CHAPTER 11 CLOAKING IN TERMS OF NONRADIATING CANCELLING CURRENTS

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1 CHAPTER 11 CLOAKING IN TERMS OF NONRADIATING CANCELLING CURRENTS Enrca Martn, Stfano Mac Dpartmnt of Informaton Engnrng, Unvrsty of Sna Va Roma 56, I-531 Sna, Italy E-mal: Artur D. Yagjan Concord MA, 1742 USA E-mal: A tory of nonradatng cancllng (NRC currnts s formulatd n ordr to drv suffcnt condtons for cloakng on t dyadc consttutv paramtrs of an ansotropc lnar mtamatral of arbtrary sap llumnatd by arbtrary sourcs. T lnk btwn t NRC currnt tory and t consttutv paramtrs of t cloak s stablsd by applyng t volumtrc quvalnc torm. T consttutv paramtrs ar vntually found as a functon of two vctor potntals satsfyng smpl boundary condtons. Two-dmnsonal and tr-dmnsonal transformaton optcs cloaks ar drvd as partcular cass of t gnral formulaton. 1. Introducton T transformaton optcs ntroducd by Pndry 1 s a powrful concptual and practcal tool to sap and addrss lctromagntc wavs troug t approprat dsgn of t consttutv paramtrs of mda. Ts mtod was appld to dsgn lctromagntc cloaks, slls of ansotropc matral capabl of rndrng any objct wtn tr ntror cavts nvsbl to dtcton from outsd t cloaks. T prfct cloak 35

2 36 E. Martn t al. nsurs tat for any ncdnt fld t lctromagntc scattrd fld vanss n t fr spac xtrnal to t cloakng sll, and t total fld vanss nsd t fr-spac cavty of t sll. Tus, any objct placd n t cavty dos not prturb t lctromagntc fld outsd t cloak and to an xtrnal obsrvr t appars as f t objct and cloak wr absnt. Suc nvsblty cloaks rqur matrals wt nomognous, ansotropc prmttvts and prmablts tat cannot b found n natur (tus bng rfrrd to as mtamatrals. An approxmaton to t dal consttutv paramtrs for a crcular cylndr was ralzd and xprmntally caractrzd by Scurg t al. 2 T formulaton proposd by Pndry 1 s basd on spatal coordnat transformatons and corrspondng transformatons of Maxwll s quatons tat provd xprssons for t rqurd nomognty and ansotropy of t prmttvty and prmablty of t cloakng mtamatral. A smlar approxmat (n t gomtrcal optcs lmt mtod for cloakng was prsntd by Lonardt 3, wr t Hlmoltz quaton s transformd by conformal mappng to produc cloakng ffcts. Subsquntly, Lonardt and Plbn dscussd t conformal mappng tory n t contxt of gnral rlatvty by analogy wt t dvaton of optcal rays clos to a gravtatonal mass 4. Mor rcntly, cloakng as bn rformulatd as a boundary valu problm wt a sngl frst-ordr Maxwll dffrntal quaton for lnar ansotropc mda 5. Ts altrnatv formulaton rvals t boundary valus of t flds at t nnr and outr surfacs of a cloak tat yld zro scattrd flds outsd t cloak and zro total flds nsd t cloak cavty. Morovr, n Rf. 5 t dffrnc btwn t 2D cas (cylndrcal cloak and t 3D cas (sprcal cloak s dscussd wt rfrnc to t bavour of t flds and polarzaton dnsts at t nnr surfac of t cloak. A mor xtnsv nvstgaton of t sngular bavour of t flds and polarzaton dnsts as bn carrd out n Rf. 6. T formulaton ntroducd n Rf. 5 as bn furtr dvlopd n Rf. 7 wt an nvstgaton of t group vlocts and t ngnrng lmtatons of broadband cloaks. Altoug t nonlnar transformaton n Rf. 1 s pyscally appalng, dffrnt typs of fld-transformatons can b adoptd tat do not rsort to any spac comprssons. In Rf. 8, gnral affn transformatons av

3 Cloakng n Trms of NonRadatng Cancllng Currnts 37 bn ntroducd wt t purpos of fndng dffrnt solutons of Maxwll s quatons n ansotropc and bansotropc mda. Basd on a smplfcaton of t gnral transformaton n Rf. 8, a lnar transformaton s usd n Rf. 9 to catgorz dffrnt typs of bansotropc mtamatrals. Dpndng on t coffcnts usd n t lnar transformaton, artfcal mda ar dfnd n Rf. 9 tat can produc t flds n a prscrbd fason n t volum occupd by t mdum. In Rf. 1, t us of dualty condtons on a lnar transformaton as tat n Rf. 9 ylds t dfnton of a nw mdum tat posssss ntrstng nvsblty and cloakng proprts. T cloakng mdum n Rf. 1 dos not nvolv any spac comprsson, and lads to fld solutons wt t Poyntng vctor tat dcrass n ampltud along t pat wtout cangng drcton nsd t mdum. Ts way, t ray vlocty matcs t spd of lgt and tus t mdum may not b subjct to t narrow bandwdt lmtaton of transformaton optcs cloaks. In ts captr, w prsnt an altrnatv approac to cloakng, wc s basd on nonradatng cancllng (NRC quvalnt currnts. Ts approac allows on to ncorporat t xstng two-dmnsonal (2D and tr-dmnsonal (3D transformaton optcs formulatons of cloakng n a gnral framwork tat provds nw pyscal nsgt nto t cloakng problm. T captr s structurd as follows. In Scton 2, t cloakng problm s ntroducd by mans of a volumtrc quvalnc prncpl. A tory of NRC quvalnt currnts s formulatd n Scton 3 and appld n Scton 4 to fnd a st of suffcnt condtons for cloakng on t prmttvty and prmablty tnsors of an ansotropc mtamatral. Scton 5 sows ow t 2D and 3D transformaton optcs cloaks can b drvd as spcal cass of t prvously ntroducd gnral formulaton. Lastly, conclusons ar drawn n Scton 6.

