Design and analytically full-wave validation of the invisibility. cloaks, concentrators, and field rotators created with a general

Transcription

1 Desig ad aalytically full-wave validatio of the ivisibility cloaks, cocetatos, ad field otatos ceated with a geeal class of tasfomatios Yu Luo, Hogsheg Che, Jigjig Zhag, Lixi a *, ad Ji Au Kog The lectomagetics Academy at Zhejiag Uivesity, Zhejiag Uivesity, Hagzhou 358, P.. Chia eseach Laboatoy of lectoics, Massachusetts Istitute of Techology, Cambidge, Massachusetts 39 Abstact We ivestigate a geeal class of electomagetic devices ceated with ay cotiuous tasfomatio fuctios by igoously calculatig the aalytical expessios of the electomagetic field i the whole space. Some iteestig pheomea associated with these tasfomatio devices, icludig the ivisibility cloaks, cocetatos, ad field otatos, ae discussed. By caefully choosig the tasfomatio fuctio, we ca ealize cloaks which ae isesitive to petubatios at both the ie ad oute boudaies. Futhemoe, we fid that whe the coatig laye of the cocetato is ealized with left-haded mateials, eegy will ciculate betwee the coatig ad the coe, ad the eegy tasmits though the coe of the cocetato ca be much bigge tha that tasmits though the cocetato. Theefoe, such cocetato is also a powe flux amplifie. Fially, we popose a spheical field otato, which fuctios as ot oly a wave vecto otato, but also a polaizatio otato, depedig o the oietatios of the spheical otato with espect to the icidet wave diectio. The fuctioality of these ovel tasfomatio devices ae all successfully cofimed by ou aalytical full wave method, which also povides a alteate computatioal efficiet validatio method i cotast to umeical validatio methods. * Autho to whom coespodece should be addessed; electoic mail: alx@zju.edu.c

2 I. INTODUCTION ecetly, cloaks of ivisibility have eceived much attetio -7. Pedy et al. fistly poposed a coodiate tasfomatio appoach to povide a ew method to cotol M fields, by which a space cosistig of the omal fee space ca be squeezed ito a ew space with diffeet volume ad space-distibuted costitutive paametes. Followig this appoach, a micowave ivisibility cloak was soo poposed ad expeimetally ealized, ad some othe ovel devices, such as the M cocetatos 3, 4, otatos 5, ad hypeles 6, 7 wee also ivestigated by simila methods. Apat fom the pue tasfomatio method, full wave simulatios 8 ad taditioal M aalysis based o Mie scatteig 9 wee also peseted ad both veified the validatio of the tasfomatio appoach. These apid pogesses make the coodiate tasfomatio appoach be a hot topic i the electomagetics commuity ad imply vey impotat futue applicatios,. The cuet discussios o the ivisibility cloak ae mostly based o a liea tasfomatio peseted by Pedy et al.. Some authos have cosideed ivisibility cloak with high ode tasfomatios, 3. I this pape, we ty to peset a geealized fomulatio o how to do the tasfomatios to obtai diffeet devices by combiig the meits of both coodiate tasfomatio ad the Mie scatteig solutios. Statig fom such a fomulatio, afoemetioed devices (ivisibility cloaks, cocetatos, ad field otatos ca be easily obtaied by simply choosig diffeet scala tasfomatio fuctios (icludig the tasfomatio o θ ad φ, ad the iteal field distibutios i these devices ca also be cotolled by tuig the shapes of the fuctios. We show that such scala tasfomatio fuctios ca be ay cotiuous fuctios, ad ca be applied to ay coodiate to ealize electomagetic devices with ovel fuctioality. This aalytical full wave method povides ot oly a global

3 physical udestadig to the effect of the coodiate tasfomatio, but also a coveiet aalysis ad desig tool fo such ovel devices due to its vey high computatioal efficiecy. II. FOMULATIONS Coside the thee dimesioal case (the followig idea is also applicable to two dimesioal cylidical case: a geeal coodiate tasfomatio betwee two spheical coodiate systems (, θ, ϕ ad (, θ, ϕ is descibed by = (,,, θ = g(, θϕ,, ϕ h(, θϕ, f θ ϕ =, ( whee, θ, ϕ epeset the coodiates i the oigial coodiate system ad f (, g ( (, h ca be abitay cotiuous fuctios. Followig the tasfomatio appoach poposed i, Maxwell equatios still emai its fom ivaiace i the ew space (, θ, ϕ but the pemittivity ad pemeability will tu ito distibuted, o space depedet tesos, i.e., ε ε = T, μ μ = T, ( whee ε ad μ epeset the scala pemittivity ad pemeability of fee space i the oigial space befoe tasfomatio. The matix T is defied by T T = J J, whee det ( J J ( f, gh, (, θ, ϕ = is the Jacobia matix 4, 5. Accodig to the Mie scatteig theoy, fo souce fee cases, we ca decompose the fields ito ad modes by itoducig the vecto potetial A ad A i the ew space ad expess the fields as

