Energy dispersion relation for negative refraction (NR) materials

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1 Enegy dispesion elaion fo negaive efacion (NR) maeials Y.Ben-Ayeh Physics Depamen, Technion Isael of Technology, Haifa 3, Isael addess: Fax: Keywods: Negaive-efacion, enegy dispesion elaion, meamaeials ABSTRACT A geneal enegy dispesion elaion is developed fo meamaeials having he negaive-efacion (NR) popey. I is shown ha absopion effecs ae involved wih NR phenomena, he condiions unde which NR occus ae discussed. Simple equaions fo NR ae developed by using Loenzian models.. 1.Inoducion We descibe biefly he idea pesened by Veselago [1] hen show he poblem elaed o enegy dispesion elaions []. Assuming a plane monochomaic wave in which all quaniies ae popoional o expikz we ge fom Maxwell equaions k E H ; k H E. (1) c c Hee E H ae he elecic magneic fields, especively, ae he elecic pemiiviy magneic pemeabiliy, especvily, is he fequency k is he wave veco. I can be seen fom hese equaions ha if hen E, H k fom a lef hed sysem diffeen fom he igh hed sysem fo he

2 common case of. Assuming ha ẑ is he uni veco along he Poyning veco, epesening he flux of enegy, one finds ha fo maeials wih n negaive values of he waveveco k zˆ c appeas wih negaive efacion index n [3,4]. As follows fom Snells law he efaced beam fom odinay maeial ( n 1) ino Negaive-efacion (NR) maeial ( n ) will emege on he ohe side of he nomal elaive o odinay maeial. While negaive pemiiviy is obained quie easily in plasma [5] i is quie difficul o obain negaive magneic pemeabiliy. Since negaive index of efacion shows ineesing opical phenomena [6-8] lage effo has been spen in obaining a negaive magneic pemeabiliy in addiion o a negaive pemiiviy in special maeials known as meamaeials [9-1] wih he NR popey. NR has been ealized expeimenally in vaious expeimens [11-13]. The eamen of NR by analyzing phase velociies does no show he complee naue of hese phenomena. Negaive values fo /o can appea in meamaeials only if a coesponding enegy dispesion elaion is valid. The dispesion fo he EM (elecomagneic) enegy W has been descibed as [1,]: 1 1 W E H, () whee he deivaives in Eq. () ae elaive o fequency. The idea following fom he use of Eq. () fo meamaeials is ha we can use in his equaion eal values fo wih negaive values Re Re, only when ( ) ( ) ae posiive so ha he oal enegy W becomes posiive. Eq. () has been deived by Lau Lifshiz [] unde he assumpion ha ( ) ( ) ae eal. Usually NR is obained nea esonances whee he eal pas of he

3 pemiiviy pemeabiliy change apidly as funcion of fequency. As is well known, he eal imaginay pas of he pemiiviy pemeabiliy ae elaed in such egions by Kames-Konig (KK) elaions such elaions ae manifesed by Hilbe ansfoms (HT) [14-18]. Theefoe, he elaion beween he imaginay eal pas of hese funcions canno be ignoed he assumpion of eal values fo is no valid fo meamaeials. Also we will show ha nea such esonances hee ae specal egions whee ae negaive including negaive values fo hei deivaives. Theefoe, in such specal egions Eq. () cano lead o posiive enegy. The possibiliy o use in Eq. () complex values i i, whee in a sho noaion he subscips denoe eal imaginay pas, especively, does no solve he poblem as W should be a eal quaniy. Theses poblems indicae ha he use of Eq.() should be genealized. Fo developing he geneal enegy dispesion elaion we descibe in Secion he dependence of on fequency show he elaions of such funcions o HT [14-18]. Then, we deive in Secion 3 he enegy dispesion elaion fo meamaeials which will genealize he use of Eq. (). We will show ha hee is a ceain heshold fo which Eq. () will give posiive enegy. In Secion 4 we analyze he implicaions of he enegy dispesion elaions o Loenzian models fo he elecic [5] magneic [19-4] suscepibiliies. In Secion 5 we summaize ou esuls conclusions.. The dependence of on fequency elaed o Hilbe ansfoms (HT) Kames-Konig (KK) elaions hei mahemaical manifesaion by HT can be applied o dielecic magneic maeials ha absob ligh. The dielecic