4 38 E. Martn t al. 2. Problm Formulaton 2.1. Ansotropc Mtamatral Cloak Consdr an annular volum of ansotropc mtamatral mmrsd n fr spac, boundd by an xtrnal surfac Σ and by an ntrnal surfac Σ ' wc rprsnts t boundary of t cavty (Fg. 1a. Dnot by V nt t cavty volum wtn Σ ', by V xt t xtrnal volum and by V t ansotropc mtamatral volum. T cavty rgon V nt s assumd to b flld wt fr spac; owvr t can b flld wt an arbtrary mdum wtout cangng t fnal rsult. In t followng, w us bold caractrs for dfnng vctors and bold caractrs wt doubl bars to dnot dyadcs; w supprss t tm dpndnc xp( jω t. { EH, } { J, M} { E, H} V xt { J, M} µ,ε V nt { EH, } ε, µ ε V, µ Σ ε, µ { E, H} ε, µ V' = V + V nt (a (b Fg. 1 Gomtry for t problm: (a ansotropc mdum and dfnton of t ntrnal flds, (b Rfrnc problm for t ncdnt fld. T ansotropc mtamatral s llumnatd by t ncdnt fld E, H producd n absnc of t mtamatral by t mprssd { } lctrc and magntc currnt dnsts {, } rgon V xt (Fg. 1b. T total lctrc and magntc flds {, } Maxwll s quatons ( jω ( ( Σ J M locatd n t xtrnal EH satsfy Er = Br M r (1 ( jω ( ( Hr = Dr J r (2 +

5 Cloakng n Trms of NonRadatng Cancllng Currnts 39 wr t magntc and lctrc nducd flds (calld t nductons B and D ar rlatd to t flds by t lnar, nomognous, ansotropc consttutv rlatons Br ( =µr ( Hr ( (3 Dr ( =εr ( Er ( (4 In t followng, t functons B and D ar ntndd to b contnuously dffrntabl functons of r xcpt possbly at t nnr boundary Σ ' wr ty may b dscontnuous but do not contan dlta functons. Possbl dlta functon sngularts n t nductons wll b xplctly ndcatd as a sparat trm. From r on w supprss and undrstand t spac dpndnc of any functon, unlss otrws ndcatd. T consttutv dyadcs smplfy outsd V to tos of t fr-spac µ µ I ; ε ε I r V, V (5 wr I s t unt dyadc. T ncdnt (unprturbd flds { E, H } oby t omognous Maxwll s quatons n fr spac nt E = jωµ H M (6 H = E J (7 jωε + T mtamatral sll cloaks t cavty f t total xtrnal fld (fld n V xt s qual to t ncdnt fld { E, H } and t total fld n t cavty vanss, tat s {, } = {, } EH E H (8 r V xt {, } = {,} r V nt EH (9 In trms of t scattrd flds {, } {, } condtons (8-(9 bcom xt E H E E H H, t cloakng s s {, } = {,} E H (1 s s r V xt {, } = {, } E H E H (11 s s r V nt

6 31 E. Martn t al. namly, t scattrd flds qual zro n t xtrnal volum and qual t ngatv of t ncdnt flds nsd t cavty. Unqunss torms nsur tat qs. (8 and (9 ar satsfd f Maxwll s quaton ar mposd n t tr rgons V xt, V and V nt, along wt propr boundary condtons on Σ and Σ '. In partcular, on Σ t tangntal componnts of t flds must b contnuous, tat s E nˆ = E nˆ ; H nˆ = H n ˆ on Σ (12 wr ˆn s t xtrnal normal to Σ. On Σ ' dffrnt sts of boundary condtons can b mposd to nsur zro total fld nsd t cavty (assumng a vansngly small loss nsd t cavty to lmnat sourcfr cavty rsonancs. In fact, unqunss of t soluton s nsurd n a fnt sourc-fr volum f t tangntal componnts of t lctrc fld, or t tangntal componnts of t magntc fld, or t normal componnts of bot t lctrc and t magntc flds ar spcfd on t volum boundary 5,11. Tus on Σ ' + w may apply altrnatvly on of t followng condtons: t tangntal componnt of lctrc fld qual to zro, t tangntal componnt of magntc fld qual to zro, or t normal componnts of bot lctrc and magntc fld qual to zro, namly ( E n ˆ ' = ; on + (13 ( H n ˆ ' = ; on + (14 ( Hn ˆ' = and En= ˆ' ; on + (15 wr n ˆ ' s t xtrnal normal to Σ ' and Σ ' + dnots t nnr sd of Σ ' (s Fg. 2a. Ts condtons on t flds at t nnr sd of ar transformd to condtons on t flds at t outr sd of Σ ' (tat s, n t cloak-matral rgon f addtonal assumptons ar mad on surfac currnts and cargs at Σ '. Assumng vansng tangntal magntc surfac currnt (or polarzaton for (, vansng tangntal lctrc surfac currnt (or polarzaton for ( and vansng magntc and lctrc surfac cargs (or polarzaton cargs for (, w obtan ( E n ˆ ' = ; on (16