4 ( A ( μ ( A ( A ( ε ( A B = i D = ω, (3 D = i B = ω whee B ad D epeset the magetic flux desity ad electic displacemet, espectively. Sice the media descibed by ε ad μ is o loge isotopic, the diectios of A ad A will ot always be alog the diectio. Fo mathematical coveiece, we let A A f f ˆ ˆ f = + θ + ˆ ϕ Φ θ siθ ϕ f f ˆ ˆ f = + θ + ˆ ϕ Φ θ siθ ϕ, (4 whee Φ ad Φ ae scala potetials fo ad cases, espectively. Note that if f ( is oly a fuctio of, fo example, a liea fuctio like f =, the the two vecto potetial A ad A will be alog ( ( ( the diectio, ad it will be educed to the case studied i ef 7. Substitutig equatio (4 ito equatio (3, we obtai the patial diffeetial equatio fo Φ : Φ ad + si g + f f si g g g f si g h + k Φ =, (5 which takes the same fom as the Helmholtz equatio, so oe of its special solutio is ˆ m Φ= B ( k f ( mh+ B mh P cos g Amcos m si, (6 whee ˆ m B ξ is iccati-bessel fuctio, P is the th odes of the associated

5 Legede polyomials of degee m, ad A m ad B m ae udetemied coefficiets. Usig equatio (4, we ca obtai the vecto potetials A ad A as follows: f f ˆ f A ˆ ˆ ˆ = am, + θ + ϕ B k f m, θ siθ ϕ A f f ˆ ˆ f = a + θ + ˆ ϕ Bˆ k f m, m, θ siθ ϕ m P ( cos g ( A cos mh+ B si mh m m m ( P cos g A cos mh+ B si mh m m, (7 whee the coefficiets ad ca be detemied by applyig coespodig a m, a m, bouday coditios. Thus all the compoets of the total fields ca be obtaied by substitutig equatio (7 ito equatios (, ad take the followig foms i f g h = ωμε f f g f h k Φ h g + si g Φ ε g si g h, (8- θ i f g h = ωμε θ f θ f g θ f h k Φ h g + si g Φ ε θ g si g θ h, (8- ϕ i f g h = ωμε siθ ϕ f ϕ f g ϕ f h k Φ h g + si g Φ εsiθ ϕ g si g ϕ h, (8-3 H g h = si g Φ μ si g h g i f g h ωμε f f g f h k Φ, (8-4 H θ g h = si g Φ μ si g θ h θ g i f g h ωμε θ f θ f g θ f h k Φ, (8-5

6 H ϕ h g = si g Φ μsiθ ϕ g si g ϕ h i f g h ωμεsiθ ϕ f ϕ f g ϕ f h k Φ whee ad H epeset the electic ad magetic fields, espectively., (8-6 III. SPHICAL CLOAKS The above fomulas ae fistly applied to the spheical cloaks. Ay cotiuous f ( g ( h ( f (, θϕ, =, fuctio,, that satisfy,, h θ ϕ = ϕ + ϕ (whee ϕ is a defiite costat ad g, θ, ϕ = θ, f,, θϕ = (these coditios ca be diectly obtaied fom the patial diffeetial equatios by settig T T the scatteig coefficiets ad to be zeo at the oute bouday ca be used to achieve a coatig the field iside which is always matched with the oute fee space at the oute bouday ad the potetial is uifom eveywhee at the ie bouday. The coatig with this quality ca ealize a pefect spheical ivisibility cloak. Detailed illustatio o it is out of the scope of this aticle ad futhe discussio, with geeal theoetical aalysis ad umeical simulatios will be give i ou othe pape. Hee we oly coside oe simple case whe = f, θ = θ, ϕ = ϕ. The associated pemittivity ad pemeability tesos ae the give by: = ( ˆˆ+ ( ˆˆ+ ( ˆˆ ϕϕ, μ μ ( ˆˆ μ ( ˆˆ θθ μ ( ε ε ε θθ ε t t = + + ˆˆ ϕϕ, t t whee ε ε =, t f ε = ε f f (, μt μ f ( ( =, ad μ = μ f f ( ( (9 Fo a abitay tasfomatio fuctio f (, we will show i detail that these paametes yield a pefect ivisibility as log as f = ad f = ae

