ScholarlyCommons. University of Pennsylvania. Andrea Alù University of Pennsylvania. Nader Engheta University of Pennsylvania,

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1 University f Pennsylvania SchlarlyCmmns Departmental Papers (ESE) Department f Electrical & Systems Engineering December Radiatin frm a Traveling-Wave Current Sheet at the Interface between a Cnventinal Material and a Metamaterial with Negative Permittivity and Permeability Andrea Alù University f Pennsylvania Nader Engheta University f Pennsylvania, engheta@ee.upenn.edu Fllw this and additinal wrks at: Recmmended Citatin Andrea Alù and Nader Engheta, "Radiatin frm a Traveling-Wave Current Sheet at the Interface between a Cnventinal Material and a Metamaterial with Negative Permittivity and Permeability",. December. Pstprint versin. Published in Micrwave and Optical Technlgy Letters, Vlume 35, Issue 6, December,, pages Publisher URL: This paper is psted at SchlarlyCmmns. Fr mre infrmatin, please cntact repsitry@pbx.upenn.edu.

2 Radiatin frm a Traveling-Wave Current Sheet at the Interface between a Cnventinal Material and a Metamaterial with Negative Permittivity and Permeability Abstract In this nte, we present the analysis fr the radiatin frm a traveling-wave infinitely-extent sheet f mnchrmatic electric current that is placed at the interface between a cnventinal lssless dielectric and a lssless material pssessing negative real permittivity and permeability. The field distributins and the directin f the Pynting vectrs in bth half spaces are discussed, and sme physical remarks are prvided. A brief nte abut launching Zenneck waves by a line current alng this interface is als mentined. Keywrds metamaterials, negative index f refractin, negative permittivity, negative permeability, left-handed medium, antenna Cmments Pstprint versin. Published in Micrwave and Optical Technlgy Letters, Vlume 35, Issue 6, December,, pages Publisher URL: This jurnal article is available at SchlarlyCmmns:

3 Radiatin frm a Traveling-Wave Current Sheet at the Interface between a Cnventinal Material and a Metamaterial with Negative Permittivity and Permeability Andrea Alu and Nader Engheta University f Pennsylvania Department f Electrical and System Engineering Philadelphia, Pennsylvania andreaal@ee.upenn.edu and engheta@ee.upenn.edu URL: Abstract In this nte, we present the analysis fr the radiatin frm a traveling-wave infinitely-extent sheet f mnchrmatic electric current that is placed at the interface between a cnventinal lssless dielectric and a lssless material pssessing negative real permittivity and permeability. The field distributins and the directin f the Pynting vectrs in bth half spaces are discussed, and sme physical remarks are prvided. A brief nte abut launching Zenneck waves by a line current alng this interface is als mentined. Keywrds: Metamaterials, negative index f refractin, negative permittivity, negative permeability, left-handed medium, antenna. Intrductin The tpic f cmplex materials in which bth permittivity and permeability pssess negative values at sme frequencies has been the subject f cnsiderable attentin recently [-]. The histry f this idea dates back t Veselag, wh in 967 theretically studied mnchrmatic plane wave prpagatin in a material whse permittivity and permeability he assumed t be simultaneusly negative at a given frequency [6]. Recently, there has been a renewed interest in this type f material since Smith, Schultz and Shelby in their research grup at UC San Dieg cnstructed such a cmpsite medium fr the micrwave regime, and experimentally shwed the presence f anmalus refractin in this medium []. There have been several names suggested fr this type f materials, such as left-handed materials (see e.g., [6]), materials with negative refractive index (see e.g., [3]), duble-negative (DNG) media [8], backward (BW) media [7], t name a few. One f the interesting electrmagnetic features f these media is the fact that fr a mnchrmatic unifrm plane wave in such a medium the directin f the Pynting vectr is antiparallel with respect t the directin f phase velcity. Owing t this feature, ne can envisin varius ptential applicatins fr DNG materials. Engheta recently intrduced theretically the idea f cmpact cavity --

