An alternative model for wave propagation in anisotropic impedance-matched metamaterials

P og ess In Elec omagne ics Resea ch, Vol. 141, 149–160, 2013 AN ALTERNATIVE MODEL FOR WAVE PROPAGATION IN ANISOTROPIC IMPEDANCE-MATCHED METAMA- TERIALS Consuelo Bell e -Ceb e os*and Ma celo Rod iguez-Dan a Depa amen o de Fisica Aplicada III, Escuela Tecnica Supe io de Ingenie ia, Uni e sidad de Se illa, Camino de los Descub imien os s/n, Se ille 41092, Spain Abs ac —The p opaga ion o ligh in an aniso opic impedance- ma ched me ama e ial is s udied in he ame o geome ical op ics. We p o e ha di ec ions o ields ¯ D,¯ Band ¯ ( ay eloci y) a e a iad o conjuga e di ec ions wi h espec o he in e se ela i e dielec ic pe mi i i y enso and cons i u es a local basis, whose ecip ocal one is o med by di ec ions o ¯ E,¯ H ields and wa e- ec o ¯ k. Consequen ly, bo h dual bases a e in insically ela ed o he physical p ope ies o medium. We ha e iden i ied hese bases wi h di ec and ecip ocal bases o a cu ilinea coo dina es sys em, showing ha physics de ines geome y. This iden i ica ion p o ides a powe ul ool o sol e wo kinds o p oblems (di ec and in e se ones) ha cu en ly a ise: In di ec p oblems, medium p ope ies a e gi en and i su ices o know ˜ε= ˜µ enso a e e y poin , o ob ain he wa e s uc u e. In in e se p oblems, medium p ope ies mus be ound o he ays o p opaga e along p esc ibed ajec o ies. The p ocedu e is applied o an illus a ing example. 1. INTRODUCTION When dealing wi h laws o p opaga ion o monoch oma ic plane wa es, he so called “op ical ans o ma ion heo y” in s a ing an equi alence p inciple be ween elec omagne ic pa ame e s o a physical medium exp essed in Ca esian coo dina es and hei analogues in acuum ob ained om a cu ilinea coo dina e ans o ma ion, pu s in co espondence he Euclidean inhomogeneous and aniso opic physical Recei ed 5 June 2013, Accep ed 6 July 2013, Scheduled 12 July 2013 * Co esponding au ho : Consuelo Bell e -Ceb e os ([email p o ec ed]). 150 Bell e -Ceb e os and Rod iguez-Dan a space, wi h ano he homogeneous, iso opic and ma e ee space, bu endowed wi h a locally la me ic. Thus, he ay ajec o y in he physical medium is associa ed wi h a geodesic cu e in i ual space. E e y analogy in Physics is a guide ha enable us o p edic hidden ela ions and ha monies. Thus, i we conside ha s eam lines in i o a ional luid mo ion a e he analogues o ays in geome ical op ics, he in isibili y p oblem in op ics becomes simila o ha o he lux o a luid s eam in he p esence o an obs acle. Analogy and p inciple o equi alence allow, in la p oblems, o eco e he old heo y o complex po en ial (con o mal mapping) and, consequen ly, ma hema ical analysis is conside ably simpli ied [1]. Ne e heless,“geome ical physics” in hese media can be unde s ood, in ou opinion, no only as a di ec consequence (“all occu s as i ”) o he in a iance o Maxwell equa ions in coo dina e ans o ma ions [2–4], bu any hing else: physical p ope ies o a medium in insically de ine he mos adequa e geome y. In gene al, a isible equencies (430–790 THz), na u al media exhibi a homogeneous and iso opic magne ic beha iou oge he wi h an inhomogeneous and aniso opic dielec ic one. Ce