7 satisfied. Suppose a x polaized plae wave with a uit amplitude ˆ ik z i xe = is icidet upo the coated sphee alog the z diectio. With the solutio of equatio (4, the vecto potetials fo the icidet fields ( >, the scatteed fields ( >, ad the fields iside the cloak laye ( < < ca be witte i the followig foms espectively: cosϕ A = ˆ aψ k A i ω i siϕ = ˆ aψ k ωη P ( cosθ ( P ( cosθ cosϕ A = ˆ a T ζ k s ω siϕ A = ˆ a T ζ k s ωη P ( cosθ P ( cs o θ ψ, (-, (- ( χ( P ( cosθ cosϕ A = ˆ f d k f + f k f c ω ( χ( ( P ( cosθ siϕ A = ˆ f d k f + f k f ψ c ωη, (-3 whee a ( ( + ( i + =, =,,3,..., ε μ η = ;,,,, T T d d f ad f ae ukow expasio coefficiets; ψ ( ξ, χ ( ξ, ζ ξ epeset the iccati-bessel fuctio of the fist, the secod, ad the thid kid, espectively 8. Sice f =, χ ( is ifiite, the fiitude of the field at the ie bouday equies that f = f = 9. By applyig the bouday coditios at the bouday of =, we ca get othe ukow coefficiets: T ( ( ψ k ψ kf ψ k ψ kf = T =, (- ζ k ψ k f ζ k ψ k f

8 d = d = ζ ( k ψ kf( ζ k ψ kf ia, (- ( Sice f =, the above equatios ca be simplified as T = T =, d = d =, ( T T The fact that coefficiets ad ae exactly equal to zeo idicates a eflectioless behavio of a pefect cloak. Substitutig equatios ( ito equatios (8, afte some algebaic maipulatios, the summatio ca be witte i closed foms. As a esult, all compoets of the electic field ae expessed as (Note that the paametes fo fee space ca be egaded as f ( =, theefoe the fields ca still be witte i the followig foms: ik f cosθ = f siθcosϕe, (3- ik f ( f θ θ ϕe cosθ = cos cos, (3- ik f ( f e cosθ ϕ = siϕ, (3-3 Theefoe, fom equatio ( ad (3, we cofimed that as log as f = ad f =, ay spheical shell with paametes defied by equatio (9 ca yield a pefect ivisibility. Diffeet f i the egio < < will oly cause diffeet field distibutio i the cloak laye, but will ot distub the field outside. The distibutio of the field i the cloak shell < < ad the sesitivity of the cloak to the petubatios at the bouday ae detemied by the tasfomatio fuctio f (. We ivestigate fou types of cloak ceated with fou diffeet tasfomatio fuctios: (Case I f =, (Case II (

9 f = ( with f ( =, (Case III f = 3 ( with f ( =, ad (Case IV f ( with f ( = ad f ( =. Fig. (a displays the cuves of the fou tasfomatio fuctios. Fig. (b shows the coespodig tagetial ad adial compoets of ε ad μ of the fou diffeet cloaks calculated fom quatio (9. Fig. (c, (d, (e, (f depict the calculated x fields distibutios ad Poytig vectos due to x polaized wave icidece oto these fou diffeet cloaks, espectively. All the cloaks have a same size of =. m, ad =. m. The wavelegth i fee space is.5 m. All the quatities ae omalized to uity i this ad the followig calculatios. With diffeet tasfomatios, the fields iside the cloaks ae diffeetly distibuted while the wave popagatig i the oute egio of the cloak emais udistubed. I Fig. (c, the field is ealy uifomly distibuted i the cloak shell with liea fuctio f = (tasfomatio fuctio used i ef ( betwee ad. Fom the esult we ca fid that this kid of cloak, which ca be called liea-tasfomed cloak, is sesitive to petubatios both at the ie bouday ad oute bouday ; I Fig. (d, the field is maily distibuted ea the oute bouday i the cloak with the covex tasfomatio fuctio f. Ad this so called covex-tasfomed cloak is ot sesitive to the petubatios at the ie bouday but much moe sesitive to petubatios at the oute bouday; I Fig. (e, the field is maily distibuted close to the ie bouday i the cloak with a cocave tasfomatio fuctio f 3. This so called cocave-tasfomed cloak is ot sesitive to the petubatios at the oute bouday, but it is sesitive to tiy