4 resnatrs in which a cmbinatin f a slab f cnventinal material and a slab f DNG material is inserted [], and in his theretical analysis he shwed that a slab f DNG metamaterial can act as a phase cmpensatr/cnjugatr, and thus by cmbining such a slab with anther slab made f a cnventinal dielectric material ne can, in principle, have a -D cavity resnatr whse dispersin relatin may nt depend n the sum f thicknesses f the interir materials filling this cavity, but instead it depends n the rati f these thicknesses []. In this brief nte, we analyze theretically the electrmagnetic radiatin frm a traveling-wave thin current sheet lcated at the interface between a cnventinal dielectric half space and a DNG material half space. This study can be infrmative twards any analysis and understanding f surce radiatin in structures invlving DNG materials. In this analysis, we cnceptually assume bth half spaces t be lssless and, therefre, the material parameters are taken t be real-valued quantities. Frmulatin f the Prblem Cnsider tw semi-infinite regins in a Cartesian crdinate system ( x, y, z ) ; ne is the half space y > which is filled with a cnventinal lssless medium with real permittivity and permeability ε > and > ; and the ther is the half space y < filled a DNG medium with parameters ε < and <. (See Fig. ) In analgy with the terminlgy DNG intrduced in [8], fr the sake f brevity we call the cnventinal medium with psitive permittivity and permeability a duble-psitive (DPS) medium. ε > > y = zˆ e j β x δ ( y) DPS x DNG ε < < Fig.. Gemetry f the prblem. The term DNG stands fr duble-negative medium [8], i.e., a medium with negative permittivity and permeability, while the term DPS, in analgy with DNG, stands fr duble-psitive medium, i.e., a medium with psitive permittivity and permeability. An infinitely extent thin sheet f surface current is lcated at the interface between the DNG and DPS media. We realize that strictly speaking, n material (except the vacuum in the classical sense) is dispersinless, and therefre the Kramers-Krnig relatins require the inclusin f dissipatin. Hwever, in ur analysis here, we assume that the dissipatin is negligible at the frequency f mnchrmatic radiatin f interest. --

5 An infinitely-extent thin sheet f mnchrmatic electric current is lcated at y =, the interface between the DNG and DPS half spaces. Assuming the time dependence t be j t e ω, the density f this current sheet can be described as ( y) j x = z ˆ e β δ () where is the strength f the surface current density, β is the rate f the linear phase change in the x directin, δ ( ) is the Dirac delta functin, and ẑ is the unit vectr in the z directin. Here we assume the directin f the current flw is alng z, while the directin f phase variatin f this traveling-wave current is alng x. (At the end f this nte, we will give the results fr the case f the current sheet with the current flw and the directin f phase variatin bth being alng the x directin.) The electrmagnetic fields radiated frm the current distributin given in Eq. () pssess the electric field E = zˆ Ez where the scalar quantity Ez satisfies the fllwing equatins in the tw semi-infinite regins ( ω ε ) + E = fr y > () ( ω ε ) z + E = fr y < (3) z with the current sheet lcated at y =. This case we refer t as the transverse electric (TE) case. Due t ur assumptin stated earlier, we have ε > and ε >, and thus the wavenumbers ω ε and ω ε are real quantities. Frm Eqs. () and (3), the expressins fr Ez can be written as E z j β x j ωε β y j β x + j ω ε β y A e e + B e e y > = j x β j ωε β y j β x + j ω ε β y A e e + B e e y < (4) The magnetic fields can be derived frm the Maxwell curl equatin E = -jω H. The fur unknwn cefficients can, in principle, be btained by applying the bundary cnditins at y = Ez( y =+ ) = Ez( y =) y j x ˆ H ( y = + ) H ( y = ) = z ˆ e β (5) and the radiatin cnditins at y =±. Hwever, care must be taken when we analyze the wave in the DNG half space, since in the DNG medium the directin f the Pynting vectr is antiparallel with respect t the directin f the phase flw. This pint is described belw. -3-