4 opical popeies of such maeials ae epesened by he equaions fo a complex displacemen elecic field D polaizaion P [5,6]: P E ; D E E ; 1 ; i, (3) coesponding o a complex pemiiviy a complex elecic suscepibiliy. The magneic opical popeies ae epesened by he equaions fo a complex magneic field B [6] magneic polaizaion M : M H ; B H H ; 1 ; i (4) coesponding o a complex pemeabiliy a complex magneic suscepibiliy. We use hee he isoopic assumpion bu fo moe geneal cases will be ensos [7]. The KK elaions fo he dielecic funcions using a linea model ae given as [15]: Im ( ) d ( ) Re ( ) 1, ( ) Im ( ) Re ( ) 1 d (5) whee ( ) ( ) ae he eal imaginay pas of he elecic suscepibiliy. In a simila way one ges [15]: Im ( ) d ( ) Re ( ) 1 ; ( ) Im ( ) Re ( ) 1 d, (6) whee ( ) ( ) ae he eal imaginay pas of he magneic suscepibiliy. One should ake ino accoun ha ( ) ( ) ae anisymmeic funcions in while ( ) ( ) ae symmeic so ha he HT manifesed in Eqs. (5) (6) elae he symmeic funcions wih he anisymmeic ones. In Eqs.

5 (5) (6) we have omied he need fo using Pincipal Value (P.V.) [6] as we have assumed ha he specal egion of is vey fa fom [15]. A simple Loenzian model of he elecic polaizaion densiy P he coesponding pemiiviy can be elaed o a damped diven hamonic oscillao [5]. Assuming a monchomaice ime dependence E( ) E exp( i), P( ) P exp( i) one ges in his model: P ( ) ; ( ) ; (1 ( )) E i (7) whee is he naual fequency of he oscillao. Such model descibes ceain oscillaions in plasma whee Ne. (8) m Hee e m ae he chage effecive mass of he elecon,especively, N is he numbe of chages pe uni volume. One should ake ino accoun ha he above funcions ae valid only fo linea sysems. Using linea models fo meamaeials a magneic Loenzian model has been developed in vaious woks [19-4] leading o he elaion: F ( ) 1 i, (9) whee is he esonance fequency fo he pemeabiliy, epesen losses F epesens he sengh of he Loenzian mode ineacions. One should noice ha he magneic suscepibiliy ( ) ( ) 1 has a vey simila fom o he elecic suscepibiliy ( ) deived by he elecic Loenzian model in Eq. (7). While he deivaion of he elecic Loenzian model fo a diven damped oscillao is vey simple, deivaions of magneic Loenzian models ae quie complicaed. [see e.g.

6 1]. In he pesen sudy we assume a esonance fequency which is usually assumed o be given fo meamaeials so ha we do no ea hee he Dude model [5] fo which. The esonance fequency in magneic Loenzian model is usually elaed o capaciance inducive elemens while he losses ae elaed o esisance elemens, such elaions can be ealized in special sucues known as spli ing esonaos [19-6]. The explici calculaions fo diffeen meamaeials give diffeen evaluaions fo he consans, F bu usually lead o magneic suscepibiliy of he fom (9) o a simila one. As ou inees in he pesen pape is in enegy dispesion elaions we will sudy in he nex Secion he geneal elaions beween magneic elecic suscepibiliies he enegy dispesion elaions. 3. Enegy dispesion elaion fo negaive efacion maeials The change of elecic magneic enegies pe uni volume uni ime U is given by [] : U 1 D D 1 B B E E H H Re E D Re H B!!!! " # " # " # " # (1) Such elaion follows fom calculaions of he Poyning veco whee he elecic magneic fields ae complex vecos [,6]. By assuming a naow b of fequencies fo he elecic magneic fields hey can be given as: E( ) E ( ) exp i, E ( ) E exp i$ d$,, $, $ $, (11) 1 E E ( ) exp i$ d

7 H ( ) H ( ) exp i ; H ( ) H exp i$ d$,, $, $ $ 1, (1) H H ( ) exp i$ d $ ; $ ; $. (13) Hee is he cenal caie fequency he inegals ove $ descibe inegals ove a naow fequency disibuion aound he cenal fequency. E $ H $ ae componens in fequency space of he Fouie ansfom of E ( ) H ( ), especively. We develop ( ) ( ) up o fis ode in % $. Then we ge: ( ) ( ) ( )! $ " #, (14) ( ) ( ) ( )! $ " #. (15) Using hese appoximaions he displaced elecic field D he induced magneic field B ae given as & ( ) D ( E$ exp i $ ( )! $ ) d$ $, " # - (16) $ we ge: & ( ) B ( H$ exp i $ ( )! $ ) d$ $, " # - Pefoming he ime deivaives D B (17) neglecing ems of ode D &. ( ). ( E$ exp i $ i ( )! $ i$ ( ) d$!), (18) $., " " # #.-