7 Cloakng n Trms of NonRadatng Cancllng Currnts 311 ( H n ˆ ' = ; on (17 ( En ˆ' = and Hn= ˆ' ; on (18 wr ( can also b wrttn as ˆ ' = B n and Dn ˆ ' =. It s assumd tat for a gvn cloak gomtry, an ( εµ, can b found tat producs a soluton to Maxwll s quatons (1-(2 wt t consttutv rlatons (3-(4 and t boundary condtons n (12 and (16-(18. Morovr, dffrnt prmttvty-prmablty functons ( εµ, may produc t sam zro scattrd flds outsd t cloak and yt dffrnt scattrd flds wtn t cloak matral Volumtrc Equvalnc Prncpl for Dscontnuous Flds From t standpont of t volumtrc quvalnc prncpl, t scattrd flds { Es, Hs} ar producd by quvalnt polarzaton currnts (wt possbl dlta functons on t boundars radatng n fr spac. Trfor, t cloakng condtons n (1 and (11 man tat t quvalnt polarzaton currnts ar nonradatng outsd t mtamatral sll and cancllng t ncdnt fld nsd t cavty of t sll. W admt r t possblty tat t flds ar dscontnuous only at t nnr surfac, wt fnt dffrnt valus on t two sds of t surfac. As a consqunc, qs. (1-(2 can b rwrttn as 12 { } j { } E + E= ω B jωµ M M (19 δ { } jω{ } H + H= D + jωp + J (2 wr M and P ar magntc and lctrc dlta-functons polarzaton δ δ dnsts tat com from t dscontnuts of E and H, rspctvly, D ndcat t rgular part of t magntc and lctrc { B} and { } nductons, rspctvly, { } dnots t curl oprator wtout ncludng t dscontnuty surfac (wc lads to a rgular functon and t oprator dnots t surfac curl oprator dfnd by 12 δ

8 312 E. Martn t al. ( n' + δ ˆ EBδ n' E E = jωµ M. (21 + δ ' δ( n' ˆ Σ HB n' H H = jωp wr t scond qualty of bot t quatons coms from t dntfcaton of t dlta currnts n (19-(2 wt t dlta dscontnuous part of t curl. In (21, n ˆ ' s t normal to drctd δ n' s a Drac dlta functon, wos argumnt s t toward t cavty, ( dstanc to along t drcton of nˆ ' and E + (E - dnots t valu of t functon E on t postv (ngatv sd of (smlar dfnton for H. Aftr subtractng (6-(7 from (19-(2 w obtan, outsd t sourcs, ( jω{ } jωµ o jωµ = + δ E E B H M (22 ( { } H H = jω D jωε E + jωp (23 Eqs. (22-(23 can b rarrangd as E = jωµ H M s o s q H = jωε E + J s s q wt t quvalnt polarzaton currnts M, J dfnd by q ( ( n' + M ˆ q = jω µ µ oi H χv + δ E E n' ( V ( n' + J ˆ q = jω ε ε I E χ + δ n' H H wr χ V s t caractrstc functon of V 1 χv = r V r V q δ (24 (25 ( In (25 E and H ar zro f w assum zro flds nsd t cavty. T quvalnt polarzaton lctrc and magntc currnts J and M ar t sourcs of t scattrd flds {, } E H n fr-spac. Ts currnts contan dlta functons f t tangntal componnts of t flds xbt a jump across. Dlta functons ar not ncssary f t jump occurs n t normal fld componnts. s s q q

9 Cloakng n Trms of NonRadatng Cancllng Currnts 313 T prfct cloakng condtons n (1-(11 ar quvalnt to t J, M n (25 ar nonradatng rqurmnt tat t currnts ( q q cancllng (NRC currnts, namly tat ty do not radat n V xt and ty cancl t ncdnt fld n V nt. T nxt scton prsnts suffcnt condtons for NRC currnts. 3. Suffcnt Condtons for NRC Currnts In ts scton w dmonstrat two torms on NRC currnts, tat wll b rfrrd to as Torm A and Torm B. Torm A can b sn as t gnralzaton to NRC currnts of t torms dmonstratd by Dvany and Wolf 13 for nonradatng lctrc currnts. Altoug t can b rgardd as a partcular cas of Torm B, t s dmonstratd sparatly, snc t s blvd tat ts dmonstraton facltats t undrstandng of Torm B. Torm A: Lt {, } F F b two vctor functons (calld r potntals dffrntabl at any pont of t volum V and { ˆn, n ˆ ' } t normal unt vctors xtrnal to Σ and ntrnal to Σ ', rspctvly. Lt t potntals satsfy t condtons wr {, } F nˆ = E nˆ ; F nˆ = H n ˆ on Σ (27 F nˆ' = ; F n ˆ' = on (28 E H ar arbtrary ncdnt flds satsfyng t fr-spac Maxwll s qs. n V'=V+V nt. Tn t currnts ( j ( jωε χv M = F ωµ F χ NRC o V J = F F NRC (29 ar NRC currnts (ty do not radat n V xt and radat mnus t ncdnt fld n V nt ; morovr, n V ty produc a fld qual to, E H = F E, F H. (Not tat t curl n (29 s not ntndd { s s} { } to ntroduc any dlta functon on t surfac wn appld to t, potntals bcaus of t assumpton of t rgularty of F. Proof: At any pont of t spac on as