6 In the DPS half space ( y > ), the phase flw vectr and the Pynting vectr are parallel, and thus bth the phase velcity vectr and the Pynting vectr are prpagating away frm the surce, when β < ω ε. Therefre, the cefficient B shuld be taken t be identically zer, and we are left with the term A e jβ x j ω ε β y e. When β > ω ε, the prper sign f square rt in the expressin ωε β shuld be chsen in rder t ensure zer fields at y =+. Therefre, fr β > ω ε we shuld have j β x y ωε β = j β ω ε, and we then have the term A e e β ω ε. In this case, the fields decay expnentially away frm the surce, as expected. In either case, n pwer will cme back twards the surce frm infinity, since there is n reflected wave. In the DNG half-space ( y < ), hwever, as was mentined earlier, the phase flw vectr and the Pynting vectr are antiparallel. As a result, when β < ω ε while the Pynting vectr shuld be pinted away frm the surce, the phase velcity vectr shuld be pinted twards the surce, as depicted in Fig.. Therefre, when β < ω ε the cefficient B shuld be taken t be zer, and we are left with the term jβ x j ωε β y A e e. It is als imprtant t nte that in the DNG half space the radiatin pwer flws away frm the surce; hwever, unlike the case f DPS medium, here the x- cmpnent f the Pynting vectr is pinted in the sense ppsite t the directin f the phase change f the current alng the x axis, as seen in Fig.. When β > ω ε we shuld have ωε β = j β ω ε in rder t guarantee zer field at y =, and we then have the term j β x y. A e e β ω ε ε > > y S k = zˆ e j β x δ ( y) DPS DNG ε < < S β k x Fig.. Schematic representatin f the directins f wave vectrs k and k, and the Pynting vectrs S and S. In the DPS medium, vectrs k and S are parallel, while in the DNG medium the vectrs k and S are antiparallel. The directins f pwer flw f the electrmagnetic radiatin frm the mnchrmatic traveling-wave current sheet shwn in Eq. () are shwn as the vectrs S and S. -4-

7 Taking int accunt the abve issues and applying the bundary cnditins, we btain the fllwing expressins fr the electric and magnetic fields and the cmplex Pynting vectrs in the tw regins: jβ x jkty ω e e y > kt kt E = z ˆ (6) jβ x jkt y ω e e y < kt kt jβ x jkty jβ x jkty e e β e e y k > t kt kt kt H = xˆ + y ˆ jβx jkt y (7) jβx jkt y e e β e e y < kt kt kt k t y Im{ k } t * y Im{ k t} βωe ωkte y > kt kt kt kt S = xˆ + y ˆ (8) y Im{ k } t * y Im{ kt } βωe ωkte y < kt kt kt kt where kt and kt are shrthand fr ωε β and ωε β when β < ω ε and β < ω ε, respectively. Nte that in the case f β > ω ε and/r β > ω ε the quantities kt and kt shuld be written as k = j β ω ε, as was discussed earlier. t kt k = j β ω ε and t It is wrth emphasizing that when β > ω ε ur chice f sign fr the term in the DNG half space leads t a physically meaningful decaying expnential as y, which is square-integrable. Hwever, if ne had chsen the ppsite sign, as was dne in Ref. [4], ne wuld have btained here a grwing expnential as y, which wuld have pssessed an x-directed real-valued Pynting vectr that wuld have grwn as y. This wuld have clearly been a nn-physical utcme, since in such -5-

8 a situatin the x-directed time-averaged pwer flux density wuld have been larger as ne wuld have mved farther away frm the surce at y =, and this is nt physically pssible. It is imprtant t nte that in the case f prpagating waves (i.e., when β < ω ε and β < ω ε ), since < the denminatrs in the field expressins, Eqs. (6) and (7), cannt becme zer. Hwever, when β > ω ε and β > ω ε, due t the chice f the square rt mentined earlier, the field quantities will becme infinitely large if the fllwing cnditin is satisfied β ω ε + β ω ε = (9) This cnceptual singularity can be explained and justified in the fllwing argument. In the case where β min { ω ε, ω ε } <, bth cmpnents f the Pynting vectr given in Eq. (8) in each regin are real-valued quantities; the y-cmpnents are pinted away frm the surce and the x-cmpnents in tw regins are antiparallel. In fact, the pwer flux density per unit area flwing ut f a clsed surface cntaining any segment f the current surce can be evaluated as P rad y> y< ω = S yˆ S y ˆ = () kt kt which, as expected, equals the pwer density emitted frm the surce, Ps * ω s = = = kt k t In the case where β max { ω ε, ω ε } P E P () >, bth kt and kt are imaginary, and thus the y-cmpnents f the Pynting vectrs are imaginary and carry n time-averaged pwer away frm the surce. The x-cmpnents f the Pynting vectrs in bth regins are, hwever, real-valued quantities, and they represent pwer f the waves mving alng the bundary and hugging the current surce. These waves pssess x-directed Pynting vectrs that are antiparallel. Frm the knwledge f Eq. (8), the ttal timeaveraged pwer flw alng the x axis, acrss any plane parallel with the y-z plane in each f the tw half spaces, can be given as rad -6-

