcatve factr ca vary large prprtsad may lead t umercal prblems whe cmputg the expresss appearg [1]. [5] Krzysztf A. Mchalsk ad Jua R. Msg. Multlayered meda gree s fucts tegral equat frmulats. IEEE Trasacts Ateas ad Prpagat, 453): , March [6] H. Meras. Radat patter cmputat f a sphercal les usg me seres. IEEE Trasacts Ateas ad Prpagat, 306): , Nvember [7] Kstata S. Nkta, Gergs S. Stamataks, Nklas K. Uzuglu, ad Aggels Karaftas. Aalyss f the teract betwee a layered sphercal huma head mdel ad a fte-legth dple. IEEE Trasacts Mcrwave Thery ad Techques, 4811): , Nvember [8] Sphcles J. Orfads. Electrmagetc Waves ad Ateas. [9] Jh R. Safrd. Scatterg by sphercally stratfed mcrwave les ateas. IEEE Trasacts Ateas ad Prpagat, 425): , May [10] J.A. Stratt. Electrmagetc thery. McGraw- Hll Bk Cmpay, We have cmpared the tw appraches a prblem represetatve f a MD abve a magetc sl fr whch a hgh rder develpmet s requred. We have shw that the TL prblem yelds accurate results whereas the drect apprach fals. ACKNOWLEDGMENTS The Authrs ackwledges the Belga Mstry f Defece whch has fuded ths wrk. They are als graceful t I. va de Bsch fr hs usefull cmmets ad suggests. REFERENCES [1] Weg Ch Chew. Waves ad Felds Ihmgeeus Meda. Va Nstrad Rehld, [2] Pascal Druyts, Chrstphe Craeye, Ygadhsh Das, ad Marc Achery. Applyg recprcty t sphercal multples. I preparat. [3] Pascal Druyts, Ygadhsh Das, Chrstphe Craeye, ad Marc Achery. Effect f the sl the metal detectr sgature f a bured me. I Prc. SPIE Defece ad Securty Sympsum, Orlad, FL, USA, Aprl [4] Herma A. Haus ad James R. Melcher. Electrmagetc Felds ad Eergy. bk/www/. 7