8 B.& ( ). ( H$ exp i $ i ( )! $ i$ ( ) d$!) $,. " " # #-.. (19) Fo he fis em on he igh side of Eqs. (18) (19) we can use especively, he elaions: / $ E exp i $ i ( ) d$ i ( ) E ( ), () $ $ H exp i $ i ( ) d$ i ( ) H ( ) / $. (1) In Eqs. () (1),, ( ) ( ) have been aken ou of he inegal Eqs. (11-1) have been used, especively. Fo he second hid ems on he igh side of Eqs. (18) (19) we use, especively, he elaions: ( ) " # i $! ( ) [ ( )] i $ i $! " #, () ( ) " # i $! ( ) [ ( )] i $ i $! " #. (3) Inseing Eqs. () () ino Eq. (18) using he elaion ( E( ) i$ ( E( ) we ge D i E ( ) E( ) [ ( )]. (4) Inseing Eqs.(1) (3) ino Eq. (19) using he elaion ( H( ) i$ ( H( ) we ge: B H ( ) i ( ) H ( ) [ ( )]. (5) Inseing Eqs. (4-5) ino Eq. (1) using Eqs. (11-1) we finally ge:

9 E( ) ( ) E( ) U 1 1 E( ) 1 ( ) H ( ) 1 ( ). (6) Hee we have used he elaions E ( ) E( ), H ( ) H ( ). We emind ha he subscips denoe he eal imaginay pas whee accoding o Eqs (3-4) we have, especively: (1 ) ; " ; (1 ) ; ". (7) The fis wo ems on he igh side of Eq. (6) epesen he change in he EM enegy which enes ino he meamaeial while he addiional wo ems epesen he absobed enegy, boh pe uni ime pe uni volume. Following he above appoximaions, in he deivaives accoding o appea only he eal pas while in he las wo ems of Eq. (6), epesening absopion, appea only he imaginay pas. One should ake cae abou he following poins. Assuming zeo elecic magneic fields a a ceain iniial ime inegaing he fis wo ems on he igh side of Eq.(3) ill ime he esul will be equivalen o ha of Eq. () unde he condiions 1 ( ) 1 which 1 ( ) 1 ( ). Fo specal egions fo ( ), Eq. () leads o negaive enegy W which canno be ue. In such cases we ge accoding o Eq. (6): E( ) H ( ) ; fo 1 ( ) ; 1 ( ). (8) We find ha unde he condiion of Eq. (8) all he EM inciden on he meamaeial is efleced,i.e., he meamaeial becomes opaque, wih vanishing EM fields in his

10 maeial. Thee ae, howeve, exensive specal egions in which he esicion (8) does no apply NR can be implemened. In such specal egions one should ake ino accoun ha a pa of he enegy supplied o he elecic magneic fields in he meamaeials is absobed as given by he addiional ems in Eq. (6). The implicaions of he moe geneal enegy dispesion elaions fo maeials saisfying he Loenzian specal pofiles will be eaed in he nex Secion 3.Enegy dispesion elaion fo meamaeials saisfying Loenzian specal pofiles In ode o use he enegy dispesion elaion o meamaeials wih Loenzian specal pofiles we need o sepaae ino hei eal imaginay pas. Using Eq. (7) he eal imaginay pas of he elecic suscepibiliy ae given as ( ) ; ( ). (9) We find ha ( ) is negaive when (1 ) becomes negaive a a ceain specal ange when 1. In ode o esimae he specal ange whee is negaive o esimae he value inoduce he appoximaions: ( ) in his specal ange we ;. (3) These appoximaions ae valid as long as. Unde hese appoximaions / / 1, (31)

11 / ( / ), (3). ( / ) / 4 /. (33) One should noice ha is symmeic in while is anisymmeic. Accoding o Eq. (31) becomes negaive when ( / ) is small elaive o /. Howeve, becomes negaive when / unde his condiions he elecic field does no peneae he meamaeial. When becomes posiive he value of inceases quie apidly as funcion of fequency NR is implemened. Pefoming simila calculaions fo he magneic Loenzian pofiles, unde he same appoximaion, we ge / ( ) 1 F, (34) ( / ) ( ) F / F ( / ) ( / ), (35) / 4 ( ) F ( / ). (36) Hee again becomes negaive when is negaive ( / ) is small elaive o F /. ( ) becomes,howeve, negaive when / unde his condiion he magneic field does no peneae he

