Scalar Hertz potentials for nonhomogeneous uniaxial dielectric magnetic mediums

Transcription

1 Inernaional Journal of Applied Eleromagneis and Mehanis IOS Press Salar Herz poenials for nonhomogeneous uniaxial dieleri magnei mediums Werner S. Weiglhofer Deparmen of Mahemais, Universiy of Glasgow, Glasgow G 8QW, UK Absra. The salar Herz poenial ehnique is applied o he field analysis in uniaxial dieleri magnei mediums. These mediums may be eiher homogeneous or nonhomogeneous wih a nonhomogeneiy of a resried form. I is shown ha he omplee eleromagnei field in a uniaxial dieleri magnei medium an be derived from wo salar Herz poenials plus, for erain ypes of problems, a small number of soure speifi auxiliary funions. The orresponding salar Green funions are expliily given for he homogenous ase and appliaions for radiaion and propagaion problems are presened.. Inroduion The salar Herz poenial ehnique is an indire soluion mehod for he eleromagnei field. In i erain salar oupled or unoupled parial differenial equaions are solved for a small number of salar funions he salar Herz poenials from whih he aual field veors follow hrough simple manipulaions suh as differeniaions. For more bakground on Herz veors and Herz poenials, inluding many hisorially relevan referenes, he reader is referred o an exellen arile by Sein. The deailed mahemaial feaures of he salar Herz poenial ehnique may be delineaed for a simple, homogeneous isoropi dieleri magnei medium; see for a omprehensive analysis. However, where he mehod speifially proves is value is in is appliaion o mediums more omplex han isoropi: mediums ha are anisoropi or bianisoropi 3,4. Appliaion of he ehnique beyond isoropy were firs repored for soure free dieleri magnei gyroropi mediums 5,6. Those resuls were hen generalized o soures in dieleri magnei gyroropi mediums 7 9. Mediums ha inorporae magneoeleri oupling were invesigaed by applying he salar Herz poenial ehnique o homogeneous and nonhomogeneous isoropi hiral mediums in 0,, respeively. In more reen years, he ehnique was used o deail he soluion of he eleromagnei field problem for uniaxial bianisoropi mediums,3 and for so alled Faraday hiral mediums 4. There are also simply moving isoropi mediums 5,4, whih are bianisoropi in a frame of referene moving wih a relaive uniform veloiy wih respe o he medium. Is onsiuive relaions appear as speial ases of hose for he Faraday hiral mediums 4 and he salar Herz poenial ehnique an hus also be suessfully applied. I was observed in ha no explii reamen of he simples ype of anisoropi medium, he uniaxial dieleri magnei medium 5 wih he salar Herz poenial ehnique has ever appeared in prin. The purpose of he presen paper is hus hreefold: i o omprehensively delineae he salar Herz poenial ehnique for nonhomogeneous uniaxial dieleri magnei mediums for presribed, general eleri and magnei soures; ii o provide explii and losed form expressions for he assoiaed salar Green funions in he homogeneous ase; and iii o highligh he wo mos imporan /00/$ IOS Press. All righs reserved