9 βω = Re ( S) xˆ dy = kt kt 4 kt P βω = Re ( S) xˆ dy = kt kt 4 kt P () and the net ttal time-averaged pwer then becmes βω P = P + P = + 4 k kt kt t kt (3) with k = j β ω ε and t t t k = j β ω ε. This net pwer may vanish if t k = k, which is different frm the cnditin given in Eq. (9). It is imprtant t nte that Eq. (9) is indeed the cnditin fr existence f surce-free Zenneck waves at the bundary between a DNG and a DPS medium, as can be deduced frm the wrk f Lindell et al. [7]. Therefre, if ne wants t launch a Zenneck wave alng this interface, ne can put a thin line current alng the z axis at the interface, i.e., ˆ line = z I δ ( x) δ ( y). Such a line current can be expanded using the fllwing Furier transfrm: I jβ x ˆ ( ) ( ) ˆ line = z I δ x δ y = z d β e δ ( y) π. (4) The fields due t this line current can then be given as I + Eline = { Eq.(6) } d β π I + H line = { Eq.(7) } dβ π (5) where Eq. (6) and Eq. (7) are field quantities due t the current sheet f j x = z ˆ e β δ y. In Eq. (5), the radiatin fields are due t the integratin ver the ( ) interval β max { ω ε, ω ε } < and can be calculated frm the knwledge f Eqs. (6) and (7). In additin t the radiatin fields, the integrals given in Eq. (5) will als prvide the surface wave due t the residue cntributin, which ccurs at the ples f Eqs. (6) and (7). These ples indeed satisfy Eq. (9), which leads t the fllwing β : -7-

10 β surf =± ω ε / ε / / /. (6) If this surf βsurf > max ω ε, ω ε, the electric field f such a surface wave can then be calculated by means f the residue therem and can be expressed as: β is real and if { } E surf = zˆ ωi jβsurf x βsurf ωε y e e y > = = zˆ ω I β k surf y t kt β ω ε + e y < β β= βsurf surf βsurf ω ε y jβ surf x e y > e βsurf ω ε y e y < + βsurf ω ε βsurf ω ε. (7) The magnetic field f such a surface wave can be calculated similarly. S frm a line current surce (Eq. (4)) this Zenneck wave is launched. Nw if we put an infinite number f such line current surces next t each ther arranged such that they frm a current sheet f Eq. () with β = βsurf, there will be an infinite number f Zenneck waves each being launched frm each line current surce, and since β = βsurf they all wuld be added cnstructively. The amplitude f such cmbined Zenneck wave wuld therefre be infinite. This justifies and describes why the cnditin given in Eq. (9) leads t singularity in Eqs. (6) and (7). It is wrth nting that Eq. (9) in general may prvide the cnditin fr a single Zenneck wave t prpagate with β surf (as given in Eq. (6)) alng the interface between ε = ε the DNG and the DPS media. Hwever, fr a particular case f, Eq. (9) is = satisfied fr any given β as lng as β > max { ω ε, ω ε }. But in this case, such surface waves will carry zer net pwer, since accrding t Eq. (3) P = in this case fr any β. In this scenari, tw ppsitely directed, but equal, pwer fluxes prpagate alng the x axis, effectively behaving similar t a standing wave in a cavity r a resnant circuit. We als nte that in this case, the denminatr f Eq. (7) vanishes, resembling a situatin in which a resnant parallel L-C circuit is excited by a mnchrmatic parallel current surce scillating at the resnant frequency ω = / LC, resulting in an infinitely large vltage in the L-C circuit. (Or equivalently a resnant series L-C circuit being excited by a mnchrmatic series vltage surce at the resnant -8-