10 petubatios at the ie bouday. I a wod, the field iside the cloak is lage i the positio whee the diffeetial of the fuctio f ( is lage. Thus by choosig a fuctio like f4 (, the diffeetial of which is zeo at both ad, we ca get a cloak i which the field is maily distibuted ea the cetal egio of the coatig ad appoaches to zeo at both boudaies. I fact, if we choose a tasfomatio fuctio f ( that satisfies f ( = ad f =, the cloak will be isesitive to petubatios at eithe the oute bouday o the ie bouday, howeve, it will be sesitive to the petubatios i the cetal egio of the cloak. IV. CONCNTATOS Fom the discussio of Pat III, we ca see if f ( is cotiuous at the bouday of ad, the scatteed field is equal to zeo, the fields i the whole egio will still take the foms of equatio (3. Fo ay medium that satisfies ε μ = = a (coespods to f ( = a, we ca cove it with a cetai coatig to ε μ make it ivisible to the detecto outside, but compaed with the cloak case whee o eegy ca be tasmitted iside, this coatig is diffeet i that eegy ca still peetate ito the coe. Fig. (a shows fou diffeet cases with fou diffeet coes (a =.5,.5,,.5 ad thei coated layes ceated with fou diffeet tasfomatio fuctios, f (, f (, f (, ad f4 3, espectively. Fo simplicity, we choose fou liea fuctios detemied by a, which ca be see i Fig. (a. is set to be. m ad is equal to. m. Fig. (b shows the coespodig costitutive paamete compoets of diffeet coatigs. Usig the afoemetioed fomulatios,

11 we ca calculate the field solutios fo these fou cases. The x field distibutios of these fou specific cases ude a x polaized plae wave icidece alog z diectio ae displayed i Fig. (c, (d, (e, ad (f, espectively. Sice the vecto potetial takes exact the same fom as equatio (, the field distibutios show i Fig. (c-f ae detemied by the elative paamete a : Whe < a <, the powe tasmitted though the coe is elative small, most powe tasmitted though the coatig laye. A example of this case is show i Fig.(c, whee a =.5. Whe a =, the coatig will educe to a pefect cloak. Whe a, the coatig ca be teated as a cocetato. Whe < a< / (hee =, most powe will tasmit though the coatig, a example of this / case is show i Fig. (d, whee a =.5. While at a, as a exteme case with = f ( / = i the coatig laye, all the powe that tasmit ito (o out fom the ie egio is alog the adii of the coatig, as show i Fig. (e. I this case, the adial compoets of ε ad μ ted to ifiite while the tagetial compoets equal zeo. As we kow, whe a wave is icidet fom vacuum to a medium, at the bouday of which the tagetial ε (μ is ifiite o the adial ε (μ is zeo, it will be totally eflected due to the suface cuet o suface voltage at the bouday 6. Ou case is diffeet i that f kt = ω εμt = k has a fiite value, which meas the wave umbe is fiite, so the electomagetic wave ca still popagate ito this media. With equatios 3, we ca see = ad H =, showig that the Poytig powe is always alog the adii of the sphee. Similaly, with equatios (9 ad equatios (3 we ca fid that the o-zeo compoets of D ad B ae always alog the adial diectio, which

12 meas the wave vecto of the electomagetic wave is always pepedicula to its Poytig powe. A moe iteestig case is whe a > /, the pemittivity ad pemeability of the coatig is egative. The powe that flows though the ie egio (coe is lage tha the powe flows though the whole cocetato, because thee is always some eegy ciculatig betwee the coatig ad the ie media, as show i Fig. (f. The popotio of the powe flow though the ie egio P i to the powe flow though the cocetato P out is: P i P out π π dϕ ˆ H siθdθ = a π π dϕ ˆ ( H siθdθ = = =, (4 whee ˆ epesets the suface omal of the ie bouday, ˆ epesets suface omal of the oute bouday. As show i Fig. (f, thee is o scatteig, but the powe is magified iside ( P i > P out. The easo of this iteestig pheomeo is that the case we coside hee is i the time hamoic state. Befoe eachig this steady time hamoic state, thee is scatteed field outside, ad the eegy is gettig stoed i the coatig. Whe the steady state is eached, the stoed eegy will ciculate betwee the coatig ad ie media. Lage moe eegy stoed i the cocetato. a ca lead to V. FILD OTATO Cylidical otatio coatig was fist poposed by H. Che ad C. T. Cha 5. I this sessio we will popose a thee-dimesioal (spheical otatio coatig, ad calculate all the compoets of the fields though ou method. The aalytical esults