10 314 E. Martn t al. wr ( n ( χ F = χ F + χ F,,, V V V ( ( n ˆ ( n' ˆ' δ( n ˆ (,,, = χv F δ n+ δ n F = = χ F n E H, V (3 δ s a Drac dlta functon, wos argumnt s t dstanc along t drcton of t unt normal ˆn to t surfac Σ. In t lattr, qualty w av usd t fact tat nˆ ' F vanss on. T currnts n (29 can b trfor rwrttn as T flds {, } ( χ jωµ ( χ δ( n ( χ jωε ( χ δ( n ˆ M = F F nˆ E NRC V o V J = F F + n H NRC V V E H radatd by ts currnts satsfy t quatons s s ( χ jωµ ( χ δ( n ( χ ' jωε ( χ ' δ( n ˆ E F = H F + nˆ E s V o s V H F = E F + n H s V s V (31 (32 Ts mans tat t flds ES χvf and HS χvf oby vrywr t Maxwll s quatons n fr-spac wt xctaton gvn by t surfac currnts M ˆ s = E n δ ( n, J ˆ s = n H δ ( n on Σ. Ts currnts ar qual to t quvalnt currnts provdd by t Lov formulaton of t quvalnc prncpl appld to t surfac Σ n fr spac. 14 Tus, ty radat mnus t ncdnt fld n V+V nt and zro fld n V xt : ( χ ( χ E F = E χ s V V+ Vnt H F = H χ s V V+ Vnt wc mans tat t flds radatd by t quvalnt sourcs n (29 ar and Torm A s provd. E = χ F E χ s V V+ Vnt H = χ F H χ s V V+ Vnt (33 (34 W

11 Cloakng n Trms of NonRadatng Cancllng Currnts 315 Rmark 1: If t ncdnt fld s zro, t currnts bcom nonradatng n V nt and V xt, and Torm A bcoms t xtnson of t torm of nonradatng currnts by Dvany and Wolf 13 to t cas of a combnaton of lctrc and magntc sourcs. Rmark 2: T statmnt of Torm A can b also rprasd by sayng F, F n V and zro n V nt (tat s, rgon V nt s tat t total fld s { } mpntrabl to any radaton from outsd. Tus, avng quvalnt currnts dfnd n (29 wt boundary condtons n (27-(28 s a suffcnt condton to cloak t rgon of spac V nt from an arbtrary ncdnt fld. T condtons n (28 can b mtgatd f on allows nonvansng surfac currnts on Σ ', as statd n t nxt Torm. (a (b Fg. 2. Dstrbuton of potntals F, F nsd V for t cas of Torm A (a and Torm B (b. T NRC currnts ar constructd va ts potntals troug (29 and (37 for (a and (b, rspctvly. In (a, t tangntal componnts of t potntals xbt at Σ a jump qual to t valu of t tangntal componnts of t ncdnt fld, and zro tangntal componnts at. In (b, t potntals av t sam bavour as n (a at Σ and prsnt a jump n t tangntal componnts at ; countr-radatng surfac currnts must b prsnt at to cancl t radaton producd by t potntal dscontnuty. Torm B: Lt {, } F F b any two dffrntabl vctor potntals dfnd at any pont of t volum V tat satsfy t condtons F nˆ = E nˆ ; F nˆ = H n ˆ on Σ (35 F nˆ' = f ; F nˆ' = f on (36

12 316 E. Martn t al. wr f, f ar arbtrary contnuous dffrntabl functons on Σ '. Tn, t currnts ( jωµ χ δ ( n' ( jωε χv δ ( n' M = F F f NRC o V J = F F + f NRC (37 ar NRC currnts (do not radat n V xt and radat mnus t ncdnt fld n V nt ; morovr, n V ty radat a fld qual to, E H = F E, F H. { s s} { } Proof: By applyng t sam stps as n t proof of Torm A w obtan ( ( χ jωµ ˆ δ ( n ˆ' δ ( n' δ ( n' ( χ jωε ˆ δ n ˆ' δ n' δ n' M = F F n E n F f NRC V o ( ( ( ( J = F F + n H + n F + f NRC V (38 Snc t last two trms cancl ac otr bcaus of t boundary condtons, (38 concds wt (31; nc, followng t sam stps as for t proof of t prvous torm, w obtan E = χ F E χ s V V+ Vnt H = χ F H χ s V V+ Vnt (39 and Torm B s provd. W Rmark 3: Torm B rducs to Torm A for vansng valus of tangntal fld componnts on Σ '. Rmark 4: If t functons f, f ar dvrgnc fr along t surfac,, namly, f =, w av s, F n ˆ ' = on (4 snc ( F nˆ'= F nˆ' + F nˆ' = F n ˆ' (41,,,, s s s