11 frequency resulting in an infinite current in the series L-C circuit.) This als smewhat resembles a situatin in which an ideal lssless cavity resnatr is excited by a frced current element, scillating at the resnant frequency f the cavity, lcated at an interir pint with a nn-zer value f the electric field f the cavity mde. In such a driven cavity, this frced scillatin leads t infinitely large values fr the electric and magnetic fields within the cavity. Similar analysis can be perfrmed fr the case f current distributin f the frm j x = x ˆ e β δ ( y). In this case, which is the transverse magnetic (TM) case, the crrespnding electrmagnetic fields and related Pynting vectrs can be given as: jβx jkty jβx jkty e e β e e y > ε ε k t ω kt k ωε ε t k t E = xˆ + y ˆ jβx jkt y (8) jβx jkt y e e β e e y < ε ε k t ω ω ε ε kt k t k t jβ x jkty e e y > ε kt εkt H = z ˆ (9), jβ x jkt y e e y εk < t εkt y Im{ k } t y Im{ k t} e βε e ktε y > kt kt ωεε k ωε ε t kt S = xˆ + y ˆ () y Im{ k } t y Im{ kt } e βε e kt ε y < kt kt ωεε ωε ε k t kt with kt = ω ε β and kt = ω ε β fr β < ω ε and β < ω ε, and kt = j β ω ε and kt = j β ω ε fr β > ω ε and β > ω ε, and ther quantities similar t the nes defined earlier fr the transverse electric (TE) case. Similar cnsideratins can be mentined in this TM case. -9-

12 In summary, we have presented ur analysis f the electrmagnetic radiatin frm an infinitely extent mnchrmatic thin sheet f current lcated at the interface between a DPS medium and a DNG medium. The results f this analysis shw that the Pynting vectrs in the tw media are pinted away frm the surce; hwever, the x-cmpnents f the tw Pynting vectrs are in ppsite directins. This is due t the fact that in the DNG medium fr a mnchrmatic unifrm plane wave prpagatin the Pynting vectr and the phase velcity vectr are antiparallel. This result is cnsistent with what Veselag has cnjectured abut the directin f the Cerenkv radiatin in the DNG media [6]. Acknwledgements Andrea Alu is supprted by the schlarship Isabella Sassi Bnadnna frm AEI (Assciazine Elettrtecnica ed Elettrnica Italiana). This wrk is supprted in part by the Fields and Waves Labratry, Department f Electrical and Systems Engineering, University f Pennsylvania. References [] R. A. Shelby, D. R. Smith, S. Schultz, Experimental verificatin f a negative index f refractin, Science, vl. 9, n. 554, pp , April 6,. [] D. R. Smith, W.. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, Cmpsite medium with simultaneusly negative permeability and permittivity, Phys. Rev. Lett., vl. 84, n. 8, pp , May. [3] D. R. Smith and N. Krll, Negative refractive index in left-handed materials, Phys. Rev. Lett., vl. 85, n. 4, pp , Octber. [4]. B. Pendry, Negative refractin makes a perfect lens, Phys. Rev. Lett., vl. 85, n. 8, pp , 3 Octber. [5] R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, Micrwave transmissin thrugh a tw-dimensinal, istrpic, left-handed metamaterial, Applied Physics Lett., vl. 78, n. 4, pp , anuary. [6] V. G. Veselag, The electrdynamics f substances with simultaneusly negative values f ε and, Sviet Physics Uspekhi, vl., n. 4, pp , 968. [Usp. Fiz. Nauk, vl. 9, pp , 967.] [7] I. V. Lindell, S. A. Tretyakv, K. I. Nikskinen, and S. Ilvnen, BW media Media with negative parameters, capable f supprting backward waves, Micrwave and Optical Technlgy Letters, Vl. 3, N., pp. 9-33, Octber. --

13 [8] R. W. Zilkwski and E. Heyman, Wave prpagatin in media having negative permittivity and permeability, Phys. Rev. E., vl. 64, n. 5, 5665, Octber. [9] M. W. McCall, A. Lakhtakia, and W. S. Weiglhfer, The negative index f refractin demystified, Eurpean urnal f Physics, Vl. 3, pp ,. [] N. Engheta, An idea fr thin subwavelength cavit y resnatrs using metamaterials with negative permittivity and permeabilit y, IEEE Antennas and Wireless Prpagatin Letters, Vl., N.,, t appear n line. [] N. Garcia and M. Niet-Vesperinas, Left-Handed Materials D Nt Make a Perfect Lens, Physical Review Letters, Vl. 88, N., 743, May,. --