13 show simila behavios with the simulatio esults i ef 5 as the electomagetic wave is popagatig i the x-y plae. Futhemoe, we demostate if the icidet wave is ot i the x-y plae, ot oly the wave fot of the wave but also the polaizatio of the fields will be otated. Coside the followig tasfomatio: =,θ = θ,ad ϕ = ϕ + g, whee g( = ad g( = ϕ. Usig equatio ( the pemittivity ad pemeability teso compoets of the coatig shell ca be give as: α ε = ε α + α, μ = μ α α + α, (5 whee α = siθ g. The above paametes ae all expessed i spheical coodiate. With this elatio, the field is matched at both the oute bouday ad the ie bouday, but the tagetial agle ϕ has bee otated by a agle ϕ fom oute bouday to ie bouday of the coatig. So the wave popagatig i x-y plae will chage its diectio by ϕ iside the eclosed domai with espect to that outside the coatig. Coside a plae wave icidet upo the coated sphee alog the ˆ ik x x diectio with uit amplitude of electic field i = ye, sice ad = = ae both matched boudaies, with equatio (7, the vecto A ad A iside of the coatig ca be expessed as e A = ˆ ψ ( k a T ϕ g( b T g o ( + + m ( ϕ+ m, m m, ω m, e A = ˆ ψ ( k a T ϕ g ( b T g o ( + + m ( ϕ+ m, m m, ωη m,, (6 e m o m whee T ( θ, ϕ = P ( cosθ cos( mϕ, T ( θ, ϕ = P ( cosθ si( m m m ϕ,, a m, b m, ik ad, b ae the spheical expasio coefficiets of si θ cos ϕ e siθ siϕ ad a m, m,

14 e ik si θ cos ϕ cosθ espectively. substitutig equatios (6 ito equatios (8, the field i the coatig ca be witte i the closed fom: ( ϕ g( + g ( cos ϕ g( = siθ si + + e ik siθ cos( ϕ+ g( (7- θ ϕ ( ϕ g( ik siθ cos( ϕ + g( = cosθsi + e (7- ( ϕ g( ik siθ cos( ϕ + g( = cos + e (7-3 H cosθ ik siθcos( ϕ + g( = e (7-4 η H θ siθ ik siθ cos( ϕ + g( = e (7-5 η H ϕ = (7-6 l l = ϕ l l If we choose the followig tasfomatio: g, the above calculatio ca be simplified 5. As a esult all the compoet of ε ad μ ae idepedet of. Fig.3 (a ad (b show the distibutio fo H z π ϕ = ad ϕ = π espectively. ˆ ik z If the wave with uit electic field = xe is icidet alog z axis π (pepedicula to the x-y plae oto the otato with ϕ =, followig the same i steps, we ca get the distibutio of x ad y, as show i Fig. 3 (c ad (d, espectively. It is iteestig to see that i this case, istead of the wave fot of the electomagetic wave, it is the polaizatio of the fields that is otated as the wave passig ito the coatig. Theefoe, this kid of spheical field otato ca fuctio as a wave vecto otato as well as a polaizatio otato. Tuig the oietatio of the spheical otato with espect to the icidet wave diectio ca cotol the

15 fuctioality of the spheical otatos. VI. CONCLUSION I this pape, we summaized a geealized fomulatio o how to use the tasfomatios to obtai diffeet devices by combiig the meits of both coodiate tasfomatio ad the Mie scatteig solutios. We show by mathematical aalysis that the coodiate tasfomatio to the Maxwell equatios ca be i a geealized fom: ay cotiuous fuctios ca be adopted i the tasfomatio, ad diffeet type of fuctios will big diffeet chaacteistics of the M behavios i the tasfomed space. Statig fom the fomulatio deduced i this pape, ivisibility cloaks, cocetatos, ad otatos ca be easily obtaied by simply selectig diffeet scala tasfomatio fuctios, ad the iteal field distibutios i these devices ca also be cotolled by tuig the shapes of the fuctios. Vaious examples fo the desig of cloaks, cocetatos, ad field otato ae give to demostate the validity of the fomulatio ad the vey high computatioal efficiecy. Ou pape pesets a vey useful tool i the aalysis ad desig fo the ivisibility cloak, M cocetatos, field otatos, ad simila ovel devices. ACKNOWLDGMNT This wok is sposoed by the Chiese Natioal Sciece Foudatio ude Gat Nos. 653 ad 6673, the NCT-7-75, the Chia Postdoctoal Sciece Foudatio ude Gat No , the ON ude Cotact No. N4---73, ad the Depatmet of the Ai Foce ude Ai Foce Cotact No. F968--C-.