2 3 W.S. Weiglhofer / Salar Herz poenials for nonhomogeneous uniaxial dieleri magnei mediums appliaions of he formalism o radiaion problems of soures parallel o he disinguished axis of he uniaxial mediums and o wave propagaion problems in soure free nonhomogeneous mediums. By doing so, a signifian gap in he researh lieraure on he opi of salar Herz poenials will be filled.. Field analysis.. Eleromagnei field represenaion The Maxwell equaions for a nonhomogeneous uniaxial dieleri magnei medium for frequeny dependen fields are given by iω ɛx Ex+ Hx =J e x, Ex iω µx Hx = J m x. In hese expressions, Ex is he eleri field and Hx is he magnei field; whereas J e x and J m x are he presribed eleri and magnei urren densiies. All hese quaniies are omplex valued field phasors and heir implii dependene on he irular frequeny ω is suppressed heneforh, as is he ime dependene of exp iω. The maerial properies of he uniaxial medium are speified by he permiiviy dyadi ɛx and he permeabiliy dyadi µx, respeively, whih are se up in he form ɛx =ɛ x I + ɛ x ɛ x, µx =µ x I + µ x µ x, where is a uni veor in he direion of he disinguished axis of he medium. The resried nonhomogeneiy is presen in he four salar onsiuive parameers of hrough heir dependene on x = x. To develop a field represenaion, a parial salarizaion of he differenial equaions wih respe o he direion speified by he uni veor is arried ou. The deomposiions E = E + E, H = H + H, 3 are inrodued, whereby E =0and H =0; E and H being alled he ransverse, E and H he longiudinal omponens of he field veors. Similarily, for he eleri and magnei urren densiy: J e = J e + J e, J m = J m + J m, 4 and also for he derivaive operaions: = + x, = + x, 5 The noaion in his paper is suh ha veors are in bold fae whereas dyadis are in normal fae and underlined wie. Conraion of indies is symbolized by a do; ha is, a b sands for aibi, whereas A = ab is a dyadi wih elemens i A ij = a i b j. The veor o he observaion poin is x and he uni dyadi is I. The supersrip indiaes inversion of dyadis and differenial operaors.

3 W.S. Weiglhofer / Salar Herz poenials for nonhomogeneous uniaxial dieleri magnei mediums 33 wherein is he Laplae operaor. The abbreviaions = / x and = / x are used subsequenly. Firs, manipulaion of he Maxwell equaions aking do produs wih leads o expressions for he omponens E and H in erms of E, H and he soure urren densiies in he form E = J e H, iωɛ H = iωµ J m + E. This leaves he following sysem of differenial equaions for he ransverse omponens as follows E iωµ H + H = J e J m, iωɛ iωɛ H + iωɛ E E = J m + J e. iωµ iωµ Insead of solving a sysem of equaions for E and H, he salarizaion proedure has lead o a siuaion whereby one needs o solve he sysem 7 for he ransverse omponens E and H only, wih E and H hen following from 6. The prie one has o pay for his reduion from he 6 omponen equaions of o he 4 omponen equaions of 7 is ha he former are parial differenial equaions of firs order while he laer are of seond order. Aing wih on 7 nex, permis he derivaion of he following sysem for E and H : + E = ɛ ɛ + k τ e + µ + k µ τ m iω J e iωµ J e + s e, ɛ H = iω µ J m iωɛ J m + s m, where he abbreviaions k = ω ɛ µ, τ e = ɛ /ɛ and τ m = µ /µ have been inrodued. 3 The soure erms s e and s m are given by s e = iω J e J m, ɛ s m = iω µ J m + J e. As for he isoropi ase, he differenial equaions for E and H are no oupled. The ruial sep in he formalism onsiss of he inroduion of salar poenials by using a wo dimensional version of he Helmholz heorem aording o E x = Φx+ Θx, H x = Πx+ Ψx Srily speaking, E, H and remain 3 veors wih null enries in heir respeive omponens for all erms in he relevan equaions o be properly defined. For all praial purposes, however, heir inerpreaion as veors is fully appropriae. 3 I should be emphasized ha beause of he dependene of he onsiuive parameers on x, he derivaive mus be viewed as an operaor in expressions suh as 8.

4 34 W.S. Weiglhofer / Salar Herz poenials for nonhomogeneous uniaxial dieleri magnei mediums In he firs insane, subsiuion of 0 ino 6 yields E = Je + Ψ, iωɛ H = Jm iωµ Θ. Nex, subsiuion of 0 ino 7 and appliaion of and gives four differenial equaions for he salar funions Φ, Θ, Π and Ψ. The four equaions ha are needed o oninue he analysis may be obained more onvenienly in an alernaive way. One akes of and hen deomposes easily o ge ɛ E = ɛ E + iω J e + J e, µ H = µ H + iω J m + J m. Using 0, hese equaions may be redued o Φ= Ψ+ iωɛ iωɛ J e, Π= Θ+ iωµ iωµ J m, 3 whih permi he omplee eliminaion of he funions Φ and Π from he field represenaion. Finally, use of in 8 yields he desired differenial equaions for E and H. They are + ɛ + k τ e Ψ= J e + iωɛ ɛ se, + µ + k τ m Θ=J m iωµ µ sm. In order o eliminae he inverse of he ransverse Laplae operaor from he preeding expressions, auxiliary funions are inrodued by virue of he definiions 4 J e = u e + v e, J m = u m + v m. 5 Consequenly, hese four auxiliary funions u e x, v e x, u m x and v m x are alulaed from he differenial equaions u p = J p, v p = J p, p = e, m. 6