13 Cloakng n Trms of NonRadatng Cancllng Currnts 317 Rmark 5: It s clar from q. (38 tat t jump n t tangntal componnts of t potntals gvs an xtra contrbuton tat as to b canclld by surfac polarzaton currnts. Addtonal surfac polarzaton currnts ar t prc tat as to b pad to rlax t boundary condtons on t potntals at. In t followng scton, Torm B s appld to fnd suffcnt condtons on t consttutv dyadcs of an ansotropc mtamatral usd for cloakng. 4. Suffcnt Cloakng Condtons on t Consttutv Dyadc Paramtrs T lctrc and magntc currnts comng from t applcaton of t quvalnc torm to t flds of a cloak sould now b dntfd wt t NRC currnts of Torm B. Ts wll lad to addtonal + + condtons on t potntals. T currnts n (25, wt E and H forcd to zro nsd t cavty bcom ( ( n' M ˆ q = jω µ µ o I H χv δ E n' ( V ( n' J ˆ q = jω ε ε I E χ δ n' H Ty can b dntfd wt t NRC currnts of Torm B suc tat ( jωµ χ δ ( n' ( jωε χv δ ( n' M = F F f NRC o V J = F F + f NRC (42 (43 wt total ntrnal flds concdnt wt t potntals (tat s, wt,, EH χ = E + E H + H χ = F, F χ f on of t addtonal { } { } { } V s s V V condtons (16-(18 s mposd at Σ '. T qualty Mq, Jq = [ MNRC, J NRC ] appld to (42-(43 lads troug smpl algbrac stps to t followng torm. Torm C: Suffcnt condtons for cloakng on t prmttvty and prmablty tnsors of a mtamatral ar t fulflmnt of t quatons

14 318 E. Martn t al. χv F = jω µ F χv χv F = jω ε F χv (44 wr F, F ar arbtrary contnuous potntals satsfyng t boundary condtons. F nˆ = E nˆ; F nˆ = H nˆ on Σ (45 and on of t altrnatv condtons ( F n ˆ ' = ; on (46 ( F n ˆ ' = ; on (47 ( F nˆ' = and F n ˆ' = ; on (48 Furtrmor, F, F rprsnt t total lctrc and magntc flds nsd t cloak. W mpasz tat t condtons (46-(48 ar altrnatv suffcnt condtons, tat s, ty do not nd to b smultanously vrfd. In partcular, t s confrmd tat t boundary condtons of nˆ B = and nˆ D = usd n Rf. 5 at t nsd nnr surfacs of cloaks ar suffcnt condtons for solutons to bot 2D and 3D cloaks to not scattr t ncdnt flds outsd t cloaks and to produc zro flds wtn t cavts of t cloaks. W also not tat (44 tlls us somtng mor tan t Maxwll s quaton. Indd, t vctor potntals ar qut arbtrary, xcpt tat ty must b dffrntabl and ty av to satsfy t boundary condton n (45 and on of t boundary condtons n (46-(48. T oprator n (44 s ntndd to b appld to a contnuous functon, possbly dfnd as rgular outsd t volum V so as not to ntroduc dlta functons on t surfacs. T suffcnt condtons (-( prscrb dffrnt bavours of t tangntal surfac polarzaton dnsts at t nnr surfac of t cloak. In fact, (, (, and ( allow on Σ ' t prsnc of lctrc surfac currnts f δ ( n', t prsnc of magntc surfac currnts f δ ( n', and t prsnc of bot lctrc and magntc surfac currnts f δ( n', f δ( n', rspctvly.

15 Cloakng n Trms of NonRadatng Cancllng Currnts 319 Torm C coms from Torm B, wc rqurs t condtons n (16-(18, wc n turn rqur vansng tangntal magntc surfac currnt, vansng tangntal lctrc surfac currnt, and vansng magntc and lctrc surfac cargs, rspctvly. Tus, t must b assumd ntally tat t cloaks also satsfy ts lattr condtons. In t nxt scton, t s confrmd tat transformaton optcs cloaks do ndd satsfy ts condtons. Eq. (44 lnks t potntals wt t consttutv tnsors of t sll matral. Snc t potntals must satsfy condtons wc dpnd on an ncdnt fld, n ordr to convrt qs. (44-(48 nto sourc-ndpndnt condtons on t consttutv paramtrs µ,ε, t vctor potntals sall ncssarly b dfnd n trms of t ncdnt fld troug a spac dpndnt transformaton tat rducs to t dntty n V xt. An xampl s provdd by t comprsson-typ transformaton of Transformaton Optcs 1 tat wll b consdrd n t followng scton. 5. Transformaton Optcs Cloaks T prsnt formulaton ncluds t Transformaton Optcs (TO cloaks as a spcal cas of Torm C. In a TO cloak, t ncdnt fld s comprssd by a coordnat transformaton nto t cloak volum V and t consttutv paramtrs of t mtamatral tat yld t sam fld comprsson ar dtrmnd n trms of t Jacoban matrx of t coordnat transformaton. In trms of ray tory, t trajctors of lctromagntc rays passng troug t rgon of comprssd spac must conform to t local mtrc. Onc t dsrd trajctors ar dtrmnd troug a conformal mappng appld to Cartsan stragt trajctors n a vrtual fr-spac, t dffrntal oprators of t Maxwll s quatons n t transformd spac lad to spac-dpndnt mtrc coffcnts tat can b rntrprtd n trms of consttutv rlatons of an ansotropc, nomognous mdum. In otr words, t local mtrc of a dformd lmntal volum s ntrprtd as a local cang of componnts of t local prmablty and prmttvty dyadcs. T local tnsors produc a comprsson or an xpanson of t local wavlngt (wt consqunt cang of local pas vlocty so as to