16 FNCS J. B. Pedy, D. Schuig, ad D.. Smith, Sciece 3, 78 (6. D. Schuig, J. J. Mock, B. J. Justice, S. A. Cumme, J. B. Pedy, A. F.Sta, ad D.. Smith, Sciece 34, 58 (6. 3 M. ahm, D. Schuig, D. A. obets, S. A. Cumme, ad D.. Smith, epit AXiv: , (7. 4 A. D. Yaghjia, S. Maci, epit AXiv: (7. 5 H. Che ad C. T. Cha, Appl. Phys. Letts, 9, 45 (7. 6 M. Tsag ad D. Psaltis, epit AXiv: 78.6 (7. 7 A. V. Kildishev ad.. Naimaov, epit AXiv: (7. 8 S. A. Cumme, B.-I. Popa, D. Schuig, D.. Smith, ad J. B. Pedy, Phys. ev., 74, 366 (6. 9 H. Che, B-I. Wu, B. Zhag, ad J. A. Kog, Phys. ev. Letts. 99, 6393 (7. J. Zhag, J. Huagfu, Y. Luo, H. Che, J. A. Kog, ad B-I. Wu, Phys. ev. B, to be published. M. ahm, S. A. Cumme, D. Schuig, J. B. Pedy, ad D.. Smith, epit AXiv: (7. W. Cai, U. K. Chettia, A. V. Kildishev, ad V. M. Shalaev, Appl. Phys. Letts, 9, 5 (7. 3. Wede, epit AXiv: , (7. 4 F. Zolla, S. Gueeau, A. Nicolet, J. B. Pedy, Opt. Letts. 3, 9 (7. 5 D. Schuig, J. B. Pedy, ad D.. Smith, Opt. xpess, 4, 9794 (6. 6 B. Zhag, H. Che, B-I. Wu, Y. Luo, L. a, ad J. A. Kog, Phys. ev. B, 76, ( (7. 7 A. Sihvola, PI 66, 9-98 (6.

17 8 H.C.va de Hulst, Light Scatteig by Small Paticles, Dove, New Yok, (957.

18 Figue Captios FIG. (a Schematic figue of the tasfomatio fuctios f ( i fou cases: (Case I f = (, (Case II f = ( with f =, (Case III f ( 3 = + with f 3 =, ad (Case IV f4 ( with f ( = ad f ( =. (b The pemittivity ad pemeability compoets 4 4 calculated fom the coespodig fou cases. (c, (d, (e, ad (f show the fields x distibutio ad Poytig vectos due to x polaized wave icidece oto a cloak ceated with the fou diffeet tasfomatio fuctios f (, f (, 3 ad f 4, espectively. f, FIG. (a Schematic figue of the fou diffeet cofiguatios: fou diffeet coes ( a =.5,.5,,.5 ad thei coated layes ceated with fou tasfomatio fuctios, f (, f (, f3 (, ad 4 f, espectively. (b The pemittivity ad pemeability compoets of the coatig i the fou cases. The adial compoet of the costitutive paamete i the coatig laye ceated with f 3 is ifiite, so it is ot plotted out hee. (c, (d, (e, ad (f ae the calculated x field distibutio ad Poytig vectos due to x polaized wave icidece oto the coated sphee. FIG. 3 (a Distibutio of H z ad Poytig vectos as the wave is icidet alog x diectio fo π ϕ = (b Distibutio of H z ad Poytig vectos as the wave is icidet alog x diectio fo ϕ = π (c Distibutio of x as a x polaized

19 π wave icidet alog z diectio fo ϕ = (d Distibutio of y as a x π polaized wave icidet alog z diectio ϕ =.

20 FIG

21 FIG.

22 FIG 3