5 W.S. Weiglhofer / Salar Herz poenials for nonhomogeneous uniaxial dieleri magnei mediums 35 One an hus summarize and represen he omplee eleromagnei field in he form E = iωɛ Ψ+ Θ + iωɛ u e, E = iωɛ Ψ+J e, H = iωµ Θ+ Ψ + iωµ u m, H = iωµ Θ J m. These formulas an be reas in ompa noaion as E = iω H = iω ɛ ɛ ɛ Ψ ɛ Ψ + Θ + J e + u e, iωɛ iωɛ µ µ µ Θ µ Θ + Ψ + iωµ J m + iωµ u m, he prime on ɛ and µ indiaing differeniaion wih respe o he argumen x. The expressions 8 show insruively how he orresponding formulas for he isoropi medium ge generalized due o i he uniaxial anisoropy and ii he nonhomogeneiy of he medium. For praial alulaions, however, 7 remain more suiable. Finally, he differenial equaions for he salar Herz poenials Ψ and Θ are given by L e Ψ= J e + ɛ u e iωɛ v m, ɛ L m Θ=J m µ u m iωµ v e, µ wih he Helmholz ype operaors L e, ; x = + ɛ + k ɛ τ e, L m, ; x = + µ + k τ m. µ I is noieable ha 9 for Ψ and Θ are no oupled. In more general ypes of mediums, for example gyroropi or uniaxial bianisoropi mediums, 9 are replaed by a sysem of differenial equaions ha ouple Ψ and Θ. Tha siuaion permis he inroduion of a so alled superpoenial, ha is a salar poenial ha fulfils a fourh order parial differenial equaion. The wo original salar Herz poenials and wih hem he eleromagnei field are hen derivable from ha one salar funion, he superpoenial; see 6,7 for more deails. Suh a possibiliy does no exis for he mediums under onsideraion here

6 36 W.S. Weiglhofer / Salar Herz poenials for nonhomogeneous uniaxial dieleri magnei mediums.. Salar Green funions Wih he eleromagnei field represenaion in plae and he problem fully salarized, he nex sep is he inroduion of salar Green funions. The definiion equaions are L p, ; x g p x, x = δx x, p = e, m. Therein, g e x, x is salar Green funion of he eleri ype, whereas g m x, x is a salar Green funion of he magnei ype and δx x is he Dira dela funion. Consequenly, from 9 and one obains { } Ψx = J e x ɛ x V ɛ x u ex + iωɛ x v m x g e x, x d 3 x, { } Θx = J m x +µ x µ x u mx + iωµ x v e x g m x, x d 3 x. V The salar Green funion equaions are no solvable unless a speifi profile for he onsiuive parameers ɛ x, ɛ x, µ x and µ x is speified. For he imporan homogeneous ase, however, when all four onsiuive parameers are onsan and do herefore no depend on x, simplifies o + τ p + k τ p gp x, x = δx x, p = e, m, 3 whih gives he losed form soluions g p x, x = exp ik D p 4πD p, p = e, m. 4 Therein, D e and D m are modified disanes aording o D e = ɛ x x ɛ x x, D m = µ x x µ x x. 5 While J e and J m, he omponens of J e x and J m x parallel o, appear expliily in he above field represenaion and differenial equaions, his is no he ase for he ransverse soures J e and J m. For hose he analysis an be pushed somewha furher. This is being done here by using aresian oordinaes x,x,x. Saring wih 6, some sraighforward manipulaions lead o u p x = hx,x,x x,x J px,x,x dx dx + hx,x,x x,x J px,x,x dx dx, v p x = 6 hx,x,x x,x J px,x,x dx dx hx,x,x x,x J px,x,x dx dx,