ain me ama e ials ( o which heo ies de eloped and desc ibed in his pape can be applied) when subjec ed o ex e nal ac ions, obey linea cons i u i e equa ions: ¯ D=ε0˜ε¯ E;¯ B=µ0˜µ¯ H, wi h equal ela i e pe mi i i y and pe meabili y enso s, (˜ε= ˜µ). The speci ici y o beha iou laws enable us o endow he space wi h a me ic biuni ocally linked o he me ama e ial physics. These p ope ies can no be applied o na u al media wi h dielec ic aniso opy and magne ic iso opy [5]. In he p esen al e na i e ea men , i has been p o ed ha in any poin o hese me ama e ials (in absence o cha ges and cu en s) he wa e on (o he ay) is ge ing dis o ed in such a way ha ¯ D,¯ B ields and ay eloci y ¯ (o ¯ E,¯ Hand ¯ k) cons i u e a local basis o conjuga ed di ec ions wi h espec o ela i e pe mi i i y o pe meabili y enso e alua ed a ha poin . 2. ALTERNATIVE ANALYTICAL MODEL In his sec ion, om ela i e pe mi i i y (o pe meabili y) enso and om laws o e olu ion in he medium, we a e in ended o es ablish ec o bases consis ing o conjuga ed di ec ions. These bases, in insically ela ed o physical p ope ies, a e pu in co espondence wi h local ec o bases ha , in cu ilinea coo dina es, a e going o de ine he medium geome y. P og ess In Elec omagne ics Resea ch, Vol. 141, 2013 151 2.1. Conjuga e Di ec ions Le ˜ Ibe a symme ic enso o second o de , wi h componen s Iij e e ed o a Ca esian e e ence ame {¯ i1,¯ i2,¯ i3}and le ˆνand ˆµbe any wo di ec ions. The bound ec o o ˆνby he enso [6] is ec o ¯ Γνde ined as ¯ Γν=˜ I·ˆν. The in insic componen , σνo ec o ¯ Γνis equal o i s p ojec ion on o ec o ˆνin such a way ha σν=¯ Γν·ˆν. The p ojec ion o ec o ¯ Γνon o ˆµis ¯ Γν·ˆµand as a consequence o Cauchy heo em [7] we a i e o: ¯ Γν·ˆµ=¯ Γµ·ˆν. Two di ec ions ˆνand ˆµa e named conjuga e i [8, 9]: ¯ Γν·ˆµ=¯ Γµ·ˆν= 0 (1) Figu e 1 p o ides a simple geome ical in e p e a ion: A ele an p ope y o Cauchy quad ic (de ined by σν=cons ) [7] is ha i s no mal a e e y poin is collinea wi h bound ec o ¯ Γν o any di ec ion ˆν. The o hogonali y o ec o s ¯ Γνand ˆµimplies he ela ion o conjuga ion. O x P H Q π π1 2µ Γν z y ν Figu e 1. Cauchy’s quad ic associa ed wi h a second-o de enso ˜ I. The o hogonali y o ec o s ¯ Γν=˜ I·ˆνand ˆµimplies ha uni ec o s ˆνand ˆµa e conjuga e di ec ions. Vec o ˆµlies in plane π2pa allel o he angen plane o he ellipsoid a “P”. I mus be no ed ha p incipal di ec ions o a enso a e also conjuga e di ec ions. Conjuga ion a ises as a b oade concep han ha o p incipal di ec ions (a pa icula case, whe e ¯ Γνis collinea wi h ˆν), bu keeping he algeb aic p ope y o diagonalisa ion o enso s, i exp essed in co a ian componen s. 152 Bell e -Ceb e os and Rod iguez-Dan a 2.2. Laws o E olu ion The double Fou ie ans o m ¯ (ω, ¯ k) o a ec o ield ¯ F( , ¯ ), unc ion o bo h ime and posi ion is: ¯ ¡ω, ¯ k¢=Zd d3¯ ¯ F( , ¯ )ei(ω −¯ k·¯ )(2) and i can be shown ha [10]: ∂¯ F ∂ → −iω (ω, ¯ k); ∇· ¯ F( , ¯ )→i¯ k· (ω, ¯ k) (3) ∇× ¯ F( , ¯ )→i¯ k× (ω, ¯ k) (4) Applying hese exp essions o Maxwell equa ions o sou ce- ee media, we ha e: ¯ kׯ E=ω¯ B;¯ kׯ H=−ω¯ D(5) ¯ k·¯ D= 0; ¯ k·¯ B= 0 (6) Bea ing in mind ha wa e ec o ¯ k(wa e- on g adien ) and ay eloci y ¯ (collinea wi h Poyn ing ec