16 32 E. Martn t al. qualz t pas dlay from an nput to an output fac of a comprssd lmntal curvlnar volum to t corrspondng pas dlay of t uncomprssd (Cartsan-coordnat lmntal volum. T fnal rsult s a mdum tat xbts prmttvty and prmablty componnts smallr (gratr tan t ons of fr spac n drctons paralll (ortogonal to t comprsson. To rconstruct t abov pyscally appalng pctur wtn t framwork of ts formulaton, lt us ntroduc n t ral spac a Cartsan rfrnc systm wt unt vctors xˆ1, xˆ ˆ 2, x. T poston 3 r= xxˆ + x xˆ + x x ˆ. Dfn a transformaton vctor s dfnd as r' = r'( r (49 tat maps t obsrvaton varabl r of t ral spac nto t obsrvaton varabl r' = x1' xˆ ˆ ˆ 1' + x2' x2' + x3' x 3' of a vrtual fr-spac, wr (x 1 ', x 2 ', x 3 ' and xˆ1', xˆ ˆ 2', x 3' ar ortogonal coordnats and unt vctors, rspctvly, of t vrtual spac. Eq. (49 can b wrttn n componnts as T nvrs transformaton s x ' = x '( x, x, x =1,2,3 ( x = x ( x ', x ', x ' =1,2,3 ( For cloakng, t transformaton n (5 must go smootly to r' = r for any r Σ blongng to t xtrnal surfac Σ ; furtrmor, t nvrs transformaton n (51 maps t orgn of t vrtual spac nto t nnr surfac Σ '. Ts condtons can b wrttn as '( Σ r r r = r (52 + ( ( r' r r = r r' = r (53 Assum also tat t transformaton bcoms t dntty outsd V xt, and nsd V nt, namly ( V ( r' r = r r ; r' r = r V (54 W ntroduc as potntals t functons xt nt

17 Cloakng n Trms of NonRadatng Cancllng Currnts 321 ( ( '( ( ( '( = F r A r E r r = F r A r H r r (55 wr E( r', H( r ' ar t ncdnt flds dscrbd n t vrtual coordnat systm. T boundary condtons (45 and (46-(48 on t potntals turn nto t followng condtons on A( r nˆ ( A( rσ I = (56 (, ( ( nˆ ' A r = (57 ( ˆ ( ˆ ( Σ ' n ' µ A r = ; n' ε A r = (58 wr I s t dntty matrx. T lattr condton s t sam appld n Rf. 5 for t spcal cas of sprcal and cylndrcal cloaks. In t vrtual spac r ', t ncdnt flds satsfy t Maxwll quatons ( jωε ( ( jωµ ( ' H r' = E r' ' E r' = H r' wr (59 = ' xˆ ' x ' + xˆ ' x ' + x ˆ ' x ', namly, dffrntaton s prformd n t coordnat of t vrtual spac. Insrtng (55 nto (44 and usng (59 lads to 1 ( ( ' Ar E r = µr ( Ar ( ' E( r ' (6 µ 1 ( ( ' Ar H r = εr ( Ar ( ' H( r ' (61 ε 1 Assumng tat a unqu soluton xsts to (6 for µ µr ( for a gvn cloak gomtry and transformaton r' ( r, tn a unqu soluton xsts 1 to (61 for ε ε( r, and t follows tat 1 1 µr ( = εr ( = αr ( (62 µ ε

18 322 E. Martn t al. Tus, (6 and (61 may b rwrttn as ( ( ( 1 1 A r α r ( ' A r E r = ' E( r' ( ( ( ( 1 1 A r α r ' A r H r = ' H( r' (63 For any vctor a, t xprsson of t curl n t ral spac can b transformd nto t curl n t vrtual spac by 1,15,16 wr M and r ' = '( T ( 1 1 a= dt( M M ' [ M ] a (64 1 M ar t Jacoban matrx of t transformaton r r and ts nvrs matrx, rspctvly (t lattr s t Jacoban matrx of t nvrs transformaton x ' 1 x M = ; M = (65 x j x j ', j= 1,3, j= 1,3 T xprsson n (64 s stll vald for a coordnat transformaton btwn two gnral curvlnar ortogonal systms, provdd tat t propr scal factors ar ncludd n t Jacoban matrx ( By usng (64 n t frst of qs. (63 w obtan A T 1 α dt( M M ' [ ] ( ' M A E r = ' E ( r ' (66 wr t dpndnc on r s supprssd. T abov xprsson s satsfd for wc may b rwrttn as ( T dt( A M = A α M M = I (67 T 1 T A= M ; ( 1 α= dt( M M M (68 T solutons for t flds and nductons wtn t cloak wll b trfor gvn by

19 Cloakng n Trms of NonRadatng Cancllng Currnts 323 ( = T ( '( ( = T ( '( ( 1 ε dt( '( ( µ dt( 1 '( Er M r E r r Hr M r H r r Dr = MM Er r Br MM H r r = (69 T boundary condtons (56-(58 may b rwrttn n trms of t Jacoban matrx as ( (, ( T ( T nˆ M r = nˆ I (7 Σ nˆ ' M r = (71 Σ ' ( 1 ˆ dt( = nm M (72 In t followng scton, t fulflmnt of ts condtons s cckd wt rfrnc to two- and tr-dmnsonal cloaks of arbtrary sap. It wll b found tat (7 and (72 ar satsfd n two-dmnsonal cloaks of arbtrary sap, wl (71 s not n gnral satsfd. On t otr and, (7, (71 and (72 ar smultanously satsfd by trdmnsonal cloaks of arbtrary sap Boundary Condtons Exprssd n Trms of Covarant and Contravarant Vctors T Jacoban matrx M and ts nvrs can b wrttn n trms of t covarant (g and contravarant (g vctors of t coordnat transformaton, wc ar dfnd by r g =, g = x ' ' (73 x Bot covarant and contravarant vctors can b usd as bass vctors for gnral curvlnar coordnat transformatons. T covarant vctors ar bult along t coordnat axs, wl t contravarant vctors ar bult to b prpndcular to t coordnat surfacs. T Jacoban matrx and ts nvrs can b wrttn n trms of covarant and contravarant vctors as follows