7 W.S. Weiglhofer / Salar Herz poenials for nonhomogeneous uniaxial dieleri magnei mediums 37 p = e, m. In hese expressions, J p and J p are he omponens of J p parallel o he direions speified by oordinaes x and x respeively. The funion hx,x,x,x is he salar Green funion of a sai line soure as defined by hx,x,x,x =δx x δx x, 7 whih has as is soluion hx,x,x,x = x 4π ln x + x x. 8 I is lear from heir defining equaions ha he auxiliary funions u e, u m, v e and v m are no medium speifi. They are generaed by he parial salarizaion approah in a naural way and are essenial o deal wih soures normal o. A deailed disussion of heir mahemaial properies an be found elsewhere 8,9. 3. Appliaions While he salar Herz poenial ehnique provides a self onsisen reamen of he field problem for general soures J e x and J m x, i is a is mos useful from an appliaional poin of view when J e x =J e x, J m x =J m x ; 9 i.e.; he soures have omponens parallel o he disinguished axis only. The resriion 9 also inludes he soure free ase. Then, as a onsequene u p x 0, v p x 0, p = e, m. 30 Therefore, 7 redue o E = Ψ+ Θ, iωɛ E = iωɛ Ψ+J e, H = Θ+ Ψ, iωµ H = iωµ Θ J m, wih Ψx = J e x g e x, x d 3 x, V Θx = J m x g m x, x d 3 x. V 3 3

8 38 W.S. Weiglhofer / Salar Herz poenials for nonhomogeneous uniaxial dieleri magnei mediums If boundaries are presen, he ehnique is one again very well adaped o geomeries whereby he boundaries oinide wih planes x = onsan. Then, he requiremen for oninuiy of he angenial omponens of E and H aross he boundary leads o Ψ+ Θ = Ψ+ Θ, iωɛ x + iωɛ x 33 Θ+ Ψ = Θ+ Ψ. iωµ iωµ x + While he salar Herz poenials are no oupled by heir differenial equaions, i is ineresing o noe ha in suh boundary value problems, a oupling beween he poenials is failiaed by he boundary ondiions. In he following, a brief exposiion is given of he wo mos imporan appliaions of he ehnique. 3.. Dipole radiaion For a homogeneous uniaxial dieleri magnei medium, le he urren densiy be speified as J e x =J 0 e δx x, J m x =0, 34 i.e., i is an eleri poin soure loaed a x = x wih J 0 e a onsan ampliude, in an infinie medium. From earlier expressions i an be found wihou diffiuly ha and hus Ψx =J 0 e g e x, x, Θx =0, 35 E = J 0 e iωɛ g e x, x, x E = J 0 e iωɛ + k H = J 0 e g e x, x, ge x, x, 36 H =0, where he remaining differeniaions an be performed easily as g e x, x is expliily given by Propagaion in nonhomogeneous mediums Consider finally a nonhomogeneous uniaxial dieleri magnei medium in he absene of soures, i.e., J e x =J m x =0. The salar Herz poenials an be speified in he form Ψx =Ψx expik s x, Θx =Θx expik s x, 37