o ), a e gi en by [11]: ¯ k=(¯ Dׯ B)ω W; ¯ =(¯ Eׯ H) W(7) (whe e Wis he elec omagne ic ene gy densi y), om Maxwell equa ions [12] o om Exp ession (7), he ollowing equa ions in ¯ - domain a ise: ¯ ׯ B=−¯ E; ¯ ׯ D=¯ H(8) ¯ ·¯ E= 0; ¯ ·¯ H= 0 (9) In o de o comple e he abo e equa ions, we add some speci ic cons i u i e ela ions o a i icial media (me ama e ials), whe e ela i e pe mi i i y enso (˜εij) and ela i e pe meabili y one (˜µij) a e iden ical when e e ed o he same ec o basis; consequen ly he iden i y holds also o in e se enso s ˜ε−1 ij = ˜µ−1 ij = ˜τij (in wha ollows, ˜τ). In o he wo ds, ¯ D=ε0˜ε·¯ E;¯ E=˜τ·¯ D ε0 ;¯ B=µ0˜ε·¯ H;¯ H=˜τ·¯ B µ0 (10) I is immedia e o s a e ha in hese media, elec omagne ic ene gy densi y is gi en by W=¯ E·¯ D=¯ B·¯ H; i.e., elec ic ene gy densi y coincides wi h magne ic one, and consequen ly, ¯ k·¯ =ω. Unde he assump ions o p opaga ion o monoch oma ic (ω = cons an ) plane wa es in he absence o cu en densi ies and cha ges, Equa ions (5), (6), (8), (9) and (10) enable us o de i e some di e en geome ical s uc u es and hei co esponding physical laws o e olu ion. P og ess In Elec omagne ics Resea ch, Vol. 141, 2013 153 2.2.1. Geome ical S uc u e o Fields a) In o de o desc ibe he laws o e olu ion o ields, we choose a local base (P, ¯ei/{i= 1,2,3}), consis ing o ec o s ¯ei, collinea wi h ¯ D,¯ B ields and ¯ ( ay eloci y), espec i ely. We will p o e ha his basis is conjuga ed wi h espec o he enso ˜τ. The chosen ec o s a e: ¯e1=¯ D |¯ D|; ¯e2=¯ B |¯ B|; ¯e3=¯ |¯ |; wi h ¯ei=hiˆui;h1=h2=h3=1 (11) and √g= (¯e1ׯe2)·¯e3. Equa ions (8) and (9) p o e ha ¯ E(¯ H) is o hogonal o ¯ B(¯ D) and ¯ , and cons i u i e laws (10) p o e ha ¯ Eis collinea wi h ¯ Γu1= ˜τ·ˆu1(and ¯ His collinea wi h ¯ Γu2= ˜τ·ˆu2). Thus, ε0¯ E |¯ D|= ˜τ·ˆu1=¯ Γu1;µ0¯ H |¯ B|= ˜τ·ˆu2=¯ Γu2(12) F om he abo e men ioned o hogonali y ela ions, Equa ion (12) and Cauchy heo em, we can deduce ha ¯ Γu1·ˆu2=¯ Γu1·ˆu3=¯ Γu2·ˆu1= ¯ Γu3·ˆu1=¯ Γu2·ˆu3=¯ Γu3·ˆu2= 0. In o he wo ds, he elec ic displacemen ield ¯ D, he magne ic induc ion ield ¯ Band ay eloci y ¯ locally p opaga e along conjuga ed di ec ions o he in e se ela i e pe mi i i y enso o he medium. F om he abo e analysis, i yields he choice o he ecip ocal local base (P, ¯ej/{j= 1,2,3}), ha consis s o ec o s collinea wi h he bound ec o s o di ec ions ˆu1, ˆu2, ˆu3and de ined as: ¯ej= ¯ Γuj hjτj =˜τ·¯ej h2 jτj−→ i yields ¯ei·¯ej=δj i(13) wi h τj=¯ Γuj·ˆuj(τjcoincide wi h he eigen alues o he enso , when ec o s ˆuia e o hogonal). In hese ames o e e ence (bases ¯eio ¯ei), ields ha e only a componen (co a ian o con a a ian ). Thus, ¯ D=D1¯e1;¯ B=B2¯e2; ¯ = 3¯e3;¯ E=E1¯e1;¯ H=H2¯e2; (14) wi h D1=|¯ D|/h1;B2=|¯ B|/h2; 3=|¯ |/h3;E1=|¯ D|h1τ1/ε0; H2=|¯ B|h2τ2/µ0, and he in e se ela i e pe mi i i y enso (in co a ian componen s) is w i en as: ˜τ=τij ¯ei⊗¯ejwi h τij =h2 iτjδj i(15) Wi h espec o he physical meaning o τ1and τ2, we e ain he exp essions o elec ic and magne ic ene gy densi ies in a linea medium [11, 13]: We=¯ E·¯ D 2;Wm=¯ H·¯ B 2(16) 154 Bell e -Ceb e os and Rod iguez-Dan a Since τj=¯ Γuj·ˆuj, we can w i e: τ1=¯ Γu1·ˆu1=ε0¯ E·¯ D |¯ D|2=We W0 e ;τ2=¯ Γu2·ˆu2=µ0¯ H·¯ B |¯ B|2=Wm W0 m (17) Coe icien τ1is he a io be ween he elec ic ene gy densi y in he medium, We, and he elec ic ene gy densi y, W0 e, ca ied by he wa e i p opaga ion we e in acuum and ob iously, i coincides wi h he in e se ela i e pe mi i i y associa ed wi h he ˆ u1di ec ion. The same applies o τ2, bu e e ed o magne ic ene gy densi y. In o de o know he physical meaning o e e y ec o o he basis ha desc ibe co a ian componen s o ields, i emains only o in es iga e wha ec o ¯e3 ep esen s. F om (7), (13), (14) and aking in o accoun he linea cha ac e o ope a o ˜τ, we can w i e, ˜τ·(¯ Eׯ H)=E1H2˜τ¡¯e1ׯe2¢=E1H2²123˜τ·¯e3=E1H2 √gh2 3τ3²312¯e1ׯe2 =h2 1h2 2h2 3τ1τ2τ3 ε0µ0gD1B2(¯e1ׯe2)= h2 1h2 2h2 3τ1τ2τ3 ε0µ0g¡¯ Dׯ B¢(18) whe e componen s ²123 =²312 o Le i-Ci i a enso a e equal o 1/√g. F om Equa ions (7), (13), (18), and aking in o accoun ha ε0µ0= 1/c2, we can w i e: ˜τ·ˆ =τ1τ2τ3h2 1h2 2h2 3c2 gω ¯ k=⇒¯e3=τ1τ2h2 1h2 2c2 gω|¯ |¯ k(19) We ha e shown ha ec o ¯e3has he di ec ion o he wa e- ec o ¯ k, and hen ¯ k=k3¯e3. To summa ize, ¯ D=D1¯e1;¯ B=B2¯e2; ¯ = 3¯e3;¯ E=E1¯e1;¯ H=H2¯e2;¯ k=k3¯e3(20) Then, ec o s o he ecip ocal base ¯eilie along ¯ E,¯ Hand ¯ k di ec ions. The e is an in insic geome ical s uc u e, associa ed wi h p opaga ion o plane wa es in hese media in such a way ha he Euclidean space is endowed, a e e y poin P, wi h wo dual bases de ined om physical p ope ies o he medium. Table 1 and Figu e 2 show his geome ical s uc u e. Table 1. Di ec base ec o s and ecip ocal base ec o s ob ained om enso ˜τ. Di ec basis Recip ocal basis ¯e1collinea wi h ¯ D¯e1colllinea wi h ¯ E ¯e2collinea wi h ¯ B¯e2collinea wi h ¯ H ¯e3collinea wi h ¯ ¯e3collinea wi h ¯ k P og ess In Elec omagne ics Resea ch, Vol. 141, 2013 155 k B ED H e1e1 e2 e3 e2 e3 P Figu e 2. Local ecip ocal bases ¯eiand ¯eiassocia ed wi h he s uc u e o a ligh plane wa e p opaga ing h ough an aniso opic impedance-ma ched medium. b) I espec i e o any hing gi en abo e, we may assume ha he space is desc ibed by a sys em o cu ilinea coo dina es, q1,q2,q3 ela ed o Ca esian ones, x1,x2,x3 h ough he equa ions: qj=qj(x1, x2, x3)⇔xj=xj(q1, q2, q3) wi h j= 1,2,3,(21) ha allow us o de ine, i he ans o ma ion is admissible [7], wo ecip ocal local bases a any poin Pas: Di ec basis: (P, ¯ej)/¯ej=∂¯ ∂qj = k=3 X k=1 ∂xk ∂qj ¯ ikwi h j={1,2,3}(22) Dual basis: (P, ¯ej)/¯ej=∇qj= k=3 X k=1 ∂qj ∂xk ¯ ikwi h j={1,2,3}(23) whe e ecip oci y ela ion ¯ei·¯ej=δj iholds. T ans o ma ion ules be ween base ec o s a e gi en by ¯ei=gij ¯ej, ¯ei=gij ¯ej, whe e gij = ¯ei·¯ejis he me ic enso . This geome y, de ined by me ic enso gij, sugges s o iden i y local bases (11) and (13), in insically ela ed o he physical p ope ies o medium, wi h (22) and (23) o he jus desc ibed a bi a y cu ilinea coo dina es sys em. F om his co espondence, physics de ines geome y. Assuming he equali y o local bases, we ha e a powe ul ool o sol e wo kinds o p oblems ha cu en ly a ise: di ec and in e se 156 Bell e -Ceb e os and Rod iguez-Dan a ones. Fo di ec p oblems, medium p ope ies a e gi en and i su ices o know ˜ε= ˜µ enso a e e y poin , because he me hod p o ides he wa e s uc u e. In in e se p oblems, we mus ind he medium p ope ies o he ays o p opaga e along p esc ibed ajec o ies. I ay ajec o y is gi en, i.e., geome y is known, we can de e mine medium p ope ies ˜τ(See Sec ion 3: Applica ion). 