20 324 E. Martn t al. M 1 g 2 = g 3 g 1 ; M [ g, g, g ] = ( T dtrmnant of t Jacoban matrx can b wrttn as ( dt M= g g g = 1 g g g ( (75 It s now sown tat t boundary condtons (52 and (53 on t coordnat transformaton mply t fulflmnt of condton (7 and at last on of t (71-(72. In partcular, q. (52 mpls M( rσ = I (76 wc drctly ylds t fulflmnt of (7. T mplcatons of condton (53 ar nvstgatd n t nxt scton sparatly n t cass of two-dmnsonal and tr-dmnsonal cloaks Tr-Dmnsonal Cloaks In t comprsson typ transformaton ladng to tr-dmnsonal cloaks, condton (53 mans tat a pont n t vrtual spac s transformd nto a fnt surfac n t ral spac. Ts mpls lm ˆ ' x ' n g = ; lm nˆ ' g = lm = b (77 r ' r' r' n' lm nˆ ' g = ; r + r Σ ' lm nˆ ' g = c (78 r + r Σ ' wr b and c ar fnt constants dffrnt from zro. Eq. (78 sows tat t componnts of t covarant vctors tangntal to Σ ' tnd toward nfnty wn approacng Σ ', wl t valu of t normal componnt rmans fnt. On t otr and, (77 mpls tat t contravarant vctors ar all ortogonal to t surfac wt fnt ampltud. Ts, n turn, mpls t fulflmnt of condton (71 snc n ˆ M n ˆ g g g (79 T lm ' = lm ' = + + r r r r

21 Cloakng n Trms of NonRadatng Cancllng Currnts 325 wc s quvalnt to ( ˆ ( nˆ' F r = n' F r = (8 Trfor, t tangntal componnts of bot t lctrc and magntc flds vans on, and tus bot (46 and (47 ar satsfd. Ts also mpls tat tr ar no surfac currnts on. Hnc, a trdmnsonal cloak obtand troug a comprsson-typ coordnat transformaton can always b cast n t framwork of Torm A. On t otr and, snc (77 mpls tat all t contravarant vctors ar algnd wt t unt vctor nˆ ' on, and nc ty ar all paralll, t follows tat Ts mpls + r r Σ ' + + r r r r lm dt M = (81 [ ] 1 lm dt( M nˆ M = lm dt( M nˆ g g g = ( and tn ylds t fulflmnt of (72. Hnc, also t normal componnts of t nductons vans on nsd t cloak, and condton (48 s also satsfd. W mpasz tat, altoug only on of t tr condtons (-( n (46-(48 s suffcnt for cloakng (wn combnd wt (45, t TO 3D cloak fulflls all t tr condtons for any sap. Ts s not tru for 2D cloaks, as sown n t nxt subscton Two-Dmnsonal Cloaks In t comprsson typ transformaton ladng to 2D cloaks nvarant along t z drcton, condton (53 mans tat t ln x= y= n t vrtual spac s transformd nto a fnt surfac n t ral spac. Ts mpls lm ˆ ˆ ' x ' z n g = lm nˆ ' g = lm = b lm zˆ g = 1 (83 r ' r' r' n ' r ' lm zˆ nˆ ' g = + r r Σ ' lm nˆ ' g = c (84 + r r Σ '

22 326 E. Martn t al. wr b and c ar fnt constants, n gnral dffrnt from. Eq. (84 sows tat t componnts of t covarant vctors tangntal to Σ ' n t xy plan tnd toward nfnty wn approacng Σ ', wl t valu of t normal componnt rmans fnt. On t otr and, (83 mpls tat t contravarant vctors tnd to av vansng tangntal componnts on Σ ' n t xy plan, but fnt tangntal componnts along z. As a consqunc, n ts cas w av z ˆ n ˆ M z ˆ n ˆ g g g (85 T lm ' = lm ' = + + r r r r wc mpls t vansng of t tangntal componnts of t fld n t xy plan ( ˆ ˆ ( zˆ nˆ' E r = z n' H r = (86 Howvr, t sam s not n gnral tru for t z-componnts of t fld, snc w av wc mpls z ˆ M z ˆ g g g (87 T lm = lm + + r r r r ( ˆ ( zˆ E r ; z H r (88 Hnc, for TO 2D cloaks not all t tangntal componnts of t flds vans on, (46 and (47 ar not n gnral satsfd and bot lctrc and magntc surfac currnts may b prsnt. 17 On t otr and, (83 mpls tat all t contravarant vctors ar coplanar on and, as a consqunc, t agan follows tat From wc + r r Σ ' + + r r r r lm dt M = (89 [ ] 1 lm dt( Mn ˆ M = lm dt( Mn ˆ g g g = ( wc lads to t fulflmnt of (72 and tn (48. Trfor, n twodmnsonal cloaks t normal componnts of t nductons do vans on wl t tangntal componnts of flds do not. Ts s ndd t sam condton usd n Rf. 5.