12 meamaeial. I is also easy o show ha beyond hese special specal egions hee ae exensive egions fo which ( ) ( ) ae negaive while become posiive elaively lage so ha in hese egions he inciden EM field peneaes ino he meamaeial shows he ineesing NR phenomena. Even in such cases a pa of he inciden enegy is absobed as descibed by Eqs. (6). 5.Summay Conclusion The usual enegy dispesion elaion, given by Eq. (), has been genealized by analyzing Poyning veco effecs fo EM signal, wih naow b of fequencies, following he use of Eq. (1). The deivaion of Eqs, (6) shows wo impoan effecs which should be aken ino accoun in he implemenaion of NR phenomena: a) Absopion effecs pe uni ime uni volume ae descibed in Eq. (6). b) I follows fom Eq.(6) ha unde he condiions ( ), ( ) 1 1 he EM fields canno peneae he meamaeials as he only soluion fo his equaion unde hese condiions is E( ) H ( ). Beyond hese specal egions posiive enegy in he meamaeials can be ealized in ageemen of Eq. () o wih he moe geneal enegy dispesion elaion of Eq.(6). In ode o ealize NR phenomena one has o apply he inciden EM waves wih fequencies nea esonances in ode o ge lage values of ( ) ( ). Such esonances can be descibed quie ofen by Loenzian models. The diffeen

13 paamees needed fo NR phenomena have been esimaed by developing appoximae equaions using Loenzian models. Refeences 1. V.G.Veselago, Sov.Phys.Usp. 1, 59 (1968). L.D. Lau, E.M. Lifshiz, Elecodynamics of Coninuous Media (Pegamon Pess,Boson,1999). 3. S.A. Ramakishna, Rep.Pog.Phys. 68, 449 (5). 4. K.Y. Bliokh, Y.P.Bliokh, Sov.Phys.Uspekhi 47, 393 (4). 5. S.A.Maie, Plasmonics: Fundamenals Applicaions (Spinge, Bah UK, 7). 6. J.B. Pendy, Phys.Rev.Le. 85, 3966 (). 7. J.Pacheco,J., T.M.Gzegoczyk, B.-I.Wu, Y.Zhang, J.A.Kong, Phys.Rev.Le. 89, 5741 (). 8. Y.Ben-Ayeh, J.Mod.Op. 5, 1871 (5). 9. N.Enghea, R.W.Ziolkowski (Edi.) Meamaeials, Physics Engineeing Exploaions (Wiley,Hoboken, NJ, 6). 1. J.B. Pendy, Conempoay Phys. 45, 191 (4). 11. A.A. Houck, J.B. Bock, I.L.Chuang, Phys.Rev.Le. 9, (3) 1. C.G.Paazzoli, R.B. Geego, K.Li, B.E.C.Kolenbah, M.Tanielian, Phys.Rev.Le. 9, 1741 (3). 13. R.A. Shelby, D.R.Smih, S.Schulz, Science 9, 77 (1). 14. G. Afken, Mahemaical Mehods fo Physiciss (Academic Pess, Boson, 1985). 15. B.E.A. Saleh, M.C.Teich, Fundamenals of Phoonics (Wiley, New Yok, 1991) 16. K-E. Peiponen, E.M. Vaiainen, T.Asakua, Dispesion, Complex Analysis Opical specoscopy: Classical Theoy (Spinge, Belin, 1999). 17. K.-E. Peiponen, V.Lucaini, E.M. Vaiainen, J.J.Saainen, Eu.Phys.J.B. 41, 61 (4). 18. P.M.Mose, H.Feshbach, Mehods of Theoeical Physics (McGaw-Hill, New Yok,1953)

14 19. D.R.Smih, N.Koll, Phys.Rev.Le. 85, 933 (). M.C.K. Wilshie, J.B.Pendy, I.R.Young, D.J.Lakman, D.J.Gildedale, J.V.Hajnal, Science 91, 849 (1) 1. S.OBien, J.B.Pendy, J.Phys.: Condens. Mae 14, 6383 (). J.B.Pendy, A.J.Holden, D.J.Robbins, W.J.Sewa, IEEE Tans. Micowave Theoy Tech. 47, 75 (1999) 3. D.R.Smih, W.J.Padilla, D.C.Vie, S.C.Nema-Nasse, S.Schulz, Phys.Rev.Le. 84, 4184 (). 4. A.Ishikawa, T.Tanaka, Opics Commun, 58, 3 (6). 5. S.G.Lipson, H.Lipson, D.S.Tannhause, Opical Physics (Cambidge Pess, Cambidge, 1995). 6. J.D. Jackson, Classical Elecodynamics (John Wiley, New Yok, 1975). 7. H.A.Haus, Elecomagneic Noise Quanum Opical Measuemens (Spinge, Belin,)

Lecture 22 Electromagnetic Waves

Lecture 22 Electromagnetic Waves Lecue Elecomagneic Waves Pogam: 1. Enegy caied by he wave (Poyning veco).. Maxwell s equaions and Bounday condiions a inefaces. 3. Maeials boundaies: eflecion and efacion. Snell s Law. Quesions you should