2.2.2. Ray E olu ion and Fe ma P inciple Unlike na u al op ical aniso opic media (wi h dielec ic aniso opy and magne ic iso opy), in hese media wi h ˜ε= ˜µ, he e is an only mode o p opaga ion and an only ay eloci y ¯ , o e e y wa e- ec o ¯ k. F om Equa ion (19) and since in his case, ¯ k·¯ =ω, we ob ain, ¯ ·˜τ·¯ =τ1τ2τ3h2 1h2 2h2 3c2 g(24) In acco dance wi h he de ini ion o ay index o e ac ion n [11] as n =c/ , and emembe ing ha ¯ is angen o he ay ajec o y: ¯ = d¯ /dl = ¯ , we can w i e: n2 =c2 2=g h2 1h2 2h2 3τ1τ2τ3 (ˆu3·˜τ·ˆu3) (25) We ha e jus ound he exp ession o he ay index as: n =√g h1h2h3µ¯ ·˜τ τ1τ2τ3·¯ ¶1/2 (26) wi h ˆu3≡¯ =d¯ dl wi h |d¯ |=dl. Since n is a unc ion bo h o posi ion ¯ and o he di ec ion o ec o d¯ /dl, Equa ion (26) shows he complexi y o ligh p opaga ion in inhomogeneous and aniso opic media, e en o he ma e ials in discussion, o which he e is an only mode o p opaga ion, and we can sol e o n . Unlike he iso opic case, he dis ance be ween wo in ini ely nea poin s depend on hei posi ion and also on hei ela i e o ien a ion. Consequen ly, equa ions o e olu ion can be ound as geodesics in a Finsle space [14], in which he line elemen is gi en by ds =n (¯ , d¯ /dl)dl. When alues o n a e subs i u ed in o he Fe ma ‘s a ia ional p inciple in o de o use Hamil onian and Lag angian o malism, a p oblem a ises because con en ional Legend e ans o m, would ob ain an iden ically anishing Hamil onian om Lag angian, which is homogeneous o deg ee one in d¯ /dl. Ne e heless, Fe ma p inciple is in a ian unde epa ame e ising, hen pa ame e “a c leng h” lmay be subs i u ed by op ical pa h, in such a way ha dl =n dσ [15], and a non-singula P og ess In Elec omagne ics Resea ch, Vol. 141, 2013 157 Lag angian a ises as L=n2 (¯ , d¯ /dσ). Consequen ly, equa ions o e olu ion can be ound as ex emals o he unc ional, δZn2 dσ = 0 (27) whe e dis ance is measu ed by a el ime (op ical pa h). 3. APPLICATION Le us illus a e he abo e heo y wi h an applica ion. Since di ec p oblems ha e been mo e discussed, we deal wi h an in e se one: Which op ical p ope ies mus he medium ha e o ligh - ays o p opaga e along a p esc ibed ajec o y? Fo ins ance, we wan o cha ac e ize medium p ope ies in such a way ha e e y plane wa e ha p opaga es along an axis, say z(wa e ec o ¯ kpa allel o zaxis), yields a ay desc ibing an helicoidal ajec o y o axis z. Fo ha pu pose, we in oduce he ollowing coo dina es sys em (See Figu e 3). q1=z θ;q2=%;q3=z(28) exp essed in cylind ical coo dina es: %,θ, z. We see ha coo dina e plane q3= cons an desc ibes he wa e- on and coo dina e line q3is along he desi ed ay. Vec o s o he di ec basis a e: ¯e1=∇q1=−z %θ2ˆuθ+1 θˆuz; ¯e2=∇q2= ˆu%; ¯e3=∇q3= ˆuz(29) 2 2 2 3 e line q line q q= cons . q= cons . 3 Figu e 3. Coo dina e su aces and coo dina e lines o cu ilinea coo dina es sys em (q1, q2, q3), de ined as q1=z/θ;q2=%;q3=z, whe e cylind ical coo dina es %,θand za e used. In his case, coo dina e plane q3= cons an desc ibes he wa e- on and coo dina e line q3is along he desi ed ay.