23 Cloakng n Trms of NonRadatng Cancllng Currnts Fld Bavour Insd t Cloak n Trms of Covarant and Contravarant Vctors T concpt of covarant and contravarant vctors can b usd to obtan a smpl dscrpton of t fld bavour nsd a gnrc TO cloak llumnatd by a plan wav. It s not rstrctv to assum tat t plan wav coms from t z ˆ drcton wt t lctrc fld polarzd along ŷ and t magntc fld polarzd along ˆx. Insd t cloak t flds ar transformd accordng to qs. (69 T ( = ( '( T ( ( '( Er M r E r r Hr M r H r r = (91 Aftr wrtng t Jacoban matrx n trms of t contravarant vctors w obtan (,, 1 jkr r Er = g g g E = ge E jk 1 E ( =,, r r Hr g g g = g ζ ζ '( jkr' ( r '( jkr' ( r (92 wr ζ s t fr-spac mpdanc. For t nductons, w av from (69 tat s 1 ( = ε dt( '( 1 ( µ dt( '( Dr MM E r r Br MM H r r = (93

24 328 E. Martn t al. Dr M g1 g2 g 3 Mg2 1 E E Br M g g g Mg jkr' ( r jkr' ( r ( = ε dt( [,, ] 1 E = ε dt( E jkr' ( r jkr' ( r ( = µ dt( [,, ] = µ dt( ζ ζ (94 Ts mans tat at any pont of t cloak t flds ar algnd wt t contravarant vctors and t nductons wt t covarant vctors. T dscrpton n trms of covarant and contravarant vctors allows on to asly vsualz t fld structur n a TO cloak obtand troug a gnrc coordnat transformaton, and provds nformaton on t fld bavour at t nnr surfac of t cloak, as sown n t prvous scton. T Poyntng vctor s algnd wt a covarant vctor, snc E E * 2 1 Sr ( = Er ( H( r = g g= dt( Mg 3 (95 2 ζ ζ Ts rsult was ndd xpctd, bcaus n TO cloaks t ray-pats conform to t local mtrc n t comprssd spac. 6. Conclusons T cloakng problm as bn rvstd n trms of nonradatng cancllng quvalnt volumtrc currnts, tat s, quvalnt currnts wc do not radat outsd t xtrnal surfac of t cloak and cancl t ncdnt fld nsd t cloak cavty. Ts formulaton lads to t dtrmnaton of suffcnt condtons for cloakng on t consttutv paramtrs of an arbtrarly sapd mtamatral sll, xprssd n trms of two vctor potntals satsfyng smpl condtons on t cloak boundary. T transformaton optcs cloak s rcovrd as a spcal cas of t proposd formulaton and t gnral boundary condtons satsfd by t flds at t nnr surfac of arbtrarly sapd trdmnsonal and two-dmnsonal cloaks av bn dtrmnd. In partcular, t s found tat all t componnts of t E, H, D, and B flds, xcpt t normal componnts of E and H, must vans at t nnr surfacs of gnrc transformaton optcs tr-dmnsonal cloaks,

25 Cloakng n Trms of NonRadatng Cancllng Currnts 329 wl n transformaton optcs two-dmnsonal cloaks tr can also b nonvansng tangntal componnts at t nnr surfacs, wc ar always assocatd wt dlta-functon sngularts n t tangntal surfac polarzaton dnsts. It s confrmd tat t boundary condtons of nˆ B = and nˆ D = usd n Rf. 5 at t nsd nnr surfacs of cloaks ar suffcnt condtons for solutons to bot 2D and 3D cloaks to not scattr t ncdnt flds outsd t cloaks and to produc zro flds wtn t cavts of t cloaks. Fnally, t covarant and contravarant vctors of t coordnat transformaton ladng to transformatonal optcs cloaks av bn usd to provd a smpl and appalng pctur of t fld bavor nsd a cloak llumnatd by a plan wav. Acknowldgmnt T work of Artur D. Yagjan was supportd n part by t US Ar Forc Offc of Scntfc Rsarc (AFOSR. Rfrncs 1. J. B. Pndry, D. Scurg, and D. R. Smt, Scnc 312, 178 ( D. Scurg, J. J. Mock, B. J. Justc, S. A. Cummr, J.B. Pndry, A. F. Starr, and D.R. Smt, Scnc 314, 977 ( U. Lonardt, Scnc 312, 1777 ( U. Lonardt and T. G. Plbn, Nw J. Pys. 8, 247 ( A. D. Yagjan and S. Mac, Nw J. Pys. 1, (28; A. D. Yagjan and S Mac, Nw J. Pys ( A. D. Yagjan, Mtamatrals 4, 7 ( A. D. Yagjan, S. Mac, and E. Martn, Nw J. Pys. 11, ( I. V. Lndll and A. H. Svola, Pys. Rv. E 79, 2664 ( S. A. Trtyakov, I. S. Nfdov, and P. Altalo, Nw J. Pys. 1, ( S. Mac, IEEE Trans Antnnas Propagat. 58, 1136 ( V. H. Rumsy, IRE Trans. Antnnas Propagat. 7, 13 ( I. Lndll, Mtods for Elctromagntc Fld Analyss (IEEE Prss, A.J. Dvany, and E. Wolf, Pys. Rv. D 8, 144 ( R. F. Harrngton, Tm Harmonc Elctromagntcs (McGraw-Hll, Nw York, Y. Luo, J. Zang, L. Ran, H. Cn, and J A Kong, IEEE Antnnas and Wrlss Propagat. Ltt. 7, 59 ( H. Cn, J. Opt. A: Pur Appl. Opt. 11, 7512 ( A. Grnlaf, Y. Kurylv, M. Lassas, and G. Ulmann, Opt. Exprss 15, (27.