9 W.S. Weiglhofer / Salar Herz poenials for nonhomogeneous uniaxial dieleri magnei mediums 39 where x = x x and s is a veor ha speifies he direion of propagaion in he plane normal o. Subsiuion of 37 ino 9 wih heir righ hand sides se o 0 as all soure erms vanish leads o Ψ ɛ Ψ + k s Ψ=0, ɛ τ e 38 Θ µ Θ + k s Θ=0, µ τ m where he prime again indiaes differeniaion wih respe o x and s = s s. Expressions 38 provide model ordinary differenial equaions ha an be solved when profiles i.e., x dependene of he onsiuive parameers are known. Two brief examples follow pursued here for Ψ only. Take ɛ ɛ 0 = ɛ ɛ 0 = ɛ 0 exp λx, µ = µ = µ 0, 39 where ɛ 0 and µ 0 are he vauum permiiviy and permeabiliy, respeively, and ɛ 0, ɛ 0, λ are onsans. The firs equaion of 38 hen beomes Ψ + a Ψ + b exp λx Ψ=0, 40 for some onsans a, b. If ɛ ɛ 0 = ɛ ɛ 0 n x = ɛ 0 x 0, µ = µ = µ 0, 4 where x 0 is some referene value. Now one has Ψ n Ψ + αx n Ψ=0, 4 x for some onsan α. Ineresingly, for n =, 4 is a Bernoulli equaion wih soluion Ψx =Ax η + Bx η, where A and B are inegraion onsans and η and η are he roos of he quadrai equaion η + η + α =0. 4. Conlusion A deailed derivaion of he salar Herz poenial ehnique for a nonhomogeneous uniaxial dieleri magnei medium has been delineaed. I was shown ha he omplee eleromagnei field an be represened by wo salar Herz poenials ha saisfy seond order parial differenial equaions ha are no oupled. Soure erms ha are based on eleri or magnei urren densiies normal o he disinguished axis of he uniaxial medium mus be aommodaed hrough four auxiliary funions. Salar Green funion represenaions were presened and explii soluions for a homogeneous medium were given. Radiaion from longiudinal soures was reaed and i was reognized ha he salar Herz poenial ehnique is well suied o suh problems. Also, he salar Herz poenial ehnique remains he only mehod ha an inorporae a resried form of nonhomogeneiy direly ino he formalism; an appliaion exemplified by he reamen of a wave propagaion problem.

10 40 W.S. Weiglhofer / Salar Herz poenials for nonhomogeneous uniaxial dieleri magnei mediums Aknowledgmen WSW is he holder of a RSE/SOEID Researh Suppor Fellowship of he Royal Soiey of Edinburgh Grea Briain. Referenes J.J. Sein, Soluions o ime harmoni Maxwell equaions wih a Herz veor, Am. J. Phys , 834. W.S. Weiglhofer, in: Eleromagnei Fields in Unonvenional Maerials and Sruures, O.N. Singh and A. Lakhakia, eds., Wiley, New York, 000,. 3 J.A. Kong, Theorems of bianisoropi media, Pro. IEEE 60 97, W.S. Weiglhofer, A perspeive of bianisoropy and Bianisoropis 97 in: speial issue Bianisoropis 97, W.S. Weiglhofer, gues ed., In. J. Appl. Eleromag. Meh , S. Przezdzieki and R.A. Hurd, A noe on salar Herz poenials for gyroropi media, Appl. Phys , S. Przezdzieki and W. Laprus, On he represenaion of eleromagnei fields in gyroropi media in erms of salar Herz poenials, J. Mah. Phys , W. Weiglhofer and W. Papousek, Skalare Herz she Poeniale für gyrorope Medien, Arh. Elekron. Überrag , W. Weiglhofer and W. Papousek, Salar Herz poenials for ransversally oriened urren densiy disribuions in gyroropi media, Arh. Elekron. Überrag , 4. 9 W. Weiglhofer, Reduion of dyadi Green s funions o salar Herz poenials for gyroropi media, Radio Si. 987, W.S. Weiglhofer, Isoropi hiral media and salar Herz poenials, J. Phys. A: Mah. Gen. 988, 49. W.S. Weiglhofer, Eleromagnei field represenaion in inhomogeneous isoropi hiral media, Eleromagneis 0 990, 7. W.S. Weiglhofer and I.V. Lindell, Fields and poenials in general uniaxial bianisoropi media I. Axial soures, In. J. Appl. Eleromagn. Maer ,. 3 W.S. Weiglhofer, Fields and poenials in general uniaxial bianisoropi media II. General soures and inhomogeneiies, In. J. Appl. Eleromagn. Meh ,. 4 W.S. Weiglhofer and S.O. Hansen, Faraday hiral media revisied I: Fields and soures, IEEE Trans. Anennas Propaga , H.C. Chen, Theory of Eleromagnei Waves, MGraw Hill, New York, W. Weiglhofer, Field represenaion in gyroropi media by one salar superpoenial, IEEE Trans. Anennas Propaga , W.S. Weiglhofer, Salar Green funions and superpoenials of a Faraday hiral medium, Arh. Elekron. Überrag , 09.